Associated production of a top-quark pair with two isolated photons at the LHC through NLO in QCD

We report on the computation of NLO QCD corrections to top-quark pair production in association with two photons at the LHC. Higher-order effects and photon bremsstrahlung are taken into account in the production and decays of the top-quark pair. Top-quark and $W$-boson decays are treated in the Narrow Width Approximation conserving spin correlations up to NLO in QCD. This is the first time that the complete set of NLO QCD corrections to the $pp \to t\bar{t}\gamma\gamma$ process including top-quark decays is calculated. We present results at the integrated and differential cross-section level in the di-lepton and lepton $+$ jet channel. In addition, we investigate the effect of photon bremsstrahlung in $t\bar{t}$ production and top-quark decays, as well as the mixed contribution. The latter contribution, in which two photons occur simultaneously in the production and decay of the $t\bar{t}$ pair, proved to be significant at both the integrated and differential cross-section level.


Introduction
The observation of the pp → t tH process at the Large Hadron Collider (LHC) reported by the CMS [1] and ATLAS [2] collaborations has launched a new endeavour to investigate the tree-level top quark Yukawa coupling (Y t ) and the CP structure of the Higgs boson.One of the most sensitive Higgs-boson decay channels for probing the pp → t tH process is H → γγ.Despite the small branching ratio the Higgs-boson signal can be extracted in this channel thanks to the excellent photon reconstruction and identification efficiency of the ATLAS and CMS detectors.Even though by probing the interactions between the H boson and electroweak W/Z gauge bosons, CMS and ATLAS have determined that the H boson quantum numbers are consistent with the Standard Model (SM) [3][4][5][6][7][8], the presence of a pseudoscalar admixture, which introduces a second coupling to the top quark, has not yet been ruled out and is worth investigating.The observation of a non-zero CP-odd coupling component would signal the existence of physics beyond the SM, and open up the possibility of CP-violation in the Higgs-boson sector, see e.g.[9][10][11][12][13][14][15][16][17][18][19][20][21] and references therein.Such a new source of CP violation could play a fundamental role in explaining the matter-antimatter asymmetry of the universe.Therefore, studies of the Y t coupling in the H → γγ channel would provide an alternative and independent path for CP tests in the Higgs-boson sector.Indeed, present analyses of the SM Higgs boson at the LHC in the pp → t tH production mode focus on the Higgs boson decaying into two photons.In fact, the first single-channel observation of the pp → t tH process by both ATLAS and CMS [22,23] has been reported in the H → γγ channel, together with the measurement of the CP structure of the Y t coupling.Despite the fact that the data disfavored the pure CP-odd model of the Ht t coupling, still only rather weak constrains exist on the possible admixture between the CP-even and CP-odd component of Y t .A close scrutiny of the backgrounds shows that the direct production of t tγγ is the most relevant, often referred to as the irreducible t tγγ background.Experimental analyses at the LHC rely on data-driven approaches to estimate this background, however, Monte Carlo simulations are also used for this purpose, but mainly with LO accuracy.
On the theory side, NLO predictions for the pp → t tγγ process with stable top quarks have already been known for some time and have been further matched to parton shower programs [24][25][26][27].In all these studies NLO QCD corrections were calculated for the pp → t tγγ production stage only.On the one hand, parton shower programs contain dominant soft-collinear logarithmic corrections and can approximate radiative effects in top-quark decays.On the other hand, such effects are described by matrix elements formally accurate only at LO.Moreover, photon radiation from the charged topquark decay products is omitted in such theoretical predictions.In addition to higher-order QCD effects also NLO EW corrections for t t production in association with two photons have recently been presented in literature [28], but again only for stable top quarks.The contribution of photons emitted after the decay of top quarks might be, however, significant and should be incorporated in theoretical predictions for this process.Consequently, a complete study with higher-order corrections for the following final state W + W − b b γγ including W decays is required to deeper understand the dynamics of the pp → t tγγ process.Similar studies for the simpler pp → t tγ process with top-quark and W decays included, either in the Narrow Width Approximation (NWA) or with full off-shell effects, are already available in literature [29][30][31][32][33].They have shown, among others, that photon radiation is distributed evenly between the t tγ production and the two top-quark decays: t → bW + (γ) → bℓ + ν ℓ γ as well as t → bW − (γ) → bℓ − νℓ γ, where ℓ = e, µ [33].
The purpose of this paper is to mitigate the current situation and to calculate for the first time NLO QCD corrections to the pp → t tγγ process taking into account higher-order effects in both the t t production and the decay of the top-quark pair.In this calculation top-quark decays are treated in the NWA.Thus, NLO t t spin correlations are preserved throughout the calculation.Furthermore, effects of photon bremsstrahlung from the charged top-quark decay products are consistently included.In detail, we calculate NLO QCD corrections to the following final states • pp → (t → W + (q q ′ ) b) × ( t → W − (ℓ − νℓ ) b) γγ, denoted as the di-lepton and lepton + jet channel, respectively.In the remainder of the paper we refer to these two processes as pp → ℓ + ν ℓ ℓ − νℓ b b γγ + X and pp → ℓ − νℓ jj b b γγ + X.In the case of the dilepton channel we consider the following leptonic combinations: (ℓ + ℓ − ) ∈ (e + e − , e + µ − , µ + e − , µ + µ − ).We do not include τ ± as most of the experimental analyses at the LHC distinguish the electron and muon from the τ channel, which is more difficult to reconstruct.For the lepton + jet decay channel we employ ℓ − = e − , µ − and W + gauge boson decays into two families of light quarks, i.e. q q ′ = u d, cs.For W + → u d and W + → cs decays QCD radiative corrections are taken into account.We examine the size of higher-order corrections and theoretical uncertainties for both decay channels.We additionally address the choice of a judicious renormalisation and factorisation scale setting and the size of parton distribution function (PDF) uncertainties.Having included photon emissions from various stages, we can assess their distribution and impact on the integrated and differential fiducial cross sections for both di-lepton and lepton + jet channels.Our results are obtained with the help of Helac-1Loop/Recola and Helac-Dipoles.For this work, the Helac-Nlo MC framework, that comprises Helac-1Loop and Helac-Dipoles, is used for the first time in NLO QCD calculations involving hadronically decaying top quarks.
The article is organized as follows: in Section 2 we outline the framework of the calculation and discuss cross-checks that have been performed.Input parameters and cuts that have been used to simulate detector response are summarised in Section 3. Numerical results for the integrated and differential cross sections for the pp → ℓ + ν ℓ ℓ − νℓ b b γγ + X process at the LHC with √ s = 13 TeV for two renormalisation (µ R ) and factorisation (µ F ) scale choices are presented in detail in Section 4. The theoretical uncertainties that are associated with neglected higher order terms in the perturbative expansion and with the different parametrisation of PDFs, are also given there.Our findings for the pp → ℓ − νℓ jj b b γγ + X process are provided in Section 5 following the same structure as for the dilepton channel.In addition, different parameter choices of the smooth photon isolation prescription are briefly discussed there as well.Lastly, in Section 6 our results for the pp → t tγγ production process are briefly summarised and conclusions are outlined.

Description of the calculation
We calculate NLO QCD corrections to the pp → t tγγ process at the LHC.In particular, we evaluate α s corrections to the Born-level process at O(α 2 s α 6 ).Unstable top quarks in the di-lepton and lepton + jet channels are considered.This leads to the following final states respectively where ℓ ± = µ ± , e ± .The decays of top quarks and W bosons are performed in the NWA, i.e. in the limit Γ t /m t → 0. In this approach all terms less singular than Γ −2 t are consistently neglected and the Breit-Wigner propagators become delta-functions which force unstable particles to be on-shell, see e.g.Refs [33][34][35][36][37][38][39][40].Thus, the differential cross section can be factorised in the production of top quarks (and photons) and top-quark decays (with photons) according to In addition, treating the W gauge boson in the NWA, the differential top-quark decay rate can be further expanded as where W + → ℓ + ν ℓ or W + → q q ′ .This leads to a total of 15 possibilities (resonant histories) from which photons can be radiated in the decay chain at LO. Furthermore, we defined in Eq. (2.2) the following three contributions: Prod., Mixed and Decay, based on the number of photons in the t t production process, in order to study the origin of photon radiation in more detail.In Figure 1 we depict a few examples of Feynman diagrams for the three contributions.At NLO, however, the number of resonant histories increases up to 45 in the di-lepton and 60 in the lepton + jet channel due to additional QCD radiation.At the Born level in the di-lepton and lepton + jet channels, we encounter the following subprocesses, respectively gg where q = u, d, c, s, b and ℓ ± = e ± , µ ± as well as q q ′ = u d, cs.Since we work in the NWA there is no cross-talk between the t t production and top-quark decays or between the t and t decay.Thus, no additional contributions arise when both leptons in the di-lepton channel or the quarks in the initial and final state in the lepton + jet decay channel are coming from the same generation.Although the contribution of bottom quarks in the initial state is only 0.1% of the integrated fiducial LO cross section in both decay channels, and therefore numerical insignificant, we still include it in our calculations.At NLO in QCD additional subprocesses, that can be constructed from Born-level subprocesses by gluon radiation and crossing, must be included.This leads to the following set of subprocesses contributing to the real emission part of the NLO QCD calculation We note that the contribution from bottom initiated subprocesses also at the NLO level is phenomenologically negligible, but we nevertheless include all subprocesses for consistency reasons.Furthermore, since subleading NLO corrections are not included in our calculation, the three contributions: Prod., Mixed and Decay do not mix with each other.Only the presence of an additional photon from the real emission part of such corrections would introduce some ambiguity and lead to different resonant structures in singular limits.The same problem also occurs in processes such as pp → t tj(j) production if higher-order QCD effects are included [37,42].Consequently, the calculation of NLO QCD corrections for the pp → t tγγ process can be performed for each resonant history, that is present at LO, independently.We work in the five-flavour scheme and keep the Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix diagonal.For the calculation of tree-level and one-loop matrix elements the program Recola [43,44] is used, together with the library Collier [45] that is employed for the numerical evaluation of one-loop scalar and tensor integrals.Because Recola is also able to provide matrix elements in the so-called double-pole approximation, see e.g.Refs.[46][47][48], it was straightforward to interface it to the Helac-NLO MC program [49] and adopt it for use in the NWA [33].In addition, we have implemented in the Recola framework the random polarisation method as introduced in Refs.[50][51][52].In that way, the polarisation state is replaced by a linear combination of helicity eigenstates while the spin summation is replaced by an integration over a phase parameter.This leads to a drastic speed improvement, especially for processes involving many final state particles.For example, for the polarisation state of a gluon, we can write where ϵ µ (k, ±) are helicity eigenstates.Then, the sum over the helicity of this gluon can be written as We do not have large differences in the values of |M ϕ | 2 as function of ϕ, since for every value of ϕ we have both helicities contributing.Consequently, a flat distribution in Monte Carlo sampling leads to satisfactory results.Notice that the range of integration could have been reduced to [0, π] with the same result.We keep 2π as the upper bound instead, to accommodate the third degree of freedom of massive gauge bosons.The latter degree of freedom can be added to Eq. (2.7) without any phase factor.For a few phase-space points the Born-level and one-loop matrix elements of all subprocesses have been cross-checked against Helac-1Loop [53].In this framework the one-loop matrix elements are reduced to scalar integrals at the integrand level using the OPP reduction technique [54] as implemented in the CutTools program [55].Furthermore, for the evaluation of one-loop scalar functions the program OneLOop [56] is employed.The calculation of the real emission part is performed with the Nagy-Soper subtraction scheme [52], which has recently been extended to also handle calculations in the NWA [42].In addition, we have used the Catani-Seymour dipole formalism [57,58] and its extension to top-quark decays [33,59] to cross-check our results for a few resonant histories.Both subtraction schemes, together with a phase-space restriction on the subtraction terms [60][61][62][63], are implemented in the Helac-Dipoles MC program [64].The phase-space integration is performed with Parni [65] and Kaleu [66].To improve the efficiency of the phase-space generation for the subtracted real emission part, additional channels have been added for the subtraction terms contributing at the different decay stages, see e.g.Ref. [67] for more details.All results are stored in modified Les Houches Event files (LHEFs) [68] as partially unweighted events [69].These files include supplementary matrix element and PDF information [70] needed for reweighting.This makes it possible to create new differential cross-section distributions, change their binning and/or ranges.Furthermore, different PDF sets as well as renormalisation/factorialisation scale settings can be used without having to perform new time-consuming calculations.

Computational setup
In this section let us define the setup for the calculations of the present work.As already mentioned we consider top-quark pair production in association with two photons at O(α 3 s α 6 ) in the di-lepton and lepton + jet decay channel.We shall provide the results for the LHC Run 2 energy of √ s = 13 TeV.As recommended by the PDF4LHC working group we employ three sets of PDFs for the use at the LHC [71].The NLO NNPDF3.1 PDF set [72], that we employ both at LO and NLO, is our default PDF set.The running of the strong coupling constant is, therefore, always performed with two-loop accuracy via the LHAPDF interface [73].Furthermore, we present additional theoretical results for the NLO MSHT20 [74] and NLO CT18 [75] PDF sets to quantify the differences between various PDF sets.The G µ -scheme is used for the derivation of the electromagnetic coupling constant α according to where m W = 80.379 GeV and m Z = 91.1876GeV.However, for the emission of the two external hard photons we use the α(0)-scheme with α −1 = α −1 (0) = 137.035999084[76].This choice corresponds to a mixed scheme where the total power of α is split into two parts according to the number of final state photons α 6 → α 2 α 4 Gµ and should be renormalised in different schemes when NLO EW corrections are included [28,77].In practise, we use α Gµ as input value and rescal the final results by α 2 /α 2 Gµ = 0.93044.... Consequently, the prediction for the pp → t tγγ cross section is decreased by about 7%.Furthermore, the following SM parameters are used m t = 172.5 GeV and Γ W = 2.0972 GeV.All other particles are considered massless.The LO and NLO top-quark widths are calculated based on Refs.[78,79] and their numerical values are given by where The width of the top quark is kept fixed during the estimation of scale uncertainties.However, the error introduced by this treatment is small.In particular, for the two scales µ R = m t /2 and µ R = 2m t is at the level of 1.5% only.The antik T jet algorithm [80] with the radius parameter R = 0.4 is used to cluster final state partons with pseudo-rapidity |η| < 5 into jets.In the di-lepton decay channel we require two opposite-sign charged leptons and exactly two b-jets.On the other hand in the lepton + jet decay channel we require one negatively charged lepton, exactly two b-jets and at least two light jets.In order to avoid QED collinear singularities in photon emission due to q → qγ splittings, a separation between quarks and photons is required.Since on the experimental side quark and gluon jets are indistinguishable, a separation between photons and gluons is additionally induced.Consequently, for a given transverse momentum of the photon (p T, γ ) an angular restriction is introduced on the phase space of the soft gluon emission.The soft divergence in the real emission part is, therefore, different from that in the virtual correction impairing the cancellation of infrared divergences.To ensure IR-safety at NLO QCD in presence of two isolated prompt photons, the smooth photon isolation prescription, as described in Ref. [81], is used.According to this prescription, the event is rejected unless the following condition is fulfilled before the jet clustering is performed for all R ≤ R γj .We use R γj = 0.4 and set ϵ γ = n = 1.Furthermore, E T i is the transverse energy of the parton i, E T γ is the transverse energy of the photon and R γi is given by At the cost of no longer reproducing the form of cone isolation applied in experimental analyses, the smooth photon isolation prescription ensures that arbitrarily soft radiation will always pass the condition, but (hard) collinear (R → 0) radiation is forbidden.It is important to quantify the differences between prediction obtained with alternative choices of ϵ γ , n and E T γ , which we also do later in this paper.Of course, the ultimate aim is to compare the results from this work with ATLAS and CMS data when they become available.In addition to the prompt photon requirements described above, the two photons must satisfy There are no restrictions on the kinematics of the extra light jet (if resolved by the jet algorithm) and the missing transverse momentum defined as p miss T = |⃗ p T, ν ℓ + ⃗ p T, νℓ |.On the other hand, both prompt photons and all jets must be well separated from each other and from charged leptons Finally, in the lepton + jet channel we require that the invariant mass of at least one light-jet pair, denoted as M jj , has to be in the following W boson mass window where m W = 80.379 GeV.Such a cut has already been applied in NLO QCD calculations involving hadronically decaying top quarks.Indeed, it has been used to suppress kinematical configurations from real radiation where the two light jets originating from the W boson are recombined into a single jet, while the extra real radiation gives rise to the presence of a second resolved jet that passes all the cuts [82].It has also been utilised to reduce photon radiation from the hadronically decaying W gauge boson [29].We employ a dynamical factorisation/renormalisation scale choice where E T is defined according to In this scale setting, that is defined on an event-by-event basis, p T, t /p T, t are the transverse momenta of the on-shell top quarks.We have checked that variations of that scale setting lead to very similar results.In detail, we have reconstructed top-quark momenta based on the MC truth information with and without photons.We have also used a more resonant aware version of this scale definition in which the transverse momentum of the photon has been included or not in the definition of E T depending whether this photon is produced in the production or decay stage.As an alternative and for comparison purposes we also employ a fixed scale setting defined as The scale uncertainties are estimated by a 7-point scale variation in which the factorisation and renormalisation scales are varied independently in the range which leads to the following pairs ) , (0.5, 1) , (1, 2) , (1, 1), (1, 0.5), (2, 2), (0.5, 0.5) . (3.14) The final scale uncertainties are obtained by finding the minimum and maximum of the resulting cross sections.

1.24
Table 1.Integrated fiducial cross sections at LO and NLO QCD for the pp → ℓ + ν ℓ ℓ − νℓ b b γγ + X process at the LHC with √ s = 13 TeV.Results are presented for two scale settings µ 0 = E T /4 and µ 0 = m t as well as for the three contributions: Prod., Mixed and Decay.The NLO NNPDF3.1 PDF set is employed.The theoretical uncertainties from the 7-point scale variation and MC integration errors (in parentheses) are also displayed.

Integrated fiducial cross sections
We begin the presentation of our results with a discussion of the integrated fiducial cross section for the process pp → ℓ + ν ℓ ℓ − νℓ b b γγ + X.In Table 1 we show our findings in the di-lepton channel at LO and NLO QCD.The results are shown for two scale settings µ 0 = E T /4 and µ 0 = m t using the NLO NNPDF3.1 PDF set in both cases.The NLO QCD corrections to the full process are rather moderate, of the order of 30% for both scale choices and thus of the same size as the LO scale uncertainties.The latter uncertainties are reduced at NLO in QCD to 6%, so by a factor of 5.At the integrated cross-section level, the difference in the theoretical results between the two scale settings is negligible and decreases from 2% at LO to 1% at NLO.In order to examine the radiation pattern of the prompt photons, the cross section is divided into the following three contributions: Prod., Mixed and Decay, as defined in Eq. (2.2).In particular, within our fiducial phase-space volume, independently of the scale choice as well as the perturbative order in QCD, the integrated cross section is dominated by the Mixed contribution (43% − 44%) rather than the Prod.one.The latter contribution is at the level of 39% − 40% only.On the other hand, the Decay contribution is about half the size (16% − 18%).Similar large effects from photon radiation from top-quark decays have already been observed for the simpler process pp → t tγ [29,33].In that case, due to the absence of the Mixed contribution, photon radiation is evenly distributed between Prod.and Decay.For the pp → t tγγ process NLO QCD corrections for the Prod.and Mixed contribution are of the order of 30%.Substantially smaller higher order effects are found for the Decay part.Indeed, they are at the level of 17% and 24% for µ 0 = E T /4 and µ 0 = m t respectively.The overall behaviour is very similar for both scale settings.The largest differences between the two scales choices can be found for the Decay contribution at LO, which are at the level of 7% while the differences at NLO are always below 1%.gg gg/pp q q q q/pp qg + qg (qg + qg)/pp σ NLO
[fb] 0.02587( 4  Results are divided into the different channels: gg, q q and qg/qg as well as given for the three contributions: Prod., Mixed and Decay.Relative contributions for these three categories are also provided.The µ 0 = E T /4 scale setting and NNPDF3.1 PDF set are employed.MC integration errors (in parentheses) are also displayed.
In the following, we examine the decomposition into different resonance configurations, Prod., Mixed and Decay, in more detail by dividing the NLO QCD cross section into the different production channels, namely: gg, q q and qg/qg.Results are shown in Table 2.They are obtained for the scale setting µ 0 = E T /4 and the NLO NNPDF3.1 PDF set.We first note that at the central scale the gg channel dominates the integrated fiducial NLO QCD cross section by about 56.4% followed by the q q channel with 24.3% and at last the qg/qg channel with 19.3%.The relative size of the gg channel decreases for the Prod.contribution alone and amounts to 36.3% which is about the same size as the q q channel with 37.5%.In absolute size the Mixed contribution of the q q and qg/qg channels is smaller compared to the Prod.contribution as expected from simple phase-space arguments.Indeed, the finite mass of the top quark and W boson limits the available phase space of the decay products.In contrast, in the gg channel the Mixed contribution increases in absolute size and becomes almost twice as large as the Prod.part.Furthermore, it dominates the overall Mixed contribution for the whole process with about 64%.We find that for all three production channels the Decay contribution is the least relevant.When comparing it to the Mixed contribution we observe that in the gg channel its contributions is reduced by a factor of 2, while in the q q and qg/qg channels we receive a factor of 6 and 9 respectively.Thus, the Decay contribution is clearly dominated by the gg channel with 86.5%.To conclude, on the one hand, we find that the gg production channel is suppressed as the number of photons in the t t production stage increases, because photons can only be radiated from the topquark line.With more photon emissions in top-quark decays, on the other hand, the available phase space is reduced and both q q and qg channels become smaller in absolute size.In the gg production channel these two effects cancel each other out and the absolute cross section can even increase with a decreasing number of photons in the t t production.This ultimately leads to the observed large Mixed and Decay contributions that we have found earlier.We note that this is the opposite behaviour to that for the pp → t tjj process at NLO in QCD, in which jet radiation in t → W b decays is much stronger suppressed, as has been shown in Ref. [42], and the two contributions, Mixed and Decay, are generally much smaller.
Finally, while employing the dynamical scale setting µ 0 = E T /4, we examine the second main source of theoretical uncertainties that comes from the choice of the PDF set.We use the corresponding prescriptions from each PDF fitting group to provide the 68% confidence level internal PDF uncertainties.Our findings are summarised in   NLO in QCD for the three PDF sets NNPDF3.1,CT18 and MSHT20 is given.The internal PDF uncertainties amount to 1% for NNPDF3.1,2.0% for CT18 and 1.4% for MSHT20.When comparing NLO theoretical predictions for NNPDF3.1,CT18 and MSHT20, the largest differences are found between the NNPDF3.1 and CT18 PDF sets of about 2.4%.They are therefore of a similar size as the internal PDF uncertainties.The theoretical uncertainties from the 7-point scale variation of about 6% are larger by a factor of 3 − 6 than the internal PDF uncertainties and remain the dominant source of theoretical uncertainties.

Differential fiducial cross sections
We turn our attention to the size of NLO QCD corrections at the differential cross-section level.Our goal here is to assess additional shape distortions on top of the NLO QCD correction of about 30%, which are already present for the normalisation.In Figure 2 we show differential cross section distributions for a few observables for the pp → ℓ + ν ℓ ℓ − νℓ b b γγ + X process at the LHC with √ s = 13 TeV.In detail, we display the transverse momentum, p T,γ 1 γ 2 , and the invariant mass, M γ 1 γ 2 , of the γ 1 γ 2 system, the scalar sum of transverse momenta of the two prompt photons, H phot T , defined according to and the scalar sum of transverse momenta of all visible particles in the final state, H vis T , given by Results are presented for µ 0 = E T /4 (blue) and µ 0 = m t (orange) at NLO (solid) and LO (dashed).Also given are the corresponding uncertainty bands resulting from the 7-point scale variation.These uncertainties are assessed on a bin-by-bin basis.The two lower panels display the differential K-factor for both scale choices with the corresponding uncertainty bands of the LO and NLO predictions.The first two observables, p T,γ 1 γ 2 and M γ 1 γ 2 , are very interesting to study as they represent an important irreducible background to the transverse momentum and invariant mass of the Higgs boson in the pp → t tH process with H → γγ.Large NLO QCD corrections up to 65% for p T,γ 1 γ 2 and 70% for M γ 1 γ 2 are found in the tail of the two distributions for the dynamical scale setting.Thus, higher-order corrections exceed the LO scale uncertainties, that are in the range of 31% − 35%, by a factor of 2.
When instead the fixed scale setting is employed, NLO QCD corrections are reduced to 22% − 32% for p T,γ 1 γ 2 and 26% − 42% for M γ 1 γ 2 leading to a better agreement with the corresponding LO predictions.On the other hand, the difference between the two NLO results for the central value of µ 0 is only about 2% for the bulk of the distribution.Towards the tails this difference increases up to 5%, showing very good agreement, which is well within the NLO theoretical uncertainties.The latter uncertainties, depending on the phase-space region, are of the order of 5% − 13%.Similar features are observed for H phot T , where the difference between the two NLO results are at most 7% while the NLO scale uncertainties are about 12% for the dynamical and 10% for the fixed scale choice.Also in this case the fixed scale setting seems to describe this differential cross-section distribution better.Indeed, NLO QCD corrections are reduced from 58% to 30% when µ 0 = m t is employed.The large differential K-factor for pure photon, dimensionful observables for µ 0 = E T /4 is driven by the LO predictions.On the other hand, we find for the H vis T observable with the fixed scale setting perturbative instabilities in the tail of the distribution.In this region the NLO scale uncertainties rapidly increase to about 50% and exceed the LO scale uncertainty bands.In that case, the dynamical scale choice at NLO in QCD is essential and leads to a reduction of the scale uncertainties from 35% at LO to 7% at NLO in QCD as well as rather flat higher-order corrections of about 30%.
We continue with the discussion of NLO QCD corrections at the differential cross-section level but this time we turn our attention to dimensionless observables.Specifically, we analyse the angular separation between the hardest b-jet and the two prompt photons in the rapidity azimuthal angle plane: In addition, we examine the angular separation between the positively charged lepton and the two prompt photons: ∆R ℓ + γ 1 and ∆R ℓ + γ 2 .These four observables, that are shown in Figure 3, provide important information on the probability of photon emission at different stages of the topquark decay chain.What all these observables have in common is a peak around ∆R ij ≈ 3, where i = b 1 , ℓ + and j = γ 1 , γ 2 , showing that photons and decay products of top quarks/W bosons are preferably produced in the back-to-back configuration.In addition, a second peak is present for small ∆R ℓ + γ 1 and ∆R ℓ + γ 2 , indicating that photon emission can also originate from the two charged leptons in more collinear configurations.As we will see later, such configurations are only present for the Mixed and Decay contributions.Even though, the second peak at ∆R ij ≈ 0.4 is absent for ∆R b 1 γ 1 and ∆R b 1 γ 2 , the ∆R b 1 γ 2 observable is enhanced with respect to ∆R b 1 γ 1 in this phase-space region, showing the increased probability that the second hardest photon can be emitted from the hardest b-jet.The size of NLO QCD corrections is very similar for all four observables and for both scale choices.The largest higher-order effects, up to 40%, can be found for ∆R b 1 γ 1 at the beginning of the spectrum, while for ∆R b 1 γ 1 ≈ 3 NLO QCD corrections of about 20% − 35% are present.NLO scale uncertainties, on the other hand, are at the level of 5% − 8% for all four observables.For large angular separation, i.e. for ∆R ij > 4, the differential K-factor rapidly rises to more than 2 and the scale uncertainties increase to 20%.These phase-space regions are, however, the least populated.We also note that the differences between theoretical predictions for the dynamical and fixed-scale setting at NLO in QCD do not exceed 2% − 3% for all four differential cross-section distributions.This can be explained by the dimensionless nature of these observables.Indeed, they receive contributions from all scales, most notably from those that are sensitive to the threshold for the top-quark pair production.For our scale settings, effects coming from the phase-space regions close to this threshold dominate and the dynamic µ 0 = E T /4 scale does not substantially alter this behavior.
In the end, we focus on observables associated with the underlying t t production.In particular, we examine the invariant mass and transverse momentum of the b 1 b 2 system, denoted as M b 1 b 2 and p T, b 1 b 2 respectively.These observables are displayed in Figure 4. Also shown are the angular separation between the two hardest b-jets, ∆R b 1 b 2 , and the angular difference between the two charged leptons in the transverse plane, ∆Φ ℓ + ℓ − .The advantage of the ∆Φ ℓ + ℓ − observable lies in the fact that measurements of charged leptons are particularly precise at the LHC due to the excellent lepton energy resolution of the ATLAS and CMS detectors.Moreover, ∆Φ ℓ + ℓ − is sensitive to t t spin correlations and can be measured with high precision since the top quarks do not even need to be reconstructed.For the M b 1 b 2 observable, the NLO QCD corrections for the dynamic scale choice are rather constant at 25% − 30%.The scale uncertainties are reduced from about 40% at LO to 8% at NLO, showing overall good perturbative convergence.On the other hand, the fixed scale setting leads to larger shape distortions (up to 40%) and increased scale uncertainties (more than 20% towards the tails).The behaviour of higher-order QCD corrections for M b 1 b 2 is very similar to the already analysed H vis T observable, where a fixed scale setting also led to deteriorated convergence in the high energy regime.For the transverse momentum of the two b-jet system, p T, b 1 b 2 , we find huge NLO QCD corrections in the tail of the distribution that are more than 300% for µ 0 = E T /4 and 200% for µ 0 = m t .Such   2 but for the observables M b1b2 , p T,b1b2 , ∆R b1b2 and ∆Φ ℓ + ℓ − .large higher-order QCD effects have already been found for the t t process and other associated t t production processes, see e.g.Refs.[40,79,[83][84][85].They are also present for similar observables like p miss T = |⃗ p T, ν ℓ + ⃗ p T, νℓ | or p T, ℓ + ℓ − .In general, they occur for observables that are constructed from the decay products of both top quarks.These huge NLO QCD corrections stem from hard jet radiation recoiling against the t t system, that lifts off the kinematical suppression present at LO.The latter is responsible for the top-quark pair being produced in the back-to-back configuration.While for the process pp → t tH or pp → t tZ a slight relaxation of the back-to-back configuration in the production of the t t system is already present at LO, leading to less significant NLO QCD corrections, such a relaxation is not apparent in our case.In addition, we have checked that these NLO QCD corrections are further enhanced for the Mixed and Decay contributions.In these two cases a differential K-factor of the order of 5 and 10 could be found, respectively.The theoretical uncertainties due to µ R and µ F scale dependence at the NLO QCD level are around 40% at high transverse momenta.In addition, the NLO QCD results for the two scale settings lead to differences exceeding 25%.NNLO QCD corrections are therefore necessary to accurately predict this observable for the phase-space region p T, b b ≳ 150 GeV, see e.g.Ref. [40].For the ∆R b 1 b 2 observable below ∆R b 1 b 2 < 1 we find NLO QCD corrections up to 45% for µ 0 = E T /4 and 55% for µ 0 = m t .In both cases, the higher-order effects exceed the size of the LO scale uncertainty bands in this region of phase space.The scale uncertainties are reduced from 28% − 40% at LO to 5% − 10% at NLO for both scale choices, where the smallest theoretical errors are found for the back-to-back configurations at around ∆R b 1 b 2 ≈ 3. Also for ∆Φ ℓ + ℓ − , yet another angular distribution that we have analysed, large NLO QCD corrections up to 80% − 90% are obtained.In that case, higher-order QCD effects are very sensitive to the particular phase-space region.Indeed, they are of the order of 80% − 90% for ∆Φ ℓ + ℓ − ≈ 0 and are reduced down to about 5% for ∆Φ ℓ + ℓ − ≈ π.The NLO scale uncertainties are of the order of 5% − 16%.Finally, at NLO in QCD the two scale settings lead to differences of at most 2% for the ∆Φ ℓ + ℓ − observable.

Distribution of prompt photons
Moving forward, we examine the impact of including photon emissions in top-quark and W gauge boson decays.In Figure 5 we present the differential cross-section distribution at NLO in QCD as a function of p T,γ 1 γ 2 , H vis T , M b 1 b 2 and p T, b 1 b 2 for the pp → ℓ + ν ℓ ℓ − νℓ b b γγ + X process at the LHC with √ s = 13 TeV.We divide our theoretical predictions into the three configurations Prod., Mixed and Decay.In each case, the lower panel displays the ratio to the full NLO QCD result.We employ the dynamical scale setting µ 0 = E T /4 and the NNPDF3.1 PDF set.At the beginning of the p T, γ 1 ,γ 2 spectrum the Prod.contribution is the smallest, at about 25% of the full result.It is followed by Decay with 32% and Mixed.with 43%.The Decay contribution rapidly decreases for larger values of transverse momenta and already at around 150 GeV it is smaller than 2%, thus, negligible when comparing to the scale uncertainties in this phase-space region.Up to p T, γ 1 ,γ 2 ≈ 150 GeV, the Mixed contribution is rather constant and for p T, γ 1 ,γ 2 > 150 GeV it slowly decreases to about 18% in the tail.This contribution is not negligible even for higher values of p T, γ 1 ,γ 2 where the scale uncertainties of about 12% are obtained.The Prod. contribution increases towards higher values of p T, γ 1 ,γ 2 and becomes the dominant one from about p T, γ 1 ,γ 2 ≈ 200 GeV.In short, we can observe a very diverse picture, which depends largely on the available phase space for the three configurations.The phase space is becoming more restricted as the number of high p T photons increases in top-quark decays.Overall, for the H vis T observable we can observe a very similar behaviour for the three resonant contributions as for the p T, γ 1 ,γ 2 spectrum.The Decay part becomes more enhanced for small values of H vis T , where it dominates this differential crosssection distribution by about 61%, however, at around 700 GeV it is negligible.On the other hand, between (250 − 450) GeV the Mixed part dominates with about 45% − 50%.Its importance decreases in the tails to about 9% where it becomes comparable in size to the NLO scale uncertainties of about 6%.For H vis T > 450 GeV the Prod.contribution starts to increase from 48% to 91% towards the tail.For the standard dimensionful observables, that are associated with the underlying pp → t t production process, like for example M b 1 b 2 and p T, b 1 b 2 , the Decay contribution is of the similar size as the Prod.contribution for the phase-space regions below ≈ 100 GeV.Even though the Decay contribution is still negligible in the tails of these distributions (2%), the decrease is drastically reduced compared to the At last, we study the composition of photon emissions in the pp → t tγγ process for the dimensionless observables ∆R b 1 γ 1 , ∆R b 1 γ 2 , ∆R ℓ + γ 1 and ∆R ℓ + γ 2 , which are all presented in Figure 6.In this way, we are able to study in more detail the probability of photon emission from different stages of the process.For the two observables ∆R b 1 γ 1 and ∆R b 1 γ 2 the size of the Mixed contribution is rather flat independently of the phase-space region and vary between 40% − 54%.An opposite trend is found for the other two contributions.The Decay configuration decreases towards larger values of ∆R ij from 31% to 5% and from 24% to 6% for ∆R b 1 γ 1 and ∆R b 1 γ 2 , respectively.This behavior is compensated by the increase of the Prod.contribution from 23% to about 50%.We observe that, for the Prod.contribution back-to-back configurations are more amplified in both cases, while as the number of photons in top-quark decays increases, photon radiation with a smaller ∆R ij becomes more and more likely.This enhancement towards smaller values of ∆R ij is even more pronounced for the ∆R ij separations between photons and leptons as demonstrated for ∆R ℓ + γ 1 and ∆R ℓ + γ 2 .In both cases the Mixed and Decay contributions have a second peak for small values of ∆R ij in addition to the one for ∆R ij ≈ 3. Consequently, the importance of Mixed and Decay is greatly increased for ∆R ij < 1.5 where the full result is dominated by these two configurations with the added contribution of about 90%.The Mixed contribution decreases towards larger values of ∆R ij from 47% to 41% and from 62% to 35% for ∆R ℓ + γ 1 and ∆R ℓ + γ 2 , respectively.On the other hand, the Prod.contribution is only about 10% for smaller values of ∆R ij .However, its importance increases substantially towards larger ∆R ij separation.Specifically, we obtain the contribution of the order of 54% for ∆R ℓ + γ 1 and 60% for ∆R ℓ + γ 2 .Also in this case the Decay contribution behaves oppositely compared to the Prod.one and decreases from 41% to 4% towards the end of the R ℓ + γ 1 spectrum (29% to 5% for ∆R ℓ + γ 2 ).Concluding, the inclusion of photon bremsstrahlung in top-quark decays is essential for a proper description of angular cross-section distributions.The shape differences between the three resonant contributions are non-trivial as new peaks can arise in the spectra.Thus, the full result cannot be obtained by simple reweighting of the Prod.contribution by some fixed factor.Due to the limited phase space for the top-quark decay products in the Decay configuration, this contribution is heavily suppressed for larger values of ∆R ij , but it is very important for the more collinear configurations.

Integrated fiducial cross sections
In this section we study the pp → t tγγ process in the lepton + jet channel at the integrated and differential cross-section level.Our main goal here is to assess the differences and similarities compared to the di-lepton channel discussed in the previous section.We can immediately notice that for the pp → ℓ − νℓ jj b b γγ + X process, the two top-quark decay chains: are no longer symmetrically treated due to the presence of the different fiducial phase-space cuts.Further differences can be expected as a result of real radiation at NLO in QCD.The real emission part of the calculation can cause large effects, especially in the tails of various differential cross-section distributions.Since the extra radiation is predominantly produced in the t t production stage, it does not suffer from a strongly limited phase space due to the finite mass of the top quark and W gauge boson.As a first step, we examine the impact of the invariant mass cut defined in Eq. (3.9) on the LO and NLO cross section for the pp → ℓ − νℓ jj b b γγ + X process at the LHC with √ s = 13 TeV.Theoretical predictions, with the corresponding scale uncertainties, are presented in Figure 7 as a function of Q cut , defined according to (5.1) We vary the Q cut parameter in the range of Q cut ∈ (5 − 50) GeV in steps of 5 GeV.We note that our final choice of Q cut = 15 GeV corresponds to the result in the third bin.We use the dynamical scale setting, µ 0 = E T /4, and the NNPDF3.1 PDF set, but similar conclusions can be drawn from the result with the fixed scale choice, µ 0 = m t .The integrated fiducial cross section is shown for the full result as well as for the three resonant contribution Prod., Mixed and Decay.The differences between the two extreme cases Q cut = 5 GeV and no cut (Q cut → ∞) is about 7% for the full integrated cross section at LO showing a rather minor dependence on this cut.At NLO in QCD, however, the situation drastically changes and huge higher-order QCD corrections are found for large values of the Q cut parameter.In particular, we find NLO QCD corrections of about 67% for Q cut = 50 GeV, which further increase up to 140% if no cut is applied.For Q cut ≤ 25 GeV the uncertainty bands of the LO and NLO QCD predictions start to overlap.Only for Q cut ≤ 15 GeV the NLO QCD prediction is within the LO scale uncertainty.These large higher-order QCD corrections are associated with kinematical configurations in which the two light jets from the hadronically decaying W gauge boson are recombined into a single jet.Such kinematical configurations are not present at LO since we are interested in the resolved topology where the two light jets are always present.On the other hand, at NLO in QCD such configurations are indeed possible due to the additional light jet from the real corrections.The latter light jet, when resolved and passes all the cuts, can act as the second decay product of the hadronically decaying W gauge boson.As demonstrated in Figure 7, the size of the real emission contribution can be drastically reduced by imposing the |m W − M jj | < 15 GeV cut.When examining the Prod.contribution separately, we note that at LO this contribution is insensitive to the M jj cut as no photons are emitted in top-quark decays, so we always have M jj = m W .At the NLO level in QCD we again observe large higher-order QCD corrections which are, however, less pronounced than for the full result.On the other hand, major corrections are visible for the Mixed and Decay contributions due to photon radiation inside the top-quark decay.Already at LO the differences between the two extreme case Q cut = 5 GeV and Q cut → ∞ amount to 9% for the Mixed configuration and to 22% for the Decay one.At NLO QCD the relaxation due to additional radiation becomes  GeV cut for two scale settings µ 0 = E T /4 and µ 0 = m t as well as for the three contributions: Prod., Mixed and Decay.The NNPDF3.1 PDF set is employed.The theoretical uncertainties from the 7-point scale variation and MC integration errors (in parenthesis) are also displayed.
even stronger compared to the Prod.case due to the more limited LO phase space caused by photon radiation in the decays.This leads to a huge increase in the integrated fiducial cross section and the K-factor.Indeed, we obtain K = 2.55 for Mixed and K = 3.16 for Decay if no M jj cut is applied.Finally, the Decay part is affected the most as the NLO QCD prediction lies within the LO scale uncertainty only for Q cut < 10 GeV.Even for small values of the Q cut cut like Q cut < 15 GeV large NLO QCD corrections are clearly noticeable.
In Table 4 the integrated fiducial cross section at LO and NLO QCD is shown for the pp → t tγγ process in the lepton + jet channel with the additional |m W − M jj | < 15 GeV cut.Similarly to the di-lepton channel, also in this case the two different scale settings µ 0 = E T /4 and µ 0 = m t are examined and the NLO NNPDF3.1 PDF set is employed.The full integrated fiducial pp cross section is dominated by the Prod.contribution with 50% at LO and 48% at NLO QCD.This is in contrast to the di-lepton channel which was dominated by the Mixed configuration.In the lepton + jet case, the Mixed contribution amounts to about 40% both at LO and NLO QCD, while the Decay part is about 10% at LO and 12% at NLO QCD.We note, however, that after omitting the cut on the invariant mass of the two light jets, the Mixed contribution becomes dominant at NLO in QCD and amounts to 43% compared to the Prod.contribution with 40%.Thus, without the M jj cut the size of the Mixed and Prod.contributions is the same as in the di-lepton channel.Returning to the results shown in Table 4, we find higher-order QCD corrections of 23% for the full result for both scale settings.For the different resonant contributions the K-factors vary widely from 1.16 to 1.44.For the Prod.and Mixed contributions the NLO QCD corrections are within the LO scale uncertainty bands.For the Decay contribution, on the other hand, where we have K = 1.38 for µ 0 = E T /4 and K = 1.44 for µ 0 = m t , higher order QCD corrections exceed the LO uncertainty bands that are of the order of 30%.The scale uncertainties of the full result are reduced by a factor of 6 from 31% at LO to 5% at NLO QCD for both scale choices.The full integrated fiducial cross section differs between the dynamical and fixed scale setting by about 1% at LO and less than 1% at NLO QCD.Similarly to the di-lepton decay channel, the largest differences between the two scale choices are found for the Decay contribution with about 7% at LO and 3% at NLO QCD.Thus, at the integrated fiducial cross-section level both scales are equivalent.
At last, in Table 5 we show the integrated fiducial cross section at NLO in QCD using different parameter choices for the smooth photon isolation prescription defined in Eq. (3.3).Results are given for the µ 0 = E T /4 scale setting and the NNPDF3.1 PDF set.In particular, the first prediction corresponds to our default choice with n = 1 and ϵ γ = 1.0.For the other two results we do not change the parameter n, but rather modify the coefficient E T γ ϵ γ in front of the right hand side of Eq. (3.3).Thus, in the second case we set ϵ γ = 0.5 and for third parameter choice we use E T γ ϵ γ = 10 GeV.It is important to address the dependence on these parameters as various values are employed in literature for processes with prompt photons, see e.g.[29,30,[86][87][88][89][90][91][92][93][94].Especially, when many photons and jets are present in the final state, the dependence on these parameters might be non-negligible and could affect comparisons between theoretical predictions and experimental results.In the lepton + jet decay channel the integrated fiducial cross section is reduced by about 5% if we set ϵ γ = 0.5.A larger reduction of about 10% is observed for the last parameter setting E T γ ϵ γ = 10 GeV.Indeed, these substantial differences are due to the high number of jets (up to 5) and/or photons (2) in the final state.Moreover, these effects are similar in size or even larger than the corresponding NLO scale uncertainties for this process and therefore of high relevance.In the di-lepton decay channel the dependence on these parameters is smaller, but still not negligible as the differences up to 3% and 6%, respectively, can be observed.Thus, these effects are at most as large as the corresponding NLO scale uncertainties, which are of the order of 6%.

Differential fiducial cross sections
We continue our discussion of the lepton + jet channel with the presentation of the results at the differential cross-section level.First, we examine the size of NLO QCD corrections for similar observables that have been studied in the di-lepton channel to directly assess the differences and similarities between the two decay channels.In Figure 8 we show the differential cross-section distributions as well as differential K-factors for the following observables: p T,b 1 , p T,b 1 b 2 , p T,γ 1 and p T,γ 2 at LO (dashed) and NLO QCD (solid) for the scales µ 0 = E T /4 (blue) and µ 0 = m t (orange) employing the NNPDF3.1 PDF set.The lower plots show again the differential K-factor for both scale choices together with the corresponding uncertainty bands of the LO and NLO QCD predictions.We find huge NLO QCD corrections for the first two observables.In particular, higher-order QCD corrections of more than 350% and 800% are found in the tails of the distributions for p T,b 1 and p T,b 1 b 2 , respectively, when the dynamical scale setting µ 0 = E T /4 is employed.The fixed scale choice µ 0 = m t does not alter this behavior.As already explained for the p T,b 1 b 2 distribution in the case of the di-lepton channel, these large NLO QCD corrections are due to real radiation recoiling against the t t system.However, in the lepton + jet channel we can have up to three hard unflavoured jets which enhance the size of NLO QCD corrections for these observables even further.Furthermore, we find that NLO QCD scale uncertainties in these high p T phase-space regions are up to 40% − 50% while the differences between the two scale choices are within the range of 35% − 45%.On the other hand, for p T < (150 − 200) GeV the NLO QCD corrections are reduced to 20% − 25% for p T,b 1 and 15% − 20% for p T,b 1 b 2 .In addition, also for the lepton + jet decay channel we find a similar behavior of higher-order QCD effects as in the di-lepton one.Indeed, also here larger differential K-factors are found when employing the dynamical scale setting.However, both µ 0 = E T /4 and µ 0 = m t lead to equivalent results at NLO in QCD.For the angular separation between the photons and the negatively charged lepton: ∆R ℓ − γ 1 and ∆R ℓ − γ 2 , we again find two distinct configurations, namely collinear and back-to-back configurations.Compared to the di-lepton channel the peak at small ∆R ℓ − γ 1 and ∆R ℓ − γ 2 is enhanced here due to the different event selection for the top-quark decay products in the two decay channels.Indeed, the definition of the fiducial phase space in the lepton + jet decay channel highly suppresses photon bremsstrahlung in the hadronically decaying W gauge boson.As in the di-lepton decay channel also here moderate NLO QCD corrections of about 15% − 25% are found for both scale choices when ∆R ij ∈ (0.4, 3).In this range of ∆R ij the scale uncertainties are below 10%.On the other hand, for ∆R ij > 3 higherorder corrections rapidly increase up to 100% − 150% and the scale uncertainties are of the order of 20% − 25%.
Finally, we examine the kinematics of the light jets.Thus, in the following we are focusing on the truly new effects that are only visible in the lepton + jet decay channel.In Figure 10 we display the transverse momentum of the hardest and the second hardest light jet, denoted as p T,j 1 and p T,j 2 respectively.Also shown are the angular separation between the first and the second hardest light jet as well as the angular difference between them in the transverse plane, denoted as ∆R j 1 j 2 and ∆Φ j 1 j 2 respectively.No ratio plots are provided for these observables due to extreme values appearing in the corresponding differential K-factors.In the case of the transverse momentum of the hardest light jet, for p T, j 1 < 150 GeV, we find NLO QCD corrections ranging from −5% to 35% when the dynamical scale setting µ 0 = E T /4 is employed.In this phase-space region the NLO scale uncertainties vary between 6% and 17%.For p T, j 1 ≥ 150 GeV the NLO QCD prediction becomes significantly harder compared to the LO distribution.In particular, the NLO QCD prediction becomes larger by up to a factor of about 30 than the LO one.The scale uncertainties in the high p T region are of the order of 50%.In addition, at the NLO QCD level the difference between the two scale settings can be even up to 20%, that is still within the large NLO uncertainty bands.As we have already discussed, at LO the phase space of the two light jets, especially in the high p T region, is restricted due to the production mechanism as both light jets are originating from the W gauge boson decay.However, at NLO QCD a light jet can also be produced in the t t production stage.This light jet, if resolved and passes all the cuts, is not affected by the kinematical restriction and can lead to huge enhancements in the tail of the p T, j 1 distribution.As this phase-space region is only LO accurate, not only large scale uncertainties but also substantial differences between the two scale choices can be observed.For the transverse momentum of the second hardest jet, p T, j 2 , this effect is even more pronounced as has already been discussed e.g. in Ref. [82] for the pp → t t + X process.Indeed, at p T, j 2 ≈ 200 GeV, the LO predictions decrease sharply to become even zero for p T, j 2 ≥ 220 GeV.Due to additional radiation at NLO QCD the restriction on the second hardest jet is lifted and thus the distribution is no longer zero for p T, j 2 > 220 GeV.For the same reason also the angular distributions ∆R j 1 j 2 and ∆Φ j 1 j 2 show significant shape distortions at NLO QCD.At LO the distribution of ∆R j 1 j 2 peaks strongly at about ∆R j 1 j 2 ≈ 1. Afterwards it rapidly decreases and is heavily suppressed for large values of ∆R j 1 j 2 .The situation drastically changes at NLO in QCD.The peak at about ∆R j 1 j 2 ≈ 1 is significantly reduced and a second peak can be found at ∆R j 1 j 2 ≈ 3.In addition, the huge suppression for large ∆R j 1 j 2 does not occur anymore.For this observable large scale uncertainties up to 50% are found for most parts of the distribution.Finally, for the ∆Φ j 1 j 2 distribution we find again large differences between the LO and NLO spectra.At LO a clear peak at π/4 is found that is caused by the production mechanism of the two light jets at this order.At NLO QCD the spectrum is rather flat over the entire range and the peak at π/4 is substantially reduced.NLO QCD scale uncertainties in most parts of the distribution are up to 28%.Concluding, all distributions based on the kinematics of the hardest (light) jets receive large contributions from real radiation at NLO in QCD and NNLO QCD corrections would be necessary for more precise predictions of these specific observables in the phase-space regions that are kinematically restricted at LO.In the absence of NNLO calculations for the pp → t tγγ process, however, other options should instead be explored.For example, a redefinition of the fiducial phase space or the inclusion of a jet veto might mitigate the large observed higher-order effects.In both cases, special and detailed studies are needed to clarify the issue.We leave such studies for future investigation.

Distribution of prompt photons
As in the case of the di-lepton channel, for the lepton + jet decay channel we also study the distribution of photons in the pp → t tγγ process.In Figure 11 we show the differential cross-section distribution at NLO QCD as a function of p T, γ 1 , p T, γ 2 , ∆R ℓ − γ 1 and ∆R ℓ − γ 2 .We employ µ 0 = E T /4 and the NNPDF3.1 PDF set.Theoretical predictions are once more divided into the following three contributions: Prod., Mixed and Decay.As expected from the di-lepton decay channel we find again a very similar picture of the size of the different resonant contributions.In particular, we observe that the Mixed contribution is the largest one in the small p T region with about 45% for p T, γ 1 and p T, γ 2 .Even the Decay contribution becomes comparable in size to the Prod.contribution for p T, γ 1 in this phasespace region and amounts to 31%.However, this contribution reduces rapidly towards larger values of p T, γ and amounts to less than 1% for p T, γ 1 > 180 GeV and p T, γ 2 > 100 GeV.Nevertheless, the Decay contribution is non-negligible in the phase space region that is relevant for current measurements at the LHC.The Mixed contribution also decreases towards the tails of both distributions but remains at the level of 17% for p T, γ 1 even when p T, γ 1 ≈ 600 GeV.Contrary, for p T,γ 2 we find that the Mixed contribution is below 1% for p T,γ 2 > 320 GeV and, therefore, phenomenologically negligible.Indeed, in the high p T, γ tails the full distribution is dominated by the Prod.contribution.Similarly to the di-lepton channel, also in this case a peak for small values of the ∆R ℓ − γ 1 and ∆R ℓ − γ 2 separation is entirely driven by the Mixed and Decay contributions, which are at the 54% − 66% and 23% − 33% level, respectively.The Prod. contribution is actually the smallest one in this phase-space region, at the level of 11% − 13% only.However, around ∆R ℓ − γ ≈ 3 it increases up to 52% and 57% for ∆R ℓ − γ 1 and ∆R ℓ − γ 2 , respectively.In this phase-space region the counterparts Mixed and Decay are at the level of 40% and 10%, respectively.

Summary
In this paper we presented the calculation of NLO QCD corrections to the pp → t tγγ process with realistic final states at the LHC with √ s = 13 TeV.In particular, we considered the di-lepton as well as lepton + jet decay channel of the top-quark pair.The decays of the top quarks and W bosons are handled in the NWA preserving spin correlations.In contrast to previous calculations of pp → t tγγ available in the literature, in this paper for the first time photon radiation and NLO QCD corrections have been consistently included in the production as well as the decays of the t t pair.An important finding of this paper is the magnitude of NLO QCD corrections for the lepton + jet decay channel due to kinematical configurations in which the two light jets from the hadronic decaying W boson are recombined into one jet.Such contributions are not present at LO in our calculation since our event selection focus on resolved topologies.They are, however, possible at NLO QCD due to extra radiation.
The second important finding of this paper comprises the decomposition of photon radiation for the pp → t tγγ process.At the integrated fiducial cross-section level the Prod.contribution amounts 40% in the di-lepton and 48% in the lepton + jet decay channel only.Thus, the incorporation of photons in top-quark and W boson decays leads to an increase of the cross section by more than a factor of two.We have found that the large size of the Mixed and Decay contributions are due to a suppression of photon emission in the gg initiated process of the Prod.part, which is partially relaxed for the other two contributions.
The third important finding of this paper is the analysis of the dependence on various parameter choices in the smooth photon isolation prescription as introduced by Frixione in Ref. [81].When many photons and jets are present in the final state, the dependence on these parameters is non-negligible and might affect comparisons between theoretical predictions and experimental results.We have found that when varying n, ϵ γ and the coefficient E T γ ϵ γ for the pp → t tγγ process in the lepton + jet decay channel, the integrated fiducial cross section has been changed by up to 10%.This change is larger than the corresponding NLO scale uncertainties for this process and therefore of high relevance.In the di-lepton decay channel the dependence on these parameters has been smaller, up to 6% only, but still as large as the corresponding NLO scale uncertainties.We plan to carry out dedicated studies on this topic not only at the integrated level, but also at the differential cross-section level in the near future.
As a second source of theoretical uncertainties, we have studied the size of internal PDF uncertainties and the relative differences among the cross-section results obtained with various PDF sets.In particular, we obtained PDF uncertainties in the range of 1% − 2% for the three PDF sets NNPDF3.1,MSHT20 and CT18.The largest differences between PDF sets have been found for NNPDF3.1 and CT18 with about 2.4%.Concluding, the size of scale uncertainties remain the dominant source of theoretical systematics.
At the differential cross-section level NLO QCD corrections of up to 70% have been observed for dimensionful observables related to photon kinematics for µ 0 = E T /4.Furthermore, these higher-order effects exceed the LO uncertainty bands.The use of the fixed scale, µ 0 = m t , leads to a reduction of higher order corrections to a moderate range and places the NLO results within the LO uncertainty bands.However, only minor differences exist between the two predictions for the central value of the scale at NLO in QCD.In addition, we have seen that in general the use of a dynamical scale setting for dimensionalful observables is essential, as the fixed scale can lead to perturbative instabilities in high p T tails of certain distributions.In the lepton + jet channel large shape differences have been found between LO and NLO predictions for several distributions describing the kinematics of the first and second hardest (light) jet.These differences are due to the additional radiation at NLO in QCD, which is not kinematically constrained due to the production mechanism, unlike the light jets that appear at LO.This becomes especially visible for p T, j 2 which is kinematically limited at LO by the finite mass of the W boson and the cut on the invariant mass of a jet pair.
In the next step, we have assessed the size of the Prod., Decay and Mixed contribution at the differential cross-section level.We have found that for dimensionful observables the Mixed contribution becomes the most important one in the small p T region.Even the Decay contribution can be comparable in size with the Prod.one in that phase-space region.However, in high p T tails the Decay contribution becomes phenomenologically negligible and the Prod.part is starting to dominate the full result.Nevertheless, the Mixed contribution is generally significant even in high p T tails, where it can amount to more than 20%.For angular distributions it is crucial to incorporate photon radiation in the decays of the top quark and W boson. Otherwise, entire new peaks may be missed, such as those clearly visible in the case of the angular separation of prompt photons and charged leptons.Finally, we conclude by saying that the inclusion of photon radiation and NLO QCD corrections in top-quark decays is essential for LHC physics.Both effects would play a crucial role in the direct measurement of the pp → t tγγ process and are equally important for reliable background modelling in the light of precise measurements of the pp → t tH signal process in the H → γγ decay channel.

Figure 1 .
Figure 1.Representative Feynman diagrams for contributions: Prod., Mixed and Decay at LO with suppressed W gauge boson decays.Feynman diagrams were produced with the help of the FeynGame program [41].

Figure 2 .
Figure 2. Differential cross-section distributions for the observables p T,γ1γ2 , M γ1γ2 , H phot T

Figure 5 .
Figure 5.Differential cross-section distributions at NLO QCD for the observables p T,γ1γ2 , H vis T , M b1b2 and p T,b1b2 for the pp → ℓ + ν ℓ ℓ − νℓ b b γγ + X process at the LHC with √ s = 13 TeV.Theoretical predictions are divided in the three contributions Prod., Mixed and Decay.They are obtained with µ 0 = E T /4 and the NNPDF3.1 PDF set.The lower panels display the ratio to the full NLO QCD result.MC integration errors are also shown.

Figure 7 .
Figure 7. Integrated fiducial cross sections at LO and NLO QCD for the pp → ℓ − νℓ jj b b γγ + X process at the LHC with √ s = 13 TeV as a function of the Q cut parameter defined as |m W − M jj | < Q cut .Results are shown for the full process and the three resonant contributions Prod., Mixed and Decay.The theoretical uncertainties from the 7-point scale variation are also provided.The NNPDF3.1 PDF set and the dynamical scale µ 0 = E T /4 are employed.The lower panel displays the K-factor with the uncertainty band and the relative scale uncertainties of the LO cross section.

Table 5 .
Integrated cross section at NLO QCD for the pp → ℓ − νℓ jj b b γγ + X process at the LHC with √ s = 13 TeV.Results are given for different parameter choices of the smooth photon isolation prescription defined in Eq. (3.3) but with n = 1.They are presented for the dynamical scale µ 0 = E T /4 employing the NNPDF3.1 PDF set.Theoretical uncertainties from the 7-point scale variation and MC integration error are also displayed.

Figure 8 .
Figure 8. Differential cross-section distributions for the observables p T,b1 , p T,b1b2 , p T,γ1 and p T,γ2 for the pp → jj ℓ − νℓ b b γγ + X process at the LHC with √ s = 13 TeV.Results are presented for µ 0 = E T /4 (blue) and µ 0 = m t (orange) at NLO (solid) and LO (dashed) using the NNPDF3.1 PDF set.The two lower panels display the differential K-factor for both scale choices with the uncertainty band and the relative scale uncertainties of the LO cross section.

Table 2 .
Integrated fiducial cross sections at NLO QCD for the pp → ℓ + ν ℓ ℓ − νℓ b b γγ + X process at the LHC with √ s = 13 TeV.

Table 3 ,
where the integrated fiducial cross section at

Table 3 .
Integrated fiducial cross sections at NLO QCD for the pp → ℓ +

Table 4 .
Integrated fiducial cross section at LO and NLO QCD for the pp → ℓ − νℓ jj b b γγ + X process at the LHC with √ s = 13 TeV.Results are given with the |m W − M jj | < 15