11institutetext: Particle Physics Group, Department of Physics and Astronomy,
University of Manchester, UK.

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

S. Dysch  T. R. Wyatt

(Received: 26/10/2019)

###### 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 $\int\\!\\!L\,\mbox{\,d}t$ = 140 fb-1 at $\sqrt{s}$ = 13 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 bib:LEP1 . For example, the ratios of the leptonic partial
widths ($\Gamma$) or branching fractions ($\mathcal{B}$) of the $Z$ boson are:

$\begin{array}[]{lllll}\displaystyle\frac{\Gamma_{\mu\mu}}{\Gamma_{ee}}&\equiv&\displaystyle\frac{\mathcal{B}(\mbox{$Z/\gamma^{*}$}\mbox{$\rightarrow$}\mbox{$\mu\mu$})}{\mathcal{B}(\mbox{$Z/\gamma^{*}$}\mbox{$\rightarrow$}\mbox{$ee$})}&=&1.0009\pm
0.0028,\\\ \\\
\displaystyle\frac{\Gamma_{\tau\tau}}{\Gamma_{ee}}&\equiv&\displaystyle\frac{\mathcal{B}(\mbox{$Z/\gamma^{*}$}\mbox{$\rightarrow$}\mbox{$\tau\tau$})}{\mathcal{B}(\mbox{$Z/\gamma^{*}$}\mbox{$\rightarrow$}\mbox{$ee$})}&=&1.0019\pm
0.0032,\end{array}$ (1)

which are consistent with LU to a precision of around three per mille bib:LEP1
. Measurements of the leptonic asymmetry parameters $\mathcal{A}_{\ell},$ from
forward-backward asymmetries, the left-right asymmetry, and the tau
polarisation and its asymmetry:

$\begin{array}[]{lll}\displaystyle\mathcal{A}_{e}&=&0.1514\pm 0.0019,\\\ \\\
\displaystyle\mathcal{A}_{\mu}&=&0.1456\pm 0.0091,\\\ \\\
\displaystyle\mathcal{A}_{\tau}&=&0.1449\pm 0.0040,\\\ \end{array}$ (2)

are also consistent with LU, albeit at the precision of a few percent bib:LEP1
.

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 bib:LEP2 :

$\frac{\mathcal{B}(\mbox{$W$}\mbox{$\rightarrow$}\mbox{$\mu\nu$})}{\mathcal{B}(\mbox{$W$}\mbox{$\rightarrow$}\mbox{$e\nu$})}=0.993\pm
0.019$ (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 bib:LEP2 :

$\mbox{$R(W)$}\equiv\frac{\mathcal{B}(\mbox{$W$}\mbox{$\rightarrow$}\mbox{$\tau\nu$})}{\mathcal{B}(\mbox{$W$}\mbox{$\rightarrow$}\mbox{$\ell\nu$})}=1.066\pm
0.025,$ (4)

where $\mathcal{B}(\mbox{$W$}\mbox{$\rightarrow$}\mbox{$\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
$\tau$-$\ell$ universality111In this paper the symbol $\ell$ is taken to
denote an electron ($e$) or a muon ($\mu$), but not a tau ($\tau$) lepton. 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 bib:pdg :

$\frac{\Gamma(\mbox{$\tau^{-}$}\mbox{$\rightarrow$}\mbox{$\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}$})}{\Gamma(\mbox{$\tau^{-}$}\mbox{$\rightarrow$}\mbox{$e^{-}\bar{\nu}_{e}\nu_{\tau}$})}=0.9762\pm
0.0028,$ (5)

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

$\frac{\Gamma(K^{-}\to
e^{-}\bar{\nu}_{e})}{\Gamma(K^{-}\to\mu^{-}\bar{\nu}_{\mu})}=(2.488\pm
0.009)\times 10^{-5},$ (6)

which is consistent with the SM prediction bib:kaon-decays  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 bib:hfag  of the results from the BaBar
bib:babar-btotau1 ; bib:babar-btotau2 , Belle bib:belle-btotau1 ; bib:belle-
btotau2 ; bib:belle-btotau3 ; bib:belle-btotau4 ; bib:belle-semilep-tagging ,
and LHCb bib:lhcb-btotau1 ; bib:lhcb-btotau2 ; bib:lhcb-btotau3  experiments
yields the results:

$\begin{array}[]{rllll}\displaystyle\mbox{$R(D)$}&\equiv&\displaystyle\frac{\mathcal{B}(\mbox{$\overline{B}^{0}$}\mbox{$\rightarrow$}\mbox{$\tau^{-}\overline{\nu}_{\tau}D^{+}$})}{\mathcal{B}(\mbox{$\overline{B}^{0}$}\mbox{$\rightarrow$}\mbox{$\ell^{-}\overline{\nu}_{\ell}D^{+}$})}&=&0.340\pm
0.027\pm 0.013,\\\ \\\
\displaystyle\mbox{$R(D^{*})$}&\equiv&\displaystyle\frac{\mathcal{B}(\mbox{$\overline{B}^{0}$}\mbox{$\rightarrow$}\mbox{$\tau^{-}\overline{\nu}_{\tau}D^{*+}$})}{\mathcal{B}(\mbox{$\overline{B}^{0}$}\mbox{$\rightarrow$}\mbox{$\ell^{-}\overline{\nu}_{\ell}D^{*+}$})}&=&0.295\pm
0.011\pm 0.008.\\\ \end{array}$ (7)

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

$\begin{array}[]{rll}\displaystyle\mbox{$R(D)$}&=&0.300\pm 0.008,\\\ \\\
\displaystyle\mbox{$R(D^{*})$}&=&0.252\pm 0.003,\\\ \end{array}$ (8)

by 1.4$\sigma$ and 2.5$\sigma$ respectively (see bib:hfag  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$\sigma$ bib:hfag .

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 bib:lq-btotau  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 bib:lq-btotau  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 bib:atlas-Wtau  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
$\mathcal{B}(\mbox{$t$}\mbox{$\rightarrow$}\mbox{$b\tau\nu$})$ may be
reinterpreted as the branching fraction
$\mathcal{B}(\mbox{$W$}\mbox{$\rightarrow$}\mbox{$\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 bib:atlas-top  in di-leptonic ${t}{t}$
events the systematic uncertainty on the branching fraction
$\mathcal{B}(\mbox{$t$}\mbox{$\rightarrow$}\mbox{$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 bib:cms-top , 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 bib:atlas-top  and CMS bib:cms-top
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:

$\mbox{$R(bbWW)$}\equiv\frac{N(\mbox{$tt$}\mbox{$\rightarrow$}\mbox{$bb\ell\tau_{\mathrm{\,had}}$})}{N(\mbox{$tt$}\mbox{$\rightarrow$}\mbox{$bbe\mu$})},$
(9)

where $N(\mbox{$tt$}\mbox{$\rightarrow$}\mbox{$bb\ell\tau_{\mathrm{\,had}}$})$
and $N(\mbox{$tt$}\mbox{$\rightarrow$}\mbox{$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:

$\mbox{$R(Z)$}\equiv\frac{N(\mbox{$Z$}\mbox{$\rightarrow$}\mbox{$\tau\tau$}\mbox{$\rightarrow$}\mbox{$\ell\tau_{\mathrm{\,had}}$})}{N(\mbox{$Z$}\mbox{$\rightarrow$}\mbox{$\tau\tau$}\mbox{$\rightarrow$}\mbox{$e\mu$})},$
(10)

where
$N(\mbox{$Z$}\mbox{$\rightarrow$}\mbox{$\tau\tau$}\mbox{$\rightarrow$}\mbox{$\ell\tau_{\mathrm{\,had}}$})$
and
$N(\mbox{$Z$}\mbox{$\rightarrow$}\mbox{$\tau\tau$}\mbox{$\rightarrow$}\mbox{$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:

$R(WZ)$ $\displaystyle\equiv$
$\displaystyle\frac{\mbox{$R(bbWW)$}}{\mbox{$R(Z)$}}$ (11)
$\displaystyle\equiv$
$\displaystyle\frac{N(\mbox{$tt$}\mbox{$\rightarrow$}\mbox{$bb\ell\tau_{\mathrm{\,had}}$})\times
N(\mbox{$Z$}\mbox{$\rightarrow$}\mbox{$\tau\tau$}\mbox{$\rightarrow$}\mbox{$e\mu$})}{N(\mbox{$tt$}\mbox{$\rightarrow$}\mbox{$bbe\mu$})\times
N(\mbox{$Z$}\mbox{$\rightarrow$}\mbox{$\tau\tau$}\mbox{$\rightarrow$}\mbox{$\ell\tau_{\mathrm{\,had}}$})}$

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 ($\eta$)
bib:coordinates  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 section 2 we describe a
Monte Carlo (MC) study of the proposed analysis method employing a simple
parameterised simulation of detector performance. In section 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 section 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 section 2.2 we describe the MC
generators used and the potential sources of phenomenological systematic
uncertainties that we have considered in our study. In section 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
section 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 TeV of $\int\\!\\!L\,\mbox{\,d}t$ =
140 fb-1, which corresponds approximately to that available for physics
analysis in ATLAS and CMS at the end of LHC run 2 bib:atlas-140 ; bib:cms-140
. In section 2.5 we evaluate the sensitivity of the measured double ratio
$R(WZ)$ to the physical quantity of interest $R(W)$. In section 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 sections 2.1 and 2.2. In section 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
bib:atlas-muonID –bib:pile-up . 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 section 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.

Figure 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 $p_{T}$.  Source of | Size of systematic uncertainty (in %, unless stated otherwise)
---|---
systematic uncertainty |  $p_{T}$/$\eta$-independent |  $p_{T}$/$\eta$-dependent
Muon ID & isolation efficiency | 2.0 |  $\displaystyle 0.5\left(\frac{80-p_{T}}{50}\right)$, 0 for $p_{T}>80\,\mathrm{GeV}$
Single-muon trigger efficiency | 1.0 | 1.0 in endcap, 0 in barrel
Muon $p_{T}$ resolution | 0.15 | 0.15 in endcap, 0 in barrel
Muon $p_{T}$ scale | 0.2 | 0.2 in endcap, 0 in barrel
Electron ID & isolation efficiency | 2.0 |  $\displaystyle 2.0\left(\frac{80-p_{T}}{50}\right)$, 0 for $p_{T}>80\,\mathrm{GeV}$
Single-electron trigger efficiency | 1.0 | 1 GeV change in $p_{T}$ threshold
Electron $p_{T}$ resolution | vary constant term in | vary resolution in endcap only
|  $\frac{\sigma_{p_{T}}}{p_{T}}$ by $\pm 0.002$ |
Electron $p_{T}$ scale | 0.2 | 0.2 in endcap, 0 in barrel
$\tau_{\mathrm{\,had}}$ efficiency | 5 |  $\displaystyle 5\left(\frac{130-p_{T}}{100}\right)$, 0 for $p_{T}>130\,\mathrm{GeV}$
$\tau_{\mathrm{\,had}}$ $p_{T}$ scale | 1 | 1 in endcap, 0 in barrel
jet energy scale | 1 | —
$b$-tag efficiency for $b$-, $c$-, light-jets | 1.5, 4, 10 | —
Misidentification rates for $\tau_{\mathrm{\,had}}$ | 10 | —
MJ background in $Z$ $\rightarrow$ $\tau\tau$ $\rightarrow$ $\ell\tau_{\mathrm{\,had}}$ sample | 5 | —
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.  Figure 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.

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 bib:powheg ,
interfaced to PYTHIAbib:pythia  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 bib:sherpa .

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 TeV have been measured for EW diboson bib:atlas-ww
and ${t}{t}$ bib:cms-tt  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 bib:ww-tt-z-ratios .

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 TeV by ATLAS bib:atlas-zptphistar . In order to evaluate
systematic uncertainties arising from $p_{T}(Z)$ we increase the weight of
events satisfying $50\ \mathrm{GeV}<\mbox{$p_{T}(Z)$}<150$ GeV by 0.5% and the
weight of events satisfying $\mbox{$p_{T}(Z)$}>150$ GeV by 1.0% bib:atlas-
zptsyst .

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 bib:atlas-top-diff-
cross-section .

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

Source of systematic uncertainty | Size of fractional systematic uncertainty (in %)
---|---
Ratio of diboson and $Z$ boson cross sections | 3
Ratio of ${t}{t}$ and $Z$ boson cross sections | 3
$p_{T}(Z)$ reweighting | 0.5 for $50<\mbox{$p_{T}(Z)$}<150$
| 1.0 for $\mbox{$p_{T}(Z)$}>150$
$m(t\bar{t})$ reweighting | $10+0.018(m(t\bar{t})-1000)$
${t}{t}$ modelling | Tested with alternative sample
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.

###  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 section 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$$-$$\mbox{$\tau_{\mathrm{\,had}}$})$,
satisfies 50 GeV $<$ $m(j$$-$$\mbox{$\tau_{\mathrm{\,had}}$})$ $<$ 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 Figure 3, which shows in the
$tt$ $\rightarrow$ $bb\ell\tau_{\mathrm{\,had}}$ candidate event sample the
distribution of $m(j$$-$$\mbox{$\tau_{\mathrm{\,had}}$})$, 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 Figure 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.

Figure 3:  The distribution of $m(j$$-$$\mbox{$\tau_{\mathrm{\,had}}$})$ 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 section 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 $<\mbox{$m^{*}_{3}$}<$ 100 GeV.

  * •

$\mbox{$\Sigma\cos\Delta\phi$}>$ $-0.1$.

  * •

$\and<60$ GeV.

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

Figure 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, $m^{*}_{3}$:

$\mbox{$m^{*}_{3}$}\equiv\frac{m_{T}(\ell,\mbox{$\tau_{\mathrm{\,had}}$},\mbox{$\not{E}_{T}$})}{\mbox{$\sin\theta^{*}_{\eta}$}}.$
(12)

Here $\not{E}_{T}$ is the missing transverse momentum and the transverse mass,
$m_{T}(\ell,\mbox{$\tau_{\mathrm{\,had}}$},\mbox{$\not{E}_{T}$})$, of the
3-body system of $\ell$, $\tau_{\mathrm{\,had}}$, and $\not{E}_{T}$ may be
defined by:

$m_{T}(\ell,\mbox{$\tau_{\mathrm{\,had}}$},\mbox{$\not{E}_{T}$})^{2}=\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$
$\\\ \qquad
m_{T}(\ell,\mbox{$\tau_{\mathrm{\,had}}$})^{2}+m_{T}(\ell,\mbox{$\not{E}_{T}$})^{2}+m_{T}(\mbox{$\tau_{\mathrm{\,had}}$},\mbox{$\not{E}_{T}$})^{2}$
(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 bib:sa-phistar  solely using the measured track directions by:

$\cos(\theta^{*}_{\eta})\equiv\tanh\left(\frac{\eta^{-}-\eta^{+}}{2}\right),$
(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 $m_{T}(\ell,\mbox{$\tau_{\mathrm{\,had}}$},\mbox{$\not{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 Figure 5.

Figure 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
$m(\ell,\mbox{$\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.

The variable $\Sigma\cos\Delta\phi$ bib:sumcos :

$\mbox{$\Sigma\cos\Delta\phi$}\equiv\cos(\Delta\phi(\ell,\mbox{$\not{E}_{T}$}))+\cos(\Delta\phi(\mbox{$\tau_{\mathrm{\,had}}$},\mbox{$\not{E}_{T}$})),$
(15)

discriminates against background events containing leptonically decaying $W$
bosons. Here $\Delta\phi(\ell,\mbox{$\not{E}_{T}$})$ is the azimuthal angle
between the $\ell$ and the $\not{E}_{T}$ and
$\Delta\phi(\mbox{$\tau_{\mathrm{\,had}}$},\mbox{$\not{E}_{T}$})$ is the
azimuthal angle between the $\tau_{\mathrm{\,had}}$ and the $\not{E}_{T}$.

The variable and bib:at  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 $a_{L}$ bib:at , the component of the
dilepton $p_{T}$ that is longitudinal to the dilepton thrust axis. The
variable and 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.

Figure 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.

As can be seen from the lower plots in Figures 4, a cut of $\and<30$ GeV would
be desirable to improve the suppression of backgrounds from ${t}{t}$ and EW
diboson events. However, a cut on and 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 and 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 Figure 7. The cut $\and<60$ 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 Figure 6.

Figure 7: Distributions of lepton $p_{T}$ in the selected $Z/\gamma^{*}$
$\rightarrow$ $\tau\tau$ candidate event samples. Three different cuts on and
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 $\int\\!\\!L\,\mbox{\,d}t$ = 140 fb-1 at $\sqrt{s}$
= 13 TeV are given in Table 3.

Process | Selected event sample
---|---
|  $tt$ $\rightarrow$ $bb\ell\tau_{\mathrm{\,had}}$ |  $tt$ $\rightarrow$ $bbe\mu$ |  $Z$ $\rightarrow$ $\tau\tau$ $\rightarrow$ $\ell\tau_{\mathrm{\,had}}$ |  $Z$ $\rightarrow$ $\tau\tau$ $\rightarrow$ $e\mu$
${t}\bar{t}$ $\rightarrow$ $b\bar{b}W^{+}W^{-}$ true | 178270 | 1092395 | 2034 | 6517
${t}\bar{t}$ $\rightarrow$ $b\bar{b}W^{+}W^{-}$ fake | 4761 | 0 | 20 | 156
$tWb$ true | 3236 | 18209 | 0 | 725
$tWb$ fake | 74 | 0 | 1 | 8
Other single top | 1731 | 0 | 2 | 42
$Z/\gamma^{*}$ $\rightarrow$ $\tau\tau$ true | 3 | 256 | 657280 | 101074
$Z$ $\rightarrow$ $ee,\mu\mu,\tau\tau$ fake | 0 | 0 | 1 | 14
$W$+jets | 61 | 0 | 1226 | 0
$WW$ true | 27 | 166 | 1555 | 6580
Other diboson | 5 | 25 | 337 | 773
Total background | 6663 | 447 | 5176 | 14815
Table 3: The expected numbers of events in the four selected event samples for
$\int\\!\\!L\,\mbox{\,d}t$ = 140 fb-1 at $\sqrt{s}$ = 13 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 $\int\\!\\!L\,\mbox{\,d}t$ = 140 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 bib:cms-taumj . 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 $R(WZ)$ to $R(W)$

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 $\mathcal{B}(\mbox{$W$}\mbox{$\rightarrow$}\mbox{$\tau\nu$})$
is multiplied by a factor $X$ relative to its SM-predicted value
$\mathcal{B}(\mbox{$W$}\mbox{$\rightarrow$}\mbox{$\tau\nu$})_{\mathrm{SM}}$,

$\mathcal{B}(\mbox{$W$}\mbox{$\rightarrow$}\mbox{$\tau\nu$})=X.\mathcal{B}(\mbox{$W$}\mbox{$\rightarrow$}\mbox{$\tau\nu$})_{\mathrm{SM}}$
(16)

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

$F=\frac{1-X.\mathcal{B}(\mbox{$W$}\mbox{$\rightarrow$}\mbox{$\tau\nu$})_{\mathrm{SM}}}{1-\mathcal{B}(\mbox{$W$}\mbox{$\rightarrow$}\mbox{$\tau\nu$})_{\mathrm{SM}}}.$
(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 $X^{n}F^{m}$, 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
Figure 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.

Figure 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 sections 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 $R(W)$.

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 Figure 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
$R(W)$.

Source of | Systematic uncertainty on measured ratios (%)
---|---
systematic |  $p_{T}$/$\eta$-independent variation |  $p_{T}$/$\eta$-dependent variation
uncertainty | $R(bbWW)$ |  $R(Z)$ |  $R(WZ)$ |  $R(bbWW)$ |  $R(Z)$ |  $R(WZ)$
Tau ID | $5.0$ | $5.0$ | $<0.1$ | $3.9$ | $4.7$ | $0.8$
Electron ID | $1.1$ | $1.3$ | $0.2$ | $0.5$ | $1.2$ | $0.7$
Electron trigger | $0.3$ | $0.2$ | $0.1$ | $0.1$ | $0.3$ | $0.2$
Muon trigger | $0.4$ | $0.5$ | $0.1$ | $0.4$ | $0.5$ | $0.1$
Muon ID | $1.0$ | $0.7$ | $0.3$ | $0.1$ | $0.2$ | $<0.1$
b-jet ID | $<0.1$ | $<0.1$ | $<0.1$ | - | - | -
Light jet mis-ID | $0.1$ | $0.1$ | $<0.1$ | - | - | -
Tau $p_{T}$ scale | $<0.1$ | $<0.1$ | $<0.1$ | $<0.1$ | $<0.1$ | $<0.1$
Electron $p_{T}$ scale | $<0.1$ | $<0.1$ | $<0.1$ | $<0.1$ | $<0.1$ | $<0.1$
Muon $p_{T}$ scale | $<0.1$ | $<0.1$ | $<0.1$ | $<0.1$ | $<0.1$ | $<0.1$
Jet energy scale | $<0.1$ | $<0.1$ | $<0.1$ | - | - | -
Electron $p_{T}$ resolution | $<0.1$ | $<0.1$ | $<0.1$ | $<0.1$ | $<0.1$ | $<0.1$
Muon $p_{T}$ resolution | $<0.1$ | $<0.1$ | $<0.1$ | $<0.1$ | $<0.1$ | $<0.1$
Fake $\tau_{\mathrm{\,had}}$ background |  |  |  |  |  |
($tt$ $\rightarrow$ $bb\ell\tau_{\mathrm{\,had}}$ sample) | 0.25 | 0 | 0.25 | - | - | -
MJ background |  |  |  |  |  |
($Z$ $\rightarrow$ $\tau\tau$ $\rightarrow$ $\ell\tau_{\mathrm{\,had}}$ sample) | 0 | 0.25 | 0.25 | - | - | -
Table 4: Changes in the ratios $R(bbWW)$, $R(Z)$, and $R(WZ)$ resulting from
the sources of experimental systematic uncertainty described in Section 2.1
and summarised in Table 1. The $p_{T}$/$\eta$-independent and
$p_{T}$/$\eta$-dependent systematic variations are considered separately.

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 $R(W)$ bib:binomial . The $m(t\bar{t})$ reweighting leads to an
uncertainty of $0.2\%$ on
$R(bbWW)$ and thus also on $R(W)$.

Source of systematic uncertainty | Systematic uncertainty on measured ratios (%)
---|---
| $R(bbWW)$ |  $R(Z)$ |  $R(WZ)$
${t}{t}$ background in $Z$ $\rightarrow$ $\tau\tau$ $\rightarrow$ $e\mu$ sample | 0 | 0.2 | 0.2
Diboson background in $Z$ $\rightarrow$ $\tau\tau$ $\rightarrow$ $e\mu$ sample | 0 | 0.2 | 0.2
$p_{T}(Z)$ reweighting | 0 | $<0.1$ | $<0.1$
$m(t\bar{t})$ reweighting | 0.2 | 0 | 0.2
${t}{t}$ modelling | 0.3 | 0 | 0.3
Table 5: Changes in the ratios $R(bbWW)$, $R(Z)$, and $R(WZ)$ resulting from
the sources of phenomenological systematic uncertainty described in Section
2.2 and summarised in Table 2.

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%.

(a) $R(bbWW)$ (b) $R(Z)$ (c) $R(WZ)$

Figure 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.

###  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 $\not{E}_{T}$ reconstruction on measurements of efficiencies,
backgrounds and systematic uncertainties published by ATLAS and CMS bib:atlas-
muonID –bib:pile-up , 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 $\int\\!\\!L\,\mbox{\,d}t$ = 140 fb-1 at $\sqrt{s}$ = 13 TeV
that is now available. It is, therefore, to be expected that the full run 2
data set will allow systematic uncertainties on 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
$\mathcal{B}(\mbox{$W$}\mbox{$\rightarrow$}\mbox{$\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.

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 Figure 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 bib:atlas-t-charged-higgs ; bib:cms-t-charged-higgs  and
neutral bib:atlas-t-neutral-higgs  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 uncorrrelated.

##  3 Summary and conclusions

A measurement of
$\mbox{$R(W)$}\equiv\mathcal{B}(\mbox{$W$}\mbox{$\rightarrow$}\mbox{$\tau\nu$})/\mathcal{B}(\mbox{$W$}\mbox{$\rightarrow$}\mbox{$\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
$\mbox{$R(WZ)$}\equiv\mbox{$R(bbWW)$}/\mbox{$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 $\not{E}_{T}$ reconstruction on measurements
of efficiencies, backgrounds and systematic uncertainties published by ATLAS
and CMS bib:atlas-muonID –bib:pile-up , having chosen identification
algorithms whose performance is suited to the needs of our analysis. For a
data set of $\int\\!\\!L\,\mbox{\,d}t$ = 140 fb-1 at $\sqrt{s}$ = 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 $e$, $\mu$, 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
$\int\\!\\!L\,\mbox{\,d}t$ = 140 fb-1 at $\sqrt{s}$ = 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 5$\sigma$.

###### Acknowledgements.

Acknowledgements: Final year undergraduate (MPhys) project students working in
Manchester with T.W. have made some important contributions to the ideas
presented in this paper. Jihyun Jeong (1993–2018) and Robin Upham made an
early feasibility study for the measurement of $R(bbWW)$ using ${t}{t}$ events
at the LHC. Vilius Cepaitis and Ricardo Wölker studied possible improvements
to the selection of $Z/\gamma^{*}$ $\rightarrow$ $\tau\tau$ 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.

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## 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 bib:atlas-muonID  and CMS bib:cms-muonID
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 bib:atlas-muonID . 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 bib:atlas-muonID , 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 bib:atlas-muonID . 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 bib:atlas-trigger ;
bib:cms-trigger . 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 bib:atlas-triggerthreshold-electron .
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 $p_{T}$ bib:atlas-muonID . We consider a
$p_{T}$/$\eta$-independent relative systematic uncertainty of 0.2% on the
$p_{T}$ scale bib:atlas-muonID . 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.

###  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 bib:atlas-electronID  and CMS
bib:cms-electronID  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 bib:atlas-elec-eta-crack . 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
bib:atlas-electronID . The efficiency is around 75% at $p_{T}=30$ GeV and
rises to around 90% for $p_{T}>80$ GeV and above bib:atlas-electronIDeff . 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 bib:atlas-muonID , 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
bib:atlas-electron-trigger . 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 bib:atlas-electronE  and CMS bib:cms-electronID  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 $\pm 0.002$ bib:atlas-electronEres . The $p_{T}$ scale is calibrated
with a precision of around 0.2% in both ATLAS and CMS bib:atlas-electronE ;
bib:cms-electronID . 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 bib:atlas-electronID2012  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 $\tau_{\mathrm{\,had}}$ 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
bib:cms-tau  describes the methods used to identify $\tau_{\mathrm{\,had}}$
candidates at $\sqrt{s}$ = 13 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
$\int\\!\\!L\,\mbox{\,d}t$ = 36 fb-1. We choose to simulate the
$\tau_{\mathrm{\,had}}$ identification efficiency and fake probabilities
corresponding to the “very-very-tight” operating point of bib:cms-tau . 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 $|\eta|$ bib:cms-taueff
. We simulate also the discriminants against electrons and muons described in
bib:cms-tau : 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 $1.4\times 10^{-3}$ bib:cms-tauemu .

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 bib:cms-tau  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 bib:cms-tau  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
$\mathcal{B}(\mbox{$W$}\mbox{$\rightarrow$}\mbox{$\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 $\frac{\sigma_{p_{T}}}{p_{T}}=0.16$ bib:cms-tau-2012 .

The relative uncertainties on the probabilities for electrons, muons, and
hadronic jets to be misidentified as a $\tau_{\mathrm{\,had}}$ candidate are
around 10% bib:cms-taumisid . The uncertainty on the $p_{T}$ scale for
$\tau_{\mathrm{\,had}}$ candidates is around 1% bib:cms-tau . 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 TeV bib:atlas-
tau2015data , using the methods described in bib:atlas-tau .

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 bib:cms-tau  for $\tau_{\mathrm{\,had}}$
identification, which has relatively low efficiency but high rejection power,
is to minimise the uncertainties arising from MJ backgrounds. From bib:cms-
muonID  and bib:cms-tau  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% bib:cms-taumj .

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
bib:antikt  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 bib:atlas-jes  and
CMS bib:cms-jes  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 bib:atlas-b-tag
and CMS bib:cms-b-tag  collaborations. We choose to simulate tag probabilities
according to those given for the “DeepCSV Loose” category of bib:cms-b-tag .
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
bib:cms-b-tag-eff .

The value of $\not{E}_{T}$ is calculated from the vector sum at particle level
of the $p_{T}$ of all neutrinos in the event. Resolution in $\not{E}_{T}$ 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 $\not{E}_{T}$ resolutions
given in bib:atlas-met  and bib:cms-met .

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 bib:pile-up . We take the performance values we have implemented to
represent averages over the pile-up conditions experienced at the LHC.