Next-to-leading-order QCD and electroweak corrections to WWW production at proton-proton colliders

Journal of High Energy Physics, Sep 2017

Triple-W-boson production in proton-proton collisions allows for a direct access to the triple and quartic gauge couplings and provides a window to the mechanism of electroweak symmetry breaking. It is an important process to test the Standard Model (SM) and might be background to physics beyond the SM. We present a calculation of the next-to-leading order (NLO) electroweak corrections to the production of WWW final states at proton-proton colliders with on-shell W bosons and combine the electroweak with the NLO QCD corrections. We study the impact of the corrections to the integrated cross sections and to kinematic distributions of the W bosons. The electroweak corrections are generically of the size of 5-10% for integrated cross sections and become more pronounced in specific phase-space regions. The real corrections induced by quark-photon scattering turn out to be as important as electroweak loops and photon bremsstrahlung corrections, but can be reduced by phase-space cuts. Considering that prior determinations of the photon parton distribution function (PDF) involve rather large uncertainties, we compare the results obtained with different photon PDFs and discuss the corresponding uncertainties in the NLO predictions. Moreover, we determine the scale and total PDF uncertainties at the LHC and a possible future 100 TeV pp collider.

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Next-to-leading-order QCD and electroweak corrections to WWW production at proton-proton colliders

Received: May Next-to-leading-order QCD and electroweak corrections to W W W production at proton-proton Stefan Dittmaier 0 2 Alexander Huss 0 1 Gernot Knippen 0 2 Zurich 0 Switzerland 0 0 D-79104 Freiburg , Germany 1 Institute for Theoretical Physics , ETH 2 Albert-Ludwigs-Universitat Freiburg, Physikalisches Institut Triple-W-boson production in proton-proton collisions allows for a direct access to the triple and quartic gauge couplings and provides a window to the mechanism of electroweak symmetry breaking. It is an important process to test the Standard Model (SM) and might be background to physics beyond the SM. We present a calculation of the next-to-leading order (NLO) electroweak corrections to the production of WWW NLO Computations - W W W nal states at proton-proton colliders with on-shell W bosons and combine the electroweak with the NLO QCD corrections. We study the impact of the corrections to the integrated cross sections and to kinematic distributions of the W bosons. The electroweak corrections are generically of the size of 5-10% for integrated cross sections and become more pronounced in speci c phase-space regions. The real corrections induced by quark-photon scattering turn out to be as important as electroweak loops and photon bremsstrahlung corrections, but can be reduced by phase-space cuts. Considering that prior determinations of the photon parton distribution function (PDF) involve rather large uncertainties, we compare the results obtained with di erent photon PDFs and discuss the corresponding uncertainties in the NLO predictions. Moreover, we determine the scale and total PDF uncertainties at the LHC and a possible future 100 TeV pp collider. 1 Introduction 2 3 Input parameter scheme 4 Numerical results 4.1 4.2 4.3 4.4 Di erential distributions NLO WWW cross sections with a jet veto Discussion of PDF uncertainties in the photon-induced channel Estimating the uncertainties of the total cross sections 5 Conclusion A Results at a single phase-space point lished in the narrow width approximation of the W bosons [6]. Our NLO calculation of EW and QCD corrections, which is based on on-shell W bosons, complements this calculation by presenting additional results and carefully assessing the impact of the uncertainties that arise from the PDFs. The issue of PDF uncertainties is particularly important for WWW production, since quark-photon induced channels have a large impact on the cross section and previous determinations of the photon PDF su er from large uncertainties. The recently released LUXqed photon PDF [7], however, is rather precise and stabilizes the prediction considerably. { 1 { This paper is structured as follows: section 2 describes the basic properties of production at proton-proton colliders and technical details of our NLO calculation. It further covers the checks and validations we have performed on our calculation. The setup of the calculation and the input parameters are summarized in section 3. We present results on total and di erential cross sections, determine the scale dependence of the NLO cross section, and assess the error induced by the uncertainty of the PDF in section 4. We conclude with section 5. 2 Triple W-boson production at NLO At leading order (LO), the production of three W bosons at proton-proton colliders is induced by the two charge-conjugated partonic subprocesses uidj ! W W+W+ and uidj ! W+W W ; (2.1) where i and j are the indices of the fermion generation. The di erent Feynman diagrams contributing to W W+W+ production at LO are shown in gure 1. As can be seen from the last diagram, the quartic WWWW coupling already enters the LO prediction. The production of three W bosons further incorporates Higgs production in association with a W boson, where the Higgs particle decays into two W bosons. However, owing to the on-shell requirement on the W bosons in our calculation, the Higgs boson is restricted to be purely o -shell. At NLO, additional partons appear in the real emission contributions, namely photons in the NLO EW real emission, gluons in the NLO QCD real emission, and quarks in the gluon-induced and photon-induced channels. A selection of NLO real emission Feynman diagrams is depicted in gure 2. In our calculation, infrared (IR) singularities, which arise due to soft and/or collinear emission, are dealt with using the dipole subtraction formalism [8{12]. In the virtual contribution, which incorporates additional closed fermion loops and virtual photon, gluon, or weak-vector-boson exchange, we encounter one-loop topologies up to pentagon diagrams. The tensor and scalar loop integrals are evaluated using the Collier library [ 13 ], which is based on the results of refs. [14{16]. Examples of NLO EW loop diagrams are shown in gure 3. We have performed two completely independent calculations: one where the amplitudes are generated and further processed with the packages FeynArts [17] and FormCalc [18], and a second calculation using in-house software based on Feynman diagrams generated with FeynArts 1 [19]. The results of the two calculations agree within the Monte Carlo integration errors. We have further checked that the results do not depend on the regularization scheme, employing either mass or dimensional regularization for the treatment of IR divergences. To check for consistency, we also compare with NLO results on WWW production available in the literature [4{6]: tables 1 and 2 show the comparison of our results to the results in the literature and agreement is found within the Monte Carlo integration errors, with the exception of the QCD result of ref. [6], where a phenomenologically insigni cant deviation is found at the few per mill level. { 2 { HJEP09(217)34 dj W+ ui dj /Z ui dj W W+ W ui dj /Z W W+ W W+ W W+ ui dj H W W W+ emission, gluonic emission, quark-photon, and quark-gluon induced channels) of the process pp ! LO [fb] W+W+ production at the LHC and a center-of-mass (CM) energy of p our results [6] de nition of the NLO cross section and the relative corrections were chosen as reported in ref. [6] with R = F = 32 MW. where the superscript OS stands for the on-shell scheme. We neglect all fermion masses except for bottom- and top-quark masses, and further ignore the negligible mixing involving the third generation quarks.1 As a result, the CKM matrix factorizes from all matrix elements and can be absorbed into the parton luminosities.2 Furthermore, the SM behaves like a CP-conserving theory in our calculation. We calculate with a block-diagonal CKM matrix, where the mixing among the rst two generations is parametrized by the Cabibbo angle Cabibbo = 0:22731, so that the relevant CKM entries are given by jVudj = jVcsj = 0:97428; jVusj = jVcdj = 0:22536: We work in the G -scheme (see, e.g., ref. [21]), where the electromagnetic coupling = G is a derived quantity and given by p 2 G = G M W2 1 M W2 MZ2 : (3.2) (3.3) 1Note that due to the negligible mixing involving quarks of the third generation, the bottom quark never 2Owing to the mass degeneracy among the quarks of the rst two generations and the absence of mixing with the third generation in our setup, the dependence on the CKM matrix drops out whenever a summation over internal and external nal-state avors is performed. The only case where the unitarity of the CKM matrix cannot be exploited in this way is when the up- and down-type quark are both in the initial state and thus receive di erent weights from the PDFs. The calculation can still be performed using a diagonal CKM by absorbing the CKM factors into the parton luminosities in this case. { 4 { The G -scheme accounts for universal corrections to the parameter and the running of from the Thomson limit to the electroweak scale. The running of the strong coupling constant s is taken from the PDF set used. We employ a dynamic renormalization and factorization scale ( R and F, respectively) given by r where pT;i is the (vectorial) transverse momentum of particle i and S is the set of all outgoing particles which carry no color. Note that this scale choice is equal to the production threshold 3MW if there are no color-charged particles in the nal state. In order to estimate the residual theory uncertainties from missing higher-order corrections, we examine the scale dependence in section 4 by varying the scales with respect to the central choice (3.4) by factors of 12 and 2. We use the LHAPDF6 library [22] to perform the convolution of the partonic cross sections with the PDFs. We calculate the pure LO cross section, denoted by LO, with the LO NNPDF3.0 set [23]. All NLO contributions, including the LO contribution to the NLO cross section 1LO, are calculated with the NLO NNPDF3.0QED [23, 24] set. The photon-induced contributions are calculated with the LUXqed set [7]. Since all PDFs in the LUXqed set, except for the photon PDF, are based on the PDF4LHC NNLO set [25], the error introduced by using di erent PDFs for the quark-photon-induced and every other channel should be negligible in the overall PDF uncertainty. We additionally provide results using the NNPDF3.0QED and the CT14QED (inclusive) [26] PDF sets in the quarkphoton-induced channels to better assess the corresponding uncertainty. Throughout this work we use PDF sets with ve active avors. 4 Numerical results EW qq0 qq0 EW correction qEqW0 at large energies, from the long-range QCD e ects and is preferable over a purely additive approach [27{29]. Note that by normalizing the QCD correction to the LO cross section evaluated with LO PDFs, the term 1+ QCD is identical to the usual de nition of the K-factor, up to small QED corrections in the PDFs. Moreover, the EW correction factor qEqW0 becomes rather insensitive to the PDF choice and the factorization scale. { 5 { qEqW0 [%] , at di erent CM energies p Carlo integration errors. s of the collider. The indicated errors are estimates for the Monte 1:7 at LHC energies of 13 TeV and 14 TeV. The photoninduced contributions are positive and overcompensate the negative EW corrections of the quark-induced channels, leading to total EW corrections of 6% and 7% at the current LHC energy of 13 TeV for the two charge-conjugated nal states, respectively. Note that this partial cancellation is not systematic in the sense that the two compensating e ects are not directly correlated. Due to the impact of the EW corrections it is important to take the EW corrections into account, when comparing data to theory. We estimate the missing higher-order EW corrections to be of the order of the squares of the individual NLO corrections, i.e. 1% for LHC energies. We observe that the EW corrections in the pure quark-induced channels, which are generically of O( 5%), show only very little sen sitivity to the collider energy. The quark-photon-induced contribution, on the other hand, rises with the pp scattering energy, reaching 40% for the scenario of a future 100 TeV collider. This demonstrates the importance of determining the photon PDF precisely for high-energy proton-proton scattering. The QCD corrections increase with growing collider CM energy owing to the higher gluon luminosity. The large K-factors of 1:7 ( 2:5) { 6 { at 13 TeV (100 TeV), which are driven by the quark-gluon-induced channels, ask for further improvements by higher-order QCD corrections. At least improvements by multi-jet merging seem mandatory. production mode among the two charge-conjugated processes. As both nal states can be easily separated, we will focus on the dominant, positively charged nal state in the following. A selection of di erential distributions including a breakdown of the corrections into the relative factors de ned in eq. (4.1) is presented in gures 4, 5, and 6. Note HJEP09(217)34 that the size of these corrections will be inherited by distributions based on decay leptons when dropping the on-shell requirement on the W bosons. As high transverse momenta of the W or high total invariant masses correspond to high partonic CM energies, the unitarizing e ect of Higgs exchanges can be seen in the drop of the associated di erential distributions shown in gures 4 and 5. In this high-energy regime, Sudakov logarithms from soft EW gauge-boson exchange are the leading contribution to the EW correction in the quark-induced channel, yielding corrections of several 10 % which can even overrule the large quark-photon-induced corrections at very high pT. At low invariant masses near the production threshold, the e ect of the Coulomb singularity, which arises due to photon exchange between W bosons, is visible. In this region the leading behavior of the NLO EW correction qEqW0 is dominated by Coul CM frame [ 30 ]. Even though the QCD corrections grow with increasing pT of the W boson they are rather independent on the total invariant mass. Figure 6 shows that the NLO QCD correction changes the shape of the distribution in the di erence of the azimuthal angle, preferring smaller angle di erences. This e ect is slightly enhanced by the total EW correction. 4.2 NLO WWW cross sections with a jet veto The large impact of the quark-photon-induced channel on the total cross section can be reduced by restricting the phase space of the additional jet in the nal state by a jet veto. To this end, we require the transverse momentum of the additional outgoing parton, which can be experimentally identi ed with a jet, to be below a certain threshold value pT;cut. This threshold should not be chosen too small in order to not a ect the e ective cancellation of IR singularities. Otherwise, large logarithms of the jet-veto cut would remain in the nal result requiring resummation [ 31 ], which however is beyond the scope of this work. As we cut on the transverse momentum of the jet alone, only the quark-photon-induced channel, the quark-gluon-induced channel and the QCD real emission contribution are a ected. The integrated cross sections for di erent values of pT;cut are presented in table 4. In gure 7 the impact of the pT-cut on the relative corrections in W W+W+ production is shown. A { 7 { 10− 4 10− 7 10− 8 50 0.1 × δ QCD δ qEqW0 NLO LO HJEP09(217)34 0 200 800 1000 400 600 curve of the QCD correction is scaled down by a factor of 0:1. NLO LO 10 5 0 √pps→ = 13 TeV 0.1 × δ QCD 1000 1500 MWWW [GeV] { 8 { 0.1 × δ QCD 1 2 π Δφ W+W+ NLO corrections. The curve of the QCD correction is scaled down by a factor of 0:1. relatively strong cut at a transverse momentum of 100 GeV reduces the total NLO cross section by 23% at the current LHC CM energy of 13 TeV. In detail, the QCD correction drops by a factor of 2 and the photon-induced channel decreases to approximately 40% of its original value. With increasing CM energy the impact of the pT-cut increases. In combination with the strong growth of the quark-photon-induced and the QCD corrections (see above) this results in a reduction of the NLO cross section by 50% for a value of pT;cut = 100 GeV. 4.3 Discussion of PDF uncertainties in the photon-induced channel The inclusion of QED corrections into the determination of PDFs was rst considered by the MRST collaboration [ 32 ], which imposed strong model assumptions on the parametrization of the photon PDF and based the t mostly on DIS data. Later, the NNPDF [24] and CT [26] collaborations provided photon PDFs as well, the former without any model assumptions and using mostly LHC data to constrain the photon PDF, the latter with similar, but less strict assumptions than the MRST collaboration and using ep ! e + X data. Recently, a new approach was put forward which made it possible to derive the photon PDF directly from the proton structure functions F2(x; Q2) and FL(x; Q2), which are well determined from ep scattering data. This new LUXqed [7] PDF set exhibits a very small uncertainty. Another approach, using the structure functions as well, was used by Harland-Lang et al. [33]. Not long ago, the xFitter collaboration published results on a photon PDF t to high-mass Drell-Yan data at the LHC [ 34 ]. { 9 { ps [TeV] pT(jet)<100 GeV pT(jet)<150 GeV pT(jet)<200 GeV momentum at di erent CM energies p resulting from the Monte Carlo integration. s of the collider. The error is an estimated integration error Until recently, the photon PDF was largely unconstrained due to the limited amount of data. Therefore, the uncertainty on quark-photon and photon-photon-induced contributions could be easily as large as the contributions themselves. The quark-photon-induced channel of WWW production constitutes the largest contribution to the NLO EW corrections. Thus, an uncertainty estimate is essential for a meaningful physical prediction. We assess the uncertainty by analyzing our results with di erent available PDF sets incorporating QED corrections: NNPDF3.0QED, CT14QEDinc, and LUXqed. We include numbers for the largely outdated MRSTQED04 set as well, but do not use it in the uncertainty estimate. The uncertainty of the photon PDF of the NNPDF set is highly non-Gaussian and only loosely constrained by data. Following the procedure used by the LHC Higgs Cross section Working Group [20] in the calculation of the Higgs production cross sections via vector-boson fusion and Higgs-strahlung, we therefore take the lower limit of the cross sections calculated separately with all NNPDF3.0QED replicas as the lower bound, the median of the cross sections as the central value, and the maximum of the 68% smallest cross sections as the upper bound. Note that obtaining an error estimate based on the uncertainty of the photon PDF alone is di cult in the case of the NNPDF3.0QED sets as there are no dedicated variations for the photon PDF. We have computed the NNPDF3.0QED errors both using the full PDF error for the quark-photon channel and xing the incoming quark to the central member PDF and only varying the photon PDF W− W+W+ + X √pps →= 7 TeV QCD correction (blue) for di erent cut values pT;cut and collider energies for the process pp ! W+W+ + X. The values at pT;cut = 1 are the relative corrections of the full integrated cross section without any jet veto. over the replicas. Both approaches result in similar error estimates. The CT14QEDinc PDF set does not give any information on a central value, only a range of their free t parameter, the initial photon momentum fraction at the t scale p0 . At the 68% con dence level, p0 is restricted to be between 0% and 0.11%. This range yields the error bar used. The error on the LUXqed PDF set was calculated as described in the corresponding paper [7]. As the LUXqed PDF set uses the Hessian method with symmetric eigenvectors to describe the uncertainties, the variance of a cross section is given by Var( ) = 2 0) ; NEV X ( j j=1 (4.3) where NEV is the number of eigenvector PDF sets, j the cross section evaluated with eigenvector set j and 0 the central value. The MRSTQED04 set does not provide any uncertainty information. Figure 8 shows the central values of the photon-induced contribution, where the error bars represent the photon-PDF uncertainty for the di erent PDF sets. In gure 9 the impact of the photon PDF uncertainty on di erential distributions is PDF set MRSTQED04 NNPDF3.0QED CT14QEDinc LUXqed 6 8 10 12 13 TeV. A central value for the CT14QEDinc PDF set was calculated by taking the midpoint of the range of cross sections calculated in the table on the left. The additional dashed gray error bar for the LUXqed PDF set shows the total PDF uncertainty for the quark-photon-induced correction. pT,W− [GeV] momentum of W QED corrections. with uncertainties due to the photon PDF for di erent PDF sets incorporating shown. We observe that the results based on the recent PDF sets are consistent with each other. Due to the limited amount of data and no model assumptions, the uncertainty of the NNPDF3.0QED set is the largest. The photon PDF of the LUXqed PDF set shows an outstanding small uncertainty, which is even less than the remaining PDF uncertainties (cf. gure 8). 4.4 Estimating the uncertainties of the total cross sections We investigate the two main sources to the uncertainty of the total cross sections: missing higher-order corrections estimated through the residual dependence on the factorization and renormalization scale, and the uncertainties of the PDFs. At LO, the production of three W bosons at a proton-proton collider is a purely electroweak process. As a consequence, there is no dependence on the renormalization scale R at LO, so that no reduction of the scale dependence when going from LO to NLO is expected. In order to estimate the scale uncertainties we vary our scale choice in eq. (3.4) up and down by a factor of 2. The scale uncertainty of the total LO and NLO cross section for di erent collider CM energies is shown in table 5. Another signi cant contribution to the total uncertainty is the overall PDF uncertainty which is given by the square root of the variance de ned by Var NLO PDF = Var This formula has to be handled with care as we chose to use di erent PDF sets for di erent contributions. We can assume that the covariance between the quark-photon-induced channel and every other contribution is independent of the choice of the PDF. This is a valid assumption as the used PDF sets agree reasonably well in their values for quark and gluon PDFs (cf. ref. [25]) and the covariance is ruled by the quark PDFs. Therefore, eq. (4.4) simpli es to + Var As the NNPDF collaboration uses Monte Carlo replicas, the variance of a cross section evaluated with the NNPDF3.0QED PDF set is given by (4.5) (4.6) Var( )jPDF = 1 Nrep 1 Nrep X ( j j=1 2 0) ; where Nrep is the number of replicas, j the cross section evaluated with replica j and 0 = h i the central value of the cross section. In contrast to the NNPDF collaboration, the LUXqed PDF set uses the Hessian method where the variance is given by the sum over the squared di erences between the central value and the contribution evaluated with each eigenvector PDF set (cf. section 4.3). The results of the PDF uncertainty estimation are shown in table 5. Note that using LUXqed the impact of the uncertainty of the photon PDF is rather small and negligible in comparison to the scale and other PDF uncertainties. Even using a more conservative approach and taking the CT14QEDinc uncertainty would insigni cantly change the total PDF uncertainty by only 5% at 13 TeV for W W+W+ production. We conclude that at past and present LHC energies, the dominant theoretical uncertainty arises from the scale dependence of the NLO prediction, given that modern and up-to-date PDFs are employed. In this case, the scale uncertainties are almost twice as large as the PDF errors and a further improvement on the prediction would require the inclusion of QCD corrections beyond NLO. (second) at di erent pp CM energies ps. Owing to its sensitivity to the mechanism of electroweak symmetry breaking and to triple and quartic gauge couplings, triple-W production is an important process to further test the validity of the SM and search for physics beyond. As precise predictions are necessary to analyze experimental data, we provide full NLO cross sections for the production of three on-shell W bosons at proton-proton colliders. We observe that NLO corrections are dominated by QCD with K-factors of 1:5 1:7. The electroweak correction are of the order of 5 10% at LHC energies. In special kinematic regimes, the electroweak corrections grow large due to high-energy logarithms. The main contribution of the electroweak corrections results from the quark-photon-induced channel, yielding corrections of 11% at 13 TeV. However, we observe large cancellations between the positive corrections from the photon-induced and the negative EW corrections to the quark-induced channels, so that the net EW corrections are at the level of 7%. The impact of the quark-photon induced channel can be e ectively suppressed by applying a veto on hard jet emissions. We estimate the impact of the uncertainty of the photon PDF on the NLO prediction by considering di erent PDF sets incorporating QED corrections. Using the recently released LUXqed PDF set, we observe a signi cant reduction in the uncertainties that originate from the photon PDF and note that the total theory error to this process is now governed by scale uncertainties. To further improve the cross section predictions, it would be necessary to perform at least some multi-jet merging, which is beyond the scope of this work. To particle improve the predictive power in the distributions one should drop the on-shell requirement and include leptonic decays of the W bosons. Acknowledgments S.D. and G.K. are supported by the Research Training Group GRK 2044 of the German Research Foundation (DFG). S.D. and G.K. acknowledge support by the state of BadenWurttemberg through bwHPC and the German Research Foundation (DFG) through grant no INST 39/963-1 FUGG. A.H. is supported by the Swiss National Science Foundation (SNF) under contract CRSII2-160814. A Results at a single phase-space point phase-space point with the four-momenta given in table 6.3 In this appendix, we provide results for the partonic process ud ! W The input parameter scheme has been de ned in section 3 and the dynamical scale setting (3.4) reduces to the production threshold, = 3MW, for the 2 ! 3 kinematics considered here. In the following, we provide the squared amplitude averaged over initialstate colors and helicities and summed over nal-state helicities. The virtual corrections are renormalized according to our input-parameter scheme with external legs renormalized on-shell. Infrared singularities are regularized using dimensional regularization (D = 4 2 ) and we further extract a factor of c = (4 ) (1 + ) from the coe cients of the poles. 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Stefan Dittmaier, Alexander Huss, Gernot Knippen. Next-to-leading-order QCD and electroweak corrections to WWW production at proton-proton colliders, Journal of High Energy Physics, 2017, 34, DOI: 10.1007/JHEP09(2017)034