A self-calibrating, double-ratio method to test tau lepton universality in W boson decays at the LHC

Abstract

Measurements in $W^+W^-$ events at LEP2 and in B hadron semileptonic decays at B factories and LHCb provide intriguing hints of a violation of lepton universality in the charged current coupling of tau leptons relative to those for electrons and muons. We propose a novel, self-calibrating method to test tau lepton universality in W boson decays at the LHC. We compare directly the ratio of the numbers of selected $\ell \tau _{\mathrm {\,had}}$ and $e\mu$ final states in di-leptonic top quark pair events with that in $Z/\gamma ^*\rightarrow \tau \tau$ events. Here $\ell =e\ {\mathrm {or}}\ \mu$ and $\tau _{\mathrm {\,had}}$ is a candidate semi-hadronic tau decay. This “double-ratio” cancels to first order sensitivity to systematic uncertainties on the reconstruction of e, $\mu$, and $\tau$ leptons, thus improving very significantly the precision to which tau lepton universality can be tested in W boson decay branching ratios at the LHC. Using particle-level Monte Carlo events, and a parameterised simulation of detector performance, we demonstrate the effectiveness of the method and estimate the most significant residual sources of uncertainty arising from experimental and phenomenological systematics. Our studies indicate that a single LHC experiment precision on the tau lepton universality test of around 1.4% is achievable with a data set of $\int \!\! L\, \hbox {\,d}t$ = 140 $\mathrm {fb}^{-1}$ at ${\sqrt{s}}~=~13~\hbox {TeV}$. This would improve significantly upon the precision of 2.5% on the four-experiment combined LEP2 measurements. If the central value of the proposed new measurement were equal to the central value of the LEP2 measurement this would yield an observation of BSM physics at a significance level of around 5$\sigma$.

1 Introduction

An important feature of the Standard Model (SM) of particle physics is Lepton Universality (LU): the idea that the Electroweak (EW) couplings of the leptons are identical in each fermion generation. For the Neutral Current (NC) interactions, mediated in the SM by the $\gamma$ and Z bosons, the validity of LU for the three flavours of charged leptons ($e, \mu , \tau )$ has been demonstrated at the scale of the Z boson mass $m_Z$ at LEP1 and SLC to high precision [1]. For example, the ratios of the leptonic partial widths ($\Gamma$) or branching fractions (${\mathcal {B}}$) of the Z boson are:

$$\begin{aligned} \begin{array}{lllll} \displaystyle \frac{\Gamma _{\mu \mu }}{\Gamma _{ee}} &{}\equiv &{} \displaystyle \frac{{\mathcal {B}}(Z/\gamma ^*\rightarrow \mu \mu )}{{\mathcal {B}}(Z/\gamma ^* \rightarrow ee)} &{}=&{} 1.0009 \pm 0.0028, \\ \displaystyle \frac{\Gamma _{\tau \tau }}{\Gamma _{ee}} &{}\equiv &{} \displaystyle \frac{{\mathcal {B}}(Z/\gamma ^*\rightarrow \tau \tau )}{{\mathcal {B}}(Z/\gamma ^* \rightarrow ee)} &{}=&{} 1.0019 \pm 0.0032, \end{array} \end{aligned}$$

(1)

which are consistent with LU to a precision of around three per mille [1]. Measurements of the leptonic asymmetry parameters ${\mathcal {A}}_\ell ,$ from forward-backward asymmetries, the left-right asymmetry, and the tau polarisation and its asymmetry:

$$\begin{aligned} \begin{array}{lll} \displaystyle {\mathcal {A}}_e &{}=&{} 0.1514 \pm 0.0019, \\ \displaystyle {\mathcal {A}}_\mu &{}=&{} 0.1456 \pm 0.0091, \\ \displaystyle {\mathcal {A}}_\tau &{}=&{} 0.1449 \pm 0.0040, \\ \end{array} \end{aligned}$$

(2)

are also consistent with LU, albeit at the precision of a few percent [1].

In contrast, for the Charged Current (CC) interactions, mediated in the SM by the $W^\pm$ bosons, the experimental measurements of leptonic branching ratios are less precise. A direct test of LU at the scale of the W boson mass $m_W$ at LEP2 can be made from the ratios of the leptonic branching fractions of the W boson. The ratio [2]:

$$\begin{aligned} \frac{{\mathcal {B}}(W\rightarrow \mu \nu )}{{\mathcal {B}}(W\rightarrow e\nu )} = 0.993 \pm 0.019 \end{aligned}$$

(3)

is consistent with e-$\mu$ universality to a precision of around two percent. However a hint at the possible violation of LU in the CC couplings of the $\tau$ is present in the ratio [2]:

$$\begin{aligned} R(W)\equiv \frac{\mathcal {B}(W\rightarrow \tau \nu )}{\mathcal {B}(W\rightarrow \ell \nu )} = 1.066 \pm 0.025, \end{aligned}$$

(4)

where ${\mathcal {B}}(W\rightarrow \ell \nu )$ is the average of the branching fractions for $W\rightarrow e\nu$ and $W\rightarrow \mu \nu$. This result deviates from the assumption of $\tau$-$\ell$ universality¹ by 2.6$\sigma$ (standard deviations).

At lower mass scales e-$\mu$ universality is tested very precisely, for example in leptonic $\tau$ decays the ratio of the muonic and electronic partial widths is measured to be [3]:

$$\begin{aligned} \frac{\Gamma (\tau ^{-}\rightarrow \mu ^{-}\bar{\nu }_\mu \nu _\tau )}{\Gamma (\tau ^{-}\rightarrow e^{-}\bar{\nu }_e\nu _\tau )} = 0.9762 \pm 0.0028, \end{aligned}$$

(5)

which is consistent with the SM prediction including mass effects of 0.9726 [3]. e-$\mu$ universality is also tested in the decays of charged kaons [3]:

$$\begin{aligned} \frac{\Gamma (K^-\rightarrow e^-{\bar{\nu }}_e)}{\Gamma (K^-\rightarrow \mu ^-{\bar{\nu }}_\mu )}=(2.488\pm 0.009)\times 10^{-5}, \end{aligned}$$

(6)

which is consistent with the SM prediction [4] of $2.477\times 10^{-5}$.

In the decay of B hadrons, $\tau$-$\ell$ universality can be tested by measurements of the branching fractions for the exclusive decays of $\overline{B}^0\rightarrow \tau ^{-}\overline{\nu }_\tau D^+$ and $\overline{B}^0\rightarrow \tau ^{-}\overline{\nu }_\tau D^{*+}$ expressed as ratios to the branching fractions for the exclusive decays $\overline{B}^0\rightarrow \ell ^{-}\overline{\nu }_\ell D^+$ and $\overline{B}^0\rightarrow \ell ^{-}\overline{\nu }_\ell D^{*+}$, respectively. Systematic uncertainties due to hadronic effects largely cancel in these ratios. A combination [5] of the results from the BaBar [6, 7], Belle [8,9,10,11,12], and LHCb [13,14,15,16] experiments yields the results:

$$\begin{aligned} \begin{array}{rllll} \displaystyle R(D)&{}\equiv &{} \displaystyle \frac{{\mathcal {B}}(\overline{B}^0\rightarrow \tau ^{-}\overline{\nu }_\tau D^+)}{{\mathcal {B}}(\overline{B}^0\rightarrow \ell ^{-}\overline{\nu }_\ell D^+)} &{}=&{} 0.340 \pm 0.027 \pm 0.013, \\ \displaystyle R(D^*)&{}\equiv &{} \displaystyle \frac{{\mathcal {B}}(\overline{B}^0\rightarrow \tau ^{-}\overline{\nu }_\tau D^{*+})}{{\mathcal {B}}(\overline{B}^0\rightarrow \ell ^{-}\overline{\nu }_\ell D^{*+})} &{}=&{} 0.295 \pm 0.011 \pm 0.008. \\ \end{array} \end{aligned}$$

(7)

These measured values exceed the SM predictions (calculated assuming LU) of:

$$\begin{aligned} \begin{array}{rll} \displaystyle R(D)&{}=&{} 0.300 \pm 0.008, \\ \\ \displaystyle R(D^*)&{}=&{} 0.252 \pm 0.003, \\ \end{array} \end{aligned}$$

(8)

by 1.4$\sigma$ and 2.5$\sigma$ respectively (see [5] and the references contained therein). Taking into account correlations between the measurements, the combined discrepancy with regard to the SM predictions corresponds to 3.1$\sigma$ [5].

Possible Beyond the Standard Model (BSM) explanations have been proposed for the potential violation of LU in $R(D)$ and $R(D^*)$. A Leptoquark (LQ) that couples more strongly to the $\tau$ than to e or $\mu$ could contribute at tree level to the decays $\overline{B}^0\rightarrow \tau ^{-}\overline{\nu }_\tau D^{(*)+}$. For example, in [17] a charge 2/3 scalar LQ with $b\tau$ and $c\nu$ Yukawa couplings is able to accommodate the measured central values of $R(D)$ and $R(D^*)$ without introducing an unacceptable level of flavour changing neutral current processes involving the first two generations of quarks and leptons. Interestingly, a possible link between the LEP2 measurement of $R(W)$ and the $\overline{B}^0\rightarrow \tau ^{-}\overline{\nu }_\tau D^{(*)+}$ excess has received little attention in the literature. We note that $R(W)$ might receive contributions at loop level from a LQ that couples preferentially to the $\mathbf {\tau }$. For example, if cs and $s\tau$ Yukawa couplings were added to the charge 2/3 LQ scenario of [17] it could produce an enhancement in ${\mathcal {B}}$($W\rightarrow \tau \nu$). Such loop-level contributions might naturally lead to a smaller fractional deviation from LU in $R(W)$ than in $R(D)$ and $R(D^*)$, but this would depend, obviously, on the sizes of the assumed couplings.

Clearly it is important to improve on the precision of the measurements of $R(W)$ [currently 2.5%], $R(D)$ [currently 11%] and $R(D^*)$ [currently 5%]. The precision of the LEP2 measurement of $R(W)$ was dominated by the limited number of available ${W}^+{W}^-$ events. A future high energy, high luminosity $e^+e^-$ collider, at which an improved $R(W)$ measurement could be performed, is likely to be decades away.

Measurements at hadron colliders that are sensitive to $R(W)$ usually rely on the identification of $\tau$ lepton decays in which the visible final state is hadronic ($\tau _{\mathrm {\,had}}$). The current best published hadron collider measurements typically have small statistical uncertainties, but are dominated by large systematic uncertainties that render them uncompetitive with the LEP2 measurement of $R(W)$. For example, a measurement of the inclusive single $W\rightarrow \tau _{\mathrm {\,had}}\nu$ cross section in pp collisions at 7 TeV by ATLAS [18] has a relative uncertainty of 15%, which is dominated by systematic uncertainties on the efficiency to trigger on and select events containing $\tau _{\mathrm {\,had}}$ candidates.

Also of interest in this context are measurements at hadron colliders of the rate of events containing top quark–antiquark pairs (${t}{t}$), especially in the di-lepton final state. For the purpose of determining $R(W)$ events containing top quarks may be regarded as a convenient source of on-mass-shell W bosons. In the absence of non-SM decay mechanisms for the top quark the branching fraction ${\mathcal {B}}(t\rightarrow b\tau \nu )$ may be reinterpreted as the branching fraction ${\mathcal {B}}(W\rightarrow \tau \nu )$.

Relative to measurements of the inclusive single $W\rightarrow \tau _{\mathrm {\,had}}\nu$ cross section, measurements using di-leptonic $tt\rightarrow bb\ell \tau _{\mathrm {\,had}}$ events eliminate systematic uncertainties associated with the trigger efficiencies for $\tau _{\mathrm {\,had}}$ candidates, because single-$\ell$ (e or $\mu$) triggers can be used. In addition, non-${t}{t}$ backgrounds can be almost completely eliminated from the $tt\rightarrow bb\ell \tau _{\mathrm {\,had}}$ final state by employing b-jet tagging and the presence of the $\ell$ candidate. This means that the background from misidentified hadronic jets to the $\tau _{\mathrm {\,had}}$ signature in the $tt\rightarrow bb\ell \tau _{\mathrm {\,had}}$ final state is likely to be significantly smaller than that in the inclusive single $W\rightarrow \tau _{\mathrm {\,had}}\nu$ final state. However, systematic uncertainties associated with the $\tau _{\mathrm {\,had}}$ candidate (identification efficiency, background and energy scale) still contribute directly to the measured rate of $tt\rightarrow bb\ell \tau _{\mathrm {\,had}}$ events. For example, in a measurement by ATLAS [19] in di-leptonic ${t}{t}$ events the systematic uncertainty on the branching fraction ${\mathcal {B}}(t\rightarrow b\tau \nu )$ is around 7.5%. In a measurement by CMS of the cross section for the $tt\rightarrow bb\ell \tau _{\mathrm {\,had}}$ final state [20], the systematic uncertainty is around 9.5%; to which the combined contribution from the identification efficiency (6.0%), background (4.3%), and energy scale (2.5%) for $\tau _{\mathrm {\,had}}$ candidates was 7.8%. In the above-mentioned best currently published measurements from ATLAS [19] and CMS [20] based on the $tt\rightarrow bb\ell \tau _{\mathrm {\,had}}$ final state the systematic uncertainties associated with the $\tau _{\mathrm {\,had}}$ candidate are around a factor of three greater than total uncertainty of 2.5% on the LEP2 measurement of $R(W)$.

We propose here a novel, self-calibrating, “double-ratio” method that will allow $R(W)$ to be measured using top quark pair (${t}\bar{t}\rightarrow b\bar{b}W^+W^{-}$) and $Z/\gamma ^*\rightarrow \tau \tau$ events at the LHC with a target precision of around 1%, which would improve significantly upon the LEP2 measurements. We define the ratio:

$$\begin{aligned} R(bbWW)\equiv \frac{N(tt\rightarrow bb\ell \tau _{\mathrm {\,had}})}{N(tt\rightarrow bbe\mu )}, \end{aligned}$$

(9)

where $N(tt\rightarrow bb\ell \tau _{\mathrm {\,had}})$ and $N(tt\rightarrow bbe\mu )$ are the numbers of observed candidate events in the $tt\rightarrow bb\ell \tau _{\mathrm {\,had}}$ and $tt\rightarrow bbe\mu$ final states, respectively. We define also the ratio:

$$\begin{aligned} R(Z)\equiv \frac{N(Z\rightarrow \tau \tau \rightarrow \ell \tau _{\mathrm {\,had}})}{N(Z\rightarrow \tau \tau \rightarrow e\mu )}, \end{aligned}$$

(10)

where $N(Z\rightarrow \tau \tau \rightarrow \ell \tau _{\mathrm {\,had}})$ and $N(Z\rightarrow \tau \tau \rightarrow e\mu )$ are the numbers of observed candidate events in the $Z\rightarrow \tau \tau \rightarrow \ell \tau _{\mathrm {\,had}}$ and $Z\rightarrow \tau \tau \rightarrow e\mu$ final states, respectively. We then define the double ratio:

$$\begin{aligned} R(WZ)\equiv & {} \frac{R(bbWW)}{R(Z)} \nonumber \\\equiv & {} \frac{N(tt\rightarrow bb\ell \tau _{\mathrm {\,had}})\times N(Z\rightarrow \tau \tau \rightarrow e\mu )}{N(tt\rightarrow bbe\mu )\times N(Z\rightarrow \tau \tau \rightarrow \ell \tau _{\mathrm {\,had}})} \end{aligned}$$

(11)

From an experimental perspective we note that the ratios $R(bbWW)$ and $R(Z)$ are designed to have approximately the same sensitivity to systematic uncertainties on the identification of e, $\mu$ and $\tau _{\mathrm {\,had}}$ candidates, and on the efficiencies of the single-$\ell$ triggers. Therefore, in the double ratio $R(WZ)$ these systematic uncertainties cancel to first order. This cancellation is not necessarily perfect for the following reasons.

• The distributions in transverse momentum ($p_T$) and pseudorapidity ($\eta$) [21] of the leptons (e, $\mu$ and $\tau$) are significantly different in the ${t}\bar{t}\rightarrow b\bar{b}W^+W^{-}$ and $Z/\gamma ^*\rightarrow \tau \tau$ signal samples. Systematic uncertainties on lepton identification and single-$\ell$ trigger efficiencies are not necessarily fully correlated across all $p_T$ and $|\eta |$ bins.

• The levels of background from misidentified hadronic jets in the $\tau _{\mathrm {\,had}}$ candidates in the $tt\rightarrow bb\ell \tau _{\mathrm {\,had}}$ and $Z\rightarrow \tau \tau \rightarrow \ell \tau _{\mathrm {\,had}}$ samples are not necessarily identical.

• The level and nature of the non-${t}{t}$ background in the $tt\rightarrow bb\ell \tau _{\mathrm {\,had}}$ and $tt\rightarrow bbe\mu$ samples will not necessarily be identical. Similarly, the level and nature of the non-Z boson background in the $Z\rightarrow \tau \tau \rightarrow \ell \tau _{\mathrm {\,had}}$ and $Z\rightarrow \tau \tau \rightarrow e\mu$ samples will not necessarily be identical.

From a phenomenological perspective the double ratio $R(WZ)$ exploits the fact that LU has been precisely verified experimentally in the Z boson branching fractions (see equation 1). Therefore, any non-SM effects should affect $R(WZ)$ primarily through $R(bbWW)$, which is sensitive to $R(W)$.

The rest of this paper is organised as follows. In Sect. 2 we describe a Monte Carlo (MC) study of the proposed analysis method employing a simple parameterised simulation of detector performance. In Sect. 3 we present our summary and conclusions.

2 A Monte Carlo study of the proposed double-ratio analysis method

Our study uses particle-level MC events for the various physics processes of relevance. In Sect. 2.1 we describe briefly the simple parameterised simulation of detector performance we use in the study of the proposed double-ratio analysis method. We describe also the variations in detector performance we consider in the study of experimental systematic uncertainties. Further details are given in the Appendix. In Sect. 2.2 we describe the MC generators used and the potential sources of phenomenological systematic uncertainties that we have considered in our study. In Sect. 2.3 we describe the candidate event selection criteria we employ for the four signal candidate event samples used in the double-ratio method. In the selection of $Z/\gamma ^*\rightarrow \tau \tau$ candidates we propose a novel selection variable, $m^*_3$, that improves the discrimination power against the dominant backgrounds, such as ${t}{t}$, diboson production, as well as events containing a leptonically decaying vector boson plus a QCD jet that is misidentified as a $\tau _{\mathrm {\,had}}$ candidate (W+jet). In Sect. 2.4 we evaluate the size and composition of the four selected candidate event samples. The expected numbers of events are given for an integrated luminosity at ${\sqrt{s}}~=~13~\hbox {TeV}$ of $\int \!\! L\, \hbox {\,d}t$ = 140 $\mathrm {fb}^{-1}$, which corresponds approximately to that available for physics analysis in ATLAS and CMS at the end of LHC run 2 [22, 23]. In Sect. 2.5 we evaluate the sensitivity of the measured double ratio $R(WZ)$ to the physical quantity of interest $R(W)$. In Sect. 2.6 we present the effect of systematic uncertainties on the ratios $R(bbWW)$, $R(Z)$, $R(WZ)$, and $R(W)$. We thus demonstrate that in the double-ratio $R(WZ)$ there is a high degree of cancellation in the experimental systematic uncertainties that have dominated previous related analyses at hadron colliders. We evaluate the residual systematic uncertainties on $R(WZ)$ arising from the effects discussed in Sects. 2.1 and 2.2. In Sect. 2.7 we discuss some limitations of this simplified study and consider some factors that will need to be taken into account in an analysis that uses detailed simulations of a specific LHC detector and is applied to the experimental data.

2.1 Simple parameterised Monte Carlo simulation of detector performance

In this section we describe briefly the simple parameterised simulation of detector performance we use to study the proposed double-ratio analysis method. We describe also the variations in detector performance we consider in the study of systematic uncertainties.

Clearly, our aim here is not to produce a completely accurate simulation of the data from either the ATLAS or CMS detectors. Nevertheless, we base our parameterisations of the detection of leptons and jets on published measurements of LHC detector performance and their associated uncertainties [24, 55]. This approach enables us to demonstrate some of the principle benefits of the proposed double-ratio method and allows us to investigate within a simple and controlled framework the principal sources of residual systematic uncertainty to which the method is sensitive.

The efficiencies associated with the reconstruction, identification, and triggering of high $p_T$, isolated lepton candidates are typically determined by the ATLAS and CMS collaborations using “tag and probe” measurements on $Z\rightarrow ee, \mu \mu , \tau \tau$ events in both MC simulations and the real data. Systematic uncertainties are usually quoted on the “scale factors” employed to correct MC simulations to provide an accurate description of the real data.

As noted in Sect. 1, the ratios $R(bbWW)$ and $R(Z)$ are designed to have approximately the same sensitivity to systematic uncertainties on the identification of e, $\mu$ and $\tau _{\mathrm {\,had}}$ candidates, and on the efficiencies of the single-$\ell$ triggers. In the double ratio $R(WZ)$, these systematic uncertainties cancel to first order. Therefore, in our study we give particular attention to the $p_T$ and $\eta$ dependence of efficiencies and to any potential $p_T$ and $\eta$ dependence in the associated systematic uncertainties. In general we expect algorithms that are designed to have $p_T$- and $\eta$-independent efficiencies to have smaller $p_T$- and $\eta$-dependent systematic uncertainties.

Figure 1 shows as a function of $p_T$ the overall efficiencies we assume for the reconstruction, identification, and isolation criteria for prompt $e, \mu$, and $\tau _{\mathrm {\,had}}$ candidates. Table 1 gives a summary of the sources of systematic uncertainty on the reconstruction of leptons and jets considered in this study, stating separately the assumed size of the $p_T$/$\eta$-independent and $p_T$/$\eta$-dependent systematic uncertainties. Figure 2 shows the $p_T$-dependent systematic uncertainties on the efficiencies we assume for the reconstruction, identification, and isolation criteria for prompt $e, \mu$, and $\tau _{\mathrm {\,had}}$ candidates.

Fig. 1

The overall efficiencies assumed for the reconstruction, identification, and isolation criteria for prompt $e, \mu$, and $\tau _{\mathrm {\,had}}$ candidates as a function of $p_T$

Table 1 Sources of experimental systematic uncertainty, together with the size (in percent, unless stated otherwise) of the $p_T$/$\eta$-independent and $p_T$/$\eta$-dependent systematic uncertainties considered in this study. In all expressions $p_T$ is given in units of GeV. We define the endcap region by $|\eta |>1.0$. See the Appendix for further detail and justification of these choices

Jets with $|\eta |$ < 2.5 are flagged as b-tagged with probabilities that depend on their flavour at truth level as follows: truth b 85%, truth c 40%, truth light quark or gluon 1%. These tag probabilities are approximately independent of $p_T$ and $|\eta |$.

A detailed description of our choices of lepton and jet identification algorithms to simulate and experimental systematic uncertainties to consider, and the reasoning behind these choices, are given in the Appendix.

2.2 Monte Carlo generators and phenomenological systematic uncertainties

Events containing W bosons, Z bosons, top quark pairs, and the EW production of single top quarks are generated using POWHEG BOX [56], interfaced to PYTHIA[57] for the simulation of parton showering and fragmentation. We thus ensure a consistent treatment of tau lepton decay in the principle sources of candidate events. EW diboson events are generated using SHERPA [58].

Significant sources of background in the selected $Z/\gamma ^*\rightarrow \tau \tau$ samples arise from EW diboson and ${t}{t}$ production. Cross sections at ${\sqrt{s}}~=~13~\hbox {TeV}$ have been measured for EW diboson [59] and ${t}{t}$ [60] production. Our estimates of the fractional composition of the selected $Z/\gamma ^*\rightarrow \tau \tau$ samples are not sensitive to systematic uncertainties on the integrated luminosity or on the predicted absolute cross sections for Z boson, ${t}{t}$, or EW diboson production. They are, however, sensitive to systematic uncertainties on the ratio of the cross sections for ${t}{t}$ and EW diboson production to that for Z bosons. We assume an uncertainty of 3% on both of these cross section ratios [61].

In $Z/\gamma ^*\rightarrow \tau \tau$ events the momentum distribution of electrons and muons produced in $\tau$ decay is softer than that for the visible $\tau _{\mathrm {\,had}}$ systems. Changes in the distribution of the transverse momentum of the produced Z bosons ($p_T(Z)$) can, therefore, affect the relative acceptance for $Z\rightarrow \tau \tau \rightarrow \ell \tau _{\mathrm {\,had}}$ and $Z\rightarrow \tau \tau \rightarrow e\mu$ candidate events. Measurements of $p_T(Z)$ have been made at ${\sqrt{s}}~=~8~\hbox {TeV}$ by ATLAS [62]. In order to evaluate systematic uncertainties arising from $p_T(Z)$ we increase the weight of events satisfying $50\ {\mathrm {GeV}}< p_T(Z)< 150$ GeV by 0.5% and the weight of events satisfying $p_T(Z)> 150$ GeV by 1.0% [63].

Fig. 2

The $p_T$-dependent systematic uncertainties on the efficiencies we assume for the reconstruction, identification, and isolation criteria for prompt $e, \mu$, and $\tau _{\mathrm {\,had}}$ candidates

Table 2 Sources of phenomenological systematic uncertainty considered in this study (given in percent). In all expressions, $p_T$ and $m(t{\bar{t}})$ are given in units of GeV. See the text for further details

In $tt\rightarrow bb\ell \tau _{\mathrm {\,had}}$ events, the majority of observed electrons and muons are from direct W boson decays and therefore have a distribution in $p_T$ that is much harder than that for the visible $\tau _{\mathrm {\,had}}$ systems. The relative acceptance for $tt\rightarrow bb\ell \tau _{\mathrm {\,had}}$ and $tt\rightarrow bbe\mu$ final states may be sensitive to the details of the modelling of ${t}{t}$ production. We investigate this sensitivity by using an alternative ${t}{t}$ generator-level sample in which the QCD factorisation and renormalisation scales are changed by a factor of two relative to the default values. In addition, we evaluate systematic uncertainties arising from simulating the distribution of the mass of the ${t}{t}$ system, $m(t\bar{t})$, by increasing the weights of events by an amount that varies linearly between 1% at $m(t\bar{t})$ = 500 GeV to 10% at $m(t\bar{t})$ = 1000 GeV [65].

Table 2 gives a summary of the sources of phenomenological systematic uncertainty considered in this study.

2.3 Candidate event selection criteria

Candidate electrons and muons are considered in the analysis if after simulation of resolution and momentum scale they satisfy $p_T > 27$ GeV. Candidate $\tau _{\mathrm {\,had}}$ and hadronic jets are considered if after simulation of resolution and momentum scale they satisfy $p_T > 25$ GeV. Hadronic jets must satisfy $|\eta |$ < 4.5. Candidate electrons, muons, $\tau _{\mathrm {\,had}}$, and b-tagged jets must satisfy $|\eta |$ < 2.5.

2.3.1 Candidate event selection criteria on leptons

In order to maximise the cancellation of systematic uncertainties between ${t}\bar{t}\rightarrow b\bar{b}W^+W^{-}$ and $Z/\gamma ^*\rightarrow \tau \tau$ event samples the same candidate event selection criteria on leptons are applied in the two event classes.

Candidate $e\tau _{\mathrm {\,had}}$ events are required to contain:

• Exactly one e candidate.

• Exactly one $\tau _{\mathrm {\,had}}$ candidate of opposite sign to the e candidate.

• No $\mu$ candidates.

• The e candidate must fire the single-e trigger.

Candidate $\mu \tau _{\mathrm {\,had}}$ events are required to contain:

• Exactly one $\mu$ candidate.

• Exactly one $\tau _{\mathrm {\,had}}$ candidate of opposite sign to the $\mu$ candidate.

• No e candidates.

• The $\mu$ candidate must fire the single-$\mu$ trigger.

Candidate $e\mu$ events are required to contain:

• Exactly one e candidate.

• Exactly one $\mu$ candidate of opposite sign to the e candidate.

• The e candidate must fire the single-e trigger, and/or the $\mu$ candidate must fire the single-$\mu$ trigger.

2.3.2 ${t}{t}$ candidate event selection criteria

In addition to the relevant criteria on leptons given in Sect. 2.3.1 above, all candidate ${t}\bar{t}\rightarrow b\bar{b}W^+W^{-}$ events ($tt\rightarrow bb\ell \tau _{\mathrm {\,had}}$ as well as $tt\rightarrow bbe\mu$) are required to contain exactly two b-tagged jets. Because the same criterion is applied in the selection of the numerator $tt\rightarrow bb\ell \tau _{\mathrm {\,had}}$ and denominator $tt\rightarrow bbe\mu$ events, we expect the ratio $R(bbWW)$ to be largely insensitive to systematic uncertainties associated with jet reconstruction, JES, JER, and b-tagging. Requiring two b-tagged jets reduces the background in the selected ${t}\bar{t}\rightarrow b\bar{b}W^+W^{-}$ event samples from non-${t}{t}$ sources; it also reduces the background from ${t}{t}$ events in which a b-quark jet is misidentified as a prompt $e, \mu$, or $\tau _{\mathrm {\,had}}$.

Candidate $tt\rightarrow bb\ell \tau _{\mathrm {\,had}}$ events are rejected if the invariant mass of the $\tau _{\mathrm {\,had}}$ candidate and the highest $p_T$ non-b-tagged jet, $m(j-\tau _{\mathrm {\,had}})$, satisfies 50 GeV < $m(j-\tau _{\mathrm {\,had}})$ < 90 GeV. This criterion reduces the background from lepton+jet ${t}{t}$ events in which the hadronically decaying W boson produces two reconstructed jets, one of which is misidentified as a $\tau _{\mathrm {\,had}}$ candidate. The effectiveness of this criterion is illustrated by Fig. 3, which shows in the $tt\rightarrow bb\ell \tau _{\mathrm {\,had}}$ candidate event sample the distribution of $m(j-\tau _{\mathrm {\,had}})$, having applied all other $tt\rightarrow bb\ell \tau _{\mathrm {\,had}}$ event selection criteria. The upper plot shows events in which the $\tau _{\mathrm {\,had}}$ candidate originates from a genuine $\tau$ decay and the lower plot shows events in which the $\tau _{\mathrm {\,had}}$ candidate originates from a misidentified hadronic jet. A clear peak at around the mass of the W boson can be seen in the lower plot. In addition to helping reject background from misidentified hadronic jets, the distributions in Fig. 3 offer the possibility to make a data-driven estimate of the background in the $\tau _{\mathrm {\,had}}$ candidate sample. This will be useful in reducing the systematic uncertainty on the background yield.

Fig. 3

The distribution of $m(j-\tau _{\mathrm {\,had}})$ in the $tt\rightarrow bb\ell \tau _{\mathrm {\,had}}$ candidate event sample. The event selection requirement on this quantity has not been applied. The upper plot shows events in which the $\tau _{\mathrm {\,had}}$ candidate originates from a genuine $\tau$ decay and the lower plot shows events in which the $\tau _{\mathrm {\,had}}$ candidate originates from a misidentified hadronic jet

2.3.3 $Z/\gamma ^*\rightarrow \tau \tau$ candidate event selection criteria

In addition to the relevant criteria on leptons given in Sect. 2.3.1 above, all candidate $Z/\gamma ^*\rightarrow \tau \tau$ events ($Z\rightarrow \tau \tau \rightarrow \ell \tau _{\mathrm {\,had}}$ as well as $Z\rightarrow \tau \tau \rightarrow e\mu$) are required to satisfy the following criteria:

• Events should contain no b-tagged jets.

• 50 GeV $<m^*_3<$ 100 GeV.

• $\Sigma \cos \Delta \phi>$ $-0.1$.

• $a_T< 60$ GeV.

The relevant variables are defined below. Distributions of each variable having applied all other selection criteria are shown in Fig. 4 for $Z\rightarrow \tau \tau \rightarrow \ell \tau _{\mathrm {\,had}}$ and $Z\rightarrow \tau \tau \rightarrow e\mu$.

Fig. 4

Distributions of variables in the selection of $Z/\gamma ^*\rightarrow \tau \tau$ candidate events having applied all other selection criteria. The left column shows $Z\rightarrow \tau \tau \rightarrow \ell \tau _{\mathrm {\,had}}$ candidate events. The right column shows $Z\rightarrow \tau \tau \rightarrow e\mu$ candidate events. The selection criteria are indicated by vertical lines

As expected, the rejected event samples containing b-tagged jets are dominated by ${t}{t}$. Because the same criterion on b-tagged jets is applied in the selection of the numerator $Z\rightarrow \tau \tau \rightarrow \ell \tau _{\mathrm {\,had}}$ and denominator $Z\rightarrow \tau \tau \rightarrow e\mu$ events, we expect the ratio $R(Z)$ to be largely insensitive to systematic uncertainties associated with jet reconstruction, JES, JER, and b-tagging. Requiring no b-tagged jets reduces the background in the selected $Z/\gamma ^*\rightarrow \tau \tau$ event samples from ${t}{t}$ and also W boson plus heavy flavour production in which a b-quark jet is misidentified as a prompt e, $\mu$, or $\tau _{\mathrm {\,had}}$. It can be seen that the other selection criteria reject a large fraction of the remaining background, principally from W+jet production (in the $Z\rightarrow \tau \tau \rightarrow \ell \tau _{\mathrm {\,had}}$ sample) and from ${t}{t}$ and diboson production (in the $Z\rightarrow \tau \tau \rightarrow e\mu$ sample).

We propose here a novel selection variable, $m^*_3$:

$$\begin{aligned} m^*_3\equiv \frac{m_T(\ell , \tau _{\mathrm {\,had}}, \,/\!\!\!E_T)}{\sin \theta ^*_\eta }. \end{aligned}$$

(12)

Here $\,/\!\!\!E_T$ is the missing transverse momentum and the transverse mass, $m_T(\ell , \tau _{\mathrm {\,had}}, \,/\!\!\!E_T)$, of the 3-body system of $\ell$, $\tau _{\mathrm {\,had}}$, and $\,/\!\!\!E_T$ may be defined by:

$$\begin{aligned}&m_T(\ell , \tau _{\mathrm {\,had}}, \,/\!\!\!E_T)^2 \nonumber \\&\qquad = m_T(\ell , \tau _{\mathrm {\,had}})^2 + m_T(\ell , \,/\!\!\!E_T)^2 + m_T(\tau _{\mathrm {\,had}}, \,/\!\!\!E_T)^2 \end{aligned}$$

(13)

$\theta ^*_\eta$ is an approximation to the scattering angle of the leptons relative to the beam direction in the dilepton rest frame. This variable is defined [66] solely using the measured track directions by:

$$\begin{aligned} \cos (\theta ^{*}_{\eta }) \equiv \tanh \left( \frac{\eta ^--\eta ^+}{2}\right) , \end{aligned}$$

(14)

where $\eta ^-$ and $\eta ^+$ are the pseudorapidities of the negatively and positively charge lepton, respectively. The division by $\sin \theta ^*_\eta$ in the definition of $m^*_3$ takes into account the relative longitudinal motion of the two leptons and, therefore, $m^*_3$ is a more closely correlated with the $\tau ^+\tau ^-$ mass than is $m_T(\ell , \tau _{\mathrm {\,had}}, \,/\!\!\!E_T)$. In the selection of $Z/\gamma ^*\rightarrow \tau \tau$ candidates this improves the discrimination power against the dominant backgrounds (such as W+jet and diboson events). A comparison of the performance of $m^*_3$ with other similar discriminating variables is shown in Fig. 5.

Fig. 5

A comparison of the signal and background efficiencies for cuts on different mass variables in the $Z\rightarrow \tau \tau \rightarrow \ell \tau _{\mathrm {\,had}}$ candidate event sample. The variable $m(\ell , \tau _{\mathrm {\,had}})$ is defined as the visible mass of the lepton and $\tau _{\mathrm {\,had}}$ candidate. The star indicates the position of the proposed cut on $m^*_3$, which outperforms the other variables over the range of interest

Fig. 6

The distributions of $p_T$ and $|\eta |$ of e, $\mu$, and $\tau _{\mathrm {\,had}}$ in the selected ${t}\bar{t}\rightarrow b\bar{b}W^+W^{-}$ and $Z/\gamma ^*\rightarrow \tau \tau$ candidate event samples. The left column shows $p_T$ and the right column shows $|\eta |$. The top row shows e and $\mu$ candidates. The bottom row shows $\tau _{\mathrm {\,had}}$ candidates

The variable $\Sigma \cos \Delta \phi$ [67]:

$$\begin{aligned} \Sigma \cos \Delta \phi \equiv \cos (\Delta \phi (\ell ,\,/\!\!\!E_T)) + \cos (\Delta \phi (\tau _{\mathrm {\,had}},\,/\!\!\!E_T)), \end{aligned}$$

(15)

discriminates against background events containing leptonically decaying W bosons. Here $\Delta \phi (\ell ,\,/\!\!\!E_T)$ is the azimuthal angle between the $\ell$ and the $\,/\!\!\!E_T$ and $\Delta \phi (\tau _{\mathrm {\,had}},\,/\!\!\!E_T)$ is the azimuthal angle between the $\tau _{\mathrm {\,had}}$ and the $\,/\!\!\!E_T$.

The variable $a_T$ [68] corresponds to the component of the $p_T$ of the dilepton system that is transverse to the dilepton thrust axis. This variable is well suited to the study of $\tau ^+\tau ^-$ final states, because it is less sensitive to any imbalance in the transverse momenta of the neutrinos produced in the tau decays than is $a_L$ [68], the component of the dilepton $p_T$ that is longitudinal to the dilepton thrust axis. The variable $a_T$ discriminates against background events containing leptonically decaying W bosons.

Figure 6 shows the distributions of $p_T$ and $|\eta |$ of e, $\mu$, and $\tau _{\mathrm {\,had}}$, in the selected ${t}\bar{t}\rightarrow b\bar{b}W^+W^{-}$ (red) and $Z/\gamma ^*\rightarrow \tau \tau$ (blue) candidate event samples. It can be seen that the $p_T$ distributions are considerably softer for the $Z/\gamma ^*\rightarrow \tau \tau$ candidate event samples than those for the selected ${t}\bar{t}\rightarrow b\bar{b}W^+W^{-}$ candidate event samples.

As can be seen from the lower plots in Fig. 4, a cut of $a_T< 30$ GeV would be desirable to improve the suppression of backgrounds from ${t}{t}$ and EW diboson events. However, a cut on $a_T$ suppresses also $Z/\gamma ^*\rightarrow \tau \tau$ events that contain initial state parton radiation. Initial state radiation broadens the distributions of lepton candidate $p_T$ in $Z/\gamma ^*\rightarrow \tau \tau$ events. A hard cut on $a_T$ would therefore suppress the high-$p_T$ regions in the distributions of lepton $p_T$ in the selected $Z/\gamma ^*\rightarrow \tau \tau$ event samples, as is illustrated in Fig. 7. The cut $a_T< 60$ GeV is chosen to reject background events from ${t}{t}$ and diboson events, without unduly biasing the lepton $p_T$ distributions and further accentuating the differences between the lepton $p_T$ distributions seen in Fig. 6.

Fig. 7

Distributions of lepton $p_T$ in the selected $Z/\gamma ^*\rightarrow \tau \tau$ candidate event samples. Three different cuts on $a_T$ are applied

2.4 Size and composition of the selected event samples

The expected numbers of selected events and the composition of the four selected event samples for $\int \!\! L\, \hbox {\,d}t$ = 140 $\mathrm {fb}^{-1}$ at ${\sqrt{s}}~=~13~\hbox {TeV}$ are given in Table 3.

Table 3 The expected numbers of events in the four selected event samples for $\int \!\! L\, \hbox {\,d}t$ = 140 $\mathrm {fb}^{-1}$ at ${\sqrt{s}}~=~13~\hbox {TeV}$. The numbers are broken down by physics production process. The numbers for ${t}{t}$, single top and Z boson production are further broken down into the two categories “true”, in which the two lepton candidates are correctly identified, and “fake”, in which at least one of the lepton candidates is incorrectly identified. Small numbers of events from ${t}\bar{t}V$ processes that contain two correctly identified leptonic W boson decays are included in the ${t}{t}$ “true’ category. Otherwise, the events from ${t}\bar{t}V$ processes are included in the ${t}{t}$ “fake” category. The numbers given in bold type represent the signal in the four selected event samples. The numbers given under “Total background” include the “fake” categories described above

As a result of this study we estimate that the fractional statistical uncertainty on $R(WZ)$ for a single LHC experiment for an integrated luminosity of $\int \!\! L\, \hbox {\,d}t$ = 140 $\mathrm {fb}^{-1}$ would be around 0.5%.

The selection requirements for $tt\rightarrow bb\ell \tau _{\mathrm {\,had}}$ and $tt\rightarrow bbe\mu$ can be seen to be very effective at removing non-${t}{t}$ sources of background, e.g., W+jet. The most significant source of non-${t}{t}$ events originates from the EW production of “single top” events in the associated production $tWb$ channel. Since these events contain two genuine leptonically decaying W bosons they can effectively be considered as contributing to the signal sample, and not as background. They are listed as “$tWb$ true” in Table 3. Single top processes that do not contain a pair of W bosons decaying to produce two correctly identified leptons are classified as background. Events which contain the associated production of either a W or Z boson with a ${t}{t}$ pair (${t}\bar{t}V$) can be considered signal if they contain at least two W bosons which decay to correctly identified leptons. Since the fractions of these events are small, $<0.1\%$, they are included in the ${t}{t}$ categories in Table 3.

The most significant source of background for $tt\rightarrow bb\ell \tau _{\mathrm {\,had}}$ originates from genuine ${t}{t}$ events in which the $\tau _{\mathrm {\,had}}$ candidate originates from a misidentified hadronic jet. The residual background from this source corresponds to about 2.5% of the selected sample of candidate $tt\rightarrow bb\ell \tau _{\mathrm {\,had}}$ events. The $tt\rightarrow bbe\mu$ candidate event sample will be selected with entirely negligible levels of background.

The most significant source of background for $Z\rightarrow \tau \tau \rightarrow \ell \tau _{\mathrm {\,had}}$ originates from QCD multijet (MJ) events (that is, events that do not contain any prompt leptons from W or Z boson decay). This background is estimated to be at the level of around 5%, as described in the Appendix and [44]. In comparison, the MC-estimated backgrounds for $Z\rightarrow \tau \tau \rightarrow \ell \tau _{\mathrm {\,had}}$ from ${t}{t}$ (0.3%) and W+jet (0.2%) are small. The most significant sources of background for $Z\rightarrow \tau \tau \rightarrow e\mu$ originate from ${t}{t}$ (5.8%) and diboson (6.3%) production.

2.5 Sensitivity of $\varvec{R(WZ)}$ to $\varvec{R(W)}$

For definiteness we make the assumption in our study that the effective $W\rightarrow \tau \nu$ coupling relevant for on-mass-shell W boson decays could be modified by some BSM effect, whilst all other W boson couplings are maintained at their SM-predicted values. Under this assumption, if the branching ratio ${\mathcal {B}}(W\rightarrow \tau \nu )$ is multiplied by a factor X relative to its SM-predicted value ${\mathcal {B}}(W\rightarrow \tau \nu )_{\mathrm {SM}}$,

$$\begin{aligned} {\mathcal {B}}(W\rightarrow \tau \nu ) = X.{\mathcal {B}}(W\rightarrow \tau \nu )_{\mathrm {SM}} \end{aligned}$$

(16)

then all other W boson branching fractions will be modified by a factor

$$\begin{aligned} F = \frac{1-X.{\mathcal {B}}(W\rightarrow \tau \nu )_{\mathrm {SM}}}{1-{\mathcal {B}}(W\rightarrow \tau \nu )_{\mathrm {SM}}}. \end{aligned}$$

(17)

In our MC study we perform a “calibration” of the double-ratio method by reweighting every simulated event containing one or more W boson decays by a factor $X^nF^m$, where n is the number of generator-level $W\rightarrow \tau \nu$ decays and m is the number of other W boson decays. This calibration procedure properly takes into account all events in the calculation of $R(bbWW)$ that contain pairs of leptonically decaying W bosons with correctly identified decay products in the “numerator” $tt\rightarrow bb\ell \tau _{\mathrm {\,had}}$ and “denominator” $tt\rightarrow bbe\mu$ samples. This includes, for example, the presence of events containing the cascade decay $W\rightarrow \tau \nu \rightarrow \ell \nu \nu$, whose presence in the “denominator” $tt\rightarrow bbe\mu$ sample slightly decreases the correlation between $R(bbWW)$ and $R(W)$. The value of $R(Z)$ is designed to be independent of any change in $R(W)$. However, the backgrounds from ${t}\bar{t}\rightarrow b\bar{b}W^+W^{-}$ and WW in the $Z\rightarrow \tau \tau \rightarrow \ell \tau _{\mathrm {\,had}}$ and $Z\rightarrow \tau \tau \rightarrow e\mu$ events contain decays of W bosons, which cause the expected background levels to alter with $R(W)$. A correction to the $R(Z)$ calculation from this effect is included. The result of this calibration is shown in Fig. 8, which shows the fractional change in $R(bbWW)$, $R(Z)$, and $R(WZ)$ as a function of the fractional change in $R(W)$. The absolute value of the ratio of $R(W)$ and $R(WZ)$ will depend on the precise experimental details specific to a given experiment and would have to be evaluated using fully simulated MC events for the specific identification criteria and event selection cuts employed in the analysis of the experimental data.

Fig. 8

The fractional change in $R(bbWW)$ (upper), $R(Z)$ (middle), and $R(WZ)$ (lower), as a function of the fractional change in $R(W)$, obtained by reweighting events at generator level as described in the text. The displayed error bars illustrate the expected experimental statistical uncertainty on the relevant quantity, and are completely correlated point to point. MC statistical uncertainties on the point-to-point variation with $R(W)$ are negligible

2.6 Evaluation of systematic uncertainties

Considering each source of systematic uncertainty described in Sects. 2.1 and 2.2, and summarised in Tables 1 and 2, we evaluate the resulting changes in the ratios $R(bbWW)$, $R(Z)$, $R(WZ)$, and $R(W)$.

Table 4 Changes in the ratios $R(bbWW)$, $R(Z)$, and $R(WZ)$ resulting from the sources of experimental systematic uncertainty described in Sect. 2.1 and summarised in Table 1. The $p_T$/$\eta$-independent and $p_T$/$\eta$-dependent systematic variations are considered separately

The $p_T$/$\eta$-independent and $p_T$/$\eta$-dependent systematic variations on the reconstruction of leptons and jets, as summarised in Table 1, are considered separately. The resultant changes in the ratios are given in Table 4. It can be seen that the $p_T$/$\eta$-independent systematic uncertainties on $R(bbWW)$ and $R(Z)$ are large, but almost exactly equal. Therefore, $p_T$/$\eta$-independent systematic uncertainties on the reconstruction of leptons and jets almost perfectly cancel in the double ratio $R(WZ)$. When considering $p_T$/$\eta$-dependent systematic uncertainties the cancellation is no longer perfect. The most significant sources of $p_T$/$\eta$-dependent systematic uncertainties are illustrated in Fig. 9. The 10% uncertainty on the 2.5% background from hadronic jets misidentified as $\tau _{\mathrm {\,had}}$ candidates in the $tt\rightarrow bb\ell \tau _{\mathrm {\,had}}$ sample leads to an uncertainty of 0.25% on $R(bbWW)$ and thus also on $R(W)$. The 5% uncertainty on the 5% background from MJ events in the $Z\rightarrow \tau \tau \rightarrow \ell \tau _{\mathrm {\,had}}$ sample leads to an uncertainty of 0.25% on $R(Z)$ and thus also on $R(W)$. Since the above two backgrounds both result from hadronic jets misidentified as $\tau _{\mathrm {\,had}}$ candidates it is conceivable that the resultant systematic uncertainties on $R(bbWW)$ and $R(Z)$ could be correlated and thus partially cancel in the calculation of $R(W)$. A realistic estimate of the degree of correlation will depend on experimental details beyond the scope of the current study and we have not taken into account any potential reduction in the systematic uncertainty on $R(W)$.

The sources of phenomenological systematic uncertainty considered in this study are summarised in Table 2 and the resultant changes in the ratios $R(bbWW)$, $R(Z)$, and $R(WZ)$ are given in Table 5. The 3% uncertainty on the backgrounds from ${t}{t}$ (5.8%) and diboson (6.3%) production in the $Z\rightarrow \tau \tau \rightarrow e\mu$ sample leads to uncertainties of 0.2% and 0.2%, respectively, on $R(Z)$ and thus also on $R(W)$. The uncertainty resulting from the $p_T(Z)$ reweighting procedure is <0.1% on $R(Z)$, and thus also on $R(W)$.

The alternative ${t}{t}$ sample with modified QCD factorisation and renormalisation scales leads to an uncertainty of 0.3% on $R(bbWW)$ and thus also on $R(W)$ [69]. The $m(t\bar{t})$ reweighting leads to an uncertainty of $0.2\%$ on $R(bbWW)$ and thus also on $R(W)$.

Fig. 9

Summary of the most significant changes in the ratios a $R(bbWW)$, b $R(Z)$, and c $R(WZ)$ arising from sources of $p_T$/$\eta$-dependent systematic uncertainty. From left to right in each sub-figure the sources are ordered in descending magnitude of the resulting systematic uncertainty

We add in quadrature the changes in the double ratio $R(WZ)$ arising from the $p_T$/$\eta$-independent and $p_T$/$\eta$-dependent systematic variations on the reconstruction of leptons and jets, together with the other considered systematic uncertainties discussed above. The total resulting systematic uncertainty on $R(WZ)$ is 1.3%.

2.7 Considerations for future measurements

Measurements of $R(W)$ on the data from ATLAS and CMS using the double ratio technique proposed here will, clearly, require the use of fully simulated MC events and will employ the sophisticated procedures developed by the individual experiments to evaluate backgrounds and experimental systematic uncertainties. Our study is based on particle-level MC events and a simple parameterised simulation of detector performance. Nevertheless, we believe our demonstration that the dominant experimental systematic uncertainties cancel in the double ratio, as well as our estimates of the major residual systematic uncertainties, to be broadly realistic. We have based our simulation of lepton, jet and $\,/\!\!\!E_T$ reconstruction on measurements of efficiencies, backgrounds and systematic uncertainties published by ATLAS and CMS [24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55], having chosen identification algorithms whose performance is suited to the needs of our analysis. The above cited performance papers are for the most part based on around a quarter of the full run 2 data set of $\int \!\! L\, \hbox {\,d}t$ = 140 $\mathrm {fb}^{-1}$ at ${\sqrt{s}}~=~13~\hbox {TeV}$ that is now available. It is, therefore, to be expected that the full run 2 data set will allow systematic uncertainties on lepton and jet identification efficiencies to be reduced by about a factor of two compared to the values we have assumed in our study. Of particular relevance to our study, it is to be hoped that the high statistics provided by the full run 2 data set will allow the $p_T$ dependence of the identification efficiency for $\tau _{\mathrm {\,had}}$ candidates to be studied over the range $25<p_T<100$ GeV, without the need to use $tt\rightarrow bb\ell \tau _{\mathrm {\,had}}$ events and implicitly assume that ${\mathcal {B}}(W\rightarrow \tau \nu )$ takes its SM value. This could be achieved, e.g., by the selection of a dedicated $Z\rightarrow \tau \tau \rightarrow \ell \tau _{\mathrm {\,had}}$ event sample in which there is a high transverse momentum initial state radiation.

Table 5 Changes in the ratios $R(bbWW)$, $R(Z)$, and $R(WZ)$ resulting from the sources of phenomenological systematic uncertainty described in Sect. 2.2 and summarised in Table 2

The residual non-cancellation in $R(WZ)$ of the $p_T$/$\eta$-dependent $\tau _{\mathrm {\,had}}$ identification uncertainty, as shown in Table 4, arises primarily from the differences in the $\tau _{\mathrm {\,had}}$ $p_T$ distributions between the $tt\rightarrow bb\ell \tau _{\mathrm {\,had}}$ and $Z\rightarrow \tau \tau \rightarrow \ell \tau _{\mathrm {\,had}}$ samples, as shown in Fig. 6. We have considered possible methods to reduce the impact of the different $\tau$ $p_T$ distributions. One simple approach would be to introduce an additional cut on $\tau _{\mathrm {\,had}}$ $p_T < 65$ GeV; this removes the high $p_T$ tail in the $tt\rightarrow bb\ell \tau _{\mathrm {\,had}}$ sample, with negligible changes to the $Z\rightarrow \tau \tau \rightarrow \ell \tau _{\mathrm {\,had}}$ sample. This has the effect of improving the cancellation in $R(WZ)$ of the $p_T$-dependent $\tau _{\mathrm {\,had}}$ identification uncertainty between the $tt\rightarrow bb\ell \tau _{\mathrm {\,had}}$ and $Z\rightarrow \tau \tau \rightarrow \ell \tau _{\mathrm {\,had}}$ samples; the residual systematic uncertainty reduces from 0.8%, as shown in Table 4, to 0.2%. More sophisticated potential methods to mitigate the effects of the $p_T$-dependent $\tau _{\mathrm {\,had}}$ identification uncertainty might be (a) to reweight the tau $p_T$ distribution in the $tt\rightarrow bb\ell \tau _{\mathrm {\,had}}$ sample to resemble that in the $Z\rightarrow \tau \tau \rightarrow \ell \tau _{\mathrm {\,had}}$ sample, or (b) to perform measurements of $R(WZ)$ in bins of $\tau _{\mathrm {\,had}}$ $p_T$ and subsequently combine the measurements, taking into account the bin-to-bin correlations in the systematic uncertainties. However, any such methods might run the risk of increasing the sensitivity of $R(WZ)$ to phenomenological uncertainties. A careful investigation of such issues will be needed in order to optimise the overall uncertainty arising from experimental and phenomenological systematic uncertainties. This will require study of the large number of fully simulated MC samples produced by the experiments, corresponding to different models of ${t}{t}$ and Z boson production and decay, and is beyond the scope of this paper.

Of course, if the measured value of $R(WZ)$ is found to disagree with that expected in the SM then further studies will be required to ascertain the nature of BSM physics that is responsible. For example, the decay $t\rightarrow bH^+$, where $H^+$ is a charged higgs boson, followed by $H^+\rightarrow \tau ^+\nu$ would modify the effective $t\rightarrow b\tau \nu$ and $t\rightarrow b\ell \nu$ branching ratios in a similar fashion to that discussed in the context of equations 16 and 17 above. Similarly, top decays via a neutral higgs boson $t\rightarrow qH$, $H\rightarrow \tau \tau$ will also increase the number of tau leptons in ${t}{t}$ events relative to the SM-expected value. Existing experimental searches for charged [70, 71] and neutral [72] higgs bosons in top quark events suffer from the large systematic uncertainties associated with $\tau _{\mathrm {\,had}}$ identification and will benefit from the double ratio method we propose in this paper to reduce experimental systematic uncertainties.

The large numbers of events from the EW production of diboson events (WW and WZ) at the LHC provide alternative samples with which to make this novel measurement. Controlling systematic uncertainties on $R(W)$ in $WW\rightarrow \ell \nu \tau _{\mathrm {\,had}}\nu$ events will be extremely challenging; we shall need extraordinarily good background rejection against fake $\tau$ candidates from misidentified jets in $W\rightarrow \ell \nu$+ jet events. One may also consider $ZW\rightarrow \ell \ell \tau _{\mathrm {\,had}}\nu$ as an alternative sample with which to perform this measurement. This channel provides lower statistics because of the lower cross section times branching fraction. However, one can veto on Z+jet backgrounds by removing events in which the $\ell ^+ \ell ^-$ momentum is back to back with the $\tau$ candidate direction. One can calibrate the residual backgrounds by looking at the back-to-back events. If the experimental measurements proposed here observe a clear violation of LU then having three channels $tt\rightarrow bb\ell \tau _{\mathrm {\,had}}$, $WW\rightarrow \ell \nu \tau _{\mathrm {\,had}}\nu$, and $ZW\rightarrow \ell \ell \tau _{\mathrm {\,had}}\nu$ could increase the significance of the observation, and could help elucidate the underlying BSM origin of the effect.

In addition to the $tt\rightarrow bb\ell \tau _{\mathrm {\,had}}$ and $Z\rightarrow \tau \tau \rightarrow \ell \tau _{\mathrm {\,had}}$ final states considered here, it may be possible to measure $R(W)$ using a similarly motivated ratio of $tt\rightarrow bb\ell \tau _{\mathrm {\,lept}}$ and $Z\rightarrow \tau \tau \rightarrow \ell \tau _{\mathrm {\,lept}}$ final states, where $\tau _{\mathrm {\,lept}}$ corresponds to a leptonic decay of the $\tau$, that may be identified using, for example, criteria based on the non-zero lifetime of the $\tau$ lepton. A combination of $R(W)$ measurements using $\tau _{\mathrm {\,had}}$ and $\tau _{\mathrm {\,lept}}$ signatures would benefit from the fact that the leading experimental uncertainties, arising from $\tau _{\mathrm {\,had}}$ and $\tau _{\mathrm {\,lept}}$ identification, would be largely uncorrelated.

3 Summary and conclusions

A measurement of $R(W)\equiv {\mathcal {B}}(W\rightarrow \tau \nu )/{\mathcal {B}}(W\rightarrow \ell \nu )$ ($\ell =e\ {\mathrm {or}}\ \mu$) represents a promising opportunity to discover a violation of lepton universality. We propose here a novel double-ratio method that will allow $R(W)$ to be measured using top quark pairs and $Z/\gamma ^*\rightarrow \tau \tau$ events at the LHC.

We define $R(bbWW)$ in di-leptonic ${t}{t}$ events to be the ratio of the numbers of $\ell \tau _{\mathrm {\,had}}$ and $e\mu$ final states (equation 9). $R(bbWW)$ is sensitive to the value of $R(W)$, but also to systematic uncertainties on the reconstruction of e, $\mu$, and $\tau$ leptons. Similarly, we define $R(Z)$ in $Z/\gamma ^*\rightarrow \tau \tau$ events to be the ratio of the numbers of $\ell \tau _{\mathrm {\,had}}$ and $e\mu$ final states (equation 10). $R(Z)$ is similarly sensitive to systematic uncertainties on the reconstruction of e, $\mu$, and $\tau$ leptons, but is insensitive to the value of $R(W)$. The double ratio $R(WZ)\equiv R(bbWW)/ R(Z)$ cancels to first order sensitivity to systematic uncertainties on the reconstruction of e, $\mu$, and $\tau$ leptons, thus improving very significantly the precision to which $R(W)$ can be measured at a hadron collider

We have performed a study of the double ratio $R(WZ)$ using particle-level MC events and a parameterised simulation of detector performance. We have based our simulation of lepton, jet and $\,/\!\!\!E_T$ reconstruction on measurements of efficiencies, backgrounds and systematic uncertainties published by ATLAS and CMS [24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55], having chosen identification algorithms whose performance is suited to the needs of our analysis. For a data set of $\int \!\! L\, \hbox {\,d}t$ = 140 $\mathrm {fb}^{-1}$ at ${\sqrt{s}}~=~13~\hbox {TeV}$ we estimate a statistical uncertainty on $R(W)$ of 0.5%. Our study confirms the almost perfect cancellation in $R(W)$ of systematic uncertainties on the reconstruction efficiencies of e, $\mu$, and $\tau$ leptons that are applied as constant factors. We find that the most significant residual sources of uncertainty on $R(W)$ arise from systematic uncertainties on the $p_T$ and $\eta$ dependence of the reconstruction efficiencies of e, $\mu$, and $\tau$ leptons, which total around $1.0\%$. We have evaluated also potential uncertainties arising from backgrounds to the selected event samples and from various phenomenological sources.

Our studies indicate that a single experiment precision on the measurement of $R(W)$ of around 1.4% is achievable with a data set of $\int \!\! L\, \hbox {\,d}t$ = 140 $\mathrm {fb}^{-1}$ at ${\sqrt{s}}~=~13~\hbox {TeV}$. This would improve significantly upon the precision of the LEP2 measurements of $R(W)$. If the central value of the new measurements were equal to the central value of the LEP2 measurements this would yield an observation of BSM physics at a significance level of around 5$\sigma$.

Appendix: Detailed description of the parameterised detector simulation

We give here a detailed description of our choices of lepton and jet identification algorithms to simulate, the experimental systematic uncertainties we consider, and the reasoning behind these choices.

1.1 Simulation of muon candidates

The efficiencies and systematic uncertainties associated with the reconstruction and identification of high $p_T$, isolated muon candidates have been presented by the ATLAS [24] and CMS [25] collaborations. In both experiments the muon reconstruction efficiency is around 99% and is independent of $p_T$ and $|\eta |$, except for some well-defined, poorly instrumented regions of both detectors that are usually excluded for precision measurements. We choose to simulate the efficiency for muon identification according to that given for the “Medium” category described in [24]. This gives an efficiency of around 96% for muons with $p_T>20$ GeV. Systematic uncertainties on the efficiency are around 0.1% for muon $p_T$ around $m_Z/2$, increasing to around 0.5% for $p_T\approx 30$ GeV and $p_T\approx 100$ GeV. We choose to simulate the “Tight” lepton isolation requirement given in [24], which is measured to have an efficiency that is independent of $p_T$ and $|\eta |$ of around 96% with a systematic uncertainty at the per mille level. We assume the efficiency of the isolation criteria for sources of non-prompt muons to be 0.03, based on the range of values given in [24]. We consider a combined systematic uncertainty on the efficiency for muon reconstruction, identification, and isolation. We consider a $p_T$/$\eta$-independent relative systematic uncertainty of 2%. We generate a $p_T$-dependent systematic uncertainty by modifying the efficiency by a relative amount that varies linearly between 0.5% at $p_T = 30$ GeV and 0% at $p_T = 80$ GeV and above.

We simulate the efficiency of the single muon trigger to be 65% (80%) in the barrel (endcap) region, as has been measured for ATLAS [26, 27]. We define the barrel region by $|\eta |<1.0$ and the endcap region by $|\eta |>1.0$. For offline $p_T$ at least 1 GeV above the trigger threshold the efficiency is almost independent of $p_T$. The trigger threshold is assumed to be similar to the electron trigger threshold of ATLAS during most of run 2 at $p_T = 26$ GeV [28]. We consider a $p_T$/$\eta$-independent relative systematic uncertainty of 1%. We generate an $\eta$-dependent systematic uncertainty by modifying the relative efficiency by 1% in the endcap region, whilst the efficiency in the barrel region remains unchanged.

We simulate the resolution in muon $p_T$ by means of a Gaussian with a width of $2.30\pm 0.15$% in the barrel region and $2.90\pm 0.15$% in the endcap region, independent of $p_T$ [24]. We consider a $p_T$/$\eta$-independent relative systematic uncertainty of 0.2% on the $p_T$ scale [24]. We generate $\eta$-dependent systematic uncertainties by modifying the resolution and $p_T$ scale only in the endcap region, whilst the values in the barrel region remain unchanged.

1.2 Simulation of electron candidates

The efficiencies and systematic uncertainties associated with the reconstruction and identification of high $p_T$, isolated electron candidates have been presented by the ATLAS [29] and CMS [30] collaborations. In both experiments the electron reconstruction efficiency is around 98% and is fairly independent of $p_T$ and $|\eta |$, except for some well-defined, poorly instrumented regions of both detectors [31]. A general feature in both ATLAS and CMS is that the efficiency of the commonly used electron identification algorithms tend to have a larger dependence on $p_T$ than is the case for muon identification. We choose to simulate approximately the efficiency for electron identification according to the “Tight likelihood” algorithm of [29]. The efficiency is around 75% at $p_T=30$ GeV and rises to around 90% for $p_T>80$ GeV and above [32]. The efficiency is determined with a systematic uncertainty of a few per mille for $p_T>30$ GeV. We choose to simulate also for electrons the “Tight” lepton isolation requirement given in [24], with an efficiency that is independent of $p_T$ and $|\eta |$ of around 96%. We consider a combined systematic uncertainty on the efficiency for electron reconstruction, identification, and isolation. We consider a $p_T$/$\eta$-independent relative systematic uncertainty of 2%. We generate a $p_T$-dependent systematic uncertainty by modifying the efficiency by a relative amount that varies linearly between 2% at $p_T = 30$ GeV and 0% at $p_T = 80$ GeV and above.

The efficiency of the single electron trigger has been measured in ATLAS to be around 90% for offline $p_T$ at least 1 GeV above the trigger threshold [33]. We consider a $p_T$/$\eta$-independent relative systematic uncertainty of 1%. We generate a $p_T$-dependent systematic uncertainty by modifying the position of the effective trigger threshold in $p_T$ by 1 GeV.

The resolution and $p_T$ scale for high $p_T$, isolated electron candidates and the associated systematic uncertainties have been measured by the ATLAS [34] and CMS [30] collaborations. We simulate the resolution in electron $p_T$ by means of a Gaussian with a width $\sigma _{p_T}$ given by $\frac{\sigma _{p_T}}{p_T} = \frac{0.16}{\sqrt{p_T}}$. We consider a $p_T$/$\eta$-independent systematic uncertainty on the resolution by varying the constant term in the formula for $\frac{\sigma _{p_T}}{p_T}$ by an amount $\pm 0.002$ [35]. The $p_T$ scale is calibrated with a precision of around 0.2% in both ATLAS and CMS [30, 34]. We consider a $p_T$/$\eta$-independent relative systematic uncertainty of 0.2% on the $p_T$ scale. We generate $\eta$-dependent systematic uncertainties by modifying the resolution and $p_T$ scale only in the endcap region, whilst the values in the barrel region remain unchanged.

Since we apply tight isolation requirements on the candidate leptons, we make the conservative assumption that the identification efficiency for non-prompt electrons from heavy flavour decays is the same as that for prompt electrons. In [36] around 2/3 of the background electrons arise from heavy flavour decays. The remaining background arises from photon conversions and misidentified hadrons, which are difficult to simulate in the context of our parameterised detector simulation. We therefore make the approximation that the total background to the sample of high $p_T$, isolated electrons in the simulated events is obtained by multiplying the number of electrons from heavy flavour decays by a factor of 1.5.

1.3 Simulation of $\tau _{\mathrm {\,had}}$ candidates

Of central importance to any measurement of $R(W)$ at a hadron collider will be an understanding of the efficiencies and backgrounds associated with the identification of $\tau$ candidates. A recent paper by the CMS Collaboration [37] describes the methods used to identify $\tau _{\mathrm {\,had}}$ candidates at ${\sqrt{s}}~=~13\hbox {TeV}$ and details the methods used to determine identification efficiencies, background rates and energy scale, and their associated systematic uncertainties, using a data set corresponding to $\int \!\! L\, \hbox {\,d}t$ = 36 $\mathrm {fb}^{-1}$. We choose to simulate the $\tau _{\mathrm {\,had}}$ identification efficiency and fake probabilities corresponding to the “very-very-tight” operating point of [37]. The efficiency is around 30% and is independent of $p_T$. The misidentification rate for hadronic jets is around $2\times 10^{-3}$ for $p_T\approx 25$ GeV decreasing to around $10^{-4}$ for $p_T\approx 100$ GeV. The misidentification rate is approximately independent of $|\eta |$ [38]. We simulate also the discriminants against electrons and muons described in [37]: the “Tight” discriminant against electrons has a $\tau _{\mathrm {\,had}}$ identification efficiency of 75% and a fake probability of $10^{-3}$; the “Tight” discriminant against muons has a $\tau _{\mathrm {\,had}}$ identification efficiency of 99% and a fake probability of $1.4\times 10^{-3}$ [39].

In our study it is particularly important to assign realistic systematic uncertainties on the identification efficiency for $\tau _{\mathrm {\,had}}$ candidates. A tag and probe analysis of $Z\rightarrow \tau \tau \rightarrow \ell \tau _{\mathrm {\,had}}$ events in [37] results in a relative uncertainty on the identification efficiency of $\tau _{\mathrm {\,had}}$ candidates of 5% for $\tau _{\mathrm {\,had}}$ $p_T$ up to 60 GeV. Samples of ${t}{t}$ events are used in [37] to cross check the efficiency for $\tau _{\mathrm {\,had}}$ $p_T$ up to 100 GeV; a relative uncertainty of 7% for $60<p_T<100$ GeV is assigned. Implicitly this latter analysis assumes that ${\mathcal {B}}(W\rightarrow \tau \nu )$ takes its SM value, because it relies on a comparison between the absolute numbers of $tt\rightarrow bb\ell \tau _{\mathrm {\,had}}$ candidate events observed in the CMS data and predicted by the MC. Clearly, for our proposed test of LU using ${t}{t}$ events it would not be legitimate to set data-MC scale factors for $\tau _{\mathrm {\,had}}$ using $tt\rightarrow bb\ell \tau _{\mathrm {\,had}}$ events in this way; this means that it will be difficult to control the $p_T$ dependence of the identification efficiency for $\tau _{\mathrm {\,had}}$ candidates beyond the range in $\tau _{\mathrm {\,had}}$ $p_T$ covered by the $Z\rightarrow \tau \tau \rightarrow \ell \tau _{\mathrm {\,had}}$ event sample. We consider a $p_T$/$\eta$-independent relative systematic uncertainty of 5%. We generate a $p_T$-dependent systematic uncertainty on the identification efficiency of $\tau _{\mathrm {\,had}}$ candidates by modifying the efficiency by a relative amount that varies linearly between 5% at $p_T = 30$ GeV and 0% at $p_T = 130$ GeV and above. The $p_T$ resolution for $\tau _{\mathrm {\,had}}$ candidates is given by $\frac{\sigma _{p_T}}{p_{T}} = 0.16$ [40].

The relative uncertainties on the probabilities for electrons, muons, and hadronic jets to be misidentified as a $\tau _{\mathrm {\,had}}$ candidate are around 10% [41]. The uncertainty on the $p_T$ scale for $\tau _{\mathrm {\,had}}$ candidates is around 1% [37]. We consider an $\eta$-dependent systematic uncertainty by modifying the $p_T$ scale by 1% only in the endcap region, whilst the scale in the barrel region remains unchanged. Preliminary systematic uncertainties of a similar magnitude have been assessed by the ATLAS Collaboration for the identification of $\tau _{\mathrm {\,had}}$ candidates at ${\sqrt{s}}~=~13~\hbox {TeV}$ [42], using the methods described in [43].

Backgrounds from MJ events are typically estimated at hadron colliders using data driven methods. Such backgrounds cannot reliably be estimated from MC. Of the four signal samples, $Z\rightarrow \tau \tau \rightarrow \ell \tau _{\mathrm {\,had}}$ will be the sample with the largest fraction of MJ background. An important motivation for our choice to simulate the “very-very-tight” operating point of [37] for $\tau _{\mathrm {\,had}}$ identification, which has relatively low efficiency but high rejection power, is to minimise the uncertainties arising from MJ backgrounds. From [25] and [37] we estimate the fraction of MJ background in the selected $Z\rightarrow \tau \tau \rightarrow \ell \tau _{\mathrm {\,had}}$ sample to be around 5%, with a relative uncertainty of around 5% [44].

The probability to mis-measure the sign of the charge of lepton candidates is expected to be less than 1% over the range of $p_T$ of relevance to our study and is neglected in our simulation.

1.4 Simulation of hadronic jets, including b-tagging

Jet finding is performed at particle-level using the anti-$k_t$ algorithm [45] with a distance parameter $R=0.4$. Determinations of the jet energy scale (JES) and jet energy resolution (JER), along with their systematic uncertainties, have been presented by the ATLAS [46] and CMS [47] collaborations. We simulate the resolution in jet $p_T$ by means of a Gaussian with a width $\sigma _{p_T}$ given by $\frac{\sigma _{p_T}}{p_T} = \frac{1.0}{\sqrt{p_T}}$. We assume an uncertainty on the jet energy scale of 1%.

The efficiencies, backgrounds and systematic uncertainties associated with the b-tagging of hadronic jets have been presented by the ATLAS [48] and CMS [49] collaborations. We choose to simulate tag probabilities according to those given for the “DeepCSV Loose” category of [49]. Jets with $|\eta |$ < 2.5 are flagged as b-tagged with probabilities that depend on their flavour at truth level as follows: truth b 85%, truth c 40%, truth light quark or gluon 1%. These tag probabilities are approximately independent of $p_T$ and $|\eta |$. The relative uncertainty in the b-jet efficiency scale factors is around 1.5% and the uncertainties in the c-jet and light-jet mis-tag scale factors are around 4% and 10% respectively [50].

The value of $\,/\!\!\!E_T$ is calculated from the vector sum at particle level of the $p_T$ of all neutrinos in the event. Resolution in $\,/\!\!\!E_T$ is taken into account by adding the difference between particle-level and detector level $p_T$ of each lepton and jet in the event. We have checked that this procedure reproduces approximately the $\,/\!\!\!E_T$ resolutions given in [52] and [54].

The effects of multiple proton–proton collisions or “pile-up” are not simulated in our study. In general, the lepton and jet identification algorithms employed by ATLAS and CMS are designed to have small pile-up dependence [55]. We take the performance values we have implemented to represent averages over the pile-up conditions experienced at the LHC.