NLO electroweak and QCD corrections to the production of a photon with three charged lepton plus missing energy at the LHC

Electroweak (EW) triboson production processes with at least one heavy gauge boson are of increasing interest at the Large Hadron Collider (LHC) as direct precision probes of one of the least-tested sectors of the Standard Model (SM), the quartic couplings of the EW gauge bosons. These processes therefore offer promising opportunities for searches for indirect signals of Beyond-the-SM (BSM) physics. In this paper, we present results for fiducial cross sections at next-to-leading-order (NLO) EW and NLO QCD to $p\;p\to e^{+}\;\nu_{e}\;\mu^{+}\;\mu^{-}\;\gamma$ at the 13 TeV LHC. This signature includes the triboson production process $p\;p \to W^{+}\;Z\;\gamma$ with leptonic decays, $W^{+} \to e^{+}\;\nu_{e}$ and $Z \to \mu^{+}\;\mu^{-}$. The computation is based on the complete set of leading-order (LO) contributions of $\mathcal{O}(\alpha^5)$ and on NLO EW and NLO QCD cross sections of $\mathcal{O}(\alpha^6)$ and $\mathcal{O}(\alpha^5 \alpha_\text{s})$, respectively, and thus off-shell effects, spin correlations and non-resonance contributions are fully taken into account. We construct a Monte Carlo framework which provides total and differential cross sections for a chosen set of basic analysis cuts. We find that while NLO EW corrections enhance the fiducial LO total cross section by only $1\%$, they can significantly change some distributions in certain kinematic regions. For example, the relative NLO EW corrections to the muon transverse momentum distribution at 500 GeV amounts to $-20\%$. To illustrate how missing NLO EW corrections could masquerade as BSM physics, we show examples for the impact of dimension-8 operators in the SM Effective Field Theory framework on selected kinematic distributions.


I. INTRODUCTION
The investigation of the self interactions of electroweak (EW) gauge bosons (γ, Z, W ± ) at an increasing level of precision offers a promising indirect window to Beyond-the-SM (BSM) physics, and is therefore an important goal of the CERN Large Hadron Collider (LHC). EW triboson production processes, p p → V V V , where at least one gauge boson is a W or Z boson, are especially interesting, since they provide direct access to the least tested EW gauge-boson interactions, the quartic-gauge-boson couplings (QGCs). For instance, W W W production at the LHC, which directly probes the W W W W QGC, was searched for by the ATLAS [1] and CMS [2] collaborations at √ s = 8 TeV and 13 TeV, respectively, and only recently was observed for the first time by the ATLAS collaboration with a significance of 8.0σ using the full Run-2 data set [3]. The first evidence for a combination of heavy triboson production processes has been reported by the ATLAS collaboration in Ref. [4] and the first observation of a combined W W W, W W Z, W ZZ, and ZZZ production signal was achieved by the CMS collaboration [5].
While heavy triboson production processes have only recently become experimentally accessible, EW triboson processes involving isolated photon(s) were among the first triboson cross section measurements performed at the LHC owing to their comparatively large cross sections (W γγ production has the largest inclusive cross section among the triboson processes with at least one heavy gauge boson). W γγ and Zγγ production cross sections have been measured by the CMS [6] and ATLAS [7,8] collaborations at √ s = 8 TeV, and by the CMS collaboration [9] at √ s = 13 TeV. Evidence for W W γ and W Zγ productions at √ s = 8 TeV has been reported by both the CMS [10] and ATLAS [11] collaborations. It is interesting to note that EW triboson production processes also allow for a study of triplegauge-boson couplings (TGCs), complementing the ones performed in diboson production processes, and constitute an important background to direct BSM searches, especially in case of leptonic decays.
Clearly, to take full advantage of the potential of EW triboson processes at the LHC to search for indirect signals of BSM physics, SM predictions for the relevant observables need to be under superb theoretical control. In particular, the extraction of information about anomalous TGCs and QGCs or higher-dimensional operators in an Effective Field Theory (EFT) framework from measurements of kinematic distributions, requires the inclusion of both QCD and EW higher-order corrections. NLO QCD predictions for EW triboson processes with at least one W or Z boson and leptonic decays have been available for many years (see, e.g., Refs [12,13] for a review), and can readily be obtained for instance from the publicly available Monte Carlo (MC) program VBFNLO [14][15][16]. In recent years, the calculation of NLO EW corrections has also seen an increased activity which is not surprising given the importance of EW triboson production to the LHC physics program 1 .
Thanks to advances in the calculation of these corrections for processes with high-particle multiplicity in final states, the NLO EW predictions are gradually becoming more sophisticated, taken into account fully decayed final states without approximations. NLO EW corrections to p p → V V V processes with on-shell EW gauge bosons have been calculated in Refs. [17] (W W Z), [18] (W W W ), [19] (W W γ, including parton shower effects), and also served as benchmarks for MadGraph5 aMC@NLOv3 [20] (W W W , W ZZ, and ZZZ).
The main focus of this paper is the calculation of NLO EW corrections to p p → e + ν e µ + µ − γ. This process includes the W + Zγ triboson process (with leptonic decays W + → e + ν e and Z → µ + µ − ) and thus is sensitive to the W W Zγ and W W γγ QGCs. The To our knowledge, the NLO EW corrections to p p → e + ν e µ + µ − γ have not yet 1 See also Ref. [13] for a discussion of the status of SM predictions and which calculations are still needed. 2 OpenLoops also uses OPP reduction methods as implemented in CutTools [35] and OneLOop [36] for the evaluation of one-loop scalar integrals. The paper is organized as follows: In Section II we describe the calculational framework underlying our MC program, separately for the calculation of the virtual and real corrections in Section II A and Section II B respectively. In Section II C we provide a detailed description of the many checks we performed to validate the results of our MC program. In Section III, after describing our choice of input parameters and of basic analysis cuts in Section III A, we provide numerical results for the total cross sections (Section III B) and kinematic distributions (Section III C) at NLO EW and NLO QCD together with a discussion of the residual theoretical uncertainty due to the factorization and renormalization scale variation at NLO QCD. The discussion of the numerical impact of NLO EW corrections closes with an example of the impact of dimension-8 operators in SMEFT in Section III D. Section IV contains a brief summary and our conclusions.

II. CALCULATIONAL FRAMEWORK
The goal of this paper is to provide precise predictions for W + Zγ production with leptonic decays at the LHC with emphasis on calculating and studying the impact of EW O(α) corrections to the process thereby taking into account the full off-shell effects, spin correlations and non-resonance contributions. We consider all fermions but the top quark to be massless. For completeness, and to study the impact of different combinations of EW and QCD corrections, we also calculated O(α s ) corrections to this process. The calculation of the hadronic cross section is based on the master formula where the sum is taken over all possible combinations of partons a, b with momenta p a,b determined by the fractions x 1,2 of the protons' momenta p 1,2 . f a,b are the parton distribution functions (PDFs) depending on the momentum fractions x 1,2 and the factorization scale µ F .σ ab denote the partonic cross sections which in case of NLO QCD also depend on the renormalization scale µ R . At NLO accuracy,σ ab consists of Born contributions (dσ B ab ), virtual one-loop corrections (dσ V ab ) and real radiation corrections (dσ R ab ): As indicated, the Born and one-loop contributions are obtained by integrating over a 2 → 5particle phase space while the real corrections require an integration over a 2 → 6-particle phase space. The partonic Born cross section is of O(α 5 ) and only receives contributions from the quark-induced processes: where q and q denote the up-type light quark (u, c) and the down-type light quark (d, s) respectively. We do not include b-quark-initiated processes, since they have a negligible effect on the hadronic cross section due to the smallness of the b-quark PDF. In FIG. 1 we show a representative set of LO Feynman diagrams which consists of topologies arising from the W + Zγ production process with subsequent leptonic decays, some featuring QGCs and TGCs, and non-resonance contributions which do not arise from the W + Zγ production process.
In our calculational framework, dσ B ab and dσ V ab are calculated by the one-loop provider RECOLA and dσ R ab is evaluated by MadDipole. In the next two sections, we will discuss in more detail the calculations and validations of virtual one-loop corrections and real radiation corrections at both NLO EW and NLO QCD accuracy, and will also address some technical issues of the implementation of these tools in our MC framework.    In the calculations of the NLO EW corrections, we chose as the EW input scheme the G µscheme [53][54][55], where the electromagnetic coupling is determined from the Fermi constant G µ and the pole masses of W and Z bosons as follows: This choice has the advantage that large logarithmic corrections associated with the running of α are absorbed into the LO cross section. However, in processes with external photons at LO such as the one of Eq. (4), and when considering light fermions to be massless throughout, the implementation of this scheme needs some care 4 . The UV counterterm contribution to the NLO EW corrections for the process of Eq. (4) contains the renormalization constant for the electric charge (δZ e ) and for the photon wave function (δZ AA ) as follows: where the term (δZ e + 1 2 δZ AA ) arises from the presence of an external photon and M LO denotes the LO matrix element. Both δZ e and δZ AA contain contributions from the derivative of the photon self energy, Π AA (0), evaluated at zero momentum, which exhibits large logarithmic contributions of the form log(m f /µ R ) when retaining the light fermion masses m f in the calculation. In dimensional regularization, these logarithms manifest as single IR poles. In the α(0)-scheme, δZ e is given by [53,54] while in the G µ -scheme it takes the form where ∆r comprises the NLO EW corrections to muon decay, and reads [57] As can be seen, in the G µ -scheme the Π AA contributions in δZ e | α(Gµ) cancel. However, if be avoided when using a mixed scheme where the electromagnetic coupling of the external photon at LO is taken as α(0), which alters the UV counterterm of Eq. (6) into Since δZ AA = −Π AA (0), the combination (δZ e | α(0) + 1 2 δZ AA ) now remains IR finite. Evaluating α at zero momentum is also a more appropriate choice for an external photon coupling.
To summarize, in this mixed scheme, the cross sections of our process are of order α 4 Gµ α(0) at LO, α 4 Gµ α(0)α s at NLO QCD and α 5 Gµ α(0) at NLO EW. To easily implement the mixed scheme in the calculation of the virtual EW corrections, we first calculated the renormalized one-loop contribution with RECOLA in the pure G µ -scheme (of order α 6 Gµ ) and then converted the results into the mixed scheme (of order α 5 Gµ α(0)) as follows: The differential dipoles dσ A qq match the singular behaviour of real corrections dσ R qq locally, and the IR poles in the virtual corrections are cancelled upon combining with the integrated dipoles dσ I qq . The collinear PDF counterterms dσ C qq absorb the residual initial-state collinear singularities into PDFs in the MS-factorization scheme. Both QCD and QED dipole contributions, dσ A,I qq , along with the collinear PDF counterterms are generated by MadDipole. In order for an exact pole cancellation to happen and a proper combination of the finite contributions, the conventions for the prefactors of the Laurent expansion about the IR poles in d dimensions in the integrated dipoles and virtual one-loop corrections have to coincide.
The tools we rely on to calculate the virtual one-loop corrections and integrated dipoles do have different conventions, and thus an additional adjustment is required. The convention used by RECOLA for the expansion is [32] (4π and the convention used by MadDipole is [42] (4π) IR e γ IR where A and B denote the double and single pole coefficients, respectively, while C and C denote the finite contributions, and γ is the Euler-Mascheroni constant with γ ≈ 0.5722.
The difference between the conventions does not affect double and single pole coefficients but finite contributions. In our calculation, we adopt the RECOLA convention and convert the finite contribution of the integrated dipoles computed by MadDipole accordingly as follows The gluon/photon-induced processes, which contribute for the first time at NLO, only exhibit a collinear singularity due to initial-state gluon/photon splitting into a qq pair. Therefore, the master formula for these processes simplifies tô and only collinear subtraction terms and PDF counterterms are needed.

C. Validations
To validate the NLO EW and NLO QCD calculations of our MC framework, we performed numerous checks. Unless noted otherwise, we used the setup described in Section III A. We compared the LO and real radiation squared amplitudes with those calculated by MadGraph5 [58] at several phase space points. The finite contributions, coefficients of double and single poles of QCD and EW one-loop corrections have been compared with those calculated by MadLoop3 [20] and OpenLoops2 [33], again at several phase space points. We found good agreement in all comparisons. To verify the proper implementation of the dipole subtraction method using MadDipole, we checked the cancellations of the double and single IR poles among the virtual contributions, the integrated dipoles and the collinear PDF counterterms.
We also checked that the real squared amplitudes in the soft/collinear limits indeed approach the value of the differential dipoles. As an example, we show in Appendix IV A (Table IV) the cancellations of the IR poles in the case of NLO EW corrections at a single phase-space point (provided in Table III). At the total hadronic cross section level, the α-parameter dependence [59] is checked. This parameter is introduced to restrict the phase space for real radiation where dipole subtraction is needed: α = 1 corresponds to no restriction, i.e. all dipoles are subtracted in the entire phase space after the application of kinematic cuts. A smaller α-parameter means that only dipoles are subtracted that mimic the singular behavior of the real corrections in this phase space region. As a result, a finite contribution is shifted between real-subtracted corrections and integrated dipoles, but their sum has to be α-independent. We checked the α-parameter about 0.05% for NLO EW corrections and about 0.2% for NLO QCD corrections. However, this effect will have no noticeable impact given the large QCD scale uncertainty which will be discussed in detail in Section III. We also performed a comparison of kinematic distributions at NLO QCD with those computed by VBFNLO [14]. We produced results for the invariant mass of the µ + µ − pair and the transverse momentum of the isolated photon. As can be seen in FIG. 7, there is good agreement within the statistical MC uncertainties of the two MC programs.
Finally, we recalculated some results for total hadronic cross sections available in the literature (adjusting our input parameters and cuts accordingly): NLO QCD corrections to on-shell W W Z production [17], NLO EW corrections to the neutral-current Drell-Yan process (δ rec qq,phot and δ qq,weak for M ll > 50 GeV of Table 1 in [60]), and NLO EW corrections to Zγ production with leptonic decays (δ CS phot , δ weak,qq and δ γγ of Table 1 in [61]). The comparison with the results obtained with our MC program is shown in Appendix IV B (Tables V,VI,VII and VIII). In general we found good agreement within the statistical uncertainties of the MC programs. Small differences are at most at the 0.9% level of the relative correction which is not surprising when comparing different MC implementations of higher-order corrections.

III. NUMERICAL RESULTS
In this section we present results for the total cross sections and kinematic distributions for p p → e + ν e µ + µ − γ at the 13 TeV LHC for a basic set of analysis cuts, and dis-cuss the impact of NLO EW and NLO QCD together with an assessment of the residual renormalization and factorization scale uncertainty. We also study the impact of different ways to combine NLO EW and NLO QCD corrections and compare the effect of NLO EW corrections with those of dimension-8 operators in SMEFT.
In order to obtain well-defined cross sections, we performed a photon-charged-lepton recombination procedure and apply a basic set of analysis cuts, loosely inspired by experimental analysis cuts for triboson production processes at the LHC. Photon-charged-lepton recombination is needed in the calculation of the EW real corrections with two photons in the final state where one photon can be collinear to a final-state charged lepton. The recombination procedure is applied so that these regions of phase space are treated fully inclusively even in the presence of lepton identification cuts. All other contributions only contain one photon in the final state which will be identified by applying a photon isolation cut and thus can never become collinear to a final-state charged lepton. When applying the recombination procedure to the real EW corrections, e.g., to ud → e + ν e µ + µ − γ γ, the separations of the photons and the charged leptons in the pseudorapidity-azimuthal-angle plane, are calculated, where j = γ and i = l ∈ {e + , µ + , µ − }, ∆η iγ = η i − η γ is the pseudorapidity difference and ∆φ iγ = φ i − φ γ is the corresponding azimuthal angle difference. The photon and charged lepton with the smallest R lγ are recombined, i.e. their four-momenta are added, as long as R lγ < 0.1. In case of R lγ < 0.1 for both photons, the event will be rejected. If no recombination takes place, the harder photon satisfying p T,γ > 15 GeV and |η γ | < 2.5 will be labeled as the identified photon.
In the gluon-induced and photon-induced processes, as well as the QCD real corrections to the quark-induced processes, the final-state parton and photon may become collinear and thus induce extra IR-singularities of QED origin. To exclude this region of phase space two methods are commonly employed: democratic clustering with the help of a quark-tophoton fragmentation function [66] or Frixione isolation [67]. In this paper, we choose using Frixione isolation to avoid having to introduce a fragmentation function dependence in our predictions. A comparison of the impact of EW and QCD corrections in l + l − γ production in Ref. [61] when using either method has shown no difference in case of EW corrections and QCD corrections only differ by about 0.5 ∼ 1%. Our results are based on the Frixione isolation cut applied as follows: The event is accepted if R iγ > δ 0 (Eq. (21) with i = q, g).
In the case of R iγ < δ 0 , the event is accepted only if where δ 0 the isolation cone size and ε the damping parameter. We chose δ 0 = 0.7 and ε = 1.
After the application of the photon recombination procedure and the Frixione isolation cut, we applied the following additional analysis cuts: For the transverse momentum and pseudorapidity of the final-state photon and charged leptons, as well as the missing transverse momentum, we require The angular separations between photon and charged leptons along with the invariant mass of the µ + µ − pair have to satisfy which ensures that there is no collinear singularities from final-state photon splitting into leptons or photons radiating off the charged leptons at LO. Here, we do not impose restrictions on the angular separations between charged leptons (R ll ) or perform recombinations for two photons (when R γγ < 0.1) given that they are not technically required.

B. Total cross sections at NLO EW and NLO QCD
We present the results for the total cross sections at LO, NLO QCD and NLO EW for the process p p → e + ν e µ + µ − γ at √ s = 13 TeV in Table I. We also provide the relative NLO However, in the next section, we will see that NLO EW corrections can have a significant impact on various kinematic distributions. In Table II     scale variation to assess the scale uncertainty of the NLO QCD total cross sections as shown in Table I. also provide the full NLO EW relative correction (blue). We note that we have investigated the potential impact of using different photon PDFs, namely NNPDF31 nlo as 0118 luxqed, MMHT2015qed nlo [68] and CT18qed [69] sets. At both total and differential cross section levels, we found agreement of the results for photon-induced processes calculated by applying these three photon PDF sets within the statistical uncertainty of the MC integration.
The results we show here have been calculated by using the NNPDF31 nlo as 0118 luxqed and multiplicative (red) approach where the combined corrections are shown as relative corrections to the NLO QCD results.
In this way, we can more clearly assess whether the effects of NLO EW corrections are visible  In FIG. 8 and FIG. 9 we show the NLO EW and NLO QCD corrections to the invariant mass distribution of the µ + µ − pair in the low-and high-invariant mass region respectively.
In the low-invariant mass region (FIG. 8)      In FIG. 14 -FIG.19 we present the NLO corrections to various transverse momentum distributions: the p T of the e + , e + ν e pair, µ + , µ + µ − pair, the photon and missing transverse momentum. The NLO EW corrections exhibit very similar behaviors among the distributions of the transverse momentum of the positron (FIG. 14), missing transverse momentum (FIG. 15) and the transverse momentum of the e + ν e system (FIG. 16). In these distributions, the photon-induced EW corrections enhance the LO distribution with increasing transverse momentum and cause a +10% ∼ 20% overall EW corrections at 500 GeV, despite

IV. SUMMARY AND CONCLUSIONS
We have calculated the NLO EW and NLO QCD corrections to the process p p → e + ν e µ + µ − γ at the 13 TeV LHC. This process includes W + Zγ production with leptonic decays (W + → e + ν e and Z → µ + µ − ), and thus is sensitive to the W W Zγ, W W γγ QGCs.
We provided results for the total cross sections and kinematic distributions for a basic set of analysis cuts and studied the impact of these corrections taking into account the theoretical uncertainty due to the factorization and renormalization scale variation at NLO QCD. We found that NLO EW corrections are small (∼ +1%) at the total cross section level and negligible in view of large NLO QCD corrections. However, in kinematic distributions their impact can be much more pronounced and visible outside the QCD scale uncertainty bands.
For example, in the case of the invariant mass of the µ + µ− pair and the transverse mass of the e + ν e pair, the NLO EW corrections reach −20% ∼ −40%, respectively, in the tail regions where they overtake the NLO QCD corrections and significantly change the shapes of the NLO QCD distributions. In other distributions we studied, the NLO EW corrections are within the NLO QCD scale uncertainties, but they still partially cancel the NLO QCD corrections. A closer look at the NLO EW corrections revealed that the photon-induced contributions largely cancel the quark-induced contributions and even become dominant in some distributions or phase space regions, such as the transverse momentum of the positron, the missing transverse momentum, and the large-angle regions of the angular separation distributions. As an illustration for how missing NLO EW corrections may be mistaken as signals of BSM physics, we studied the LO effects of two dimension-8 operators in SMEFT, O M,5 or O T,1 . For their coefficients we chose values inspired by experimental constraints and observed that their impact on certain kinematic distributions can be similar to the one caused by NLO EW corrections to the SM predictions. While this is just a first look, we think it still motivates a comprehensive study of this interplay which is however outside the scope of this paper. To conclude, we hope that this study of NLO EW corrections emphasizes again the importance of including these corrections in the interpretation of EW triboson data at the LHC, especially when placing constraints on QGCs and TGCs or dimension-8 operators in SMEFT. Our MC framework, based on RECOLA and MadDipole, has been constructed sufficiently flexible, so that it can be readily adjusted to provide LHC predictions for other SM processes at NLO EW and NLO QCD accuracy up to the same level of final-state particle multiplicity.      p p → e + e − γ at √ s = 14 TeV, calculated by our in-house MC program and in Ref. [61].