NLO QCD corrections to W ± Zγ production with leptonic decays
aa r X i v : . [ h e p - ph ] N ov FTUV–10–1109KA–TP–33–2010SFB/CPP-10-96IFUM-967-FT
NLO QCD corrections to W ± Z γ productionwith leptonic decays
G. Bozzi , F. Campanario , M. Rauch , H. Rzehak and D. Zeppenfeld Dipartimento di Fisica, Universita’ di Milano and INFN, Sezione di Milano,20133 Milan, Italy Institut f¨ur Theoretische Physik, Karlsruhe Institute of Technology,Universit¨at Karlsruhe, 76128 Karlsruhe, Germany
Abstract
We present a computation of the O ( α s ) QCD corrections to W ± Zγ productionat the Large Hadron Collider. The photon is considered as real, and we include fullleptonic decays for the W and Z bosons. Based on the structure of the VBFNLOprogram package, we obtain numerical results through a fully flexible Monte Carloprogram, which allows to implement general cuts and distributions of the final-stateparticles. The NLO QCD corrections are sizable and strongly exceed the theory errorobtained by a scale variation of the leading-order result. Also, the shapes of relevantobservables are significantly altered. + ν e µ + µ − γZ/γ ∗ W + ¯ q ′ q e + ν e µ + µ − γZ/γ ∗ W + W + ¯ q ′ q e + ν e µ + µ − γZ/γ ∗ W + W + ¯ q ′ q Figure 1: Examples of the three topologies of Feynman diagrams contributing to the process pp → W ± Zγ + X at tree-level. In the right-hand diagram, the quartic coupling is markedwith a dot. At the LHC a new energy frontier is reached, which allows for new searches of unknownparticles and further tests of the Standard Model. From the theory side this also requiresvery precise predictions of cross sections and distributions, exceeding the precision of aleading-order approximation. We present a calculation of W ± Zγ production including QCDcorrections and the leptonic decays of the W - and the Z -boson with off-shell effects, pp, p ¯ p → W ± Zγ + X → ℓ ± ( − ) ν ℓ +2 ℓ − γ + X . (1.1)This process with leptons, photons and missing energy in the final state provides a back-ground to new physics searches (see, for example, Ref. [1]). Also, this process offers thepossibility to study the quartic vector-boson couplings
W W Zγ and
W W γγ (see right dia-gram in Fig. 1) [2] and test the Standard Model. It is one of the missing pieces for a fullknowledge of triple vector boson production at next-to leading order (NLO) QCD. Therehas been a strong effort for the calculation of these processes. The processes with only mas-sive vector bosons in the final state have been completely calculated [3–6]. Rather recently,also the NLO QCD calculation of the processes
W W γ and
ZZγ have been completed [7]and first results of the NLO QCD computation of the process
W γγ have been presented [8].Vector-boson pair production accompanied with one jet has also been studied including QCDcorrections for
W W j , W γj , W Zj and
ZZj [9–13].
We calculate all contributions to the processes (1.1) up to order α s α in the limit whereall fermions are massless. At leading order, 71 distinct diagrams appear, which we group asthree different topologies, according to the number of vector bosons attached to the quarkline. An example of each class is depicted in Fig.1. For the last two topologies we alsoinclude the cases where vector bosons are radiated off the lepton lines.To speed up the calculation, invariant subparts, which appear multiple times in differentFeynman graphs, are computed only once per phase-space point and independently of the2est of the cross-section. Hereby, we use the procedure of leptonic tensors , as first describedin Ref. [14]. This greatly reduces the computation time needed. For the computation ofthe matrix elements, we use the helicity method introduced in Ref. [15]. Furthermore, bycharge conservation, a W boson must always couple to the quark line. Hence, we only needto compute the left-handed chirality part.At NLO QCD, virtual and real emission diagrams contribute to the cross section. Bothcontain infrared divergences, which must cancel in the sum according to the Kinoshita-Lee-Nauenberg (KLN) theorem [16]. To handle this cancellation in a numerically stable way, weuse the Catani-Seymour dipole subtraction method [17]. Initial-state collinear singularitiesare partly factorized into the parton-distribution functions. This leads to additional so-called“finite collinear terms”.The NLO real corrections are given by diagrams where an additional gluon is attachedto the quark line which is possible in two different ways. Either this gluon is a final-stateparticle and considered as radiated off the quark line, or an initial-state gluon which splitsinto a q ¯ q pair and we have an emission of a quark. With 194 different Feynman diagrams,the use of leptonic tensors proves to be an advantage in this case.The presence of isolated on-shell photons requires extra care in the case of real emissions.An additional singularity arises from photon emission collinear to a massless quark. Requir-ing a simple separation cut between photon and jet is not allowed, since the cancellationof the gluonic infrared divergences between virtual and real emission processes would bespoiled. The phase space for soft-gluon emission would be reduced while leaving untouchedthe virtual counterpart, leading to the non-cancellation. In principle, this problem can besolved by adding processes with quark fragmentation to photons. Here, we use a simplerapproach by requiring a specially crafted cut first suggested in Ref. [18]. We will discuss thedetails of this cut in section 3.Virtual NLO QCD corrections arise from the insertion of gluon lines into the topologiesof Fig. 1 in every possible way. Therefore, the left topology gives rise to loops with up to fiveexternal particles, i.e. pentagon diagrams. For the middle and right ones at most box andtriangle diagrams, respectively, appear. Up to the box level, we compute the loop integralsusing Passarino-Veltman reduction [19]; additionally we avoid the explicit calculation of theinverse of the Gram matrix. Instead, we solve a system of linear equations, which is numer-ically more stable close to the singular points. For the pentagon contributions, we apply themethod of Ref. [20]. This circumvents the appearance of small Gram determinants in planarconfigurations of the external momenta altogether. The complete virtual corrections M V = f M V + α S π C F (cid:18) πµ Q (cid:19) ǫ Γ(1 + ǫ ) (cid:20) − ǫ − ǫ − π (cid:21) M B , (2.1)can be separated into a part which is proportional to the Born matrix element, M B , anda remainder, f M V . Here, Q denotes the partonic center-of-mass energy, which correspondsto the invariant mass of the W Zγ final state. For diagrams with only a single W attachedto the quark line, f M V vanishes, i.e. the virtual corrections completely factorize to the Bornamplitude. 3or this factorization formula to hold, the transversality property of the photon ǫ ( p γ ) · p γ =0 must be used [7]. Then we can reuse the general results for the finite remainder alreadyobtained in Refs. [5, 7], adapting the attached vector bosons to our case. Another opti-mization can be performed by shifting parts from the pentagon diagrams into box contribu-tions [4, 5, 7, 14]. To do this, we split the polarization vector of the vector bosons into a partproportional to the four-momentum of the boson q V and a remainder ˜ ǫ µV , which is chosensuch that ˜ ǫ V · ( q W + q Z ) = 0 . (2.2)This part leads to a reduction in the size of the pentagon contributions, which we cantherefore compute with lower statistics. Contracting the pentagons with q µV , they can beexpressed as a difference of two box diagrams which are numerically faster to compute.Hence, we can gain speed while keeping the total error the same. This shift also serves asa consistency check between the box and pentagon routines, as the total result must stayunchanged.To verify the correctness of our calculation, we have performed several checks. First, wehave compared all tree-level amplitudes against matrix elements generated by MadGraph [21]and find an agreement of at least 14 digits, which is at the level of the machine precision.Additionally, we have also compared the integrated cross section for both pp → W ± Zγ and pp → W ± Zγj against
MadEvent and
Sherpa [22]. We find an agreement at the per milllevel, which is compatible with the integration error. Furthermore, we have checked theimplementation of the Catani-Seymour subtraction scheme. We have verified that for thereal emission part the ratio between the differential real-emission cross section and the dipoleapproaches − → → → → → Results
We perform the numerical evaluation of our calculation with an NLO Monte Carlo pro-gram based on the structure of the
VBFNLO program package [23]. As input parameters inthe electroweak sector we take the W and Z boson masses and the Fermi constant. Theweak mixing angle and the electromagnetic coupling constant are computed from these usingtree-level relations: m W = 80 .
398 GeV sin ( θ W ) = 0 . m Z = 91 . α − = 132 . G F = 1 . · − GeV − . (3.1)Top-quark effects are not considered and all other quarks are taken massless. Effectsfrom generation mixing are neglected, as we set the CKM matrix to the identity matrix. Asthe central value for factorization and renormalization scales we choose the invariant massof the leptons and the photon µ F = µ R = µ = m W Zγ ≡ q ( p ℓ + p ν + p ℓ + p ¯ ℓ + p γ ) . (3.2)For the parton distribution functions, we choose CTEQ6L1 at LO and the CTEQ6M setwith α S ( m Z ) = 0 . y and the transverse mo-mentum, p T , of the final-state photon and charged leptons p T γ >
10 GeV p T ℓ >
20 GeV | y γ | < . | y ℓ | < . . (3.3)These take into account typical requirements of the experimental detectors. Furthermore,leptons, photon and jet need to be well separated in order to avoid divergences from collinearphotons and to be able to identify them as separate objects in the detectors. Therefore, weimpose R ℓℓ > . R ℓγ > . R jℓ > . R jγ > . m ℓℓ >
15 GeV . (3.4)Here, a jet refers to a final-state quark or gluon in the NLO real emission contribution with p T j >
20 GeV and | y j | < .
5. The last cut in Eq. (3.4) eliminates the singularity from avirtual photon γ ∗ → ℓ + ℓ − by requiring that the invariant mass of each pair of oppositelycharged leptons is larger than 15 GeV. For treating the collinear singularity between thephoton and a parton i , we use the procedure of Ref. [18]. The event is accepted only if p T i ≤ p T γ − cos R γi − cos δ or R γi > δ , (3.5)where δ is a fixed separation parameter which we set to 0 .
7. Eq. (3.5) allows final-statepartons arbitrarily close to the photon axis as long as they are soft enough. Thereby, it5HC √ s =7 TeV √ s =14 TeVLO[fb] NLO [fb] K-factor LO[fb] NLO [fb] K-factor4 σ (” W + Zγ ” → e + ν e µ + µ − γ ) p T γ ( ℓ ) > p T γ ( ℓ ) > σ (” W − Zγ ” → e − ¯ ν e µ + µ − γ ) p T γ ( ℓ ) > p T γ ( ℓ ) > pp → W ± Zγ + X with leptonic decays, at LO and NLO, and for two sets of cuts. Relativestatistical errors of the Monte Carlo are at the per mill level. The factor 4 accounts for allcombinations of final-state first- and second-generation leptons.Tevatron ( √ s = 1 .
96 TeV) LO[ab] NLO [ab] K-factor4 σ (” W ± Zγ ” → e ± ( − ) ν e µ + µ − γ ) p T γ ( ℓ ) > p T γ ( ℓ ) > p ¯ p → W + Zγ + X or p ¯ p → W − Zγ + X including leptonic decays, at LO and NLO, and for two sets of cuts. Relative statistical errorsof the Monte Carlo are below the per mill level. The factor 4 accounts for all combinationsof final-state first- and second-generation leptons.6 σ [fb] ξ solid: µ F = µ R = ξ µ dashed: µ F = ξ µ , µ R = µ dotted: µ F = µ , µ R = ξ µ LO NLO σ [f b ] ξ Total NLOVirtual-bornVirtual-boxVirtual-pentagons Real
Figure 2:
Left:
Scale dependence of the total LHC cross section at √ s = 14 TeV for pp → W + Zγ + X → ℓ +1 ℓ +2 ℓ − γ + p/ T + X at LO and NLO within the cuts of Eqs. (3.3, 3.4,3.5). The factorization and renormalization scales are together or independently varied inthe range from . · µ to · µ . Right:
Same as in the left panel but for the different NLOcontributions at µ F = µ R = ξµ with µ = m W Zγ . retains the full QCD pole, which cancels against the virtual part, but it does not introducedivergences from the electroweak sector.In Tables 1 and 2, we present results for the integrated cross section of W ± Zγ productionfor the LHC with a center-of mass energy of both 7 and 14 TeV and for the Tevatron withits center-of-mass energy of 1 .
96 TeV. Note, for the Tevatron, the cross section for W + Zγ and W − Zγ production is the same; the given number is the individual result of one of them.Besides the standard cut on p T γ of 10 GeV we also show results for 20 GeV. For the Tevatron,we have reduced the cut on p T ℓ to 10 GeV throughout. The cross sections shown correspondto the production of both electrons and muons for all leptons. Interference effects fromidentical leptons in the final state are neglected, since their contribution is small.From hereon, we will focus on the LHC with a center-of-mass energy of 14 TeV. InFigs. 2 and 3 we show the dependence of the cross section for W + Zγ and W − Zγ production,respectively, when varying the renormalization and factorization scale in the interval µ F , µ R = ξ · µ (0 . < ξ <
10) (3.6)around the central scale µ given in Eq. (3.2). We see that the variation of the LO crosssection with the scale strongly underestimates the size of the NLO contributions. At thecentral scale, we obtain a K-factor of 1 .
84 for W + Zγ and 1 .
94 for W − Zγ . The dependenceon the factorization scale slightly reduces when we move from a LO calculation to NLO asexpected. On the other hand, the dependence on the renormalization scale shows a largevariation. This is due to the fact that α s enters the cross section only at NLO, where weobserve the typical leading renormalization scale dependence. When varying the factorizationand the renormalization scale jointly by a factor 2 around the central scale µ , we see a changeof 7.5% at LO and 6.7% at NLO for W + Zγ . For W − Zγ , the numbers change only slightlyto 7.7% and 7.3% for LO and NLO, respectively.7 σ [fb] ξ solid: µ F = µ R = ξ µ dashed: µ F = ξ µ , µ R = µ dotted: µ F = µ , µ R = ξ µ LO NLO σ [f b ] ξ Total NLOVirtual-bornVirtual-boxVirtual-pentagons Real
Figure 3:
Left:
Scale dependence of the total LHC cross section at √ s = 14 TeV for pp → W − Zγ + X → ℓ − ℓ +2 ℓ − γ + p/ T + X + X at LO and NLO within the cuts of Eqs. (3.3,3.4, 3.5). The factorization and renormalization scales are together or independently variedin the range from . · µ to · µ . Right:
Same as in the left panel but for the differentNLO contributions at µ F = µ R = ξµ with µ = m W Zγ . On the right-hand side of Figs. 2 and 3, we show the combined factorization and renor-malization scale dependence of the NLO cross section split into the individual contributions.Almost the entire scale dependence is given by the real-emission part, which contains the truereal-emission cross section, the dipole terms from the Catani-Seymour subtraction schemeand the finite collinear terms. We obtain the bulk of the NLO contribution from the Bornmatrix element and the virtual corrections proportional to it. This includes the terms fromboxes and pentagons factored out in Eq. (2.1). At the central scale, it is more than twice aslarge as the real part. The finite virtual remainders due to box and pentagon corrections,which are shifted using Eq. (2.2), only yield a small contribution. The shape of their scaledependence is similar to the one of the total cross section.In Figs. 4, 5 and 6, we show the distribution of the transverse momentum of the photonand the hardest lepton as well as of the missing transverse momentum originating from theneutrino, respectively. On the left-hand side of each figure, we depict the differential crosssection at LO and NLO both for W + Zγ and W − Zγ , and on the right-hand side we plot thedifferential K-factor, defined in the following way: K = dσ NLO /dxdσ LO /dx . (3.7)We present results which do not include any cut on the additional jet, as well as includinga veto on jets with p T j >
50 GeV. As previously, a jet is defined as a final-state parton with | y j | < . d σ / dp T γ [f b ] p T γ [GeV]W - Z γ W + Z γ K - f a c t o r p T γ [GeV]W - Z γ W + Z γ Figure 4:
Left:
Transverse-momentum distribution of the photon in W + Zγ and W − Zγ production with leptonic decays of the W- and the Z-boson at the LHC with √ s = 14 TeV.LO (dashed blue line) and NLO (solid red) results are shown for µ F = µ R = µ = m W Zγ andthe cuts of Eqs. (3.3, 3.4, 3.5).
Right:
K-factor for the transverse-momentum distributionof the photon as defined in Eq. (3.7) without (solid red) and including a jet veto of 50 GeV(dashed blue). d σ / dp T l m a x [f b ] p T l max [GeV]W - Z γ W + Z γ W - Z γ W + Z γ W - Z γ W + Z γ K - f a c t o r p T l max [GeV]W - Z γ W + Z γ W - Z γ W + Z γ W - Z γ W + Z γ Figure 5:
Left:
Transverse-momentum distribution of the lepton with largest transversemomentum in W + Zγ and W − Zγ production with leptonic decays of the W- and the Z-boson at the LHC with √ s = 14 TeV. LO (dashed blue line) and NLO (solid red) results areshown for µ F = µ R = µ = m W Zγ and the cuts of Eqs. (3.3, 3.4, 3.5).
Right:
K-factor forthe maximal-transverse-momentum distribution of the lepton as defined in Eq. (3.7) without(solid red) and including a jet veto of 50 GeV (dashed blue). d σ / dp T m i ss [f b ] p Tmiss [GeV]W - Z γ W + Z γ W - Z γ W + Z γ K - f a c t o r p Tmiss [GeV]W - Z γ W + Z γ W - Z γ W + Z γ Figure 6:
Left:
Distribution of missing transverse momentum in W + Zγ and W − Zγ pro-duction with leptonic decays of the W- and the Z-boson at the LHC with √ s = 14 TeV.LO (dashed blue line) and NLO (solid red) results are shown for µ F = µ R = µ = m W Zγ and the cuts of Eqs. (3.3, 3.4, 3.5).
Right:
K-factor for the missing transverse-momentumdistribution as defined in Eq. (3.7) without (solid red) and including a jet veto of 50 GeV(dashed blue). d σ / d R γ l [f b ] R γ l W - Z γ W + Z γ W - Z γ W + Z γ W - Z γ W + Z γ W - Z γ W + Z γ K - f a c t o r R γ l W - Z γ W + Z γ W - Z γ W + Z γ W - Z γ W + Z γ W - Z γ W + Z γ Figure 7:
Left:
Separation of the photon and the hardest lepton in W + Zγ and W − Zγ production with leptonic decays of the W- and the Z-boson at the LHC with √ s = 14 TeV.LO (dashed blue line) and NLO (solid red) results are shown for µ F = µ R = µ = m W Zγ and the cuts of Eqs. (3.3, 3.4, 3.5).
Right:
K-factor for the distribution of the photon-leptonseparation as defined in Eq. (3.7) without (solid red) and including a jet veto of 50 GeV(dashed blue). W − Zγ for example (Fig. 5), we obtain K-factors between 1.10and 1.33 over the plotted momentum range. The large differential K-factors are thereforecaused by the emission of the additional jet, where the leptonic system recoils against the jet.The integrated K-factors are also reduced, namely to 1 .
23 for W + Zγ and 1 .
29 for W − Zγ production.In Fig. 7, we show the separation in the rapidity–azimuthal-angle plane ( R separation)between the photon and the lepton with the largest transverse momentum. Again, we observea significant dependence of the K-factor on the value of the R separation, varying between1 . .
55. Once we include the jet veto, this dependence is again much smaller.Therefore, a simple approximation of rescaling the leading-order cross section with theintegrated K-factor does not hold. It is necessary to perform a full NLO calculation toreproduce the correct shape of the distributions.
We have calculated the NLO QCD corrections to the processes pp, p ¯ p → W ± Zγ + X including full leptonic decays of the W and Z boson. With three leptons, a photon andmissing transverse energy as signature, it is an important background for searches for newphysics, in particular supersymmetry. Additionally, it can serve as a signal process formeasuring the quartic gauge couplings W W Zγ and
W W γγ at the LHC.We find that the corrections yield a sizable increase of the cross section with respect to theleading-order result, with integrated K-factors typically around 1 .
9. The LO scale variationstrongly underestimates these contributions with a variation below the 10% level. Varyingfactorization and renormalization scale µ F = µ R = µ by a factor 2 around the central value µ = m W Zγ , we obtain a remaining scale dependence at NLO of about 7%.The NLO corrections also show a significant dependence on the observable and on dif-ferent phase-space regions. Therefore, it is important to have a dedicated fully-exclusiveNLO parton-level Monte Carlo code available for W ± Zγ production. This process will beincluded into a future version of the VBFNLO program package.
Acknowledgments
We would like to thank Vera Hankele for helpful discussions. This research was supportedin part by the Deutsche Forschungsgemeinschaft via the Sonderforschungsbereich/TransregioSFB/TR-9 “Computational Particle Physics” and the Initiative and Networking Fund of theHelmholtz Association, contract HA-101 (“Physics at the Terascale”). F.C. acknowledgespartial support by European FEDER and Spanish MICINN under grant FPA2008-02878.11he Feynman diagrams in this paper were drawn using Jaxodraw [25].
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