NLO QCD+EW predictions for $2\ell2\nu$ diboson signatures at the LHC

We present next-to-leading order (NLO) calculations including QCD and electroweak (EW) corrections for $2\ell2\nu$ diboson signatures with two opposite-charge leptons and two neutrinos. Specifically, we study the processes $pp\to e^+ \mu^- \nu_{e}\bar\nu_{\mu}$ and $pp\to e^+ e^-\nu\bar\nu$, including all relevant off-shell diboson channels, $W^+W^-$, $ZZ$, $\gamma Z$, as well as non-resonant contributions. Photon-induced processes are computed at NLO EW, and we discuss subtle differences related to the definition and the renormalisation of the coupling $\alpha$ for processes with initial- and final-state photons. All calculations are performed within the automated Munich/Sherpa+OpenLoops frameworks, and we provide numerical predictions for the LHC at 13 TeV. The behaviour of the corrections is investigated with emphasis on the high-energy regime, where NLO EW effects can amount to tens of percent due to large Sudakov logarithms. The interplay between $WW$ and $ZZ$ contributions to the same-flavour channel, $pp\to e^+ e^-\nu\bar\nu$, is discussed in detail, and a quantitative analysis of photon-induced contributions is presented. Finally, we consider approximations that account for all sources of large logarithms, at high and low energy, by combining virtual EW corrections with a YFS soft-photon resummation or a QED parton shower.


Introduction
The production of vector-boson pairs, W + W − , W ± Z and ZZ, plays an important role in various areas of the LHC physics programme. Experimental studies of this family of processes permit to test key aspects of the Standard Model (SM) at energies that range from the EW scale up to the TeV regime. In particular, due to the high sensitivity to anomalous trilinear couplings, differential measurements at high transverse momentum allow one to test the gauge symmetry structure of EW interactions and to search for indirect effects of physics Beyond the Standard Model (BSM). Diboson final states are widely studied also in the context of direct BSM searches. Moreover, they play the role of nontrivial backgrounds in a broad range of measurements and searches. Most notably, they represent the irreducible background to Higgs-boson analyses in the H → W + W − and H → ZZ decay modes. These motivations, together with the increasing level of accuracy of experimental measurements, call for continuous improvements in the theoretical description of diboson production at the LHC.
Leptonically decaying vector-boson pairs yield clean experimental signatures with charged leptons and neutrinos. In this paper we focus on final states with two opposite-charge leptons and two neutrinos, generically denoted as 2 2ν. Their production is dominated by W + W − resonances, resulting in the highest cross sections among the various channels with dibosons decaying into charged leptons and neutrinos. The resonant structure of pp → 2 2ν depends on the lepton-flavour configuration, and we consider both the case of different and same charged-lepton flavours. In the different-flavour case, + besides revisiting the different-flavour e + µ − ν eνµ channel, for the first time we also study the sameflavour e + e − νν channel at NLO EW, including all relevant off-shell and non-resonant effects, as well as interferences and spin correlations. In the same-flavour channel, we investigate the relative importance of W + W − and ZZ resonances and of their interference. In particular, while ZZ resonances are generally subdominant, we point out that for certain distributions they can play a significant role. Third, at variance with previous studies, we treat qqand γγ-induced channels on the same footing, including NLO EW corrections throughout, and not only for the qq channel. In this respect, we note that the EW corrections to the qq channel involve qγ-induced processes that are related-via cancellations of collinear singularities-to the EW corrections to the γγ channel. Thus, the EW corrections to the γγ channel are mandatory for a fully consistent treatment of pp → 2 2ν at NLO EW. Fourth, we assess the importance of photon-induced contributions and related uncertainties based on various state-of-the-art PDFs and their comparison. Fifth, we study a convenient approximation of the EW corrections amenable to a simplified form of matching to parton showers and multi-jet merging at NLO QCD+EW [24]. Specifically, we consider IR regularised virtual EW corrections supplemented with QED radiation as described by YFS softphoton resummation or, alternatively, by a QED parton shower.
Finally, motivated by subtleties that arise from photon-induced processes at NLO EW, we present a complete O(α) analysis of the interplay between the definition of the electromagnetic coupling and the renormalisation of the photon wave function and of the γPDF in processes with external photons. In particular, we demonstrate that, in order to avoid large logarithms of the lightquark and lepton masses associated with ∆α(M 2 Z ), the coupling of initial-state photons should be defined at at the scale µ 2 F or at the EW scale, using, for instance, the G µ scheme or α(M Z ) scheme. This was first pointed out in [25], based on considerations related to the evolution of the γPDF at LO. In contrast, as is well known, for final-state photons α(0) should be used.
This paper is organised as follows: In Section 2 we introduce general features and ingredients of pp → 2 2ν, while technical aspects of the calculations are detailed in Section 3. Numerical predictions for the 13 TeV LHC are presented in Section 4, with emphasis on the behaviour of QCD and EW corrections, and our findings are summarised in Section 5. In Appendix A we document the implementation of Catani-Seymour subtraction at O(α) in Sherpa and Munich, and we discuss the issue of the definition and renormalisation of α for processes with external photons. Technical details related to the separation of single-top contamination at NLO QCD are addressed in Appendix B. Appendix C details a breakdown of the electroweak corrections presented in Section 4 by flavour channels. Finally, in Appendix D we present benchmark cross sections for pp → 2 2ν in various fiducial regions.

Categorisation of 2 2ν final states
In the Standard Model, the signature of two opposite-charged leptons and missing energy is dominantly produced through pp → W + W − /ZZ → 2 2ν, i.e. with two types of diboson resonances that decay into two leptons and two neutrinos. Such signatures can be categorized according to the flavour of the two charged leptons into a different-flavour (DF) mode and a same-flavour (SF) mode, with different implications on the underlying production mechanisms. We restrict our discussion to final states with electrons and muons, and we focus on pp → 2 2ν processes with DF and SF final states corresponding, respectively, to e + µ − + E T and e + e − + E T . Note that such processes Figure 1. Sample of Born diagrams contributing to 2 2ν production in the different-flavour case ( = ) and in the same-flavour case ( = ). Both double-resonant (a,b) and single-resonant (c) diagrams are shown. are invariant with respect to e ↔ µ interchange. More precisely, taking into account appropriate momentum mappings, we have dσ(pp → µ + µ − ν e/τνe/τ ) = dσ(pp → e + e − ν µ/τνµ/τ ) . (2.1) In our calculation we do not apply any resonance approximation, but include the full set of Feynman diagrams that contribute to pp → 2 2ν at each perturbative order, thereby including all sub-dominant contributions with single-and non-resonant diagrams besides the dominant doubleresonant ones. All off-shell effects, interferences and spin correlations are consistently taken into account, treating resonances in the complex-mass scheme [34] throughout.
At LO, the DF process pp → e + µ − ν eνµ , is dominated by resonant W + W − production in the qq channel and subsequent decays. The full set of Feynman diagrams contributing to pp → e + µ − ν eνµ will be referred to as DF WW channel. Representative tree-level diagrams both for double-resonant and sub-leading contributions are shown in Fig. 1.
The situation in the SF case is more involved since its signature can be produced by different partonic processes, pp → e + e − ν µ/τνµ/τ and pp → e + e − ν eνe . Their final states are indistinguishable on an event-wise level, as the produced neutrinos can only be detected as missing transverse energy and their flavours cannot be resolved. Consequently, predictions for e + e − + E T production originate as the incoherent sum over all three possible neutrino-flavour contributions.
The SF process pp → e + e − ν µ/τνµ/τ is dominated by resonant ZZ production in qq annihilation and subsequent Z → e + e − and Z → νν decays. Such double-resonant contributions are accompanied by all allowed topologies with sub-leading resonance structures, including diagrams with γ * → e + e − subtopologies, as well as other single-and non-resonant topologies. The full set of Feynman diagrams contributing to pp → e + e − ν µ/τνµ/τ will be referred to as SF ZZ channel. Sample tree-level diagrams are depicted in Fig. 2.
Finally, the SF process pp → e + e − ν eνe proceeds both via W + W − and ZZ diboson resonances. The corresponding amplitudes are built by coherently summing over all diagrams entering the Figure 3. Sample of photon-induced Born diagrams contributing to 2 2ν production in the differentflavour case ( = ) and in the same-flavour case ( = ). Double-resonant (a,b), single-resonant (c) and non-resonant (d) diagrams are shown. two previously discussed DF WW and SF ZZ channels. Consequently, this channel is referred to as SF WW/ZZ channel, and all diagrams shown in Figs. 1-2 are representatives of the tree-level diagrams contributing here. Due to the fact that the phase-space regions with resonant intermediate W + W − and ZZ states are typically distinct, the assumption is justified that the SF WW/ZZ cross section is dominated by the incoherent sum of double-resonant contributions of one and the other type, while the effect of quantum interferences is small. It is, however, not obvious if this assumption still holds in phasespace regions away from such double-resonant topologies. Interference effects are studied in detail in Section 4.2 by comparing exact predictions in the SF WW/ZZ channel against the incoherent sum of the W + W − and ZZ channels.

Photon-induced production
Besides the dominant qq production mode, 2 2ν final states can also be produced in photonphoton scattering. As we do not count the photon PDF as an O(α) suppressed quantity, such γγ → 2 2ν processes contribute already at the LO, i.e. at O(α 4 ). Their quantitative relevance varies significantly between the channels. Photon-induced contributions to the DF channel are dominated by γγ → W + W − → e + µ − ν eνµ topologies, which are accompanied by single-resonant topologies involving t-channel lepton-pair production with an emission of a W boson off one of the produced leptons, and non-resonant diagrams with multiperipheral topologies. Sample tree diagrams for the described DF topologies are collected in Fig. 3. Due to a t-channel pole, regulated by the W mass, the contribution of the double-resonant diagram depicted in Fig. 3(a) is enhanced for large invariant masses of the intermediate W + W − pair [9,10]. In fact, for on-shell W + W − pair production the contribution of the γγ channel was found to increase beyond 10% of the LO qq annihilation mode for m W W > 800 GeV [9]. In this paper we investigate the significance of the γinduced production mode using state-of-the-art PDFs and taking into account NLO EW corrections, as well as realistic selection cuts on the 2 2ν final state.
The DF channel γγ → e + e − ν µ/τνµ/τ does not involve any double-resonant topology due the lack of triple and quartic gauge couplings among neutral EW bosons. Similarly, non-resonant multiperipheral topologies do not exist due to lepton-flavour conservation. Thus, lepton-pair production Figure 5. Sample of one loop diagrams contributing to 2 2ν production in the different-flavour case ( = ) and in the same-flavour case ( = ) in the quark-induced (a-d) and photon-induced (e-h) channels. in t-channel topologies with subsequent emission of a Z boson with Z → νν is the only photoninduced production mechanism at LO, as shown in the sample diagrams of Fig. 4. Consequently, the invariant mass of the charged-lepton pair does not show a Breit-Wigner peak around M Z . Similarly as for quark-antiquark annihilation, the γγ → e + e − ν eνe channel is build from the coherent sum of all diagrams entering γγ → e + µ − ν eνµ and γγ → e + e − ν µ/τνµ/τ .

Ingredients of QCD and EW corrections
At NLO QCD all O(α s α 4 ) contributions to pp → 2 2ν are taken into account. In the qq channel, the only QCD loop corrections arise from virtual-gluon exchange, while the real corrections result from real-gluon emission and crossed topologies describing (anti-)quark-gluon channels. The infrared divergences separately arising in these two contributions are mediated by the standard dipolesubtraction approach [35,36]. We note that the γγ channels do not receive QCD corrections at NLO, due to the absence of any QCD partons in all tree-level diagrams.
At NLO EW we include the full set of O(α 5 ) contributions to pp → 2 2ν. At this order both the qq and γγ channels receive corrections from virtual EW bosons and from closed fermion loops, cf. Figs. 5-6. These corrections include Higgs resonances with decay into four fermions coupled to weak bosons (in the qq channel) or coupled to a heavy-fermion loop (in the γγ channel). The real corrections in the qq channel can be split into real-photon emission channels and γq → 2 2νq channels 1 with initial-state γ → qq splittings. The γγ channel also receives real corrections from Figure 7. Sample of real emission diagrams contributing to 2 2ν production in the different-flavour case ( = ) and in the same-flavour case ( = ), in the quark-antiquark channel (a), the (anti-)quark-photon channel (b,c) and the photon-photon channel (d). photon bremsstrahlung, and also from γq → 2 2νq channels with initial-state q → qγ splittings, cf. Figs. 7-8. While the separation into qq and γγ channels can still be preserved for virtual and photon-bremsstrahlung contributions, such separation is no longer meaningful for the qγ-initiated channels due to their singularity structure: both above-mentioned splittings result in infrareddivergent configurations, and these qγ channels simultaneously cancel infrared poles arising in qq and γγ channels. This situation demands the inclusion of the full NLO EW corrections to the qq and γγ Born processes to guarantee infrared safety and consistency. To deal with the mediation of these divergences between virtual and real corrections the QED extension of the dipole-subtraction method [37][38][39] is applied (see Appendix A). Instead of a separation of NLO contributions into qq and γγ channels, we quantify the impact of photon-induced processes by considering contributions involving at least one photon PDF factor and all other contributions that are also present under the assumption of vanishing photon PDFs. At LO this distinction coincides with the splitting according to production modes, while at NLO EW it combines γγ and γq channels in spite of the fact that the latter involves qq-related contributions.

Tools
The calculations presented in this paper have been performed with the automated frameworks Mu-nich+OpenLoops and Sherpa+OpenLoops. They automate the full chain of all operationsfrom process definition to collider observables-that enter NLO QCD+EW simulations at parton level. The recently achieved automation of EW corrections [24,26] is based on the well established QCD implementations and allows for NLO QCD+EW simulations for a vast range of SM processes, up to high particle multiplicities, at current and future colliders.
In these frameworks virtual amplitudes are provided by the OpenLoops program [28], which is based on the open-loops algorithm [27] -a fast numerical recursion for the evaluation of one-loop scattering amplitudes. Combined with the Collier tensor reduction library [40], which imple-ments the Denner-Dittmaier reduction techniques [41,42] and the scalar integrals of [43], or with CutTools [44], which implements the OPP method [45], together with the OneLOop library [46], the employed recursion permits to achieve very high CPU performance and a high degree of numerical stability. We validated phase-space point wise the contributing tree amplitudes between Sherpa and OpenLoops, and the one-loop amplitudes between OpenLoops and an in-house algebraic amplitude generator and also against Recola [47]. All remaining tasks, i.e. the bookkeeping of partonic subprocesses, phase-space integration, and the subtraction of QCD and QED bremsstrahlung, are supported by the two independent and fully automated Monte Carlo generators, Munich [29] and Sherpa [30][31][32][33]. These two tools have been validated extensively against each other. As a further validation of the Monte Carlo integration employed for the results presented here, we want to note the perfect agreement between Sherpa and the results of [16] for the related process pp → 4 presented in [48].

YFS soft-photon resummation and QED parton shower
As discussed in Section 1, the NLO EW corrections to pp → 2 2ν are dominated by EW Sudakov logarithms of virtual origin and QED logarithms stemming from photon radiation off leptons. In [24] it was shown that, for observables that are sufficiently inclusive with respect to photon radiation, full NLO EW results can be reproduced with good accuracy by an approximation consisting only of virtual EW corrections upon appropriate subtraction of IR singularities. This approximation, which was dubbed EW VI , is defined through Therein, dσ LO is the leading order differential cross section, while dσ V EW and dσ I EW are the NLO EW virtual correction and the endpoint part of the integrated Catani-Seymour subtraction terms, ensuring a finite result by construction. In practice, a logarithmic approximation over the real photon emission phase space is added to the virtual corrections. This approach captures all Sudakov effects at NLO EW [49] and is very suitable for a combination of QCD and EW higher-order effects through a simplified multi-jet merging approach at NLO QCD+EW [24]. As a further possible step towards a fully consistent implementation of matching and merging at NLO QCD+EW, in this paper we investigate the possibility of supplementing the EW VI approximation with QED radiation effects by means of naive matching to QED parton showers or QED resummation. Specifically, we consider a soft-photon resummation in the Yennie-Frautschi-Suura (YFS) scheme [50] and, alternatively, the Csshower QED parton shower [51,52] based on Catani-Seymour splitting kernels. Combined with the EW VI approximation and a differentially applied NLO QCD K-factor, the NLO QCD×EW VI ⊗ YFS and NLO QCD×EW VI ⊗ CSS approximations are defined. As in the fixed-order calculation, both qq and γγ channels are taken into account on the same footing.
The original YFS scheme resums real and virtual soft-photon corrections to arbitrary scattering processes. The implementation in Sherpa [53] is specialised to correct decays of massive resonances, and for both cases relevant in this paper, i.e. for W and Z resonances, the resummation of soft logarithms is matched to exact O(α) corrections. 2 Its accuracy in charged-and neutral-current Drell-Yan processes has been validated in [54,55]. As neither photon emissions off the initial states nor γ → ff splittings are included, it is worth noting that no γq channels, occurring in the real emission correction of the full calculation, are accounted for.
The YFS implementation in Sherpa includes a generic resonance identification, ensuring that collective multipole radiation off the charged lepton ensemble preserves all resonance structures present in the event. To this end, first the final state of a scattering process is analysed, and possible resonances decaying into leptons and neutrinos are identified on the basis of event kinematics and existing vertices in the model. For the process studied in this paper, pp → 2 2ν, multiple resonance structures are possible. They are disentangled on the basis of the distance measures 3 ∆ Z = |m − M Z |/Γ Z and ∆ W = |m ν − M W |/Γ W . In 2 2ν production this leads to three distinct cases: (a) two pairs of leptons are identified to come from a specific resonance; (b) one pair of leptons is identified to come from a specific resonance, the other is classified as non-resonant; (c) all leptons are classified as non-resonant. Subsequently, identified resonant-production subprocesses are separated from the rest of the event, and the emerging decay is dressed with photon radiation respecting the Breit-Wigner distribution of the resonance, i.e. preserving the original virtuality of the off-shell lepton/neutrino system. Finally, all left-over non-resonantly produced leptons are grouped in a fictitious X → n + mν process, with suitably adjusted charges and masses for X. In this case, resummed real and virtual radiative QED corrections are applied in the soft limit only, including however hard collinear real-emission corrections through suitably subtracted Catani-Seymour dipole splitting functions [53].
In the Csshower, the construction of the emitting dipoles follows the subtraction terms used in the fixed-order calculation. Owing to the unitary nature of all parton showers, dipoles whose splitting functions are negative, i.e. all dipoles formed by partons with like-sign electric charges, are inactive and do not contribute. 4 In the QCD case this corresponds to the leading-colour limit, and keeping C F and C A at their N C = 3 values guarantees a full-colour treatment of the collinear limit, while the soft-limit remains at N C → ∞. No such limit is meaningful in QED. Consequently, the absence of the like-signed dipoles has a degrading impact both on the description of the collinear and the soft limit. Moreover, the Csshower has no knowledge of the internal resonance structure of the Born process. Thus, dipoles of charged particles spanning across one or multiple resonances will inevitably distort their line shape through their recoil assignments. 5 At the same time, however, all processes including photon radiation off the initial state quarks and γ → ff splittings are present. Thus, every channel occurring in the fixed-order calculation is described in its respective soft-collinear limits.

Input parameters, scale choices and variations
The input parameters for the NLO QCD+EW calculations of pp → 2 2ν presented in Section 4 are summarised in Table 1. All unstable particles are treated in the complex-mass scheme [34], where width effects are absorbed into the complex-valued renormalised squared masses As top-quark and Higgs-boson contributions enter only at loop level, the dependence of our results on Γ t and Γ H is completely negligible. The CKM matrix is assumed to be diagonal. In fact, due to the negligible mixing of the first two and the third quark generations and because all quarks of the first two quark generations are taken to be massless, the unitarity of the CKM matrix ensures the independence of all physical results from the values of its matrix elements. The EW couplings are derived from the gauge-boson masses and the Fermi constant using  Table 1. Numerical values of all input parameters. The gauge boson masses are taken from [59], while their widths are obtained from state-of the art calculations. The Higgs mass and width are taken from [60]. The top quark mass is taken from [59] while its width has been calculated at NLO QCD. The electron and muon masses as well as the electromagnetic coupling in the Thomson limit, α(0), are only relevant for calculations involving YFS soft-photon resummation and the Csshower.
where the W -boson mass and the squared sine of the mixing angle, are complex-valued. The G µ -scheme guarantees an optimal description of pure SU(2) interactions at the EW scale. It is used for all channels, including photon-induced ones. In this respect, while it is well known that the coupling of final-state photons should be parametrised in terms of α(0), in Appendix A analysing the interplay between the counterterms associated with the renormalisation of α, the photon wave function, and the γPDF, we demonstrate that the coupling of initial-state photons cannot be parametrised in terms of α(0). Instead a high-energy definition of α, for example in the α(M Z )or the G µ -scheme, for the coupling of initial-state photons should be employed. In all fixed-order results the renormalisation scale µ R and factorisation scale µ F are set to Therein, H lep T is the scalar sum of the transverse momenta of all charged final-state leptons plus the missing transverse momentum, In order to guarantee infrared safety at NLO EW, the scale of (3.6) must be insensitive to collinear photon emissions off charged leptons. To this end, any charged leptons are dressed with collinear photons with ∆R γ < 0.1. Our default scale choice corresponds to ξ R = ξ F = 1, and theoretical uncertainties are assessed by applying the scale variations (ξ R , ξ F ) = (2, 2), (2, 1), . For all considered processes at the inclusive level the difference with respect to a fixed scale choice µ 0 = M W is below 2% at NLO QCD, while inclusive NLO EW corrections agree at the level of one permille.

PDFs
For the calculation of hadron-level cross sections we employ the CT14qed parton distributions [22], which include NLO QCD and LO QED effects, 6 with the corresponding α S (M Z ) = 0.118. The NLO PDF set is used for LO computations as well as for NLO QCD and NLO EW predictions. In order to assess the potentially large uncertainties stemming from photon-induced processes, two alternative sets based on different determinations of the photon PDF are considered, namely the recently calculated LUXqed PDFs [23] and the data driven fit of NNPDF3.0qed [21,62]. Specifically, we replace the photon PDF of the default set by the alternative parametrisations, while using CT14qed quark and gluon PDFs throughout. This is justified by the negligible dependence of the quark and gluon densities on the γPDF.
The three considered sets implement different treatments of the photon PDF. The CT14qed PDFs assume as initial condition for the γPDF at Q 0 = 1.295 GeV an inelastic contribution that results from the convolution of primordial quark distributions with QED splitting functions. This ansatz involves a free normalisation parameter, which is traded for the inelastic photon momentum fraction, , and fitted to DIS data with isolated photons. For our default predictions we use a CT14qed set corresponding to the best fit value, p γ 0 = 0.05%. The inelastic component, which describes processes where the proton breaks, is complemented by an elastic component, corresponding to the case where the proton remains intact. The latter is determined at the scale Q 0 using the equivalent photon approximation (EPA) [63]. The sum of inelastic and elastic contributions at Q 0 is evolved as a single photon density 7 through coupled DGLAP equations for photons, quarks and gluons at NLO QCD + LO QED.
In the LUXqed approach, the usual description of ep → e + X data, where a virtual photon radiated from the electron beam probes quarks inside the proton via γ * q scattering, is related to an alternative interpretation, where the lepton beam probes the photon content of the proton via γ scattering. In this way, the photon density can be derived from proton structure functions in a model-independent way, and building on available global fits of QCD PDFs, parametrisations of ep data at low Q 2 , and elastic contributions, one arrives at an accurate determination of the γPDF. Then, starting at Q 0 = 10 GeV, the photon density is evolved with all other QCD partons through DGLAP equations including QED corrections up to O(α S α).
The NNPDF3.0qed photon PDF is based on a much more general multiparameter neuralnetwork parametrisation, which can naturally account for both the elastic and inelastic components. Thus the NNPDF3.0qed photon density is much more receptive to the poor sensitivity of current data to photon-induced processes. This leads to much larger admissible photon densities combined with much bigger uncertainties as compared to the other PDF sets. The resulting photon density is evolved at NLO QCD + LO QED.
In order to avoid undesired contaminations from single-top contributions of type pp → W t → W W b → 2 2ν in the NLO QCD and NLO EW corrections, in our calculations we apply a full veto against final-state b-quarks. Since such a veto would jeopardize IR cancellations for m b = 0, we consider the b-quark to be massive, i.e. we assume the presence of only four light flavours. In order to reconcile this choice with the fact that the employed PDFs involve five active flavours, an appropriate PDF-scheme conversion [66] is applied. As discussed in Appendix B, this transformation is almost trivial for the process at hand. At LO, pp → 2 2ν comprises neither gluon channels nor α S terms. Thus, only the γγ channel requires a correction related to the scheme dependence of the γPDF. Taking this into account, we can safely perform our calculations using five-flavour PDFs, 7 Note that, in contrast to "inelastic photons", which are inherently off-shell, "elastic" photons as obtained form the EPA at Q 2 0 are exactly on-shell, even when they enter hard-scattering processes at Q 2 m 2 p . Nevertheless, also elastic photons can undergo γ → qq splittings at arbitrary Q 2 . Thus, elastic and inelastic photons contribute to the PDF evolution towards high Q 2 on the same footing. In practice, the photon PDF at high-Q 2 receives contributions form the elastic and inelastic γPDF at Q 2 0 , both decreased due to γ → qq splittings, and positive contributions from (anti)quark distributions via q → γq splittings. It turns out that, due to the much larger quark density, the latter contributions dominate by far. Thus, the details of the evolution of the elastic and inelastic γPDFs play only a marginal role [64,65].
omitting initial-and final-state b-quarks, and using m b > 0 in the loops. Up to terms beyond NLO QCD+EW, this approach is perfectly consistent with a conventional calculation in the 4F scheme.

Results
In this section we present numerical predictions for the DF and SF processes, pp → e + µ − ν eνµ and pp → e + e − νν, at √ s = 13 TeV. The impact of NLO corrections is illustrated by comparing against LO predictions, which include qq and γγ-induced processes at O(α 4 ). For the combination of QCD and EW higher-order effects we consider both an additive and a multiplicative approach, defined, respectively, as and Therein, the relative QCD and EW corrections are defined as where the dσ (i,j) are the cross section contributions of O(α i S α j ), thus dσ (0,4) ≡ dσ LO . In order to illustrate the interplay of the various partonic channels in the multiplicative QCD×EW combination, we write each dσ (i,j) as a sum over contributions dσ ab (i,j) where a and b are the proton constituents initiating the subprocess at the given order. At LO, for the decomposition into qq and γγ channels and their relative weights we write .

(4.5)
At NLO, the QCD correction factor in (4.3) corresponds to where the relative correction is restricted to the qq channel. Finally, for the EW correction in (4.3) we have Here, since the newly emerging γq and γq channels act as real emission corrections to both the LO qq and γγ channels, it is not possible to unambiguously split the full EW correction into two parts associated with the qq and γγ channels. 8 Therefore, our definition of δ EW amounts to choosing not to assign arbitrary fractions of the γqand γq-corrections to act as corrections to the LO qq and γγ channels, but to define an overall NLO EW correction factor. With the above definitions the multiplicative combination (4.2) can be cast in the form where the relative weight of QCD corrections in the different partonic channels is manifestly respected. In particular, the γγ channel remains free of QCD correction effects, consistent with its behaviour at NLO QCD. Alternatively, as is often done in the literature, one may choose to regard the combination of the γqand γq-induced NLO EW effects as a correction to the qq channel, and to attribute the remnant NLO EW corrections to the γγ chanel. With this ad hoc splitting, it is natural to adopt a channel-by-channel factorisation of EW and QCD corrections, While one may debate if (4.2) is more or less motivated than (4.11), we observe that, using the difference between the two prescriptions can be cast in the form This indicates that the two prescriptions tend to coincide if either one LO channel dominates, the QCD correction is small, or both channels' EW corrections are of the same size. In large regions of the phase space these conditions are simultaneously satisfied, and for all observables studied in the following the scheme dependence (4.13) is found to be smaller that 5 % , in most cases even below 0.5 % , of the LO cross section. Both the size of the EW corrections contributed by the three individual channels and the above scheme dependence are detailed in Appendix C.
In the multiplicative approach, which we deem our best prediction, the uncertainties are estimated by scaling the NLO QCD predictions with the relative NLO EW correction, evaluated at the central scale. This is justified by the fact that δ EW (µ R , µ F ) is independent of µ R and involves only a very weak µ F dependence of O(α), while the LO QCD µ F -dependence cancels out in the ratio. As discussed in Section 2.1, we include photon-induced contributions throughout, including γγ → 2 2ν, γγ → 2 2νγ and γq → 2 2νq channels at NLO EW. To assess the uncertainty arising from the choice of photon PDF we vary their parametrisation from their default (CT14qed) to that of LUXqed and NNPDF3.0qed, while keeping the quark and gluon PDFs fixed, cf. Section 3.4. The overall impact of photon-induced processes is illustrated by switching off the photon PDF, both at LO and NLO EW.
Additionally, as discussed in Section 3.2, we investigate to which degree exact NLO QCD×EW results can be reproduced by approximations based on the combination of IR-subtracted virtual Table 2. Inclusive selection cuts for off-shell vector-boson pair production in the 2 2ν channel. The missing transverse momentum ET is calculated from the vector sum of neutrino momenta.
EW corrections (EW VI ) with YFS QED resummation or, alternatively, with the Csshower. Such approximation, denoted as NLO QCD×EW VI ⊗ YFS and NLO QCD×EW VI ⊗ CSS, can be realised in realistic particle-level simulations using currently public tools, and can be regarded as a first step towards NLO QCD+EW matching and merging.
In the following, we study various fiducial cross sections and differential distributions. Physical observables involving charged leptons are known to be highly sensitive to QED radiative corrections. This should be avoided by using dressed leptons. To this end we recombine all leptons with nearly collinear photons that lie within a cone This dressing procedure captures the bulk of the collinear final-state radiation, while keeping contamination from large-angle photon radiation at a negligible level. In our analysis we apply a set of acceptance cuts, as listed in Table 2, on the transverse momentum, pseudo-rapidity and angular separation of the dressed charged leptons and on the missing transverse momentum calculated based on the neutrino momenta, E T = p T,νν .
Inclusive vector-boson pair production receives large NLO QCD corrections in kinematic regions where one of the vector bosons might become soft. This effect is a variant of the well known 'giant K-factors' [67]. In order to suppress these large QCD corrections that spoil the perturbative convergence we veto events with where H jet T = i∈jets p T,i based on anti-k T jets with R = 0.4 and p T > 30 GeV. In practice, H jet T = p T,j at NLO QCD. A reliable inclusive prediction without such a jet veto requires the merging of pp → 2 2ν + 0, 1 jets at NLO QCD+EW, but goes beyond the scope of the present paper. The complete analysis has been implemented in Rivet [68]. For reference, we present the cross sections of the inclusive as well as three more exclusive event selections for both channels in Appendix D.
Differential distributions for pp → e + µ − ν eνµ are presented in Figs. 9-15. In every figure, the left plot shows absolute predictions as well as relative NLO corrections with scale-variation bands. The upper-right ratio plot quantifies the importance of photon-induced contributions as well as the effect of using different γPDFs, while the lower-right ratio plot compares exact NLO results against the NLO QCD×EW VI ⊗ YFS and NLO QCD×EW VI ⊗ CSS approximations. For reference, we also show the pure fixed-order NLO QCD×EW VI approximation, which includes only the IR-subtracted part of virtual EW corrections and lacks any differential description of QED real corrections.
In Figs. 9-12 we present distributions in the transverse momenta of the leading and subleading leptons, p T, 1 and p T, 2 , the total missing transverse momentum, E T , and the invariant mass of  the e + µ − pair, m . The leading and subleading lepton are defined by their ordering in transverse momentum, irrespective of their charge. EW corrections to these observables feature the typical Sudakov behaviour, with small effects below 100 GeV and large negative corrections at the TeV scale. In the tails of the lepton-p T and m distributions NLO EW corrections can reach and even largely exceed −50%. The dominant effects originate from qq → W + W − topologies with resonant W bosons, and the strong enhancement of EW Sudakov corrections is induced by the high p T and the large SU(2) charges of the W bosons. In the presence of EW corrections of several tens of percent, fixed-order NLO predictions should be supplemented by a resummation of Sudakov logarithms. As a rough indication of the possible magnitude of higher-order EW effects, we observe that naïve exponentiation can turn NLO EW corrections of −50-80% into an overall all-order EW correction of −40-55%. We also note that EW corrections of this magnitude appear in a kinematic range that cannot be probed with decent statistics at the LHC. Nevertheless, such phase-space regions would play an important role at a 100 TeV pp collider [69].
Due to the presence of the jet veto (4.16), the impact of QCD corrections in Figs. 9-12 is rather mild at energies below M W , and grows only up to +10-40% in the tails. While the actual size of QCD K-factors depends on the scale choice, we recall that, in general, QCD corrections to pp → e + µ − ν eνµ receive sizeable real-emission contributions in the absence of jet vetoes [2]. Scale uncertainties at NLO QCD are rather constant and somewhat below 10%.
Due to their opposite sign, QCD and EW corrections cancel against each other to a certain extent. At the same time, in regions where both QCD and EW corrections are well beyond 10%,  Figure 10. Distribution in the transverse momentum of the subleading lepton, p T, 2 , for pp → e + µ − νeνµ at 13 TeV. Details as in Fig. 9.
contributions of relative O(α S α) become relevant. Such NNLO QCD×EW effects are estimated in our predictions by means of the multiplicative combination of NLO corrections (4.2), which is well justified if EW corrections are dominated by Sudakov logarithms and QCD radiation is softer than the characteristic scale of the qq → 2 2ν EW subprocess. Comparing the additive and multiplicative combination of QCD and EW corrections in Figs. 9-12, we find that contributions of relative O(α S α) can exceed 10% in the tails. Among the virtues of a multiplicative combination of QCD and EW corrections, it is worth pointing out that NLO EW corrections are implicitly supplemented by QCD radiation, resulting, for instance, in a reasonable behaviour with respect to possible jet vetoes. At the same time, it should be stressed that, for a more reliable assessment of O(αα S ) corrections, an approach like NLO QCD+EW merging [24] is certainly preferable.
The behaviour of the E T distribution (Fig. 11) deserves a few additional comments. First, in the tail of this distribution we observe that Sudakov EW effects are less pronounced than in other observables. This is due to the fact that requiring a high-p T ν eνµ pair forces the W bosons into the off-shell regime. As a result, Sudakov logarithms arise only from EW interactions between the on-shell final-state leptons and, like in Drell-Yan processes, they turn out to be less enhanced than in pp → W + W − . Second, the QCD K-factor features a sizeable enhancement characterised by a rather sharp threshold at E T ∼ M W . This is related to the fact that, in pp → W + W − at LO, p T,W + W − = 0 strongly disfavours the production of a ν eνµ pair with p T > M W . Therefore, the W + W − transverse momentum induced by NLO QCD radiation results in a sizeable enhancement in the E T > M W region (see [2]).
Photon-induced contributions in Figs. 9-12 can reach up to 5-20%, depending on the observable. The largest effects are typically observed in the TeV tails. The γ-induced contributions to the E T distribution, however, approach 10% already at 200 GeV, an effect that can already be observed at LO. This is due to the presence of non-resonant diagrams that are absent in the qq channels. They can populate this phase-space region, which is disfavoured as soon as the neutrinos need to  Figure 11. Distribution of the missing transverse momentum, ET, for pp → e + µ − νeνµ at 13 TeV. Details as in Fig. 9.
be produced through an s-channel W propagator. This effect is further increased by real-emission channels of type γq → 2 2νq, which are strongly enhanced at E T ≥ M W , similarly as for QCD radiative effects.
Comparing different photon PDFs, for all observables we find a fairly good agreement between CT14qed and LUXqed γPDFs, with differences that never exceed the level of five percent. Conversely, the usage of the NNPDF3.0qed γPDF yields similar inclusive cross sections as CT14qed and LUXqed, but much bigger γ-induced contributions in the tails. Nevertheless the differences are consistent with the large uncertainty of the photon density in NNPDF3.0qed, while using the other PDF sets leads to a γPDF uncertainty well below the overall QCD scale uncertainty. The largest γ-induced effects are observed in the tail of the m distribution, where the dominant contribution originates from γγ → W + W − topologies with t-channel poles in the forward/backward regions. We note that relaxing rapidity cuts on charged leptons, which act as a cut-off on t-channel poles, would further enhance γγ → W + W − contributions.
Comparing the NLO QCD×EW VI ⊗YFS and NLO QCD×EW VI ⊗CSS approximations against exact NLO QCD×EW results, in Figs. 9-12 we observe agreement at the few-percent level for p T, 1 and m , while in the tails of the p T, 2 and E T distributions the error of the NLO QCD×EW VI ⊗YFS approximation can exceed 10%. This can be attributed to the fact that the YFS resummation as implemented in Sherpa does not account for initial-state QED radiation in the qq channels and neglects the γq channels. The Csshower, on the other hand, describes these configurations, but lacks accuracy due to its dipole structure. In any case, both approximations improve the pure fixed-order approximation of NLO QCD×EW VI .
Figs. 13-14 illustrate distributions in the W -boson mass, m ν , and in the W + W − invariant mass, m νν . Such observables are not experimentally accessible, but they provide valuable insights into the resonance structure of pp → e + µ − ν eνµ and into the behaviour of EW corrections. Focussing on the m ν and m νν regions of the W → ν and Z → 2 2ν peaks and the W + W − → νν  Figure 12. Distribution in the invariant mass of the e + µ − pair, m , for pp → e + µ − νeνµ at 13 TeV. Details as in Fig. 9.
threshold, we observe that QCD corrections are almost insensitive to the presence of EW resonances and thresholds. Photon-induced contributions are on the level of 1-3%, while EW corrections feature sizeable shape distortions due to γ bremsstrahlung off the charged leptons. Such shape corrections can be understood as a net migration of events from the peak and threshold regions towards the lowmass tails and, in the case of the m νν distribution, towards the local minimum above the Z → νν peak. In these observables, apart from the region of very high m νν , the NLO QCD×EW VI ⊗ YFS approximation is found to reproduce exact results with fairly good accuracy. In particular, in the off-shell regime, i.e. for m ν < M W or m νν < 2M W , the offset between NLO QCD×EW VI approximation and exact results indicates the presence of QED radiation effects beyond 10%, which turn out to be well described by the YFS approach. The remaining differences are below 5% or so. They can be attributed to higher-order corrections, missing in the fixed-order calculations, and to ambiguities related to the YFS resummation for highly off-shell decays. In contrast, QED radiative corrections to m ν and m νν are strongly overestimated in the NLO QCD×EW VI ⊗ CSS approach. This is most likely due to the fact that the Csshower is unaware of resonance structures.
Fig. 14 also displays the multi-TeV region of the m νν distribution, where large negative EW Sudakov corrections are observed, as well as γ-induced contributions beyond 10%, with large deviations between the different γPDF sets. At the same time the NLO QCD×EW VI ⊗ YFS and NLO QCD×EW VI ⊗ CSS predictions grow gradually worse when compared with the exact NLO QCD×EW calculation due, respectively, to the missing or limited accuracy in the description of γ → qq-splittings in the initial state.
Finally, in Fig. 15 we show the distribution in the azimuthal separation of the e + µ − pair, ∆φ . For this observable, EW corrections and γ-induced effects are almost flat and similarly small as for the integrated cross section. As for QCD corrections, we observe a pronounced kinematic dependence for ∆φ → π. This can be understood as a statistical effect related to the migration of events form highly populated to poorly populated bins.  Figure 13. Distribution in the invariant mass of the matching lepton-neutrino pair, m ν , for pp → e + µ − νeνµ at 13 TeV. Details as in Fig. 9.  Figure 14. Distribution in the invariant mass of all four final state leptons and neutrinos, m νν , for pp → e + µ − νeνµ at 13 TeV. Details as in Fig. 9.   and  where dσ int stands for the interference between topologies of W W and ZZ type. As we will see, the splitting of the DF cross section into a W W and a ZZ channel (and the interference of the two) is very instructive in order to understand the shapes and higher-order corrections of certain kinematic distributions, which are affected to a different extent by these two dominant contributions in different regions of phase space. A selection of differential distributions is presented in Figs. [16][17][18][19][20][21][22]. Similarly as in Section 4.1, in every figure we illustrate NLO QCD and EW predictions with corresponding K-factors (left plot) as well as γ-induced effects (upper-right plot). Since the NLO QCD×EW VI ⊗ YFS and NLO QCD×EW VI ⊗ CSS approximations behave similarly as for the different-flavour process, we do not show corresponding plots. 9 Instead, in the lower-right panels we quantify the relative importance of the W W and ZZ contributions defined in (4.18)-(4.19), as well as their incoherent sum,  Figure 17. Distribution in the transverse momentum of the subleading lepton, p T, 2 , for pp → e + e − νν at 13 TeV. Details as in Fig. 16.
The most striking evidence emerging from Figs. [16][17][18][19] is that the incoherent sum (4.22) provides an excellent approximation of the full e + e − νν cross section at NLO QCD×EW level. In fact, in all considered observables, apart from the far off-shell tail of the four-lepton invariant mass distribution m νν shown in Fig. 22, interference effects are so suppressed that they cannot be resolved at all with the available Monte Carlo statistics.
The integrated e + e − νν cross section, the distributions in p T, 1 (Fig. 16), p T, 2 (Fig. 17), m (Fig. 19), and ∆φ (Fig. 22) are dominated by W W resonances in the majority of the plotted range. In those regions it is not surprising to observe that QCD and EW corrections behave very similarly as in the different-flavour case discussed in Section 4.1. Vice versa, in the presence of sizeable ZZ contributions, radiative corrections can behave in a different manner as compared to the differentflavour case. For example, this is observed in the tail of the p T, 2 distribution beyond 1 TeV. There, ZZ resonance contributions become as important as W W ones, resulting in a reduction of the magnitude and a change in shape of the EW corrections. Similarly, the size of the contributions from γ-induced processes is reduced as compared to pp → e + µ − ν eνµ (Fig. 10).
In the E T distribution (Fig. 18) we observe a more intriguing interplay between W W and ZZ resonances. While W W topologies represent the main contribution at low and very high E T , the region between 100 GeV and 1 TeV is dominated by ZZ resonances. This is related to the fact-already observed in the different-flavour case-that the production of a νν system via W W resonances is strongly suppressed for E T > M W . In the pp → e + e − νν channel, this suppression manifests itself through the enhancement of ZZ contributions, where large E T can directly arise through a boosted Z boson decaying to νν. In contrast, due to the absence of ZZ resonances, in the e + µ − ν eνµ channel the suppression of W W resonances leads to the enhancement of radiative effects at NLO QCD and NLO EW. Vice versa, due to the opening of ZZ resonances, in the e + e − νν channel we observe smaller NLO QCD and photon-induced contributions and larger negative NLO EW corrections.
dσ/dσ NLO QCD×EW Figure 18. Distribution in the missing transverse momentum, ET, for pp → e + e − νν at 13 TeV. Details as in Fig. 16.  Figure 19. Distribution in the e + e − pair, m , for pp → e + e − νν at 13 TeV. Details as in Fig. 16.
The invariant mass of the e + e − pair (Fig. 19) represents a powerful discriminant between W W and ZZ channels. On the one hand, most of the spectrum is driven by W W contributions and behaves very similarly as for the corresponding different-flavour observable shown in Fig. 12. On the other hand, in the vicinity of m ≈ M Z , the ZZ channel gives rise to a sharp Z → e + e − peak well above the W W continuum. In this region photon radiation off the charged leptons induces  significant distortions of the Z line shape that are obviously not present in the DF case. Such shape corrections are qualitatively similar to those in Fig. 13 for the m ν spectrum. However, since Z → e + e − decays involve two charged leptons, we find an even more significant reduction of the peak cross section. Moreover, due to the presence of a large W W background, the positive NLO EW K-factor below the peak turns out to be much less pronounced than in m ν . Similarly, the experimentally unobservable m ν distribution shown in Fig. 20 for the SF case, while dominated by W W resonant channels near the W resonance, receives large contributions from ZZ channels on either side. Consequently, the large NLO EW corrections below the W peak in the W W channel, dominated by real photon radiation, cf. Fig. 13, are much smaller as they are diluted by the very small corrections for the ZZ channel.
The equally unobservable four-lepton invariant mass, m νν , displayed in Fig. 21, shows similar features as its DF counterpart. Again, the reason is the dominance of the W W channels over much of its range. Only at very low invariant masses, near the the Z-pole, the importance of the ZZ channels increases up to becoming dominant. This is the only regime out of all observables considered in this paper, where a visible interference effect between the W W and ZZ channels can be observed, reaching up to -25% on the Z-pole itself. This observation can be explained by the fact that this is the only region where at least one of the gauge bosons is forced off shell and both the W W and ZZ channels populate the same surviving resonance, cf. the diagrams of Fig. 1c and Fig. 2b.
Finally, Fig. 22 shows the azimuthal separation of both charged leptons. Here, due to the dominance of the W W channel throughout we observe very similar effects as already documented in Fig. 15 Figure 22. Distribution in the azimuthal separation of the e + e − pair, ∆φ , for pp → e + e − νν at 13 TeV. Details as in Fig. 16.

Summary and conclusions
We have presented NLO QCD and EW predictions for e + µ − ν eνµ and e + e − νν production at the LHC. These reactions are representative of all possible diboson processes pp → W W/ZZ → + i − j νν, which lead to signatures with two leptons of opposite charge plus missing transverse energy. Due to the large SU(2) charges of the intermediate W and Z bosons, the underlying qq → V V subprocesses induce huge EW Sudakov effects at high energy. As a result, in various observables we find negative EW corrections beyond −50% at the TeV scale. Also QCD corrections can be sizeable, and in order to account for unknown NNLO contributions of O(α S α) in an approximate way, we have advocated a factorised combination of NLO QCD and EW corrections.
Photon-induced processes have been computed at NLO EW, taking into account the channels of type γγ → 2 2ν, γγ → 2 2νγ, and γq → 2 2νq. In the tails of the m and leading-lepton p T distributions, such contributions can become important. In particular, the E T distribution receives large γ-induced corrections starting already from about 100 GeV due to a suppression of the LO qq → W W process in this region. With the poorly constrained photon density of the NNPDF3.0qed fit, γ-induced processes can be strong enough to compensate the negative corrections of Sudakov type. However, based on the more precise photon densities in the CT14qed and LUXqed PDFs, γ-induced contributions can reach at most 10-20% and remain clearly subleading with respect to EW Sudakov logarithms.
For observables that are inclusive with respective to QED radiation, NLO EW corrections can be described with good accuracy by a so-called EW VI approximation, which includes only IRsubtracted virtual EW corrections and is particularly suitable in the context of multi-jet merging. However, for observables depending on charged leptons, also radiative QED effects can play an important role. Thus we have studied the possibility of augmenting the EW VI approximation through QED radiation effects generated via YFS soft-photon resummation or, alternatively, by the Catani-Seymour dipole-based DGLAP-type resummation of the Csshower. In general, both approaches provide reasonably accurate results. More precisely, both approaches describe the highenergy regions on a similar level with deviations being typically smaller than 10%, while the YFS resummation implementation in Sherpa also preserves the existing resonance structure.
Radiative corrections in 2 2ν channels with same and opposite lepton flavour behave in a fairly similar way. This can be understood in the light of the respective resonance structures. On the one hand, pp → e + µ − ν eνµ solely contains W + W − resonances, while pp → e + e − νν involves both W + W − and ZZ resonances. On the other hand, possible interferences between W + W − and ZZ topologies turn out to be completely negligible for all relevant observables. Moreover, W + W − contributions to pp → e + e − νν are widely dominant with respect to ZZ ones. This is the reason why QCD and EW corrections behave very similarly in e + e − νν and e + µ − ν eνµ production. Nevertheless, we have pointed out that NLO effects can still be quite sensitive to the flavour structure in certain observables. This can for example be observed in the m and E T distributions in correspondence to the occurrence of ZZ dominated regions that originate, respectively, from the Z → + − resonance and due to the suppression of W + W − topologies for E T > M W .
Concerning the treatment of hard scattering processes with external photons at NLO EW, in Appendix A we have presented a general analysis of the interplay between the definition of the coupling α for external photons, the renormalisation of the photon wave function, and the renormalisation of the γPDF. In particular, we have pointed out that, in order to avoid large logarithms associated with ∆α(M 2 Z ), the coupling α for final-and initial-state photons should be defined, respectively, at high energy and in the Thomson limit, Q 2 → 0. In practice, at energies of the order of the EW scale or above, initial-state photon couplings can be parametrised in the G µ scheme or in the α(M Z ) scheme, while α(0) should not be used.
The tools that have been used in this project-Sherpa, Munich, and OpenLoops-implement automated NLO QCD+EW algorithms that are applicable to any Standard Model process and will be made publicly available in the near future.
A Infrared subtraction, γPDF renormalisation and definition of α This Appendix starts, in Section A.1, with a general documentation of the implementation of Catani-Seymour subtraction at NLO EW in Sherpa and Munich. This serves as a basis for the discussion, in Sections A.2-A.3, of the cancellation of light-fermion mass singularities in processes with external photons. Such cancellations involve a subtle interplay between the definition of the coupling α, the renormalisation of the γPDF, and the photon wave-function renormalisation.
In particular we point out that, in order to avoid a logarithmic sensitivity to light-quark and lepton masses, the coupling of on-shell final-state photons should be defined in the limit of vanishing Q 2 , while for initial-state photons a definition of α at the EW scale or at µ 2 F ∼ŝ should be used. This was first noticed in [25], based on arguments related to the evolution of the γPDF at LO, and is confirmed by our explicit analysis at NLO EW.

A.1 Catani-Seymour subtraction at O(α)
In this section we present the implementation of Catani-Seymour subtraction at NLO EW in Sherpa and Munich. While the construction of Catani-Seymour dipoles for QED radiation has been discussed in detail in [37][38][39], our implementation relies on the direct transposition of the original O(α S ) subtraction formalism [35,36] to O(α). In the following, we provide the complete set of formulae that permits to obtain O(α) dipoles from the results of [35] for massless partons, thereby extending the schematic description given in [26]. Moreover, we point out some subtle aspects related to leptonic contributions and external photons, which are relevant for the cancellation of fermion-mass logarithms discussed in Sections A.2-A.3.
Let us consider the O(α) corrections to a 2 → m hard-scattering process. The subtraction term for the singularities stemming from photon-or fermion-bremsstrahlung in the (m + 1)-parton phase space has the general form where S in+out = S in ∪ S out is the full set of initial-state (S in ) and final-state (S out ) partons. Each term in the triple sum over I, J and K describes 1/(p I p J ) singularities arising from the exchange of a soft parton J between an emitter I and a spectator K, as well as collinear singularities involving the partons I, J. The relevant splitting kernelsV IJ,K in (A.1) are convoluted with the reduced Born cross section dσ B | IJ→ IJ , where the partons I and J are clustered into a single parton IJ according to the respective splitting process. 10 The various types of splitting kernels are listed in Table 3. In general we consider O(α) emissions off quarks and leptons, generically denoted as f = q,q, − , + , dipole type Table 3. Correspondence between the generic splitting kernelsV IJ,K of (A.1) and the kernels V of [35]. For initial-state and final-state partons we use specific indices a, b ∈ Sin and i, j, k ∈ Sout. Moreover, since the emittee J ∈ Sout we identify J = j. The terms with initial-state emitters, I = a ∈ Sin, describe splittings a → (aj) + j, while final-state emitters, I = i ∈ Sout, corresponds to splittings (ij) → i + j. The spectators K can be either initial-state (K = b) or final-state (K = k) partons.
as well as photon splittings. Explicit expressions forV IJ,K corresponding to f → f γ and γ → ff splittings can be obtained from the corresponding QCD kernels [35] for q → qg,q →qg, and g → qq splittings by replacing where N c,f = 1 for leptons and 3 for quarks. As discussed in Appendix A.2, all terms f N C,f Q 2 f , which arise from massless fermion-loop insertions in the photon propagator or related real-emission contributions, should include both quarks and charged leptons. The matrix Q IJ,K in (A.1) collects the charge factors of the partons IJ and K. It is related to the colour-insertion operators of [35] via If emitter IJ is a charged fermion, we simply have where Q is the incoming charge, e.g. Q = ∓1 (±1) for an incoming (outgoing) ∓ . For a photon emitter, IJ = γ, (A.4) is not applicable due to Q γ = 0. This situation occurs for final-state γ → ff splittings and initial-state f → γf splittings, which involve only collinear singularities that are insensitive to the electromagnetic charge of the spectator K. In fact, the role of the spectator is merely to absorb the recoil resulting form the splitting process, and the matrix Q IJ,K in (A.1) distributes the recoil to the various spectators based on the identity which is a manifestation of the charge-conservation relation K Q K = 0. Since Q γ,K does not need to be related to the actual charges of the spectators K, any matrix Q γ,K that obeys (A.5) guarantees a consistent IR subtraction. The choice implemented in Sherpa and Munich for initialstate photon emitters reads Here, a and b denote the two initial-state partons, i.e. the recoil of initial-state γ → ff splittings is absorbed by the initial-state partner b of the emitter photon a.
Final-state γ → ff splittings should not be considered for processes with identified on-shell photons. However, they should be taken into account when photons are not distinguished from ff pairs. In order to account for both cases in a flexible way, we introduce a discriminator FS,γi for every final-state photon, defined as The FS,γi can be set individually for each photon, taking care that the prescription is infrared safe. Of course, if multiple photons fulfil the identification criteria simultaneously the assignment has to be properly symmetrised. 11 The charge correlation matrix in Sherpa and Munich is then chosen as 12 Q γi,K = − 1 2 FS,γi δ a,K + δ b,K for every γ i ∈ S out . (A.8) In this way, when final-state γ i → ff splittings are allowed the resulting recoil is shared by the two initial-state partons a and b. The cancellation of soft and collinear singularities against virtual corrections requires the analytic integrals of the dipole terms (A.1) supplemented by PDF-factorisation counterterms. This leads to [35] (A.9) 11 One example may be the production of an isolated photon accompanied by a jet, which at O(α 2 ) can be described by qq → γγ. Now, once an isolated photon is found, for which we set FS,γ i = 0, the remaining photon forming the jet at LO is allowed to split, thus its FS,γ i = 1. 12 Other recoil strategies for initial and final state photon splittings are possible and a number of generic choices is implemented in Sherpa.
where the various dσ B ab terms denote reduced Born cross sections that result from the clustering of an unresolved parton. All IR divergences are captured by the I operator. For massless fermions at O(α) it reads and for f = q,q, − , + . The fermion sum in (A.13) runs over massless fermions and includes a single term per fermion-antifermion pair. As discussed in Appendix A.2, all massless leptons and quarks should be taken into account, i.e.
where N 0, , N 0,u and N 0,d are the number of massless leptons and quarks of type up and down, respectively. The P and K operators in (A.9) read P a,a ({p}; x; µ 2 where b stands for the initial-state partner of a. All relevant ingredients are specified in Table 4. Note that S in+out = S in ∪ S out in (A.10) and (A.15) should be understood as the incoming and outgoing partons of the relevant Born sub-process. The P and K operators are free from soft and collinear singularities. The former depends on the factorisation-scale µ F introduced via the PDF counterterm, while the latter depends on the factorisation scheme. The result (A.15) corresponds to the case of two initial-state hadrons in the MS scheme and can be easily translated to the DIS scheme [35].

Processes with resolved photons
For hard processes with resolved photons in the final state, real-emission processes corresponding to final-state γ → ff splittings and related subtraction terms should be omitted at O(α). This is achieved by setting FS,γi = 0 in (A.8). Consequently, in the subtraction term (A.1) we have Thus, external photons contribute to (A.1) only through IJ → I + J final-state splittings of type f → f γ, while they can contribute to all types of I → IJ + J initial-state splittings, i.e. γ → ff , f → γf , and f → f γ. In analogy to (A.16), for the matrix (identical) Q I,J that enters the I, K and P operators we have Thus, resolved final-state photons can be completely excluded from the sums over I ∈ S out and K ∈ S out in (A.10) and (A.15), and external photon contributions to I, K and P arise only through i.e. from dipoles with initial-state emitters a = γ and initial-state spectators b.

Processes with unresolved photons
For hard processes with unresolved photons in the final state, the I, P and K operators in (A.10) and (A.15), contain, as compared to the case of resolved photons, the following additional contributions from final-state γ → ff splittings, (A.21) Here, µ D is the scale of the dimensional regularisation. The term proportional to P γf absorbs collinear singularities arising from real-emission processes where the (off-shell) initial-state photon originates from f → γf splittings. The remaining term is due to the γγ splitting function (see Table 4), which consists only of virtual fermion-loop contributions associated with the photon wave function renormalisation. It can be understood as a negative correction to the γPDF that compensates real γ → ff splittings. The corresponding splitting functions are related via the momentum sum rule Including also the logarithmic dependence on µ F , which appears in the P operator in (A.15), the effect of the γPDF renormalisation can be summarised through an overall renormalisation factor, for each initial-state photon.

Contributions form γ → + − splittings
Photon splittings into qq and + − should be included on the same footing at O(α). Thus, as pointed out above, the photon anomalous dimension of (A.14) should include both quark and lepton contributions. This should be clear, since γ γ represents contributions of virtual type, and different kinds of fermion loops are indistinguishable. Moreover, omitting leptonic contributions to γ γ would jeopardise the cancellations of fermion-mass singularities between (A.24) and the virtual corrections to the hard cross section (see Appendix A.3).
Since γ γ in (A.24) arises from the renormalisation of the γPDF of (A.21), virtual γ → + − splittings should be taken into account also in the evolution of γ(x, µ F ). In addition, for consistency with the sum rule (A.23), also real γ → + − splittings and thus lepton distributions should be included in the PDF evolution. While this is desirable from the theoretical viewpoint, the effect of γ → + − splittings hardly exceeds 1% in the photon PDF [22] and is completely negligible in the quark PDFs. Moreover, lepton-induced processes are extremely suppressed at the LHC [70]. Thus, excluding γ → + − splittings from the PDF evolution, as in the CT14qed set used in the nominal predictions in this paper, is well justified.

A.3 Definition and renormalisation of α in processes with external photons
The collinear singularities in (A.24) have to be combined with corresponding singularities that arise from the 1-loop counterterms associated with the renormalisation of the photon wave function (δZ AA ) and of the electromagnetic coupling α. Such counterterms yield a universal correction factor for each external (incoming or outgoing) photon in the hard scattering process. In the following, in order to articulate the interplay between the renormalisation of α and the cancellation of collinear singularities, we will focus on the contributions from light fermions with 0 ≤ m f < M Z , which can be either treated in dimensional regularisation or using finite fermion masses. While all massless and massive fermions are assumed to contribute to the virtual corrections and to the ultraviolet renormalisation, only massless fermions are assumed to be included in the Catani-Seymour subtraction and in the γPDF renormalisation. The photon wave function counterterm reads, represents the UV divergent piece, while all collinear singularities are contained in where the anomalous dimension γ γ , defined in (A.14), accounts for all massless fermion loops, while the sum over f ∈ F m includes all light fermions with 0 < m f < M Z . As is well known, ∆α(M 2 Z ) is associated with the running of α from Q 2 = 0 to Q 2 = M 2 Z . In order to arrive at a finite expression for ∆α(M 2 Z ), all fermions could be treated as massive, in which case γ γ = 0. Alternatively, hadronic contributions to ∆α(M 2 Z ) can be obtained via dispersion relations. However, we advocate the approach of choosing an appropriate definition of α, such as to cancel all singularities associated with ∆α(M 2 Z ) in the final result. As detailed in the following, such a definition depends on the presence of resolved external photons in the processes at hand.

Resolved final-state photons
In processes with resolved on-shell photons that do not split into ff pairs the collinear singularity from δZ AA remains uncancelled unless the electromagnetic coupling is renormalised in the on-shell scheme. Thus, α should be defined as the photon coupling in the on-shell limit q 2 → 0. The resulting counterterm is related to the photon wave-function renormalisation via [71] δα(0) where θ w is the weak mixing angle. Light-fermion contributions to (A. Thus, as is well known, in order to avoid fermion-mass singularities from ∆α(M 2 Z ) in the hard cross section, the couplings of on-shell (resolved) final-state photons should be parametrised in terms of α(0).

Initial-state photons and unresolved final-state photons
In the case of initial-state photons, virtual contributions to the γPDF renormalisation (A.21) are designed such as to absorb the collinear singularity of δZ AA . Thus, by construction, the combination is free from 1/ mass singularities, and there is no need to adopt the α(0) scheme. In fact, expressing the coupling of initial-state photons in terms of results in the overall initial-state photon factor which is manifestly free from 1/ fermion-mass singularities, while, as usual, those degrees of freedom that do not contribute as active fermions in the PDF evolution give rise to logarithms of m f in the hard-scattering cross section. Vice versa, using the α(0) scheme for initial-state photons would lead to the divergent result δZ γ,virt OS,light + δZ γ,PDF = δα(0) α(0) A fully analogous cancellation mechanism applies also to unresolved final-state photons, where the term proportional to n in (A.20), which originates from final-state γ → ff splittings, plays a similar role as the γPDF counterterm for initial-state photons.
Thus, in order to avoid fermion-mass singularities in the hard cross section, the couplings of initial-state photons and unresolved final-state photons should be parametrised in terms of α(M 2 Z ) or any other scheme where α is defined at a hard scale, such as the G µ -scheme or a running α(µ 2 R ) with µ 2 R ∼ŝ. For the case of initial-state photons, this was first pointed out in [25] based on arguments related to the PDF evolution.

B Flavour-number scheme conversion
As discussed in Section 3.4, in order to avoid single-top contributions, we compute parton-level cross sections using m b > 0 and omitting external b-quarks, both in the initial and in the final state.
This approach corresponds to the four-flavour scheme, and can be consistently used in combination with five-flavour PDFs by applying a simple scheme conversion [66], which amounts to the following substitution at the level of squared Born matrix elements, Here ij ∈ {qq, gq, gq, gg} are the initial-state QCD partons, while n (αS) ij and n (g) ij are, respectively, the power of α S and the number of initial-state gluons in the channel at hand. For the process of interest in this paper, pp → 2 2ν, initial-state gluons do not contribute at Born level, and in the qq channel we have n (αS) qq = n (g) qq = 0. Thus, as far as QCD partons are concerned, the scheme conversion of (B.1) is trivial. However, the γγ → 2 2ν channel requires a non-zero scheme transformation, which involves a single term, related to the scheme dependence of the photon PDF. Note that there is no scheme-conversion term associated with the electromagnetic coupling required since, usually, α is not defined in the MS scheme.

C Electroweak corrections by parton luminosity
Figs. 23-24 and 25-26 detail the relative electroweak correction induced by each parton luminosity for the DF and SF pp → 2 2ν production processes, respectively. Here, only the CT14qed PDF is used for the photon density. As described in Section 2, the pp → 2 2ν production process at LO receives contributions from the qq and γγ channels, while at NLO EW also the qγ andqγ channels arise. It needs to be noted that the relative contributions from different parton luminosities are factorisation scale dependent. In each figure, the upper panel shows the relative correction of the γγ-induced production process at LO in addition to the relative size of the NLO EW corrections in the qq-, γq/γqand γγ-induced channels relative to the LO qq-induced process. In addition it quantifies the size of the scheme conversion term (4.13) that originates when relating our definition of the multiplicative combination of NLO QCD and EW correction to another definition based upon individual corrections to both LO production channels. This scheme dependence is of relative O(α S α) and contributes generally 0.5 % , rising to 5 % in extreme regions. The lower panel compares the size of both electroweak Sudakov-like corrections to their respective Born process. Naturally, the LO γγ-induced correction is small but positive throughout. The NLO EW corrections are dominated by the qq-channel exhibiting the usual Sudakov suppression at large transverse momenta. The distribution in the missing transverse momentum in the DF case provides an exception, the origins of which and its specific characteristics have been discussed in detail in Section 4.1.
The γγ-induced NLO EW corrections are detailed both in the upper panel, showing their relative size in comparison to the LO qq channel, and the lower panel, showing their relative size in comparison to the LO γγ channel. While they contribute only small amounts to the total NLO EW correction, the comparison against the LO γγ channel clearly exhibits their Sudakov-like behaviour as transverse momenta are increasing. Despite similar shapes, the size of this Sudakovtype correction is found to be slightly larger in the γγ channel than in the qq channel.
The NLO EW γqand γq-induced corrections that are associated with both LO processes show a different behaviour. At this order, no one-loop diagrams contribute and, thus, the Sudakov-type behaviour is absent. Instead, the corrections are positive and of a similar magnitude as the LO γγ channel. Please note, since these two channels exhibit a final state quark or anti-quark, their precise magnitudes are strongly dependent on the chosen form of the jet veto. Choosing a tighter veto, e.g. by applying a strict veto against any jet activity above 30 GeV, decreases their contribution, while loosening it, e.g. by not vetoing jets altogether, increases it.
In conclusion, while the NLO EW correction in the γγ channel is dominated by EW Sudakov logarithms whose magnitude in the TeV region balances the additional power of α to arrive at a result of the same magnitude as the LO γγ-induced contribution, the NLO EW correction in the γq and γq channels uses the replacement of one of its PDF by a quark or anti-quark PDF (relative to the LO γγ-induced process) to cancel the additional power in α. Thus, when summing all contributions that depend on the photon density in the proton, LO γγ and NLO EW γq-, γqand γγ channels, there are sizeable cancellations between the different contributions to the cross sections at NLO EW accuracy.

D Cross section tables
This last section compiles, for reference, a list of cross sections and corrections with different phasespace cuts applied. Tables 5-6 detail the cross sections for DF and SF 2 2ν production, while Tables 7-8 show the contribution from the SF WW/ZZ and one of the two SF ZZ channels making up the SF signature. We list cross sections and corrections for the inclusive fiducial phase space as well as three more exclusive phase-space regions focussing on various high-p T scenarios. In each case, the LO cross section serves as a reference to define the NLO QCD, NLO EW, NLO QCD+EW and NLO QCD×EW corrections, computed in our default setup using the CT14qed PDFs, cf. Section 3. The latter we consider our best prediction for each particle selection.
pp → e + µ − ν eνµ inclusive p T, 1 > 500 GeV E T > 500 GeV m >  Table 5. Cross-sections for pp → e + µ − νeνµ at 13 TeV with CT14qed PDFs and fiducial cuts of Table 2 (1 st column) plus one additional cut on p T, 1 (2 nd column), ET (3 rd column), or m (4 th column). The top row lists LO cross sections, while the following four rows give the relative change induced by the NLO QCD, EW, QCD+EW and QCD×EW corrections. The sub-and superscripts give their respective relative uncertainties determined through customary µR and µF variations, while keeping the reference LO cross section fixed in the ratios. The impact of alternative descriptions of the photon density are explored by neglecting it entirely (no γPDF) or using the densities provided by the LUXqed and NNPDF3.0qed sets. For quarks and gluons always the central PDF set CT14qed is chosen.
pp → e + e − νν inclusive p T, 1 > 500 GeV E T > 500 GeV m > 1 TeV −1 % 7 % 3 % 4 % Table 6. Cross-sections for pp → e + e − νν at 13 TeV including all neutrino flavours. Higher-order corrections, scale uncertainties and photon-induced contributions are presented as in Table 5.  Figure 23. Relative corrections in the transverse momentum of the leading and subleading lepton, p T, 1 and p T, 2 , for pp → e + µ − νeνµ at 13 TeV. In the upper panel we show the relative corrections to the LO qq channel, induced by the LO γγ channel (δ qq LO (γγ)), and the NLO EW qq-, qγ/qγ-and γγ-induced processes (δ qq NLO EW (qq), δ qq NLO EW (qγ +qγ), δ qq NLO EW (γγ), respectively). We further also show the relative size of the scheme conversion term of (4.13) with respect to the LO qq channel (δ qq scheme ). The lower panel shows the qq-and γγ-induced NLO EW corrections relative to the LO cross section in the qq and γγ channel, respectively. At large transverse momenta, this corresponds to the channels' respective electroweak Sudakov corrections. pp → e + e − ν eνe inclusive p T, 1 > 500 GeV E T > 500 GeV m > 1 TeV  Table 7. Cross-sections for pp → e + e − νeνe at 13 TeV. Higher-order corrections, scale uncertainties and photon-induced contributions are presented as in Table 5.