$k_t$-factorization vs collinear factorization in $W^{+}W^{-}$ pair production at the LHC

In this paper, we calculate the inclusive rate of $W^{+} W^{-}$ pair production through leptonic decay channels $W^+W^- \to l^+\nu_l + l^{\prime -} \nu_{l'}$ in the $k_t$-factorization framework. We also consider the exclusive $W^{+} W^{-}$ pair production through the one-loop induced $gg \to H \to W^+W^-$ channel that is important for the study of new physics beyond the Standard Model. The results are compared with predictions from the Herwig 7 event generator in the collinear factorization framework and with the experimental data from the ATLAS and the CMS collaborations. It will be shown that our predictions for the $W^+W^-$ boson pair production signals are in agreement with the experimental data as well as the collinear results.


I. INTRODUCTION
The production of the W + W − pairs is one of the most important electroweak (EW) processes at the CERN Large Hadron Collider (LHC), that provides a strong test for the viability of the Standard Model (SM) . Moreover, these processes have played a key role in precision measurements at the LHC and also for the estimation of the irreducible backgrounds in Higgs boson searches. Furthermore, detecting any deviation from the theoretical predictions of the SM could automatically signal the existence of new physics beyond the Standard Model (BSM). This is even more relevant for heavy gauge vector boson (HGVB) pair production events (e.g. γ/H/Z 0 → W + W − ), due to their sensitivity to BSM modifications, e.g. via an extended Higgs sector [22].
To the date, a number of theoretical calculations for the prediction of the W + W − pair production rate have been done in different frameworks, to LO [23], NLO [24][25][26][27] and NNLO [28,29] in QCD accuracies.
Here, we calculate the inclusive rate of W + W − pair production through leptonic decay channels W + W − → l + ν l + l − ν l in two different approaches: (i) The k t -factorization framework where we use the transverse momentum dependent parton distribution functions (TMD PDFs) of Kimber-Martin-Ryskin (KMR) [30,31]. In this framework, un-integrated parton distributions (UPDFs) are used to weight the relevant partonic sub-processes.
These transverse-momentum-dependent parton distribution functions (TMD PDFs) are defined based on the solutions of the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations [32][33][34][35], and the last-step evolution approximation. The latter, directly introduces transverse momentum dependency into the partonic densities by softening the strong ordering assumption, i.e. k 2 t,1 · · · k 2 t,semi-hard ∼ µ 2 hard . This formalism suppresses the soft gluon singularities that arise from color coherence interference effects, by employing the angular ordering constraint (AOC) [36] and effectively limits its evolution to a singularity-free phase space. Hence, the choice of AOC has a definitive effect on the characteristics of the resulting UPDF. Employing different AOC visualizations have resulted in different formalisms, namely the KMR, LO Martin-Ryskin-Watt (MRW) and NLO MRW UPDFs [31]. The capability of these phenomenological assets in correctly describing various physical observables have been already investigated, see e.g. [22,[37][38][39][40][41].
Additionally, we calculate the exclusive W + W − pair production through the one-loop induced gg → H → W + W − channel in both k t -factorization and collinear frameworks. This channel is particularly important in searching for BSM signal at the LHC. We will report our study on exclusive W + W − pair production in BSM within an upcoming paper. There we will use the BSM sensitive gg → H i → W + W − vertices, with H i being either the SM Higgs boson or extra Higgs scalars in the Maximally Symmetric Two Higgs Doublet Model (MS-2HDM) [55] and in the 2HDM type II with cancellation quadratic divergences [56]. We will compare our results for both inclusive and exclusive W + W − pair production with the existing experimental data from ATLAS [17][18][19] and CMS [20].
The outline of this paper is as follows: Section II includes the details of partonic interactions for W + W − pair production at the LHC. In Section III, we present a brief description of our calculations in the k t -factorization framework and will show the setups for our numerical analysis. Section IV contains our predictions for the inclusive W + W − pair production as well as the Higgs-decay-driven W + W − production. The results of both frameworks are compared with the experimental data from the ATLAS and the CMS collaborations. These comparisons will show that our predictions for the W + W − boson pair production signals are in agreement with the experimental data. Finally, the section V will outline our conclusions.
We consider the inclusive production of the W + W − pairs at the LHC that decay to two charged leptons and two neutrinos. In such processes, the general hadronic scattering is with A(P 1 ) and B(P 2 ) being the colliding beam protons, a(k 1 ) and b(k 2 ) denote the partons that initiate any of the involved sub-processes and l ± = e ± , µ ± are the resulting final-state fermions. The 4-momenta of the beam protons, the initial partons, the exchanged W -bosons and the final state 1 The Herwig 7 multi-purpose event generator provides a versatile platform for performing this class of analysis.
Such calculations can be made in different setups, depending on the requirements, for instance, see [27].   leptons are denoted by P i , k j , q m and p n , respectively. The hadronic scattering (1) is dominated by the following partonic channels: These sub-processes are depicted in Figure 1 and Figure 2.
The relevant leading-order (LO) sub-processes are comprised only of the quark-quark tree-level contributions (2) and the LO Drell-Yan type processes (3), which are the dominant contributions into the inclusive W + W − pair production events at the LHC current center-of-mass energies. These sub-processes are shown as the diagrams (a) and (b) of the Figure 1. Additionally, the relevant tree-level plus one jet sub-processes are the P 1 + P 2 → W + + W − + J contributions shown in processes (4), (5),(6) and (7) and correspondingly in the diagrams (c) through (f) of the Figure 1.
In the case of the W + W − pair production via QCD one-loop channels, one can argue the importance of counting the gluon-gluon fusion contributions via single QCD loop diagrams as shown in sub-processes (8) and (9) and correspondingly in the diagrams (a) and (b) of the Figure 2 [28, 29,57]. This, despite being valid in the case of the collinear framework, will cause irreducible double-counting while working within the k t -factorization formalism, directly due to the use of the box and crossed-box diagrams in the definition of the k t -factorization partonic densities 2 [38,58,59].
The production of W + W − pairs through Higgs decay is predominantly dominated by the gluongluon fusion channel as shown in the sub-process (10) and in Figure 2c. Note that the higher-order (QCD and radiative) corrections to this channel are considerable. However, it has been shown that by using the K-factor approximation one can capture a very good description of the event, without the need for adding higher-order calculations [37,60].
The existence of a leptonic final-state for a W + W − pair production event provides a clean experimental signature while preventing reconstruction of the H → W + W − resonance. The latter, in turn, causes some difficulty for the detection of the background signals and lowers the sensitivity of the experimental measurements [29]. Therefore, it would be more important to increase the precision of the theoretical calculations to enhance our understanding of the H → W + W − signal [61,62]. Meanwhile, since it is not possible to calculate the invariant masses of the HGVB in a pair-production event, the process (1) can be considered as an irreducible background signal for many LHC measurements.

III. CALCULATION FRAMEWORK
At the hadronic level, the differential cross-section for the production of W + W − pairs (via leptonic decay) in the k t -factorization framework can be written as with the particle phase space, dφ, and the flux factor, F , defined as Here, y i and φ i are the pseudorapidities and the azimuthal angles of emission of the final-state particles, respectively, while s in the Eq. (13) is the center-of-mass energy squared in the infinite Also, x 1 and x 2 are defined as with m i,t = (m 2 i +p 2 i,t ) 1/2 being the transverse mass of the final state particles. The matrix elements of the relevant sub-processes, M, are calculated using the Feynman rules in combination with the Eikonal approximation for the incoming quark spin densities and the non-sense polarization approximation for the polarization vectors of the incoming gluons [37,[63][64][65][66][67][68]. To generate the analytic expressions for these matrix elements, we have used the algebraic manipulation package FORM [69] and checked our results, independently, by MATHEMATICA.
The differential cross-section for W + W − pair production of hadronic collisions can be derived for the sub-processes (2) and (3) as 2,t dp 2 1,t dp 2 2,t dp 2 3,t dy 1 dy 2 dy 3 dy 4 dψ 1 2π and for the sub-processes (4) through (7) as 2,t dp 2 1,t dp 2 2,t dp 2 3,t dp 2 4,t dy 1 dy 2 dy 3 dy 4 dy 5 Finally, for the sub-process (10) we have 2,t dp 2 1,t dp 2 2,t dp 2 3,t dy 1 dy 2 dy 3 dy 4 where ψ i are the azimuthal angles of the initial partons and µ 2 c = m H,t with m H and p H,t being the mass and the transverse momentum of the exchanged Higgs boson in the sub-process (10). The functions f a (x, k 2 t , µ 2 ) in the Eqs. (16), (17) and (18) are the KMR UPDFs. These double-scaled TMD PDFs depend on the fraction of the longitudinal momentum of the parent hadron carried by the incoming parton, x, transverse momenta k t and the hard-scale µ.
The KMR UPDFs are defined as [30] f a (x, with the Sudakov form factor, with α S ≡ α S (k 2 ) as the running coupling of the strong interaction and P ab (z) as the LO splitting functions for b → a + X partonic splittings [30,70]. The upper bounds of the integrations in the Eqs. (19) and (20), i.e. µ/(µ + k t ), are the manifestations of the AOC that characterize the kinematics of the KMR UPDFs. Furthermore, b(x, k 2 t ) are the solutions of the DGLAP evolution equations. For the purpose of our calculations, these PDFs are obtained from the MMHT2014-LO libraries [71].
Here, we numerically calculate the differential cross-section for the production of W + W − pairs by choosing the hard-scale of the process as and therefore lim The corresponding event selection constraints are chosen by the specifications of the existing experimental measurements as shown in Table I.

IV. NUMERICAL RESULTS AND DISCUSSION
In this section, we present our results for the calculation of inclusive W + W − pair production at √ s = 8 TeV and 13 TeV, through leptonic decay channels W + W − → l + ν l + l − ν l in two In the figure 3 the differential cross-section for the production of W + W − pairs are plotted as a function of cos * θ, with ∆y being the difference between the rapidities of the produced leptons. As expected, it can be seen that the LO channels have the largest contributions into the production rate (∼62.0%) with the t-channel q + q → W + + W − having the greatest impact (∼42.3%). On the other hand, the LO plus 1 jet channels contribute upto about 27.6% towards the total cross-section while the one-loop induced g + g → H → W + + W − (enhanced by the use of the K-factor approximation) contributes about 10.4%. The smallest contributions are coming from the s-channel LO plus 1 jet sub-processes that are associated with a γ → W + W − decay vertex. These include a 0.5% share for the quark-quark channel and a 0.6% for the quark-gluon channel.
The left panel of Figure 3 illustrates a comparison between the k t -factorization predictions, within the corresponding uncertainty bounds, to similar predictions from Herwig 7 and the experimental measurements of the ATLAS collaboration [18]. In comparison to the experimental measurements, as well as the collinear predictions, it can be observed that the k t -factorization results within their uncertainty bounds produce a relatively good description of the data. The overall behaviour of our predictions in these plots is similar to our previous description for to the experimental measurements as well as the collinear results. What is significant here is the behaviour of the g + g → H → W + W − histograms in the high-p t regions, i.e. for the regions equivalent to p t > 150 GeV (similarly for the region m > 250 GeV). In these domains, the g + g → H → W + + W − contributions into the production event become dominant, lifting the high-p t tails of these histograms and ensuring that the total predictions are comparable to the experimental data.
Besides the above-mentioned indirect deviance for the correct behaviour of our g + g → H → Although in this case, the precision of the data is not as good as the inclusive case, our usual comparisons show a sound behaviour for our predictions within their uncertainty bounds.
From the above comparisons, it appears that the SM predictions are satisfactory for describing   Herwig 7 [42,43] and to the experimental measurements of the ATLAS collaboration [17]. The left panel also contains a similar comparison for W + W − pair production via Higgs boson decay and ATLAS [18].
the experimental image of W + W − pair production at the LHC, at least for the √ s = 8 TeV centerof-mass energies. However, one can expect the higher center-of-mass energies to provide a better chance for observing the contributions from BSM, e.g. with extended Higgs sectors. Recently, the ATLAS collaboration has published another set of measurements for the inclusive production of In Figures 10, 11 [42,43] and to the experimental data from ATLAS and CMS [18,20].
k t -factorization in describing the experimental data. This is since the AOC and an intrinsically embedded initial state real emission 3 are inherently higher-order effects and are more relevant in higher energy contents.
Overall, the above comparisons demonstrate that our SM base-line calculations in the k tfactorization formalism are satisfactory, in agreement with the conventional theoretical approaches of the same level of QCD accuracy and adequate to describe the experimental data.

V. CONCLUSIONS
We have calculated the inclusive rate of W + W − pair production through leptonic decay channels within the corresponding uncertainty bounds, to the similar predictions from Herwig 7 [42,43] and to the experimental data from ATLAS [19].
Here we considered the relevant partonic sub-processes up to a NLO-equivalent-level QCD  11. Differential cross-section for the production of W + W − pairs as a function of the azimuthal angle between the produced leptons, ∆φ , at √ s = 13 TeV. The notation of the figure is the same as Figure 10.  FIG. 12. Differential cross-section for the production of W + W − pairs as a function of the mass of the produced lepton pair, m , at √ s = 13 TeV. The notation of the figure is the same as Figure 10.
particle interactions, hadronization, etc. Nevertheless, we have shown that our k t -factorization framework, despite its simplicity, is adequate for describing the existing experimental data of   FIG. 14. Differential cross-section for the production of W + W − pairs as a function of the transverse momentum of the produced lepton pair, p t , at √ s = 13 TeV. The notation of the figure is the same as   15. Differential cross-section for the production of W + W − pairs as a function of the pseudorapidity of the produced lepton pair, y , at √ s = 13 TeV. The notation of the figure is the same as Figure 10.
the fiducial cross-section for the production of W + W − pairs in both inclusive and Higgs-decayoriginated cases.
The large uncertainty of the k t -factorization framework is a consequence of putting the last-step evolution approximation on top of the intrinsic collinear uncertainties and forcing an additional controlling evolution scale (k t ) into the calculation. This problem can be solved by performing a global fit for these UPDFs to the Hadron-Hadron scattering data, as well as including higherorder effects in their respective calculations. We hope that the promising results such as those presented in this paper would provide an incentive for the particle physics community to move in this direction.
Additionally, we consider the exclusive W + W − pair production through gg → H → W + W − channel that is important for the study of the BSM physics. Our framework can successfully provide the necessary SM base-line for the on-going search for BSM signal in the LHC run 2 data, which would be the subject of our next paper.