††institutetext:  Institute for Theoretical Particle Physics and Cosmology,
RWTH Aachen University,
D-52056 Aachen, Germany

# Complete NLO corrections to top-quark pair production with isolated photons

Daniel Stremmer  and Malgorzata Worek<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We compute for the first time the so-called complete NLO corrections to top-
quark pair production with one and two isolated photons in the di-lepton top-
quark decay channel. The Narrow Width Approximation is used for the modeling
of unstable top quarks and $W$ bosons. Higher-order QCD and EW effects as well
as photon bremsstrahlung are consistently included at all stages: in
production and top-quark decays. We present results at the integrated and
differential fiducial cross-section level for both processes for the LHC Run
II center-of-mass energy of $\sqrt{s}=13$ TeV. In addition, we investigate the
scale choice in photonic observables. Finally, the individual size of each
subleading contribution is discussed in detail and the origin of the main
subleading corrections is scrutinised. For the latter case, alternative
calculations are performed in which the subleading NLO corrections are
included only in the production of $t\bar{t}\gamma$ and
$t\bar{t}\gamma\gamma$.

###### Keywords:

Higher-Order Perturbative Calculations, Top Quark, Specific QCD Phenomenology

††dedication: TTK-24-06, P3H-24-005

##  1 Introduction

Of all the associated production of a top-quark pair with one electroweak
gauge boson $(pp\to t\bar{t}V,\,V=Z,W^{\pm},\gamma)$ at the LHC, the $pp\to
t\bar{t}\gamma$ process has the largest cross section Maltoni:2015ena ;
Schwienhorst:2022yqu  and is the most interesting to model. First of all,
already at the lowest order in the perturbative expansion, the $pp\to
t\bar{t}\gamma$ process requires various cuts to keep the prompt photon well
isolated and render the whole process infrared (IR) finite, even in the
presence of stable top quarks. Secondly, due to its massless nature, the
photon can be emitted from all stages of the process, i.e. in the production
of $t\bar{t}$ and top-quark decays. In the latter case, prompt photons can
originate not only from top quarks, but also from their decay products,
including the bottom quarks, $W^{\pm}$ gauge bosons and charged leptons. In
addition, they can be radiated from incoming partons. It is a well-known fact
that a large fraction of isolated photons comes from radiative decays of top
quarks Melnikov:2011ta ; Bevilacqua:2019quz . With fairly inclusive cuts
applied on the finale states, that are currently used in measurements of
inclusive and differential cross sections of $t\bar{t}\gamma$ production, the
contribution of photons at the decay stage reaches almost $50\%$.
Consequently, proper modeling of $pp\to t\bar{t}\gamma$ is very challenging.
The $pp\to t\bar{t}\gamma$ process probes the $t-\gamma$ electroweak coupling
and provides a direct way to measure the top quark electric charge Baur:2001si
. The latter is known to be consistent with the Standard Model (SM), although
it was measured only indirectly in the production of $t\bar{t}$ D0:2014oid ;
ATLAS:2013mkl . Precise measurements of the $t-\gamma$ coupling at the LHC
serve as an additional test of the SM. However, any deviations from the SM
prediction, for example in the $p_{T}$ spectrum of the photon, could point to
new physics through anomalous dipole moments of the top quark Baur:2004uw ;
Bouzas:2012av ; Rontsch:2015una ; Schulze:2016qas ; BessidskaiaBylund:2016jvp
. In addition, the $pp\to t\bar{t}\gamma$ process plays an important role in
studies of the top-quark charge asymmetry $(A_{C}^{t})$ Aguilar-
Saavedra:2014vta ; Aguilar-Saavedra:2018gfv ; Bergner:2018lgm ; Pagani:2021iwa
. Indeed, it provides complementary information to the measured asymmetries in
$t\bar{t}$ production, where $A_{C}^{t}$ appears for the first time at next-
to-leading order (NLO) only. Contrary, for the $pp\to t\bar{t}\gamma$ process
this asymmetry is present already at LO in quark-induced subprocesses due to
the interference effects between Feynman diagrams in which the photon is
emitted from quarks in the initial state and diagrams in which it is emitted
from quarks in the final state. The overall asymmetry in the $pp\to
t\bar{t}\gamma$ process at the LHC with $\sqrt{s}=13$ TeV has a negative value
and is of the order of $1\%-2\%$ depending on the fiducial phase-space regions
that are scrutinised Bergner:2018lgm ; Pagani:2021iwa . However, $A_{C}^{t}$
can be modified by beyond the SM (BSM) physics. Indeed, substantial deviations
from the SM prediction can be expected in the case of BSM models with a light
colour octet Barcelo:2011vk ; MarquesTavares:2011spp ; Alvarez:2011hi ;
Aguilar-Saavedra:2011lzv  or an additional $Z^{\prime}$ Alvarez:2013jqa . In
both BSM cases, the absolute value of $A_{C}^{t}$ is predicted to be much
smaller Aguilar-Saavedra:2014vta .

In the case of the $pp\to t\bar{t}\gamma\gamma$ process, the situation is only
somewhat similar to the case of $t\bar{t}\gamma$. Among all the associated
production of a top-quark pair with two gauge vector bosons, i.e.
$t\bar{t}W^{+}W^{-}$, $t\bar{t}ZZ$, $t\bar{t}\gamma\gamma$,
$t\bar{t}W^{\pm}\gamma$, $t\bar{t}W^{\pm}Z$ and $t\bar{t}Z\gamma$, the $pp\to
t\bar{t}\gamma\gamma$ process is the second largest Maltoni:2015ena . The
importance of $pp\to t\bar{t}\gamma\gamma$ stems from the fact that it is the
main (irreducible) background process in SM Higgs-boson studies for the $pp\to
t\bar{t}H$ signal process in the $H\to\gamma\gamma$ decay channel. However,
the similarities to the production of $t\bar{t}\gamma$ are due to the
distribution of photons in the $pp\to t\bar{t}\gamma\gamma$ process and the
importance of the contribution, which is not entirely related to the
production stage of the process. The effect of photon bremsstrahlung in the
$pp\to t\bar{t}\gamma\gamma$ process has recently been studied in detail
Stremmer:2023kcd . It has been shown that the so-called mixed contribution, in
which two photons occur simultaneously in the production and decay of the
$t\bar{t}$ pair, is the dominant contribution at the integrated and
differential fiducial cross-section level.

Evidence for the production of a top-quark pair in association with an
isolated photon has already been reported in $p\bar{p}$ collisions at the
Tevatron collider at $\sqrt{s}=1.96$ TeV by the CDF Collaboration CDF:2011ccg
. The production of $t\bar{t}\gamma$ final states has been observed for the
first time at the LHC at $\sqrt{s}=7$ TeV by the ATLAS Collaboration
ATLAS:2015jos . To date, both ATLAS and CMS have observed the production of
$t\bar{t}\gamma$ at the LHC at $\sqrt{s}=8$ TeV and $\sqrt{s}=13$ TeV
ATLAS:2017yax ; CMS:2017tzb ; CMS:2021klw ; ATLAS:2018sos ; ATLAS:2020yrp ;
CMS:2022lmh , respectively. So far, no significant deviations from the SM
predictions have been found, even though the measured cross sections are
larger than theoretical predictions. In addition, within the current and still
rather large uncertainties also, the analysed differential cross-section
distributions have been fairly well described by the NLO theory predictions.
The measurements in the $pp\to t\bar{t}\gamma$ process have also been
interpreted in the framework of the standard model effective field theory,
where rather stringent limits on the two relevant Wilson coefficients have
been found. Furthermore, the measurement of the top-quark charge asymmetry in
$pp\to t\bar{t}\gamma$ has recently been performed by the ATLAS collaboration
ATLAS:2022wec . Also in this case the measurement is compatible with the SM
predictions within the present uncertainties. However, in all these cases,
current precision is still limited, mainly due to statistical and various
systematic uncertainties, leaving room for potential future improvements.
Finally, the $pp\to t\bar{t}\gamma\gamma$ process has not yet been
experimentally observed at the LHC.

On the theory side, NLO QCD corrections to the $pp\to t\bar{t}\gamma$
production process with stable top quarks have been calculated over ten years
ago Duan:2009kcr ; Duan:2011zxb  and calculated afresh in Ref. Maltoni:2015ena
. The results with NLO electroweak (EW) corrections have also been delivered
later Duan:2016qlc . The results presented there have shown that for
differential cross-section distributions the NLO EW corrections are
significant in the high energy region due to the EW Sudakov effect. Recently,
the so-called complete NLO predictions for $pp\to t\bar{t}\gamma$ have been
calculated Pagani:2021iwa , again for stable top quarks only. In addition to
the NLO QCD and EW corrections at ${\cal O}(\alpha_{s}^{3}\alpha)$ and ${\cal
O}(\alpha_{s}^{2}\alpha^{2})$ respectively, various subleading contributions,
along with their higher-order effects, have been taken into account to form
the complete NLO result. In detail, the set of all the possible contributions
at ${\cal O}({\alpha_{s}^{2}\alpha})$, ${\cal O}({\alpha_{s}\alpha^{2}})$ and
${\cal O}({\alpha_{s}^{0}\alpha^{3}})$ at LO, where the middle one results
from the interference of the other two results at the amplitude level and from
the photon initiated partonic process $g\gamma\to t\bar{t}\gamma$, as well as
at ${\cal O}({\alpha_{s}^{3}\alpha})$, ${\cal O}({\alpha_{s}^{2}\alpha^{2}})$,
${\cal O}({\alpha_{s}\alpha^{3}})$ and ${\cal O}({\alpha_{s}^{0}\alpha^{4}})$
at NLO is what is denoted as the complete NLO result. Finally, the approximate
NNLO cross section, with second-order soft-gluon corrections added to the NLO
result including QCD and EW corrections, has been calculated in Ref.
Kidonakis:2022qvz . On top of the stable top-quark approximation, various
predictions that take into account top-quark decays are available in the
literature. First, NLO QCD theoretical predictions for $t\bar{t}\gamma$ have
been matched with the Pythia parton shower program Kardos:2014zba . In this
approach top-quark decays have been treated in the parton-shower approximation
omitting $t\bar{t}$ spin correlations and photon emission in the parton-shower
evolution. More realistic predictions at NLO in QCD have been presented in
Ref. Melnikov:2011ta . In this case, top-quark decays in the Narrow Width
Approximation (NWA) have been included, maintaining $t\bar{t}$ spin
correlations in the top-quark decay products. In addition, photon radiation
off charged top-quark decay products has been incorporated. A complete
description of the $pp\to t\bar{t}\gamma$ process in the di-lepton top-quark
decay channel at NLO in QCD has also been finally provided Bevilacqua:2018woc
. This calculation is based on matrix elements for the
$e^{+}\nu_{e}\mu^{-}\bar{\nu}_{\mu}b\bar{b}\gamma$ final state including all
resonant and non-resonant Feynman diagrams, interferences and off-shell
effects of the top quarks and $W$ gauge bosons. Finally, a dedicated
comparison between the full off-shell calculation and the results in the NWA
has also been carried out Bevilacqua:2019quz . For the $pp\to
t\bar{t}\gamma\gamma$ process the situation is much more simpler. NLO QCD
predictions for the process with stable top quarks have already been known for
some time and have been further matched to parton shower programs
Alwall:2014hca ; Kardos:2014pba ; Maltoni:2015ena ; vanDeurzen:2015cga . In
addition to higher-order QCD effects, NLO EW corrections for
$t\bar{t}\gamma\gamma$ have recently been reported Pagani:2021iwa , but again
only for stable top quarks. Finally, very recently NLO QCD corrections in the
NWA for the di-lepton and lepton $+$ jet top-quark decay channels have been
calculated for the $pp\to t\bar{t}\gamma\gamma$ process Stremmer:2023kcd .

From the above description, we can quite clearly see a rather diverse picture
for $pp\to t\bar{t}\gamma$ and $pp\to t\bar{t}\gamma\gamma$. On the one hand,
we already have the complete NLO result for $t\bar{t}\gamma$, albeit only for
stable top quarks. On the other hand, no such predictions exist for
$t\bar{t}\gamma\gamma$. In both cases, the complete NLO result with more
realistic final states is still missing. With this article, we want to
mitigate the current situation and to calculate for the first time the
complete NLO predictions for both $pp\to t\bar{t}\gamma$ and $pp\to
t\bar{t}\gamma\gamma$ in the di-lepton top-quark decay channel.

The aim of our work is, therefore, manifold. Firstly, we perform a detailed
study of the predictions with complete NLO corrections included for $pp\to
t\bar{t}\gamma$ and $pp\to t\bar{t}\gamma\gamma$ in the di-lepton top-quark
decay channel. Secondly, as we calculate these higher order effects within the
same framework, while using the same input parameters, we want to examine
whether there are common features or significant differences between the
theoretical predictions for $pp\to t\bar{t}\gamma$ and $pp\to
t\bar{t}\gamma\gamma$. Thirdly, we plan to analyse the impact of subleading
NLO corrections on integrated and differential fiducial cross sections and
systematically study the theoretical uncertainties due to missing higher
orders. Finally, we are going to scrutinise the individual size of each
subleading contribution and investigate the origin of the main subleading
corrections. For this purpose, we perform alternative calculations for both
processes in which the subleading NLO corrections are included only in the
production of $t\bar{t}\gamma(\gamma)$.

The rest of the paper is organised as follows. All leading and subleading
contributions at LO and NLO are carefully defined in Section 2. In Section 3
we describe the framework of the calculations and discuss various changes as
well as cross-checks that have been performed. All input parameters and
fiducial cuts that have been used to simulate detector response are outlined
in Section 4. Numerical results for the integrated and differential cross
sections for the $pp\to t\bar{t}\gamma$ process at the LHC are presented in
detail in Section 5. Our results for the $pp\to t\bar{t}\gamma\gamma$ process,
on the other hand, are discussed in Section 6. We summarise our findings in
Section 7.

##  2 Definition of the leading and subleading contributions

We calculate the full set of leading and subleading LO contributions and NLO
corrections to top-quark pair production with one and two isolated photons at
the LHC. We consistently include photon bremsstrahlung as well as QCD and EW
corrections in both the production and decay of the top-quark pair. Decays of
unstable top quarks and $W$ gauge bosons are treated in the NWA preserving
spin correlations, i.e. in the limit when $\Gamma_{t}/m_{t}\to 0$
$(\Gamma_{W}/m_{W}\to 0)$. In this approximation all contributions without two
resonant top quarks (and $W$ gauge bosons) are neglected and the Breit-Wigner
propagators lead to delta-functions which force the unstable particles to be
on-shell, see e.g. Ref. Denner:1999gp ; Denner:2005fg ; Melnikov:2009dn ;
Melnikov:2011qx ; Campbell:2012uf ; Behring:2019iiv ; Bevilacqua:2019quz ;
Czakon:2020qbd  for more details. We consider the di-lepton top-quark decay
channel leading to the following decay chains

$\begin{split}pp&\to t\bar{t}(\gamma)\to
W^{+}W^{-}\,b\bar{b}(\gamma)\to\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma+X,\\\\[5.69046pt]
pp&\to t\bar{t}(\gamma\gamma)\to
W^{+}W^{-}\,b\bar{b}(\gamma\gamma)\to\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma\gamma+X,\end{split}$
(1)

with $\ell^{\pm}=\mu^{\pm},e^{\pm}$ and where the brackets indicate that
photon bremsstrahlung is allowed at each stage of the process. For brevity, we
refer to these processes as $pp\to t\bar{t}\gamma$ and $pp\to
t\bar{t}\gamma\gamma$ in the di-lepton top-quark decay channel, respectively.
At LO there are in total $5$ ($15$) possibilities or resonance histories from
which the photons can be radiated off in the decay chain of
$t\bar{t}\gamma(\gamma)$. At NLO, the number of resonance histories increases
to $15$ ($35$) for real radiation with an additional photon and to $15$ ($45$)
for additional QCD radiation.

###  2.1 LO contributions

Figure 1:  Example Feynman diagrams for $pp\to t\bar{t}\gamma$ contributing to
${\rm LO}_{1}$. All Feynman diagrams in this paper were produced with the help
of the FeynGame program Harlander:2020cyh .

At LO, both processes get contributions from three different orders of
$\alpha_{s}$ and $\alpha$. The dominant contribution is at the order
$\mathcal{O}(\alpha_{s}^{2}\alpha^{4+n_{\gamma}})$, where $n_{\gamma}$ is the
number of photons appearing in the Born-level process, and we call it ${\rm
LO}_{1}$, following the notation in Ref. Frederix:2017wme ; Frederix:2018nkq .
At this order, we encounter the typical QCD production of a top-quark pair,
which leads to the following partonic subprocesses

$\begin{array}[]{c}gg\to\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma(\gamma)\,,\\\\[5.69046pt]
\begin{array}[]{clllcll}q\bar{q}/\bar{q}q&\to&\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma(\gamma)\,,&&b\bar{b}/\bar{b}b&\to&\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma(\gamma)\,,\end{array}\end{array}$
(2)

with $q=u,d,c,s$. Example Feynman diagrams for the ${\rm LO}_{1}$ contribution
are shown in Figure 1. We work in the five-flavour scheme and consistently
include in the initial state all PDF suppressed channels from bottom quarks as
well as photons in all subleading contributions. The ${\rm LO}_{1}$
contribution can be obtained from the square of matrix elements at the order
$\mathcal{O}(g_{s}^{2}g^{4+n_{\gamma}})$. The $q\bar{q}$ and $b\bar{b}$
subprocesses provide additional contributions to the amplitudes at the order
$\mathcal{O}(g^{6+n_{\gamma}})$. The interference of both types of matrix
elements gives rise to the first subleading LO contribution at
$\mathcal{O}(\alpha_{s}^{1}\alpha^{5+n_{\gamma}})$ which we refer to as ${\rm
LO}_{2}$.

Figure 2:  Example Feynman diagrams for $pp\to t\bar{t}\gamma$ contributing to
${\rm LO}_{2}$.

This interference is exactly zero for the $q\bar{q}$ initial state due to
color algebra, but does not vanish for the $b\bar{b}$ subprocess due to
additional $t$-channel Feynman diagrams with an intermediate $W$ boson, as
shown on the right in Figure 2. Similar Feynman diagrams do not exist for
$q\bar{q}$ because we keep the Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix
diagonal. In addition, at this order we encounter for the first time photon-
induced channels with the $g\gamma$ initial state, as illustrated on the left
in Figure 2. Thus, the partonic subprocesses of ${\rm LO}_{2}$ can be
summarised as

$\begin{array}[]{c}\begin{array}[]{clllcll}g\gamma/\gamma
g&\to&\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma(\gamma)\,,&&b\bar{b}/\bar{b}b&\to&\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma(\gamma)\,.\end{array}\end{array}$
(3)

The third and last subleading LO contribution, ${\rm LO}_{3}$, is the purely
EW induced production of a top-quark pair at the order
$\mathcal{O}(\alpha^{6+n_{\gamma}})$. Example diagrams are depicted in Figure
3. Again, only the $q\bar{q}$ and $b\bar{b}$ channels are present, as well as
the highly suppressed $\gamma\gamma$ channel. Therefore, the following
reactions must be taken into account for ${\rm LO}_{3}$

Figure 3:  Example Feynman diagrams for $pp\to t\bar{t}\gamma$ contributing to
${\rm LO}_{3}$.

$\begin{array}[]{c}\gamma\gamma\to\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma(\gamma)\,,\\\\[5.69046pt]
\begin{array}[]{clllcll}q\bar{q}/\bar{q}q&\to&\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma(\gamma)\,,&&b\bar{b}/\bar{b}b&\to&\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma(\gamma)\,.\end{array}\end{array}$
(4)

With respect to ${\rm LO}_{1}$, this contribution is not only suppressed by
the power coupling, but also the gluon PDF does not enter this contribution at
all. Finally, we denote as ${\rm LO}$ the the sum of all three LO
contributions

${\rm LO}={\rm LO}_{1}+{\rm LO}_{2}+{\rm LO}_{3}\,.$ (5)

###  2.2 NLO contributions

Figure 4:  LO and NLO contributions for $pp\to t\bar{t}\gamma(\gamma)$ with
$n_{\gamma}=1(2)$.

At NLO, we obtain corrections of QCD and/or EW nature to all LO contributions,
as demonstrated in Figure 4. The dominant higher-order corrections at NLO,
denoted as ${\rm NLO}_{1}$, arise from the QCD corrections to ${\rm LO}_{1}$
at order $\mathcal{O}(\alpha_{s}^{3}\alpha^{4+n_{\gamma}})$. We call the sum
of the two ${\rm NLO}_{\rm QCD}$ with

${\rm NLO}_{\rm QCD}={\rm LO}_{1}+{\rm NLO}_{1}\,,$ (6)

and use it as a starting point to quantify the size of all subleading LO and
NLO contributions. While the partonic subprocesses for the virtual corrections
in ${\rm NLO}_{1}$ are identical to those of ${\rm LO}_{1}$, for the real
emission part additional reactions due to extra QCD radiation have to be taken
into account. In particular, all partonic subprocesses can be obtained by
adding an additional gluon and all possible crossings of partons, leading to

$\begin{array}[]{c}gg\to\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma(\gamma)\,g\,,\\\\[5.69046pt]
\begin{array}[]{clllcll}q\bar{q}/\bar{q}q&\to&\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma(\gamma)\,g\,,&&b\bar{b}/\bar{b}b&\to&\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma(\gamma)\,g\,,\\\\[5.69046pt]
gq/qg&\to&\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma(\gamma)\,q\,,&&g\bar{q}/\bar{q}g&\to&\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma(\gamma)\,\bar{q}\,,\\\\[5.69046pt]
gb/bg&\to&\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma(\gamma)\,b\,,&&g\bar{b}/\bar{b}g&\to&\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma(\gamma)\,\bar{b}\,.\end{array}\end{array}$
(7)

Example Feynman diagrams for the real emission part of ${\rm NLO}_{1}$ are
shown in Figure 5. The real corrections also consistently include additional
photon and gluon radiation in all stages, i.e. in the $t\bar{t}$ production
and the decays of top quarks and $W$ gauge bosons.

Figure 5:  Example Feynman diagrams for $pp\to t\bar{t}\gamma$ contributing to
${\rm NLO}_{1}$.

The next contribution, ${\rm NLO}_{2}$, at the order
$\mathcal{O}(\alpha_{s}^{2}\alpha^{5+n_{\gamma}})$, cannot be completely
separated into parts with only QCD or EW corrections. When calculating the
corresponding virtual corrections we have on the one hand, the interference of
tree-level diagrams at $\mathcal{O}(g^{6+n_{\gamma}})$ with one-loop diagrams
at $\mathcal{O}(g_{s}^{4}g^{4+n_{\gamma}})$ and on the other hand, the
interference between tree-level diagrams at
$\mathcal{O}(g_{s}^{2}g^{4+n_{\gamma}})$ and one-loop diagrams at
$\mathcal{O}(g_{s}^{2}g^{6+n_{\gamma}})$. The first contribution can be seen
as a part of the NLO QCD corrections to ${\rm LO}_{2}$. For the second
contribution, no clear distinction is possible, since it can be seen as either
the NLO QCD corrections to ${\rm LO}_{2}$ or the NLO EW corrections to ${\rm
LO}_{1}$. Finally, for the $g\gamma$ channel, we have the interference of
tree-level diagrams at $\mathcal{O}(g_{s}^{1}g^{5+n_{\gamma}})$ with one-loop
diagrams at $\mathcal{O}(g_{s}^{3}g^{5+n_{\gamma}})$. Compared to ${\rm
NLO}_{1}$, the real corrections in ${\rm NLO}_{2}$ include more partonic
subprocesses, involving QCD and QED-like singularities. The first set of real
corrections is obtained by additional QCD radiation to ${\rm LO}_{2}$, which
leads to

$\begin{array}[]{c}\begin{array}[]{clllcll}q\bar{q}/\bar{q}q&\to&\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma(\gamma)\,g\,,&&b\bar{b}/\bar{b}b&\to&\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma(\gamma)\,g\,,\\\\[5.69046pt]
gq/qg&\to&\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma(\gamma)\,q\,,&&g\bar{q}/\bar{q}g&\to&\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma(\gamma)\,\bar{q}\,,\\\\[5.69046pt]
gb/bg&\to&\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma(\gamma)\,b\,,&&g\bar{b}/\bar{b}g&\to&\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma(\gamma)\,\bar{b}\,.\end{array}\end{array}$
(8)

In this case, we have the interference of tree-level diagrams at order
$\mathcal{O}(g_{s}^{3}g^{4+n_{\gamma}})$ and
$\mathcal{O}(g_{s}^{1}g^{6+n_{\gamma}})$. Although the $q\bar{q}$ channel is
exactly zero in ${\rm LO}_{2}$, its NLO QCD corrections no longer vanish, and
the corresponding virtual and real corrections must be taken properly into
account. In particular, the interference of initial- and final-state gluon
radiation gives rise to the non-vanishing contribution, as explained in Refs.
Denner:2016jyo ; Denner:2023eti . The latter contribution induces
singularities that are of soft nature only. Consequently, there are no
collinear singularities in the $q\bar{q}$ channel and gluon radiation still
vanishes in top-quark decays. It also follows that the $gq$ and $g\bar{q}$
channels contain no collinear singularities and turn out to be IR finite.
Therefore, no dipole subtraction is required for these channels and the naive
usage of it can even affect the efficiency of the phase-space integration. In
the Nagy-Soper Bevilacqua:2013iha  subtraction scheme, which is used in our
calculation, the corresponding collinear subtraction terms vanish by
construction if the underlying Born-level matrix element is exactly zero,
since they are directly proportional to it. This is, however, not a general
feature of dipole subtraction schemes. For example, in the Catani-Seymour
subtraction scheme Catani:1996vz ; Catani:2002hc , the corresponding
subtraction term is a sum of the dipole terms $\mathcal{D}_{k}^{ai}$ involving
color-correlated matrix elements. These dipole terms and the momentum mapping
to the underlying Born-level process depend on the spectator parton $k$.
Therefore, the sum of the dipole terms vanishes only in the exact collinear
limit, where the dependence on the spectator parton $k$ vanishes and the sum
becomes proportional to the Born-level matrix element. In principle, it is
possible for such initial-state singularities to choose exactly one spectator
parton and to substitute the color correlator
$\mathbf{T}_{k}\cdot\mathbf{T}_{ai}\to-\mathbf{T}_{ai}^{2}$ in
$\mathcal{D}_{k}^{ai}$. In this way the corresponding subtraction term is
always proportional to the Born-level matrix element and would vanish as in
the Nagy-Soper case. On the other hand, collinear singularities are present in
the bottom-quark induced channels. In this case, gluon radiation in top-quark
decays must be consistently included. Furthermore, additional partonic
subprocesses, which originate from ${\rm LO}_{1}$ with additional photon
radiation, have to be included, leading to

$\begin{array}[]{c}gg\to\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma\gamma(\gamma)\,,\\\\[5.69046pt]
\begin{array}[]{clllcll}q\bar{q}/\bar{q}q&\to&\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma\gamma(\gamma)\,,&&b\bar{b}/\bar{b}b&\to&\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma\gamma(\gamma)\,.\end{array}\end{array}$
(9)

Similar to the calculation of NLO QCD corrections to the $pp\to t\bar{t}j(j)$
process in the NWA Melnikov:2011qx ; Bevilacqua:2022ozv , photons can appear
simultaneously in the production and decays, as illustrated in Figure 6.
Therefore, the relevant dipoles must be consistently included at each stage of
the decay chain. Finally, we have new photon-induced partonic subprocesses
which are constructed by gluon radiation from the $g\gamma$ channel in ${\rm
LO}_{2}$ and by crossing from the processes listed in Eq. (9), resulting in

Figure 6:  Example Feynman diagrams for $pp\to t\bar{t}\gamma$ contributing to
${\rm NLO}_{2}$.

$\begin{array}[]{c}g\gamma/\gamma
g\to\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma(\gamma)\,g\,,\\\\[5.69046pt]
\begin{array}[]{clllcll}\gamma
q/q\gamma&\to&\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma(\gamma)\,q\,,&&\gamma\bar{q}/\bar{q}\gamma&\to&\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma(\gamma)\,\bar{q}\,,\\\\[5.69046pt]
\gamma
b/b\gamma&\to&\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma(\gamma)\,b\,,&&\gamma\bar{b}/\bar{b}\gamma&\to&\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma(\gamma)\,\bar{b}\,.\end{array}\end{array}$
(10)

While the $g\gamma$ channel clearly contains only QCD singularities due to
gluon radiation, for the remaining partonic processes we simultaneously
encounter QCD and QED initial-state singularities arising from the $\gamma\to
q\bar{q}$ and $q\to gq$ splittings, as shown in Figure 6. This clearly
illustrates that even real corrections cannot be divided into those of QCD or
EW origin, as we have already seen in the calculation of virtual corrections.

The next subleading NLO contribution at the order
$\mathcal{O}(\alpha_{s}^{1}\alpha^{6+n_{\gamma}})$, ${\rm NLO}_{3}$, features
a similarly wide range of virtual and real corrections with respect to ${\rm
NLO}_{2}$. This contribution is obtained from EW corrections to ${\rm LO}_{2}$
and QCD corrections to ${\rm LO}_{3}$. Thus, EW corrections to the $q\bar{q}$
channel in ${\rm LO}_{2}$ vanish again, since with respect to the Born level
the color structure remains unchanged by these corrections. The virtual
corrections consist of the interference of tree-level diagrams at
$\mathcal{O}(g_{s}^{2}g^{4+n_{\gamma}})$ with the one-loop amplitude at
$\mathcal{O}(g^{8+n_{\gamma}})$ and the tree-level matrix element at
$\mathcal{O}(g^{6+n_{\gamma}})$ with one-loop diagrams at
$\mathcal{O}(g_{s}^{2}g^{6+n_{\gamma}})$. The first contribution can be seen
as part of the NLO EW corrections to ${\rm LO}_{2}$, and thus it vanishes for
$q\bar{q}$ as explained before, and only has to be included for the $b\bar{b}$
initial state. In the second term again no distinction between QCD and EW
corrections is possible for the $q\bar{q}$ and $b\bar{b}$ channels, while for
the $\gamma\gamma$ channel this contribution consists of pure QCD corrections
to ${\rm LO}_{3}$. Finally, for $g\gamma$ we have the interference of tree-
level diagrams at $\mathcal{O}(g_{s}^{1}g^{5+n_{\gamma}})$ with one-loop
diagrams at $\mathcal{O}(g_{s}^{1}g^{7+n_{\gamma}})$. The real corrections can
be summarised as follows. First, we have the NLO QCD corrections to ${\rm
LO}_{3}$, which can be obtained by additional gluon radiation, bringing about
the following contributions

$\begin{array}[]{c}\gamma\gamma\to\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma(\gamma)g\,,\\\\[5.69046pt]
\begin{array}[]{clllcll}q\bar{q}/\bar{q}q&\to&\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma(\gamma)g\,,&&b\bar{b}/\bar{b}b&\to&\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma(\gamma)\,g\,.\end{array}\end{array}$
(11)

The crossing of initial- and final-state partons leads to the additional set
of partonic subprocesses

$\begin{array}[]{c}\begin{array}[]{clllcll}gq/qg&\to&\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma(\gamma)\,q\,,&&g\bar{q}/\bar{q}g&\to&\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma(\gamma)\,\bar{q}\,,\\\\[5.69046pt]
gb/bg&\to&\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma(\gamma)\,b\,,&&g\bar{b}/\bar{b}g&\to&\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma(\gamma)\,\bar{b}\,,\end{array}\end{array}$
(12)

which again contain simultaneously QCD and QED initial-state singularities, as
illustrated in Figure 7. Next, we obtain additional contributions from photon
radiation off the partonic subprocesses in ${\rm LO}_{2}$, as exemplified in
Figure 7. Together with all possible crossings of partons and photons we
obtain

Figure 7:  Example Feynman diagrams for $pp\to t\bar{t}\gamma$ contributing to
${\rm NLO}_{3}$.

$\begin{array}[]{c}\begin{array}[]{clllcll}g\gamma/\gamma
g&\to&\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma\gamma(\gamma)&&b\bar{b}/\bar{b}b&\to&\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma\gamma(\gamma)\,,\\\\[5.69046pt]
\gamma
b/b\gamma&\to&\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma(\gamma)\,b&&\gamma\bar{b}/\bar{b}\gamma&\to&\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma(\gamma)\,\bar{b}\,.\end{array}\end{array}$
(13)

In addition, $q\bar{q}$ and $\gamma q/\gamma\bar{q}$ subprocesses vanish due
to color, as additional photon radiation does not change the color structure
already present in ${\rm LO}_{2}$.

Since in ${\rm NLO}_{3}$ we find for the first time EW 1-loop corrections in
photon-initiated subprocesses, in particular in the $g\gamma$ channel, we
briefly mention here the choice of the renormalisation scheme for the
electromagnetic coupling $\alpha$, which will be explained in detail later in
the article. As we are interested in the experimental signature with at least
one (two) isolated photon(s), we do not consider the $\gamma\to f\bar{f}$
splitting for final state photons. Therefore, the electromagnetic coupling
constant connected with final state photons has to be renormalised in the on-
shell scheme to obtain IR finite results, see e.g. Refs. Denner:2019vbn ;
Pagani:2021iwa . However, the electromagnetic coupling associated with initial
state photons must be renormalised in a $\overline{\rm MS}$-like scheme,
because $\gamma\to f\bar{f}$ splittings are included in the evolution of the
proton PDF. Thus, we use the $G_{\mu}$ scheme in this case and the remaining
IR $\epsilon$ poles from the sum of real and virtual corrections are absorbed
into a redefinition of PDFs, as for all other QCD and QED collinear initial
state singularities. This leads to a collinear factorisation counterterm of
the form

$\frac{\alpha}{2\pi}\frac{1}{\Gamma(1-\epsilon)}\frac{1}{\epsilon}\left(\frac{4\pi\mu_{R}^{2}}{\mu_{F}^{2}}\right)^{\epsilon}P_{\gamma\gamma}(x),$
(14)

which has to be convoluted with the corresponding Born-level cross section.
The one-loop Altarelli-Parisi QED kernel $P_{\gamma\gamma}(x)$, see e.g. Ref.
Frederix:2018nkq  for more details, is given by

$P_{\gamma\gamma}(x)=-\frac{2}{3}\,\delta(1-x)\,\sum_{f}N_{c,f}\,Q_{f}^{2},$
(15)

where the summation runs over all massless, charged fermions, $N_{c,f}$ is a
color factor, which is $3$ for quarks and $1$ for leptons, and $Q_{f}$ is the
charge of the fermion $f$. Unlike all other QCD and QED splitting functions,
$P_{\gamma\gamma}(x)$ contains only a virtual contribution given by the delta
distribution.

Finally, we have the NLO EW corrections to ${\rm LO}_{3}$, called ${\rm
NLO}_{4}$, and all new partonic subprocesses can simply be obtained from
additional photon radiation and crossing, leading to

$\begin{array}[]{c}\gamma\gamma\to\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma\gamma(\gamma)\,,\\\\[5.69046pt]
\begin{array}[]{clllcll}q\bar{q}/\bar{q}q&\to&\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma\gamma(\gamma)\,,&&b\bar{b}/\bar{b}b&\to&\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma\gamma(\gamma)\,,\\\\[5.69046pt]
\gamma
q/q\gamma&\to&\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma(\gamma)\,q\,,&&\gamma\bar{q}/\bar{q}\gamma&\to&\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma(\gamma)\,\bar{q}\,,\\\\[5.69046pt]
\gamma
b/b\gamma&\to&\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma(\gamma)\,b\,,&&\gamma\bar{b}/\bar{b}\gamma&\to&\ell^{+}\nu_{\ell}\,\ell^{-}\bar{\nu}_{\ell}\,b\bar{b}\,\gamma(\gamma)\,\bar{b}\,,\end{array}\end{array}$
(16)

where example Feynman diagrams are shown in Figure 8.

Figure 8:  Example Feynman diagrams for $pp\to t\bar{t}\gamma$ contributing to
${\rm NLO}_{4}$.

We denote the complete result at NLO, including all subleading contributions,
as ${\rm NLO}$. The latter result is given by the following sum

${\rm NLO}={\rm LO}_{1}+{\rm LO}_{2}+{\rm LO}_{3}+{\rm NLO}_{1}+{\rm
NLO}_{2}+{\rm NLO}_{3}+{\rm NLO}_{4}.$ (17)

We refrain from dividing our complete NLO calculation into different resonant
contributions based on the origin of photon emission, as already done at NLO
QCD for the $pp\to t\bar{t}\gamma$ process in Refs. Melnikov:2011ta ;
Bevilacqua:2019quz  and for $pp\to t\bar{t}\gamma\gamma$ in Ref.
Stremmer:2023kcd . It is because additional photon radiation introduces
further complications due to mixing of Born-level processes with different
photon radiation patterns at LO. This issue has already been examined in
detail in the case of the calculation of NLO QCD corrections to $pp\to
t\bar{t}jj$ Bevilacqua:2022ozv . Instead, we perform an alternative
calculation, labelled as ${\rm NLO}_{\rm prd}$, which we define according to

${\rm NLO}_{\rm prd}={\rm LO}_{1}+{\rm LO}_{2}+{\rm LO}_{3}+{\rm NLO}_{1}+{\rm
NLO}_{2,{\rm prd}}+{\rm NLO}_{3,{\rm prd}}+{\rm NLO}_{4,{\rm prd}},$ (18)

where the subscript prd indicates that photon bremsstrahlung and subleading
NLO corrections are only included in the production stage of the $pp\to
t\bar{t}\gamma(\gamma)$ process. In detail, in ${\rm NLO}_{\rm prd}$, all LO
(${\rm LO}_{1}$, ${\rm LO}_{2}$, ${\rm LO}_{3}$) contributions as well as
${\rm NLO}_{1}$ are fully included in both the $t\bar{t}\gamma(\gamma)$
production and decays of the top-quark pair. On the other hand, the subleading
NLO corrections (${\rm NLO}_{2}$, ${\rm NLO}_{3}$, ${\rm NLO}_{4}$) are only
included in the production stage of the $t\bar{t}\gamma(\gamma)$ process when
the photon (the two photons) are also present there. In this way, we
investigate the origin of the main subleading corrections. This approximation
is motivated by the fact that, on the one hand, it leads to an enormous
simplification of the calculations, especially for the real emission part, and
ultimately in the matching to parton shower programs. On the other hand, the
largest subleading NLO contributions are expected to originate from EW Sudakov
logarithms in the tails of dimensionful observables. However, it is known that
high $p_{T}$ phase-space regions are dominated by the case where all photons
are produced in the $t\bar{t}$ production Melnikov:2011ta ; Bevilacqua:2019quz
; Stremmer:2023kcd , which should directly reduce the relevance of including
subleading NLO corrections in other radiation patterns. With the calculation
of ${\rm NLO}_{\rm prd}$ we would like to confirm that this is the case for
the $pp\to t\bar{t}\gamma(\gamma)$ process for various differential cross-
section distributions. For the sake of completeness, we will also show the
${\rm NLO}_{\rm prd}$ result for the $t\bar{t}\gamma(\gamma)$ integrated
fiducial cross section.

##  3 Computational framework

As we have already alluded to in Section 2 the calculation of real corrections
is performed with the Nagy-Soper subtraction scheme Bevilacqua:2013iha , which
has recently been extended to NLO QCD calculations in the NWA involving
internal on-shell resonances Bevilacqua:2022ozv . To check the validity of the
results, we also use the Catani-Seymour subtraction scheme Catani:1996vz ;
Catani:2002hc . Furthermore, for additional cross-checks in both cases we
utilize the phases-space restriction on the subtraction terms Nagy:1998bb ;
Nagy:2003tz ; Bevilacqua:2009zn ; Czakon:2015cla . For the purposes of this
study, both subtraction schemes, that are implemented in the Helac-Dipoles MC
program Czakon:2009ss , have been extended to deal with soft and collinear
singularities of QED origin that plague the real radiation contributions of
NLO EW calculations. In the following we describe both extensions in detail
and discuss how the calculation of virtual corrections is organised for the
purposes of our study.

###  3.1 Real corrections

At NLO QCD, we have to deal with contributions of different origin, on the one
hand with virtual corrections due to the interference of one-loop and tree-
level amplitudes, and on the other hand with real corrections arising from the
emission of additional QCD partons. However, both contributions themselves are
divergent, and only the sum of them is finite for IR safe observables. A
standard approach nowadays is the use of a subtraction scheme, where an
auxiliary cross section is introduced, which is designed to locally mimic the
singular limits of the real-correction cross section. Moreover, this auxiliary
cross section has to be simple enough so that the integration over the one-
parton subspaces, which cause the soft and collinear divergences, is possible
either analytically or numerically, leading to the so-called integrated
subtration terms and/or dipoles. These can then be combined with the virtual
corrections to cancel all IR divergences. In general, the subtraction term in
QCD can be written as

$\mathcal{A}^{\textrm{D}}=\sum_{i,j,k=1}^{n+1}\mathcal{A}^{\textrm{B}}(\\{\tilde{p}\\}_{n}^{ijk})\otimes\mathcal{D}^{(ijk)}(\\{\tilde{p}\\}_{n}^{(ijk)},\\{p\\}_{n+1})\,\left(\mathbf{T}_{ij}\cdot\mathbf{T}_{k}\right),$
(19)

where $\mathcal{A}_{\textrm{B}}$ is the tree-level matrix element of the
underlying Born-level process which results from the recombination of the two
partons $i$ and $j$. The terms $\mathcal{D}^{(ijk)}$ are the so-called dipoles
and $\\{\tilde{p}\\}_{n}^{(ijk)}$ is the new set of momenta for the underlying
Born-level process, which is obtained by a momentum mapping from the initial
momenta $\\{p\\}_{n+1}$. Spin correlations between the squared Born-level
matrix element and the dipoles are indicated by the symbol $\otimes$. Finally,
$\mathbf{T}_{k}$ are color operators according to the definitions given in
Ref. Catani:1996vz . The indices $i,j,k$ indicate that in general the dipoles
as well as the momentum mapping depend on the two splitting partons $i,j$ and
potentially also on the spectator parton $k$. The QED subtraction scheme can
be obtained from the QCD one by simple substitutions. In particular, the color
operators have to be replaced by charge operators according to

$\mathbf{T}_{ij}\cdot\mathbf{T}_{k}\rightarrow\mathbf{Q}_{ij}\mathbf{Q}_{k},$
(20)

where $\mathbf{Q}_{k}$ is the charge of the particle with a relative minus
sign between final- and initial-state particles, and possible further simple
replacements of color factors. However, this cannot be applied to the case
where the splitting pair is recombined into a photon ($Q_{ij}=0$). Instead, in
this case we perform the following substitution

$\mathbf{T}_{ij}\cdot\mathbf{T}_{k}\rightarrow-
w_{k}\mathbf{Q}_{i}^{2}\;\;\;\;\;\;\;\;\textrm{with}\;\;\;\;\;\;\;\;\sum_{k}w_{k}=1.$
(21)

The particular choice of these weights $w_{k}$ is completely arbitrary and, in
our case we simply set it to $\delta_{k,k_{0}}$, which implies that we choose
exactly one spectator particle. In general, this spectator particle must be
the same in the actual subtraction and in the calculation of the integrated
dipoles, since the momentum mapping and dipoles terms depend on $k$. We note
that in the Nagy-Soper subtraction scheme, both the momentum mapping and the
collinear dipole terms are independent of the spectator particle, so that the
choice of $k_{0}$ plays no role in the calculation. Finally, the structure of
Helac-Dipoles has been modified to simultaneously keep track of different
orders in $\alpha_{s}$ and $\alpha$ and to allow the reweighting to different
renormalisation and factorisation scale settings on the fly. To verify the new
modifications, we have reproduced the results of Ref. Denner:2016jyo  at the
integrated and differential cross section level. In detail, we have calculated
afresh NLO EW corrections to the $pp\to t\bar{t}+X$ process in the di-lepton
top-quark decay channel for the LHC Run II center-of-mass energy of
$\sqrt{s}=13$ TeV. In this calculation all finite-width effects, resonant and
non-resonant contributions of the top quark and $W^{\pm}/Z$ gauge bosons as
well as interference effects have been included.

Next, we discuss the changes in the Nagy-Soper subtraction scheme for
calculations involving resonant particles. These modifications are to a large
extent independent of QCD or QED-like singularities. To this end, we first
briefly recall the notation and main aspects of the Nagy-Soper subtraction
scheme which were presented in detail in Ref. Bevilacqua:2013iha . We focus on
the case of final-state splittings since initial-state splittings remain
unaffected. We first introduce the following definitions

$\displaystyle Q=p_{1}+p_{2}=\sum_{l=3}^{n+1}p_{l},$ (22) $\displaystyle
P_{ij}=p_{i}+p_{j}\qquad\qquad\textrm{and}\qquad\qquad K=Q-P_{ij},$ (23)

where $Q$ is the total momentum of the process, $P_{ij}$ is the momentum sum
of the splitting partons and $K$ is the so-called collective spectator. The
momentum mapping from $\\{p\\}_{n+1}$ to $\\{\tilde{p}\\}_{n}^{(ij)}$ is then
defined by the condition that the momentum of the splitting parton
$\tilde{p}_{i}$ should lie in the $Q$-$P_{ij}$ plane according to

$P_{ij}=\beta\tilde{p}_{i}+\gamma Q,$ (24)

where $\beta$ and $\gamma$ are uniquely fixed by the following two conditions

$\displaystyle\tilde{Q}$ $\displaystyle=Q\,,$ (25)
$\displaystyle\widetilde{K}^{2}$ $\displaystyle=K^{2}\,.$ (26)

From Eq. (25) it is clear that the momentum mapping from $K$ to
$\widetilde{K}$ is given by a Lorentz transformation
($k_{i}^{\mu}=\tensor{\Lambda(K,\widetilde{K})}{{}^{\mu}_{\nu}}\,\widetilde{k}_{i}^{\nu}$),
which is defined as Nagy:2007ty ; Bevilacqua:2013iha

$\tensor{\Lambda(K,\widetilde{K})}{{}^{\mu}_{\nu}}=\tensor{g}{{}^{\mu}_{\nu}}-\frac{2(K+\widetilde{K})^{\mu}(K+\widetilde{K})_{\nu}}{(K+\widetilde{K})^{2}}+\frac{2K^{\mu}\widetilde{K}_{\nu}}{K^{2}}.$
(27)

Next, the dipole terms $\mathcal{D}^{(ijk)}$ in Eq. (19) are decomposed into
diagonal terms $W^{(ii,j)}$ containing soft and collinear singularities and
interference terms $W^{(ik,j)}$ including only soft singularities according to

$\mathcal{D}^{(ijk)}_{\tilde{s}_{1}\tilde{s}_{2}}=W^{(ii,j)}_{\tilde{s}_{1}\tilde{s}_{2}}\delta_{ik}+W^{(ik,j)}_{\tilde{s}_{1}\tilde{s}_{2}}(1-\delta_{ik})\,,$
(28)

where $\tilde{s}_{1}$ and $\tilde{s}_{2}$ are the helicity values of the
splitting parton. The diagonal and interference terms are further split into a
sum of so-called splitting functions $v^{(ij)}$ as specified by

$\displaystyle W^{(ii,j)}_{\tilde{s}_{1}\tilde{s}_{2}}$
$\displaystyle=\sum_{s_{i},s_{j}}v^{(ij)}_{\tilde{s}_{1}s_{i}s_{j}}\left(v^{(ij)}_{\tilde{s}_{2}s_{i}s_{j}}\right)^{*},$
(29) $\displaystyle W^{(ik,j)}_{\tilde{s}_{1}\tilde{s}_{2}}$
$\displaystyle=\sum_{s_{i},s_{j}}v^{(ij),eik}_{\tilde{s}_{1}s_{i}s_{j}}\left(v^{(kj),eik}_{\tilde{s}_{2}s_{i}s_{j}}\right)^{*}\,,$
(30)

where $v^{(ij),eik}$ is the eikonal approximation of these splitting
functions. In QCD the $v^{(ij),eik}$ function reduces to a much simpler
expression given by

$\displaystyle
v^{(ij),eik}_{\tilde{s}_{i}s_{i}s_{j}}=\sqrt{4\pi\alpha_{s}}\,\delta_{\tilde{s}_{i}s_{i}}\frac{\epsilon(p_{j},s_{j})^{*}\cdot
p_{i}}{p_{i}\cdot p_{j}}.$ (31)

The splitting functions are obtained by factorising the splitting
$\tilde{p}_{i}\to p_{i}+p_{j}$ from the divergent matrix element in the
following way $\mathcal{M}_{n+1}^{\rm div}=\mathcal{M}_{n}\otimes v^{(ij)}$.
The exact form for the different QCD splittings for final- and initial-state
splittings can be found in Ref. Nagy:2007ty .

The original Nagy-Soper subtraction scheme can directly be used in the case of
unresolved partons in the production process. However, if an unstable particle
is part of the emitter pair, then we use the spin-averaged versions of the
dipole terms $\mathcal{D}^{(ijk)}$ in Eq. (19). In addition, we use the
eikonal approximation for the splitting functions in the diagonal term
$W^{(ii,j)}$ given in Eq. (29) when a $W$ gauge boson is part of the emitter
pair. These modifications are also applied to unstable particles in nested
decay chains, such as $W$ gauge bosons in radiative top-quark decays. Further
modifications are required for the dipole subtraction in decay processes. The
first change concerns the momentum mapping which is required to preserve the
total momentum of the decay process, i.e. the momentum of the decaying mother
particle. This is achieved by replacing the total momentum $Q$ in the original
momentum mapping with the momentum of the mother particle, which is called
$Q_{\rm dec}$ in analogy. By definition $Q_{\rm dec}$ is then preserved under
the momentum mapping according to the condition defined in Eq. (25). The
dipole terms $\mathcal{D}^{(ijk)}$ also have an explicit and an additional
implicit dependence on $Q$ by the axial gauge with $Q$ as reference vector
used for the polarisation vectors of gluons and photons. Thus, to reuse all
the results of the original subtraction scheme, we have to perform the
replacement $Q\to Q_{\rm dec}$ also in all the dipole terms
$\mathcal{D}^{(ijk)}$ and apply a gauge transformation to the external
polarisation vectors entering the splitting functions $v^{(ij)}$. The mother
particle is excluded from the list of possible emitters in the dipole
subtraction, but it is still included as a possible spectator particle. To
properly account for all soft singularities associated with the mother
particle, we replace $W^{(ik,j)}$, defined in Eq. (29), in the case where the
mother particle is the spectator $k$, by $W^{(ik,j)}_{\rm dec}$, which we
define as

$\displaystyle W_{\rm dec}^{(ik,j)}=W^{(ik,j)}+W^{(ki,j)}-W^{(kk,j),eik}.$
(32)

For the calculation of the corresponding integrated dipoles we closely follow
the semi-numerical approach of Ref. Bevilacqua:2013iha . In this case, the
spin-averaged diagonal and interference terms are cast in the following form

$\displaystyle\overline{W}^{(ii,j)}-\overline{W}_{\rm
dec}^{(ik,j)}=\underbrace{\left(\overline{W}^{(ii,j)}-\overline{W}^{(ii,j),eik}\right)}_{\mathcal{D}_{ii}}+\underbrace{\left(\overline{W}^{(ii,j),eik}-\overline{W}_{\rm
dec}^{(ik,j)}\right)}_{\mathcal{I}_{ik,{\rm dec}}},$ (33)

where $\mathcal{D}_{ii}$ can only have collinear singularities by construction
and are already known from the original formulation. The second term can have
soft and collinear singularities and can be further simplified to

$\displaystyle\mathcal{I}_{ik,{\rm dec}}=4\pi\alpha_{s}\frac{-((p_{j}\cdot
p_{k})p_{i}-(p_{i}\cdot p_{j})p_{k})^{2}}{(p_{i}\cdot p_{j})^{2}(p_{j}\cdot
p_{k})^{2}}.$ (34)

We have calculated and implemented this new dipole in the case of massless and
massive splitting partons $\tilde{p}_{i}$. The latter case is required for the
calculation of EW corrections to top-quark decays, i.e. for $W^{\pm}\to
W^{\pm}\gamma$ splittings. In addition, it can also be used for e.g. NLO QCD
corrections with massive bottom quarks inside top-quark decays. In summary,
our implementation of the Nagy-Soper subtraction scheme allows us to calculate
QCD and EW radiative corrections for any number of coloured, charged, massive
and massless particles in nested decay chains.

We have carried out several tests to confirm the correctness of the
modifications we have made. First of all, we have already used this new
extension for the calculation of the NLO QCD corrections to the $pp\to
t\bar{t}jj$ process in the di-lepton top-quark decay channel in the NWA
Bevilacqua:2022ozv . These results, were cross-checked with an alternative
calculation using the Catani-Seymour subtraction scheme and its extension to
the NWA case Campbell:2004ch ; Melnikov:2011qx ; Bevilacqua:2019quz . In
addition, we have recalculated the NLO QCD corrections of the top-quark width
with massive bottom quarks Jezabek:1988iv ; Czarnecki:1990kv  as well as the
corresponding real corrections, which are presented in the appendix of Ref.
Campbell:2012uf . Finally, we have reproduced the calculation of the NLO
electroweak corrections to the top-quark decay width while treating the
internal $W$ boson in the NWA Basso:2015gca .

In our current calculation of the $pp\to t\bar{t}\gamma(\gamma)$ process in
the NWA, we have computed the real correction part of the ${\rm NLO}$
calulations with the Nagy-Soper subtraction scheme. We have utilised two
different values of the phase-space restriction on the subtraction terms and
found excellent agreement for the obtained results within the corresponding MC
errors. Finally, we have reproduced the results of ${\rm NLO}_{i,{\rm prd}}$
with $i=1,2,3,4$ using the Catani-Seymour subtraction scheme. A full
comparison with the Catani-Seymour subtraction is not yet possible as our
implementation is currently limited to massless emitters in decay processes.

###  3.2 Virtual corrections

The calculation of the virtual corrections is organized using reweighting
techniques Binoth:2008kt ; Bevilacqua:2009zn . In practice, partially
unweighted events Bevilacqua:2016jfk  that are generated at the Born level,
are then used to calculate the one-loop corrections. The phase-space
integration is performed using the programs Parni vanHameren:2007pt  and Kaleu
vanHameren:2010gg . The combined tree-level and one-loop results are then
stored in modified Les Houches Event Files (LHEFs) Alwall:2006yp ;
Bern:2013zja , which have been extended to save, for each event, the necessary
information for all contributions at different orders of $\alpha_{s}$ and
$\alpha$. This allows us, among other things, to reweight our results to
different renormalisation and factorisation scale settings. Tree-level and
one-loop matrix elements are obtained with the help of the Recola matrix
element generator Actis:2016mpe ; Actis:2012qn . We have modified the
calculation of the matrix elements in Recola to incorporate the random
polarisation method Draggiotis:1998gr ; Draggiotis:2002hm ; Bevilacqua:2013iha
which leads to significant performance improvements in the phase-space
integration. Following the notation of Ref. Actis:2016mpe , every one-loop
amplitude can be written as a linear combination of tensor integrals

$\mathcal{A}_{1-{\rm
loop}}=\sum_{t}c_{\mu_{1}\dots\mu_{r_{t}}}^{(t)}\,T^{\mu_{1}\dots\mu_{r_{t}}}_{(t)}+\mathcal{A}_{\rm
CT}\,,$ (35)

where $\mathcal{A}_{\rm CT}$ include the counterterms,
$c_{\mu_{1}\dots\mu_{r_{t}}}^{(t)}$ are the tensor coefficients that do not
depend on the loop momentum $q$, $t$ classifies the different tensor integrals
needed for the process and $r_{t}$ labels the rank of the tensor integral.
Furthermore, the tensor integrals are given by

$T^{\mu_{1}\dots\mu_{r_{t}}}_{(t)}=\frac{(2\pi\mu)^{4-D}}{i\pi^{2}}\int
d^{D}q\frac{q^{\mu_{1}}\dots q^{\mu_{r_{t}}}}{D_{0}^{(t)}\dots
D_{k_{t}}^{(t)}}\,,\quad\quad\quad\quad
D_{i}^{(t)}=(q+p_{i}^{(t)})^{2}-(m_{i}^{(t)})^{2}\,,$ (36)

where $k_{t}$ is the number of propagators. The tensor coefficients are
calculated by Recola similarly to the tree-level amplitudes by recursive
construction of loop off-shell currents vanHameren:2009vq ; Actis:2012qn . The
relevant tensor integrals are computed with the Collier library Denner:2016kdg
which implements the reduction techniques of Denner:2005nn ; Denner:2002ii
and the scalar integrals of Denner:2010tr . Alternatively, we have implemented
the reduction to scalar integrals with CutTools Ossola:2007ax . The resulting
scalar integrals are computed with OneLOop vanHameren:2010cp . The CutTools
program implements the OPP reduction method Ossola:2006us , which is based on
double, triple and quadruple cuts. A similar approach has already been
employed in the OpenLoops program Buccioni:2017yxi . In our case, the OPP
reduction and the evaluation of the scalar integrals are performed with
quadruple precision, while the tensor coefficients are calculated with double
precision. Not only do we use this second reduction scheme for additional
cross-checks of our framework, but also for phase-space points containing
tensor integrals that have been flagged by Collier as possibly unstable.

As explained in Ref. Actis:2016mpe  the Recola program offers the possibility
to select the resonant parts of the amplitude, and can therefore be employed
to calculate matrix elements in the pole approximation. In this approximation,
only the resonant parts of the amplitude are kept, thus, the complex squared
mass of the resonant particle $(\mu^{2}=m^{2}-i\Gamma m)$ is replaced by its
real part $(\Re(\mu^{2})=m^{2})$ everywhere in the amplitude except for the
denominators of resonant propagators where the width $\Gamma$ of the unstable
particle is still present. The NWA matrix element is then obtained by
performing the usual limit $\Gamma/m\to 0$ in the Breit-Wigner propagators.
Finally, we correct the symmetry factor calculated in Recola for processes
involving identical particles in the final state but originating from
different subprocesses in the decay chain, as explained in Ref. Denner:2020orv
.

As an alternative, we have implemented a second fully automatic method for the
construction of one-loop matrix elements in the NWA, which is based on the
calculation of the on-shell amplitudes for the individual contributions needed
in e.g. the $pp\to t\bar{t}(\gamma\gamma)$ process in the di-lepton top-quark
decay channel. To this end, we compute separately the subamplitudes for $pp\to
t\bar{t}(\gamma\gamma)$, $t\to W^{+}b(\gamma\gamma)$, $\bar{t}\to
W^{-}\bar{b}(\gamma\gamma)$, $W^{+}\to\ell^{+}\nu_{\ell}(\gamma\gamma)$ and
$W^{-}\to\ell^{-}\bar{\nu}_{\ell}(\gamma\gamma)$ with Recola and then combine
them appropriately in color and helicity space to obtain the full one-loop
amplitude. This alternative implementation is particularly useful for cross-
checking because in this case we can switch on and off one-loop corrections
for specific contributions. We note here, that we have already used both
approaches in Ref. Bevilacqua:2022ozv  where NLO QCD calculations for the
$pp\to t\bar{t}jj$ process in the di-lepton top-quark decay channel have been
carried out.

We have performed various cross-checks between Recola and Recola2
Denner:2017wsf  for all partonic subprocesses. Specifically, for each
subprocess and for several phase-space points we checked the finite remainders
as well as the coefficients of the poles in $\epsilon$ of the virtual
amplitudes. For testing purposes we utilised the Background-Field-Method (BFM)
as implemented in Recola2 Denner:1994xt ; Denner:2017vms ; Denner:2019vbn .
The BFM, which is an alternative formulation for the quantisation of gauge
fields that can be seen as a different gauge, can be used as a complementary
method in the calculation of one-loop matrix elements besides the usual
formulation.

The proper description of unstable particles like $W^{\pm}$ gauge bosons and
top quarks in perturbation theory requires the introduction of the complex-
mass scheme (CMS), which was first proposed for lowest-order calculations in
Ref. Denner:1999gp  and then generalised to higher orders in Refs.
Denner:2005fg ; Denner:2006ic . In this scheme, complex masses are introduced
at the level of the Lagrangian by splitting the bare masses into complex
renormalised masses and complex counterterms. This corresponds to a
rearrangement of the perturbative expansion without changing the underlying
theory and preserving gauge invariance. Unitarity is respected up to higher
orders as long as unstable particles are excluded from the final states.
Renormalisation must nevertheless be treated carefully as not to spoil these
properties of the bare theory. To this end, the standard on-shell scheme has
been generalised in Ref. Denner:2005fg  to the complex-mass case by requiring
unit residues of propagators at complex poles and by taking into account the
imaginary parts of the self-energies in the renormalisation constants. This
necessitates, however, the calculation of the self-energies for complex
squared momenta. This complication can be avoided by expanding the self-
energies present in the renormalisation constants about real arguments such
that one-loop accuracy is preserved. In case of the $W^{\pm}$ gauge boson and
top-quark, extra constants must be added to the expanded self-energies in
order not to spoil the one-loop accuracy of the results due to the presence of
branch cuts introduced by infrared divergences. The simplified version of the
complex renormalisation is described in detail in Refs. Denner:2005fg ;
Denner:2019vbn , where all renormalisation constants are provided as well.
These renormalisation constants are implemented in Recola and employed in our
calculation of NLO QCD and EW corrections to the $pp\to
t\bar{t}\gamma(\gamma)$ process in the di-lepton top-quark decay channel
without any modifications.

As already mentioned in Section 2.1, the presence of final-state photons in
the Born-level process requires special care in the renormalisation of the
electromagnetic coupling constant $\alpha$. In particular, we split the total
power of $\alpha^{n}$ into
$\alpha_{G_{\mu}}^{n-n_{\gamma}}\alpha(0)^{n_{\gamma}}$ where $n_{\gamma}=1$
($n_{\gamma}=2$) for the underlying Born-level process $pp\to
t\bar{t}\gamma(\gamma)$. The powers of the electromagnetic coupling constant
associated with final-state photons are renormalised in the on-shell scheme
(the $\alpha(0)$ scheme) Denner:2005fg ; Denner:2019vbn ; Pagani:2021iwa . The
remaining powers of $\alpha$ are renormalised in the $G_{\mu}$ scheme, where
the renormalized electromagnetic coupling $\alpha$ is derived from the Fermi
constant $G_{F}$, measured in muon decays, and the masses of the $W$ and $Z$
bosons. The renormalisation in this mixed scheme is realised by first
performing the complete renormalisation of all powers of $\alpha$ in the
$G_{\mu}$ scheme. The renormalisation scheme is then changed for
$\alpha(0)^{n_{\gamma}}$ by introducing a new counterterm which is given by

$2\,n_{\gamma}\,{\rm Re}\left(\delta Z_{e}\big{|}_{\alpha(0)}-\delta
Z_{e}\big{|}_{G_{\mu}}\right)d\sigma^{\rm LO}=n_{\gamma}\,{\rm Re}\left(\Delta
r^{(1)}\right)d\sigma^{\rm LO},$ (37)

where $\Delta r^{(1)}$ are the NLO EW corrections to the muon decay
Sirlin:1980nh ; Denner:1991kt ; Hollik:1988ii ; Marciano:1980pb  and $\delta
Z_{e}\big{|}_{\alpha(0)}$ as well as $\delta Z_{e}\big{|}_{G_{\mu}}$ are the
renormalisation constants of the electric charge $e$ in the on-shell and
$G_{\mu}$ scheme, respectively. We note that in Eq. (37) and in general in the
whole calculation, the electromagnetic coupling is always set to
$\alpha=\alpha_{G_{\mu}}$ and the final result is then rescaled by
$(\alpha(0)/\alpha_{G_{\mu}})^{n_{\gamma}}$. This also implies that the
relative EW corrections are always calculated with $\alpha=\alpha_{G_{\mu}}$.
In order to test the consistency of the calculation in this mixed approach we
have explicitly checked the cancellation of all IR singularities in the sum of
virtual corrections and integrated dipoles for a few phase-space points for
all partonic subprocesses.

##  4 LHC Setup for numerical predictions

The calculations of complete NLO corrections for $pp\to t\bar{t}\gamma$ and
$pp\to t\bar{t}\gamma\gamma$ in the di-lepton top-quark decay channel are
performed for proton-proton collisions at the LHC with a center of mass energy
of ${\sqrt{s}=13}$ TeV. When both higher-order QCD and EW corrections as well
as photon-initiated subprocesses are considered, the PDFs with QED effects and
the photon content of the proton are essential. To this end, we use the NLO
NNPDF3.1luxQED PDF set Manohar:2016nzj ; NNPDF:2017mvq ; Manohar:2017eqh ;
Bertone:2017bme  with $\alpha_{s}(m_{Z})=0.118$, where photons are properly
taken into account in the PDF evolution, at LO and NLO. The PDF set as well as
the running of the strong coupling constant with two-loop accuracy is obtained
using the LHAPDF interface Buckley:2014ana . As already explained in the last
section, for $\alpha$ a mixed scheme is considered. In particular, $\alpha$ is
first calculated in the $G_{\mu}$-scheme according to

$\alpha_{G_{\mu}}=\frac{\sqrt{2}}{\pi}\,G_{\mu}\,m_{W}^{2}\,\left(1-\frac{m_{W}^{2}}{m_{Z}^{2}}\right)\,,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}G_{\mu}=1.1663787\cdot
10^{-5}\textrm{ GeV}^{-2}\,,$ (38)

with $m_{W}=80.379$ GeV and $m_{Z}=91.1876$ GeV. However, $\alpha$ associated
with Born-level final-state photons is treated in the on-shell scheme, where
we use $\alpha^{-1}=\alpha^{-1}(0)=137.035999084$ Workman:2022ynf  as the
input parameter. Additional photon radiation from the real emission part of
the NLO calculation is evaluated with $\alpha_{G_{\mu}}$. We use the following
input values for the masses and widths of the unstable particles

$\begin{array}[]{lll}m_{t}=172.5~{}{\rm GeV}\,,&&\Gamma_{W}=2.0972~{}{\rm
GeV}\,,\vspace{0.2cm}\\\ m_{H}=125~{}{\rm GeV}\,,&&\Gamma_{H}=4.07\cdot
10^{-3}~{}{\rm GeV}\,,\\\ \end{array}$

The width of the $Z$ boson is set to zero to avoid artificial higher-order
terms that would otherwise appear due to the complex-mass scheme, since the
width of the $W$ boson is assumed to be zero everywhere except in the
propagators, as explained in Ref. Denner:2016jyo . All other particles are
assumed to be massless. We use the same top-quark width at LO and NLO. Taking
into account NLO QCD and EW corrections, the latter width is given by

$\Gamma_{t}^{\rm NLO}=1.3735727~{}{\rm GeV},$ (39)

where we adapt the conventions given in Ref. Denner:2012yc , while the NLO QCD
corrections are obtained from Ref. Jezabek:1988iv  for $\mu_{R}=m_{t}$. We
note here that, contrary to what is usually done when calculating NLO QCD
corrections, in this study we use the same PDF set as well as top-quark width
for both LO and NLO predictions. This should help us to directly examine the
relative magnitude of the subleading contributions, while simultaneously
separating out the effects coming from the use of different settings at LO and
NLO. The NLO EW corrections are computed numerically with the help of Helac-
Dipoles and Recola. As already indicated in Eq. (18), in this scheme, the LO
and NLO contributions can simply be added to give the complete result.

The event selection and scale choice are based on our previous work on NLO QCD
corrections to the $pp\to t\bar{t}\gamma\gamma$ process Stremmer:2023kcd . All
final-state partons with pseudo-rapidity $|\eta|<5$ are clustered into jets
with separation $R=0.4$ in the rapidity azimuthal-angle plane via the IR-safe
$anti$-$k_{T}$ jet algorithm Cacciari:2008gp . We require two oppositely
charged leptons, at least two $b$-jets and at least one (two) photon(s) for
$t\bar{t}\gamma(\gamma)$, respectively. The IR safety of the
$t\bar{t}\gamma(\gamma)$ cross section involving jets and prompt photon(s) is
ensured by using the smooth photon isolation prescription as described in Ref.
Frixione:1998jh . Therefore, the event is rejected unless the following
condition is fulfilled before the jet algorithm is applied

$\sum_{i}E_{T\,i}\,\Theta(R-R_{\gamma
i})\leq\epsilon_{\gamma}\,E_{T\,\gamma}\left(\frac{1-\cos(R)}{1-\cos(R_{\gamma
j})}\right)^{n}\,,$ (40)

for all $R\leq R_{\gamma j}$ with $R_{\gamma j}=0.4$ and
$\epsilon_{\gamma}=n=1$. The transverse energy of the parton $i$/photon is
denoted by $E_{T\,i}$/$E_{T\,\gamma}$ and $R_{\gamma i}$ is given by

$R_{\gamma i}=\sqrt{(y_{\gamma}-y_{i})^{2}+(\phi_{\gamma}-\phi_{i})^{2}}\,.$
(41)

In subleading NLO corrections we also encounter partonic processes with an
additional photon in the final state with respect to the Born-level
subprocesses. In this case, the smooth photon isolation prescription can be
extended to additionally require the isolation of photons with charged leptons
and photons, as suggested in Ref. Pagani:2021iwa . In this approach, after the
smooth photon isolation, charged leptons and partons are recombined with only
non-isolated photons based on a jet clustering algorithm. In general, the use
of the smooth photon isolation prescription is not required in this case.
Alternatively, it is sufficient to perform the photon recombination directly
with charged leptons and partons, where additionally the following
recombination rule $\gamma+\gamma\to\gamma$ is allowed. Similar schemes have
already been applied in associated photon production processes, see e.g. Refs.
Denner:2014bna ; Denner:2015fca . We have used both approaches for the
calculation of higher-order corrections to the $pp\to t\bar{t}\gamma$ process,
where the photon recombination has also been performed with the $anti$-$k_{T}$
jet algorithm with the radius parameter $R=0.4$. We have found no
phenomenologically relevant differences even at the differential cross-section
level. Therefore, we remain with the second approach, not using the smooth
isolation prescription in these contributions. After the jet algorithm/photon
recombination, the prompt photons are defined with the following conditions

$\begin{array}[]{lll}p_{T,\,\gamma}>25~{}{\rm
GeV}\,,&\quad\quad\quad\quad\quad|y_{\gamma}|<2.5\,,&\quad\quad\quad\quad\quad\Delta
R_{\gamma\gamma}>0.4\,.\end{array}$ (42)

In order to ensure well-observable isolated $b$-jets and charged leptons in
the central-rapidity region, we require

$\begin{array}[]{lll}p_{T,\,b}>25~{}{\rm
GeV}\,,&\quad\quad\quad\quad\quad|y_{b}|<2.5\,,&\quad\quad\quad\quad\quad\Delta
R_{bb}>0.4\,,\\\\[5.69046pt] p_{T,\,\ell}>25~{}{\rm
GeV}\,,&\quad\quad\quad\quad\quad|y_{\ell}|<2.5\,,&\quad\quad\quad\quad\quad\Delta
R_{\ell\ell}>0.4\,.\end{array}$ (43)

In addition, charged leptons and photons need to be well separated from any
$b$-jet in the rapidity-azimuthal angle plane as well as from each other

$\begin{array}[]{lll}\Delta R_{l\gamma}>0.4\,,&\quad\quad\quad\quad\quad\Delta
R_{lb}>0.4\,,&\quad\quad\quad\quad\quad\Delta R_{b\gamma}>0.4\,.\end{array}$
(44)

Finally, there are no restrictions on the kinematics of the extra light
jet/photon (if resolved by the jet algorithm) and the missing transverse
momentum,
$p_{T}^{miss}=|\vec{p}_{T,\,\nu_{\ell}}+\vec{p}_{T,\,\bar{\nu}_{\ell}}|$.
Based on the findings in Ref. Stremmer:2023kcd  we set the factorisation and
renormalisation scale to a common dynamical scale setting, $\mu_{0}$, given by

$\mu_{R}=\mu_{F}=\mu_{0}=\frac{E_{T}}{4}\,,$ (45)

with

$E_{T}=\sqrt{m^{2}_{t}+p_{T,\,t}^{2}}+\sqrt{m^{2}_{t}+p_{T,\,\bar{t}}^{2}}+\sum_{i=1}^{n_{\gamma}}p_{T,\,\gamma_{i}}\,,$
(46)

where $p_{T,\,t}$ and $p_{T,\,\bar{t}}$ are the transverse momenta of the on-
shell top quarks. In addition, for dimensionful cross-section distributions
constructed from the photon kinematics, we present results for the fixed scale
setting, $\mu_{0}=m_{t}$. Theoretical uncertainties due to missing higher
orders are estimated by a 7-point scale variation, where the factorisation and
renormalisation scales are independently varied in the range

$\frac{1}{2}\,\mu_{0}\leq\mu_{R}\,,\mu_{F}\leq
2\,\mu_{0}\,,\quad\quad\quad\quad\quad\quad\quad\quad\frac{1}{2}\leq\frac{\mu_{R}}{\mu_{F}}\leq
2\,,$ (47)

which leads to the following pairs

$\left(\frac{\mu_{R}}{\mu_{0}}\,,\frac{\mu_{F}}{\mu_{0}}\right)=\Big{\\{}\left(2,1\right),\left(0.5,1\right),\left(1,2\right),(1,1),(1,0.5),(2,2),(0.5,0.5)\Big{\\}}\,.$
(48)

By searching for the minimum and maximum of the resulting cross sections we
obtain the displayed uncertainty bands. We use the 7-point scale variation
both at the integrated and differential cross-section level.

##  5 Top-quark pair production with one isolated photon

###  5.1 Integrated fiducial cross sections

Now we turn to the discussion of the numerical results. We begin with our
findings for the $pp\to t\bar{t}\gamma$ process at the integrated fiducial
cross-section level. In particular, in Table 1 we present the results for
${\rm LO}$, ${\rm NLO}$, ${\rm NLO}_{\rm QCD}$ and ${\rm NLO}_{\rm prd}$, as
defined in the last section, with the corresponding theoretical uncertainties
obtained by scale variation. In addition, we display the decomposition of the
LO and NLO results into the individual ${\rm LO}_{i}$ and ${\rm NLO}_{i}$
contributions calculated at the different orders in $\alpha_{s}$ and $\alpha$.
Finally, the relative size of all results with respect to the dominant LO
contribution, ${\rm LO}_{1}$, is given in the last column.

|  |  $\sigma_{i}$ [fb] | Ratio to ${\rm LO}_{1}$
---|---|---|---
${\rm LO}_{1}$ |  $\mathcal{O}(\alpha_{s}^{2}\alpha^{5})$ |  $55.604(8)^{+31.4\%}_{-22.3\%}$ |  $1.00$
${\rm LO}_{2}$ |  $\mathcal{O}(\alpha_{s}^{1}\alpha^{6})$ |  $0.18775(5)^{+20.1\%}_{-15.4\%}$ |  $+0.34\%$
${\rm LO}_{3}$ |  $\mathcal{O}(\alpha_{s}^{0}\alpha^{7})$ |  $0.26970(4)^{+14.3\%}_{-16.9\%}$ |  $+0.49\%$
${\rm NLO}_{1}$ |  $\mathcal{O}(\alpha_{s}^{3}\alpha^{5})$ |  $+3.44(5)$ |  $+6.19\%$
${\rm NLO}_{2}$ |  $\mathcal{O}(\alpha_{s}^{2}\alpha^{6})$ |  $-0.1553(9)$ |  $-0.28\%$
${\rm NLO}_{3}$ |  $\mathcal{O}(\alpha_{s}^{1}\alpha^{7})$ |  $+0.2339(3)$ |  $+0.42\%$
${\rm NLO}_{4}$ |  $\mathcal{O}(\alpha_{s}^{0}\alpha^{8})$ |  $+0.001595(8)$ |  $+0.003\%$
${\rm LO}$ |  |  $56.061(8)^{+31.2\%}_{-22.1\%}$ |  $1.0082$
${\rm NLO}_{\rm QCD}$ |  |  $59.05(5)^{+1.6\%}_{-5.9\%}$ |  $1.0620$
${\rm NLO}_{\rm prd}$ |  |  $59.08(5)^{+1.5\%}_{-5.9\%}$ |  $1.0626$
${\rm NLO}$ |  |  $59.59(5)^{+1.6\%}_{-5.9\%}$ |  $1.0717$
Table 1:  Integrated fiducial cross section for the $pp\to t\bar{t}\gamma+X$
process in the di-lepton top-quark decay channel at the LHC with $\sqrt{s}=13$
TeV. Results are shown for ${\rm LO}$, ${\rm NLO}$, ${\rm NLO}_{\rm QCD}$ and
${\rm NLO}_{\rm prd}$ with the corresponding scale uncertainties.
Additionally, all ${\rm LO}_{i}$ and ${\rm NLO}_{i}$ contributions and the
ratio with respect to ${\rm LO}_{1}$ are displayed. MC integration errors are
given in parentheses. Results are provided for $\mu_{0}=E_{T}/4$ and the NLO
NNPDF3.1luxQED PDF set.

We find that both subleading LO contributions, ${\rm LO}_{2}$ and ${\rm
LO}_{3}$, are below $0.5\%$ of ${\rm LO}_{1}$. In addition, ${\rm LO}_{3}$ is
slightly larger than ${\rm LO}_{2}$ due to cancellations in ${\rm LO}_{2}$
between the $g\gamma$ and $b\bar{b}$ subprocesses. It turns out that the
latter channel gives a negative result that can be attributed to the
interference between the different orders in $\alpha_{s}$ and $\alpha$.
Although, the $b\bar{b}$ initial state is PDF suppressed with respect to the
$q\bar{q}$ one, it provides the dominant contribution to ${\rm LO}_{3}$, and
is more than three times larger than the $q\bar{q}$ channel, due to the
presence of additional $t$-channel Feynman diagrams that do not appear in the
$q\bar{q}$ subprocess. The $\gamma\gamma$ initiated contribution is smaller
than the MC integration error of the ${\rm LO}$ result and thus completely
negligible. Consequently, at the integrated fiducial cross-section level, the
subleading LO contributions are phenomenologically insignificant, especially
when compared to the scale uncertainties of the complete LO result (or the
dominant ${\rm LO}_{1}$ contribution for that matter). The theoretical errors
of both LO and ${\rm LO}_{1}$ are of similar size and amount to $31\%$.

As expected, the dominant NLO corrections originate from the NLO QCD
corrections (${\rm NLO}_{1}$) to ${\rm LO}_{1}$ and lead to an increase of the
integrated fiducial cross section by about $6.2\%$. All subleading NLO
contributions are again found to be less than $0.5\%$ of ${\rm LO}_{1}$. In
more detail, ${\rm NLO}_{2}$ and ${\rm NLO}_{3}$ are of the same order of
magnitude and ${\rm NLO}_{4}$ is completely negligible compared to the NLO
scale uncertainties and even MC integration errors. We note that in all the
contributions we used the same top-quark width ($\Gamma_{t}^{\rm
NLO,QCD+EW}=1.3735727$ GeV). In particular, if the LO top-quark width
($\Gamma_{t}^{\rm LO}=1.4806842$ GeV) has been used instead for ${\rm LO}_{1}$
and the NLO QCD one ($\Gamma_{t}^{\rm NLO,QCD}=1.3535983$ GeV) for ${\rm
NLO}_{\rm QCD}$, as it is often done in higher-order QCD calculations, we
would obtain NLO QCD corrections of about $27\%$. Furthermore, we note that
the sum of all subleading LO and NLO contributions increases the ${\rm
NLO}_{\rm QCD}$ result by about $1\%$ and therefore plays only a minor role at
the integrated fiducial cross-section level compared to the ${\rm NLO}_{\rm
QCD}$ scale uncertainties that are of the order of $6\%$. The scale
uncertainties themselves remain unchanged by the inclusion of the subleading
higher-order contributions, and are thus completely driven by the ${\rm
NLO}_{\rm QCD}$ part. Finally, we can observe that ${\rm NLO}_{\rm prd}$ and
${\rm NLO}_{\rm QCD}$ agree within the MC errors, which results from the
cancellation of subleading LO contributions ($0.8\%$) and the subleading NLO
corrections to the production of $pp\to t\bar{t}\gamma$ ($-0.7\%$). In
addition, the difference between ${\rm NLO}$ and ${\rm NLO}_{\rm prd}$, and
thus the subleading NLO corrections involving radiative top-quark decays
(including photon radiation and/or QCD/EW corrections), is of similar
(absolute) size as the other two effects $(0.9\%)$.

###  5.2 Differential fiducial cross sections

Figure 9:  Differential cross-section distributions for the observables
$p_{T,\gamma_{1}}$, $M_{b_{1}b_{2}}$, $p_{T,b_{1}}$ and $p_{T,b_{2}}$ for the
$pp\to t\bar{t}\gamma+X$ process in the di-lepton top-quark decay channel at
the LHC with $\sqrt{s}=13$ TeV. The upper panels present absolute predictions
for ${\rm LO}$, ${\rm NLO}$, ${\rm NLO}_{\rm QCD}$ and ${\rm NLO}_{\rm prd}$
together with the ${\rm NLO}_{\rm prd}$ uncertainty bands. Middle panels show
the ratio of all NLO results with respect to ${\rm NLO}_{\rm QCD}$ and the
lower panels display the relative size of all subleading ${\rm LO}_{i}$ and
${\rm NLO}_{i}$ contributions compared to ${\rm LO}_{1}$. Results are provided
for $\mu_{0}=E_{T}/4$ and the NLO NNPDF3.1luxQED PDF set. Figure 10:  Same as
Figure 9 but for the observable $p_{T,\gamma_{1}}$. The result is shown for
$\mu_{0}=m_{t}$.

Figure 11:  Same as Figure 9 but for the observables $p_{T,b_{1}b_{2}}$,
$p_{T,\ell^{+}\ell^{-}}$, $\Delta R_{\ell^{+}\gamma_{1}}$ and $\Delta
R_{b_{1}\gamma_{1}}$.

While at the integrated fiducial cross-section level the effects of the
subleading LO and NLO contributions are small, it is expected that these
effects might be enhanced for particular observables in certain phase-space
regions. To this end, in Figure 9 we present the differential cross-section
distributions for the transverse momentum of the hardest photon,
$p_{T,\gamma_{1}}$, the invariant mass of the two hardest $b$-jets,
$M_{b_{1}b_{2}}$, as well as their transverse momenta, denoted as
$p_{T,b_{1}}$ and $p_{T,b_{2}}$, respectively. In the upper panels we provide
the absolute predictions for ${\rm LO}$, ${\rm NLO}$, ${\rm NLO}_{\rm QCD}$
and ${\rm NLO}_{\rm prd}$. The middle panels show the ratio of the three NLO
results to ${\rm NLO}_{\rm QCD}$. In addition, both panels display the ${\rm
NLO}_{\rm QCD}$ theoretical uncertainties obtained from scale variation.
Finally, the lower panels present the relative size of all subleading ${\rm
LO}_{i}$ and ${\rm NLO}_{i}$ contributions with respect to ${\rm LO}_{1}$.

For $p_{T,\gamma_{1}}$ we find that the largest NLO corrections arise from
${\rm NLO}_{1}$ and lead in the tail of the distribution to an increase of the
${\rm LO}_{1}$ cross section by about $25\%$. The inclusion of all subleading
LO and NLO contributions reduces the ${\rm NLO}_{\rm QCD}$ prediction by up to
$4\%$ due to EW Sudakov logarithms induced by ${\rm NLO}_{2}$. All other
subleading contributions remain below $1\%$. The ${\rm NLO}_{\rm prd}$
contribution, where the subleading NLO corrections are only included in the
production stage of $t\bar{t}\gamma$, underestimates the full calculation by
about $2\%$. The overall picture remains the same for the $M_{b_{1}b_{2}}$
observable. In particular, we again find that ${\rm NLO}_{2}$ is the dominant
subleading contribution, leading to a reduction in the ${\rm NLO}_{\rm QCD}$
prediction of about $5\%$. This reduction is about half the size of the
corresponding scale uncertainties. On the other hand, ${\rm NLO}_{\rm prd}$
recovers the full calculation correctly in the tails and only small
differences up to $1.5\%$ are found at the beginning of the spectrum. In the
case of the transverse momenta of the hardest and second hardest $b$-jet,
$p_{T,b_{1}}$ and $p_{T,b_{2}}$, the relative size of ${\rm NLO}_{2}$ with
respect to ${\rm LO}_{1}$ is increased to $10\%$. On the contrary, the impact
of all subleading contributions on the final NLO result seems to be very
different. Indeed, we find for $p_{T,b_{1}}$ that ${\rm NLO}_{3}$ is enhanced
in the tail and amounts to $3\%-4\%$, thus partially cancelling the EW Sudakov
logarithms in ${\rm NLO}_{2}$. Moreover, we find large NLO QCD corrections to
${\rm LO}_{1}$ of up to $95\%$, which further suppress the importance of the
subleading contributions. However, both effects, the enhancement of ${\rm
NLO}_{3}$ and the large ${\rm NLO}_{1}$ contribution, have the same origin and
arise from large NLO QCD real corrections to ${\rm LO}_{1}$ and ${\rm
LO}_{3}$, induced by hard jet recoil against the $t\bar{t}$ system. This can
be seen by inspecting the size of the NLO scale uncertainties, which increase
continuously towards the tail of the distribution and become closer in size to
those at LO. Thus, the relative size of the subleading corrections is highly
dependent on the fiducial phase space and might be enhanced if a more
exclusive event selection and/or jet vetos are applied. On the other hand, for
$p_{T,b_{2}}$ the subleading contributions have a more significant impact on
its spectrum, reducing the ${\rm NLO}_{\rm QCD}$ prediction by up to $10\%$.
The scale uncertainties of ${\rm NLO}_{\rm QCD}$ are about $15\%$ and
therefore comparable in size. Moreover, we find that the larger size of the
subleading contributions can also affect the scale uncertainties, which in
this case increase to about $20\%$ for the complete ${\rm NLO}$ case. Finally,
we can observe that, for both $b$-jet transverse momentum spectra the ${\rm
NLO}_{\rm prd}$ predictions fully recover the complete NLO result.

Following the findings of our previous work on the calculation of the NLO QCD
corrections to the $pp\to t\bar{t}\gamma\gamma$ process in the di-lepton top-
quark decay channel Stremmer:2023kcd , where a traditional fixed scale
setting, $\mu_{0}=m_{t}$, led to a reduction of NLO QCD corrections and scale
uncertainties for photonic observables, in the following we reexamine the
$p_{T,\gamma_{1}}$ observable also for this scale choice. To this end, in
Figure 10 we present $p_{T,\gamma_{1}}$ afresh for $\mu_{0}=m_{t}$. We find
that for this fixed scale choice the ${\rm NLO}_{1}$ contribution is reduced
from $25\%$ to $13\%$. Moreover, while for the default scale setting,
$\mu_{0}=E_{T}/4$, the largest NLO QCD corrections are present in the tail,
for $\mu_{0}=m_{t}$ the ${\rm LO}$ and ${\rm NLO}_{\rm QCD}$ predictions
coincide in this phase-space region. Furthermore, the differences of up to
$13\%$ are found at the beginning of the spectrum. The relative size of the
subleading LO and NLO contributions is basically unchanged and leads to a
reduction of the complete calculation of about $5\%$ compared to $4\%$ for
$\mu_{0}=E_{T}/4$. In addition, the scale uncertainties are slightly reduced
to $7\%-9\%$ compared to $10\%-11\%$. Thus, for the scale setting
$\mu_{0}=m_{t}$, the subleading contributions become more important, since the
subleading effects and the scale uncertainties are of similar size.

We continue the discussion and present in Figure 11 other differential cross-
section distributions. In detail, we display the transverse momentum of the
$b_{1}b_{2}$ and $\ell^{+}\ell^{-}$ systems, denoted as $p_{T,b_{1}b_{2}}$ and
$p_{T,\ell^{+}\ell^{-}}$, respectively. Furthermore, we show the angular
separation between $\gamma_{1}$ and $\ell^{+}$ as well as $\gamma_{1}$ and
$b_{1}$, denoted as $\Delta R_{\ell^{+}\gamma_{1}}$ and $\Delta
R_{b_{1}\gamma_{1}}$, respectively. The two dimensionful observables,
$p_{T,b_{1}b_{2}}$ and $p_{T,\ell^{+}\ell^{-}}$, are affected by huge NLO QCD
corrections to ${\rm LO}_{1}$ of about $450\%$ and $200\%$. These corrections
arise from a kinematical suppression at LO Denner:2012yc ; Denner:2015yca ;
Bevilacqua:2019cvp ; Czakon:2020qbd ; Stremmer:2021bnk . Indeed, at the LO
level the two top quarks are predominantly produced in the back-to-back
configuration, so that the transverse momentum of both $b$-jets or leptons may
separately be large, but they largely cancel each other out in the combined
system. At NLO, however, this suppression is weakened due to hard jet recoil
against the $t\bar{t}$ system. This leads, on the one hand, to huge NLO QCD
corrections to ${\rm LO}_{1}$, and on the other hand, also to an enhancement
of the NLO QCD corrections with respect to ${\rm LO}_{3}$ as part of ${\rm
NLO}_{3}$. In particular, for $p_{T,b_{1}b_{2}}$ we find that ${\rm NLO}_{3}$
is about $10\%$ of ${\rm LO}_{1}$. The ${\rm NLO}_{2}$ contribution is of
similar magnitude but has the opposite sign. Consequently, ${\rm NLO}_{3}$ and
${\rm NLO}_{2}$ cancel each other out, and the ${\rm NLO}_{\rm QCD}$ and ${\rm
NLO}$ predictions coincide. Since the two NLO subleading contributions are of
different origins, this cancellation is too a large extend accidental and
highly dependent on the event selection and for example jet vetos. Especially,
vetoing undesired additional jets would drastically reduce the size of ${\rm
NLO}_{1}$ and ${\rm NLO}_{3}$. Also for $p_{T,\ell^{+}\ell^{-}}$ we find a
similar but less pronounced cancellation of ${\rm NLO}_{2}$ ($-5\%$) and ${\rm
NLO}_{3}$ ($2\%$). The ${\rm NLO}_{\rm prd}$ approximation is again able to
fully recover the complete NLO calculation.

For angular distributions such as regular rapidity distributions, $\Delta R$
separations or (azimuthal) opening angles between two final states, we find no
significant effects from any of the subleading LO or NLO contributions.
Indeed, they are individually in the range of $0.5\%-1.0\%$ when compared to
${\rm LO}_{1}$. Only for the $\Delta R$ separation between a photon and a
$b$-jet or charged lepton, such as $\Delta R_{\ell^{+}\gamma_{1}}$ and $\Delta
R_{b_{1}\gamma_{1}}$, we find a small enhancement for the ${\rm NLO}_{3}$
contribution when comparing to ${\rm LO}_{1}$. This enhancement, of about
$2\%$, is visible for large values of the $\Delta R$ separation. However,
these phase-space regions are generally not only less populated but also
characterized by substantial NLO scale uncertainties up to $15\%$. Such
theoretical uncertainties are induced by large NLO QCD corrections to ${\rm
LO}_{1}$. Thus, this small enhancement of ${\rm NLO}_{3}$ is still negligible.

In summary, subleading LO and NLO contributions are generally only important
in the tails of dimensionful observables. In particular, the presence of EW
Sudakov logarithms in ${\rm NLO}_{2}$ can reduce differential predictions by
up to $10\%$. In addition, we have found that for observables affected by
large NLO QCD corrections due to real radiation, the ${\rm NLO}_{3}$
contribution can be enhanced. This can lead to an accidental cancellation with
${\rm NLO}_{2}$ since the origin of both contributions is vastly different.
Furthermore, the cancellation might be heavily influenced by the event
selection and a possible jet veto. Finally, the ${\rm NLO}_{\rm prd}$
prediction is able to mimic the complete NLO calculation for hadronic
observables, while it underestimates the complete prediction by up to
$1\%-2\%$ for non-hadronic ones. Nevertheless, such small effects are well
within the theoretical uncertainties due to scale dependence. Thus, the ${\rm
NLO}_{\rm prd}$ theoretical prediction can be safely used to model various
differential cross-section distributions in phenomenological studies at the
LHC for the $pp\to t\bar{t}\gamma+X$ process in the di-lepton top-quark decay
channel taking into account the current theoretical precision for this
process.

##  6 Top-quark pair production with two isolated photons

###  6.1 Integrated fiducial cross sections

Next we continue with the $pp\to t\bar{t}\gamma\gamma$ process and start with
the discussion of the integrated fiducial cross section, again focusing our
attention on the di-lepton top-quark decay channel. In particular, we are
interested in any differences between this process and $pp\to t\bar{t}\gamma$.

|  |  $\sigma_{i}$ [fb] | Ratio to ${\rm LO}_{1}$
---|---|---|---
${\rm LO}_{1}$ |  $\mathcal{O}(\alpha_{s}^{2}\alpha^{6})$ |  $0.15928(3)^{+31.3\%}_{-22.1\%}$ |  $1.00$
${\rm LO}_{2}$ |  $\mathcal{O}(\alpha_{s}^{1}\alpha^{7})$ |  $0.0003798(2)^{+25.8\%}_{-19.2\%}$ |  $+0.24\%$
${\rm LO}_{3}$ |  $\mathcal{O}(\alpha_{s}^{0}\alpha^{8})$ |  $0.0010991(2)^{+10.6\%}_{-13.1\%}$ |  $+0.69\%$
${\rm NLO}_{1}$ |  $\mathcal{O}(\alpha_{s}^{3}\alpha^{6})$ |  $+0.0110(2)$ |  $+6.89\%$
${\rm NLO}_{2}$ |  $\mathcal{O}(\alpha_{s}^{2}\alpha^{7})$ |  $-0.00233(2)$ |  $-1.46\%$
${\rm NLO}_{3}$ |  $\mathcal{O}(\alpha_{s}^{1}\alpha^{8})$ |  $+0.000619(1)$ |  $+0.39\%$
${\rm NLO}_{4}$ |  $\mathcal{O}(\alpha_{s}^{0}\alpha^{9})$ |  $-0.0000166(2)$ |  $-0.01\%$
${\rm LO}$ |  |  $0.16076(3)^{+30.9\%}_{-21.9\%}$ |  $1.0093$
${\rm NLO}_{\rm QCD}$ |  |  $0.1703(2)^{+1.9\%}_{-6.2\%}$ |  $1.0690$
${\rm NLO}_{\rm prd}$ |  |  $0.1694(2)^{+1.7\%}_{-5.9\%}$ |  $1.0637$
${\rm NLO}$ |  |  $0.1700(2)^{+1.8\%}_{-6.0\%}$ |  $1.0674$
Table 2:  Same as Table 1 but for the $pp\to t\bar{t}\gamma\gamma+X$ process
in the di-lepton top-quark decay channel.

In Table 2 we present the integrated fiducial cross section for ${\rm LO}$,
${\rm NLO}$, ${\rm NLO}_{\rm QCD}$ and ${\rm NLO}_{\rm prd}$ as well as the
corresponding scale uncertainties. We also present the numerical results for
all individual ${\rm LO}_{i}$ and ${\rm NLO}_{i}$ contributions. We find again
that all subleading LO contributions amount to less than $1\%$ of ${\rm
LO}_{1}$ and are therefore negligible when comparing to the LO scale
uncertainties. The ${\rm NLO}_{1}$ contribution is similar in size with
$6.9\%$ compared to $6.2\%$ for $pp\to t\bar{t}\gamma$. As expected, this
contribution yields the largest higher-order corrections at the NLO level. The
biggest difference between the two processes is found for ${\rm NLO}_{2}$.
Indeed, the ${\rm NLO}_{2}$ contribution increases from $-0.3\%$ to $-1.5\%$,
which is consistent with the findings in the literature for the process at
hand but with stable top quarks Pagani:2021iwa . The ${\rm NLO}_{3}$
contribution remains at the level of $0.4\%$. Finally, ${\rm NLO}_{4}$
contributes at the level of $0.01\%$ only. This contribution is therefore
smaller than the MC integration error and phenomenologically completely
irrelevant. Due to the increase in size of ${\rm NLO}_{2}$, there are larger
cancellations between the individual contributions, so that the difference
between the ${\rm NLO}_{\rm QCD}$ and ${\rm NLO}$ predictions decreases and
both results agree within their coresponding MC errors. As for the $pp\to
t\bar{t}\gamma$ process, also in this case the scale uncertainties at the
integrated fiducial cross-section level are barely affected by the subleading
contributions. Including all subleading LO contributions and subleading NLO
corrections to the production of $pp\to t\bar{t}\gamma\gamma$ in ${\rm
NLO}_{\rm prd}$ leads to a decrease of the ${\rm NLO}_{\rm QCD}$ result by
about $-0.5\%$. Similar to the previous process, we again have cancellations
between the subleading LO contributions ($0.9\%$) and the subleading NLO
corrections ($-1.4\%$). The differences between ${\rm NLO}$ and ${\rm
NLO}_{\rm prd}$ are reduced to $0.4\%$ compared to $0.9\%$ for the $pp\to
t\bar{t}\gamma$ process, implying that subleading NLO corrections involving
radiative top-quark decays are less important, at least at the integrated
fiducial cross-section level.

###  6.2 Differential fiducial cross sections

Similarly to $t\bar{t}\gamma$ production, also for the $pp\to
t\bar{t}\gamma\gamma+X$ process we are investigating the impact of all
subleading contributions and subleading higher-order corrections on various
differential cross-section distributions. In Figure 12 we show the observables
$p_{T,\gamma_{1}\gamma_{2}}$, $M_{\gamma_{1}\gamma_{2}}$, $p_{T,\gamma_{1}}$
and $p_{T,b_{1}}$. As in the case of $pp\to t\bar{t}\gamma+X$, here we again
present in the upper panels the absolute predictions for ${\rm LO}$, ${\rm
NLO}$, ${\rm NLO}_{\rm QCD}$ and ${\rm NLO}_{\rm prd}$. The middle panels
display the ratio with respect to ${\rm NLO}_{\rm QCD}$, including the scale
uncertainties of ${\rm NLO}_{\rm QCD}$, and the lower panels show the size of
the subleading contributions compared to ${\rm LO}_{1}$.

Figure 12:  Same as Figure 9 but for the $pp\to t\bar{t}\gamma\gamma+X$
process in the di-lepton top-quark decay channel and for the observables
$p_{T,\gamma_{1}\gamma_{2}}$, $M_{\gamma_{1}\gamma_{2}}$, $p_{T,\gamma_{1}}$
and $p_{T,b_{1}}$.

Figure 13:  Same as Figure 9 but for the $pp\to t\bar{t}\gamma\gamma+X$
process in the di-lepton top-quark decay channel and for the observables
$p_{T,\gamma_{1}\gamma_{2}}$, $M_{\gamma_{1}\gamma_{2}}$ and
$p_{T,\gamma_{1}}$. Results are shown for $\mu_{0}=m_{t}$.

First, we focus on $p_{T,\gamma_{1}\gamma_{2}}$ and
$M_{\gamma_{1}\gamma_{2}}$. Their importance stems from the fact that both
differential cross-section distributions can be seen as the direct and
irreducible background to the kinematics of the Higgs boson in $pp\to
t\bar{t}H$ production with the $H\to\gamma\gamma$ decay. For both observables
we find that ${\rm NLO}_{1}$ is the dominant source of NLO corrections,
leading to an increase of ${\rm LO}_{1}$ of up to $30\%$ in the tails. The
subleading LO and NLO contributions are dominated by ${\rm NLO}_{2}$, due to
EW Sudakov logarithms, which amounts to $-8\%$ for
$p_{T,\gamma_{1}\gamma_{2}}$ and $-5\%$ for $M_{\gamma_{1}\gamma_{2}}$ with
respect to ${\rm LO}_{1}$. This results in a decrease of the full calculation
by about $5\%$ and $3\%$, respectively. At the same time, in these phase-space
regions, the ${\rm NLO}$ scale uncertainties are larger by a factor of $2$ for
$p_{T,\gamma_{1}\gamma_{2}}$ and $4$ for $M_{\gamma_{1}\gamma_{2}}$. The ${\rm
NLO}_{\rm prd}$ prediction fully recovers the complete calculation for
$p_{T,\gamma_{1}\gamma_{2}}$ and differences of only about $1\%$ can be found
for $M_{\gamma_{1}\gamma_{2}}$. Thus, ${\rm NLO}_{\rm prd}$ is more than
sufficient to properly describe the shape of the two observables in the
presence of subleading LO contributions and subleading NLO corrections.

We then turn to the $p_{T,\gamma_{1}}$ differential cross-section
distribution, where NLO QCD corrections up to $25\%-30\%$ can be found. The
subleading contributions reduce the ${\rm NLO}$ calculation by about $5\%-6\%$
and are therefore slightly larger than those found for the $pp\to
t\bar{t}\gamma$ process. Indeed, in the latter case the difference is about
$4\%$. This small rise is due to the increase in the ${\rm NLO}_{2}$
contribution that we have already observed at the integrated cross-section
level. In the case of $p_{T,b_{1}}$ the difference between ${\rm NLO}_{\rm
QCD}$ and ${\rm NLO}$ is somewhat enlarged in the tails from $4\%$ for $pp\to
t\bar{t}\gamma$ to $6\%$ for the $pp\to t\bar{t}\gamma\gamma$ process. This
can again be attributed to the increase of the ${\rm NLO}_{2}$ corrections
from $-10\%$ to $-13\%$, while the size of ${\rm NLO}_{3}$ is slight reduced
to $2\%-3\%$ from to $3\%-4\%$ for $pp\to t\bar{t}\gamma$. In general, the
presence of a second photon already at LO affects the kinematics of the
$t\bar{t}$ system in such a way that the large NLO QCD corrections in ${\rm
NLO}_{1}$ are reduced for several observables such as $p_{T,b_{1}}$ or
$p_{T,b_{1}b_{2}}$ compared to the $pp\to t\bar{t}\gamma$ process. In
particular, for $p_{T,b_{1}}$ we find a reduction from $95\%$ to $55\%$. It
follows directly that the enhancement of ${\rm NLO}_{3}$ in such observables
is also reduced. Therefore, the accidental cancellations between ${\rm
NLO}_{2}$ and ${\rm NLO}_{3}$, which occurred in the $pp\to t\bar{t}\gamma$
case, although still present, are substantially reduced.

For comparison purposes and similar to what we have done for photonic
observables in case of the $pp\to t\bar{t}\gamma$ process, also here we show
differential cross-section distributions with the alternative scale choice. In
detail, in Figure 13 we display $p_{T,\gamma_{1}\gamma_{2}}$,
$M_{\gamma_{1}\gamma_{2}}$ and $p_{T,\gamma_{1}}$ for $\mu_{R}=\mu_{F}=m_{t}$.
For $p_{T,\gamma_{1}\gamma_{2}}$ and $M_{\gamma_{1}\gamma_{2}}$, we again find
that this scale choice leads to a reduction in the size of the NLO QCD
corrections in ${\rm NLO}_{1}$ from about $30\%$ to $10\%-12\%$. In addition,
the scale uncertainties are changed from $12\%-13\%$ to $8\%-10\%$. Thus, the
fixed scale setting not only decreases higher-order corrections in ${\rm
NLO}_{1}$, but also provides improved scale uncertainties. The reduction of
higher-order effects in ${\rm NLO}_{1}$ has also a direct impact on the
significance of the subleading NLO corrections. In particular, because the
relative size of the subleading ${\rm NLO}_{2}$ corrections with respect to
${\rm LO}_{1}$ does not change for $\mu_{0}=m_{t}$, the overall ${\rm
NLO}_{2}$ contribution becomes larger with respect to the complete ${\rm NLO}$
result. Indeed, we observe the rise of ${\rm NLO}_{2}$ from $5\%$ to $8\%$ as
well as from $3\%$ to $4\%$ for $p_{T,\gamma_{1}\gamma_{2}}$ and
$M_{\gamma_{1}\gamma_{2}}$, respectively. Furthermore, for
$p_{T,\gamma_{1}\gamma_{2}}$ the subleading NLO corrections become as large as
the corresponding scale uncertainties of ${\rm NLO}_{\rm QCD}$, which are also
equal in size to those of the complete ${\rm NLO}$ result. The same can be
observed for the third differential cross-section distribution, namely for
$p_{T,\gamma_{1}}$. In this case the ${\rm NLO}_{1}$ contribution is reduced
in the tails from $20\%$ for $\mu_{0}=E_{T}/4$ to $5\%$ for $\mu_{0}=m_{t}$.
This increases the importance of the ${\rm NLO}_{2}$ contribution again and
leads to a reduction of the difference between ${\rm NLO}_{2}$ and ${\rm NLO}$
by about $8\%$ compared to $5\%-6\%$ for $\mu_{0}=E_{T}/4$. Consequently, the
subleading contributions are of similar size as the scale uncertainties of the
${\rm NLO}_{\rm QCD}$ result, and are at the level of $10\%$. Lastly, for all
three photonic observables, the complete ${\rm NLO}$ predictions for the two
scale settings differ by at most $4\%$ in the tails and are therefore within
the respective scale uncertainties.

##  7 Summary

We have presented the first calculation of the complete set of NLO corrections
to $pp\to t\bar{t}\gamma+X$ and $pp\to t\bar{t}\gamma\gamma+X$ including top-
quark decays at the LHC with $\sqrt{s}=13$ TeV. In order to study the di-
lepton top-quark decay channel we have employed the Narrow Width Approximation
that works in the $\Gamma/m\to 0$ limit and preserves spin correlations. In
this approximation all contributions without two resonant top quarks and $W$
gauge bosons are simply neglected. In our calculations we have included the
dominant LO contribution at ${\cal O}(\alpha_{s}^{2}\alpha^{5})$ for $pp\to
t\bar{t}\gamma+X$ and ${\cal O}(\alpha_{s}^{2}\alpha^{6})$ for $pp\to
t\bar{t}\gamma\gamma+X$ as well as the corresponding NLO QCD corrections.
Furthermore, all subleading LO contributions and all remaining NLO corrections
are also taken into account. In addition, NLO corrections, as well as photon
radiation are consistently included in the production phase as well as in all
decay stages of the process. Even if we have considerd the di-lepton decay
channel of the top quark, the extension to hadronic decays of the $W$ boson
can be straightforwardly incorporated into our framework.

On the technical side, to perform these calculations, we have extended the
Nagy-Soper subtraction scheme as implemented in the Helac-Dipoles Monte Carlo
program for calculations with QED-like singularities and modified the
structure of the program to allow the simultaneous calculation of all
contributions at different orders in $\alpha_{s}$ and $\alpha$.

The main findings of this paper apply to both processes and can be summarised
as follows. At the integrated fiducial cross-section level, as well as for all
the angular distributions we examined, the subleading LO contributions and
subleading NLO corrections are negligibly small with respect to the ${\rm
NLO}$ scale uncertainties of the complete result. On the other hand, the
dominant ${\rm NLO}_{1}$ contribution is essential for precise predictions, as
it leads to a reduction in scale uncertainties from about $30\%$ to $6\%$.
Furthermore, it also introduces a change in the normalisation as well as in
the shape of various differential cross-section distributions. In general, the
inclusion of the subleading contributions has been found to be essential only
in the tails of dimensionful observables due to one-loop EW Sudakov
logarithms. Indeed, the presence of the EW Sudakov logarithms in ${\rm
NLO}_{2}$ leads to a reduction in the tails of up to $10\%$ compared to the
${\rm NLO}_{\rm QCD}$ result. Moreover, the ${\rm NLO}_{2}$ contribution can
be as large as the ${\rm NLO}_{\rm QCD}$ scale uncertainties, potentially
affecting the comparison between theoretical predictions and experimental
measurements. In addition, for certain observables affected by large higher-
order real-emission QCD corrections, e.g. for $p_{T,b_{1}b_{2}}$, the ${\rm
NLO}_{3}$ contribution can be enhanced from a few percent up to $10\%$ with
respect to ${\rm LO}_{1}$. As the origins of the subleading ${\rm NLO}_{2}$
and ${\rm NLO}_{3}$ contributions are very different and there are random
cancellations between them, they should always be considered together. We note
here, that both the accidental cancellations between ${\rm NLO}_{2}$ and ${\rm
NLO}_{3}$ as well as the size of real-emission QCD corrections depend on the
exact event selection and can be substantially affected, for example, by
applying a veto on an additional jet. Finally, the subleading LO contributions
and the ${\rm NLO}_{4}$ contribution are negligibly small at the integrated
and differential fiducial cross-section level with respect to the NLO scale
uncertainties.

An important finding of the paper is that the ${\rm NLO}_{\rm prd}$
approximation models the complete ${\rm NLO}$ result very well. Indeed,
differences of up to $2\%$ only are found for some leptonic and/or photonic
observables, but these are negligible compared to the NLO scale uncertainties
of the complete result. We remind the readers that in ${\rm NLO}_{\rm prd}$,
all LO contributions as well as ${\rm NLO}_{1}$ are fully included in both the
$t\bar{t}\gamma(\gamma)$ production and decays of the top-quark pair. However,
the subleading NLO corrections (${\rm NLO}_{2}$, ${\rm NLO}_{3}$, ${\rm
NLO}_{4}$) are only included in the production stage of the
$t\bar{t}\gamma(\gamma)$ process. The same applies to photon radiation for
these three NLO contributions. The ${\rm NLO}_{\rm prd}$ approximation not
only provides a great simplification of higher-order calculations, especially
when it comes to the real emission corrections, but will ultimately also be of
great benefit when matching ${\rm NLO}_{\rm prd}$ predictions to parton shower
programs.

###### Acknowledgements.

We thank Ansgar Denner and Mathieu Pellen for providing us with the results
for the NLO EW corrections to the $pp\to
e^{+}\nu_{e}\mu^{-}\bar{\nu}_{\mu}b\bar{b}+X$ process that helped to cross
check our implementation of the EW corrections in the Helac-Dipoles framework.
This work was supported by the Deutsche Forschungsgemeinschaft (DFG) under
grant 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 authors gratefully
acknowledge the computing time provided to them at the NHR Center NHR4CES at
RWTH Aachen University (project number p0020216). This is funded by the
Federal Ministry of Education and Research, and the state governments
participating on the basis of the resolutions of the GWK for national high
performance computing at universities.

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