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

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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) interaca e-mail: twyatt@fnal.gov (corresponding author) tions, mediated in the SM by the γ and Z bosons, the validity of LU for the three flavours of charged leptons (e, μ, τ ) 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 ( ) or branching fractions (B) of the Z boson are: which are consistent with LU to a precision of around three per mille [1]. Measurements of the leptonic asymmetry parameters A , from forward-backward asymmetries, the left-right asymmetry, and the tau polarisation and its asymmetry: A e = 0.1514 ± 0.0019, A μ = 0.1456 ± 0.0091, 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 ± 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]: is consistent with e-μ universality to a precision of around two percent. However a hint at the possible violation of LU in the CC couplings of the τ is present in the ratio [2]: where B(W → ν) is the average of the branching fractions for W → eν and W → μν. This result deviates from the assumption of τ -universality 1 by 2.6σ (standard deviations). At lower mass scales e-μ universality is tested very precisely, for example in leptonic τ decays the ratio of the muonic and electronic partial widths is measured to be [3]: which is consistent with the SM prediction including mass effects of 0.9726 [3]. e-μ universality is also tested in the decays of charged kaons [3]: which is consistent with the SM prediction [4] of 2.477 × 10 −5 .
In the decay of B hadrons, τ -universality can be tested by measurements of the branching fractions for the exclusive decays of B 0 → τ − ν τ D + and B 0 → τ − ν τ D * + expressed as ratios to the branching fractions for the exclusive decays B 0 → − ν D + and B 0 → − ν 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: These measured values exceed the SM predictions (calculated assuming LU) of: by 1.4σ and 2.5σ 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 3.1σ [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 τ than to e or μ could contribute at tree level to the decays B 0 → τ − ν τ D ( * )+ . For example, in [17] a charge 2/3 scalar LQ with bτ and cν 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 B 0 → τ − ν τ 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 τ . For example, if cs and sτ Yukawa couplings were added to the charge 2/3 LQ scenario of [17] it could produce an enhancement in B(W → τ ν). 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 τ lepton decays in which the visible final state is hadronic (τ 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 → τ had ν 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 τ had candidates.
Also of interest in this context are measurements at hadron colliders of the rate of events containing top quark-antiquark pairs (tt), 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 B(t → bτ ν) may be reinterpreted as the branching fraction B(W → τ ν).
Relative to measurements of the inclusive single W → τ had ν cross section, measurements using di-leptonic tt → bb τ had events eliminate systematic uncertainties associated with the trigger efficiencies for τ had candidates, because single-(e or μ) triggers can be used. In addition, non-tt backgrounds can be almost completely eliminated from the tt → bb τ had final state by employing b-jet tagging and the presence of the candidate. This means that the background from misidentified hadronic jets to the τ had signature in the tt → bb τ had final state is likely to be significantly smaller than that in the inclusive single W → τ had ν final state. However, systematic uncertainties associated with the τ had candidate (identification efficiency, background and energy scale) still contribute directly to the measured rate of tt → bb τ had events. For example, in a measurement by ATLAS [19] in dileptonic tt events the systematic uncertainty on the branching fraction B(t → bτ ν) is around 7.5%. In a measurement by CMS of the cross section for the tt → bb τ 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 τ had candidates was 7.8%. In the above-mentioned best currently published measurements from ATLAS [19] and CMS [20] based on the tt → bb τ had final state the systematic uncertainties associated with the τ 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 (tt → bbW + W − ) and Z /γ * → τ τ events at the LHC with a target precision of around 1%, which would improve significantly upon the LEP2 measurements. We define the ratio: where N (tt → bb τ had ) and N (tt → bbeμ) are the numbers of observed candidate events in the tt → bb τ had and tt → bbeμ final states, respectively. We define also the ratio: where N (Z → τ τ → τ had ) and N (Z → τ τ → eμ) are the numbers of observed candidate events in the Z → τ τ → τ had and Z → τ τ → eμ final states, respectively. We then define the double ratio: From an experimental perspective we note that the ratios R(bbW W ) and R(Z ) are designed to have approximately the same sensitivity to systematic uncertainties on the identification of e, μ and τ had candidates, and on the efficiencies of the single-triggers. Therefore, in the double ratio R(W Z) 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 (η) [21] of the leptons (e, μ and τ ) are significantly different in the tt → bbW + W − and Z /γ * → τ τ signal samples. Systematic uncertainties on lepton identification and single-trigger efficiencies are not necessarily fully correlated across all p T and |η| bins. -The levels of background from misidentified hadronic jets in the τ had candidates in the tt → bb τ had and Z → τ τ → τ had samples are not necessarily identical. -The level and nature of the non-tt background in the tt → bb τ had and tt → bbeμ samples will not necessarily be identical. Similarly, the level and nature of the non-Z boson background in the Z → τ τ → τ had and Z → τ τ → eμ samples will not necessarily be identical.
From a phenomenological perspective the double ratio R(W Z) 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(W Z) primarily through R(bbW W ), 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.

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 /γ * → τ τ candidates we propose a novel selection variable, m * 3 , that improves the discrimination power against the dominant backgrounds, such as tt, diboson production, as well as events containing a leptonically decaying vector boson plus a QCD jet that is misidentified as a τ 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 √ s = 13 TeV of L dt = 140 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(W Z) to the physical quantity of interest R(W ). In Sect. 2.6 we present the effect of systematic uncertainties on the ratios R(bbW W ), R(Z ), R(W Z), and R(W ). We thus demonstrate that in the double-ratio R(W Z) 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(W Z) 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.

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 → ee, μμ, τ τ 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(bbW W ) and R(Z ) are designed to have approximately the same sensitivity to systematic uncertainties on the identification of e, μ and τ had candidates, and on the efficiencies of the single-triggers. In the double ratio R(W Z), these systematic uncertainties cancel to first order. Therefore, in our study we give particular attention to the p T and η dependence of efficiencies and to any potential p T and η dependence in the associated system-  Fig. 1 The overall efficiencies assumed for the reconstruction, identification, and isolation criteria for prompt e, μ, and τ had candidates as a function of p T atic uncertainties. In general we expect algorithms that are designed to have p T -and η-independent efficiencies to have smaller p T -and η-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, μ, and τ 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 /ηindependent and p T /η-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, μ, and τ had candidates.
Jets with |η| < 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 |η|.
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.

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 /γ * → τ τ samples arise from EW diboson and tt production. Cross sections at √ s = 13 TeV have been measured for EW diboson [59] and tt [60] production. Our estimates of the fractional composition of the selected Z /γ * → τ τ samples are not sensitive to systematic uncertainties on the integrated luminosity or on the predicted absolute cross sections for Z boson, tt, or EW diboson production. They are, however, sensitive to systematic uncertainties on the ratio of the cross sections for tt 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 /γ * → τ τ events the momentum distribution of electrons and muons produced in τ decay is softer than that for the visible τ 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 → τ τ → τ had and Z → τ τ → eμ candidate events. Measurements of p T (Z ) have been made at √ s = 8 TeV by ATLAS [62]. In order to evaluate systematic uncertainties arising from p T (Z ) we increase the weight of events satisfying 50 GeV < p T (Z ) < 150 GeV by 0.5% and the weight of events satisfying p T (Z ) > 150 GeV by 1.0% [63].
In tt → bb τ 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 The p T -dependent systematic uncertainties on the efficiencies we assume for the reconstruction, identification, and isolation criteria for prompt e, μ, and τ had candidates for the visible τ had systems. The relative acceptance for tt → bb τ had and tt → bbeμ final states may be sensitive to the details of the modelling of tt production. We investigate this sensitivity by using an alternative tt 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 tt sys-

Ratio of diboson and Z boson cross sections 3
Ratio of tt and Z boson cross sections 3 tt modelling Tested with alternative sample tem, m(tt), by increasing the weights of events by an amount that varies linearly between 1% at m(tt) = 500 GeV to 10% at m(tt) = 1000 GeV [65]. Table 2 gives a summary of the sources of phenomenological systematic uncertainty considered in this study.

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 τ had and hadronic jets are considered if after simulation of resolution and momentum scale they satisfy p T > 25 GeV. Hadronic jets must satisfy |η| < 4.5. Candidate electrons, muons, τ had , and b-tagged jets must satisfy |η| < 2.5.

Candidate event selection criteria on leptons
In order to maximise the cancellation of systematic uncertainties between tt → bbW + W − and Z /γ * → τ τ event samples the same candidate event selection criteria on leptons are applied in the two event classes. Candidate eτ had events are required to contain: -Exactly one e candidate. -Exactly one τ had candidate of opposite sign to the e candidate. -No μ candidates.
-The e candidate must fire the single-e trigger.
Candidate μτ had events are required to contain: -Exactly one μ candidate. -Exactly one τ had candidate of opposite sign to the μ candidate.
-No e candidates.
-The μ candidate must fire the single-μ trigger.
Candidate eμ events are required to contain: -Exactly one e candidate.
-Exactly one μ candidate of opposite sign to the e candidate. -The e candidate must fire the single-e trigger, and/or the μ candidate must fire the single-μ trigger.

tt candidate event selection criteria
In addition to the relevant criteria on leptons given in Sect. 2.3.1 above, all candidate tt → bbW + W − events (tt → bb τ had as well as tt → bbeμ) are required to contain exactly two b-tagged jets. Because the same criterion is applied in the selection of the numerator tt → bb τ had and denominator tt → bbeμ events, we expect the ratio R(bbW W ) to be largely insensitive to systematic uncertainties associated with jet reconstruction, JES, JER, and btagging. Requiring two b-tagged jets reduces the background in the selected tt → bbW + W − event samples from non-tt sources; it also reduces the background from tt events in which a b-quark jet is misidentified as a prompt e, μ, or τ had . Candidate tt → bb τ had events are rejected if the invariant mass of the τ had candidate and the highest p T non-btagged jet, m( j − τ had ), satisfies 50 GeV < m( j − τ had ) < 90 GeV. This criterion reduces the background from lep-ton+jet tt events in which the hadronically decaying W boson produces two reconstructed jets, one of which is misidentified as a τ had candidate. The effectiveness of this criterion is illustrated by Fig. 3, which shows in the tt → bb τ had candidate event sample the distribution of m( j − τ had ), having applied all other tt → bb τ had event selection criteria. The upper plot shows events in which the τ had candidate originates from a genuine τ decay and the lower plot shows events in which the τ 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 τ had candidate sample. This will be useful in reducing the systematic uncertainty on the background yield.  The distribution of m( j − τ had ) in the tt → bb τ had candidate event sample. The event selection requirement on this quantity has not been applied. The upper plot shows events in which the τ had candidate originates from a genuine τ decay and the lower plot shows events in which the τ had candidate originates from a misidentified hadronic jet

Z /γ * → τ τ candidate event selection criteria
In addition to the relevant criteria on leptons given in Sect. 2.3.1 above, all candidate Z /γ * → τ τ events (Z → τ τ → τ had as well as Z → τ τ → eμ) are required to satisfy the following criteria: The relevant variables are defined below. Distributions of each variable having applied all other selection criteria are shown in Fig. 4 for Z → τ τ → τ had and Z → τ τ → eμ.
As expected, the rejected event samples containing btagged jets are dominated by tt. Because the same criterion on b-tagged jets is applied in the selection of the numerator Z → τ τ → τ had and denominator Z → τ τ → eμ 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 /γ * → τ τ event samples from tt and also W boson plus heavy flavour production in which a b-quark jet is misidentified as a prompt e, μ, or τ 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 → τ τ → τ had sample) and from tt and diboson production (in the Z → τ τ → eμ sample).
We propose here a novel selection variable, m * 3 : Here / E T is the missing transverse momentum and the transverse mass, m T ( , τ had , / E T ), of the 3-body system of , τ had , and / E T may be defined by: θ * η 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: where η − and η + are the pseudorapidities of the negatively and positively charge lepton, respectively. The division by sin θ * η 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 τ + τ − mass than is m T ( , τ had , / E T ). In the selection of Z /γ * → τ τ 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.
The variable cos φ [67]: discriminates against background events containing leptonically decaying W bosons. Here φ( , / E T ) is the azimuthal angle between the and the / E T and φ(τ had , / E T ) is the azimuthal angle between the τ 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 τ + τ − 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 Events  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 |η| of e, μ, and τ had , in the selected tt → bbW + W − (red) and Z /γ * → τ τ (blue) candidate event samples. It can be seen that the p T distributions are considerably softer for the Z /γ * → τ τ candidate event samples than those for the selected tt → bbW + 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 tt and EW diboson events. However, a cut on a T suppresses also Z /γ * → τ τ events that contain initial state parton radiation. Initial state radiation broadens the distributions of lepton candidate p T in Z /γ * → τ τ events. A hard cut on a T would therefore suppress the highp T regions in the distributions of lepton p T in the selected Z /γ * → τ τ event samples, as is illustrated in Fig. 7. The cut a T < 60 GeV is chosen to reject background events from tt 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.

Size and composition of the selected event samples
The expected numbers of selected events and the composition of the four selected event samples for L dt = 140 fb −1 at √ s = 13 TeV are given in Table 3. As a result of this study we estimate that the fractional statistical uncertainty on R(W Z) for a single LHC experiment for an integrated luminosity of L dt = 140 fb −1 would be around 0.5%.
The selection requirements for tt → bb τ had and tt → bbeμ can be seen to be very effective at removing non-tt sources of background, e.g., W +jet. The most significant source of non-tt events originates from the EW production of "single top" events in the associated production t W b 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 "t W b 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 tt pair (tt 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 tt categories in Table 3.
The most significant source of background for tt → bb τ had originates from genuine tt events in which the τ 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 → bb τ had events. The tt → bbeμ candidate event sample will be selected with entirely negligible levels of background.
The most significant source of background for Z → τ τ → τ 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 → τ τ → τ had from tt (0.3%) and W +jet (0.2%) are small. The most significant sources of background for Z → τ τ → eμ originate from tt (5.8%) and diboson (6.3%) production.

Sensitivity of R(W Z) to R(W )
For definiteness we make the assumption in our study that the effective W → τ ν 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 Table 3 The expected numbers of events in the four selected event samples for L dt = 140 fb −1 at √ s = 13 TeV. The numbers are broken down by physics production process. The numbers for tt, 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 tt V processes that contain two correctly identified leptonic W boson decays are included in the tt "true' category. Otherwise, the events from tt V processes are included in the tt "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

Process
Selected event sample then all other W boson branching fractions will be modified by a factor In our MC study we perform a "calibration" of the doubleratio method by reweighting every simulated event containing one or more W boson decays by a factor X n F m , where n is the number of generator-level W → τ ν 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(bbW W ) that contain pairs of leptonically decaying W bosons with correctly identified decay products in the "numerator" tt → bb τ had and "denominator" tt → bbeμ samples. This includes, for example, the presence of events containing the cascade decay W → τ ν → νν, whose presence in the "denominator" tt → bbeμ sample slightly decreases the correlation between R(bbW W ) and R(W ). The value of R(Z ) is designed to be independent of any change in R(W ). However, the backgrounds from tt → bbW + W − and W W in the Z → τ τ → τ had and Z → τ τ → eμ 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(bbW W ), R(Z ), and R(W Z) as a function of the fractional change in R(W ). The absolute value of the ratio of R(W ) and R(W Z) 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.

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(bbW W ),

R(Z ), R(W Z), and R(W ).
The p T /η-independent and p T /η-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 /η-independent systematic uncertainties on R(bbW W ) and R(Z ) are large, but almost exactly equal.  Therefore, p T /η-independent systematic uncertainties on the reconstruction of leptons and jets almost perfectly cancel in the double ratio R(W Z). When considering p T /η-dependent systematic uncertainties the cancellation is no longer perfect. The most significant sources of p T /η-dependent systematic uncertainties are illustrated in Fig. 9. The 10% uncertainty on the 2.5% background from hadronic jets misidentified as τ had candidates in the tt → bb τ had sample leads to an uncertainty of 0.25% on R(bbW W ) and thus also on R(W ). The 5% uncertainty on the 5% background from MJ events in the Z → τ τ → τ had sample leads to an uncertainty of Systematic p T /η-independent variation p T /η-dependent variation 25% on R(Z ) and thus also on R(W ). Since the above two backgrounds both result from hadronic jets misidentified as τ had candidates it is conceivable that the resultant systematic uncertainties on R(bbW W ) 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(bbW W ), R(Z ), and R(W Z) are given in Table 5. The 3% uncertainty on the backgrounds from tt (5.8%) and diboson (6.3%) production in the Z → τ τ → eμ 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 tt sample with modified QCD factorisation and renormalisation scales leads to an uncertainty of 0.3% on R(bbW W ) and thus also on R(W ) [69]. The m(tt) reweighting leads to an uncertainty of 0.2% on R(bbW W ) and thus also on R(W ).
We add in quadrature the changes in the double ratio R(W Z) arising from the p T /η-independent and p T /ηdependent systematic variations on the reconstruction of lep-tons and jets, together with the other considered systematic uncertainties discussed above. The total resulting systematic uncertainty on R(W Z) is 1.3%.

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 [24-55], 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 L dt = 140 fb −1 at √ s = 13 TeV that is now available. It is, therefore, to be expected that the full run 2 data set will allow systematic uncertainties on lep-   Table 2 Source of systematic uncertainty Systematic uncertainty on measured ratios (%) ton 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 τ had candidates to be studied over the range 25 < p T < 100 GeV, without the need to use tt → bb τ had events and implicitly assume that B(W → τ ν) takes its SM value. This could be achieved, e.g., by the selection of a dedicated Z → τ τ → τ had event sample in which there is a high transverse momentum initial state radiation. The residual non-cancellation in R(W Z) of the p T /ηdependent τ had identification uncertainty, as shown in Table 4, arises primarily from the differences in the τ had p T distributions between the tt → bb τ had and Z → τ τ → τ had samples, as shown in Fig. 6. We have considered possible methods to reduce the impact of the different τ p T distributions. One simple approach would be to introduce an additional cut on τ had p T < 65 GeV; this removes the high p T tail in the tt → bb τ had sample, with negligible changes to the Z → τ τ → τ had sample. This has the effect of improving the cancellation in R(W Z) of the p T -dependent τ had identification uncertainty between the tt → bb τ had and Z → τ τ → τ 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 τ had identification uncertainty might be (a) to reweight the tau p T distribution in the tt → bb τ had sample to resemble that in the Z → τ τ → τ had sample, or (b) to perform measurements of R(W Z) in bins of τ 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(W Z) 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 tt and Z boson production and decay, and is beyond the scope of this paper.
Of course, if the measured value of R(W Z) 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 → bH + , where H + is a charged higgs boson, followed by H + → τ + ν would modify the effective t → bτ ν and t → b ν 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 → q H, H → τ τ will also increase the number of tau leptons in tt 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 τ 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 (W W and W Z) at the LHC provide alternative samples with which to make this novel measurement. Controlling systematic uncertainties on R(W ) in W W → ντ had ν events will be extremely challenging; we shall need extraordinarily good background rejection against fake τ candidates from misidentified jets in W → ν+ jet events. One may also consider Z W → τ had ν 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 + − momentum is back to back with the τ 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 → bb τ had , W W → ντ had ν, and Z W → τ had ν could increase the significance of the observation, and could help elucidate the underlying BSM origin of the effect.
In addition to the tt → bb τ had and Z → τ τ → τ had final states considered here, it may be possible to measure R(W ) using a similarly motivated ratio of tt → bb τ lept and Z → τ τ → τ lept final states, where τ lept corresponds to a leptonic decay of the τ , that may be identified using, for example, criteria based on the non-zero lifetime of the τ lepton. A combination of R(W ) measurements using τ had and τ lept signatures would benefit from the fact that the leading experimental uncertainties, arising from τ had and τ lept identification, would be largely uncorrelated.

Summary and conclusions
A measurement of R(W ) ≡ B(W → τ ν)/B(W → ν) ( = e or μ) 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 /γ * → τ τ events at the LHC.
We define R(bbW W ) in di-leptonic tt events to be the ratio of the numbers of τ had and eμ final states (equation 9). R(bbW W ) is sensitive to the value of R(W ), but also to systematic uncertainties on the reconstruction of e, μ, and τ leptons. Similarly, we define R(Z ) in Z /γ * → τ τ events to be the ratio of the numbers of τ had and eμ final states (equation 10). R(Z ) is similarly sensitive to systematic uncertainties on the reconstruction of e, μ, and τ leptons, but is insensitive to the value of R(W ). The double ratio R(W Z) ≡ R(bbW W )/R(Z ) cancels to first order sensitivity to systematic uncertainties on the reconstruction of e, μ, and τ 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(W Z) 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 [24-55], having chosen identification algorithms whose performance is suited to the needs of our analysis. For a data set of L dt = 140 fb −1 at √ s = 13 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, μ, and τ 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 η dependence of the reconstruction efficiencies of e, μ, and τ 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 L dt = 140 fb −1 at √ s = 13 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 5σ .
Z /γ * → τ τ events at the LHC from which the variable m * 3 arose. We are very grateful to our colleague Chris Parkes for his useful suggestions for the improvement of this paper.

Data Availability Statement
This manuscript has no associated data or the data will not be deposited. [Authors' comment: It presents a feasibility study using generator-level MC events with parameterised simulation of detector performance. All experimental data is taken from published sources.] Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecomm ons.org/licenses/by/4.0/. Funded by SCOAP 3 .

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.

A.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 |η|, 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 ≈ 30 GeV and p T ≈ 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 |η| 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 /η-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 |η| < 1.0 and the endcap region by |η| > 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 /η-independent relative systematic uncertainty of 1%. We generate an η-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±0.15% in the barrel region and 2.90 ± 0.15% in the endcap region, independent of p T [24]. We consider a p T /η-independent relative systematic uncertainty of 0.2% on the p T scale [24]. We generate η-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.

A.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 |η|, 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 |η| of around 96%. We consider a combined systematic uncertainty on the efficiency for electron reconstruction, identification, and isolation. We consider a p T /η-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 /ηindependent 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 σ p T given by We consider a p T /η-independent systematic uncertainty on the resolution by varying the constant term in the formula for We consider a p T /η-independent relative systematic uncertainty of 0.2% on the p T scale. We generate η-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.

A.3 Simulation of τ 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 τ candidates. A recent paper by the CMS Collaboration [37] describes the methods used to identify τ had candidates at √ s = 13TeV 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 L dt = 36 fb −1 . We choose to simulate the τ 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 × 10 −3 for p T ≈ 25 GeV decreasing to around 10 −4 for p T ≈ 100 GeV. The misidentification rate is approximately independent of |η| [38]. We simulate also the discriminants against electrons and muons described in [37]: the "Tight" discriminant against electrons has a τ had identification efficiency of 75% and a fake probability of 10 −3 ; the "Tight" discriminant against muons has a τ had identification efficiency of 99% and a fake probability of 1.4 × 10 −3 [39].
In our study it is particularly important to assign realistic systematic uncertainties on the identification efficiency for τ had candidates. A tag and probe analysis of Z → τ τ → τ had events in [37] results in a relative uncertainty on the identification efficiency of τ had candidates of 5% for τ had p T up to 60 GeV. Samples of tt events are used in [37] to cross check the efficiency for τ 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 B(W → τ ν) takes its SM value, because it relies on a comparison between the absolute numbers of tt → bb τ had candidate events observed in the CMS data and predicted by the MC. Clearly, for our proposed test of LU using tt events it would not be legitimate to set data-MC scale factors for τ had using tt → bb τ had events in this way; this means that it will be difficult to control the p T dependence of the identification efficiency for τ had candidates beyond the range in τ had p T covered by the Z → τ τ → τ had event sample. We consider a p T /η-independent relative systematic uncertainty of 5%. We generate a p T -dependent systematic uncertainty on the identification efficiency of τ 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 τ had candidates is given by The relative uncertainties on the probabilities for electrons, muons, and hadronic jets to be misidentified as a τ had candidate are around 10% [41]. The uncertainty on the p T scale for τ had candidates is around 1% [37]. We consider an η-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 τ had candidates at √ s = 13 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 → τ τ → τ 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 τ 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 → τ τ → τ 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.

A.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 σ p T given by 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 |η| < 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 |η|. 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 "pileup" 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.  . https://doi.org/10.1088/1748-0221/10/06/P06005 31. For example, the region 1.37<|η|<1.52 is usually excluded for precision measurements involving electrons or τ had . We simulate this gap in this detector parameterisation 32. The efficiency for electron identification according for the "Tight likelihood" category is shown as functions of p T and η in Figure  8 of[28] 33. We simulate approximately the efficiency for the single-electron trigger according to that given as a function of p T in Figure 26 Table 3. The "Very tight likelihood" criterion in this paper has a very similar efficiency to "Tight likelihood" criterion in [28]. We, therefore, take the background probabilities given here for "Very tight likelihood" as being representative of those for the "Tight likelihood" in  Figure 7 (left) of [36] the fraction of MJ background in the signal region of visible mass in the selected τ had sample is seen to be around 20%, when using the "tight" operating point of the MVA-based τ had identification algorithm and applying the "tight" isolation requirement to the muon. It can be seen from Figure 4 (left) of [36] that the signal efficiency for τ had decreases by a factor of around 1.5 between the "tight" and "very-very-tight" operating points and from Figure 4 (right) of [36] that the background rejection improves by a factor of three. Therefore background to signal ratio improves by roughly a factor of two between the "tight" τ had operating points. Similarly, in section 6.2 of [24] it is stated that the probability for a muon produced in a hadronic jet to satisfy tight isolation requirements is about 0.05 in the barrel, and goes up to about 0.15 in the endcap. This probability is around a factor of two larger than the probability of 0.03 that we have assumed for the probability for a muon produced in a hadronic jet. Taking these two factors of two into account we can, therefore, estimate the fraction of MJ background in the Z → τ τ → τ had sample selected using the "very-very-tight" operating point of the MVA-based τ had identification algorithm and our chosen electron/muon isolation to be around 20/(2 × 2) = 5%. The relative uncertainty on the MJ background quoted in section 9.