Test of the universality of τ and μ lepton couplings in W-boson decays with the ATLAS detector

The standard model of particle physics encapsulates our best current understanding of physics at the smallest scales. A fundamental axiom of this theory is the universality of the couplings of the different generations of leptons to the electroweak gauge bosons. The measurement of the ratio of the decay rate of W bosons to τ leptons and muons, R(τ/μ), constitutes an important test of this axiom. Using 139 fb−1 of proton–proton collisions recorded with the ATLAS detector at a centre-of-mass energy of 13 TeV, we report a measurement of this quantity from di-leptonic tt¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\overline{t}$$\end{document} events where the top quarks decay into a W boson and a bottom quark. We can distinguish muons originating from W bosons and those originating from an intermediate τ lepton through the muon transverse impact parameter and differences in the muon transverse momentum spectra. The measured value of R(τ/μ) is 0.992 ± 0.013 [± 0.007(stat) ± 0.011(syst)] and is in agreement with the hypothesis of universal lepton couplings as postulated in the standard model. This is the only such measurement from the Large Hadron Collider, so far, and obtains twice the precision of previous measurements. The ATLAS Collaboration reports a measurement of the ratio of the decay rates of W bosons to τ leptons and muons, in agreement with universal lepton couplings as postulated in the standard model of particle physics.


Experimental set-up
The ATLAS experiment [26][27][28] at the LHC is a multipurpose particle detector with nearly hermetic coverage for recording particles produced in LHC collisions through a combination of particle position and energy measurements. ATLAS has a forward-backward symmetric cylindrical geometry and a near 4π coverage in solid angle. More details of the ATLAS coordinate system are provided in the Methods. ATLAS consists of an inner tracking detector surrounded by a thin superconducting solenoid providing an axial magnetic field of 2 T, electromagnetic and hadron calorimeters, and a muon spectrometer. The inner tracking detector covers the pseudorapidity range |η| < 2.5. It consists of silicon pixel, silicon microstrip, and transition radiation tracking detectors, and the innermost layer of the pixel detector is at a radius of 33 mm from the beamline, providing a precise measurement of track impact parameters. Lead/liquid-argon (LAr) sampling calorimeters provide electromagnetic energy measurements with high granularity. A steel/scintillator-tile hadron calorimeter covers the central pseudorapidity range (|η| < 1.7). The endcap and forward regions of the detector are instrumented with LAr calorimeters for both electromagnetic and hadronic energy measurements up to |η| = 4.9. The muon spectrometer surrounds the calorimeters and is based on three large air-core toroidal superconducting magnets with eight coils each. The field integral of the toroids ranges between 2.0 and 6.0 T m across most of the detector. The muon spectrometer includes a system of precision tracking chambers and fast detectors for triggering. A two-level trigger system is used to select events. The first-level trigger is implemented in hardware and uses a subset of the detector information to accept events at a rate of 100 kHz. This is followed by a software-based trigger that reduces

Test of the universality of τ and μ lepton couplings in W-boson decays with the ATLAS detector ATLAS Collaboration*
The standard model of particle physics encapsulates our best current understanding of physics at the smallest scales. A fundamental axiom of this theory is the universality of the couplings of the different generations of leptons to the electroweak gauge bosons. The measurement of the ratio of the decay rate of W bosons to τ leptons and muons, R(τ/μ), constitutes an important test of this axiom. Using 139 fb −1 of proton-proton collisions recorded with the ATLAS detector at a centre-of-mass energy of 13 TeV, we report a measurement of this quantity from di-leptonic tt events where the top quarks decay into a W boson and a bottom quark. We can distinguish muons originating from W bosons and those originating from an intermediate τ lepton through the muon transverse impact parameter and differences in the muon transverse momentum spectra. The measured value of R(τ/μ) is 0.992 ± 0.013 [± 0.007(stat) ± 0.011(syst)] and is in agreement with the hypothesis of universal lepton couplings as postulated in the standard model. This is the only such measurement from the Large Hadron Collider, so far, and obtains twice the precision of previous measurements.
the accepted event rate to 1 kHz, on average, depending on the data-taking conditions.
The analysed pp collision data were recorded with the ATLAS detector from 2015 to 2018 at a centre-of-mass energy of √ s = 13 TeV. Owing to the conditions required to achieve high instantaneous luminosity, in these data there are additional collisions in the same and neighbouring LHC proton bunch crossings (pile-up). This resulted in an average of 34 collisions per bunch crossing.
Events in this measurement were selected by single-lepton triggers 29-31 requiring a single high-p T isolated electron or muon. After the application of data-quality requirements 32 , the data sample corresponds to an integrated luminosity of 139 fb −1 with an uncertainty of 1.7% 33 , obtained using the LUCID-2 detector 34 for the primary luminosity measurements.
Samples of simulated events were produced using Monte Carlo techniques to model the different SM processes. After event generation of the process of interest for each sample, the detector response was modelled using a simulation based on GEANT4 35 . The data and Monte Carlo simulated events were passed through the same reconstruction and analysis procedures. Samples were simulated for the signal processes, the production of tt and single top quarks in association with a W boson (Wt), as well as the different backgrounds. Precise details of the theoretical predictions and event generators used to create the simulated events are provided in the Methods.

Selecting a pure and unbiased sample of W bosons
Collisions are selected to obtain a high-purity sample of tt events in which the top quarks decay into a W boson and a b quark, and the two W bosons then decay to leptons. This is referred to as di-leptonic tt. The two leptonic W-boson decays are exploited in a tag and probe approach: in each event, tag leptons (electrons or muons) are used to select the event, after which a test is performed to determine, in an unbiased way, whether the probe muon was directly produced by a W boson, W → μν μ , or via an intermediate τ lepton, W → τν τ → μν μ ν τ ν τ . Events are categorized into signal regions, used to extract R(τ/μ), and additional control regions, used to constrain the normalization of the major backgrounds.
The selections rely on reconstructed muons, electrons and hadronic jets. Details of the physics object reconstruction definitions used are provided in the Methods. Events entering the signal region are required to contain either one electron and one muon of opposite electric charge (e-μ channel) or two muons of opposite electric charge (μ-μ channel). In the e-μ channel, the electron is required to pass the trigger, and in the μ-μ channel, the tag muon is required to pass the trigger. This ensures that the probe muons have no trigger bias. If both leptons in the μ-μ channel satisfy the tag and probe criteria, both muons in turn are used as probes. Events with more than two leptons are rejected. In addition, events must have at least two reconstructed hadronic jets that are identified as containing a b hadron. Finally, to reduce the backgrounds from Z bosons and hadron decays, events where the invariant mass of the two muons, m μμ , satisfies 85 < m μμ < 95 GeV are excluded in the μ-μ channel, and events with di-lepton mass m ℓℓ < 15 GeV are excluded in both channels.
This selection results in a sample of approximately half a million collision events, with a di-leptonic tt purity of over 95% in the e-μ channel and 85% in the μ-μ channel. This sample is used to test the origin of the probe muons and extract the measurement of R(τ/μ). This is extracted from the ratio of the number of events in which the probe muon orginates from the process W → τν τ → μν μ ν τ ν τ , referred to as μ τ(→μ) , to those that come from the process W → μν μ , referred to as μ prompt . A fit is performed that exploits the difference in shape between these two components and the backgrounds of the distributions of the probe muon's transverse impact parameter |d μ 0 | and its transverse momentum p μ T .

Muon transverse impact parameter distribution calibration
The muon's transverse impact parameter |d μ 0 | has particular importance for this analysis and requires careful treatment. It is measured in the x-y plane as the closest distance of approach of the track to the beamline. The shape of the |d μ 0 | distribution of prompt muons is determined using a Z → μμ calibration region to create templates that are used to predict the distribution in the signal region.
The calibration region is defined by requiring that two muons satisfying the same kinematic criteria as in the signal region, but with the di-muon mass requirement changed to 85 < m μμ < 100 GeV. No requirements on hadronic jets are applied. This gives a sample of ~95 million prompt muons with a purity of >99.9%.
Templates of the shape of the |d μ 0 | distribution are then taken from this data sample after subtracting the expected contributions from the simulation of processes with significant muon parent lifetimes, primarily Z → ττ. These |d μ 0 | templates are extracted in 33 bins in p μ T and |η μ | to capture the dependence of the distribution on these variables. Separate templates are used for 2015 + 2016, 2017 and 2018 data to account for differences in the beam conditions and in the alignment of the inner detector. Additionally, using this calibration region, the Gaussian part of the |d μ 0 | resolution is estimated in data and simulation by fitting the |d μ 0 | distribution in the range |d μ 0 | < 0.02 mm. For p μ T = 20 GeV, the resolution is ~14 μm. Corrections to account for differences in the resolution of the detector between the data and simulation are applied to the muons from τ decays and hadron decays. For the range of |d μ 0 | values considered in this analysis, |d μ 0 | < 0.5 mm, the resolution measured from prompt muons is applicable to those with significant displacement.

Background normalization estimation
The two largest backgrounds are Z(→μμ) + jets and events in which the probe muon does not originate from a W-boson decay. Three dedicated control regions are used to extract the normalization of these backgrounds.
The Z(→μμ) + jets background is important at small values of |d μ 0 |. The normalization of the Z(→μμ) + jets background in the μ-μ channel is extracted from the data in a control region where the same event selection is applied, including the hadronic jet requirements, but without the m μμ criterion, and is then extrapolated to the signal region using simulation. In the control region, the peak of the invariant mass distribution of the di-muon system is fitted over the range 50 < m μμ < 140 GeV. A Voigt profile 36 is used for the Z → μμ resonance and a third-order Chebychev polynomial for all non-resonant processes, which provides a good description of the data. Other functions were tested to provide a systematic uncertainty, which is combined with the statistical uncertainties. The normalization factor required to scale the simulated sample to data is found to be 1.36 ± 0.01. The di-muon invariant mass in the control region is shown in Fig. 1 after this normalization is applied.
This normalization factor is also applied to the small Z(→ττ) + jets background.
The most important background at large values of |d μ 0 | is from events in which the probe muon originates from the decay of b or c hadrons, or more rarely from in-flight decays of π ± and K ± . This occurs primarily in tt events where one W boson decays leptonically and the other hadronically, referred to as semi-leptonic tt. These muons are referred to as μ had . A data-driven method is used to determine the normalization of this background from two control regions, one each for the e-μ and μ-μ channels. The control regions have the same event selection as the signal regions, but the two leptons are required to have same-sign electric charge. This results in a sample with a high purity of this μ had background.The largest source of μ had is from decays of b hadrons, and this contributes equally to same-sign and opposite-sign selections, while the other substantial source, c hadrons, has a component in both selections, but they are not equal. The extrapolation from same-sign control region to opposite-sign signal region is estimated from simulation. In the same-sign control region there are two backgrounds to μ had at high p μ T : tt + V, and tt, which occurs through electron charge misidentification in the e-μ channel. A normalization correction factor is applied to these processes based on the number of events observed with a probe muon with p μ T > 30 GeV. This is done before extracting the normalization of the μ had background. The normalization factors to scale the simulation to data for the μ had background are found to be 1.39 and 1.37 in the e-μ and μ-μ, channels, respectively. Figure 2 shows that the simulation and data are consistent within uncertainties in the μ-μ channel same-sign control region, providing confidence that the differential distributions of p μ T and |d μ 0 | are well-modelled.   A region enriched in μ had events is defined by selecting events with two same-sign muons. This control region is used to extract normalization factors to correct the μ had prediction to match the data. The probe muon tranverse impact parameter (left, |d μ 0 |) and transverse momentum (right, p μ T ) distributions in this region are shown. The data are shown by black markers and the different components contributing to this region, taken from simulation, are shown by stacked histograms. The different contributions are the primary process of interest, μ had (μ (hadron decay)), along with the main backgrounds from events involving a top quark pair produced in association with a W or Z boson (tt + V processes), a pair of W and/or Z bosons (Di-boson processes) and the grouping of all remaining SM processes (Other SM processes). The y-axis label includes the value that defines the scaling of the variable bin width histogram. The extracted normalization factor is applied to the μ had prediction, along with the effect of any constraints and pulls on the systematic uncertainties from the fit to the signal region data. The bottom panels show the ratio of the data to the predicted expectation after the fit. The uncertainties on the data are the Poisson errors due to the limited size of the data sample. Blue bands indicate the ±1σ systematic uncertainties on the prediction with the constraints from the analysis fit applied. The open blue arrowheads in the ratio panel indicate points where the ratio values lie outside the y-axis range shown. The signal region used to extract R(τ/μ) is enriched in di-lepton tt events. The |d μ 0 | distributions for each signal region (left, e-μ channel; right, μ-μ channel) and probe muon p μ T bin (top, 5 < p μ T < 10 GeV; middle, 10 < p μ T < 20 GeV; bottom, 20 < p μ T < 250 GeV) used in the analysis are shown. The data are represented by black markers and the different components contributing to this region, taken from simulation, are shown by stacked histograms. The different contributions are the two primary processes of interest used to extract R(τ/μ), μ prompt from top quark decays (Prompt μ (top)) and μ τ(→μ) from top decays (τ → μ(top)). The main backgrounds are also shown: events with a μ had (μ (hadron decay)), events with a Z boson decaying to a di-muon pair (Z → μμ), events with a Z boson decaying to a di-τ pair (Z → ττ) and the grouping of all remaining SM processes (Other SM processes). Distributions are shown after the fit has been performed. The y-axis label includes the value that defines the scaling of the variable bin width histogram. The bottom panels show the ratio of the data to the predicted expectation after the fit. The uncertainties on the data are the Poisson errors due to the limited size of the data sample. Blue bands indicate the ±1σ systematic uncertainties on the prediction with the constraints from the analysis fit applied. The contribution from 'Other SM processes' is dominated by di-boson and tt + V production. The chi-squared statistic values range from 3.5 to 10.2 for eight degrees of freedom for the distributions.

Fit configuration and systematic uncertainties
A two-dimensional profile likelihood fit 37 is performed in the |d μ 0 | and p μ T distributions. The bin boundaries were chosen to provide the best separation between the different μ prompt , μ τ(→μ) and μ had processes given the available data. This resulted in three bins in p μ T (boundaries of 5, 10, 20 and 250 GeV) and eight bins in the transverse impact parameter |d μ 0 | (boundaries of 0, 0.01, 0.02, 0.03, 0.04, 0.06, 0.09, 0.15 and 0.5 mm), of the probe muon for both the e-μ and μ-μ channels, making 48 bins in total.
The ratio of the number of μ τ(→μ) events to the number μ prompt is fitted by minimizing the negative log-likelihood, to measure R(τ/μ). More than 100 (nuisance) parameter values representing the statistical and systematic uncertainties, which can be modified by the fit, are included. The relative uncertainty of 0.23% in the τ → μν μ ν τ branching ratio is also included in the measured value of R(τ/μ) and is a subdominant component of the overall uncertainty. As both the tt and Wt processes contain two W bosons, both are treated as signal. Two fit parameters are allowed to float freely: R(τ/μ) and k(tt). The k(tt) parameter is a constant scaling factor applied to the normalization of both the μ prompt and μ τ(→μ) components of the signal, whereas R(τ/μ) only affects the μ τ(→μ) components. Both are applied across all bins and in both channels. R(τ/μ) is the parameter of interest and is not affected by the overall normalization scaling factors of the tt and Wt processes. The fit is performed after applying the normalization scaling factors derived in the control regions. Other processes are normalized to their theoretical cross-sections, taking into account the uncertainty in these predictions.
Because many systematic uncertainties are correlated between the μ prompt and μ τ(→μ) templates, they cancel out in the R(τ/μ) ratio, minimizing their impact on the precision of the result. These include uncertainties related to jet reconstruction, flavour tagging and trigger efficiencies. The remaining dominant uncertainties are the uncertainties in the data-driven methods, the theoretical modelling uncertainties and the reconstruction uncertainties; these are described in the following and further details are provided in the Methods.
An uncertainty associated with the data-driven templates for the |d μ 0 | distribution of μ prompt is derived to account for the differences between the Z → μμ calibration region where they were derived and the signal region.
Uncertainties in the data-driven normalization of the μ had background due to the size of the same-sign dataset, the choice of Monte Carlo generators used and the uncertainty in the subtraction of the other processes in the same-sign control region are included in the fit. The uncertainties associated with the Z + jets normalization derived from data are also applied. Uncertainties on the cross-section calculations and the integrated luminosity are applied to all other backgrounds estimated from simulation, but have a minor impact on the result. Uncertainties in the shape of the μ prompt , μ τ(→μ) and μ had distributions due to the modelling of the simulated tt samples are derived. The combined yield of the μ prompt and μ τ(→μ) templates is allowed to float in the fit, but changes in generator configuration choices can result in modifications to the muon p μ T and subsequently the |d μ 0 | distribution, such that there can be relative changes in the μ prompt and μ τ(→μ) template yields in each bin, leading to an uncertainty in R(τ/μ). For the μ had background, in addition to the uncertainties in the normalization, there can be changes to the muon p μ T modelling and the relative fractions of muons from different sources, both of which can change the shape of the |d μ 0 | distribution. The most important of these variations is in the parton shower and hadronization model.
The muon reconstruction and isolation efficiencies are determined in di-muon (Z → μμ and J/ψ → μμ) data and simulation using a tag and probe method 38 . Corrections are applied to the simulated samples to account for the differences between data and simulation, and the uncertainties on these correction factors are included in the analysis. Additionally, an uncertainty due to the modelling of pile-up is obtained by reweighting the simulated events 39 . Figure 3 shows the differential distributions of |d μ 0 | in the six signal regions for the data and the expectation after the fit to data. Good agreement is observed between the corrected simulation samples and the data. The global goodness of fit when fitting the expectation from simulation, defined as twice the change in negative  The main sources of uncertainty on the measured value of R(τ/μ) are shown. The size of the impact each uncertainty has on R(τ/μ) is assessed by fixing the relevant fit parameters for a given uncertainty and refitting to data. The reduction in the total uncertainty in this modified fit gives the quoted impact. Different individual components used in the fit are combined into categories.

Extraction of R(τ/μ)
log-likelihood relative to a fit performed assuming the pre-fit expectation per degree of freedom, has a value of 1.11 (P value of 0.29). The separation between the μ prompt and μ τ(→μ) processes can be seen clearly. The μ prompt processes dominate at low |d μ 0 |, whereas μ τ(→μ) dominates at high |d μ 0 |. The μ had background is also important at high |d μ 0 |, but contributes most significantly at low p μ T . The analysis was finalised before looking at the value of R(τ/μ) in the data to minimize any bias. It was also checked that the result is consistent with respect to different channels, kinematic bins, data-taking periods and the charge of the probe lepton.
The total systematic uncertainty is 0.011, including the uncertainty in the τ → μν μ ν τ branching ratio, and the statistical uncertainty is 0.007. Table 1 lists the different contributions of systematic uncertainty grouped into categories. The leading contributions come from the imperfect knowledge of the tail of the |d μ 0 | distribution, the parton shower and hadronization model uncertainty, and the muon selection uncertainties.
The measured value of R(τ/μ) is exceeding the precision from LEP which measured 1.070 ± 0.026. The result is shown in Fig. 4 alongside the combination of LEP measurements. The present result agrees with the SM expectation of equal couplings for different lepton flavours and the hypothesis of lepton-flavour universality. This result surpasses the precision of the previous LEP result and resolves the tension they observed with the SM prediction of lepton-flavour universality. This precise measurement of R(τ/μ) achieved so far, this is an example of the ability of the ATLAS experiment to perform high-precision measurements.

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ATLAS coordinate system and nomenclature.
ATLAS uses a right-handed coordinate system with its origin at the nominal interaction point (IP) in the centre of the detector and the z axis along the beam pipe. The x axis points from the IP to the centre of the LHC ring, and the y axis points upwards. Cylindrical coordinates (r, ϕ) are used in the transverse plane, ϕ being the azimuthal angle around the z axis. The pseudorapidity is defined in terms of the polar angle θ as η = −ln tan(θ/2). Angular distance is measured in units of ΔR ≡ √ (Δη) 2 + (Δϕ) 2 . The beamline is the central (x-y) axis of the luminous region. The transverse impact parameter, |d μ 0 |, is measured in the x-y plane as the closest distance of approach of the track to the beamline. It is defined relative to the beamline rather than the primary vertex so that the resolution of |d μ 0 | is independent of the vertex (x-y) resolution, which depends on the physics process.
Monte Carlo simulation and theoretical predictions. The top-pair and single-top-quark events, including Wt-, t-and s-channel production, were generated using the POWHEGBOX 40-44 generator interfaced to the PYTHIA8 45-47 parton shower and hadronization model (more details are available in ref. 48 ). The decays of bottom and charm hadrons are important for backgrounds in the analysis and were modelled using the EvtGen 49 program. The tt and single-top processes were normalized to the inclusive cross-section calculation of the highest available precision 50-58 and tt events additionally have a differential reweighting applied to match the next-to-next-to-leading order (NNLO) in quantum chromodynamics (QCD) top quark p T calculation 59 . For single-top quark production in the Wt channel, the diagram removal scheme 60 was used to remove overlap with tt production. The tt + W and tt + Z events were simulated with Sherpa and MadGraph5_aMC@NLO 61 interfaced with Pythia8, respectively. The background from V(= W, Z) + jets and VV + jets events was simulated with the Sherpa 44,62-71 generator (more details can be found in refs. 72,73 ). All processes were normalized to their highest-order available cross-sections 74,75 .
Simulated inelastic pp collisions 39 were overlaid on events in all samples to model the observed data distribution of pile-up from addition collisions in the same and neighbouring bunch crossings.
Object identification in ATLAS. Muons are reconstructed using combined fits of inner detector 76,77 and muon spectrometer tracks 38 . They are required to satisfy the 'medium' identification critera of ref. 38 . They are also required to be strictly isolated from other activity by requiring that the sum of the p T of other tracks within a surrounding cone of size ΔR = 0.3 and the sum of p T calculated from calorimeter energy deposits within a cone of size ΔR = 0.2 around the muons are below certain thresholds. Tag muons are required to have p μ T > 27.3 GeV to pass the trigger thresholds, while probe muons are required to have p μ T > 5 GeV. Both the tag and probe muons are required to have |η| < 2.5 and to originate close to the primary vertex with a distance of closest approach in the r-z plane of less than 0.3 mm and a transverse impact parameter relative to the beamline, |d μ 0 |, of less than 0.5 mm. The primary vertex is defined as the vertex with the highest Σp 2 T of the tracks associated with it. Additional criteria are applied to test the compatibility of the momenta measured separately in the inner detector and the muon spectrometer, to remove reconstructed muons that result from in-flight decays of charged kaons and pions.
Electrons are reconstructed from inner detector tracks matched to clusters of calorimeter-cell energy clusters 78 . They are required to satisfy the 'tight' identification criteria of ref. 78 and the same strict isolation criteria as applied to muons. Tag electrons are required to have p e T > 27 GeV to pass the trigger requirements and satisfy |η| < 2.47, excluding the transition region between the barrel and end-cap calorimeter, 1.37 < |η| < 1.52. They must also satisfy the same criteria as for muons for their distance of closest approach to the primary vertex in the transverse and r-z plane.
Hadronic jets are built from the energy in calorimeter cells at the electromagnetic energy scale 79 , using the anti-k t algorithm 80 with a radius parameter of 0.4. They are then calibrated to the energy scale of jets created from stable generator-level particles excluding muons and neutrinos, using the same jet algorithm 81 . For jets with 25 < p T < 60 GeV and |η| < 2.4, pile-up suppression requirements in the form of a jet vertex tagger 82 are applied. Only jets with p T > 25 GeV and |η| < 2.5 are considered in the analysis. To classify jets as containing a b hadron, the 70% efficiency working point of the MV2c10 b-tagging algorithm 83,84 is used.
An overlap removal procedure as described in ref. 85 is applied to resolve the ambiguity if lepton signals in the calorimeter are also reconstructed as hadronic jets.
Systematic uncertainties. Several of the most important systematic uncertainties in the measurement of R(τ/μ) merit a more detailed discussion, which is provided here.
Owing to differences in the hadronic environment around the lepton between the Z and tt final states and the coarse binning in p μ T and |η|, which may not be able to encapsulate the full shape information, small biases can exist in the data-driven μ prompt template distributions. The size of such a possible bias is estimated from the full difference between μ prompt |d μ 0 | templates from Z and tt in simulation. This uncertainty is split into two components corresponding to the tail, |d μ 0 | 0.05 mm, and core, d μ 0 | 0.05 mm, to prevent the data from constraining the uncertainty by using the full |d μ 0 | distribution. The breakdown of uncertainties contributing to the μ had background control region normalization in the e-μ (μ-μ) channels is as follows: 4% (4%) due to the size of the same-sign dataset; 8% (3%) due to the choice of Monte Carlo generators used; 1.0% (1.3%) due the uncertainty in the subtraction of the other processes in the same-sign control region.
The uncertainties due to the choice of Monte Carlo event generator for the μ prompt , μ τ(→μ) and μ had processes are estimated by varying different components of the modelling in a factorized way. The following variations are considered targeting different sources of uncertainty: • Inital-and final-state radiation: A14 eigen-tune variations 46 of the strong coupling (α s ) • Missing higher-order QCD corrections: factorization and renormalization scales simultaneously varied up and down by a factor of two • Resummation scale uncertainty: POWHEG h damp parameter varied from 1.5 to 3 m top (symmetrized) • Parton shower and hadronization model: Herwig 7 86,87 , H7UE tune 87,88 (symmetrized) • Higher-order correction to top p T spectrum: not applying the NNLO top p T reweighting 59 (symmetrized) In the cases where only a single alternative is given, the uncertainty is taken to be the deviation from the nominal result and then symmetrized, as indicated above. The effects on μ prompt and μ τ(→μ) (collectively referred to as 'signal') are treated as correlated and the effects on μ had are treated separately. The parton shower and hadronization uncertainty is separated into four nuisance parameters: one each corresponding to low and middle probe muon p μ T bins used in the fit, and two corresponding to the high p μ T bin where the uncertainty is further separated into components related to normalization and shape differences.

Data availability
The experimental data that support the findings of this study are available in HEPData with the identifier https://www.hepdata.net/record/100232.

Code availability
The ATLAS software is available at the following link: https://gitlab.cern.ch/ atlas/athena.