Electroweak corrections to W-boson pair production at the LHC

Electroweak corrections to W-boson pair production at the LHC


Vector-boson pair production ranks among the most important Standard-Model benchmark processes at the LHC, not only in view of on-going Higgs analyses. These processes may also help to gain a deeper understanding of the electroweak interaction in general, and to test the validity of the Standard Model at highest energies. In this work, the first calculation of the full one-loop electroweak corrections to on-shell W-boson pair production at hadron colliders is presented. We discuss the impact of the corrections on the total cross section as well as on relevant differential distributions. We observe that corrections due to photon-induced channels can be amazingly large at energies accessible at the LHC, while radiation of additional massive vector bosons does not influence the results significantly.

SFB/CPP-12-58 TTP12-027 LPN12-088

a]Anastasiya Bierweiler, a]Tobias Kasprzik, a]Johann H. Kühn, b]Sandro Uccirati \affiliation[a] Karlsruhe Institute of Technology (KIT), Institut für Theoretische Teilchenphysik,
D-76128 Karlsruhe, Germany \affiliation[b]Universität Würzburg, Institut für Theoretische Physik und Astrophysik,
D-97074 Würzburg, Germany \emailAddkasprzik@particle.uni-karlsruhe.de

1 Introduction

With the recent start of LHC operation hard scattering processes became accessible with parton energies up to several TeV and, indeed, jet-jet invariant masses up to 5 TeV have been observed [1, 2] in the first round of data taking. With the significant increase of the integrated luminosity and the doubling of the beam energy anticipated in the next few years, electroweak processes such as the production of lepton or gauge-boson pairs with invariant masses of several TeV will become accessible in the near future. These reactions may on the one hand allow for precise tests of the Standard Model (SM) and on the other hand potential deviations may point to “physics beyond the SM”. Indeed, the observation of anomalous couplings of quarks, leptons or gauge bosons might well be the first signal of “New Physics”. As an alternative one may look for peaks or distortions in the W, Z or fermion spectra resulting from the decays of new massive particles. Clearly, any well-founded claim of “physics beyond the SM” must be based on precise measurements combined with similarly precise calculations, possibly at the level of several percent.

Obviously this requires theory predictions which must include next-to-leading order (NLO), possibly even next-to-next-to-leading order (NNLO) perturbative QCD corrections. Considering the smallness of the weak coupling, , the need for electroweak (EW) corrections is less obvious. In a first step one might try to absorb the dominant corrections arising from the running of the fine-structure constant from low scales to and from the -parameter [3] in properly chosen effective couplings. However, this approach is only justified for energies of order or . As is well known from earlier investigations [4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 8, 9], EW corrections increase with the squared logarithm of the energy, and may reach several tens of percent for energies accessible at the LHC. In view of their strong dependence both on energy and scattering angle, they will induce significant distortions in transverse-momentum and rapidity distributions and consequently may well mimic “New Physics” and hence must be carefully taken into account.

Most of the investigations along these lines have concentrated on the issue of Sudakov logarithms, i.e. terms that are at loops enhanced proportional to with , corresponding to leading, next-to-leading etc. logarithmic (NLL) enhancement. This line of research was motivated by the necessity of including at least the dominant two-loop terms, once one-loop corrections exceed the level. Using evolution equations, originally derived in the context of QED [20, 21, 22, 23] and QCD, four-fermion processes have been studied up to NLL [7, 13, 14, 15, 16], W-pair production in electron–positron and quark–antiquark annihilation up to NLL [24, 25]. Employing diagrammatic methods or the framework of soft-collinear effective field theory most of these results were confirmed in the NLL and NNLL approximation [10, 12, 18, 26].

As stated above, the intermediate energy region, up to approximately 1 TeV, will be explored with high statistics, and the complete one-loop corrections may become relevant. This potentially includes real photon radiation, virtual photon exchange and terms suppressed by , where denotes the partonic center-of-mass (CM) energy squared. Up to now these one-loop corrections have been evaluated for various SM benchmark processes, including production [27, 28, 29] (for earlier studies see ref. [30]), production [31], SM-Higgs production [32, 33, 34, 35] and decay [36, 37, 38, 39], and dijet production, where the purely weak corrections are known [40, 41]. For the Drell–Yan process, the one-loop EW corrections have been studied in great detail by many authors (see, e.g., Ref. [42] and references therein) as well as for gauge-boson plus jet production, the latter in the on-shell approximation [43, 44, 45, 46, 47, 48] and including W- or Z-decays to leptons with all off-shell effects consistently taken into account [49, 50].

So far, EW corrections to vector-boson pair production at the LHC are only known in the high-energy approximation [51, 52, 53], while the full one-loop EW corrections are not yet available, despite their great phenomenological interest. These processes contribute the most important irreducible background to the production of a SM Higgs boson in the intermediate mass range. A profound theoretical understanding of the underlying physics will also allow a precise analysis of the non-abelian structure of the EW sector, in particular of the vector boson self interactions. At the LHC these will be explored with high precision, exploiting the high luminosity in combination with the highest possible CM energies. Moreover, vector-boson pair production processes are well suited for putting experimental constraints on the existence of anomalous trilinear and quartic gauge couplings, since their phenomenological effects are expected to be sizable at large invariant masses of the vector-boson pairs accessible at the LHC. Considering searches for supersymmetry (SUSY) at the LHC, all vector-boson pair production channels constitute important backgrounds to signals with leptons and missing transverse energy, as e.g. predicted for chargino–neutralino pair production.

During recent years, a great effort has been made to push the accuracy of the theoretical predictions to a new level. The NLO QCD corrections, including the leptonic decays of the vector bosons, have been studied by several authors and are implemented in Monte Carlo programs [54, 55, 56, 57, 58, 59, 60, 61, 62]. The results have been matched with parton showers and combined with soft gluon resummations [63] to improve the predictions for vector bosons produced at small transverse momenta. The loop-induced channels , formally a second order effect in QCD, nevertheless play an important role at the LHC due to the enhancement of the gluon luminosity and have also been studied extensively [64, 65, 66, 67, 68]. Specifically, concerning background estimates to Higgs production at the LHC, the contribution of the gluon-induced channel to W-boson pair production amounts to 30% after experimental cuts [67].

As stated before, EW corrections to gauge-boson pair production at hadron colliders have only been evaluated in the high energy limit [51, 52, 53]. This is in contrast to the reaction , where the complete one-loop calculation has long been available [69, 70, 71, 72], and, following the demands of LEP experiments, the full one-loop correction for electron–positron annihilation into four fermions (including resonant and non-resonant amplitudes) has been calculated [73, 74, 75, 76].

To arrive at predictions which satisfy the needs of the next round LHC experiments, we embark on the full one-loop corrections for W-pair production in proton–proton collisions, where the gauge bosons are treated as stable particles. For completeness we also recalculate the well-known NLO QCD corrections and, furthermore, discuss two competing processes: W-pair production through collisions, and through the quark-loop-induced gluon-fusion process. Employing the MRST2004QED PDF set [77] for the luminosity, we observe a surprisingly large contribution at large invariant W-pair mass and a pronounced peaking at small scattering angles. In fact, for small angles and high energies this parametrically suppressed reaction is comparable to the induced reaction. Even for large angles (i.e. large ) it still amounts to order of 5% and is therefore comparable with the genuine EW corrections. Gluon fusion, in contrast, exhibits a fairly smooth angular distribution and is typically of order 5 to 10 percent in the whole kinematic region of interest. Obviously all these effects must be taken into consideration.

The layout of this paper is as follows: In Section 2 we start with a qualitative discussion of W-pair production at leading order (LO). We introduce the kinematic variables and recall the dominant features of W production: the dependence on the W transverse momentum, W-pair invariant mass and rapidity. These results will illustrate the large kinematic region accessible at the LHC and the marked differences between the -, - and -induced channels, as far as transverse-momentum and angular distributions are concerned. Section 3 will be devoted to a detailed discussion of one-loop EW corrections to the reaction . We specify the renormalization scheme, our input parameters, the treatment of QED corrections with the associated soft and collinear singularities, their absorption in the PDFs, and the numerical evaluation of hard photon radiation. As stated above the rate of W-pair production from collisions is surprisingly large and the resulting rates and distributions are studied in Section 4. The cross sections for including EW and QCD corrections are convoluted with PDFs, and predictions for various distributions are shown and compared with those from and gluon-fusion processes. For future tests and cross checks tables are presented which precisely specify the results at various kinematic points. In Section 5, we discuss the phenomenological impact of additional massive vector bosons in the final state. Section 6 contains our conclusions.

2 W-pair production: the leading order

At lowest order, , W-boson pair production at the LHC is dominated by quark–antiquark annihilation,


where the denote one of the light quark flavours, and . The corresponding diagrams are shown in Fig. 1 for the channel.


(120,40)(3,1) \FADiagram \FAProp(0.,15.)(6.,10.)(0.,)/Straight1 \FALabel(2.48771,11.7893)[tr] \FAProp(0.,5.)(6.,10.)(0.,)/Straight-1 \FALabel(3.51229,6.78926)[tl] \FAProp(20.,15.)(14.,10.)(0.,)/Sine-1 \FALabel(16.4877,13.2107)[br] \FAProp(20.,5.)(14.,10.)(0.,)/Sine1 \FALabel(17.5123,8.21074)[bl] \FAProp(6.,10.)(14.,10.)(0.,)/Sine0 \FALabel(10.,8.93)[t] \FAVert(6.,10.)0 \FAVert(14.,10.)0


(0.,15.)(10.,14.)(0.,)/Straight1 \FALabel(4.84577,13.4377)[t] \FAProp(0.,5.)(10.,6.)(0.,)/Straight-1 \FALabel(5.15423,4.43769)[t] \FAProp(20.,15.)(10.,14.)(0.,)/Sine-1 \FALabel(14.8458,15.5623)[b] \FAProp(20.,5.)(10.,6.)(0.,)/Sine1 \FALabel(15.1542,6.56231)[b] \FAProp(10.,14.)(10.,6.)(0.,)/Straight1 \FALabel(9.18,10.)[r] \FAVert(10.,14.)0 \FAVert(10.,6.)0


(0.,15.)(10.,14.)(0.,)/Straight1 \FALabel(4.84577,13.4377)[t] \FAProp(0.,5.)(10.,6.)(0.,)/Straight-1 \FALabel(5.15423,4.43769)[t] \FAProp(20.,15.)(10.,6.)(0.,)/Sine-1 \FALabel(16.8128,13.2058)[br] \FAProp(20.,5.)(10.,14.)(0.,)/Sine1 \FALabel(17.6872,8.20582)[bl] \FAProp(10.,14.)(10.,6.)(0.,)/Straight1 \FALabel(9.28,10.)[r] \FAVert(10.,14.)0 \FAVert(10.,6.)0

Figure 1: Tree-level diagrams for the LO process .



(0.,15.)(10.,10.)(0.,)/Sine0 \FALabel(4.78682,11.5936)[tr] \FAProp(0.,5.)(10.,10.)(0.,)/Sine0 \FALabel(5.21318,6.59364)[tl] \FAProp(20.,15.)(10.,10.)(0.,)/Sine-1 \FALabel(14.7868,13.4064)[br] \FAProp(20.,5.)(10.,10.)(0.,)/Sine1 \FALabel(15.2132,8.40636)[bl] \FAVert(10.,10.)0


(0.,15.)(10.,14.)(0.,)/Sine0 \FALabel(4.84577,13.4377)[t] \FAProp(0.,5.)(10.,6.)(0.,)/Sine0 \FALabel(5.15423,4.43769)[t] \FAProp(20.,15.)(10.,14.)(0.,)/Sine-1 \FALabel(14.8458,15.5623)[b] \FAProp(20.,5.)(10.,6.)(0.,)/Sine1 \FALabel(15.1542,6.56231)[b] \FAProp(10.,14.)(10.,6.)(0.,)/Sine-1 \FALabel(8.93,10.)[r] \FAVert(10.,14.)0 \FAVert(10.,6.)0


(0.,15.)(10.,14.)(0.,)/Sine0 \FALabel(4.84577,13.4377)[t] \FAProp(0.,5.)(10.,6.)(0.,)/Sine0 \FALabel(5.15423,4.43769)[t] \FAProp(20.,15.)(10.,6.)(0.,)/Sine-1 \FALabel(16.8128,13.2058)[br] \FAProp(20.,5.)(10.,14.)(0.,)/Sine1 \FALabel(17.6872,8.20582)[bl] \FAProp(10.,14.)(10.,6.)(0.,)/Sine1 \FALabel(9.03,10.)[r] \FAVert(10.,14.)0 \FAVert(10.,6.)0

Figure 2: Tree-level diagrams for the LO processes .
Figure 3: Left: Total cross sections as a function of the cut on the W-boson transverse momentum at the LHC14 (top), LHC8 (center) and the Tevatron (bottom). The corresponding relative rates w.r.t. to the channel are shown on the r.h.s., together with the rates of QCD and EW corrections. See text for details.
Figure 4: Left: Total cross sections as a function of the cut on the WW-invariant mass at the LHC14 (top), LHC8 (center) and the Tevatron (bottom). The corresponding relative rates w.r.t. to the channel are shown on the r.h.s., together with the rates of QCD and EW corrections. See text for details.
Figure 5: Differential LO cross sections for the W-boson rapidity gap with default cuts (top) and with a minimal invariant mass of 1000 GeV (bottom) at the LHC14. On the right-hand-side, the corresponding relative rates due to photon- and gluon-induced channels w.r.t. the -contributions are shown, as well as the EW corrections. See text for details.
Figure 6: Differential LO cross sections for the W-boson rapidity gap with default cuts (top) and with a minimal invariant mass of 500 GeV (bottom) at the LHC8. On the right-hand-side, the corresponding relative rates due to photon- and gluon-induced channels w.r.t. the -contributions are shown, as well as the EW corrections. See text for details.

Since there is a finite probability of finding a photon in a proton we in addition take into account the photon-induced tree-level contributions


(see Fig. 2). In the numerical analysis the MRST2004QED PDF set [77] is used for the photon density. Although suppressed by two photon-PDF factors, this channel may give rise to potentially large contributions at high invariant masses due to peculiar features of the corresponding partonic cross section, as will be pointed out later.

A detailed phenomenological discussion of the specific properties of process (2) at photon colliders can be found in Ref. [78], where also the computation of the full corresponding EW corrections in the SM was presented. Also the photon-induced contributions to W-pair production in elastic hadron–hadron scattering have been studied in an equivalent-photon approximation [79, 80], assuming a tagging of the forward-scattered hadrons. In our approach, however, we quantify for the first time the purely inelastic contributions which cannot be separated from the dominating -induced contributions.

For completeness, we also re-calculate the effects due to the loop-induced channel


which adds a relative correction of about to the LO cross section [65].

Due to the large CM energies accessible at the Tevatron and the LHC, we can safely neglect the lepton and light quark masses (i.e. all but the top-quark mass) unless they are needed to regularize infrared (IR) singularities in the real and virtual contributions, as will be discussed later. However, the channel receives contributions from top-quark exchange in the channel that for consistency have to be taken into account in the Born approximation and in the NLO calculation. In this work we assume a block-diagonal CKM matrix with . Therefore, the cross sections (1) are proportional to , and it is sufficient to only consider flavour-diagonal processes. Consequently, at the parton level the complete one-loop calculation has to be carried out independently for the following three different channels,


Let us, in a first step, evaluate the LO prediction for -, - and -induced processes. The cross sections for W-pairs as a function of , corresponding to a cut on the transverse momenta of the W bosons, and as a function of , corresponding to a cut on the mass of the W-pair, are shown in Figs. 3 and 4, respectively. The input values and setup we use for our computation will be specified in Section 4.1, together with the rapidity and cuts employed throughout. Given an integrated luminosity of at 8 TeV and at , transverse momenta (invariant masses) up to () can be explored at 8 TeV and up to () at 14 TeV. Here, we assume that events with one W boson decaying into or and one decaying hadronically can be detected with 30% efficiency and require more than 100 detected events.

When comparing the and dependence of -, - and -induced processes, marked differences can be observed. For large the rates of - and -induced reactions are quite comparable. However, the cut on reduces the relative effect of the photon-induced channels significantly. This is shown on the right hand sides of the respective figures, where the cross sections of the - and -induced reactions relative to the -induced process (denoted as and , respectively) are plotted. Let us first concentrate on the LHC at 14 TeV (LHC14). The relative rate for W-pairs from gluon fusion amounts to less than 10% for small and decreases with increasing (Fig. 3). The parametric suppression of by is compensated by the large gluon luminosity for small which, however, dies out with increasing . A qualitatively similar behaviour is observed for a cut on (Fig. 4). The relative importance of the photon-induced process, in contrast, increases with increasing , and even more so with . This behaviour can be traced to the energy dependence of the cross section for the partonic subprocess , which approaches a constant value in the high-energy limit, , with an extremely strong peaking in the forward and backward directions. The cut on , when compared to the one on thus leads to a significantly stronger reduction of the event rate. Nevertheless, a 10% admixture of photon-induced W-pairs remains for above 800 GeV. Qualitatively similar statements are applicable to the LHC at 8 TeV (LHC8). However, the relative contributions of both - and -induced reactions are significantly smaller, and become completely negligible for the Tevatron. In the right three plots of Figs. 3 and 4 we also anticipate some of the results on one-loop QCD () and EW () corrections. For transverse momenta around 800 GeV, accessible at LHC14, large negative EW corrections amount to nearly and are the main subject of this paper. In fact, they are quite comparable to the positive QCD corrections, which are displayed in the same figures. Note that the latter strongly depend on the cuts on real jet radiation, as will be discussed at the end of Section 3.3.

As indicated above, the difference between the and dependence of the cross sections can be traced to the different angular distributions of the W bosons in the W-pair rest frame, which for is strongly peaked in the forward and backward directions. The marked differences in the angular distributions are illustrated in Fig. 5, where the distributions in the rapidity difference (which, for fixed , corresponds to the angular distribution in the W-pair rest frame) are shown for the mass intervals and , respectively. In the first case, the cross section is dominated by central events with W-pairs of low invariant mass and relatively isotropic angular distributions. All three components are peaked at small . Considering their ratios, displayed on the r.h.s. one observes the stronger preference of the process for small and, conversely, the enhancement of the process for large . Considering events with large invariant mass, , the distribution of the three components exhibits a drastically different behaviour. The rate of the gluon-fusion process is fairly small and its angular distribution does not exhibit any pronounced structure, resulting in a flat rapidity distribution. As expected, W-pair production is dominated by annihilation with its strong peaking at small , i.e. in the forward and backward directions. This is reflected in the peaks of around . The contribution from fusion remains sizable, with a very pronounced peaking for large rapidity difference, corresponding to a highly anisotropic angular distribution in the rest frame. The EW corrections exhibit a completely different behaviour. For large , corresponding to small they are negative but less than 10%. In the Sudakov regime, however, corresponding to small and thus large , the Sudakov effect comes into play and negative corrections of more than 30% are observed. In total, the sum of these corrections, denoted in the right displays of Fig. 5, will lead to a dramatic distortion of with corrections varying between and for large and small , respectively. Note that such a behaviour could well be misinterpreted as a signal for anomalous couplings.1 A less pronounced, but qualitatively similar behaviour is observed for LHC8 with the mass intervals and (Fig. 6).

3 Radiative corrections

In addition to the tree-level contributions, a full analysis of the W-boson pair production cross section requires the inclusion of the one-loop EW corrections to the -induced processes as well as the contributions due to radiation of one additional bremsstrahlung photon. Thus, the corresponding total partonic cross section at NLO may be written as


with , and the different partonic contributions are given by


Here, and denote the corresponding Feynman amplitudes of the tree-level and one-loop contributions, respectively, is the Lorentz invariant phase-space measure of the final-state particles and is the partonic CM energy. The normalization factors are given by , and .

Although the photon-induced process (2) is considered at leading order, we emphasize that it formally contributes at next-to-next-to-leading order in the QED coupling constant according to Eq. (5) of Ref. [77]. Thus, we do not include the EW corrections to the process (including the channel) in our analysis. Since the corresponding relative EW corrections are at the level of 10% at high energies and small scattering angles (see Fig. 9 of Ref. [78]), the related uncertainties w.r.t. the channel are never exceeding 3% even at high invariant masses and may safely be neglected. For a similar reasoning we also do not take into account the real-radiation processes . The dominating contribution from this process class, attributed to the collinear splitting, is already included in the definition of the photon PDFs, while the configuration corresponding to collinear and splittings can be interpreted as a QED correction to the (anti)quark PDF, which is estimated to be small [82]. The residual contributions, namely photon-induced WW + jet production again gives a genuine NNLO contribution. Moreover, including all these contributions (which of course would be possible), to properly account for all one-loop QED effects would require the rigorous application of the out-dated MRST2004QED PDF set which is discouraged by the authors of Ref. [77].

Defining the momentum fractions and of the initial-state hadrons carried by the incoming partons, the hadronic NLO cross sections at the LHC is given by a convolution of the partonic cross section with the parton distribution functions (PDFs),


where the hadronic CM energy is related to via , with . The kinematic production threshold of a W-boson pair in the final state is reflected in the choice of the lower integration boundary , corresponding to a minimal partonic CM energy of , and denotes the factorization scale.

We have performed two completely independent calculations of the EW corrections, and the results displayed in Tables 14 have been reproduced by both of them.

3.1 Virtual corrections and renormalization

The virtual one-loop corrections receive contributions from self-energies, triangles and box diagrams. In the first approach the diagrams are automatically generated with FeynArts 3.5 [83, 84] and FormCalc 6.1 [85, 86] is used to calculate and algebraically simplify the corresponding amplitudes. Afterwards, the multiplication with the Born-level amplitude, as well as the summation (averaging) over polarisations, spins and colours is performed completely analytically within the FormCalc framework.

In an alternative approach we used the program QGraf [87] to generate Feynman diagrams. The implementation of Feynman rules, the reconstruction of the Dirac structure (as well as the evaluation of Dirac traces) and the calculation of squared matrix elements is carried out analytically using the computer algebra program FORM [88]. To avoid numerical instabilities, potentially small Gram-determinants, which occur in the tensor reduction of particular four-point-functions are cancelled at the analytical level using FORM.

We use two different analytical implementations of the Passarino–Veltman algorithm [89] (based on Mathematica and FORM, respectively) to reduce tensor coefficients to scalar integrals. The reduction was also tested against the numerical approach implemented in the LoopTools 2.5 [85, 90] library, which is used for the evaluation of the scalar one-loop integrals.

For the calculation of the gluon-induced process (3) we use the fully automated setup of FeynArts and FormCalc, where the computation of the squared one-loop amplitude as well as the summation over polarisations are carried out numerically.

The ultraviolet (UV) divergences that arise in the computation of the one-loop diagrams are treated in dimensional regularization going from 4 to space-time dimensions, where the UV divergences appear as single poles in the small complex parameter . After adding the counterterms in a proper renormalization procedure, the poles vanish, and the limit can be taken to obtain physical results.

Our results are based on the on-shell renormalization scheme defined in the following.

On-shell scheme

Our choice for the renormalization prescription is the on-shell (OS) renormalization scheme as specified in Ref. [91]. Instead of defining the electromagnetic coupling constant in the Thomson-limit, however, we work in the scheme where is derived from the Fermi-constant via


In this scheme, the weak corrections to muon decay are included in the charge renormalization constant by the replacement


in the calculation of the counterterm contributions (see, e.g., Ref. [92]). As a consequence, the EW corrections are independent of logarithms of the light-quark masses. Moreover, this definition effectively resums the contributions associated with the running of from zero to the weak scale and absorbs some leading universal corrections from the  parameter into the LO amplitude.

3.2 Real corrections




(0.,15.)(5.5,10.)(0.,)/Straight1 \FALabel(-0.5,16)[tr] \FAProp(0.,5.)(5.5,10.)(0.,)/Straight-1 \FALabel(-0.5,6)[tr] \FAProp(20.,17.)(12.5,10.)(0.,)/Sine-1 \FALabel(19,16)[br] \FAProp(20.,10.)(12.5,10.)(0.,)/Sine1 \FALabel(17.6,11.07)[b] \FAProp(20.,3.)(12.5,10.)(0.,)/Sine0 \FALabel(19,7)[tr] \FAProp(5.5,10.)(12.5,10.)(0.,)/Sine0 \FALabel(9.,8.93)[t] \FAVert(5.5,10.)0 \FAVert(12.5,10.)0


(0.,15.)(4.5,10.)(0.,)/Straight1 \FALabel(1.57789,11.9431)[tr] \FAProp(0.,5.)(4.5,10.)(0.,)/Straight-1 \FALabel(2.92211,6.9431)[tl] \FAProp(20.,17.)(13.,14.5)(0.,)/Sine-1 \FALabel(15.9787,16.7297)[b] \FAProp(20.,10.)(10.85,8.4)(0.,)/Sine1 \FALabel(18.4569,10.7663)[b] \FAProp(20.,3.)(13.,14.5)(0.,)/Sine0 \FALabel(17.7665,4.80001)[tr] \FAProp(4.5,10.)(10.85,8.4)(0.,)/Sine0 \FALabel(7.29629,8.17698)[t] \FAProp(13.,14.5)(10.85,8.4)(0.,)/ScalarDash-1 \FALabel(11.1789,11.8821)[r] \FAVert(4.5,10.)0 \FAVert(13.,14.5)0 \FAVert(10.85,8.4)0


(0.,15.)(4.5,10.)(0.,)/Straight1 \FALabel(1.57789,11.9431)[tr] \FAProp(0.,5.)(4.5,10.)(0.,)/Straight-1 \FALabel(2.92211,6.9431)[tl] \FAProp(20.,17.)(13.,14.5)(0.,)/Sine-1 \FALabel(15.9787,16.7297)[b] \FAProp(20.,10.)(10.85,8.4)(0.,)/Sine1 \FALabel(18.4569,10.7663)[b] \FAProp(20.,3.)(13.,14.5)(0.,)/Sine0 \FALabel(17.7665,4.80001)[tr] \FAProp(4.5,10.)(10.85,8.4)(0.,)/Sine0 \FALabel(7.29629,8.17698)[t] \FAProp(13.,14.5)(10.85,8.4)(0.,)/Sine-1 \FALabel(10.9431,11.9652)[r] \FAVert(4.5,10.)0 \FAVert(13.,14.5)0 \FAVert(10.85,8.4)0


(0.,15.)(5.5,10.)(0.,)/Straight1 \FALabel(2.18736,11.8331)[tr] \FAProp(0.,5.)(5.5,10.)(0.,)/Straight-1 \FALabel(3.31264,6.83309)[tl] \FAProp(20.,17.)(11.5,10.)(0.,)/Sine-1 \FALabel(15.2447,14.2165)[br] \FAProp(20.,10.)(15.5,6.5)(0.,)/Sine1 \FALabel(17.2784,8.9935)[br] \FAProp(20.,3.)(15.5,6.5)(0.,)/Sine0 \FALabel(18.2216,5.4935)[bl] \FAProp(5.5,10.)(11.5,10.)(0.,)/Sine0 \FALabel(8.5,11.07)[b] \FAProp(11.5,10.)(15.5,6.5)(0.,)/ScalarDash-1 \FALabel(13.1239,7.75165)[tr] \FAVert(5.5,10.)0 \FAVert(11.5,10.)0 \FAVert(15.5,6.5)0


(0.,15.)(5.5,10.)(0.,)/Straight1 \FALabel(2.18736,11.8331)[tr] \FAProp(0.,5.)(5.5,10.)(0.,)/Straight-1 \FALabel(3.31264,6.83309)[tl] \FAProp(20.,17.)(11.5,10.)(0.,)/Sine-1 \FALabel(15.2447,14.2165)[br] \FAProp(20.,10.)(15.5,6.5)(0.,)/Sine1 \FALabel(17.2784,8.9935)[br] \FAProp(20.,3.)(15.5,6.5)(0.,)/Sine0 \FALabel(18.2216,5.4935)[bl] \FAProp(5.5,10.)(11.5,10.)(0.,)/Sine0 \FALabel(8.5,11.07)[b] \FAProp(11.5,10.)(15.5,6.5)(0.,)/Sine-1 \FALabel(12.9593,7.56351)[tr] \FAVert(5.5,10.)0 \FAVert(11.5,10.)0 \FAVert(15.5,6.5)0


(0.,15.)(10.,14.5)(0.,)/Straight1 \FALabel(5.0774,15.8181)[b] \FAProp(0.,5.)(10.,5.5)(0.,)/Straight-1 \FALabel(5.0774,4.18193)[t] \FAProp(20.,17.)(10.,5.5)(0.,)/Sine-1 \FALabel(17.2913,15.184)[br] \FAProp(20.,10.)(10.,14.5)(0.,)/Sine1 \FALabel(17.8274,10.002)[tr] \FAProp(20.,3.)(10.,10.)(0.,)/Sine0 \FALabel(14.5911,5.71019)[tr] \FAProp(10.,14.5)(10.,10.)(0.,)/Straight1 \FALabel(9.18,12.25)[r] \FAProp(10.,5.5)(10.,10.)(0.,)/Straight-1 \FALabel(9.18,7.75)[r] \FAVert(10.,14.5)0 \FAVert(10.,5.5)0 \FAVert(10.,10.)0


(0.,15.)(10.,5.5)(0.,)/Straight1 \FALabel(3.19219,13.2012)[bl] \FAProp(0.,5.)(10.,14.5)(0.,)/Straight-1 \FALabel(3.10297,6.58835)[tl] \FAProp(20.,17.)(10.,14.5)(0.,)/Sine-1 \FALabel(14.6241,16.7737)[b] \FAProp(20.,10.)(10.,10.)(0.,)/Sine1 \FALabel(15.95,11.07)[b] \FAProp(20.,3.)(10.,5.5)(0.,)/Sine0 \FALabel(14.6241,3.22628)[t] \FAProp(10.,5.5)(10.,10.)(0.,)/Straight-1 \FALabel(10.82,7.75)[l] \FAProp(10.,14.5)(10.,10.)(0.,)/Straight1 \FALabel(10.82,12.25)[l] \FAVert(10.,5.5)0 \FAVert(10.,14.5)0 \FAVert(10.,10.)0


(0.,15.)(10.,7.)(0.,)/Straight1 \FALabel(2.82617,13.9152)[bl] \FAProp(0.,5.)(10.,14.5)(0.,)/Straight-1 \FALabel(3.04219,6.54875)[tl] \FAProp(20.,17.)(10.,14.5)(0.,)/Sine-1 \FALabel(14.6241,16.7737)[b] \FAProp(20.,10.)(15.5,6.5)(0.,)/Sine1 \FALabel(17.2784,8.9935)[br] \FAProp(20.,3.)(15.5,6.5)(0.,)/Sine0 \FALabel(17.2784,4.0065)[tr] \FAProp(10.,7.)(10.,14.5)(0.,)/Straight1 \FALabel(10.82,10.75)[l] \FAProp(10.,7.)(15.5,6.5)(0.,)/Sine0 \FALabel(12.6097,5.68637)[t] \FAVert(10.,7.)0 \FAVert(10.,14.5)0 \FAVert(15.5,6.5)0


(0.,15.)(10.,14.5)(0.,)/Straight1 \FALabel(5.0774,15.8181)[b] \FAProp(0.,5.)(10.,5.5)(0.,)/Straight-1 \FALabel(5.0774,4.18193)[t] \FAProp(20.,17.)(10.,10.)(0.,)/Sine-1 \FALabel(17.4388,16.3076)[br] \FAProp(20.,10.)(10.,14.5)(0.,)/Sine1 \FALabel(17.2774,10.302)[tr] \FAProp(20.,3.)(10.,5.5)(0.,)/Sine0 \FALabel(14.6241,3.22628)[t] \FAProp(10.,14.5)(10.,10.)(0.,)/Straight1 \FALabel(9.18,12.25)[r] \FAProp(10.,5.5)(10.,10.)(0.,)/Straight-1 \FALabel(9.18,7.75)[r] \FAVert(10.,14.5)0 \FAVert(10.,5.5)0 \FAVert(10.,10.)0


(0.,15.)(10.,14.5)(0.,)/Straight1 \FALabel(5.0774,15.8181)[b] \FAProp(0.,5.)(10.,5.5)(0.,)/Straight-1 \FALabel(5.0774,4.18193)[t] \FAProp(20.,17.)(15.5,8.)(0.,)/Sine-1 \FALabel(18.0564,15.0818)[br] \FAProp(20.,10.)(10.,14.5)(0.,)/Sine1 \FALabel(12.6226,14.248)[bl] \FAProp(20.,3.)(15.5,8.)(0.,)/Sine0 \FALabel(18.4221,6.0569)[bl] \FAProp(10.,14.5)(10.,5.5)(0.,)/Straight1 \FALabel(9.18,10.)[r] \FAProp(10.,5.5)(15.5,8.)(0.,)/Sine0 \FALabel(12.9114,5.81893)[tl] \FAVert(10.,14.5)0 \FAVert(10.,5.5)0 \FAVert(15.5,8.)0


(0.,15.)(10.,5.5)(0.,)/Straight1 \FALabel(3.19219,13.2012)[bl] \FAProp(0.,5.)(10.,13.)(0.,)/Straight-1 \FALabel(3.17617,6.28478)[tl] \FAProp(20.,17.)(16.,13.5)(0.,)/Sine-1 \FALabel(17.4593,15.9365)[br] \FAProp(20.,10.)(16.,13.5)(0.,)/Sine1 \FALabel(17.4593,11.0635)[tr] \FAProp(20.,3.)(10.,5.5)(0.,)/Sine0 \FALabel(14.6241,3.22628)[t] \FAProp(10.,5.5)(10.,13.)(0.,)/Straight1 \FALabel(10.82,9.25)[l] \FAProp(10.,13.)(16.,13.5)(0.,)/Sine0 \FALabel(12.8713,14.3146)[b] \FAVert(10.,5.5)0 \FAVert(10.,13.)0 \FAVert(16.,13.5)0


(0.,15.)(10.,13.)(0.,)/Straight1 \FALabel(5.30398,15.0399)[b] \FAProp(0.,5.)(10.,5.5)(0.,)/Straight-1 \FALabel(5.0774,4.18193)[t] \FAProp(20.,17.)(15.5,13.5)(0.,)/Sine-1 \FALabel(17.2784,15.9935)[br] \FAProp(20.,10.)(15.5,13.5)(0.,)/Sine1 \FALabel(18.2216,12.4935)[bl] \FAProp(20.,3.)(10.,5.5)(0.,)/Sine0 \FALabel(15.3759,5.27372)[b] \FAProp(10.,13.)(10.,5.5)(0.,)/Straight1 \FALabel(9.18,9.25)[r] \FAProp(10.,13.)(15.5,13.5)(0.,)/Sine0 \FALabel(12.8903,12.1864)[t] \FAVert(10.,13.)0 \FAVert(10.,5.5)0 \FAVert(15.5,13.5)0

Figure 7: Generic bremsstrahlung diagrams for the process .

In a second step, the diagrams contributing to the bremsstrahlung amplitude (see Fig. 7) must be considered. In the first approach the amplitudes are generated with FeynArts and analytically squared within Mathematica using FeynCalc [93]. We use MadGraph [94] and FormCalc for internal checks, however, the computational performance of the FeynArts/FeynCalc-based code turns out to be more efficient. Alternatively, the corresponding Feynman-diagrams are generated with QGraf and the corresponding amplitudes, as well as the squared matrix elements, are evaluated analytically with FORM. In both approaches, the numerical evaluation of the bremsstrahlung contributions is carried out in FORTRAN using the VEGAS [95] algorithm.

In the computation of the real corrections, care has to be taken since the phase-space integral over exhibits IR singularities in phase-space regions where the photon is radiated collinear to an initial-state quark or becomes soft, i.e. the energy of the photon goes to zero. We apply the well-known technique of the two–cut-off phase-space slicing [96] to analytically carry out the phase-space integration over the soft and collinear singularities by using small mass regulators and for the light quarks and the photon, respectively, and exploiting universal factorization properties for the squared bremsstrahlung amplitudes in the soft and collinear limit (see, e.g., Ref. [97]). Accordingly, the IR singularities appear as and terms in this particular mass regularisation scheme. Note that the hierarchy has to be respected carefully in the evaluation of the IR-singular terms. To separate the soft and collinear phase-space configurations from the hard bremsstrahlung, one imposes a cut on the photon energy, and a cut on the angle between incoming (anti-)quark and photon, . The residual phase-space integration corresponding to hard, non-collinear photon emission, i.e.  and , can safely be carried out without IR regulators, allowing for an efficient numerical evaluation using Vegas. Adding the soft, collinear and hard contributions, the dependence on the slicing parameters and cancels out in the computation of IR-safe observables.2 For a more detailed description of the phase-space slicing method as applied in our computation, including all relevant formulae and further useful references, see e.g. Ref. [32].

According to the Bloch–Nordsieck theorem [98], the soft singularities emerging from real radiation cancel against corresponding contributions from the virtual corrections related to photon exchange between on-shell legs in loop diagrams. However, the initial-state collinear singularities survive and have to be absorbed in the renormalized PDFs, where we apply the factorization scheme as described in Ref. [99]. Since we do not include the leptonic decays of the final-state W bosons in this work, we do not have to deal with subtleties concerning infrared safety arising from the event-selection procedure for photon radiation off charged leptons in the final state.

3.3 NLO QCD contributions

In addition to the EW corrections, we have also recalculated the NLO QCD corrections to W-boson pair production, using the same setup as detailed above. The corresponding real corrections exhibit additional partonic channels with one gluon in the initial state, namely


where the contributions and formally contribute to associated production of a W-boson and a potentially resonant top-quark at leading order and thus have to be treated separately. We exclude those channels from our analysis by discarding events with a -quark in the final state, assuming a 100% tagging efficiency.

In the numerical analysis one finds that the distribution of the W-boson is plagued by huge NLO QCD -factors of a few hundreds of percent, resulting in potentially large uncertainties in the theory predictions. To circumvent this problem, we follow the strategy proposed in Ref. [49] and veto events with a hard jet recoiling against a hard , because these signatures belong to jet production at LO rather than being a correction to W-pair production. Technically, we discard events where the transverse momentum of a visible jet (with and ) is larger than half of the highest W-boson . (The important issue of giant QCD -factors has also been discussed in Ref. [100].)

4 Numerical results

In this section we present numerical results for total cross sections and differential distributions for the process at the Fermilab Tevatron for a total CM energy of , as well as for the process at the CERN LHC with CM energies of (LHC8) and (LHC14), respectively. We discuss in detail the phenomenological implications of the EW corrections. In the following the relative corrections are defined through . The photon- and gluon-induced channels (2) and (3), although formally contributing at leading order, are also considered as relative corrections and , respectively.

4.1 Input parameters and event selection

We use the following SM input parameters for the numerical analysis,


In the on-shell scheme applied in our computation, the weak mixing angle is a derived quantity. For the computation of the process and its EW radiative corrections, we use the MSTW2008LO PDF set [101] in the LHAPDF setup [102]. In order to consistently include corrections, in particular real radiation with the resulting collinear singularities, PDFs should take in principle these QED effects into account. Such a PDF analysis has been performed in Ref. [77], and the effects are known to be small, as far as their effect on the quark distribution is concerned [82]. In addition, the currently available PDFs incorporating corrections [77] include QCD effects at NLO, whereas our EW analysis is LO with respect to perturbative QCD only. For these reasons, the MSTW2008LO set is used as our default choice for the process.

W-pair production through gluon fusion is considered at leading order, hence the same leading-order set [101] is implemented. For W-pair production through the partonic subprocess we use the MRST2004QED set [77]. Note that this set has been also used in Ref. [103] to predict the contamination of the Drell–Yan process by the contribution.

For the computation of the QCD corrections, however, the MSTW2008NLO set is adopted, corresponding to a value of . As explained above, the CKM matrix can be assumed to be diagonal. The renormalization and factorization scales are always identified, our default scale choice being the phase-space dependent average of the W-boson transverse masses


for the evaluation of the QCD as well as the EW corrections.3 A similar scale choice was taken in Ref. [52] for the computation of the EW corrections to four-lepton production at the LHC. In our default setup, we require a minimum transverse momentum and a maximum rapidity for the final-state W bosons,


to exclude events where the bosons are emitted collinear to the initial-state partons. As far as QCD corrections are concerned, the jet veto introduced before is only applied to visible jets with


to ensure infrared safety.

4.2 Integrated cross sections

We start the discussion of our numerical results with the presentation of integrated cross sections at the LHC and Tevatron, evaluated with our default setup. Table 1 shows integrated LO cross sections, together with the (relative) EW corrections, contributions from , gluon fusion and QCD corrections. Moreover, relative corrections due to massive-boson radiation (denoted as ) are included, which will be discussed in the following chapter. In addition, the corresponding results for the total cross sections without any event-selection cuts and without a dedicated jet veto are shown in Table 2 as a benchmark. While our default cuts reduce the LO -induced cross section by 44% (34%) at the LHC14 (LHC8), the Tevatron cross section is only marginally affected. The relative EW corrections are changed significantly but remain small as expected. In contrast, the relative QCD corrections to the total cross sections are reduced from 40% (30%) at the LHC (Tevatron) to almost zero (10%) after introducing the jet veto proposed at the end of Section 4.1.

Table 1: Integrated leading-order cross sections and relative corrections for the LHC and the Tevatron evaluated with the default setup defined in Section 4.1.
Table 2: Total leading-order cross sections and corresponding relative corrections for the LHC and the Tevatron without any phase-space cuts.

Predictions for the production cross section with cuts on the transverse momenta or the invariant mass of the W-pair have been presented already in Section 2. For cross checks and to simplify numerical comparisons we show in Tables 3 and 4 the cross sections and the corresponding relative corrections for different cut values on the W transverse momenta (Table 3) and on the invariant mass of the W-boson pair (Table 4). While the relative corrections due to process (3) are negligible at high invariant masses and transverse momenta, we recall that the photon-induced channels lead to surprisingly large relative contributions of at , compensating a significant part of the moderate negative Sudakov-enhanced genuine EW contributions. This behaviour may be understood qualitatively recalling that the partonic cross section is, for sufficiently high energies, given by a constant, , and strongly peaked at small angles, while the corresponding cross section for large decreases as . Although the NLO QCD corrections turn out to be accidentally small in our default setup, they still amount to for large and , respectively, exhibiting a similar behaviour at the LHC and the Tevatron (see Figs. 3, 4 and Tables 3, 4).

Table 3: Integrated leading-order cross sections and relative corrections for different values of at the LHC at , evaluated with the default setup defined in Section 4.1.
Table 4: Integrated leading-order cross sections and relative corrections for different values of at the LHC at , evaluated with the default setup defined in Section 4.1.

4.3 Transverse-momentum, invariant-mass and rapidity distributions

Figure 8: Distributions of the transverse momentum at the LHC14 (top), LHC8 (center) and the Tevatron (bottom). On the left-hand side, LO contributions due to processes (1)(),  (2)(), and (3)() are shown. On the right-hand side, corresponding relative corrections are presented, normalized to the dominating LO channel (1). See text for details.
Figure 9: Distributions of the invariant mass of the W-boson pair at the LHC14 (top), LHC8 (center) and the Tevatron (bottom). On the left-hand side, LO contributions due to processes (1)(),  (2)(), and (3)() are shown. On the right-hand side, corresponding relative corrections are presented, normalized to the dominating LO channel (1). See text for details.
Figure 10: Distributions of the rapidities of the W-bosons at the LHC14 (top), LHC8 (center) and the Tevatron (bottom). On the left-hand side, LO contributions due to processes (1)(),  (2)(), and (3)() are shown. On the right-hand side, corresponding relative corrections are presented, normalized to the dominating LO channel (1). See text for details.
Figure 11: Distributions of the rapidities of the W-bosons at the LHC14 (top) and LHC8 (bottom) for . On the left-hand side, LO contributions due to processes (1)(),  (2)(), and (3)() are shown. On the right-hand side, corresponding relative corrections are presented, normalized to the dominating LO channel (1). See text for details.
Figure 12: Distributions of the rapidities of the W-bosons at the LHC14 (top) and LHC8 (bottom) for . On the left-hand side, LO contributions due to processes (1)(),  (2)(), and (3)() are shown. On the right-hand side, corresponding relative corrections are presented, normalized to the dominating LO channel (1). See text for details.
Figure 13: Distributions of the W-boson rapidity gap at the LHC14 for . On the left-hand side, LO contributions due to processes (1)(),  (2)(), and (3)() are shown. On the right-hand side, corresponding relative corrections are presented, normalized to the dominating LO channel (1). See text for details.
Figure 14: Distributions of the W-boson rapidity gap at the LHC8 for . On the left-hand side, LO contributions due to processes (1)(),  (2)(), and (3)() are shown. On the right-hand side, corresponding relative corrections are presented, normalized to the dominating LO channel (1). See text for details.

In a first step we show the transverse-momentum distributions of and (Fig. 8) and the distributions of the invariant mass (Fig. 9) for LHC14, LHC8 and the Tevatron. The interpretation of the results is similar to that of the partially integrated cross sections presented in Figs. 3 and 4 in Section 2. In particular, we again observe the large Sudakov logarithms resulting in corrections of for values around 800 GeV. Again, the correction for the invariant-mass distribution is smaller, since large may still involve small momentum transfer . At the LHC, the QCD corrections evaluated with the jet veto proposed in Section 4.1 still reach 50% (20%) at high (), while at the Tevatron they do not exceed 20% even at large transverse momenta.

Rapidity distributions of and individually are shown in Fig. 10. Note that the rapidity distributions of and are different at the LHC, as a consequence of the asymmetric piece of the differential cross section for and , and the difference between valence- and sea-quark distributions. For the Tevatron one finds . At the LHC8 (LHC14), gluon fusion and the process increase the W-pair production rate by (10%) and (2%), respectively. The EW corrections are , reflecting the small of typical events and the absence of the large negative Sudakov logarithms. At the Tevatron, only QCD corrections must be considered for this particular observable; EW corrections, and fusion can safely be neglected.

The rapidity distributions for the LHC14 and the LHC8 for the subsamples with invariant mass and are shown in Figs. 11 and 12, respectively. In these cases EW corrections and the process are significantly more important.

Let us finally consider W-pair production at the highest energies accessible at the LHC. As already mentioned in Section 2, the distribution in the rapidity difference (for fixed ) between and bosons can be directly related to the angular distribution in the rest frame and thus can be used to search for anomalous couplings at the TeV scale. In Figs. 13 and 14 we therefore show these distributions for for LHC14 and for LHC8, respectively. At the LHC14, large contributions are visible both from the process () and from the weak corrections (), leading to a strong distortion of this particular distribution. As already discussed in Section 2, the additional contributions from and fusion are small in the region accessible to the Tevatron and hence will not be discussed further.

Finally, we compare our predictions to older results obtained in the high-energy approximation. Unfortunately, a tuned comparison of our results with the ones presented in Ref. [52] is not possible, since in that paper dedicated event-selection cuts on the leptonic decay products of the W bosons are applied. However, comparing Fig. 7 (bottom, left) from Ref. [52] with Fig. 5 (bottom, right), we find a reasonable agreement for the relative EW corrections within a few percent. Here, we assume that each of the charged leptons on average carries away 50% of the momentum of the decaying W, and a strong correlation between and , which seems to be justified in case of strongly-boosted Ws.

5 Radiation of massive vector bosons

Figure 15: Left: Total LO cross sections for , () and production; Right: Corrections due to and relative to quark-induced W-pair production. See text for details.

Finally, we discuss the phenomenological effects of additional massive vector bosons in the final state produced in the partonic processes


Although these channels are parametrically suppressed by one order of the EW coupling, collinear or soft massive boson radiation may potentially lead to logarithmically enhanced contributions and thus needs further investigation. In the numerical analysis we use a simplified approach to conservatively estimate the pollution of W-pair production which can be expected from 3-boson final states; we treat the additional boson completely inclusively, whereas our default cuts are applied to those opposite-sign W-bosons with highest . In Fig. 15 we present the corresponding numerical results for the () final state as well as the contribution from hard-photon radiation ( GeV, ) relative to quark-induced production at leading order. (The corresponding numbers can be found in Tables 3 and 4, respectively.) The relative corrections due to massive boson radiation are below 5% even for large transverse momenta and invariant masses and therefore of minor importance. Remarkably enough the contribution from hard photon radiation with these cuts (which is included in the cross sections discussed in the previous chapter) is numerically close to the one from additional massive boson radiation.

6 Conclusions

We have calculated the full NLO EW corrections to -boson pair production at hadron colliders and present predictions valid in the full energy reach of the LHC. At large parton CM energies, the relative corrections are dominated by the well-known universal Sudakov logarithms that lead to substantial negative contributions, while the QCD corrections turn out to be moderate after application of a dynamic jet veto.

As a surprising new result we find that photon-induced contributions can be of the same size as the genuine EW corrections at high energies and moderate scattering angles and thus must not be neglected when predicting W-boson pair production at highest energies. At low scattering angles, however, the channel dominates the EW contributions even at moderate energies. In the future, the leptonic decays of the W bosons should be included properly, allowing for a realistic event definition to match the increasing accuracy of future LHC measurements.


We would like to thank Peter Marquard for fruitful discussions. This work has been supported by “Strukturiertes Promotionskolleg Elementarteilchen- und Astroteilchenphysik”, SFB TR9 “Computational and Particle Physics” and BMBF Contract 05HT4VKATI3.


  1. The interplay of logarithmic EW corrections and anomalous trilinear gauge-boson couplings has been studied in Ref. [81] for WZ and WW production.
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  3. Although the relative EW corrections hardly depend on the choice of the factorization scale, the QCD corrections can be significantly stabilized using a scale which is adjusted to the kinematics of the underlying hard process. (For a detailed discussion of this important issue, see, e.g., Ref. [50].)


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