Automated EW corrections with isolated photons: tt¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overline{t} $$\end{document}γ, tt¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overline{t} $$\end{document}γγ and tγj as case studies

In this work we compute for the first time the so-called Complete-NLO predictions for top-quark pair hadroproduction in association with at least one isolated photon (tt¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overline{t} $$\end{document}γ). We also compute NLO QCD+EW predictions for the similar case with at least two isolated photons (tt¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overline{t} $$\end{document}γγ) and for single-top hadroproduction in association with at least one isolated photon. In addition, we complement our results with NLO QCD+EW predictions of the hadronic and leptonic decays of top-quark including an isolated photon. All these results have been obtained in a completely automated approach, by extending the capabilities of the MadGraph5_aMC@NLO framework and enabling the Complete-NLO predictions for processes with isolated photons in the final state. We discuss the technical details of the implementation, which involves a mixed EW renormalisation scheme for such processes.


Introduction
After the first ten years of operation of the Large Hadron Collider (LHC), our knowledge of the fundamental interactions of elementary particles has been tremendously improved. At the LHC, the long-searched Higgs boson has been observed [1,2], and its properties have also been studied in details and found to be compatible with those predicted by the Standard Model (SM) [3]. Moreover, the SM itself, our current best understanding of elementary particles and fundamental interactions, has been stress-tested not only for what concerns the Higgs sector, but in almost all other aspects, e.g., electroweak (EW) interactions, QCD dynamics and flavour physics. So far, no clear and unambiguous sign of beyond-the-SM (BSM) physics has been found at colliders, but the BSM search programme at the LHC is still only at the initial phase, since 20 times more data will be collected in the coming years, large part of it during the High-Luminosity (HL) runs [4][5][6][7][8][9].

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The success of this ambitious research programme relies on the ability of providing precise and reliable SM predictions. For this reason, in the past years, a plethora of new calculations and techniques have appeared in the literature, aiming to improve SM (but also BSM) predictions. On the one hand, a lot of efforts have been put in improving the calculations of purely QCD radiative corrections, going from Next-to-Leading-Order (NLO) to Next-to-NLO (NNLO) or even Next-to-NNLO (N 3 LO) predictions and in parallel improving the resummation of large logarithms appearing at fixed order. On the other hand, a lot of work has been done for the calculations of NLO QCD and EW corrections for processes with high-multiplicity final states. To this purpose, NLO QCD and EW corrections have been implemented in Monte Carlo generators and, at different levels in the different frameworks, they have been even automated [10][11][12][13][14][15].
As explained in detail in ref. [15], the automation of NLO corrections of QCD and EW origins has been implemented in the MadGraph5_aMC@NLO framework [16]. Thanks to this, not only NLO QCD+EW corrections for many new hadroproduction processes have become available [11,13,15,[17][18][19][20][21][22][23], but also the subleading NLO effects can be systematically assessed. In other words, all perturbative orders arising from tree-level diagrams and their interference with their one-loop counterparts in the SM can be computed in the automated means. This has lead to the discovery that, in some cases, the subleading NLO corrections (those beyond the standard NLO QCD O(α s ) and NLO EW O(α) corrections) can be much larger than their naive estimates based on the power counting of α s and α [19,22,24]. In particular, this has been observed in the context of top-quark physics for the production of a top-quark pair in association with a W boson (ttW ) [19,20,[25][26][27][28] and for the production of four top quarks (tttt) [19].
While the theoretical framework of ref. [15] is general and applicable to any possible final state, the formalism regarding fragmentation functions has not been implemented yet in the MadGraph5_aMC@NLO code. In particular, this formalism is necessary when measurable quantities are not defined only by means of massive particles and/or after a jet-clustering(-like) algorithm. Consequently, so far, one of the main limitations of the code has been the impossibility of calculating NLO EW corrections and Complete-NLO predictions for processes with tagged photons.
The aim of this paper is twofold. First, we want amend to the aforementioned limitations, allowing NLO EW and Complete-NLO calculations for processes involving photons that are tagged by applying an isolation algorithm [29]. Second, we want to exploit the new capabilities of the code for computing NLO EW and Complete-NLO predictions involving both photons and top quarks. Since, as previously mentioned, unexpected large NLO EW radiative effects have been observed in similar processes involving top quarks and massive vector bosons, it is natural to check the same thing for processes involving top quarks and isolated photons.
The automation of EW corrections involving isolated photons has been achieved by using the so-called α(0) renormalisation scheme in the MadGraph5_aMC@NLO framework. Its implementation is in fact performed via a mixed-scheme approach [30,31], which is based on the idea that α should be renormalised in the α(0)-scheme only for (final-state) isolated photons, while other EW interactions should be renormalised in the α(m Z ) or JHEP09(2021)155 like, 2 such as the MS scheme itself but also, for the EW sector, the more commonly used α(m Z ) and especially G µ -scheme. Using this class of renormalisation schemes, if there is a massless particle p i that can spilt into two massless particles p j and p k via an EW splitting p i −→ p j p k , an NLO EW calculation cannot be straightforwardly carried out for a final state including p i as a physical object. The calculation would be IR divergent. In the SM, photons and all the charged massless fermions, can precisely split via QED interactions into two massless particles. IR safety can be in general achieved via two different solutions: clustering massless particles into fully-democratic jets 3 or using fragmentation functions. The definition and the scope of the former has been extensively explored in ref. [13], while the description of the necessary theoretical framework for the implementation of fragmentation functions in MadGraph5_aMC@NLO has be presented in ref. [15]. The usage of fragmentation function in the context of an NLO EW calculation has also been exploited in ref. [46].
On the other hand, for the case of photons in the final state, a very well known and (probably much) simpler solution than the usage of fragmentation functions exists: performing perturbative calculations in the α(0)-scheme. In fact, concerning the purely QED part of NLO EW corrections, besides effects that are formally beyond NLO and related to the fragmentation-function evolution, a calculation performed in an MS-like renormalisation scheme employing the photon fragmentation function and a calculation performed in the α(0)-scheme with isolated-photons lead to the same result, as shown, e.g., in ref. [13] and discussed also in section 2.2. In the following context, we will in general understand that the Frixione isolation algorithm [29] is employed for isolating the photon.
In the following of this section we describe the modifications and extensions, w.r.t. the theoretical framework in ref. [15], that we have implemented in MadGraph5_aMC@NLO. Via these new features, NLO EW corrections are enabled for any process involving isolated photons in the final state. Moreover, Complete-NLO predictions can be calculated for any process, besides some cases involving simultaneously both isolated photons and jets. We will return to it later, at the end of section 2.2.2, on this limitation and explain the reason behind it.
There are three main improvements w.r.t the framework described in ref. [15]. They concern the following three aspects: • the renormalisation conditions, • the FKS counter-terms, • the photon isolation together with democratic jets.
Before describing the technical part of these aspects in section 2.2, we define the necessary notations and describe the general approach to the problem in section 2.1. Besides, we also show the syntax that should be used in order to run the code. 4

Notation
Adopting the notations already used in refs. [11,13,15,[17][18][19][20][21][22][23]47] the different contributions from the expansion in powers of α s and α of any differential or inclusive cross section Σ at LO (Σ LO ) and at NLO (Σ NLO ) can be denoted as: where k ≥ 1 and the specific value of k is process dependent. Each Σ LO i denotes a different α n s α m perturbative order stemming at LO, i.e., from Born diagrams only. In a given process, both the values of n and m are different for each Σ LO i , but the sum n + m is fixed. If where each Σ NLO i denotes a different NLO perturbative order stemming from the interference between Born and one-loop diagrams.
The quantity denoted as Σ LO 1 is what is commonly referred as LO in the literature, while here "LO" denotes the sum of all the possible Σ LO i . We will also use the standard notations "NLO QCD " and "NLO QCD+EW " for the quantities Σ LO 1 + Σ NLO 1 and Σ LO 1 + Σ NLO 1 + Σ NLO 2 , respectively. The Σ NLO 1 and Σ NLO 2 terms are in other words the NLO QCD and NLO EW corrections, respectively. We will also use in general the alias "(N)LO i " in order to indicate the quantity Σ (N)LO i . The set of all the possible contributions of O(α n s α m ) at LO and NLO is what is denoted as "Complete-NLO".

Calculation set-up with isolated photons
Following the strategy described in refs. [30,31], for processes including isolated photons in the final state we perform the renormalisation of EW corrections in a mixed scheme. Any NLO i term with i ≥ 2 for a process including isolated photons involves the renormalisation of EW interactions, or equivalently of the powers of α that are present in the LO i−1 . The standard case of NLO EW corrections correspond to i = 2. For a general process involving n γ isolated photons, pp−→n γ γ iso + X , (2.3) if Σ LO i ∝ α n s α m , then m ≥ n γ and there are n γ powers of α related to the vertices with final-state external photons and m − n γ ones of a different kind. While MS-like schemes like the α(m Z ) and especially the G µ -scheme are in general superior to the α(0)-scheme for the calculation of the EW corrections, in the case of final-state legs associated to isolated photons it is the opposite (see e.g. the aforementioned refs. [30,31] for more details about it). Actually, in modern calculations where light-fermion masses are set equal to zero, this choice of scheme is not only superior but also necessary to achieve IR safety. We will show this in more details in section 2.2.1. Therefore, we renormalise n γ powers of α in the α(0)-scheme, while m − n γ powers in an other MS-like scheme, which in the rest of the article will be, if not differently specified, the G µ -scheme. 5 It is worth to stress that JHEP09(2021)155 the α(0)-scheme should not be adopted for initial-state photons [48,49]. The usage of the α(0)-scheme also implies that the standard procedure for the generation of real-radiation diagrams described in ref. [15] has to be modified. In particular, since the final-state photon in a diagram coincides with a physical object, the isolated-photon, final-state QED γ−→ff splittings, where f is a charged massless fermion, should be vetoed. In other words, for the process in eq. (2.3) the real radiation diagrams with final state (n γ − 1)γ iso + ff + X leading to the same perturbative order of NLO i+1 should not be taken into account in the calculation. For this reason, similarly to the virtual contribution, the divergencies arising from real radiation are different than in an MS-like scheme. Thus, the definition of FKS counterterms should be amended too (see details in section 2.2.2).
When performing a calculation, the differences between two specific renormalisation schemes do not only consist of the different renormalisation counter terms. Indeed, also the numerical values that should be used for the input parameters are different. In the case of the α(0)-scheme and G µ -scheme, the numerical values for the QED fine structure constant α are different, namely α = α(0) and α = α Gµ with Since for a process with n γ isolated photons in the final state and with Σ LO i ∝ α n s α m we renormalise n γ powers of α in the α(0)-scheme and m − n γ powers in the G µ -scheme, we consistently set the input parameters according to the rule At NLO accuracy, for what concerns the input parameters, it is instead necessary to differentiate two separate cases on the basis of the power of α s in Σ LO i : n > 0, which allows for the presence of Σ LO i+1 , and n = 0, which in the SM model implies that Σ LO i+1 is not present. If n > 0, at variance with the LO case, in Σ NLO i+1 there is in general no freedom of choice in the numerical value of the additional power of α without spoiling the cancellation of UV and/or IR divergencies. The numerical value of the additional power of α, which we denote asᾱ, has to be set equal to α Gµ , i.e., Indeed, the α s and α expansion of Σ LO is actually an expansion in α s and α Gµ , as can be seen in (2.5).
the only choice ofᾱ that in general preserves the exact cancellation of both UV and IR divergencies isᾱ = α Gµ . If instead n = 0, namely no QCD interactions in Σ LO i , the quantity Σ LO i+1 does not exist and therefore eq. (2.7) does not imply that the relationᾱ = α Gµ must be true in order to preserve the exact cancellation of both UV and IR divergencies. In other words,

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and the choice of the numerical value ofᾱ is in principle arbitrary, being an NNLO O(α 2 ) effect w.r.t. Σ LO i . The same argument could be repeated for any other MS-like scheme in the place of the G µ -scheme. On the other hand, also for n = 0, the choiceᾱ = α Gµ should be in general preferred and regarded as superior thanᾱ = α(0). Indeed, NLO corrections do not contain any additional isolated photon, since the additional photons appearing via real radiation are unresolved. 6 The only case in whichᾱ = α(0) is preferable is when Σ NLO i+1 predictions are used for observables involving n γ + 1 isolated photons, but for this case a calculation involving n γ + 1 isolated photons already at the tree level, therefore proportional to α(0) nγ +1 according to (2.5), should be preferred. Still, it is important to note that the choice of the value ofᾱ, as already said, formally affects NNLO O(α 2 ) corrections and especially its impact in (2.8) is typically at the permille and more often at sub-permille level on the prediction of an observable. Indeed, if we consider for instance the NLO EW corrections (NLO 2 ), the impact of this choice w.r.t. the dominant LO, the LO 1 , is of the order (NLO 2 /LO 1 )∆ᾱ, with ∆ᾱ ≡ (α Gµ − α(0))/ᾱ 0.04, with the quantity (NLO 2 /LO 1 ) being also at the percent level.
Finally, we notice something quite counterintuitive that is a consequence of settinḡ α = α Gµ . Even with n γ = m (all EW interactions being associated to vertices involving isolated photons), the mixed scheme is not fully equivalent to the pure α(0) scheme precisely becauseᾱ = α Gµ . Not only, for the same reasons already explained before, the mixed scheme is also in this case superior to the pure α(0) scheme. Moreover, if on top of that n > 0, although all the EW final-state objects are isolated photons, the conditionᾱ = α Gµ is still in general necessary for IR-safety and UV-finiteness, due to eq. (2.7). 7 We want to stress that all this discussion onᾱ is particularly relevant in the context of a fully-fledged automation. If analytical expressions are available and/or is possible to separate subsets of diagrams or contributions that are separately IR and UV finite, further sophistications employing separate and optimised input values for α, as well as other parameters, can be performed. On the other hand, with the previous discussion we want to emphasise that in an automated calculation the only safe procedure is setting a common valueᾱ for all the contributions to the Σ NLO i+1 , and especially to show to which valueᾱ should or can be set.

Generation syntax
After having specified the notation and the general aspects of the theoretical set-up, we now illustrate the commands that have to be used in the MadGraph5_aMC@NLO frame- 6 One can understand this also from the fact that, e.g., NNLO O(α 2 ) corrections would involve both additional single and double real radiation of light particles. In the former class, one-photon emissions at one-loop would be present. In the latter class, tree-level one-photon emission with further γ−→ff splitting would be also present and not vetoed. Therefore the single emission should be parametrised byᾱ = αG µ rather thanᾱ = α(0). 7 In principle, one could setᾱ = α(0), but it would be necessary to alter eq. (2.5) by using only α(0) as the input parameter for all powers of α, which is clearly not a good choice. On the other hand, this procedure would lead for processes with nγ = m to the exactly the same results obtainable with the pure α(0) scheme, also with n > 0.

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work in order to calculate EW corrections for processes involving isolated photons. We have introduced the notation !a! for an isolated photon in the generation syntax of the framework. Let us present a few concrete examples used in this paper. First of all, the correct model have to be imported. One can choose either 8 import model loop_qcd_qed_sm_a0-Gmu or import model loop_qcd_qed_sm_Gmu-a0 Both of them work accordingly to the mixed scheme described in the previous section. However, the former corresponds to the choiceᾱ = α(0) in (2.8), while the latter to the choiceᾱ = α Gµ . As explained, the second option is the only one that, starting from (2.5), in general satisfies the necessary condition from eq. (2.7). Moreover, as also already explained, is superior from a formal point of view and should be in general preferred. In the results presented in section 3 we will use this option, unless differently specified. Then, if we want to calculate a single top associated hadroproduction process at NLO QCD+EW accuracy we use the following syntax: where we have taken an example studied in this paper. If one is only interested in NLO QCD or NLO EW corrections, the QED or QCD flag in the squared bracket should be respectively omitted. In general, in order to select a LO contribution proportional to ∝ α n s α m the tag QCDˆ2 should be set to 2n and the tag QEDˆ2 should be set to 2m. 9 We want to stress that the most important point at the generation level is the usage of !a! for isolated photons, which is not equivalent to the simple a (non-isolated photon). Only the former prevents the γ−→ff splitting, which is necessary for the consistency in the NLO calculation with isolated photons.
Before providing the technical details we want to mention that we have cross-checked results obtained in a completely automated way against calculations already present in the literature for the hadroproduction of the + − /νν +γ [50], γγ [51] and γγγ [52] final states, finding perfect agreements.

Technical details
We now provide the technical details of the calculation set-up outlined in section 2.1.2.

Renormalisation and its implementation
The renormalisation of UV divergent amplitudes involves the transition from bare to renormalised quantities, which in the EW sector involves e → e(1 + δZ e ) or equivalently α → α(1 + 2δZ e ). The α(0)-scheme corresponds to the definition where δZ AA is the wave-function renormalisation constant of the photon, with δZ AA = −Π AA (0), i.e., the vacuum polarisation at virtuality equal to zero. Similarly, δZ ZA is the non-diagonal entry of the (A, Z) wave-function renormalisation. The terms s W and c W are the sine and cosine of the Weinberg angle, respectively. With this definition, if we would retain the masses m f of all the charged fermions in the SM, and µ is the regularisation scale. Q f is the charge of the fermion and N f C is the corresponding colour factor (N f C = 1 for the leptons, N f C = 3 for the quarks). UV divergencies correspond to the ∆ term, while the logarithms in eq. (2.10) corresponds to IR divergencies in the massless limit m f = 0, which would lead to extra 1/ poles. These are precisely the poles that would not be present in an MS-like scheme. The symbol ". . ." stands for all the remaining terms of δZ e | α(0) : weak contributions and QED terms that are neither 1/ poles nor logarithms involving m f .
In a process like the one in (2.3), the external final-state photons are on-shell, exactly as in the kinematic configuration for which α(0) is defined and eq. (2.9) is derived. Therefore, δZ e | α(0) cancels exactly the (UV and IR) poles emerging from one-loop corrections connected to the vertex where the external photon is attached to the full process. On the contrary, in an MS-like scheme, the UV poles would be canceled but the IR ones would not; only by combining the renormalised one-loop contribution with the integrated real-emission (n γ − 1)γ iso + ff + X final state the IR divergencies would be canceled. For all the other vertices in the processes, the situation is opposite. A renormalisation in an MS-like scheme leads to the cancellation of UV divergencies, but in the α(0)-scheme it introduces also a term of order α log(Q 2 /m 2 f ), where Q is the scale associated with the specific interaction vertex. With massive fermions, this is a sign of a wrong choice of the renormalisation scheme, leading to artificially enhanced corrections at large energies. With massless fermions, the calculation is simply IR divergent. To overcome these problems, our solution is precisely the mixed-scheme described in section 2.1.1.
The procedure for automating the implementation of mixed renormalisation is the following. First, we start with the case of (2.8) and then we move to the case of (2.6), which in this context is a simplified version of (2.8). Let us consider process with n γ JHEP09(2021)155 isolated photons in the final state and with Σ LO i ∝ α m , therefore n γ powers of α in the α(0)-scheme and m − n γ powers in the G µ -scheme. First, one has to perform the calculation in either the α(0)-scheme or the G µ -scheme. After that, one has to either add the quantity (m−n γ ) ∆ Gµ,α(0) Σ LO i to the virtual contribution or subtract n γ ∆ Gµ,α(0) Σ LO i to it, respectively, where (2.11) After that, one can rescale both the LO i and NLO i contributions in order to achieve the prescription in (2.8), namely, multiplying both results by either (α Gµ /α(0)) m−nγ or (α(0)/α Gµ ) nγ , respectively. We notice that while in the latter case the rescaling factor is only depending on the number of isolated photons, in the former it also depends on the considered QED perturbative order m. The choice of which scheme to start with corresponds to the choice of the value ofᾱ in (2.8), eitherᾱ = α(0) orᾱ = α Gµ , respectively, and in turn on which model is imported when performing the calculation in is in general inferior toᾱ = α Gµ and especially possible only if n = 0, as shown in (2.6). The procedure for automating the implementation of mixed renormalisation according to (2.6) is actually the same than in the case (2.8), but limited toᾱ = α Gµ and the usage of only the model loop_qcd_qed_sm_Gmu-a0.
At this point, it is worth to briefly remind the relations among the different renormalisation conditions. If we consider the α(0), the α(m Z ) and the G µ schemes, these are the relations: [53][54][55]. The quantity ∆α(m 2 Z ) is of purely QED origin and it takes into account the contribution of light fermions to the run of α from the scale Q = 0 to Q = m Z , namely, When fermions are treated as massless, ∆α(m 2 Z ) exactly cancels the IR divergence in δZ e | α(0) . The remaining components of ∆r are not IR sensitive and mainly concern the purely weak part of the renormalisation of α. In particular, ∆ρ corresponds to the top-mass-enhanced corrections to the ρ parameter. After having recalled these differences we want to mention a possible discrepancy that may be left between a calculation in the G µ -scheme together with the photon fragmentation function and the mixed scheme with α(0) and G µ . While the running of the fragmentation function can naturally compensate for the effect of ∆α(m 2 Z ), the term ∆r − ∆α(m 2 Z ) has to be removed "by hand" in order to avoid its contribution to vertices with external photons.

Modification of the FKS counterterms
Before discussing the technical details concerning the IR counterterms for the integration of the separately divergent contributions of virtual and real emission diagrams, we remind the reader that the MadGraph5_aMC@NLO framework [16] deals with IR singularities via the FKS method [56,57], which has been automated for the first time in MadFKS [58,59]. We also recall that one-loop amplitudes can be evaluated via different types of integralreduction techniques, the OPP method [60] or the Laurent-series expansion [61], and techniques for tensor-integral reduction [62][63][64]. All these techniques are automated in the module MadLoop [65], which on top of generating the amplitudes switches dynamically among them. The codes CutTools [66], Ninja [67,68] and Collier [69] are employed within MadLoop, which has been optimised by taking inspiration from OpenLoops [70] for the integrand evaluation.
We can now discuss the aforementioned IR counterterms. Since IR-divergent γ−→ff splittings for isolated photons are vetoed and the IR structure of the renormalised one-loop amplitudes is altered when using the α(0)-scheme, both the counterterms for regularising virtual and real contributions have to be altered. Regarding virtual contributions, in the FKS language this means modifying the term dσ (C,n) (defined, e.g., in eq. 3.26 of ref. [15]), which collects the Born-like remainders of the final-and initial-state collinear subtractions. 10 Part of the modifications are due to the finite part V DIV of the one-loop matrix elements. As we have already discussed in details, by employing the α(0)scheme the IR-pole structure is altered w.r.t. an MS-like scheme. Therefore also V (n,1) DIV has to be modified. Regarding the real radiation, on the other hand, nothing needs to be modified. Indeed, thanks to the implementation of the FKS subtraction method in MadFKS [58], by vetoing matrix elements stemming from the QED splitting of isolated photons, the corresponding real emission counterterm is not generated.
The quantity dσ (C,n) is defined at eqs. (3.26-3.27) of ref. [15], where NLO EW corrections and more in general Complete-NLO predictions have been automated, while V (n,1) DIV at eqs. (3.30-3.32) of the same reference. If I k is an isolated photon (I k = γ iso ) then in the aforementioned equations (2.16) While the condition C QED (γ iso ) = 0 is unchanged w.r.t. ordinary photons, for which C QED (γ) = 0 already holds, it is especially important to note that (the quantities on the r.h.s. are defined in the appendix A of ref. [15]).
We can now address a point that has been ignored so far in our discussion. Not all the vertices connected to photons in the final state have to be renormalised in the α(0)-scheme.

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Indeed, this has to be done only if the photon is considered as a physical object, namely an isolated photon. If for example one considers the process (2.3) with X containing jets, those have to be in general defined as democratic-jets and therefore photons can be part of them. At LO, this means that each of those jets can be in principle formed by a single photon in the final state. It is very important to note that such photons are not isolated photons; they can split into fermions and especially their interactions with the rest of the process are renormalised in the G µ -scheme. However, for hadronic collisions, the presence of final-state photons that can be tagged as a democratic jet is very uncommon at LO i for the case i = 1. Indeed, since non-isolated photons and gluons are treated in the same way by the democratic-jet clustering, given a partonic process with a non-isolated photon in the final state a similar one with such a photon replaced by a gluon almost always exists. 11 Therefore, if the latter appears at LO i for the final-state signature that is considered, the former appears at LO i+1 . On the other hand, this also means that in hadronic processes non-isolated photons can be in principle present at LO i with i > 1. Especially, albeit being not very frequently, both isolated and non-isolated photons can be present at LO i with i > 1, for instance in signatures featuring both isolated photons and jets. We leave this case for future work.

Simultaneous photon isolation and democratic-jet clustering
While the simultaneous presence of both isolated and non-isolated photons is very uncommon at LO 1 and not so frequent at LO i with i > 1, if isolated photons are present at LO i , both isolated and non-isolated photons are always present at the same time at NLO i+1 . Indeed, as soon as one external line or propagator in the process is electrically charged, the real emission of QED includes the process pp−→n γ γ iso + X + γ . (2.19) This also means that, if light particles are part of X, democratic jets have to be in general employed in order to achieve IR safety. If there are only leptons among the light particles of X, dressed leptons may be sufficient, but in general the main point is that non-isolated photons have to be recombined with massless particles when they get close to be collinear. When both democratic jets and isolated photons are the physical objects appearing in the final state, the two algorithmic procedures for identifying them, isolation and clustering, do not commute. If X contains n j jets, the procedure that has to be followed for the inclusive cross section at NLO i with i > 2 for a process as defined as in (2.3), and therefore including also real radiation process as defined as in (2.19), is the following: 1. Run the photon-isolation algorithm, isolating photons from QCD-interacting particles as well as QED interacting particles, including photons themselves. 12 2. If at least n γ photons are identified as isolated photons proceed, otherwise the event is rejected. 11 This is not possible only if the process does not contain coloured particles in the final state and cannot be initiated by coloured partons. 12 For IR safety, the isolation of photons from photons is actually necessary only at NNLO and beyond.

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3. Run the jet clustering algorithm including all the QCD and QED interacting particles, but among the photons only those that have not been tagged as isolated.
4. If less than n j jets have been tagged, reject the event.
If dressed leptons are part of the final-state physical objects, the recombination of bare leptons and non-isolated photons is done at the third step of the previous list. If both jet clustering and lepton recombination is performed, and the jet clustering involves non-isolated photons, leptons and jets have to be separated, e.g., in the (η, φ) plane of the pseudorapidities and azimuthal angles.

Common set-up
In this section we describe the calculation setup, which is common for the processes we have considered in this work: Unless it is differently specified, in the following with the notation tγj we will understand both tγj andtγj production. Also, we will understand that γ is an isolated photon, without specifying γ iso as in the previous sections. We provide results for proton-proton collisions at the LHC, with a centre-of-mass energy of 13 TeV. In our calculation, we employ the complex mass scheme [15,71,72], using the following on-shell input parameters We have set Γ t = 0, since at least one external top quark is always present. In the case of ttγ and ttγγ production, also the widths of the W and Z bosons are set equal to zero. All the calculations are performed in the five-flavour scheme (5FS), besides the case of tγj production, where also the NLO QCD calculation in the four-flavour scheme (4FS) is considered for estimating the flavour-scheme uncertainty. The value m b = 4.92 GeV directly enters the calculation only in this specific case and has been chosen in order to be consistent with the corresponding calculation in the 5FS. Following the same argument of ref. [23], we choose the set NNPDF3.1 [73,74] for all our calculations. In this set, the value of m b used in the PDF evolution is precisely m b = 4.92 GeV.

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As we have discussed in section 2, we renormalised EW interactions in a mixed scheme. The input values for G µ and α(0) are: QCD interactions are instead renormalised in the MS-scheme, with the (renormalisationgroup running) value of α s directly taken from the PDF sets used in the calculation. We estimate QCD scale uncertainties by independently varying by a factor of two both the renormalisation scale µ r and the factorisation scale µ f around the central value µ 0 defined as follows, 3) for the other production processes .
The quantity m T,i is the transverse mass of the particle i. The scale definition in eq. (3.3), where with j b we denote the b-jet, is analogue to the one of ref. [23], which is based on the findings of refs. [75,76]. The definition in eq. (3.

4) is instead the default option in
MadGraph5_aMC@NLO, with the sum running over the final-state-particle momenta, including those from real emissions. Finally, we specify the parameters related to procedure explained in section 2.2.3 for the isolation of photons and the clustering of democratic jets or dressed leptons. Photon isolation is performed à la Frixione [29], with the parameters After this, we cluster jets via the anti-k T algorithm [77] as implemented in FastJet We remind the reader that in our calculation a jet can correspond to a single non-isolated photon. 13 When we will consider b-jets, in the case of tγj production, we will simply mean jets containing a bottom (anti)quark; no restrictions on their pseudorapidity are imposed. 14 Also, for this process, the jet definition is relevant only for differential distributions and not for total cross sections; single-top photon is properly defined and IR finite without tagging any jet. In section 3.5 we will also deal with leptons in the final state, which have to be dressed with photons in order to achieve IR safety. Since in this work lepton-photon recombination concerns only the case of top-quark decays in their rest frame, a dressed lepton is obtained by recombining a bare lepton with any non-isolated photon γ satisfying the condition

7)
13 LHC analyses typically defines jets with up to 99% of their energy of electromagnetic origin. Up to 90% can even be associated to a single photon. See also ref. [13]. 14 In our calculation, no γ, g−→bb splittings are involved in the final state and in turn, b-jets cannot include more than one bottom (anti)quark. Therefore, no IR safety problems are present in this b-jet definition even if we use the 5FS.

Top-quark pair and one photon associated production: ttγ
The NLO EW corrections to top-quark pair hadroproduction in association with a single photon (ttγ) have already been calculated in ref. [79], by using the α(0)-scheme. We repeat the calculation, in a completely automated way, by employing the mixed renormalisation scheme discussed in section 2.1.2 and providing for the first time Complete-NLO predictions. For this process, according to eq. (2.2), k = 3 and therefore not only NLO EW and NLO QCD corrections are present (NLO 1 and NLO 2 in our notation), but also the LO 2 , LO 3 , NLO 3 and NLO 4 contributions, where the LO 1 is proportional to α 2 s α. We remind the reader that NLO QCD corrections to ttγ production have been already calculated in refs. [80][81][82][83], and in particular in refs. [82,83] it has been shown their large impact in reducing the top-quark charge asymmetry at the LHC. This last aspect has also been investigated in ref. [84]. The matching with QCD parton shower, besides being in general available in the MadGraph5_aMC@NLO framework and taken into account in ref. [82], has been studied in ref. [85], without spin correlations, via the PowHel framework [86], which in turn relies on the Powheg-Box system [87,88]. NLO QCD corrections including top-quark decays have been presented for the first time in ref. [33] in the narrow width approximation (NWA), and for the complete non-resonant e + ν e µ − ν µ bbγ leptonic signature in ref. [89]. Comparison among the different NLO QCD approximations has been carried out in ref. [90].
This process has already been observed at the LHC [91], and further measurements have been performed [92][93][94][95], showing so far no sign of deviations from the SM predictions.

Numerical results
In table 1 we provide results for the total cross section and the charge asymmetry A C , with different cuts on the transverse momentum and rapidity of the photon. We remind the reader that the charge asymmetry is defined as In all cases results are provided in different approximations, namely,   Table 1. Cross sections and charge asymmetries for ttγ production. The uncertainties are respectively the scale and the PDF ones in the form: ± absolute size (± relative size). The first number in parentheses after the central value is the absolute statistical error.
where in eqs. (3.9)- (3.12) there are the precise definitions of the quantities entering tables and plots in this section. The Complete-NLO is therefore simply denoted as "NLO". Moreover, for the case with the cut p T (γ) ≥ 25 GeV, we show in table 2 the ratio of the contribution of each separate perturbative order with the LO QCD .
As can be seen in table 1, for all the four phase-space cuts choices, NLO EW corrections are negative and ∼ −2% of the LO QCD or equivalently ∼ −1% of the NLO QCD , i.e., one order of magnitude smaller than QCD scale uncertainties at NLO accuracy. Moreover, the difference between the Complete-NLO prediction, NLO in the table, and the NLO QCD one is even smaller. Indeed, as can be seen in table 2 for the case p T (γ) ≥ 25 GeV, the LO 2 , the LO 3 and the NLO 3 all together largely cancel the impact of the NLO 2 , the NLO EW corrections. Our conclusion is that, given the current QCD uncertainties (scale+PDF), which are dominated by the scale dependence at NLO, at the inclusive level the impact of EW corrections on the ttγ cross section is negligible. We remind the reader that this conclusion could be drawn only after having performed a complete calculation. Moreover, it is in contrast to what has been observed for other processes involving top quarks, such as ttW and tttt production [19].
The impact of the NLO corrections is different in the case of the charge asymmetry A C . First of all, the NLO QCD corrections strongly decrease the LO QCD predictions, as already discussed in refs. [82,84], with the NLO QCD /LO QCD ratio ranging from 0.51 to 0.57 in the four phase-space cuts choices of table 1. Moreover, since we evaluate scale uncertainties by keeping scales correlated in the numerator and denominator of A C (see eq. (3.8)), LO QCD scale uncertainties are very small. However, NLO QCD corrections induce an additional negative term to the numerator of A C , which therefore has a scale dependence that is anti-correlated with the one of the denominator. The net effect is an increment, but also a more realistic estimate, of the scale uncertainty for A C . The impact of NLO EW corrections is also not negligible, with the NLO QCD+EW /NLO QCD ratio ranging from 0.89 to 0.92 in the four phase-space cuts choices. The additional terms in the Complete-NLO, (NLO−NLO QCD+EW ), further reduce the predictions, with the NLO/NLO QCD ratio ranging from 0.84 to 0.88 in the four phase-space cuts choices. Overall, the Complete-NLO predictions reduce the NLO QCD ones by shifting their central values to (almost) the lower edge of the scale-uncertainty bands.
In figure 1 we show differential distributions for ttγ production. In particular we show the transverse momentum distributions (p T ) of the hardest isolated photon (γ 1 ), the top quark and the top-quark pair, the invariant mass of the top-quark pair and of the entire ttγ system, and the rapidity of the top-quark. For each plot we show in the main panel the central value of the LO, NLO QCD and NLO predictions. In the first inset we separately show the relative scale and PDF uncertainties of the NLO prediction together with their sum in quadrature, the total uncertainty. In the last inset we show again the relative total uncertainty, but now for the NLO QCD prediction, together with the NLO QCD+EW /NLO QCD and NLO/NLO QCD ratios.
For the p T (γ) and p T (t) distributions, the NLO EW corrections are negative and grow in absolute value in the tail. This effect is expected and due to the EW Sudakov logarithms. For these two observables, the impact of the NLO EW corrections cannot be neglected, especially for p T (t), where in the tail the term NLO 2 = (NLO QCD+EW − NLO QCD ) is almost as large as the total NLO QCD uncertainty, which in turn, as for any other observable considered here, is numerically as large as the total NLO uncertainty. We also notice that the impact of the (NLO − NLO QCD+EW ) term is on the other hand negligible. The case of p T (tt) is special. As discussed in detail in refs. [82,83]   the p T (tt) plot in figure 1, the NLO QCD corrections scale as α s log 2 (p T (tt)/Q) where Q is a scale that increases by increasing R 0 (γ) or p min T (γ), the isolation parameters of eq. (3.5). This effect underlies the large increase of scale and PDF uncertainties and the smallness of the (NLO QCD+EW /NLO QCD − 1) and (NLO/NLO QCD − 1) terms. For what concerns the m(tt) and m(ttγ) distributions, we see similar effects as in the p T (γ) and p T (t) ones for the NLO QCD+EW /NLO QCD ratio, although with smaller deviations from unity. On the other hand, especially for m(tt), the effect is largely compensated by the additional terms in (NLO − NLO QCD+EW ). The y(t) rapidity does not show large EW effects, similarly to the inclusive rates. The only effects that are not flat are the relative size of the uncertainties, growing in the peripheral region.

Top-quark pair and two photons associated production: ttγγ
The calculation of NLO EW corrections to top-quark pair hadroproduction in association with two photons (ttγγ) is presented for the first time here. We perform the calculation, in a completely automated way, and we limit ourselves to the case of NLO EW and NLO QCD corrections. However, also for this process, according to eq. (2.2), k = 3 and therefore not only NLO EW and NLO QCD corrections are present (NLO 1 and NLO 2 in our notation). We leave the Complete-NLO study to future work, but given what has been observed in the case of ttγ production, we do not expect large effects in comparison to the QCD uncertainties.

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ttγγ Cuts Order  Table 3. Cross sections and charge asymmetries for ttγγ production. The uncertainties are respectively the scale and the PDF ones in the form: ± absolute size (± relative size). The first number in parentheses after the central value is the absolute statistical error.
We remind the readers that NLO QCD corrections to ttγγ production have been calculated for the first time in ref. [96], matched to parton shower effects in ref. [97] and thoroughly studied together with all the other ttV V processes in ref. [82]. The last two references have also investigated its impact in the ttH searches where the Higgs boson decays into two photons, which is one of the main motivations to study ttγγ production at the LHC.

Numerical results
Similarly to the case of ttγ in table 1, in table 3 we provide results for the total cross section and the charge asymmetry A C for ttγγ production, with different cuts on the transverse momenta, the rapidities and the ∆R(γ 1 , γ 2 ) distance of the two hardest isolated photons. As for ttγ production, NLO EW corrections are well within the total uncertainty of NLO QCD predictions, although their relative impact is slightly larger for this process: ∼ −3% of the LO QCD prediction or equivalently ∼ −2% of the NLO QCD one. We want to stress again that only after performing an exact calculation such as the one presented here we can claim that at the inclusive level NLO EW corrections are negligible in comparison to the total QCD uncertainty (scale+PDF).
In the case of A C , the most striking difference with the case of ttγ is its absolute size, which is roundabout five times larger. On the other hand, total rates are, depending on  the cuts, hundreds to thousand times smaller than for ttγ production. One should also not forget that with a 100 TeV collider, rates will increase by roughly a factor fifty [82], but the value of A C will also decrease. Indeed with higher hadronic energies the relative contribution of gluon-gluon initiated processes increases, but being completely symmetric it enters only the denominator of A C (see eq. (3.8)). The same effects can be seen in ttγ by comparing results in table 1 with those in ref. [83], which are for 100 TeV collisions. Thus, while the measurement of A C in ttγ hadroproduction is achievable in the next future [84], in the case of ttγγ its feasibility still remains an open question. Nevertheless it is important to notice the impact of NLO corrections. NLO QCD corrections decrease the size of A C , with the NLO QCD /LO QCD ratio ranging from 0.64 to 0.75 in the four phase-space cuts choices of table 3. Similarly to ttγ production, LO QCD scale uncertainties are very small, but they are much larger when NLO QCD corrections are taken into account. The effect of NLO EW is also not negligible, being the NLO QCD+EW /NLO QCD ratio ∼ 0.96 for all the four phase-space cuts choices. Still, it is well within the total QCD uncertainties (scale+PDF), but it may be further reduced by the missing (NLO − NLO QCD+EW ) term. As already mentioned, we leave this calculation for future work. We now move to the case of differential distributions. In figure 2 we show distributions for the transverse momentum of the first and second hardest isolated photons and their invariant mass, the transverse momentum and rapidity of the top quark, and the invariant mass of the top-quark pair. The layout of the plots is very similar to the one of the plots JHEP09(2021)155 displayed in figure 1 and described in section 3.2; the only difference is that Complete-NLO predictions are not present. Most of the features described for the plots in figure 1 apply also for the corresponding ones presented in figure 1, therefore we do not repeat them here. We notice that the largest effect of NLO EW corrections is present for the case of the p T (t) distributions, reaching in the tail almost the lower edge of total QCD uncertainties (scale+PDF). In the case of m(γ 1 γ 2 ), which clearly was not present in figure 1, effects of NLO EW corrections are well within the total QCD uncertainties.

Single-top photon associated production: tγj
For the calculation of NLO QCD and EW corrections of single-top photon associated hadroproduction (tγj), we closely follow the approach of ref. [23], where the same kind of calculation has been performed for the single-top and H or Z boson associated hadroproduction. For the first time we provide NLO QCD+EW predictions for tγj production, together with an estimate of the flavour-scheme uncertainties, based on the procedure that has been presented, motivated and explained in details in ref. [23]. Here we will not repeat the details; we invite the interested reader to look for them in ref. [23]. As for any other process, NLO QCD corrections to tγj production can be calculated since a few years ago in a completely automated way via the MadGraph5_aMC@NLO framework [16]. On the other hand, at least to the best of our knowledge, so far a dedicated study of tγj production has never been performed even at NLO QCD accuracy. 15 Therefore, results in this section are new not only for what concerns NLO EW corrections but also at NLO QCD accuracy. We remind the reader that the CMS collaboration has already found the evidence for tγj production [100], and searches in the context of flavour-changing neutral currents have been performed for this process by the ATLAS collaboration [101].
At variance with ttγ and ttγγ production, according to eq. (2.2), k = 1 for tγj production and therefore at LO only the LO QCD , also denoted LO 1 , contribution is present and at NLO only the NLO EW and NLO QCD corrections are present, NLO 1 and NLO 2 in our notation. This also means that the Complete-NLO and the NLO QCD+EW predictions coincide (NLO = NLO QCD+EW ). However, again at variance with ttγ and ttγγ production, since n γ = 1 and LO 1 ∝ α 3 , the use of the mixed scheme in principle allows for both the casesᾱ = α(0) andᾱ = α Gµ , as shown in (2.8). We will therefore comment more on the choice of the value ofᾱ for this process.
Before moving to numerical results, we want to summarise very briefly the approach of ref. [23], which is used also here for estimating flavour-scheme uncertainties. First of all, it is important to note that tγj process involves at LO a bottom quark in the initial state. As very well known, similarly to the case of single-top production without photons in the final state [75,[102][103][104], this implies that the calculation can be performed in the 4FS or 5FS. We perform our calculation in the 5FS, without selecting any particular channels (s-, t or tW associated), but we want also to take into account the uncertainty due the choice of the 5FS instead of the 4FS, for which the calculation is more cumbersome. In ref. [23] JHEP09(2021)155 we have motivated why the following approach should be preferred for this purpose. First, the t-channel only production mode is identified both in the 4FS and 5FS at NLO QCD accuracy and denoted as NLO 4FS QCD,t−ch. and NLO 5FS QCD,t−ch. , respectively. Then, the scale uncertainties for these two quantities are evaluated via the nine-point independent variation of the renormalisation and factorisation scales, around a common central value. Next, a combined scale+flavour uncertainty band is identified as the envelope of the previous two and denoted as 5FS scale 4−5 , with the central value equal to the one in the 5FS. Finally the relative upper and lower uncertainty induced by the 5FS scale 4−5 is then propagated to the entire NLO QCD and NLO QCD+EW prediction, without selecting the t-channel only. All the motivations for this approach, can be found in ref. [23], where all the argument underlying this procedure do not depend on the presence of the Z or Higgs boson in the final state, which can therefore be substituted with the photon.

Numerical results
For the definition of the phase-space cuts we follow the analysis performed by the CMS collaboration [100], which has led to the evidence for tγj production in proton-proton collisions. Events are required to satisfy the following cuts: Based on this we define two phase-space regions: Inclusive (only the first cut applied) and Fiducial (all cuts applied).
In table 4 we report Inclusive and Fiducial results for different approximations. In the upper half of the table there are results at NLO QCD accuracy in the 4FS and 5FS for the t-channel mode only, together with the 5FS scale 4−5 prediction, whose definition has been introduced before in this section. In the lower part of the table there are results obtained without selecting the t-channel only, for both NLO QCD and NLO QCD+EW predictions. In both cases we display the pure 5FS and the 5FS scale 4−5 prediction, which is derived via the procedure introduced in the previous section and based on ref. [23]. The predictions dubbed as 5FS scale 4−5 , including all channels and flavour+scale uncertainties, are the most precise and reliable, especially the one at NLO QCD+EW accuracy, which taking into account both NLO QCD and EW corrections has to be considered as our best prediction for tγj production. The label tW h in the table refers to those diagrams consisting of tW γ associated production with subsequent W decay into quarks (h for hadronic), which appear both via NLO QCD and EW corrections.
First of all, by comparing results in the upper and lower half of table 4, it is evident how the sum of the contributions of the s-channel and tW h modes exceeds the total uncertainty of the t-channel alone. Thus, these two contributions cannot be ignored in the comparisons between data and the SM predictions.   Table 4. Cross section for tγj production. The uncertainties are respectively the (flavour+)scale and the PDF ones in the form: ± absolute size (± relative size). The first number in parentheses after the central value is the absolute statistical error.
the corresponding 4FS and 5FS results. For both cuts, the NLO EW corrections are ∼ −3% of the NLO QCD predictions, therefore well within the 5FS scale 4−5 uncertainty. On the other hand, we notice that in the pure 5FS the lower edge of the NLO QCD band would be much closer to the NLO QCD+EW central prediction, for both the phase-space cuts. This fact supports the relevance of employing the 5FS scale 4−5 approach for obtaining reliable results. The relevance of the 5FS scale 4−5 approach and the importance of the NLO EW corrections can be better appreciated with differential distributions, which we are going to describe in the following.
In figure 3 we show differential distributions for tγj production, without selecting the tchannel. In particular, we show the pseudorapidity and transverse-momentum distributions of the hardest light-jet (j l 1 ), and the transverse momentum of the top (anti)quark and hardest isolated-photon. The plot on the left are obtained with the Inclusive cuts, while those on the right with the Fiducial one. For each plot we show in the main panel the central value of the LO, NLO QCD and NLO QCD+EW predictions in the 5FS. In the first inset we separately show the relative scale+flavour and PDF uncertainty of the NLO QCD+EW prediction together with their sum in quadrature, the total uncertainty. In the last inset we show the 5FS scale and 5FS scale 4−5 scale+flavour relative uncertainties for the NLO QCD prediction, together with the NLO QCD+EW /NLO QCD ratio.
First of all we see that plots for Inclusive and Fiducial cuts are almost identical, besides their normalisations. The only exception is the threshold region for the p T (t) distribution. Thus, the following considerations apply to both cases. The large difference between LO and NLO QCD or NLO QCD+EW predictions in the central region of the η(j l 1 ) distributions is due to the opening of the tW h channel via the NLO corrections, which, as explained in refs. [21,23], is not enhanced for large η(j l 1 ) values and therefore populates the central region of this distribution. In the peripheral region, if we did not take into account flavour

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uncertainties, namely in the 5FS, NLO EW corrections would be larger than the QCD scale-uncertainty band; only with the 5FS scale 4−5 approach are within it. The same argument applies to the tail of the p T (γ 1 ) distribution, where although NLO EW corrections reach the size of ∼ −15% of the NLO QCD prediction, 16 they are still within 5FS scale 4−5 total uncertainty. The situation is instead different in the tail of the p T (j l 1 ) and p T (t) distributions, where NLO EW corrections are larger than 5FS scale 4−5 uncertainties, which on the other hand almost overlap with the 5FS ones.
In conclusion, no sizeable differences have been observed between results for the Inclusive and Fiducial regions, besides the total rates, and the 5FS scale 4−5 approach should be preferred both for total and differential rates. Only following this approach, NLO EW corrections are in general within the total uncertainty, but also in this case exceptions are present in the tail of distributions. We also have compared results obtained withᾱ = α(0) andᾱ = α Gµ in order to assess how large is the numerical impact of the choice of the value of α, where the latter choice is superior from a formal point of view. As expected, even in the tail of the p T (γ 1 ) distribution, where corrections have been found to be sizeable, the choice of the value forᾱ had an impact below the percent level. In general, results obtained via the two different choices ofᾱ have been found compatible within their numerical accuracy.

Top-quark decay involving photons: t → b + ν γ and t → bjjγ
As discussed in e.g., refs. [33,90] for the case of ttγ production, when top quark decays are taken into account, the contribution of photons radiated via the top decay is sizeable. The predictions for ttγ, ttγγ, and tγj that we have discussed in the previous sections do not include this contribution, being the top quark stable. For each of the previous processes, if top decays were considered, an important contribution would be given by the same process without one isolated photon in the final state (tt, ttγ and tj respectively) and the subsequent t−→bW γ decay for one of the top quarks. On the other hand, the focus of this work is the calculation of EW corrections. We have shown that, besides in the tails of the distributions, NLO EW corrections are in general within the QCD uncertainties for the case with photon emitted by the hard process. In NWA, the case of the photons emitted from the top decay depends on two factors. First, the NLO EW corrections to the tt, ttγ and tj production processes. Second, the NLO EW corrections to the top-quark decay t−→bW γ. The former are documented in the literature [15,18] or discussed in this work in the case of ttγ. The latter are calculated for the first time in this section.
As already mentioned, we calculate the NLO QCD+EW predictions for the leptonic and hadronic top-quark decays t → b + ν γ and t → bjjγ. In the case of t → b + ν γ we actually select the channel t → µ + ν µ bγ for the calculation, although all the others leptonic channels are equivalent, assuming massless τ leptons. The case with top antiquarks gives the same results. For this process, according to eq. (2.2), k = 1 and therefore only NLO EW and NLO QCD corrections are present (NLO 1 and NLO 2 in our notation), where the LO 1 is proportional to α 3 . 16 For this process, the large size of the EW corrections in the tail is partially due to the fact that we require exactly one isolated photon. Indeed, since part of the photon radiation with pT > 25 GeV is vetoed, an additional negative correction that grows in absolute size in the tail is present.   For this calculation part of the settings listed in section 3.1 are modified. First, the central value of the renormalisation scale is set to m t , then the Frixione isolation algorithm is adapted to the case of a decay process in its rest frame. The isolation is performed by using, instead of R 0 (γ) = 0.4 and p min T (γ) > 25 GeV like in (3.5), the parameters: E min (γ) > 25 GeV and θ 0 (γ) = 0.1, where the separation of the photons and hadronic or electromagnetic activities is performed by looking at the separation angle.
In table 5 we report LO, NLO QCD and NLO QCD+EW predictions for the partial widths of t → bjjγ and t → b + ν γ decays. The NLO QCD corrections reduce the LO prediction by −23% and −11% for the hadronic and leptonic case, respectively. In both cases, NLO QCD scale uncertainties are only a few percents of the absolute value. NLO EW corrections are in both cases smaller than 1% of the LO prediction.
In figure 4 we show for the leptonic case the energy spectrum of the hardest photon, E(γ 1 ), and of the lepton, E( + ). In the main panel we show LO, NLO QCD and NLO QCD+EW predictions, in the first inset we show the NLO QCD /LO ratio and in the second inset the NLO QCD+EW /NLO QCD ratio. As can be seen, while the relative impact of NLO QCD corrections varies a lot in both distributions, the NLO EW corrections remain JHEP09(2021)155 at or below the percent level in the full spectrum, with the exception of the tail of the distribution in the case of E( + ). Needless to say, although the choiceᾱ = α Gµ in (2.8) is formally superior, in practice the choice of the value ofᾱ is completely negligible.
The results obtained for the t → bjjγ and t → b + ν γ decays point to the fact that, when looking at ttγ, ttγγ or tγj production, although the contribution of photons emitted after the top decay is in general sizeable, the size of the NLO EW corrections to the topquark decay in association with photons is negligible. A study of the NLO EW corrections for the complete final state W + bW −b γ including W decays in NWA or with full off-shell effects, as already done for NLO QCD in respectively ref. [33] and ref. [89], is definitely worth to be considered, but beyond the scope of this paper. The same applies to the W + bW −b γγ and W bjγ final states.

Conclusions
In this paper we have calculated for the first time: • the Complete-NLO predictions for top-quark pair production in association with at least one photon (ttγ), • the NLO QCD+EW corrections for top-quark pair production in association with at least two photons (ttγγ), • the NLO QCD+EW corrections for single-top production in association with one photon (tγj), together with a 4FS and 5FS comparison, • the NLO QCD+EW corrections for leptonic (t → b + ν γ) and hadronic (t → bjjγ) decays.
In the case of cross sections, we find that EW corrections are in general within QCD uncertainties. For tγj production, that is true only if the uncertainty due to the flavour scheme is taken into account. Moreover, for this process, in the tail of the distributions EW corrections are sizeable and of the same size of (or larger than) QCD uncertainties. We also have analysed the top-quark charge asymmetry A C for ttγ and ttγγ production and found sizeable effects for NLO QCD and NLO EW corrections and as well for subleading NLO orders. Therefore, unlike other processes involving top quarks (ttW and tttt), EW corrections are under control for this class of processes and have a size that is of the order estimated from the naive α s and α power counting. We want to stress that this conclusion can be drawn only after having performed an exact calculation of NLO corrections, as done here in this work.
All these calculations have been performed in a completely automated approach via the MadGraph5_aMC@NLO framework, without any dedicated customisation for the processes considered. In order to achieve this, we have extended the capabilities of the JHEP09(2021)155 scheme (α(0) and G µ or α(m Z )) for this class of processes. We have also discussed the issues related to the choice of the numerical value of α in the O(α) corrections and the subtleties related to this aspect in the context of automated calculations.