TTK-20-43, P3H-20-078, CAVENDISH-HEP-20/14

# NLO QCD corrections to off-shell ${t\bar{t}W^{\pm}}$ production at the LHC:
Correlations and Asymmetries

Giuseppe Bevilacqua(a)111e-mail<EMAIL_ADDRESS>Huan-
Yu Bi(b)222e-mail<EMAIL_ADDRESS>Heribertus Bayu
Hartanto(c)333e-mail<EMAIL_ADDRESS>Manfred Kraus(d)444e-mail:
<EMAIL_ADDRESS>Jasmina Nasufi(b)555e-mail<EMAIL_ADDRESS>and Malgorzata Worek(b)666e-mail<EMAIL_ADDRESS>(corresponding
author) .
(a)MTA-DE Particle Physics Research Group, University of Debrecen, H-4010
Debrecen, PBox 105, Hungary
(b)Institute for Theoretical Particle Physics and Cosmology, RWTH Aachen
University, D-52056 Aachen, Germany
(c)Cavendish Laboratory, University of Cambridge, J.J. Thomson Avenue,
Cambridge CB3 0HE, United Kingdom
(d)Physics Department, Florida State University, Tallahassee, FL 32306-4350,
USA

###### Abstract

Recent discrepancies between theoretical predictions and experimental data in
multi-lepton plus $b$-jets analyses for the $t\bar{t}W^{\pm}$ process, as
reported by the ATLAS collaboration, have indicated that more accurate
theoretical predictions and high precision observables are needed to constrain
numerous new physics scenarios in this channel. To this end we employ NLO QCD
computations with full off-shell top quark effects included to provide
theoretical predictions for the ${\cal
R}=\sigma_{t\bar{t}W^{+}}/\sigma_{t\bar{t}W^{-}}$ cross section ratio at the
LHC with $\sqrt{s}=13$ TeV. Depending on the transverse momentum cut on the
$b$-jet we obtain $2\%-3\%$ theoretical precision on ${\cal R}$, which should
help to shed some light on new physics effects that can reveal themselves only
once sufficiently precise Standard Model theoretical predictions are
available. Furthermore, triggered by these discrepancies we reexamine the
charge asymmetry of the top quark and its decay products in the
$t\bar{t}W^{\pm}$ production process. In the case of charge asymmetries, that
are uniquely sensitive to the chiral nature of possible new physics in this
channel, theoretical uncertainties below $15\%$ are obtained. Additionally,
the impact of the top quark decay modelling is scrutinised by explicit
comparison with predictions in the narrow-width approximation.

##  I Introduction

The Large Hadron Collider (LHC) with the Run II energy of $\sqrt{s}=13$ TeV
has opened up the possibility of studying various top quark production and
decay mechanisms at larger mass scales than previously explored in any
experiment. The $t\bar{t}$ pair production associated with the $W^{\pm}$ gauge
boson is among the most interesting signatures that can be studied with high
precision at the LHC. It is a key process to constrain top quark intrinsic
properties, which might be modified in the presence of new physics. Moreover,
the process can be used in the framework of the Standard Model Effective Field
Theory (SMEFT), where the effects of potential new particles can be
systematically included in terms of higher-dimensional operators, see e.g.
Hagiwara et al. (1993); Giudice et al. (2007); Grzadkowski et al. (2010);
Corbett et al. (2013); Brivio and Trott (2019). The latter are suppressed by a
sufficiently large new physics energy scale $\Lambda$. The framework relies on
the idea that new physics is too heavy to be directly produced and observed at
the LHC, thus, only deviations from the Standard Model (SM) can be probed in
various ATLAS and CMS top quark measurements. Compared with top quark pair
production and single top quark production, the associated $t\bar{t}W^{\pm}$
process does not bring sensitivity to new operators, however, it helps to
resolve blind directions in the SMEFT parameter space that occur in the
current LHC fits. On top of that $t\bar{t}W^{\pm}$ can probe operators that
are difficult to access in other channels. For example, since the $W^{\pm}$
gauge boson is radiated from the initial state, $t\bar{t}W^{\pm}$ is sensitive
to a subset of the possible four-quark operators only. In the SM,
$t\bar{t}W^{\pm}$ is dominated by quark-antiquark interactions, while
$t\bar{t}$ is dominated by the $gg$ initial state. This means that relative to
the SM contribution the four-quark operators would give sizeable effects in
the $t\bar{t}W^{\pm}$ production process. Consequently, $t\bar{t}W^{\pm}$
production is often included in the global SMEFT analysis of LHC top quark
measurements, see e.g. Ref. Brivio et al. (2020).

In addition, the $t\bar{t}W^{\pm}$ process plays an important role in studies
of the top quark charge asymmetry denoted as $A_{c}^{t}$ Maltoni et al. (2014,
2016). Also in this case the lack of the symmetric $gg$ initial state and the
emission of the $W^{\pm}$ gauge boson from the initial states contribute to a
substantially larger top quark charge asymmetry than that measured in the
$t\bar{t}$ process. Furthermore, the asymmetry of the top quark decay
products, i.e the charged lepton $(A_{c}^{\ell})$ and the $b$-jet
$(A_{c}^{b})$ are very large and already present at the LO due to the
polarisation of the initial fermionic line by the $W^{\pm}$ emission. These
asymmetries are an interesting playground for various beyond the SM (BSM)
theories, as $A_{c}^{t}$, $A_{c}^{\ell}$ and $A_{c}^{b}$ are uniquely
sensitive to the chiral nature of possible new physics that might directly
affect such measurements.

Last but not least, $t\bar{t}W^{\pm}$ production is a background process in
the multi-lepton final state with two same-sign leptons, accompanied by
missing transverse momentum and $b$-jets Aad et al. (2016a); Khachatryan et
al. (2016a); Sirunyan et al. (2017); Aaboud et al. (2017a). Even though same-
sign leptons are a relatively rare phenomenon in the SM, as they only appear
in processes with a rather small cross section, they have been extensively
exploited in various models of new physics. The same-sign lepton signature is
present, among others, in models with supersymmetry, universal extra
dimensions, top-quark partners and an extended Higgs boson sector Barnett et
al. (1993); Guchait and Roy (1995); Baer et al. (1996); Maalampi and Romanenko
(2002); Cheng et al. (2002); Dreiner et al. (2007); Contino and Servant
(2008); von Buddenbrock et al. (2016, 2018, 2019); Buddenbrock et al. (2019).
Besides, same-sign leptons are considered a key feature in searches for heavy
Majorana neutrinos as well as for $tt$ and $\bar{t}\bar{t}$ resonances Almeida
et al. (1997); Bai and Han (2009).

Finally, the $pp\to t\bar{t}W^{\pm}$ process is the main background in SM
measurements involving final states with multiple leptons and $b$-jets. This
is the case, for example, for the measurement of the associated production of
the SM Higgs boson with top quarks ATL (2019a). The $pp\to t\bar{t}W^{\pm}$
process has also played a crucial role in the announcement of strong evidence
for the production of four top quarks, an analysis, which has been recently
performed by the ATLAS Collaboration Aad et al. (2020).

The direct measurement of $pp\to t\bar{t}W^{\pm}$ production in multi-lepton
final states has already been carried out at $\sqrt{s}=13$ TeV by the ATLAS
and CMS collaborations Aaboud et al. (2017b); Sirunyan et al. (2018); Aaboud
et al. (2019). In the recent measurements of $t\bar{t}H$ and $t\bar{t}W^{\pm}$
production in multi-lepton final states ATL (2019a) the resulting
$t\bar{t}W^{\pm}$ normalisation has been found to be higher than the
theoretical prediction provided by multipurpose Monte Carlo (MC) generators,
which are currently employed by the ATLAS collaboration. Apart from the
$t\bar{t}W^{\pm}$ normalisation, a tension in the modelling of the final state
kinematics in the phase space regions dominated by $t\bar{t}W^{\pm}$
production, has been observed. From the experimental point of view such an
accurate study of $pp\to t\bar{t}W^{\pm}$ production in the same-sign lepton
final state has become feasible thanks to the increasing amount of data
collected at the LHC with $\sqrt{s}=13$ TeV. This increased integrated
luminosity has significantly raised the need for more precise theoretical
predictions. The latter should include higher order QCD corrections both to
the production and decays of top quarks and $W$ gauge bosons as well as
$t\bar{t}$ spin correlations at the same level of accuracy.

The first calculations for the $pp\to t\bar{t}W^{\pm}$ process, that meet the
mentioned conditions, have been carried out in the narrow-width approximation
(NWA) within the Mcfm framework Campbell and Ellis (2012). The first full NLO
QCD computations, which include complete top quark off-shell effects for the
$pp\to t\bar{t}W^{\pm}$ process in the multi-lepton channel, have been
recently presented in Ref. Bevilacqua et al. (2020a). In these computations,
obtained with the help of Helac-NLO, off-shell top quarks have been described
by Breit-Wigner propagators, furthermore, double-, single- as well as non-
resonant top-quark contributions along with all interference effects have been
consistently incorporated at the matrix element level. Independent
computations for $t\bar{t}W^{+}$ production have been obtained very recently
within the MoCaNLO+Recola framework Denner and Pelliccioli (2020). They not
only confirmed the results presented in Ref. Bevilacqua et al. (2020a) but
also performed a comparison between the full results and those obtained with
the help of the double-pole approximation.

In Ref. Bevilacqua et al. (2020a) results at NLO QCD accuracy have been
presented in the form of fiducial integrated and differential cross sections
for two selected renormalisation and factorisation scale choices (a fixed and
a dynamical one) and three different PDF sets. Detailed studies of the scale
dependence of the NLO predictions have been carried out together with
calculations of PDF uncertainties. Furthermore, the impact of the top quark
off-shell effects on the $pp\to t\bar{t}W^{\pm}$ cross section has been
examined by an explicit comparison with the results in the NWA. In the current
paper we will move away from the technical aspects of higher order
calculations and the estimation of the residual theoretical uncertainties and
go towards more phenomenological studies for the $pp\to t\bar{t}W^{\pm}$
process. Specifically, the purpose of this paper is twofold. First, we would
like to provide a systematic analysis of the two processes $pp\to
t\bar{t}W^{+}$ and $pp\to t\bar{t}W^{-}$ in the multi-lepton decay channel to
extract the most accurate NLO QCD predictions for the ${\cal R}=\sigma^{\rm
NLO}_{t\bar{t}W^{+}}/\sigma^{\rm NLO}_{t\bar{t}W^{-}}$ cross section ratio.
Generally, cross section ratios are more stable against radiative corrections
than absolute cross sections, assuming that the two processes are correlated.
They have smaller theoretical uncertainties as various uncertainties tend to
cancel in a cross section ratio. Consequently, such precise theoretical
predictions have enhanced predictive power and should be used in indirect
searches for new physics at the LHC. Let us add here, that the ${\cal
R}=\sigma^{\rm NLO}_{t\bar{t}W^{+}}/\sigma^{\rm NLO}_{t\bar{t}W^{-}}$ cross
section ratio has recently been studied in Ref. Frederix and Tsinikos (2020)
in the context of parton shower. The NLO QCD and subleading electroweak
corrections for the $t\bar{t}W^{\pm}$ process were matched, using the MC@NLO
matching scheme Frixione and Webber (2002); Frixione et al. (2003), to the
parton shower using the PYTHIA8 framework Sjöstrand et al. (2015) and the
MadGraph5-aMC@NLO system Alwall et al. (2014); Frederix et al. (2018a). The
top quark and the $W^{\pm}$ gauge boson decays were realised within the
MadSpin framework Artoisenet et al. (2013) in order to fully keep the LO spin
correlations. The ${\cal R}$ ratio has been considered for the two following
signatures: the same sign di-lepton and the multi lepton channels. The scale
uncertainties were taken to be correlated, although, correlations were not
studied in any detail. In addition, the PDF uncertainties were not discussed
at all.

The second goal of the paper is to study separately the intrinsic properties
of $t\bar{t}W^{+}$ and $t\bar{t}W^{-}$ production. More specifically, we shall
use state-of-the-art NLO QCD theoretical predictions for the $t\bar{t}W^{\pm}$
process to re-examine the top quark charge asymmetry and asymmetries of the
top quark decay products both at the integrated and differential level.
Likewise, in this case, the polarisation and asymmetry effects in the $pp\to
t\bar{t}W^{\pm}$ production process can be employed to constrain new physics
effects that might occur in this channel. Furthermore, for both the cross
section ratio and the top quark (decay products) charge asymmetry, the impact
of the modelling of top quark production and decays will be studied.

We note here, that state-of-the art theoretical predictions at NLO in QCD with
complete top quark off-shell effects included are also available for other
processes at the LHC. Such effects, for example, have been incorporated for
$pp\to t\bar{t}$ Denner et al. (2011); Bevilacqua et al. (2011); Denner et al.
(2012); Denner and Pellen (2018), $pp\to t\bar{t}j$ Bevilacqua et al. (2016a,
b), $pp\to t\bar{t}H$ Denner and Feger (2015), $pp\to t\bar{t}\gamma$
Bevilacqua et al. (2018) and for $t\bar{t}Z(Z\to\nu_{\ell}\nu_{\ell})$
Bevilacqua et al. (2019). They have also been incorporated for the $pp\to
t\bar{t}b\bar{b}$ process Denner et al. (2020). We additionally add, that
continuous efforts have been devoted to improve the theoretical modeling of
hadronic observables for $t\bar{t}W^{\pm}$ at NLO through matching with parton
shower and multi-jet merging Garzelli et al. (2012); Frederix and Tsinikos
(2020); von Buddenbrock et al. (2020); Cordero et al. (2021). A further step
towards a more precise modelling of on-shell $t\bar{t}W^{\pm}$ production with
stable top quarks and $W^{\pm}$ gauge boson has been achieved by including
either NLO electroweak corrections Frixione et al. (2015) and the subleading
electroweak corrections Dror et al. (2016); Frederix et al. (2018b) or by
incorporating soft gluon resummation effects with next-to-next-to-leading
logarithmic accuracy Li et al. (2014); Broggio et al. (2016); Kulesza et al.
(2019); Broggio et al. (2019); Kulesza et al. (2020). Very recently, NLO QCD
and electroweak corrections to the full off-shell $t\bar{t}W^{+}$ production
at the LHC have been combined for the first time for the three-charged-lepton
channel Denner and Pelliccioli (2021).

The paper is organised as follows. In section II the Helac-NLO computational
framework and input parameters used in our studies are briefly described. In
section III correlations between $t\bar{t}W^{+}$ and $t\bar{t}W^{-}$ are
examined. The results for the cross section ratio ${\cal R}=\sigma^{\rm
NLO}_{t\bar{t}W^{+}}/\sigma^{\rm NLO}_{t\bar{t}W^{-}}$ are provided in section
IV. The integrated top quark charge asymmetry as well as asymmetries of the
top quark decay products are studied in section V. Results for the
differential and cumulative $A_{c}^{\ell}$ asymmetry are provided in section
VI. In section VII the results are summarised and our conclusions are
provided. Finally, in appendix A we discuss the Kolmogorov-Smirnov test that
may be used to provide a quantitative measure of similarity between the two
given processes.

##  II Computational Framework and Input Parameters

All our results both for the full off-shell and NWA computations have been
obtained with the help of the Helac-NLO Monte Carlo framework Bevilacqua et
al. (2013a). The calculation was performed using Helac-1Loop van Hameren et
al. (2009); Ossola et al. (2008) for the virtual corrections and Helac-Dipoles
Czakon et al. (2009); Bevilacqua et al. (2013b) for the real emission part.
The integration over the phase space has been achieved with the help of Kaleu
van Hameren (2010). In our studies we keep the Cabibbo-Kobayashi-Maskawa
mixing matrix diagonal and neglect the Higgs boson contributions. Following
recommendations of the PDF4LHC Working Group for the usage of PDFs suitable
for applications at the LHC Run II Butterworth et al. (2016) we employ the
NNPDF3.0 PDF set Ball et al. (2015). In particular, we use NNPDF30-nlo-as-0118
with $\alpha_{s}(m_{Z})=0.118$ (NNPDF30-lo-as-0130 with
$\alpha_{s}(m_{Z})=0.130$) at NLO (LO). The running of the strong coupling
constant $\alpha_{s}$ with two-loop accuracy at NLO is provided by the LHAPDF
interface Buckley et al. (2015). The number of active flavours is set to
$N_{F}=5$. We employ the following SM input parameters

$\begin{array}[]{lll}G_{\mu}=1.166378\cdot 10^{-5}~{}{\rm
GeV}^{-2}\,,&&m_{t}=172.5~{}{\rm GeV}\,,\vspace{0.2cm}\\\ m_{W}=80.385~{}{\rm
GeV}\,,&&\Gamma^{\rm NLO}_{W}=2.09767~{}{\rm GeV}\,,\vspace{0.2cm}\\\
m_{Z}=91.1876~{}{\rm GeV}\,,&&\Gamma^{\rm NLO}_{Z}=2.50775~{}{\rm
GeV}\,,\vspace{0.2cm}\\\ \Gamma_{t}^{\rm NLO}=1.33247~{}{\rm
GeV}\,,&&\Gamma^{\rm NLO}_{t,{\rm NWA}}=1.35355~{}{\rm GeV}\,.\end{array}$ (1)

For the $W$ and $Z$ gauge boson widths we use the NLO QCD values as calculated
for $\mu_{R}=m_{W}$ and $\mu_{R}=m_{Z}$ respectively. All other partons,
including bottom quarks, as well as leptons are treated as massless particles.
The LO and NLO top quark widths are calculated according to Ref. Denner et al.
(2012). The top quark width is treated as a fixed parameter throughout this
work. Its value corresponds to a fixed scale $\mu_{R}=m_{t}$. The
electromagnetic coupling $\alpha$ is calculated from the Fermi constant
$G_{\mu}$, i.e. in the $G_{\mu}-$scheme, via

$\alpha_{G_{\mu}}=\frac{\sqrt{2}}{\pi}\,G_{\mu}\,m_{W}^{2}\,\sin^{2}\theta_{W}\,,$
(2)

where $\sin^{2}\theta_{W}$ is defined according to

$\sin^{2}\theta_{W}=1-\frac{m_{W}^{2}}{m_{Z}^{2}}\,.$ (3)

We use kinematic-dependent factorisation and renormalisation scales
$\mu_{R}=\mu_{F}=\mu_{0}$ with the central value $\mu_{0}=H_{T}/3$ where
$H_{T}$ is the scalar sum of all transverse momenta in the event, including
the missing transverse momentum. The latter is constructed from the three
neutrinos $\nu_{e}$, $\nu_{e}$ and $\nu_{\mu}$. The additional light jet, if
resolved, is not included in the definition of $H_{T}$. In various comparisons
we also use a fixed scale defined as $\mu_{0}=m_{t}+m_{W}/2$. Jets are
constructed out of all final-state partons with pseudo-rapidity $|\eta|<5$ via
the anti$-k_{T}$ jet algorithm Cacciari et al. (2008) with the separation
parameter $R=0.4$. We require exactly two $b$-jets and three charged leptons,
two of which are same-sign leptons. All final states have to fulfil the
following selection criteria that mimic very closely the ATLAS detector
response ATL (2019a)

$\begin{array}[]{ll l l }p_{T,\,\ell}>25~{}{\rm GeV}\,,&&p_{T,\,b}>25~{}{\rm
GeV}\,,\vspace{0.2cm}\\\ |y_{\ell}|<2.5\,,&&|y_{b}|<2.5\,,\vspace{0.2cm}\\\
\Delta R_{\ell\ell}>0.4\,,&&\Delta R_{\ell b}>0.4\,,\end{array}$ (4)

where $\ell$ stands for the charged lepton. We do not impose any restrictions
on the kinematics of the additional light jet and the missing transverse
momentum.

##  III Correlations between $\boldsymbol{t\bar{t}W^{+}}$ and
$\boldsymbol{t\bar{t}W^{-}}$

We start with the NLO QCD differential cross sections for $pp\to
e^{+}\nu_{e}\,\mu^{-}\bar{\nu}_{\mu}\,e^{+}\nu_{e}\,b\bar{b}\,+X$ and $pp\to
e^{-}\bar{\nu}_{e}\,\mu^{+}\nu_{\mu}\,e^{-}\bar{\nu}_{e}\,b\bar{b}\,+X$. They
are obtained for the LHC Run II energy of $\sqrt{s}=13$ TeV. For brevity, we
will refer to these reactions as $pp\to t\bar{t}W^{+}$ and $pp\to
t\bar{t}W^{-}$. In Ref. Bevilacqua et al. (2020a) we have shown that the NLO
QCD effects for $t\bar{t}W^{+}$ and $t\bar{t}W^{-}$ are very similar. Indeed,
both processes show alike ${\cal K}$-factors and dependencies on the
perturbative scales. Furthermore, the off-shell effects for both processes are
of the same order. Thus, it is highly probable that some of the uncertainties
cancel in the ratio of $t\bar{t}W^{+}$ and $t\bar{t}W^{-}$ cross sections, and
${\cal R}=\sigma^{\rm NLO}_{t\bar{t}W^{+}}/\sigma^{\rm NLO}_{t\bar{t}W^{-}}$
might exhibit an enhanced perturbative stability. In the following we would
like to understand similarities and potential differences between the two
processes even further. We note that, at the leading order the production
mechanism for $t\bar{t}W^{+}$ $(t\bar{t}W^{-})$ is via the scattering of up-
type quark (anti-quark) and the corresponding down-type anti-quark (quark),
i.e. $u\bar{d}$ and $c\bar{s}$ for $pp\to t\bar{t}W^{+}$ as well as $\bar{u}d$
and $\bar{c}s$ for $pp\to t\bar{t}W^{-}$. The quark-gluon initial state opens
up only at NLO in QCD. Similarities with respect to higher order QCD
corrections, in the production mechanisms as well as in the kinematics of the
final states suggest that the two processes may be treated as correlated as
far as the choice of scales is concerned. In any case, as long as one is
interested in some specific observables listed below. To show this we examine
the common features in the kinematics of the final states. Since we are
interested in the shape differences/similarities only and because the fiducial
cross section for $pp\to t\bar{t}W^{-}$ is about a factor of two smaller than
the one for the $pp\to t\bar{t}W^{+}$ process we concentrate on the normalised
NLO QCD differential cross sections.

Figure 1:  Comparison of the normalised NLO QCD differential cross sections
for $pp\to t\bar{t}W^{\pm}$ in the multi-lepton final state at the LHC with
$\sqrt{s}=13$ TeV. The transverse momentum of the hardest same-sign lepton
$(p_{T,\,e_{1}})$ and the invariant mass of the two same-sign leptons
$(M_{e_{1}e_{2}})$ are presented. Also given are the scalar sum of the
transverse momenta of the leptons $(H_{T}^{lep})$ and the distance in the
azimuthal angle rapidity plane between the two same-sign leptons $(\Delta
R_{e_{1}e_{2}})$. The lower panels display the ratio of the normalised
distributions $t\bar{t}W^{+}/t\bar{t}W^{-}$. The NLO NNPDF3.0 PDF set is
employed and $\mu_{R}=\mu_{F}=H_{T}/3$ is used.

Figure 2:  As in Figure 1 but for the transverse momentum of the hardest
$b$-jet $(p_{T,\,b_{1}})$, the invariant mass of the two $b$-jets
$(M_{b_{1}b_{2}})$, the scalar sum of the transverse momenta of the visible
final states $(H_{T}^{vis})$ and the distance in the azimuthal angle rapidity
plane between the two $b$-jets $(\Delta R_{b_{1}b_{2}})$.

In the following the collection of leptonic observables will be examined. In
the $pp\to t\bar{t}W^{\pm}$ process same-sign charged leptons $e^{\pm}e^{\pm}$
occur. In the case of final states with identical leptons the ordering in
$p_{T}$ has to be introduced to label the particles. To this end, we denote
the first and the second hardest same-sign charged lepton as $e_{1}^{\pm}$ and
$e_{2}^{\pm}$ respectively. In Figure 1 we present the NLO QCD differential
cross sections for $pp\to t\bar{t}W^{+}$ and $pp\to t\bar{t}W^{-}$ as a
function of the transverse momentum of $e^{\pm}_{1}$ $(p_{T,\,e_{1}})$, the
invariant mass of the $e_{1}^{\pm}e_{2}^{\pm}$ system $(M_{e_{1},\,e_{2}})$
and the scalar sum of the transverse momenta of the charged leptons available
in the given process $(H_{T}^{lep})$. The latter is defined as

$H_{T}^{lep}=p_{T,\,\mu^{\mp}}+p_{T,\,e^{\pm}_{1}}+p_{T,\,e^{\pm}_{2}}\,.$ (5)

Also shown in Figure 1 is the distance in the azimuthal angle rapidity plane
between $e_{1}^{\pm}$ and $e_{2}^{\pm}$ $(\Delta R_{e_{1},\,e_{2}})$. All
differential cross sections shown are, indeed, rather similar. This is
particularly true for $H_{T}^{lep}$, $\Delta R_{e_{1},\,e_{2}}$ and
$p_{T,\,e_{1}}$ but also for $M_{e_{1},\,e_{2}}$ at the beginning of the
spectrum. Even-though the latter distribution diverges substantially in the
tails, the contribution from these particular phase-space regions to the
integrated fiducial $pp\to t\bar{t}W^{\pm}$ cross section is negligible. To
contrast these results, we refer the reader to Ref. Bevilacqua and Worek
(2014) where the processes $pp\to t\bar{t}b\bar{b}$ and $pp\to t\bar{t}jj$
demonstrate sizeable dissimilarities over the whole kinematics range.

In the next step we look at the $b$-jet kinematics. The two $b$-jets are
ordered according to their $p_{T}$. The hardest $(b_{1})$ and the softest
$b$-jet $(b_{2})$ kinematics are exhibited in Figure 2. We note here, however,
that the charge identification of the $b$-jets is possible at the LHC, see e.g
Krohn et al. (2013); Waalewijn (2012); Tok (2017); ATL (2018). Thus, one can
distinguish between $b$-jets initiated by $b$ and $\bar{b}$. In this work,
however, we do not perform such $b$-jet identification. We depict the NLO QCD
differential cross sections as a function of the transverse momentum of
$b_{1}$ $(p_{T,\,b_{1}})$, the invariant mass of the two $b$-jet system
$(M_{b_{1},\,b_{2}})$ and the distance in the azimuthal angle rapidity plane
between $b_{1}$ and $b_{2}$ $(\Delta R_{b_{1},\,b_{2}})$. Also presented in
Figure 2 is the scalar sum of the transverse momenta of all the visible final
states, denoted as $H_{T}^{vis}$. The latter is given by

$H_{T}^{vis}=p_{T,\,b_{1}}+p_{T,\,{b_{2}}}+p_{T,\,\mu^{\mp}}+p_{T,\,e^{\pm}_{1}}+p_{T,\,e^{\pm}_{2}}\,.$
(6)

An interesting comment can be made here. Namely, that the $b$-jets are
preferably produced in back-to-back configurations. Hereby, $b$-jets come more
often from top quark decays rather than from the $g\to b\bar{b}$ splitting.
The latter configuration, which is produced in the off-shell case where no
top-quark resonances are present, would manifest itself in the enhancement
close to $\Delta R_{b_{1},\,b_{2}}\approx 0.4$. In the case of $b$-jet
kinematics and for the $H_{T}^{vis}$ observable we can see similarities
between $pp\to t\bar{t}W^{+}$ and $pp\to t\bar{t}W^{-}$.

To summarise this part, as anticipated both $t\bar{t}W^{+}$ and
$t\bar{t}W^{-}$ production processes show a good level of similarity. In
addition to what has already been demonstrated in Ref. Bevilacqua et al.
(2020a), namely that the dominant higher order QCD effects are alike for
$t\bar{t}W^{+}$ and $t\bar{t}W^{-}$, we have shown here that the kinematics
(shapes of various differential cross sections) of the two processes is much
the same. Such similarities are there because the differences in the PDFs for
the valence quarks do not manifest themselves so much for the chosen
observables. The fact that the QCD corrections are not flavour sensitive does
not destroy the picture when NLO corrections in QCD are added. Furthermore, it
justifies the use of correlated scales later on. For both processes, our
findings are not modified when the fixed scale choice
$\mu_{R}=\mu_{F}=\mu_{0}=m_{t}+m_{W}/2$ is used instead or when different PDF
sets are employed. We further note that, the ratio is built of integrated
cross section and the peak region is very similar in both distributions. One
could still point out that the normalisation matters. On the other hand, the
normalization is driven by the same power of $\alpha_{s}$ in both processes,
which necessarily cancels. Hence, it matters very much that the NLO
corrections may be calculated with correlated scales.

Finally, in the appendix we review the Kolmogorov-Smirnov (KS) test, that may
be used to provide a quantitative measure of similarity between two given
processes. Not only do we argue there for the usefulness of the KS test at
great length but we also discuss its shortcomings and advantages. This novel
approach is not used as a theoretical argument for the correlation of the
tested processes. It is rather an interesting way of showing that two
processes are similar in various differential distributions. Thus, used alone
the KS test is not a sufficient argument to prove the correlation between
$t\bar{t}W^{+}$ and $t\bar{t}W^{-}$. However, used together with all other
arguments mentioned in this section it increases the confidence that
$t\bar{t}W^{+}$ and $t\bar{t}W^{-}$ are indeed very similar.

##  IV Cross Section Ratios

Once we established that $pp\to t\bar{t}W^{+}$ and $t\bar{t}W^{-}$ are
correlated we can look at their ratio with the goal of increasing the
precision of NLO QCD predictions for both processes. We take the ratio of
$t\bar{t}W^{+}$ and $t\bar{t}W^{-}$ at the next-to-leading order in QCD. We do
not concentrate on ${\cal R}=\sigma^{\rm LO}_{t\bar{t}W^{+}}/\sigma^{\rm
LO}_{t\bar{t}W^{-}}$, as in the lowest order in perturbation expansion the
dependence on the strong coupling constant cancels out completely resulting in
highly underestimated theoretical uncertainties for the cross section ratio.
Consequently, the NLO in QCD is the first order where the theoretical
uncertainties are meaningful for the ${\cal R}$ observable. Indeed, for the
fixed scale choice, $\mu_{0}=m_{t}+m_{W}/2$, we have

$\begin{split}\sigma^{\rm
LO}_{t\bar{t}W^{+}}&=106.88^{\,+27.75\,(26\%)}_{\,-20.53\,(19\%)}{}^{\,+4.45\,(4\%)}_{\,-4.45\,(4\%)}\,{\rm[ab]},\\\
\sigma_{t\bar{t}W^{-}}^{\rm
LO}&={57.24}^{\,+14.92\,(26\%)}_{\,-11.04\,(19\%)}{}^{\,+2.79\,(5\%)}_{\,-2.79\,(5\%)}\,{\rm[ab]}.\end{split}$
(7)

The first sub- and super-scripts indicate the scale variation while the second
ones the PDF uncertainties. The LO cross section ratio reads

${\cal R}=\sigma^{\rm LO}_{t\bar{t}W^{+}}/\sigma_{t\bar{t}W^{-}}^{\rm
LO}=1.867^{\,+0.002\,(\,\,\,0.1\%)}_{\,-0.001\,(0.05\%)}{}^{\,+0.057\,(3\%)}_{\,-0.057\,(3\%)}.$
(8)

Thus, the scale dependence is at the $0.1\%$ level. The NLO QCD corrections to
${\cal R}$ are negative and of the order of $3\%$. Consequently, the LO scale
uncertainties underestimate the size of the NLO corrections by a factor of
$30$. Similar behaviour is observed for the dynamical scale setting.
Specifically, for $\mu_{0}=H_{T}/3$ we obtain

$\begin{split}\sigma^{\rm
LO}_{t\bar{t}W^{+}}&=115.10^{\,+30.50\,(26\%)}_{\,-22.45\,(20\%)}{}^{\,+4.80\,(4\%)}_{\,-4.80\,(4\%)}\,{\rm[ab]},\\\
\sigma_{t\bar{t}W^{-}}^{\rm
LO}&={62.40}^{\,+16.67\,(27\%)}_{\,-12.27\,(20\%)}{}^{\,+3.05\,(5\%)}_{\,-3.05\,(5\%)}\,{\rm[ab]}.\end{split}$
(9)

The LO cross section ratio for the dynamical scale setting yields

${\cal R}=\sigma^{\rm LO}_{t\bar{t}W^{+}}/\sigma_{t\bar{t}W^{-}}^{\rm
LO}=1.844^{\,+0.004\,(0.2\%)}_{\,-0.003\,(0.2\%)}{}^{\,+0.056\,(3\%)}_{\,-0.056\,(3\%)}.$
(10)

In this case the NLO QCD corrections to ${\cal R}$ are also negative but
slightly smaller at only $2\%$.

We choose the renormalisation and factorisation scales in the numerator and
denominator in a correlated way and we always choose the same values for the
scales in both the numerator and the denominator. This approach is justified
by the outcomes of the previous section, i.e. by observing that various NLO
distributions for $t\bar{t}W^{+}$ and $t\bar{t}W^{-}$ have very similar
shapes. In other words, the ratio ${\cal R}$ is rather constant across the
dominant parts of phase space. This form of kinematic correlation indicates
that NLO predictions for the two processes are dominated by the same
topologies.

$\mu_{0}=m_{t}+m_{W}/2$ | $\sigma^{\rm NLO}_{t\bar{t}W^{+}}\pm\delta_{\rm scale}\pm\delta_{\rm PDF}$ | $\sigma^{\rm NLO}_{t\bar{t}W^{-}}\pm\delta_{\rm scale}\pm\delta_{\rm PDF}$ | ${\cal R}\pm\delta_{\rm scale}\pm\delta_{\rm PDF}$
---|---|---|---
NNPDF3.0 | [ab] | [ab] | ${\cal R}=\sigma^{\rm NLO}_{t\bar{t}W^{+}}/\sigma^{\rm NLO}_{t\bar{t}W^{-}}$
$p_{T,\,b}>25$ GeV | $123.2^{\,+6.3\,(5\%)}_{\,-8.7\,(7\%)}{}^{\,+2.1\,(2\%)}_{\,-2.1\,(2\%)}$ | $68.0^{\,+4.8\,(7\%)}_{\,-5.5\,(8\%)}{}^{\,+1.2\,(2\%)}_{\,-1.2\,(2\%)}$ | $1.81^{\,+0.02\,(1\%)}_{\,-0.03\,(2\%)}{}^{\,+0.03\,(2\%)}_{\,-0.03\,(2\%)}$
$p_{T,\,b}>30$ GeV | $113.1^{\,+5.4\,(5\%)}_{\,-7.8\,(7\%)}{}^{\,+1.9\,(2\%)}_{\,-1.9\,(2\%)}$ | $62.3^{\,+4.2\,(7\%)}_{\,-4.9\,(8\%)}{}^{\,+1.1\,(2\%)}_{\,-1.1\,(2\%)}$ | $1.81^{\,+0.02\,(1\%)}_{\,-0.04\,(2\%)}{}^{\,+0.03\,(2\%)}_{\,-0.03\,(2\%)}$
$p_{T,\,b}>35$ GeV | $102.6^{\,+4.7\,(5\%)}_{\,-6.8\,(7\%)}{}^{\,+1.7\,(2\%)}_{\,-1.7\,(2\%)}$ | $56.3^{\,+3.7\,(7\%)}_{\,-4.4\,(8\%)}{}^{\,+1.0\,(2\%)}_{\,-1.0\,(2\%)}$ | $1.82^{\,+0.02\,(1\%)}_{\,-0.04\,(2\%)}{}^{\,+0.03\,(2\%)}_{\,-0.03\,(2\%)}$
$p_{T,\,b}>40$ GeV | $92.0^{\,+4.0\,(4\%)}_{\,-6.1\,(7\%)}{}^{\,+1.6\,(2\%)}_{\,-1.6\,(2\%)}$ | $50.3^{\,+3.3\,(6\%)}_{\,-3.9\,(8\%)}{}^{\,+0.9\,(2\%)}_{\,-0.9\,(2\%)}$ | $1.83^{\,+0.02\,(1\%)}_{\,-0.04\,(2\%)}{}^{\,+0.03\,(2\%)}_{\,-0.03\,(2\%)}$
Table 1:  NLO QCD integrated fiducial cross sections for $pp\to t\bar{t}W^{\pm}$ in the multi-lepton final state at the LHC with $\sqrt{s}=13$ TeV. Also shown are results for ${\cal R}=\sigma^{\rm NLO}_{t\bar{t}W^{+}}/\sigma^{\rm NLO}_{t\bar{t}W^{-}}$. Theoretical uncertainties as estimated from the scale variation and from the PDFs are listed as well. Four different values of the $p_{T,\,b}$ cut are used. The NNPDF3.0 PDF set is employed and $\mu_{R}=\mu_{F}=\mu_{0}$ where $\mu_{0}=m_{t}+m_{W}/2$. $\mu_{0}=H_{T}/3$ | $\sigma^{\rm NLO}_{t\bar{t}W^{+}}\pm\delta_{\rm scale}\pm\delta_{\rm PDF}$ | $\sigma^{\rm NLO}_{t\bar{t}W^{-}}\pm\delta_{\rm scale}\pm\delta_{\rm PDF}$ | ${\cal R}\pm\delta_{\rm scale}\pm\delta_{\rm PDF}$
---|---|---|---
NNPDF3.0 | [ab] | [ab] | ${\cal R}=\sigma^{\rm NLO}_{t\bar{t}W^{+}}/\sigma^{\rm NLO}_{t\bar{t}W^{-}}$
$p_{T,\,b}>25$ GeV | $124.4^{\,+4.3\,(3\%)}_{\,-7.7\,(6\%)}{}^{\,+2.1\,(2\%)}_{\,-2.1\,(2\%)}$ | $68.6^{\,+3.5\,(5\%)}_{\,-4.8\,(7\%)}{}^{\,+1.2\,(2\%)}_{\,-1.2\,(2\%)}$ | $1.81^{\,+0.02\,(1\%)}_{\,-0.03\,(2\%)}{}^{\,+0.03\,(2\%)}_{\,-0.03\,(2\%)}$
$p_{T,\,b}>30$ GeV | $113.9^{\,+3.5\,(3\%)}_{\,-6.8\,(6\%)}{}^{\,+1.9\,(2\%)}_{\,-1.9\,(2\%)}$ | $62.7^{\,+3.0\,(5\%)}_{\,-4.3\,(7\%)}{}^{\,+1.1\,(2\%)}_{\,-1.1\,(2\%)}$ | $1.82^{\,+0.02\,(1\%)}_{\,-0.03\,(2\%)}{}^{\,+0.03\,(2\%)}_{\,-0.03\,(2\%)}$
$p_{T,\,b}>35$ GeV | $103.1^{\,+3.1\,(3\%)}_{\,-6.0\,(6\%)}{}^{\,+1.7\,(2\%)}_{\,-1.7\,(2\%)}$ | $56.5^{\,+2.6\,(5\%)}_{\,-3.8\,(7\%)}{}^{\,+1.0\,(2\%)}_{\,-1.0\,(2\%)}$ | $1.82^{\,+0.02\,(1\%)}_{\,-0.03\,(2\%)}{}^{\,+0.03\,(2\%)}_{\,-0.03\,(2\%)}$
$p_{T,\,b}>40$ GeV | $92.3^{\,+2.8\,(3\%)}_{\,-5.3\,(6\%)}{}^{\,+1.5\,(2\%)}_{\,-1.5\,(2\%)}$ | $50.4^{\,+2.3\,(5\%)}_{\,-3.4\,(7\%)}{}^{\,+0.9\,(2\%)}_{\,-0.9\,(2\%)}$ | $1.83^{\,+0.02\,(1\%)}_{\,-0.03\,(2\%)}{}^{\,+0.03\,(2\%)}_{\,-0.03\,(2\%)}$
Table 2:  As in Table 1 but for $\mu_{0}=H_{T}/3$.

In Tables 1 and 2 we present integrated fiducial cross sections at NLO in QCD
for $pp\to t\bar{t}W^{+}$ and $t\bar{t}W^{-}$ in the multi-lepton decay
channel together with the theoretical uncertainties due to scale dependence
for $\mu_{F}=\mu_{R}=\mu_{0}$ where $\mu_{0}=m_{t}+m_{W}/2$ or
$\mu_{0}=H_{T}/3$. The uncertainty on higher orders for the integrated
fiducial cross sections is estimated by varying $\mu_{R}$ and $\mu_{F}$
independently around a central scale $\mu_{0}$ in the range
$1/2\leq\mu_{R}/\mu_{0},\mu_{F}/\mu_{0}\leq 2$ with the additional condition
$1/2\leq\mu_{R}/\mu_{F}\leq 2$. As it is always done we search for the minimum
and maximum of the resulting cross sections. For the PDF error we use the
corresponding prescription from the NNPDF3.0 group to provide the $68\%$
confidence level (C.L.) PDF uncertainties. Specifically, the NNPDF3.0 group
uses the Monte Carlo sampling method in conjunction with neural networks and
the PDF uncertainties are obtained with the help of the replicas method, see
e.g. Buckley et al. (2015). Also given in Tables 1 and 2 is the ${\cal R}$
cross section ratio and its systematic uncertainties $\delta_{\rm scale}$ and
$\delta_{\rm PDF}$. To properly account for the cross-correlations between the
two processes the latter is evaluated in the similar fashion as $\delta_{\rm
PDF}$ for $\sigma^{\rm NLO}_{t\bar{t}W^{+}}$ and $\sigma^{\rm
NLO}_{t\bar{t}W^{-}}$. First we examine the stability of ${\cal R}$ with
respect to the $p_{T,\,b}$ cut. To this end, we show results for four
different values of the $p_{T,\,b}$ cut. We observe very stable cross section
ratio results both in terms of the central value and theoretical
uncertainties. Furthermore, we notice that the scale choice does not play any
role for such an inclusive observable. The PDF uncertainties, which for $pp\to
t\bar{t}W^{+}$ and $pp\to t\bar{t}W^{-}$ are consistently at the $2\%$ level,
do not cancel out substantially in the cross section ratio. The final
theoretical error receives similar contributions from $\delta_{\rm scale}$ and
$\delta_{\rm PDF}$. These uncertainties are both at the $2\%$ level. Such
precise theoretical predictions have normally been obtained only once the NNLO
QCD corrections are incorporated. Thus, ${\cal R}$ at NLO in QCD represents a
very precise observable to be measured at the LHC. A few comments are in order
here. NLO correlations for $t\bar{t}W^{+}$ and $t\bar{t}W^{-}$ are completely
insensitive to potentially large $gg$-induced NNLO corrections to the ${\cal
R}=\sigma^{\rm NLO}_{t\bar{t}W^{+}}/\sigma^{\rm NLO}_{t\bar{t}W^{-}}$ ratio.
For example, due to the fact that the $gg$ initial state is the same for the
two processes we can assume that the $t\bar{t}W^{+}$ and $t\bar{t}W^{-}$ cross
sections receive the same $gg$-channel correction, which we denote as
$\delta\sigma_{gg}^{\rm NNLO}$. As a result, the ratio would be shifted by a
relative factor $\delta{\cal R}/{\cal R}\approx({\cal
R}-1)\delta\sigma_{gg}^{\rm NNLO}/\sigma^{\rm
NLO}_{t\bar{t}W^{+}}\approx\delta\sigma_{gg}^{\rm NNLO}/\sigma^{\rm
NLO}_{t\bar{t}W^{+}}$, which could in principle amount to several percent.
Therefore, it could be well above the reported uncertainty estimate of
$1\%-2\%$. The judgment of the scale variation prescription for ${\cal R}$
would of course be much easier in the presence of NNLO QCD calculations.
Unfortunately such calculations are out of reach even for the simplest case of
$pp\to t\bar{t}W^{\pm}$ production with stable top quarks and $W^{\pm}$ gauge
bosons. Adding decays of the unstable particles and incorporating the complete
off-shell effects is simply difficult to imagine at the current stage of NNLO
QCD calculations. Nevertheless, in the absence of the NNLO calculations for
the process at hand we can assess the $gg$-channel correction
$\delta\sigma_{gg}^{\rm NNLO}$ by performing a LO study for $gg\to
e^{+}\nu_{e}\,\mu^{-}\bar{\nu}_{\mu}\,e^{+}\nu_{e}\,b\bar{b}\,\bar{u}d$ and
$gg\to e^{+}\nu_{e}\,\mu^{-}\bar{\nu}_{\mu}\,e^{+}\nu_{e}\,b\bar{b}\,\bar{c}s$
with the same input parameters, cuts and for example by employing the fixed
scale choice. The size of $\delta\sigma_{gg}^{\rm NNLO}/\sigma^{\rm
NLO}_{t\bar{t}W^{+}}$ correction estimated in this way amounts to $0.3\%$.
Similar studies can be performed for $gg\to
e^{-}\bar{\nu}_{e}\,\mu^{+}\nu_{\mu}\,e^{-}\bar{\nu}_{e}\,b\bar{b}\,u\bar{d}$
and $gg\to
e^{-}\bar{\nu}_{e}\,\mu^{+}\nu_{\mu}\,e^{-}\bar{\nu}_{e}\,b\bar{b}\,c\bar{s}$.
To evaluate LO cross sections, however, we have used cuts on light jets that
are not there when the true NNLO QCD corrections are calculated. To remedy
this in the next step we have used the SecToR Improved Phase sPacE for real
Radiation (Stripper) library Czakon (2010, 2011); Czakon and Heymes (2014);
Czakon et al. (2019) that implements a general subtraction scheme for the
evaluation of NNLO QCD contributions from double-real radiation to processes
with at least two particles in the final state at LO. By employing
Stripper777Courtesy of M. Czakon. we were able to calculate the actual
contribution $\delta\sigma_{gg}^{\rm NNLO}/\sigma^{\rm NLO}_{t\bar{t}W^{+}}$
for the $pp\to t\bar{t}W^{+}$ process with stable top quarks and $W^{+}$ gauge
boson. This contribution is of the order of $0.2\%$, thus, it is similar in
size to $\delta\sigma_{gg}^{\rm NNLO}/\sigma^{\rm NLO}_{t\bar{t}W^{+}}$
estimated by the LO studies with the complete top quark and $W^{\pm}$ off-
shell effects included. Notably, it is also well below the reported
uncertainty estimate for the ${\cal R}=\sigma^{\rm
NLO}_{t\bar{t}W^{+}}/\sigma^{\rm NLO}_{t\bar{t}W^{-}}$ ratio $(1\%-2\%)$.
Because of that, the impact of the $gg$-channel on the ratio at hand is quite
small.

We note here, that there are different approaches in the literature for
handling of uncertainties in ratios. For example one can take the relative
size of the last considered order compared to the previous one as an estimate
of the perturbative uncertainty, see e.g. Ref. Duhr et al. (2020).
Specifically, following Duhr et al. (2020) one can define

$\delta_{pert.}=\pm\left|1-\frac{{\cal R}^{\rm NLO}(\mu_{0})}{{\cal R}^{\rm
LO}(\mu_{0})}\right|\times 100\%\,,$ (11)

where the values of $\mu_{0}$ are chosen in a correlated way in the numerator
and the denominator of ${\cal R}$. This error estimator assumes that the sub-
leading terms should be smaller than the last known correction. The obvious
downside to this approach is that it gives a vanishing result whenever two
consecutive perturbative orders provide identical numerical predictions.
Furthermore, this prescription leads to the rather suspicious shapes of the
uncertainty bands for the differential cross section ratios. Thus, as clearly
stated in Ref. Duhr et al. (2020) this approach on its own can not serve as a
good estimator of perturbative uncertainties. Nevertheless, it can be used in
conjunction with traditional methods to further ensure that the correlated
scale dependence is indeed a reasonable approach. Had we used the prescription
from Eq. (11) we would obtain $\delta_{pert.}=3\%$ for $\mu_{0}=m_{t}+m_{W}/2$
and $\delta_{pert.}=2\%$ for the dynamical scale setting. Thus, the
uncertainty estimate for the ${\cal R}=\sigma^{\rm
NLO}_{t\bar{t}W^{+}}/\sigma^{\rm NLO}_{t\bar{t}W^{-}}$ ratio would rather be
$\delta_{pert.}=2\%-3\%$, which is only slightly larger than $\delta_{\rm
scale}=1\%-2\%$.

In the next step we examine the impact of the top quark production and decay
modelling on the cross section ratio. To this end we present results for the
full NWA and for the NWALOdecay case. The former comprises NLO QCD corrections
to the production and to the subsequent top quark decays, the latter NLO QCD
corrections to the production of $t\bar{t}W^{\pm}$ and LO top quark decays.
Should we use the NLO QCD results in the full NWA for the $pp\to
t\bar{t}W^{\pm}$ process our findings for $\mu_{0}=m_{t}+m_{W}/2$ would be as
follows

${\cal R}=\frac{\sigma^{\rm NLO,\,NWA}_{t\bar{t}W^{+}}}{\sigma^{\rm
NLO,\,NWA}_{t\bar{t}W^{-}}}=1.81\pm 0.04\,(2\%)\,,$ (12)

where the quoted theoretical error results only from the scale dependence as
the PDF uncertainties will not be affected by changes in the modeling of the
top quark decays. On the other hand for the dynamical scale choice
$\mu_{0}=H_{T}/3$ we would obtain

${\cal R}=\frac{\sigma^{\rm NLO,\,NWA}_{t\bar{t}W^{+}}}{\sigma^{\rm
NLO,\,NWA}_{t\bar{t}W^{-}}}=1.81\pm 0.03\,(2\%)\,.$ (13)

We can observe that the full NWA approach does not modify either the value or
the size of the theoretical error for the integrated cross section ratio. The
latter result is not surprising taking into account that the impact of the top
quark off-shell effects on the integrated fiducial $t\bar{t}W^{\pm}$ cross
section is negligible. Furthermore, theoretical uncertainties for the full NWA
and full off-shell case are similar independently of the scale choice
Bevilacqua et al. (2020a).

Finally, we have employed the NWALOdecay case. For $\mu_{0}=m_{t}+m_{W}/2$ we
obtained

${\cal R}=\frac{\sigma^{\rm NLO,\,NWA_{LOdecay}}_{t\bar{t}W^{+}}}{\sigma^{\rm
NLO,\,NWA_{LOdecay}}_{t\bar{t}W^{-}}}=1.82\pm 0.02\,(1\%)\,,$ (14)

whereas for $\mu_{0}=H_{T}/3$ we can report

${\cal R}=\frac{\sigma^{\rm NLO,\,NWA_{LOdecay}}_{t\bar{t}W^{+}}}{\sigma^{\rm
NLO,\,NWA_{LOdecay}}_{t\bar{t}W^{-}}}=1.81\pm 0.02\,(1\%)\,.$ (15)

Even for this case the cross section ratios are very stable and rather
insensitive to the details of the modelling of the top quark production and
decays. Let us note here, that for the absolute $pp\to t\bar{t}W^{\pm}$
integrated cross sections the difference between the NWALOdecay approach and
the full off-shell one is at the level of $5\%$. In addition, theoretical
uncertainties due to the scale dependence are higher in the former case, up to
$11\%-13\%$ Bevilacqua et al. (2020a). Yet in the cross section ratio these
differences cancel out making ${\cal R}=\sigma^{\rm
NLO}_{t\bar{t}W^{+}}/\sigma^{\rm NLO}_{t\bar{t}W^{-}}$ very precise and an
extremely interesting theoretical observable to be measured at the LHC.

To conclude this part, we note that for the cross section ratio at NLO in QCD
the residual perturbative uncertainties are reduced to $1\%-2\%$ and are
similar in size to the PDF uncertainties. The theoretical uncertainties
associated with the top quark decay modelling are negligible. This suggests
that the ${\cal R}=\sigma^{\rm NLO}_{t\bar{t}W^{+}}/\sigma^{\rm
NLO}_{t\bar{t}W^{-}}$ observable can be employed either for the precision SM
measurements or to shed some light on possible new physics scenarios that
might reveal themselves only once sufficiently precise theoretical predictions
are available. For example in the case of BSM physics the presence of two
same-sign leptons in the final state, a relatively rare phenomenon at the LHC,
constitutes an optimal signature for many new physics models from
supersymmetry, supergravity and Majorana neutrinos to models with the modified
Higgs boson sector. Given the final accuracy of ${\cal R}$, it should be used
to put more stringent constrains on the parameter space of these models.

##  V Charge Asymmetries in $\boldsymbol{t\bar{t}W^{\pm}}$ Production

The $pp$ initial state at the LHC is expected to produce top quark and
antiquark rapidity distributions that are symmetric about $y=0$ in $t\bar{t}$
production. However, since the quarks in the initial state can be from valence
quarks, while the antiquarks are from the sea, the larger average momentum-
fraction of quarks leads to an excess of top quarks produced in the forward
directions. The rapidity distribution of top quarks in the SM is therefore
broader than that of the more centrally produced top antiquarks. This suggests
that $\Delta|y|=|y_{t}|-|y_{\bar{t}}|$, which is the difference between the
absolute value of the top quark rapidity $|y_{t}|$ and the absolute value of
the anti-top quark rapidity $|y_{\bar{t}}|$, is a suitable observable to
measure the top quark charge asymmetry at the LHC. This asymmetry is
nevertheless very small, see e.g. Bernreuther and Si (2012); Czakon et al.
(2018). For the $pp\to t\bar{t}W^{\pm}$ process the presence of the $W^{\pm}$
gauge boson results in the top quark charge asymmetry that is significantly
larger than in $t\bar{t}$ production Maltoni et al. (2014). The main reason
behind this is the absence of the symmetric $gg$ channel that is not
accessible until NNLO. Furthermore, the emission of the $W^{\pm}$ gauge boson
from the initial state polarises the initial quark line and in turn the
$t\bar{t}$ pair. As a result, the charge asymmetries for the top quark decay
products are large and already present at the leading order. In the following
we calculate afresh the top quark charge asymmetry in the $t\bar{t}W^{\pm}$
process in the multi-lepton final state using the state-of-the-art NLO QCD
calculations with the full top quark off-shell effects included. Additionally,
the asymmetries for the top quark decay products, $A^{b}_{c}$ and
$A_{c}^{\ell}$, will be examined. In this part of the paper, one of our main
goals is to carefully assess the impact of the top quark modelling on
$A_{c}^{t},A^{b}_{c}$ and $A_{c}^{\ell}$. We start with asymmetries at the
integrated level albeit in the fiducial regions of the phase space as defined
in Section II. For $A_{c}^{\ell}$ we will additionally calculate the
differential and cumulative asymmetry with respect to the following
observables: $p_{T}(\ell_{t}\ell_{\bar{t}})$, $|y(\ell_{t}\ell_{\bar{t}})|$
and $M({\ell_{t}\ell_{\bar{t}}})$, where $\ell_{t},\ell_{\bar{t}}$ stands for
the charged leptons stemming from the top and anti-top quark decay
respectively. For the two processes under consideration $pp\to
e^{+}\nu_{e}\,\mu^{-}\bar{\nu}_{\mu}\,e^{+}\nu_{e}\,b\bar{b}$ and $pp\to
e^{-}\bar{\nu}_{e}\,\mu^{+}\nu_{\mu}\,e^{-}\bar{\nu}_{e}\,b\bar{b}$ the
reconstruction of the (anti-)top quark momentum through its decay products is
required. As we are dealing with identical leptons in the final state,
however, we must employ an additional mechanism to properly assign the
positron (electron) and the corresponding neutrino $\nu_{e}$ (anti-neutrino
$\bar{\nu}_{e}$) to the correct top (anti-top) quark. In the case of
$t\bar{t}W^{+}$ production we use the following four different resonance
histories (a similar procedure is applied in the $t\bar{t}W^{-}$ case)

$\begin{split}t&\to e^{+}_{1}\,\nu_{e,1}\,b\quad\quad\quad\quad{\rm
and}\quad\quad\quad\quad\bar{t}\to\mu^{-}\,\bar{\nu}_{\mu}\,\bar{b}\,,\\\\[5.69046pt]
t&\to e^{+}_{1}\,\nu_{e,2}\,b\quad\quad\quad\quad{\rm
and}\quad\quad\quad\quad\bar{t}\to\mu^{-}\,\bar{\nu}_{\mu}\,\bar{b}\,,\\\\[5.69046pt]
t&\to e^{+}_{2}\,\nu_{e,1}\,b\quad\quad\quad\quad{\rm
and}\quad\quad\quad\quad\bar{t}\to\mu^{-}\,\bar{\nu}_{\mu}\,\bar{b}\,,\\\\[5.69046pt]
t&\to e^{+}_{2}\,\nu_{e,2}\,b\quad\quad\quad\quad{\rm
and}\quad\quad\quad\quad\bar{t}\to\mu^{-}\,\bar{\nu}_{\mu}\,\bar{b}\,.\end{split}$
(16)

These four resonant histories are not sufficient if NLO QCD calculations are
considered. In the case of the subtracted real emission part we additionally
take into account the extra light jet if resolved. Specifically, to closely
mimic what is done on the experimental side only the light jet that passes all
the cuts, that are also required for the two $b$-jets, is added to the
resonance history. Thus, in such a case a total of twelve different resonant
histories have to be considered. We compute for each history the following
quantity, see Ref. Bevilacqua et al. (2020b)

Figure 3:  Reconstructed invariant mass of the top quark and anti-top quark at
NLO in QCD for $pp\to t\bar{t}W^{+}$ and $pp\to t\bar{t}W^{-}$ in the multi-
lepton final state. Results are given for the LHC with $\sqrt{s}=13$ TeV. The
NLO NNPDF3.0 PDF set is employed and $\mu_{R}=\mu_{F}=\mu_{0}$ where
$\mu_{0}=H_{T}/3$.

Figure 4:  Comparison of the rapidity and pseudo-rapidity distributions of the
$t$ and $\bar{t}$ quarks at the NLO QCD level for $pp\to t\bar{t}W^{+}$ and
$pp\to t\bar{t}W^{-}$ in the multi-lepton final state. Also shown are rapidity
distributions of the charged leptons and $b$-jets from $t$ and $\bar{t}$
decays. Results are given for the LHC with $\sqrt{s}=13$ TeV. The NLO NNPDF3.0
PDF set is employed and $\mu_{R}=\mu_{F}=\mu_{0}$ where $\mu_{0}=H_{T}/3$ is
used.

${\cal Q}=|M_{t}-m_{t}|+|M_{\bar{t}}-m_{t}|\,,$ (17)

where $M_{t}$ and $M_{\bar{t}}$ are the (reconstructed) invariant masses of
the top and anti-top quark respectively and $m_{t}=172.5$ GeV. For each phase
space point we pick the history that minimises the ${\cal Q}$ value. In this
way all the (anti-)top quark decay products are identified. They are employed
in the definition of $A_{c}^{t}$, $A_{c}^{\ell}$ and $A_{c}^{b}$. To show how
well such a reconstruction works in Figure 3 we display the reconstructed
invariant mass of the top (anti-top) quark at NLO in QCD for the $pp\to
t\bar{t}W^{+}$ ($pp\to t\bar{t}W^{-}$) process in the multi-lepton channel.
Out of all twelve histories the four histories with the smallest ${\cal Q}$
value are shown. Clearly one can see that the reconstruction works very well.

Using the notation of Ref. Czakon et al. (2015, 2016, 2018) we define the top
quark charge asymmetry as follows

$A_{c}^{t}=\frac{\sigma_{\rm bin}^{+}-\sigma_{\rm bin}^{-}}{\sigma_{\rm
bin}^{+}+\sigma_{\rm bin}^{-}}\,,\quad\quad\quad\quad\sigma^{\pm}_{\rm
bin}=\int\theta(\pm\Delta|y|)\,\theta_{\rm bin}\,d\sigma\,,$ (18)

where $\Delta|y|=|y_{t}|-|y_{\bar{t}}|$ and $d\sigma$ is the differential
fiducial $t\bar{t}W^{\pm}$ cross section calculated at NLO in QCD. The binning
function $\theta_{\rm bin}$ can take the values zero or one. Its purpose is to
restrict to a given bin the kinematics of the $t\bar{t}W^{\pm}$ process in one
of the kinematic variables that is considered. The integrated asymmetry is
obtained by setting $\theta_{\rm bin}=1$. We note here that the charge-
symmetric $gg$ initial state, that is the dominant mechanism for $t\bar{t}$
production at the LHC, is not present for $t\bar{t}W^{\pm}$ production.
Therefore, unlike for $pp\to t\bar{t}$, it will not contribute to the
denominator of Eq. (18) to dilute the asymmetry. The LHC measurements for the
top quark charge asymmetry in $pp\to t\bar{t}$ production have been carried
out in terms of rapidity as well as pseudorapidity differences, see e.g.
Chatrchyan et al. (2012); Aad et al. (2012); Khachatryan et al. (2016b); Aad
et al. (2016b); Aaboud et al. (2018); ATL (2019b). Even-though, the top quark
charge asymmetry based on rapidity and pseudorapidity has the same features
its value can differ quite substantially. Consequently, we shall provide
results for $A^{t}_{c}$ for both cases. In the case of the top quark decay
products $A_{c}^{\ell}$ and $A_{c}^{b}$ are based on
$\Delta|y|=|y_{\ell_{t}}|-|y_{\ell_{\bar{t}}}|$ and
$\Delta|y|=|y_{b}|-|y_{\bar{b}}|$ respectively.

The top quark charge asymmetry can be visualised by superimposing the rapidity
(or the pseudo-rapidity) of $t$ and $\bar{t}$ for the $t\bar{t}W^{+}$ process.
The same can be done separately for $t\bar{t}W^{-}$. Similarly, we can plot
together the top and anti-top quark decay products, $b$ and $\bar{b}$ as well
as $\ell_{t}$ and $\ell_{\bar{t}}$. In Figure 4 we present such a comparison
at the NLO QCD level for the $t\bar{t}W^{\pm}$ process. We can observe that
all spectra are symmetric about $y=0$ $(\eta=0)$, as it should be, and that
the anti-top quark is more central with respect to the top quark. The same is
visible for the $b$-jet. This can be directly translated into the positive
value of $A_{c}^{t}$ and $A_{c}^{b}$. The situation is reversed for the
charged leptons. In the later case the charged lepton from the top quark decay
is more central, which will manifest itself in the negative value of
$A_{c}^{\ell}$.

In Tables 3 and 4 we present our findings for $A_{c}^{t}$, $A_{c}^{\ell}$ and
$A_{c}^{b}$ at NLO QCD for $t\bar{t}W^{+}$ production in the multi-lepton
channel at the LHC with $\sqrt{s}=13$ TeV. Results are given for the fixed
scale choice $\mu_{R}=\mu_{F}=m_{t}+m_{W}/2$. The top quark charge asymmetry
calculated in terms of rapidities (pseudo-rapidities) is denoted as
$A^{t}_{c,y}$ $(A^{t}_{c,\eta})$. We present the results with the full off-
shell effects included as well as for the full NWA and for the NWALOdecay
case. For all three approaches theoretical uncertainties due to the scale
dependence are also given. They are estimated by varying the renormalisation
and factorisation scales in $\alpha_{s}$ and PDFs up and down by a factor of
$2$ around the central scale of the process $\mu_{0}$. We show theoretical
predictions for the unexpanded and expanded version of the asymmetry. As the
ratio in Eq. (18) generates contributions of ${\cal O}(\alpha_{s}^{2})$ and
higher, which in principle can be affected by the unknown NNLO contributions,
we expand $A^{i}_{c}$ to first order in $\alpha_{s}$. The expanded version of
$A^{i}_{c}$ at NLO in QCD, where $i$ stands for $i=t,\ell,b$, is defined as

$A^{i}_{c,exp}=\frac{\sigma_{\rm LO}^{-}}{\sigma_{\rm
LO}^{+}}\left(1+\frac{\delta\sigma^{-}_{\rm NLO}}{\sigma^{-}_{\rm
LO}}-\frac{\delta\sigma_{\rm NLO}^{+}}{\sigma_{\rm LO}^{+}}\right)\,.$ (19)

where $\sigma^{\pm}$ stands for $\sigma^{\pm}=\sigma^{+}_{\rm
bin}\pm\sigma^{-}_{\rm bin}$ and $\delta\sigma^{\pm}_{\rm NLO}$ are the NLO
contributions to the fiducial cross section. Furthermore, $\sigma^{\pm}_{\rm
LO}$ are evaluated with NLO input parameters. In Tables 3 and 4 we include in
parenthesis the Monte Carlo (MC) integration errors to show that the latter
are smaller than or at least similar in size to the theoretical errors from
the scale dependence. Since the PDF dependence of the asymmetry is very small
(at the per-mill level) we do not quote the PDF errors in our predictions.

Before we analyse our results we remind the reader that the results presented
in Ref. Maltoni et al. (2014) were generated for NLO+PS and for a different
setup and input parameters. Thus, a direct comparison for the absolute values
for $A^{t}_{c,\eta}$, $A^{b}_{c,y}$, $A^{\ell}_{c,y}$ is not possible, but we
are rather interested in the relative size of their theoretical uncertainties.
In Ref. Maltoni et al. (2014) LO spin correlations are incorporated with the
help of MadSpin. The latter is employed before events can be passed to Herwig6
Corcella et al. (2001). The two charged leptons coming from top and anti-top
quark decays are chosen to be respectively positrons and electrons, while the
extra $W^{\pm}$ bosons decay into muons. Consequently, leptons and $b$-jets
coming from the top and anti-top quark decays can be uniquely identified.
Issues related to the top quark reconstruction are, therefore, not considered.
The results given in Ref. Maltoni et al. (2014) will serve us as a guideline
in the study of the impact of the top quark reconstruction. In particular for
$A^{t}_{c,\eta}$ theoretical uncertainties quoted in Ref. Maltoni et al.
(2014) are of the order of $\delta_{\rm scale}={}^{+19\%}_{-14\%}$. For
$A^{b}_{c,y}$ and $A^{\ell}_{c,y}$, on the other hand, they are $\delta_{\rm
scale}={}^{+2.5\%}_{-2.2\%}$ and $\delta_{\rm scale}={}^{+8.5\%}_{-6.0\%}$
respectively. The results for $A^{t}_{c,\eta}$, $A^{b}_{c,y}$ and
$A^{\ell}_{c,y}$ have been given for the combined $pp\to t\bar{t}W^{\pm}$ case
and for $\mu_{R}=\mu_{F}=2m_{t}$. Furthermore, for the computation of
$A^{b}_{c,y}$, events that do not feature two $b$-jets coming from the top
quark decays were discarded.

$t\bar{t}W^{+}$ | Off-shell | Full NWA | $\textsc{NWA}_{\scriptsize\mbox{LOdecay}}$
---|---|---|---
$\mu_{0}=m_{t}+m_{W}/2$ |  |  |
$A_{c,y}^{t}$ $[\%]$ | $2.09(8)^{+1.06\,(51\%)}_{-0.70\,(33\%)}$ | $1.68(4)^{+1.00(60\%)}_{-0.67(40\%)}{}$ | $0.86(3)^{+0.66\,(77\%)}_{-0.43\,(50\%)}$
$A_{c,exp,y}^{t}$ $[\%]$ | $2.62(10)^{+0.39\,(15\%)}_{-0.34\,(13\%)}$ | $2.19(4)^{+0.38(17\%)}_{-0.34(16\%)}{}$ | $1.94(5)^{+0.46\,(24\%)}_{-0.32\,(16\%)}$
$A_{c,\eta}^{t}$ $[\%]$ | $3.10(8)^{+1.21\,(39\%)}_{-0.80\,(26\%)}$ | $2.58(4)^{+1.31(51\%)}_{-0.75(29\%)}$ | $1.16(4)^{+0.71\,(61\%)}_{-0.44\,(38\%)}$
$A_{c,exp,\eta}^{t}$ $[\%]$ | $3.70(10)^{+0.46\,(12\%)}_{-0.40\,(11\%)}$ | $3.18(5)^{+0.56(18\%)}_{-0.34(11\%)}$ | $2.25(5)^{+0.51\,(23\%)}_{-0.32\,(14\%)}$
Table 3: Unexpanded and expanded $A_{c}^{t}$ asymmetry at NLO in QCD for
$pp\to t\bar{t}W^{+}$ in multi-lepton channel at the LHC with $\sqrt{s}=13$
TeV. Various approaches for the modelling of the top quark production and
decays are considered: the full off-shell case, the full NWA and the
NWALOdecay case. Also given are Monte Carlo (in parenthesis) integration and
theoretical errors. The NNPDF3.0 PDF set is employed and
$\mu_{R}=\mu_{F}=\mu_{0}$ where $\mu_{0}=m_{t}+m_{W}/2$.

Figure 5:  Unexpanded and expanded $A^{t}_{c}$ asymmetry at NLO in QCD for
$pp\to t\bar{t}W^{+}$ in the multi-lepton channel at the LHC with
$\sqrt{s}=13$ TeV. Various approaches for the modelling of the top quark
production and decays are considered: the full off-shell case, the full NWA
and the NWALOdecay case. The NNPDF3.0 PDF set is employed and
$\mu_{R}=\mu_{F}=\mu_{0}$ where $\mu_{0}=m_{t}+m_{W}/2$.

In the following we analyse our findings for $A^{t}_{c}$ as reconstructed from
the top quark decay products for $\mu_{R}=\mu_{F}=m_{t}+m_{W}/2$. They are
presented in Table 3 and graphically depicted in Figure 5. First we notice
that the difference between the unexpanded NLO asymmetry $(A^{t}_{c})$ and the
one with a consistent expansion in $\alpha_{s}$ $(A^{t}_{c,exp})$ expressed
either in terms of rapidity $(A^{t}_{c,y})$ or pseudo-rapidity differences
$(A^{t}_{c,\eta})$, is rather moderate for the full off-shell and the full NWA
case. Specifically, the absolute value of $A^{t}_{c}$ increases by
$0.5\%-0.6\%$ (by about $20\%-30\%$ in the relative terms) when the expansion
is introduced. For the NWALOdecay approach, on the other hand, we observe a
larger increase by $1.1\%$. In addition, theoretical uncertainties due to the
scale dependence are substantially reduced for the expanded version of
$A^{t}_{c}$. In that case we have estimated that the uncertainties are of the
order of $15\%$, $18\%$ and $24\%$ depending on the approach used. These
uncertainties are similar in size to uncertainties given in Ref. Maltoni et
al. (2014) for $A^{t}_{c,\eta}$. Therefore, the two definitions, $A^{t}_{c}$
and $A^{t}_{c,exp}$, give reasonably consistent results especially in the full
off-shell and full NWA case as it can be nicely visualised in Figure 5. We can
also note that the results for $A^{t}_{c,y}$ differ by almost $1\%$ (by about
$40\%-50\%$ in the relative terms) from those for $A^{t}_{c,\eta}$. Only in
the case of NWALOdecay is the difference smaller, i.e. $0.3\%$ ($15\%-35\%$ in
the relative terms). Finally, similar remarks apply to the charge asymmetry in
$t\bar{t}W^{-}$ production.

A few comments are in order here. The $A^{t}_{c}$ asymmetry in
$t\bar{t}W^{\pm}$ production is rather small so any effect can have
substantial influence on its absolute value. From Ref. Broggio et al. (2019)
we already know, that the inclusion of the electroweak corrections increases
the asymmetry only by a small amount, i.e. by about $0.16\%$. On the other
hand, NNLO QCD corrections, which will stabilise the size of the theoretical
uncertainties, can play a crucial role for $A^{t}_{c}$. Although NNLO QCD
corrections to $t\bar{t}W^{\pm}$ production with $gg\to
t\bar{t}W^{\pm}q\bar{q}^{\prime}$ processes are completely symmetric and can
contribute only to the denominator of $A^{t}_{c}$, they might still alter the
value of $A^{t}_{c}$. First studies presented in Ref. Broggio et al. (2019),
with the approximate NNLO888The approximate NNLO predictions from Ref. Broggio
et al. (2019) are evaluated by adding to the NLO results the ${\cal
O}(\alpha_{s}^{2})$ term of the expansion of the NNLL soft gluon resummation
of the cross section. The soft gluon resummation concerns, however, only the
Born process. to $t\bar{t}W^{\pm}$ where stable top quarks and $W^{\pm}$ gauge
bosons are considered, have shown, however, that $A^{t}_{c}$ increases by less
by $0.6\%$ and $0.8\%$ percent for the $m(t\bar{t}W^{\pm})$ and $H_{T}$ based
scale choices. By including these higher order effects the scale dependence is
reduced to $\delta_{\rm scale}={}^{+6.0\%}_{-3.0\%}$ Broggio et al. (2019).
Finally, similar effects have been observed for the $pp\to t\bar{t}$ process
at the LHC for $\sqrt{s}=8$ TeV Czakon et al. (2018) where among others the
inclusive top quark charge asymmetry $A_{c,y}^{t}$ and $A_{c,exp,y}^{t}$ at
NLO QCD and NNLO QCD has been studied together with NLO electroweak
corrections. Even-though there is no close analogy between the higher order
corrections to $t\bar{t}$ and $t\bar{t}W^{\pm}$ production as these processes
are dominated by different partonic channels and should thus be regarded as
uncorrelated/dissimilar, it is interesting to see the similar (relative) size
of the various effects for $A^{t}_{c,y}$ and $A^{t}_{c,exp,y}$. The absolute
value of the top quark charge asymmetry for the $pp\to t\bar{t}$ process at
the LHC is of course much smaller.

In Table 4 and in Figure 6 we present results for the charge asymmetries of
the top quark decay products $A^{\ell}_{c}$ and $A^{b}_{c}$ for
$\mu_{R}=\mu_{F}=m_{t}+m_{W}/2$. Not only are these asymmetries much larger,
but the reconstruction of the top quarks is also not required. Moreover, the
advantage of the $A^{\ell}_{c}$ observable in comparison to $A^{b}_{c}$ lies
in the fact that measurements of leptons are particularly precise at the LHC
due to the excellent lepton energy resolution of the ATLAS and CMS detectors.
For $A^{b}_{c}$, on the other hand, good $b$-jet tagging efficiency and low
light jet misstag rate would be mandatory. For the full off-shell and full NWA
case the difference between $A^{\ell}_{c}$ and $A^{\ell}_{c,exp}$ is $0.9\%$
in absolute terms, thus within theoretical uncertainties of $A^{\ell}_{c,exp}$
. Only in the NWALOdecay case does it increase to $1.9\%$, which is above the
theoretical scale uncertainty $\delta_{scale}$ even for $A^{\ell}_{c}$.
Overall, theoretical uncertainties for $A^{\ell}_{c,exp}$ are below $15\%$
independently of the approach employed. Thus, they are slightly higher than in
Ref. Maltoni et al. (2014). For $A^{b}_{c}$ the situation is very stable. We
observe small $0.1\%-0.2\%$ changes between $A^{b}_{c}$ and $A^{b}_{c,exp}$.
Theoretical uncertainties of the order of $1\%$ are estimated, which is
similar in size to theoretical errors quoted in Maltoni et al. (2014). We can
conclude this part by saying that the full NWA description is sufficient to
describe $A^{t}_{c}$, $A^{b}_{c}$ and $A^{\ell}_{c}$. The inclusion of the
complete off-shell effects for the $pp\to t\bar{t}W^{+}$ process increases the
central values of the asymmetries while at the same time the theoretical
errors are kept almost unchanged. We have also shown that in the case of the
top quark charge asymmetry and the asymmetries of the top quark decay products
the NLO QCD corrections to the top quark decays play a crucial role.

$t\bar{t}W^{+}$ | Off-shell | Full NWA | $\textsc{NWA}_{\scriptsize\mbox{LOdecay}}$
---|---|---|---
$\mu_{0}=m_{t}+m_{W}/2$ |  |  |
$A_{c,y}^{b}$ $[\%]$ | $6.46(8)^{+0.05\,(0.8\%)}_{-0.05\,(0.8\%)}$ | $6.18(4)^{+0.13(2.1\%)}_{-0.05(0.8\%)}{}$ | $5.99(3)^{+0.10\,(1.7\%)}_{-0.01\,(0.2\%)}$
$A_{c,exp,y}^{b}$ $[\%]$ | $6.56(10)^{+0.02\,(0.3\%)}_{-0.07\,(1.1\%)}$ | $6.28(4)^{+0.03(0.5\%)}_{-0.01\,(0.1\%)}$ | $6.21(5)^{+0.06\,(1.0\%)}_{-0.01\,(0.2\%)}$
$A_{c,y}^{\ell}$ $[\%]$ | $-7.90(10)^{+2.15\,(27\%)}_{-1.39\,(17\%)}$ | $-8.43(4)^{+2.10(25\%)}_{-1.37(16\%)}{}$ | $-10.11(3)^{+1.36\,(13\%)}_{-0.95\,(9.4\%)}$
$A_{c,exp,y}^{\ell}$ $[\%]$ | $-7.00(12)^{+1.00\,(14\%)}_{-0.80\,(11\%)}$ | $-7.52(4)^{+0.95(13\%)}_{-0.78(10\%)}$ | $-8.23(5)^{+1.01\,(12\%)}_{-0.79\,(9.6\%)}$
Table 4: Unexpanded and expanded $A_{c}^{\ell}$ and $A_{c}^{b}$ asymmetries at
NLO in QCD for $pp\to t\bar{t}W^{+}$ in multi-lepton channel at the LHC with
$\sqrt{s}=13$ TeV. Various approaches for the modelling of the top quark
production and decays are considered: the full off-shell case, the full NWA
and the NWALOdecay case. Also given are Monte Carlo (in parenthesis)
integration and theoretical errors. The NNPDF3.0 PDF set is employed and
$\mu_{R}=\mu_{F}=\mu_{0}$ where $\mu_{0}=m_{t}+m_{W}/2$.

Figure 6:  As in Figure 5 but for $A^{b}_{c}$ and $A^{\ell}_{c}$.

Our conclusions are not changed when the dynamical scale choice,
$\mu_{0}=H_{T}/3$, is employed instead. Results for $A^{t}_{c,}$,
$A^{\ell}_{c}$ and $A^{b}_{c}$ at NLO in QCD with $\mu_{0}=H_{T}/3$ are shown
in Table 5 and Table 6. When comparing to the theoretical predictions for
$\mu_{0}=m_{t}+m_{W}/2$ we can notice an overall agreement, within
$0.1\sigma-0.7\sigma$, between all central values of the asymmetries. In
addition, similar theoretical uncertainties due to the scale dependence are
estimated for both scale choices.

$t\bar{t}W^{+}$ | Off-shell | Full NWA | $\textsc{NWA}_{\scriptsize\mbox{LOdecay}}$
---|---|---|---
$\mu_{0}=H_{T}/3$ |  |  |
$A_{c,y}^{t}$ $[\%]$ | $2.36(8)^{+1.19\,(50\%)}_{-0.77\,(33\%)}$ | $1.93(5)^{+1.23\,(64\%)}_{-0.72\,(37\%)}$ | $1.11(3)^{+0.55\,(49\%)}_{-0.53\,(48\%)}$
$A_{c,exp,y}^{t}$ $[\%]$ | $2.66(10)^{+0.38\,(14\%)}_{-0.34\,(13\%)}$ | $2.20(5)^{+0.45(20\%)}_{-0.31(14\%)}$ | $2.08(5)^{+0.24\,(11\%)}_{-0.40\,(19\%)}$
$A_{c,\eta}^{t}$ $[\%]$ | $3.46(9)^{+1.41\,(41\%)}_{-0.90\,(26\%)}$ | $3.02(5)^{+1.44(48\%)}_{-0.93(31\%)}$ | $1.42(4)^{+0.59\,(41\%)}_{-0.56\,(39\%)}$
$A_{c,exp,\eta}^{t}$ $[\%]$ | $3.81(10)^{+0.46\,(12\%)}_{-0.40\,(10\%)}$ | $3.36(5)^{+0.48(14\%)}_{-0.43(13\%)}$ | $2.42(5)^{+0.27\,(11\%)}_{-0.44\,(18\%)}$
Table 5: As in Table 3 but for $\mu_{0}=H_{T}/3$. $t\bar{t}W^{+}$ | Off-shell | Full NWA | $\textsc{NWA}_{\scriptsize\mbox{LOdecay}}$
---|---|---|---
$\mu_{0}=H_{T}/3$ |  |  |
$A_{c,y}^{b}$ $[\%]$ | $6.48(9)^{+0.04\,(0.6\%)}_{-0.05\,(0.8\%)}$ | $6.16(4)^{+0.07(1.1\%)}_{-0.01\,(0.2\%)}$ | $6.05(3)^{+0.02\,(0.3\%)}_{-0.01\,(0.2\%)}$
$A_{c,exp,y}^{b}$ $[\%]$ | $6.53(10)^{+0.03\,(0.4\%)}_{-0.08\,(1.2\%)}$ | $6.21(5)^{+0.09(1.4\%)}_{-0.05(0.8\%)}$ | $6.23(5)^{+0.02\,(0.3\%)}_{-0.04\,(0.6\%)}$
$A_{c,y}^{\ell}$ $[\%]$ | $-7.46(11)^{+2.46\,(33\%)}_{-1.55\,(21\%)}$ | $-7.94(4)^{+2.45(31\%)}_{-1.54(19\%)}$ | $-9.81(4)^{+1.46\,(15\%)}_{-1.03\,(10\%)}$
$A_{c,exp,y}^{\ell}$ $[\%]$ | $-6.93(13)^{+1.01\,(14\%)}_{-0.81\,(12\%)}$ | $-7.43(5)^{+0.99(13\%)}_{-0.79(11\%)}{}$ | $-8.14(5)^{+1.00\,(12\%)}_{-0.81\,(10\%)}$
Table 6: As in Table 4 but for $\mu_{0}=H_{T}/3$.

Our state-of-the art results for the top quark charge asymmetry and for the
charge asymmetries of the top quark decay products are summarised in Table 7.
We provide the NLO QCD results for $A_{c,exp}^{t}$, $A_{c,exp}^{\ell}$ and
$A^{b}_{c,exp}$. They are calculated from the theoretical predictions, which
include the full top quark off-shell effects. We present results for $pp\to
t\bar{t}W^{+}$ and $pp\to t\bar{t}W^{-}$ in the multi-lepton channel at the
LHC with $\sqrt{s}=13$ TeV. We additionally present the combined results for
the $pp\to t\bar{t}W^{\pm}$ process. Also in this case the results for the top
quark charge asymmetry are given in terms of rapidities,
$\Delta|y|=|y_{t}|-|y_{\bar{t}}|$, and pseudo-rapidities,
$\Delta|\eta|=|\eta_{t}|-|\eta_{\bar{t}}|$. A comment on the difference in
size of asymmetries for $pp\to t\bar{t}W^{+}$ and $pp\to t\bar{t}W^{-}$ is in
order. The asymmetries are larger for $pp\to t\bar{t}W^{+}$ than for $pp\to
t\bar{t}W^{-}$. Otherwise, however, they behave similarly. As pointed out in
Ref. Maltoni et al. (2014) this can be understood by applying an argument
based on parton luminosities. At the LO the $t\bar{t}W^{+}$ process is
produced predominantly via $u\bar{d}$ whereas for $t\bar{t}W^{-}$ the
$\bar{u}d$ subprocess is the most relevant one. The longitudinal momenta of
the initial partons are on average $p_{u}>p_{d}>p_{\bar{u}}\approx
p_{\bar{d}}$. In both cases the momentum of the top and anti-top quarks is
connected to the momentum of the $q$ and $\bar{q}$ respectively. The large
longitudinal momentum transferred to the top quark from the initial $u$ quark
in the $t\bar{t}W^{+}$ case increases the corresponding $|y_{t}|$ value.
Consequently, the charge asymmetry of the top quark is enhanced compared to
the one calculated for $t\bar{t}W^{-}$. When analysing the combined results
for $pp\to t\bar{t}W^{\pm}$ we can observe that the theoretical uncertainties
due to the scale dependence reach up to $15\%$. The scale choice plays no role
here as for $\mu_{0}=m_{t}+m_{W}/2$ and $\mu_{0}=H_{T}/3$ similar results are
obtained.

$\mu_{0}=m_{t}+m_{W}/2$ | $t\bar{t}W^{+}$ | $t\bar{t}W^{-}$ | $t\bar{t}W^{\pm}$
---|---|---|---
$A_{c,exp,y}^{t}$ $[\%]$ | $~{}~{}~{}2.62^{+0.39\,(15\%)}_{-0.34\,(13\%)}$ | $~{}~{}~{}1.97^{+0.31\,(16\%)}_{-0.25\,(13\%)}$ | $~{}~{}~{}2.40^{+0.37\,(15\%)}_{-0.31\,(13\%)}$
$A_{c,exp,\eta}^{t}$ $[\%]$ | $~{}~{}~{}3.70^{+0.46\,(12\%)}_{-0.40\,(11\%)}$ | $~{}~{}~{}1.31^{+0.32\,(24\%)}_{-0.25\,(19\%)}$ | $~{}~{}~{}2.87^{+0.41\,(14\%)}_{-0.35\,(12\%)}$
$A_{c,exp,y}^{b}$ $[\%]$ | $~{}~{}~{}6.56^{+0.02\,(0.3\%)}_{-0.07\,(1.1\%)}$ | $~{}~{}~{}4.80^{+0.05\,(1.0\%)}_{-0.05\,(1.0\%)}$ | $~{}~{}5.93^{+0.03\,(0.5\%)}_{-0.08\,(1.3\%)}$
$A_{c,exp,y}^{\ell}$ $[\%]$ | $-7.00^{+1.00\,(14\%)}_{-0.80\,(11\%)}$ | $-5.68^{+0.78\,(14\%)}_{-0.61\,(11\%)}$ | $-6.51^{+0.93\,(14\%)}_{-0.74\,(11\%)}$
$\mu_{0}=H_{T}/3$ | $t\bar{t}W^{+}$ | $t\bar{t}W^{-}$ | $t\bar{t}W^{\pm}$
$A_{c,exp,y}^{t}$ $[\%]$ | $~{}~{}~{}2.66^{+0.38\,(14\%)}_{-0.34\,(13\%)}$ | $~{}~{}~{}2.05^{+0.33\,(16\%)}_{-0.27\,(13\%)}$ | $~{}~{}~{}2.45^{+0.37\,(15\%)}_{-0.31\,(13\%)}$
$A_{c,exp,\eta}^{t}$ $[\%]$ | $~{}~{}~{}3.81^{+0.46\,(12\%)}_{-0.40\,(10\%)}$ | $~{}~{}~{}1.31^{+0.33\,(25\%)}_{-0.26\,(20\%)}$ | $~{}~{}~{}2.94^{+0.42\,(14\%)}_{-0.35\,(12\%)}$
$A_{c,exp,y}^{b}$ $[\%]$ | $~{}~{}~{}6.53^{+0.03\,(0.4\%)}_{-0.08\,(1.2\%)}$ | $~{}~{}~{}4.80^{+0.06\,(1.2\%)}_{-0.11\,(2.3\%)}$ | $~{}~{}~{}5.91^{+0.04\,(0.7\%)}_{-0.09\,(1.5\%)}$
$A_{c,exp,y}^{\ell}$ $[\%]$ | $-6.93^{+1.01\,(14\%)}_{-0.81\,(12\%)}$ | $-5.67^{+0.81\,(14\%)}_{-0.63\,(11\%)}$ | $-6.46^{+0.95\,(15\%)}_{-0.75\,(12\%)}$
Table 7: Expanded $A_{c}^{t}$, $A_{c}^{\ell}$ and $A_{c}^{b}$ at NLO in QCD
for $pp\to t\bar{t}W^{+}$ and $pp\to t\bar{t}W^{-}$ in the multi-lepton
channel at the LHC with $\sqrt{s}=13$ TeV. Results are obtained with the full
off-shell effects included. Also given are combined results for $pp\to
t\bar{t}W^{\pm}$ and theoretical uncertainties. The NNPDF3.0 PDF set is
employed and $\mu_{R}=\mu_{F}=\mu_{0}$ where $\mu_{0}=m_{t}+m_{W}/2$ and
$\mu_{0}=H_{T}/3$.

##  VI Differential and cumulative asymmetry

Figure 7:  The $p_{T}(\ell_{t}\ell_{\bar{t}})$-dependent differential (left
panel) and cumulative (right panel) $A_{c}^{\ell}$ asymmetry at NLO QCD for
$pp\to t\bar{t}W^{\pm}$ in the multi-lepton channel at the LHC with
$\sqrt{s}=13$ TeV. Various approaches for the modelling of the top quark
production and decays are considered. Also given are theoretical uncertainties
for the full off-shell case. For all approaches Monte Carlo errors are
provided for both differential and cumulative asymmetries. The NNPDF3.0 PDF
set is employed and $\mu_{R}=\mu_{F}=\mu_{0}$ where $\mu_{0}=H_{T}/3$.

Figure 8:  As in Figure 7 but for $|y(\ell_{t}\ell_{\bar{t}})|$.

Figure 9:  As in Figure 7 but for $M(\ell_{t}\ell_{\bar{t}})$.

In this part of the paper we present predictions for differential
$A_{c}^{\ell}$ asymmetry with respect to the following observables: transverse
momentum of the two charged leptons, $p_{T}(\ell_{t}\ell_{\bar{t}})$, rapidity
of the two charged leptons, $|y(\ell_{t}\ell_{\bar{t}})|$, and invariant mass
of the two charged leptons, $M({\ell_{t}\ell_{\bar{t}}})$, where
$\ell_{t}\ell_{\bar{t}}$ originate from the $t\bar{t}$ pair. The differential
results are given using the unexpanded definition from Eq. (18). We also
present predictions for cumulative asymmetries, that are closely related to
the corresponding differential asymmetries. One can employ the same definition
as in Eq. (18), however, this time for a given value of the kinematic variable
for which we compute the asymmetry the bin ranges from zero to that value.
Even though differential and cumulative asymmetries contain the same
information, the latter one behaves better simply because it is more
inclusive, i.e. the higher order corrections are distributed more uniformly
over the whole kinematic range. In addition, the cumulative asymmetry should
give the integrated one in the last bin assuming that the plotted range of the
corresponding differential distribution covers the whole available phase
space. In practice, we shall see that if the left-over phase space region is
negligible the integrated asymmetry can be recovered very accurately. Let us
note that differential asymmetries have been studied at the LHC for the $pp\to
t\bar{t}$ production process by both experimental collaborations ALTAS and
CMS, see e.g. Khachatryan et al. (2016c); Aaboud et al. (2018).

In Figure 7 the $p_{T}(\ell_{t}\ell_{\bar{t}})$-dependent differential and
cumulative $A_{c}^{\ell}$ asymmetry at NLO QCD for $pp\to t\bar{t}W^{\pm}$ in
multi-lepton channel at the LHC with $\sqrt{s}=13$ TeV is displayed. Various
approaches for the modelling of the top quark production and decays are
considered. Also given are theoretical uncertainties for the full off-shell
case. For all approaches Monte Carlo integration errors are provided for both
differential and cumulative asymmetries. For the
$p_{T}(\ell_{t}\ell_{\overline{t}})$-dependent differential asymmetry the MC
error is smaller than the theoretical one in all bins but the last. In the
last bin both uncertainties are comparable in size. The NNPDF3.0 PDF set is
employed and $\mu_{R}$ as well as $\mu_{F}$ are set to the common value
$\mu_{R}=\mu_{F}=\mu_{0}=H_{T}/3$. For the differential $A_{c}^{\ell}$
asymmetry the difference between the full off-shell result and the full NWA
case is in the $5\%-30\%$ range depending on the bin, yet within theoretical
uncertainties, that are of the order of $30\%$. We notice that this is not the
case for the last bin where the top quark off-shell effects affect
$A_{c}^{\ell}$ substantially. Specifically, they are above $60\%$. Also
theoretical uncertainties increase in that bin and are of the order of $50\%$.
Both values, however, are harder to specify more precisely due to the large
statistical errors. The NWALOdecay case, on the other hand, is outside the
scale dependence bands almost in the whole plotted range. The difference to
the full off-shell approach is larger, even up to $70\%$. A similar effect is
also visible for the cumulative asymmetry where a rather constant $30\%$
difference is noted for the NWALOdecay case. Finally, we note that the last
bin of the cumulative $A_{c}^{\ell}$ asymmetry gives $A_{c}^{\ell}=-7.02(8)$
for the complete off-shell case, $A_{c}^{\ell}=-7.30(4)$ for the full NWA and
$A_{c}^{\ell}=-8.94(3)$ for the NWALOdecay case, where in parentheses the MC
error is displayed. All three results are indeed in perfect agreement within
the MC errors with the corresponding results for the unexpanded leptonic
charge asymmetry for the combined $pp\to t\bar{t}W^{\pm}$ process.

Similar observations can be made for the other two differential and cumulative
$A_{c}^{\ell}$ asymmetries. The $|y(\ell_{t}\ell_{\bar{t}})|$\- and
$M(\ell_{t}\ell_{\bar{t}})$-dependent versions of $A_{c}^{\ell}$ are exhibited
in Figure 8 and Figure 9 respectively. Since in each case the $y$ axis is
chosen to be the same for the differential and cumulative version of
$A_{c}^{\ell}$ we can distinctly observe that the cumulative asymmetry has
smaller fluctuations of the theoretical errors and it is smoother due to
better statistical errors. Taking the differential
$M(\ell_{t}\ell_{\bar{t}})$-dependent version of $A_{c}^{\ell}$ as an example
we observe large, of the order of $100\%$, theoretical uncertainties at the
tails. On the other hand, the cumulative $M(\ell_{t}\ell_{\bar{t}})$-dependent
$A_{c}^{\ell}$ asymmetry has stable theoretical uncertainties of the order of
$30\%$ in the whole plotted range.

We summarise by noting, that several processes beyond the SM can alter
$A_{c}$, see e.g. Jung et al. (2011); Ferrario and Rodrigo (2008); Aguilar-
Saavedra and Perez-Victoria (2011a); Maltoni et al. (2014); Alvarez et al.
(2020), either with anomalous vector or axial-vector couplings or via
interference with SM processes. Different models also predict different
asymmetries as a function of the invariant mass and the transverse momentum,
see e.g. Aguilar-Saavedra and Perez-Victoria (2011b). Of course due to the
much smaller cross section the $pp\to t\bar{t}W^{\pm}$ process will not
replace the use of the asymmetries in $t\bar{t}$ production, however, it can
provide a complementary tool as it is uniquely sensitive to the chiral nature
of possible new physics that might manifest itself in this channel. This
motivates our interest in the top quark charge asymmetry as well as the
asymmetries of its decay products and their sensitivity to the top quark
production and decay modelling. Furthermore, using our NLO QCD results with
the full off-shell effects included, we are able to provide more precise
theoretical predictions for $A_{c}^{t}$, $A_{c}^{\ell}$ and $A_{c}^{b}$ in the
$t\bar{t}W^{\pm}$ production process at the LHC with $\sqrt{s}=13$ TeV.
Finally, having at hand the full theory with no approximations included we are
able to study the real size of theoretical uncertainties due to the scale
dependence. In other words to verify whether they are under- or overestimated
in the presence of various approximations.

##  VII Summary

In this paper we provided the state-of-the-art theoretical predictions for
observables, which might be used to constrain numerous new physics scenarios
in the $t\bar{t}W^{\pm}$ channel. We considered the $t\bar{t}W^{\pm}$
production process in the multi-lepton decay channel for the LHC Run II energy
of $\sqrt{s}=13$ TeV for which discrepancies in the overall normalisation and
in the modelling of the top quark decays have been recently reported by the
ATLAS collaboration. Without the need of including terms beyond NLO in the
perturbative expansion in $\alpha_{s}$ we obtained $1\%-2\%$ theoretical
uncertainties due to the scale dependence for this process by calculating the
following cross section ratio ${\cal
R}=\sigma_{t\bar{t}W^{+}}/\sigma_{t\bar{t}W^{-}}$. The PDF uncertainties for
${\cal R}$ are similar in size. Fully realistic NLO QCD calculations have been
employed in our studies for both $t\bar{t}W^{+}$ and $t\bar{t}W^{-}$.
Specifically, we use
$e^{+}\nu_{e}\,\mu^{-}\bar{\nu}_{\mu}\,e^{+}\nu_{e}\,b\bar{b}$ and
$e^{-}\bar{\nu}_{e}\,\mu^{+}{\nu}_{\mu}\,e^{-}\bar{\nu}_{e}\,b\bar{b}$ matrix
elements in our NLO QCD calculations. They include all resonant and non-
resonant top quark and $W$ gauge boson Feynman diagrams, their interference
effects as well as off-shell effects of $t$ and $W$. We examined the fixed and
dynamical scale choice for $\mu_{R}$ and $\mu_{F}$ to assess their impact on
the cross section ratio. We noticed that the scale choice does not play any
role for such an inclusive observable. Indeed, although the scale variation is
taken correlated in both cases, the fact that the errors come out the same in
both cases means that tails of distributions, where the processes show more
differences do not matter for the analysis. Otherwise, the error estimate for
the fixed scale should be much larger, or the value of the ratio and
asymmetries should be shifted outside the error bands. In the next step we
examined the impact of the top quark production and decay modelling on the
cross section ratio. We observed that the full NWA approach does not modify
either the value or the size of the theoretical error for the integrated cross
section ratio. Even for the simplified version of the NWA, i.e. for the
NWALOdecays case, no changes have been observed. Thus, the ${\cal
R}=\sigma_{t\bar{t}W^{+}}/\sigma_{t\bar{t}W^{-}}$ observable is very stable
and insensitive to the details of the modelling of the top quark decay chain.
As such, it can be safely exploited at the LHC either for the precision SM
measurements or in searches for BSM physics. The ${\cal R}$ observable can be
used, for example, to provide valuable input for the up and down quark parton
distribution functions. In the case of new physics searches the presence of
two same-sign leptons in the final state offers a very interesting signature,
that has been highly scrutinised in many new physics models. The latter range
from supersymmetry and supergravity to the more specific scenarios with the
Majorana neutrinos and the modified Higgs boson sector. Given the final
accuracy of ${\cal R}$ and its insensitivity to the top quark modelling, the
${\cal R}$ observable might be used at the LHC, for example to achieve more
stringent limits on the parameter space of these models.

In the second part of the paper we reexamined the top quark charge asymmetry
and the charge asymmetries of the top quark decay products for
$t\bar{t}W^{+}$, $t\bar{t}W^{-}$ and $t\bar{t}W^{\pm}$ production in the
fiducial phase-space regions. Also in this case theoretical predictions with
the full off-shell effects were utilised. We presented predictions for the
expanded and unexpanded asymmetries. Overall, good agreement has been found
for $A^{t}_{c}$, $A^{\ell}_{c}$ and $A^{b}_{c}$ when comparing the full off-
shell case with the full NWA approach. For the NWALOdecay case, however,
discrepancies between central values of the asymmetries even up to $2\sigma$
have been found. The later fact indicates that NLO QCD corrections to the top
quark decays play a crucial role here. Generally, the inclusion of the
complete description for the $pp\to t\bar{t}W^{\pm}$ process in the multi-
lepton final state has increased the central values of the asymmetries keeping
at the same time the theoretical errors unchanged and below $15\%$. The scale
choice has played no role as for $\mu_{0}=m_{t}+m_{W}/2$ and $\mu_{0}=H_{T}/3$
similar results have been obtained for $A_{c}^{t}$, $A_{c}^{\ell}$ and
$A_{c}^{b}$.

As a bonus of our study, we presented predictions for the differential and
cumulative $A_{c}^{\ell}$ asymmetry with respect to
$p_{T}(\ell_{t}\ell_{\bar{t}})$, $|y(\ell_{t}\ell_{\bar{t}})|$ and
$M(\ell_{t}\ell_{\bar{t}})$. The advantage of choosing $A_{c}^{\ell}$ lies in
the fact that the measurements of the charged leptons are particularly precise
at the LHC due to the excellent lepton energy resolution of the ATLAS and CMS
detectors. We note here that for these studies the unexpanded version of
$A_{c}^{\ell}$ has been examined. Depending on the bin the differences between
the full off-shell results and the full NWA ones have been in the $5\%-30\%$
range. However, this is well within theoretical uncertainties, that are of the
order of $30\%$. On the other hand, large differences have been noticed for
the NWALOdecay case even up to $70\%$. Similarly for the cumulative asymmetry
the NWALOdecay curves are lying outside the uncertainty bands independently of
the observable and the considered bin. We would like to add here that even
though differential and cumulative asymmetries contain the same information,
the latter behaves better simply because it is more inclusive. In other words,
the higher order corrections are distributed more uniformly over the whole
kinematic range.

Last but not least, we would like to mention at this point that, several BSM
physics scenarios can alter the top quark charge asymmetry. Thus, theoretical
predictions for the $A_{c}^{t}$, $A_{c}^{\ell}$ and $A_{c}^{b}$ observables
should be as accurate as possible. Using our NLO QCD results with the full
off-shell effects included not only are we able to provide the state-of-the-
art theoretical predictions for $A_{c}^{t}$, $A_{c}^{\ell}$ and $A_{c}^{b}$ in
the $t\bar{t}W^{\pm}$ production process but also by the explicit comparison
to various NWA approaches we could carefully examine the impact of different
top-quark decay modelling accuracies on the scale uncertainties.

###### Acknowledgements.

This research of H.Y.B., J.N. and M.W was supported by the Deutsche
Forschungsgemeinschaft (DFG) under the following grants: 400140256 - GRK 2497:
The physics of the heaviest particles at the Large Hardon Collider and
396021762 - TRR 257: P3H - Particle Physics Phenomenology after the Higgs
Discovery. Support by a grant of the Bundesministerium für Bildung und
Forschung (BMBF) is additionally acknowledged. The work of G.B. was supported
by grant K 125105 of the National Research, Development and Innovation Office
in Hungary. H.B.H. has received funding from the European Research Council
(ERC) under the European Union’s Horizon 2020 Research and Innovation
Programme (grant agreement no. 683211 and 772099). Furthermore, the work of
H.B.H has been partially supported by STFC consolidated HEP theory grant
ST/T000694/1. Simulations were performed with computing resources granted by
RWTH Aachen University under project rwth0414.

##  Appendix A Kolmogorov-Smirnov test

In section III we have only used visual inspection to see whether two given
one-dimensional normalised cross section distributions are similar or not. We
stress that this is independent of other arguments that we provided to argue
for the similarity of the processes. Even though this is an excellent place to
start with, we would like to find a more quantitative approach to analyse the
issue. In statistics literature several standard procedures exist for this
task. Typically, the similarity of histograms is measured by a test statistic.
The latter provides the quantitative expression of the distance between the
two histograms that are compared. The smaller the distance the more similar
are the compared histograms. There are several definitions of the test
statistics in specialist literature on statistical methods. In the following
we shall concentrate on the Kolmogorov-Smirnov test (KS test) statistics. The
purpose of the (two-sample) KS test is to look for differences in the shape of
two one-dimensional probability distributions. It is based on comparing two
cumulative distribution functions (CDFs). The KS test reports on the maximum
difference between the two CDFs and calculates a $p$-value from that and the
sample sizes. If the two tested histograms are indeed identical then they
would have the same CDF. However, in reality two samples that are compared are
randomly taken from their corresponding probability distributions. Therefore,
even for the two truly identical histograms the corresponding CDFs will be
slightly different. We can use this fact to test the two distribution equality
by comparing the KS test statistic to $0$. If the latter is significantly
larger than $0$ and close to $1$, then we might conclude that the
distributions are not equal and the two processes considered are not similar.
We begin with the differential cross section distribution for $pp\to
t\bar{t}W^{+}$ and $pp\to t\bar{t}W^{-}$ as a function of the variable $x$,
where $x$ for example is $x=p_{T,\,e_{1}},M_{e_{1}e_{2}}$. When comparing both
histograms, we use the same number of bins. We would like to verify the
hypothesis that the two histograms are similar. To this end we calculate the
KS test statistics according to

${\rm KS}_{\rm statistic}=\sup_{x}|F^{1}_{n_{1}}(x)-F^{2}_{n_{2}}(x)|\,,$ (20)

where $F^{1}_{n_{1}}$ and $F^{2}_{n_{2}}$ are the CDFs, $n_{1}$ and $n_{2}$
are the sizes of the first and second sample respectively and $\sup$ is the
supremum function. We assume approximately $2000$ events for $pp\to
t\bar{t}W^{+}$ and about $1000$ for $pp\to t\bar{t}W^{-}$, which correspond to
the integrated LHC luminosity of ${\cal L}=500$ fb-1 including a lepton-
flavour factor of $8$. After finding the maximum distance, we use the
following condition

Figure 10:  The distribution of the Kolmogorov-Smirnov test statistic
(distance) for the null hypothesis of equality of the histogram shapes. NLO
QCD differential cross section distributions for $pp\to t\bar{t}W^{+}$ and
$pp\to t\bar{t}W^{-}$ in the multi-lepton final state are employed as a
function of $p_{T,\,e_{1}}$, $M_{e_{1}e_{2}}$, $H_{T}^{lep}$ and $\Delta
R_{e_{1}e_{2}}$ for the LHC with $\sqrt{s}=$13 TeV. Also shown are the
distributions of the corresponding $p$-values. The total number of $N_{tries}$
is set to 1000.

Figure 11:  The distribution of $p$-values for the Kolmogorov-Smirnov test
statistic for the NLO QCD differential cross section for $pp\to t\bar{t}W^{+}$
and $pp\to t\bar{t}W^{-}$ in the multi-lepton final state as a function of
$H_{T}^{vis}$ for the LHC with $\sqrt{s}=13$ TeV. A different number of bins
is assumed for each plot, however, the integrated luminosity is kept fixed.
Specifically, we use the following four cases: 5 bins (upper left), 10 bins
(upper right), 20 bins (lower left) and 40 bins (lower right). The total
number of $N_{tries}$ is set to 1000.

$\sqrt{n}\,\,{\rm KS}_{\rm statistic}\,>\lambda(\alpha)\,,$ (21)

where

$n=\frac{n_{1}\cdot n_{2}}{n_{1}+n_{2}}\,,$ (22)

with $n_{1}=2000$, $n_{2}=1000$ and $\lambda(\alpha)$ is the threshold value
that depends on the level of significance $\alpha$. It can be found from the
following condition

${\cal P}\left(\sqrt{n}\,\,{\rm KS}_{\rm
statistic}>\lambda(\alpha)\right)=1-{\cal Q}_{\rm
KS}\left(\lambda(\alpha)\right)=\alpha\,,$ (23)

where ${\cal P}$ denotes probability and ${\cal Q}_{\rm KS}(x)$ stands for the
Kolmogorov-Smirnov distribution. We reject the hypothesis that the two
distributions are similar if

$\sqrt{n}\,\,{\rm KS}_{\rm statistic}>\lambda(\alpha)\,,$ (24)

and accept it when

$\sqrt{n}\,\,{\rm KS}_{\rm statistic}\leq\lambda(\alpha)\,.$ (25)

We would normally start to question the hypothesis of the similarity of the
histograms only if we find a difference larger than $2\sigma$ (the $p$-value
smaller than $0.0455$). If the difference is smaller than $2\sigma$ (the
$p$-value larger than $0.0455$) then we assume that the two tested
distributions are indeed similar. Results that differ more than $3\sigma$ (the
$p$-value smaller than $0.0027$) can be directly translated into having enough
evidence to reject the hypothesis, i.e. saying that there is a real difference
between the two samples that are being studied. Note that the KS test does not
identify the source of the difference between histograms. It is a robust way
of saying that there is a difference, however, the origin of such a difference
must be identified by other means.

As the example in Figure 10 we present the distribution of the KS test
statistic for the following NLO QCD differential cross sections:
$p_{T,\,e_{1}}$, $M_{ee}$, $H_{T}^{lep}$ and $\Delta R_{ee}$ for $pp\to
t\bar{t}W^{+}$ and $pp\to t\bar{t}W^{-}$. The total number of tries is set to
$N_{tries}=1000$. All KS test statistic values are distributed within the
$0.01-0.07$ range, i.e. very close to zero, which suggests that $pp\to
t\bar{t}W^{+}$ and $pp\to t\bar{t}W^{-}$ are indeed correlated. Also shown in
Figure 10 are the distributions of the corresponding $p$-values for the KS
test statistic. We can observe that the $p$-values are mostly distributed in
the vicinity of $1$, again supporting the hypothesis that $pp\to
t\bar{t}W^{+}$ and $pp\to t\bar{t}W^{-}$ are highly correlated. We note here,
that similar results have been obtained for the kinematics of the $b$-jet and
for the $H_{T}^{vis}$ observable.

We would like to stress at this point, that for higher integrated luminosity
or for an increased number of bins, the sensitivity of the KS test increases
as well. As an example we present in Figure 11 the distribution of $p$-values
for the KS test statistic for the $H_{T}^{vis}$ observable for $pp\to
t\bar{t}W^{+}$ and $pp\to t\bar{t}W^{-}$. We use four different values for the
number of histogram bins, keeping the number of total events fixed for both
processes. Specifically, we employ 5, 10, 20 and 40 bins respectively. We can
observe that the percentage of $N_{tries}$ with the $p$-value close to $1$ is
getting lower as the number of bins increases.

We summarise this part by noting, that there are many test statistics for the
comparison of the shapes of two one-dimensional histograms. The most popular
are: the Pearson-$\chi^{2}$ test, the Anderson-Darling test or the Cramer-von-
Mises test, see e.g. Porter (2008). Each of these tests has its pros and cons
and it is not possible to choose the one test that is the best for all
applications. Overall, the more we know about what we really want to compare
and test, the more reliable the test we can choose for our particular problem.
We have examined all the above-mentioned tests and have decided to use the
Kolmogorov-Smirnov test of the equality. The two sample KS test assumes
continuous distributions. It is one of the most general nonparametric999The
nonparametric test does not assume that data points are sampled from the
Gaussian distribution or any other defined distribution for that matter. tests
for comparing two samples, as it is sensitive to differences in shape of the
empirical cumulative distribution functions of the two samples. It is also the
most robust test as it tests for any violation of the null hypothesis.
However, it requires a relatively large number of data points in each bin. We
further notice, that the KS test is more sensitive to the regions near the
peak of the tested distributions rather than to their tails. For the latter
the Anderson-Darling test would do a better job. This observation is very
useful in our case as for many dimensionful observables tails are usually
plagued by larger statistical fluctuations and are, therefore, not really
reliable for such comparisons.

Finally, we stress that the distributions of observables observed in the two
processes are not identical. Hence, the outcome of the test statistic does
depend on the number of events. With large numbers of events, the test
statistic would obviously discover that the distributions are different.
However, we are here interested in the question of ”how similar are the
distributions?” and not ”are the distributions identical?”. Quantifying
similarity can therefore be done by choosing a number of events. Had we taken
processes with very dissimilar distributions the $p$-value for the same number
of events as chosen here, would be much smaller, and we would conclude that
the distributions are less similar than in this particular case.

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