Averages of b-hadron, c-hadron, and $\tau$-lepton properties as of summer 2016

Abstract

This article reports world averages of measurements of b-hadron, c-hadron, and $\tau$-lepton properties obtained by the Heavy Flavor Averaging Group using results available through summer 2016. For the averaging, common input parameters used in the various analyses are adjusted (rescaled) to common values, and known correlations are taken into account. The averages include branching fractions, lifetimes, neutral meson mixing parameters, $C\!P$  violation parameters, parameters of semileptonic decays, and Cabbibo–Kobayashi–Maskawa matrix elements.

1 Introduction

Flavor dynamics plays an important role in elementary particle interactions. The accurate knowledge of properties of heavy flavor hadrons, especially b hadrons, plays an essential role for determining the elements of the Cabibbo–Kobayashi–Maskawa (CKM) quark-mixing matrix [1, 2]. The operation of the Belle and BaBar $e^+e^-$ B factory experiments led to a large increase in the size of available B-meson, D-hadron and $\tau$-lepton samples, enabling dramatic improvement in the accuracies of related measurements. The CDF and D0 experiments at the Fermilab Tevatron have also provided important results in heavy flavour physics, most notably in the $B^0_s$ sector. In the D-meson sector, the dedicated $e^+e^-$ charm factory experiments CLEO-c and BESIII have made significant contributions. Run I of the CERN Large Hadron Collider delivered high luminosity, enabling the collection of even larger samples of b and c hadrons, and thus a further leap in precision in many areas, at the ATLAS, CMS, and (especially) LHCb experiments. With the LHC Run II ongoing, further improvements are keenly anticipated.

The Heavy Flavor Averaging Group (HFLAV)¹ was formed in 2002 to continue the activities of the LEP Heavy Flavor Steering Group [3]. This group was responsible for calculating averages of measurements of b-flavor related quantities. HFLAV has evolved since its inception and currently consists of seven subgroups:

• the “B Lifetime and Oscillations” subgroup provides averages for b-hadron lifetimes, b-hadron fractions in $\varUpsilon (4S)$ decay and pp or $p{\bar{p}}$ collisions, and various parameters governing $B^{0}$–${\overline{B}} ^0$ and ${B^{0}_s}$–${\overline{B}} _s^0$ mixing;

• the “Unitarity Triangle Parameters” subgroup provides averages for parameters associated with time-dependent $C\!P$ asymmetries and $B \rightarrow DK$ decays, and resulting determinations of the angles of the CKM unitarity triangle;

• the “Semileptonic B Decays” subgroup provides averages for inclusive and exclusive B-decay branching fractions, and subsequent determinations of the CKM matrix element magnitudes $|V_{cb}|$ and $|V_{ub}|$;

• the “B to Charm Decays” subgroup provides averages of branching fractions for B decays to final states involving open charm or charmonium mesons;

• the “Rare Decays” subgroup provides averages of branching fractions and $C\!P$ asymmetries for charmless, radiative, leptonic, and baryonic B-meson and $b$-baryon decays;

• the “Charm Physics” subgroup provides averages of numerous quantities in the charm sector, including branching fractions; properties of excited $D^{**}$ and $D^{}_{sJ}$ mesons; properties of charm baryons; $D^0$–${\overline{D}} ^0$ mixing, $C\!P$, and T violation parameters; and $D^+$ and $D^+_s$ decay constants $f^{}_{D}$ and $f^{}_{D_s}$.

• the “Tau Physics” subgroup provides averages for $\tau$ branching fractions using a global fit and elaborates the results to test lepton universality and to determine the CKM matrix element magnitude $|V_{us}|$; furthermore, it lists the $\tau$ lepton-flavor-violating upper limits and computes the combined upper limits.

Subgroups consist of representatives from experiments producing relevant results in that area, i.e., representatives from BaBar, Belle, BESIII, CDF, CLEO(c), D0, and LHCb.

This article is an update of the last HFLAV preprint, which used results available by summer 2014 [5]. Here we report world averages using results available by summer 2016. In some cases, important new results made available in the latter part of 2016 have been included, or there have been minor revisions in the averages since summer 2016. All plots carry a timestamp indicating when they were produced. In general, we use all publicly available results that are supported by written documentation, including preliminary results presented at conferences or workshops. However, we do not use preliminary results that remain unpublished for an extended period of time, or for which no publication is planned. Close contacts have been established between representatives from the experiments and members of subgroups that perform averaging to ensure that the data are prepared in a form suitable for combinations.

Section 2 describes the methodology used for calculating averages. In the averaging procedure, common input parameters used in the various analyses are adjusted (rescaled) to common values, and, where possible, known correlations are taken into account. Sections 3–9 present world average values from each of the subgroups listed above. A brief summary of the averages presented is given in Sect. 10. A complete listing of the averages and plots, including updates since this document was prepared, are also available on the HFLAV web site: http://www.slac.stanford.edu/xorg/hflav.

2 Averaging methodology

The main task of HFLAV is to combine independent but possibly correlated measurements of a parameter to obtain the world’s best estimate of that parameter’s value and uncertainty. These measurements are typically made by different experiments, or by the same experiment using different data sets, or sometimes by the same experiment using the same data but using different analysis methods. In this section, the general approach adopted by HFLAV is outlined. For some cases, somewhat simplified or more complex algorithms are used; these are noted in the corresponding sections.

Our methodology focuses on the problem of combining measurements obtained with different assumptions about external (or “nuisance”) parameters and with potentially correlated systematic uncertainties. It is important for any averaging procedure that the quantities measured by experiments be statistically well-behaved, which in this context means having a (one- or multi-dimensional) Gaussian likelihood function that is described by the central value(s) $\varvec{x}_i$ and covariance matrix $\varvec{V}_{\!i}$. In what follows we assume $\varvec{x}$ does not contain redundant information, i.e., if it contains n elements then n is the number of parameters being determined. A $\chi ^2$ statistic is constructed as

$$\begin{aligned} \chi ^2(\varvec{x}) = \sum _i^N \left( \varvec{x}_i - \varvec{x} \right) ^\mathrm{T} \varvec{V}_{\!i}^{-1} \left( \varvec{x}_i - \varvec{x} \right) , \end{aligned}$$

(1)

where the sum is over the N independent determinations of the quantities $\varvec{x}$. These are typically from different experiments; possible correlations of the systematic uncertainties are discussed below. The results of the average are the central values $\varvec{\hat{x}}$, which are the values of $\varvec{x}$ at the minimum of $\chi ^2(\varvec{x})$, and their covariance matrix

$$\begin{aligned} \varvec{\hat{V}}^{-1} = \sum _i^N \varvec{V}_{\!i}^{-1}. \end{aligned}$$

(2)

We report the covariance matrices or the correlation matrices derived from the averages whenever possible. In some cases where the matrices are large, it is inconvenient to report them in this document; however, all results can be found on the HFLAV web pages.

The value of $\chi ^2(\varvec{\hat{x}})$ provides a measure of the consistency of the independent measurements of $\varvec{x}$ after accounting for the number of degrees of freedom ($\mathrm{dof}$), which is the difference between the number of measurements and the number of fitted parameters: $N\cdot n - n$. The values of $\chi ^2(\varvec{\hat{x}})$ and $\mathrm{dof}$ are typically converted to a confidence level (C.L.) and reported together with the averages. In cases where $\chi ^2/\mathrm{dof}> 1$, we do not usually scale the resulting uncertainty, in contrast to what is done by the Particle Data Group [6]. Rather, we examine the systematic uncertainties of each measurement to better understand them. Unless we find systematic discrepancies among the measurements, we do not apply any additional correction to the calculated error. If special treatment is necessary to calculate an average, or if an approximation used in the calculation might not be sufficiently accurate (e.g., assuming Gaussian errors when the likelihood function exhibits non-Gaussian behavior), we include a warning message. Further modifications to the averaging procedures for non-Gaussian situations are discussed in Sect. 2.2.

For observables such as branching fractions, experiments typically report upper limits when the signal is not significant. Sometimes there is insufficient information available to combine upper limits on a parameter obtained by different experiments; in this case we usually report only the most restrictive upper limit. For branching fractions of lepton-flavor-violating decays of tau leptons, we calculate combined upper limits as discussed in Sect. 9.6.

2.1 Treatment of correlated systematic uncertainties

Consider two hypothetical measurements of a parameter x, which can be summarized as

$$\begin{aligned}&x_1 \,\pm \, \delta x_1 \,\pm \, \Delta x_{1,1} \,\pm \, \Delta x_{1,2} \ldots \\&x_2 \,\pm \, \delta x_2 \,\pm \, \Delta x_{2,1} \,\pm \, \Delta x_{2,2} \ldots , \end{aligned}$$

where the $\delta x_k$ are statistical uncertainties and the $\Delta x_{k,i}$ are contributions to the systematic uncertainty. The simplest approach is to combine statistical and systematic uncertainties in quadrature:

$$\begin{aligned}&x_1 \,\pm \, \left( \delta x_1 \oplus \Delta x_{1,1} \oplus \Delta x_{1,2} \oplus \cdots \right) \\&x_2 \,\pm \, \left( \delta x_2 \oplus \Delta x_{2,1} \oplus \Delta x_{2,2} \oplus \cdots \right) , \end{aligned}$$

and then perform a weighted average of $x_1$ and $x_2$ using their combined uncertainties, treating the measurements as independent. This approach suffers from two potential problems that we try to address. First, the values $x_k$ may have been obtained using different assumptions for nuisance parameters; e.g., different values of the $B^{0}$lifetime may have been used for different measurements of the oscillation frequency $\mathrm{\Delta }m_d$. The second potential problem is that some systematic uncertainties may be correlated between measurements. For example, different measurements of $\mathrm{\Delta }m_d$ may depend on the same branching fraction used to model a common background.

The above two problems are related, as any quantity $y_i$ upon which $x_k$ depends gives a contribution $\Delta x_{k,i}$ to the systematic error that reflects the uncertainty $\Delta y_i$ on $y_i$. We thus use the values of $y_i$ and $\Delta y_i$ assumed by each measurement in our averaging (we refer to these values as $y_{k,i}$ and $\Delta y_{k,i}$). To properly treat correlated systematic uncertainties among measurements requires decomposing the overall systematic uncertainties into correlated and uncorrelated components. As different measurements often quote different types of systematic uncertainties, achieving consistent definitions in order to properly treat correlations requires close coordination between HFLAV and the experiments. In some cases, a group of systematic uncertainties must be combined into a coarser description in order to obtain an average that is consistent among measurements. Systematic uncertainties that are uncorrelated with any other source of uncertainty are combined together with the statistical error, so that the only systematic uncertainties treated explicitly are those that are correlated with at least one other measurement via a consistently-defined external parameter $y_i$. When asymmetric statistical or systematic uncertainties are quoted by experiments, we symmetrize them since our combination method implicitly assumes Gaussian likelihoods (or parabolic log likelihoods) for each measurement.

The fact that a measurement of x is sensitive to $y_i$ indicates that, in principle, the data used to measure x could also be used for a simultaneous measurement of x and $y_i$. This is illustrated by the large contour in Fig. 1a. However, there often exists an external constraint $\Delta y_i$ on $y_i$ (represented by the horizontal band in Fig. 1a) that is more precise than the constraint $\sigma (y_i)$ from the x data alone. In this case one can perform a simultaneous fit to x and $y_i$, including the external constraint, and obtain the filled (x, y) contour and dashed one-dimensional estimate of x shown in Fig. 1a. For this procedure one usually takes the external constraint $\Delta y_i$ to be Gaussian.

Fig. 1

Illustration of the possible dependence of a measured quantity x on a nuisance parameter $y_i$. The left-hand plot a compares the 68% confidence level contours of a hypothetical measurement’s unconstrained (large ellipse) and constrained (filled ellipse) likelihoods, using the Gaussian constraint on $y_i$ represented by the horizontal band. The solid error bars represent the statistical uncertainties $\sigma (x)$ and $\sigma (y_i)$ of the unconstrained likelihood. The dashed error bar shows the statistical error on x from a constrained simultaneous fit to x and $y_i$. The right-hand plot b illustrates the method described in the text of performing fits to x with $y_i$ fixed at different values. The dashed diagonal line between these fit results has the slope $\rho (x,y_i)\sigma (y_i)/\sigma (x)$ in the limit of an unconstrained parabolic log likelihood. The result of the constrained simultaneous fit from a is shown as a dashed error bar on x

When the external constraints $\Delta y_i$ are significantly more precise than the sensitivity $\sigma (y_i)$ of the data alone, the additional complexity of a constrained fit with extra free parameters may not be justified by the resulting increase in sensitivity. In this case the usual procedure is to perform a baseline fit with all $y_i$ fixed to nominal values $y_{i,0}$, obtaining $x = x_0 \,\pm \, \delta x$. This baseline fit neglects the uncertainty due to $\Delta y_i$, but this error is subsequently recovered by repeating the fit separately for each external parameter $y_i$, with its value fixed to $y_i = y_{i,0}\,\pm \, \Delta y_i$. This gives the result $x = \tilde{x}_{0,i} \,\pm \, \delta \tilde{x}$ as illustrated in Fig. 1b. The shift in the central value $\Delta x_i = \tilde{x}_{0,i} - x_0$ is usually quoted as the systematic uncertainty due to the unknown value of $y_i$. If the unconstrained data can be represented by a Gaussian likelihood function, the shift will equal

$$\begin{aligned} \Delta x_i = \rho (x,y_i)\frac{\sigma (x)}{\sigma (y_i)}\,\Delta y_i, \end{aligned}$$

(3)

where $\sigma (x)$ and $\rho (x,y_i)$ are the statistical uncertainty on x and the correlation between x and $y_i$ in the unconstrained data, respectively. This procedure gives very similar results to that of the constrained fit with extra parameters: the central values $x_0$ agree to $\mathcal{O}(\Delta y_i/\sigma (y_i))^2$, and the uncertainties $\delta x \oplus \Delta x_i$ agree to $\mathcal{O}(\Delta y_i/\sigma (y_i))^4$.

To combine two or more measurements that share systematic uncertainty due to the same external parameter(s) $y_i$, we try to perform a constrained simultaneous fit of all measurements to obtain values of x and $y_i$. When this is not practical, e.g. if we do not have sufficient information to reconstruct the likelihoods corresponding to each measurement, we perform the two-step approximate procedure described below.

Consider two statistically-independent measurements, $x_1 \,\pm \, (\delta x_1 \oplus \Delta x_{1,i})$ and $x_2\,\pm \,(\delta x_2\oplus \Delta x_{2,i})$, of the quantity x as shown in Fig. 2a, b. For simplicity we consider only one correlated systematic uncertainty for each external parameter $y_i$. As our knowledge of the $y_i$ improves, the measurements of x will shift to different central values and uncertainties. The first step of our procedure is to adjust the values of each measurement to reflect the current best knowledge of the external parameters $y_i'$ and their ranges $\Delta y_i'$, as illustrated in Fig. 2c, d. We adjust the central values $x_k$ and correlated systematic uncertainties $\Delta x_{k,i}$ linearly for each measurement (indexed by k) and each external parameter (indexed by i):

$$\begin{aligned}&x_k' = x_k + \sum _i\,\frac{\Delta x_{k,i}}{\Delta y_{k,i}}\bigg (y_i'-y_{k,i}\bigg )\end{aligned}$$

(4)

$$\begin{aligned}&\Delta x_{k,i}'= \Delta x_{k,i} \frac{\Delta y_i'}{\Delta y_{k,i}}. \end{aligned}$$

(5)

This procedure is exact in the limit that the unconstrained likelihood of each measurement is Gaussian.

Fig. 2

Illustration of the HFLAV combination procedure for correlated systematic uncertainties. Upper plots a, b show examples of two individual measurements to be combined. The large (filled) ellipses represent their unconstrained (constrained) likelihoods, while horizontal bands indicate the different assumptions about the value and uncertainty of $y_i$ used by each measurement. The error bars show the results of the method described in the text for obtaining x by performing fits with $y_i$ fixed to different values. Lower plots c, d illustrate the adjustments to accommodate updated and consistent knowledge of $y_i$. Open circles mark the central values of the unadjusted fits to x with y fixed; these determine the dashed line used to obtain the adjusted values

The second step is to combine the adjusted measurements, $x_k'\,\pm \, (\delta x_k\oplus \Delta x_{k,1}'\oplus \Delta x_{k,2}'\oplus \cdots )$ by constructing the goodness-of-fit statistic

$$\begin{aligned}&\chi ^2_{\text {comb}}(x,y_1,y_2,\ldots ) \nonumber \\&\quad \equiv \sum _k\, \frac{1}{\delta x_k^2}\times \bigg [ x_k' - \bigg (x + \sum _i\,(y_i-y_i')\frac{\Delta x_{k,i}'}{\Delta y_i'}\bigg ) \bigg ]^2\nonumber \\&\quad \quad + \sum _i\, \bigg (\frac{y_i - y_i'}{\Delta y_i'}\bigg )^2. \end{aligned}$$

(6)

We minimize this $\chi ^2$ to obtain the best values of x and $y_i$ and their uncertainties, as shown in Fig. 3. Although this method determines new values for the $y_i$, we typically do not report them.

For comparison, the exact method we perform if the unconstrained likelihoods $\mathcal{L}_k(x,y_1,y_2,\ldots )$ are available is to minimize the simultaneous likelihood

$$\begin{aligned} \mathcal{L}_{\text {comb}}(x,y_1,y_2,\ldots ) \equiv \prod _k\,\mathcal{L}_k(x,y_1,y_2,\ldots )\,\prod _{i}\,\mathcal{L}_i(y_i),\nonumber \\ \end{aligned}$$

(7)

with an independent Gaussian constraint for each $y_i$:

$$\begin{aligned} \mathcal{L}_i(y_i) = \exp \bigg [-\frac{1}{2}\,\bigg (\frac{y_i-y_i'}{\Delta y_i'}\bigg )^2\bigg ] \; . \end{aligned}$$

(8)

The results of this exact method agree with those of the approximate method when the $\mathcal{L}_k$ are Gaussian and $\Delta y_i' \ll \sigma (y_i)$. If the likelihoods are non-Gaussian,, experiments need to provide $\mathcal{L}_k$ in order to perform a combination. If $\sigma (y_i)\approx \Delta y_i'$, experiments are encouraged to perform a simultaneous measurement of x and $y_i$ so that their data will improve the world knowledge of $y_i$.

Fig. 3

Illustration of the combination of two hypothetical measurements of x using the method described in the text. The ellipses represent the unconstrained likelihoods of each measurement, and the horizontal band represents the latest knowledge about $y_i$ that is used to adjust the individual measurements. The filled small ellipse shows the result of the exact method using $\mathcal{L}_{\text {comb}}$, and the hollow small ellipse and dot show the result of the approximate method using $\chi ^2_{\text {comb}}$

For averages where common sources of systematic uncertainty are important, central values and uncertainties are rescaled to a common set of input parameters following the prescription above. We use the most up-to-date values for common inputs, consistently across subgroups, taking values from within HFLAV or from the Particle Data Group when possible. The parameters and values used are listed in each subgroup section.

2.2 Treatment of non-Gaussian likelihood functions

For measurements with Gaussian errors, the usual estimator for the average of a set of measurements is obtained by minimizing

$$\begin{aligned} \chi ^2(x) = \sum _k^N \frac{\bigg (x_k-x\bigg )^2}{\sigma ^2_k}, \end{aligned}$$

(9)

where $x_k$ is the kth measured value of x and $\sigma _k^2$ is the variance of the distribution from which $x_k$ was drawn. The value $\hat{x}$ at minimum $\chi ^2$ is the estimate for the parameter x. The true $\sigma _k$ are unknown but typically the error as assigned by the experiment $\sigma _k^\mathrm{raw}$ is used as an estimator for it. However, caution is advised when $\sigma _k^\mathrm{raw}$ depends on the measured value $x_k$. Examples of this are multiplicative systematic uncertainties such as those due to acceptance, or the $\sqrt{N}$ dependence of Poisson statistics for which $x_k \propto N$ and $\sigma _k \propto \sqrt{N}$. Failing to account for this type of dependence when averaging leads to a biased average. Such biases can be avoided by minimizing

$$\begin{aligned} \chi ^2(x) = \sum _k^N \frac{(x_k-x)^2}{\sigma ^2_k(\hat{x})}, \end{aligned}$$

(10)

where $\sigma _k(\hat{x})$ is the uncertainty on $x_k$ that includes the dependence of the uncertainty on the value measured. As an example, consider the error due to acceptance for which $\sigma _k(\hat{x}) = (\hat{x} / x_k)\times \sigma _k^\mathrm{raw}$. Inserting this into Eq. (10) leads to

$$\begin{aligned} \hat{x} = \frac{\sum _k^N x_k^3/(\sigma _k^\mathrm{raw})^2}{\sum _k^N x_k^2/(\sigma _k^\mathrm{raw})^2}, \end{aligned}$$

which is the correct behavior, i.e., weighting by the inverse square of the fractional uncertainty $\sigma _k^\mathrm{raw}/x_k$. It is sometimes difficult to assess the dependence of $\sigma _k^\mathrm{raw}$ on $\hat{x}$ from the errors quoted by the experiments.

Another issue that needs careful treatment is that of correlations among measurements, e.g., due to using the same decay model for intermediate states to calculate acceptances. A common practice is to set the correlation coefficient to unity to indicate full correlation. However, this is not necessarily conservative and can result in underestimated uncertainty on the average. The most conservative choice of correlation coefficient between two measurements i and j is that which maximizes the uncertainty on $\hat{x}$ due to the pair of measurements,

$$\begin{aligned} \sigma _{\hat{x}(i,j)}^2 = \frac{\sigma _i^2\,\sigma _j^2\,(1-\rho _{ij}^2)}{\sigma _i^2 + \sigma _j^2 - 2\,\rho _{ij}\,\sigma _i\,\sigma _j}, \end{aligned}$$

(11)

namely

$$\begin{aligned} \rho _{ij} =\mathrm {min}\bigg (\frac{\sigma _i}{\sigma _j},\frac{\sigma _j}{\sigma _i}\bigg ). \end{aligned}$$

(12)

This corresponds to setting $\sigma _{\hat{x}(i,j)}^2=\mathrm {min}(\sigma _i^2,\sigma _j^2)$. Setting $\rho _{ij}=1$ when $\sigma _i\ne \sigma _j$ can lead to a significant underestimate of the uncertainty on $\hat{x}$, as can be seen from Eq. (11).

Finally, we carefully consider the various errors contributing to the overall uncertainty of an average. The covariance matrix describing the uncertainties of different measurements and their correlations is constructed, i.e., $\varvec{V} = \varvec{V}_\mathrm{stat} + \varvec{V}_\mathrm{sys} + \varvec{V}_\mathrm{theory}$. If the measurements are from independent data samples, then $\varvec{V}_\mathrm{stat}$ is diagonal, but $\varvec{V}_\mathrm{sys}$ and $\varvec{V}_\mathrm{theory}$ may contain correlations. The variance on the average $\hat{x}$ can be written

$$\begin{aligned} \sigma ^2_{\hat{x}}= & {} \frac{ \sum _{i,j} \bigg (\varvec{V}^{-1}\, \bigg [ \varvec{V}_\mathrm{stat}+ \varvec{V}_\mathrm{sys}+ \varvec{V}_\mathrm{theory} \bigg ] \, \varvec{V}^{-1}\bigg )_{ij} }{\left( \sum _{i,j} \varvec{V}^{-1}_{ij}\right) ^2} \nonumber \\= & {} \sigma ^2_{\text {stat}} + \sigma ^2_{\text {sys}} + \sigma ^2_{\text {th}}. \end{aligned}$$

(13)

This breakdown of uncertainties is used in certain cases, but usually only a single, total uncertainty is quoted for an average.

3 Production fractions, lifetimes and mixing parameters of $b$ hadrons

Quantities such as $b$-hadron production fractions, $b$-hadron lifetimes, and neutral $B$-meson oscillation frequencies have been studied in the nineties at LEP and SLC ($e^{+}e^{-}$colliders at $\sqrt{s}=m_{Z}$) as well as at the first version of the Tevatron ($p{\bar{p}}$ collider at $\sqrt{s}=1.8~\mathrm TeV$). This was followed by precise measurements of the $B^0$ and $B^+$ mesons performed at the asymmetric $B$ factories, KEKB and PEPII ($e^{+}e^{-}$colliders at $\sqrt{s}=m_{\varUpsilon (4S)}$), as well as measurements related to the other $b$ hadrons, in particular ${B^{0}_s}$, $B^+_c$ and $\varLambda _b^0$, performed at the upgraded Tevatron ($\sqrt{s}=1.96~\mathrm TeV$). Since a few years, the most precise measurements are coming from the LHC (pp collider at $\sqrt{s}=7$ and $8~\mathrm TeV$), in particular the LHCb experiment.

In most cases, these basic quantities, although interesting by themselves, became necessary ingredients for the more refined measurements, such as those of decay-time dependent $C\!P$-violating asymmetries. It is therefore important that the best experimental values of these quantities continue to be kept up-to-date and improved.

In several cases, the averages presented in this chapter are needed and used as input for the results given in the subsequent chapters. Within this chapter, some averages need the knowledge of other averages in a circular way. This coupling, which appears through the $b$-hadron fractions whenever inclusive or semi-exclusive measurements have to be considered, has been reduced drastically in the past several years with increasingly precise exclusive measurements becoming available and dominating practically all averages.

Table 1 Published measurements of the $B^+/B^0$ production ratio in $\varUpsilon (4S)$ decays, together with their average (see text). Systematic uncertainties due to the imperfect knowledge of $\tau (B^+)/\tau (B^0)$ are included

In addition to $b$-hadron fractions, lifetimes and oscillation frequencies, this chapter also deals with $C\!P$ violation in the $B^0$ and ${B^{0}_s}$ mixing amplitudes, as well as the $C\!P$-violating phase $\phi _s^{c{{\bar{c}}}s} \simeq -2\beta _s$, which is the phase difference between the ${B^{0}_s}$ mixing amplitude and the $b\rightarrow c{\bar{c}}s$ decay amplitude. The angle $\beta$, which is the equivalent of $\beta _s$ for the $B^0$ system, is discussed in Sect. 4.

Throughout this chapter published results that have been superseded by subsequent publications are ignored (i.e., excluded from the averages) and are only referred to if necessary.

3.1 $b$-hadron production fractions

We consider here the relative fractions of the different $b$-hadron species found in an unbiased sample of weakly decaying $b$ hadrons produced under some specific conditions. The knowledge of these fractions is useful to characterize the signal composition in inclusive $b$-hadron analyses, to predict the background composition in exclusive analyses, or to convert (relative) observed rates into (relative) branching fraction measurements. We distinguish here the following three conditions: $\varUpsilon (4S)$ decays, $\varUpsilon (5S)$ decays, and high-energy collisions (including $Z^0$ decays).

3.1.1 $b$-hadron production fractions in $\varUpsilon (4S)$ decays

Only pairs of the two lightest (charged and neutral) $B$ mesons can be produced in $\varUpsilon (4S)$ decays. Therefore only the following two branching fractions must be considered:

$$\begin{aligned} f^{+-}= & {} \Gamma (\varUpsilon (4S) \rightarrow B^+B^-)/ \Gamma _\mathrm{tot}(\varUpsilon (4S)), \end{aligned}$$

(14)

$$\begin{aligned} f^{00}= & {} \Gamma (\varUpsilon (4S) \rightarrow B^0{\bar{B}}^0)/ \Gamma _\mathrm{tot}(\varUpsilon (4S)). \end{aligned}$$

(15)

In practice, most analyses measure their ratio

$$\begin{aligned} R^{+-/00}= & {} f^{+-}/f^{00}\nonumber \\= & {} \Gamma (\varUpsilon (4S) \rightarrow B^+B^-)/\Gamma (\varUpsilon (4S) \rightarrow B^0{\bar{B}}^0),\nonumber \\ \end{aligned}$$

(16)

which is easier to access experimentally. Since an inclusive (but separate) reconstruction of $B^+$ and $B^0$ is difficult, exclusive decay modes to specific final states f, ${B^+} \rightarrow f^+$ and ${B^0} \rightarrow f^0$, are usually considered to perform a measurement of $R^{+-/00}$, whenever they can be related by isospin symmetry (for example $B^+ \rightarrow {J/\psi } K^+$ and $B^0 \rightarrow {J/\psi } K^0$). Under the assumption that $\Gamma (B^+ \rightarrow f^+) = \Gamma (B^0 \rightarrow f^0)$, i.e., that isospin invariance holds in these $B$ decays, the ratio of the number of reconstructed $B^+ \rightarrow f^+$ and $B^0 \rightarrow f^0$ mesons, after correcting for efficiency, is proportional to

$$\begin{aligned} \frac{f^{+-}\, {\mathcal {B}}(B^+ \rightarrow f^+)}{f^{00}\, {\mathcal {B}}(B^0 \rightarrow f^0)}= & {} \frac{f^{+-}\, \Gamma ({B^+}\rightarrow f^+)\, \tau (B^+)}{f^{00}\, \Gamma ({B^0}\rightarrow f^0)\,\tau (B^0)} \nonumber \\= & {} \frac{f^{+-}}{f^{00}} \, \frac{\tau (B^+)}{\tau (B^0)}, \end{aligned}$$

(17)

where $\tau (B^+)$ and $\tau (B^0)$ are the $B^+$ and $B^0$ lifetimes respectively. Hence the primary quantity measured in these analyses is $R^{+-/00} \, \tau (B^+)/\tau (B^0)$, and the extraction of $R^{+-/00}$ with this method therefore requires the knowledge of the $\tau (B^+)/\tau (B^0)$ lifetime ratio.

The published measurements of $R^{+-/00}$ are listed in Table 1 ² together with the corresponding assumed values of $\tau (B^+)/\tau (B^0)$. All measurements are based on the above-mentioned method, except the one from Belle, which is a by-product of the $B^0$ mixing frequency analysis using dilepton events (but note that it also assumes isospin invariance, namely $\Gamma (B^+ \rightarrow \ell ^+\mathrm{X}) = \Gamma (B^0 \rightarrow \ell ^+\mathrm{X})$). The latter is therefore treated in a slightly different manner in the following procedure used to combine these measurements:

• each published value of $R^{+-/00}$ from CLEO and BaBar is first converted back to the original measurement of $R^{+-/00} \, \tau (B^+)/\tau (B^0)$, using the value of the lifetime ratio assumed in the corresponding analysis;

• a simple weighted average of these original measurements of $R^{+-/00} \, \tau (B^+)/\tau (B^0)$ from CLEO and BaBar is then computed, assuming no statistical or systematic correlations between them;

• the weighted average of $R^{+-/00} \, \tau (B^+)/\tau (B^0)$ is converted into a value of $R^{+-/00}$, using the latest average of the lifetime ratios, $\tau (B^+)/\tau (B^0)={{1.076}}{{{\,\pm \,}}0.004}$ (see Sect. 3.2.3);

• the Belle measurement of $R^{+-/00}$ is adjusted to the current values of $\tau (B^0)=1.520{{\,\pm \,}}0.004~\mathrm ps$ and $\tau (B^+)/\tau (B^0)={{1.076}}{{{\,\pm \,}}0.004}$ (see Sect. 3.2.3), using the quoted systematic uncertainties due to these parameters;

• the combined value of $R^{+-/00}$ from CLEO and BaBar is averaged with the adjusted value of $R^{+-/00}$ from Belle, assuming a 100% correlation of the systematic uncertainty due to the limited knowledge on $\tau (B^+)/\tau (B^0)$; no other correlation is considered.

Table 2 Published measurements of $f^{\varUpsilon (5S)}_{s}$, obtained assuming $f^{\varUpsilon (5S)}_{B\!\!\!\!/}=0$ and quoted as in the original publications, except for the 2010 Belle measurement, which is quoted as $1-f^{\varUpsilon (5S)}_{u,d}$ with $f^{\varUpsilon (5S)}_{u,d}$ from Ref. [1]. Our average of $f^{\varUpsilon (5S)}_{s}$ assuming $f^{\varUpsilon (5S)}_{B\!\!\!\!/}=0$, given on the penultimate line, does not include the most recent Belle result quoted on the last line (see footnote 4)

The resulting global average,

$$\begin{aligned} R^{+-/00} = \frac{f^{+-}}{f^{00}} = {1.059}{{{\,\pm \,}}0.027}, \end{aligned}$$

(18)

is consistent with equal production rate of charged and neutral $B$ mesons, although only at the $2.2\,\sigma$ level.

On the other hand, the BaBar collaboration has performed a direct measurement of the $f^{00}$ fraction using an original method, which neither relies on isospin symmetry nor requires the knowledge of $\tau (B^+)/\tau (B^0)$. Its analysis, based on a comparison between the number of events where a single $B^0 \rightarrow D^{*-} \ell ^+ \nu$ decay could be reconstructed and the number of events where two such decays could be reconstructed, yields [12]

$$\begin{aligned} f^{00}= 0.487 \,\pm \, 0.010\,\text{(stat) } \,\pm \, 0.008\,\text{(syst) }. \end{aligned}$$

(19)

The two results of Eqs. (18) and (19) are of very different natures and completely independent of each other. Their product is equal to $f^{+-} = 0.516{\,\pm \,}0.019$, while another combination of them gives $f^{+-} + f^{00}= 1.003{\,\pm \,}0.029$, compatible with unity. Assuming³ $f^{+-}+f^{00}= 1$, also consistent with CLEO’s observation that the fraction of $\varUpsilon (4S)$ decays to $B{\overline{B}}$ pairs is larger than 0.96 at 95% CL [16], the results of Eqs. (18) and (19) can be averaged (first converting Eq. (18) into a value of $f^{00}=1/(R^{+-/00}+1)$) to yield the following more precise estimates:

$$\begin{aligned}&f^{00} = 0.486\,{{\,\pm \,}}0.006,\quad f^{+-} = 1 -f^{00} = 0.514\,{\,\pm \,}0.006,\nonumber \\&\quad \frac{f^{+-}}{f^{00}} = 1.058\,{\,\pm \,}0.024. \end{aligned}$$

(20)

The latter ratio differs from one by $2.4\,\sigma$.

3.1.2 $b$-hadron production fractions in $\varUpsilon (5S)$ decays

Hadronic events produced in $e^+e^-$ collisions at the $\varUpsilon (5S)$ (also known as $\varUpsilon (10860)$) energy can be classified into three categories: light-quark (u, d, s, c) continuum events, $b{\bar{b}}$ continuum events, and $\varUpsilon (5S)$ events. The latter two cannot be distinguished and will be called $b{\bar{b}}$ events in the following. These $b{\bar{b}}$ events, which also include $b{\bar{b}}\gamma$ events because of possible initial-state radiation, can hadronize in different final states. We define $f^{\varUpsilon (5S)}_{u,d}$ as the fraction of $b{\bar{b}}$ events with a pair of non-strange bottom mesons ($B{\bar{B}}$, $B{\bar{B}}^*$, $B^*{\bar{B}}$, $B^*{\bar{B}}^*$, $B{\bar{B}}\pi$, $B{\bar{B}}^*\pi$, $B^*{\bar{B}}\pi$, $B^*{\bar{B}}^*\pi$, and $B{\bar{B}}\pi \pi$ final states, where B denotes a $B^0$ or $B^+$ meson and ${\bar{B}}$ denotes a ${\bar{B}}^0$ or $B^-$ meson), $f^{\varUpsilon (5S)}_{s}$ as the fraction of $b{\bar{b}}$ events with a pair of strange bottom mesons ($B_s^0{\bar{B}}_s^0$, $B_s^0{\bar{B}}_s^{*0}$, $B_s^{*0}{\bar{B}}_s^0$, and $B_s^{*0}{\bar{B}}_s^{*0}$ final states), and $f^{\varUpsilon (5S)}_{B\!\!\!\!/}$ as the fraction of $b{\bar{b}}$ events without any bottom meson in the final state. Note that the excited bottom-meson states decay via $B^* \rightarrow B \gamma$ and $B_s^{*0} \rightarrow B_s^0 \gamma$. These fractions satisfy

$$\begin{aligned} f^{\varUpsilon (5S)}_{u,d}+ f^{\varUpsilon (5S)}_{s}+ f^{\varUpsilon (5S)}_{B\!\!\!\!/}= 1. \end{aligned}$$

(21)

Table 3 External inputs on which the $f^{\varUpsilon (5S)}_{s}$ averages are based

The CLEO and Belle collaborations have published measurements of several inclusive $\varUpsilon (5S)$ branching fractions, ${\mathcal {B}}(\varUpsilon (5S) \rightarrow D_s X)$, ${\mathcal {B}}(\varUpsilon (5S) \rightarrow \phi X)$ and ${\mathcal {B}}(\varUpsilon (5S) \rightarrow D^0 X)$, from which they extracted the model-dependent estimates of $f^{\varUpsilon (5S)}_{s}$ reported in Table 2. This extraction was performed under the implicit assumption $f^{\varUpsilon (5S)}_{B\!\!\!\!/}=0$, using the relation

$$\begin{aligned}&\frac{1}{2}{\mathcal {B}}(\varUpsilon (5S) \rightarrow D_s X)=f^{\varUpsilon (5S)}_{s}\times {\mathcal {B}}(B_s^0\rightarrow D_s X) \nonumber \\&\quad + \bigg (1-f^{\varUpsilon (5S)}_{s}-f^{\varUpsilon (5S)}_{B\!\!\!\!/}\bigg )\times {\mathcal {B}}(B\rightarrow D_s X), \end{aligned}$$

(22)

and similar relations for ${\mathcal {B}}(\varUpsilon (5S) \rightarrow D^0 X)$ and ${\mathcal {B}}(\varUpsilon (5S) \rightarrow \phi X)$. In Table 2 we list also the values of $f^{\varUpsilon (5S)}_{s}$ derived from measurements of $f^{\varUpsilon (5S)}_{u,d}={\mathcal {B}}(\varUpsilon (5S) \rightarrow B{\bar{B}}X)$ [17, 18], as well as our average value of $f^{\varUpsilon (5S)}_{s}$, all obtained under the assumption $f^{\varUpsilon (5S)}_{B\!\!\!\!/}=0$.

However, the assumption $f^{\varUpsilon (5S)}_{B\!\!\!\!/}=0$ is known to be invalid since the observation of the following final states in $e^+e^-$ collisions at the $\varUpsilon (5S)$ energy: $\varUpsilon (1S)\pi ^+\pi ^-$, $\varUpsilon (2S)\pi ^+\pi ^-$, $\varUpsilon (3S)\pi ^+\pi ^-$ and $\varUpsilon (1S)K^+K^-$ [22, 23], $h_b(1P)\pi ^+\pi ^-$ and $h_b(2P)\pi ^+\pi ^-$ [24], and more recently $\varUpsilon (1S)\pi ^0\pi ^0$, $\varUpsilon (2S)\pi ^0\pi ^0$ and $\varUpsilon (3S)\pi ^0\pi ^0$ [25]. The sum of the measurements of the corresponding visible cross-sections, adding also the contributions of the unmeasured $\varUpsilon (1S)K^0{\bar{K}}^0$, $h_b(1P)\pi ^0\pi ^0$ and $h_b(2P)\pi ^0\pi ^0$ final states assuming isospin conservation, amounts to

$$\begin{aligned}&\sigma ^\mathrm{vis}(e^+e^-\rightarrow (b {\bar{b}})hh) = 13.2\,\pm \,1.4~\mathrm{pb},\nonumber \\&\quad ~~\text{ for }~(b {\bar{b}})=\varUpsilon (1S,2S,3S),h_b(1P,2P)~\mathrm{and}~ hh=\pi \pi ,KK. \end{aligned}$$

We divide this by the $b {\bar{b}}$ production cross section, $\sigma (e^+e^- \rightarrow b {\bar{b}} X) = 337 \,\pm \, 15$ pb, obtained as the average of the CLEO [21] and Belle [20]$^{4}$ measurements, to obtain

$$\begin{aligned}&{\mathcal {B}}(\varUpsilon (5S) \rightarrow (b {\bar{b}})hh) = 0.039\,\pm \,0.004,\nonumber \\&\quad \text{ for }~ (b {\bar{b}})=\varUpsilon (1S,2S,3S),h_b(1P,2P)~\mathrm{and}~hh=\pi \pi ,KK, \end{aligned}$$

which is to be considered as a lower bound for $f^{\varUpsilon (5S)}_{B\!\!\!\!/}$.

Following the method described in Ref. [1], we perform a $\chi ^2$ fit of the original measurements of the $\varUpsilon (5S)$ branching fractions of Refs.  [17,18,19],⁴ using the inputs of Table 3, the relations of Eqs. (21) and (22) and the one-sided Gaussian constraint $f^{\varUpsilon (5S)}_{B\!\!\!\!/}\ge {\mathcal {B}}(\varUpsilon (5S) \rightarrow (b {\bar{b}}) hh)$, to simultaneously extract $f^{\varUpsilon (5S)}_{u,d}$, $f^{\varUpsilon (5S)}_{s}$ and $f^{\varUpsilon (5S)}_{B\!\!\!\!/}$. Taking all known correlations into account, the best fit values are

$$\begin{aligned} f^{\varUpsilon (5S)}_{u,d}= & {} 0.761^{+0.027}_{-0.042}, \end{aligned}$$

(23)

$$\begin{aligned} f^{\varUpsilon (5S)}_{s}= & {} 0.200^{+0.030}_{-0.031}, \end{aligned}$$

(24)

$$\begin{aligned} f^{\varUpsilon (5S)}_{B\!\!\!\!/}= & {} 0.039^{+0.050}_{-0.004}, \end{aligned}$$

(25)

where the strongly asymmetric uncertainty on $f^{\varUpsilon (5S)}_{B\!\!\!\!/}$ is due to the one-sided constraint from the observed $(b {\bar{b}}) hh$ decays. These results, together with their correlation, imply

$$\begin{aligned} f^{\varUpsilon (5S)}_{s}/f^{\varUpsilon (5S)}_{u,d}= & {} 0.263^{+0.052}_{-0.044}, \end{aligned}$$

(26)

in fair agreement with the results of a BaBar analysis [27], performed as a function of centre-of-mass energy.⁵

The production of $B^0_s$ mesons at the $\varUpsilon (5S)$ is observed to be dominated by the $B_s^{*0}{\bar{B}}_s^{*0}$ channel, with $\sigma (e^+e^- \rightarrow B_s^{*0}{\bar{B}}_s^{*0})/ \sigma (e^+e^- \rightarrow B_s^{(*)0}{\bar{B}}_s^{(*)0}) = (87.0\,\pm \, 1.7)\%$ [28, 29]. The proportions of the various production channels for non-strange B mesons have also been measured [17].

3.1.3 $b$-hadron production fractions at high energy

At high energy, all species of weakly decaying $b$ hadrons may be produced, either directly or in strong and electromagnetic decays of excited $b$ hadrons. It is often assumed that the fractions of these different species are the same in unbiased samples of high-$p_\mathrm{T}$ $b$ jets originating from $Z^0$ decays, from $p{\bar{p}}$ collisions at the Tevatron, or from $p p$ collisions at the LHC. This hypothesis is plausible under the condition that the square of the momentum transfer to the produced $b$ quarks, $Q^2$, is large compared with the square of the hadronization energy scale, $Q^2 \gg \varLambda _\mathrm{QCD}^2$. On the other hand, there is no strong argument that the fractions at different machines should be strictly equal, so this assumption should be checked experimentally. The available data show that the fractions depend on the kinematics of the produced $b$ hadron. A simple phenomenological model appears to agree with all data and indicates that the fractions are constant if the $b$ hadron is produced with sufficiently high transverse momentum from any collider. Unless otherwise indicated, these fractions are assumed to be equal at all high-energy colliders until demonstrated otherwise by experiment. Both CDF and LHCb report a $p_\mathrm{T}$ dependence for $\varLambda _b^0$ production relative to $B^+$ and $B^0$; the number of $\varLambda _b^0$ baryons observed at low $p_\mathrm{T}$ is enhanced with respect to that seen at LEP’s higher $p_\mathrm{T}$. Therefore we present three sets of complete averages: one set including only measurements performed at LEP, a second set including only measurements performed at the Tevatron, a third set including measurements performed at LEP, Tevatron and LHC. The LHCb production fractions results by themselves are still incomplete, lacking measurements of the production of weakly-decaying baryons heavier than $\varLambda _b^0$.

Fig. 4

Ratio of production fractions $f_{\varLambda _b^0}/f_{d}$ as a function of $p_\mathrm{T}$ of the $b$ hadron from LHCb data for $b$ hadrons decaying semileptonically [46] and fully reconstructed in hadronic decays [48]. The curve represents a fit to the LHCb hadronic data [48]. The computed LEP ratio is included at an approximate $p_\mathrm{T}$ in Z decays, but does not participate in any fit

Contrary to what happens in the charm sector where the fractions of $D^+$ and $D^0$ are different, the relative amount of $B^+$ and $B^0$ is not affected by the electromagnetic decays of excited $B^{*+}$ and $B^{*0}$ states and strong decays of excited $B^{**+}$ and $B^{**0}$ states. Decays of the type $B_s^{**0} \rightarrow B^{(*)}K$ also contribute to the $B^+$ and $B^0$ rates, but with the same magnitude if mass effects can be neglected. We therefore assume equal production of $B^+$ and $B^0$ mesons. We also neglect the production of weakly decaying states made of several heavy quarks (like $B^+_c$ and doubly heavy baryons) which is known to be very small. Hence, for the purpose of determining the $b$-hadron fractions, we use the constraints

$$\begin{aligned} f_{u} = f_{d} \quad \text{ and }\quad f_{u} + f_{d} + f_{s} + f_\mathrm{baryon} = 1, \end{aligned}$$

(27)

where $f_{u}$, $f_{d}$, $f_{s}$ and $f_\mathrm{baryon}$ are the unbiased fractions of $B^+$, $B^0$, ${B^{0}_s}$ and $b$ baryons, respectively.

We note that there are many measurements of the production cross-sections of different species of $b$ hadrons. In principle these could be included in a global fit to determine the production fractions. We do not perform such a fit at the current time, and instead average only the explicit measurements of the production fractions.

The LEP experiments have measured $f_{s} \times {\mathcal {B}}({B^{0}_s} \rightarrow D_s^- \ell ^+ \nu _\ell {X})$ [30,31,32], ${\mathcal {B}}(b \rightarrow \varLambda _b^0) \times {\mathcal {B}}(\varLambda _b^0 \rightarrow \varLambda _c^+ \ell ^-{\bar{\nu }}_\ell {X})$ [33, 34] and ${\mathcal {B}}(b \rightarrow \varXi _b ^-) \times {\mathcal {B}}(\varXi _b^- \rightarrow \varXi ^-\ell ^-\overline{\nu }_\ell {X})$ [35, 36] from partially reconstructed final states including a lepton, $f_\mathrm{baryon}$ from protons identified in $b$ events [37], and the production rate of charged $b$ hadrons [38]. Ratios of $b$-hadron fractions have been measured by CDF using lepton+charm final states [39,40,41]⁶ and double semileptonic decays with $K^*\mu \mu$ and $\phi \mu \mu$ final states [42]. Measurements of the production of other heavy flavour baryons at the Tevatron are included in the determination of $f_\mathrm{baryon}$  [43,44,45]⁷ using the constraint

$$\begin{aligned} f_\mathrm{baryon}= & {} f_{\varLambda _b^0} + f_{\varXi _b^0} + f_{\varXi _b^-} + f_{\Omega _b^-} \nonumber \\= & {} f_{\varLambda _b^0}\bigg (1 + 2\frac{f_{\varXi _b^-}}{f_{\varLambda _b^0}} + \frac{f_{\Omega _b^-}}{f_{\varLambda _b^0}}\bigg ), \end{aligned}$$

(28)

where isospin invariance is assumed in the production of $\varXi _b^0$ and $\varXi _b^-$. Other $b$ baryons are expected to decay strongly or electromagnetically to those baryons listed. For the production measurements, both CDF and D0 reconstruct their $b$ baryons exclusively to final states which include a ${J/\psi }$ and a hyperon ($\varLambda _b^0 \rightarrow {J/\psi } \varLambda$, $\varXi _b^- \rightarrow {J/\psi } \varXi ^-$ and $\Omega _b^- \rightarrow {J/\psi } \Omega ^-$). We assume that the partial decay width of a $b$ baryon to a ${J/\psi }$ and the corresponding hyperon is equal to the partial width of any other $b$ baryon to a ${J/\psi }$ and the corresponding hyperon. LHCb has also measured ratios of $b$-hadron fractions in charm+lepton final states [46] and in fully reconstructed hadronic two-body decays $B^0 \rightarrow D^-\pi ^+$, ${B^{0}_s} \rightarrow D_s^- \pi ^+$ and $\varLambda _b^0 \rightarrow \varLambda _c^+ \pi ^-$ [47, 48].

Both CDF and LHCb observe that the ratio $f_{\varLambda _b^0}/f_{d}$ depends on the $p_\mathrm{T}$ of the charm+lepton system [41, 46].⁸ CDF chose to correct an older result to account for the $p_\mathrm{T}$ dependence. In a second result, CDF binned their data in $p_\mathrm{T}$ of the charm+electron system [40]. The more recent LHCb measurement using hadronic decays [48] obtains the scale for $R_{\varLambda _b^0} = f_{\varLambda _b^0}/f_{d}$ from their previous charm + lepton data [46]. The LHCb measurement using hadronic data also bins the same data in pseudorapidity ($\eta$) and sees a linear dependence of $R_{\varLambda _b^0}$. Since $\eta$ is not entirely independent of $p_\mathrm{T}$ it is impossible to tell at this time whether this dependence is just an artifact of the $p_\mathrm{T}$ dependence. Figure 4 shows the ratio $R_{\varLambda _b^0}$ as a function of $p_\mathrm{T}$ for the $b$ hadron, as measured by LHCb. LHCb fits their scaled hadronic data to obtain

$$\begin{aligned} R_{\varLambda _b^0}= & {} (0.151\,\pm \, 0.030) + \exp \bigg \{-(0.57\,\pm \, 0.11)\nonumber \\&- (0.095\,\pm \, 0.016)[{\mathrm{GeV/}c} ]^{-1} \times p_\mathrm{T}\bigg \}. \end{aligned}$$

(29)

A value of $R_{\varLambda _b^0}$ is also calculated for LEP and placed at the approximate $p_\mathrm{T}$ for the charm+lepton system, but this value does not participate in any fit.⁹ Because the two LHCb results for $R_{\varLambda _b^0}$ are not independent, we use only their semileptonic data for the averages. Note that the $p_\mathrm{T}$ dependence of $R_{\varLambda _b^0}$ combined with the constraint from Eq. (27) implies a compensating $p_\mathrm{T}$ dependence in one or more of the production fractions, $f_{u}$, $f_{d}$, or $f_{s}$.

Fig. 5

Ratio of production fractions $f_{s}/f_{d}$ as a function of $p_\mathrm{T}$ of the reconstructed $b$ hadrons for the LHCb [47] and ATLAS [49] data. Note the suppressed zero for the vertical axis. The curves represent fits to the data: a linear fit (solid), and an exponential fit described in the text (dotted). The $p_\mathrm{T}$ independent value average of $R_s$ (dashed) is shown for comparison. The computed LEP ratio is included at an approximate $p_\mathrm{T}$ in Z decays, but does not participate in any fit

LHCb and ATLAS have investigated the $p_\mathrm{T}$ dependence of $f_{s}/f_{d}$ using fully reconstructed ${B^{0}_s}$ and $B^0$ decays. LHCb reported $3\sigma$ evidence that the ratio $R_s = f_{s}/f_{d}$ decreases with $p_\mathrm{T}$ using fully reconstructed ${B^{0}_s}$ and $B^0$ decays and theoretical predictions for branching ratios [47]. Data from the ATLAS experiment [49] using decays of ${B^{0}_s}$ and $B^0$ to $J/\psi$ final states and using theoretical predictions for branching ratios [50] indicates that $R_s$ is consistent with no $p_T$ dependence. Figure 5 shows the ratio $R_s$ as a function of $p_\mathrm{T}$ measured by LHCb and ATLAS. Two fits are performed. The first fit, using a linear parameterization, yields $R_s = (0.2701\,\pm \, 0.0058) - (0.00139\,\pm \, 0.00044)[{\mathrm{GeV/}c} ]^{-1} \times p_\mathrm{T}$. A second fit, using a simple exponential, yields $R_s = \exp \bigg \{(-1.304\,\pm \, 0.024) - (0.0058\,\pm \, 0.0019)[{\mathrm{GeV/}c} ]^{-1} \times p_\mathrm{T}\bigg \}$. The two fits are nearly indistinguishable over the $p_\mathrm{T}$ range of the results, but the second gives a physical value for all $p_\mathrm{T}$. $R_s$ is also calculated for LEP and placed at the approximate $p_\mathrm{T}$ for the $b$ hadron, though the LEP result doesn’t participate in the fit. Our world average for $R_s$ is also included in the figure for reference.

In order to combine or compare LHCb results with other experiments, the $p_\mathrm{T}$-dependent $f_{\varLambda _b^0}/(f_{u} + f_{d})$ is weighted by the $p_\mathrm{T}$ spectrum.¹⁰ Table 4 compares the $p_\mathrm{T}$-weighted LHCb data with comparable averages from CDF. The average CDF and LHCb data are in agreement despite the $b$ hadrons being produced in different kinematic regimes.

Table 4 Comparison of average production fraction ratios from CDF [40, 41] and LHCb [46]. The kinematic regime of the charm+lepton system reconstructed in each experiment is also shown

Ignoring $p_\mathrm{T}$ dependence, all these published results have been adjusted to the latest branching fraction averages [6] and combined following the procedure and assumptions described in Ref. [1], to yield $f_{u} =f_{d} =0.404{\,\pm \,}0.006$, $f_{s} =0.102{\,\pm \,}0.005$ and $f_\mathrm{baryon} =0.090{\,\pm \,}0.012$ under the constraints of Eq. (27). Repeating the combinations for LEP and the Tevatron, we obtain $f_{u} =f_{d} =0.412{\,\pm \,}0.008$, $f_{s} =0.088{\,\pm \,}0.013$ and $f_\mathrm{baryon} =0.089{\,\pm \,}0.012$ when using the LEP data only, and $f_{u} =f_{d} =0.340{\,\pm \,}0.021$, $f_{s} =0.101{\,\pm \,}0.015$ and $f_\mathrm{baryon} = 0.218{\,\pm \,}0.047$ when using the Tevatron data only. As noted previously, the LHCb data are insufficient to determine a complete set of $b$-hadron production fractions. The world averages (LEP, Tevatron and LHC) for the various fractions are presented here for comparison with previous averages. Significant differences exist between the LEP and Tevatron fractions, therefore use of the world averages should be taken with some care. For these combinations other external inputs are used, e.g., the branching ratios of $B$ mesons to final states with a $D$ or $D^*$ in semileptonic decays, which are needed to evaluate the fraction of semileptonic ${B^{0}_s}$ decays with a $D_s^-$ in the final state.

Table 5 Time-integrated mixing probability $\overline{\chi }$ (defined in Eq. (30)), and fractions of the different $b$-hadron species in an unbiased sample of weakly decaying $b$ hadrons, obtained from both direct and mixing measurements. The correlation coefficients between the fractions are also given. The last column includes measurements performed at LEP, Tevatron and LHC

Time-integrated mixing analyses performed with lepton pairs from $b{\bar{b}}$ events produced at high-energy colliders measure the quantity

$$\begin{aligned} \overline{\chi } = f'_{d} \,\chi _{d} + f'_{s} \,\chi _{s}, \end{aligned}$$

(30)

where $f'_{d}$ and $f'_{s}$ are the fractions of $B^0$ and ${B^{0}_s}$ hadrons in a sample of semileptonic $b$-hadron decays, and where $\chi _{d}$ and $\chi _{s}$ are the $B^0$ and ${B^{0}_s}$ time-integrated mixing probabilities. Assuming that all $b$ hadrons have the same semileptonic decay width implies $f'_i = f_i R_i$, where $R_i = \tau _i/\tau _{b}$ is the ratio of the lifetime $\tau _i$ of species i to the average $b$-hadron lifetime $\tau _{b} = \sum _i f_i \tau _i$. Hence measurements of the mixing probabilities $\overline{\chi }$, $\chi _{d}$ and $\chi _{s}$ can be used to improve our knowledge of $f_{u}$, $f_{d}$, $f_{s}$ and $f_\mathrm{baryon}$. In practice, the above relations yield another determination of $f_{s}$ obtained from $f_\mathrm{baryon}$ and mixing information,

$$\begin{aligned} f_{s} = \frac{1}{R_{s}} \frac{(1+r)\overline{\chi }-(1-f_\mathrm{baryon} R_\mathrm{baryon}) \chi _{d}}{(1+r)\chi _{s}- \chi _{d}}, \end{aligned}$$

(31)

where $r=R_{u}/R_{d} = \tau (B^+)/\tau (B^0)$.

The published measurements of $\overline{\chi }$ performed by the LEP experiments have been combined by the LEP Electroweak Working Group to yield $\overline{\chi } = 0.1259{\,\pm \,}0.0042$ [51].¹¹ This can be compared with the Tevatron average, $\overline{\chi } = 0.147{{\,\pm \,}}0.011$, obtained from D0  [52] and CDF [53]. The two averages deviate from each other by $1.8\,\sigma$; this could be an indication that the production fractions of $b$ hadrons at the $Z$ peak or at the Tevatron are not the same. We choose to combine these two results in a simple weighted average, assuming no correlations, and, following the PDG prescription, we multiply the combined uncertainty by 1.8to account for the discrepancy. Our world average is then $\overline{\chi } = 0.1284{{\,\pm \,}}0.0069$.

Introducing the $\overline{\chi }$ average in Eq. (31), together with our world average $\chi _{d} = 0.1860{{\,\pm \,}}0.0011$ [see Eq. (67) of Sect. 3.3.1], the assumption $\chi _{s} = 1/2$ [justified by Eq. (76) in Sect. 3.3.2], the best knowledge of the lifetimes (see Sect. 3.2) and the estimate of $f_\mathrm{baryon}$ given above, yields $f_{s} = 0.118{{\,\pm \,}}0.018$ (or $f_{s} = 0.111{{\,\pm \,}}0.011$ using only LEP data, or $f_{s} = 0.166{\,\pm \,}0.029$ using only Tevatron data), an estimate dominated by the mixing information. Taking into account all known correlations (including that introduced by $f_\mathrm{baryon}$), this result is then combined with the set of fractions obtained from direct measurements (given above), to yield the improved estimates of Table 5, still under the constraints of Eq. (27). As can be seen, our knowledge on the mixing parameters reduces the uncertainty on $f_{s}$, quite substantially in the case of LEP data. It should be noted that the results are correlated, as indicated in Table 5.

3.2 $b$-hadron lifetimes

In the spectator model the decay of $b$ hadrons $H_b$ is governed entirely by the flavour changing $b\rightarrow Wq$ transition ($q =c,u$). For this very reason, lifetimes of all $b$ hadrons are the same in the spectator approximation regardless of the (spectator) quark content of the $H_b$. In the early 1990’s experiments became sophisticated enough to start seeing the differences of the lifetimes among various $H_b$ species. The first theoretical calculations of the spectator quark effects on $H_b$ lifetime emerged only few years earlier [55].

Since then, such calculations are performed in the framework of the Heavy Quark Expansion (HQE) [55,56,57], using as most important assumption that of quark-hadron duality [58, 59]. Since a few years, possible quark-hadron duality violating effects are severely constrained by experiments [60]. In these calculations, the total decay rate of an $H_b$ is expressed as the sum of a series of expectation values of operators of increasing dimension, multiplied by the correspondingly higher powers of $\varLambda _\mathrm{QCD}/m_b$:

$$\begin{aligned} \Gamma _{H_b} = |\mathrm{CKM}|^2 \sum _n c_n \bigg (\frac{\varLambda _\mathrm{QCD}}{m_b}\bigg )^n \langle H_b|O_n|H_b\rangle , \end{aligned}$$

(32)

where $|\mathrm{CKM}|^2$ is the relevant combination of CKM matrix elements. The coefficients $c_n$ of this expansion, known as the Operator Product Expansion [61], can be calculated perturbatively. Hence, the HQE predicts $\Gamma _{H_b}$ in the form of an expansion in both $\varLambda _\mathrm{QCD}/m_{b}$ and $\alpha _s(m_{b})$. The precision of current experiments requires an expansion up to the next-to-leading order in QCD, i.e., the inclusion of corrections of the order of $\alpha _s(m_{b})$ to the $c_n$ terms. The non-perturbative parts of the calculation are grouped into the expectation values $\langle H_b|O_n|H_b\rangle$ of operators $O_n$. These can be calculated using lattice QCD or QCD sum rules, or can be related to other observables via the HQE. One may reasonably expect that powers of $\varLambda _\mathrm{QCD}/m_{b}$ provide enough suppression that only the first few terms of the sum in Eq. (32) matter.

Theoretical predictions are usually made for the ratios of the lifetimes (with $\tau (B^0)$ often chosen as the common denominator) rather than for the individual lifetimes, for this allows several uncertainties to cancel. The precision of the HQE calculations (see Refs. [1], and Refs. [1] for the latest updates) is in some instances already surpassed by the measurements, e.g., in the case of $\tau (B^+)/\tau (B^0)$. More accurate predictions are now a matter of progress in the evaluation of the non-perturbative hadronic matrix elements, in particular using lattice QCD where significant advances were made in the last decade. However, the following important conclusions can be drawn from the HQE, even in its present state, which are in agreement with experimental observations:

• The heavier the mass of the heavy quark, the smaller is the variation in the lifetimes among different hadrons containing this quark, which is to say that as $m_{b}\rightarrow \infty$ we retrieve the spectator picture in which the lifetimes of all $H_b$ states are the same. This is well illustrated by the fact that lifetimes are rather similar in the $b$ sector, while they differ by large factors in the charm sector ($m_{c}<m_{b}$).

• The non-perturbative corrections arise only at the order of $\varLambda _\mathrm{QCD}^2/m_{b}^2$, which translates into differences among $H_b$ lifetimes of only a few percent.

• It is only the difference between meson and baryon lifetimes that appears at the $\varLambda _\mathrm{QCD}^2/m_{b}^2$ level. The splitting of the meson lifetimes occurs at the $\varLambda _\mathrm{QCD}^3/m_{b}^3$ level, yet it is enhanced by a phase space factor $16\pi ^2$ with respect to the leading free $b$ decay.

To ensure that certain sources of systematic uncertainty cancel, lifetime analyses are sometimes designed to measure ratios of lifetimes. However, because of the differences in decay topologies, abundance (or lack thereof) of decays of a certain kind, etc., measurements of the individual lifetimes are also common. In the following section we review the most common types of lifetime measurements. This discussion is followed by the presentation of the averaging of the various lifetime measurements, each with a brief description of its particularities.

3.2.1 Lifetime measurements, uncertainties and correlations

In most cases, the lifetime of an $H_b$ state is estimated from a flight distance measurement and a $\beta \gamma$ factor which is used to convert the geometrical distance into the proper decay time. Methods of accessing lifetime information can roughly be divided in the following five categories:

• 1. Inclusive (flavour-blind) measurements. These early measurements were aimed at extracting the lifetime from a mixture of $b$-hadron decays, without distinguishing the decaying species. Often the knowledge of the mixture composition was limited, which made these measurements experiment-specific. Also, these measurements had to rely on Monte Carlo simulation for estimating the $\beta \gamma$ factor, because the decaying hadrons are not fully reconstructed. These were usually the largest statistics $b$-hadron lifetime measurements accessible to a given experiment, and could therefore serve as an important performance benchmark.

• 2. Measurements in semileptonic decays of a specific $\varvec{H}_{\varvec{b}}$. The $W$ boson from $b \rightarrow Wc$ produces a $\ell \nu _l$ pair ($\ell =e,\mu$) in about 21% of the cases. The electron or muon from such decays provides a clean and efficient trigger signature. The $c$ quark from the $b\rightarrow Wc$ transition and the other quark(s) making up the decaying $H_b$ combine into a charm hadron, which is reconstructed in one or more exclusive decay channels. Knowing what this charmed hadron is allows one to separate, at least statistically, different $H_b$ species. The advantage of these measurements is in the sample size, which is usually larger than in the case of exclusively reconstructed $H_b$ decays. Some of the main disadvantages are related to the difficulty of estimating the lepton+charm sample composition and to the Monte Carlo reliance for the momentum (and hence $\beta \gamma$ factor) estimate.

• 3. Measurements in exclusively reconstructed hadronic decays. These have the advantage of complete reconstruction of the decaying $H_b$ state, which allows one to infer the decaying species as well as to perform precise measurement of the $\beta \gamma$ factor. Both lead to generally smaller systematic uncertainties than in the above two categories. The downsides are smaller branching ratios and larger combinatorial backgrounds, especially in $H_b\rightarrow H_c\pi (\pi \pi )$ and multi-body $H_c$ decays, or in a hadron collider environment with non-trivial underlying event. Decays of the type $H_b\rightarrow {J/\psi } H_s$ are relatively clean and easy to trigger, due to the ${J/\psi } \rightarrow \ell ^+\ell ^-$ signature, but their branching fraction is only about 1%.

• 4. Measurements at asymmetric B factories. In the $\varUpsilon (4S) \rightarrow B {\bar{B}}$ decay, the $B$ mesons ($B^+$ or $B^0$) are essentially at rest in the $\varUpsilon (4S)$ frame. This makes direct lifetime measurements impossible in experiments at symmetric colliders producing $\varUpsilon (4S)$ at rest. At asymmetric $B$ factories the $\varUpsilon (4S)$ meson is boosted resulting in $B$ and ${\bar{B}}$ moving nearly parallel to each other with the same boost. The lifetime is inferred from the distance $\Delta z$ separating the $B$ and ${\bar{B}}$ decay vertices along the beam axis and from the $\varUpsilon (4S)$ boost known from the beam energies. This boost is equal to $\beta \gamma \approx 0.55$ (0.43) in the BaBar (Belle) experiment, resulting in an average $B$ decay length of approximately 250 (190) $\upmu$m. In order to determine the charge of the $B$ mesons in each event, one of them is fully reconstructed in a semileptonic or hadronic decay mode. The other $B$ is typically not fully reconstructed, only the position of its decay vertex is determined from the remaining tracks in the event. These measurements benefit from large sample sizes, but suffer from poor proper time resolution, comparable to the $B$ lifetime itself. This resolution is dominated by the uncertainty on the decay vertices, which is typically 50 (100) $\upmu$m for a fully (partially) reconstructed $B$ meson. With much larger samples in the future, the resolution and purity could be improved (and hence the systematics reduced) by fully reconstructing both $B$ mesons in the event.

• 5. Direct measurement of lifetime ratios. This method, initially applied in the measurement of $\tau (B^+)/\tau (B^0)$, is now also used for other $b$-hadron species at the LHC. The ratio of the lifetimes is extracted from the proper time dependence of the ratio of the observed yields of two different $b$-hadron species, both reconstructed in decay modes with similar topologies. The advantage of this method is that subtle efficiency effects (partially) cancel in the ratio.

In some of the latest analyses, measurements of two (e.g., $\tau (B^+)$ and $\tau (B^+)/\tau (B^0)$) or three (e.g. $\tau (B^+)$, $\tau (B^+)/\tau (B^0)$, and $\Delta m_{d}$) quantities are combined. This introduces correlations among measurements. Another source of correlations among the measurements are the systematic effects, which could be common to an experiment or to an analysis technique across the experiments. When calculating the averages, such known correlations are taken into account.

3.2.2 Inclusive $b$-hadron lifetimes

The inclusive $b$-hadron lifetime is defined as $\tau _{b} = \sum _i f_i \tau _i$ where $\tau _i$ are the individual species lifetimes and $f_i$ are the fractions of the various species present in an unbiased sample of weakly decaying $b$ hadrons produced at a high-energy collider.¹² This quantity is certainly less fundamental than the lifetimes of the individual species, the latter being much more useful in comparisons of the measurements with the theoretical predictions. Nonetheless, we perform the averaging of the inclusive lifetime measurements for completeness and because they might be of interest as “technical numbers.”

In practice, an unbiased measurement of the inclusive lifetime is difficult to achieve, because it would imply an efficiency which is guaranteed to be the same across species. So most of the measurements are biased. In an attempt to group analyses that are expected to select the same mixture of $b$ hadrons, the available results (given in Table 6) are divided into the following three sets:

• 1. measurements at LEP and SLD that include any $b$-hadron decay, based on topological reconstruction (secondary vertex or track impact parameters);

• 2. measurements at LEP based on the identification of a lepton from a $b$ decay; and

• 3. measurements at hadron colliders based on inclusive $H_b\rightarrow {J/\psi } X$ reconstruction, where the ${J/\psi }$ is fully reconstructed.

Table 6 Measurements of average $b$-hadron lifetimes

The measurements of the first set are generally considered as estimates of $\tau _{b}$, although the efficiency to reconstruct a secondary vertex most probably depends, in an analysis-specific way, on the number of tracks coming from the vertex, thereby depending on the type of the $H_b$. Even though these efficiency variations can in principle be accounted for using Monte Carlo simulations (which inevitably contain assumptions on branching fractions), the $H_b$ mixture in that case can remain somewhat ill-defined and could be slightly different among analyses in this set.

On the contrary, the mixtures corresponding to the other two sets of measurements are better defined in the limit where the reconstruction and selection efficiency of a lepton or a ${J/\psi }$ from an $H_b$ does not depend on the decaying hadron type. These mixtures are given by the production fractions and the inclusive branching fractions for each $H_b$ species to give a lepton or a ${J/\psi }$. In particular, under the assumption that all $b$ hadrons have the same semileptonic decay width, the analyses of the second set should measure $\tau (b \rightarrow \ell ) = (\sum _i f_i \tau _i^3) /(\sum _i f_i \tau _i^2)$ which is necessarily larger than $\tau _{b}$ if lifetime differences exist. Given the present knowledge on $\tau _i$ and $f_i$, $\tau (b \rightarrow \ell )-\tau _{b}$ is expected to be of the order of 0.003 $\mathrm ps$. On the other hand, the third set measuring $\tau (b \rightarrow {J/\psi } )$ is expected to give an average smaller than $\tau _{b}$ because of the $B^+_c$ meson, which has a significantly larger probability to decay to a ${J/\psi }$ than other $b$-hadron species.

Measurements by SLC and LEP experiments are subject to a number of common systematic uncertainties, such as those due to (lack of knowledge of) $b$ and c fragmentation, $b$ and c decay models, ${\mathcal {B}}(B\rightarrow \ell )$, ${\mathcal {B}}(B\rightarrow c\rightarrow \ell )$, ${\mathcal {B}}(c\rightarrow \ell )$, $\tau _{{c}}$, and $H_b$ decay multiplicity. In the averaging, these systematic uncertainties are assumed to be 100% correlated. The averages for the sets defined above (also given in Table 6) are

$$\begin{aligned}&\tau (b ~\text{ vertex }) = 1.572{\,\pm \,}0.009~\mathrm ps, \end{aligned}$$

(33)

$$\begin{aligned}&\tau (b \rightarrow \ell ) = 1.537{\,\pm \,}0.020~\mathrm ps, \end{aligned}$$

(34)

$$\begin{aligned}&\tau (b \rightarrow {J/\psi } ) = 1.533{\,\pm \,}0.036~\mathrm ps. \end{aligned}$$

(35)

The differences between these averages are consistent both with zero and with expectations within less than $2\,\sigma$.

3.2.3 $B^0$ and $B^+$ lifetimes and their ratio

After a number of years of dominating these averages the LEP experiments yielded the scene to the asymmetric $B$  factories and the Tevatron experiments. The $B$  factories have been very successful in utilizing their potential – in only a few years of running, BaBar and, to a greater extent, Belle, have struck a balance between the statistical and the systematic uncertainties, with both being close to (or even better than) an impressive 1% level. In the meanwhile, CDF and D0 have emerged as significant contributors to the field as the Tevatron Run II data flowed in. Recently, the LHCb experiment reached a further step in precision, improving by a factor $\sim$2 over the previous best measurements.

At the present time we are in an interesting position of having three sets of measurements (from LEP/SLC, $B$ factories and Tevatron/LHC) that originate from different environments, are obtained using substantially different techniques and are precise enough for incisive comparison.

Table 7 Measurements of the $B^0$ lifetime

Table 8 Measurements of the $B^+$ lifetime

Table 9 Measurements of the ratio $\tau (B^+)/\tau (B^0)$

Table 10 Measurements of the effective ${B^{0}_s}$ lifetimes obtained from single exponential fits

The averaging of $\tau (B^+)$, $\tau (B^0)$ and $\tau (B^+)/\tau (B^0)$ measurements is summarized in Tables , , and . For $\tau (B^+)/\tau (B^0)$ we average only the measurements of this quantity provided by experiments rather than using all available knowledge, which would have included, for example, $\tau (B^+)$ and $\tau (B^0)$ measurements which did not contribute to any of the ratio measurements.

The following sources of correlated (within experiment/machine) systematic uncertainties have been considered:

• for SLC/LEP measurements – $D^{**}$ branching ratio uncertainties [3], momentum estimation of $b$ mesons from $Z^0$ decays ($b$-quark fragmentation parameter $\langle X_E \rangle = 0.702 \,\pm \, 0.008$ [3]), ${B^{0}_s}$ and $b$-baryon lifetimes (see Sects. , ), and $b$-hadron fractions at high energy (see Table 5);

• for $B$-factory measurements – alignment, z scale, machine boost, sample composition (where applicable);

• for Tevatron/LHC measurements – alignment (separately within each experiment).

The resultant averages are:

$$\begin{aligned}&\tau (B^0) = 1.520{{\,\pm \,}}0.004~\mathrm ps, \end{aligned}$$

(36)

$$\begin{aligned}&\tau (B^+) = 1.638{{\,\pm \,}}0.004~\mathrm ps, \end{aligned}$$

(37)

$$\begin{aligned}&\tau (B^+)/\tau (B^0) = {{1.076}}{{{\,\pm \,}}0.004}. \end{aligned}$$

(38)

3.2.4 ${B^{0}_s}$ lifetimes

Like neutral kaons, neutral $B$ mesons contain short- and long-lived components, since the light (L) and heavy (H) eigenstates differ not only in their masses, but also in their total decay widths. Neglecting $C\!P$ violation in ${B^{0}_s}-{\bar{B}}^0_s$ mixing, which is expected to be very small [60, 107,108,109,110] (see also Sect. 3.3.3), the mass eigenstates are also $C\!P$ eigenstates, with the light state being $C\!P$-even and the heavy state being $C\!P$-odd. While the decay width difference $\Delta \Gamma _{\mathrm{d}}$ can be neglected in the $B^0$ system, the ${B^{0}_s}$ system exhibits a significant value of $\Delta \Gamma _{s} = \Gamma _{s\mathrm L} - \Gamma _{s\mathrm H}$, where $\Gamma _{s\mathrm L}$ and $\Gamma _{s\mathrm H}$ are the total decay widths of the light eigenstate $B ^0_{s\mathrm L}$ and the heavy eigenstate $B ^0_{s\mathrm H}$, respectively. The sign of $\Delta \Gamma _{s}$ is known to be positive [111], i.e., $B ^0_{s\mathrm H}$ lives longer than $B ^0_{s\mathrm L}$. Specific measurements of $\Delta \Gamma _{s}$ and $\Gamma _{s} = (\Gamma _{s\mathrm L} + \Gamma _{s\mathrm H})/2$ are explained and averaged in Sect. 3.3.2, but the results for $1/\Gamma _{s\mathrm L} = 1/(\Gamma _{s} +\Delta \Gamma _{s} /2)$, $1/\Gamma _{s\mathrm H}= 1/(\Gamma _{s}-\Delta \Gamma _{s} /2)$ and the mean ${B^{0}_s}$ lifetime, defined as $\tau ({B^{0}_s}) = 1/\Gamma _{s}$, are also quoted at the end of this section.

Many ${B^{0}_s}$ lifetime analyses, in particular the early ones performed before the non-zero value of $\Delta \Gamma _{s}$ was firmly established, ignore $\Delta \Gamma _{s}$ and fit the proper time distribution of a sample of ${B^{0}_s}$ candidates reconstructed in a certain final state f with a model assuming a single exponential function for the signal. We denote such effective lifetime measurements [112] as $\tau _\mathrm{single}({B^{0}_s} \rightarrow f)$; their true values may lie a priori anywhere between $1/\Gamma _{s\mathrm L}$ and $1/\Gamma _{s,\mathrm H}$, depending on the proportion of $B^0_{s\mathrm L}$ and $B^0_{s\mathrm H}$ in the final state f. More recent determinations of effective lifetimes may be interpreted as measurements of the relative composition of $B^0_{s\mathrm L}$ and $B^0_{s\mathrm H}$ decaying to the final state f. Table 10 summarizes the effective lifetime measurements.

Averaging measurements of $\tau _\mathrm{single}({B^{0}_s} \rightarrow f)$ over several final states f will yield a result corresponding to an ill-defined observable when the proportions of $B^0_{s\mathrm L}$ and $B^0_{s\mathrm H}$ differ. Therefore, the effective ${B^{0}_s}$ lifetime measurements are broken down into several categories and averaged separately.

• ${{B^{0}_s}} {\rightarrow } {D}_s^{{{\mp }}} {X}$ decays include mostly flavour-specific decays but also decays with an unknown mixture of light and heavy components. Measurements performed with such inclusive states are no longer used in averages.

• Decays to flavour-specific final states, i.e., decays to final states f with decay amplitudes satisfying $A({B^{0}_s} \rightarrow f) \ne 0$, $A({\bar{B}}^0_s \rightarrow {\bar{f}}) \ne 0$, $A({B^{0}_s} \rightarrow {\bar{f}}) = 0$ and $A({\bar{B}}^0_s \rightarrow f)=0$, have equal fractions of $B^0_{s\mathrm L}$ and $B^0_{s\mathrm H}$ at time zero. Their total untagged time-dependent decay rates $\Gamma _s(t)$ have a mean value $\int _0^\infty t\Gamma _s(t)dt/\int _0^\infty \Gamma _s(t)dt$, called the flavour-specific lifetime, equal to [131]

  $$\begin{aligned}&\tau _\mathrm{single}({B^{0}_s} \rightarrow \text{ flavour } \text{ specific }) \nonumber \\&\quad = \frac{1/\Gamma _{s\mathrm L}^2+1/\Gamma _{s\mathrm H}^2}{1/\Gamma _{s\mathrm L}+1/\Gamma _{s\mathrm H}} = \frac{1}{\Gamma _{s}} \, \frac{{1+\bigg (\frac{\Delta \Gamma _{s} }{2\Gamma _{s}}\bigg )^2}}{{1-\bigg (\frac{\Delta \Gamma _{s} }{2\Gamma _{s}}\bigg )^2} }. \quad \end{aligned}$$

  (39)

  Because of the fast ${B^{0}_s}-{\bar{B}}^0_s$ oscillations, possible biases of the flavour-specific lifetime due to a combination of ${B^{0}_s}/{\bar{B}}^0_s$ production asymmetry, $C\!P$ violation in the decay amplitudes ($|A({B^{0}_s} \rightarrow f)| \ne |A({\bar{B}}^0_s \rightarrow {\bar{f}})|$), and $C\!P$ violation in ${B^{0}_s}-{\bar{B}}^0_s$ mixing ($|q_{s}/p_{s}| \ne 1$) are strongly suppressed, by a factor ${\sim }$ $x_s^2$ (given in Eq. (75)). The ${B^{0}_s}/{\bar{B}}^0_s$ production asymmetry at LHCb and the $C\!P$ asymmetry due to mixing have been measured to be compatible with zero with a precision below 3% [132] and 0.3% [see Eq. (83)], respectively. The corresponding effects on the flavour-specific lifetime, which therefore have a relative size of the order of $10^{-5}$ or smaller, can be neglected at the current level of experimental precision. Under the assumption of no production asymmetry and no $C\!P$ violation in mixing, Eq. (39) is exact even for a flavour-specific decay with $C\!P$ violation in the decay amplitudes. Hence any flavour-specific decay mode can be used to measure the flavour-specific lifetime. The average of all flavour-specific ${B^{0}_s}$ lifetime measurements [95, 104, 116,117,118,119,120,121,122] is

  $$\begin{aligned} \tau _\mathrm{single}({B^{0}_s} \rightarrow \text{ flavour } \text{ specific }) = 1.516{{\,\pm \,}}0.014~\mathrm ps. \end{aligned}$$

  (40)

• $\varvec{{B^{0}_s}} \varvec{\rightarrow } \varvec{{J/\psi }} \varvec{\phi }$ decays contain a well-measured mixture of $C\!P$-even and $C\!P$-odd states. There are no known correlations between the existing ${B^{0}_s} \rightarrow {J/\psi } \phi$ effective lifetime measurements; these are combined into the average $\tau _\mathrm{single}({B^{0}_s} \rightarrow {J/\psi } \phi ) = 1.479{{\,\pm \,}}0.012~\mathrm ps$. A caveat is that different experimental acceptances may lead to different admixtures of the $C\!P$-even and $C\!P$-odd states, and simple fits to a single exponential may result in inherently different values of $\tau _\mathrm{single}({B^{0}_s} \rightarrow {J/\psi } \phi )$. Analyses that separate the $C\!P$-even and $C\!P$-odd components in this decay through a full angular study, outlined in Sect. 3.3.2, provide directly precise measurements of $1/\Gamma _{s}$ and $\Delta \Gamma _{s}$ (see Table 21).

• Decays to CP eigenstates have also been measured, in the $C\!P$-even modes ${B^{0}_s} \rightarrow D_s^{(*)+}D_s^{(*)-}$ by ALEPH [124], ${B^{0}_s} \rightarrow K^+ K^-$ by LHCb [104, 125], ${B^{0}_s} \rightarrow D_s^+D_s^-$ by LHCb [121] and ${B^{0}_s} \rightarrow J/\psi \eta$ by LHCb [126], as well as in the $C\!P$-odd modes ${B^{0}_s} \rightarrow {J/\psi } f_0(980)$ by CDF [128] and D0  [129], ${B^{0}_s} \rightarrow {J/\psi } \pi ^+\pi ^-$ by LHCb [130] and ${B^{0}_s} \rightarrow {J/\psi } K^0_\mathrm{S}$ by LHCb [127]. If these decays are dominated by a single weak phase and if $C\!P$ violation can be neglected, then $\tau _\mathrm{single}({B^{0}_s} \rightarrow C\!P \text{-even }) = 1/\Gamma _{s\mathrm L}$ and $\tau _\mathrm{single}({B^{0}_s} \rightarrow C\!P \text{-odd }) = 1/\Gamma _{s\mathrm H}$ [see Eqs. (70) and (71) for approximate relations in presence of mixing-induced $C\!P$ violation]. However, not all these modes can be considered as pure $C\!P$ eigenstates: a small $C\!P$-odd component is most probably present in ${B^{0}_s} \rightarrow D_s^{(*)+}D_s^{(*)-}$ decays. Furthermore the decays ${B^{0}_s} \rightarrow K^+ K^-$ and ${B^{0}_s} \rightarrow {J/\psi } K^0_\mathrm{S}$ may suffer from direct $C\!P$ violation due to interfering tree and loop amplitudes. The averages for the effective lifetimes obtained for decays to pure $C\!P$-even ($D_s^+D_s^-$, ${J/\psi } \eta$) and $C\!P$-odd (${J/\psi } f_0(980)$, ${J/\psi } \pi ^+\pi ^-$) final states, where $C\!P$ conservation can be assumed, are

  $$\begin{aligned}&\tau _{\mathrm{single}}({B^{0}_s} \rightarrow C\!P \text{-even }) = 1.422{{\,\pm \,}}0.023~\mathrm ps, \end{aligned}$$

  (41)
  $$\begin{aligned}&\tau _{\mathrm{single}}({B^{0}_s} \rightarrow C\!P \text{-odd }) = 1.658{{\,\pm \,}}0.032~\mathrm ps. \end{aligned}$$

  (42)

Table 11 Measurements of the $B^+_c$ lifetime

As described in Sect. 3.3.2, the effective lifetime averages of Eqs. (40), (41) and (42) are used as ingredients to improve the determination of $1/\Gamma _{s}$ and $\Delta \Gamma _{s}$ obtained from the full angular analyses of ${B^{0}_s} \rightarrow {J/\psi } \phi$ and ${B^{0}_s} \rightarrow {J/\psi } K^+K^-$ decays. The resulting world averages for the ${B^{0}_s}$ lifetimes are

$$\begin{aligned}&\tau (B^0_{s\mathrm L}) = \frac{1}{\Gamma _{s\mathrm L}} = \frac{1}{\Gamma _{s} +\Delta \Gamma _{s} /2} = 1.413{{\,\pm \,}}0.006~\mathrm ps, \end{aligned}$$

(43)

$$\begin{aligned}&\tau (B^0_{s\mathrm H}) = \frac{1}{\Gamma _{s\mathrm H}} = \frac{1}{\Gamma _{s}-\Delta \Gamma _{s} /2} = 1.609{{\,\pm \,}}0.010~\mathrm ps, \end{aligned}$$

(44)

$$\begin{aligned}&\tau ({B^{0}_s}) = \frac{1}{\Gamma _{s}} = \frac{2}{\Gamma _{s\mathrm L}+\Gamma _{s\mathrm H}} = 1.505{{\,\pm \,}}0.005~\mathrm ps. \end{aligned}$$

(45)

3.2.5 $B^+_c$ lifetime

Early measurements of the $B^+_c$ meson lifetime, from CDF [133, 134] and D0  [135], use the semileptonic decay mode $B^+_c \rightarrow {J/\psi } \ell ^+ \nu$ and are based on a simultaneous fit to the mass and lifetime using the vertex formed with the leptons from the decay of the ${J/\psi }$ and the third lepton. Correction factors to estimate the boost due to the missing neutrino are used. Correlated systematic errors include the impact of the uncertainty of the $B^+_c$ $p_T$ spectrum on the correction factors, the level of feed-down from $\psi (2S)$ decays, Monte Carlo modeling of the decay model varying from phase space to the ISGW model, and mass variations. With more statistics, CDF2 was able to perform the first $B^+_c$ lifetime based on fully reconstructed $B^+_c \rightarrow J/\psi \pi ^+$ decays [136], which does not suffer from a missing neutrino. Recent measurements from LHCb, both with $B^+_c \rightarrow {J/\psi } \mu ^+ \nu$  [137] and $B^+_c \rightarrow {J/\psi } \pi ^+$  [138] decays, achieve the highest level of precision.

All the measurements are summarized in Table 11 and the world average, dominated by the LHCb measurements, is determined to be

$$\begin{aligned} \tau (B^+_c) = 0.507{{\,\pm \,}}0.009~\mathrm ps. \end{aligned}$$

(46)

3.2.6 $\varLambda _b^0$ and $b$-baryon lifetimes

The first measurements of $b$-baryon lifetimes, performed at LEP, originate from two classes of partially reconstructed decays. In the first class, decays with an exclusively reconstructed $\varLambda _c^+$ baryon and a lepton of opposite charge are used. These products are more likely to occur in the decay of $\varLambda _b^0$ baryons. In the second class, more inclusive final states with a baryon ($p$, ${\bar{p}}$, $\varLambda$, or ${\bar{\varLambda }}$) and a lepton have been used, and these final states can generally arise from any $b$ baryon. With the large $b$-hadron samples available at the Tevatron and the LHC, the most precise measurements of $b$ baryons now come from fully reconstructed exclusive decays.

The following sources of correlated systematic uncertainties have been considered: experimental time resolution within a given experiment, $b$-quark fragmentation distribution into weakly decaying $b$ baryons, $\varLambda _b^0$ polarisation, decay model, and evaluation of the $b$-baryon purity in the selected event samples. In computing the averages the central values of the masses are scaled to $M(\varLambda _b^0) = 5619.51 \,\pm \, 0.23~\mathrm MeV/{c}^2$ [6].

For measurements with partially reconstructed decays, the meaning of the decay model systematic uncertainties and the correlation of these uncertainties between measurements are not always clear. Uncertainties related to the decay model are dominated by assumptions on the fraction of n-body semileptonic decays. To be conservative, it is assumed that these are 100% correlated whenever given as an error. DELPHI varies the fraction of four-body decays from 0.0 to 0.3. In computing the average, the DELPHI result is corrected to a value of $0.2 \,\pm \, 0.2$ for this fraction. Furthermore the semileptonic decay results from LEP are corrected for a $\varLambda _b^0$ polarisation of $-0.45^{+0.19}_{-0.17}$ [3] and a b fragmentation parameter $\langle x_E \rangle _b =0.702\,\pm \, 0.008$ [51].

Table 12 Measurements of the $b$-baryon lifetimes

The list of all measurements are given in Table 12. We do not attempt to average measurements performed with $p\ell$ or $\varLambda \ell$ correlations, which select unknown mixtures of b baryons. Measurements performed with $\varLambda _c^+ \ell$ or $\varLambda \ell ^+\ell ^-$ correlations can be assumed to correspond to semileptonic $\varLambda _b^0$ decays. Their average (${1.247_{0.069}^{+0.071}}$ ps)) is significantly different from the average using only measurements performed with exclusively reconstructed hadronic $\varLambda _b^0$ decays (1.470${{\,\pm \,}}0.010$ $\mathrm ps$). The latter is much more precise and less prone to potential biases than the former. The discrepancy between the two averages is at the level of $3.1\,\sigma$ and assumed to be due to an experimental systematic effect in the semileptonic measurements or to a rare statistical fluctuation. The best estimate of the $\varLambda _b^0$ lifetime is therefore taken as the average of the exclusive measurements only. The CDF $\varLambda _b^0 \rightarrow {J/\psi } \varLambda$ lifetime result [145] is larger than the average of all other exclusive measurements by $2.4\,\sigma$. It is nonetheless kept in the average without adjustment of input errors. The world average $\varLambda _b^0$ lifetime is then

$$\begin{aligned} \tau (\varLambda _b^0) = 1.470{{\,\pm \,}}0.010~\mathrm ps. \end{aligned}$$

(47)

For the strange $b$ baryons, we do not include the measurements based on inclusive $\varXi ^{{\mp }} \ell ^{{\mp }}$ [35, 36, 148] final states, which consist of a mixture of $\varXi _b^-$ and $\varXi _b^0$ baryons. Instead we only average results obtained with fully reconstructed $\varXi _b^-$, $\varXi _b^0$ and $\Omega _b^-$ baryons, and obtain

$$\begin{aligned} \tau (\varXi _b^-)= & {} 1.571{{\,\pm \,}}0.040~\mathrm ps, \end{aligned}$$

(48)

$$\begin{aligned} \tau (\varXi _b^0)= & {} 1.479{{\,\pm \,}}0.031~\mathrm ps, \end{aligned}$$

(49)

$$\begin{aligned} \tau (\Omega _b^-)= & {} 1.64^{+0.18}_{-0.17}~\mathrm ps. \end{aligned}$$

(50)

It should be noted that several b-baryon lifetime measurements from LHCb  [147, 150,151,152] were made with respect to the lifetime of another b hadron (i.e., the original measurement is that of a decay width difference). Before these measurements are included in the averages quoted above, we rescale them according to our latest lifetime average of that reference b hadron. This introduces correlations between our averages, in particular between the $\varXi _b^-$ and $\varXi _b^0$ lifetimes. Taking this correlation into account leads to

$$\begin{aligned} \tau (\varXi _b^0) / \tau (\varXi _b^-) = 0.929{{\,\pm \,}}0.028. \end{aligned}$$

(51)

3.2.7 Summary and comparison with theoretical predictions

Averages of lifetimes of specific $b$-hadron species are collected in Table 13. As described in the introduction to Sect. 3.2, the HQE can be employed to explain the hierarchy of $\tau (B^+_c) \ll \tau (\varLambda _b^0)< \tau ({B^{0}_s}) \approx \tau (B^0) < \tau (B^+)$, and used to predict the ratios between lifetimes. Recent predictions are compared to the measured lifetime ratios in Table 14.

The predictions of the ratio between the $B^+$ and $B^0$ lifetimes, $1.06 \,\pm \, 0.02$ [65, 66] or $1.04 ^{+0.05}_{-0.01} \,\pm \, 0.02 \,\pm \, 0.01$ [68, 69], are in good agreement with experiment.

The total widths of the ${B^{0}_s}$ and $B^0$ mesons are expected to be very close and differ by at most 1% [67,68,69, 153, 154]. This prediction is consistent with the experimental ratio $\tau ({B^{0}_s})/\tau (B^0)=\Gamma _{\mathrm{d}}/\Gamma _{s}$, which is smaller than 1 by $(1.0{{\,\pm \,}}0.4)\%$. The authors of Refs. [60, 107] predict $\tau ({B^{0}_s})/\tau (B^0) = 1.00050 \,\pm \, 0.00108 \,\pm \, 0.0225\times \delta$, where $\delta$ quantifies a possible breaking of the quark-hadron duality. In this context, they interpret the $2.5\sigma$ difference between theory and experiment as being due to either new physics or a sizable duality violation. The key message is that improved experimental precision on this ratio is very welcome.

The ratio $\tau (\varLambda _b^0)/\tau (B^0)$ has particularly been the source of theoretical scrutiny since earlier calculations using the HQE [55,56,57, 155] predicted a value larger than 0.90, almost $2\,\sigma$ above the world average at the time. Many predictions cluster around a most likely central value of 0.94 [159]. Calculations of this ratio that include higher-order effects predict a lower ratio between the $\varLambda _b^0$ and $B^0$ lifetimes [65,66,67] and reduce this difference. Since then the experimental average is now definitely settling at a value significantly larger than initially, in agreement with the latest theoretical predictions. A recent review [68, 69] concludes that the long-standing $\varLambda _b^0$ lifetime puzzle is resolved, with a nice agreement between the precise experimental determination of $\tau (\varLambda _b^0)/\tau (B^0)$ and the less precise HQE prediction which needs new lattice calculations. There is also good agreement for the $\tau (\varXi _b^0)/\tau (\varXi _b^-)$ ratio.

Table 13 Summary of the lifetime averages for the different $b$-hadron species

Table 14 Experimental averages of $b$-hadron lifetime ratios and Heavy-Quark Expansion (HQE) predictions [68, 69]

The lifetimes of the most abundant $b$-hadron species are now all known to sub-percent precision. Neglecting the contributions of the rarer species ($B^+_c$ meson and $b$ baryons other than the $\varLambda _b^0$), one can compute the average $b$-hadron lifetime from the individual lifetimes and production fractions as

$$\begin{aligned} \tau _b = \frac{f_{d} \tau (B^0)^2+ f_{u} \tau (B^+)^2+0.5 f_{s} \tau (B^0_{s\mathrm H})^2+0.5 f_{s} \tau (B^0_{s\mathrm L})^2+ f_\mathrm{baryon} \tau (\varLambda _b^0)^2}{f_{d} \tau (B^0) + f_{u} \tau (B^+) +0.5 f_{s} \tau (B^0_{s\mathrm H}) +0.5 f_{s} \tau (B^0_{s\mathrm L}) + f_\mathrm{baryon} \tau (\varLambda _b^0) }. \end{aligned}$$

(52)

Using the lifetimes of Table 13 and the fractions in Z decays of Table 5, taking into account the correlations between the fractions (Table 5) as well as the correlation between $\tau (B_{s\mathrm H})$ and $\tau (B_{s\mathrm L})$ ($-$0.398), one obtains

$$\begin{aligned} \tau _b(Z) = 1.566{{\,\pm \,}}0.003~\mathrm ps. \end{aligned}$$

(53)

This is in very good agreement with (and three times more precise than) the average of Eq. (33 34 35) for the inclusive measurements performed at LEP.

3.3 Neutral $B$-meson mixing

The $B^0-{\bar{B}}^0$ and ${B^{0}_s}-{\bar{B}}^0_s$ systems both exhibit the phenomenon of particle-antiparticle mixing. For each of them, there are two mass eigenstates which are linear combinations of the two flavour states, $B^0_q$ and ${\bar{B}}^0_q$,

$$\begin{aligned} | B^0_{q\mathrm L}\rangle= & {} p_q |B^0_q \rangle + q_q |{\bar{B}}^0_q \rangle , \end{aligned}$$

(54)

$$\begin{aligned} | B^0_{q\mathrm H}\rangle= & {} p_q |B^0_q \rangle - q_q |{\bar{B}}^0_q \rangle , \end{aligned}$$

(55)

where the subscript $q=d$ is used for the $B^0_d$ ($=B^0$) meson and $q=s$ for the ${B^{0}_s}$ meson. The heaviest (lightest) of these mass states is denoted $B^0_{q\mathrm H}$ ($B^0_{q\mathrm L}$), with mass $m_{q\mathrm H}$ ($m_{q\mathrm L}$) and total decay width $\Gamma _{q\mathrm H}$ ($\Gamma _{q\mathrm L}$). We define

$$\begin{aligned} \Delta m_q= & {} m_{q\mathrm H} - m_{q\mathrm L},\quad x_q = \Delta m_q/\Gamma _q, \end{aligned}$$

(56)

$$\begin{aligned} \Delta \Gamma _q \,= & {} \Gamma _{q\mathrm L} - \Gamma _{q\mathrm H},\quad y_q= \Delta \Gamma _q/(2\Gamma _q), \end{aligned}$$

(57)

where $\Gamma _q = (\Gamma _{q\mathrm H} + \Gamma _{q\mathrm L})/2 =1/\bar{\tau }(B^0_q)$ is the average decay width. $\Delta m_q$ is positive by definition, and $\Delta \Gamma _q$ is expected to be positive within the Standard Model.¹³

There are four different time-dependent probabilities describing the case of a neutral $B$ meson produced as a flavour state and decaying without $C\!P$ violation to a flavour-specific final state. If $C\!PT$ is conserved (which will be assumed throughout), they can be written as

$$\begin{aligned} \left\{ \begin{array}{lll} \mathcal{P}(B^0_q \rightarrow B^0_q) &{} = &{} \frac{1}{2} e^{-\Gamma _q t} \bigg [ \cosh \!\bigg (\frac{1}{2}\Delta \Gamma _q t\bigg ) + \cos \!\bigg (\Delta m_q t\bigg )\bigg ] \\ \mathcal{P}(B^0_q \rightarrow {\bar{B}}^0_q) &{} = &{} \frac{1}{2} e^{-\Gamma _q t} \bigg [ \cosh \!\bigg (\frac{1}{2}\Delta \Gamma _q t\bigg ) - \cos \!\bigg (\Delta m_q t\bigg )\bigg ]\bigg |q_q/p_q\bigg |^2 \\ \mathcal{P}({\bar{B}}^0_q \rightarrow B^0_q) &{} = &{} \frac{1}{2} e^{-\Gamma _q t} \bigg [ \cosh \!\bigg (\frac{1}{2}\Delta \Gamma _q t\bigg ) - \cos \!\bigg (\Delta m_q t\bigg )\bigg ]\bigg |p_q/q_q\bigg |^2 \\ \mathcal{P}({\bar{B}}^0_q \rightarrow {\bar{B}}^0_q) &{} = &{} \frac{1}{2} e^{-\Gamma _q t} \bigg [ \cosh \!\bigg (\frac{1}{2}\Delta \Gamma _q t\bigg ) + \cos \!\bigg (\Delta m_q t\bigg )\bigg ] \end{array} \right. ,\nonumber \\ \end{aligned}$$

(58)

where t is the proper time of the system (i.e., the time interval between the production and the decay in the rest frame of the $B$ meson). At the $B$ factories, only the proper-time difference $\Delta t$ between the decays of the two neutral $B$ mesons from the $\varUpsilon (4S)$ can be determined, but, because the two $B$ mesons evolve coherently (keeping opposite flavours as long as neither of them has decayed), the above formulae remain valid if t is replaced with $\Delta t$ and the production flavour is replaced by the flavour at the time of the decay of the accompanying $B$ meson in a flavour-specific state. As can be seen in the above expressions, the mixing probabilities depend on three mixing observables: $\Delta m_q$, $\Delta \Gamma _q$, and $|q_q/p_q|^2$, which signals $C\!P$ violation in the mixing if $|q_q/p_q|^2 \ne 1$. Another (non independent) observable often used to characterize $C\!P$ violation in the mixing is the so-called semileptonic asymmetry, defined as

$$\begin{aligned} \mathcal{A}_\mathrm{SL}^q = \frac{|p_{q}/q_{q}|^2 - |q_{q}/p_{q}|^2}{|p_{q}/q_{q}|^2 + |q_{q}/p_{q}|^2}. \end{aligned}$$

(59)

All mixing observables depend on two complex numbers, $M^q_{12}$ and $\Gamma ^q_{12}$, which are the off-diagonal elements of the mass and decay $2\times 2$ matrices describing the evolution of the $B^0_q-{\bar{B}}^0_q$ system. In the Standard Model the quantity $|\Gamma ^q_{12}/M^q_{12}|$ is small, of the order of $(m_b/m_t)^2$ where $m_b$ and $m_t$ are the bottom and top quark masses. The following relations hold, to first order in $|\Gamma ^q_{12}/M^q_{12}|$:

$$\begin{aligned} \Delta m_q= & {} 2 |M^q_{12}| \bigg [1 + \mathcal{O} \bigg (|\Gamma ^q_{12}/M^q_{12}|^2 \bigg ) \bigg ], \end{aligned}$$

(60)

$$\begin{aligned} \Delta \Gamma _q= & {} 2 |\Gamma ^q_{12}| \cos \phi ^q_{12} \bigg [1 + \mathcal{O} \bigg (|\Gamma ^q_{12}/M^q_{12}|^2 \bigg ) \bigg ], \end{aligned}$$

(61)

$$\begin{aligned} \mathcal{A}_\mathrm{SL}^q= & {} \mathrm{Im}\bigg (\Gamma ^q_{12}/M^q_{12} \bigg ) + \mathcal{O} \bigg (|\Gamma ^q_{12}/M^q_{12}|^2 \bigg )\nonumber \\= & {} \frac{\Delta \Gamma _{s} }{\Delta m_{s}}\tan \phi ^q_{12} + \mathcal{O} \bigg (|\Gamma ^q_{12}/M^q_{12}|^2 \bigg ), \end{aligned}$$

(62)

where

$$\begin{aligned} \phi ^q_{12} = \arg \bigg ( -{M^q_{12}}/{\Gamma ^q_{12}} \bigg ) \end{aligned}$$

(63)

is the observable phase difference between $-M^q_{12}$ and $\Gamma ^q_{12}$ (often called the mixing phase). It should be noted that the theoretical predictions for $\Gamma ^q_{12}$ are based on the same HQE as the lifetime predictions.

In the next sections we review in turn the experimental knowledge on the $B^0$ decay-width and mass differences, the ${B^{0}_s}$ decay-width and mass differences, $C\!P$ violation in $B^0$ and ${B^{0}_s}$ mixing, and mixing-induced $C\!P$ violation in ${B^{0}_s}$ decays.

3.3.1 $B^0$ mixing parameters $\Delta \Gamma _{\mathrm{d}}$ and $\Delta m_{d}$

Table 15 Time-dependent measurements included in the $\Delta m_{d}$ average. The results obtained from multi-dimensional fits involving also the $B^0$ (and $B^+$) lifetimes as free parameter(s) [98, 100, 101] have been converted into one-dimensional measurements of $\Delta m_{d}$. All the measurements have then been adjusted to a common set of physics parameters before being combined

A large number of time-dependent $B^0$–${\bar{B}}^0$ oscillation analyses have been performed in the past 20 years by the ALEPH, DELPHI, L3, OPAL, CDF, D0, BaBar, Belle and LHCb collaborations. The corresponding measurements of $\Delta m_{d}$ are summarized in Table 15. Although a variety of different techniques have been used, the individual $\Delta m_{d}$ results obtained at different colliders have remarkably similar precision. The systematic uncertainties are comparable to the statistical uncertainties; they are often dominated by sample composition, mistag probability, or $b$-hadron lifetime contributions. Before being combined, the measurements are adjusted on the basis of a common set of input values, including the averages of the $b$-hadron fractions and lifetimes given in this report (see Sects. , ). Some measurements are statistically correlated. Systematic correlations arise both from common physics sources (fractions, lifetimes, branching ratios of $b$ hadrons), and from purely experimental or algorithmic effects (efficiency, resolution, flavour tagging, background description). Combining all published measurements listed in Table 15 and accounting for all identified correlations as described in Ref. [1] yields $\Delta m_{d} = 0.5065{\,\pm \,}0.0016{\,\pm \,}0.0011~\mathrm ps^{-1}$.

On the other hand, ARGUS and CLEO have published measurements of the time-integrated mixing probability $\chi _{d}$  [184, 186, 187], which average to $\chi _{d} =0.182{\,\pm \,}0.015$. Following Ref. [1], the decay width difference $\Delta \Gamma _{\mathrm{d}}$ could in principle be extracted from the measured value of $\Gamma _{\mathrm{d}} =1/\tau (B^0)$ and the above averages for $\Delta m_{d}$ and $\chi _{d}$ (provided that $\Delta \Gamma _{\mathrm{d}}$ has a negligible impact on the $\Delta m_{d}$ and $\tau (B^0)$ analyses that have assumed $\Delta \Gamma _{\mathrm{d}} =0$), using the relation

$$\begin{aligned} \chi _{d} = \frac{x_{d} ^2+y_{d} ^2}{2(x_{d} ^2+1)}. \end{aligned}$$

(64)

However, direct time-dependent studies provide much stronger constraints: $|\Delta \Gamma _{\mathrm{d}} |/\Gamma _{\mathrm{d}} < 18\%$ at 95% CL from DELPHI [165], $-6.8\%< \mathrm{sign}(\mathrm{Re} \lambda _{C\!P}) \Delta \Gamma _{\mathrm{d}} /\Gamma _{\mathrm{d}} < 8.4\%$ at 90% CL from BaBar [188], and $\mathrm{sign}(\mathrm{Re} \lambda _{C\!P})\Delta \Gamma _{\mathrm{d}} /\Gamma _{\mathrm{d}} = (1.7 \,\pm \, 1.8 \,\pm \, 1.1)\%$ [190] from Belle, where $\lambda _{C\!P} = (q/p)_{d} ({\bar{A}}_{C\!P}/A_{C\!P})$ is defined for a $C\!P$-even final state (the sensitivity to the overall sign of $\mathrm{sign}(\mathrm{Re} \lambda _{C\!P}) \Delta \Gamma _{\mathrm{d}} /\Gamma _{\mathrm{d}}$ comes from the use of $B^0$ decays to $C\!P$ final states). In addition LHCb has obtained $\Delta \Gamma _{\mathrm{d}} /\Gamma _{\mathrm{d}} =(-4.4 \,\pm \, 2.5 \,\pm \, 1.1)\%$ [103] by comparing measurements of the $B^0 \rightarrow {J/\psi } K^{*0}$ and $B^0 \rightarrow {J/\psi } K^0_\mathrm{S}$ decays, following the method of Ref. [191]. More recently ATLAS has measured $\Delta \Gamma _{\mathrm{d}} /\Gamma _{\mathrm{d}} =(-0.1 \,\pm \, 1.1 \,\pm \, 0.9)\%$ [192] using a similar method. Assuming $\mathrm{Re} \lambda _{C\!P} > 0$, as expected from the global fits of the Unitarity Triangle within the Standard Model [193], a combination of these five results (after adjusting the DELPHI and BaBar results to $1/\Gamma _{\mathrm{d}} =\tau (B^0)=1.520{{\,\pm \,}}0.004~\mathrm ps$) yields

$$\begin{aligned} \Delta \Gamma _{\mathrm{d}} /\Gamma _{\mathrm{d}} = -0.002{\,\pm \,}0.010, \end{aligned}$$

(65)

an average consistent with zero and with the Standard Model prediction of $(3.97\,\pm \,0.90)\times 10^{-3}$ [107]. An independent result, $\Delta \Gamma _{\mathrm{d}} /\Gamma _{\mathrm{d}} =(0.50 \,\pm \, 1.38)\%$ [195], was obtained by the D0 collaboration from their measurements of the single muon and same-sign dimuon charge asymmetries, under the interpretation that the observed asymmetries are due to $C\!P$ violation in neutral B-meson mixing and interference. This indirect determination was called into question [196] and is therefore not included in the above average, as explained in Sect. 3.3.3 (see Footnote 17).

Assuming $\Delta \Gamma _{\mathrm{d}} =0$ and using $1/\Gamma _{\mathrm{d}} =\tau (B^0)=1.520{{\,\pm \,}}0.004~\mathrm ps$, the $\Delta m_{d}$ and $\chi _{d}$ results are combined through Eq. (64) to yield the world average

$$\begin{aligned} \Delta m_{d} = 0.5064{\,\pm \,}0.0019~\mathrm ps^{-1}, \end{aligned}$$

(66)

or, equivalently,

$$\begin{aligned} x_{d} = 0.770{\,\pm \,}0.004\quad \text{ and }\quad \chi _{d} =0.1860{{\,\pm \,}}0.0011. \end{aligned}$$

(67)

Figure 6 compares the $\Delta m_{d}$ values obtained by the different experiments.

Fig. 6

The $B^0$–${\bar{B}}^0$ oscillation frequency $\Delta m_{d}$ as measured by the different experiments. The averages quoted for ALEPH, L3 and OPAL are taken from the original publications, while the ones for DELPHI, CDF, BaBar, Belle and LHCb are computed from the individual results listed in Table 15 without performing any adjustments. The time-integrated measurements of $\chi _{d}$ from the symmetric $B$ factory experiments ARGUS and CLEO are converted to a $\Delta m_{d}$ value using $\tau (B^0)=1.520{{\,\pm \,}}0.004~\mathrm ps$. The two global averages are obtained after adjustments of all the individual $\Delta m_{d}$ results of Table 15 (see text)

The $B^0$ mixing averages given in Eqs. (66) and (67) and the $b$-hadron fractions of Table 5 have been obtained in a fully consistent way, taking into account the fact that the fractions are computed using the $\chi _{d}$ value of Eq. (67) and that many individual measurements of $\Delta m_{d}$ at high energy depend on the assumed values for the $b$-hadron fractions. Furthermore, this set of averages is consistent with the lifetime averages of Sect. 3.2.

3.3.2 ${B^{0}_s}$ mixing parameters $\Delta \Gamma _{s}$ and $\Delta m_{s}$

The best sensitivity to $\Delta \Gamma _{s}$ is currently achieved by the recent time-dependent measurements of the ${B^{0}_s} \rightarrow {J/\psi } \phi$ (or more generally ${B^{0}_s} \rightarrow (c{\bar{c}}) K^+K^-$) decay rates performed at CDF [197], D0  [198], ATLAS [199, 200] CMS [201] and LHCb [202, 203], where the $C\!P$-even and $C\!P$-odd amplitudes are statistically separated through a full angular analysis. These studies use both untagged and tagged ${B^{0}_s}$ candidates and are optimized for the measurement of the $C\!P$-violating phase $\phi _s^{c{{\bar{c}}}s}$, defined later in Sect. 3.3.4. The LHCb collaboration analyzed the ${B^{0}_s} \rightarrow {J/\psi } K^+K^-$ decay, considering that the $K^+K^-$ system can be in a P-wave or S-wave state, and measured the dependence of the strong phase difference between the P-wave and S-wave amplitudes as a function of the $K^+K^-$ invariant mass [111]. This allowed, for the first time, the unambiguous determination of the sign of $\Delta \Gamma _{s}$, which was found to be positive at the $4.7\,\sigma$ level. The following averages present only the $\Delta \Gamma _{s} > 0$ solutions.

The published results [197,198,199,200,201,202,203] are shown in Table 16. They are combined taking into account, in each analysis, the correlation between $\Delta \Gamma _{s}$ and $\Gamma _{s}$. The results, displayed as the red contours labelled “${B^{0}_s} \rightarrow (c{\bar{c}}) KK$ measurements” in the plots of Fig. , are given in the first column of numbers of Table 17.

Table 16 Measurements of $\Delta \Gamma _{s}$ and $\Gamma _{s}$ using ${B^{0}_s} \rightarrow {J/\psi } \phi$, ${B^{0}_s} \rightarrow {J/\psi } K^+K^-$ and ${B^{0}_s} \rightarrow \psi (2S)\phi$ decays. Only the solution with $\Delta \Gamma _{s} > 0$ is shown, since the two-fold ambiguity has been resolved in Ref. [1]. The first error is due to statistics, the second one to systematics. The last line gives our average

Fig. 7

Contours of $\Delta \ln L = 0.5$ (39% CL for the enclosed 2D regions, 68% CL for the bands) shown in the $(\Gamma _{s},\,\Delta \Gamma _{s} )$ plane on the left and in the $(1/\Gamma _{s\mathrm L},\,1/\Gamma _{s\mathrm H})$ plane on the right. The average of all the ${B^{0}_s} \rightarrow {J/\psi } \phi$, ${B^{0}_s} \rightarrow {J/\psi } K^+K^-$ and ${B^{0}_s} \rightarrow \psi (2S)\phi$ results is shown as the red contour, and the constraints given by the effective lifetime measurements of ${B^{0}_s}$ to flavour-specific, pure $C\!P$-odd and pure $C\!P$-even final states are shown as the blue, green and purple bands, respectively. The average taking all constraints into account is shown as the grey-filled contour. The yellow band is a theory prediction $\Delta \Gamma _{s} = 0.088 \,\pm \, 0.020~\hbox {ps}^{-1}$ [60, 107] that assumes no new physics in ${B^{0}_s}$ mixing

Table 17 Averages of $\Delta \Gamma _{s}$, $\Gamma _{s}$ and related quantities, obtained from ${B^{0}_s} \rightarrow {J/\psi } \phi$, ${B^{0}_s} \rightarrow {J/\psi } K^+K^-$ and ${B^{0}_s} \rightarrow \psi (2S)\phi$ alone (first column), adding the constraints from the effective lifetimes measured in pure $C\!P$ modes ${B^{0}_s} \rightarrow D_s^+D_s^-,J/\psi \eta$ and ${B^{0}_s} \rightarrow {J/\psi } f_0(980), {J/\psi } \pi ^+\pi ^-$ (second column), and adding the constraint from the effective lifetime measured in flavour-specific modes ${B^{0}_s} \rightarrow D_s^-\ell ^+\nu X, \, D_s^-\pi ^+, \, D_s^-D^+$ (third column, recommended world averages)

An alternative approach, which is directly sensitive to first order in $\Delta \Gamma _{s} /\Gamma _{s}$, is to determine the effective lifetime of untagged ${B^{0}_s}$ candidates decaying to pure $C\!P$ eigenstates; we use here measurements with ${B^{0}_s} \rightarrow D_s^+D_s^-$ [121], ${B^{0}_s} \rightarrow J/\psi \eta$ [126], ${B^{0}_s} \rightarrow {J/\psi } f_0(980)$ [128, 129] and ${B^{0}_s} \rightarrow {J/\psi } \pi ^+\pi ^-$ [130] decays. The precise extraction of $1/\Gamma _{s}$ and $\Delta \Gamma _{s}$ from such measurements, discussed in detail in Ref. [1], requires additional information in the form of theoretical assumptions or external inputs on weak phases and hadronic parameters. If f designates a final state in which both ${B^{0}_s}$ and ${\bar{B}}^0_s$ can decay, the ratio of the effective ${B^{0}_s}$ lifetime decaying to f relative to the mean ${B^{0}_s}$ lifetime is [112]¹⁴

$$\begin{aligned} \frac{\tau _\mathrm{single}({B^{0}_s} \rightarrow f)}{\tau ({B^{0}_s})} = \frac{1}{1-y_s^2} \bigg [ \frac{1 - 2A_f^{\Delta \Gamma } y_s + y_s^2}{1 - A_f^{\Delta \Gamma } y_s}\bigg ], \end{aligned}$$

(68)

where

$$\begin{aligned} A_f^{\Delta \Gamma } = -\frac{2 \mathrm{Re}(\lambda _f)}{1+|\lambda _f|^2}. \end{aligned}$$

(69)

To include the measurements of the effective ${B^{0}_s} \rightarrow D_s^+D_s^-$ ($C\!P$-even), ${B^{0}_s} \rightarrow {J/\psi } f_0(980)$ ($C\!P$-odd) and ${B^{0}_s} \rightarrow {J/\psi } \pi ^+\pi ^-$ ($C\!P$-odd) lifetimes as constraints in the $\Delta \Gamma _{s}$ fit,¹⁵ we neglect sub-leading penguin contributions and possible direct $C\!P$ violation. Explicitly, in Eq. (69), we set $A_{C\!P \text{-even }}^{\Delta \Gamma } = \cos \phi _s^{c{{\bar{c}}}s}$ and $A_{C\!P \text{-odd }}^{\Delta \Gamma } = -\cos \phi _s^{c{{\bar{c}}}s}$. Given the small value of $\phi _s^{c{{\bar{c}}}s}$, we have, to first order in $y_s$:

$$\begin{aligned} \tau _\mathrm{single}({B^{0}_s} \rightarrow C\!P \text{-even })\approx & {} \frac{1}{\Gamma _{s\mathrm L}} \bigg (1 + \frac{(\phi _s^{c{{\bar{c}}}s})^2 y_s}{2} \bigg ), \end{aligned}$$

(70)

$$\begin{aligned} \tau _\mathrm{single}({B^{0}_s} \rightarrow bad hbox)\approx & {} \frac{1}{\Gamma _{s\mathrm H}} \bigg (1 - \frac{(\phi _s^{c{{\bar{c}}}s})^2 y_s}{2} \bigg ). \end{aligned}$$

(71)

The numerical inputs are taken from Eqs. (41) and (42) and the resulting averages, combined with the ${B^{0}_s} \rightarrow {J/\psi } K^+K^-$ information, are indicated in the second column of numbers of Table 17. These averages assume $\phi _s^{c{{\bar{c}}}s} = 0$, which is compatible with the $\phi _s^{c{{\bar{c}}}s}$ average presented in Sect. 3.3.4.

Information on $\Delta \Gamma _{s}$ can also be obtained from the study of the proper time distribution of untagged samples of flavour-specific ${B^{0}_s}$ decays [131], where the flavour (i.e., ${B^{0}_s}$ or ${\bar{B}}^0_s$) at the time of decay can be determined by the decay products. In such decays, e.g. semileptonic ${B^{0}_s}$ decays, there is an equal mix of the heavy and light mass eigenstates at time zero. The proper time distribution is then a superposition of two exponential functions with decay constants $\Gamma _{s\mathrm L}$ and $\Gamma _{s\mathrm H}$. This provides sensitivity to both $1/\Gamma _{s}$ and $(\Delta \Gamma _{s} /\Gamma _{s})^2$. Ignoring $\Delta \Gamma _{s}$ and fitting for a single exponential leads to an estimate of $\Gamma _{s}$ with a relative bias proportional to $(\Delta \Gamma _{s} /\Gamma _{s})^2$, as shown in Eq. (39). Including the constraint from the world-average flavour-specific ${B^{0}_s}$ lifetime, given in Eq. (40), leads to the results shown in the last column of Table 17. These world averages are displayed as the grey contours labelled “Combined” in the plots of Fig. . They correspond to the lifetime averages $1/\Gamma _{s} =1.505{{\,\pm \,}}0.005~\mathrm ps$, $1/\Gamma _{s\mathrm L}=1.413{{\,\pm \,}}0.006~\mathrm ps$, $1/\Gamma _{s\mathrm H}=1.609{{\,\pm \,}}0.010~\mathrm ps$, and to the decay-width difference

$$\begin{aligned}&\Delta \Gamma _{s} = +0.086{\,\pm \,}0.006~\mathrm ps^{-1} \quad \text{ and }\nonumber \\&\Delta \Gamma _{s} /\Gamma _{s} = +0.130{\,\pm \,}0.009. \end{aligned}$$

(72)

The good agreement with the Standard Model prediction $\Delta \Gamma _{s} = 0.088 \,\pm \, 0.020~\hbox {ps}^{-1}$ [60, 107] excludes significant quark-hadron duality violation in the HQE [204].

Estimates of $\Delta \Gamma _{s} /\Gamma _{s}$ obtained from measurements of the ${B^{0}_s} \rightarrow D_s^{(*)+} D_s^{(*)-}$ branching fraction [124, 205,206,207] have not been used, since they are based on the questionable [208, 209] assumption that these decays account for all $C\!P$-even final states. The results of early lifetime analyses attempting to measure $\Delta \Gamma _{s} /\Gamma _{s}$ [79, 85, 114, 118] have not been used either.

The strength of ${B^{0}_s}$ mixing has been known to be large for more than 20 years. Indeed the time-integrated measurements of $\overline{\chi }$ (see Sect. 3.1.3), when compared to our knowledge of $\chi _{d}$ and the $b$-hadron fractions, indicated that $\chi _{s}$ should be close to its maximal possible value of 1 / 2. Many searches of the time dependence of this mixing were performed by ALEPH [210], DELPHI [114, 118, 165, 211], OPAL [212, 213], SLD [214, 215], CDF (Run I) [216] and D0  [217] but did not have enough statistical power and proper time resolution to resolve the small period of the ${B^{0}_s}$ oscillations.

${B^{0}_s}$ oscillations have been observed for the first time in 2006 by the CDF collaboration [218], based on samples of flavour-tagged hadronic and semileptonic ${B^{0}_s}$ decays (in flavour-specific final states), partially or fully reconstructed in $1bad hbox^{-1}$ of data collected during Tevatron’s Run II. More recently the LHCb collaboration obtained the most precise results using fully reconstructed ${B^{0}_s} \rightarrow D_s^-\pi ^+$ and ${B^{0}_s} \rightarrow D_s^-\pi ^+\pi ^-\pi ^+$ decays at the LHC [219, 220]. LHCb has also observed ${B^{0}_s}$ oscillations with ${B^{0}_s} \rightarrow {J/\psi } K^+K^-$ decays [202] and with semileptonic ${B^{0}_s} \rightarrow D_s^-\mu ^+ X$ decays [182]. The measurements of $\Delta m_{s}$ are summarized in Table 18.

Table 18 Measurements of $\Delta m_{s}$

Fig. 8

Published measurements of $\Delta m_{s}$, together with their average

A simple average of the CDF and LHCb results, taking into account the correlated systematic uncertainties between the three LHCb measurements, yields

$$\begin{aligned} \Delta m_{s}= & {} 17.757{\,\pm \,}0.020{\,\pm \,}0.007~\mathrm ps^{-1} \nonumber \\= & {} 17.757{\,\pm \,}0.021~\mathrm ps^{-1} \end{aligned}$$

(73)

and is illustrated in Fig. 8. The Standard Model prediction $\Delta m_{s} = 18.3 \,\pm \, 2.7~\hbox {ps}^{-1}$ [60, 107] is consistent with the experimental value, but has a much larger error dominated by the uncertainty on the hadronic matrix elements. The ratio $\Delta \Gamma _{s} /\Delta m_{s}$ can be predicted more accurately, $0.0048 \,\pm \, 0.0008$ [60, 107], and is in good agreement with the experimental determination of

$$\begin{aligned} \Delta \Gamma _{s} /\Delta m_{s} = 0.00486{\,\pm \,}0.00034. \end{aligned}$$

(74)

Multiplying the $\Delta m_{s}$ result of Eq. (73) with the mean ${B^{0}_s}$ lifetime of Eq. (45), $1/\Gamma _{s} =1.505{{\,\pm \,}}0.005~\mathrm ps$, yields

$$\begin{aligned} x_{\mathrm{s}} = 26.72{\,\pm \,}0.09. \end{aligned}$$

(75)

With $2y_{\mathrm{s}} =+0.130{\,\pm \,}0.009$ [see Eq. (72)] and under the assumption of no $C\!P$ violation in ${B^{0}_s}$ mixing, this corresponds to

$$\begin{aligned} \chi _{s} = \frac{x_{s} ^2+y_{s} ^2}{2(x_{s} ^2+1)} = 0.499304{\,\pm \,}0.000005. \end{aligned}$$

(76)

The ratio of the $B^0$ and ${B^{0}_s}$ oscillation frequencies, obtained from Eqs. (66) and (73),

$$\begin{aligned} \frac{\Delta m_{d}}{\Delta m_{s}} = 0.02852{\,\pm \,}0.00011, \end{aligned}$$

(77)

can be used to extract the following magnitude of the ratio of CKM matrix elements,

$$\begin{aligned} \bigg |\frac{V_{td}}{V_{ts}}\bigg | = \xi \sqrt{\frac{\Delta m_{d}}{\Delta m_{s}}\frac{m({B^{0}_s})}{m(B^0)}} = 0.2053{\,\pm \,}0.0004{\,\pm \,}0.0032,\nonumber \\ \end{aligned}$$

(78)

where the first quoted error is from experimental uncertainties (with the masses $m({B^{0}_s})$ and $m(B^0)$ taken from Ref. [1]), and where the second quoted error is from theoretical uncertainties in the estimation of the SU(3) flavour-symmetry breaking factor $\xi = 1.206{\,\pm \,}0.018{\,\pm \,}0.006$, obtained from recent three-flavour lattice QCD calculations [221, 222]. Note that Eq. (78) assumes that $\Delta m_{s}$ and $\Delta m_{d}$ only receive Standard Model contributions.

Table 19 Measurements$^{16}$ of $C\!P$ violation in $B^0$ mixing and their average in terms of both $\mathcal{A}_\mathrm{SL}^\mathrm{d}$ and $|q_{d}/p_{d}|$. The individual results are listed as quoted in the original publications, or converted$^{18}$ to an $\mathcal{A}_\mathrm{SL}^\mathrm{d}$ value. When two errors are quoted, the first one is statistical and the second one systematic. The ALEPH and OPAL results assume no $C\!P$ violation in ${B^{0}_s}$ mixing

3.3.3 $C\!P$ violation in $B^0$ and ${B^{0}_s}$ mixing

Evidence for $C\!P$ violation in $B^0$ mixing has been searched for, both with flavour-specific and inclusive $B^0$ decays, in samples where the initial flavour state is tagged. In the case of semileptonic (or other flavour-specific) decays, where the final state tag is also available, the asymmetry

$$ \mathcal{A}_\mathrm{SL}^\mathrm{d} = \frac{ N({{\bar{B}}^0}(t) \rightarrow \ell ^+ \nu _{\ell } X) - N(\hbox {B}^0(t) \rightarrow \ell ^- {\bar{\nu }}_{\ell } X) }{N({{\bar{B}}^0}(t) \rightarrow \ell ^+ \nu _{\ell } X) + N(\hbox {B}^0(t) \rightarrow \ell ^- {\bar{\nu }}_{\ell } X)}$$

(79)

has been measured, either in decay-time-integrated analyses at CLEO [187, 223], BaBar [224], CDF [225] and D0  [195], or in decay-time-dependent analyses at OPAL [168], ALEPH [226], BaBar  [188, 227, 228] and Belle  [229]. Note that the asymmetry of time-dependent decay rates in Eq. (79) is related to $|q_d/p_d|$ through Eq. (59) and is therefore time-independent. In the inclusive case, also investigated and published by ALEPH [226] and OPAL [87], no final state tag is used, and the asymmetry [230, 231]

$$\begin{aligned}&\frac{ N(\hbox {B}^0(t) \rightarrow \mathrm{all}) - N({{\bar{B}}^0}(t) \rightarrow \mathrm{all}) }{ N(\hbox {B}^0(t) \rightarrow \mathrm{all}) + N({{\bar{B}}^0}(t) \rightarrow \mathrm{all}) } \nonumber \\&\quad \simeq \mathcal{A}_\mathrm{SL}^\mathrm{d} \bigg [ \frac{\Delta m_{d}}{2\Gamma _{\mathrm{d}}} \sin (\Delta m_{d} \,t) - \sin ^2\bigg (\frac{\Delta m_{d} \,t}{2}\bigg )\bigg ] \end{aligned}$$

(80)

must be measured as a function of the proper time to extract information on $C\!P$ violation.

On the other hand, D0  [232] and LHCb [233] have studied the time-dependence of the charge asymmetry of $B^0 \rightarrow D^{(*)-}\mu ^+\nu _{\mu }X$ decays without tagging the initial state, which would be equal to

$$\begin{aligned}&\frac{N(D^{(*)-}\mu ^+\nu _{\mu }X)-N(D^{(*)+}\mu ^-{\bar{\nu }}_{\mu }X)}{N(D^{(*)-}\mu ^+\nu _{\mu }X)+N(D^{(*)+}\mu ^-{\bar{\nu }}_{\mu }X)} \nonumber \\&\quad = \mathcal{A}_\mathrm{SL}^\mathrm{d} \frac{1- \cos (\Delta m_{d} \,t)}{2} \end{aligned}$$

(81)

in absence of detection and production asymmetries.

Table 19 summarizes the different measurements¹⁶ of $\mathcal{A}_\mathrm{SL}^\mathrm{d}$ and $|q_{\mathrm{d}}/p_{\mathrm{d}}|$: in all cases asymmetries compatible with zero have been found, with a precision limited by the available statistics.

Table 20 Measurements of $C\!P$ violation in ${B^{0}_s}$ and $B^0$ mixing, together with their correlations $\rho (\mathcal{A}_\mathrm{SL}^\mathrm{s},\mathcal{A}_\mathrm{SL}^\mathrm{d})$ and their two-dimensional average. Only total errors are quoted

A simple average of all measurements performed at the $B$ factories [187, 188, 223, 224, 227, 229] yields $\mathcal{A}_\mathrm{SL}^\mathrm{d} = -0.0019{\,\pm \,}0.0027$. Adding also the D0  [232] and LHCb [233] measurements obtained with reconstructed semileptonic $B^0$ decays yields $\mathcal{A}_\mathrm{SL}^\mathrm{d} = +0.0001{\,\pm \,}0.0020$. As discussed in more detail later in this section, the D0 analysis with single muons and like-sign dimuons [195] separates the $B^0$ and ${B^{0}_s}$ contributions by exploiting the dependence on the muon impact parameter cut; including the $\mathcal{A}_\mathrm{SL}^\mathrm{d}$ result quoted by D0 in the average yields $\mathcal{A}_\mathrm{SL}^\mathrm{d} = -0.0010{\,\pm \,}0.0018$. All the other $B^0$ analyses performed at high energy, either at LEP or at the Tevatron, did not separate the contributions from the $B^0$ and ${B^{0}_s}$ mesons. Under the assumption of no $C\!P$ violation in ${B^{0}_s}$ mixing ($\mathcal{A}_\mathrm{SL}^\mathrm{s} =0$), a number of these early analyses [52, 87, 168, 226] quote a measurement of $\mathcal{A}_\mathrm{SL}^\mathrm{d}$ or $|q_{d}/p_{d}|$ for the $B^0$ meson. However, these imprecise determinations no longer improve the world average of $\mathcal{A}_\mathrm{SL}^\mathrm{d}$. The latter assumption makes sense within the Standard Model, since $\mathcal{A}_\mathrm{SL}^\mathrm{s}$ is predicted to be much smaller than $\mathcal{A}_\mathrm{SL}^\mathrm{d}$  [60, 107], but may not be suitable in the presence of new physics.

The Tevatron experiments have measured linear combinations of $\mathcal{A}_\mathrm{SL}^\mathrm{d}$ and $\mathcal{A}_\mathrm{SL}^\mathrm{s}$ using inclusive semileptonic decays of $b$ hadrons, $\mathcal{A}_\mathrm{SL}^\mathrm{b} = +0.0015 \,\pm \, 0.0038 \text{(stat) } \,\pm \, 0.0020 \text{(syst) }$ [225] and $\mathcal{A}_\mathrm{SL}^\mathrm{b} = -0.00496 \,\pm \, 0.00153 \text{(stat) } \,\pm \, 0.00072 \text{(syst) }$ [195], at CDF1 and D0 respectively. While the imprecise CDF1 result is compatible with no $C\!P$ violation, the D0 result, obtained by measuring the single muon and like-sign dimuon charge asymmetries, differs by 2.8 standard deviations from the Standard Model expectation of $\mathcal{A}_\mathrm{SL}^{b,\mathrm SM} = (-2.3\,\pm \, 0.4) \times 10^{-4}$  [195, 208]. With a more sophisticated analysis in bins of the muon impact parameters, D0 conclude that the overall deviation of their measurements from the SM is at the level of $3.6\,\sigma$. Interpreting the observed asymmetries in bins of the muon impact parameters in terms of $C\!P$ violation in B-meson mixing and interference, and using the mixing parameters and the world $b$-hadron fractions of Ref. [1], the D0 collaboration extracts [195] values for $\mathcal{A}_\mathrm{SL}^\mathrm{d}$ and $\mathcal{A}_\mathrm{SL}^\mathrm{s}$ and their correlation coefficient,¹⁷ as shown in Table 20. However, the various contributions to the total quoted errors from this analysis and from the external inputs are not given, so the adjustment of these results to different or more recent values of the external inputs cannot (easily) be done.

Finally, direct determinations of $\mathcal{A}_\mathrm{SL}^\mathrm{s}$, also shown in Table 20, are obtained by D0  [235] and LHCb [236] from the time-integrated charge asymmetry of untagged ${B^{0}_s} \rightarrow D_s^- \mu ^+\nu X$ decays.

Fig. 9

Measurements of $\mathcal{A}_\mathrm{SL}^\mathrm{s}$ and $\mathcal{A}_\mathrm{SL}^\mathrm{d}$ listed in Table 20 ($B$-factory average as the grey band, D0 measurements as the green ellipses, LHCb measurements as the blue ellipse) together with their two-dimensional average (red hatched ellipse). The red point close to (0, 0) is the Standard Model prediction of Refs. [60, 107] with error bars multiplied by 10. The prediction and the experimental world average deviate from each other by $0.5\,\sigma$

Using a two-dimensional fit, all measurements of $\mathcal{A}_\mathrm{SL}^\mathrm{s}$ and $\mathcal{A}_\mathrm{SL}^\mathrm{d}$ obtained by D0 and LHCb are combined with the $B$-factory average of Table 19. Correlations are taken into account as shown in Table 20. The results, displayed graphically in Fig. 9, are

$$\begin{aligned}&\mathcal{A}_\mathrm{SL}^\mathrm{d} = -0.0021{\,\pm \,}0.0017\nonumber \\&~~~~~~~\Longleftrightarrow |q_{d}/p_{d}| = 1.0010{\,\pm \,}0.0008, \end{aligned}$$

(82)

$$\begin{aligned}&\mathcal{A}_\mathrm{SL}^\mathrm{s} = -0.0006{\,\pm \,}0.0028\nonumber \\&~~~~~~~\Longleftrightarrow |q_{s}/p_{s}| = 1.0003{\,\pm \,}0.0014, \end{aligned}$$

(83)

$$\begin{aligned}&\rho (\mathcal{A}_\mathrm{SL}^\mathrm{d}, \mathcal{A}_\mathrm{SL}^\mathrm{s}) = -0.054, \end{aligned}$$

(84)

where the relation between $\mathcal{A}_\mathrm{SL}^q$ and $|q_{q}/p_{q}|$ is given in Eq. (59).¹⁸ However, the fit $\chi ^2$ probability is only $4.5\%$. This is mostly due to an overall discrepancy between the D0 and LHCb averages at the level of $2.2\,\sigma$. Since the assumptions underlying the inclusion of the D0 muon results in the average$^{17}$ are somewhat controversial [237], we also provide in Table 20 an average excluding these results.

The above averages show no evidence of $C\!P$ violation in $B^0$ or ${B^{0}_s}$ mixing. They deviate by $0.5\,\sigma$ from the very small predictions of the Standard Model (SM), $\mathcal{A}_\mathrm{SL}^{d,\mathrm SM} = -(4.7\,\pm \, 0.6)\times 10^{-4}$ and $\mathcal{A}_\mathrm{SL}^{s,\mathrm SM} = +(2.22\,\pm \, 0.27)\times 10^{-5}$ [60, 107]. Given the current size of the experimental uncertainties, there is still significant room for a possible new physics contribution, in particular in the ${B^{0}_s}$ system. In this respect, the deviation of the D0 dimuon asymmetry [195] from expectation has generated a lot of excitement. However, the recent $\mathcal{A}_\mathrm{SL}^\mathrm{s}$ and $\mathcal{A}_\mathrm{SL}^\mathrm{d}$ results from LHCb are not precise enough yet to settle the issue. It was pointed out [238] that the D0 dimuon result can be reconciled with the SM expectations of $\mathcal{A}_\mathrm{SL}^\mathrm{s}$ and $\mathcal{A}_\mathrm{SL}^\mathrm{d}$ if there were non-SM sources of $C\!P$ violation in the semileptonic decays of the b and c quarks. A recent Run 1 ATLAS study [239] of charge asymmetries in muon+jets $t{\bar{t}}$ events, in which a $b$-hadron decays semileptonically to a soft muon, yields results with limited statistical precision, compatible both with the D0 dimuon asymmetry and with the SM predictions. More experimental data, especially from Run 2 of LHC, is awaited eagerly.

At the more fundamental level, $C\!P$ violation in ${B^{0}_s}$ mixing is caused by the weak phase difference $\phi ^s_{12}$ defined in Eq. (63). The SM prediction for this phase is tiny [60, 107],

$$\begin{aligned} \phi _{12}^{s,\mathrm SM} = 0.0046\,{\,\pm \,}0.0012; \end{aligned}$$

(85)

however, new physics in ${B^{0}_s}$ mixing could change this observed phase to

$$\begin{aligned} \phi ^s_{12} = \phi _{12}^{s,\mathrm SM} + \phi _{12}^{s,\mathrm NP}. \end{aligned}$$

(86)

Using Eq. (62), the current knowledge of $\mathcal{A}_\mathrm{SL}^\mathrm{s}$, $\Delta \Gamma _{s}$ and $\Delta m_{s}$, given in Eqs. (83), (72) and (73) respectively, yields an experimental determination of $\phi ^s_{12}$,

$$\begin{aligned} \tan \phi ^s_{12} = \mathcal{A}_\mathrm{SL}^\mathrm{s} \frac{\Delta m_{s}}{\Delta \Gamma _{s} } = -0.1{\,\pm \,}0.6, \end{aligned}$$

(87)

which represents only a very weak constraint at present.

3.3.4 Mixing-induced $C\!P$ violation in ${B^{0}_s}$ decays

$C\!P$ violation induced by ${B^{0}_s}-{\bar{B}}^0_s$ mixing has been a field of very active study and fast experimental progress in the past few years. The main observable is the $C\!P$-violating phase $\phi _s^{c{{\bar{c}}}s}$, defined as the weak phase difference between the ${B^{0}_s}-{\bar{B}}^0_s$ mixing amplitude $M^s_{12}$ and the $b \rightarrow c{\bar{c}}s$ decay amplitude.

The golden mode for such studies is ${B^{0}_s} \rightarrow {J/\psi } \phi$, followed by ${J/\psi } \rightarrow \mu ^+\mu ^-$ and $\phi \rightarrow K^+K^-$, for which a full angular analysis of the decay products is performed to separate statistically the $C\!P$-even and $C\!P$-odd contributions in the final state. As already mentioned in Sect. 3.3.2, CDF [197], D0  [198], ATLAS [199, 200], CMS [201] and LHCb [202, 203] have used both untagged and tagged ${B^{0}_s} \rightarrow {J/\psi } \phi$ (and more generally ${B^{0}_s} \rightarrow (c{\bar{c}}) K^+K^-$) events for the measurement of $\phi _s^{c{{\bar{c}}}s}$. LHCb [240] has used ${B^{0}_s} \rightarrow {J/\psi } \pi ^+\pi ^-$ events, analyzed with a full amplitude model including several $\pi ^+\pi ^-$ resonances (e.g., $f_0(980)$), although the ${J/\psi } \pi ^+\pi ^-$ final state had already been shown to be almost $C\!P$ pure with a $C\!P$-odd fraction larger than 0.977 at 95% CL [241]. In addition, LHCb has used the ${B^{0}_s} \rightarrow D^+_s D^-_s$ channel [242] to measure $\phi _s^{c{{\bar{c}}}s}$.

All CDF, D0, ATLAS and CMS analyses provide two mirror solutions related by the transformation $(\Delta \Gamma _{s} , \phi _s^{c{{\bar{c}}}s}) \rightarrow (-\Delta \Gamma _{s} , \pi -\phi _s^{c{{\bar{c}}}s})$. However, the LHCb analysis of ${B^{0}_s} \rightarrow {J/\psi } K^+K^-$ resolves this ambiguity and rules out the solution with negative $\Delta \Gamma _{s}$  [111], a result in agreement with the Standard Model expectation. Therefore, in what follows, we only consider the solution with $\Delta \Gamma _{s} > 0$.

We perform a combination of the CDF [197], D0  [198], ATLAS [199, 200], CMS [201] and LHCb [202, 203, 240] results summarized in Table 21. This is done by adding the two-dimensional log profile-likelihood scans of $\Delta \Gamma _{s}$ and $\phi _s^{c{{\bar{c}}}s}$ from all ${B^{0}_s} \rightarrow \ (c{\bar{c}}) K^+K^-$ analyses and a one-dimensional log profile-likelihood of $\phi _s^{c{{\bar{c}}}s}$ from the ${B^{0}_s} \rightarrow {J/\psi } \pi ^+\pi ^-$ and ${B^{0}_s} \rightarrow D_s^+ D_s^-$ analyses; the combined likelihood is then maximized with respect to $\Delta \Gamma _{s}$ and $\phi _s^{c{{\bar{c}}}s}$.

In the ${B^{0}_s} \rightarrow {J/\psi } \phi$ and ${B^{0}_s} \rightarrow {J/\psi } K^+K^-$ analyses, $\phi _s^{c{{\bar{c}}}s}$ and $\Delta \Gamma _{s}$ come from a simultaneous fit that determines also the ${B^{0}_s}$ lifetime, the polarisation amplitudes and strong phases. While the correlation between $\phi _s^{c{{\bar{c}}}s}$ and all other parameters is small, the correlations between $\Delta \Gamma _{s}$ and the polarisation amplitudes are sizable. However, since the various experiments use different conventions for the amplitudes and phases, a full combination including all correlations is not performed. Instead, our average only takes into account the correlation between $\phi _s^{c{{\bar{c}}}s}$ and $\Delta \Gamma _{s}$.

In the recent LHCb ${B^{0}_s} \rightarrow {J/\psi } K^+K^-$ analysis [202], the $\phi _s^{c{{\bar{c}}}s}$ values are measured for the first time for each polarisation of the final state. Since those values are compatible within each other, we still use the unique value of $\phi _s^{c{{\bar{c}}}s}$ for our world average, corresponding to the one measured by the other-than-LHCb analyses. In the same analysis, the statistical correlation coefficient between $\phi _s^{c{{\bar{c}}}s}$ and $|\lambda |$ (which signals $C\!P$ violation in the decay if different from unity) is measured to be very small ($-0.02$). We neglect this correlation in our average. Furthermore, the statistical correlation coefficient between $\phi _s^{c{{\bar{c}}}s}$ and $\Delta \Gamma _{s}$ is measured to be small $(-0.08)$. When averaging LHCb results of ${B^{0}_s} \rightarrow {J/\psi } K^+K^-$, ${B^{0}_s} \rightarrow {J/\psi } \pi ^+\pi ^-$ and ${B^{0}_s} \rightarrow D_s^+ D_s^-$, we neglect this correlation coefficient (putting it to zero). Given the increasing experimental precision, we have also stopped using the two-dimensional $\Delta \Gamma _{s} -\phi _s^{c{{\bar{c}}}s}$ histograms provided by the CDF and D0 collaborations: we are now approximating those with two-dimensional Gaussian likelihoods.

Fig. 10

68% CL regions in ${B^{0}_s}$ width difference $\Delta \Gamma _{s}$ and weak phase $\phi _s^{c{{\bar{c}}}s}$ obtained from individual and combined CDF [197], D0  [198], ATLAS [199, 200], CMS [201] and LHCb [202, 203, 240, 242] likelihoods of ${B^{0}_s} \rightarrow {J/\psi } \phi$, ${B^{0}_s} \rightarrow {J/\psi } K^+K^-$, ${B^{0}_s} \rightarrow \psi (2S) \phi$, ${B^{0}_s} \rightarrow {J/\psi } \pi ^+\pi ^-$ and ${B^{0}_s} \rightarrow D_s^+D_s^-$ samples. The expectation within the Standard Model [60, 107, 193] is shown as the black rectangle

Table 21 Direct experimental measurements of $\phi _s^{c{{\bar{c}}}s}$, $\Delta \Gamma _{s}$ and $\Gamma _{s}$ using ${B^{0}_s} \rightarrow {J/\psi } \phi$, ${J/\psi } K^+K^-$, $\psi (2S)\phi$, ${J/\psi } \pi ^+\pi ^-$ and $D_s^+D_s^-$ decays. Only the solution with $\Delta \Gamma _{s} > 0$ is shown, since the two-fold ambiguity has been resolved in Ref. [1]. The first error is due to statistics, the second one to systematics. The last line gives our average

We obtain the individual and combined contours shown in Fig. 10. Maximizing the likelihood, we find, as summarized in Table 21:

$$\begin{aligned} \Delta \Gamma _{s}= & {} +0.085\,{\,\pm \,}0.007~\mathrm ps^{-1}, \end{aligned}$$

(88)

$$\begin{aligned} \phi _s^{c{{\bar{c}}}s}= & {} -0.030\,{\,\pm \,}0.033. \end{aligned}$$

(89)

The above $\Delta \Gamma _{s}$ average is consistent, but highly correlated with the average of Eq. (72). Our final recommended average for $\Delta \Gamma _{s}$ is the one of Eq. (72), which includes all available information on $\Delta \Gamma _{s}$.

In the Standard Model and ignoring sub-leading penguin contributions, $\phi _s^{c{{\bar{c}}}s}$ is expected to be equal to $-2\beta _s$, where $\beta _s = \arg \left[ -\left( V_{ts}V^*_{tb}\right) /\left( V_{cs}V^*_{cb}\right) \right]$ is a phase analogous to the angle $\beta$ of the usual CKM unitarity triangle (aside from a sign change). An indirect determination via global fits to experimental data gives [193]

$$\begin{aligned} (\phi _s^{c{{\bar{c}}}s})^\mathrm{SM} = -2\beta _s = -0.0370\,{\,\pm \,}0.0006. \end{aligned}$$

(90)

The average value of $\phi _s^{c{{\bar{c}}}s}$ from Eq. (89) is consistent with this Standard Model expectation.

From its measurements of time-dependent $C\!P$ violation in ${B^{0}_s} \rightarrow K^+K^-$ decays, the LHCb collaboration has determined the ${B^{0}_s}$ mixing phase to be $-2\beta _s = -0.12^{+0.14}_{-0.12}$ [243], assuming a U-spin relation (with up to 50% breaking effects) between the decay amplitudes of ${B^{0}_s} \rightarrow K^+K^-$ and $B^0 \rightarrow \pi ^+\pi ^-$, and a value of the CKM angle $\gamma$ of $(70.1 \,\pm \,7.1)^{\circ }$. This determination is compatible with, and less precise than, the world average of $\phi _s^{c{{\bar{c}}}s}$ from Eq. (89).

New physics could contribute to $\phi _s^{c{{\bar{c}}}s}$. Assuming that new physics only enters in $M^s_{12}$ (rather than in $\Gamma ^s_{12}$), one can write [208, 209]

$$\begin{aligned} \phi _s^{c{{\bar{c}}}s} = - 2\beta _s + \phi _{12}^{s,\mathrm NP}, \end{aligned}$$

(91)

where the new physics phase $\phi _{12}^{s,\mathrm NP}$ is the same as that appearing in Eq. (86). In this case

$$\begin{aligned} \phi ^s_{12} = \phi _{12}^{s,\mathrm SM} +2\beta _s + \phi _s^{c{{\bar{c}}}s} = 0.012\,{\,\pm \,}0.033, \end{aligned}$$

(92)

where the numerical estimation was performed with the values of Eqs. (85), (89) and (90). Keeping in mind the approximation and assumption mentioned above, this can serve as a reference value to which the measurement of Eq. (87) can be compared.

4 Measurements related to unitarity triangle angles

The charge of the “$C\!P (t)$ and Unitarity Triangle angles” group is to provide averages of measurements obtained from analyses of decay-time-dependent asymmetries, and other quantities that are related to the angles of the Unitarity Triangle (UT). In cases where considerable theoretical input is required to extract the fundamental quantities, no attempt to do so is made. However, straightforward interpretations of the averages are given, where possible.

In Sect. 4.1 a brief introduction to the relevant phenomenology is given. In Sect. 4.2 an attempt is made to clarify the various different notations in use. In Sect. 4.3 the common inputs to which experimental results are rescaled in the averaging procedure are listed. We also briefly introduce the treatment of experimental errors. In the remainder of this section, the experimental results and their averages are given, divided into subsections based on the underlying quark-level decays. All the measurements reported are quantities determined from decay-time-dependent analyses, with the exception of several in Sect. 4.14, which are related to the UT angle $\gamma$ and are obtained from decay-time-integrated analyses. In the compilations of measurements, indications of the sizes of the data samples used by each experiment are given. For the $e^+e^-$ B factory experiments, this is quoted in terms of the number of $B{\bar{B}}$ pairs in the data sample, while the integrated luminosity is given for experiments at hadron colliders.

4.1 Introduction

The Standard Model Cabibbo–Kobayashi–Maskawa (CKM) quark mixing matrix ${V}$ must be unitary. The CKM matrix has four free parameters and these are conventionally written by the product of three (complex) rotation matrices [244], where the rotations are characterised by the Euler mixing angles between the generations, $\theta _{12}$, $\theta _{13}$ and $\theta _{23}$, and one overall phase $\delta$,

$$\begin{aligned} {V}= & {} \bigg ( \begin{array}{ccc} V_{ud} &{} V_{us} &{} V_{ub} \\ V_{cd} &{} V_{cs} &{} V_{cb} \\ V_{td} &{} V_{ts} &{} V_{tb} \\ \end{array} \bigg )\nonumber \\= & {} \left( \begin{array}{ccc} c_{12}c_{13} &{} s_{12}c_{13} &{} s_{13}e^{-i\delta } \\ -s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta } &{} c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta } &{} s_{23}c_{13} \\ s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta } &{} -c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta } &{} c_{23}c_{13} \end{array}\right) \nonumber \\ \end{aligned}$$

(93)

where $c_{ij}=\cos \theta _{ij}$, $s_{ij}=\sin \theta _{ij}$ for $i<j=1,2,3$.

Following the observation of a hierarchy between the different matrix elements, the Wolfenstein parametrisation [245] is an expansion of ${V}$ in terms of the four real parameters $\lambda$ (the expansion parameter), A, $\rho$ and $\eta$. Defining to all orders in $\lambda$ [246]

$$\begin{aligned} s_{12}\equiv & {} \lambda ,\nonumber \\ s_{23}\equiv & {} A\lambda ^2, \nonumber \\ s_{13}e^{-i\delta }\equiv & {} A\lambda ^3(\rho -i\eta ), \end{aligned}$$

(94)

and inserting these into the representation of Eq. (93), unitarity of the CKM matrix is achieved to all orders. A Taylor expansion of ${V}$ leads to the familiar approximation

$$\begin{aligned} {V}= & {} \left( \begin{array}{ccc} 1 - \lambda ^2/2 &{} \lambda &{} A \lambda ^3 ( \rho - i \eta ) \\ - \lambda &{} 1 - \lambda ^2/2 &{} A \lambda ^2 \\ A \lambda ^3 ( 1 - \rho - i \eta ) &{} - A \lambda ^2 &{} 1 \\ \end{array} \right) \nonumber \\&+ \mathcal{O}\bigg ( \lambda ^4 \bigg ). \end{aligned}$$

(95)

At order $\lambda ^{5}$, the obtained CKM matrix in this extended Wolfenstein parametrisation is:

$$\begin{aligned} {V}= & {} \left( \begin{array}{l@{\quad }l@{\quad }l} 1 - \frac{1}{2}\lambda ^{2} - \frac{1}{8}\lambda ^4 &{} \lambda &{} A \lambda ^{3} (\rho - i \eta ) \\ - \lambda + \frac{1}{2} A^2 \lambda ^5 \bigg [ 1 - 2 (\rho + i \eta ) \bigg ] &{} 1 - \frac{1}{2}\lambda ^{2} - \frac{1}{8}\lambda ^4 (1+4A^2) A \lambda ^{2} \\ A \lambda ^{3} \bigg [ 1 - (1-\frac{1}{2}\lambda ^2)(\rho + i \eta ) \bigg ] &{} -A \lambda ^{2} + \frac{1}{2}A\lambda ^4 \bigg [ 1 - 2(\rho + i \eta ) \bigg ] &{} 1 - \frac{1}{2}A^2 \lambda ^4 \end{array} \right) + \mathcal{O}\bigg ( \lambda ^{6} \bigg ). \end{aligned}$$

(96)

A non-zero value of $\eta$ implies that the CKM matrix is not purely real in this, or any, parametrisation, and is the origin of $C\!P$ violation in the Standard Model. This is encapsulated in a parametrisation-invariant way through the Jarlskog parameter $J = \mathrm{Im}\left( V_{us}V_{cb}V^*_{ub}V^*_{cs}\right)$ [247].

The unitarity relation ${V}^\dagger {V}= { 1 }$ results in a total of nine expressions, that can be written as $\sum _{i=u,c,t} V_{ij}^*V_{ik} = \delta _{jk}$, where $\delta _{jk}$ is the Kronecker symbol. Of the off-diagonal expressions ($j \ne k$), three can be transformed into the other three (under $j \leftrightarrow k$, corresponding to complex conjugation). This leaves three relations in which three complex numbers sum to zero, which therefore can be expressed as triangles in the complex plane, together with three relations in which the squares of the elements in each column of the CKM matrix sum to unity. Similar relations are obtained for the rows of the matrix from ${V}{V}^\dagger = { 1 }$, so there are in total six triangle relations and six sums to unity. More details about unitarity triangles can be found in Refs. [248,249,250,251].

One of the triangle relations,

$$\begin{aligned} V_{ud}V_{ub}^* + V_{cd}V_{cb}^* + V_{td}V_{tb}^* = 0, \end{aligned}$$

(97)

is of particular importance to the $B$ system, being specifically related to flavour-changing neutral-current $b \rightarrow d$ transitions. The three terms in Eq. (97) are of the same order, $\mathcal{O}( \lambda ^3 )$, and this relation is commonly known as the Unitarity Triangle. For presentational purposes, it is convenient to rescale the triangle by $(V_{cd}V_{cb}^*)^{-1}$, as shown inFig. 11.

Fig. 11

The unitarity triangle

Two popular naming conventions for the UT angles exist in the literature:

$$\begin{aligned} \alpha\equiv & {} \phi _2 = \arg \bigg [ - \frac{V_{td}V_{tb}^*}{V_{ud}V_{ub}^*} \bigg ], \nonumber \\ \beta\equiv & {} \phi _1 = \arg \bigg [ - \frac{V_{cd}V_{cb}^*}{V_{td}V_{tb}^*} \bigg ], \nonumber \\ \gamma\equiv & {} \phi _3 = \arg \bigg [ - \frac{V_{ud}V_{ub}^*}{V_{cd}V_{cb}^*} \bigg ]. \end{aligned}$$

(98)

In this document the $( \alpha , \beta , \gamma )$ set is used. The sides $R_u$ and $R_t$ of the Unitarity Triangle (the third side being normalised to unity) are given by

$$\begin{aligned} R_u= & {} \bigg |\frac{V_{ud}V_{ub}^*}{V_{cd}V_{cb}^*} \bigg | = \sqrt{\overline{\rho } ^2+\overline{\eta } ^2}, \nonumber \\ R_t= & {} \bigg |\frac{V_{td}V_{tb}^*}{V_{cd}V_{cb}^*}\bigg | = \sqrt{(1-\overline{\rho })^2+\overline{\eta } ^2}. \end{aligned}$$

(99)

where $\overline{\rho }$ and $\overline{\eta }$ define the apex of the Unitarity Triangle [246]

$$\begin{aligned} \overline{\rho } + i\overline{\eta }\equiv & {} -\frac{V_{ud}V_{ub}^*}{V_{cd}V_{cb}^*} \equiv 1 + \frac{V_{td}V_{tb}^*}{V_{cd}V_{cb}^*} \nonumber \\= & {} \frac{\sqrt{1-\lambda ^2}\,(\rho + i \eta )}{\sqrt{1-A^2\lambda ^4}+\sqrt{1-\lambda ^2}A^2\lambda ^4(\rho +i\eta )}. \end{aligned}$$

(100)

The exact relation between $( \rho , \eta )$ and $( \overline{\rho }, \overline{\eta })$ is

$$\begin{aligned} \rho + i\eta = \frac{ \sqrt{ 1-A^2\lambda ^4 }(\overline{\rho } +i\overline{\eta }) }{ \sqrt{ 1-\lambda ^2 } \bigg [ 1-A^2\lambda ^4(\overline{\rho } +i\overline{\eta }) \bigg ] }. \end{aligned}$$

(101)

By expanding in powers of $\lambda$, several useful approximate expressions can be obtained, including

$$\begin{aligned} \overline{\rho }= & {} \rho \left( 1 - \frac{1}{2}\lambda ^{2}\right) + \mathcal{O}(\lambda ^4), \nonumber \\ \overline{\eta }= & {} \eta \left( 1 - \frac{1}{2}\lambda ^{2}\right) + \mathcal{O}(\lambda ^4), \nonumber \\ V_{td}= & {} A \lambda ^{3} (1-\overline{\rho }-i\overline{\eta }) + \mathcal{O}(\lambda ^6). \end{aligned}$$

(102)

Recent world average values for the Wolfenstein parameters, evaluated using many of the measurements reported in this document, are [252]

$$\begin{aligned} A= & {} 0.8227 \,^{+0.0066}_{-0.0136}, \quad \lambda = 0.22543 \,^{+0.00042}_{-0.00031}, \nonumber \\ \overline{\rho }= & {} 0.1504 \,^{+0.0121}_{-0.0062}, \quad \overline{\eta } = 0.3540 \,^{+0.0069}_{-0.0076}. \end{aligned}$$

(103)

The relevant unitarity triangle for the $b \rightarrow s$ transition is obtained by replacing $d \leftrightarrow s$ in Eq. (97). Definitions of the set of angles $( \alpha _s, \beta _s, \gamma _s )$ can be obtained using equivalent relations to those of Eq. (98). However, this gives a value of $\beta _s$ that is negative in the Standard Model, so that the sign is usually flipped in the literature; this convention, i.e. $\beta _s = \arg \left[ - (V_{ts}V_{tb}^*)/(V_{cs}V_{cb}^*) \right]$, is also followed here and in Sect. 3. Since the sides of the $b \rightarrow s$ unitarity triangle are not all of the same order of $\lambda$, the triangle is squashed and $\beta _s \sim \lambda ^2\eta$.

4.2 Notations

Several different notations for $C\!P$ violation parameters are commonly used. This section reviews those found in the experimental literature, in the hope of reducing the potential for confusion, and to define the frame that is used for the averages.

In some cases, when $B$ mesons decay into multibody final states via broad resonances ($\rho$, $K^*$, etc.), the experimental analyses ignore the effects of interference between the overlapping structures. This is referred to as the quasi-two-body (Q2B) approximation in the following.

4.2.1 $C\!P$ asymmetries

The $C\!P$ asymmetry is defined as the difference between the rate involving a b quark and that involving a ${\bar{b}}$ quark, divided by the sum. For example, the partial rate (or charge) asymmetry for a charged $B$ decay would be given as

$$\begin{aligned} \mathcal{A}_{f} \;\equiv \; \frac{\Gamma (B^{-}\rightarrow f)-\Gamma (B^{+}\rightarrow {\bar{f}})}{\Gamma (B^{-}\rightarrow f)+\Gamma (B^{+}\rightarrow {\bar{f}})}. \end{aligned}$$

(104)

4.2.2 Time-dependent $C\!P$ asymmetries in decays to $C\!P$ eigenstates

If the amplitudes for $B^{0}$ and ${\overline{B}} ^0$ to decay to a final state f, which is a $C\!P$ eigenstate with eigenvalue ${\eta _f}$, are given by $A_f$ and $\overline{A}_f$, respectively, then the decay distributions for neutral $B$ mesons, with known (i.e. “tagged”) flavour at time $\Delta t =0$, are given by

$$\begin{aligned} \Gamma _{{\overline{B}} ^0 \rightarrow f} (\Delta t)= & {} \frac{e^{-| \Delta t | / \tau (B^{0})}}{4\tau (B^{0})} \bigg [ 1 + \frac{2\, \mathrm{Im}(\lambda _f)}{1 + |\lambda _f|^2} \sin (\Delta m \Delta t)\nonumber \\&- \frac{1 - |\lambda _f|^2}{1 + |\lambda _f|^2} \cos (\Delta m \Delta t) \bigg ], \end{aligned}$$

(105)

$$\begin{aligned} \Gamma _{B^{0}\rightarrow f} (\Delta t)= & {} \frac{e^{-| \Delta t | / \tau (B^{0})}}{4\tau (B^{0})} \bigg [ 1 - \frac{2\, \mathrm{Im}(\lambda _f)}{1 + |\lambda _f|^2} \sin (\Delta m \Delta t) \nonumber \\&+ \frac{1 - |\lambda _f|^2}{1 + |\lambda _f|^2} \cos (\Delta m \Delta t) \bigg ]. \end{aligned}$$

(106)

Here $\lambda _f = \frac{q}{p} \frac{\overline{A}_f}{A_f}$ contains terms related to $B^{0}$–${\overline{B}} ^0$ mixing and to the decay amplitude (the eigenstates of the $B^0 {{\overline{B}} ^0}$ system with physical masses and lifetimes are $| B_{\pm } \rangle = p | B^{0}\rangle \pm q | {\overline{B}} ^0 \rangle$). This formulation assumes $C\!PT$ invariance, and neglects possible lifetime differences between the two physical states (see Sect. 3.3 where the mass difference $\Delta m$ is also defined) in the neutral $B$ meson system. The case where non-zero lifetime differences are taken into account is discussed in Sect. 4.2.3.

The notation and normalisation used in Eqs. (105) and (106) are relevant for the $e^+e^-$ B factory experiments. In this case, neutral B mesons are produced via the $e^+e^- \rightarrow \varUpsilon (4S) \rightarrow B {\overline{B}}$ process, and the wavefunction of the produced $B {\overline{B}}$ pair evolves coherently until one meson decays. When one of the pair decays into a final state that tags its flavour, the flavour of the other at that instant is known. The evolution of the other neutral B meson is therefore described in terms of $\Delta t$, the difference between the decay times of the two mesons in the pair. At hadron collider experiments, t is usually used in place of $\Delta t$ since the flavour tagging is done at production ($t = 0$); due to the nature of the production in hadron colliders (incoherent $b{\bar{b}}$ quark pair production with many additional associated particles), very different methods are used for tagging compared to those in $e^+e^-$ experiments. Moreover, since negative values of t are not possible, the normalisation is such that $\int _0^{+\infty } ( \Gamma _{{\overline{B}} ^0 \rightarrow f} (t) + \Gamma _{B^{0}\rightarrow f} (t) ) dt = 1$, rather than $\int _{-\infty }^{+\infty } ( \Gamma _{{\overline{B}} ^0 \rightarrow f} (\Delta t) + \Gamma _{B^{0}\rightarrow f} (\Delta t) ) d(\Delta t) = 1$, as in Eqs. (105) and (106).

The time-dependent $C\!P$ asymmetry, again defined as the difference between the rate involving a b quark and that involving a ${\bar{b}}$ quark, is then given by

$$\begin{aligned} \mathcal{A}_{f} (\Delta t)\equiv & {} \frac{ \Gamma _{{\overline{B}} ^0 \rightarrow f} (\Delta t) - \Gamma _{B^{0}\rightarrow f} (\Delta t) }{ \Gamma _{{\overline{B}} ^0 \rightarrow f} (\Delta t) + \Gamma _{B^{0}\rightarrow f} (\Delta t) } \; \nonumber \\= & {} \frac{2\, \mathrm{Im}(\lambda _f)}{1 + |\lambda _f|^2} \sin (\Delta m \Delta t)\nonumber \\&- \frac{1 - |\lambda _f|^2}{1 + |\lambda _f|^2} \cos (\Delta m \Delta t). \end{aligned}$$

(107)

While the coefficient of the $\sin (\Delta m \Delta t)$ term in Eq. (107) is customarily¹⁹ denoted $S_f$:

$$\begin{aligned} S_f \;\equiv \; \frac{2\, \mathrm{Im}(\lambda _f)}{1 + |\lambda _f|^2}, \end{aligned}$$

(108)

different notations are in use for the coefficient of the $\cos (\Delta m \Delta t)$ term:

$$\begin{aligned} C_f \;\equiv \; - A_f \;\equiv \; \frac{1 - |\lambda _f|^2}{1 + |\lambda _f|^2}. \end{aligned}$$

(109)

The C notation has been used by the BaBar collaboration (see e.g. Ref. [253]), and is also adopted in this document. The A notation has been used by the Belle collaboration (see e.g. Ref. [254]). When the final state is a $C\!P$ eigenstate, the notation $S_{C\!P}$ and $C_{C\!P}$ is widely used, including in this document, instead of specifying the final state f. In addition, particularly when grouping together measurements with different final states mediated by the same quark-level transition, the S, C notation with a subscript indicating the transition is used.

Neglecting effects due to $C\!P$ violation in mixing (by taking $|q/p| = 1$), if the decay amplitude contains terms with a single weak (i.e. $C\!P$ violating) phase then $|\lambda _f| = 1$ and one finds $S_f = -{\eta _f}\sin (\phi _\mathrm{mix} + \phi _\mathrm{dec})$, $C_f = 0$, where $\phi _\mathrm{mix}=\arg (q/p)$ and $\phi _\mathrm{dec}=\arg (\overline{A}_f/A_f)$. Note that the $B^{0}$–${\overline{B}} ^0$ mixing phase $\phi _\mathrm{mix}$ is approximately equal to $2\beta$ in the Standard Model (in the usual phase convention) [255, 256].

If amplitudes with different weak phases contribute to the decay, no clean interpretation of $S_f$ is possible without further input. In this document, only the theoretically cleanest channels are interpreted as measurements of the weak phase (e.g. $b \rightarrow c{\bar{c}}s$ transitions for $\sin (2\beta )$), though even in these cases some care is necessary. In channels in which the second amplitude is expected to be suppressed, the concept of an effective weak phase difference is sometimes used, e.g. $\sin (2\beta ^\mathrm{eff})$ in $b \rightarrow q\bar{q}s$ transition.

If, in addition to having a weak phase difference, the decay amplitudes have different $C\!P$ conserving strong phases, then $| \lambda _f | \ne 1$. Additional input is required for interpretation of the results. The coefficient of the cosine term becomes non-zero, indicating $C\!P$ violation in decay.

Due to the fact that $\sin (\Delta m \Delta t)$ and $\cos (\Delta m \Delta t)$ are respectively odd and even functions of $\Delta t$, only small correlations (that can be induced by backgrounds, for example) between $S_f$ and $C_f$ are expected at an $e^+e^-$ B factory experiment, where the range of $\Delta t$ is $-\infty< \Delta t < +\infty$. The situation is different for measurements at hadron collider experiments, where the range of the time variable is $0< t < +\infty$, so that more sizable correlations can be expected. We include the correlations in the averages where available.

Frequently, we are interested in combining measurements governed by similar or identical short-distance physics, but with different final states (e.g., $B^{0}\rightarrow {J/\psi } K^0_{S}$ and $B^{0}\rightarrow {J/\psi } K^0_{L}$). In this case, we remove the dependence on the $C\!P$ eigenvalue of the final state by quoting $-{\eta }S_f$. In cases where the final state is not a $C\!P$ eigenstate but has an effective $C\!P$ content (see below), the reported $-{\eta }S$ is corrected by the effective $C\!P$.

4.2.3 Time-dependent distributions with non-zero decay width difference

A complete analysis of the time-dependent decay rates of neutral B mesons must also take into account the difference in lifetimes, denoted $\Delta \Gamma$, between the mass eigenstates. This is particularly important in the ${B^{0}_s}$ system, since a non-negligible value of $\Delta \Gamma _s$ has been established (see Sect. 3.3 for the latest experimental constraints). The formalism given here is therefore appropriate for measurements of ${B^{0}_s}$ decays to a $C\!P$ eigenstate f as studied as hadron colliders, but appropriate modifications for $B^{0}$ mesons or for the $e^+e^-$ environment are straightforward to make.

Neglecting $C\!P$ violation in mixing, the relevant replacements for Eqs. (105) and (106) are [257]

$$\begin{aligned} \Gamma _{{\bar{B}}^0_s \rightarrow f} (t)= & {} \mathcal{N} \frac{e^{-t/\tau ({B^{0}_s})}}{2\tau ({B^{0}_s})} \Bigg [ \cosh \Big (\frac{\Delta \Gamma _s t}{2}\Big ) + S_f \sin (\Delta m_s t) \nonumber \\&- C_f \cos \Big (\Delta m_s t\Big ) + A^{\Delta \Gamma }_f \sinh \Big (\frac{\Delta \Gamma _s t}{2}\Big ) \Bigg ],\nonumber \\ \end{aligned}$$

(110)

and

$$\begin{aligned} \Gamma _{{B^{0}_s} \rightarrow f} (t)= & {} \mathcal{N} \frac{e^{-t/\tau ({B^{0}_s})}}{2\tau ({B^{0}_s})} \Bigg [ \cosh \Big (\frac{\Delta \Gamma _s t}{2}\Big ) - S_f \sin \Big (\Delta m_s t\Big ) \nonumber \\&+ C_f \cos \Big (\Delta m_s t\Big ) + A^{\Delta \Gamma }_f \sinh \Big (\frac{\Delta \Gamma _s t}{2}\Big ) \Bigg ],\nonumber \\ \end{aligned}$$

(111)

where $S_f$ and $C_f$ are as defined in Eqs. (108) and (109), respectively, $\tau ({B^{0}_s}) = 1/\Gamma _s$ is defined in Sect. 3.2.4, and the coefficient of the $\sinh$ term is²⁰

$$\begin{aligned} A^{\Delta \Gamma }_f = - \frac{2\, \mathrm{Re}(\lambda _f)}{1 + |\lambda _f|^2}. \end{aligned}$$

(112)

With the requirement $\int _{0}^{+\infty } [ \Gamma _{{\bar{B}}^0_s \rightarrow f} (t) + \Gamma _{{B^{0}_s} \rightarrow f} (t) ] dt = 1$, the normalisation factor is fixed to $\mathcal{N} = (1 - (\frac{\Delta \Gamma _s}{2\Gamma _s})^2)/(1 +\frac{A^{\Delta \Gamma }_f \Delta \Gamma _s}{2\Gamma _s})$.²¹

A time-dependent analysis of $C\!P$ asymmetries in flavour-tagged ${B^{0}_s}$ decays to a $C\!P$ eigenstate f can thus determine the parameters $S_f$, $C_f$ and $A^{\Delta \Gamma }_f$. Note that, by definition,

$$\begin{aligned} ( S_f )^2 + ( C_f )^2 + ( A^{\Delta \Gamma }_f )^2 = 1, \end{aligned}$$

(113)

and this constraint can be imposed or not in the fits. Since these parameters have sensitivity to both $\mathrm{Im}(\lambda _f)$ and $\mathrm{Re}(\lambda _f)$, alternative choices of parametrisation, including those directly involving $C\!P$ violating phases (such as $\beta _s$), are possible. These can also be adopted for vector-vector final states.

The untagged time-dependent decay rate is given by

$$\begin{aligned}&\Gamma _{{\bar{B}}^0_s \rightarrow f} (t) + \Gamma _{{B^{0}_s} \rightarrow f} (t) = \mathcal{N} \frac{e^{-t/\tau ({B^{0}_s})}}{\tau ({B^{0}_s})} \nonumber \\&\quad \times \Big [ \cosh \bigg (\frac{\Delta \Gamma _s t}{2}\bigg ) + A^{\Delta \Gamma }_f \sinh \bigg (\frac{\Delta \Gamma _s t}{2}\bigg ) \Big ]. \end{aligned}$$

(114)

Thus, an untagged time-dependent analysis can probe $\lambda _f$, through the dependence of $A^{\Delta \Gamma }_f$ on $\mathrm{Re}(\lambda _f)$, when $\Delta \Gamma _s \ne 0$. This is equivalent to determining the “effective lifetime” [112], as discussed in Sect. 3.2.4. The analysis of flavour-tagged ${B^{0}_s}$ mesons is, of course, more sensitive.

The discussion in this and the previous section is relevant for decays to $C\!P$ eigenstates. In the following sections, various cases of time-dependent $C\!P$ asymmetries in decays to non-$C\!P$ eigenstates are considered. For brevity, these will be given assuming that the decay width difference $\Delta \Gamma$ is negligible. Modifications similar to those described here can be made to take into account a non-zero decay width difference.

4.2.4 Time-dependent $C\!P$ asymmetries in decays to vector-vector final states

Consider $B$ decays to states consisting of two spin-1 particles, such as ${J/\psi } K^{*0}(\rightarrow K^0_{S} \pi ^0)$, ${J/\psi } \phi$, $D^{*+}D^{*-}$ and $\rho ^+\rho ^-$, which are eigenstates of charge conjugation but not of parity.²² For such a system, there are three possible final states: in the helicity basis these can be written $h_{-1}, h_0, h_{+1}$. The $h_0$ state is an eigenstate of parity, and hence of $C\!P$, however $C\!P$ transforms $h_{+1} \leftrightarrow h_{-1}$ (up to an unobservable phase). In the transversity basis, these states are transformed into $h_\parallel = (h_{+1} + h_{-1})/2$ and $h_\perp = (h_{+1} - h_{-1})/2$. In this basis all three states are $C\!P$ eigenstates, and $h_\perp$ has the opposite $C\!P$ to the others.

The amplitude for decays to the transversity basis states are usually given by $A_{0,\perp ,\parallel }$, with normalisation such that $| A_0 |^2 + | A_\perp |^2 + | A_\parallel |^2 = 1$. Given the relation between the $C\!P$ eigenvalues of the states, the effective $C\!P$ content of the vector-vector state is known if $| A_\perp |^2$ is measured. An alternative strategy is to measure just the longitudinally polarised component, $| A_0 |^2$ (sometimes denoted by $f_\mathrm{long}$), which allows a limit to be set on the effective $C\!P$ since $| A_\perp |^2 \le | A_\perp |^2 + | A_\parallel |^2 = 1 - | A_0 |^2$. The value of the effective $C\!P$ content can be used to treat the decay with the same formalism as for $C\!P$ eigenstates. The most complete treatment for neutral $B$ decays to vector-vector final states is, however, time-dependent angular analysis (also known as time-dependent transversity analysis). In such an analysis, interference between $C\!P$-even and $C\!P$-odd states provides additional sensitivity to the weak and strong phases involved.

In most analyses of time-dependent $C\!P$ asymmetries in decays to vector-vector final states carried out to date, an assumption has been made that each helicity (or transversity) amplitude has the same weak phase. This is a good approximation for decays that are dominated by amplitudes with a single weak phase, such $B^{0}\rightarrow {J/\psi } K^{*0}$, and is a reasonable approximation in any mode for which only very limited statistics are available. However, for modes that have contributions from amplitudes with different weak phases, the relative size of these contributions can be different for each helicity (or transversity) amplitude, and therefore the time-dependent $C\!P$ asymmetry parameters can also differ. The most generic analysis, suitable for modes with sufficient statistics, allows for this effect; such an analysis has been carried out by LHCb for the $B^{0}\rightarrow {J/\psi } \rho ^0$ decay [258]. An intermediate analysis can allow different parameters for the $C\!P$-even and $C\!P$-odd components; such an analysis has been carried out by BaBar for the decay $B^{0}\rightarrow D^{*+}D^{*-}$ [259]. The independent treatment of each helicity (or transversity) amplitude, as in the latest result on ${B^{0}_s} \rightarrow {J/\psi } \phi$ [202] (discussed in Sect. 3), becomes increasingly important for high precision measurements.

4.2.5 Time-dependent asymmetries: self-conjugate multiparticle final states

Amplitudes for neutral $B$ decays into self-conjugate multiparticle final states such as $\pi ^+\pi ^-\pi ^0$, $K^+K^-K^0_{S}$, $\pi ^+\pi ^-K^0_{S}$, ${J/\psi } \pi ^+\pi ^-$ or $D\pi ^0$ with $D \rightarrow K^0_{S} \pi ^+\pi ^-$ may be written in terms of $C\!P$-even and $C\!P$-odd amplitudes. As above, the interference between these terms provides additional sensitivity to the weak and strong phases involved in the decay, and the time-dependence depends on both the sine and cosine of the weak phase difference. In order to perform unbinned maximum likelihood fits, and thereby extract as much information as possible from the distributions, it is necessary to choose a model for the multiparticle decay, and therefore the results acquire some model dependence. In certain cases, model-independent methods are also possible, but the resulting need to bin the Dalitz plot leads to some loss of statistical precision. The number of observables depends on the final state (and on the model used); the key feature is that as long as there are regions where both $C\!P$-even and $C\!P$-odd amplitudes contribute, the interference terms will be sensitive to the cosine of the weak phase difference. Therefore, these measurements allow distinction between multiple solutions for, e.g., the two values of $2\beta$ from the measurement of $\sin (2\beta )$.

We now consider the various notations that have been used in experimental studies of time-dependent asymmetries in decays to self-conjugate multiparticle final states.

$B^{0}\rightarrow D^{(*)}h^0$ with $D \rightarrow K^0_{S} \pi ^+\pi ^-$

The states $D\pi ^0$, $D^*\pi ^0$, $D\eta$, $D^*\eta$, $D\omega$ are collectively denoted $D^{(*)}h^0$. When the D decay model is fixed, fits to the time-dependent decay distributions can be performed to extract the weak phase difference. However, it is experimentally advantageous to use the sine and cosine of this phase as fit parameters, since these behave as essentially independent parameters, with low correlations and (potentially) rather different uncertainties. A parameter representing $C\!P$ violation in the B decay can be simultaneously determined. For consistency with other analyses, this could be chosen to be $C_f$, but could equally well be $| \lambda _f |$, or other possibilities.

Belle performed an analysis of these channels with $\sin (2\phi _1)$ and $\cos (2\phi _1)$ as free parameters [260]. BaBar has performed an analysis in which $| \lambda _f |$ was also determined [261]. Belle has in addition performed a model-independent analysis [262] using as input information about the average strong phase difference between symmetric bins of the Dalitz plot determined by CLEO-c [263].²³ The results of this analysis are measurements of $\sin (2\phi _1)$ and $\cos (2\phi _1)$.

$B^{0}\rightarrow D^{*+}D^{*-}K^0_{S}$

The hadronic structure of the $B^{0}\rightarrow D^{*+}D^{*-}K^0_{S}$ decay is not sufficiently well understood to perform a full time-dependent Dalitz plot analysis. Instead, following Ref. [264], BaBar [265] and Belle  [266] divide the Dalitz plane in two region: $m(D^{*+}K^0_{S})^2 > m(D^{*-}K^0_{S})^2$ $(\eta _y = +1)$ and $m(D^{*+}K^0_{S})^2 < m(D^{*-}K^0_{S})^2$ $(\eta _y = -1)$; and then fit to a decay time distribution with asymmetry given by

$$\begin{aligned} \mathcal{A}_{f} (\Delta t)= & {} \eta _y \frac{J_c}{J_0} \cos (\Delta m \Delta t) - \bigg [ \frac{2J_{s1}}{J_0} \sin (2\beta )\nonumber \\&+ \eta _y \frac{2J_{s2}}{J_0} \cos (2\beta ) \bigg ] \sin (\Delta m \Delta t) \,. \end{aligned}$$

(115)

The fitted observables are $\frac{J_c}{J_0}$, $\frac{2J_{s1}}{J_0} \sin (2\beta )$ and $\frac{2J_{s2}}{J_0} \cos (2\beta )$, where the parameters $J_0$, $J_c$, $J_{s1}$ and $J_{s2}$ are the integrals over the half Dalitz plane $m(D^{*+}K^0_{S})^2 < m(D^{*-}K^0_{S})^2$ of the functions $|a|^2 + |{\bar{a}}|^2$, $|a|^2 - |{\bar{a}}|^2$, $\mathrm{Re}({\bar{a}}a^*)$ and $\mathrm{Im}({\bar{a}}a^*)$ respectively, where a and ${\bar{a}}$ are the decay amplitudes of $B^{0}\rightarrow D^{*+}D^{*-}K^0_{S}$ and ${\overline{B}} ^0 \rightarrow D^{*+}D^{*-}K^0_{S}$ respectively. The parameter $J_{s2}$ (and hence $J_{s2}/J_0$) is predicted to be positive; assuming this prediction to be correct, it is possible to determine the sign of $\cos (2\beta )$.

$B^{0}\rightarrow {J/\psi } \pi ^+ \pi ^-$

Amplitude analyses of $B^{0}\rightarrow {J/\psi } \pi ^+ \pi ^-$ decays [258, 267] show large contributions from the $\rho (770)^0$ and $f_0(500)$ states, together with smaller contributions from higher resonances. Since modelling the $f_0(500)$ structure is challenging [268], it is difficult to determine reliably its associated $C\!P$ violation parameters. Corresponding parameters for the ${J/\psi } \rho ^0$ decay can, however, be determined. In the LHCb analysis [258], $2\beta ^\mathrm{eff}$ is determined from the fit; results are then converted into values for $S_{C\!P}$ and $C_{C\!P}$ to allow comparison with other modes. Here, the notation $S_{C\!P}$ and $C_{C\!P}$ denotes parameters obtained for the ${J/\psi } \rho ^0$ final state accounting for the composition of $C\!P$-even and $C\!P$-odd amplitudes (while assuming that all amplitudes involve the same phases), so that no dilution occurs. Possible $C\!P$ violation effects in the other amplitudes contributing to the Dalitz plot are treated as a source of systematic uncertainty.

Amplitude analyses have also been done for the ${B^{0}_s} \rightarrow {J/\psi } \pi ^+ \pi ^-$ decay, where the final state is dominated by scalar resonances including the $f_0(980)$ [240, 241]. Time-dependent analyses of this ${B^{0}_s}$ decay allow a determination of $2\beta _s$, as discussed in Sect. 3.

$B^{0}\rightarrow K^+K^-K^0$

Studies of $B^{0}\rightarrow K^+K^-K^0$ [269,270,271] and of the related decay $B^{+}\rightarrow K^+K^-K^+$ [271,272,273], show that the decay is dominated by a large nonresonant contribution with significant components from the intermediate $K^+K^-$ resonances $\phi (1020)$, $f_0(980)$, and other higher resonances, as well a contribution from $\chi _{c0}$.

The full time-dependent Dalitz plot analysis allows the complex amplitudes of each contributing term to be determined from data, including $C\!P$ violation effects (i.e. allowing the complex amplitude for the $B^{0}$ decay to be independent from that for ${\overline{B}} ^0$ decay), although one amplitude must be fixed to serve as a reference. There are several choices for parametrisation of the complex amplitudes (e.g. real and imaginary part, or magnitude and phase). Similarly, there are various approaches to include $C\!P$ violation effects. Note that positive definite parameters such as magnitudes are disfavoured in certain circumstances (they inevitably lead to biases for small values). In order to compare results between analyses, it is useful for each experiment to present results in terms of the parameters that can be measured in a Q2B analysis (such as $\mathcal{A}_{f}$, $S_f$, $C_f$, $\sin (2\beta ^\mathrm{eff})$, $\cos (2\beta ^\mathrm{eff})$, etc.)

In the BaBar analysis of the $B^{0}\rightarrow K^+K^-K^0$ decay [271], the complex amplitude for each resonant contribution is written as

$$\begin{aligned} A_f= & {} c_f ( 1 + b_f ) e^{i ( \phi _f + \delta _f )},\,\nonumber \\ {\bar{A}}_f= & {} c_f ( 1 - b_f ) e^{i ( \phi _f - \delta _f )}, \end{aligned}$$

(116)

where $b_f$ and $\delta _f$ introduce $C\!P$ violation in the magnitude and phase respectively. Belle [270] use the same parametrisation but with a different notation for the parameters.²⁴ (The weak phase in $B^0$–${\bar{B}}^0$ mixing ($2\beta$) also appears in the full formula for the time-dependent decay distribution.) The Q2B parameter of $C\!P$ violation in decay is directly related to $b_f$,

$$\begin{aligned} \mathcal{A}_{f} = \frac{-2b_f}{1+b_f^2} \approx C_f, \end{aligned}$$

(117)

and the mixing-induced $C\!P$ violation parameter can be used to obtain $\sin (2\beta ^\mathrm{eff})$,

$$\begin{aligned} -\eta _f S_f \approx \frac{1-b_f^2}{1+b_f^2}\sin (2\beta ^\mathrm{eff}_f), \end{aligned}$$

(118)

where the approximations are exact in the case that $| q/p | = 1$.

Both BaBar [271] and Belle  [270] present results for $c_f$ and $\phi _f$, for each resonant contribution, and in addition present results for $\mathcal{A}_{f}$ and $\beta ^\mathrm{eff}_{f}$ for $\phi (1020) K^0$, $f_0(980) K^0$ and for the remainder of the contributions to the $K^+K^-K^0$ Dalitz plot combined. BaBar also present results for the Q2B parameter $S_{f}$ for these channels. The models used to describe the resonant structure of the Dalitz plot differ, however. Both analyses suffer from symmetries in the likelihood that lead to multiple solutions, from which we select only one for averaging.

$B^{0}\rightarrow \pi ^+\pi ^-K^0_{S}$

Studies of $B^{0}\rightarrow \pi ^+\pi ^-K^0_{S}$ [274, 275] and of the related decay $B^{+}\rightarrow \pi ^+\pi ^-K^+$ [272, 276,277,278] show that the decay is dominated by components from intermediate resonances in the $K\pi$ ($K^*(892)$, $K^*_0(1430)$) and $\pi \pi$ ($\rho (770)$, $f_0(980)$, $f_2(1270)$) spectra, together with a poorly understood scalar structure that peaks near $m(\pi \pi ) \sim 1300 \ \mathrm{MeV}/c^2$ and is denoted $f_X$ ²⁵ and a large nonresonant component. There is also a contribution from the $\chi _{c0}$ state.

The full time-dependent Dalitz plot analysis allows the complex amplitudes of each contributing term to be determined from data, including $C\!P$ violation effects. In the BaBar analysis [274], the magnitude and phase of each component (for both $B^{0}$ and ${\overline{B}} ^0$ decays) are measured relative to $B^{0}\rightarrow f_0(980)K^0_{S}$, using the following parametrisation

$$\begin{aligned} A_f = | A_f | e^{i\,\mathrm{arg}(A_f)}\quad {\bar{A}}_f = | {\bar{A}}_f | e^{i\,\mathrm{arg}({\bar{A}}_f)} . \end{aligned}$$

(119)

In the Belle analysis [275], the $B^{0}\rightarrow K^{*+}\pi ^-$ amplitude is chosen as the reference, and the amplitudes are parametrised as

$$\begin{aligned} A_f = a_f ( 1 + c_f ) e^{i ( b_f + d_f )}\quad {\bar{A}}_f = a_f ( 1 - c_f ) e^{i ( b_f - d_f )}.\nonumber \\ \end{aligned}$$

(120)

In both cases, the results are translated into Q2B parameters such as $2\beta ^\mathrm{eff}_f$, $S_f$, $C_f$ for each $C\!P$ eigenstate f, and parameters of $C\!P$ violation in decay for each flavour-specific state. Relative phase differences between resonant terms are also extracted.

Table 22 Definitions of the U and I coefficients. Modified from Ref. [281]

$B^{0}\rightarrow \pi ^+\pi ^-\pi ^0$

The $B^{0}\rightarrow \pi ^+\pi ^-\pi ^0$ decay is dominated by intermediate $\rho$ resonances. Though it is possible, as above, to determine directly the complex amplitudes for each component, an alternative approach [279, 280] has been used by both BaBar [281, 282] and Belle  [283, 284]. The amplitudes for $B^{0}$ and ${\overline{B}} ^0$ decays to $\pi ^+\pi ^-\pi ^0$ are written as

$$\begin{aligned} A_{3\pi }= & {} f_+ A_+ + f_- A_- + f_0 A_0, \nonumber \\ {\bar{A}}_{3\pi }= & {} f_+ {\bar{A}}_+ + f_- {\bar{A}}_- + f_0 {\bar{A}}_0, \end{aligned}$$

(121)

respectively. The symbols $A_+$, $A_-$ and $A_0$ represent the complex decay amplitudes for $B^{0}\rightarrow \rho ^+\pi ^-$, $B^{0}\rightarrow \rho ^-\pi ^+$ and $B^{0}\rightarrow \rho ^0\pi ^0$ while ${\bar{A}}_+$, ${\bar{A}}_-$ and ${\bar{A}}_0$ represent those for ${\overline{B}} ^0 \rightarrow \rho ^+\pi ^-$, ${\overline{B}} ^0 \rightarrow \rho ^-\pi ^+$ and ${\overline{B}} ^0 \rightarrow \rho ^0\pi ^0$ respectively. The terms $f_+$, $f_-$ and $f_0$ incorporate kinematic and dynamical factors and depend on the Dalitz plot coordinates. The full time-dependent decay distribution can then be written in terms of 27 free parameters, one for each coefficient of the form factor bilinears, as listed in Table 22. These parameters are sometimes referred to as “the Us and Is”, and can be expressed in terms of $A_+$, $A_-$, $A_0$, ${\bar{A}}_+$, ${\bar{A}}_-$ and ${\bar{A}}_0$. If the full set of parameters is determined, together with their correlations, other parameters, such as weak and strong phases, parameters of $C\!P$ violation in decay, etc., can be subsequently extracted. Note that one of the parameters (typically $U_+^+$) is often fixed to unity to provide a reference; this does not affect the analysis.

4.2.6 Time-dependent $C\!P$ asymmetries in decays to non-$C\!P$ eigenstates

Consider a non-$C\!P$ eigenstate f, and its conjugate ${\bar{f}}$. For neutral $B$ decays to these final states, there are four amplitudes to consider: those for $B^{0}$ to decay to f and ${\bar{f}}$ ($A_f$ and $A_{{\bar{f}}}$, respectively), and the equivalents for ${\overline{B}} ^0$ ($\overline{A}_f$ and $\overline{A}_{{{\bar{f}}}}$). If $C\!P$ is conserved in the decay, then $A_f= \overline{A}_{{{\bar{f}}}}$ and $A_{{\bar{f}}}= \overline{A}_f$.

The time-dependent decay distributions can be written in many different ways. Here, we follow Sect. 4.2.2 and define $\lambda _f = \frac{q}{p}\frac{\overline{A}_f}{A_f}$ and $\lambda _{\bar{f}} = \frac{q}{p}\frac{\overline{A}_{{{\bar{f}}}}}{A_{{\bar{f}}}}$. The time-dependent $C\!P$ asymmetries that are sensitive to mixing-induced $C\!P$ violation effects then follow Eq. (107):

$$\begin{aligned} \mathcal{A}_f (\Delta t)\equiv & {} \frac{ \Gamma _{{\overline{B}} ^0 \rightarrow f} (\Delta t) - \Gamma _{B^{0}\rightarrow f} (\Delta t) }{ \Gamma _{{\overline{B}} ^0 \rightarrow f} (\Delta t) + \Gamma _{B^{0}\rightarrow f} (\Delta t) } \nonumber \\= & {} S_f \sin (\Delta m \Delta t) - C_f \cos (\Delta m \Delta t), \end{aligned}$$

(122)

$$\begin{aligned} \mathcal{A}_{{\bar{f}}} (\Delta t)\equiv & {} \frac{ \Gamma _{{\overline{B}} ^0 \rightarrow {\bar{f}}} (\Delta t) - \Gamma _{B^{0}\rightarrow {\bar{f}}} (\Delta t) }{ \Gamma _{{\overline{B}} ^0 \rightarrow {\bar{f}}} (\Delta t) + \Gamma _{B^{0}\rightarrow {\bar{f}}} (\Delta t) } \nonumber \\= & {} S_{{\bar{f}}} \sin (\Delta m \Delta t) - C_{{\bar{f}}} \cos (\Delta m \Delta t), \end{aligned}$$

(123)

with the definitions of the parameters $C_f$, $S_f$, $C_{{\bar{f}}}$ and $S_{{\bar{f}}}$, following Eqs. (108) and (109).

The time-dependent decay rates are given by

$$\begin{aligned} \Gamma _{{\overline{B}} ^0 \rightarrow f} (\Delta t)= & {} \frac{e^{-| \Delta t | / \tau (B^{0})}}{8\tau (B^{0})} ( 1 + \langle \mathcal{A}_{f{\bar{f}}}\rangle )\nonumber \\&\times \bigg [ 1 + S_f \sin (\Delta m \Delta t) - C_f \cos (\Delta m \Delta t) \bigg ],\nonumber \\ \end{aligned}$$

(124)

$$\begin{aligned} \Gamma _{B^{0}\rightarrow f} (\Delta t)= & {} \frac{e^{-| \Delta t | / \tau (B^{0})}}{8\tau (B^{0})} ( 1 + \langle \mathcal{A}_{f{\bar{f}}}\rangle )\nonumber \\&\times \bigg [ 1 - S_f \sin (\Delta m \Delta t) + C_f \cos (\Delta m \Delta t) \bigg ],\nonumber \\ \end{aligned}$$

(125)

$$\begin{aligned} \Gamma _{{\overline{B}} ^0 \rightarrow {\bar{f}}} (\Delta t)= & {} \frac{e^{-| \Delta t | / \tau (B^{0})}}{8\tau (B^{0})} ( 1 - \langle \mathcal{A}_{f{\bar{f}}}\rangle ) \nonumber \\&\times \bigg [ 1 + S_{{\bar{f}}} \sin (\Delta m \Delta t) - C_{{\bar{f}}} \cos (\Delta m \Delta t) \bigg ],\nonumber \\ \end{aligned}$$

(126)

$$\begin{aligned} \Gamma _{B^{0}\rightarrow {\bar{f}}} (\Delta t)= & {} \frac{e^{-| \Delta t | / \tau (B^{0})}}{8\tau (B^{0})} ( 1 - \langle \mathcal{A}_{f{\bar{f}}}\rangle )\nonumber \\&\times \bigg [ 1 - S_{{\bar{f}}} \sin (\Delta m \Delta t) + C_{{\bar{f}}} \cos (\Delta m \Delta t) \bigg ],\nonumber \\ \end{aligned}$$

(127)

where the time-independent parameter $\langle \mathcal{A}_{f{\bar{f}}}\rangle$ represents an overall asymmetry in the production of the f and ${\bar{f}}$ final states,²⁶

$$\begin{aligned} \langle \mathcal{A}_{f{\bar{f}}}\rangle = \frac{ \bigg ( | A_f|^2 + | \overline{A}_f|^2 \bigg ) - \bigg ( | A_{{\bar{f}}}|^2 + | \overline{A}_{{{\bar{f}}}}|^2 \bigg ) }{ \bigg ( | A_f|^2 + | \overline{A}_f|^2 \bigg ) + \bigg ( | A_{{\bar{f}}}|^2 + | \overline{A}_{{{\bar{f}}}}|^2 \bigg )}.\nonumber \\ \end{aligned}$$

(128)

Assuming $|q/p| = 1$, i.e. absence of $C\!P$ violation in mixing, the parameters $C_f$ and $C_{{\bar{f}}}$ can also be written in terms of the decay amplitudes as follows:

$$\begin{aligned} C_f = \frac{ | A_f|^2 - | \overline{A}_f|^2 }{ | A_f|^2 + | \overline{A}_f|^2 } \quad \mathrm{and} \quad C_{{\bar{f}}} = \frac{ | A_{{\bar{f}}}|^2 - | \overline{A}_{{{\bar{f}}}}|^2 }{ | A_{{\bar{f}}}|^2 + | \overline{A}_{{{\bar{f}}}}|^2 },\nonumber \\ \end{aligned}$$

(129)

giving asymmetries in the decay amplitudes of $B^{0}$ and ${\overline{B}} ^0$ to the final states f and ${\bar{f}}$ respectively. In this notation, the conditions for absence of $C\!P$ violation in decay are $\langle \mathcal{A}_{f{\bar{f}}}\rangle = 0$ and $C_f = - C_{{\bar{f}}}$. Note that $C_f$ and $C_{{\bar{f}}}$ are typically non-zero; e.g., for a flavour-specific final state, $\overline{A}_f= A_{{\bar{f}}}= 0$ ($A_f= \overline{A}_{{{\bar{f}}}}= 0$), they take the values $C_f = - C_{{\bar{f}}} = 1$ ($C_f = - C_{{\bar{f}}} = -1$).

The coefficients of the sine terms contain information about the weak phase. In the case that each decay amplitude contains only a single weak phase (i.e., no $C\!P$ violation in decay as well as none in mixing), these terms can be written as

$$\begin{aligned} S_f= & {} \frac{ - 2 | A_f| | \overline{A}_f| \sin ( \phi _\mathrm{mix} + \phi _\mathrm{dec} - \delta _f ) }{ | A_f|^2 + | \overline{A}_f|^2 } \quad \mathrm{and}\nonumber \\ S_{{\bar{f}}}= & {} \frac{ - 2 | A_{{\bar{f}}}| | \overline{A}_{{{\bar{f}}}}| \sin ( \phi _\mathrm{mix} + \phi _\mathrm{dec} + \delta _f ) }{ | A_{{\bar{f}}}|^2 + | \overline{A}_{{{\bar{f}}}}|^2 }, \end{aligned}$$

(130)

where $\delta _f$ is the strong phase difference between the decay amplitudes. If there is no $C\!P$ violation, the condition $S_f = - S_{{\bar{f}}}$ holds. If decay amplitudes with different weak and strong phases contribute, no clean interpretation of $S_f$ and $S_{{\bar{f}}}$ is possible.

The conditions for $C\!P$ invariance $C_f = - C_{{\bar{f}}}$ and $S_f = - S_{{\bar{f}}}$ motivate a rotation of the parameters:

$$\begin{aligned} S_{f{\bar{f}}}= & {} \frac{S_{f} + S_{{\bar{f}}}}{2},\quad \Delta S_{f{\bar{f}}} = \frac{S_{f} - S_{{\bar{f}}}}{2}\nonumber \\ C_{f{\bar{f}}}= & {} \frac{C_{f} + C_{{\bar{f}}}}{2},\quad \Delta C_{f{\bar{f}}} = \frac{C_{f} - C_{{\bar{f}}}}{2}. \end{aligned}$$

(131)

With these parameters, the $C\!P$ invariance conditions become $S_{f{\bar{f}}} = 0$ and $C_{f{\bar{f}}} = 0$. The parameter $\Delta C_{f{\bar{f}}}$ gives a measure of the “flavour-specificity” of the decay: $\Delta C_{f{\bar{f}}}=\,\pm \,1$ corresponds to a completely flavour-specific decay, in which no interference between decays with and without mixing can occur, while $\Delta C_{f{\bar{f}}} = 0$ results in maximum sensitivity to mixing-induced $C\!P$ violation. The parameter $\Delta S_{f{\bar{f}}}$ is related to the strong phase difference between the decay amplitudes of the $B^{0}$ meson to the f and to ${\bar{f}}$ final states. We note that the observables of Eq. (131) exhibit experimental correlations (typically of $\sim 20\%$, depending on the tagging purity, and other effects) between $S_{f{\bar{f}}}$ and $\Delta S_{f{\bar{f}}}$, and between $C_{f{\bar{f}}}$ and $\Delta C_{f{\bar{f}}}$. On the other hand, the final state specific observables of Eqs. (124)–(127) tend to have low correlations.

Alternatively, if we recall that the $C\!P$ invariance conditions at the decay amplitude level are $A_f= \overline{A}_{{{\bar{f}}}}$ and $A_{{\bar{f}}}= \overline{A}_f$, we are led to consider the parameters [252]

$$\begin{aligned} \mathcal{A}_{f{\bar{f}}} = \frac{ | \overline{A}_{{{\bar{f}}}}|^2 - | A_f|^2 }{ | \overline{A}_{{{\bar{f}}}}|^2 + | A_f|^2 } \quad \mathrm{and} \quad \mathcal{A}_{{\bar{f}}f} \!=\! \frac{ | \overline{A}_f|^2 - | A_{{\bar{f}}}|^2 }{ | \overline{A}_f|^2 + | A_{{\bar{f}}}|^2 }.\nonumber \\ \end{aligned}$$

(132)

These are sometimes considered more physically intuitive parameters since they characterise $C\!P$ violation in decay in decays with particular topologies. For example, in the case of $B^{0}\rightarrow \rho ^{\pm } \,\pi ^{{\mp }}$ (choosing $f = \rho ^+\pi ^-$ and ${\bar{f}} = \rho ^-\pi ^+$), $\mathcal{A}_{f{\bar{f}}}$ (also denoted $\mathcal{A}^{+-}_{\rho \pi }$) parametrises $C\!P$ violation in decays in which the produced $\rho$ meson does not contain the spectator quark, while $\mathcal{A}_{{\bar{f}}f}$ (also denoted $\mathcal{A}^{-+}_{\rho \pi }$) parametrises $C\!P$ violation in decays in which it does. Note that we have again followed the sign convention that the asymmetry is the difference between the rate involving a b quark and that involving a ${\bar{b}}$ quark, cf. Eq. (104). Of course, these parameters are not independent of the other sets of parameters given above, and can be written

$$\begin{aligned} \mathcal{A}_{f{\bar{f}}}= & {} - \frac{ \langle \mathcal{A}_{f{\bar{f}}}\rangle + C_{f{\bar{f}}} + \langle \mathcal{A}_{f{\bar{f}}}\rangle \Delta C_{f{\bar{f}}} }{ 1 + \Delta C_{f{\bar{f}}} + \langle \mathcal{A}_{f{\bar{f}}}\rangle C_{f{\bar{f}}} } \quad \mathrm{and} \nonumber \\ \mathcal{A}_{{\bar{f}}f}= & {} \frac{ - \langle \mathcal{A}_{f{\bar{f}}}\rangle + C_{f{\bar{f}}} + \langle \mathcal{A}_{f{\bar{f}}}\rangle \Delta C_{f{\bar{f}}} }{ - 1 + \Delta C_{f{\bar{f}}} + \langle \mathcal{A}_{f{\bar{f}}}\rangle C_{f{\bar{f}}} }. \end{aligned}$$

(133)

They usually exhibit strong correlations.

We now consider the various notations used in experimental studies of time-dependent $C\!P$ asymmetries in decays to non-$C\!P$ eigenstates.

$B^{0}\rightarrow D^{*\,\pm \,}D^{\mp }$

The ($\langle \mathcal{A}_{f{\bar{f}}}\rangle$, $C_f$, $S_f$, $C_{{\bar{f}}}$, $S_{{\bar{f}}}$) set of parameters was used in early publications by both BaBar [285] and Belle  [286] (albeit with slightly different notations) in the $D^{*\,\pm \,}D^{{\mp }}$ system ($f = D^{*+}D^-$, ${\bar{f}} = D^{*-}D^+$). In their most recent paper on this topic Belle  [287] instead used the parametrisation ($A_{D^*D}$, $S_{D^*D}$, $\Delta S_{D^*D}$, $C_{D^*D}$, $\Delta C_{D^*D}$), while BaBar [259] give results in both sets of parameters. We therefore use the ($A_{D^*D}$, $S_{D^*D}$, $\Delta S_{D^*D}$, $C_{D^*D}$, $\Delta C_{D^*D}$) set.

$B^{0}\rightarrow \rho ^{\,\pm \,}\pi ^{\mp }$

In the $\rho ^{\pm } \,\pi ^{\mp }$ system, the ($\langle \mathcal{A}_{f{\bar{f}}}\rangle$, $C_{f{\bar{f}}}$, $S_{f{\bar{f}}}$, $\Delta C_{f{\bar{f}}}$, $\Delta S_{f{\bar{f}}}$) set of parameters has been used originally by BaBar [288] and Belle  [289], in the Q2B approximation; the exact names²⁷ used in this case are $( \mathcal{A}_{C\!P}^{\rho \pi }, C_{\rho \pi }, S_{\rho \pi }, \Delta C_{\rho \pi }, \Delta S_{\rho \pi } )$, and these names are also used in this document.

Table 23 Conversion between the various notations used for $C\!P$ violation parameters in the $D^{\,\pm \,}\pi ^{{\mp }}$, $D^{*\,\pm \,}\pi ^{{\mp }}$ and $D^{\,\pm \,}\rho ^{{\mp }}$ systems. The $b_i$ terms used by BaBar have been neglected. Recall that $( \alpha , \beta , \gamma ) \equiv ( \phi _2, \phi _1, \phi _3 )$

Table 24 Translations used to convert the parameters measured by Belle to the parameters used for averaging in this document. The angular momentum factor L is $-1$ for $D^* \pi$ and $+1$ for $D\pi$. Recall that $( \alpha , \beta , \gamma ) \equiv ( \phi _2, \phi _1, \phi _3 )$

Since $\rho ^{\pm } \,\pi ^{\mp }$ is reconstructed in the final state $\pi ^+\pi ^-\pi ^0$, the interference between the $\rho$ resonances can provide additional information about the phases (see Sect. 4.2.5). Both BaBar [281] and Belle  [283, 284] have performed time-dependent Dalitz plot analyses, from which the weak phase $\alpha$ is directly extracted. In such an analysis, the measured Q2B parameters are also naturally corrected for interference effects.

$B^{0}\rightarrow D^{{\mp }}\pi ^{\,\pm \,}, D^{*{\mp }}\pi ^{\,\pm \,}, D^{{\mp }}\rho ^{\,\pm \,}$

Time-dependent $C\!P$ analyses have also been performed for the final states $D^{{\mp }}\pi ^{\,\pm \,}$, $D^{*{\mp }}\pi ^{\,\pm \,}$ and $D^{{\mp }}\rho ^{\,\pm \,}$. In these theoretically clean cases, no penguin contributions are possible, so there is no $C\!P$ violation in decay. Furthermore, due to the smallness of the ratio of the magnitudes of the suppressed ($b \rightarrow u$) and favoured ($b \rightarrow c$) amplitudes (denoted $R_f$), to a very good approximation, $C_f = - C_{{\bar{f}}} = 1$ (using $f = D^{(*)-}h^+$, ${\bar{f}} = D^{(*)+}h^-$ $h = \pi ,\rho$), and the coefficients of the sine terms are given by

$$\begin{aligned} S_f= & {} - 2 R_f \sin ( \phi _\mathrm{mix} + \phi _\mathrm{dec} - \delta _f ) \quad \text {and} \nonumber \\ S_{{\bar{f}}}= & {} - 2 R_f \sin ( \phi _\mathrm{mix} + \phi _\mathrm{dec} + \delta _f ). \end{aligned}$$

(134)

Thus weak phase information can be cleanly obtained from measurements of $S_f$ and $S_{{\bar{f}}}$, although external information on at least one of $R_f$ or $\delta _f$ is necessary. (Note that $\phi _\mathrm{mix} + \phi _\mathrm{dec} = 2\beta + \gamma \equiv 2\phi _1 + \phi _3$ for all the decay modes in question, while $R_f$ and $\delta _f$ depend on the decay mode.)

Again, different notations have been used in the literature. BaBar [290, 291] defines the time-dependent probability function by

$$\begin{aligned} f^{\pm } \, (\eta , \Delta t)= & {} \frac{e^{-|\Delta t|/\tau }}{4\tau } \nonumber \\&\times \bigg [ 1 {\mp } S_\zeta \sin (\Delta m \Delta t) {\mp } \eta C_\zeta \cos (\Delta m \Delta t) \bigg ],\nonumber \\ \end{aligned}$$

(135)

where the upper (lower) sign corresponds to the tagging meson being a $B^{0}$ (${\overline{B}} ^0$). The parameters $\eta$ and $\zeta$ take the values $+1$ and $+$ ($-1$ and −) when the final state is, e.g., $D^-\pi ^+$ ($D^+\pi ^-$). However, in the fit, the substitutions $C_\zeta = 1$ and $S_\zeta = a {\mp } \eta b_i - \eta c_i$ are made, where the subscript i denotes tagging category. Neglecting b terms,

$$\begin{aligned}&S_+ = a - c \quad \text {and} \quad S_- = a + c \ \Leftrightarrow \nonumber \\&a = (S_+ + S_-)/2 \quad \text {and} \nonumber \\&c = (S_- - S_+)/2 , \end{aligned}$$

(136)

in analogy to the parameters of Eq. (131). These are motivated by the possibility of $C\!P$ violation on the tag side [292], which is absent for semileptonic $B$ decays (mostly lepton tags). The parameter a is not affected by tag side $C\!P$ violation. The parameter b only depends on tag side $C\!P$ violation parameters and is not directly useful for determining UT angles. A clean interpretation of the c parameter is only possible for lepton-tagged events, so the BaBar measurements report c measured with those events only.

The parameters used by Belle in the analysis using partially reconstructed $B$ decays [293], are similar to the $S_\zeta$ parameters defined above. However, in the Belle convention, a tagging $B^{0}$ corresponds to a $+$ sign in front of the sine coefficient; furthermore the correspondence between the super/subscript and the final state is opposite, so that $S_{\pm } \,$ (BaBar) = $- S^{\mp }$ (Belle). In this analysis, only lepton tags are used, so there is no effect from tag side $C\!P$ violation. In the Belle analysis using fully reconstructed $B$ decays [294], this effect is measured and taken into account using $D^* \ell \nu$ decays; in neither Belle analysis are the a, b and c parameters used. In the latter case, the measured parameters are $2 R_{D^{(*)}\pi } \sin ( 2\phi _1 + \phi _3 \,\pm \, \delta _{D^{(*)}\pi } )$; the definition is such that $S^{\pm } \,$ (Belle) = $- 2 R_{D^* \pi } \sin ( 2\phi _1 + \phi _3 \,\pm \, \delta _{D^* \pi } )$. However, the definition includes an angular momentum factor $(-1)^L$ [295], and so for the results in the $D\pi$ system, there is an additional factor of $-1$ in the conversion.

Explicitly, the conversion then reads as given in Table 23, where we have neglected the $b_i$ terms used by BaBar (which are zero in the absence of tag side $C\!P$ violation). For the averages in this document, we use the a and c parameters, and give the explicit translations used in Table 24. It is to be fervently hoped that the experiments will converge on a common notation in future.

${B^{0}_s} \rightarrow D_s^{{\mp }}K^{\pm } \,$

The phenomenology of ${B^{0}_s} \rightarrow D_s^{{\mp }}K^{\pm } \,$ decays is similar to that for $B^{0}\rightarrow D^{{\mp }}\pi ^{\,\pm \,}$, with some important caveats. The two amplitudes $b \rightarrow u$ and $b \rightarrow c$ amplitudes have the same level of Cabibbo-suppression (i.e. are of the same order in $\lambda$) though the former is suppressed by $\sqrt{\rho ^2+\eta ^2}$. The large value of the ratio R of their magnitudes allows it to be determined from data, as the deviation of $C_f$ and $C_{{\bar{f}}}$ from unity (in magnitude) can be observed. Moreover, the non-zero value of $\Delta \Gamma _s$ allows the determination of additional terms, $A^{\Delta \Gamma }_f$ and $A^{\Delta \Gamma }_{{\bar{f}}}$ (see Sect. 4.2.3), that break ambiguities in the solutions for $\phi _\mathrm{mix} + \phi _\mathrm{dec}$, which for ${B^{0}_s} \rightarrow D_s^{{\mp }}K^{\pm } \,$ decays is equal to $\gamma -2\beta _s$.

LHCb [296] has performed such an analysis with ${B^{0}_s} \rightarrow D_s^{{\mp }}K^{\pm } \,$ decays. The absence of $C\!P$ violation in decay is assumed, and the parameters that are determined from the fit are labelled C, $A^{\Delta \Gamma }$, ${\bar{A}}{}^{\Delta \Gamma }$, S, ${\bar{S}}$. These are trivially related to the definitions used in this section.

Time-dependent asymmetries in radiative $B$ decays

As a special case of decays to non-$C\!P$ eigenstates, let us consider radiative $B$ decays. Here, the emitted photon has a distinct helicity, which is in principle observable, but in practice is not usually measured. Thus the measured time-dependent decay rates for $B^{0}$ decays are given by [297, 298]

$$\begin{aligned} \Gamma _{{\overline{B}} ^0 \rightarrow X \gamma } (\Delta t)= & {} \Gamma _{{\overline{B}} ^0 \rightarrow X \gamma _L} (\Delta t) + \Gamma _{{\overline{B}} ^0 \rightarrow X \gamma _R} (\Delta t)\nonumber \\= & {} \frac{e^{-| \Delta t | / \tau (B^{0})}}{4\tau (B^{0})} \left[ 1 + \left( S_L + S_R \right) \times \sin (\Delta m \Delta t)\right. \nonumber \\&\left. - \left( C_L + C_R \right) \cos (\Delta m \Delta t) \right] , \end{aligned}$$

(137)

$$\begin{aligned} \Gamma _{B^{0}\rightarrow X \gamma } (\Delta t)= & {} \Gamma _{B^{0}\rightarrow X \gamma _L} (\Delta t) + \Gamma _{B^{0}\rightarrow X \gamma _R} (\Delta t) \nonumber \\= & {} \frac{e^{-| \Delta t | / \tau (B^{0})}}{4\tau (B^{0})} \bigg [ 1 - ( S_L + S_R )\times \sin (\Delta m \Delta t) \nonumber \\&+ ( C_L + C_R ) \cos (\Delta m \Delta t) \bigg ], \end{aligned}$$

(138)

where in place of the subscripts f and ${\bar{f}}$ we have used L and R to indicate the photon helicity. In order for interference between decays with and without $B^{0}$-${\overline{B}} ^0$ mixing to occur, the X system must not be flavour-specific, e.g., in case of $B^{0}\rightarrow K^{*0}\gamma$, the final state must be $K^0_{S} \pi ^0 \gamma$. The sign of the sine term depends on the C eigenvalue of the X system. At leading order, the photons from $b \rightarrow q \gamma$ (${\bar{b}} \rightarrow \bar{q} \gamma$) are predominantly left (right) polarised, with corrections of order of $m_q/m_b$, thus interference effects are suppressed. Higher-order effects can lead to corrections of order $\varLambda _\mathrm{QCD}/m_b$ [299, 300], though explicit calculations indicate that such corrections may be small for exclusive final states [301, 302]. The predicted smallness of the S terms in the Standard Model results in sensitivity to new physics contributions.

The formalism discussed above is valid for any radiative decay to a final state where the hadronic system is an eigenstate of C. In addition to $K^0_{S} \pi ^0 \gamma$, experiments have presented results using $B^{0}$ decays to $K^0_{S} \eta \gamma$, $K^0_{S} \rho ^0\gamma$ and $K^0_{S} \phi \gamma$. For the case of the $K^0_{S} \rho ^0\gamma$ final state, particular care is needed, as due to the non-negligible width of the $\rho ^0$ meson, decays selected as $B^{0}\rightarrow K^0_{S} \rho ^0\gamma$ can include a significant contribution from $K^{*\,\pm \,}\pi ^{\mp } \gamma$ decays, which are flavour-specific and do not have the same oscillation phenomenology. It is therefore necessary to correct the fitted asymmetry parameter for a “dilution factor”.

In the case of radiative ${B^{0}_s}$ decays, the time-dependent decay rates of Eqs. (137) and (138) must be modified, in a similar way as discussed in Sect. 4.2.3, to account for the non-zero value of $\Delta \Gamma _{s}$. Thus, for decays such as ${B^{0}_s} \rightarrow \phi \gamma$, there is an additional observable, $A^{\Delta \Gamma }_{\phi \gamma }$, which can be determined from an untagged effective lifetime measurement [303].

4.2.7 Asymmetries in $B \rightarrow D^{(*)} K^{(*)}$ decays

$C\!P$ asymmetries in $B \rightarrow D^{(*)} K^{(*)}$ decays are sensitive to $\gamma$. The neutral $D^{(*)}$ meson produced is an admixture of $D^{(*)0}$ (produced by a $b \rightarrow c$ transition) and ${\overline{D}} ^{(*)0}$ (produced by a colour-suppressed $b \rightarrow u$ transition) states. If the final state is chosen so that both $D^{(*)0}$ and ${\overline{D}} ^{(*)0}$ can contribute, the two amplitudes interfere, and the resulting observables are sensitive to $\gamma$, the relative weak phase between the two $B$ decay amplitudes [304]. Various methods have been proposed to exploit this interference, including those where the neutral D meson is reconstructed as a $C\!P$ eigenstate (GLW) [305, 306], in a suppressed final state (ADS) [307, 308], or in a self-conjugate three-body final state, such as $K^0_{S} \pi ^+\pi ^-$ (GGSZ or Dalitz) [309, 310]. It should be emphasised that while each method differs in the choice of D decay, they are all sensitive to the same parameters of the B decay, and can be considered as variations of the same technique.

Consider the case of $B^{\mp } \rightarrow D K^{\mp }$, with D decaying to a final state f, which is accessible from both $D^0$ and ${\overline{D}} ^0$. We can write the decay rates for $B^{-}$ and $B^{+}$ ($\Gamma _{\mp }$), the charge averaged rate [$\Gamma = (\Gamma _- + \Gamma _+)/2$) and the charge asymmetry ($A = (\Gamma _- - \Gamma _+)/(\Gamma _- + \Gamma _+)$, see Eq. (104)] as

$$\begin{aligned} \Gamma _{\mp }\propto & {} r_B^2 + r_D^2 + 2 r_B r_D \cos \left( \delta _B + \delta _D {\mp } \gamma \right) , \end{aligned}$$

(139)

$$\begin{aligned} \Gamma\propto & {} r_B^2 + r_D^2 + 2 r_B r_D \cos \left( \delta _B + \delta _D \right) \cos \left( \gamma \right) , \end{aligned}$$

(140)

$$\begin{aligned} A= & {} \frac{ 2 r_B r_D \sin \left( \delta _B + \delta _D \right) \sin \left( \gamma \right) }{ r_B^2 + r_D^2 + 2 r_B r_D \cos \left( \delta _B + \delta _D \right) \cos \left( \gamma \right) }, \end{aligned}$$

(141)

where the ratio of $B$ decay amplitudes²⁸ is usually defined to be less than one,

$$\begin{aligned} r_B = \left| \frac{A( B^{-}\rightarrow {\overline{D}} ^0 K^- )}{A( B^{-}\rightarrow D^0 K^- )} \right| = \left| \frac{A( B^{+}\rightarrow D^0 K^+ )}{A( B^{+}\rightarrow {\overline{D}} ^0 K^+ )} \right| ,\nonumber \\ \end{aligned}$$

(142)

and the ratio of D decay amplitudes is correspondingly defined by

$$\begin{aligned} r_D = \left| \frac{A( D^0 \rightarrow f )}{A( {\overline{D}} ^0 \rightarrow f )} \right| . \end{aligned}$$

(143)

The relation between $B^{-}$ and $B^{+}$ amplitudes given in Eq. (142) is a result of their being only one weak phase contributing to each amplitude in the Standard Model, which is the source of the theoretical cleanliness of this approach to measure $\gamma$ [311]. The strong phase differences between the $B$ and D decay amplitudes are given by $\delta _B$ and $\delta _D$, respectively. The values of $r_D$ and $\delta _D$ depend on the final state f: for the GLW analysis, $r_D = 1$ and $\delta _D$ is trivial (either zero or $\pi$); for other modes, values of $r_D$ and $\delta _D$ are not trivial and for multibody final states they vary across the phase space. This can be quantified either by an explicit D decay amplitude model or by model-independent information. In the case that the multibody final state is treated inclusively, the formalism is modified by the inclusion of a coherence factor, usually denoted $\kappa$, while $r_D$ and $\delta _D$ become effectively parameters corresponding to amplitude-weighted averages across the phase space.

Note that, for given values of $r_B$ and $r_D$, the maximum size of A (at $\sin ( \delta _B + \delta _D ) = 1$) is $2 r_B r_D \sin ( \gamma ) / ( r_B^2 + r_D^2 )$. Thus even for D decay modes with small $r_D$, large asymmetries, and hence sensitivity to $\gamma$, may occur for B decay modes with similar values of $r_B$. For this reason, the ADS analysis of the decay $B^{\mp } \rightarrow D \pi ^{\mp }$ is also of interest.

The expressions of Eqs. (139)–(143) are for a specific point in phase space, and therefore are relevant where both B and D decays are to two-body final states. Additional coherence factors enter the expressions when the B decay is to a multibody final state (further discussion of multibody D decays can be found below). In particular, experiments have studied $B^+ \rightarrow DK^*(892)^+$, $B^0 \rightarrow DK^*(892)^0$ and $B^+ \rightarrow DK^+\pi ^+\pi ^-$ decays. Considering, for concreteness, the $B \rightarrow DK^*(892)$ case, the non-negligible width of the $K^*(892)$ resonance implies that contributions from other $B \rightarrow DK\pi$ decays can pass the selection requirements. Their effect on the Q2B analysis can be accounted for with a coherence factor [312], usually denoted $\kappa$, which tends to unity in the limit that the $K^*(892)$ resonance is the only signal amplitude contributing in the selected region of phase space. In this case, the hadronic parameters $r_B$ and $\delta _B$ become effectively weighted averages across the selected phase space of the magnitude ratio and relative strong phase between the CKM-suppressed and -favoured amplitudes; these effective parameters are denoted ${\bar{r}}_B$ and ${\bar{\delta }}_B$ (the notations $r_s$, $\delta _s$ and $r_S$, $\delta _S$ are also found in the literature). An alternative, and in certain cases more advantageous, approach is Dalitz plot analysis of the full $B \rightarrow DK\pi$ phase space [313, 314].

$B \rightarrow D^{(*)} K^{(*)}$ with $D \rightarrow$ $C\!P$ eigenstate decays

In the GLW analysis, the measured quantities are the partial rate asymmetry and the charge averaged rate, which are measured both for $C\!P$-even and $C\!P$-odd D decays. The latter is defined as

$$\begin{aligned} R_{C\!P} = \frac{2 \, \Gamma \big ( B^{+}\rightarrow D_{C\!P} K^+ \big )}{\Gamma \big ( B^{+}\rightarrow {\overline{D}} ^0 K^+ \big )}. \end{aligned}$$

(144)

It is experimentally convenient to measure $R_{C\!P}$ using a double ratio,

$$\begin{aligned} R_{C\!P} = \frac{ \Gamma \big ( B^{+}\rightarrow D_{C\!P} K^+ \big ) \, / \, \Gamma \big ( B^{+}\rightarrow {\overline{D}} ^0 K^+ \big ) }{ \Gamma \big ( B^{+}\rightarrow D_{C\!P} \pi ^+ \big ) \, / \, \Gamma \big ( B^{+}\rightarrow {\overline{D}} ^0 \pi ^+ \big )}\quad \end{aligned}$$

(145)

that is normalised both to the rate for the favoured ${\overline{D}} ^0 \rightarrow K^+ \pi ^-$ decay, and to the equivalent quantities for $B^{+}\rightarrow D\pi ^+$ decays (charge conjugate processes are implicitly included in Eqs. (144) and (145)). In this way the constant of proportionality drops out of Eq. (140). Equation (145) is exact in the limit that the contribution of the $b \rightarrow u$ decay amplitude to $B^{+}\rightarrow D \pi ^+$ vanishes and when the flavour-specific rates $\Gamma ( B^{+}\rightarrow {\overline{D}} ^0 h^+ )$ ($h = \pi ,K$) are determined using appropriately flavour-specific D decays. In reality, the decay $D \rightarrow K\pi$ is used, leading to a small source of systematic uncertainty. The $C\!P$ asymmetry is defined as

$$\begin{aligned} A_{C\!P} = \frac{ \Gamma (B^{-}\rightarrow D_{C\!P}K^-) - \Gamma (B^{+}\rightarrow D_{C\!P}K^+) }{ \Gamma (B^{-}\rightarrow D_{C\!P}K^-) + \Gamma (B^{+}\rightarrow D_{C\!P}K^+) }.\nonumber \\ \end{aligned}$$

(146)

$B \rightarrow D^{(*)} K^{(*)}$ with $D \rightarrow$ non-$C\!P$ eigenstate two-body decays

For the ADS analysis, based on a suppressed $D \rightarrow f$ decay, the measured quantities are again the partial rate asymmetry, and the charge averaged rate. In this case it is sufficient to measure the rate in a single ratio (normalised to the favoured $D \rightarrow {\bar{f}}$ decay) since potential systematic uncertainties related to detection cancel naturally; the observed quantity is then

$$\begin{aligned} R_\mathrm{ADS} \!=\! \frac{ \Gamma ( B^{-}\rightarrow [\,f\,]_D K^-) \!+\! \Gamma ( B^{+}\rightarrow [\,{\bar{f}}\,]_D K^+) }{ \Gamma ( B^{-}\rightarrow [\,{\bar{f}}\,]_D K^-) \!+\! \Gamma ( B^{+}\rightarrow [\,f\,]_D K^+)},\nonumber \\ \end{aligned}$$

(147)

where the inclusion of charge-conjugate modes has been made explicit. The $C\!P$ asymmetry is defined as

$$\begin{aligned} A_\mathrm{ADS} \!=\! \frac{ \Gamma (B^{-}\rightarrow [\,f\,]_DK^-)\!-\! \Gamma (B^{+}\rightarrow [\,f\,]_DK^+) }{ \Gamma (B^{-}\rightarrow [\,f\,]_DK^-)\!+\! \Gamma (B^{+}\rightarrow [\,f\,]_DK^+)}.\nonumber \\ \end{aligned}$$

(148)

Since the uncertainty of $A_\mathrm{ADS}$ depends on the central value of $R_\mathrm{ADS}$, for some statistical treatments it is preferable to use an alternative pair of parameters [315]

$$\begin{aligned} R_-= & {} \frac{ \Gamma ( B^{-}\rightarrow [\,f\,]_D K^-) }{ \Gamma ( B^{-}\rightarrow [\,{\bar{f}}\,]_D K^-) } \,\nonumber \\ R_+= & {} \frac{ \Gamma ( B^{+}\rightarrow [\,{\bar{f}}\,]_D K^+) }{ \Gamma ( B^{+}\rightarrow [\,f\,]_D K^+) }, \end{aligned}$$

(149)

where there is no implied inclusion of charge-conjugate processes. These parameters are statistically uncorrelated but may be affected by common sources of systematic uncertainty. We use the $(R_\mathrm{ADS}, A_\mathrm{ADS})$ set in our compilation where available.

In the ADS analysis, there are two additional unknowns ($r_D$ and $\delta _D$) compared to the GLW case. However, the value of $r_D$ can be measured using decays of D mesons of known flavour, and $\delta _D$ can be measured from interference effects in decays of quantum-correlated $D{\bar{D}}$ pairs produced at the $\psi (3770)$ resonance. More generally, one needs access to two different linear admixtures of $D^0$ and ${\bar{D}}{}^0$ states in order to determine the relative phase: one such sample can be flavour tagged D mesons, which are available in abundant quantities in many experiments; the other can be $C\!P$-tagged D mesons from $\psi (3770)$ decays or could be mixed D mesons (or could be the combination of $D^0$ and ${\bar{D}}{}^0$ that is found in $B \rightarrow DK$ decays). In fact, the most precise information on both $r_D$ and $\delta _D$ currently comes from global fits on charm mixing parameters, as discussed in Sect. 8.1.

Table 25 Summary of relations between measured and physical parameters in GLW, ADS and Dalitz analyses of $B \rightarrow D^{(*)} K^{(*)}$ decays

The relation of $A_\mathrm{ADS}$ to the underlying parameters given in Eq. (141) and Table 25 is exact for a two-body D decay. For multibody decays, a similar formalism can be used with the introduction of a coherence factor [316]. This is most appropriate for doubly-Cabibbo-suppressed decays to non-self-conjugate final states, but can also be modified for use with singly-Cabibbo-suppressed decays [317]. For multibody self-conjugate final states, such as $K^0_{S} \pi ^+\pi ^-$, a Dalitz plot analysis (discussed below) is often more appropriate. However, in certain cases where the final state can be approximated as a $C\!P$ eigenstate, a modified version of the GLW formalism can be used [318]. In such cases the observables are denoted $A_\mathrm{qGLW}$ and $R_\mathrm{qGLW}$ to indicate that the final state is not a pure $C\!P$ eigenstate.

$B \rightarrow D^{(*)} K^{(*)}$ with $D \rightarrow$ multibody final state decays

In the Dalitz plot (or GGSZ) analysis of D decays to multibody self-conjugate final states, once a model is assumed for the D decay, which gives the values of $r_D$ and $\delta _D$ across the Dalitz plot, it is possible to perform a simultaneous fit to the $B^+$ and $B^-$ samples and directly extract $\gamma$, $r_B$ and $\delta _B$. However, the uncertainties on the phases depend approximately inversely on $r_B$. Furthermore, $r_B$ is positive definite and therefore tends to be overestimated (unless $\sigma (r_B) \ll r_B$), which leads to an underestimation of the uncertainty on $\gamma$ that must be corrected statistically. An alternative approach is to extract from the data the “Cartesian” variables

$$\begin{aligned} ( x_{\pm } \,, y_{\pm } \, )= & {} ( \mathrm{Re}(r_B e^{i(\delta _B\,\pm \,\gamma )}), \mathrm{Im}(r_B e^{i(\delta _B\,\pm \,\gamma )}) )\nonumber \\= & {} ( r_B \cos (\delta _B\,\pm \,\gamma ), r_B \sin (\delta _B\,\pm \,\gamma ) ). \end{aligned}$$

(150)

These variables tend to be statistically well-behaved, and are therefore appropriate for combination of results. The pairs of variables $( x_{\pm } \,, y_{\pm })$ can be extracted from independent fits of the $B^{\pm } \,$ data samples.

The assumption of a model for the D decay leads to a non-negligible, and hard to quantify, source of uncertainty. To obviate this, it is possible to use instead a model-independent approach, in which the Dalitz plot (or, more generally, the phase space) is binned [309, 319, 320]. In this case, hadronic parameters describing the average strong phase difference in each bin between the suppressed and favoured decay amplitudes enter the equations. These parameters can be determined from interference effects in decays of quantum-correlated $D{\bar{D}}$ pairs produced at the $\psi (3770)$ resonance. Measurements of such parameters have been made for various different hadronic D decays by CLEO-c and BESIII.

If a multibody decay is dominated by one $C\!P$ state, there will be additional sensitivity to $\gamma$ in the numbers of events in the $B^{\pm } \,$ data samples. This can be taken into account in various ways. One possibility is to perform a GLW-like analysis, as mentioned above. An alternative approach proceeds by defining $z_{\pm } \, = x_{\pm } \, + i y_{\pm } \,$ and $x_0 = - \int \mathrm{Re}[ f(s_1,s_2)f^*(s_2,s_1) ] ds_1ds_2$, where $s_1, s_2$ are the coordinates of invariant mass squared that define the Dalitz plot and f is the complex amplitude for D decay as a function of the Dalitz plot coordinates.²⁹ The fitted parameters ($\rho ^{\pm } \,, \theta ^{\pm } \,$) are then defined by

$$\begin{aligned} \rho ^{\pm } \, e^{i \theta ^{\pm } \,} = z_{\pm } \, - x_0. \end{aligned}$$

(151)

Note that the yields of $B^{\pm } \,$ decays are proportional to $1 + (\rho ^{\pm } \,)^2 - (x_0)^2$. This choice of variables has been used by BaBar in the analysis of $B^{+}\rightarrow DK^+$ with $D \rightarrow \pi ^+\pi ^-\pi ^0$ [322]; for this D decay, and with the assumed amplitude model, a value of $x_0 = 0.850$ is obtained.

The relations between the measured quantities and the underlying parameters are summarised in Table 25. It must be emphasised that the hadronic factors $r_B$ and $\delta _B$ are different, in general, for each $B$ decay mode.

Table 26 Common inputs used in calculating the averages

4.3 Common inputs and error treatment

The common inputs used for rescaling are listed in Table 26. The $B^{0}$ lifetime ($\tau (B^{0})$), mixing parameter ($\Delta m_d$) and relative width difference ($\Delta \Gamma _d / \Gamma _d$) averages are provided by the HFLAV Lifetimes and Oscillations subgroup (Sect. 3). The fraction of the perpendicularly polarised component ($| A_{\perp } |^2$) in $B \rightarrow {J/\psi } K^* (892)$ decays, which determines the $C\!P$ composition in these decays, is averaged from results by BaBar [323], Belle  [324], CDF [325], D0 [93] and LHCb [326]. See also the HFLAV B to Charm Decay Parameters subgroup (Sect. 6).

At present, we only rescale to a common set of input parameters for modes with reasonably small statistical errors ($b \rightarrow c{\bar{c}}s$ transitions of $B^{0}$ mesons). Correlated systematic errors are taken into account in these modes as well. For all other modes, the effect of such a procedure is currently negligible.

As explained in Sect. 1, we do not apply a rescaling factor on the error of an average that has $\chi ^2/\mathrm{dof}> 1$ (unlike the procedure currently used by the PDG [327]). We provide a confidence level of the fit so that one can know the consistency of the measurements included in the average, and attach comments in case some care needs to be taken in the interpretation. Note that, in general, results obtained from data samples with low statistics will exhibit some non-Gaussian behaviour. We average measurements with asymmetric errors using the PDG [327] prescription. In cases where several measurements are correlated (e.g. $S_f$ and $C_f$ in measurements of time-dependent $C\!P$ violation in B decays to a particular $C\!P$ eigenstate) we take these into account in the averaging procedure if the uncertainties are sufficiently Gaussian. For measurements where one error is given, it represents the total error, where statistical and systematic uncertainties have been added in quadrature. If two errors are given, the first is statistical and the second systematic. If more than two errors are given, the origin of the additional uncertainty will be explained in the text.

Table 27 Results and averages for $S_{b \rightarrow c{\bar{c}} s}$ and $C_{b \rightarrow c{\bar{c}} s}$. The result marked ($^{*}$) uses “hadronic and previously unused muonic decays of the $J/\psi$”. We neglect a small possible correlation of this result with the main BaBar result [331] that could be caused by reprocessing of the data

4.4 Time-dependent asymmetries in $b \rightarrow c{\bar{c}}s$ transitions

4.4.1 Time-dependent $C\!P$ asymmetries in $b \rightarrow c{\bar{c}}s$ decays to $C\!P$ eigenstates

In the Standard Model, the time-dependent parameters for $B^{0}$ decays governed by $b \rightarrow c{\bar{c}} s$ transitions are predicted to be $S_{b \rightarrow c{\bar{c}} s} = - {\eta }\sin (2\beta )$ and $C_{b \rightarrow c{\bar{c}} s} = 0$ to very good accuracy. Deviations from this relation are currently limited to the level of $\lesssim 1^{\circ }$ on $2\beta$ [328,329,330]. The averages for $-{\eta }S_{b \rightarrow c{\bar{c}} s}$ and $C_{b \rightarrow c{\bar{c}} s}$ are provided in Table 27. The averages for $-{\eta }S_{b \rightarrow c{\bar{c}} s}$ are shown in Fig. 12.

Fig. 12

(Left) Average of measurements of $S_{b \rightarrow c{\bar{c}} s}$, interpreted as $\sin (2\beta )$. (Right) Constraints on the $(\overline{\rho },\overline{\eta })$ plane, obtained from the average of $-{\eta }S_{b \rightarrow c{\bar{c}} s}$ and Eq. (152). Note that the solution with the smaller (larger) value of $\beta$ has $\cos (2\beta )>0$ ($<\!\!0$)

Both BaBar and Belle have used the ${\eta }= -1$ modes ${J/\psi } K^0_{S}$, $\psi (2S) K^0_{S}$, $\chi _{c1} K^0_{S}$ and $\eta _c K^0_{S}$, as well as ${J/\psi } K^0_{L}$, which has ${\eta }= +1$ and ${J/\psi } K^{*0}(892)$, which is found to have ${\eta }$ close to $+1$ based on the measurement of $| A_\perp |$ (see Sect. 4.3). The most recent Belle result does not use $\eta _c K^0_{S}$ or ${J/\psi } K^{*0}(892)$ decays.³⁰ ALEPH, OPAL, CDF and LHCb have used only the ${J/\psi } K^0_{S}$ final state. BaBar has also determined the $C\!P$ violation parameters of the $B^{0}\rightarrow \chi _{c0} K^0_{S}$ decay from the time-dependent Dalitz plot analysis of the $B^{0}\rightarrow \pi ^+\pi ^-K^0_{S}$ mode (see Sect. 4.7.2). In addition, Belle has performed a measurement with data accumulated at the $\varUpsilon (5S)$ resonance, using the ${J/\psi } K^0_{S}$ final state – this involves a different flavour tagging method compared to the measurements performed with data accumulated at the $\varUpsilon (4S)$ resonance. A breakdown of results in each charmonium-kaon final state is given in Table 28.

Table 28 Breakdown of B factory results on $S_{b \rightarrow c{\bar{c}} s}$ and $C_{b \rightarrow c{\bar{c}} s}$

Table 29 Averages from $B^{0}\rightarrow J/\psi K^{*0}$ transversity analyses

It should be noted that, while the uncertainty in the average for $-{\eta }S_{b \rightarrow c{\bar{c}} s}$ is still limited by statistics, the precision for $C_{b \rightarrow c{\bar{c}} s}$ is close to being dominated by the systematic uncertainty, particularly for measurements from the $e^+e^-$ B factory experiments. This occurs due to the possible effect of tag side interference [292] on the $C_{b \rightarrow c{\bar{c}} s}$ measurement, an effect which is correlated between different $e^+e^- \rightarrow \varUpsilon (4S) \rightarrow B{\bar{B}}$ experiments. Understanding of this effect may continue to improve in future, allowing the uncertainty to reduce.

Fig. 13

Averages of (left) $\sin (2\beta ) \equiv \sin (2\phi _1)$ and (right) $\cos (2\beta ) \equiv \cos (2\phi _1)$ from time-dependent analyses of $B^{0}\rightarrow {J/\psi } K^{*0}$ decays

From the average for $-{\eta }S_{b \rightarrow c{\bar{c}} s}$ above, we obtain the following solutions for $\beta$ (in $\left[ 0, \pi \right]$):

$$\begin{aligned} \beta = ( 21.9 \,\pm \, 0.7 )^\circ \quad \mathrm{or} \quad \beta = ( 68.1 \,\pm \, 0.7 )^\circ . \end{aligned}$$

(152)

In radians, these values are $\beta = ( 0.382 \,\pm \, 0.012 )$, $\beta = ( 1.189 \,\pm \, 0.012 )$.

This result gives a precise constraint on the $(\overline{\rho },\overline{\eta })$ plane, as shown in Fig. 12. The measurement is in remarkable agreement with other constraints from $C\!P$ conserving quantities, and with $C\!P$ violation in the kaon system, in the form of the parameter $\epsilon _K$. Such comparisons have been performed by various phenomenological groups, such as CKMfitter [252] and UTFit [339] (see also Refs. [340, 341]).

4.4.2 Time-dependent transversity analysis of $B^{0}\rightarrow J/\psi K^{*0}$ decays

$B$ meson decays to the vector-vector final state $J/\psi K^{*0}$ are also mediated by the $b \rightarrow c {\bar{c}} s$ transition. When a final state that is not flavour-specific ($K^{*0} \rightarrow K^0_{S} \pi ^0$) is used, a time-dependent transversity analysis can be performed allowing sensitivity to both $\sin (2\beta )$ and $\cos (2\beta )$ [342]. Such analyses have been performed by both $B$ factory experiments. In principle, the strong phases between the transversity amplitudes are not uniquely determined by such an analysis, leading to a discrete ambiguity in the sign of $\cos (2\beta )$. The BaBar collaboration resolves this ambiguity using the known variation [343] of the P-wave phase (fast) relative to the S-wave phase (slow) with the invariant mass of the $K\pi$ system in the vicinity of the $K^*(892)$ resonance. The result is in agreement with the prediction from s quark helicity conservation, and corresponds to Solution II defined by Suzuki [344]. We include only the solutions consistent with this phase variation in Table 29 and Fig. 13.

At present the results are dominated by large and non-Gaussian statistical errors, and exhibit significant correlations. We perform uncorrelated averages, the interpretation of which has to be done with the greatest care. Nonetheless, it is clear that $\cos (2\beta )>0$ is preferred by the experimental data in $J/\psi K^{*0}$ (for example, BaBar [345] find a confidence level for $\cos (2\beta )>0$ of $89\%$).

4.4.3 Time-dependent $C\!P$ asymmetries in $B^{0}\rightarrow D^{*+}D^{*-}K^0_{S}$ decays

Both BaBar [265] and Belle  [266] have performed time-dependent analyses of the $B^{0}\rightarrow D^{*+}D^{*-}K^0_{S}$ decay, to obtain information on the sign of $\cos (2\beta )$. More information can be found in Sect. 4.2.5. The results are given in Table 30, and shown in Fig. 14.

Table 30 Results from time-dependent analysis of $B^{0}\rightarrow D^{*+}D^{*-}K^0_{S}$

Fig. 14

Averages of (left) $(J_c/J_0)$, (middle) $(2J_{s1}/J_0) \sin (2\beta )$ and (right) $(2J_{s2}/J_0) \cos (2\beta )$ from time-dependent analyses of $B^{0}\rightarrow D^{*+}D^{*-}K^0_{S}$ decays

From the above result and the assumption that $J_{s2}>0$, BaBar infer that $\cos (2\beta )>0$ at the $94\%$ confidence level [265].

4.4.4 Time-dependent analysis of ${B^{0}_s}$ decays through the $b \rightarrow c{\bar{c}}s$ transition

As described in Sect. 4.2.3, time-dependent analysis of decays such as ${B^{0}_s} \rightarrow J/\psi \phi$ probes the $C\!P$ violating phase of ${B^{0}_s}$–${\bar{B}}^0_s$ oscillations, $\phi _s$.³¹ The combination of results on ${B^{0}_s} \rightarrow {J/\psi } \phi$ decays, including also results from ${B^{0}_s} \rightarrow {J/\psi } \pi ^+\pi ^-$ and ${B^{0}_s} \rightarrow D_s^+D_s^-$ decays, is performed by the HFLAV Lifetimes and Oscillations subgroup, see Sect. 3.

Table 31 Results from analyses of $B^{0}\rightarrow D^{(*)}h^0$, $D \rightarrow C\!P$ eigenstates decays

4.5 Time-dependent $C\!P$ asymmetries in colour-suppressed $b \rightarrow c{\bar{u}}d$ transitions

4.5.1 Time-dependent $C\!P$ asymmetries: $b \rightarrow c{\bar{u}}d$ decays to $C\!P$ eigenstates

Decays of $B$ mesons to final states such as $D\pi ^0$ are governed by $b \rightarrow c{\bar{u}}d$ transitions. If the final state is a $C\!P$ eigenstate, e.g. $D_{C\!P}\pi ^0$, the usual time-dependence formulae are recovered, with the sine coefficient sensitive to $\sin (2\beta )$. Since there is no penguin contribution to these decays, there is even less associated theoretical uncertainty than for $b \rightarrow c{\bar{c}}s$ decays such as $B \rightarrow {J/\psi } K^0_{S}$. Such measurements therefore allow to test the Standard Model prediction that the $C\!P$ violation parameters in $b \rightarrow c{\bar{u}}d$ transitions are the same as those in $b \rightarrow c{\bar{c}}s$ [346]. Although there is an additional contribution from CKM suppressed $b \rightarrow u {\bar{c}} d$ amplitudes, which have a different weak phase compared to the leading $b \rightarrow c{\bar{u}}d$ transition, the effect is small and can be taken into account in the analysis [347, 348].

Results are available from a joint analysis of BaBar and Belle data [349]. The following $C\!P$-even final states are included: $D\pi ^0$ and $D\eta$ with $D \rightarrow K^0_{S} \pi ^0$ and $D \rightarrow K^0_{S} \omega$; $D\omega$ with $D \rightarrow K^0_{S} \pi ^0$; $D^* \pi ^0$ and $D^* \eta$ with $D^* \rightarrow D\pi ^0$ and $D \rightarrow K^+ K^-$. The following $C\!P$-odd final states are included: $D\pi ^0$, $D\eta$ and $D\omega$ with $D \rightarrow K^+ K^-$, $D^* \pi ^0$ and $D^* \eta$ with $D^* \rightarrow D\pi ^0$ and $D \rightarrow K^0_{S} \pi ^0$. All $B^0 \rightarrow D^{(*)} h^0$ decays are analysed together, taking into account the different $C\!P$ factors (denoted $D^{(*)} _{C\!P} h^0$). The results are summarised in Table 31.

Table 32 Averages from $B^{0}\rightarrow D^{(*)}h^0$, $D \rightarrow K^0_{S} \pi ^+\pi ^-$ analyses

Fig. 15

Averages of (left) $\sin (2\beta )$ and (right) $\cos (2\beta )$ measured in colour-suppressed $b \rightarrow c{\bar{u}}d$ transitions

4.5.2 Time-dependent Dalitz plot analyses of $b \rightarrow c{\bar{u}}d$ decays

When multibody D decays, such as $D \rightarrow K^0_{S} \pi ^+\pi ^-$ are used, a time-dependent analysis of the Dalitz plot of the neutral D decay allows for a direct determination of the weak phase $2\beta$. (Equivalently, both $\sin (2\beta )$ and $\cos (2\beta )$ can be measured.) This information can be used to resolve the ambiguity in the measurement of $2\beta$ from $\sin (2\beta )$ [350].

Results of such analyses are available from both Belle  [260] and BaBar [261]. The decays $B \rightarrow D\pi ^0$, $B \rightarrow D\eta$, $B \rightarrow D\omega$, $B \rightarrow D^*\pi ^0$ and $B \rightarrow D^*\eta$ are used. (This collection of states is denoted by $D^{(*)}h^0$.) The daughter decays are $D^* \rightarrow D\pi ^0$ and $D \rightarrow K^0_{S} \pi ^+\pi ^-$. The results are given in Table 32, and shown in Fig. 15. Note that BaBar quote uncertainties due to the D decay model separately from other systematic errors as a third source of uncertainty, while Belle do not.

Again, it is clear that the data prefer $\cos (2\beta )>0$. Indeed, Belle  [260] determine the sign of $\cos (2\phi _1)$ to be positive at $98.3\%$ confidence level, while BaBar [261] favour the solution of $\beta$ with $\cos (2\beta )>0$ at $87\%$ confidence level. Note, however, that the Belle measurement has strongly non-Gaussian behaviour. Therefore, we perform uncorrelated averages, from which any interpretation has to be done with the greatest care.

A model-independent time-dependent analysis of $B^0 \rightarrow D^{(*)} h^0$ decays, with $D \rightarrow K^0_{S} \pi ^+ \pi ^-$, has been performed by Belle  [262]. The decays $B^0 \rightarrow D\pi ^0$, $B^0 \rightarrow D\eta$, $B^0 \rightarrow D\eta ^{\prime }$, $B^0 \rightarrow D\omega$, $B^0 \rightarrow D^* \pi ^0$ and $B^0 \rightarrow D^* \eta$ are used. The results are also included in Table 32. From these results, Belle disfavour the solution with the value of $\sin (2\phi _1)$ from $b \rightarrow c{\bar{c}}s$ transitions but a negative value for $\cos (2\phi _1)$, at $5.1\,\sigma$ significance. The solution with the $b \rightarrow c{\bar{c}}s$ value of $\sin (2\phi _1)$ and positive $\cos (2\phi _1)$ is consistent with the data at the level of $1.3\,\sigma$. Note that due to the strong statistical and systematic correlations, model-dependent results and model-independent results from the same experiment cannot be combined.

Table 33 Averages for the $b \rightarrow c{\bar{c}}d$ modes, $B^{0}\rightarrow J/\psi \pi ^{0}$ and $D^+D^-$

4.6 Time-dependent $C\!P$ asymmetries in $b \rightarrow c{\bar{c}}d$ transitions

The transition $b \rightarrow c{\bar{c}} d$ can occur via either a $b \rightarrow c$ tree or a $b \rightarrow d$ penguin amplitude. The flavour changing neutral current $b \rightarrow d$ penguin can be mediated by any up-type quark in the loop, and hence the amplitude can be written as

$$\begin{aligned} A_{b \rightarrow d}= & {} {F_u V_{ub}V^*_{ud} + F_c V_{cb}V^*_{cd} + F_t V_{tb}V^*_{td}} \nonumber \\= & {} (F_u - F_c) V_{ub}V^*_{ud} + (F_t - F_c) V_{tb}V^*_{td}, \end{aligned}$$

(153)

where $F_{u,c,t}$ describe all factors except CKM suppression entering each quark loop diagram. In the last line, both terms are $\mathcal{O}(\lambda ^3)$, so it can be seen that the $b \rightarrow d$ penguin amplitude contains terms with different weak phases at the same order of CKM suppression.

Table 34 Averages for the $b \rightarrow c{\bar{c}}d$ modes, ${J/\psi } \rho ^0$, $D^{*+} D^{*-}$ and $D^{*\,\pm \,}D^{\mp }$

Fig. 16

Averages of (left) $S_{b \rightarrow c{\bar{c}} d}$ and (right) $C_{b \rightarrow c{\bar{c}} d}$ for the mode $B^{0}\rightarrow J/ \psi \pi ^0$

Fig. 17

Averages of (left) $S_{b \rightarrow c{\bar{c}} d}$ and (right) $C_{b \rightarrow c{\bar{c}} d}$ for the mode $B^{0}\rightarrow D^+D^-$

In the above, we have chosen to eliminate the $F_c$ term using unitarity. However, we could equally well write

$$\begin{aligned} \begin{array}{ccccc} A_{b \rightarrow d} &{} = &{} (F_u - F_t) V_{ub}V^*_{ud} &{} + &{} (F_c - F_t) V_{cb}V^*_{cd} \\ &{} = &{} (F_c - F_u) V_{cb}V^*_{cd} &{} + &{} (F_t - F_u) V_{tb}V^*_{td}. \\ \end{array} \end{aligned}$$

(154)

Since the $b \rightarrow c{\bar{c}}d$ tree amplitude has the weak phase of $V_{cb}V^*_{cd}$, either of the above expressions allow the penguin amplitude to be decomposed into a part with weak phase the same as the tree amplitude and another part with a different weak phase, which can be chosen to be either $\beta$ or $\gamma$. The choice of parametrisation cannot, of course, affect the physics [351]. In any case, if the tree amplitude dominates, there is little sensitivity to any phase other than that from $B^{0}$–${\overline{B}} ^0$ mixing.

The $b \rightarrow c{\bar{c}}d$ transitions can be investigated with studies of various different final states. Results are available from both BaBar and Belle using the final states ${J/\psi } \pi ^0$, $D^+D^-$, $D^{*+}D^{*-}$ and $D^{*\,\pm \,}D^{{\mp }}$, and from LHCb using the final states ${J/\psi } \rho ^0$ and $D^+D^-$; the averages of these results are given in Tables 33 and 34. The results using the $C\!P$ eigenstate (${\eta }= +1$) modes ${J/\psi } \pi ^0$ and $D^+D^-$ are shown in Fig. 16 and Fig. 17 respectively, with two-dimensional constraints shown in Fig. 18.

Fig. 18

Averages of two $b \rightarrow c{\bar{c}} d$ dominated channels, for which correlated averages are performed, in the $S_{C\!P}$ $vs.$ $C_{C\!P}$ plane. (Left) $B^{0}\rightarrow J/ \psi \pi ^0$ and (right) $B^{0}\rightarrow D^+D^-$

Results for the vector-vector mode ${J/\psi } \rho ^0$ are obtained from a full time-dependent amplitude analysis of $B^{0}\rightarrow {J/\psi } \pi ^+ \pi ^-$ decays. LHCb [258] find a ${J/\psi } \rho ^0$ fit fraction of $65.6 \,\pm \, 1.9\%$ and a longitudinal polarisation fraction of $56.7 \,\pm \, 1.8\%$ (uncertainties are statistical only; both results are consistent with those from a time-integrated amplitude analysis [267] where systematic uncertainties were also evaluated). Fits are performed to obtain $2\beta ^\mathrm{eff}$ in the cases that all transversity amplitudes are assumed to have the same $C\!P$ violation parameter. A separate fit is performed allowing different parameters. The results in the former case are presented in terms of $S_{C\!P}$ and $C_{C\!P}$ in Table 34.

The vector-vector mode $D^{*+}D^{*-}$ is found to be dominated by the $C\!P$-even longitudinally polarised component; BaBar measures a $C\!P$-odd fraction of $0.158 \,\pm \, 0.028 \,\pm \, 0.006$ [259] while Belle measures a $C\!P$-odd fraction of $0.138 \,\pm \, 0.024 \,\pm \, 0.006$ [352]. These values, listed as $R_\perp$, are included in the averages, which ensures that the correlations are taken into account.³² BaBar has also performed an additional fit in which the $C\!P$-even and $C\!P$-odd components are allowed to have different $C\!P$ violation parameters S and C. These results are included in Table 34. Results using $D^{*+}D^{*-}$ are shown in Fig. 19.

As discussed in Sect. 4.2.6, the most recent papers on the non-$C\!P$ eigenstate mode $D^{*\,\pm \,}D^{{\mp }}$ use the (A, S, $\Delta S$, C, $\Delta C$) set of parameters, and we therefore perform the averages with this choice.

Fig. 19

Averages of (left) $S_{b \rightarrow c{\bar{c}} d}$ and (right) $C_{b \rightarrow c{\bar{c}} d}$ for the mode $B^{0}\rightarrow D^{*+}D^{*-}$

Fig. 20

Averages of (left) $-{\eta }S_{b \rightarrow c{\bar{c}} d}$ interpreted as $\sin (2\beta ^\mathrm{eff})$ and (right) $C_{b \rightarrow c{\bar{c}} d}$. The $-{\eta }S_{b \rightarrow c{\bar{c}} d}$ figure compares the results to the world average for $-{\eta }S_{b \rightarrow c{\bar{c}} s}$ (see Sect. 4.4.1)

In the absence of the penguin contribution (tree dominance), the time-dependent parameters would be given by $S_{b \rightarrow c{\bar{c}} d} = - {\eta }\sin (2\beta )$, $C_{b \rightarrow c{\bar{c}} d} = 0$, $S_{+-} = \sin (2\beta + \delta )$, $S_{-+} = \sin (2\beta - \delta )$, $C_{+-} = - C_{-+}$ and $\mathcal{A} = 0$, where $\delta$ is the strong phase difference between the $D^{*+}D^-$ and $D^{*-}D^+$ decay amplitudes. In the presence of the penguin contribution, there is no clean interpretation in terms of CKM parameters; however, direct $C\!P$ violation may be observed through any of $C_{b \rightarrow c{\bar{c}} d} \ne 0$, $C_{+-} \ne - C_{-+}$ or $A_{+-} \ne 0$.

The averages for the $b \rightarrow c{\bar{c}} d$ modes are shown in Figs. 20 and 21. Results are consistent with tree dominance, and with the Standard Model, though the Belle results in $B^{0}\rightarrow D^+D^-$ [357] show an indication of $C\!P$ violation in decay, and hence a non-zero penguin contribution. The average of $S_{b \rightarrow c{\bar{c}} d}$ in each of the $J/\psi \pi ^{0}$, $D^+ D^-$ and $D^{*+}D^{*-}$ final states is more than $5\sigma$ from zero, corresponding to observations of $C\!P$ violation in these decay channels. Possible non-Gaussian effects due to some of the inputs measurements being outside the physical region ($S_{C\!P}^2 + C_{C\!P}^2 \le 1$) should, however, be borne in mind.

Fig. 21

Compilation of constraints in the $-{\eta }S_{b \rightarrow c{\bar{c}} d}$, interpreted as $\sin (2\beta ^\mathrm{eff})$, $vs.$ $C_{b \rightarrow c{\bar{c}} d}$ plane

4.6.1 Time-dependent $C\!P$ asymmetries in ${B^{0}_s}$ decays mediated by $b \rightarrow c{\bar{c}}d$ transitions

Time-dependent $C\!P$ asymmetries in ${B^{0}_s}$ decays mediated by $b \rightarrow c{\bar{c}}d$ transitions provide a determination of $2\beta _s^\mathrm{eff}$ where possible effects from penguin amplitudes may cause a shift from the value of $2\beta _s$ seen in $b \rightarrow c{\bar{c}}s$ transitions. Results in the $b \rightarrow c{\bar{c}}d$ case, with larger penguin effects, can be used together with flavour symmetries to derive limits on the possible size of penguin effects in the $b \rightarrow c{\bar{c}}s$ transitions [358, 359].

The parameters have been measured in ${B^{0}_s} \rightarrow {J/\psi } K^0_{S}$ decays by LHCb, as summarised in Table 35. The results supersede an earlier measurement of the effective lifetime, which is directly related to $A^{\Delta \Gamma }$, in the same mode [127], which is discussed in Sect. 3.

4.7 Time-dependent $C\!P$ asymmetries in charmless $b \rightarrow q\bar{q}s$ transitions

Similarly to Eq. (153), the $b \rightarrow s$ penguin amplitude can be written as

$$\begin{aligned} A_{b \rightarrow s}= & {} {F_{u} V_{ub}V^*_{us} + F_{c} V_{cb}{V^*}_{cs} + F_{t} V_{tb}V^*_{ts}} \nonumber \\= & {} (F_{u} - F_{c}) V_{ub}V^*_{us} + (F_t - F_c) V_{tb}V^*_{ts}, \end{aligned}$$

(155)

using the unitarity of the CKM matrix to eliminate the $F_c$ term. In this case, the first term in the last line is $\mathcal{O}(\lambda ^4)$ while the second is $\mathcal{O}(\lambda ^2)$. Therefore, in the Standard Model, this amplitude is dominated by $V_{tb}V^*_{ts}$, and to within a few degrees ($| \delta \beta ^\mathrm{eff} | \equiv | \beta ^\mathrm{eff}-\beta | \lesssim 2^\circ$ for $\beta \approx 20^\circ$) the time-dependent parameters can be written³³ $S_{b \rightarrow q{\bar{q}} s} \approx - {\eta }\sin (2\beta )$, $C_{b \rightarrow q{\bar{q}} s} \approx 0$, assuming $b \rightarrow s$ penguin contributions only ($q = u,d,s$).

Due to the suppression of the Standard Model amplitude, contributions of additional diagrams from physics beyond the Standard Model, with heavy virtual particles in the penguin loops, may have observable effects. In general, these contributions will affect the values of $S_{b \rightarrow q{\bar{q}} s}$ and $C_{b \rightarrow q{\bar{q}} s}$. A discrepancy between the values of $S_{b \rightarrow c{\bar{c}} s}$ and $S_{b \rightarrow q{\bar{q}} s}$ can therefore provide a clean indication of new physics [346, 361,362,363].

Table 35 Measurements of $C\!P$ violation parameters from ${B^{0}_s} \rightarrow {J/\psi } K^0_{S}$

Table 36 Averages of $-{\eta }S_{b \rightarrow q{\bar{q}} s}$ and $C_{b \rightarrow q{\bar{q}} s}$. Where a third source of uncertainty is given, it is due to model uncertainties arising in Dalitz plot analyses

Table 37 Averages of $-{\eta }S_{b \rightarrow q{\bar{q}} s}$ and $C_{b \rightarrow q{\bar{q}} s}$ (continued). Where a third source of uncertainty is given, it is due to model uncertainties arising in Dalitz plot analyses

However, there is an additional consideration to take into account. The above argument assumes that only the $b \rightarrow s$ penguin contributes to the $b \rightarrow q{\bar{q}} s$ transition. For $q = s$ this is a good assumption, which neglects only rescattering effects. However, for $q = u$ there is a colour-suppressed $b \rightarrow u$ tree diagram (of order $\mathcal{O}(\lambda ^4)$), which has a different weak (and possibly strong) phase. In the case $q = d$, any light neutral meson that is formed from $d \bar{d}$ also has a $u {\bar{u}}$ component, and so again there is “tree pollution”. The $B^{0}$decays to $\pi ^0 K^0_{S}$, $\rho ^0K^0_{S}$ and $\omega K^0_{S}$ belong to this category. The mesons $\phi$, $f_0$ and $\eta ^{\prime }$ are expected to have predominant $s\bar{s}$ composition, which reduces the relative size of the possible tree pollution. If the inclusive decay $B^{0}\rightarrow K^+ K^- K^0$ (excluding $\phi K^0$) is dominated by a nonresonant three-body transition, an Okubo–Zweig–Iizuka-suppressed [364,365,366] tree-level diagram can occur through insertion of an $s\overline{s}$ pair. The corresponding penguin-type transition proceeds via insertion of a $u\overline{u}$ pair, which is expected to be favoured over the $s\overline{s}$ insertion by fragmentation models. Neglecting rescattering, the final state $K^0 \overline{K} ^0 K^0$ (reconstructed as $K^0_{S} K^0_{S} K^0_{S}$) has no tree pollution [367]. Various estimates, using different theoretical approaches, of the values of $\Delta S = S_{b \rightarrow q{\bar{q}} s} - S_{b \rightarrow c{\bar{c}} s}$ exist in the literature [368,369,370,371,372,373,374,375,376,377,378,379,380,381]. In general, there is agreement that the modes $\phi K^0$, $\eta ^{\prime } K^0$ and $K^0 \overline{K} ^0 K^0$ are the cleanest, with values of $| \Delta S |$ at or below the few percent level ($\Delta S$ is usually predicted to be positive). Nonetheless, the uncertainty is sufficient that interpretation is given in terms of $\sin (2\beta ^\mathrm{eff})$.

4.7.1 Time-dependent $C\!P$ asymmetries: $b \rightarrow q\bar{q}s$ decays to $C\!P$ eigenstates

The averages for $-{\eta }S_{b \rightarrow q{\bar{q}} s}$ and $C_{b \rightarrow q{\bar{q}} s}$ can be found in Tables 36 and 37, and are shown in Figs. 22, 23 and 24. Results from both BaBar and Belle are averaged for the modes $\eta ^{\prime } K^0$ ($K^0$ indicates that both $K^0_{S}$ and $K^0_{L}$ are used) $K^0_{S} K^0_{S} K^0_{S}$, $\pi ^0 K^0_{S}$ and $\omega K^0_{S}$.³⁴ Results on $\phi K^0_{S}$ and $K^+ K^- K^0_{S}$ (implicitly excluding $\phi K^0_{S}$ and $f_0K^0_{S}$) are taken from time-dependent Dalitz plot analyses of $K^+ K^- K^0_{S}$; results on $\rho ^0K^0_{S}$, $f_2K^0_{S}$, $f_XK^0_{S}$ and $\pi ^+ \pi ^- K^0_{S}$ nonresonant are taken from time-dependent Dalitz plot analyses of $\pi ^+ \pi ^- K^0_{S}$ (see Sect. 4.7.2). The results on $f_0K^0_{S}$ are from combinations of both Dalitz plot analyses. BaBar has also presented results with the final states $\pi ^0\pi ^0K^0_{S}$ and $\phi K^0_{S} \pi ^0$.

Fig. 22

(Top) Averages of (left) $-{\eta }S_{b \rightarrow q{\bar{q}} s}$, interpreted as $\sin (2\beta ^\mathrm{eff}$ and (right) $C_{b \rightarrow q{\bar{q}} s}$. The $-{\eta }S_{b \rightarrow q{\bar{q}} s}$ figure compares the results to the world average for $-{\eta }S_{b \rightarrow c{\bar{c}} s}$ (see Sect. 4.4.1). (Bottom) Same, but only averages for each mode are shown. More figures are available from the HFLAV web pages

Fig. 23

Averages of four $b \rightarrow q{\bar{q}} s$ dominated channels, for which correlated averages are performed, in the $S_{C\!P}$ $vs.$ $C_{C\!P}$ plane, where $S_{C\!P}$ has been corrected by the $C\!P$ eigenvalue to give $\sin (2\beta ^\mathrm{eff})$. (Top left) $B^{0}\rightarrow \phi K^0$, (top right) $B^{0}\rightarrow \eta ^\prime K^0$, (bottom left) $B^{0}\rightarrow K^0_{S} K^0_{S} K^0_{S}$, (bottom right) $B^{0}\rightarrow \pi ^0K^0_{S}$. More figures are available from the HFLAV web pages

Of these final states, $\phi K^0_{S}$, $\eta ^{\prime } K^0_{S}$, $\pi ^0 K^0_{S}$, $\rho ^0K^0_{S}$, $\omega K^0_{S}$ and $f_0K^0_{L}$ have $C\!P$ eigenvalue ${\eta }= -1$, while $\phi K^0_{L}$, $\eta ^{\prime } K^0_{L}$, $K^0_{S} K^0_{S} K^0_{S}$, $f_0K^0_{S}$, $f_2K^0_{S}$, $f_XK^0_{S}$, $\pi ^0\pi ^0K^0_{S}$ and $\pi ^+ \pi ^- K^0_{S}$ nonresonant have ${\eta }= +1$. The final state $K^+K^-K^0_{S}$ (with $\phi K^0_{S}$ and $f_0K^0_{S}$ implicitly excluded) is not a $C\!P$ eigenstate, but the $C\!P$-content can be absorbed in the amplitude analysis to allow the determination of a single effective S parameter. (In earlier analyses of the $K^+K^-K^0$ final state, its $C\!P$ composition was determined using an isospin argument [383] and a moments analysis [384].) Throughout this section, $f_0 \equiv f_0(980)$ and $f_2 \equiv f_2(1270)$. Details of the assumed lineshapes of these states, and of the $f_X$ (which is taken to have even spin), can be found in the relevant experimental papers [270, 271, 274, 275].

The final state $\phi K^0_{S} \pi ^0$ is also not a $C\!P$ eigenstate but its $C\!P$-composition can be determined from an angular analysis. Since the parameters are common to the $B^{0}\rightarrow \phi K^0_{S} \pi ^0$ and $B^{0}\rightarrow \phi K^+ \pi ^-$ decays (because only $K\pi$ resonances contribute), BaBar perform a simultaneous analysis of the two final states [391] (see Sect. 4.7.3).

It must be noted that Q2B parameters extracted from Dalitz plot analyses are constrained to lie within the physical boundary ($S_{C\!P}^2 + C_{C\!P}^2 < 1$) and consequently the obtained errors are highly non-Gaussian when the central value is close to the boundary. This is particularly evident in the BaBar results for $B^{0}\rightarrow f_0K^0$ with $f_0 \rightarrow \pi ^+\pi ^-$ [274]. These results must be treated with extreme caution.

As explained above, each of the modes listed in Tables 36 and 37 has potentially different subleading contributions within the Standard Model, and thus each may have a different value of $-{\eta }S_{b \rightarrow q{\bar{q}} s}$. Therefore, there is no strong motivation to make a combined average over the different modes. We refer to such an average as a “naïve s-penguin average”. It is naïve not only because of the theoretical uncertainties are neglected, but also since possible correlations of systematic effects between different modes are not included. In spite of these caveats there remains interest in the value of this quantity and therefore it is given here: $\langle -{\eta }S_{b \rightarrow q{\bar{q}} s} \rangle = 0.655 \,\pm \, 0.032$, with confidence level $0.77~(0.3\sigma )$. This value is in agreement with the average $-{\eta }S_{b \rightarrow c{\bar{c}} s}$ given in Sect. 4.4.1. (The average for $C_{b \rightarrow q{\bar{q}} s}$ is $\langle C_{b \rightarrow q{\bar{q}} s} \rangle = -0.006 \,\pm \, 0.026$ with confidence level $0.53~(0.6\sigma )$.)

From Table 36 it may be noted that the averages for $-{\eta }S_{b \rightarrow q{\bar{q}} s}$ in $\phi K^0_{S}$, $\eta ^{\prime } K^0$, $f_0K^0_{S}$ and $K^+ K^- K^0_{S}$ are all now more than $5\sigma$ away from zero, so that $C\!P$ violation in these modes can be considered well established. There is no evidence (above $2\sigma$) for $C\!P$ violation in any $b \rightarrow q {\bar{q}} s$ decay.

4.7.2 Time-dependent Dalitz plot analyses: $B^{0}\rightarrow K^+K^-K^0$ and $B^{0}\rightarrow \pi ^+\pi ^-K^0_{S}$

As mentioned in Sect. 4.2.5 and above, both BaBar and Belle have performed time-dependent Dalitz plot analysis of $B^{0}\rightarrow K^+K^-K^0$ and $B^{0}\rightarrow \pi ^+\pi ^-K^0_{S}$ decays. The results are summarised in Tables 38 and 39. Averages for the $B^{0}\rightarrow f_0 K^0_{S}$ decay, which contributes to both Dalitz plots, are shown in Fig. 25. Results are presented in terms of the effective weak phase (from mixing and decay) difference $\beta ^\mathrm{eff}$ and the parameter of $C\!P$ violation in decay $\mathcal{A}$ ($\mathcal{A}= -C$) for each of the resonant contributions. Note that Dalitz plot analyses, including all those included in these averages, often suffer from ambiguous solutions – we quote the results corresponding to those presented as solution 1 in all cases. Results on flavour specific amplitudes that may contribute to these Dalitz plots (such as $K^{*+}\pi ^-$) are averaged by the HFLAV Rare Decays subgroup (Sect. 7).

For the $B^{0}\rightarrow K^+K^-K^0$ decay, both BaBar and Belle measure the $C\!P$ violation parameters for the $\phi K^0$, $f_0K^0$ and “other $K^+ K^- K^0$” amplitudes, where the latter includes all remaining resonant and nonresonant contributions to the charmless three-body decay. For the $B^{0}\rightarrow \pi ^+\pi ^-K^0_{S}$ decay, BaBar reports $C\!P$ violation parameters for all of the $C\!P$ eigenstate components in the Dalitz plot model ($\rho ^0K^0_{S}$, $f_0K^0_{S}$, $f_2K^0_{S}$, $f_XK^0_{S}$ and nonresonant decays; see Sect. 4.2.5), while Belle reports the $C\!P$ violation parameters for only the $\rho ^0K^0_{S}$ and $f_0K^0_{S}$ amplitudes, although the used Dalitz plot model is rather similar.

4.7.3 Time-dependent analyses of $B^{0}\rightarrow \phi K^0_{S} \pi ^0$

The final state in the decay $B^{0}\rightarrow \phi K^0_{S} \pi ^0$ is a mixture of $C\!P$-even and $C\!P$-odd amplitudes. However, since only $\phi K^{*0}$ resonant states contribute (in particular, $\phi K^{*0}(892)$, $\phi K^{*0}_0(1430)$ and $\phi K^{*0}_2(1430)$ are seen), the composition can be determined from the analysis of $B \rightarrow \phi K^+ \pi ^-$ decays, assuming only that the ratio of branching fractions ${\mathcal {B}}(K^{*0} \rightarrow K^0_{S} \pi ^0)/{\mathcal {B}}(K^{*0} \rightarrow K^+ \pi ^-)$ is the same for each excited kaon state.

BaBar [391] has performed a simultaneous analysis of $B^{0}\rightarrow \phi K^0_{S} \pi ^0$ and $B^{0}\rightarrow \phi K^+ \pi ^-$ decays that is time-dependent for the former mode and time-integrated for the latter. Such an analysis allows, in principle, all parameters of the $B^{0}\rightarrow \phi K^{*0}$ system to be determined, including mixing-induced $C\!P$ violation effects. The latter is determined to be $\Delta \phi _{00} = 0.28 \,\pm \, 0.42 \,\pm \, 0.04$, where $\Delta \phi _{00}$ is half the weak phase difference between $B^{0}$ and ${\overline{B}} ^0$ decays to the $\phi K^{*0}_0(1430)$ final state. As discussed above, this can also be presented in terms of the Q2B parameter $\sin (2\beta ^\mathrm{eff}_{00}) = \sin (2\beta +2\Delta \phi _{00}) = 0.97 \,^{+0.03}_{-0.52}$. The highly asymmetric uncertainty arises due to the conversion from the phase to the sine of the phase, and the proximity of the physical boundary.

Fig. 24

Compilation of constraints in the $-{\eta }S_{b \rightarrow q{\bar{q}} s}$, interpreted as $\sin (2\beta ^\mathrm{eff})$, $vs.$ $C_{b \rightarrow q{\bar{q}} s}$ plane

Table 38 Results from time-dependent Dalitz plot analyses of the $B^{0}\rightarrow K^+K^-K^0$ decay. Correlations (not shown) are taken into account in the average

Table 39 Results from time-dependent Dalitz plot analysis of the $B^{0}\rightarrow \pi ^+\pi ^-K^0_{S}$ decay. Correlations (not shown) are taken into account in the average

Similar $\sin (2\beta ^\mathrm{eff})$ parameters can be defined for each of the helicity amplitudes for both $\phi K^{*0}(892)$ and $\phi K^{*0}_2(1430)$. However, the relative phases between these decays are constrained due to the nature of the simultaneous analysis of $B^{0}\rightarrow \phi K^0_{S} \pi ^0$ and $B^{0}\rightarrow \phi K^+ \pi ^-$, decays and therefore these measurements are highly correlated. Instead of quoting all these results, BaBar provide an illustration of their measurements with the following differences:

$$\begin{aligned}&\sin (2\beta - 2\Delta \delta _{01}) - \sin (2\beta ) = -0.42\,^{+0.26}_{-0.34}, \end{aligned}$$

(156)

$$\begin{aligned}&\sin (2\beta - 2\Delta \phi _{\parallel 1}) - \sin (2\beta ) = -0.32\,^{+0.22}_{-0.30}, \end{aligned}$$

(157)

$$\begin{aligned}&\sin (2\beta - 2\Delta \phi _{\perp 1}) - \sin (2\beta ) = -0.30\,^{+0.23}_{-0.32}, \end{aligned}$$

(158)

$$\begin{aligned}&\sin (2\beta - 2\Delta \phi _{\perp 1}) - \sin (2\beta - 2\Delta \phi _{\parallel 1}) = 0.02 \,\pm \, 0.23,\nonumber \\ \end{aligned}$$

(159)

$$\begin{aligned}&\sin (2\beta - 2\Delta \delta _{02}) - \sin (2\beta ) = -0.10\,^{+0.18}_{-0.29}, \end{aligned}$$

(160)

where the first subscript indicates the helicity amplitude and the second indicates the spin of the kaon resonance. For the complete definitions of the $\Delta \delta$ and $\Delta \phi$ parameters, please refer to the BaBar paper [391].

Parameters of $C\!P$ violation in decay for each of the contributing helicity amplitudes can also be measured. Again, these are determined from a simultaneous fit of $B^{0}\rightarrow \phi K^0_{S} \pi ^0$ and $B^{0}\rightarrow \phi K^+ \pi ^-$ decays, with the precision being dominated by the statistics of the latter mode. Measurements of $C\!P$ violation in decay, obtained from decay-time-integrated analyses, are tabulated by the HFLAV Rare Decays subgroup (Sect. 7).

4.7.4 Time-dependent $C\!P$ asymmetries in ${B^{0}_s} \rightarrow K^+ K^-$

The decay ${B^{0}_s} \rightarrow K^+ K^-$ involves a $b \rightarrow u{\bar{u}}s$ transition, and hence has both penguin and tree contributions. Both mixing-induced and $C\!P$ violation in decay effects may arise, and additional input is needed to disentangle the contributions and determine $\gamma$ and $\beta _s^\mathrm{eff}$. For example, the observables in $B^0 \rightarrow \pi ^+ \pi ^-$ can be related using U-spin, as proposed in Refs. [392, 393].

The observables are $A_\mathrm{mix} = S_{C\!P}$, $A_\mathrm{dir} = -C_{C\!P}$, and $A_{\Delta \Gamma }$. They can all be treated as free parameters, but are physically constrained to satisfy $A_\mathrm{mix}^2 + A_\mathrm{dir}^2 + A_{\Delta \Gamma }^2 = 1$. Note that the untagged decay distribution, from which an “effective lifetime” can be measured, retains sensitivity to $A_{\Delta \Gamma }$; measurements of the ${B^{0}_s} \rightarrow K^+ K^-$ effective lifetime have been made by LHCb [104, 125]. Compilations and averages of effective lifetimes are performed by the HFLAV Lifetimes and Oscillations subgroup, see Sect. 3.

The observables in ${B^{0}_s} \rightarrow K^+ K^-$ have been measured by LHCb [394], who do not impose the constraint mentioned above to eliminate $A_\mathrm{\Delta \Gamma }$. The results are shown in Table 40, and correspond to evidence for $C\!P$ violation both in the interference between mixing and decay, and in the ${B^{0}_s} \rightarrow K^+ K^-$ decay.

Fig. 25

Averages of (left) $\beta ^\mathrm{eff} \equiv \phi _1^\mathrm{eff}$ and (right) $A_{C\!P}$ for the $B^{0}\rightarrow f_0K^0_{S}$ decay including measurements from Dalitz plot analyses of both $B^{0}\rightarrow K^+K^-K^0_{S}$ and $B^{0}\rightarrow \pi ^+\pi ^-K^0_{S}$

Table 40 Results from time-dependent analysis of the ${B^{0}_s} \rightarrow K^{+} K^{-}$ decay

Table 41 Results for $B^{0}\rightarrow K^0_{S} K^0_{S}$

Interpretations of an earlier set of results [395], in terms of constraints on $\gamma$ and $2\beta _s$, have been separately published by LHCb [243].

4.7.5 Time-dependent $C\!P$ asymmetries in ${B^{0}_s} \rightarrow \phi \phi$

The decay ${B^{0}_s} \rightarrow \phi \phi$ involves a $b \rightarrow s\bar{s}s$ transition, and hence is a “pure penguin” mode (in the limit that the $\phi$ meson is considered a pure $s\bar{s}$ state). Since the mixing phase and the decay phase are expected to cancel in the Standard Model, the prediction for the phase from the interference of mixing and decay is predicted to be $\phi _s(\phi \phi ) = 0$ with low uncertainty [396]. Due to the vector-vector nature of the final state, angular analysis is needed to separate the $C\!P$-even and $C\!P$-odd contributions. Such an analysis also makes it possible to fit directly for $\phi _s(\phi \phi )$.

A constraint on $\phi _s(\phi \phi )$ has been obtained by LHCb using $3.0 \,\mathrm{fb}^{-1}$ of data [397]. The result is $\phi _s(\phi \phi ) = -0.17 \,\pm \, 0.15 \,\pm \, 0.03 \, \mathrm{rad}$ where the first uncertainty is statistical and the second is systematic.

4.8 Time-dependent $C\!P$ asymmetries in $b \rightarrow q\bar{q}d$ transitions

Decays such as $B^{0}\rightarrow K^0_{S} K^0_{S}$ are pure $b \rightarrow q\bar{q}d$ penguin transitions. As shown in Eq. (153), this diagram has different contributing weak phases, and therefore the observables are sensitive to their difference (which can be chosen to be either $\beta$ or $\gamma$). Note that if the contribution with the top quark in the loop dominates, the weak phase from the decay amplitudes should cancel that from mixing, so that no $C\!P$ violation (neither mixing-induced nor in decay) occurs. Non-zero contributions from loops with intermediate up and charm quarks can result in both types of effect (as usual, a strong phase difference is required for $C\!P$ violation in decay to occur).

Both BaBar [398] and Belle  [399] have performed time-dependent analyses of $B^{0}\rightarrow K^0_{S} K^0_{S}$ decays. The results are given in Table 41 and shown in Fig. 26.

Fig. 26

Averages of (left) $S_{C\!P}$ and (right) $C_{C\!P}$ for the mode $B^{0}\rightarrow K^0_{S} K^0_{S}$

4.9 Time-dependent asymmetries in $b \rightarrow s\gamma$ transitions

The radiative decays $b \rightarrow s\gamma$ produce photons which are highly polarised in the Standard Model. The decays $B^{0}\rightarrow F \gamma$ and ${\overline{B}} ^0 \rightarrow F \gamma$ produce photons with opposite helicities, and since the polarisation is, in principle, observable, these final states cannot interfere. The finite mass of the s quark introduces small corrections to the limit of maximum polarisation, but any large mixing-induced $C\!P$ violation would be a signal for new physics. Since a single weak phase dominates the $b \rightarrow s \gamma$ transition in the Standard Model, the cosine term is also expected to be small.

Atwood et al. [298] have shown that an inclusive analysis of $K^0_{S} \pi ^0\gamma$ can be performed, since the properties of the decay amplitudes are independent of the angular momentum of the $K^0_{S} \pi ^0$ system. However, if non-dipole operators contribute significantly to the amplitudes, then the Standard Model mixing-induced $C\!P$ violation could be larger than the naïve expectation $S \simeq -2 (m_s/m_b) \sin \left( 2\beta \right)$ [299, 300]. In this case, the $C\!P$ parameters may vary over the $K^0_{S} \pi ^0\gamma$ Dalitz plot, for example as a function of the $K^0_{S} \pi ^0$ invariant mass.

With the above in mind, we quote two averages: one for $K^*(892)$ candidates only, and the other one for the inclusive $K^0_{S} \pi ^0\gamma$ decay (including the $K^*(892)$). If the Standard Model dipole operator is dominant, both should give the same quantities (the latter naturally with smaller statistical error). If not, care needs to be taken in interpretation of the inclusive parameters, while the results on the $K^*(892)$ resonance remain relatively clean. Results from BaBar and Belle are used for both averages; both experiments use the invariant mass range $0.60< M_{K^0_{S} \pi ^0} < 1.80~{\mathrm{GeV/}c^2}$ in the inclusive analysis.

Table 42 Averages for $b \rightarrow s \gamma$ modes

In addition to the $K^0_{S} \pi ^0\gamma$ decay, both BaBar and Belle have presented results using the $K^0_{S} \rho \gamma$ mode, while BaBar (Belle) has in addition presented results using the $K^0_{S} \eta \gamma$ ($K^0_{S} \phi \gamma$) channel. For the $K^0_{S} \rho \gamma$ case, due to the non-negligible width of the $\rho ^0$ meson, decays selected as $B^{0}\rightarrow K^0_{S} \rho ^0\gamma$ can include a significant contribution from $K^{*\,\pm \,}\pi ^{\mp }\gamma$ decays, which are flavour-specific and do not have the same oscillation phenomenology. Both BaBar and Belle measure $S_\mathrm{eff}$ for all $B$ decay candidates with the $\rho ^0$ selection being $0.6< m(\pi ^+ \pi ^-) < 0.9~{\mathrm{GeV/}c^2}$, obtaining $0.14 \,\pm \, 0.25 \,^{+0.04}_{-0.03}$ (BaBar) and $0.09 \,\pm \, 0.27 \,^{+0.04}_{-0.07}$ (Belle). These values are then corrected for a “dilution factor”, that is evaluated with different methods in the two experiments: BaBar [400, 401] obtains a dilution factor of $-0.78 \,^{+0.19}_{-0.17}$ while Belle  [402] obtains $+0.83 \,^{+0.19}_{-0.03}$. Until the discrepancy between these values is understood, the average of the results should be treated with caution.

Fig. 27

Averages of (left) $S_{b \rightarrow s \gamma }$ and (right) $C_{b \rightarrow s \gamma }$. Recall that the data for $K^*\gamma$ is a subset of that for $K^0_{S} \pi ^0\gamma$

The results are given in Table 42, and shown in Figs. 27 and 28. No significant $C\!P$ violation results are seen; the results are consistent with the Standard Model and with other measurements in the $b \rightarrow s\gamma$ system (see Sect. 7).

A similar analysis can be performed for radiative ${B^{0}_s}$ decays to, for example, the $\phi \gamma$ final state. As for other observables determined with self-conjugate final states produced in ${B^{0}_s}$ decays, the effective lifetime also provides sensitivity, and can be determined without tagging the initial flavour of the decaying meson. The LHCb collaboration has determined the associated parameter $A_{\Delta \Gamma }(\phi \gamma ) = -0.98 \,^{+0.46}_{-0.52}\,^{+0.23}_{-0.20}$ [407].

Fig. 28

Averages of three $b \rightarrow s\gamma$ dominated channels, for which correlated averages are performed, in the $S_{C\!P}$ $vs.$ $C_{C\!P}$ plane. (Left) $B^{0}\rightarrow K^*\gamma$, (middle) $B^{0}\rightarrow K^0_{S} \pi ^0\gamma$ (including $K^*\gamma$), (right) $B^{0}\rightarrow K^0_{S} \rho \gamma$

Table 43 Averages for $B^{0}\rightarrow \rho ^{0} \gamma$

4.10 Time-dependent asymmetries in $b \rightarrow d\gamma$ transitions

The formalism for the radiative decays $b \rightarrow d\gamma$ is much the same as that for $b \rightarrow s\gamma$ discussed above. Assuming dominance of the top quark in the loop, the weak phase in decay should cancel with that from mixing, so that the mixing-induced $C\!P$ violation parameter $S_{C\!P}$ should be very small. Corrections due to the finite light quark mass are smaller compared to $b \rightarrow s\gamma$, since $m_d < m_s$, but QCD corrections of $\mathcal{O}\left( \varLambda _\mathrm{QCD}/m_b\right)$ may be sizable [299]. Large $C\!P$ violation effects could be seen through a non-zero value of $C_{b \rightarrow d \gamma }$ since the top loop is not the only contribution.

Results using the mode $B^{0}\rightarrow \rho ^0\gamma$ are available from Belle and are given in Table 43.

4.11 Time-dependent $C\!P$ asymmetries in $b \rightarrow u{\bar{u}}d$ transitions

The $b \rightarrow u \bar{u} d$ transition can be mediated by either a $b \rightarrow u$ tree amplitude or a $b \rightarrow d$ penguin amplitude. These transitions can be investigated using the time dependence of $B^{0}$ decays to final states containing light mesons. Results are available from both BaBar and Belle for the $C\!P$ eigenstate (${\eta }= +1$) $\pi ^+\pi ^-$ final state and for the vector-vector final state $\rho ^+\rho ^-$, which is found to be dominated by the $C\!P$-even longitudinally polarised component (BaBar measures $f_\mathrm{long} = 0.992 \,\pm \, 0.024 \, ^{+0.026}_{-0.013}$ [409] while Belle measures $f_\mathrm{long} = 0.988 \,\pm \, 0.012 \,\pm \, 0.023$ [410]). BaBar has also performed a time-dependent analysis of the vector-vector final state $\rho ^0\rho ^0$ [411], in which they measure $f_\mathrm{long} = 0.70 \,\pm \, 0.14 \,\pm \, 0.05$; Belle measure a smaller branching fraction than BaBar for $B^{0}\rightarrow \rho ^0\rho ^0$ [412] with corresponding signal yields too small to perform a time-dependent analysis; for the longitudinal polarisation they measure $f_\mathrm{long} = 0.21 \,^{+0.18}_{-0.22} \,\pm \, 0.13$. LHCb has measured the branching fraction and longitudinal polarisation for $B^{0}\rightarrow \rho ^0\rho ^0$, and for the latter finds $f_\mathrm{long} = 0.745 \,^{+0.048}_{-0.058} \,\pm \, 0.034$ [413], but has not yet performed a time-dependent analysis of this decay. The Belle measurement for $f_\mathrm{long}$ is thus in some tension with the other results. BaBar has furthermore performed a time-dependent analysis of the $B^{0}\rightarrow a_1^{\pm } \, \pi ^{\mp }$ decay [414]; further experimental input for the extraction of $\alpha$ from this channel is reported in a later publication [415].

Results, and averages, of time-dependent $C\!P$ violation parameters in $b \rightarrow u \bar{u} d$ transitions are listed in Table 44. The averages for $\pi ^+\pi ^-$ are shown in Fig. 29, and those for $\rho ^+\rho ^-$ are shown in Fig. 30, with the averages in the $S_{C\!P}$ $vs.$ $C_{C\!P}$ plane shown in Fig. 31 and averages of $C\!P$ violation parameters in $B^{0}\rightarrow a_1^{\pm } \, \pi ^{\mp }$ decay shown in Fig. 32.

Table 44 Averages for $b \rightarrow u \bar{u} d$ modes

Fig. 29

Averages of (left) $S_{C\!P}$ and (right) $C_{C\!P}$ for the mode $B^{0}\rightarrow \pi ^+\pi ^-$

Fig. 30

Averages of (left) $S_{C\!P}$ and (right) $C_{C\!P}$ for the mode $B^{0}\rightarrow \rho ^+\rho ^-$

Fig. 31

Averages of $b \rightarrow u\bar{u} d$ dominated channels, for which correlated averages are performed, in the $S_{C\!P}$ $vs.$ $C_{C\!P}$ plane. (Left) $B^{0}\rightarrow \pi ^+\pi ^-$ and (right) $B^{0}\rightarrow \rho ^+\rho ^-$

If the penguin contribution is negligible, the time-dependent parameters for $B^{0}\rightarrow \pi ^+\pi ^-$ and $B^{0}\rightarrow \rho ^+\rho ^-$ are given by $S_{b \rightarrow u\bar{u} d} = {\eta }\sin (2\alpha )$ and $C_{b \rightarrow u\bar{u} d} = 0$. In the presence of the penguin contribution, $C\!P$ violation in decay may arise, and there is no straightforward interpretation of $S_{b \rightarrow u\bar{u} d}$ and $C_{b \rightarrow u\bar{u} d}$. An isospin analysis [419] can be used to disentangle the contributions and extract $\alpha$.

For the non-$C\!P$ eigenstate $\rho ^{\,\pm \,}\pi ^{{\mp }}$, both BaBar [281] and Belle  [283, 284] have performed time-dependent Dalitz plot analyses of the $\pi ^+\pi ^-\pi ^0$ final state [279]; such analyses allow direct measurements of the phases. Both experiments have measured the U and I parameters discussed in Sect. 4.2.5 and defined in Table 22. We have performed a full correlated average of these parameters, the results of which are summarised in Fig. 33.

Both experiments have also extracted the Q2B parameters. We have performed a full correlated average of these parameters, which is equivalent to determining the values from the averaged U and I parameters. The results are given in Table 45.³⁵ Averages of the $B^{0}\rightarrow \rho ^0\pi ^0$ Q2B parameters are shown in Figs. 34 and 35.

With the notation described in Sect. 4.2 (Eq. (131)), the time-dependent parameters for the Q2B $B^{0}\rightarrow \rho ^{\pm } \,\pi ^{\mp }$ analysis are, neglecting penguin contributions, given by

$$\begin{aligned} S_{\rho \pi }= & {} \sqrt{1 - \bigg (\frac{\Delta C}{2}\bigg )^2}\sin (2\alpha )\cos (\delta ) \, \nonumber \\ \Delta S_{\rho \pi }= & {} \sqrt{1 - \bigg (\frac{\Delta C}{2}\bigg )^2}\cos (2\alpha )\sin (\delta ) \end{aligned}$$

(161)

and $C_{\rho \pi } = \mathcal{A}_{C\!P}^{\rho \pi } = 0$, where $\delta =\arg (A_{-+}A^*_{+-})$ is the strong phase difference between the $\rho ^-\pi ^+$ and $\rho ^+\pi ^-$ decay amplitudes. In the presence of the penguin contribution, there is no straightforward interpretation of the Q2B observables in the $B^{0}\rightarrow \rho ^{\pm } \,\pi ^{\mp }$ system in terms of CKM parameters. However, $C\!P$ violation in decay may arise, resulting in either or both of $C_{\rho \pi } \ne 0$ and $\mathcal{A}_{C\!P}^{\rho \pi } \ne 0$. Equivalently, $C\!P$ violation in decay may be seen by either of the decay-type-specific observables $\mathcal{A}^{+-}_{\rho \pi }$ and $\mathcal{A}^{-+}_{\rho \pi }$, defined in Eq. (132), deviating from zero. Results and averages for these parameters are also given in Table 45. Averages of $C\!P$ violation parameters in $B^{0}\rightarrow \rho ^{\pm } \,\pi ^{\mp }$ decays are shown in Fig. 36, both in $\mathcal{A}^{\rho \pi }_{C\!P}$ $vs.$ $C_{\rho \pi }$ space and in $\mathcal{A}^{-+}_{\rho \pi }$ $vs.$ $\mathcal{A}^{+-}_{\rho \pi }$ space.

The averages for $S_{b \rightarrow u\bar{u} d}$ and $C_{b \rightarrow u\bar{u} d}$ in $B^{0}\rightarrow \pi ^+\pi ^-$ decays are both more than $5\sigma$ away from zero, suggesting that both mixing-induced and $C\!P$ violation in decay are well-established in this channel. The discrepancy between results from BaBar and Belle that used to exist in this channel (see, for example, Ref. [420]) is no longer apparent, and the results from LHCb are also fully consistent with other measurements. Some difference is, however, seen between the BaBar and Belle measurements in the $a_1^{\pm } \,\pi ^{\mp }$ system. The confidence level of the five-dimensional average is 0.03, which corresponds to a $2.1\sigma$ discrepancy. As seen in Table 44, this discrepancy is primarily in the values of $S_{a_1\pi }$, and is not evident in the $\mathcal{A}^{-+}_{a_1\pi }$ $vs.$ $\mathcal{A}^{+-}_{a_1\pi }$ projection shown in Fig. 32. Since there is no evidence of systematic problems in either analysis, we do not rescale the errors of the averages.

Fig. 32

Averages of $C\!P$ violation parameters in $B^{0}\rightarrow a_1^{\pm } \,\pi ^{\mp }$ in $\mathcal{A}^{-+}_{a_1\pi }$ $vs.$ $\mathcal{A}^{+-}_{a_1\pi }$ space

Fig. 33

Summary of the U and I parameters measured in the time-dependent $B^{0}\rightarrow \pi ^+\pi ^-\pi ^0$ Dalitz plot analysis

Table 45 Averages of quasi-two-body parameters extracted from time-dependent Dalitz plot analysis of $B^{0}\rightarrow \pi ^+\pi ^-\pi ^0$

In $B^{0}\rightarrow \rho ^{\pm } \,\pi ^{\mp }$ decays, both experiments see an indication of $C\!P$ violation in the $\mathcal{A}^{\rho \pi }_{C\!P}$ parameter (as seen in Fig. 36). The average is more than $3\sigma$ from zero, providing evidence of direct $C\!P$ violation in this channel. In $B^{0}\rightarrow \rho ^+\rho ^-$ decays there is no evidence for $C\!P$ violation, either mixing-induced or in decay. The absence of evidence of penguin contributions in this mode leads to strong constraints on $\alpha \equiv \phi _2$.

4.11.1 Constraints on $\alpha \equiv \phi _2$

The precision of the measured $C\!P$ violation parameters in $b \rightarrow u{\bar{u}}d$ transitions allows constraints to be set on the UT angle $\alpha \equiv \phi _2$. Constraints have been obtained with various methods:

• Both BaBar [416] and Belle  [417] have performed isospin analyses in the $\pi \pi$ system. Belle exclude $23.8^\circ< \phi _2 < 66.8^\circ$ at 68% CL while BaBar give a confidence level interpretation for $\alpha$, and constrain $\alpha \in \left[ 71^\circ , 109^\circ \right]$ at 68% CL. Values in the range $\left[ 23^\circ , 67^\circ \right]$ are excluded at 90% CL. In both cases, only solutions in $0^\circ$–$180^\circ$ are quoted.

• Both experiments have also performed isospin analyses in the $\rho \rho$ system. The most recent result from BaBar is given in an update of the measurements of the $B^+\rightarrow \rho ^+\rho ^0$ decay [421], and sets the constraint $\alpha = \left( 92.4 \,^{+6.0}_{-6.5}\right) ^\circ$. The most recent result from Belle is given in their paper on time-dependent $C\!P$ violation parameters in $B^{0}\rightarrow \rho ^+\rho ^-$ decays, and sets the constraint $\phi _2 = \left( 93.7 \,\pm \, 10.6 \right) ^\circ$ [410].

• The time-dependent Dalitz plot analysis of the $B^{0}\rightarrow \pi ^+\pi ^-\pi ^0$ decay allows a determination of $\alpha$ without input from any other channels. BaBar [282] present a scan, but not an interval, for $\alpha$, since their studies indicate that the scan is not statistically robust and cannot be interpreted as 1−CL. Belle  [283, 284] has obtained a constraint on $\alpha$ using additional information from the SU(2) partners of $B \rightarrow \rho \pi$, which can be used to constrain $\alpha$ via an isospin pentagon relation [422]. With this analysis, Belle obtains the constraint $\phi _2 = (83 \, ^{+12}_{-23})^\circ$ (where the errors correspond to $1\sigma$, i.e. $68.3\%$ confidence level).

• The results from BaBar on $B^{0}\rightarrow a_1^{\pm } \, \pi ^{\mp }$ [414] can be combined with results from modes related by flavour symmetries ($a_1K$ and $K_1\pi$) [423]. This has been done by BaBar [415], resulting in the constraint $\alpha = \left( 79 \,\pm \, 7 \,\pm \, 11 \right) ^\circ$, where the first uncertainty is from the analysis of $B^{0}\rightarrow a_1^{\pm } \, \pi ^{\mp }$ that obtains $\alpha ^\mathrm{eff}$, and the second is due to the constraint on $| \alpha ^\mathrm{eff} - \alpha |$. This approach gives a result with several ambiguous solutions; that consistent with other determinations of $\alpha$ and with global fits to the CKM matrix parameters is quoted here.

• The CKMfitter [252] and UTFit [339] groups use the measurements from Belle and BaBar given above with other branching fractions and $C\!P$ asymmetries in $B \rightarrow \pi \pi$, $\rho \pi$ and $\rho \rho$ modes, to perform isospin analyses for each system, and to obtain combined constraints on $\alpha$.

• The BaBar and Belle collaborations have combined their results on $B \rightarrow \pi \pi$, $\pi \pi \pi ^0$ and $\rho \rho$ decays to obtain [424]

  $$\begin{aligned} \alpha \equiv \phi _2 = (88 \,\pm \, 5)^\circ \,. \end{aligned}$$

  (162)

  The above solution is that consistent with the Standard Model (an ambiguous solution shifted by $180^\circ$ exists). The strongest constraint currently comes from the $B \rightarrow \rho \rho$ system. The inclusion of results from $B^{0}\rightarrow a_1^{\pm } \, \pi ^{\mp }$ does not significantly affect the average.

Fig. 34

Averages of (left) $S_{b \rightarrow u\bar{u} d}$ and (right) $C_{b \rightarrow u\bar{u} d}$ for the mode $B^{0}\rightarrow \rho ^0\pi ^0$

Fig. 35

Averages of $b \rightarrow u\bar{u} d$ dominated channels, for the mode $B^{0}\rightarrow \rho ^0\pi ^0$ in the $S_{C\!P}$ $vs.$ $C_{C\!P}$ plane

Fig. 36

$C\!P$ violation in $B^{0}\rightarrow \rho ^{\pm } \,\pi ^{\mp }$ decays. (Left) $\mathcal{A}^{\rho \pi }_{C\!P}$ $vs.$ $C_{\rho \pi }$ space, (right) $\mathcal{A}^{-+}_{\rho \pi }$ $vs.$ $\mathcal{A}^{+-}_{\rho \pi }$ space

Note that methods based on isospin symmetry make extensive use of measurements of branching fractions and $C\!P$ asymmetries, as averaged by the HFLAV Rare Decays subgroup (Sect. 7). Note also that each method suffers from discrete ambiguities in the solutions. The model assumption in the $B^{0}\rightarrow \pi ^+\pi ^-\pi ^0$ analysis helps resolve some of the multiple solutions, and results in a single preferred value for $\alpha$ in $\left[ 0, \pi \right]$. All the above measurements correspond to the choice that is in agreement with the global CKM fit.

At present we make no attempt to provide an HFLAV average for $\alpha \equiv \phi _2$. More details on procedures to calculate a best fit value for $\alpha$ can be found in Refs. [252, 339].

4.12 Time-dependent $C\!P$ asymmetries in $b \rightarrow c{\bar{u}}d / u{\bar{c}}d$ transitions

Non-$C\!P$ eigenstates such as $D^{\mp }\pi ^{\pm } \,$, $D^{*{\mp }}\pi ^{\pm } \,$ and $D^{\mp }\rho ^{\pm } \,$ can be produced in decays of $B^{0}$ mesons either via Cabibbo favoured ($b \rightarrow c$) or doubly Cabibbo suppressed ($b \rightarrow u$) tree amplitudes. Since no penguin contribution is possible, these modes are theoretically clean. The ratio of the magnitudes of the suppressed and favoured amplitudes, R, is sufficiently small (predicted to be about 0.02), that $\mathcal{O}(R^2)$ terms can be neglected, and the sine terms give sensitivity to the combination of UT angles $2\beta +\gamma$.

As described in Sect. 4.2.6, the averages are given in terms of the parameters a and c of Eq. (136). $C\!P$ violation would appear as $a \ne 0$. Results are available from both BaBar and Belle in the modes $D^{\mp }\pi ^{\pm } \,$ and $D^{*{\mp }}\pi ^{\pm } \,$; for the latter mode both experiments have used both full and partial reconstruction techniques. Results are also available from BaBar using $D^{\mp }\rho ^{\pm } \,$. These results, and their averages, are listed in Table 46, and are shown in Fig. 37. The constraints in c $vs.$ a space for the $D\pi$ and $D^*\pi$ modes are shown in Fig. 38. It is notable that the average value of a from $D^*\pi$ is more than $3\sigma$ from zero, providing evidence of $C\!P$ violation in this channel.

Table 46 Averages for $b \rightarrow c{\bar{u}}d / u{\bar{c}}d$ modes

Fig. 37

Averages for $b \rightarrow c{\bar{u}}d / u{\bar{c}}d$ modes

For each mode, $D\pi$, $D^*\pi$ and $D\rho$, there are two measurements (a and c, or $S^+$ and $S^-$) that depend on three unknowns (R, $\delta$ and $2\beta +\gamma$), of which two are different for each decay mode. Therefore, there is not enough information to solve directly for $2\beta +\gamma$. However, for each choice of R and $2\beta +\gamma$, one can find the value of $\delta$ that allows a and c to be closest to their measured values, and calculate the separation in terms of numbers of standard deviations. (We currently neglect experimental correlations in this analysis.) These values of $N(\sigma )_\mathrm{min}$ can then be displayed as a function of R and $2\beta +\gamma$ (and can trivially be converted to confidence levels). These plots are given for the $D\pi$ and $D^*\pi$ modes in Fig. 38; the uncertainties in the $D\rho$ mode are currently too large to give any meaningful constraint.

The constraints can be tightened if one is willing to use theoretical input on the values of R and/or $\delta$. One popular choice is the use of SU(3) symmetry to obtain R by relating the suppressed decay mode to $B$ decays involving $D_s$ mesons. More details can be found in Refs. [295, 425,426,427,428].

Fig. 38

Results from $b \rightarrow c{\bar{u}}d / u{\bar{c}}d$ modes. (Top) Constraints in c vs. a space. (Bottom) Constraints in $2\beta +\gamma$ vs. R space. (Left) $D^*\pi$ and (right) $D\pi$ modes

4.13 Time-dependent $C\!P$ asymmetries in $b \rightarrow c{\bar{u}}s / u{\bar{c}}s$ transitions

4.13.1 Time-dependent $C\!P$ asymmetries in $B^{0}\rightarrow D^{\mp } K^0_{S} \pi ^{\pm } \,$

Time-dependent analyses of transitions such as $B^{0}\rightarrow D^{\mp } K^0_{S} \pi ^{\pm } \,$ can be used to probe $\sin (2\beta +\gamma )$ in a similar way to that discussed above (Sect. 4.12). Since the final state contains three particles, a Dalitz plot analysis is necessary to maximise the sensitivity. BaBar [429] has carried out such an analysis. They obtain $2\beta +\gamma = \left( 83 \,\pm \, 53 \,\pm \, 20 \right) ^\circ$ (with an ambiguity $2\beta +\gamma \leftrightarrow 2\beta +\gamma +\pi$) assuming the ratio of the $b \rightarrow u$ and $b \rightarrow c$ amplitude to be constant across the Dalitz plot at 0.3.

4.13.2 Time-dependent $C\!P$ asymmetries in ${B^{0}_s} \rightarrow D_s^{\mp } K^{\pm } \,$

Time-dependent analysis of ${B^{0}_s} \rightarrow D_s^{\mp } K^{\pm } \,$ decays can be used to determine $\gamma -2\beta _s$ [430, 431]. Compared to the situation for $B^{0}\rightarrow D^{(*){\mp }} \pi ^{\pm } \,$ decays discussed in Sect. 4.12, the larger value of the ratio R of the magnitudes of the suppressed and favoured amplitudes allows it to be determined from the data. Moreover, the non-zero value of $\Delta \Gamma _s$ allows the determination of additional terms, labelled $A^{\Delta \Gamma }$ and ${\bar{A}}{}^{\Delta \Gamma }$, that break ambiguities in the solutions for $\gamma -2\beta _s$.

LHCb [296, 432] has measured the time-dependent $C\!P$ violation parameters in ${B^{0}_s} \rightarrow D_s^{\mp } K^{\pm } \,$ decays, using $3.0 \ \mathrm{fb}^{-1}$ of data. The results are given in Table 47, and correspond to $3.6\,\sigma$ evidence for $C\!P$ violation in the interference between mixing and ${B^{0}_s} \rightarrow D_s^{\mp } K^{\pm } \,$ decays. From these results, and a constraint on $2\beta _s$ from independent LHCb measurements [202], LHCb determine $\gamma = (127 \,^{+17}_{-22})^\circ$, $\delta _{D_sK} = (358 \,^{+15}_{-16})^\circ$ and $R_{D_sK} = 0.37 \,^{+0.10}_{-0.09}$.

Table 47 Results for ${B^{0}_s} \rightarrow D_s^{\mp } K^{\pm } \,$

4.14 Rates and asymmetries in $B \rightarrow D^{(*)} K^{(*)}$ decays

As explained in Sect. 4.2.7, rates and asymmetries in $B^{+}\rightarrow D^{(*)} K^{(*)+}$ decays are sensitive to $\gamma$, and have negligible theoretical uncertainty [311]. Various methods using different $D^{(*)}$ final states have been used.

4.14.1 D decays to $C\!P$ eigenstates

Results are available from BaBar, Belle, CDF and LHCb on GLW analyses in the decay mode $B^{+}\rightarrow DK^+$. All experiments use the $C\!P$-even D decay final states $K^+K^-$ and $\pi ^+\pi ^-$; BaBar and Belle in addition use the $C\!P$-odd decay modes $K^0_{S} \pi ^0$, $K^0_{S} \omega$ and $K^0_{S} \phi$, though care is taken to avoid statistical overlap with the $K^0_{S} K^+K^-$ sample used for Dalitz plot analyses (see Sect. 4.14.4). BaBar and Belle also have results in the decay mode $B^{+}\rightarrow D^* K^+$, using both the $D^* \rightarrow D\pi ^0$ decay, which gives $C\!P (D^*) = C\!P (D)$, and the $D^* \rightarrow D\gamma$ decays, which gives $C\!P (D^*) = -C\!P (D)$. In addition, BaBar and LHCb have results in the decay mode $B^{+}\rightarrow DK^{*+}$, and LHCb has results in the decay mode $B^{+}\rightarrow DK^+ \pi ^+\pi ^-$. The results and averages are given in Table 48 and shown in Fig. 39.

Table 48 Averages from GLW analyses of $b \rightarrow c{\bar{u}}s / u{\bar{c}}s$ modes. The sample size is given in terms of number of $B{\bar{B}}$ pairs, $N(B{\bar{B}})$, for the $e^+e^-$ B factory experiments BaBar and Belle, and in terms of integrated luminosity, $\int \mathcal{L}\,dt$, for the hadron collider experiments CDF and LHCb

Fig. 39

Averages of $A_{C\!P}$ and $R_{C\!P}$ from GLW analyses

LHCb has performed a GLW analysis using the $B^0 \rightarrow DK^{*0}$ decay with the $C\!P$-even $D \rightarrow K^+K^-$ and $D \rightarrow \pi ^+\pi ^-$ channels [441]. The results are presented separately to allow for possible $C\!P$ violation effects in the charm decays, which are, however, known to be small. The results are given in Table 49 where an average is also reported.

As pointed out in Refs. [313, 314], a Dalitz plot analysis of $B^0 \rightarrow DK^+\pi ^-$ decays provides more sensitivity to $\gamma \equiv \phi _3$ than the quasi-two-body $DK^{*0}$ approach. The analysis provides direct sensitivity to the hadronic parameters $r_B$ and $\delta _B$ associated with the $B^0 \rightarrow DK^{*0}$ decay amplitudes, rather than effective hadronic parameters averaged over the $K^{*0}$ selection window as in the quasi-two-body case.

Table 49 Results from GLW analysis of $B^{0}\rightarrow DK^{*0}$

Such an analysis has been performed by LHCb. A simultaneous fit is performed to the $B^0 \rightarrow DK^+\pi ^-$ Dalitz plots with the neutral D meson reconstructed in the $K^+\pi ^-$, $K^+K^-$ and $\pi ^+\pi ^-$ final states. The reported results in Table 50 are for the Cartesian parameters, defined in Eq. (150) associated with the $B^0 \rightarrow DK^*(892)^0$ decay. Note that, since the measurements use overlapping data samples, these results cannot be combined with the LHCb results for GLW observables in $B^0 \rightarrow DK^*(892)^0$ decays reported in Table 49.

Table 50 Results from Dalitz plot analysis of $B ^0 \rightarrow DK^+\pi ^-$ decays with $D \rightarrow K^+ K^-$ and $\pi ^+ \pi ^-$

LHCb use these results to obtain confidence levels for $\gamma$, $r_B(DK^{*0})$ and $\delta _B(DK^{*0})$. In addition, results are reported for the hadronic parameters needed to relate these results to quasi-two-body measurements of $B^0 \rightarrow DK^*(892)^0$ decays, where a selection window of $m(K^+\pi ^-)$ within $50~{\mathrm{MeV/}c^2}$ of the pole mass and helicity angle satisfying $|\cos (\theta _{K^{*0}})|>0.4$ is assumed. These parameters are the coherence factor $\kappa$, the ratio of quasi-two-body and amplitude level $r_B$ values, $\bar{R}_B = {\bar{r}}_B/r_B$, and the difference between quasi-two-body and amplitude level $\delta _B$ values, $\Delta {\bar{\delta }}_B = {\bar{\delta }}_B-\delta _B$. LHCb [442] obtain

$$\begin{aligned} \gamma= & {} 0.958 \,^{+0.005}_{-0.010}\,^{+0.002}_{-0.045},\nonumber \\ \bar{R}_B= & {} 1.02 \,^{+0.03}_{-0.01} \,\pm \, 0.06,\nonumber \\ \Delta {\bar{\delta }}_B= & {} 0.02 \,^{+0.03}_{-0.02} \,\pm \, 0.11. \end{aligned}$$

(163)

Table 51 Averages from GLW-like analyses of $b \rightarrow c{\bar{u}}s / u{\bar{c}}s$ modes

Table 52 Averages from ADS analyses of $b \rightarrow c{\bar{u}}s / u{\bar{c}}s$ modes

4.14.2 D decays to quasi-$C\!P$ eigenstates

As discussed in Sect. 4.2.7, if a multibody neutral D meson decay can be shown to be dominated by one $C\!P$ eigenstate, it can be used in a “GLW-like” (sometimes called “quasi-GLW”) analysis [318]. The same observables $R_{C\!P}$, $A_{C\!P}$ as for the GLW case are measured, but an additional factor of $(2F_+-1)$, where $F_+$ is the fractional $C\!P$-even content, enters the expressions relating these observables to $\gamma \equiv \phi _3$. The $F_+$ factors have been measured using CLEO-c data to be $F_+(\pi ^+\pi ^-\pi ^0) = 0.973 \,\pm \, 0.017$, $F_+(K^+K^-\pi ^0) = 0.732 \,\pm \, 0.055$, $F_+(\pi ^+\pi ^-\pi ^+\pi ^-) = 0.737 \,\pm \, 0.028$ [443].

The GLW-like observables for $D\rightarrow \pi ^+\pi ^-\pi ^0$, $K^+K^-\pi ^0$ and $D\rightarrow \pi ^+\pi ^-\pi ^+\pi ^-$ have been measured by LHCb. The $A_\mathrm{qGLW}$ observable for $D\rightarrow \pi ^+\pi ^-\pi ^0$ was measured in an earlier analysis by BaBar, from which additional observables, discussed in Sect. 4.2.7 and reported in Table 55 below, were reported. The results are given in Table 51.

4.14.3 D decays to suppressed final states

For ADS analyses, all of BaBar, Belle, CDF and LHCb have studied the modes $B^{+}\rightarrow DK^+$ and $B^{+}\rightarrow D\pi ^+$. BaBar has also analysed the $B^{+}\rightarrow D^* K^+$ mode. There is an effective shift of $\pi$ in the strong phase difference between the cases that the $D^*$ is reconstructed as $D\pi ^0$ and $D\gamma$ [315], therefore these modes are studied separately. In addition, BaBar has studied the $B^{+}\rightarrow DK^{*+}$ mode, where $K^{*+}$ is reconstructed as $K^0_{S} \pi ^+$, and LHCb has studied the $B^{+}\rightarrow DK^+ \pi ^+ \pi ^-$ mode. In all the above cases the suppressed decay $D \rightarrow K^-\pi ^+$ has been used. BaBar, Belle and LHCb also have results using $B^{+}\rightarrow DK^+$ with $D \rightarrow K^-\pi ^+\pi ^0$, while LHCb has results using $B^{+}\rightarrow DK^+$ with $D \rightarrow K^-\pi ^+\pi ^+\pi ^-$. The results and averages are given in Table 52 and shown in Fig. 40.

Similar phenomenology as for $B \rightarrow DK$ decays holds for $B \rightarrow D\pi$ decays, though in this case the interference is between $b \rightarrow c{\bar{u}}d$ and $b \rightarrow u{\bar{c}}d$ transitions, and the ratio of suppressed to favoured amplitudes is expected to be much smaller, $\mathcal{O}(1\%)$. For most D meson final states this implies that the interference effect is too small to be of interest, but in the case of ADS analysis it is possible that effects due to $\gamma$ may be observable. Accordingly, the experiments now measure the corresponding observables in the $D\pi$ final states. The results and averages are given in Table 53 and shown in Fig. 41.

BaBar, Belle and LHCb have also presented results from a similar analysis method with self-tagging neutral B decays: $B^{0}\rightarrow DK^{*0}$ with $D \rightarrow K^-\pi ^+$ (all), $D \rightarrow K^-\pi ^+\pi ^0$ and $D \rightarrow K^-\pi ^+\pi ^+\pi ^-$ (BaBar only). All these results are obtained with the $K^{*0} \rightarrow K^+\pi ^-$ decay. Effects due to the natural width of the $K^{*0}$ are handled using the parametrisation suggested by Gronau [312].

Fig. 40

Averages of $R_\mathrm{ADS}$ and $A_\mathrm{ADS}$ for $B \rightarrow D^{(*)}K^{(*)}$ decays

Table 53 Averages from ADS analyses of $b \rightarrow c{\bar{u}}d / u{\bar{c}}d$ modes

Fig. 41

Averages of $R_\mathrm{ADS}$ and $A_\mathrm{ADS}$ for $B \rightarrow D^{(*)}\pi$ decays

Table 54 Results from ADS analysis of $B^{0}\rightarrow DK^{*0}$, $D \rightarrow K^-\pi ^+$

The following 95% CL limits are set by BaBar [450]:

$$\begin{aligned}&R_\mathrm{ADS}(K\pi )< 0.244 \quad R_\mathrm{ADS}(K\pi \pi ^0)< 0.181 \nonumber \\&R_\mathrm{ADS}(K\pi \pi \pi ) < 0.391, \end{aligned}$$

(164)

while Belle  [451] obtain

$$\begin{aligned} R_\mathrm{ADS}(K\pi ) < 0.16. \end{aligned}$$

(165)

The results from LHCb, which are presented in terms of the parameters $R_+$ and $R_-$ instead of $R_\mathrm{ADS}$ and $A_\mathrm{ADS}$, are given in Table 54.

Combining the results and using additional input from CLEO-c [452, 453] a limit on the ratio between the $b \rightarrow u$ and $b \rightarrow c$ amplitudes of ${\bar{r}}_B(DK^{*0}) \in \left[ 0.07,0.41 \right]$ at 95% CL limit is set by BaBar. Belle set a limit of ${\bar{r}}_B < 0.4$ at 95% CL. LHCb take input from Sect. 8 and obtain ${\bar{r}}_B = 0.240 \,^{+0.055}_{-0.048}$ (different from zero with $2.7\sigma$ significance).

4.14.4 D decays to multiparticle self-conjugate final states (model-dependent analysis)

For the model-dependent Dalitz plot analysis, both BaBar and Belle have studied the modes $B^{+}\rightarrow DK^+$, $B^{+}\rightarrow D^* K^+$ and $B^{+}\rightarrow DK^{*+}$. For $B^{+}\rightarrow D^* K^+$, both experiments have used both $D^*$ decay modes, $D^* \rightarrow D\pi ^0$ and $D^* \rightarrow D\gamma$, taking the effective shift in the strong phase difference into account.³⁶ In all cases the decay $D \rightarrow K^0_{S} \pi ^+\pi ^-$ has been used. BaBar also used the decay $D \rightarrow K^0_{S} K^+K^-$. LHCb has also studied $B^{+}\rightarrow DK^+$ decays with $D \rightarrow K^0_{S} \pi ^+\pi ^-$. BaBar has also performed an analysis of $B^{+}\rightarrow DK^+$ with $D \rightarrow \pi ^+\pi ^-\pi ^0$. Results and averages are given in Table 55, and shown in Figs. 42 and 43. The third error on each measurement is due to D decay model uncertainty.

Table 55 Averages from model-dependent Dalitz plot analyses of $b \rightarrow c{\bar{u}}s / u{\bar{c}}s$ modes. Note that the uncertainties assigned to the averages do not include model errors

The parameters measured in the analyses are explained in Sect. 4.2.7. All experiments measure the Cartesian variables, defined in Eq. (150), and perform frequentist statistical procedures, to convert these into measurements of $\gamma$, $r_B$ and $\delta _B$. In the $B^{+}\rightarrow DK^+$ with $D \rightarrow \pi ^+\pi ^-\pi ^0$ analysis, the parameters $(\rho ^{\,\pm \,}, \theta ^{\pm } \,)$ are used instead.

Fig. 42

Contours in the $(x_{\pm } \,, y_{\pm })$ from model-dependent analysis of $B^{+}\rightarrow D^{(*)}K^{(*)+}$, $D \rightarrow K^0_{S} h^+ h^-$ ($h = \pi ,K$). (Left) $B^{+}\rightarrow DK^+$, (middle) $B^{+}\rightarrow D^* K^+$, (right) $B^{+}\rightarrow DK^{*+}$. Note that the uncertainties assigned to the averages given in these plots do not include model errors

Fig. 43

Averages of $(x_{\pm } \,, y_{\pm })$ from model-dependent analyses of $B^{+}\rightarrow D^{(*)}K^{(*)+}$ with $D \rightarrow K^0_{S} h^+h^-$ ($h=\pi ,K$). (Top left) $x_+$, (top right) $x_-$, (bottom left) $y_+$, (bottom right) $y_-$. The top plots include constraints on $x_{\,\pm \,}$ obtained from GLW analyses (see Sect. 4.14.1). Note that the uncertainties assigned to the averages given in these plots do not include model errors

Both experiments reconstruct $K^{*+}$ as $K^0_{S} \pi ^+$, but the treatment of possible nonresonant $K^0_{S} \pi ^+$ differs: Belle assign an additional model uncertainty, while BaBar use a parametrisation suggested by Gronau [312]. The parameters $r_B$ and $\delta _B$ are replaced with effective parameters $\kappa {\bar{r}}_B$ and ${\bar{\delta }}_B$; no attempt is made to extract the true hadronic parameters of the $B^{+}\rightarrow DK^{*+}$ decay.

We perform averages using the following procedure, which is based on a set of reasonable, though imperfect, assumptions.

• It is assumed that effects due to the different D decay models used by the two experiments are negligible. Therefore, we do not rescale the results to a common model.

• It is further assumed that the model uncertainty is $100\%$ correlated between experiments, and therefore this source of error is not used in the averaging procedure. (This approximation is compromised by the fact that the BaBar results include $D \rightarrow K^0_{S} K^+K^-$ decays in addition to $D \rightarrow K^0_{S} \pi ^+\pi ^-$.)

• We include in the average the effect of correlations within each experiment’s set of measurements.

• At present it is unclear how to assign an average model uncertainty. We have not attempted to do so. Our average includes only statistical and systematic errors. An unknown amount of model uncertainty should be added to the final error.

• We follow the suggestion of Gronau [312] in making the $DK^*$ averages. Explicitly, we assume that the selection of $K^{*+} \rightarrow K^0_{S} \pi ^+$ is the same in both experiments (so that $\kappa$, ${\bar{r}}_B$ and ${\bar{\delta }}_B$ are the same), and drop the additional source of model uncertainty assigned by Belle due to possible nonresonant decays.

• We do not consider common systematic errors, other than the D decay model.

Constraints on $\gamma \equiv \phi _3$

The measurements of $(x_{\pm } \,, y_{\pm })$ can be used to obtain constraints on $\gamma \equiv \phi _3$, as well as the hadronic parameters $r_B$ and $\delta _B$. BaBar [455], Belle  [454, 457] and LHCb [456] have all done so using a frequentist procedure (there are some differences in the details of the techniques used).

• BaBar obtain $\gamma = (68 \,^{+15}_{-14} \,\pm \, 4 \,\pm \, 3)^\circ$ from $DK^+$, $D^* K^+$ and $DK^{*+}$.

• Belle obtain $\phi _3 = (78 \,^{+11}_{-12} \,\pm \, 4 \,\pm \, 9)^\circ$ from $DK^+$ and $D^* K^+$.

• LHCb obtain $\gamma = (84 \,^{+49}_{-42})^\circ$ from $DK^+$ using 1 $\mathrm{fb}^{-1}$ of data (a more precise result using 3 $\mathrm{fb}^{-1}$ and the model-independent method is reported below).

• The experiments also obtain values for the hadronic parameters as detailed in Table 56.

• In the BaBar analysis of $B^{+}\rightarrow DK^+$ with $D \rightarrow \pi ^+\pi ^-\pi ^0$ decays [322], a constraint of $-30^\circ< \gamma < 76^\circ$ is obtained at the 68% confidence level.

• The results discussed here are included in the HFLAV combination to obtain a world average value for $\gamma \equiv \phi _3$, as discussed in Sect. 4.14.7.

Table 56 Summary of constraints on hadronic parameters from model-dependent analyses of $B^{+}\rightarrow D^{(*)} K^{(*)+}$ and $B^{0}\rightarrow \mathcal{D}K^{*0}$ decays. Note the alternative parametrisation of the hadronic parameters used by BaBar in the $DK^{*+}$ mode

BaBar and LHCb have performed a similar analysis using the self-tagging neutral B decay $B^{0}\rightarrow DK^{*0}$ (with $K^{*0} \rightarrow K^+\pi ^-$). Effects due to the natural width of the $K^{*0}$ are handled using the parametrisation suggested by Gronau [312]. LHCb give results in terms of the Cartesian parameters, as shown in Table 55. BaBar [459] present results only in terms of $\gamma$ and the hadronic parameters. The obtained constraints on $\gamma \equiv \phi _3$ are

• BaBar obtain $\gamma = (162 \,\pm \, 56)^\circ$

• LHCb obtain $\gamma = (80 \,^{+21}_{-22})^\circ$

• Values for the hadronic parameters are given in Table 56.

4.14.5 D decays to multiparticle self-conjugate final states (model-independent analysis)

A model-independent approach to the analysis of $B^{+}\rightarrow D^{(*)} K^+$ with multibody D decays was proposed by Giri, Grossman, Soffer and Zupan [309], and further developed by Bondar and Poluektov [319, 320]. The method relies on information on the average strong phase difference between $D^0$ and ${\overline{D}} ^0$ decays in bins of Dalitz plot position that can be obtained from quantum-correlated $\psi (3770) \rightarrow D^0 {\overline{D}} ^0$ events. This information is measured in the form of parameters $c_i$ and $s_i$ that are the amplitude weighted averages of the cosine and sine of the strong phase difference in a Dalitz plot bin labelled by i, respectively. These quantities have been obtained for $D \rightarrow K^0_{S} \pi ^+\pi ^-$ (and $D \rightarrow K^0_{S} K^+K^-$) decays by CLEO-c [263, 460].

Belle  [461] and LHCb [462] have used the model-independent Dalitz plot analysis approach to study the mode $B^{+}\rightarrow DK^+$. Both Belle  [463] and LHCb [464] have also used this approach to study $B^{0}\rightarrow DK^*(892)^0$ decays. In both cases, the experiments use $D \rightarrow K^0_{S} \pi ^+\pi ^-$ decays while LHCb has also included the $D \rightarrow K^0_{S} K^+K^-$ decay. The Cartesian variables $(x_{\pm } \,, y_{\pm } )$, defined in Eq. (150), are determined from the data. Note that due to the strong statistical and systematic correlations with the model-dependent results given in Sect. 4.14.4, these results cannot be combined.

The results and averages are given in Table 57, and shown in Fig. 44. Most results have three sets of errors, which are statistical, systematic, and uncertainty coming from the knowledge of $c_i$ and $s_i$ respectively. To perform the average, we remove the last uncertainty, which should be 100% correlated between the measurements. Since the size of the uncertainty from $c_i$ and $s_i$ is found to depend on the size of the $B \rightarrow DK$ data sample, we assign the LHCb uncertainties (which are mostly the smaller of the Belle and LHCb values) to the averaged result. This procedure should be conservative. In the LHCb $B^{0}\rightarrow DK^*(892)^0$ results [464], the values of $c_i$ and $s_i$ are constrained to their measured values within uncertainties in the fit to data, and hence the effect is absorbed in their statistical uncertainties. The $B^{0}\rightarrow DK^*(892)^0$ average is performed neglecting the model uncertainties on the Belle results.

Table 57 Averages from model-independent Dalitz plot analyses of $b \rightarrow c{\bar{u}}s / u{\bar{c}}s$ modes

Constraints on $\gamma \equiv \phi _3$

The measurements of $(x_{\pm } \,, y_{\pm })$ can be used to obtain constraints on $\gamma$, as well as the hadronic parameters $r_B$ and $\delta _B$. The experiments have done so using frequentist procedures (there are some differences in the details of the techniques used) (Table 58).

• From $B^{+}\rightarrow DK^+$, Belle  [461] obtain $\phi _3 = (77.3 \,^{+15.1}_{-14.9} \pm \, 4.1 \,\pm \, 4.3)^\circ$.

• From $B^{+}\rightarrow DK^+$, LHCb [462] obtain $\gamma = (62 \,^{+15}_{-14})^\circ$.

• From $B^{0}\rightarrow DK^*(892)^0$, LHCb [464] obtain $\gamma = (71 \,\pm \, 20)^\circ$.

• The experiments also obtain values for the hadronic parameters as detailed in Table 59.

• The results discussed here are included in the HFLAV combination to obtain a world average value for $\gamma \equiv \phi _3$, as discussed in Sect. 4.14.7.

4.14.6 D decays to multiparticle non-self-conjugate final states (model-independent analysis)

Following the original suggestion of Grossman, Ligeti and Soffer [317], decays of D mesons to $K^0_{S} K^{\,\pm \,} \pi ^{\mp }$ can be used in a similar approach to that discussed above to determine $\gamma \equiv \phi _3$. Since these decays are less abundant, the event samples available to date have not been sufficient for a fine binning of the Dalitz plots, but the analysis can be performed using only an overall coherence factor and related strong phase difference for the decay. These quantities have been determined by CLEO-c [466] both for the full Dalitz plots and in a restricted region $\,\pm \, 100 \ \mathrm{MeV}/c^2$ around the peak of the $K^*(892)^{\pm } \,$ resonance.

LHCb [465] has reported results of an analysis of $B^+\rightarrow D K^+$ and $B^+ \rightarrow D \pi ^+$ decays with $D \rightarrow K^0_{S} K^{\,\pm \,} \pi ^{\mp }$. The decays with different final states of the D meson are distinguished by the charge of the kaon from the decay of the D meson relative to the charge of the B meson, and are labelled “same sign” (SS) and “opposite sign” (OS). Six observables potentially sensitive to $\gamma \equiv \phi _3$ are measured: two ratios of rates for DK and $D\pi$ decays (one each for SS and OS) and four asymmetries (for DK and $D\pi$, SS and OS). This is done both for the full Dalitz plot of the D decay and for the $K^*(892)^{\pm } \,$-dominated region (with the same boundaries as used by CLEO-c). Note that there is a significant overlap of events between the two samples. The results, shown in Table 58 do not yet have sufficient precision to set significant constraints on $\gamma \equiv \phi _3$.

4.14.7 Combinations of results on rates and asymmetries in $B \rightarrow D^{(*)} K^{(*)}$ decays to obtain constraints on $\gamma \equiv \phi _3$

BaBar and LHCb have both produced constraints on $\gamma \equiv \phi _3$ from combinations of their results on $B^+ \rightarrow DK^+$ and related processes. The experiments use a frequentist procedure (there are some differences in the details of the techniques used).

Fig. 44

Contours in the $(x_{\pm } \,, y_{\pm })$ plane from model-independent analysis of $B^{+}\rightarrow DK^+$ with $D \rightarrow K^0_{S} h^+ h^-$ ($h = \pi ,K$)

Table 58 Results from model-independent Dalitz plot analysis of $B^+ \rightarrow DK^+$, $D \rightarrow K^0_{S} K^{\,\pm \,} \pi ^{\mp }$

• BaBar [467] use results from DK, $D^*K$ and $DK^*$ modes with GLW, ADS and GGSZ analyses, to obtain $\gamma = (69 \,^{+17}_{-16})^\circ$.

• LHCb [468] use results from the $DK^+$ mode with GLW, GLW-like, ADS, GGSZ ($K^0_{S} h^+h^-$) and GLS ($K^0_{S} K^{\,\pm \,} \pi ^{\mp }$) analyses, as well as $DK^{*0}$ with GLW, ADS and GGSZ analyses, $DK^+ \pi ^-$ GLW Dalitz plot analysis, $DK^+ \pi ^- \pi ^+$ with GLW and ADS analyses and ${B^{0}_s} \rightarrow D_s^{\mp }K^{\,\pm \,}$ decays. The LHCb combination takes into account subleading effects due to charm mixing and $C\!P$ violation [469]. The result is $\gamma = (72.2 \,^{+6.8}_{-7.3})^\circ$.

• All the combinations use inputs determined from $\psi (3770)\rightarrow D^0 {\overline{D}} ^0$ data samples (and/or from the HFLAV Charm Physics subgroup global fits on charm mixing parameters; see Sect. 8.1) to constrain the hadronic parameters in the charm system.

• Constraints are also obtained on the hadronic parameters involved in the decays. A summary of these is given in Table 60.

• The CKMfitter [252] and UTFit [339] groups perform similar combinations of all available results to obtain combined constraints on $\gamma \equiv \phi _3$.

Independently from the constraints on $\gamma \equiv \phi _3$ obtained by the experiments, the results summarised in Sect. 4.14 are statistically combined to produce world average constraints on $\gamma \equiv \phi _3$ and the hadronic parameters involved. The combination is performed with the GammaCombo framework [470] and follows a frequentist procedure, similar to those used by the experiments [467, 468, 471].

The input measurements used in the combination are listed in Table 61. Individual measurements are used as inputs, rather than the averages presented in Sect. 4.14, in order to facilitate cross-checks and to ensure the most appropriate treatment of correlations. A combination based on the our averages for each of the quantities measured by experiments gives consistent results.

All results from GLW and GLW-like analyses of $B^{+}\rightarrow D^{(*)} K^{(*)+}$ modes, as listed in Tables 48 and 51, are used. All results from ADS analyses of $B^{+}\rightarrow D^{(*)} K^{(*)+}$ as listed in Table 52 are also used. Regarding $B^0 \rightarrow DK^{*0}$ decays, the results of the $B^0 \rightarrow DK^+ \pi ^-$ GLW–Dalitz analysis (Table 50) are included, as are the LHCb results of the ADS analysis of $B^0 \rightarrow DK^{*0}$ (Table 54). Concerning results of GGSZ analyses of $B^{+}\rightarrow D^{(*)} K^{(*)+}$ with $D\rightarrow K^0_{S} h^{+}h^{-}$, the model-dependent results, as listed in Table 55, are used for the BaBar and Belle experiments, whilst the model-independent results, as listed in Table 57, are used for LHCb. This choice is made in order to maintain consistency of the approach across experiments whilst maximising the size of the samples used to obtain inputs for the combination. For GGSZ analyses of $B^0 \rightarrow DK^{*0}$ with $D\rightarrow K^0_{S} h^{+}h^{-}$ the model-independent result from LHCb (given in Table 57) is used for consistency with the treatment of the LHCb $B^{+}\rightarrow DK^+$ GGSZ result; the model-independent result by Belle is also included. The result of the GLS analysis of $B^{+}\rightarrow D K^+$ with $D\rightarrow K^{*{\,\pm \,}} K^{\mp }$ from LHCb (Table 58) are used. Finally, results from the time-dependent analysis of ${B^{0}_s} \rightarrow D_{s}^{{\mp }} K^{\,\pm \,}$ from LHCb (Table 47) are used.

Table 59 Summary of constraints on hadronic parameters from model-independent analyses of $B^{+}\rightarrow DK^+$ and $B^{0}\rightarrow DK^{*0}$, $D \rightarrow K^0_{S} h^+h^-$ ($h=\pi ,K$) decays

Several results with sensitivity to $\gamma$ are not included in the combination. Results from time-dependent analyses of $B^{0}\rightarrow D^{(*){\mp }}\pi ^{\pm } \,$ and $D^{\mp }\rho ^{\pm } \,$ (Table 46) are not used as there are insufficient constraints on the associated hadronic parameters. Similarly, results from $B^{0}\rightarrow D^{\mp } K^0_{S} \pi ^{\,\pm \,}$ (Sect. 4.13.1) are not used. Results from the LHCb $B^{0}\rightarrow DK^{*0}$ GLW analysis (Table 49) are not used because of the statistical overlap with the GLW–Dalitz analysis which is used instead. Limits on ADS parameters reported in Sect. 4.14.3 are not used. Results on $B^{+}\rightarrow D\pi ^+$ decays, given in Table 53, are not used since the small value of $r_B(D\pi ^+)$ means these channels have less sensitivity to $\gamma$ and are more vulnerable to biases due to subleading effects [468]. Results from the BaBar Dalitz plot analysis of $B^{+}\rightarrow DK^+$ with $D \rightarrow \pi ^+ \pi ^- \pi ^0$ (given in Table 55) are not included due to their limited sensitivity. Results from the $B^{+}\rightarrow DK^+$, $D \rightarrow K^0_{S} \pi ^+ \pi ^-$ GGSZ model-dependent analysis by LHCb (given in Table 55), and of the model-independent analysis of the same decay by Belle (given in Table 57) are not included due to the statistical overlap with results from model-(in)dependent analyses of the same data.

Auxiliary inputs are used in the combination in order to constrain the D system parameters and subsequently improve the determination of $\gamma \equiv \phi _3$. These include the ratio of suppressed to favoured decay amplitudes and the strong phase difference for $D\rightarrow K^{\,\pm \,} \pi ^{\mp }$ decays, taken from the HFLAV Charm Physics subgroup global fits (see Sect. 8). The amplitude ratios, strong phase differences and coherence factors of $D\rightarrow K^{\,\pm \,} \pi ^{\mp } \pi ^0$, $D\rightarrow K^{\,\pm \,} \pi ^{\mp } \pi ^+ \pi ^-$ and $D\rightarrow K^0_{S} K^{\,\pm \,} \pi ^{\,\pm \,}$ decays are taken from CLEO-c and LHCb measurements [466, 472, 473]. The fraction of $C\!P$-even content for quasi-GLW $D\rightarrow \pi ^+ \pi ^- \pi ^+ \pi ^-$, $D\rightarrow K^+ K^- \pi ^0$ and $D\rightarrow \pi ^+ \pi ^- \pi ^0$ decays are taken from CLEO-c measurements [443]. Constraints required to relate the hadronic parameters of the $B^0 \rightarrow DK^{*0}$ GLW–Dalitz analysis to the effective hadronic parameters of the quasi-two-body approaches are taken from LHCb measurements [442]. Finally, the value of $-2\beta _{s}$ is taken from the HFLAV Lifetimes and Oscillations subgroup (see Sect. 3); this is required to obtain sensitivity to $\gamma \equiv \phi _3$ from the time-dependent analysis of ${B^{0}_s} \rightarrow D_{s}^{{\mp }} K^{\,\pm \,}$ decays. A summary of the auxiliary constraints is given in Table 62.

The following reasonable, although imperfect, assumptions are made when performing the averages.

Table 60 Summary of constraints on hadronic parameters obtained from global combinations of results in $B^{+}\rightarrow D^{(*)} K^{(*)+}$ and $B^{0}\rightarrow DK^{*0}$ decays

• $C\!P$ violation in $D\rightarrow K^+ K^-$ and $D\rightarrow \pi ^+ \pi ^-$ decays is assumed to be zero. The results of Sect. 8 anyhow suggest such effects to be negligible.

• The combination is potentially sensitive to subleading effects from $D^0$–${\overline{D}} ^0$ mixing which is not accounted for [469, 474, 475]. The effect is expected to be small given that $r_{B} \gtrsim 0.1$ (for all included modes) whilst $r_{D} \approx 0.05$.

• All $B^{+}\rightarrow D K^{*+}$ modes are treated as two-body decays. In other words any dilution caused by non-$K^{*+}$ contributions in the selected regions of the $DK^0_{S} \pi ^+$ or $DK^+ \pi ^0$ Dalitz plots is assumed to be negligible. As a check of this assumption, it was found that including a coherence factor for $B^{+}\rightarrow D K^{*+}$ modes, $\kappa _{B}(DK^{*+}) = 0.9$, had negligible impact on the results.

• All of the inputs are assumed to be completely uncorrelated. Whilst this is true of the statistical uncertainties, it is not necessarily the case for systematic uncertainties. In particular, the model uncertainties for different model-dependent GGSZ analyses are fully correlated (when the same model is used) and similarly the model-independent GGSZ analyses have correlated systematic uncertainties originating from the knowledge of the strong phase variation across the Dalitz plot. The effect of including these correlations is estimated to be $<\!\!1^{\circ }$.

Table 61 List of measurements used in the $\gamma$ combination

Table 62 List of the auxiliary inputs used in the combinations

In total, there are 116 observables and 33 free parameters. The combination has a $\chi ^2$ value of 95.5, which corresponds to a global p-value of 0.164. The obtained world average for the Unitarity Triangle angle $\gamma \equiv \phi _3$ is

$$\begin{aligned} \gamma \equiv \phi _3 = (74.0\,^{+5.8}_{-6.4})^{\circ }. \end{aligned}$$

(166)

An ambiguous solution at $\gamma \equiv \phi _3 \longrightarrow \gamma \equiv \phi _3+\pi$ also exists. The results for the hadronic parameters are listed in Table 63. Results for input analyses as split by $B$ meson decay mode are shown in Table 64 and Fig. 45. Results for input analyses as split by the method are shown in Table 65 and Fig. 46. Results for the hadronic ratios, $r_{B}$, are shown in Fig. 47. A demonstration of how the various analyses contribute to the combination is shown in Fig. 48.

Table 63 Averages values obtained for the hadronic parameters in $B \rightarrow D^{(*)} K^{(*)}$ decays

5 Semileptonic B decays

This section contains our averages for semileptonic B meson decays, i.e. decays of the type $B\rightarrow X\ell \nu _\ell$, where X refers to one or more hadrons, $\ell$ to a charged lepton and $\nu _\ell$ to its associated neutrino. Unless otherwise stated, $\ell$ stands for an electron or a muon, lepton universality is assumed, and both charge conjugate states are combined. Some averages assume isospin symmetry, this will be explicitly mentioned at every instance.

Table 64 Averages of $\gamma \equiv \phi _3$ split by $B$ meson decay mode

Fig. 45

World average of $\gamma \equiv \phi _{3}$, in terms of 1−CL, split by decay mode

Averages are presented separately for CKM favored $b\rightarrow c$ quark transitions and CKM suppressed $b\rightarrow u$ transitions. Among these transitions we distinguish exclusive decays involving a specific meson ($X=D, D^*, \pi , \rho ,\ldots$) from inclusive decay modes, i.e. the sum over all possible hadronic states, one or more mesons and baryons. Semileptonic decays proceed via first order weak interactions and are well described in the framework of the standard model (SM). Their decay rates are sensitive to the magnitude squared of the CKM elements $V_{cb}$ and $V_{ub}$, and their determination is one of the primary goals for the study of these decays. Semileptonic decays involving the $\tau$ lepton might be sensitive to beyond SM processes because of the high $\tau$ mass, which might result in enhanced couplings to a hypothetical charged Higgs boson or leptoquarks.

The technique for obtaining the averages follows the general HFLAV procedure (Sect. 2) unless otherwise stated. More information on the averages, in particular on the common input parameters is available on the HFLAV semileptonic webpage.

Table 65 Averages of $\gamma \equiv \phi _3$ split by method. For GLW method only the solution nearest the combined average is shown

Fig. 46

World average of $\gamma \equiv \phi _{3}$, in terms of 1−CL, split by analysis method

Fig. 47

World averages for the hadronic parameters $r_{B}$ in the different decay modes, in terms of 1−CL

Fig. 48

Contributions to the combination from different input measurements, shown in the plane of the relevant $r_{B}$ parameter $vs.$ $\gamma \equiv \phi _3$. From left to right, top to bottom: $B^{+}\rightarrow D K^+$, $B^{+}\rightarrow D^* K^+$, $B^{+}\rightarrow D K^{*+}$ and $B^0 \rightarrow D K^{*0}$. Contours show the two-dimensional $68\,\%$ and $95\,\%$ CL regions

5.1 Exclusive CKM-favoured decays

5.1.1 $\bar{B}\rightarrow D^*\ell ^-{\bar{\nu }}_\ell$

The recoil variable w used to describe $\bar{B}\rightarrow D^*\ell ^-{\bar{\nu }}_\ell$ decays is the product of the four-velocities of the initial and final state mesons, $w=v_B\cdot v_{D^{(*)}}$. The differential decay rate for massless fermions as a function of w is given by (see, e.g., [476])

$$\begin{aligned} \frac{d\Gamma (\bar{B}\rightarrow D^*\ell ^-{\bar{\nu }}_\ell )}{dw}= & {} \frac{G^2_\mathrm {F} m^3_{D^*}}{48\pi ^3}(m_B-m_{D^*})^2\nonumber \\&\times \chi (w)\eta _\mathrm {EW}^2\mathcal {F}^2(w)|V_{cb}|^2, \end{aligned}$$

(167)

Table 66 Measurements of the Caprini, Lellouch and Neubert (CLN) [478] form factor parameters in $\bar{B}\rightarrow D^*\ell ^-{\bar{\nu }}_\ell$ before and after rescaling. Most analyses (except [483, 484]) measure only $\eta _\mathrm {EW}\mathcal{F}(1)|V_{cb}|$, and $\rho ^2$, so only these two parameters are shown here

where $G_\mathrm {F}$ is Fermi’s constant, $m_B$ and $m_{D^*}$ are the B and $D^*$ meson masses, $\chi (w)$ is a known expression of w and $\eta _\mathrm {EW}$ is a small electroweak correction [477]. Some authors also include a long-distance EM radiation effect (Coulomb correction) in this factor. The form factor $\mathcal {F}(w)$ for the $\bar{B}\rightarrow D^*\ell ^-{\bar{\nu }}_\ell$ decay contains three independent functions, $h_{A_1}(w)$, $R_1(w)$ and $R_2(w)$,

$$\begin{aligned} \chi (w)\mathcal {F}^2(w)= & {} h_{A_1}^2(w)\sqrt{w^2-1}(w+1)^2\nonumber \\&\times \bigg \{2\bigg [\frac{1-2wr+r^2}{(1-r)^2}\bigg ]\bigg [1+R^2_1(w)\frac{w^2-1}{w+1}\bigg ] \nonumber \\&+\bigg [1+(1-R_2(w))\frac{w-1}{1-r}\bigg ]^2\bigg \}, \end{aligned}$$

(168)

where $r=m_{D^*}/m_B$.

To extract $|V_{cb}|$, the experimental analyses we consider in this section use the parametrization of these form factor functions by Caprini, Lellouch and Neubert (CLN) [478],

$$\begin{aligned} h_{A_1}(w)= & {} h_{A_1}(1)\big [1-8\rho ^2z+(53\rho ^2-15)z^2\nonumber \\&-(231\rho ^2-91)z^3\big ], \end{aligned}$$

(169)

$$\begin{aligned}&R_1(w) = R_1(1)-0.12(w-1)+0.05(w-1)^2, \end{aligned}$$

(170)

$$\begin{aligned}&R_2(w) = R_2(1)+0.11(w-1)-0.06(w-1)^{2}, \end{aligned}$$

(171)

where $z=(\sqrt{w+1}-\sqrt{2})/(\sqrt{w+1}+\sqrt{2})$. The form factor $\mathcal{F}(w)$ is thus described by the slope $\rho ^2$ and the ratios $R_1(1)$ and $R_2(1)$.

We use the measurements of these form factor parameters shown in Table 66 and rescale them to the latest values of the input parameters (mainly branching fractions of charmed mesons) [479]. Most of the measurements in Table 66 are based on the decay $\bar{B}^0\rightarrow D^{*+}\ell ^-{\bar{\nu }}_\ell$. Some measurements [480, 481] are sensitive also to the $B^-\rightarrow D^{*0}\ell ^-{\bar{\nu }}_\ell$, and one measurement [482] is based on the decay $B^-\rightarrow D^{*0}\ell ^-{\bar{\nu }}_\ell$. Isospin symmetry is assumed in this average. The earlier results for the LEP experiments and CLEO have significantly rescaled results, and significantly larger uncertainties than the recent measurements by the B-factories Belle and BaBar.

In the next step, we perform a four parameter fit of $\eta _\mathrm {EW}\mathcal{F}(1)|V_{cb}|$, $\rho ^2$, $R_1(1)$ and $R_2(1)$ to the rescaled measurements, taking into account correlated statistical and systematic uncertainties. Only two measurements constrain all four parameters [483, 484], the remaining measurements determine only the normalization $\eta _\mathrm {EW}\mathcal{F}(1)|V_{cb}|$ and the slope $\rho ^2$. The result of the fit is

$$\begin{aligned}&\eta _\mathrm {EW}\mathcal{F}(1)|V_{cb}|= (35.61\,\pm \, 0.43)\times 10^{-3}, \end{aligned}$$

(172)

$$\begin{aligned}&\rho ^2 = 1.205\,\pm \, 0.026,\end{aligned}$$

(173)

$$\begin{aligned}&R_1(1) = 1.404\,\pm \, 0.032, \end{aligned}$$

(174)

$$\begin{aligned}&R_2(1) = 0.854\,\pm \, 0.020, \end{aligned}$$

(175)

and the correlation coefficients are

$$\begin{aligned}&\rho _{\eta _\mathrm {EW}\mathcal{F}(1)|V_{cb}|,\rho ^2} = 0.338,\end{aligned}$$

(176)

$$\begin{aligned}&\rho _{\eta _\mathrm {EW}\mathcal{F}(1)|V_{cb}|,R_1(1)} = -0.104,\end{aligned}$$

(177)

$$\begin{aligned}&\rho _{\eta _\mathrm {EW}\mathcal{F}(1)|V_{cb}|,R_2(1)} = -0.071,\end{aligned}$$

(178)

$$\begin{aligned}&\rho _{\rho ^2,R_1(1)} = 0.570,\end{aligned}$$

(179)

$$\begin{aligned}&\rho _{\rho ^2,R_2(1)} = -0.810,\end{aligned}$$

(180)

$$\begin{aligned}&\rho _{R_1(1),R_2(1)} = -0.758. \end{aligned}$$

(181)

The uncertainties and correlations quoted here include both statistical and systematic contributions. The $\chi ^2$ of the fit is 30.2 for 23 degrees of freedom, which corresponds to a confidence level of 14.4%. An illustration of this fit result is given in Fig. 49.

Fig. 49

Illustration of the a the average and b the dependence of $\eta _\mathrm {EW}\mathcal{F}(1)|V_{cb}|$ on $\rho ^2$. The error ellipses correspond to $\Delta \chi ^2 = 1$ (CL = 39%)

Using the latest update from the Fermilab Lattice and MILC Collaborations [489], the form factor normalization $\eta _\mathrm {EW}\mathcal{F}(1)$ is

$$\begin{aligned} \eta _\mathrm {EW}\mathcal{F}(1) = 0.912\,\pm \, 0.014, \end{aligned}$$

(182)

where $\eta _\mathrm {EW}=1.0066\,\pm \, 0.0050$ has been used. The central value of this number corresponds to the electroweak correction only. The uncertainty has been increased to accommodate the Coulomb effect. Based on Eq. (172), this results in

$$\begin{aligned} |V_{cb}|= (39.05\,\pm \, 0.47_\mathrm{exp}\,\pm \, 0.58_\mathrm{th})\times 10^{-3}, \end{aligned}$$

(183)

where the first uncertainty is experimental and the second error is theoretical (lattice QCD calculation and electro-weak correction).

Table 67 ${\overline{B}} ^0 \rightarrow D^{*+}\ell ^{-}\overline{\nu } _\ell$ branching fractions calculated from the rescaled CLN parameters in Table 66. For Ref. [482] the published value of ${\mathcal {B}}(B^-\rightarrow D^{*0}\ell ^-{\bar{\nu }}_\ell )$ has been rescaled by the factor $\tau (B^0)/\tau (B^+)$ for comparison to the other measurements

Table 68 Average of the $B^-\rightarrow D^{*0}\ell ^-{\bar{\nu }}_\ell$ branching fraction measurements

From each rescaled measurements in Table 66, we calculate the $\bar{B}\rightarrow D^*\ell ^-{\bar{\nu }}_\ell$ form factor $\eta _\mathrm {EW}\mathcal{F}(w)$ and, by numerical integration, the branching ratio of the decay $\bar{B}^0\rightarrow D^{*+}\ell ^-{\bar{\nu }}_\ell$. For measurements that do not determine the parameters $R_1(1)$ and $R_2(1)$ we assume the average values given in Eqs. (174) and (175). The results are quoted in Table 67. The branching ratio is

$$\begin{aligned} {\mathcal {B}}({\overline{B}} ^0 \rightarrow D^{*+}\ell ^{-}\overline{\nu } _\ell )=(4.88\,\pm \, 0.10)\%. \end{aligned}$$

(184)

We have also performed a one-dimensional average of measurements of the decay $B^-\rightarrow D^{*0}\ell ^-{\bar{\nu }}_\ell$, which is shown in Table 68. The result of this average, $(5.59\,\pm \, 0.02\,\pm \, 0.19)$%, is consistent with the average including both charge states given in Eq. (184) rescaled by the lifetime ratio $\tau (B^+)/\tau (B^0)$, $(5.26\,\pm \, 0.11)$% (Fig. 50).

Fig. 50

a Branching fractions of exclusive semileptonic B decays: a ${\overline{B}} ^0 \rightarrow D^{*+}\ell ^{-}\overline{\nu } _\ell$ (Table 67) and b $B^-\rightarrow D^{*0}\ell ^-{\bar{\nu }}_\ell$ (Table 68)

5.1.2 $\bar{B}\rightarrow D\ell ^-{\bar{\nu }}_\ell$

The differential decay rate for massless fermions as a function of w (introduced in the previous section) is given by (see, e.g., [476])

$$\begin{aligned} \frac{\bar{B}\rightarrow D\ell ^-{\bar{\nu }}_\ell }{dw}= & {} \frac{G^2_\mathrm {F} m^3_D}{48\pi ^3}(m_B+m_D)^2(w^2-1)^{3/2}\nonumber \\&\times \eta _\mathrm {EW}^2\mathcal {G}^2(w)|V_{cb}|^2, \end{aligned}$$

(185)

where $G_\mathrm {F}$ is Fermi’s constant, and $m_B$ and $m_D$ are the B and D meson masses. Again, $\eta _\mathrm {EW}$ is the electroweak correction introduced in the previous section. In contrast to $\bar{B}\rightarrow D^*\ell ^-{\bar{\nu }}_\ell$, $\mathcal {G}(w)$ contains a single form-factor function $f_+(w)$,

$$\begin{aligned} \mathcal {G}^2(w) = \frac{4r}{(1+r)^2} f^2_+(w), \end{aligned}$$

(186)

where $r=m_D/m_B$.

As for $\bar{B}\rightarrow D^*\ell ^-{\bar{\nu }}_\ell$ decays, we adopt the prescription by Caprini, Lellouch and Neubert [478], which describes the shape and normalization of the measured decay distributions in terms of two parameters: the normalization $\mathcal{G}(1)$ and the slope $\rho ^2$,

$$\begin{aligned} \mathcal {G}(z)= & {} \mathcal {G}(1)\big [1 - 8 \rho ^2 z + (51 \rho ^2 - 10 ) z^2 \nonumber \\&- (252 \rho ^2 - 84 ) z^3\big ], \end{aligned}$$

(187)

where $z=(\sqrt{w+1}-\sqrt{2})/(\sqrt{w+1}+\sqrt{2})$.

Table 69 shows experimental measurements of the two CLN parameters, which are corrected to match the latest values of the input parameters [479]. Both measurements of ${\overline{B}} ^0 \rightarrow D^{+}\ell ^{-}\overline{\nu } _\ell$ and $B^-\rightarrow D^0\ell ^-{\bar{\nu }}_\ell$ are used and isospin symmetry is assumed in the analysis.

Table 69 Measurements of the Caprini, Lellouch and Neubert (CLN) [478] form factor parameters in $\bar{B}\rightarrow D\ell ^-{\bar{\nu }}_\ell$ before and after rescaling

Fig. 51

Illustration of a the average and b dependence of $\eta _\mathrm {EW}\mathcal{G}(w)|V_{cb}|$ on $\rho ^2$. The error ellipses correspond to $\Delta \chi ^2 = 1$ (CL = 39%)

Table 70 $\bar{B}^0\rightarrow D^+\ell ^-{\bar{\nu }}_\ell$ branching fractions calculated from the rescaled CLN parameters in Table 69, which are based on both charged and neutral B decays, combined under the assumption of isospin symmetry

The form factor parameters are extracted by a two-parameter fit to the rescaled measurements of $\eta _\mathrm {EW}\mathcal{G}(1)|V_{cb}|$ and $\rho ^2$ taking into account correlated statistical and systematic uncertainties. The result of the fit is

$$\begin{aligned}&\eta _\mathrm {EW}\mathcal{G}(1)|V_{cb}|= (41.57\,\pm \, 1.00)\times 10^{-3}, \end{aligned}$$

(188)

$$\begin{aligned}&\rho ^2 = 1.128 \,\pm \, 0.033, \end{aligned}$$

(189)

with a correlation of

$$\begin{aligned} \rho _{\eta _\mathrm {EW}\mathcal{G}(1)|V_{cb}|,\rho ^2} = 0.751. \end{aligned}$$

(190)

The uncertainties and the correlation coefficient include both statistical and systematic contributions. The $\chi ^2$ of the fit is 4.7 for 8 degrees of freedom, which corresponds to a probability of 79.3%. An illustration of this fit result is given in Fig. 51.

The most recent lattice QCD result obtained for the form factor normalization is [494]

$$\begin{aligned} \mathcal{G}(1) = 1.0541\,\pm \, 0.0083. \end{aligned}$$

(191)

Using again $\eta _\mathrm {EW}=1.0066\,\pm \, 0.0050$, we determine $|V_{cb}|$ from Eq. (188),

$$\begin{aligned} |V_{cb}|= (39.18\,\pm \, 0.94_\mathrm{exp}\,\pm \, 0.36_\mathrm{th})\times 10^{-3}, \end{aligned}$$

(192)

where the first error is experimental and the second theoretical. This number is in excellent agreement with $|V_{cb}|$ obtained from $\bar{B}\rightarrow D^*\ell ^-{\bar{\nu }}_\ell$ decays given in Eq. (183).

From each rescaled measurement in Table 69, we have calculated the $\bar{B}\rightarrow D\ell ^-{\bar{\nu }}_\ell$ form factor $\mathcal{G}(w)$ and, by numerical integration, the branching ratio of the decay ${\overline{B}} ^0 \rightarrow D^{+}\ell ^{-}\overline{\nu } _\ell$. The results are quoted in Table 70 and illustrated in Fig. 52. The branching ratio for the average values of $\eta _\mathrm {EW}\mathcal{G}(1)|V_{cb}|$ and $\rho ^2$ is

$$\begin{aligned} {\mathcal {B}}({\overline{B}} ^0 \rightarrow D^{+}\ell ^{-}\overline{\nu } _\ell )=(2.13\,\pm \, 0.07)\%. \end{aligned}$$

(193)

Fig. 52

Illustration of Table 70

Table 71 Average of ${\overline{B}} ^0 \rightarrow D^{+}\ell ^{-}\overline{\nu } _\ell$ branching fraction measurements

We have also performed one-dimensional averages of measurements of ${\overline{B}} ^0 \rightarrow D^{+}\ell ^{-}\overline{\nu } _\ell$ and $B^-\rightarrow D^0\ell ^-{\bar{\nu }}_\ell$ decays. The results are shown in Tables 71 and 72. The ${\overline{B}} ^0 \rightarrow D^{+}\ell ^{-}\overline{\nu } _\ell$ average, $(2.20\,\pm \, 0.04\,\pm \, 0.09)$%, is consistent with the result in Eq. (193), $(2.13\,\pm \, 0.07)$%. The $B^-\rightarrow D^0\ell ^-{\bar{\nu }}_\ell$ average, $(2.33\,\pm \, 0.04\,\pm \, 0.09)$%, also compares well to the result in Eq. (193), rescaled by the lifetime ratio $\tau (B^+)/\tau (B^0)$, $(2.30\,\pm \, 0.07)$%.

Table 72 Average of $B^-\rightarrow D^0\ell ^-{\bar{\nu }}_\ell$ branching fraction measurements

Table 73 Averages of the $B \rightarrow D^{(*)} \pi ^- \ell ^-{\bar{\nu }}_{\ell }$ branching fractions and individual results

5.1.3 ${\bar{B}} \rightarrow D^{(*)}\pi \ell ^-{\bar{\nu }}_{\ell }$

The average inclusive branching fractions for ${\bar{B}} \rightarrow D^{(*)}\pi \ell ^-{\bar{\nu }}_{\ell }$ decays, where no constraint is applied to the $D^{(*)}\pi$ system, are determined by the combination of the results provided in Table 73 for ${\bar{B}}^0 \rightarrow D^0 \pi ^+ \ell ^-{\bar{\nu }}_{\ell }$, ${\bar{B}}^0 \rightarrow D^{*0} \pi ^+ \ell ^-{\bar{\nu }}_{\ell }$, $B^- \rightarrow D^+ \pi ^- \ell ^-{\bar{\nu }}_{\ell }$, and $B^- \rightarrow D^{*+} \pi ^- \ell ^-{\bar{\nu }}_{\ell }$ decays. The measurements included in the average are scaled to a consistent set of input parameters and their uncertainties [479]. For both the BaBar and Belle results, the B semileptonic signal yields are extracted from a fit to the missing mass squared distribution for a sample of fully reconstructed $B{\overline{B}}$ events. Figure 53 shows the measurements and the resulting average for the four decay modes.

5.1.4 ${\bar{B}} \rightarrow D^{**} \ell ^-{\bar{\nu }}_{\ell }$

$D^{**}$ mesons contain one charm quark and one light anti-quark with relative angular momentum $L=1$. According to Heavy Quark Symmetry (HQS) [496], they form one doublet of states with angular momentum $j \equiv s_q + L= 3/2$ $\left[ D_1(2420), D_2^*(2460)\right]$ and another doublet with $j=1/2$ $\left[ D^*_0(2400), D_1'(2430)\right]$, where $s_q$ is the light quark spin. Parity and angular momentum conservation constrain the decays allowed for each state. The $D_1$ and $D_2^*$ states decay via a D-wave to $D^*\pi$ and $D^{(*)}\pi$, respectively, and have small decay widths, while the $D_0^*$ and $D_1'$ states decay via an S-wave to $D\pi$ and $D^*\pi$ and are very broad. For the narrow states, the averages are determined by the combination of the results provided in Tables 74 and 75 for ${\mathcal {B}}(B^- \rightarrow D_1^0\ell ^-{\bar{\nu }}_{\ell }) \times {\mathcal {B}}(D_1^0 \rightarrow D^{*+}\pi ^-)$ and ${\mathcal {B}}(B^- \rightarrow D_2^0\ell ^-{\bar{\nu }}_{\ell }) \times {\mathcal {B}}(D_2^0 \rightarrow D^{*+}\pi ^-)$. For the broad states, the averages are determined by the combination of the results provided in Tables 76 and 77 for ${\mathcal {B}}(B^- \rightarrow D_1'^0\ell ^-{\bar{\nu }}_{\ell }) \times {\mathcal {B}}(D_1'^0 \rightarrow D^{*+}\pi ^-)$ and ${\mathcal {B}}(B^- \rightarrow D_0^{*0}\ell ^-{\bar{\nu }}_{\ell }) \times {\mathcal {B}}(D_0^{*0} \rightarrow D^{+}\pi ^-)$. The measurements are scaled to a consistent set of input parameters and their uncertainties [479].

Fig. 53

Average branching fraction of exclusive semileptonic B decays a ${\bar{B}}^0 \rightarrow D^0 \pi ^+ \ell ^-{\bar{\nu }}_{\ell }$, b ${\bar{B}}^0 \rightarrow D^{*0} \pi ^+ \ell ^-{\bar{\nu }}_{\ell }$, c $B^- \rightarrow D^+ \pi ^- \ell ^-{\bar{\nu }}_{\ell }$, and d $B^- \rightarrow D^{*+} \pi ^- \ell ^-{\bar{\nu }}_{\ell }$. The corresponding individual results are also shown

For both the B-factory and the LEP and Tevatron results, the B semileptonic signal yields are extracted from a fit to the invariant mass distribution of the $D^{(*)+}\pi ^-$ system. Apart for the CLEO, Belle and BaBar results, the other measurements are for the ${\bar{B}} \rightarrow D^{**}(D^*\pi ^-)X \ell ^- {\bar{\nu }}_{\ell }$ final state and we assume that no particles are left in the X system. The BaBar tagged ${\bar{B}} \rightarrow D_2^* \ell ^- {\bar{\nu }}_{\ell }$ has been measured selecting $D_s^*\rightarrow D\pi$ final state and it has been translated in a result on $D_2^*\rightarrow D^*\pi$ decay mode, assuming ${\mathcal {B}}(D_2^*\rightarrow D\pi )/{\mathcal {B}}(D_2^*\rightarrow D^*\pi )=1.54\,\pm \, 0.15$ [327]. Figures 54 and  55 show the measurements and the resulting averages.

5.2 Inclusive CKM-favored decays

5.2.1 Global analysis of $\bar{B}\rightarrow X_c\ell ^-{\bar{\nu }}_\ell$

The semileptonic decay width $\Gamma (\bar{B}\rightarrow X_c\ell ^-{\bar{\nu }}_\ell )$ has been calculated in the framework of the operator production expansion (OPE) [55,56,57]. The result is a double-expansion in $\varLambda _\mathrm{QCD}/m_b$ and $\alpha _s$, which depends on a number of non-perturbative parameters. These parameters describe the dynamics of the b-quark inside the B hadron and can be measured using observables in $\bar{B}\rightarrow X_c\ell ^-{\bar{\nu }}_\ell$ decays, such as the moments of the lepton energy and the hadronic mass spectrum.

Two renormalization schemes are commonly used to defined the b-quark mass and other theoretical quantities: the kinetic [504,505,506,507] and the 1S [508] schemes. An independent set of theoretical expressions is available for each, with several non-perturbative parameters. The non-perturbative parameters in the kinetic scheme are: the quark masses $m_b$ and $m_c$, $\mu ^2_\pi$ and $\mu ^2_G$ at $O(1/m^2_b)$, and $\rho ^3_D$ and $\rho ^3_{LS}$ at $O(1/m^3_b)$. In the 1S scheme, the parameters are: $m_b$, $\lambda _1$ at $O(1/m^2_b)$, and $\rho _1$, $\tau _1$, $\tau _2$ and $\tau _3$ at $O(1/m^3_b)$. Note that the numerical values of the kinetic and 1S b-quark masses cannot be compared without converting one or the other, or both, to the same renormalization scheme.

Table 74 Published and rescaled individual measurements and their averages for of the branching fraction ${\mathcal {B}}(B^- \rightarrow D_1^0\ell ^-{\bar{\nu }}_{\ell })\times {\mathcal {B}}(D_1^0 \rightarrow D^{*+}\pi ^-)$

Table 75 Published and rescaled individual measurements and their averages for ${\mathcal {B}}(B^- \rightarrow D_2^0\ell ^-{\bar{\nu }}_{\ell })\times {\mathcal {B}}(D_2^0 \rightarrow D^{*+}\pi ^-)$

Table 76 Published and rescaled individual measurements and their averages for ${\mathcal {B}}(B^- \rightarrow D_1^{'0}\ell ^-{\bar{\nu }}_{\ell })\times {\mathcal {B}}(D_1^{'0} \rightarrow D^{*+}\pi ^-)$

Table 77 Published and rescaled individual measurements and their averages for ${\mathcal {B}}(B^- \rightarrow D_0^{*0}\ell ^-{\bar{\nu }}_{\ell })\times {\mathcal {B}}(D_0^{*0} \rightarrow D^{+}\pi ^-)$

We used two kinematic distributions for $\bar{B}\rightarrow X_c\ell ^-{\bar{\nu }}_\ell$ decays, the hadron effective mass to derive moments $\langle M^n_X\rangle$ of order $n=2,4,6$, and the charged lepton momentum to derive moments $\langle E^n_\ell \rangle$ of order $n=0,1,2,3$. Moments are determined for different values of $E_\mathrm {cut}$, the lower limit on the minimum lepton momentum. The moments derived from the same distributions with different value of $E_\mathrm {cut}$ are highly correlated. The list of measurements is given in Table 78. The only input is the average lifetime $\tau _B$ of neutral and charged B mesons, taken to be $(1.579\,\pm \, 0.004)$ ps (Sect. 3).

In the kinetic and 1S schemes, the moments in $\bar{B}\rightarrow X_c\ell ^-{\bar{\nu }}_\ell$ are not sufficient to determine the b-quark mass precisely. In the kinetic scheme analysis we constrain the c-quark mass (defined in the $\overline{\mathrm{MS}}$ scheme) to the value of Ref. [515],

$$\begin{aligned} m_c^{\overline{\mathrm{MS}}}(3~\mathrm{GeV})=0.986\,\pm \, 0.013~\mathrm{GeV}. \end{aligned}$$

(194)

In the 1S scheme analysis, the b-quark mass is constrained by measurements of the photon energy moments in $B\rightarrow X_s\gamma$ [516,517,518,519].

Fig. 54

Rescaled individual measurements and their averages for a ${\mathcal {B}}(B^- \rightarrow D_1^0\ell ^-{\bar{\nu }}_{\ell }) \times {\mathcal {B}}(D_1^0 \rightarrow D^{*+}\pi ^-)$ and b ${\mathcal {B}}(B^- \rightarrow D_2^0\ell ^-{\bar{\nu }}_{\ell }) \times {\mathcal {B}}(D_2^0 \rightarrow D^{*+}\pi ^-)$

Fig. 55

Rescaled individual measurements and their averages for a ${\mathcal {B}}(B^- \rightarrow D_1'^0\ell ^-{\bar{\nu }}_{\ell }) \times {\mathcal {B}}(D_1'^0 \rightarrow D^{*+}\pi ^-)$ and b ${\mathcal {B}}(B^- \rightarrow D_0^{*0}\ell ^-{\bar{\nu }}_{\ell }) \times {\mathcal {B}}(D_0^{*0} \rightarrow D^{+}\pi ^-)$

5.2.2 Analysis in the kinetic scheme

The fit relies on the calculations of the lepton energy and hadron mass moments in $\bar{B}\rightarrow X_c\ell ^-{\bar{\nu }}_\ell$ decays described in Refs. [506, 507] and closely follows the procedure of Ref. [520]. The analysis determines $|V_{cb}|$ and the six non-perturbative parameters mentioned above.

Table 78 Experimental inputs used in the global analysis of $\bar{B}\rightarrow X_c\ell ^-{\bar{\nu }}_\ell$. n is the order of the moment, c is the threshold value of the lepton momentum in GeV. In total, there are 23 measurements from BaBar, 15 measurements from Belle and 12 from other experiments

The detailed fit result and the matrix of the correlation coefficients is given in Table 79. The fit to the lepton energy and hadronic mass moments is shown in Figs. 56 and 57, respectively. The result in terms of the main parameters is

$$\begin{aligned}&|V_{cb}|= (42.19\,\pm \, 0.78)\times 10^{-3}, \end{aligned}$$

(195)

$$\begin{aligned}&m_b^\mathrm{kin} = 4.554\,\pm \, 0.018~\mathrm{GeV}, \end{aligned}$$

(196)

$$\begin{aligned}&\mu ^2_\pi = 0.464\,\pm \, 0.076~\mathrm{GeV^2}, \end{aligned}$$

(197)

with a $\chi ^2$ of 15.6 for 43 degrees of freedom. The scale $\mu$ of the quantities in the kinematic scheme is 1 GeV.

Table 79 Fit result in the kinetic scheme, using a precise c-quark mass constraint. The error matrix of the fit contains experimental and theoretical contributions. In the lower part of the table, the correlation matrix of the parameters is given. The scale $\mu$ of the quantities in the kinematic scheme is 1 GeV

Fig. 56

Fit to the inclusive partial semileptonic branching ratios and to the lepton energy moments in the kinetic mass scheme. In all plots, the grey band is the theory prediction with total theory error. BaBar data are shown by circles, Belle by squares and other experiments (DELPHI, CDF, CLEO) by triangles. Filled symbols mean that the point was used in the fit. Open symbols are measurements that were not used in the fit

Fig. 57

Same as Fig. 56 for the fit to the hadronic mass moments in the kinetic mass scheme

The inclusive $\bar{B}\rightarrow X_c\ell ^-{\bar{\nu }}_\ell$ branching fraction determined by this analysis is

$$\begin{aligned} {\mathcal {B}}(\bar{B}\rightarrow X_c\ell ^-{\bar{\nu }}_\ell )=(10.65\,\pm \, 0.16)\%. \end{aligned}$$

(198)

Including the rate of charmless semileptonic decays (Sect. 5.4), ${\mathcal {B}}(\bar{B}\rightarrow X_u\ell ^-{\bar{\nu }}_\ell )=(2.13\,\pm \, 0.31)\times 10^{-3}$, we obtain the semileptonic branching fraction,

$$\begin{aligned} {\mathcal {B}}(\bar{B}\rightarrow X\ell ^-{\bar{\nu }}_\ell )=(10.86\,\pm \, 0.16)\%. \end{aligned}$$

(199)

5.2.3 Analysis in the 1S scheme

The fit relies on the same set of moment measurements and the calculations of the spectral moments described in Ref. [508]. The theoretical uncertainties are estimated as explained in Ref. [521]. Only trivial theory correlations, i.e. between the same moment at the same threshold are included in the analysis. The fit determines $|V_{cb}|$ and the six non-perturbative parameters mentioned above.

The detailed result of the fit using the $B\rightarrow X_s\gamma$ constraint is given in Table 80. The result in terms of the main parameters is

$$\begin{aligned}&|V_{cb}|= (41.98\,\pm \, 0.45)\times 10^{-3}, \end{aligned}$$

(200)

$$\begin{aligned}&m_b^{1S} = 4.691\,\pm \, 0.037~\mathrm{GeV}, \end{aligned}$$

(201)

$$\begin{aligned}&\lambda _1 = -0.362\,\pm \, 0.067~\mathrm{GeV^2}, \end{aligned}$$

(202)

with a $\chi ^2$ of 23.0 for 59 degrees of freedom. We find a good agreement in the central values of $|V_{cb}|$ between the kinetic and 1S scheme analyses. No conclusion should, however, been drawn regarding the uncertainties in $|V_{cb}|$ as the two approaches are not equivalent in the number of higher-order corrections included.

5.3 Exclusive CKM-suppressed decays

In this section, we give results on exclusive charmless semileptonic branching fractions and the determination of $|V_{ub}|$ based on $B \rightarrow \pi l\nu$ decays. The measurements are based on two different event selections: tagged events, in which the second B meson in the event is fully (or partially) reconstructed, and untagged events, for which the momentum of the undetected neutrino is inferred from measurements of the total momentum sum of the detected particles and the knowledge of the initial state. The LHCb experiment recently reported a direct measurement of $|V_{ub}|/|V_{cb}|$ [522] reconstructing the $\varLambda _b^0 \rightarrow p\mu \nu$ decays and normalizing the branching fraction to the $\varLambda _b^0 \rightarrow \varLambda _c^+ (\rightarrow pK\pi )\mu \nu$ decays. We show a combination of $|V_{ub}|-|V_{cb}|$ using the LHCb constraint on $|V_{ub}|/|V_{cb}|$, the exclusive determination of $|V_{ub}|$ from $B \rightarrow \pi l\nu$ and $|V_{cb}|$ from both $B\rightarrow D^*\ell \nu$ and $B\rightarrow D\ell \nu$. We also present branching fraction averages for $B^{0}\rightarrow \rho \ell ^+\nu$, $B^{+}\rightarrow \omega \ell ^+\nu$, $B^{+}\rightarrow \eta \ell ^+\nu$ and $B^{+}\rightarrow \eta ^{\prime } \ell ^+\nu$.

5.3.1 $B \rightarrow \pi l\nu$ branching fraction and $q^2$ spectrum

Currently, the four most precise measurements of the differential $B \rightarrow \pi l\nu$ decay rate as a function of the four-momentum transfer squared,  $q^2$, from BaBar and Belle [523,524,525,526] are used to obtain an average $q^2$ spectrum and an average for the total branching fraction. The measurements are presented in Fig. 58. From the two untagged BaBar analyses [525, 526], the combined results for $B^0 \rightarrow \pi ^- \ell ^+ \nu$ and $B^+ \rightarrow \pi ^0 \ell ^+ \nu$ decays based on isospin symmetry are used. The hadronic-tag analysis by Belle [524] provides results for $B^0 \rightarrow \pi ^- \ell ^+ \nu$ and $B^+ \rightarrow \pi ^0 \ell ^+ \nu$ separately, but not for the combination of both channels. In the untagged analysis by Belle [523], only $B^0 \rightarrow \pi ^- \ell ^+ \nu$ decays were measured. The experimental measurements use different binnings in $q^2$, but have matching bin edges, which allows them to be easily combined.

To arrive at an average $q^2$ spectrum, a binned maximum-likelihood fit to determine the average partial branching fraction in each $q^2$ interval is performed differentiating between common and individual uncertainties and correlations for the various measurements. Shared sources of systematic uncertainty of all measurements are included in the likelihood as nuisance parameters constrained using standard normal distributions. The most important shared sources of uncertainty are due to continuum subtraction, branching fractions, the number of B-meson pairs (only correlated among measurement by the same experiment), tracking efficiency (only correlated among measurements by the same experiment), uncertainties from modelling the $b \rightarrow u \, \ell \, {\bar{\nu }}_\ell$ contamination, modelling of final state radiation, and contamination from $b \rightarrow c \, \ell \bar{\nu }_\ell$ decays.

The averaged $q^2$ spectrum is shown in Fig. 58. The probability of the average is computed as the $\chi ^2$ probability quantifying the agreement between the input spectra and the averaged spectrum and amounts to $6\%$. The partial branching fractions and the full covariance matrix obtained from the likelihood fit are given in Tables 81 and  82. The average for the total $B^0 \rightarrow \pi ^- \ell ^+ \nu _\ell$ branching fraction is obtained by summing up the partial branching fractions:

$$\begin{aligned} {\mathcal {B}}(B^0 \rightarrow \pi ^- \ell ^+ \nu _\ell ) = (1.50 \,\pm \, 0.02_\mathrm{stat} \,\pm \, 0.06_\mathrm{syst}) \times 10^{-4}.\!\! \end{aligned}$$

(203)

Table 80 Fit result in the 1S scheme, using $B\rightarrow X_s\gamma$ moments as a constraint. In the lower part of the table, the correlation matrix of the parameters is given

5.3.2 $|V_{ub}|$ from $B \rightarrow \pi l\nu$

The $|V_{ub}|$ average can be determined from the averaged $q^2$ spectrum in combination with a prediction for the normalization of the $B \rightarrow \pi$ form factor. The differential decay rate for light leptons (e, $\mu$) is given by

$$\begin{aligned} \Delta \Gamma&= \Delta \Gamma (q^2_\mathrm{low}, q^2_\mathrm{high})\nonumber \\&= \int _{q^2_\mathrm{low}}^{q^2_\mathrm{high}} \text {d} q^2 \left[ \frac{8 | \vec p_\pi | }{3} \frac{ G_F^2 \, | V_{ub} |^2 q^2 }{256 \, \pi ^3 \, m_B^2} H_0^2(q^2) \right] , \end{aligned}$$

(204)

where $G_F$ is Fermi’s constant, $| \vec p_\pi |$ is the absolute four-momentum of the final state $\pi$ (a function of $q^2$), $m_B$ the $B^0$-meson mass, and $H_0(q^2)$ the only non-zero helicity amplitude. The helicity amplitude is a function of the form factor $f_+$,

$$\begin{aligned} H_0 = \frac{2 m_B \, | \vec p_\pi | }{\sqrt{q^2}}, f_+(q^2). \end{aligned}$$

(205)

The form factor $f_{+}$ can be calculated with non-perturbative methods, but its general form can be constrained by the differential $B \rightarrow \pi l\nu$ spectrum. Here, we parametrize the form factor using the BCL parametrization [527].

The decay rate is proportional to $|V_{ub}| ^2 |f_+(q^2)|^2$. Thus to extract $|V_{ub}|$ one needs to determine $f_+(q^2)$ (at least at one value of $q^2$). In order to enhance the precision, a binned $\chi ^2$ fit is performed using a $\chi ^2$ function of the form

$$\begin{aligned} \chi ^2&= \bigg ( \vec {\mathcal {B}} - \Delta \vec {\Gamma } \, \tau \bigg )^T C^{-1} \bigg ( \vec {\mathcal {B}} - \Delta \vec {\Gamma } \, \tau \bigg ) + \chi ^2_\mathrm{LQCD} + \chi ^2_\mathrm{LCSR} \end{aligned}$$

(206)

Fig. 58

The $B \rightarrow \pi l\nu$ $q^2$ spectrum measurements and the average spectrum obtained from the likelihood combination (shown in black)

Table 81 Partial $B^0 \rightarrow \pi ^- \ell ^+ \nu _\ell$ branching fractions per GeV$^2$ for the input measurements and the average obtained from the likelihood fit. The uncertainties are the combined statistical and systematic uncertainties

with C denoting the covariance matrix given in Table 82, $\vec {\mathcal {B}}$ the vector of averaged branching fractions and $\Delta \vec {\Gamma } \, \tau$ the product of the vector of theoretical predictions of the partial decay rates and the $B^0$-meson lifetime. The form factor normalization is included in the fit by the two extra terms in Eq. (206): $\chi _\mathrm{LQCD}$ uses the latest FLAG lattice average [222] from two state-of-the-art unquenched lattice QCD calculations [528, 529]. The resulting constraints are quoted directly in terms of the coefficients $b_j$ of the BCL parameterization and enter Eq. (206) as

$$\begin{aligned} \chi ^2_\mathrm{LQCD}&= \bigg ( {\vec {b}} - {\vec {b}_\mathrm{LQCD}} \bigg )^T \, C_\mathrm{LQCD}^{-1} \, \bigg ( {\vec {b}} - {\vec {b}_\mathrm{LQCD}} \bigg ), \end{aligned}$$

(207)

Table 82 Covariance matrix of the averaged partial branching fractions per GeV$^2$ in units of $10^{-14}$

with ${\vec {b}}$ the vector containing the free parameters of the $\chi ^2$ fit constraining the form factor, ${\vec {b}_\mathrm{LQCD}}$ the averaged values from Ref. [222], and $C_\mathrm{LQCD}$ their covariance matrix. Additional information about the form factor can be obtained from light-cone sum rule calculations. The state-of-the-art calculation includes up to two-loop contributions [530]. It is included in Eq. (206) via

$$\begin{aligned} \chi ^2_\mathrm{LQCR}&= \bigg ( f_+^\mathrm{LCSR} - f_+(q^2 = 0; {\vec {b}}) \bigg )^2 / \sigma _{f_+^\mathrm{LCSR} }^2. \end{aligned}$$

(208)

The $|V_{ub}|$ average is obtained for two versions: the first combines the data with the LQCD constraints and the second additionally includes the information from the LCSR calculation. The resulting values for $|V_{ub}|$ are

$$\begin{aligned} |V_{ub}|&= \bigg ( 3.70 \,\pm \, 0.10 \, (\text {exp}) \,\pm \, 0.12 \, (\text {theo}) \bigg )\nonumber \\&\quad \times 10^{-3} \, \mathrm (data+LQCD), \end{aligned}$$

(209)

$$\begin{aligned} |V_{ub}|&= \bigg ( 3.67 \,\pm \, 0.09 \, (\text {exp}) \,\pm \, 0.12 \, (\text {theo}) \bigg ) \nonumber \\&\quad \times 10^{-3} \, \mathrm (data+LQCD+LCSR), \end{aligned}$$

(210)

for the first and second fit version, respectively. The result of the fit including both LQCD and LCSR is shown in Fig. 59. The $\chi ^2$ probability of the fit is $47\%$. We quote the result of the fit including both LQCD and LCSR calculations as our average for $|V_{ub}|$. The best fit values for $|V_{ub}|$ and the BCL parameters and their covariance matrix are given in Tables 83 and  84.

5.3.3 Combined extraction of $|V_{ub}|$ and $|V_{cb}|$

The LHCb experiment reported the first observation of the CKM suppressed decay $\varLambda _b^0 \rightarrow p\mu \nu$ [522] and the measurement of the ratio of partial branching fractions at high $q^2$ for $\varLambda _b^0 \rightarrow p\mu \nu$ and $\varLambda _b^0 \rightarrow \varLambda _c^+ (\rightarrow pK\pi )\mu \nu$ decays

$$\begin{aligned} R&= \dfrac{{\mathcal {B}}(\varLambda _b^0 \rightarrow p\mu \nu )_{q^2>15~GeV^2} }{{\mathcal {B}}(\varLambda _b^0 \rightarrow \varLambda _c^+ \mu \nu )_{q^2>7~GeV^2} }\nonumber \\&=(1.00\,\pm \, 0.04\,\pm \, 0.08)\times 10^{-2}. \end{aligned}$$

(211)

Fig. 59

Fit of the BCL parametrization to the averaged $q^2$ spectrum from BaBar and Belle and the LQCD and LCSR calculations. The error bands represent the $1~\sigma$ (dark green) and $2~\sigma$ (light green) uncertainties of the fitted spectrum

Table 83 Best fit values and uncertainties for the combined fit to data, LQCD and LCSR results

Table 84 Covariance matrix for the combined fit to data, LQCD and LCSR results

The ratio R is proportional to $(|V_{ub}|/|V_{cb}|)^2$ and sensitive to the form factors of $\varLambda _b^0 \rightarrow p$ and $\varLambda _b^0 \rightarrow \varLambda _c^+$ transitions that have to be computed with non-perturbative methods, like lattice QCD. The uncertainty on ${\mathcal {B}}(\varLambda _c^+ \rightarrow p K \pi )$ is the largest source of systematic uncertainties on R. Using the recent HFLAV average ${\mathcal {B}}(\varLambda _c^+ \rightarrow p K \pi )=(6.46\,\pm \, 0.24)\%$ reported in Table 296, which includes the recent BESIII measurements  [531], the rescaled value for R is

$$\begin{aligned} R = (0.95\,\pm \, 0.04\,\pm \, 0.07)\times 10^{-2} \end{aligned}$$

(212)

With the precise lattice QCD prediction [532] of the form factors in the experimentally interesting $q^2$ region considered, results in

$$\begin{aligned} \dfrac{|V_{ub}|}{|V_{cb}|} = 0.080\,\pm \, 0.004_{Exp.} \,\pm \, 0.004_{F.F.} \end{aligned}$$

(213)

where the first uncertainty is the total experimental error and the second one is due to the knowledge of the form factors. A combined fit for $|V_{ub}|$ and $|V_{cb}|$ that includes the constraint from LHCb, and the determination of $|V_{ub}|$ and $|V_{cb}|$ from exclusive B meson decays, results in

$$\begin{aligned}&| V_{ub} | = ( 3.50 \,\pm \, 0.13 ) \times 10^{-3}\, \end{aligned}$$

(214)

$$\begin{aligned}&| V_{cb} | = ( 39.13 \,\pm \, 0.59 ) \times 10^{-3} \, \end{aligned}$$

(215)

$$\begin{aligned}&\rho (|V_{ub}| |,|V_{cb}|) =0.14 \end{aligned}$$

(216)

where the uncertainties are considered uncorrelated. The $\chi ^2$ of the fit is 4.4 for 2 d.o.f corresponding to a $P(\chi ^2)$ of 11.0%. The fit result is shown in Fig. 60, where both the $\Delta \chi ^2$ and the two-dimensional $68\%$ C.L. contours are indicated. The $|V_{ub}|/|V_{cb}|$ value extracted from R is more compatible with the exclusive determinations of $|V_{ub}|$. Another recent calculation, by Faustov and Galkin [533], based on a relativistic quark model, gives a value of $|V_{ub}|/|V_{cb}|$ closer to the inclusive determinations.

5.3.4 Other exclusive charmless semileptonic $B$ decays

We report the branching fraction average for $B^{0}\rightarrow \rho \ell ^+\nu$, $B^{+}\rightarrow \omega \ell ^+\nu$, $B^{+}\rightarrow \eta \ell ^+\nu$ and $B^{+}\rightarrow \eta ^{\prime } \ell ^+\nu$ decays. The measurements and their averages are listed in Tables 85, 86, 87, 88, and presented in Figs.  61 and  62. In the $B^{0}\rightarrow \rho ^-\ell ^+\nu$ average, both the $B^{0}\rightarrow \rho ^-\ell ^+\nu$ and $B^{+}\rightarrow \rho ^0\ell ^+\nu$ decays are used, where the $B^{+}\rightarrow \rho ^0\ell ^+\nu$ are rescaled by $2\tau _{B^0}/\tau _{B^+}$ assuming the isospin symmetry. For $B^{+}\rightarrow \omega \ell ^+\nu$ and $B^{+}\rightarrow \eta \ell ^+\nu$ decays, the agreement between the different measurements is good. $B^{+}\rightarrow \eta ^{\prime } \ell ^+\nu$ shows a discrepancy between the old CLEO measurement and the BaBar untagged analysis, but the statistical uncertainties of the CLEO measurement are large. The $B^{0}\rightarrow \rho \ell ^+\nu$ results, instead, show significant differences, in particular the BaBar untagged analysis gives a branching fraction significantly lower (by about 2$\sigma$) that the Belle measurement based on the hadronic-tag. A possible reason for such discrepancy could be the broad nature of the $\rho$ resonance that makes the control of the background under the $\rho$ mass peak more difficult in the untagged analysis than in the hadronic-tag analysis.

We do not report $|V_{ub}|$  for these exclusive charmless decays, because the form factor calculations have not yet reached the precision achieved for $B\rightarrow \pi \ell \nu$ decays. Unquenched lattice QCD calculations of the form factors are not available for these decays, but LCSR calculations exist for all these decay modes. The most recent of these calculations for the $B\rightarrow \rho \ell \nu$ and $B\rightarrow \omega \ell \nu$ decays are reported in Refs. [534, 535].

Fig. 60

$|V_{ub}|$-$|V_{cb}|$ combined average including the LHCb measurement of $|V_{ub}|/|V_{cb}|$, the exclusive $|V_{ub}|$ measurement from $B \rightarrow \pi l\nu$, and $|V_{cb}|$ measurements from both $B\rightarrow D^*\ell \nu$ and $B\rightarrow D\ell \nu$. The point with the error bars corresponds to the inclusive $|V_{cb}|$ from the kinetic scheme Sect. 5.2.2, and the inclusive $|V_{ub}|$ from GGOU calculation Sect. 5.4.3

Table 85 Summary of exclusive determinations of $B^{0}\rightarrow \rho \ell ^+\nu$. The errors quoted correspond to statistical and systematic uncertainties, respectively

Table 86 Summary of exclusive determinations of $B^{+}\rightarrow \omega \ell ^+\nu$. The errors quoted correspond to statistical and systematic uncertainties, respectively

Table 87 Summary of exclusive determinations of $B^{+}\rightarrow \eta \ell ^+\nu$. The errors quoted correspond to statistical and systematic uncertainties, respectively

Table 88 Summary of exclusive determinations of $B^{+}\rightarrow \eta '\ell ^+\nu$. The errors quoted correspond to statistical and systematic uncertainties, respectively

5.4 Inclusive CKM-suppressed decays

Measurements of $B \rightarrow X_u \ell ^+ \nu$ decays are very challenging because of the fifty times larger rates Cabibbo-favoured $B \rightarrow X_c \ell ^+ \nu$ decays. Cuts designed to suppress this dominant background severely complicate the perturbative QCD calculations required to extract $|V_{ub}|$. For strict phase space limitations, parameterizations of the so-called shape functions are required to describe the unmeasured regions of the phase space. In this update, we use several theoretical calculations to extract $|V_{ub}|$  and do not advocate the use of one method over another. The authors of the different calculations have provided codes to compute the partial rates in limited regions of phase space covered by the measurements. Latest results by Belle [545] and BaBar [546] explore large portions of phase space, with a consequent reduction of the theoretical uncertainties.

Fig. 61

a Summary of exclusive determinations of ${\mathcal {B}}(B^{0}\rightarrow \rho \ell ^+\nu )$ and their average. Measurements of $B^+ \rightarrow \rho ^0\ell ^+\nu$ branching fractions have been multiplied by $2\tau _{B^0}/\tau _{B^+}$ in accordance with isospin symmetry. b Summary of exclusive determinations of $B^{+}\rightarrow \omega \ell ^+\nu$ and their average

Fig. 62

a Summary of exclusive determinations of ${\mathcal {B}}(B^{+}\rightarrow \eta \ell ^+\nu )$ and their average. b Summary of exclusive determinations of ${\mathcal {B}}(B^{+}\rightarrow \eta ^{\prime } \ell ^+\nu )$ and their average

In the averages, the systematic errors associated with the modeling of $B \rightarrow X_c\ell ^+\nu _\ell$ and $B \rightarrow X_u\ell ^+\nu _\ell$ decays and the theoretical uncertainties are taken as fully correlated among all measurements. Reconstruction-related uncertainties are taken as fully correlated within a given experiment. Measurements of partial branching fractions for $B \rightarrow X_u\ell ^+\nu _\ell$ transitions from $\varUpsilon (4S)$ decays, together with the corresponding selected region, are given in Table 89. The signal yields for all the measurements shown in Table 89 are not rescaled to common input values of the B meson lifetime (see Sect. 3) and the semileptonic width [327]. We use all results published by BaBar in Ref. [546], since the statistical correlations are given. To make use of the theoretical calculations of Ref. [547], we restrict the kinematic range of the invariant mass of the hadronic system, $M_X$, and the square of the invariant mass of the lepton pair, $q^2$. This reduces the size of the data sample significantly, but also the theoretical uncertainty, as stated by the authors [547]. The dependence of the quoted error on the measured value for each source of uncertainty is taken into account in the calculation of the averages.

It has been first suggested by Neubert [548] and later detailed by Leibovich, Low, and Rothstein (LLR) [549] and Lange, Neubert and Paz (LNP) [550], that the uncertainty of the leading shape functions can be eliminated by comparing inclusive rates for $B \rightarrow X_u\ell ^+\nu _\ell$ decays with the inclusive photon spectrum in $B \rightarrow X_s\gamma$, based on the assumption that the shape functions for transitions to light quarks, u or s, are the same at first order. However, shape function uncertainties are only eliminated at the leading order and they still enter via the signal models used for the determination of efficiency.

Table 89 Summary of measurements of partial branching fractions for $B\rightarrow X_u \ell ^+ \nu _{\ell }$ decays. The errors quoted on $\Delta {\mathcal {B}}$ correspond to statistical and systematic uncertainties. $E_e$ is the electron energy in the B rest frame, $p^*$ the lepton momentum in the B frame and $m_X$ is the invariant mass of the hadronic system. The light-cone momentum $P_+$ is defined in the B rest frame as $P_+=E_X-|\mathbf {p}_X|$. The $s_\mathrm {h}^{\mathrm {max}}$ variable is described in Refs. [552, 553]

In the following, the different theoretical methods and the resulting averages are described. A recent BaBar measurement of the inclusive electron spectrum [551] was released at the time of this writing and could not be included in the averages.

5.4.1 BLNP

Bosch, Lange, Neubert and Paz (BLNP) [559,560,561,562] provide theoretical expressions for the triple differential decay rate for $B\rightarrow X_u \ell ^+ \nu _\ell$ events, incorporating all known contributions, whilst smoothly interpolating between the “shape-function region” of large hadronic energy and small invariant mass, and the “OPE region” in which all hadronic kinematical variables scale with the b-quark mass. BLNP assign uncertainties to the b-quark mass, which enters through the leading shape function, to sub-leading shape function forms, to possible weak annihilation contribution, and to matching scales. The BLNP calculation uses the shape function renormalization scheme; the heavy quark parameters determined from the global fit in the kinetic scheme, described in Sect. 5.2.2, were therefore translated into the shape function scheme by using a prescription by Neubert [563, 564]. The resulting parameters are $m_b(\mathrm{SF})=(4.582 \,\pm \, 0.023 \,\pm \, 0.018)~\mathrm{GeV}$, $\mu _\pi ^2(\mathrm{SF})=(0.202 \,\pm \, 0.089 ^{+0.020}_{-0.040})~{\mathrm{GeV/}c^2}$, where the second uncertainty is due to the scheme translation. The extracted values of $|V_{ub}|$  for each measurement along with their average are given in Table 90 and illustrated in Fig. 63a. The total uncertainty is $^{+5.8}_{-6.0}\%$ and is due to: statistics ($^{+2.1}_{-2.1}\%$), detector effects ($^{+1.7}_{-1.8}\%$), $B\rightarrow X_c \ell ^+ \nu _\ell$ model ($^{+1.2}_{-1.2}\%$), $B\rightarrow X_u \ell ^+ \nu _\ell$ model ($^{+1.8}_{-1.7}\%$), heavy quark parameters ($^{+2.6}_{-2.6}\%$), SF functional form ($^{+0.2}_{-0.3}\%$), sub-leading shape functions ($^{+0.6}_{-0.7}\%$), BLNP theory: matching scales $\mu ,\mu _i,\mu _h$ ($^{+3.8}_{-3.7}\%$), and weak annihilation ($^{+0.0}_{-1.4}\%$). The error assigned to the matching scales is the source of the largest uncertainty, while the uncertainty due to HQE parameters (b-quark mass and $\mu _\pi ^2)$ is second. The uncertainty due to weak annihilation has been assumed to be asymmetric, i.e. it only tends to decrease $|V_{ub}|$.

Table 90 Summary of input parameters used by the different theory calculations, corresponding inclusive determinations of $|V_{ub}|$ and their average. The errors quoted on $|V_{ub}|$ correspond to experimental and theoretical uncertainties, respectively

Fig. 63

Measurements of $|V_{ub}|$ from inclusive semileptonic decays and their average based on the BLNP (a) and DGE (b) prescription. The labels indicate the variables and selections used to define the signal regions in the different analyses

5.4.2 DGE

Andersen and Gardi (Dressed Gluon Exponentiation, DGE) [565] provide a framework where the on-shell b-quark calculation, converted into hadronic variables, is directly used as an approximation to the meson decay spectrum without the use of a leading-power non-perturbative function (or, in other words, a shape function). The on-shell mass of the b-quark within the B-meson ($m_b$) is required as input. The DGE calculation uses the $\overline{MS}$ renormalization scheme. The heavy quark parameters determined from the global fit in the kinetic scheme, described in Sect. 5.2.2, were therefore translated into the $\overline{MS}$ scheme by using a calculation by Gardi, giving $m_b({\overline{MS}})=(4.188 \,\pm \, 0.043)~\mathrm{GeV}$. The extracted values of $|V_{ub}|$  for each measurement along with their average are given in Table 90 and illustrated in Fig. 63b. The total error is $^{+4.8}_{-4.8}\%$, whose breakdown is: statistics ($^{+1.9}_{-1.9}\%$), detector effects ($^{+1.7}_{-1.7}\%$), $B\rightarrow X_c \ell ^+ \nu _\ell$ model ($^{+1.3}_{-1.3}\%$), $B\rightarrow X_u \ell ^+ \nu _\ell$ model ($^{+2.1}_{-1.7}\%$), strong coupling $\alpha _s$ ($^{+0.5}_{-0.5}\%$), $m_b$ ($^{+3.2}_{-2.9}\%$), weak annihilation ($^{+0.0}_{-1.8}\%$), matching scales in DGE ($^{+0.5}_{-0.4}\%$). The largest contribution to the total error is due to the effect of the uncertainty on $m_b$. The uncertainty due to weak annihilation has been assumed to be asymmetric, i.e. it only tends to decrease $|V_{ub}|$.

5.4.3 GGOU

Gambino, Giordano, Ossola and Uraltsev (GGOU) [567] compute the triple differential decay rates of $B \rightarrow X_u \ell ^+ \nu _\ell$, including all perturbative and non–perturbative effects through $O(\alpha _s^2 \beta _0)$ and $O(1/m_b^3)$. The Fermi motion is parameterized in terms of a single light–cone function for each structure function and for any value of $q^2$, accounting for all subleading effects. The calculations are performed in the kinetic scheme, a framework characterized by a Wilsonian treatment with a hard cutoff $\mu \sim 1~\mathrm{GeV}$. GGOU have not included calculations for the “($E_e,s^{max}_h$)” analysis [553]. The heavy quark parameters determined from the global fit in the kinetic scheme, described in Sect. 5.2.2, are used as inputs: $m_b^{kin}=(4.554 \,\pm \, 0.018)~\mathrm{GeV}$, $\mu _\pi ^2=(0.464 \,\pm \, 0.076)~{\mathrm{GeV/}c^2}$. The extracted values of $|V_{ub}|$  for each measurement along with their average are given in Table 90 and illustrated in Fig. 64a. The total error is $^{+4.2}_{-4.6}\%$ whose breakdown is: statistics ($^{+2.0}_{-2.0}\%$), detector effects ($^{+1.7}_{-1.7}\%$), $B\rightarrow X_c \ell ^+ \nu _\ell$ model ($^{+1.3}_{-1.3}\%$), $B\rightarrow X_u \ell ^+ \nu _\ell$ model ($^{+1.8}_{-1.8}\%$), $\alpha _s$, $m_b$ and other non-perturbative parameters ($^{+1.4}_{-1.4}\%$), higher order perturbative and non-perturbative corrections ($^{+1.5}_{-1.5}\%$), modelling of the $q^2$ tail ($^{+1.2}_{-1.2}\%$), weak annihilations matrix element ($^{+0.0}_{-1.9}\%$), functional form of the distribution functions ($^{+0.2}_{-0.2}\%$). The leading uncertainties on $|V_{ub}|$ are both from theory, and are due to perturbative and non-perturbative parameters and the modelling of the $q^2$ tail. The uncertainty due to weak annihilation has been assumed to be asymmetric, i.e. it only tends to decrease $|V_{ub}|$.

Fig. 64

Measurements of $|V_{ub}|$ from inclusive semileptonic decays and their average based on the GGOU (a) and ADFR (b) prescription. The labels indicate the variables and selections used to define the signal regions in the different analyses

5.4.4 ADFR

Aglietti, Di Lodovico, Ferrera and Ricciardi (ADFR) [568] use an approach to extract $|V_{ub}|$, that makes use of the ratio of the $B \rightarrow X_c \ell ^+ \nu _\ell$ and $B \rightarrow X_u \ell ^+ \nu _\ell$ widths. The normalized triple differential decay rate for $B \rightarrow X_u \ell ^+ \nu _\ell$ [566, 569,570,571] is calculated with a model based on (i) soft-gluon resummation to next-to-next-leading order and (ii) an effective QCD coupling without Landau pole. This coupling is constructed by means of an extrapolation to low energy of the high-energy behaviour of the standard coupling. More technically, an analyticity principle is used. The lower cut on the electron energy for the endpoint analyses is 2.3 GeV [566]. The ADFR calculation uses the $\overline{MS}$ renormalization scheme; the heavy quark parameters determined from the global fit in the kinetic scheme, described in Sect. 5.2.2, were therefore translated into the $\overline{MS}$ scheme by using a calculation by Gardi, giving $m_b({\overline{MS}})=(4.188 \,\pm \, 0.043)~\mathrm{GeV}$. The extracted values of $|V_{ub}|$  for each measurement along with their average are given in Table 90 and illustrated in Fig. 64b. The total error is $^{+5.5}_{-5.5}\%$ whose breakdown is: statistics ($^{+1.9}_{-1.9}\%$), detector effects ($^{+1.7}_{-1.7}\%$), $B\rightarrow X_c \ell ^+ \nu _\ell$ model ($^{+1.3}_{-1.3}\%$), $B\rightarrow X_u \ell ^+ \nu _\ell$ model ($^{+1.3}_{-1.3}\%$), $\alpha _s$ ($^{+1.1}_{-1.0}\%$), $|V_{cb}|$ ($^{+1.9}_{-1.9}\%$), $m_b$ ($^{+0.7}_{-0.7}\%$), $m_c$ ($^{+1.3}_{-1.3}\%$), semileptonic branching fraction ($^{+0.8}_{-0.7}\%$), theory model ($^{+3.6}_{-3.6}\%$). The leading uncertainty is due to the theory model.

5.4.5 BLL

Bauer, Ligeti, and Luke (BLL) [547] give a HQET-based prescription that advocates combined cuts on the dilepton invariant mass, $q^2$, and hadronic mass, $m_X$, to minimise the overall uncertainty on $|V_{ub}|$. In their reckoning a cut on $m_X$ only, although most efficient at preserving phase space ($\sim$80%), makes the calculation of the partial rate untenable due to uncalculable corrections to the b-quark distribution function or shape function. These corrections are suppressed if events in the low $q^2$ region are removed. The cut combination used in measurements is $M_x<1.7~{\mathrm{GeV/}c^2}$ and $q^2 > 8~{{\mathrm{GeV}^2\!/}c^4}$. The extracted values of $|V_{ub}|$  for each measurement along with their average are given in Table 90 and illustrated in Fig. 65. The total error is $^{+7.7}_{-7.7}\%$ whose breakdown is: statistics ($^{+3.3}_{-3.3}\%$), detector effects ($^{+3.0}_{-3.0}\%$), $B\rightarrow X_c \ell ^+ \nu _\ell$ model ($^{+1.6}_{-1.6}\%$), $B\rightarrow X_u \ell ^+ \nu _\ell$ model ($^{+1.1}_{-1.1}\%$), spectral fraction ($m_b$) ($^{+3.0}_{-3.0}\%$), perturbative approach: strong coupling $\alpha _s$ ($^{+3.0}_{-3.0}\%$), residual shape function ($^{+2.5}_{-2.5}\%$), third order terms in the OPE ($^{+4.0}_{-4.0}\%$). The leading uncertainties, both from theory, are due to residual shape function effects and third order terms in the OPE expansion. The leading experimental uncertainty is due to statistics.

Fig. 65

Measurements of $|V_{ub}|$ from inclusive semileptonic decays and their average in the BLL prescription

5.4.6 Summary

The averages presented in several different frameworks are presented in Table 91. In summary, we recognize that the experimental and theoretical uncertainties play out differently between the schemes and the theoretical assumptions for the theory calculations are different. Therefore, it is difficult to perform an average between the various determinations of $|V_{ub}|$. Since the methodology is similar to that used to determine the inclusive $|V_{cb}|$ average, we choose to quote as reference value the average determined by the GGOU calculation, which gives $|V_{ub}|$ $= (4.52 \,\pm \, 0.15 ^{+0.11}_{-0.14}) \times 10^{-3}$.

5.5 $B\rightarrow D^{(*)}\tau \nu _\tau$ decays

In the SM the semileptonic decay are tree level processes which proceed via coupling to the $W^{\,\pm \,}$ boson. These couplings are assumed to be universal for all leptons and are well understood theoretically, (see Sects. 5.1, 5.2). This universality has be tested in purely leptonic and semileptonic B meson decays involving a $\tau$ lepton, which might be sensitive to a hypothetical charged Higgs boson or other non-SM processes.

Compared to $B^+\rightarrow \tau \nu _\tau$, the $B\rightarrow D^{(*)}\tau \nu _\tau$ decay has advantages: the branching fraction is relatively high, because it is not Cabibbo-suppressed, and it is a three-body decay allowing access to many observables besides the branching fraction, such as $D^{(*)}$ momentum, $q^2$ distributions, and measurements of the $D^*$ and $\tau$ polarisations (see Ref. [572] and references therein for recent calculations).

Table 91 Summary of inclusive determinations of $|V_{ub}|$. The errors quoted on $|V_{ub}|$ correspond to experimental and theoretical uncertainties

Table 92 Measurements of $\mathcal{R}(D^*)$ and $\mathcal{R}(D)$, their correlations and the combined average

Experiments have measured two ratios of branching fractions defined as

$$\begin{aligned} \mathcal{R}(D)= & {} \dfrac{ {\mathcal {B}}(B\rightarrow D\tau \nu _\tau ) }{ {\mathcal {B}}(B\rightarrow D\ell \nu _\ell ) },\end{aligned}$$

(217)

$$\begin{aligned} \mathcal{R}(D^*)= & {} \dfrac{ {\mathcal {B}}(B\rightarrow D^*\tau \nu _\tau ) }{ {\mathcal {B}}(B\rightarrow D^*\ell \nu _\ell ) } \end{aligned}$$

(218)

where $\ell$ refers either to electron or $\mu$. These ratios are independent of $|V_{cb}|$ and to a large extent, also of the $B\rightarrow D^{(*)}$ form factors. As a consequences the SM predictions for these ratios are quite precise:

• $\mathcal{R}(D)=0.300\,\pm \, 0.008$, which is an average obtained by FLAG [222] by combining the most recent lattice calculations of the $B\rightarrow D\ell \nu$ form factors [494, 573];

• $\mathcal{R}(D^*)=0.252\,\pm \, 0.003$, which is a prediction, [574, 575] that updates recent QCD calculations  [576, 577] based on the recent $B\rightarrow D^{*}$ measurements from the B-Factories.

Recently, in Ref. [578] Bigi and Gambino re-analysed the recent experimental results and theoretical calculation of $B\rightarrow D\ell \nu$ obtaining $\mathcal{R}(D)=0.299\,\pm \, 0.003$, compatible with the predictions reported above but with a total error reduced by a factor three.

From the experimental side, in the case of the leptonic $\tau$ decay, the ratios $\mathcal{R}(D^{(*)})$ can be directly measured, and many systematic uncertainties cancel in the measurement. The $B^0\rightarrow D^{*+}\tau \nu _\tau$ decay was first observed by Belle [579] performing an “inclusive” reconstruction, which is based on the reconstruction of the $B_{tag}$ from all the particles of the events, other than the $D^{(*)}$ and the lepton candidate, without looking for any specific $B_{tag}$ decay chain. Since then, both BaBar and Belle have published improved measurements and have found evidence for the $B\rightarrow D\tau \nu _\tau$ decays [580, 581].

The most powerful way to study these decays at the B-Factories exploits the hadronic $B_{tag}$. Using the full dataset and an improved $B_{tag}$ selection, BaBar measured [574]:

$$\begin{aligned}&\mathcal{R}(D)=0.440\,\pm \, 0.058\,\pm \, 0.042,\nonumber \\&\mathcal{R}(D^*)=0.332\,\pm \, 0.024\,\pm \, 0.018 \end{aligned}$$

(219)

where decays to both $e^{\pm } \,$ and $\mu ^{\pm } \,$ were summed and $B^0$ and $B^-$ were combined in a isospin-constrained fit. The fact that the BaBar result exceeded SM predictions by $3.4\sigma$, raised considerable interest.

Belle published various measurements using different techniques, and LHCb also joined the effort with a measurement of $R(D^*)$. The most important sources of systematic uncertainties correlated for the different measurement is due to the $B\rightarrow D^{**}$ background components that are difficult to disentangle from the signal. In the average the systematic uncertainties due to the $B\rightarrow D^{**}$ composition and kinematics are considered fully correlated among the measurements.

The results of the individual measurements, their averages and correlations are presented in Table 92 and Fig. 66. The combined results, projected separately on $\mathcal{R}(D)$ and $\mathcal{R}(D^*)$, are reported in Fig. 67a, b respectively.

Fig. 66

Measurement of $\mathcal{R}(D)$ and $\mathcal{R}(D^*)$ and their average compared with the prediction for $\mathcal{R}(D^*)$  [577] and $\mathcal{R}(D)$  [494, 573]. The dashed ellipses corresponds to the 2 and 4 $\sigma$ contours

Fig. 67

a Measurement of $\mathcal{R}(D)$ and b $\mathcal{R}(D^*)$. The average is the projection of the average obtained from the combined fit

The averaged $\mathcal{R}(D)$ and $\mathcal{R}(D^*)$ exceed the SM predictions by 2.2$\sigma$ and 3.4$\sigma$ respectively. Considering the $\mathcal{R}(D)$ and $\mathcal{R}(D^*)$ total correlation of $-0.23$, the difference with respect to the SM is about 3.9 $\sigma$, the combined $\chi ^2=18.83$ for 2 degrees of freedom corresponds to a p-value of $8.3\times 10^{-5}$, assuming Gaussian error distributions.

6 Decays of b-hadrons into open or hidden charm hadrons

Ground state B mesons and b baryons dominantly decay to particles containing a charm quark via the $b \rightarrow c$ quark transition. In this section, measurements of such decays to hadronic final states are considered; semileptonic decay modes, which are usually used to determine the strength of the $b \rightarrow c$ transition as quantified in the magnitude of the CKM matrix element $|V_{cb}|$, are discussed in Sect. 5. Some B meson decays to open or hidden charm hadrons that are fundamental for the measurements of $C\!P$-violation phases – like $\phi _s^{c{\bar{c}}s}$ (Sect. 3), $\beta \equiv \phi _1$ and $\gamma \equiv \phi _3$ (Sect. 4) – are discussed elsewhere in this report. Similarly, the use of $b \rightarrow c$ decay modes for the determination of important properties of b-hadrons, like their masses or (absolute, relative or effective) lifetimes, is discussed in Sect. 3. The properties of certain b hadron decays to open or hidden charm hadrons, such as small Q values and similar topologies for different modes, allow the minimization of systematic uncertainties in these measurements.

The fact that decays to final states containing open or hidden charm hadrons dominate the b-hadron widths makes them a very important part of the experimental programme in heavy flavor physics. Understanding the rate of charm production in b-hadron decays is crucial to validate the HQE that underpins much of the theoretical framework for b physics (see, for example, Ref. [586] for a review). Moreover, such decays are often used as normalization modes for measurements of rarer decays. In addition, they are the dominant background in many analyses. To model accurately such backgrounds with simulated data, it is essential to have precise knowledge of the contributing decay modes. In particular, with the expected increase in the data samples at LHCb and Belle II, the enhanced statistical sensitivity has to be matched by low systematic uncertainties due to knowledge of the dominant b-hadron decay modes. For multibody decays, knowledge of the distribution of decays across the phase-space (e.g., the Dalitz plot density for three-body decays or the polarization amplitudes for vector-vector final states) is required in addition to the total branching fraction.

The large yields of $b \rightarrow c$ decays to multibody final states make them ideal to study the spectroscopy of both open and hidden charm hadrons. In particular, they have been used to both discover, and measure the properties of, exotic particles such as the X(3872) [587, 588], $Z(4430)^+$ [589, 590] and $P_c(4450)^+$ [591] states. The large yields available similarly make decays involving $b \rightarrow c$ transitions very useful to study baryon-antibaryon pair production.

In addition to the dominant b-hadron decays to final states containing charmed hadrons, there are several decays in this category that are expected to be highly suppressed in the Standard Model. These are of interest to probe particular decay topologies (e.g., the $B^- \rightarrow D^-_s \phi$ decay, which is dominated by the annihilation diagram) and thereby constrain effects in other hadronic decays or to search for new physics. There are also other decays involving $b \rightarrow c$ transitions, such as ${\overline{B}} ^0 \rightarrow D^-_s \pi ^+$, that are mediated by the W emission involving the $|V_{ub}|$ CKM matrix element. Finally, some $b \rightarrow c$ decays involving lepton flavour or number violation are extremely suppressed in the Standard Model, and therefore provide highly sensitive null tests.

In this section, we give an exhaustive list of measured branching ratios of decay modes to hadrons containing charm quarks. The averaging procedure follows the methodology described in Sect. 2. Where available, correlations between measurements are taken into account. If an insignificant measurement and a limit for the same parameter are provided the former is taken so that it can be included in averages. The confidence level of an average is quoted if it is below 1%. We provide averages of the polarization amplitudes of B meson decays to vector-vector states, but we do not currently provide detailed averages of quantities obtained from Dalitz plot analyses, due to the complications arising from the dependence on the model used.

The results are presented in subsections organized according to the type of decaying bottom hadron: ${\overline{B}} ^0$ (Sect. 6.1), $B^-$ (Sect. 6.2), ${\overline{B}} ^0/B^-$ admixture (Sect. 6.3), ${\overline{B}} _s^0$ (Sect. 6.4), $B_c^-$ (Sect. 6.5), b baryons (Sect. 6.6). For each subsection the measurements are arranged according to the final state into the following groups: a single charmed meson, two charmed mesons, a charmonium state, a charm baryon, or other states, like for example the X(3872) meson. The individual measurements and averages are shown as numerical values in tables followed by a graphical representation of the averages. The symbol $\mathcal {B}$ is used for branching ratios, f for production fractions (see Sect. 3), and $\sigma$ for cross sections. The decay amplitudes for longitudinal, parallel, and perpendicular transverse polarization in pseudoscalar to vector-vector decays are denoted ${\mathcal {A}}_0$, ${\mathcal {A}}_\parallel$, and ${\mathcal {A}}_\perp$, respectively, and the definitions $\delta _\parallel = \arg ({\mathcal {A}}_\parallel /{\mathcal {A}}_0)$ and $\delta _\perp = \arg ({\mathcal {A}}_\perp /{\mathcal {A}}_0)$ are used for their relative phases. The inclusion of charge conjugate modes is always implied.

Following the approach used by the PDG [327], for decays that involve neutral kaons we mainly quote results in terms of final states including either a $K^0$ or $\overline{K} ^0$ meson (instead of a $K^0_{S}$ or $K^0_{L}$). In some cases where the decay is not flavour-specific and the final state is not self-conjugate, the inclusion of the conjugate final state neutral kaon is implied – in fact, the flavour of the neutral kaon is never determined experimentally, and so the specification as $K^0$ or $\overline{K} ^0$ simply follows the quark model expectation for the dominant decay. An exception occurs for some ${B^{0}_s}$ decays, specifically those to $C\!P$ eigenstates, where the width difference between the mass eigenstates (see Sect. 3) means that the measured branching fraction, integrated over decay time, is specific to the studied final state [592]. Therefore it is appropriate to quote the branching fraction for, e.g., ${\overline{B}} _s^0 \rightarrow {J/\psi } K^0_{S}$ instead of ${\overline{B}} _s^0 \rightarrow {J/\psi } \overline{K} ^0$.

Several measurements assume $\Gamma (\varUpsilon (4S) \rightarrow B^+B^-) = \Gamma (\varUpsilon (4S) \rightarrow B^0{\bar{B}}^0)$. While there is no evidence for isospin violation in $\varUpsilon (4S)$ decays, deviations from this assumptions can be of the order of a few percent, see Sect. 3.1.1 and Ref. [593]. As the effect is negligible for many averages, we do not apply a correction or additional systematic uncertainty, but we point out that it can be relevant for averages with a percent level uncertainty.

6.1 Decays of ${\bar{B}}^0$ mesons

Measurements of ${\bar{B}}^0$ decays to charmed hadrons are summarized in Sects. 6.1.1–6.1.5.

6.1.1 Decays to a single open charm meson

Averages of ${\bar{B}}^0$ decays to a single open charm meson are shown in Tables 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105 and Figs. 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80. In this section $D^{**}$ refers to the sum of all the non-strange charm meson states with masses in the range 2.2–2.8 ${\mathrm{GeV/}c^2}$.

Table 93 Decays to a $D^{(*)}$ meson and one or more pions I $[10^{-3}]$

Table 94 Decays to a $D^{(*)}$ meson and one or more pions II $[10^{-4}]$

Table 95 Decays to a $D^{(*)0}$ meson and a light meson $[10^{-4}]$

Table 96 Decays to a $D^{(*)+}$ meson and one or more kaons $[10^{-3}]$

Table 97 Decays to a $D^{(*)0}$ meson and a kaon $[10^{-4}]$

Table 98 Decays to a $D_s^{(*)}$ meson $[10^{-4}]$

Table 99 Relative decay rates I

Table 100 Relative decay rates II

Table 101 Absolute product decay rates to excited D mesons I $[10^{-4}]$

Table 102 Absolute product decay rates to excited D mesons II $[10^{-5}]$

Table 103 Absolute and relative decay rates to excited D mesons $[10^{-2}]$

Table 104 Baryonic decays I $[10^{-4}]$

Table 105 Baryonic decays II $[10^{-5}]$

Fig. 68

Summary of the averages from Table 93

Fig. 69

Summary of the averages from Table 94

Fig. 70

Summary of the averages from Table 95

Fig. 71

Summary of the averages from Table 96

Fig. 72

Summary of the averages from Table 97

Fig. 73

Summary of the averages from Table 98

Fig. 74

Summary of the averages from Table 99

Fig. 75

Summary of the averages from Table 100

Fig. 76

Summary of the averages from Table 101

Fig. 77

Summary of the averages from Table 102

Fig. 78

Summary of the averages from Table103

Fig. 79

Summary of the averages from Table 104

Fig. 80

Summary of the averages from Table 105

6.1.2 Decays to two open charm mesons

Averages of ${\bar{B}}^0$ decays to two open charm mesons are shown in Tables 106, 107, 108, 109, 110, 111, 112, 113, 114, 115 and Figs. 81, 82, 83, 84, 85, 86, 87, 88, 89.

Table 106 Decays to $D^{(*)+}D^{(*)-}$ $[10^{-3}]$

Table 107 Decays to two D mesons and a kaon I $[10^{-3}]$

Table 108 Decays to two D mesons and a kaon II $[10^{-4}]$

Table 109 Decays to $D_s^{(*)-}D^{(*)+}$ I $[10^{-3}]$

Table 110 Decays to $D_s^{(*)-}D^{(*)+}$ II $[10^{-4}]$

Table 111 Decays to $D_s^{(*)+}D_s^{(*)-}$ $[10^{-3}]$

Table 112 Relative decay rates $[10^{-3}]$

Table 113 Absolute decay rates to excited $D_s$ mesons $[10^{-3}]$

Table 114 Product decay rates to excited $D_s$ mesons I $[10^{-3}]$

Table 115 Product decay rates to excited $D_s$ mesons II $[10^{-4}]$

Fig. 81

Summary of the averages from Table 106

Fig. 82

Summary of the averages from Table 107

Fig. 83

Summary of the averages from Table 108

Fig. 84

Summary of the averages from Table 109

Fig. 85

Summary of the averages from Table 110

Fig. 86

Summary of the averages from Table 111

Fig. 87

Summary of the averages from Table 113

Fig. 88

Summary of the averages from Table 114

Fig. 89

Summary of the averages from Table 115

6.1.3 Decays to charmonium states

Averages of ${\bar{B}}^0$ decays to charmonium states are shown in Tables 116, 117, 118, 119, 120, 121, 122, 123, 124 and Figs. 90, 91, 92, 93, 94, 95, 96, 97, 98.

Table 116 Decays to $J/\psi$ and one kaon $[10^{-3}]$

Table 117 Decays to charmonium other than $J/\psi$ and one kaon I $[10^{-3}]$

Table 118 Decays to charmonium other than $J/\psi$ and one kaon II $[10^{-4}]$

Table 119 Decays to charmonium and light mesons I $[10^{-5}]$

Table 120 Decays to charmonium and light mesons II $[10^{-5}]$

Table 121 Decays to $J/\psi$ and photons, baryons, or heavy mesons $[10^{-5}]$

Table 122 Relative decay rates I

Table 123 Relative decay rates II

Table 124 Polarization fractions

Fig. 90

Summary of the averages from Table 116

Fig. 91

Summary of the averages from Table 117

Fig. 92

Summary of the averages from Table 118

Fig. 93

Summary of the averages from Table 119

Fig. 94

Summary of the averages from Table 120

Fig. 95

Summary of the averages from Table 121

Fig. 96

Summary of the averages from Table 122

Fig. 97

Summary of the averages from Table 123

Fig. 98

Summary of the averages from Table 124

6.1.4 Decays to charm baryons

Averages of ${\bar{B}}^0$ decays to charm baryons are shown in Tables 125, 126, 127, 128 and Figs. 99, 100, 101.

Table 125 Absolute decay rates to charm baryons I $[10^{-4}]$

Table 126 Absolute decay rates to charm baryons II $[10^{-5}]$

Table 127 Absolute decay rates to charm baryons III $[10^{-6}]$

Table 128 Relative decay rates to charm baryons $[10^{-3}]$

Fig. 99

Summary of the averages from Table 125

Fig. 100

Summary of the averages from Table 126

Fig. 101

Summary of the averages from Table 127

6.1.5 Decays to other (XYZ) states

Averages of ${\bar{B}}^0$ decays to other (XYZ) states are shown in Tables 129, 130, 131, 132, 133, 134 and Figs. 102, 103, 104, 105.

Table 129 Decays to X(3872) $[10^{-5}]$

Table 130 Decays to X(3872) with $X(3872) \rightarrow D{\bar{D}}$ $[10^{-4}]$

Table 131 Decays to neutral states other than X(3872) $[10^{-4}]$

Table 132 Decays to charged states I $[10^{-4}]$

Table 133 Decays to charged states II $[10^{-5}]$

Table 134 Relative decay rates

Fig. 102

Summary of the averages from Table 129

Fig. 103

Summary of the averages from Table 131

Fig. 104

Summary of the averages from Table 133

Fig. 105

Summary of the averages from Table 134

6.2 Decays of $B^-$ mesons

Measurements of $B^-$ decays to charmed hadrons are summarized in Sects. 6.2.1–6.2.5.

6.2.1 Decays to a single open charm meson

Averages of $B^-$ decays to a single open charm meson are shown in Tables 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149 and Figs. 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119. In this section $D^{**}$ refers to the sum of all the non-strange charm meson states with masses in the range 2.2–2.8 ${\mathrm{GeV/}c^2}$.

Table 135 Decays to a $D^{(*)}$ meson and one or more pions $[10^{-2}]$

Table 136 Decays to a $D^{(*)0}$ meson and one or more kaons $[10^{-3}]$

Table 137 Decays to a $D^{(*)-}$ meson and a neutral kaon or a kaon and a pion $[10^{-6}]$

Table 138 Relative decay rates to $D^{0}$ mesons I

Table 139 Relative decay rates to $D^{0}$ mesons II $[10^{-2}]$

Table 140 Absolute decay rates to excited D mesons $[10^{-3}]$

Table 141 Absolute product decay rates to excited D mesons I $[10^{-3}]$

Table 142 Absolute product decay rates to excited D mesons II $[10^{-5}]$

Table 143 Relative decay rates to excited D mesons

Table 144 Relative product decay rates to excited D mesons

Table 145 Decays to $D_s^{(*)}$ mesons I $[10^{-4}]$

Table 146 Decays to $D_s^{(*)}$ mesons I $[10^{-5}]$

Table 147 Baryonic decays I $[10^{-4}]$

Table 148 Baryonic decays II $[10^{-5}]$

Table 149 Lepton number violating decays $[10^{-6}]$

Fig. 106

Summary of the averages from Table 135

Fig. 107

Summary of the averages from Table 136

Fig. 108

Summary of the averages from Table 137

Fig. 109

Summary of the averages from Table 138

Fig. 110

Summary of the averages from Table 139

Fig. 111

Summary of the averages from Table 141

Fig. 112

Summary of the averages from Table 142

Fig. 113

Summary of the averages from Table 143

Fig. 114

Summary of the averages from Table 144

Fig. 115

Summary of the averages from Table 145

Fig. 116

Summary of the averages from Table 146

Fig. 117

Summary of the averages from Table 147

Fig. 118

Summary of the averages from Table 148

Fig. 119

Summary of the averages from Table 149

6.2.2 Decays to two open charm mesons

Averages of $B^-$ decays to two open charm mesons are shown in Tables 150, 151, 152, 153, 154, 155, 156, 157, 158 and Figs. 120, 121, 122, 123, 124, 125, 126, 127.

Table 150 Decays to $D^{(*)-}D^{(*)0}$ $[10^{-3}]$

Table 151 Decays to two D mesons and a kaon I $[10^{-2}]$

Table 152 Decays to two D mesons and a kaon II $[10^{-3}]$

Table 153 Decays to $D_s^{(*)-}D^{(*)+}$ $[10^{-2}]$

Table 154 Product decays rates to $D_s^{(*)-}D^{(*)+}$ $[10^{-4}]$

Table 155 Relative decay rates

Table 156 Absolute decays rates to excited $D_s$ mesons $[10^{-3}]$

Table 157 Product decays rates to excited $D_s$ mesons I $[10^{-3}]$

Table 158 Product decays rates to excited $D_s$ mesons II $[10^{-3}]$

Fig. 120

Summary of the averages from Table 150

Fig. 121

Summary of the averages from Table 151

Fig. 122

Summary of the averages from Table 152

Fig. 123

Summary of the averages from Table 153

Fig. 124

Summary of the averages from Table 154

Fig. 125

Summary of the averages from Table 156

Fig. 126

Summary of the averages from Table 157

Fig. 127

Summary of the averages from Table 158

6.2.3 Decays to charmonium states

Averages of $B^-$ decays to charmonium states are shown in Tables 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169 and Figs. 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138.

Table 159 Decays to $J/\psi$ and one kaon I $[10^{-3}]$

Table 160 Decays to $J/\psi$ and one kaon II $[10^{-4}]$

Table 161 Decays to charmonium other than $J/\psi$ and one kaon I $[10^{-3}]$

Table 162 Decays to charmonium other than $J/\psi$ and one kaon II $[10^{-5}]$

Table 163 Decays to charmonium other than $J/\psi$ and one kaon III $[10^{-6}]$

Table 164 Decays to charmonium and light mesons $[10^{-5}]$

Table 165 Decays to $J/\psi$ and a heavy mesons $[10^{-4}]$

Table 166 Decays with baryons I $[10^{-5}]$

Table 167 Decays with baryons II $[10^{-6}]$

Table 168 Relative decay rates I

Table 169 Relative decay rates II

Fig. 128

Summary of the averages from Table 159

Fig. 129

Summary of the averages from Table 160

Fig. 130

Summary of the averages from Table 161

Fig. 131

Summary of the averages from Table 162

Fig. 132

Summary of the averages from Table 163

Fig. 133

Summary of the averages from Table 164

Fig. 134

Summary of the averages from Table 165

Fig. 135

Summary of the averages from Table 166

Fig. 136

Summary of the averages from Table 167

Fig. 137

Summary of the averages from Table 168

Fig. 138

Summary of the averages from Table 169

6.2.4 Decays to charm baryons

Averages of $B^-$ decays to charm baryons are shown in Tables 170, 171, 172 and Figs. 139, 140.

Table 170 Absolute (product) decay rates $[10^{-4}]$

Table 171 Relative decay rates I

Table 172 Relative decay rates II

Fig. 139

Summary of the averages from Table 170

Fig. 140

Summary of the averages from Table 172

6.2.5 Decays to other (XYZ) states

Averages of $B^-$ decays to other (XYZ) states are shown in Tables 173, 174, 175, 176, 177, 178 and Figs. 141, 142, 143, 144, 145.

Table 173 Absolute decay rates $[10^{-4}]$

Table 174 Product decay rates to X(3872) I $[10^{-4}]$

Table 175 Product decay rates to X(3872) II $[10^{-5}]$

Table 176 Product decay rates to neutral states other than X(3872) $[10^{-5}]$

Table 177 Relative product decay rates to states with $s\bar{s}$ component

Table 178 Product decay rates to charged states $[10^{-5}]$

Fig. 141

Summary of the averages from Table 174

Fig. 142

Summary of the averages from Table 175

Fig. 143

Summary of the averages from Table 176

Fig. 144

Summary of the averages from Table 177

Fig. 145

Summary of the averages from Table 178

6.3 Decays of admixtures of ${\bar{B}}^0$ / $B^-$ mesons

Measurements of ${\bar{B}}^0$ / $B^-$ decays to charmed hadrons are summarized in Sects. 6.3.1–6.3.3.

6.3.1 Decays to two open charm mesons

Averages of ${\bar{B}}^0$ / $B^-$ decays to two open charm mesons are shown in Table 179.

Table 179 B decays to double charm $[10^{-4}]$

6.3.2 Decays to charmonium states

Averages of ${\bar{B}}^0$ / $B^-$ decays to charmonium states are shown in Tables 180, 181, 182, 183, 184 and Figs. 146, 147, 148, 149, 150.

Table 180 Decay amplitudes for parallel transverse polarization

Table 181 Decay amplitudes for perpendicular transverse polarization

Table 182 Decay amplitudes for longitudinal polarization

Table 183 Relative phases of parallel transverse polarization decay amplitudes

Table 184 Relative phases of perpendicular transverse polarization decay amplitudes

Fig. 146

Summary of the averages from Table 180

Fig. 147

Summary of the averages from Table 181

Fig. 148

Summary of the averages from Table 182

Fig. 149

Summary of the averages from Table 183

Fig. 150

Summary of the averages from Table 184

Table 185 Absolute decay rates to X/Y states $[10^{-4}]$

Fig. 151

Summary of the averages from Table 185

6.3.3 Decays to other (XYZ) states

Averages of ${\bar{B}}^0$ / $B^-$ decays to other (XYZ) states are shown in Table 185 and Fig. 151.

Table 186 Decays to a $D_s^{(*)}$ and a light meson I $[10^{-3}]$

Table 187 Decays to a $D_s^{(*)}$ and a light meson II $[10^{-4}]$

6.4 Decays of ${\bar{B}}_s^0$ mesons

Measurements of ${\bar{B}}_s^0$ decays to charmed hadrons are summarized in Sects. 6.4.1–6.4.4. These measurements require knowledge of the production rates of ${\bar{B}}_s^0$ mesons, usually measured relative to those of ${\bar{B}}^0$ and $B^-$ mesons, in the appropriate experimental environment. Since these production fractions are reasonably well known, see Sect. 3.1, they can be corrected for allowing the results to be presented in terms of the absolute ${\bar{B}}_s^0$ branching fraction, or the relative branching fraction to a lighter B meson decay mode. This is usually done in the publications; we do not make any attempt to rescale results according to more recent determinations of the relative production fractions. Ratios of branching fractions of two decays of the same hadron do not require any such correction.

6.4.1 Decays to a single open charm meson

Averages of ${\bar{B}}_s^0$ decays to a single open charm meson are shown in Tables 186, 187, 188, 189, 190, 191 and Figs. 152, 153, 154, 155, 156, 157.

Table 188 Decays to a $D^{(*)}$ and a light meson I $[10^{-4}]$

Table 189 Decays to a $D^{(*)}$ and a light meson II $[10^{-6}]$

Table 190 Relative decay rates I

Table 191 Relative decay rates II $[10^{-2}]$

Fig. 152

Summary of the averages from Table 186

Fig. 153

Summary of the averages from Table 187

Fig. 154

Summary of the averages from Table 188

Fig. 155

Summary of the averages from Table 189

Fig. 156

Summary of the averages from Table 190

Fig. 157

Summary of the averages from Table 191

6.4.2 Decays to two open charm mesons

Averages of ${\bar{B}}_s^0$ decays to two open charm mesons are shown in Tables 192, 193, 194 and Figs. 158, 159, 160.

Table 192 Absolute decay rates $[10^{-2}]$

Table 193 Relative decay rates I

Table 194 Relative decay rates II $[10^{-2}]$

Fig. 158

Summary of the averages from Table 192

Fig. 159

Summary of the averages from Table 193

Fig. 160

Summary of the averages from Table 194

6.4.3 Decays to charmonium states

Averages of ${\bar{B}}_s^0$ decays to charmonium states are shown in Tables 195, 196, 197, 198, 199 and Figs. 161, 162, 163, 164, 165.

Table 195 Absolute decay rates I $[10^{-4}]$

Table 196 Absolute decay rates II $[10^{-5}]$

Table 197 Relative decay rates I

Table 198 Relative decay rates II

Table 199 Relative decay rates III $[10^{-2}]$

Fig. 161

Summary of the averages from Table 195

Fig. 162

Summary of the averages from Table 196

Fig. 163

Summary of the averages from Table 197

Fig. 164

Summary of the averages from Table 198

Fig. 165

Summary of the averages from Table 199

6.4.4 Decays to charm baryons

Averages of ${\bar{B}}_s^0$ decays to charm baryons are shown in Tables 200 and 201.

Table 200 Decays to one charm baryon $[10^{-4}]$

Table 201 Decays to two charm baryons

6.5 Decays of $B_c^-$ mesons

Measurements of $B_c^-$ decays to charmed hadrons are summarized in Sects. 6.5.1–6.5.2. Since the absolute cross-section for $B_c^-$ meson production in any production environment is currently not known, it is not possible to determine absolute branching fractions. Instead, results are presented either as ratios of branching fractions of different $B_c^-$ decays, or are normalised to the branching fraction of the decay of a lighter B meson (usually $B^-$). In the latter case the measured quantity is the absolute or relative $B_c^-$ branching fraction multiplied by the ratio of cross-sections (or, equivalently, production fractions) of the $B_c^-$ and the lighter B meson.

It should be noted that the ratio of cross-sections for different b hadron species can depend on production environment, and on the fiducial region accessed by each experiment. While this has been studied for certain b hadron species (see Sect. 3.1), there is currently little published data that would allow to investigate the effect for $B_c^-$ mesons. Therefore, we do not attempt to apply any correction for this effect.

Table 202 Relative decay rates I

Table 203 Relative decay rates II

Table 204 Relative production times decay rates $[10^{-3}]$

6.5.1 Decays to charmonium states

Averages of $B_c^-$ decays to charmonium states are shown in Tables 202, 203, 204, 205 and Figs. 166, 167.

Table 205 Decay rates times relative production rates $[10^{-6}]$

Fig. 166

Summary of the averages from Table 202

Fig. 167

Summary of the averages from Table 203

6.5.2 Decays to a B meson

Averages of $B_c^-$ decays to a B meson are shown in Table 206.

Table 206 Decays to $B_s^{0}$ meson $[10^{-3}]$

Table 207 Relative decay rates to $D^{0}$ mesons

6.6 Decays of b baryons

Measurements of b baryons decays to charmed hadrons are summarized in Sects. 6.6.1–6.6.3. Comments regarding the production rates of ${\bar{B}}_s^0$ and $B_c^-$ mesons relative to lighter B mesons, in Sects. 6.4 and 6.5 respectively, are also appropriate here. Specifically, since the cross-section for production of $\varLambda _b^0$ baryons is reasonably well-known, it is possible to determine absolute or relative branching fractions for its decays (although some older measurements are presented as products involving the cross-section). The cross-sections for production of heavier b baryons are not known, and therefore measured quantities are presented as absolute or relative branching fraction multiplied by a ratio of cross-sections (or, equivalently, production fractions).

6.6.1 Decays to a single open charm meson

Averages of b baryons decays to a single open charm meson are shown in Table 207 and Fig. 168.

Fig. 168

Summary of the averages from Table 207

6.6.2 Decays to charmonium states

Averages of b baryons decays to charmonium states are shown in Tables 208, 209, 210, 211, 212 and Figs. 169, 170, 171.

Table 208 $\varLambda _b^{0}$ decays to charmonium $[10^{-4}]$

Table 209 $f_b$ times $\varLambda _b^{0}$ decay to charmonium $[10^{-5}]$

Table 210 Relative $\varLambda _b^{0}$ decay rates

Table 211 $\varXi _b^{-}$ and $\Omega _b^{-}$ decays to charmonium

Table 212 Parity violation in $\varLambda _b^{0}$ decays to charmonium

Fig. 169

Summary of the averages from Table 208

Fig. 170

Summary of the averages from Table 210

Fig. 171

Summary of the averages from Table 211

6.6.3 Decays to charm baryons

Averages of b baryons decays to charm baryons are shown in Tables 213, 214, 215, 216, 217 and Figs. 172, 173, 174, 175.

Table 213 Absolute decay rates $[10^{-2}]$

Table 214 Relative decay rates to $\varLambda _c$ I

Table 215 Relative decay rates to $\varLambda _c$ II $[10^{-2}]$

Table 216 Relative decay rates to excited or $\Sigma _c$ states

Table 217 $\varXi _b$ decay rates $[10^{-4}]$

Fig. 172

Summary of the averages from Table 213

Fig. 173

Summary of the averages from Table 214

Fig. 174

Summary of the averages from Table 215

Fig. 175

Summary of the averages from Table 216

7 B decays to charmless final states

This section provides branching fractions (BF), polarization fractions, partial rate asymmetries ($A_{C\!P}$) and other observables of B decays to final states that do not contain charm hadrons or charmonia mesons. The order of entries in the tables corresponds to that in PDG2014 [327], and the quoted RPP numbers are the PDG numbers of the corresponding branching fractions. The asymmetry is defined as

$$\begin{aligned} A_{C\!P} = \frac{N_b - N_{\bar{b}}}{N_b + N_{\bar{b}}}, \end{aligned}$$

(220)

where $N_b$ ($N_{\bar{b}}$) is the number of hadrons containing a b (${\bar{b}}$) quark decaying into a specific final state. This definition is consistent with that of Eq. (104) in Sect. 4.2.1. Four different $B^{0}$ and $B^{+}$ decay categories are considered: charmless mesonic (i.e., final states containing only mesons), baryonic (only hadrons, but including a baryon-antibaryon pair), radiative (including a photon or a lepton-antilepton pair) and semileptonic/leptonic (including/only leptons). We also include measurements of ${B^{0}_s}$, $B^+_c$ and b-baryon decays. Measurements supported with written documents are accepted in the averages; written documents include journal papers, conference contributed papers, preprints or conference proceedings. In all the tables of this section, values in red (blue) are new published (preliminary) results since PDG2014. Results from $A_{C\!P}$ measurements obtained from time-dependent analyses are listed and described in Sect. 4.

Most of the branching fractions from BaBar and Belle assume equal production of charged and neutral B pairs. The best measurements to date show that this is still a reasonable approximation (see Sect. 3). For branching fractions, we provide either averages or the most stringent upper limits. If one or more experiments have measurements with $>\!\!4 \sigma$ for a decay channel, all available central values for that channel are used in the averaging. We also give central values and errors for cases where the significance of the average value is at least $3 \sigma$, even if no single measurement is above $4 \sigma$. For $A_{C\!P}$ we provide averages in all cases. At the end of some of the tables we give a list of results that were not included. Typical cases are the measurements of distributions, such as differential branching fractions or longitudinal polarizations, which are measured in different binning schemes by the different collaborations, and thus cannot be directly used to obtain averages.

Our averaging is performed by maximizing the likelihood, $\mathcal {L}= \mathop \prod \nolimits _{i}\mathcal {P}_{i}(x)$, where ${\mathcal P_i}$ is the probability density function (PDF) of the $i\mathrm{th}$ measurement, and x is, e.g., the branching fraction or $A_{C\!P}$. The PDF is modelled by an asymmetric Gaussian function with the measured central value as its most probable value and the quadratic sum of the statistical and systematic errors as the standard deviation. The experimental uncertainties are considered to be uncorrelated with each other when the averaging is performed. As mentioned in Sect. 2, no error scaling is applied when the fit $\chi ^2$ is greater than 1, except for cases of extreme disagreement (at present we have no such cases).

The largest improvement since the last report has come from the inclusion of a variety of new measurements from the LHC, especially LHCb. The measurements of ${B^{0}_s}$ decays are particularly noteworthy.

Sections 7.1 and 7.2 provide compilations of branching fractions of $B^{0}$ and $B^{+}$ to mesonic and baryonic charmless final states, respectively, while Sect. 7.3 gives branching fractions of b-baryon decays. In Sects. 7.4 and 7.5 various observables of interest are given in addition to branching fractions: in the former, branching fractions of ${B^{0}_s}$-meson charmless decays, and in the latter observables related to leptonic and radiative $B^{0}$ and $B^{+}$ meson decays, including processes in which the photon yields a pair of charged or neutral leptons. Section 7.5 also reports limits from searches for lepton-flavor/number-violating decays. Sections 7.6 and 7.7 give $C\!P$ asymmetries and results of polarization measurements, respectively, in various b-hadron charmless decays. Finally, Sect. 7.8 gives branching fractions of $B^+_c$ meson decays to charmless final states.

7.1 Mesonic decays of $B^{0}$and $B^{+}$ mesons

This section provides branching fractions of charmless mesonic decays: Tables 218, 219 and 220 for $B^{+}$ and Tables 221, 222, 223 and 224 for $B^{0}$ mesons. The tables are separated according to the presence or absence of kaons in the final state. Finally, Table 225 details several relative branching fractions of $B^{0}$decays.

Figure 176 gives a graphic representation of a selection of high-precision branching fractions given in this section. Footnote symbols indicate that the footnote in the corresponding table should be consulted.

7.2 Baryonic decays of $B^{+}$ and $B^{0}$mesons

This section provides branching fractions of charmless baryonic decays of $B^{+}$ and $B^{0}$mesons in Tables 226 and 227, respectively. Relative branching fractions are given in Table 228.

Figures 177 and 178 show graphic representations of a selection of results given in this section. Footnote symbols indicate that the footnote in the corresponding table should be consulted.

7.3 Decays of $b$ baryons

A compilation of branching fractions of $\varLambda _b^0$ baryon decays is given in Table 229. Table 230 provides the partial branching fractions of $\varLambda _b^0 \rightarrow \varLambda \mu ^+ \mu ^-$ decays. A compilation of branching fractions of $\varXi ^{0}_{b}$ baryon decays is given in Table 231.

Figure 179 shows a graphic representation of branching fractions of $\varLambda _b^0$ decays. Footnote symbols indicate that the footnote in the corresponding table should be consulted.

List of other measurements that are not included in the tables:

• In Ref. [928], LHCb provides a measurement of the differential $\varLambda _b^0 \rightarrow \varLambda \mu ^+\mu ^-$ branching fraction. It is given in bins of $m^2(\mu ^+\mu ^-)$ that are different from those used in the past by the LHCb and CDF collaborations (see table of differential branching fractions).

• In the paper [929], LHCb measures the ratios

  $$\begin{aligned} \frac{\sigma (pp\rightarrow \varXi ^{\prime -}_{b}X) {\mathcal {B}}(\varXi ^{\prime -}_{b}\rightarrow \varXi ^{0}_{b}\pi ^-)}{\sigma (pp\rightarrow \varXi ^{0}_{b}X)},\nonumber \\ \frac{\sigma (pp\rightarrow \varXi ^{\prime -}_{b}X) {\mathcal {B}}(\varXi ^{*-}_{b}\rightarrow \varXi ^{0}_{b}\pi ^-)}{\sigma (pp\rightarrow \varXi ^{\prime -}_{b}X) {\mathcal {B}}(\varXi ^{\prime -}_{b}\rightarrow \varXi ^{0}_{b}\pi ^-)}. \end{aligned}$$

• In the paper [930], LHCb measures the ratio

  $$\begin{aligned} \frac{\sigma (pp\rightarrow \varXi ^{*-}_{b}X) {\mathcal {B}}(\varXi ^{*-}_{b}\rightarrow \varXi ^{0}_{b}\pi ^-)}{\sigma (pp\rightarrow \varXi ^{0}_{b}X)}. \end{aligned}$$

7.4 Decays of ${B^{0}_s}$ mesons

Tables 232 and 233 detail branching fractions and relative branching fractions of ${B^{0}_s}$ meson decays, respectively.

Figures 180 and 181 show graphic representations of a selection of results given in this section. Footnote symbols indicate that the footnote in the corresponding table should be consulted.

List of other measurements that are not included in the tables:

• ${B^{0}_s} \rightarrow \phi \mu ^+\mu ^-$ : LHCb measures the differential BF in bins of $m^2(\mu ^+\mu ^-)$. It also performs an angular analysis and measures $F_L$, $S_3$, $S_4$, $S_7$, $A_5$, $A_6$, $A_8$ and $A_9$ in bins of $m^2(\mu ^+\mu ^-)$  [948].

• ${B^{0}_s} \rightarrow \phi \gamma$ : LHCb has measured the photon polarization  [407].

7.5 Radiative and leptonic decays of $B^{0}$and $B^{+}$ mesons.

This section reports different observables for leptonic and radiative $B^{0}$ and $B^{+}$ meson decays, including processes in which the photon yields a pair of charged or neutral leptons. Tables 234 and 235 provide compilations of branching fractions of $B^{+}$mesons to radiative, and lepton-flavor/number-violating final states, respectively. Tables 236 and 237 provide compilations of branching fractions of $B^{0}$mesons, and $B^{\,\pm \,}/B^{0}$ meson admixture, respectively. Table 238 contains branching fractions of leptonic and radiative-leptonic $B^{+}$ and $B^{0}$ decays. It is followed by Tables 239 and 240, which give relative branching fractions of $B^{+}$ decays and a compilations of inclusive decays, respectively. Table 241 contains isospin asymmetry measurements.

Figures 182, 183, 184, 185, 186 and 187 show graphic representations of a selection of results given in this section. Footnote symbols indicate that the footnote in the corresponding table should be consulted.

List of other measurements that are not included in the tables:

• $B^+ \rightarrow K^+ \pi ^- \pi ^+ \gamma$ : LHCb has measured the up-down asymmetries in bins of the $K\pi \pi \gamma$ mass  [1022].

• In Ref.  [1023], LHCb has also measured the branching fraction of $B^+ \rightarrow K^+ e^- e^+$ in the $m^2(\ell \ell )$ bin $[1, 6]~{{\mathrm{GeV}^2\!/}c^4}$.

• In the $B^+ \rightarrow \pi ^+ \mu ^+ \mu ^-$ paper  [966], LHCb has also measured the differential branching fraction in bins of $m^2(\ell \ell )$.

• For $B \rightarrow K \ell ^- \ell ^+$, LHCb has measured $F_H$ and $A_\mathrm{FB}$ in 17 (5) bins of $m^2(\ell \ell )$ for the $K^+$ ($K^0_{S}$) final state  [1024]. Belle has measured $F_L$ and $A_\mathrm{FB}$ in 6 $m^2(\ell \ell )$ bins  [1025].

• For the $B \rightarrow K^{*} \ell ^- \ell ^+$ analyses, partial branching fractions and angular observables in bins of $m^2(\ell \ell )$ are also available:

Table 218 Branching fractions of charmless mesonic $B^+$ decays with kaons (part 1) in units of $\times 10^{-6}$. Upper limits are at 90% CL. Where values are shown in red (blue), this indicates that they are new published (preliminary) results since PDG2014

Table 219 Branching fractions of charmless mesonic $B^+$ decays with kaons (part 2) in units of $\times 10^{-6}$. Upper limits are at 90% CL. Where values are shown in red (blue), this indicates that they are new published (preliminary) results since PDG2014

Table 220 Branching fractions of charmless mesonic $B^+$ decays without kaons in units of $\times 10^{-6}$. Upper limits are at 90% CL. Where values are shown in red (blue), this indicates that they are new published (preliminary) results since PDG2014

Table 221 Branching fractions of charmless mesonic $B^{0}$ decays with kaons (part 1) in units of $\times 10^{-6}$. Upper limits are at 90% CL. Where values are shown in red (blue), this indicates that they are new published (preliminary) results since PDG2014

Table 222 Branching fractions of charmless mesonic $B^{0}$ decays with kaons (part 2) in units of $\times 10^{-6}$. Upper limits are at 90% CL. Where values are shown in red (blue), this indicates that they are new published (preliminary) results since PDG2014.

Table 223 Branching fractions of charmless mesonic $B^{0}$ decays with kaons (part 3) in units of $\times 10^{-6}$. Upper limits are at 90% CL. Where values are shown in red (blue), this indicates that they are new published (preliminary) results since PDG2014

• $B^0 \rightarrow K^{*0} e^- e^+$ : LHCb has measured $F_L$, $A_T^{(2)}$, $A_T^\mathrm{Im}$, $A_T^\mathrm{Re}$ in the $[0.002, 1.120] {{\mathrm{GeV}^2\!/}c^4}$ bin of $m^2(\ell \ell )$ [1026], and has also determined the branching fraction in the dilepton mass region $[10,1000] {\mathrm{MeV/}c^2}$ [1023].

• $B \rightarrow K^{*} \ell ^- \ell ^+$ : Belle has measured $F_L$, $A_\mathrm{FB}$, isospin asymmetry in 6 $m^2(\ell \ell )$ bins  [970] and $P_4'$, $P_5'$, $P_6'$, $P_8'$ in 4 $m^2(\ell \ell )$ bins  [1025]. BaBar has measured $F_L$, $A_\mathrm{FB}$, $P_2$ in 5 $m^2(\ell \ell )$ bins  [1027].

• $B^0 \rightarrow K^{*0} \mu ^- \mu ^+$ : LHCb has measured $F_L$, $A_\mathrm{FB}$, $S_3-S_9$, $A_3-A_9$, $P_1-P_3$, $P_4'-P_8'$ in 8 $m^2(\ell \ell )$ bins  [1028]. CMS has measured $F_L$ and $A_\mathrm{FB}$ in 7 $m^2(\ell \ell )$ bins [1029].

• For $B \rightarrow X_s \ell ^- \ell ^+$ ($X_s$ is a hadronic system with an s quark), Belle has measured $A_\mathrm{FB}$ in bins of $m^2(\ell \ell )$ with a sum of 10 exclusive final states  [1030].

  Fig. 176

  

  Selection of high-precision charmless mesonic B meson branching fraction measurements

• $B^0 \rightarrow K^+ \pi ^- \mu ^+ \mu ^-$, with $1330< m(K^+ \pi ^-) < 1530~{\mathrm{GeV/}c^2}$: LHCb has measured the partial branching fraction in bins of $m^2(\mu ^- \mu ^+)$ in the range $[0.1,8.0] {{\mathrm{GeV}^2\!/}c^4}$, and has also determined angular moments  [1031].

7.6 Charge asymmetries in $b$-hadron decays

This section contains, in Tables 242, 243, 244, 245, 246 247, compilations of $C\!P$ asymmetries in decays of various $b$-hadrons: $B^{+}$, $B^{0}$mesons, $B^{\,\pm \,}/B^{0}$ admixtures, ${B^{0}_s}$ mesons and finally $\varLambda _b^0$ baryons. Measurements of time-dependent $C\!P$ asymmetries are not listed here but are discussed in Sect. 4.

Figure 188 shows a graphic representation of a selection of results given in this section. Footnote symbols indicate that the footnote in the corresponding table should be consulted.

List of other measurements that are not included in the tables:

• In the paper [1044], LHCb has measured the triple-product asymmetries for the decays $\varLambda _b^0 \rightarrow p\pi ^-\pi ^+\pi ^-$ and $\varLambda _b^0 \rightarrow p\pi ^- K^+ K^-$.

7.7 Polarization measurements in $b$-hadron decays

In this section, compilations of polarization measurements in $b$-hadron decays are given. Tables 248 (249) details measurements of the longitudinal fraction, $f_L$, in $B^{+}$ ($B^{0}$) decays, and Tables 250 (251) the results of the full angular analyses of $B^{+}$ ($B^{0}$) $\rightarrow \phi K^*$ decays. Table 252 gives results of the full angular analysis of $B^{0}\rightarrow \phi K_2^{*0}(1430)$ decays. Tables 253, 254 and 255 detail quantities of ${B^{0}_s}$ decays: $f_L$ measurements, and observables from full angular analyses of decays to $\phi \phi$ and $\phi \overline{K} ^{*0}$.

Table 224 Branching fractions of charmless mesonic $B^{0}$ decays without kaons in units of $\times 10^{-6}$. Upper limits are at 90% CL. Where values are shown in red (blue), this indicates that they are new published (preliminary) results since PDG2014.

Table 225 Relative branching fractions of charmless mesonic $B^{0}$ decays. Upper limits are at 90% CL. Where values are shown in red (blue), this indicates that they are new published (preliminary) results since PDG2014

Figures 189 and 190 show graphic representations of a selection of results shown in this section. Footnote symbols indicate that the footnote in the corresponding table should be consulted (Table 256).

7.8 Decays of $B^+_c$ mesons

Table 257 details branching fractions of $B^+_c$ meson decays to charmless hadronic final states.

Table 226 Branching fractions of charmless baryonic $B^{+}$ decays in units of $\times 10^{-6}$. Upper limits are at 90% CL. Where values are shown in red (blue), this indicates that they are new published (preliminary) results since PDG2014

Table 227 Branching fractions of charmless baryonic $B^{0}$ decays in units of $\times 10^{-6}$. Upper limits are at 90% CL. Where values are shown in red (blue), this indicates that they are new published (preliminary) results since PDG2014

Table 228 Relative branching fractions of charmless baryonic $B$ decays. Where values are shown in red (blue), this indicates that they are new published (preliminary) results since PDG2014

8 Charm physics

8.1 $D^0$-$\overline{D}{}^{\,0}$ mixing and $C\!P$ violation

8.1.1 Introduction

In 2007 Belle [1048] and BaBar [1049] obtained the first evidence of $D^0$-$\overline{D}{}^{\,0}$ mixing, for which experiments had searched for more than two decades. These results were later confirmed by CDF [1050] and more recently by LHCb [1051]. There are now numerous measurements of $D^0$-$\overline{D}{}^{\,0}$ mixing with various levels of sensitivity. All measurements are input into a global fit to determine world average values of mixing parameters, $C\!P$-violation ($CPV$) parameters, and strong phase differences.

Our notation is as follows. The mass eigenstates are denoted

$$\begin{aligned} D^{}_1= & {} p|D^0\rangle -q|\overline{D}{}^{\,0}\rangle \end{aligned}$$

(221)

$$\begin{aligned} D^{}_2= & {} p|D^0\rangle +q|\overline{D}{}^{\,0}\rangle , \end{aligned}$$

(222)

where we use the convention [1052] $C\!P |D^0\rangle =-|\overline{D}{}^{\,0}\rangle$ and $C\!P |\overline{D}{}^{\,0}\rangle =-|D^0\rangle$. Thus in the absence of $C\!P$ violation, $D^{}_1$ is $C\!P$-even and $D^{}_2$ is $C\!P$-odd. The weak phase $\phi$ is defined as $\mathrm{Arg}(q/p)$. The mixing parameters are defined as $x\equiv (m^{}_1-m^{}_2)/\Gamma$ and $y\equiv (\Gamma ^{}_1-\Gamma ^{}_2)/(2\Gamma )$, where $m^{}_1,\,m^{}_2$ and $\Gamma ^{}_1,\,\Gamma ^{}_2$ are the masses and decay widths for the mass eigenstates, and $\Gamma \equiv (\Gamma ^{}_1+\Gamma ^{}_2)/2$.

The global fit determines central values and errors for ten underlying parameters. These consist of the mixing parameters x and y; indirect $CPV$ parameters |q / p| and $\phi$; the ratio of decay rates $R^{}_D\equiv [\Gamma (D^0\!\rightarrow \!K^+\pi ^-)+\Gamma (\overline{D}{}^{\,0}\!\rightarrow \!K^-\pi ^+)]/ [\Gamma (D^0\!\rightarrow \!K^-\pi ^+)+\Gamma (\overline{D}{}^{\,0}\!\rightarrow \!K^+\pi ^-)]$; direct $CPV$ asymmetries $A^{}_D$, $A^{}_K$, and $A^{}_\pi$ in $D^0\!\rightarrow \!K^+\pi ^-$, $K^+K^-$, and $\pi ^+\pi ^-$ decays, respectively; the strong phase difference $\delta$ between $\overline{D}{}^{\,0}\!\rightarrow \!K^-\pi ^+$ and $D^0\!\rightarrow \!K^-\pi ^+$ amplitudes; and the strong phase difference $\delta ^{}_{K\pi \pi }$ between $\overline{D}{}^{\,0}\!\rightarrow \!K^-\rho ^+$ and $D^0\!\rightarrow \!K^-\rho ^+$ amplitudes.

Fig. 177

Branching fractions of charmless baryonic modes with non-strange baryons

Fig. 178

Branching fractions of charmless baryonic modes with strange baryons

The fit uses 49 observables from measurements of $D^0\!\rightarrow \!K^+\ell ^-\nu$, $D^0\!\rightarrow \!K^+K^-$, $D^0\!\rightarrow \!\pi ^+\pi ^-$, $D^0\!\rightarrow \!K^+\pi ^-$, $D^0\!\rightarrow \!K^+\pi ^-\pi ^0$, $D^0\!\rightarrow \!K^0_S\,\pi ^+\pi ^-$, $D^0\!\rightarrow \!\pi ^0\,\pi ^+\pi ^-$, $D^0\!\rightarrow \!K^0_S\,K^+ K^-$, and $D^0\!\rightarrow \!K^+\pi ^-\pi ^+\pi ^-$ decays,³⁷ and from double-tagged branching fractions measured at the $\psi (3770)$ resonance. The relationships between the measured observables and the fitted parameters are given in Table 258. Correlations among observables are accounted for by using covariance matrices provided by the experimental collaborations. Errors are assumed to be Gaussian, and systematic errors among different experiments are assumed to be uncorrelated unless specific correlations have been identified. We have checked this method with a second method that adds together three-dimensional log-likelihood functions for x, y, and $\delta$ obtained from several analyses; this combination accounts for non-Gaussian errors. When both methods are applied to the same set of measurements, equivalent results are obtained.

Table 229 Branching fractions of charmless $\varLambda _b^0$ decays in units of $\times 10^{-6}$. Upper limits are at 90% CL. Where values are shown in red (blue), this indicates that they are new published (preliminary) results since PDG2014

Table 230 Partial branching fractions of $\varLambda _b^0 \rightarrow \mu ^+\mu ^-$ decays in intervals of $q^2=m^2(\mu ^+\mu ^-)$ in units of $\times 10^{-6}$. Where values are shown in red (blue), this indicates that they are new published (preliminary) results since PDG2014

Mixing in the $D^0$, $B^0$, and $B^0_s$ heavy flavor systems is governed by a short-distance box diagram. In the $D^0$ system, this diagram is doubly-Cabibbo-suppressed relative to amplitudes dominating the decay width. In addition, because the d and s quark masses are sufficiently close, this diagram is also GIM-suppressed. Thus the short-distance mixing rate is extremely small, and $D^0$-$\overline{D}{}^{\,0}$ mixing is expected to be dominated by long-distance processes. These are difficult to calculate, and theoretical estimates for x and y range over three orders of magnitude (up to the percent level) [1053,1054,1055,1056].

Almost all methods besides that of the $\psi (3770)\!\rightarrow \!\overline{D}D$ measurements [1057] identify the flavor of the $D^0$ or $\overline{D}{}^{\,0}$ when produced by reconstructing the decay $D^{*+}\!\rightarrow \!D^0\pi ^+$ or $D^{*-}\!\rightarrow \!\overline{D}{}^{\,0}\pi ^-$. The charge of the pion, which has low momentum relative to that of the $D^0$ and is often referred to as the “soft” pion, identifies the D flavor. For this decay $M^{}_{D^*}-M^{}_{D^0}-M^{}_{\pi ^+}\equiv Q\approx 6~\mathrm{MeV}$, which is close to the kinematic threshold; thus analyses typically require that the reconstructed Q be small to suppress backgrounds. An LHCb measurement [1058] of the difference between time-integrated $C\!P$ asymmetries $A_{C\!P}(K^+K^-) - A_{C\!P}(\pi ^+\pi ^-)$ identifies the flavor of the $D^0$ by partially reconstructing $\overline{B}\!\rightarrow \!D^0\mu ^- X$ decays (and charge-conjugates); in this case the charge of the muon originating from the B decay identifies the flavor of the $D^0$.

Table 231 Branching fractions of charmless $\varXi ^{0}_{b}$ decays in units of $\times 10^{-6}$. Upper limits are at 90% CL. Where values are shown in red (blue), this indicates that they are new published (preliminary) results since PDG2014

For time-dependent measurements, the $D^0$ decay time is calculated as $M^{}_{D^0}\times (\vec {\mathbf{d}}\cdot \vec {\mathbf{p}})/(cp^2)$, where $\vec {\mathbf{d}}$ is the displacement vector between the $D^*$ and $D^0$ decay vertices, $\vec {\mathbf{p}}$ is the reconstructed $D^0$ momentum, and p and $M^{}_{D^0}$ are in GeV. The $D^*$ vertex position is taken to be the intersection of the $D^0$ momentum vector with the beamspot profile for $e^+e^-$ experiments, and at the primary interaction vertex for ${\bar{p}}p$ and pp experiments [1050, 1051].

8.1.2 Input observables

The global fit determines central values and errors for ten underlying parameters using a $\chi ^2$ statistic. The fitted parameters are x, y, $R^{}_D$, $A^{}_D$, |q / p|, $\phi$, $\delta$, $\delta ^{}_{K\pi \pi }$, $A^{}_K$ and $A^{}_\pi$. In the $D\!\rightarrow \!K^+\pi ^-\pi ^0$ Dalitz plot analysis [1059], the phases of intermediate resonances in the $\overline{D}{}^{\,0}\!\rightarrow \!K^+\pi ^-\pi ^0$ decay amplitude are determined relative to the phase for $\mathcal{A}(\overline{D}{}^{\,0}\!\rightarrow \!K^+\rho ^-)$, and the phases of intermediate resonances for $D^0\!\rightarrow \!K^+\pi ^-\pi ^0$ are determined relative to the phase for $\mathcal{A}(D^0\!\rightarrow \!K^+\rho ^-)$. As the $\overline{D}{}^{\,0}$ and $D^0$ Dalitz plots are fitted independently, the phase difference $\delta ^{}_{K\pi \pi }$ between the two reference amplitudes cannot be determined from these fits. However, the phase difference can be constrained in the global fit and thus is included as a fitted parameter.

Fig. 179

Branching fractions of charmless $\varLambda _b^0$ decays

Table 232 Branching fractions of charmless ${B^{0}_s}$ decays in units of $\times 10^{-6}$. Upper limits are at 90% CL. Where values are shown in red (blue), this indicates that they are new published (preliminary) results since PDG2014

All input measurements are listed in Tables 259, 260 and 261. The observable $R^{}_M=(x^2+y^2)/2$ is measured in both $D^0\!\rightarrow \!K^+\pi ^-\pi ^+\pi ^-$ [1061] and $D^0\!\rightarrow \!K^+\ell ^-\nu$ decays. In the case of the latter, the HFLAV world average [1060] is used in the global fit. The inputs used for this average [1062,1063,1064,1065] are plotted in Fig. 191. The observables

$$\begin{aligned} y^{}_{C\!P}= & {} \frac{1}{2}\bigg (\bigg |\frac{q}{p}\bigg | + \bigg |\frac{p}{q}\bigg |\bigg ) y\cos \phi - \frac{1}{2}\bigg (\bigg |\frac{q}{p}\bigg | - \bigg |\frac{p}{q}\bigg |\bigg ) x\sin \phi \nonumber \\\end{aligned}$$

(223)

$$\begin{aligned} A^{}_\Gamma= & {} \frac{1}{2}\bigg (\bigg |\frac{q}{p}\bigg | - \bigg |\frac{p}{q}\bigg |\bigg ) y\cos \phi - \frac{1}{2}\bigg (\bigg |\frac{q}{p}\bigg | + \bigg |\frac{p}{q}\bigg |\bigg ) x\sin \phi \nonumber \\ \end{aligned}$$

(224)

Table 233 Relative branching fractions of charmless ${B^{0}_s}$ decays. Upper limits are at 90% CL. Where values are shown in red (blue), this indicates that they are new published (preliminary) results since PDG2014

are also HFLAV world average values [1060]; the inputs used for these averages are plotted in Figs. 192 and 193, respectively. The $D^0\!\rightarrow \!K^+\pi ^-$ measurements used are from Belle [1066, 1067], BaBar [1049], CDF [1068], and more recently LHCb [1051, 1069]; earlier measurements have much less precision and are not used. The observables from $D^0\!\rightarrow \!K^0_S\,\pi ^+\pi ^-$ decays are measured in two ways: assuming $C\!P$ conservation ($D^0$ and $\overline{D}{}^{\,0}$ decays combined), and allowing for $C\!P$ violation ($D^0$ and $\overline{D}{}^{\,0}$ decays fitted separately). The no-$CPV$ measurements are from Belle [1070], BaBar [1071], and LHCb [1072], but for the $CPV$-allowed case only Belle measurements [1070] are available. The $D^0\!\rightarrow \!K^+\pi ^-\pi ^0, D^0\rightarrow K^0_S K^+ K^-$ and $D^0\!\rightarrow \!\pi ^0\,\pi ^+\pi ^-$ results are from BaBar [1059, 1073], the $D^0\!\rightarrow \!K^+\pi ^-\pi ^+\pi ^-$ results are from LHCb [1061], and the $\psi (3770)\!\rightarrow \!\overline{D}D$ results are from CLEOc [1057].

Fig. 180

Branching fractions of charmless non-leptonic ${B^{0}_s}$ decays

Fig. 181

Branching fractions of charmless leptonic ${B^{0}_s}$ decays

The relationships between the observables and the fitted parameters are listed in Table 258. For each set of correlated observables we construct a difference vector $\vec {V}$ between measured values and those calculated from fitted parameters using these relations; e.g., for $D^0\!\rightarrow \!K^0_S\,\pi ^+\pi ^-$ decays, $\vec {V}=(\Delta x,\Delta y,\Delta |q/p|,\Delta \phi )$. The contribution of a set of observables to the fit $\chi ^2$ is calculated as $\vec {V}\cdot (M^{-1})\cdot \vec {V}^T$, where $M^{-1}$ is the inverse of the covariance matrix for the measurement. Covariance matrices are constructed from the correlation coefficients among the measured observables. These correlation coefficients are also listed in Tables 259, 260 and 261.

Table 234 Branching fractions of charmless semileptonic and radiative $B^+$ decays in units of $\times 10^{-6}$. Upper limits are at 90% CL. Where values are shown in red (blue), this indicates that they are new published (preliminary) results since PDG2014

Table 235 Branching fractions of charmless semileptonic $B^+$ decays to LFV and LNV final states in units of $\times 10^{-6}$. Upper limits are at 90% CL. Where values are shown in red (blue), this indicates that they are new published (preliminary) results since PDG2014

Table 236 Branching fractions of charmless semileptonic and radiative $B^{0}$ decays in units of $\times 10^{-6}$. Upper limits are at 90% CL. Where values are shown in red (blue), this indicates that they are new published (preliminary) results since PDG2014

8.1.3 Fit results

The global fit uses MINUIT with the MIGRAD minimizer, and all errors are obtained from MINOS [1090]. Four separate fits are performed:

• 1. assuming $C\!P$ conservation, i.e., fixing $A^{}_D\!=\!0$, $A_K\!=\!0$, $A^{}_\pi \!=\!0$, $\phi \!=\!0$, and $|q/p|\!=\!1$;

• 2. assuming no direct $CPV$ in doubly Cabibbo-suppressed (DCS) decays ($A^{}_D\!=\!0$) and fitting for parameters (x, y, |q / p|) or $(x,y,\phi )$;

• 3. assuming no direct $CPV$ in DCS decays and fitting for alternative parameters [1091, 1092] $x^{}_{12}= 2|M^{}_{12}|/\Gamma$, $y^{}_{12}= |\Gamma ^{}_{12}|/\Gamma$, and $\phi ^{}_{12}= \mathrm{Arg}(M^{}_{12}/\Gamma ^{}_{12})$, where $M^{}_{12}$ and $\Gamma ^{}_{12}$ are the off-diagonal elements of the $D^0$-$\overline{D}{}^{\,0}$ mass and decay matrices, respectively. The parameter $\phi ^{}_{12}$ is a weak phase that is responsible for $C\!P$ violation in mixing.

• 4. allowing full $CPV$ (floating all parameters).

For the fits assuming no-direct-$CPV$ in DCS decays, in addition to $A^{}_D\!=\!0$ we impose other constraints that reduce four independent parameters to three.³⁸ For fit (b) we impose the relation [1092, 1093] $\tan \phi = (1-|q/p|^2)/(1+|q/p|^2)\times (x/y)$ in two ways: first we float parameters x, y, and $\phi$ and from these derive |q / p|; we then repeat the fit floating x, y, and |q / p| and from these derive $\phi$. The central values returned by the two fits are identical, but the first fit yields MINOS errors for $\phi$, while the second fit yields MINOS errors for |q / p|. For no-direct-$CPV$ fit (c), we fit for underlying parameters $x^{}_{12}$, $y^{}_{12}$, and $\phi ^{}_{12}$, and from these calculate x, y, |q / p|, and $\phi$ to which measured observables are compared. All fit results are listed in Table 262. For the $CPV$-allowed fit, individual contributions to the $\chi ^2$ are listed in Table 263. The total $\chi ^2$ is 76.8 for $49-10=39$ degrees of freedom.

Table 237 Branching fractions of charmless semileptonic and radiative decays of $B^{\,\pm \,}/B^{0}$ admixture in units of $\times 10^{-6}$. Upper limits are at 90% CL. Where values are shown in red (blue), this indicates that they are new published (preliminary) results since PDG2014

Table 238 Branching fractions of leptonic and radiative-leptonic $B^{+}$ and $B^{0}$ decays in units of $\times 10^{-6}$. Upper limits are at 90% CL. Where values are shown in red (blue), this indicates that they are new published (preliminary) results since PDG2014

Table 239 Relative branching fractions of semileptonic and radiative $B^{+}$ decays. Where values are shown in red (blue), this indicates that they are new published (preliminary) results since PDG2014

Table 240 Branching fractions of $B^{+}/B^{0}\rightarrow \bar{q}$ gluon decays in units of $\times 10^{-6}$. Upper limits are at 90% CL. Where values are shown in red (blue), this indicates that they are new published (preliminary) results since PDG2014

Confidence contours in the two dimensions (x, y) or $(|q/p|,\phi )$ are obtained by allowing, for any point in the two-dimensional plane, all other fitted parameters to take their preferred values. The resulting $1\sigma -5\sigma$ contours are shown in Fig. 194 for the $C\!P$-conserving case, in Fig. 195 for the no-direct-$CPV$ case, and in Fig. 196 for the $CPV$-allowed case. The contours are determined from the increase of the $\chi ^2$ above the minimum value. One observes that the (x, y) contours for the no-$CPV$ fit are very similar to those for the $CPV$-allowed fit. In the latter fit, the $\chi ^2$ at the no-mixing point $(x,y)\!=\!(0,0)$ is 450 units above the minimum value, which, for two degrees of freedom, corresponds to a confidence level³⁹ (C.L.) $>11.5\sigma$. Thus, no mixing is excluded at this high level. In the $(|q/p|,\phi )$ plot, the point (1, 0) is within the $1\sigma$ contour; thus the data is consistent with $C\!P$ conservation.

One-dimensional likelihood curves for individual parameters are obtained by allowing, for a fixed value of a selected parameter, all other fitted parameters to take their preferred values. The resulting functions $\Delta \chi ^2=\chi ^2-\chi ^2_\mathrm{min}$ ($\chi ^2_\mathrm{min}$ is the minimum value) are shown in Fig. 197. The points where $\Delta \chi ^2=3.84$ determine 95% C.L. intervals for the parameters. These intervals are listed in Table 262.

Table 241 Isospin asymmetry in radiative and semileptonic B meson decays. The notations are those adopted by the PDG. Where values are shown in red (blue), this indicates that they are new published (preliminary) results since PDG2014

Fig. 182

Branching fractions of $b\rightarrow s\ell ^{+}\ell ^{-}$ decays

Fig. 183

Branching fractions of $b\rightarrow \ell ^{+}\ell ^{-}(\ell ^{+}\ell ^{-}), \ell ^{+}\ell ^{-}\gamma$ and $b\rightarrow d \ell ^{+}\ell ^{-}$ decays

Fig. 184

$X \ell ^+ \ell ^-$ modes with an inclusive lepton pair and/or inclusive X

Fig. 185

Limits on lepton-flavor-violating decays

Fig. 186

Limits on lepton-number-violating decays

Fig. 187

Branching fractions of charmless B decays with neutrinos

8.1.4 Conclusions

From the fit results listed in Table 262 and shown in Figs. 196 and 197, we conclude that:

• the experimental data consistently indicate that $D^0$ mesons mix. The no-mixing point $x=y=0$ is excluded at $>11.5\sigma$. The parameter x differs from zero by $1.9\sigma$, and y differs from zero by $9.4\sigma$. This mixing is presumably dominated by long-distance processes, which are difficult to calculate. Thus unless it turns out that $|x|\gg |y|$ [1053] (which is not indicated), it will be difficult to identify new physics from (x, y) alone.

• Since $y^{}_{C\!P}$ is positive, the $C\!P$-even state is shorter-lived as in the $K^0$-$\bar{K}{}^{\,0}$ system. However, since x also appears to be positive, the $C\!P$-even state is heavier, unlike in the $K^0$-$\overline{K}{}^{\,0}$ system.

• There is no evidence for $CPV$ arising from $D^0$-$\overline{D}{}^{\,0}$ mixing ($|q/p|\ne 1$) or from a phase difference between the mixing amplitude and a direct decay amplitude ($\phi \ne 0$). The CDF experiment (and initially LHCb) measured a time-integrated asymmetry in $D^0\!\rightarrow \!K^+K^-, \pi ^+\pi ^-$ decays that hints at direct $CPV$ (see Table 261); however, recent measurements from LHCb with higher statistics disfavor this hypothesis and are consistent with zero.

8.2 $C\!P$ asymmetries

One way $C\!P$ violation manifests itself is if the decay rate for a particle differs from that of its $C\!P$-conjugate[1095]. Such phenomena can be classified into two broad categories, termed direct $C\!P$ violation and indirect $C\!P$ violation [1096]. Direct $C\!P$ violation refers to charm changing, $\Delta C\!=\!1$, processes and can occur in both charged and neutral charm hadron decays. It results from interference between two different decay amplitudes (e.g., a penguin and tree amplitude) that have different weak (CKM) and strong phases.⁴⁰ In the Standard Model a difference in strong phases may arise for example due to final-state interactions (FSI)[1097], different isospin amplitudes, intermediate resonance contributions, or different partial waves. A difference in weak phases arises from different CKM vertex couplings, as is often the case for tree and penguin diagrams. Within the SM direct $C\!P$ violation is expected only in Singly Cabibbo Suppressed (SCS) charm decays, as only these decays receive a contribution from the penguin amplitude. This type of $C\!P$ violation depends on the decay mode, the SM asymmetries may reach a percent level. Indirect $C\!P$ violation refers to $\Delta C=2$ processes and arises in $D^0$ decays due to $D^0-\overline{D}{}^{\,0}$ mixing. It can occur as an asymmetry in the mixing itself, or it can result from interference between a decay amplitude following mixing and a non-mixed amplitude. Within the SM charm indirect $C\!P$ asymmetry is expected to be universal.

The $C\!P$ asymmetry is defined as the difference between D and $\overline{D}$ partial widths divided by their sum:

$$\begin{aligned} A_{C\!P}= & {} \frac{\Gamma (D)-\Gamma (\overline{D})}{\Gamma (D)+\Gamma (\overline{D})}. \end{aligned}$$

(225)

In the case of $D^+$ and $D^+_s$ decays, $A^{}_{C\!P}$ measures direct $C\!P$ violation; in the case of $D^0$ decays, $A^{}_{C\!P}$ measures direct and indirect $C\!P$ violation combined (see also Sect. 8.4).

In each experiment, care must be taken to correct for production and detection asymmetries. To take into account differences in production rates between D and $\overline{D}$ (which would affect the number of respective decays observed), some experiments (like FOCUS and E791) normalize to a Cabibbo-favored mode. In this case there is the additional benefit that most corrections due to reconstruction inefficiencies cancel out, reducing systematic uncertainties. An implicit assumption is that there is no measurable $C\!P$ violation in the Cabibbo-favored normalizing mode. The $C\!P$ asymmetry is calculated as

$$\begin{aligned} A_{C\!P}= & {} \frac{\eta (D)-\eta (\overline{D})}{\eta (D)+\eta (\overline{D})}, \end{aligned}$$

(226)

where (considering, for example, $D^0 \rightarrow K^-K^+$)

$$\begin{aligned} \eta (D)= & {} \frac{N(D^0 \rightarrow K^-K^+)}{N(D^0 \rightarrow K^-\pi ^+)}, \end{aligned}$$

(227)

$$\begin{aligned} \eta (\overline{D})= & {} \frac{N(\overline{D}{}^{\,0}\rightarrow K^-K^+)}{N(\overline{D}{}^{\,0}\rightarrow K^+\pi ^-)}. \end{aligned}$$

(228)

Other experiments (like LHCb) determine $A_{C\!P}$ through the relation:

$$\begin{aligned} A_\mathrm{meas}= & {} A_{C\!P} + A_\mathrm{prod} + A_\mathrm{det}, \end{aligned}$$

(229)

where $A_\mathrm{meas}$ is the measured asymmetry, $A_\mathrm{prod}$ is the asymmetry in the charm meson production, and $A_\mathrm{det}$ is due to difference in detection efficiencies between positively and negatively charged hadrons.

Values of $A^{}_{C\!P}$ for $D^+$, $D^0$ and $D_s^+$ decays are listed in Tables 264, 265, 266, 267 and 268 respectively. In these tables we report asymmetries for the actual final state, i.e., resonant substructure is implicitly included but not considered separately. The high accuracy of these measurements allows one to see and correct for $C\!P$ violation due to the CPV in $K^0$-$\overline{K}{}^{\,0}$ mixing [1098]. For example, the decay modes $D^+ \rightarrow (\overline{K^0}/K^0)K^+$ and $D_s^+ \rightarrow (\overline{K^0}/K^0)\pi ^+$ (shown in Tables 264 and 268, respectively) are the modes $D^+ \rightarrow K^0_s K^+$ and $D_s^+ \rightarrow K^0_s \pi ^+$ after subtracting for this effect. For multi-body decays some experiments use model independent techniques to reveal local $C\!P$ asymmetry. The first technique (Miranda method) [1099] uses a binned $\chi ^2$ approach to compare the relative density in a bin of phase space of a decay with that of its $C\!P$ conjugate. In the Energy Test technique [1100] two event samples are compared and a test statistic variable (T) is used to determine the average distances of events in phase space. If the distributions of events in both samples are identical, T will randomly fluctuate around a value close to zero.

Overall, $C\!P$ asymmetry measurements have been carried out for 49 charm decay modes, and in several modes the sensitivity is well below $5 \times 10^{-3}$. There is currently no evidence for $C\!P$ violation in the charm meson sector. The $C\!P$ asymmetry observed in the mode $D^+ \rightarrow K^0_s\pi ^+$ is consistent with what expected from the $K^0-\overline{K}{}^{\,0}$ system [1098], and thus it is not attributed to charm.

Neither in the charm baryon sector there is evidence of $C\!P$ asymmetry. These are just two measurements on $\varLambda _C^+$, with limited sensitivity, done by FOCUS [1101] and by CLEO [1102].

Taken together, the limits obtained for $C\!P$ asymmetries in the charm sector pose tight constraints on new physics models.

8.3 T-odd asymmetries

Measuring T-odd asymmetries provides an alternative way to search for $C\!P$ violation in the charm sector, due to ${C\!P}T$ invariance. T-odd asymmetries are measured using triple-product correlations of the form $\vec {a}\cdot (\vec {b}\times \vec {c})$, where a, b, and c are spins or momenta; this combination is odd under time reversal (T). If the triple-product is formed using both spin and momenta, i.e.,

$$\begin{aligned} \vec {s_1} \cdot (\vec {p_2} \times \vec {p}_3), \end{aligned}$$

(230)

then it is even for P-conjugation. However, if only momenta are used, then it is odd for P-conjugation. In this case the asymmetry allows one to probe $C\!P$ violation occurring via P-violation. This may arise in P-odd amplitudes, which are allowed in decays to final states with 4 spinless particles.

Taking as an example the decay mode $D^0 \rightarrow K^+K^-\pi ^+\pi ^-$, one forms the triple-product correlation using the momenta of the final state particles. We note that when using only momenta, at least four daughter particles are required to give a nonzero correlation (as three daughters decay in a plane). Defining for $D^0$ the T-odd correlation

$$\begin{aligned} C_T \equiv \vec {p}^{}_{K^+}\cdot (\vec {p}_{\pi ^+}\times \vec {p}_{\pi ^-}), \end{aligned}$$

(231)

and the corresponding quantity for $\overline{D}{}^{\,0}$

$$\begin{aligned} \overline{C}_T \equiv \vec {p}^{}_{K^-}\cdot (\vec {p}_{\pi ^-}\times \vec {p}_{\pi ^+}), \end{aligned}$$

(232)

one constructs the asymmetry

$$\begin{aligned} A_{T}= & {} \frac{\Gamma (C_T>0)-\Gamma (C_T<0)}{\Gamma (C_T>0)+\Gamma (C_T<0)} \end{aligned}$$

(233)

for $D^0$ decays and

$$\begin{aligned} \overline{A}_{T}= & {} \frac{\Gamma (-\overline{C}_T>0)-\Gamma (-\overline{C}_T<0)}{\Gamma (-\overline{C}_T>0)+\Gamma (-\overline{C}_T<0)} \end{aligned}$$

(234)

for $\overline{D}{}^{\,0}$ decays. In these expressions, $\Gamma$ represents a partial width. The asymmetries $A_T$ and $\overline{A}_T$ depend on the angular distribution of the daughter particles and may be nonzero due to final state interactions or P-violation in weak decays.

Since $P(C_T)=-C_T$ and $C(C_T)=\overline{C_T}$, $C\!P(A_T) = \overline{A}_T$. One can thus construct the $C\!P$-odd (and P-odd, T-odd) quantity

$$\begin{aligned} \mathcal{A}^{}_{T}\equiv & {} \frac{A_{T}-\overline{A}_{T}}{2}\,; \end{aligned}$$

(235)

a nonzero value indicates $C\!P$ violation (see Refs. [1140,1141,1142,1143,1144,1145]).

Recently, this topic has been revisited (see Refs. [1146, 1147]) with the suggestion to use other asymmetries constructed from triple products in multi-body decays to probe C, P, and $C\!P$ symmetries. Up until now, experiments have measured only the asymmetry $\mathcal{A}^{}_T$ defined in Eq. (235). (Note that this asymmetry is referred to in the literature by several names: $A^{}_{T\,\mathrm{viol}}$, $a^P_{C\!P}$, and $a^{T-\mathrm{odd}}_{C\!P}$.)

Values of $\mathcal{A}^{}_{T}$ for $D^+$, $D^+_s$, and $D^0$ decay modes are listed in Table 269. The first measurements were made by FOCUS, and subsequent BaBar measurements reached a sensitivity of $\sim 1\%$. Currently the best sensitivity is from LHCb.

However, despite relatively high precision ($<\!\!1\%$), there is no evidence for $C\!P$ violation.

8.4 Interplay of direct and indirect $C\!P$ violation

In decays of $D^0$ mesons, $C\!P$ asymmetry measurements have contributions from both direct and indirect $C\!P$ violation as discussed in Sect. 8.1. The contribution from indirect $C\!P$ violation depends on the decay-time distribution of the data sample [1092]. This section describes a combination of measurements that allows the extraction of the individual contributions of the two types of $C\!P$ violation. At the same time, the level of agreement for a no-$C\!P$-violation hypothesis is tested. The observables are:

$$\begin{aligned} A_{\Gamma } \equiv \frac{\tau (\overline{D}{}^{\,0}\!\rightarrow \!h^+ h^-) - \tau (D^0 \!\rightarrow \!h^+ h^- )}{\tau (\overline{D}{}^{\,0}\!\rightarrow \!h^+ h^-) + \tau (D^0 \!\rightarrow \!h^+ h^- )}, \end{aligned}$$

(236)

Table 242 $C\!P$ asymmetries of charmless hadronic $B^{+}$ decays (part 1). Where values are shown in red (blue), this indicates that they are new published (preliminary) results since PDG2014

Table 243 $C\!P$ asymmetries of charmless hadronic $B^{+}$ decays (part 2). Where values are shown in red (blue), this indicates that they are new published (preliminary) results since PDG2014

Table 244 $C\!P$ asymmetries of charmless hadronic $B^{0}$ decays. Where values are shown in red (blue), this indicates that they are new published (preliminary) results since PDG2014

Table 245 $C\!P$ asymmetries of charmless hadronic decays of $B^{\pm } \,/B^0$ admixture. Where values are shown in red (blue), this indicates that they are new published (preliminary) results since PDG2014

Table 246 $C\!P$ asymmetries of charmless hadronic ${B^{0}_s}$ decays. Where values are shown in red (blue), this indicates that they are new published (preliminary) results since PDG2014

Table 247 $C\!P$ asymmetries of charmless hadronic $\varLambda _b^0$ decays. Where values are shown in red (blue), this indicates that they are new published (preliminary) results since PDG2014

Fig. 188

$A_{C\!P}$ of most precisely measured modes

where $h^+ h^-$ can be $K^+ K^-$ or $\pi ^+\pi ^-$, and

$$\begin{aligned} \Delta A_{C\!P} \equiv A_{C\!P}(K^+K^-) - A_{C\!P}(\pi ^+\pi ^-), \end{aligned}$$

(237)

where $A_{C\!P}$ are time-integrated $C\!P$ asymmetries. The underlying theoretical parameters are:

$$\begin{aligned} a_{C\!P}^\mathrm{dir}\equiv & {} \frac{|\mathcal{A}_{D^0\rightarrow f} |^2 - |\mathcal{A}_{\overline{D}{}^{\,0}\rightarrow f} |^2}{|\mathcal{A}_{D^0\rightarrow f} |^2 + |\mathcal{A}_{\overline{D}{}^{\,0}\rightarrow f} |^2},\nonumber \\ a_{C\!P}^\mathrm{ind}\equiv & {} \frac{1}{2} \bigg [ \bigg (\bigg |\frac{q}{p}\bigg | + \bigg |\frac{p}{q}\bigg |\bigg ) x \sin \phi - \bigg (\bigg |\frac{q}{p}\bigg | - \bigg |\frac{p}{q}\bigg |\bigg ) y \cos \phi \bigg ],\nonumber \\ \end{aligned}$$

(238)

where $\mathcal{A}_{D\rightarrow f}$ is the amplitude for $D\!\rightarrow \!f$ [1151]. We use the following relations between the observables and the underlying parameters [1152]:

$$\begin{aligned} A_{\Gamma }= & {} - a_{C\!P}^\mathrm{ind} - a_{C\!P}^\mathrm{dir} y_{C\!P},\end{aligned}$$

(239)

$$\begin{aligned} \Delta A_{C\!P}= & {} \Delta a_{C\!P}^\mathrm{dir} \bigg (1 + y_{C\!P} \frac{\overline{\langle t\rangle }}{\tau } \bigg ) + a_{C\!P}^\mathrm{ind} \frac{\Delta \langle t\rangle }{\tau } + \overline{a_{C\!P}^\mathrm{dir}} y_{C\!P} \frac{\Delta \langle t\rangle }{\tau },\nonumber \\\approx & {} \Delta a_{C\!P}^\mathrm{dir} \bigg (1 + y_{C\!P} \frac{\overline{\langle t\rangle }}{\tau } \bigg ) + a_{C\!P}^\mathrm{ind} \frac{\Delta \langle t\rangle }{\tau }. \end{aligned}$$

(240)

Equation (239) constrains mostly indirect $C\!P$ violation, and the direct $C\!P$ violation contribution can differ for different final states. In Eq. (240), $\langle t\rangle /\tau$ denotes the mean decay time in units of the $D^0$ lifetime; $\Delta X$ denotes the difference in quantity X between $K^+K^-$ and $\pi ^+\pi ^-$ final states; and $\overline{X}$ denotes the average for quantity X. We neglect the last term in this relation as all three factors are $\mathcal {O}(10^{-2})$ or smaller, and thus this term is negligible with respect to the other two terms. Note that $\Delta \langle t\rangle /\tau \ll \langle t\rangle /\tau$, and it is expected that $|a_{C\!P}^\mathrm{dir}| < |\Delta a_{C\!P}^\mathrm{dir}|$ because $a_{C\!P}^\mathrm{dir}(K^+K^-)$ and $a_{C\!P}^\mathrm{dir}(\pi ^+\pi ^-)$ are expected to have opposite signs in the Standard Model [1151].

A $\chi ^2$ fit is performed in the plane $\Delta a_{C\!P}^\mathrm{dir}$ vs. $a_{C\!P}^\mathrm{ind}$. For the BaBar result the difference of the quoted values for $A_{C\!P}(K^+K^-)$ and $A_{C\!P}(\pi ^+\pi ^-)$ is calculated, adding all uncertainties in quadrature. This may overestimate the systematic uncertainty for the difference as it neglects correlated errors; however, the result is conservative and the effect is small as all measurements are statistically limited. For all measurements, statistical and systematic uncertainties are added in quadrature when calculating the $\chi ^2$. We use the current world average value $y_{C\!P} = (0.835 \,\pm \, 0.155)\%$ (see Sect. 8.1) and the measurements listed inTable 270.

In this fit, $A_\Gamma (KK)$ and $A_\Gamma (\pi \pi )$ are assumed to be identical. This assumption (expected to hold in the Standard Model) is supported by all measurements to date. A significant relative shift due to final-state dependent $A_\Gamma$ values between $\Delta A_{C\!P}$ measurements with different mean decay times is excluded by these measurements.

The combination plot (see Fig. 198) shows the measurements listed in Table 270 for $\Delta A_{C\!P}$ and $A_\Gamma$.

Table 248 Longitudinal polarization fraction $f_L$ for $B^+$ decays. Where values are shown in red (blue), this indicates that they are new published (preliminary) results since PDG2014

Table 249 Longitudinal polarization fraction $f_L$ for $B^{0}$ decays. Where values are shown in red (blue), this indicates that they are new published (preliminary) results since PDG2014

Table 250 Results of the full angular analyses of $B^+ \rightarrow \phi K^{*+}$ decays. Where values are shown in red (blue), this indicates that they are new published (preliminary) results since PDG2014

Table 251 Results of the full angular analyses of $B^0 \rightarrow \phi K^{*0}$ decays. Where values are shown in red (blue), this indicates that they are new published (preliminary) results since PDG2014

Table 252 Results of the full angular analyses of $B^0 \rightarrow \phi K_2^{*0}(1430)$ decays. Where values are shown in red (blue), this indicates that they are new published (preliminary) results since PDG2014

Table 253 Longitudinal polarization fraction $f_L$ for ${B^{0}_s}$ decays. Where values are shown in red (blue), this indicates that they are new published (preliminary) results since PDG2014

Table 254 Results of the full angular analyses of ${B^{0}_s} \rightarrow \phi \phi$ decays. Where values are shown in red (blue), this indicates that they are new published (preliminary) results since PDG2014

From the fit, the change in $\chi ^2$ from the minimum value for the no-$CPV$ point (0,0) is 4.7, which corresponds to a C.L. of $9.3\times 10^{-2}$ for two degrees of freedom. Thus the data are consistent with the no-$C\!P$-violation hypothesis at $9.3\%$ C.L. This p-value corresponds to $1.7\sigma$. The central values and $\,\pm \,1\sigma$ uncertainties for the individual parameters are

$$\begin{aligned} a_{C\!P}^\mathrm{ind}= & {} (+0.030 \,\pm \, 0.026 )\% \nonumber \\ \Delta a_{C\!P}^\mathrm{dir}= & {} (-0.134 \,\pm \, 0.070 )\%. \end{aligned}$$

(241)

Compared to the previous average, the tension in the difference between direct $C\!P$ violation in the two final states is reduced, while the common indirect $C\!P$ violation moved away from the no-$C\!P$-violation point by about one standard deviation.

8.5 Semileptonic decays

8.5.1 Introduction

Semileptonic decays of D mesons involve the interaction of a leptonic current with a hadronic current. The latter is nonperturbative and cannot be calculated from first principles; thus it is usually parameterized in terms of form factors. The transition matrix element is written

$$\begin{aligned} \mathcal{M}= & {} -i\,\frac{G_F}{\sqrt{2}}\,V^{}_{cq}\,L^\mu H_\mu , \end{aligned}$$

(242)

where $G_F$ is the Fermi constant and $V^{}_{cq}$ is a CKM matrix element. The leptonic current $L^\mu$ is evaluated directly from the lepton spinors and has a simple structure; this allows one to extract information about the form factors (in $H^{}_\mu$) from data on semileptonic decays [1154]. Conversely, because there are no strong final-state interactions between the leptonic and hadronic systems, semileptonic decays for which the form factors can be calculated allow one to determine $V^{}_{cq}$ [2].

8.5.2 $D\!\rightarrow \!P \overline{\ell }\nu _{\ell }$ decays

When the final state hadron is a pseudoscalar, the hadronic current is given by

$$\begin{aligned} H_\mu= & {} \langle P(p) | \bar{q}\gamma _\mu c | D(p') \rangle \nonumber \\= & {} f_+(q^2)\times \bigg [ (p' + p)_\mu -\frac{m_D^2-m_P^2}{q^2}q_\mu \bigg ] \nonumber \\&+ f_0(q^2)\frac{m_D^2-m_P^2}{q^2}q_\mu , \end{aligned}$$

(243)

where $m_D$ and $p'$ are the mass and four momentum of the parent D meson, $m_P$ and p are those of the daughter meson, $f_+(q^2)$ and $f_0(q^2)$ are form factors, and $q = p' - p$. Kinematics require that $f_+(0) = f_0(0)$. The contraction $q_\mu L^\mu$ results in terms proportional to $m^{}_\ell$[1155], and thus for $\ell =e$ the terms proportionals to $q_\mu$ in Eq. (243) are negligible. For light leptons only the $f_+(q^2)$ vector form factor is relevant and the differential partial width is

$$\begin{aligned} \frac{d\Gamma (D \rightarrow P \bar{\ell }\nu _\ell )}{dq^2\, d\cos \theta _\ell }= & {} \frac{G_F^2|V_{cq}|^2}{32\pi ^3} p^{*\,3}|f_{+}(q^2)|^2\sin \theta ^2_\ell , \end{aligned}$$

(244)

where ${p^*}$ is the magnitude of the momentum of the final state hadron in the D rest frame, and $\theta _\ell$ is the angle of the lepton in the $\ell \nu$ rest frame with respect to the direction of the pseudoscalar meson in the D rest frame.

Table 255 Results of the full angular analyses of ${B^{0}_s} \rightarrow \phi \overline{K}^{*0}$ decays. Where values are shown in red (blue), this indicates that they are new published (preliminary) results since PDG2014

Fig. 189

Longitudinal polarization fraction in charmless B decays

Fig. 190

Longitudinal polarization fraction in charmless ${B^{0}_s}$ decays

8.5.3 Form factor parameterizations

The form factor is traditionally parameterized with an explicit pole and a sum of effective poles:

$$\begin{aligned} f_+(q^2)= & {} \frac{f_+(0)}{(1-\alpha )}\Bigg [ \bigg (\frac{1}{1- q^2/m^2_\mathrm{pole}}\bigg )\nonumber \\&+\ \sum _{k=1}^{N}\frac{\rho _k}{1- q^2/(\gamma _k\,m^2_\mathrm{pole})}\Bigg ], \end{aligned}$$

(245)

where $\rho _k$ and $\gamma _k$ are expansion parameters and $\alpha$ is a parameter that normalizes the form factor at $q^2=0$, $f_+(0)$. The parameter $m_{\mathrm{pole}}$ is the mass of the lowest-lying $c\bar{q}$ resonance with the vector quantum numbers; this is expected to provide the largest contribution to the form factor for the $c\!\rightarrow \!q$ transition. The sum over N gives the contribution of higher mass states. For example, for $D\rightarrow \pi$ transitions the dominant resonance is expected to be $D^*(2010)$, and thus $m^{}_\mathrm{pole}=m^{}_{D^*(2010)}$. For $D\rightarrow K$ transitions, the dominant contribution is expected from $D^{*}_s(2112)$, with $m^{}_\mathrm{pole}=m^{}_{D^{*}_{s}(2112)}$.

Table 256 Results of the full angular analyses of ${B^{0}_s} \rightarrow {K}^{*0}\overline{K}^{*0}$ decays. Where values are shown in red (blue), this indicates that they are new published (preliminary) results since PDG2014

Table 257 Relative branching fractions of $B^+_c$ decays. Where values are shown in red (blue), this indicates that they are new published (preliminary) results since PDG2014

Table 258 Left: decay modes used to determine fitted parameters $x,\,y,\,\delta ,\,\delta ^{}_{K\pi \pi },\,R^{}_D,\,A^{}_D,\,|q/p|$, and $\phi$. Middle: the observables measured for each decay mode. Right: the relationships between the observables measured and the fitted parameters. $\langle t\rangle$ is the mean reconstructed decay time for $D^0\!\rightarrow \!K^+K^-$ or $D^0\!\rightarrow \!\pi ^+\pi ^-$ decays

Table 259 Observables used in the global fit except those from time-dependent $D^0\!\rightarrow \!K^+\pi ^-$ measurements, and those from direct $CPV$ measurements. The $D^0\!\rightarrow \!K^+\pi ^-\pi ^0$ observables are $x'' \equiv x\cos \delta ^{}_{K\pi \pi } + y\sin \delta ^{}_{K\pi \pi }$ and $y'' \equiv -x\sin \delta ^{}_{K\pi \pi } + y\cos \delta ^{}_{K\pi \pi }$

Table 260 Time-dependent $D^0\!\rightarrow \!K^+\pi ^-$ observables used for the global fit. The observables $R^+_D$ and $R^-_D$ are related to parameters $R^{}_D$ and $A^{}_D$ via $R^{\pm } \,_D = R^{}_D (1\,\pm \, A^{}_D)$

Table 261 Measurements of time-integrated $C\!P$ asymmetries. The observable $A^{}_{C\!P}(f)\equiv [\Gamma (D^0\!\rightarrow \!f)-\Gamma (\overline{D}{}^{\,0}\!\rightarrow \!f)]/ [\Gamma (D^0\!\rightarrow \!f)+\Gamma (\overline{D}{}^{\,0}\!\rightarrow \!f)]$, and $\Delta \langle t\rangle$ is the difference between mean reconstructed decay times for $D^0\!\rightarrow \!K^+K^-$ and $D^0\!\rightarrow \!\pi ^+\pi ^-$ decays (due to different trigger and reconstruction efficiencies)

8.5.4 Simple pole

Equation (245) can be simplified by neglecting the sum over effective poles, leaving only the explicit vector meson pole. This approximation is referred to as “nearest pole dominance” or “vector-meson dominance.” The resulting parameterization is

$$\begin{aligned} f_+(q^2)= & {} \frac{f_+(0)}{(1-q^2/m^2_\mathrm{pole})}. \end{aligned}$$

(246)

However, values of $m_{\mathrm{pole}}$ that give a good fit to the data do not agree with the expected vector meson masses [1156]. To address this problem, the “modified pole” or Becirevic–Kaidalov (BK) parameterization [1157] was introduced. $m_\mathrm{pole} /\sqrt{\alpha _\mathrm{BK}}$ is interpreted as the mass of an effective pole, higher than $m_\mathrm{pole}$, thus it is expected that $\alpha _\mathrm{BK}<1$.

The parameterization takes the form

$$\begin{aligned} f_+(q^2)= & {} \frac{f_+(0)}{(1-q^2/m^2_\mathrm{pole})} \frac{1}{\bigg (1-\alpha ^{}_\mathrm{BK}\frac{q^2}{m^2_\mathrm{pole}}\bigg )}. \end{aligned}$$

(247)

These parameterizations are used by several experiments to determine form factor parameters. Measured values of $m^{}_\mathrm{pole}$ and $\alpha ^{}_\mathrm{BK}$ are listed in Tables 271 and 272 for $D\rightarrow K\ell \nu _{\ell }$ and $D\rightarrow \pi \ell \nu _{\ell }$ decays, respectively.

8.5.5 z expansion

An alternative series expansion around some value $q^2=t_0$ to parameterize $f^{}_+(q^2)$ can be used [1154, 1158,1159,1160]. This parameterization is model independent and satisfies general QCD constraints, being suitable for fitting experimental data. The expansion is given in terms of a complex parameter z, which is the analytic continuation of $q^2$ into the complex plane:

$$\begin{aligned} z(q^2,t_0)= & {} \frac{\sqrt{t_+ - q^2} - \sqrt{t_+ - t_0}}{\sqrt{t_+ - q^2} + \sqrt{t_+ - t_0}}, \end{aligned}$$

(248)

where $t_{\pm } \, \equiv (m_D \,\pm \, m_P)^2$ and $t_0$ is the (arbitrary) $q^2$ value corresponding to $z=0$. The physical region corresponds to $\,\pm \,|z|_{max} = \,\pm \, 0.051$ for $D\rightarrow K \ell \nu _\ell$ and $= \,\pm \, 0.17$ for $D\rightarrow \pi \ell \nu _\ell$, using $t_{0}= t_{+} (1-\sqrt{1-t_{-}/t_{+}})$.

The form factor is expressed as

$$\begin{aligned} f_+(q^2)= & {} \frac{1}{P(q^2)\,\phi (q^2,t_0)}\sum _{k=0}^\infty a_k(t_0)[z(q^2,t_0)]^k, \end{aligned}$$

(249)

Fig. 191

World average value of $R^{}_M=(x^2+y^2)/2$ from Ref. [1060], as calculated from $D^0\!\rightarrow \!K^+\ell ^-\nu$ measurements [1062,1063,1064,1065]

where the $P(q^2)$ factor accommodates sub-threshold resonances via

$$\begin{aligned} P(q^2)\equiv & {} {\left\{ \begin{array}{ll} 1 &{}\quad (D\rightarrow \pi ) \\ z(q^2,M^2_{D^*_s}) &{}\quad (D\rightarrow K). \end{array}\right. } \end{aligned}$$

(250)

The “outer” function $\phi (t,t_0)$ can be any analytic function, but a preferred choice (see, e.g. Refs. [1158, 1159, 1161]) obtained from the Operator Product Expansion (OPE) is

$$\begin{aligned} \phi (q^2,t_0)= & {} \alpha \bigg (\sqrt{t_+ - q^2}+\sqrt{t_+ - t_0}\bigg )\nonumber \\&\times \frac{t_+ - q^2}{(t_+ - t_0)^{1/4}}\ \frac{\left( \sqrt{t_+ - q^2}\ +\ \sqrt{t_+ - t_-}\right) ^{3/2}}{\left( \sqrt{t_+ - q^2}+\sqrt{t_+}\right) ^5}, \nonumber \\ \end{aligned}$$

(251)

with $\alpha = \sqrt{\pi m_c^2/3}$. The OPE analysis provides a constraint upon the expansion coefficients, $\sum _{k=0}^{N}a_k^2 \le 1$. These coefficients receive $1/M_D$ corrections, and thus the constraint is only approximate. However, the expansion is expected to converge rapidly since $|z|<0.051\ (0.17)$ for $D\!\rightarrow \!K$ ($D\!\rightarrow \!\pi$) over the entire physical $q^2$ range, and Eq. (249) remains a useful parameterization. The main disadvantage as compared to phenomenological approaches is that there is no physical interpretation of the fitted coefficients $a_K$.

Fig. 192

World average value of $y^{}_{C\!P}$ from Ref. [1060], as calculated from $D^0\!\rightarrow \!K^+ K^-\!\!,\,\pi ^+\pi ^-$ measurements [1079,1080,1081,1082,1083,1084,1085,1086]

Fig. 193

World average value of $A^{}_\Gamma$ from Ref. [1060], as calculated from $D^0\!\rightarrow \!K^+K^-$ and $D^0\!\rightarrow \!\pi ^+\pi ^-$ measurements [1084, 1085, 1087,1088,1089]

8.5.6 Three-pole formalism

An update of the vector pole dominance model has been developed for the $D \rightarrow \pi \ell \nu _\ell$ channel [1162]. It uses information of the residues of the semileptonic form factor at its first two poles, the $D^*(2010)$ and $D^{*'}(2600)$ resonances. The form factor is expressed as an infinite sum of residues from $J^P =1^-$ states with masses $m_{D^*_n}$:

$$\begin{aligned} f_+(q^2) = \sum _{n=0}^\infty \frac{\displaystyle {\underset{q^2= m_{D^*_n}^2}{\mathrm{Res}}} f_+(q^2)}{m_{D^*_n}^2-q^2}, \end{aligned}$$

(252)

with the residues given by

$$\begin{aligned} \displaystyle {\underset{q^2=m_{D_n^*}^2}{\mathrm{Res}}} f_+(q^2)= \frac{1}{2} m_{D_n^*} f_{D_n^*} g_{D_n^*D\pi }. \end{aligned}$$

(253)

Values of the $f_{D^*}$ and $f_{D^{*'}}$ decay constants have been obtained by lattice QCD calculations, relative to $f_{D}$, with 2$\%$ and 28$\%$ precision, respectively [1162]. The couplings to the $D\pi$ state, $g_{D^*D\pi }$ and $g_{D^{*'} D\pi }$, are extracted from measurements of the $D^*(2010)$ and $D^{*'}(2600)$ widths by BaBar and LHCb experiments [1163,1164,1165]. Thus the contribution from the first pole is known with a $3\%$ accuracy. The contribution from the $D^{*'}(2600)$ is determined with poorer accuracy, $\sim$30%, mainly due to lattice uncertainties. A superconvergence condition [1166] is applied:

$$\begin{aligned} \sum _{n=0}^\infty {\displaystyle {\underset{q^2=m_{D^*_n}^2}{\mathrm{Res}}} f_+(q^2) }= 0, \end{aligned}$$

(254)

protecting the form factor behavior at large $q^2$. Within this model the first two poles are not sufficient to describe the data, and a third effective pole needs to be included.

One of the advantages of this phenomenological model is that it can be extrapolated outside the charm physical region, providing a method to extract the CKM matrix element $V_{ub}$ using the ratio of the form factors of the $D\rightarrow \pi \ell \nu$ and $B\rightarrow \pi \ell \nu$ decay channels. It will be used once lattice calculations provide the form factor ratio $f^{+}_{B\pi }(q^2)/f^{+}_{D\pi }(q^2)$ at the same pion energy.

This form factor description can be extended to the $D\rightarrow K \ell \nu$ decay channel, considering the contribution of several $c\bar{s}$ resonances with $J^P = 1^-$. The first two pole masses contributing to the form factor correspond to the $D^{*}_s(2112)$ and $D^{*}_{s1}(2700)$ resonant states [327]. A constraint on the first residue can be obtained using information of the $f_K$ decay constant [327] and the g coupling extracted from the $D^{*+}$ width [1163]. The contribution from the second pole can be evaluated using the decay constants from [1167], the measured total width and the ratio of $D^{*} K$ and DK decay branching fractions [327].

8.5.7 Experimental techniques and results

Different techniques by several experiments are used to measure D meson semileptonic decays having a pseudoscalar particle in the final state. The most recent results are provided by the BaBar [1168] and BES III [1169, 1170] collaborations. Belle [1171], BaBar [1172], and CLEO-c [1173, 1174] have all previously reported results. Belle fully reconstructs $e^+e^- \rightarrow D \bar{D} X$ events from the continuum under the $\varUpsilon (4S)$ resonance, achieving very good $q^2$ resolution (15$\mathrm{~MeV}^2$) and a low background level but with a low efficiency. Using 282 $bad hbox ^{-1}$ of data, about 1300 and 115 signal semileptonic decays are isolated for both lepton channels together ($e+\mu$), for the Cabibbo-favored and Cabibbo-suppressed modes, respectively. The BaBar experiment uses a partial reconstruction technique in which the semileptonic decays are tagged via $D^{*+}\rightarrow D^0\pi ^+$ decays. The D direction and neutrino energy are obtained using information from the rest of the event. With 75 $bad hbox ^{-1}$ of data, 74,000 signal events in the $D^0 \rightarrow {K}^- e^+ \nu$ mode are obtained. This technique provides a large signal yield but also a high background level and a poor $q^2$ resolution (ranging from 66 to 219 MeV$^2$). In this case the measurement of the branching fraction is obtained by normalizing to the $D^0 \rightarrow K^- \pi ^+$ decay channel; thus the measurement would benefit from future improvements in the determination of this reference channel. The Cabibbo-suppressed mode has been recently measured using the same technique and 350 fb$^{-1}$ data. For this measurement, 5000 $D^0 \rightarrow {\pi }^- e^+ \nu$ signal events were reconstructed [1168].

The CLEO-c experiment uses two different methods to measure charm semileptonic decays. Tagged analyses [1173] rely on the full reconstruction of $\Psi (3770)\rightarrow D {\overline{D}}$ events. One of the D mesons is reconstructed in a hadronic decay mode, the other in the semileptonic channel. The only missing particle is the neutrino so the $q^2$ resolution is very good and the background level very low. With the entire CLEO-c data sample, 818 $\mathrm \,pb ^{-1}$, 14,123 and 1374 signal events are reconstructed for the $D^0 \rightarrow K^{-} e^+\nu$ and $D^0\rightarrow \pi ^{-} e^+\nu$ channels, and 8467 and 838 for the $D^+\rightarrow {\overline{K}}^{0} e^+\nu$ and $D^+\rightarrow \pi ^{0} e^+\nu$ decays, respectively. Another technique without tagging the D meson in a hadronic mode (“untagged” in the following) has been also used by CLEO-c [1174]. In this method, the entire missing energy and momentum in an event are associated with the neutrino four momentum, with the penalty of larger background as compared to the tagged method. Using the “tagged” method the BES III experiment measures the $D^0 \rightarrow {K}^- e^+ \nu$ and $D^0 \rightarrow {\pi }^- e^+ \nu$ decay channels. With 2.9 fb$^{-1}$ they fully reconstruct 70,700 and 6300 signal events for each channel, respectively[1169]. In a separated analysis the BES III experiment measures also the $D^+$ decay mode into $D^+ \rightarrow K^{0}_{L} e^+ \nu$ [1170]. Using several tagged hadronic events they reconstruct 20,100 semileptonic candidates.

Previous measurements were also performed by several experiments. Events registered at the $\varUpsilon (4S)$ energy corresponding to an integrated luminosity of 7 $bad hbox ^{-1}$ were analyzed by CLEO III [1175]. Fixed target photo-production experiments performed also measurements of the normalized form factor distribution (FOCUS [1176]) and total decay rates (Mark-III [1177], E653 [1178, 1179], E687 [1180, 1181], E691 [1182], BES II [1183, 1184], CLEO II [1185]). In the FOCUS fixed target photo-production experiment, $D^0$ semimuonic events were obtained from the decay of a $D^{*+}$, with a kaon or a pion detected.

Table 262 Results of the global fit for different assumptions concerning $CPV$

Table 263 Individual contributions to the $\chi ^2$ for the $CPV$-allowed fit

Results of the hadronic form factor parameters, $m_{pole}$ and $\alpha _{BK}$, obtained from the measurements discussed above, are given in Tables 271 and 272.

The z-expansion formalism has been used by BaBar [1168, 1172], BES III[1186] and CLEO-c [1173], [1174]. Their fits use the first three terms of the expansion, and the results for the ratios $r_1\equiv a_1/a_0$ and $r_2\equiv a_2/a_0$ are listed in Tables 273 and 274.

8.5.8 Combined results for the $D\rightarrow K\ell \nu _\ell$ channel

The $q^2$ distribution provided by each individual measurement is used to determine a combined result by performing a fit to the z-expansion formalism at second order. Results for the form factor normalization $f_+^K(0)|V_{cs}|$ and the shape parameters $r_1$ and $r_2$ for each individual measurement and for the combination are presented in Table 275. Measurements have been corrected with respect to the original ones using recent values from PDG [327]. This includes updated branching fractions of normalization channels, corrected CKM matrix elements and the D meson lifetime. The BaBar measurement has been corrected accounting for final-state radiation. The result for the $D^+\rightarrow K_{L}^{0} e^+ \nu _e$ decay channel from BES III [1170] is included as a constraint in the combined result since correlation matrices are not provided. Correlation coefficients of the parameters are quoted in the last column of Table 275. The $\chi ^2$ per degree of freedom is 114.7/101. Results are shown in Fig. 201.

In the combination of the electron and muon channels, the measurements with muons are corrected for the reduction of phase space and for the $f_0(q^2)$ contribution [1187]. Channels with a $D^0$ or a $D^+$ are combined assuming isospin invariance and using physical meson and lepton masses. These combined results are noted as $D \rightarrow K \ell \nu _\ell$ in the following. Hadronic form factors are assumed to be the same for charged and neutral D mesons. Separate results for the $D^0 \rightarrow K^- \ell ^+ \nu _\ell$ and $D^+ \rightarrow \bar{K}^0 \ell ^+ \nu _\ell$ decay channels are shown in Table 276 and Fig. 200. Using the fitted parameters and integrating over the full $q^2$ range, the combined semileptonic branching fraction, expressed in terms of the $D^0$ decay channel gives:

$$\begin{aligned} {\mathcal {B}}(D^0 \rightarrow K^- \ell ^+ \nu _\ell ) = (3.490\,\pm \, 0.011\,\pm \,0.020)\% \end{aligned}$$

(255)

Data from the different experiments are also fitted within the three-pole form factor formalism. Constraints on the first and second poles are imposed using information of the $D^{*}_s(2112)$ and $D^{*}_{s1}(2700)$ resonances. Results are presented in Table 277. Fitted parameters are the first two residues $\gamma _{0}^{K}=\underset{ q^2=m_{D^{*}_s(2112)}^2}{\mathrm{Res}} f_{+}^{K}(q^2)$ and $\gamma ^{\pi }_{1}=\underset{q^2=m_{D^{*}_{s1}(2700)}^2}{\mathrm{Res}} f_{+}^{K}(q^2)$ and an effective mass, $m_{D^{*''}_{s\;\mathrm{eff}}}$, accounting for higher mass hadronic contributions. It is found that the fitted effective third pole mass is larger than the mass of the second radial excitation, around $3.2~{\mathrm{GeV/}c^2}$, as expected. The contribution to the form factor by only the $D^{*}_{s}$ resonance is disfavoured by the data. Figure 199 (left) shows the result of the fitted form factors for the z-expansion and three-pole parametrizations (Figs. 200, 201).

8.5.9 Combined results for the $D\rightarrow \pi \ell \nu _\ell$ channel

The combined result for the $D\rightarrow \pi \ell \nu _{\ell }$ decay channel is obtained from a fit to BaBar, Belle, BES III, and CLEO-c data, with updated input values from [327]. The available measurements are fitted in bins of $q^2$ to the z-expansion model at second order. Results of the individual fits for each experiment and the combined result are shown in Table 278. The $\chi ^2$ per degree of freedom of the combined fit is 51 / 55.

Using the fitted parameters and integrating over the full $q^2$ range, the combined semileptonic branching fraction, expressed in terms of the $D^0$ decay channel gives:

$$\begin{aligned} {\mathcal {B}}(D^0 \rightarrow \pi ^- \ell ^+ \nu _\ell ) = (2.891\,\pm \, 0.030\,\pm \,0.022)\times 10^{-3}\nonumber \\ \end{aligned}$$

(256)

Results of the three-pole model to the $D \rightarrow \pi \ell \nu _\ell$ data are shown in Table 279. Fitted parameters are the first two residues $\gamma _{0}^{\pi }=\underset{ q^2=m_{D^{*}}^2}{\mathrm{Res}} f^{\pi }_{+}(q^2)$ and $\gamma ^{\pi }_{1}=\underset{q^2=m_{D^{*'}}^2}{\mathrm{Res}} f^{\pi }_{+}(q^2)$ (which are constrained using present measurements of masses and widths of the $D^*(2010)$ and $D^{*'}(2600)$ mesons, and lattice computations of decay constants, following [1162]), and an effective mass, $m_{D^{*''}_\mathrm{eff}}$, accounting for higher mass hadronic contributions. The $V_{cd}$ value entering in the fit is given in Eq. (257). The $\chi ^2$ per degree of freedom of the combined fit is 57.5 / 57.

The effective mass $m_{D^{*''}_\mathrm{eff}}$ is larger than the predicted mass of the second radially excited state with $J^P = 1^{-}$ ($\sim$ 3.11 $\mathrm{GeV}$), indicating that more contributions are needed to explain the form factor. Figure 199 (right) shows the result of the combined form factor for the z-expansion and three-pole parameterizations.

Fig. 194

Two-dimensional contours for mixing parameters (x, y), for no $CPV$

Fig. 195

Two-dimensional contours for theoretical parameters $(x^{}_{12},y^{}_{12})$ (top left), $(x^{}_{12},\phi ^{}_{12})$ (top right), and $(y^{}_{12},\phi ^{}_{12})$ (bottom), for no direct $CPV$

Fig. 196

Two-dimensional contours for parameters (x, y) (top) and $(|q/p|,\phi )$ (bottom), allowing for $CPV$

8.5.10 $V_{cs}$ and $V_{cd}$ determination

Assuming unitarity of the CKM matrix, the values of the CKM matrix elements entering in charm semileptonic decays are evaluated from the $V_{ud}$, $V_{td}$ and $V_{cb}$ elements [327]:

$$\begin{aligned} \begin{array}{ll} |V_{cs}| &{} = 0.97343 \,\pm \, 0.00015,\\ |V_{cd}| &{} = 0.22521 \,\pm \, 0.00061. \end{array} \end{aligned}$$

(257)

Using the combined values of $f_+^K(0)|V_{cs}|$ and $f_+^{\pi }(0)|V_{cd}|$ in Tables 275 and 278, leads to the form factor values:

$$\begin{aligned} \begin{array}{ll} f_+^K(0) &{} = 0.7423 \,\pm \, 0.0035, \\ f_+^{\pi }(0) &{}= 0.6327 \,\pm \, 0.0086, \end{array} \end{aligned}$$

which are in agreement with present lattice QCD computations [222]: $f_+^K(0) = 0.747 \,\pm \, 0.019$ and $f_+^\pi (0) = 0.666 \,\pm \, 0.029$. The experimental accuracy is at present higher than the one from lattice calculations. If instead one assumes the lattice QCD form factor values, one obtains for the CKM matrix elements using the combined results in Tables 275 and 278:

$$\begin{aligned} \begin{array}{ll} |V_{cs}| &{}= 0.967 \,\pm \, 0.025, \\ |V_{cd}| &{}= 0.2140 \,\pm \, 0.0097, \end{array} \end{aligned}$$

still compatible with unitarity of the CKM matrix.

8.5.11 $D\!\rightarrow \!V\overline{\ell }\nu _\ell$ decays

When the final state hadron is a vector meson, the decay can proceed through both vector and axial vector currents, and four form factors are needed. The hadronic current is $H^{}_\mu = V^{}_\mu + A^{}_\mu$, where [1155]

$$\begin{aligned} V_\mu= & {} \langle V(p,\varepsilon ) | \bar{q}\gamma _\mu c | D(p') \rangle \ \nonumber \\= & {} \frac{2V(q^2)}{m_D+m_V} \varepsilon _{\mu \nu \rho \sigma }\varepsilon ^{*\nu }p^{\prime \rho }p^\sigma \end{aligned}$$

(258)

$$\begin{aligned} A_\mu= & {} \langle V(p,\varepsilon ) | -\bar{q}\gamma _\mu \gamma _5 c | D(p') \rangle \nonumber \\= & {} -i\,(m_D+m_V)A_1(q^2){\varepsilon _{\mu }^{*}} \nonumber \\&+\ i \frac{A_2(q^2)}{m_D+m_V}(\varepsilon ^*\cdot q)(p' + p)_\mu \nonumber \\&+\, i\,\frac{2m_V}{q^2}\bigg (A_3(q^2)-A_0(q^2)\bigg )[\varepsilon ^*\cdot (p' + p)] q_\mu .\nonumber \\ \end{aligned}$$

(259)

In this expression, $m_V$ is the daughter meson mass and

$$\begin{aligned} A_3(q^2) = \frac{m_D + m_V}{2m_V}A_1(q^2)\ -\ \frac{m_D - m_V}{2m_V}A_2(q^2). \end{aligned}$$

(260)

Kinematics require that $A_3(0) = A_0(0)$. Terms proportional to $q_\mu$ are only important for the case of $\tau$ leptons. Thus, only three form factors are relevant in the decays involving $\mu$ or e: $A_1(q^2)$, $A_2(q^2)$ and $V(q^2)$. The differential partial width is

$$\begin{aligned} \frac{d\Gamma (D \rightarrow V \overline{\ell }\nu _\ell )}{dq^2\, d\cos \theta _\ell }= & {} \frac{G_F^2\,|V_{cq}|^2}{128\pi ^3m_D^2}\,p^*\,q^2\times \bigg [\frac{(1-\cos \theta _\ell )^2}{2}|H_-|^2\ \nonumber \\&+ \frac{(1+\cos \theta _\ell )^2}{2}|H_+|^2 + \sin ^2\theta _\ell |H_0|^2\bigg ],\nonumber \\ \end{aligned}$$

(261)

where $H^{}_{\pm } \,$ and $H^{}_0$ are helicity amplitudes, corresponding to helicities of the V meson or virtual W, given by

$$\begin{aligned} H_{\pm } \,= & {} \frac{1}{m_D + m_V}\bigg [(m_D + m_V)^2A_1(q^2){\mp } 2m^{}_D\,p^* V(q^2)\bigg ] \nonumber \\\end{aligned}$$

(262)

$$\begin{aligned} H_0= & {} \frac{1}{|q|}\frac{m_D^2}{2m_V(m_D + m_V)} \times \ \bigg [ \bigg (1- \frac{m_V^2 - q^2}{m_D^2}\bigg ) \nonumber \\&\times (m_D + m_V)^2 A_1(q^2)- 4{p^*}^2 A_2(q^2) \bigg ]. \end{aligned}$$

(263)

Fig. 197

The function $\Delta \chi ^2=\chi ^2-\chi ^2_\mathrm{min}$ for fitted parameters $x,\,y,\,\delta ,\,\delta ^{}_{K\pi \pi },\,|q/p|$, and $\phi$. The points where $\Delta \chi ^2=3.84$ (denoted by dashed horizontal lines) determine 95% C.L. intervals

$p^*$ is the magnitude of the three-momentum of the V system, measured in the D rest frame, and $\theta _\ell$ is the angle of the lepton momentum, in the W rest frame, with respect to the opposite direction of the D meson [see Fig. 202 for the electron case ($\theta _e$)]. The left-handed nature of the quark current manifests itself as $|H_-|>|H_+|$. The differential decay rate for $D\!\rightarrow \!V\ell \nu$ followed by the vector meson decaying into two pseudoscalars is

$$\begin{aligned}&\frac{d\Gamma (D\!\rightarrow \!V \overline{\ell }\nu , V\!\rightarrow \!P_1P_2)}{dq^2 d\cos \theta _V d\cos \theta _\ell d\chi } \nonumber \\&\quad = \frac{3G_F^2}{2048\pi ^4} |V_{cq}|^2 \frac{p^*(q^2)q^2}{m_D^2} {\mathcal {B}}(V\rightarrow P_1P_2)\ \nonumber \\&\qquad \times \Big \{ (1 + \cos \theta _\ell )^2 \sin ^2\theta _V |H_+(q^2)|^2 \nonumber \\&\qquad +\ (1 - \cos \theta _\ell )^2 \sin ^2\theta _V |H_-(q^2)|^2 \nonumber \\&\qquad +\ 4\sin ^2\theta _\ell \cos ^2\theta _V|H_0(q^2)|^2 \nonumber \\&\qquad -\ 4\sin \theta _\ell (1 + \cos \theta _\ell ) \nonumber \\&\qquad \times \sin \theta _V \cos \theta _V \cos \chi H_+(q^2) H_0(q^2) \nonumber \\&\qquad +\ 4\sin \theta _\ell (1 - \cos \theta _\ell )\nonumber \\&\qquad \times \sin \theta _V \cos \theta _V \cos \chi H_-(q^2) H_0(q^2) \nonumber \\&\qquad -\ 2\sin ^2\theta _\ell \sin ^2\theta _V \cos 2\chi H_+(q^2) H_-(q^2) \Big \}, \end{aligned}$$

(264)

where the helicity angles $\theta ^{}_\ell$, $\theta ^{}_V$, and acoplanarity angle $\chi$ are defined in Fig. 202.

Table 264 $C\!P$ asymmetries $A^{}_{C\!P}= [\Gamma (D^+)-\Gamma (D^-)]/[\Gamma (D^+)+\Gamma (D^-)]$ for two-body $D^{\pm } \,$ decays

Table 265 $C\!P$ asymmetries $A^{}_{C\!P}= [\Gamma (D^+)-\Gamma (D^-)]/[\Gamma (D^+)+\Gamma (D^-)]$ for three- and four-body $D^{\pm } \,$ decays

Table 266 $C\!P$ asymmetries $A^{}_{C\!P}=[\Gamma (D^0)-\Gamma (\overline{D}{}^{\,0})]/[\Gamma (D^0)+\Gamma (\overline{D}{}^{\,0})]$ for two-body $D^0,\overline{D}{}^{\,0}$ decays

Table 267 $C\!P$ asymmetries $A^{}_{C\!P}=[\Gamma (D^0)-\Gamma (\overline{D}{}^{\,0})]/[\Gamma (D^0)+\Gamma (\overline{D}{}^{\,0})]$ for three- and four-body $D^0,\overline{D}{}^{\,0}$ decays

Table 268 $C\!P$ asymmetries $A^{}_{C\!P}= [\Gamma (D_s^+)-\Gamma (D_s^-)]/[\Gamma (D_s^+)+\Gamma (D_s^-)]$ for $D_s^{\pm } \,$ decays

Table 269 Measurements of the T-odd asymmetry $\mathcal{A}^{}_{T} = (A_{T}-\overline{A}_{T})/2$

Ratios between the values of the hadronic form factors expressed at $q^2=0$ are usually introduced:

$$\begin{aligned} r_V \equiv V(0) / A_1(0),\quad r_2 \equiv A_2(0) / A_1(0). \end{aligned}$$

(265)

8.5.12 Form factor measurements

In 2002 FOCUS reported [1188] an asymmetry in the observed $\cos (\theta _V)$ distribution in $D^+\rightarrow K^-\pi ^+\mu ^+\nu$ decays. This is interpreted as evidence for an S-wave $K^-\pi ^+$ component in the decay amplitude. Since $H_0$ typically dominates over $H_{\,\pm \,}$, the distribution given by Eq. (264) is, after integration over $\chi$, roughly proportional to $\cos ^2\theta _V$. Inclusion of a constant S-wave amplitude of the form $A\,e^{i\delta }$ leads to an interference term proportional to $|A H_0 \sin \theta _\ell \cos \theta _V|$; this term causes an asymmetry in $\cos (\theta _V)$. When FOCUS fit their data including this S-wave amplitude, they obtained $A = 0.330 \,\pm \, 0.022 \,\pm \, 0.015~\mathrm{GeV} ^{-1}$ and $\delta = 0.68 \,\pm \, 0.07 \,\pm \, 0.05$ [1189]. Both BaBar [1190] and CLEO-c [1191] have also found evidence for an $f^{}_0 \rightarrow K^+ K^-$ component in semileptonic $D^{}_s$ decays.

The CLEO-c collaboration extracted the form factors $H_+(q^2)$, $H_-(q^2)$, and $H_0(q^2)$ from 11000 $D^+ \rightarrow K^- \pi ^+ \ell ^+ \nu _\ell$ events in a model-independent fashion directly as functions of $q^2$ [1192]. They also determined the S-wave form factor $h_0(q^2)$ via the interference term, despite the fact that the $K\pi$ mass distribution appears dominated by the vector $K^*(892)$ state. It is observed that $H_0(q^2)$ dominates over a wide range of $q^2$, especially at low $q^2$. The transverse form factor $H_t(q^2)$, which can be related to $A_3(q^2)$, is small compared to lattice gauge theory calculations and suggests that the form factor ratio $r_3 \equiv A_3(0) / A_1(0)$ is large and negative.

BaBar [1193] selected a large sample of $244\times 10^3$ $D^+ \rightarrow K^- \pi ^+ e^+ \nu _e$ candidates with a ratio $S/B\sim 2.3$ from an analyzed integrated luminosity of $347~bad hbox ^{-1}$. With four particles emitted in the final state, the differential decay rate depends on five variables. In addition to the four variables defined in previous sections there is also $m^2$, the mass squared of the $K\pi$ system. To analyze the $D^+ \rightarrow K^- \pi ^+ e^+ \nu _e$ decay channel it is assumed that all form factors have a $q^2$ variation given by the simple pole model and the effective pole mass value, $m_A=(2.63 \,\pm \, 0.10 \,\pm \, 0.13)~{\mathrm{GeV/}c^2}$, is fitted for the axial vector form factors. This value is compatible with expectations when comparing with the mass of $J^P=1^+$ charm mesons. For the mass dependence of the form factors, a Breit–Wigner with a mass dependent width and a Blatt–Weisskopf damping factor is used. For the S-wave amplitude, considering what was measured in $D^+ \rightarrow K^- \pi ^+\pi ^+$ decays, a polynomial variation below the $\overline{K}^*_0(1430)$ and a Breit–Wigner distribution, above are assumed. For the polynomial part, a linear term is sufficient to fit data. It is verified that the variation of the S-wave phase is compatible with expectations from elastic $K\pi$  [343, 1194] (after correcting for $\delta ^{3/2}$) according to the Watson theorem. At variance with elastic scattering, a negative relative sign between the S- and P-waves is measured; this is compatible with the previous theorem. Contributions from other spin-1 and spin-2 resonances decaying into $K^-\pi ^+$ are considered.

In Fig. 203, measured values from CLEO-c of the products $q^2H_0^2(q^2)$ and $q^2h_0(q^2)H_0(q^2)$ are compared with corresponding results from BaBar illustrating the difference in behavior of the scalar $h_0$ component and the $H_0$ form factor. For this comparison, the plotted values from BaBar for the two distributions are fixed to 1 at $q^2=0$. The different behavior of $h_0(q^2)$ and $H_0(q^2)$ can be explained by their different dependence in the $p^*$ variable.

Table 280 lists measurements of $r_V$ and $r_2$ from several experiments. Most of the measurements assume that the $q^2$ dependence of hadronic form factors is given by the simple pole ansatz. Some of these measurements do not consider a S-wave contribution and it is included in the measured values. The measurements are plotted in Fig. 204, which shows that they are all consistent.

8.6 Leptonic decays

Purely leptonic decays of $D^+$ and $D_{s}^+$ mesons are among the simplest and theoretically cleanest probes of $c\rightarrow d$ and $c\rightarrow s$ quark flavor-changing transitions. The branching fraction of leptonic decays that proceed via the annihilation of the initial quark-antiquark pair ($c\overline{d}$ or $c\overline{s}$) into a virtual $W^+$ that finally materializes as an antilepton-neutrino pair ($\ell ^+\nu _{\ell }$). Their Standard Model branching fraction is given by

$$\begin{aligned}&{\mathcal {B}}(D_{q}^+\rightarrow \ell ^+\nu _{\ell })\nonumber \\&\quad = \frac{G_F^2}{8\pi }\tau _{D_q}f_{D_{q}}^2|V_{cq}|^2m_{D_{q}}m_{\ell }^2 \bigg (1-\frac{m_{\ell }^2}{m_{D_{q}}^2} \bigg )^2. \end{aligned}$$

(266)

Here, $m_{D_{q}}$ is the $D_{q}$ meson mass, $\tau _{D_q}$ is its lifetime, $m_{\ell }$ is the charged lepton mass, $|V_{cq}|$ is the magnitude of the relevant CKM matrix element, and $G_F$ is the Fermi coupling constant. The parameter $f_{D_{q}}$ is the $D_q$ meson decay constant and is related to the wave-function overlap of the meson’s constituent quark and anti-quark. The decay constants have been predicted using several methods, the most accurate and robust being the lattice gauge theory (LQCD) calculations. The Flavor Lattice Averaging Group [222] combines all LQCD calculations and provides averaged values for $f_D$ and $f_{D_s}$ (see Table 281) that are used within this section to extract the magnitudes of the $V_{cd}$ and $V_{cs}$ CKM matrix elements from experimentally measured branching fractions of $D^+\rightarrow \ell ^+\nu _{\ell }$ and $D_s^+\rightarrow \ell ^+\nu _{\ell }$ decays, respectively.

The leptonic decays of pseudoscalar mesons are helicity-suppressed and their decay rates are thus proportional to the square of the charged lepton mass. Leptonic decays into electrons, with expected ${\mathcal {B}}\lesssim 10^{-7}$, are not experimentally observable yet, whereas decays to taus are favored over decays to muons. In particular, the ratio of the latter decays is equal to $R^{D_q}_{\tau /\mu }\equiv {\mathcal {B}}(D^+_q\rightarrow \tau ^+\nu _{\tau })/{\mathcal {B}}(D^+_q\rightarrow \mu ^+\nu _{\mu })=m_{\tau }^2/m_{\mu }^2\cdot (1-m^2_{\tau }/m^2_{D_q})^2/(1-m^2_{\mu }/m^2_{D_q})^2$, and amounts to $9.76\,\pm \,0.03$ in the case of $D_s^+$ decays and to $2.67\,\pm \,0.01$ in the case of $D^+$ decays based on the world average values of masses of the muon, tau and $D_q$ meson given in Ref. [6]. Any deviation from this expectation could only be interpreted as violation of lepton universality in charged currents and would hence point to NP effects [1203].

Averages presented within this subsection are weighted averages, in which correlations between measurements and dependencies on input parameters are taken into account. There is only one new experimental result on leptonic charm decays since our last report from 2014 – the measurements of ${\mathcal {B}}(D_s^+\rightarrow \mu ^+\nu _{\mu })$ and ${\mathcal {B}}(D_s^+\rightarrow \tau ^+\nu _{\tau })$ by BESIII collaboration [1204]. The Lattice QCD calculations of the D and $D_s$ meson decay constants have improved significantly since our last report and we use the latest averages of $N_f=2+1+1$ calculations provided by the Flavour Lattice Averaging Group [222] in our determinations of the CKM matrix elements $|V_{cd}|$ and $|V_{cs}|$.

8.6.1 $D^+\rightarrow \ell ^+\nu _{\ell }$ decays and $|V_{cd}|$

We use measurements of the branching fraction ${\mathcal {B}}(D^+\rightarrow \mu ^+\nu _{\mu })$ from CLEO-c [1103] and BESIII [1205] to calculate the world average (WA) value. We obtain

$$\begin{aligned} {\mathcal {B}}^\mathrm{WA}(D^+\rightarrow \mu ^+\nu _{\mu }) = (3.74\,\pm \,0.17)\times 10^{-4}, \end{aligned}$$

(267)

from which we determine the product of the decay constant and the CKM matrix element to be

$$\begin{aligned} f_{D}|V_{cd}| = (45.9\,\pm \,1.1)~\text{ MeV }, \end{aligned}$$

(268)

where the uncertainty includes the uncertainty on ${\mathcal {B}}^\mathrm{WA}(D^+\rightarrow \mu ^+\nu _{\mu })$ and external inputs⁴¹ needed to extract $f_{D}|V_{cd}|$ from the measured branching fraction using Eq. (266). Using the LQCD value for $f_D$ from Table 281 we finally obtain the CKM matrix element $V_{cd}$ to be

$$\begin{aligned} |V_{cd}| = 0.2164\,\pm \,0.0050(\mathrm exp.)\,\pm \,0.0015(\mathrm LQCD), \end{aligned}$$

(269)

where the uncertainties are from the experiments and lattice calculations, respectively. All input values and the resulting world averages are summarized in Table 282 and plotted in Fig. 205.

Table 270 Inputs to the fit for direct and indirect $C\!P$ violation. The first uncertainty listed is statistical, and the second is systematic

Fig. 198

Plot of all data and the fit result. Individual measurements are plotted as bands showing their $\,\pm \,1\sigma$ range. The no-$CPV$ point (0,0) is shown as a filled circle, and the best fit value is indicated by a cross showing the one-dimensional uncertainties. Two-dimensional $68\%$ C.L., $95\%$ C.L., and $99.7\%$ C.L. regions are plotted as ellipses

The upper limit on the ratio of branching fractions is found to be $R_{\tau /\mu }^D<3.2$ at 90% C.L., which is just slightly above the SM expected value.

8.6.2 $D_s^+\rightarrow \ell ^+\nu _{\ell }$ decays and $|V_{cs}|$

We use measurements of the absolute branching fraction ${\mathcal {B}}(D_s^+\rightarrow \mu ^+\nu _{\mu })$ from CLEO-c [1137], BaBar [1206], Belle [1207], and BESIII [1204], and obtain a WA value of

$$\begin{aligned} {\mathcal {B}}^\mathrm{WA}(D_{s}^+\rightarrow \mu ^+\nu _{\mu }) = (5.54\,\pm \,0.23)\times 10^{-3}. \end{aligned}$$

(270)

Table 271 Results for $m_\mathrm{pole}$ and $\alpha _\mathrm{BK}$ from various experiments for $D^0\rightarrow K^-\ell ^+\nu$ and $D^+\rightarrow \bar{K}^0\ell ^+\nu$ decays

Table 272 Results for $m_\mathrm{pole}$ and $\alpha _\mathrm{BK}$ from various experiments for $D^0\rightarrow \pi ^-\ell ^+\nu$ and $D^+\rightarrow \pi ^0\ell ^+\nu$ decays

Table 273 Results for $r_1$ and $r_2$ from various experiments for the $D\rightarrow K\ell \nu _{\ell }$ decay channel. The correlation coefficient between these parameters is larger than 0.9

Table 274 Results for $r_1$ and $r_2$ from various experiments, for $D\rightarrow \pi \ell \nu _{\ell }$. The correlation coefficient between these parameters is larger than 0.9

Table 275 Results of the fits to $D\rightarrow K \ell \nu _{\ell }$ measurements from several experiments, using the z-expansion. External inputs have been updated to PDG [327]. The correlation coefficients listed in the last column refer to $\rho _{12} \equiv \rho _{|V_{cs}| f_{+}^{K}(0),r_1}$, $\rho _{13} \equiv \rho _{|V_{cs}| f_{+}^{K}(0),r_2}$, and $\rho _{23} \equiv \rho _{r_1,r_2}$ and are for the total uncertainties (statistical $\oplus$ systematic). The result for the $D^+\rightarrow K_{L}^{0} e^+ \nu _e$ decay channel from BES III [1170] is included in the combined results as a constraint on the normalization, $|V_{cs}| f_{+}^{K}(0)$. The entry others refers to total decay rates measured by Mark-III [1177], E653 [1178, 1179], E687[1180, 1181], E691 [1182], BES II [1183, 1184] and CLEO II [1185]

Table 276 Results for the $D^0 \rightarrow K^- \ell ^+ \nu _\ell$ and $D^+ \rightarrow \bar{K}^0 \ell ^+ \nu _\ell$ decays channels using the z-expansion formalism at second order

Table 277 Results of the three-pole model form factors obtained from a fit to all measurements. Fitted parameters are the first two residues $\gamma _{0}^{K}$ and $\gamma _{1}^{K}$, which are constrained using present measurements of masses and widths of the $D^{*}_{s}$ and $D^{*}_{s1}$ mesons, and lattice computations of decay constants, and the effective mass, $m_{D^{*''}_{s\;\mathrm{eff}}}$, accounting for higher mass hadronic contributions

The WA value for ${\mathcal {B}}(D_s^+\rightarrow \tau ^+\nu _{\tau })$ is also calculated from CLEO-c, BaBar, Belle, and BESIII measurements. CLEO-c made separate measurements for $\tau ^+\rightarrow e^+\nu _e\overline{\nu }{}_{\tau }$ [1208], $\tau ^+\rightarrow \pi ^+\overline{\nu }{}_{\tau }$ [1137], and $\tau ^+\rightarrow \rho ^+\overline{\nu }{}_{\tau }$ [1209], BaBarmade separate measurements for $\tau ^+\rightarrow e^+\nu _e\overline{\nu }{}_{\tau }$ [1206] and $\tau ^+\rightarrow \mu ^+\nu _{\mu }\overline{\nu }{}_{\tau }$, Belle made separate measurements for $\tau ^+\rightarrow e^+\nu _e\overline{\nu }{}_{\tau }$, $\tau ^+\rightarrow \mu ^+\nu _{\mu }\overline{\nu }{}_{\tau }$, and $\tau ^+\rightarrow \pi ^+\overline{\nu }{}_{\tau }$ [1207], and BESIII made measurements using $\tau ^+\rightarrow \pi ^+\overline{\nu }{}_{\tau }$ [1204] decays. Combining all of them we obtain the WA value of

$$\begin{aligned} {\mathcal {B}}^\mathrm{WA}(D_{s}^+\rightarrow \tau ^+\nu _{\tau }) = (5.51\,\pm \,0.24)\times 10^{-2}. \end{aligned}$$

(271)

The ratio of branching fractions is found to be

$$\begin{aligned} R_{\tau /\mu }^{D_{s}} = 9.95\,\pm \,0.57, \end{aligned}$$

(272)

and is consistent with the value expected in the SM.

From the average values of branching fractions of muonic and tauonic decays we determine⁴² the product of $D_s$ meson decay constant and the $|V_{cs}|$ CKM matrix element to be

$$\begin{aligned} f_{D_s} |V_{cs}|=(250.3\,\pm \,4.5)~\text{ MeV }, \end{aligned}$$

(273)

where the uncertainty is due to the uncertainties on ${\mathcal {B}}^\mathrm{WA}(D_s^+\rightarrow \mu ^+\nu _{\mu })$ and ${\mathcal {B}}^\mathrm{WA}(D_s^+\rightarrow \tau ^+\nu _{\tau })$ and the external inputs. All input values and the resulting world averages are summarized in Table 283 and plotted in Fig. 206. To obtain the averages given within this subsection and in Table 283 we have taken into account the correlations within each experiment⁴³ for the uncertainties related to: normalization, tracking, particle identification, signal and background parameterizations, and peaking background contributions.

Using the LQCD value for $f_{D_s}$ from Table 281, we finally obtain the magnitude of the CKM matrix element $V_{cs}$ to be

$$\begin{aligned} |V_{cs}| = 1.006\,\pm \,0.018(\mathrm exp.)\,\pm \,0.005(\mathrm LQCD), \end{aligned}$$

(274)

where the uncertainties are from the experiments and lattice calculations, respectively.

8.6.3 Comparison with other determinations of $|V_{cd}|$ and $|V_{cs}|$

Table 284 summarizes and Fig. 207 shows all determinations of the CKM matrix elements $|V_{cd}|$ and $|V_{cs}|$. As can be seen, the most precise direct determinations of these CKM matrix elements are those from leptonic and semileptonic $D_{(s)}$ decays. The values are in agreement within uncertainties with the values obtained from the global fit assuming CKM matrix unitarity.

Fig. 199

Form factors as function of $q^2$ for the $D \rightarrow K \ell \nu _\ell$ (left) and $D \rightarrow \pi \ell \nu _\ell$ (right) channels, obtained from a fit to all experimental data. Central values (central lines) and uncertainties (one $\sigma$ deviation) are shown for the z-expansion and the 3-pole parameterization

Fig. 200

Results of the combined fit shown separately for the $D^0\rightarrow K^-\ell ^+\nu$ and $D^+\rightarrow \bar{K}^0 \ell ^+\nu$ decay channels. Ellipses are shown for 68% C.L

Fig. 201

The $D^0\rightarrow K^-\ell ^+\nu$ (left) and $D^0\rightarrow \pi ^-\ell ^+\nu$ (right) 68% C.L. error ellipses from the average fit of the 3-parameter z-expansion results

Table 278 Results of the fits to $D\rightarrow \pi \ell \nu _{\ell }$ measurements from several experiments, using the z-expansion. External inputs are updated to PDG [327]. The correlation coefficients listed in the last column refer to $\rho _{12} \equiv \rho _{|V_{cd}| f_{+}^{\pi }(0),r_1}$, $\rho _{13} \equiv \rho _{|V_{cd}| f_{+}^{\pi }(0),r_2}$, and $\rho _{23} \equiv \rho _{r_1,r_2}$ and are for the total uncertainties (statistical $\oplus$ systematic)

Table 279 Results of the three-pole model to BaBar, Belle, BES III and CLEO-c (tagged and untagged) data. Fitted parameters are the first two residues $\gamma _{0}^{\pi }$ and $\gamma _{1}^{\pi }$, which are constrained using present measurements of masses and widths of the $D^*$ and $D^{*'}$ mesons, and lattice computations of decay constants, and the effective mass, $m_{D^{*''}_\mathrm{eff}}$, accounting for higher mass hadronic contributions

8.6.4 Extraction of $D_{(s)}$ meson decay constants

Assuming unitarity of the CKM matrix, the values of the elements relevant in the case of (semi-)leptonic charm decays are known from the global fit of the CKM matrix and are given in Table 284. These values can be used to extract the D and $D_s$ meson decay constants from the experimentally measured products $f_{D}|V_{cd}|$ and $f_{D_s}|V_{cs}|$ using Eq. (268) and Eq. (273), respectively. This leads to the experimentally measured $D_{(s)}$ meson decay constants to be:

$$\begin{aligned} f_D^\mathrm{exp}= & {} (203.7\,\pm \,4.9)~\mathrm{MeV,}\end{aligned}$$

(275)

$$\begin{aligned} f_{D_s}^\mathrm{exp}= & {} (257.1\,\pm \,4.6)~\mathrm{MeV,} \end{aligned}$$

(276)

and the ratio of the constants is determined to be

$$\begin{aligned} f_{D_s}^\mathrm{exp}/f_D^\mathrm{exp} = 1.262\,\pm \,0.037. \end{aligned}$$

(277)

The values are in agreement with the LQCD determinations given in Table 281 within the uncertainties. The largest discrepancy is in the determinations of the ratio of the decay constants where the agreement is only at the level of $2.4\sigma$.

Fig. 202

Decay angles $\theta _V$, $\theta _\ell$ and $\chi$. Note that the angle $\chi$ between the decay planes is defined in the D-meson reference frame, whereas the angles $\theta ^{}_V$ and $\theta ^{}_\ell$ are defined in the V meson and W reference frames, respectively

Fig. 203

Comparison between CLEO-c and BaBar results for the quantities $q^2 H_0^2(q^2)$ and $q^2 h_0(q^2) H_0(q^2)$

Table 280 Results for $r_V$ and $r_2$ from various experiments

8.7 Hadronic decays of $D_s$ mesons

BaBar, CLEO-c and Belle collaborations have measured the absolute branching fractions of hadronic decays, $D_{s}^+\rightarrow K^-K^+\pi ^+$, $D_{s}^+\rightarrow \overline{K}{}^0\pi ^+$, and $D_{s}^+\rightarrow \eta \pi ^+$. The first two decay modes are the reference modes for the measurements of branching fractions of the $D_{s}^+$ decays to any other final state. Table 285 and Fig. 208 summarise the individual measurements and averaged values, which are found to be

$$\begin{aligned}&{\mathcal {B}}^\mathrm{WA}(D_{s}^+\rightarrow K^-K^+\pi ^+) = (5.44\,\pm \,0.14)\%,\end{aligned}$$

(278)

$$\begin{aligned}&{\mathcal {B}}^\mathrm{WA}(D_{s}^+\rightarrow \overline{K}{}^0\pi ^+) = (3.00\,\pm \,0.09)\%,\end{aligned}$$

(279)

$$\begin{aligned}&{\mathcal {B}}^\mathrm{WA}(D_{s}^+\rightarrow \eta \pi ^+) = (1.71\,\pm \,0.08)\%, \end{aligned}$$

(280)

where the uncertainties are total uncertainties. These averages are the same as in our previous report from 2014. The ${\mathcal {B}}(D_{s}^+\rightarrow K^-K^+\pi ^+)$ is for a phase space integrated decay and therefore includes all intermediate resonances.

Fig. 204

A comparison of $r_2$ (filled inverted triangles) and $r_V$ (open triangles) values from various experiments. The first seven measurements are for $D^+\rightarrow K^-\pi ^+ l^+\nu _l$ decays. Also shown as a line with 1-$\sigma$ limits is the average of these. The last two points are $D_s^+$ and $D^0$ decays

8.8 Two-body hadronic $D^0$ decays and final state radiation

Measurements of the branching fractions for the decays $D^0\rightarrow K^-\pi ^+$, $D^0\rightarrow \pi ^+\pi ^-$, and $D^0\rightarrow K^+ K^-$ have reached sufficient precision to allow averages with $\mathcal{O}(1\%)$ relative uncertainties. At these precisions, Final State Radiation (FSR) must be treated correctly and consistently across the input measurements for the accuracy of the averages to match the precision. The sensitivity of measurements to FSR arises because of a tail in the distribution of radiated energy that extends to the kinematic limit. The tail beyond $\sum {E_\gamma } \approx 30$ MeV causes typical selection variables like the hadronic invariant mass to shift outside the selection range dictated by experimental resolution, as shown in Fig. 209. While the differential rate for the tail is small, the integrated rate amounts to several percent of the total $h^+ h^-(n\gamma )$ rate because of the tail’s extent. The tail therefore translates directly into a several percent loss in experimental efficiency.

All measurements that include an FSR correction have a correction based on the use of PHOTOS [1210,1211,1212,1213] within the experiment’s Monte Carlo simulation. PHOTOS itself, however, has evolved, over the period spanning the set of measurements. In particular, the incorporation of interference between radiation off the two separate mesons has proceeded in stages: it was first available for particle–antiparticle pairs in version 2.00 (1993), extended to any two-body, all-charged, final states in version 2.02 (1999), and further extended to multi-body final states in version 2.15 (2005). The effects of interference are clearly visible, as shown in Fig. 209, and cause a roughly 30% increase in the integrated rate into the high energy photon tail. To evaluate the FSR correction incorporated into a given measurement, we must therefore note whether any correction was made, the version of PHOTOS used in correction, and whether the interference terms in PHOTOS were turned on.

Table 281 The LQCD average for D and $D_s$ meson decay constants and their ratio from the Flavor Lattice Averaging Group [222]

Table 282 Experimental results and world averages for ${\mathcal {B}}(D^+\rightarrow \ell ^+\nu _{\ell })$ and $f_{D}|V_{cd}|$. The first uncertainty is statistical and the second is experimental systematic. The third uncertainty in the case of $f_{D^+}|V_{cd}|$ is due to external inputs (dominated by the uncertainty of $\tau _D$)

Fig. 205

WA value for $f_{D}|V_{cd}|$. For each point, the first error listed is the statistical and the second error is the systematic error

Table 283 Experimental results and world averages for ${\mathcal {B}}(D_{s}^+\rightarrow \ell ^+\nu _{\ell })$ and $f_{D_s}|V_{cs}|$. The first uncertainty is statistical and the second is experimental systematic. The third uncertainty in the case of $f_{D_s}|V_{cs}|$ is due to external inputs (dominated by the uncertainty of $\tau _{D_s}$). We have recalculated ${\mathcal {B}}(D_{s}^+\rightarrow \tau ^+\nu _{\tau })$ quoted by CLEO-c and BaBar using the latest values for branching fractions of $\tau$ decays to electron, muon, or pion and neutrinos [6]. CLEO-c and BaBar include statistical uncertainty of number of $D_{s}$ tags (denominator in the calculation of branching fraction) in the statistical uncertainty of measured ${\mathcal {B}}$. We subtract this uncertainty from the statistical one and add it to the systematic uncertainty

Fig. 206

WA value for $f_{D_s}|V_{cs}|$. For each point, the first error listed is the statistical and the second error is the systematic error

Table 284 Average of the magnitudes of the CKM matrix elements $|V_{cd}|$ and $|V_{cs}|$ determined from the leptonic and semileptonic D and $D_s$ decays. In the calculation of average values we assume 100% correlations in uncertainties due to LQCD. The values determined from neutrino scattering or W decays and determination from the global fit to the CKM matrix are given for comparison as well

8.8.1 Branching fraction corrections

Before averaging the measured branching fractions, the published results are updated, as necessary, to the FSR prediction of PHOTOS 2.15 with interference included. The correction will always shift a branching fraction to a higher value: with no FSR correction or with no interference term in the correction, the experimental efficiency determination will be biased high, and therefore the branching fraction will be biased low.

Most of the branching fraction analyses used the kinematic quantity sensitive to FSR in the candidate selection criteria. For the analyses at the $\psi (3770)$, this variable was $\Delta E$, the difference between the candidate $D^0$ energy and the beam energy (e.g., $E_K + E_\pi - E_\mathrm{beam}$ for $D^0\rightarrow K^-\pi ^+$). In the remainder of the analyses, the relevant quantity was the reconstructed hadronic two-body mass $m_{h^+h^-}$. To make the correction, we only need to evaluate the fraction of decays that FSR moves outside of the range accepted for the analysis.

The corrections were evaluated using an event generator (EvtGen [1214]) that incorporates PHOTOS to simulate the portions of the decay process most relevant to the correction. We compared corrections determined both with and without smearing to account for experimental resolution. The differences were negligible, typically of $\mathcal{O}(1\%)$ of the correction itself. The immunity of the correction to resolution effects comes about because most of the long FSR-induced tail in, for example, the $m_{h^+h^-}$ distribution resides well away from the selection boundaries. The smearing from resolution, on the other hand, mainly affects the distribution of events right at the boundary.

For measurements incorporating an FSR correction that did not include interference, we update by assessing the FSR-induced efficiency loss for both the PHOTOS version and configuration used in the analysis and our nominal version 2.15 with interference. For measurements that published their sensitivity to FSR, our generator-level predictions for the original efficiency loss agreed to within a few percent of the correction. This agreement lends additional credence to the procedure.

Once the event loss from FSR in the most sensitive kinematic quantity is accounted for, the event loss in other quantities is very small. For example, analyses using $D^{*+}$ tags show little sensitivity to FSR in the reconstructed $D^{*+}\!-D^0$ mass difference, i.e., in $m^{}_{K^-\pi ^+\pi ^+}\!-m^{}_{K^-\pi ^+}$. In this case the effect of FSR tends to cancel in the difference of reconstructed masses. In the $\psi (3770)$ analyses, the beam-constrained mass distributions ($\sqrt{E_\mathrm{beam}^2 - |\vec {p}_K + \vec {p}_\pi |^2}$) also show much smaller sensitivity than does the two-body mass.

Fig. 207

Comparison of magnitudes of the CKM matrix elements $|V_{cd}|$ (left) and $|V_{cs}|$ (right) determined from the (semi-)leptonic charm decays and from neutrino scattering data or W decays and determination from the global fit assuming CKM unitarity [252]

Fig. 208

WA values for ${\mathcal {B}}(D_{s}^+\rightarrow K^-K^+\pi ^+)$ (top), ${\mathcal {B}}(D_{s}^+\rightarrow \overline{K}{}^0\pi ^+)$ (middle), ${\mathcal {B}}(D_{s}^+\rightarrow \eta \pi ^+)$ (bottom). For each point, the first error listed is the statistical and the second error is the systematic error

The FOCUS [1215] analysis of the branching fraction ratios ${\mathcal {B}}(D^0\rightarrow \pi ^+\pi ^-)/{\mathcal {B}}(D^0\rightarrow K^-\pi ^+)$ and ${\mathcal {B}}(D^0\rightarrow K^+ K^-)/{\mathcal {B}}(D^0\rightarrow K^-\pi ^+)$ obtained yields using fits to the two-body mass distributions. FSR will both distort the low end of the signal mass peak, and will contribute a signal component to the low side tail used to estimate the background. The fitting procedure is not sensitive to signal events out in the FSR tail, which would be counted as part of the background.

A more complex toy Monte Carlo procedure was required to analyze the effect of FSR on the fitted yields, which were published with no FSR corrections applied. A detailed description of the procedure and results is available on the HFLAV web site, and a brief summary is provided here. Determining the correction involved an iterative procedure in which samples of similar size to the FOCUS sample were generated and then fit using the FOCUS signal and background parameterizations. The MC parameterizations were tuned based on differences between the fits to the toy MC data and the FOCUS fits, and the procedure was repeated. These steps were iterated until the fit parameters matched the original FOCUS parameters.

Table 285 Experimental results and world averages for branching fractions of $D_{s}^+\rightarrow K^-K^+\pi ^+$, $D_{s}^+\rightarrow \overline{K}{}^0K^+$, and $D_{s}^+\rightarrow \eta \pi ^+$ decays. The first uncertainty is statistical and the second is experimental systematic. CLEO-c reports in Ref. [1138] ${\mathcal {B}}(D_{s}^+\rightarrow K^0_SK^+)$. We include it in the average of ${\mathcal {B}}(D_{s}^+\rightarrow \overline{K}{}^0K^+)$ by using the relation ${\mathcal {B}}(D_{s}^+\rightarrow \overline{K}{}^0K^+)\equiv 2{\mathcal {B}}(D_{s}^+\rightarrow K^0_SK^+)$

The toy MC samples for the first iteration were based on the generator-level distribution of $m_{K^-\pi ^+}$, $m_{\pi ^+\pi ^-}$, and $m_{K^+K^-}$, including the effects of FSR, smeared according to the original FOCUS resolution function, and on backgrounds generated using the parameterization from the final FOCUS fits. For each iteration, 400–1600 individual data-sized samples were generated and fit. The means of the parameters from these fits determined the corrections to the generator parameters for the following iteration. The ratio between the number of signal events generated and the final signal yield provides the required FSR correction in the final iteration. Only a few iterations were required in each mode. Figure 210 shows the FOCUS data, the published FOCUS fits, and the final toy MC parameterizations. The toy MC provides an excellent description of the data.

The corrections obtained to the individual FOCUS yields were $1.0298\,\pm \, 0.0001$ for $K^-\pi ^+$, $1.062 \,\pm \, 0.001$ for $\pi ^+\pi ^-$, and $1.0183 \,\pm \, 0.0003$ for $K^+K^-$. These corrections tend to cancel in the branching ratios, leading to corrections of $1.031\,\pm \, 0.001$ for ${\mathcal {B}}(D^0\rightarrow \pi ^+\pi ^-)/{\mathcal {B}}(D^0\rightarrow K^-\pi ^+)$, and $0.9888\,\pm \, 0.0003$ for ${\mathcal {B}}(D^0\rightarrow K^+ K^-)/{\mathcal {B}}(D^0\rightarrow K^-\pi ^+)$.

Table 286 summarizes the corrected branching fractions. The published FSR-related modeling uncertainties have been replaced by with a new, common, estimate based on the assumption that the dominant uncertainty in the FSR corrections comes from the fact that the mesons are treated like structureless particles. No contributions from structure-dependent terms in the decay process (e.g., radiation off individual quarks) are included in PHOTOS. Internal studies done by various experiments have indicated that in $K\pi$ decays, the PHOTOS corrections agree with data at the 20–30% level. We therefore attribute a 25% uncertainty to the FSR prediction from potential structure-dependent contributions. For the other two modes, the only difference in structure is the final state valence quark content. While radiative corrections typically come in with a 1 / M dependence, one would expect the additional contribution from the structure terms to come in on time scales shorter than the hadronization time scale. In this case, you might expect $\mathrm \varLambda _\mathrm{QCD}$ to be the relevant scale, rather than the quark masses, and therefore that the amplitude is the same for the three modes. In treating the correlations among the measurements this is what we assume. We also assume that the PHOTOS amplitudes and any missing structure amplitudes are relatively real with constructive interference. The uncertainties largely cancel in the branching fraction ratios. For the final average branching fractions, the FSR uncertainty on $K\pi$ dominates. Note that because of the relative sizes of FSR in the different modes, the $\pi \pi /K\pi$ branching ratio uncertainty from FSR is positively correlated with that for the $K\pi$ branching fraction, while the $KK/K\pi$ branching ratio FSR uncertainty is negatively correlated.

Fig. 209

The $K\pi$ invariant mass distribution for $D^0\rightarrow K^-\pi ^+ (n\gamma )$ decays. The three curves correspond to three different configurations of PHOTOS for modeling FSR: version 2.02 without interference (blue/grey), version 2.02 with interference (red dashed) and version 2.15 with interference (black). The true invariant mass has been smeared with a typical experimental resolution of 10 MeV${}/c^2$. Inset The corresponding spectrum of total energy radiated per event. The arrow indicates the $\sum E_\gamma$ value that begins to shift kinematic quantities outside of the range typically accepted in a measurement

Fig. 210

FOCUS data (dots), original fits (blue) and toy MC parameterization (red) for $D^0\rightarrow K^-\pi ^+$ (left), $D^0\rightarrow \pi ^+\pi ^-$ (center), and $D^0\rightarrow \pi ^+\pi ^-$ (right)

Table 286 The experimental measurements relating to ${\mathcal {B}}(D^0\rightarrow K^-\pi ^+)$, ${\mathcal {B}}(D^0\rightarrow \pi ^+\pi ^-)$, and ${\mathcal {B}}(D^0\rightarrow K^+ K^-)$ after correcting to the common version and configuration of PHOTOS. The uncertainties are statistical and total systematic, with the FSR-related systematic estimated in this procedure shown in parentheses. Also listed are the percent shifts in the results from the correction, if any, applied here, as well as the original PHOTOS and interference configuration for each publication

The ${\mathcal {B}}(D^0\rightarrow K^-\pi ^+)$ measurement of reference [1223], the ${\mathcal {B}}(D^0\rightarrow \pi ^+\pi ^-)/{\mathcal {B}}(D^0\rightarrow K^-\pi ^+)$ measurements of references [1122] and [1081], and the ${\mathcal {B}}(D^0\rightarrow K^+ K^-)/{\mathcal {B}}(D^0\rightarrow K^-\pi ^+)$ measurement of reference [1081] are excluded from the branching fraction averages presented here. These measurements appear not to have incorporated any FSR corrections, and insufficient information is available to determine the 2–3% corrections that would be required.

Table 287 The correlation matrix corresponding to the full covariance matrix. Subscripts h denote which of the $D^0 \rightarrow h^+ h^-$ decay results from a single experiment is represented in that row or column

8.8.2 Average branching fractions

The average branching fractions for $D^0\rightarrow K^-\pi ^+$, $D^0\rightarrow \pi ^+\pi ^-$ and $D^0\rightarrow K^+ K^-$ decays are obtained from a single $\chi ^2$ minimization procedure, in which the three branching fractions are floating parameters. The central values are obtained from a fit in which the full covariance matrix – accounting for all statistical, systematic (excluding FSR), and FSR measurement uncertainties – is used. Table 287 presents the correlation matrix for this nominal fit. We then obtain the three reported uncertainties on those central values as follows: The statistical uncertainties are obtained from a fit using only the statistical covariance matrix. The systematic uncertainties are obtained by subtracting (in quadrature) the statistical uncertainties from the uncertainties determined via a fit using a covariance matrix that accounts for both statistical and systematic measurement uncertainties. The FSR uncertainties are obtained by subtracting (in quadrature) the uncertainties determined via a fit using a covariance matrix that accounts for both statistical and systematic measurement uncertainties from the uncertainties determined via the fit using the full covariance matrix.

In forming the full covariance matrix, the FSR uncertainties are treated as fully correlated (or anti-correlated) as described above. For the covariance matrices involving systematic measurement uncertainties, ALEPH’s systematic uncertainties in the $\theta _{D^*}$ parameter are treated as fully correlated between the ALEPH 97 and ALEPH 91 measurements. Similarly, the tracking efficiency uncertainties in the CLEO II 98 and the CLEO II 93 measurements are treated as fully correlated.

The averaging procedure results in a final $\chi ^2$ of 11.0 for 10 ($13-3$) degrees of freedom. The branching fractions obtained are

$$\begin{aligned}&{\mathcal {B}}(D^0\rightarrow K^-\pi ^+) = ( 3.962 \,\pm \, 0.017 \,\pm \, 0.038 \,\pm \, 0.027 )\,\%, \nonumber \\\end{aligned}$$

(281)

$$\begin{aligned}&{\mathcal {B}}(D^0\rightarrow \pi ^+\pi ^-) = ( 0.144 \,\pm \, 0.002 \,\pm \, 0.002 \,\pm \, 0.002 )\,\%, \nonumber \\\end{aligned}$$

(282)

$$\begin{aligned}&{\mathcal {B}}(D^0\rightarrow K^+ K^-) = ( 0.399 \,\pm \, 0.003 \,\pm \, 0.005 \,\pm \, 0.002 )\,\%.\nonumber \\ \end{aligned}$$

(283)

The uncertainties, estimated as described above, are statistical, systematic (excluding FSR), and FSR modeling. The correlation coefficients from the fit using the total uncertainties are

$$\begin{aligned} \begin{array}{llll} &{} K^-\pi ^+ &{} \pi ^+\pi ^- &{} K^+ K^- \\ K^-\pi ^+ &{} 1.00 &{} 0.71 &{} 0.76 \\ \pi ^+\pi ^- &{} 0.71 &{} 1.00 &{} 0.53 \\ K^+ K^- &{} 0.76 &{} 0.53 &{} 1.00 \\ \end{array} \end{aligned}$$

As the $\chi ^2$ would suggest and Fig. 211 shows, the average value for ${\mathcal {B}}(D^0\rightarrow K^-\pi ^+)$ and the input branching fractions agree very well. With the estimated uncertainty in the FSR modeling used here, the FSR uncertainty dominates the statistical uncertainty in the average, suggesting that experimental work in the near future should focus on verification of FSR with $\sum E_\gamma \gtrsim 100$ MeV. Note that the systematic uncertainty excluding FSR is still larger than the FSR uncertainty; in the most precise measurements of these branching fractions, the largest systematic uncertainty is the uncertainty on the tracking efficiency. The ${\mathcal {B}}(D^0\rightarrow K^+K^-)$ and ${\mathcal {B}}(D^0\rightarrow \pi ^+\pi ^-)$ measurements inferred from the branching ratio measurements also agree well (Fig. 212).

The ${\mathcal {B}}(D^0\rightarrow K^-\pi ^+)$ average obtained here is approximately two statistical standard deviations higher than the 2016 PDG update average [6]. Table 288 shows the evolution from a fit similar to the PDG’s (no FSR corrections or correlations, reference [1223] included) to the average presented here. There are two main contributions to the difference. The branching fraction in reference [1223] is low, and its exclusion shifts the result upwards. The dominant shift ($+0.017\%$) is due to the FSR corrections, which as expected shift the result upwards.

Fig. 211

Comparison of measurements of ${\mathcal {B}}(D^0\rightarrow K^-\pi ^+)$ (blue) with the average branching fraction obtained here (red, and yellow band)

Fig. 212

The ${\mathcal {B}}(D^0\rightarrow K^+K^-)$ (left) and ${\mathcal {B}}(D^0\rightarrow \pi ^+\pi ^-)$ (right) values obtained by scaling the measured branching ratios with the ${\mathcal {B}}(D^0\rightarrow K^-\pi ^+)$ branching fraction average obtained here. For the measurements (blue points), the error bars correspond to the statistical, systematic and $K\pi$ normalization uncertainties. The average obtained here (red point, yellow band) lists the statistical, systematics excluding FSR, and the FSR systematic

Table 288 Evolution of the $D^0\rightarrow K^-\pi ^+$ branching fraction from a fit with no FSR corrections or correlations (similar to the average in the PDG 2016 update [6]) to the nominal fit presented here

There is no reason to presume that the effects of FSR should be different in $D^0\rightarrow K^+\pi ^-$ and $D^0\rightarrow K^-\pi ^+$ decays, as both decay to one charged kaon and one charged pion. Measurements of the relative branching fraction ratio between the doubly Cabibbo-suppressed decay $D^0\rightarrow K^+\pi ^-$ and the Cabibbo-favored decay $D^0\rightarrow K^-\pi ^+$ ($R_D$, determined in Sect. 8.1) are now approaching $\mathcal{O}(1\%)$ relative uncertainties. This makes it worthwhile to combine our $R_D$ average with the ${\mathcal {B}}(D^0\rightarrow K^-\pi ^+)$ average obtained in Eq. (281), to provide measurements of the branching fraction:

$$\begin{aligned} {\mathcal {B}}(D^0\rightarrow K^+\pi ^-)= & {} (1.379 \pm 0.023) \nonumber \\&\times 10^{-4}~({\mathrm{assuming~no}~CPV}), \end{aligned}$$

(284)

$$\begin{aligned} {\mathcal {B}}(D^0\rightarrow K^+\pi ^-)= & {} (1.383 \pm 0.023) \nonumber \\&\times 10^{-4}~({CPV~\mathrm{allowed}}). \end{aligned}$$

(285)

Note that, by definition of $R_D$, these branching fractions do not include any contribution from Cabibbo-favored ${\overline{D}} ^0 \rightarrow K^+\pi ^-$ decays.

Table 289 Recent measurements of mass and width for different excited $D_{s}$ mesons. The column $J^{P}$ list the most significant assignment of spin and parity. If possible an average mass or width is calculated

Table 290 Recent measurements of mass and width for different excited D mesons. The column $J^{P}$ list the most significant assignment of spin and parity. If possible an average mass or width is calculated

8.9 Excited $D_{(s)}$ mesons

Excited charm meson states have received increased attention since the first observation of states that could not be accommodated by QCD predictions [1224,1225,1226,1227]. Tables 289, 290 and 291 summarize recent measurements of the masses and widths of excited D and $D_{s}$ mesons, respectively. If a preferred assignment of spin and parity was measured it is listed in the column $J^{P}$, where the label natural denotes $J^{P}=0^{-},1^{+},2^{-}\ldots$ and unnatural $J^{P}=0^{+},1^{-},2^{+}\ldots$. If possible, an average mass and width are calculated, which is listed in the gray shaded row. The calculation of the averages assumes no correlation between individual measurements. A summary of the averaged masses and widths is shown in Fig. 213.

The masses and widths of narrow ($\Gamma <50$ MeV) orbitally excited D mesons (1P states, denoted $D^{**}$), both neutral and charged, are well-established. Measurements of broad states ($\Gamma \sim$ 200–400 MeV) are less abundant, as identifying the signal is more challenging. There is a slight discrepancy between the $D_0^*(2400)^0$ masses measured by the Belle [710] and FOCUS [1228] experiments. No data exist yet for the $D_1(2430)^{\,\pm \,}$ state. Dalitz plot analyses of $B\rightarrow \overline{D}{}^{(*)}\pi \pi$ decays strongly favor the assignments $0^+$ and $1^+$ for the spin-parity quantum numbers of the $D_0^*(2400)^0/D_0^*(2400)^{\pm } \,$ and $D_1(2430)^{0}$ states, respectively. The measured masses and widths, as well as the $J^P$ values, are in agreement with theoretical predictions based on potential models [496, 1229,1230,1231].

Tables 292 and 293 summarize the branching fractions of B meson decays to excited D and $D_{s}$ states, respectively. It is notable that the branching fractions for B mesons decaying to a narrow $D^{**}$ state and a pion are similar for charged and neutral B initial states, while the branching fractions to a broad $D^{**}$ state and $\pi ^+$ are much larger for $B^+$ than for $B^0$. This may be due to the fact that color-suppressed amplitudes contribute only to the $B^+$ decay and not to the $B^0$ decay (for a theoretical discussion, see Refs. [1232, 1233]). Measurements of individual branching fractions of D mesons are difficult due to the unknown fragmentation of a c quark to $D^{**}$ or due to the unknown $B \rightarrow D^{**} X$ branching fractions.

The discoveries of the $D_{s0}^*(2317)^{\,\pm \,}$ and $D_{s1}(2460)^{\,\pm \,}$ have triggered increased interest in properties of, and searches for, excited $D_s$ mesons (here generically denoted $D_s^{**}$). While the masses and widths of $D_{s1}(2536)^{\,\pm \,}$ and $D_{s2}^{*}(2573)^{\,\pm \,}$ states are in relatively good agreement with potential model predictions, the masses of $D_{s0}^*(2317)^{\,\pm \,}$ and $D_{s1}(2460)^{\,\pm \,}$ states are significantly lower than expected (see Ref. [1234] for a discussion of $c\bar{s}$ models). Moreover, the mass splitting between these two states greatly exceeds that between the $D_{s1}(2536)^{\,\pm \,}$ and $D_{s2}(2573)^{\,\pm \,}$. These unexpected properties have led to interpretations of the $D_{s0}^*(2317)^{\,\pm \,}$ and $D_{s1}(2460)^{\,\pm \,}$ as exotic four-quark states [1235, 1236].

While there are few measurements of the $J^P$ values of $D_{s0}^*(2317)^{\,\pm \,}$ and $D_{s1}(2460)^{\,\pm \,}$, the available data favor $0^+$ and $1^+$, respectively. A molecule-like (DK) interpretation of the $D_{s0}^*(2317)^{\,\pm \,}$ and $D_{s1}(2460)^{\,\pm \,}$ [1235, 1236] that can account for their low masses and isospin-breaking decay modes is tested by searching for charged and neutral isospin partners of these states; thus far such searches have yielded negative results. Therefore the subset of models that predict equal production rates for different charged states is excluded. The molecular picture can also be tested by measuring the rates for the radiative processes $D_{s0}^*(2317)^{\,\pm \,}/D_{s1}(2460)^{\,\pm \,}\rightarrow D_s^{(*)}\gamma$ and comparing to theoretical predictions. The predicted rates, however, are below the sensitivity of current experiments.

Table 291 Recent measurements of mass and width for different excited D mesons. The column $J^{P}$ list the most significant assignment of spin and parity. If possible an average mass or width is calculated

Fig. 213

Averaged masses for excited $D_{s}$ mesons are shown in (a) and for D mesons in (b). The average widths for excited $D_{s}$ mesons are shown in (c) and for excited D mesons in (d). The vertical shaded regions distinguish between different spin parity states

Another model successful in explaining the total widths and the $D_{s0}^*(2317)^{\,\pm \,}$ – $D_{s1}(2460)^{\,\pm \,}$ mass splitting is based on the assumption that these states are chiral partners of the ground states $D_{s}^{+}$ and $D_{s}^{*}$ [1237]. While some measured branching fraction ratios agree with predicted values, further experimental tests with better sensitivity are needed to confirm or refute this scenario. A summary of the mass difference measurements is given in Table 294.

Measurements by BaBar [1238] and LHCb [1239] first indicated the existence of a strange-charm $D^{*}_{sJ}(2860)^{\,\pm \,}$ meson. An LHCb study of $B_{s}^{0}\rightarrow \overline{D}{}^{0}K^{-}\pi ^{+}$ decays, in which they searched for excited $D_{s}$ mesons [1240], showed with $10\sigma$ significance that this state is comprised of two different particles, one of spin 1 and one of spin 3. This represents the first measurement of a heavy flavored spin-3 particle, and the first observation B meson decays to spin 3 particles. A subsequent study of $D_{sJ}$ mesons by the LHCb collaboration [1241] supports the natural parity assignment for this state ($J^P=3^-$). This study also shows weak evidence for a further structure at a mass around 3040 MeV$/c^2$ with unnatural parity, which was first hinted at by a BaBar analysis [1238].

Recent evidence shows that the 1D family of charm resonances can be explored in the Dalitz plot analyses of B-meson decays in the same way as seen for the charm-strange resonances. The LHCb collaboration performed an analysis of $B^0 \rightarrow \overline{D}{}^0\pi ^+\pi ^-$ decays, in which they measured the spin-parity assignment of the state $D^*_3(2760)^{\,\pm \,}$, which was observed previously by BaBar [1164] and LHCb [1165], to be $J^P=3^-$. The measurement suggests a spectroscopic assignment of ${}^3D_3$. This is the second observation of a spin-3 charm meson.

Other observed excited $D_s$ states include $D_{s1}^{*}(2700)^{\,\pm \,}$ and $D_{s2}^{*}(2573)^{\,\pm \,}$. The properties of both (mass, width, $J^P$) have been measured and determined in several analyses. A theoretical discussion [1242] investigates the possibility that the $D_{s1}(2700)^{\,\pm \,}$ could represent radial excitations of the $D_s^{*\,\pm \,}$. Similarly, the $D_{s1}^{*}(2860)^{\,\pm \,}$ and $D_{sJ}(3040)^{\,\pm \,}$ could be excitations of $D_{s0}^*(2317)^{\,\pm \,}$ and $D_{s1}(2460)^{\,\pm \,}$ or $D_{s1}(2536)^{\,\pm \,}$, respectively.

Table 295 summarizes measurements of the helicity parameter $A_{D}$ (also referred to as polarization amplitude). In $D^{**}$ meson decays to $D^{**} \rightarrow D^{*}\pi$, $D^{*} \rightarrow D \pi$, the helicity distribution varies like $1 + A_{D}\cos ^{2}\theta _{H}$, where $\theta _{H}$ is the angle in the $D^{*}$ rest frame between the two pions emitted by decay $D^{**} \rightarrow D^{*}\pi$ and the $D^{*} \rightarrow D \pi$. The parameter is sensitive to possible S-wave contributions in the decay. In the case of a D meson decay decaying purely via D-wave, the helicity parameter is predicted to give $A_{D}=3$. Studies of the $D_{1}(2420)^{0}$ meson by the ZEUS and BaBar collaborations suggest that there is an S-wave admixture in the decay, which is contrary to Heavy Quark Effective Theory calculations [476, 1243].

Table 292 Product of B meson branching fraction and (daughter) excited D meson branching fraction

Table 293 Product of B meson branching fraction and (daughter) excited $D^{}_s$ meson branching fraction

Table 294 Mass difference measurements for excited D mesons

8.10 $\varLambda _c^+$ branching fractions

Charmed baryon decays play an important role in studies of weak and strong interactions. For example, they provide crucial input for measurements of exclusive and inclusive decay rates of b-flavored mesons and baryons, and also for measurements of fragmentation fractions of charm and bottom quarks. In spite of this importance, experimental data on $\varLambda _c^+$ baryon decays was scarce until 2014, when Belle published the first model-independent measurement of the branching fraction for $\varLambda _c^+\rightarrow pK^-\pi ^+$ [1271]. This measurement improved upon the precision of previous (model-dependent) measurements by a factor of five. Since then the precision of other $\varLambda _c^+$ branching fractions has improved due to measurements based on threshold data performed by BESIII [531]. BESIII also reported the first measurement of the branching fraction for the semileptonic decay $\varLambda _c^+\rightarrow \varLambda e^+ \nu _e$ [1272]. Here we present a global fit for branching fractions of Cabbibo-favored $\varLambda _c^+$ decays, taking into account all relevant experimental measurements and their correlations. All measurements used assume unpolarised production of the $\varLambda _c^+$.

The measurements listed in Table 296 are input to a least-squares fit minimizing a $\chi ^2$ statistic. The fitted quantities are the $\varLambda _c^+$ branching fractions for twelve hadronic modes and one semileptonic mode. The measurements are labelled using the $\Gamma _n$ notation employed by the Particle Data Group [6], where n is an integer that specifies the decay mode. The fitted output consists of 13 quantities – twelve hadronic and one semileptonic branching fraction. The advantage of our fit is that it takes into account correlations among measurements from the same experiment, i.e., systematic uncertainties related to normalization, track-finding efficiency, particle identification efficiency, and $\pi ^0$, $K^0_S$, and $\varLambda$ reconstruction efficiencies. For the twelve hadronic branching fractions measured by BESIII, we use BESIII’s published correlation matrix [531].

The resulting fitted values for the branching fractions are given in Table 296. The overall $\chi ^2$ of the fit is 30.0 for 23 degrees of freedom, which corresponds to a p value of 0.149. The correlation matrix for the fitted branching fractions is shown in Fig. 214, and constraints from individual measurements for pairs of fitted branching fractions are shown in Fig. 215. The branching fraction of the normalisation decay $\varLambda _c^+\rightarrow pK^-\pi ^+$ is found to be

$$\begin{aligned} {\mathcal {B}}(\varLambda _c^+\rightarrow pK^-\pi ^+)=(6.46\,\pm \, 0.24)\%. \end{aligned}$$

8.11 Excited charm baryons

In this section we summarize the present status of excited charmed baryons, decaying strongly or electromagnetically. We list their masses (or the mass difference between the excited baryon and the corresponding ground state), natural widths, decay modes, and assigned quantum numbers. The present ground-state measurements are: $M(\varLambda _c^+)=2286.46\,\pm \,0.14$ MeV/$c^2$ measured by BaBar [1283], $M(\varXi _c^0)=(2470.85^{+0.28}_{-0.04}$) MeV/$c^2$ and $M(\varXi _c^+)=(2467.93^{+0.28}_{-0.40}$) MeV/$c^2$, both dominated by CDF [145], and $M(\Omega _c^0)=(2695.2\,\pm \,1.7$) MeV/$c^2$, dominated by Belle [1284]. Should these values change, so will some of the values for the masses of the excited states.

Table 297 summarizes the excited $\varLambda _c^+$ baryons. The first two states listed, namely the $\varLambda _c(2595)^+$ and $\varLambda _c(2625)^+$, are well-established. The measured masses and decay patterns suggest that they are orbitally excited $\varLambda _c^+$ baryons with total angular momentum of the light quarks $L=1$. Thus their quantum numbers are assigned to be $J^P=(\frac{1}{2})^-$ and $J^P=(\frac{3}{2})^-$, respectively. Their mass measurements are dominated by CDF [1285]: $M(\varLambda _c(2595)^+)=(2592.25\,\pm \, 0.24\,\pm \, 0.14$) MeV/$c^2$ and $M(\varLambda _c(2625)^+)=(2628.11\,\pm \, 0.13\,\pm \, 0.14$) MeV/$c^2$. Earlier measurements did not fully take into account the restricted phase-space of the $\varLambda _c(2595)^+$ decays.

The next two states, $\varLambda _c(2765)^+$ and $\varLambda _c(2880)^+$, were discovered by CLEO [1286] in the $\varLambda _c^+\pi ^+\pi ^-$ final state. CLEO found that a significant fraction of the $\varLambda _c(2880)^+$ decays proceeds via an intermediate $\Sigma _c(2445)^{++/0}\pi ^{-/+}$. Later, BaBar [1287] observed that this state has also a $D^0 p$ decay mode. This was the first example of an excited charmed baryon decaying into a charm meson plus a baryon; previously all excited charmed baryon were found in their hadronic transitions into lower lying charmed baryons. In the same analysis, BaBarobserved for the first time an additional state, $\varLambda _c(2940)^+$,

Table 295 Measurements of polarization amplitudes for excited D mesons

Table 296 Experimental results and world averages for branching fractions of twelve hadronic and one semileptonic $\varLambda _c^+$ decay. The first uncertainty is statistical and the second is systematic

decaying into $D^0 p$. Studying the $D^+ p$ final state, BaBar found no signal; this implies that the $\varLambda _c(2880)^+$ and $\varLambda _c(2940)^+$ are $\varLambda _c^+$ excited states rather than $\Sigma _c$ excitations. Belle reported the result of an angular analysis that favors 5 / 2 for the $\varLambda _c(2880)^+$ spin hypothesis. Moreover, the measured ratio of branching fractions ${\mathcal {B}}(\varLambda _c(2880)^+\rightarrow \Sigma _c(2520)\pi ^{\,\pm \,})/{\mathcal {B}}(\varLambda _c(2880)^+\rightarrow \Sigma _c(2455)\pi ^{\,\pm \,})=(0.225\,\pm \, 0.062\,\pm \, 0.025)$, combined with theoretical predictions based on HQS [496, 1288], favor even parity. However this prediction is only valid if the P-wave portion of $\Sigma _c(2520)\pi$ is suppressed. The current open questions in the excited $\varLambda _c^+$ family include the determination of quantum numbers for the other states, and the nature of the $\varLambda _c(2765)^+$ state, in particular whether it is an excited $\Sigma _c^+$ or $\varLambda _c^+$. However, there is no doubt that the state exists, as it is clearly visible in Belle data.

Table 298 summarizes the excited $\Sigma _c^{++,+,0}$ baryons. The ground iso-triplets of $\Sigma _c(2455)^{++,+,0}$ and $\Sigma _c(2520)^{++,+,0}$ baryons are well-established. Belle [1289]precisely measured the mass differences and widths of the doubly charged and neutral members of this triplet. The short list of excited $\Sigma _c$ baryons is completed by the triplet of $\Sigma _c(2800)$ states observed by Belle [1290]. Based on the measured masses and theoretical predictions [1291, 1292], these states are assumed to be members of the predicted $\Sigma _{c2}$ $3/2^-$ triplet. From a study of resonant substructure in $B^-\rightarrow \varLambda _c^+{\bar{p}}\pi ^-$ decays, BaBar found a significant signal in the $\varLambda _c^+\pi ^-$ final state with a mean value higher than measured for the $\Sigma _c(2800)$ by Belle by about $3\sigma$ (Table 298). The decay widths measured by Belle and BaBar are consistent, but it is an open question if the observed state is the same as the Belle state.

Table 299 summarizes the excited $\varXi _c^{+,0}$ and $\Omega _c^0$ baryons. The list of excited $\varXi _c$ baryons has several states, of unknown quantum numbers, having masses above 2900 MeV/$c^2$ and decaying into three different types of decay modes: $\varLambda _c/\Sigma _c n\pi$, $\varXi _c n\pi$ and the most recently observed $\varLambda D$. Some of these states ($\varXi _c(2970)^+$, $\varXi _c(3055)$ and $\varXi _c(3080)^{+,0}$) have been observed by both Belle [1293,1294,1295] and BaBar [694] and are considered well-established. The $\varXi _c(2930)^0$ state decaying into $\varLambda _c^+ K^-$ is seen only by BaBar [1296] and needs confirmation. The $\varXi _c(3123)^+$ observed by BaBar [694] in the $\Sigma _c(2520)^{++}\pi ^-$ final state has not been confirmed by Belle [1294] with twice the statistics; thus its existence in doubt and it is omitted from Table 299.

Fig. 214

Correlation coefficients between averaged $\varLambda _c^+$ branching fractions

Fig. 215

Plots of all individual measurements and the fitted averages. Individual measurements are plotted as bands (ellipses) showing their $\,\pm \,1\sigma$, $\,\pm \,2\sigma$, and $\,\pm \,3\sigma$ ranges. The best fit value is indicated by a cross showing the one-dimensional errors. In cases where multiple ratio measurements exists ($\Gamma _i/\Gamma _j$), only the most precise one is plotted

Several of the width and mass measurements for the $\varXi _c(3055)$ and $\varXi _c(3080)$ iso-doublets are only in marginal agreement between experiments and decay modes. However, there seems little doubt that the differing measurements are of the same particle.

Belle [1297] has recently analyzed large samples of $\varXi _c^\prime$, $\varXi _c(2645)$, $\varXi _c(2790)$, $\varXi _c(2815)$ and $\varXi _c(2970)$ decays. From this analysis they obtain the most precise mass measurements of all five iso-doublets, and the first significant width measurements of the $\varXi _c(2645)$, $\varXi _c(2790)$ and $\varXi _c(2815)$. The level of agreement in the different measurements of the mass and width of the $\varXi _c(2970)$, formerly named by the PDG as the $\varXi _c(2980)$, is not satisfactory. This leaves open the possibility of there being other resonances nearby or that threshold effects have not been fully understood. The present situation in the excited $\varXi _c$ sector is summarized in Table 299.

The excited $\Omega _c^0$ doubly strange charmed baryon has been seen by both BaBar [1298] and Belle [1284]. The mass differences $\delta M=M(\Omega _c^{*0})-M(\Omega _c^0)$ measured by the experiments are in good agreement and are also consistent with most theoretical predictions [1299,1300,1301,1302]. No higher mass $\Omega _c$ states have yet been observed.

Figure 216 shows the levels of excited charm baryons along with corresponding transitions between them, and also transitions to the ground states.

We note that Belle and BaBar recently discovered that transitions between families are possible, i.e., between the $\varXi _c$ and $\varLambda _c^+$ families of excited charmed baryons [694, 1293] and that highly excited states are found to decay into a non-charmed baryons and a D meson[1287, 1295].

8.12 Rare and forbidden decays

This section provides a summary of searches for rare and forbidden charm decays in tabular form. The decay modes can be categorized as flavor-changing neutral currents, lepton-flavor-violating, lepton-number-violating, and both baryon- and lepton-number-violating decays. Figures 217, 218 and 219 plot the upper limits for $D^0$, $D^+$, $D_s^+$, and $\varLambda _c^+$ decays. Tables 300, 301, 302 and 303 give the corresponding numerical results. Some theoretical predictions are given in Refs. [1303,1304,1305,1306,1307,1308,1309,1310].

In several cases the rare-decay final states have been observed with the di-lepton pair being the decay product of a vector meson. For these measurements the quoted limits are those expected for the non-resonant di-lepton spectrum. For the extrapolation to the full spectrum a phase-space distribution of the non-resonant component has been assumed. This applies to the CLEO measurement of the decays $D_{(s)}^+\rightarrow (K^+,\pi ^+)e^+e^-$ [1311], to the D0 measurements of the decays $D_{(s)}^+\rightarrow \pi ^+\mu ^+\mu ^-$ [1312], and to the BaBar measurements of the decays $D_{(s)}^+\rightarrow (K^+,\pi ^+)e^+e^-$ and $D_{(s)}^+\rightarrow (K^+,\pi ^+)\mu ^+\mu ^-$, where the contribution from $\phi \rightarrow l^+l^-$ ($l=e,\mu$) has been excluded. In the case of the LHCb measurements of the decays $D^0\rightarrow \pi ^+\pi ^-\mu ^+\mu ^-$ [1313] as well as the decays $D_{(s)}^+\rightarrow \pi ^+\mu ^+\mu ^-$ [1314] the contributions from $\phi \rightarrow l^+l^-$ as well as from $\rho ,\omega \rightarrow l^+l^-$ ($l=e,\mu$) have been excluded.

9 Tau lepton properties

Table 297 Summary of excited $\varLambda _c^+$ baryons

Table 298 Summary of the excited $\Sigma _c^{++,+,0}$ baryon family

Table 299 Summary of excited $\varXi _c^{+,0}$ and $\Omega _c^0$ baryon families. For the first four iso-doublets, the mass difference with respect to the ground state is given, as the uncertainties are dominated by the uncertainty in the ground state mass. In the remaining cases, the uncertainty on the measurement of the excited state itself dominates

We present averages of a selection of $\tau$ lepton quantities with the goal to provide the best tests of the universality of the charged-current weak interaction (Sect. 9.2) and of the Cabibbo–Kobayashi–Maskawa (CKM) matrix coefficient $|V_{us}|$ from $\tau$ decays (Sect. 9.4). We focus on the averages that benefit most from the adoption of the HFLAV methodology [420], namely a global fit of the $\tau$ branching fractions that best exploits the available experimental information. Since the 2016 edition, the HFLAV-Tau group has collaborated to the determination of the $\tau$-lepton branching fractions based on a global fit and to the related mini-review that are included in the “Review of particle physics” [6]. The differences between the PDG 2016 fit and the fit presented here are detailed in Sect. 9.1.4.

All relevant published statistical correlations are used, and a selection of measurements, particularly the most precise and the most recent ones, was studied to take into account the significant systematic dependencies from external parameters and common sources of systematic uncertainty.

Fig. 216

Level diagram for multiplets and transitions for excited charm baryons

Finally, we report in Sect. 9.5 the latest limits on the lepton-flavour-violating $\tau$ branching fractions and in Sect. 9.6 we determine the combined upper limits for the branching fractions that have multiple experimental results.

The $\tau$ lepton results are obtained from inputs available through summer 2016 and have been published on the web in 2016 with the label “Summer 2016”. However, there have been minor revisions since then, and we have updated tables and plots in this report with the label “Spring 2017”.

9.1 Branching fraction fit

A global fit of the available experimental measurements is used to determine the $\tau$ branching fractions, together with their uncertainties and statistical correlations. The $\tau$ branching fractions provide a test for theory predictions based on the Standard Model (SM) EW and QCD interactions and can be further elaborated to test the EW charged-current universality for leptons, to determine the CKM matrix coefficient $|V_{us}|$ (and the QCD coupling constant $\alpha _s$ at the $\tau$ mass).

Fig. 217

Upper limits at $90\%$ C.L. for $D^0$ decays. The top plot shows flavor-changing neutral current decays, and the bottom plot shows lepton-flavor-changing (LF), lepton-number-changing (L), and both baryon- and lepton-number-changing (BL) decays

The measurements used in the fit are listed in Table 304 and consist of either $\tau$ decay branching fractions, labelled as $\Gamma _{i}$, or ratios of two $\tau$ decay branching fractions, labelled as $\Gamma _{i}/\Gamma _{j}$. A minimum $\chi ^2$ fit is performed for all the measured quantities and for some additional branching fractions and ratios of branching fractions, and all fit results are listed in Table 304. Some fitted quantities are equal to the ratio of two other fitted quantities, as documented with the notation $\Gamma _{i}/\Gamma _{j}$ in Table 304. Some fitted quantities are sums of other fitted quantities, for instance $\Gamma _8 = B(\tau \rightarrow h^{-} \nu _{tau})$ is the sum of $\Gamma _9 = B(\tau \rightarrow \pi ^{-}\nu _{\tau })$ and $\Gamma _{10} = B(\tau \rightarrow K^{-}\nu _{\tau })$. The symbol h is used to mean either a $\pi$ or K. Section 9.1.7 lists all equations relating one quantity to the sum of other quantities. In the following, we refer to both types of relations between fitted quantities collectively as constraint equations or constraints. The fit $\chi ^2$ is minimized subject to all the above mentioned constraints, listed in Table 304 and Sect. 9.1.7. The fit procedure is equivalent to that employed in the previous HFLAV reports [5, 234, 420].

Fig. 218

Upper limits at $90\%$ C.L. for $D^+$ (top) and $D_s^+$ (bottom) decays. Each plot shows flavor-changing neutral current decays, lepton-flavor-changing decays (LF), and lepton-number-changing (L) decays

Fig. 219

Upper limits at $90\%$ C.L. for $\varLambda _c^+$ decays. Shown are flavor-changing neutral current decays, lepton-flavor-changing (LF) decays, and lepton-number-changing (L) decays

9.1.1 Technical implementation of the fit procedure

The fit computes the quantities $q_i$ by minimizing a $\chi ^2$ while respecting a series of equality constraints on the $q_i$. The $\chi ^2$ is computed using the measurements $x_i$ and their covariance matrix $V_{ij}$ as

$$\begin{aligned} \chi ^2 = (x_i - A_{ik}q_k)^t V_{ij}^{-1} (x_j - A_{jl}q_l), \end{aligned}$$

(286)

where the model matrix $A_{ij}$ is used to get the vector of the predicted measurements $x_i^\prime$ from the vector of the fit parameters $q_j$ as $x_i^\prime = A_{ij}q_j$. In this particular implementation, the measurements are grouped according to the measured quantity, and all quantities with at least one measurement correspond to a fit parameter. Therefore, the matrix $A_{ij}$ has one row per measurement $x_i$ and one column per fitted quantity $q_j$, with unity coefficients for the rows and column that identify a measurement $x_i$ of the quantity $q_j$. In summary, the $\chi ^2$ given in Eq. (286) is minimized subject to the constraints

$$\begin{aligned} f_r(q_s) - c_r = 0, \end{aligned}$$

(287)

where Eq. (287) corresponds to the constraint equations, written as a set of “constraint expressions” that are equated to zero. Using the method of Lagrange multipliers, a set of equations is obtained by taking the derivatives with respect to the fitted quantities $q_k$ and the Lagrange multipliers $\lambda _r$ of the sum of the $\chi ^2$ and the constraint expressions multiplied by the Lagrange multipliers $\lambda _r$, one for each constraint:

$$\begin{aligned}&\text {min} \bigg [ (A_{ik}q_k {-} x_i)^t V_{ij}^{-1} (A_{jl}q_l {-} x_j) + 2\lambda _r(f_r(q_s) - c_r) \bigg ] \end{aligned}$$

(288)

$$\begin{aligned}&(\partial /\partial q_k, \partial /\partial \lambda _r) \, [\text {expression above}] = 0. \end{aligned}$$

(289)

Equation (289) defines a set of equations for the vector of the unknowns $(q_k, \lambda _r)$, some of which may be non-linear, in case of non-linear constraints. An iterative minimization procedure approximates at each step the non-linear constraint expressions by their first order Taylor expansion around the current values of the fitted quantities, $\bar{q}_s$:

$$\begin{aligned} f_r(q_s) - c_r \simeq f_r(\bar{q}_s) + \frac{\partial f_r(q_s)}{\partial q_s}\bigg |_{\bar{q}_s} (q_s - \bar{q}_s) - c_r, \end{aligned}$$

(290)

which can be written as

$$\begin{aligned} B_{rs} q_s - c^\prime _r, \end{aligned}$$

(291)

where $c^\prime _r$ are the resulting constant known terms, independent of $q_s$ at first order. After linearization, the differentiation by $q_k$ and $\lambda _r$ is trivial and leads to a set of linear equations

$$\begin{aligned}&A_{ki}^t V_{ij}^{-1} A_{jl} q_l + B^t_{kr} \lambda _r = A_{ki}^t V_{ij}^{-1} x_j \end{aligned}$$

(292)

$$\begin{aligned}&B_{rs} q_s = c^\prime _r, \end{aligned}$$

(293)

which can be expressed as:

$$\begin{aligned} F_{ij} u_j = v_i, \end{aligned}$$

(294)

where $u_j = (q_k, \lambda _r)$ and $v_i$ is the vector of the known constant terms running over the index k and then r in the right terms of Eqs. (292) and (293). Solving the equation set in Eq. (294) gives the fitted quantities and their covariance matrix, using the measurements and their covariance matrix.

Table 300 Upper limits at $90\%$ C.L. for $D^0$ decays

Table 301 Upper limits at $90\%$ C.L. for $D^+$ decays

Table 302 Upper limits at $90\%$ C.L. for $D_s^+$ decays

Table 303 Upper limits at $90\%$ C.L. for $\varLambda _c^+$ decays

The fit procedure starts by computing the linear approximation of the non-linear constraint expressions around the quantities seed values. With an iterative procedure, the unknowns are updated at each step by solving the equations and the equations are then linearized around the updated values, until the RMS average of relative variation of the fitted unknowns is reduced below $10^{-12}$.

9.1.2 Fit results

The fit output consists of 135 fitted quantities that correspond to either branching fractions or ratios of branching fractions. The fitted quantities values and uncertainties are listed in Table 304. The off-diagonal statistical correlation terms between a subset of 47 “basis quantities” are listed in Sect. 9.1.6. All the remaining statistical correlation terms can be obtained using the constraint equations listed in Table 304 and Sect. 9.1.7.

The fit has $\chi ^2/\text {d.o.f.} = 137/123$, corresponding to a confidence level $\text {CL} = 17.84 \%$. We use a total of 170 measurements to fit the above mentioned 135 quantities subjected to 88 constraints. Although the unitarity constraint is not applied, the fit is statistically consistent with unitarity, where the residual is $\Gamma _{998} = 1- \Gamma _\mathrm{All} = (0.0355 \,\pm \, 0.1031)\cdot 10^{-2}$.

A scale factor of 5.44 (as in the three previous reports [5, 234, 420]) has been applied to the published uncertainties of the two severely inconsistent measurements of $\Gamma _{96} = \tau \rightarrow KKK\nu$ by BaBar and Belle. The scale factor has been determined using the PDG procedure, i.e., to the proper size in order to obtain a reduced $\chi ^2$ equal to 1 when fitting just the two $\Gamma _{96}$ measurements.

For several old results, for historical reasons, the table reports the total error (statistical plus systematic) in the position of the statistical error and zero in the position of the systematic error. Since the fit depends only on the total errors, the results are unaffected.

9.1.3 Changes with respect to the previous report

The following changes have been introduced with respect to the previous HFLAV report [5].

Two old preliminary results have been removed:

• $\Gamma _{35} = B(\tau \rightarrow \pi K_S\nu )$, BaBar [1340],

• $\Gamma _{40} = B(\tau \rightarrow \pi K_S\pi ^0\nu )$, BaBar [1341].

They were announced in 2008 and 2009 but have not been published.

In the 2014 report, for several BaBar and Belle experimental results we used more precise numerical values than the published ones, using internal information from the Collaborations. We revert to the published figures in this report, as the improvements in the fit results were negligible. In so doing, we use in this report the same values that are used in the PDG 2016 fit.

The Belle result on $\tau ^- \rightarrow {K_{S}^{0}}(\mathrm{particles})^{-} \nu _{\tau }$ [1342] has been discarded, because it was determined that the published information does not permit a reliable determination of the correlations with the other results in the same paper. The correlations estimated for the HFLAV 2014 report were inconsistent. As a result, both the covariance matrix of the Belle results and the overall correlation matrix for the branching ratio fit results were non-positive-definite. It has been found that the inconsistency had negligible impact on lepton universality tests and on the $|V_{us}|$ measurements.

Table 304 HFLAV Spring 2017 branching fractions fit results

The ALEPH result on $\Gamma _{46}(\tau ^{-} \rightarrow \pi ^{-}K^{0} {\bar{K}}^{0} \nu _{\tau })$ [1329] has been removed from the fit inputs, since it is simply the sum of twice $\Gamma _{47}= \pi ^{-}K_S^{0}K_S^{0}\nu _{\tau }$ and $\Gamma _{48}= \pi ^{-}K_S^{0}K_L^{0}\nu _{\tau }$ from the same paper, hence 100% correlated with them.

Several minor corrections have been applied to the constraints. The list of constraints included in the following fully documents the changes when compared with the same list in the 2014 edition. In some cases the relation equating one decay mode to a sum of modes included some minor terms that did not match the mode definitions. In other cases, the sum included modes with overlapping components. The effects on the 2014 fit results have been found to be modest with respect to the quoted uncertainties. For instance, the definition of the total branching fraction has been updated as follows:

$$\begin{aligned} {\Gamma _{\text {All}}}= & {} {\Gamma _{3}} + \Gamma _{5} + \Gamma _{9} + \Gamma _{10} + \Gamma _{14} + \Gamma _{16} + \Gamma _{20} + \Gamma _{23}\\&+ \Gamma _{27} + \Gamma _{28} + \Gamma _{30} + \Gamma _{35} + \Gamma _{37} + \Gamma _{40} + \Gamma _{42} \\&+ \Gamma _{47}\cdot {}(1 + ((\Gamma _{<K^0|K_L>}\cdot {}\Gamma _{<{\bar{K}}^0|K_L>}) / (\Gamma _{<K^0|K_S>}\\&\cdot {}\Gamma _{<{\bar{K}}^0|K_S>})))+ \Gamma _{48} + \Gamma _{62} \\&+ \Gamma _{70} + \Gamma _{77} + \Gamma _{811} + \Gamma _{812} + \Gamma _{93} \\&+ \Gamma _{94} + \Gamma _{832} + \Gamma _{833} + \Gamma _{126} + \Gamma _{128} + \Gamma _{802} + \Gamma _{803} \\&+ \Gamma _{800} + \Gamma _{151} + \Gamma _{130} + \Gamma _{132} + \Gamma _{44} + \Gamma _{53}\\&+ \Gamma _{50}\cdot {}(1 + ((\Gamma _{<K^0|K_L>}\cdot {}\Gamma _{<{\bar{K}}^0|K_L>}) / (\Gamma _{<K^0|K_S>}\\&\cdot {}\Gamma _{<{\bar{K}}^0|K_S>})))+\Gamma _{51} \\&+\Gamma _{167}\cdot {}(\Gamma _{\phi \rightarrow K^+K^-}+\Gamma _{\phi \rightarrow K_S K_L}) + \Gamma _{152} + \Gamma _{920} \\&+ \Gamma _{821} + \Gamma _{822} + \Gamma _{831} + \Gamma _{136} + \Gamma _{945} + \Gamma _{805} \end{aligned}$$

In the 2014 definition, the term $\Gamma _{78} = h^{-}h^{-}h^{+}3 \pi ^{0}\nu _{\tau }$ included the contributions of $\Gamma _{50} = \pi ^{-}\pi ^{0}K_{S}^{0}K_S^{0}\nu _{\tau }$ and $\Gamma _{132}=\pi ^{-}{\bar{K}}^{0}\eta \nu _{\tau }$, which were already included explicitly in ${\Gamma _\mathrm{All}}$. In the present definition, $\Gamma _{78}$ has been replaced with modes whose sum corresponds to

$$\begin{aligned} {\Gamma _{810}}= {2\pi ^- \pi ^+ 3\pi ^0 \nu _\tau ~(\text {ex.~}K^0)} \end{aligned}$$

Table 305 Basis quantities correlation coefficients in percent, subtable 1

Table 306 Basis quantities correlation coefficients in percent, subtable 2

Table 307 Basis quantities correlation coefficients in percent, subtable 3

Table 308 Basis quantities correlation coefficients in percent, subtable 4

Table 309 Basis quantities correlation coefficients in percent, subtable 5

Table 310 Basis quantities correlation coefficients in percent, subtable 6

Table 311 Basis quantities correlation coefficients in percent, subtable 7

Table 312 Basis quantities correlation coefficients in percent, subtable 8

Table 313 Basis quantities correlation coefficients in percent, subtable 9

As in 2014, the total $\tau$ branching fraction ${\Gamma _\mathrm{All}}$ definition includes two modes that have overlapping final states, to a minor extent, which we consider negligible:

$$\begin{aligned} {\Gamma _{50}}&= {\pi ^- \pi ^0 K_S^0 K_S^0 \nu _\tau }\\ {\Gamma _{132}}&={{\pi ^- {\bar{K}}^0 \eta \nu _\tau }}. \end{aligned}$$

Finally, we updated to the PDG 2015 results [327] all the parameters corresponding to the measurements’ systematic biases and uncertainties and all the parameters appearing in the constraint equations in Sect. 9.1.7 and Table 304.

9.1.4 Differences between the HFLAV Spring 2017 fit and the PDG 2016 fit

As is st andard for the PDG branching fraction fits, the PDG 2016 $\tau$ branching fraction fit is unitarity constrained, while the HFLAV 2016 fit is unconstrained.

The HFLAV-Tau fit uses the ALEPH measurements of branching fractions defined according to the final state content of “hadrons” and kaons, where a “hadron” corresponds to either a pion or a kaon, since this set of results is closer to the actual experimental measurements and facilitates a more comprehensive treatment of the experimental results correlations [420]. The PDG 2016 fit on the other hand continues to use – as in the past editions – the ALEPH measurements of modes with pions and kaons, which correspond to the final set of published measurements of the collaboration. It is planned eventually to update the PDG fit to use the same ALEPH measurement set that is used by HFLAV.

The HFLAV Spring 2017 fit, as in 2014, uses the ALEPH estimate for $\Gamma _{805} = B(\tau \rightarrow a_{1}^{-}( \rightarrow \pi ^{-}\gamma ) \nu _{\tau })$, which is not a direct measurement. The PDG 2016 fit uses the PDG average of $B(a_1\rightarrow \pi \gamma )$ as a parameter and defines $\Gamma _{805}= B(a_1\rightarrow \pi \gamma )\times B(\tau \rightarrow 3\pi \nu )$. As a consequence, the PDG fit procedure does not take into account the large uncertainty on $B(a_1\rightarrow \pi \gamma )$, resulting in an underestimated fit uncertainty on $\Gamma _{805}$. Therefore, in this case an appropriate correction has to be applied after the fit.

9.1.5 Branching ratio fit results and experimental inputs

Table 304 reports the $\tau$ branching ratio fit results and experimental inputs.

9.1.6 Correlation terms between basis branching fractions uncertainties

The following tables report the correlation coefficients between basis quantities, in percent (Tables 305, 306, 307, 308, 309, 310, 311, 312, 313, 314).

Table 314 Basis quantities correlation coefficients in percent, subtable 10

9.1.7 Equality constraints

We list in the following the equality constraints that relate a branching fraction to a sum of branching fractions. The constraint equations include as coefficients the values of some non-tau branching fractions, denoted e.g., with the self-describing notation $\Gamma _{K_S \rightarrow \pi ^0\pi ^0}$. Some coefficients are probabilities corresponding to modulus square amplitudes describing quantum mixtures of states such as $K^0$, ${\bar{K}}^0$, $K_S$, $K_L$, denoted with e.g., $\Gamma _{<K^0|K_S>} = |{<}K^0|K_S{>}|^2$. All non-tau quantities are taken from the PDG 2015 [327] fits (when available) or averages, and are used without accounting for their uncertainties, which are however in general small with respect to the uncertainties on the $\tau$ branching fractions.

The following list does not include the constraints listed in Table 304, where some measured ratios of branching fractions are expressed as ratios of two branching fractions.

$$\begin{aligned} {\Gamma _{1}}&=\Gamma _{3} + \Gamma _{5} + \Gamma _{9} + \Gamma _{10} + \Gamma _{14} + \Gamma _{16} \\ {}&\quad + \Gamma _{20} + \Gamma _{23} + \Gamma _{27} + \Gamma _{28} + \Gamma _{30} + \Gamma _{35} \\ {}&\quad + \Gamma _{40} + \Gamma _{44} + \Gamma _{37} + \Gamma _{42} + \Gamma _{47} + \Gamma _{48} \\ {}&\quad + \Gamma _{804} + \Gamma _{50} + \Gamma _{51} + \Gamma _{806} + \Gamma _{126}\cdot {}\Gamma _{\eta \rightarrow \text {neutral}} \\ {}&\quad + \Gamma _{128}\cdot {}\Gamma _{\eta \rightarrow \text {neutral}} \\&\quad + \Gamma _{130}\cdot {}\Gamma _{\eta \rightarrow \text {neutral}} + \Gamma _{132}\cdot {}\Gamma _{\eta \rightarrow \text {neutral}} \\ {}&\quad + \Gamma _{800}\cdot {}\Gamma _{\omega \rightarrow \pi ^0\gamma } + \Gamma _{151}\cdot {}\Gamma _{\omega \rightarrow \pi ^0\gamma }\\&\quad + \Gamma _{152}\cdot {}\Gamma _{\omega \rightarrow \pi ^0\gamma } \\ {}&\quad + \Gamma _{167}\cdot {}\Gamma _{\phi \rightarrow K_S K_L}\\ {\Gamma _{2}}&=\Gamma _{3} + \Gamma _{5} \\&\quad + \Gamma _{9} + \Gamma _{10} + \Gamma _{14} + \Gamma _{16} \\ {}&\quad + \Gamma _{20} + \Gamma _{23} + \Gamma _{27} + \Gamma _{28} + \Gamma _{30} \\&\quad + \Gamma _{35}\cdot {}(\Gamma _{<{\bar{K}}^0|K_S>}\cdot {}\Gamma _{K_S\rightarrow \pi ^0\pi ^0}\\ {}&\quad +\Gamma _{<{\bar{K}}^0|K_L>}) + \Gamma _{40}\cdot {}(\Gamma _{<{\bar{K}}^0|K_S>}\cdot {}\Gamma _{K_S\rightarrow \pi ^0\pi ^0}\\&\quad +\Gamma _{<{\bar{K}}^0|K_L>})+ \Gamma _{44}\cdot {}(\Gamma _{<{\bar{K}}^0|K_S>}\cdot {}\Gamma _{K_S\rightarrow \pi ^0\pi ^0}\\ {}&\quad +\Gamma _{<{\bar{K}}^0|K_L>}) + \Gamma _{37}\cdot {}(\Gamma _{<{\bar{K}}^0|K_S>}\cdot {}\Gamma _{K_S\rightarrow \pi ^0\pi ^0}\\&\quad +\Gamma _{<{\bar{K}}^0|K_L>})+ \Gamma _{42}\cdot {}(\Gamma _{<{\bar{K}}^0|K_S>}\cdot {}\Gamma _{K_S\rightarrow \pi ^0\pi ^0} \\ {}&\quad +\Gamma _{<{\bar{K}}^0|K_L>}) + \Gamma _{47}\cdot {}(\Gamma _{K_S\rightarrow \pi ^0\pi ^0}\cdot {}\Gamma _{K_S\rightarrow \pi ^0\pi ^0})\\&\quad + \Gamma _{48}\cdot {}\Gamma _{K_S\rightarrow \pi ^0\pi ^0} \\ {}&\quad + \Gamma _{804} + \Gamma _{50}\cdot {}(\Gamma _{K_S\rightarrow \pi ^0\pi ^0}\cdot {}\Gamma _{K_S\rightarrow \pi ^0\pi ^0})\\&\quad + \Gamma _{51}\cdot {}\Gamma _{K_S\rightarrow \pi ^0\pi ^0} \\ {}&\quad + \Gamma _{806} + \Gamma _{126}\cdot {}\Gamma _{\eta \rightarrow \text {neutral}} + \Gamma _{128}\cdot {}\Gamma _{\eta \rightarrow \text {neutral}} \\&\quad + \Gamma _{130}\cdot {}\Gamma _{\eta \rightarrow \text {neutral}} \\ {}&\quad + \Gamma _{132}\cdot {}(\Gamma _{\eta \rightarrow \text {neutral}}\cdot {}(\Gamma _{<{\bar{K}}^0|K_S>}\cdot {}\Gamma _{K_S\rightarrow \pi ^0\pi ^0}\\&\quad +\Gamma _{<{\bar{K}}^0|K_L>}))+ \Gamma _{800}\cdot {}\Gamma _{\omega \rightarrow \pi ^0\gamma } \\ {}&\quad + \Gamma _{151}\cdot {}\Gamma _{\omega \rightarrow \pi ^0\gamma } + \Gamma _{152}\cdot {}\Gamma _{\omega \rightarrow \pi ^0\gamma } \\&\quad + \Gamma _{167}\cdot {}(\Gamma _{\phi \rightarrow K_S K_L}\cdot {}\Gamma _{K_S\rightarrow \pi ^0\pi ^0})\\ {\Gamma _{7}}&= \Gamma _{35}\cdot {}\Gamma _{<{\bar{K}}^0|K_L>} + \Gamma _{9} + \Gamma _{804} + \Gamma _{37}\cdot {}\Gamma _{<K^0|K_L>}\\ {}&\quad + \Gamma _{10}\\ {\Gamma _{8}}&={\Gamma _{9} + \Gamma _{10}} \end{aligned}$$

$$\begin{aligned} {\Gamma _{11}}=&\Gamma _{14} + \Gamma _{16} + \Gamma _{20} + \Gamma _{23} + \Gamma _{27} + \Gamma _{28}\\ {}&\quad + \Gamma _{30} + \Gamma _{35}\cdot {}(\Gamma _{<K^0|K_S>}\cdot {}\Gamma _{K_S\rightarrow \pi ^0\pi ^0})\\&\quad + \Gamma _{37}\cdot {}(\Gamma _{<K^0|K_S>}\cdot {}\Gamma _{K_S\rightarrow \pi ^0\pi ^0})\\ {}&\quad + \Gamma _{40}\cdot {}(\Gamma _{<K^0|K_S>}\cdot {}\Gamma _{K_S\rightarrow \pi ^0\pi ^0})\\&\quad + \Gamma _{42}\cdot {}(\Gamma _{<K^0|K_S>}\cdot {}\Gamma _{K_S\rightarrow \pi ^0\pi ^0}) \\ {}&\quad + \Gamma _{47}\cdot {}(\Gamma _{K_S\rightarrow \pi ^0\pi ^0}\cdot {}\Gamma _{K_S\rightarrow \pi ^0\pi ^0})\\&\quad + \Gamma _{50}\cdot {}(\Gamma _{K_S\rightarrow \pi ^0\pi ^0}\cdot {}\Gamma _{K_S\rightarrow \pi ^0\pi ^0})\\ {}&\quad + \Gamma _{126}\cdot {}\Gamma _{\eta \rightarrow \text {neutral}} + \Gamma _{128}\cdot {}\Gamma _{\eta \rightarrow \text {neutral}} \\&\quad + \Gamma _{130}\cdot {}\Gamma _{\eta \rightarrow \text {neutral}} \\ {}&\quad + \Gamma _{132}\cdot {}(\Gamma _{<K^0|K_S>}\cdot {}\Gamma _{K_S\rightarrow \pi ^0\pi ^0}\cdot {}\Gamma _{\eta \rightarrow \text {neutral}})\\&\quad + \Gamma _{151}\cdot {}\Gamma _{\omega \rightarrow \pi ^0\gamma } \\ {}&\quad + \Gamma _{152}\cdot {}\Gamma _{\omega \rightarrow \pi ^0\gamma } + \Gamma _{800}\cdot {}\Gamma _{\omega \rightarrow \pi ^0\gamma } \end{aligned}$$

$$\begin{aligned} {\Gamma _{12}}&=\Gamma _{128}\cdot {}\Gamma _{\eta \rightarrow 3\pi ^0} + \Gamma _{30} + \Gamma _{23} + \Gamma _{28} + \Gamma _{14}\\&\quad + \Gamma _{16} + \Gamma _{20} + \Gamma _{27} \\&\quad +\Gamma _{126}\cdot {}\Gamma _{\eta \rightarrow 3\pi ^0} + \Gamma _{130}\cdot {}\Gamma _{\eta \rightarrow 3\pi ^0} \end{aligned}$$

$$\begin{aligned} {\Gamma _{13}}&= {\Gamma _{14}}+{\Gamma _{16}} \end{aligned}$$

$$\begin{aligned} {\Gamma _{17}}= & {} \Gamma _{128}\cdot {}\Gamma _{\eta \rightarrow 3\pi ^0} + \Gamma _{30} + \Gamma _{23} + \Gamma _{28} \\&+ \Gamma _{35}\cdot {}(\Gamma _{<K^0|K_S>}\cdot {}\Gamma _{K_S\rightarrow \pi ^0\pi ^0})\\&+ \Gamma _{40}\cdot {}(\Gamma _{<K^0|K_S>}\cdot {}\Gamma _{K_S\rightarrow \pi ^0\pi ^0})\\&+ \Gamma _{42}\cdot {}(\Gamma _{<K^0|K_S>}\cdot {}\Gamma _{K_S\rightarrow \pi ^0\pi ^0})\\&+ \Gamma _{20} + \Gamma _{27} + \Gamma _{47}\cdot {}(\Gamma _{K_S\rightarrow \pi ^0\pi ^0}\cdot {}\Gamma _{K_S\rightarrow \pi ^0\pi ^0})\\&+ \Gamma _{50}\cdot {}(\Gamma _{K_S\rightarrow \pi ^0\pi ^0}\cdot {}\Gamma _{K_S\rightarrow \pi ^0\pi ^0})\\&+ \Gamma _{126}\cdot {}\Gamma _{\eta \rightarrow 3\pi ^0} + \Gamma _{37}\cdot {}(\Gamma _{<K^0|K_S>}\cdot {}\Gamma _{K_S\rightarrow \pi ^0\pi ^0})\\&+ \Gamma _{130}\cdot {}\Gamma _{\eta \rightarrow 3\pi ^0} \end{aligned}$$

$$\begin{aligned} {\Gamma _{18}}= & {} \Gamma _{23} + \Gamma _{35}\cdot {}(\Gamma _{<K^0|K_S>}\cdot {}\Gamma _{K_S\rightarrow \pi ^0\pi ^0})\\&+ \Gamma _{20} + \Gamma _{37}\cdot {}(\Gamma _{<K^0|K_S>}\cdot {}\Gamma _{K_S\rightarrow \pi ^0\pi ^0}) \end{aligned}$$

$$\begin{aligned} {\Gamma _{19}}={\Gamma _{23} + \Gamma _{20}} \end{aligned}$$

$$\begin{aligned} {\Gamma _{24}}= & {} \Gamma _{27} + \Gamma _{28} + \Gamma _{30} \nonumber \\&+ \Gamma _{40}\cdot {}(\Gamma _{<K^0|K_S>}\cdot {}\Gamma _{K_S\rightarrow \pi ^0\pi ^0})\\&+ \Gamma _{42}\cdot {}(\Gamma _{<K^0|K_S>}\cdot {}\Gamma _{K_S\rightarrow \pi ^0\pi ^0})\\&+ \Gamma _{47}\cdot {}(\Gamma _{K_S\rightarrow \pi ^0\pi ^0}\cdot {}\Gamma _{K_S\rightarrow \pi ^0\pi ^0})\\&+ \Gamma _{50}\cdot {}(\Gamma _{K_S\rightarrow \pi ^0\pi ^0}\cdot {}\Gamma _{K_S\rightarrow \pi ^0\pi ^0})\\&+ \Gamma _{126}\cdot {}\Gamma _{\eta \rightarrow 3\pi ^0} + \Gamma _{128}\cdot {}\Gamma _{\eta \rightarrow 3\pi ^0} \\&+ \Gamma _{130}\cdot {}\Gamma _{\eta \rightarrow 3\pi ^0} \\&+ \Gamma _{132}\cdot {}(\Gamma _{<K^0|K_S>}\cdot {}\Gamma _{K_S\rightarrow \pi ^0\pi ^0}\cdot {}\Gamma _{\eta \rightarrow 3\pi ^0}) \end{aligned}$$

$$\begin{aligned} {\Gamma _{25}}= & {} \Gamma _{128}\cdot {}\Gamma _{\eta \rightarrow 3\pi ^0} + \Gamma _{30} + \Gamma _{28} \\&+ \Gamma _{27} + \Gamma _{126}\cdot {}\Gamma _{\eta \rightarrow 3\pi ^0}\\&+ \Gamma _{130}\cdot {}\Gamma _{\eta \rightarrow 3\pi ^0}\\&+ \Gamma _{30} + \Gamma _{28} + \Gamma _{27} + \Gamma _{126}\cdot {}\Gamma _{\eta \rightarrow 3\pi ^0} \\ {}&+ \Gamma _{130}\cdot {}\Gamma _{\eta \rightarrow 3\pi ^0} \end{aligned}$$

$$\begin{aligned} {\Gamma _{26}}= & {} \Gamma _{128}\cdot {}\Gamma _{\eta \rightarrow 3\pi ^0}+ \Gamma _{27}\\&+ \Gamma _{28} + \Gamma _{40}\cdot {}(\Gamma _{<K^0|K_S>}\cdot {}\Gamma _{K_S\rightarrow \pi ^0\pi ^0})\\&+ \Gamma _{42}\cdot {}(\Gamma _{<K^0|K_S>}\cdot {}\Gamma _{K_S\rightarrow \pi ^0\pi ^0}) \end{aligned}$$

$$\begin{aligned} {\Gamma _{29}}= & {} {\Gamma _{30} + \Gamma _{126}\cdot {}\Gamma _{\eta \rightarrow 3\pi ^0} + \Gamma _{130}\cdot {}\Gamma _{\eta \rightarrow 3\pi ^0}} \end{aligned}$$

$$\begin{aligned} {\Gamma _{31}}= & {} \Gamma _{128}\cdot {}\Gamma _{\eta \rightarrow \text {neutral}} + \Gamma _{23} \\&+ \Gamma _{28} + \Gamma _{42} + \Gamma _{16}\\&+ \Gamma _{37} + \Gamma _{10} + \Gamma _{167}\cdot {}(\Gamma _{\phi \rightarrow K_S K_L}\cdot {}\Gamma _{K_S\rightarrow \pi ^0\pi ^0}) \end{aligned}$$

$$\begin{aligned} {\Gamma _{32}}= & {} \Gamma _{16} + \Gamma _{23} + \Gamma _{28} + \Gamma _{37} \\&+ \Gamma _{42} + \Gamma _{128}\cdot {}\Gamma _{\eta \rightarrow \text {neutral}} \\&+ \Gamma _{130}\cdot {}\Gamma _{\eta \rightarrow \text {neutral}} \\&+ \Gamma _{167}\cdot {}(\Gamma _{\phi \rightarrow K_S K_L}\cdot {}\Gamma _{K_S\rightarrow \pi ^0\pi ^0}) \end{aligned}$$

$$\begin{aligned} {\Gamma _{33}}= & {} \Gamma _{35}\cdot {}\Gamma _{<{\bar{K}}^0|K_S>} + \Gamma _{40}\cdot {}\Gamma _{<{\bar{K}}^0|K_S>} \\&+ \Gamma _{42}\cdot {}\Gamma _{<K^0|K_S>} \\&+ \Gamma _{47} + \Gamma _{48} + \Gamma _{50} + \Gamma _{51} \\&+ \Gamma _{37}\cdot {}\Gamma _{<K^0|K_S>} \\&+ \Gamma _{132}\cdot {}(\Gamma _{<{\bar{K}}^0|K_S>}\cdot {}\Gamma _{\eta \rightarrow \text {neutral}})\\&+ \Gamma _{44}\cdot {}\Gamma _{<{\bar{K}}^0|K_S>} + \Gamma _{167}\cdot {}\Gamma _{\phi \rightarrow K_S K_L} \end{aligned}$$

$$\begin{aligned} {\Gamma _{34}}= & {} {\Gamma _{35} + \Gamma _{37}}\\ {\Gamma _{38}}= & {} {\Gamma _{42} + \Gamma _{37}}\\ {\Gamma _{39}}= & {} {\Gamma _{40} + \Gamma _{42}}\\ {\Gamma _{43}}= & {} {\Gamma _{40} + \Gamma _{44}}\\ {\Gamma _{46}}= & {} {\Gamma _{48} + \Gamma _{47} + \Gamma _{804}}\\ {\Gamma _{49}}= & {} {\Gamma _{50} + \Gamma _{51} + \Gamma _{806}} \end{aligned}$$

$$\begin{aligned} {\Gamma _{54}}= & {} \Gamma _{35}\cdot {}(\Gamma _{<K^0|K_S>}\cdot {}\Gamma _{K_S\rightarrow \pi ^+\pi ^-})\\&+ \Gamma _{37}\cdot {}(\Gamma _{<K^0|K_S>}\cdot {}\Gamma _{K_S\rightarrow \pi ^+\pi ^-})\\&+ \Gamma _{40}\cdot {}(\Gamma _{<K^0|K_S>}\cdot {}\Gamma _{K_S\rightarrow \pi ^+\pi ^-})\\&+ \Gamma _{42}\cdot {}(\Gamma _{<K^0|K_S>}\cdot {}\Gamma _{K_S\rightarrow \pi ^+\pi ^-})\\&+ \Gamma _{47}\cdot {}(2\cdot {}\Gamma _{K_S\rightarrow \pi ^+\pi ^-}\cdot {}\Gamma _{K_S\rightarrow \pi ^0\pi ^0})\\&+ \Gamma _{48}\cdot {}\Gamma _{K_S\rightarrow \pi ^+\pi ^-} \\&+ \Gamma _{50}\cdot {}(2\cdot {}\Gamma _{K_S\rightarrow \pi ^+\pi ^-}\cdot {}\Gamma _{K_S\rightarrow \pi ^0\pi ^0})\\&+ \Gamma _{51}\cdot {}\Gamma _{K_S\rightarrow \pi ^+\pi ^-} \\&+ \Gamma _{53}\cdot {}(\Gamma _{<{\bar{K}}^0|K_S>}\cdot {}\Gamma _{K_S\rightarrow \pi ^0\pi ^0}+\Gamma _{<{\bar{K}}^0|K_L>})\\&+ \Gamma _{62} + \Gamma _{70}\\&+ \Gamma _{77} + \Gamma _{78} + \Gamma _{93} \\&+ \Gamma _{94} + \Gamma _{126}\cdot {}\Gamma _{\eta \rightarrow \text {charged}}\\&+\Gamma _{128}\cdot {}\Gamma _{\eta \rightarrow \text {charged}} + \Gamma _{130}\cdot {}\Gamma _{\eta \rightarrow \text {charged}} \\&+ \Gamma _{132}\cdot {}(\Gamma _{<{\bar{K}}^0|K_L>}\cdot {}\Gamma _{\eta \rightarrow \pi ^+\pi ^-\pi ^0}\\&+ \Gamma _{<{\bar{K}}^0|K_S>}\cdot {}\Gamma _{K_S\rightarrow \pi ^0\pi ^0}\cdot {}\Gamma _{\eta \rightarrow \pi ^+\pi ^-\pi ^0}\\&+ \Gamma _{<{\bar{K}}^0|K_S>}\cdot {}\Gamma _{K_S\rightarrow \pi ^+\pi ^-}\cdot {}\Gamma _{\eta \rightarrow 3\pi ^0})\\&+ \Gamma _{151}\cdot {}(\Gamma _{\omega \rightarrow \pi ^+\pi ^-\pi ^0}+\Gamma _{\omega \rightarrow \pi ^+\pi ^-})\\&+ \Gamma _{152}\cdot {}(\Gamma _{\omega \rightarrow \pi ^+\pi ^-\pi ^0}+\Gamma _{\omega \rightarrow \pi ^+\pi ^-})\\&+ \Gamma _{167}\cdot {}(\Gamma _{\phi \rightarrow K^+K^-} + \Gamma _{\phi \rightarrow K_S K_L}\cdot {}\Gamma _{K_S\rightarrow \pi ^+\pi ^-})\\&+ \Gamma _{802} + \Gamma _{803} \\&+ \Gamma _{800}\cdot {}(\Gamma _{\omega \rightarrow \pi ^+\pi ^-\pi ^0}+\Gamma _{\omega \rightarrow \pi ^+\pi ^-}) \end{aligned}$$

$$\begin{aligned} {\Gamma _{55}}= & {} \Gamma _{128}\cdot {}\Gamma _{\eta \rightarrow \text {charged}} \\&+ \Gamma _{152}\cdot {}(\Gamma _{\omega \rightarrow \pi ^+\pi ^-\pi ^0}+\Gamma _{\omega \rightarrow \pi ^+\pi ^-})+ \Gamma _{78} \\&+ \Gamma _{77} + \Gamma _{94} + \Gamma _{62} + \Gamma _{70} + \Gamma _{93} \\&+ \Gamma _{126}\cdot {}\Gamma _{\eta \rightarrow \text {charged}} \\&+ \Gamma _{802} + \Gamma _{803} + \Gamma _{800}\cdot {}(\Gamma _{\omega \rightarrow \pi ^+\pi ^-\pi ^0}+\Gamma _{\omega \rightarrow \pi ^+\pi ^-})\\&+ \Gamma _{151}\cdot {}(\Gamma _{\omega \rightarrow \pi ^+\pi ^-\pi ^0}\\&+\Gamma _{\omega \rightarrow \pi ^+\pi ^-})\\&+ \Gamma _{130}\cdot {}\Gamma _{\eta \rightarrow \text {charged}} + \Gamma _{168} \end{aligned}$$

$$\begin{aligned} {\Gamma _{56}}= & {} \Gamma _{35}\cdot {}(\Gamma _{<K^0|K_S>}\cdot {}\Gamma _{K_S\rightarrow \pi ^+\pi ^-})\\&+ \Gamma _{62} + \Gamma _{93} + \Gamma _{37}\cdot {}(\Gamma _{<K^0|K_S>}\cdot {}\Gamma _{K_S\rightarrow \pi ^+\pi ^-})\\&+ \Gamma _{802} + \Gamma _{800}\cdot {}\Gamma _{\omega \rightarrow \pi ^+\pi ^-} \\&+ \Gamma _{151}\cdot {}\Gamma _{\omega \rightarrow \pi ^+\pi ^-} + \Gamma _{168}\\ \end{aligned}$$

$$\begin{aligned} {\Gamma _{57}}= & {} \Gamma _{62} + \Gamma _{93} + \Gamma _{802}\\&+ \Gamma _{800}\cdot {}\Gamma _{\omega \rightarrow \pi ^+\pi ^-}\\&+ \Gamma _{151}\cdot {}\Gamma _{\omega \rightarrow \pi ^+\pi ^-}\\&+ \Gamma _{167}\cdot {}\Gamma _{\phi \rightarrow K^+K^-} \end{aligned}$$

$$\begin{aligned} {\Gamma _{58}}= & {} \Gamma _{62} + \Gamma _{93} + \Gamma _{802} + \Gamma _{167}\cdot {}\Gamma _{\phi \rightarrow K^+K^-} \end{aligned}$$

$$\begin{aligned} {\Gamma _{59}}= & {} \Gamma _{35}\cdot {}(\Gamma _{<K^0|K_S>}\cdot {}\Gamma _{K_S\rightarrow \pi ^+\pi ^-})\\&+ \Gamma _{62} + \Gamma _{800}\cdot {}\Gamma _{\omega \rightarrow \pi ^+\pi ^-} \end{aligned}$$

$$\begin{aligned} {\Gamma _{60}}= & {} \Gamma _{62} + \Gamma _{800}\cdot {}\Gamma _{\omega \rightarrow \pi ^+\pi ^-} \end{aligned}$$

$$\begin{aligned} {\Gamma _{63}}= & {} \Gamma _{40}\cdot {}(\Gamma _{<K^0|K_S>}\cdot {}\Gamma _{K_S\rightarrow \pi ^+\pi ^-})\\&+ \Gamma _{42}\cdot {}(\Gamma _{<K^0|K_S>}\cdot {}\Gamma _{K_S\rightarrow \pi ^+\pi ^-})\\&+ \Gamma _{47}\cdot {}(2\cdot {}\Gamma _{K_S\rightarrow \pi ^+\pi ^-}\cdot {}\Gamma _{K_S\rightarrow \pi ^0\pi ^0})\\&+ \Gamma _{50}\cdot {}(2\cdot {}\Gamma _{K_S\rightarrow \pi ^+\pi ^-}\cdot {}\Gamma _{K_S\rightarrow \pi ^0\pi ^0})\\&+ \Gamma _{70} + \Gamma _{77} + \Gamma _{78}\\&+ \Gamma _{94} + \Gamma _{126}\cdot {}\Gamma _{\eta \rightarrow \text {charged}}\\&+ \Gamma _{128}\cdot {}\Gamma _{\eta \rightarrow \text {charged}} + \Gamma _{130}\cdot {}\Gamma _{\eta \rightarrow \text {charged}} \\&+ \Gamma _{132}\cdot {}(\Gamma _{<{\bar{K}}^0|K_S>}\cdot {}\Gamma _{K_S\rightarrow \pi ^+\pi ^-}\cdot {}\Gamma _{\eta \rightarrow \text {neutral}}\\&+ \Gamma _{<{\bar{K}}^0|K_S>}\cdot {}\Gamma _{K_S\rightarrow \pi ^0\pi ^0}\cdot {}\Gamma _{\eta \rightarrow \text {charged}})\\&+ \Gamma _{151}\cdot {}\Gamma _{\omega \rightarrow \pi ^+\pi ^-\pi ^0} +\Gamma _{152}\cdot {}(\Gamma _{\omega \rightarrow \pi ^+\pi ^-\pi ^0}\\&+\Gamma _{\omega \rightarrow \pi ^+\pi ^-}) + \Gamma _{800}\cdot {}\Gamma _{\omega \rightarrow \pi ^+\pi ^-\pi ^0} + \Gamma _{803} \end{aligned}$$

$$\begin{aligned} {\Gamma _{64}}= & {} \Gamma _{78} + \Gamma _{77} + \Gamma _{94} + \Gamma _{70}\\&+ \Gamma _{126}\cdot {}\Gamma _{\eta \rightarrow \pi ^+\pi ^-\pi ^0}\\&+ \Gamma _{128}\cdot {}\Gamma _{\eta \rightarrow \pi ^+\pi ^-\pi ^0} \\&+\Gamma _{130}\cdot {}\Gamma _{\eta \rightarrow \pi ^+\pi ^-\pi ^0}\\&+ \Gamma _{800}\cdot {}\Gamma _{\omega \rightarrow \pi ^+\pi ^-\pi ^0} \\&+ \Gamma _{151}\cdot {}\Gamma _{\omega \rightarrow \pi ^+\pi ^-\pi ^0} \\&+ \Gamma _{152}\cdot {}(\Gamma _{\omega \rightarrow \pi ^+\pi ^-\pi ^0}+\Gamma _{\omega \rightarrow \pi ^+\pi ^-})\\&+ \Gamma _{803} \end{aligned}$$

$$\begin{aligned} {\Gamma _{65}}= & {} \Gamma _{40}\cdot {}(\Gamma _{<K^0|K_S>}\cdot {}\Gamma _{K_S\rightarrow \pi ^+\pi ^-})\\&+ \Gamma _{42}\cdot {}(\Gamma _{<K^0|K_S>}\cdot {}\Gamma _{K_S\rightarrow \pi ^+\pi ^-}) + \Gamma _{70} + \Gamma _{94} \\&+ \Gamma _{128}\cdot {}\Gamma _{\eta \rightarrow \pi ^+\pi ^-\pi ^0} \\&+ \Gamma _{151}\cdot {}\Gamma _{\omega \rightarrow \pi ^+\pi ^-\pi ^0} \\&+ \Gamma _{152}\cdot {}\Gamma _{\omega \rightarrow \pi ^+\pi ^-} \\&+ \Gamma _{800}\cdot {}\Gamma _{\omega \rightarrow \pi ^+\pi ^-\pi ^0} + \Gamma _{803} \end{aligned}$$

$$\begin{aligned} {\Gamma _{66}}= & {} \Gamma _{70} + \Gamma _{94} + \Gamma _{128}\cdot {}\Gamma _{\eta \rightarrow \pi ^+\pi ^-\pi ^0} \\&+ \Gamma _{151}\cdot {}\Gamma _{\omega \rightarrow \pi ^+\pi ^-\pi ^0} \\&+ \Gamma _{152}\cdot {}\Gamma _{\omega \rightarrow \pi ^+\pi ^-} \\&+ \Gamma _{800}\cdot {}\Gamma _{\omega \rightarrow \pi ^+\pi ^-\pi ^0} + \Gamma _{803} \end{aligned}$$

$$\begin{aligned} {\Gamma _{67}}= & {} \Gamma _{70} + \Gamma _{94} \\&+ \Gamma _{128}\cdot {}\Gamma _{\eta \rightarrow \pi ^+\pi ^-\pi ^0} + \Gamma _{803} \end{aligned}$$

$$\begin{aligned} {\Gamma _{68}}= & {} \Gamma _{40}\cdot {}(\Gamma _{<K^0|K_S>}\cdot {}\Gamma _{K_S\rightarrow \pi ^+\pi ^-})\\&+ \Gamma _{70} + \Gamma _{152}\cdot {}\Gamma _{\omega \rightarrow \pi ^+\pi ^-} \\&+ \Gamma _{800}\cdot {}\Gamma _{\omega \rightarrow \pi ^+\pi ^-\pi ^0} \end{aligned}$$

$$\begin{aligned} {\Gamma _{69}}= & {} \Gamma _{152}\cdot {}\Gamma _{\omega \rightarrow \pi ^+\pi ^-} + \Gamma _{70} \\&+ \Gamma _{800}\cdot {}\Gamma _{\omega \rightarrow \pi ^+\pi ^-\pi ^0} \end{aligned}$$

$$\begin{aligned} {\Gamma _{74}}= & {} \Gamma _{152}\cdot {}\Gamma _{\omega \rightarrow \pi ^+\pi ^-\pi ^0} + \Gamma _{78} \\&+ \Gamma _{77} + \Gamma _{126}\cdot {}\Gamma _{\eta \rightarrow \pi ^+\pi ^-\pi ^0} \\&+ \Gamma _{130}\cdot {}\Gamma _{\eta \rightarrow \pi ^+\pi ^-\pi ^0} \end{aligned}$$

$$\begin{aligned} {\Gamma _{75}}= & {} \Gamma _{152}\cdot {}\Gamma _{\omega \rightarrow \pi ^+\pi ^-\pi ^0} \\&+ \Gamma _{47}\cdot {}(2\cdot {}\Gamma _{K_S\rightarrow \pi ^+\pi ^-}\cdot {}\Gamma _{K_S\rightarrow \pi ^0\pi ^0})\\&+ \Gamma _{77} + \Gamma _{126}\cdot {}\Gamma _{\eta \rightarrow \pi ^+\pi ^-\pi ^0} \\&+ \Gamma _{130}\cdot {}\Gamma _{\eta \rightarrow \pi ^+\pi ^-\pi ^0} \end{aligned}$$

$$\begin{aligned} {\Gamma _{76}}= & {} \Gamma _{152}\cdot {}\Gamma _{\omega \rightarrow \pi ^+\pi ^-\pi ^0} + \Gamma _{77} \\&+ \Gamma _{126}\cdot {}\Gamma _{\eta \rightarrow \pi ^+\pi ^-\pi ^0}\\&+ \Gamma _{130}\cdot {}\Gamma _{\eta \rightarrow \pi ^+\pi ^-\pi ^0} \end{aligned}$$

$$\begin{aligned} {\Gamma _{78}}= & {} \Gamma _{810} + \Gamma _{50}\cdot {}(2\cdot {}\Gamma _{K_S\rightarrow \pi ^+\pi ^-}\cdot {}\Gamma _{K_S\rightarrow \pi ^0\pi ^0})\\&+ \Gamma _{132}\cdot {}(\Gamma _{<{\bar{K}}^0|K_S>}\cdot {}\Gamma _{K_S\rightarrow \pi ^+\pi ^-}\cdot {}\Gamma _{\eta \rightarrow 3\pi ^0}) \end{aligned}$$

$$\begin{aligned} {\Gamma _{79}}= & {} \Gamma _{37}\cdot {}(\Gamma _{<K^0|K_S>}\cdot {}\Gamma _{K_S\rightarrow \pi ^+\pi ^-})\\&+ \Gamma _{42}\cdot {}(\Gamma _{<K^0|K_S>}\cdot {}\Gamma _{K_S\rightarrow \pi ^+\pi ^-})\\&+ \Gamma _{93} + \Gamma _{94} + \Gamma _{128}\cdot {}\Gamma _{\eta \rightarrow \text {charged}} \\&+ \Gamma _{151}\cdot {}(\Gamma _{\omega \rightarrow \pi ^+\pi ^-\pi ^0}\\&+\Gamma _{\omega \rightarrow \pi ^+\pi ^-}) + \Gamma _{168} + \Gamma _{802} + \Gamma _{803} \end{aligned}$$

$$\begin{aligned} {\Gamma _{80}}= & {} \Gamma _{93} + \Gamma _{802} + \Gamma _{151}\cdot {}\Gamma _{\omega \rightarrow \pi ^+\pi ^-} \end{aligned}$$

$$\begin{aligned} {\Gamma _{81}}= & {} \Gamma _{128}\cdot {}\Gamma _{\eta \rightarrow \pi ^+\pi ^-\pi ^0} + \Gamma _{94} \\&+ \Gamma _{803} + \Gamma _{151}\cdot {}\Gamma _{\omega \rightarrow \pi ^+\pi ^-\pi ^0} \end{aligned}$$

$$\begin{aligned} {\Gamma _{82}}= & {} \Gamma _{128}\cdot {}\Gamma _{\eta \rightarrow \text {charged}} \\&+ \Gamma _{42}\cdot {}(\Gamma _{<K^0|K_S>}\cdot {}\Gamma _{K_S\rightarrow \pi ^+\pi ^-}) + \Gamma _{802} \\&+ \Gamma _{803} + \Gamma _{151}\cdot {}(\Gamma _{\omega \rightarrow \pi ^+\pi ^-\pi ^0}+\Gamma _{\omega \rightarrow \pi ^+\pi ^-})\\&+ \Gamma _{37}\cdot {}(\Gamma _{<K^0|K_S>}\cdot {}\Gamma _{K_S\rightarrow \pi ^+\pi ^-}) \end{aligned}$$

$$\begin{aligned} {\Gamma _{83}}= & {} \Gamma _{128}\cdot {}\Gamma _{\eta \rightarrow \pi ^+\pi ^-\pi ^0} + \Gamma _{802} \\&+ \Gamma _{803} + \Gamma _{151}\cdot {}(\Gamma _{\omega \rightarrow \pi ^+\pi ^-\pi ^0}\\&+\Gamma _{\omega \rightarrow \pi ^+\pi ^-}) \end{aligned}$$

$$\begin{aligned} {\Gamma _{84}}= & {} \Gamma _{802} + \Gamma _{151}\cdot {}\Gamma _{\omega \rightarrow \pi ^+\pi ^-} \\&+ \Gamma _{37}\cdot {}(\Gamma _{<K^0|K_S>}\cdot {}\Gamma _{K_S\rightarrow \pi ^+\pi ^-}) \end{aligned}$$

$$\begin{aligned} {\Gamma _{85}}= & {} \Gamma _{802} + \Gamma _{151}\cdot {}\Gamma _{\omega \rightarrow \pi ^+\pi ^-} \end{aligned}$$

$$\begin{aligned} {\Gamma _{87}}= & {} \Gamma _{42}\cdot {}(\Gamma _{<K^0|K_S>}\cdot {}\Gamma _{K_S\rightarrow \pi ^+\pi ^-})\\&+ \Gamma _{128}\cdot {}\Gamma _{\eta \rightarrow \pi ^+\pi ^-\pi ^0} \\&+ \Gamma _{151}\cdot {}\Gamma _{\omega \rightarrow \pi ^+\pi ^-\pi ^0} + \Gamma _{803} \end{aligned}$$

$$\begin{aligned} {\Gamma _{88}}= & {} \Gamma _{128}\cdot {}\Gamma _{\eta \rightarrow \pi ^+\pi ^-\pi ^0} + \Gamma _{803} + \Gamma _{151}\cdot {}\Gamma _{\omega \rightarrow \pi ^+\pi ^-\pi ^0} \end{aligned}$$

$$\begin{aligned} {\Gamma _{89}}= & {} \Gamma _{803} + \Gamma _{151}\cdot {}\Gamma _{\omega \rightarrow \pi ^+\pi ^-\pi ^0} \end{aligned}$$

$$\begin{aligned} {\Gamma _{92}}= & {} \Gamma _{94} + \Gamma _{93} \end{aligned}$$

$$\begin{aligned} {\Gamma _{96}}= & {} {\Gamma _{167}\cdot {}\Gamma _{\phi \rightarrow K^+K^-}} \end{aligned}$$

$$\begin{aligned} {\Gamma _{102}}= & {} {\Gamma _{103} + \Gamma _{104}} \end{aligned}$$

$$\begin{aligned} {\Gamma _{103}}= & {} {\Gamma _{820} + \Gamma _{822} + \Gamma _{831}\cdot {}\Gamma _{\omega \rightarrow \pi ^+\pi ^-}} \end{aligned}$$

$$\begin{aligned} {\Gamma _{104}}= & {} {\Gamma _{830} + \Gamma _{833}} \end{aligned}$$

$$\begin{aligned} {\Gamma _{106}}= & {} \Gamma _{30} + \Gamma _{44}\cdot {}\Gamma _{<{\bar{K}}^0|K_S>} \\&+ \Gamma _{47} + \Gamma _{53}\cdot {}\Gamma _{<K^0|K_S>}\\&+ \Gamma _{77} + \Gamma _{103} + \Gamma _{126}\cdot {}(\Gamma _{\eta \rightarrow 3\pi ^0}+\Gamma _{\eta \rightarrow \pi ^+\pi ^-\pi ^0})\\&+ \Gamma _{152}\cdot {}\Gamma _{\omega \rightarrow \pi ^+\pi ^-\pi ^0} \end{aligned}$$

$$\begin{aligned} {\Gamma _{110}}= & {} \Gamma _{10} + \Gamma _{16} + \Gamma _{23} + \Gamma _{28} + \Gamma _{35} + \Gamma _{40}\\&+ \Gamma _{128} + \Gamma _{802} + \Gamma _{803} + \Gamma _{151} + \Gamma _{130} + \Gamma _{132} \\&+ \Gamma _{44} + \Gamma _{53} + \Gamma _{168} + \Gamma _{169}\\&+ \Gamma _{822} + \Gamma _{833} \end{aligned}$$

$$\begin{aligned} {\Gamma _{149}}= & {} \Gamma _{152} + \Gamma _{800} + \Gamma _{151} \end{aligned}$$

$$\begin{aligned} {\Gamma _{150}}= & {} {\Gamma _{800} + \Gamma _{151}} \end{aligned}$$

$$\begin{aligned} {\Gamma _{168}}= & {} {\Gamma _{167}\cdot {}\Gamma _{\phi \rightarrow K^+K^-}} \end{aligned}$$

$$\begin{aligned} {\Gamma _{169}}= & {} \Gamma _{167}\cdot {}\Gamma _{\phi \rightarrow K_S K_L} \end{aligned}$$

$$\begin{aligned} {\Gamma _{804}}= & {} \Gamma _{47} \cdot {} ((\Gamma _{<K^0|K_L>}\cdot {}\Gamma _{<{\bar{K}}^0|K_L>}) /\\&(\Gamma _{<K^0|K_S>}\cdot {}\Gamma _{<{\bar{K}}^0|K_S>})) \end{aligned}$$

$$\begin{aligned} {\Gamma _{806}}= & {} \Gamma _{50} \cdot {} ((\Gamma _{<K^0|K_L>}\cdot {}\Gamma _{<{\bar{K}}^0|K_L>}) /\\&(\Gamma _{<K^0|K_S>}\cdot {}\Gamma _{<{\bar{K}}^0|K_S>})) \end{aligned}$$

$$\begin{aligned} {\Gamma _{810}}= & {} \Gamma _{910} + \Gamma _{911} + \Gamma _{811}\cdot {}\Gamma _{\omega \rightarrow \pi ^+\pi ^-\pi ^0} + \Gamma _{812} \end{aligned}$$

$$\begin{aligned} {\Gamma _{820}}= & {} {\Gamma _{920} + \Gamma _{821}} \end{aligned}$$

$$\begin{aligned} {\Gamma _{830}}= & {} {\Gamma _{930} + \Gamma _{831}\cdot {}\Gamma _{\omega \rightarrow \pi ^+\pi ^-\pi ^0} + \Gamma _{832}} \end{aligned}$$

$$\begin{aligned} {\Gamma _{910}}= & {} {\Gamma _{136}\cdot {}\Gamma _{\eta \rightarrow 3\pi ^0}} \end{aligned}$$

$$\begin{aligned} {\Gamma _{911}}= & {} {\Gamma _{945}\cdot {}\Gamma _{\eta \rightarrow \pi ^+\pi ^-\pi ^0}} \end{aligned}$$

$$\begin{aligned} {\Gamma _{930}}= & {} {\Gamma _{136}\cdot {}\Gamma _{\eta \rightarrow \pi ^+\pi ^-\pi ^0}} \end{aligned}$$

$$\begin{aligned} {\Gamma _{944}}= & {} {\Gamma _{136}\cdot {}\Gamma _{\eta \rightarrow \gamma \gamma }} \end{aligned}$$

$$\begin{aligned} {\Gamma _{\text {All}}}= & {} \Gamma _{3} + \Gamma _{5} + \Gamma _{9} + \Gamma _{10} + \Gamma _{14} + \Gamma _{16}\\&+ \Gamma _{20} + \Gamma _{23} + \Gamma _{27} + \Gamma _{28} + \Gamma _{30} + \Gamma _{35} \\&+ \Gamma _{37} + \Gamma _{40} + \Gamma _{42} \\&+ \Gamma _{47}\cdot {}(1 + ((\Gamma _{<K^0|K_L>}\\&\cdot {}\Gamma _{<{\bar{K}}^0|K_L>}) /(\Gamma _{<K^0|K_S>}\cdot {}\Gamma _{<{\bar{K}}^0|K_S>}))) \\&+ \Gamma _{48} + \Gamma _{62} + \Gamma _{70} + \Gamma _{77} + \Gamma _{811} + \Gamma _{812} \\&+ \Gamma _{93} + \Gamma _{94} + \Gamma _{832} + \Gamma _{833} + \Gamma _{126} + \Gamma _{128} \\&+ \Gamma _{802} + \Gamma _{803} + \Gamma _{800} + \Gamma _{151} + \Gamma _{130} + \Gamma _{132} \\&+ \Gamma _{44} + \Gamma _{53}+ \Gamma _{50}\cdot {}(1 + ((\Gamma _{<K^0|K_L>}\cdot {}\Gamma _{<{\bar{K}}^0|K_L>}) /\\&(\Gamma _{<K^0|K_S>}\cdot {}\Gamma _{<{\bar{K}}^0|K_S>}))) + \Gamma _{51} + \Gamma _{167}\\&\cdot {}(\Gamma _{\phi \rightarrow K^+K^-}+\Gamma _{\phi \rightarrow K_S K_L}) + \Gamma _{152} + \Gamma _{920} \\&+ \Gamma _{821} + \Gamma _{822} + \Gamma _{831} + \Gamma _{136} + \Gamma _{945} + \Gamma _{805} \end{aligned}$$

9.2 Tests of lepton universality

Lepton universality tests probe the Standard Model prediction that the charged weak current interaction has the same coupling for all lepton generations. The precision of such tests has been significantly improved since the 2014 edition by the addition of the Belle $\tau$ lifetime measurement [1402], while improvements from the $\tau$ branching fraction fit are negligible. We compute the universality tests as in the previous report by using ratios of the partial widths of a heavier lepton $\lambda$ decaying to a lighter lepton $\rho$ [1403],

$$\begin{aligned} \Gamma (\lambda \rightarrow \nu _{\lambda } \rho \overline{\nu } _{\rho } (\gamma ))&= \frac{\mathcal {B}(\lambda \rightarrow \nu _{\lambda } \rho \overline{\nu } _{\rho })}{\tau _{\lambda }}\\&= \frac{G_{\lambda } G_{\rho } m^5_{\lambda }}{192 \pi ^3}\, f\bigg (\frac{m^2_{\rho }}{m^2_{\lambda }}\bigg ) R^{\lambda }_W R^{\lambda }_\gamma , \end{aligned}$$

where

$$\begin{aligned}&G_{\rho } = \frac{g^2_{\rho }}{4 \sqrt{2} M^2_W},\\&f(x) = 1 -8x +8x^3 -x^4 -12x^2 \text {ln}x, \\&R^{\lambda }_W = 1 + \frac{3}{5} \frac{m^2_{\lambda }}{M^2_W} + \frac{9}{5} \frac{m^2_{\rho }}{M^2_W}~ [1390-1392], \\&R^{\lambda }_\gamma = 1+\frac{\alpha (m_{\lambda })}{2\pi } \bigg (\frac{25}{4}-\pi ^2\bigg ). \end{aligned}$$

We use $R^\tau _\gamma =1-43.2\cdot 10^{-4}$ and $R^\mu _\gamma =1-42.4\cdot 10^{-4}$ [1403] and $M_W$ from PDG 2015 [327]. We use HFLAV Spring 2017 averages and PDG 2015 for the other quantities. Using pure leptonic processes we obtain

$$\begin{aligned} \bigg ( \frac{g_\tau }{g_\mu } \bigg )&= 1.0010 \,\pm \, 0.0015,\\ \bigg ( \frac{g_\tau }{g_e} \bigg )&= 1.0029 \,\pm \, 0.0015,\\ \bigg ( \frac{g_\mu }{g_e} \bigg )&= 1.0019 \,\pm \, 0.0014. \end{aligned}$$

Using the expressions for the $\tau$ semi-hadronic partial widths, we obtain

$$\begin{aligned} \bigg ( \frac{g_\tau }{g_\mu } \bigg )^2 = \frac{B({\tau \rightarrow h \nu _\tau })}{B({h \rightarrow \mu {\bar{\nu }}_\mu })} \frac{2m_h m^2_{\mu }\tau _h}{(1 + \delta R_{\tau /h})m^3_{\tau }\tau _{\tau }} \bigg ( \frac{1-m^2_{\mu }/m^2_h}{1-m^2_h/m^2_{\tau }} \bigg )^2, \end{aligned}$$

where h = $\pi$ or K and the radiative corrections are $\delta R_{\tau /\pi } = (0.16 \,\pm \, 0.14) \%$ and $\delta R_{\tau /K} = (0.90 \,\pm \, 0.22)\%$ [1407,1408,1409,1410]. We measure:

$$\begin{aligned} \bigg ( \frac{g_\tau }{g_\mu } \bigg )_\pi&= 0.9961 \,\pm \, 0.0027,\\ \bigg ( \frac{g_\tau }{g_\mu } \bigg )_K&= 0.9860 \,\pm \, 0.0070. \end{aligned}$$

Similar tests could be performed with decays to electrons, however they are less precise because the hadron two body decays to electrons are helicity-suppressed. Averaging the three $g_\tau /g_\mu$ ratios we obtain

$$\begin{aligned} \bigg ( \frac{g_\tau }{g_\mu } \bigg )_{\tau {+}\pi {+}K}&= 1.0000 \,\pm \, 0.0014, \end{aligned}$$

accounting for statistical correlations. Table 315 reports the statistical correlation coefficients for the fitted coupling ratios.

Table 315 Universality coupling ratios correlation coefficients (%)

Since there is 100% correlation between $g_\tau/g_\mu$, $g_\tau/g_e$ and $g_\mu/g_e$, the correlation matrix is expected to be positive semi-definite, with one eigenvalue equal to zero. Due to numerical inaccuracies, one eigenvalue is expected to be close to zero rather than exactly zero.

9.3 Universality improved $B(\tau \rightarrow e \nu {\bar{\nu }})$ and $R_{\text {had}}$

We compute two quantities that are used in this report and that have been traditionally used for further elaborations and tests involving the $\tau$ branching fractions:

• the “universality improved” experimental determination of $B_e = B(\tau \rightarrow e \nu {\bar{\nu }})$, which relies on the assumption that the Standard Model and lepton universality hold;

• the ratio $R_{\text {had}}$ between the total branching fraction of the $\tau$ to hadrons and the universality improved $\mathcal {B}_e$, which is the same as the ratio of the two respective partial widths.

Following Ref. [1411], we obtain a more precise experimental determination of $B_e$ using the $\tau$ branching fraction to $\mu \nu {\bar{\nu }}$, $B_\mu$, and the $\tau$ lifetime. We average:

• the $B_e$ fit value $\Gamma _{5}$,

• the $B_e$ determination from the $B_\mu = B(\tau \rightarrow \mu \nu {\bar{\nu }})$ fit value ${\Gamma _{3}}$ assuming that $g_\mu /g_e = 1$, hence (see also Sect. 9.2) $B_e$ = $B_\mu \cdot f(m^2_e/m^2_\tau )/f(m^2_\mu /m^2_\tau )$,

• the $B_e$ determination from the $\tau$ lifetime assuming that $g_\tau /g_\mu =1$, hence $B_e = B(\mu \rightarrow e {\bar{\nu }}_e \nu _\mu )\cdot (\tau _\tau / \tau _\mu ) \cdot (m_\tau /m_\mu )^5 \cdot f(m^2_e/m^2_\tau )/f(m^2_e/m^2_\mu ) \cdot (\delta _\gamma ^\tau \delta _W^\tau )/(\delta _\gamma ^\mu \delta _W^\mu )$ where $B(\mu \rightarrow e {\bar{\nu }}_e \nu _\mu ) = 1$.

Accounting for statistical correlations, we obtain

$$\begin{aligned} B_e^{\text {uni}} = (17.815 \,\pm \, 0.023)\%. \end{aligned}$$

We use $B_e^{\text {uni}}$ to obtain the ratio

$$\begin{aligned} R_{\text {had}} = \frac{\Gamma (\tau \rightarrow \text {hadrons})}{\Gamma (\tau \rightarrow e\nu {\bar{\nu }})} = \frac{\Gamma _{\text {hadrons}}}{B_e^{\text {uni}}} = 3.6349\,\pm \, 0.0082, \end{aligned}$$

where $\Gamma (\tau \rightarrow \text {hadrons})$ and $\Gamma (\tau \rightarrow e\nu {\bar{\nu }})$ indicate the partial widths and $\Gamma _{\text {hadrons}}$ is the total branching fraction of the $\tau$ to hadrons, or the total branching fraction in any measured final state minus the leptonic branching fractions, i.e., with our notation $\Gamma _{\text {hadrons}} = {\Gamma _\mathrm{All}} - {\Gamma _{3}} - {\Gamma _{5}} = (64.76 \,\pm \, 0.10)\%$ (see Sect. 9.1 and Table 304 for the definitions of ${\Gamma _\mathrm{All}}, {\Gamma _{3}}, {\Gamma _{5}}$). We underline that this report’s definition of $\Gamma _{\text {hadrons}}$ corresponds to summing all $\tau$ hadronic decay modes, like in the previous report, rather than – as done elsewhere – subtracting the leptonic branching fractions from unity, i.e., $\Gamma _{\text {hadrons}} = 1 - {\Gamma _{3}} - {\Gamma _{5}}$.

9.4 $|V_{us}|$ measurement

The CKM matrix element magnitude $|V_{us}|$ is most precisely determined from kaon decays [1412] (see Fig. 220), and its precision is limited by the uncertainties of the lattice QCD estimates of the meson decay constants $f_+^{K\pi }(0)$ and $f_K/f_\pi$. Using the $\tau$ branching fractions, it is possible to determine $|V_{us}|$ in an alternative way [1413, 1414] that does not depend on lattice QCD and has small theory uncertainties (as discussed in Sect. 9.4.1). Moreover, $|V_{us}|$ can be determined using the $\tau$ branching fractions similarly to the kaon case, using the same meson decay constants from Lattice QCD.

Fig. 220

$|V_{us}|$ averages

9.4.1 $|V_{us}|$ from $B(\tau \rightarrow X_s\nu )$

The $\tau$ hadronic partial width is the sum of the $\tau$ partial widths to strange and to non-strange hadronic final states, $\Gamma _{\text {had}} = \Gamma _s + \Gamma _{\text {VA}}$. The suffix “VA” traditionally denotes the sum of the $\tau$ partial widths to non-strange final states, which proceed through either vector or axial-vector currents.

Dividing any partial width $\Gamma _x$ by the electronic partial width, $\Gamma _e$, we obtain partial width ratios $R_x$ (which are equal to the respective branching fraction ratios $B_x/B_e$) for which $R_{\text {had}} = R_s + R_{\text {VA}}$. In terms of such ratios, $|V_{us}|$ can be measured as [1413, 1414]

$$\begin{aligned} |V_{us}| _{\tau s}&= \sqrt{R_s/\bigg [\frac{R_{\text {VA}}}{|V_{ud}| ^2} - \delta R_{\text {theory}}\bigg ]}, \end{aligned}$$

where $\delta R_{\text {theory}}$ can be determined in the context of low energy QCD theory, partly relying on experimental low energy scattering data. The literature reports several calculations [1415,1416,1417]. In this report we use Ref. [1415], whose estimated uncertainty size is intermediate between the two other ones. We use the information in that paper and the PDG 2015 value for the s-quark mass $m_s = 95.00 \,\pm \, 5.00 \text {MeV}$ [327] to calculate $\delta R_{\text {theory}} = 0.242 \,\pm \, 0.032.$.

We proceed following the same procedure of the 2012 HFLAV report [234]. We sum the relevant $\tau$ branching fractions to compute $B_{\text {VA}}$ and $B_s$ and we use the universality improved $B_e^{\text {uni}}$ (see Sect. 9.3) to compute the $R_{\text {VA}}$ and $R_s$ ratios. In past determinations of $|V_{us}|$, for example in the 2009 HFLAV report [420], the total hadronic branching fraction has been computed using unitarity as $B_{\text {had}} ^{\text {uni}} = 1 - B_e -B_\mu$, obtaining then $B_s$ from the sum of the strange branching fractions and $B_{\text {VA}}$ from $B_{\text {had}} ^{\text {uni}} - B_s$. We prefer to use the more direct experimental determination of $B_{\text {VA}}$ for two reasons. First, both methods result in comparable uncertainties on $|V_{us}|$, since the better precision on $B_{\text {had}} ^{\text {uni}} = 1 - B_e -B_\mu$ is vanified by increased statistical correlations in the expressions $(1-B_e -B_\mu )/B_e^{\text {univ}}$ and $B_s/(B_{\text {had}}-B_s)$ in the $|V_{us}|$ calculation. Second, if there are unobserved $\tau$ hadronic decay modes, they would affect $B_{\text {VA}}$ and $B_s$ in a more asymmetric way when using unitarity.

Using the $\tau$ branching fraction fit results with their uncertainties and correlations (Sect. 9.1), we compute $B_s = (2.909 \,\pm \, 0.048)\%$ (see also Table 316) and $B_{\text {VA}} = B_{\text {hadrons}} - B_s = (61.85 \,\pm \, 0.10))\%$, where $B_{\text {hadrons}}$ is equal to $\Gamma _{\text {hadrons}}$ defined in Sect. 9.3. PDG 2015 averages are used for non-$\tau$ quantities, and $|V_{ud}| = 0.97417 \,\pm \, 0.00021$ [1418].

We obtain $|V_{us}| _{\tau s} = 0.2186 \,\pm \, 0.0021$, which is $3.1 \sigma$ lower than the unitarity CKM prediction $|V_{us}| _{\text {uni}} = 0.22582 \,\pm \, 0.00089$, from $(|V_{us}| _{\text {uni}})^2 = 1 - |V_{ud}| ^2$. The $|V_{us}| _{\tau s}$ uncertainty includes a systematic error contribution of 0.47% from the theory uncertainty on $\delta R_{\text {theory}}$. There is no significant change with respect to the previous HFLAV report.

Table 316 HFLAV Spring 2017 $\tau$ branching fractions to strange final states

9.4.2 $|V_{us}|$ from $B(\tau \rightarrow K\nu ) / B(\tau \rightarrow \pi \nu )$

We compute $|V_{us}|$ from the ratio of branching fractions $B(\tau \rightarrow K^{-}\nu _{\tau }) / B(\tau \rightarrow \pi ^{-}\nu _{\tau }) = (6.438 \,\pm \, 0.094)$ from the equation [1404]:

$$\begin{aligned} \frac{B(\tau \rightarrow K^{-}\nu _{\tau })}{B(\tau \rightarrow \pi ^{-}\nu _{\tau })}&= \frac{f_K^2 |V_{us}| ^2}{f_\pi ^2 |V_{ud}| ^2}\\&\times \frac{( m_\tau ^2 - m_K^2 )^2}{( m_\tau ^2 - m_\pi ^2 )^2} \frac{1+\delta R_{\tau /K}}{1+\delta R_{\tau /\pi }}(1+\delta R_{K/\pi }) \end{aligned}$$

We use $f_K/f_\pi = 1.1930 \,\pm \, 0.0030$ from the FLAG 2016 Lattice averages with $N_f=2+1+1$ [222],

$$\begin{aligned}&\frac{1+\delta R_{\tau /K}}{1+\delta R_{\tau /\pi }} = \frac{1+(1.1930 \,\pm \, 0.0030)\%}{1+(0.16 \,\pm \, 0.14)\%} ~\mathrm{[1393-1396]},\\&1+\delta R_{K/\pi } = 1 + (-1.13 \,\pm \, 0.23)\%\, [1390, 1405, 1406]. \end{aligned}$$

We compute $|V_{us}| _{\tau K/\pi } = 0.2236 \,\pm \, 0.0018$, $1.1 \sigma$ below the CKM unitarity prediction.

9.4.3 $|V_{us}|$ from $\tau$ summary

We summarize the $|V_{us}|$ results reporting the values, the discrepancy with respect to the $|V_{us}|$ determination from CKM unitarity, and an illustration of the measurement method:

$$\begin{aligned}&|V_{us}| _{\text {uni}} = 0.22582 \,\pm \, 0.00089 \\&\quad [\text {from } \sqrt{1 - |V_{ud}| ^2} \quad \text {(CKM unitarity)}], \\&|V_{us}| _{\tau s} = 0.2186 \,\pm \, 0.0021\quad -3.1 \sigma \\&\quad [\text {from } \Gamma (\tau ^- \rightarrow X_s^- \nu _\tau )], \\ |V_{us}| _{\tau K/\pi }&= 0.2236 \,\pm \, 0.0018 \quad -1.1 \sigma \\&\quad [\text {from } \Gamma (\tau ^{-} \rightarrow K^{-} \nu _\tau )/\Gamma (\tau ^{-} \rightarrow \pi ^{-} \nu _\tau )]. \end{aligned}$$

Averaging the two above $|V_{us}|$ determinations that rely on the $\tau$ branching fractions (taking into account all correlations due to the $\tau$ HFLAV and other mentioned inputs) we obtain, for $|V_{us}|$ and its discrepancy:

$$\begin{aligned} |V_{us}| _\tau&= 0.2216 \,\pm \, 0.0015 -2.4 \sigma \\&\quad [\text {average of 2} |V_{us}| \tau \text {measurements}]. \end{aligned}$$

All $|V_{us}|$ determinations based on measured $\tau$ branching fractions are lower than both the kaon and the CKM-unitarity determinations. This is correlated with the fact that the direct measurements of the three major $\tau$ branching fractions to kaons [$B(\tau \rightarrow K^{-}\nu _{\tau })$, $B(\tau \rightarrow K^{-}\pi ^{0}\nu _\tau )$ and $B(\tau \rightarrow \pi ^{-} {\bar{K}}^{0}\nu _\tau )$] are lower than their determinations from the kaon branching fractions into final states with leptons within the SM [1404, 1421, 1422].

A recent determination of $|V_{us}|$  [1423, 1424] that relies on the $\tau$ spectral functions in addition to the inclusive $\tau \rightarrow X_s \nu$ branching fraction reports a $|V_{us}|$ value about $1\sigma$ lower than the CKM-unitarity determination. This determination uses inputs that are partially different from the ones used in this report. Specifically, the HFLAV average of $B(\tau \rightarrow K^{-}\nu _{\tau })$ has been replaced with the SM prediction based on the measured $B(K^- \rightarrow \mu ^- {\bar{\nu }}_\mu )$ and the HFLAV average of $B(\tau \rightarrow K^{-}\pi ^{0}\nu _\tau )$ has been replaced with an in-progress BaBar measurement that is published in a PhD thesis. Both changes increase the resulting $\tau \rightarrow X_s \nu$ inclusive branching fraction. This study claims that the newly proposed $|V_{us}|$ calculation has a more stable and reliable theory uncertainty, which could possibly have been underestimated in former studies, which are used for the HFLAV $|V_{us}|$ average.

In previous editions of the HFLAV report, we also computed $|V_{us}|$ using the branching fraction $B(\tau \rightarrow K\nu )$ and without taking the ratio with $B(\tau \rightarrow \pi \nu )$. We do not report this additional determination because it did not include the long-distance radiative corrections in addition to the short-distance contribution, and because it had a negligible effect on the overall precision of the $|V_{us}|$ calculation with $\tau$ data.

Figure 220 reports the HFLAV $|V_{us}|$ determinations that use the $\tau$ branching fractions, compared to two $|V_{us}|$ determinations based on kaon data [6] and to $|V_{us}|$ obtained from $|V_{ud}|$ and the CKM matrix unitarity [6].

9.5 Upper limits on $\tau$ lepton-flavour-violating branching fractions

The Standard Model predicts that the $\tau$ lepton-flavour-violating (LFV) branching fractions are too small to be measured with the available experimental precision. We report in Table 317 and Fig. 221 the experimental upper limits on these branching fractions that have been published by the B-factories BaBar and Belle and later experiments. We omit previous weaker upper limits (mainly from CLEO) and all preliminary results presented several years ago. The previous HFLAV report [5] still included a few preliminary results, which have all been removed now.

Table 317 Experimental upper limits on lepton flavour violating $\tau$ decays. The modes are grouped according to the properties of their final states. Modes with baryon number violation are labelled with “BNV”

Fig. 221

Tau lepton-flavor-violating branching fraction upper limits summary plot. In order to appreciate the physics reach improvement over time, theplot includes also the CLEO upper limits reported by PDG 2016 [6]

Table 318 Combinations of upper limits on lepton flavour violating $\tau$ decay modes. The modes are grouped according to the properties of their final states. Modes with baryon number violation are labelled with “BNV”

Fig. 222

Tau lepton-flavour-violating branching fraction upper limits combinations summary plot. For each channel we report the HFLAV combined limit, and the experimental published limits. In some cases, the combined limit is weaker than the limit published by a single experiment. This arises since the $\text {CL}_s$ method used in the combination can be more conservative compared to other legitimate methods, especially when the number of observed events fluctuates below the expected background

9.6 Combination of upper limits on $\tau$ lepton-flavour-violating branching fractions

Combining upper limits is a delicate issue, since there is no standard and generally agreed procedure. Furthermore, the $\tau$ LFV searches published limits are extracted from the data with a variety of methods, and cannot be directly combined with a uniform procedure. It is however possible to use a single and effective upper limit combination procedure for all modes by re-computing the published upper limits with just one extraction method, using the published information that documents the upper limit determination: number of observed candidates, expected background, signal efficiency and number of analyzed $\tau$ decays.

We chose to use the $\text {CL}_s$ method [1443] to re-compute the $\tau$ LFV upper limits, since it is well known and widely used (see the Statistics review of PDG 2013 [6]), and since the limits computed with the $\text {CL}_s$ method can be combined in a straightforward way (see below). The $\text {CL}_s$ method is based on two hypotheses: signal plus background and background only. We calculate the observed confidence levels for the two hypotheses:

$$\begin{aligned} \text {CL}_{s+b}&= P_{s+b}(Q \le Q_{obs}) = \int _{- \infty }^{Q_{obs}} \frac{dP_{s+b}}{dQ} dQ, \end{aligned}$$

(295)

$$\begin{aligned} \text {CL}_b&= P_{b}(Q \le Q_{obs}) = \int _{- \infty }^{Q_{obs}} \frac{dP_{b}}{dQ} dQ, \end{aligned}$$

(296)

where $\text {CL}_{s+b}$ is the confidence level observed for the signal plus background hypotheses, $\text {CL}_b$ is the confidence level observed for the background only hypothesis, $\frac{dP_{s+b}}{dQ}$ and $\frac{dP_{b}}{dQ}$ are the probability distribution functions (PDFs) for the two corresponding hypothesis and Q is called the test statistic. The $\text {CL}_s$ value is defined as the ratio between the confidence level for the signal plus background hypothesis and the confidence level for the background hypothesis:

$$\begin{aligned} \text {CL}_s = \dfrac{\text {CL}_{s+b}}{\text {CL}_b}. \end{aligned}$$

(297)

When multiple results are combined, the PDFs in Eqs. (295) and (296) are the product of the individual PDFs,

$$\begin{aligned} \text {CL}_s&= \frac{\prod _{i=1}^{N}\sum _{n=0}^{n_i} \dfrac{e^{-(s_i+b_i)} (s_i+b_i)^{n}}{n!}}{\prod _{i=1}^{N} \sum _{n=0}^{n_i} \dfrac{e^{-b_i} b_i^{n}}{n!}}\nonumber \\&\times \frac{\prod _{j=1}^{N} \bigg [s_iS_i(x_{ij})+b_iB_i(x_{ij})\bigg ]}{\prod _{j=1}^{N}B_i(x_{ij})}, \end{aligned}$$

(298)

where N is the number of results (or channels), and, for each channel i, $n_i$ is the number of observed candidates, $x_{ij}$ are the values of the discriminating variables (with index j), $s_i$ and $b_i$ are the number of signal and background events and $S_i$, $B_i$ are the probability distribution functions of the discriminating variables. The discriminating variables $x_{ij}$ are assumed to be uncorrelated. The expected signal $s_i$ is related to the $\tau$ lepton branching fraction $B(\tau \rightarrow f_i)$ into the searched final state $f_i$ by $s_i = N_i\epsilon _iB(\tau \rightarrow f_i)$, where $N_i$ is the number of produced $\tau$ leptons and $\epsilon _i$ is the detection efficiency for observing the decay $\tau \rightarrow f_i$. For $e^+ e^-$ experiments, $N_i = 2\mathcal {L}_i\sigma _{\tau \tau }$, where $\mathcal {L}_i$ is the integrated luminosity and $\sigma _{\tau \tau }$ is the $\tau$ pair production cross section $\sigma (e^+ e^- \rightarrow \tau ^+ \tau ^-)$ [1444]. In experiments where $\tau$ leptons are produced in more complex multiple reactions, the effective $N_i$ is typically estimated with Monte Carlo simulations calibrated with related data yields.

The extraction of the upper limits is performed using the code provided by Tom Junk [1445]. The systematic uncertainties are modeled in the Monte Carlo toy experiments by convolving the $S_i$ and $B_i$ PDFs with Gaussian distributions corresponding to the nuisance parameters.

Table 318 reports the HFLAV combinations of the $\tau$ LFV limits. Since there is negligible gain in combining limits of very different strength, the combinations do not include the CLEO searches and do not include results where the single event sensitivity is more than a factor of 5 lower than the value for the search with the best limit.

Figure 222 reports a graphical representation of the limits in Table 318.

10 Summary

This article provides updated world averages of measurements of b-hadron, c-hadron, and $\tau$-lepton properties using results available through Summer 2016. A small selection of highlights of the results described in Sects. 3–9 is given in Table 320.

Since the previous version of this document [5], the $b$-hadron lifetime and mixing averages have mostly made gradual progress in precision. Notable exceptions with significant improvement are the averages for the mass difference in the $B^0$–${\bar{B}} ^0$ system ($\Delta m_{d}$) and the $C\!P$ violation parameter in ${B^{0}_s}$–${\bar{B}} _s^0$ system ($|q_{s}/p_{s}|$). In total eleven new results (of which ten from the LHC Run 1 data and one from the Tevatron data) have been incorporated in these averages. On the other hand, all results that remained unpublished and for which there is no publication plan, have been removed from the averages.

The lifetime hierarchy for the most abundant weakly decaying $b$-hadron species is well established, with impressive precisions of 5 fs or less for the most common $B^0$, $B^+$ and ${B^{0}_s}$ mesons, and compatible with the expectations from the Heavy Quark Expansion. However, statistics are still lacking for $b$ baryons heavier than $\varLambda _b^0$ ($\varXi _b^-$, $\varXi _b^0$, $\Omega _b$, and all other yet-to-be-discovered $b$ baryons), but this will surely come from the LHC with sufficient time. A sizable value of the decay width difference in the ${B^{0}_s}-{\bar{B}} _s^0$ system is measured with a relative precision of 7% and is well predicted by the Standard Model (SM). In contrast, the experimental results for the decay width difference in the $B^0-{\bar{B}} ^0$ system are not yet precise enough to distinguish the small (expected) value

Table 319 Published information that has been used to re-compute upper limits with the $\text {CL}_s$ method, i.e. the number of $\tau$ leptons produced, the signal detection efficiency and its uncertainty, the number of expected background events and its uncertainty, and the number of observed events. The uncertainty on the efficiency includes the minor uncertainty contribution on the number of $\tau$ leptons (typically originating on the uncertainties on the integrated luminosity and on the production cross-section). The additional limit used in the combinations (from LHCb) has been originally determined with the $\text {CL}_s$ method

from zero. The mass differences in both systems are known very accurately, to the (few) per mil level. On the other hand, $C\!P$ violation in the mixing of either system has not been observed yet, with asymmetries known within a couple per mil but still consistent both with zero and their SM predictions. A similar conclusion holds for the $C\!P$ violation induced by ${B^{0}_s}$ mixing in the $b\rightarrow c{\bar{c}}s$ transition, although in this case the experimental precision on the corresponding weak phase is an order of magnitude larger, but now becoming just smaller than the SM central value. Many measurements are still dominated by statistical uncertainties and will improve once new results from the LHC Run 2 become available.

The measurement of $\sin 2\beta \equiv \sin 2\phi _1$ from $b \rightarrow c{\bar{c}}s$ transitions such as $B^{0}\rightarrow {J/\psi } K^0_{S}$ has reached $<\!\!2.5\,\%$ precision: $\sin 2\beta \equiv \sin 2\phi _1 = 0.691 \,\pm \, 0.017$. Measurements of the same parameter using different quark-level processes provide a consistency test of the Standard Model and allow insight into possible new physics. All results among hadronic $b \rightarrow s$ penguin dominated decays of $B^{0}$mesons are currently consistent with the Standard Model expectations. Measurements of $C\!P$ violation parameters in ${B^{0}_s} \rightarrow \phi \phi$ allow a similar comparison to the value of $\phi _s^{c{{\bar{c}}}s}$; again, results are consistent with the SM expectation (which in this case is very close to zero). Among measurements related to the Unitarity Triangle angle $\alpha \equiv \phi _2$, results from the $\rho \rho$ system allow constraints at the level of $\approx 6^\circ$. These remain the strongest constraints, although results from all of BaBar, Belle and LHCb lead to good precision on the $C\!P$ violation parameters in $B^{0}\rightarrow \pi ^+ \pi ^-$ decays. Knowledge of the third angle $\gamma \equiv \phi _3$ also continues to improve, with the current world average being $(74.0\,^{+5.8}_{-6.4})^{\circ }$. The precision is expected to improve further as more data becomes available at LHCb and Belle II.

In semileptonic B meson decays, the anomalies reported in the last version of the document have remained: The discrepancy between $|V_{cb}|$ measured with inclusive and exclusive decays is of the order of $3\sigma$ ($3.2\sigma$ for $|V_{cb}|$ from ${\bar{B}} \rightarrow D^*\ell ^-{\bar{\nu }}_\ell$, $2.4\sigma$ for $|V_{cb}|$ from ${\bar{B}} \rightarrow D\ell ^-{\bar{\nu }}_\ell$). The difference between $|V_{ub}|$ measured with inclusive decays ${\bar{B}} \rightarrow X_u\ell ^-\overline{\nu } _\ell$ and $|V_{ub}|$ from ${\bar{B}} \rightarrow \pi \ell ^-\overline{\nu } _\ell$ has risen to $3.6\sigma$. An important new contribution to the determination of the values of $|V_{ub}|$ and $|V_{cb}|$ comes from exclusive b-baryon decays. The largest anomaly however is observed in $B\rightarrow D^{(*)}\tau \nu _\tau$ decays: The combined discrepancy of the measured values of $\mathcal{R}(D^*)$ and $\mathcal{R}(D)$ to their standard model expectations is found to be $3.9\sigma$.

The most important new measurements of rare b-hadron decays are coming from the LHC. Precision measurements of ${B^{0}_s}$ decays are particularly noteworthy, including several measurements of the longitudinal polarisation fraction from LHCb. ATLAS, CMS and LHCb have significantly improved the sensitivity to the $B^0_{(s)}\rightarrow \mu ^+\mu ^-$ decays. Recently, CMS and LHCb published a combined analysis that allowed the first observation of the ${B^{0}_s} \rightarrow \mu ^+\mu ^-$ decay to be obtained, and provided three standard deviations evidence of the $B^{0}\rightarrow \mu ^+\mu ^-$ decay. The results are compatible with the SM predictions, and yield constrains on the parameter space of new physics models. CMS and LHCb have also performed angular analyses of the $B^{0}\rightarrow K^{*0} \mu ^+\mu ^-$ decay, complementing, extending and improving on the precision of results from BaBar and Belle. One of the observables measured by LHCb, $P_5^{\prime }$, differs from the SM prediction by $3.7\sigma$ in one of the $m^2_{\mu ^+\mu ^-}$ intervals; results from Belle on this observable are consistent but less precise. Improved measurements from LHCb and other experiments are keenly anticipated. A measurement of the ratio of branching fractions of $B^{+}\rightarrow K^+\mu ^+\mu ^-$ and $B^{+}\rightarrow K^+e^+e^-$ decays ($R_K$) has been made by LHCb. In the low $m^2_{\ell ^+\ell ^-}$ region, it differs from the standard model prediction by $2.6\sigma$. Among the $C\!P$ violating observables in rare decays, the “$K\pi$ puzzle” persists, and important new results have appeared in three-body decays. LHCb has produced many other results on a wide variety of decays, including b-baryon and $B^+_c$-meson decays. Belle and BaBar continue to produce new results though their output rates are dwindling. It will still be some years before we see new results from the upgraded SuperKEKB B factory and the Belle II experiment.

About 800 b to charm results from BaBar, Belle, CDF, D0, LHCb, CMS, and ATLAS reported in more than 200 papers are compiled in a list of over 600 averages. The huge samples of b hadrons that are available in contemporary experiments allows measurements of decays to states with open or hidden charm content with unprecedented precision. In addition to improvements in precision for branching fractions of ${\bar{B}} ^0$ and $B^-$ mesons, many new decay modes have been discovered. In addition, there is a rapidly increasing set of measurements available for ${\bar{B}} _s^0$ and $B_c^-$ mesons as well as for b baryon decays.

Table 320 Selected world averages. Where two uncertainties are given the first is statistical and the second is systematic, except where indicated otherwise

In the charm sector, $D^0$–${\bar{D}} ^0$ mixing is now well-established and the emphasis has shifted to searching for $C\!P$ violation. Measurements of 49 observables from the E791, FOCUS, Belle, BaBar, CLEO, BESIII, CDF, and LHCb experiments are input into a global fit for 10 underlying parameters, and the no-mixing hypothesis is excluded at a confidence level $>\!\!11.5\sigma$. The mixing parameters x and y individually differ from zero by $1.9\sigma$ and $9.4\sigma$, respectively. The world average value for the observable $y_{C\!P}$ is positive, indicating that the $C\!P$-even state is shorter-lived as in the $K^0$–$\overline{K} ^0$ system. The $C\!P$ violation parameters |q / p| and $\phi$ are consistent with the no-$C\!P$ violation hypothesis within $1\sigma$. Thus there is no evidence for $C\!P$ violation arising from mixing ($|q/p|\ne 1$) or from a phase difference between the mixing amplitude and a direct decay amplitude ($\phi \ne 0$). In addition, the most recent data indicates no direct $C\!P$ violation in $D^0\!\rightarrow \!K^+K^-/\pi ^+\pi ^-$ decays; performing a global fit to all relevant measurements gives $\Delta a^\mathrm{dir}_{C\!P}= (-0.134\,\pm \, 0.070)\%$. The world’s most precise measurements of $|V^{}_{cd}|$ and $|V^{}_{cs}|$ are obtained from leptonic $D^+\!\rightarrow \!\mu ^+\nu$ and $D^+_s\!\rightarrow \!\mu ^+\nu /\tau ^+\nu$ decays, respectively. These measurements have theoretical uncertainties arising from decay constants. However, calculations of decay constants within lattice QCD have improved such that the theory error is $<\!\!1/3$ the experimental errors of the measurements.

Since 2016, HFLAV provides the $\tau$ branching fraction fit averages for the PDG Review of Particle Physics. For the PDG, a unitarity constrained variant of the fit is performed, using only inputs that are published and included in the PDG. Two preliminary results used in the HFLAV 2014 report have been removed both in the HFLAV and in the PDG variants of the fit. A few minor imperfections of the 2014 fit have been corrected. There are no non-negligible changes to the lepton universality tests and to the $|V_{us}|$ determinations from the $\tau$ branching fractions. There is still a large discrepancy between $|V_{us}|$ from $\tau$, $|V_{us}|$ from kaons and $|V_{us}|$ from $|V_{ud}|$ and CKM matrix unitarity. On this topic, recent studies [1423, 1424] claim to get a more reliable theory uncertainty on $|V_{us}|$ with a revised calculation method that uses also the $\tau$ spectral functions. Just one more $\tau$ lepton-flavour-violating branching fraction upper limit has been published, which does not change the computed combined related limit. The list of limits and their combinations has been revised to remove old preliminary results.