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.

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.

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.

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.

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.

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.

$b$-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).

$b$-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 $2.4\,\sigma$.

$b$-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].

$b$-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.

$b$-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.

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.

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 $2\,\sigma$.

$B^0$ 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)

${B^{0}_s}$ 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)

$B^+_c$ 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)

$\varLambda _b^0$ 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)

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.

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.

$B^0$ 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.

${B^{0}_s}$ 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

$C\!P$ 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.

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.

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.

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 $\beta _s \sim \lambda ^2\eta$.

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.

$C\!P$ 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)

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 $C\!P$.

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.

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.

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.

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