aa r X i v : . [ h e p - ph ] J u l Vector-boson pair production at NNLO ∗ Massimiliano Grazzini † Physik-Institut, Universit¨at Z¨urich, CH-8057 Z¨urich, Switzerland
Abstract
We consider the inclusive production of vector-boson pairs in hadron collisions. Wereview the theoretical progress in the computation of radiative corrections to thisprocess up to next-to-next-to-leading order in QCD perturbation theory.The production of vector-boson pairs is a relevant process for physics studies within andbeyond the Standard Model (SM). First of all, this process can be used to measure the vectorboson trilinear couplings. Any deviation from the pattern predicted by SU (2) ⊗ U (1) gaugeinvariance would be a signal of new physics. The Tevatron collaborations have measured W W , ZZ , W Z , Zγ and W γ cross sections at invariant masses larger than those probed at LEP2, settinglimits on the corresponding anomalous couplings, and the LHC experiments are now continuingthis research program [1]. Furthermore, vector boson pairs are an important background for newphysics searches. Although the recently discovered Higgs resonance is well below the
W W and ZZ threshold, the off-shell W W and ZZ backgrounds are crucial both in the extraction of the Higgssignal and in a measurement of the Higgs boson width [2–4]. Possible charged Higgs bosons fromnon standard Higgs sectors could decay into W Z final states. Typical signals of supersymmetry,e.g. three charged leptons plus missing energy, receive an important background in
W Z and
W γ production. In this contribution we review the current status of theoretical predictions for vector-boson pair production, with emphasis on QCD radiative corrections, and focusing on NNLO QCDeffects in Zγ [5], W γ [6] and ZZ [7] production (NNLO corrections to γγ production have beenpresented in Ref. [8]).The theoretical efforts for a precise prediction of vector-boson pair production in the SMstarted more than 20 years ago, with the first NLO QCD calculations [9–15] with stable vector ∗ Based on talks given at the XLIXth Rencontres de Moriond, QCD and High energy interactions, La Thuile,march 2014, and at Loops and Legs in Quantum Field Theory, Weimar, april 2014. † On leave of absence from INFN, Sezione di Firenze, Sesto Fiorentino, Florence, Italy.
W W , ZZ and Zγ production the loop-induced gluon fusion contribution, which is formallynext-to-next-to-leading order (NNLO), has been computed in Refs. [20–24]. NLO predictions forvector-boson pair production including the gluon-induced contribution, the leptonic decay of thevector boson with spin correlations and off-shell effects have been presented in Ref. [25]. Electro-weak corrections to vector boson pair production have been considered in Refs. [26–31].The NNLO QCD computation of V V ′ production requires the evaluation of the tree-levelscattering amplitudes with two additional (unresolved) partons, of the one-loop amplitudes withone additional parton, and of the one-loop-squared and two-loop corrections to the Born subprocess q ¯ q → V V ′ . Up to now, the bottleneck for the NNLO calculation has been the knowledge ofthe relevant two-loop amplitudes. The two-loop helicity amplitudes for W γ and Zγ productionhave been presented in Ref. [32]. Recently, a major step forward has been carried out, with theevaluation of all the two-loop planar [33, 34] and non planar [35, 36] master integrals relevant forthe production of off-shell vector boson pairs, and the calculation of the corresponding helicityamplitudes is now feasible.Even having all the relevant amplitudes, the computation of the NNLO corrections is still anon-trivial task, due to the presence of infrared (IR) singularities at intermediate stages of thecalculation that prevent a straightforward application of numerical techniques. To handle andcancel these singularities at NNLO the q T subtraction formalism [37] is particularly suitable, sinceit is fully developed [38] to work in the hadronic production of heavy colourless final states.In the following we present a selection of numerical results for Zγ [5], W γ [6] and ZZ [7]production at the LHC. In the above applications the required tree-level and one-loop ampli-tudes were obtained with the OpenLoops [39] generator, which employs the Denner-Dittmaieralgorithm [40] for the numerical evaluation of one-loop integrals and implements a fast numericalrecursion for the calculation of NLO scattering amplitudes within the SM.We use the MSTW 2008 [41] sets of parton distributions, with densities and α S evaluatedat each corresponding order (i.e., we use ( n + 1)-loop α S at N n LO, with n = 0 , , N f = 5 massless quarks/antiquarks and gluons in the initial state. As for the electroweakcouplings, we use the so called G µ scheme, where the input parameters are G F , m W , m Z . Inparticular we use the values G F = 1 . × − GeV − , m W = 80 .
398 GeV, Γ W = 2 . m Z = 91 . Z = 2 . V γ final state ( V = W, Z ), besides the direct production in the hardsubprocess, the photon can also be produced through the fragmentation of a QCD parton, andthe evaluation of the ensuing contribution to the cross section requires the knowledge of a non-perturbative photon fragmentation function, which typically has large uncertainties. The fragmen-tation contribution is significantly suppressed by the photon isolation criteria that are necessarilyapplied in hadron-collider experiments in order to suppress the large backgrounds. The standardcone isolation, which is usually applied in the experiments, suppresses a large fraction of thefragmentation component. The smooth cone isolation completely suppresses the fragmentationcontribution [42], and is used in the following with parameters R = 0 . ǫ = 0 . Zγ production [5] and we use the cuts that are applied by the ATLAScollaboration [43]. The default renormalization ( µ R ) and factorization ( µ F ) scales are set to µ R = µ F = µ ≡ p m Z + ( p γT ) . We require the photon to have a transverse momentum p γT > | η γ | < .
37. The charged leptons are required to have p lT >
25 GeVand | η l | < .
47, and their invariant mass m ll must be m ll >
40 GeV. We require the separationin rapidity and azimuth ∆ R between the leptons and the photon to be ∆ R ( l, γ ) > .
7. Jets arereconstructed with the anti- k T algorithm [44] with radius parameter D = 0 .
4. A jet must have E jet T >
30 GeV and | η jet | < .
4. We require the separation ∆ R between the leptons (photon)and the jets to be ∆ R ( l/γ, jet) > .
3. Our results for the corresponding cross sections are σ LO =850 . ± . σ NLO = 1226 . ± . σ NNLO = 1305 ± gg contribution amounts to 8% of the full NNLOcorrection and thus to less than 1% of σ NNLO . The corresponding fiducial cross section measuredby ATLAS is σ = 1 . ± .
02 (stat) ± .
11 (syst) ± .
05 (lumi) pb. We see that the NNLO effectsimprove the agreement of the QCD prediction with the data, which, however, still have relativelylarge uncertainties.We now move to consider
W γ production [6], and we still use the cuts that are applied bythe ATLAS collaboration [43]. The default renormalization and factorization scales are set to µ R = µ F = µ ≡ p m W + ( p γT ) . The cuts are identical to those used for Zγ except that theinvariant mass cut is replaced by a cut on the missing transverse momentum, p miss T : we require p miss T >
35 GeV. Our results for the corresponding
W γ cross sections are σ LO = 906 . ± . σ NLO = 2065 . ± . σ NNLO = 2456 ± W γ production the QCD radiative corrections are rather large: the NLO corrections increase the LOresult by more than a factor of two. The NNLO corrections are thus larger than those found for Zγ and are in fact about 19% for central values of the scales. The QCD predictions can be comparedto the LHC data: the corresponding fiducial cross section measured by the ATLAS collaborationis [43] σ = 2770 ± ± ± ZZ production (see Ref. [7] for moredetails). In this case the default renormalization and factorization scales are set to µ R = µ F = m Z .In Fig. 2 we show the cross section computed at LO, NLO and NNLO as a function of the centre-of-mass energy √ s . For comparison, we also show the NLO result supplemented with the loop-induced gluon fusion contribution (“NLO+gg”) computed with NNLO PDFs. The lower panelin Fig. 1 shows the NNLO and NLO+gg predictions normalized to the NLO result. The NLOcorrections increase the LO result by about 45%. The impact of NNLO corrections with respectto the NLO result ranges from 11% ( √ s = 7 TeV) to 17% ( √ s = 14 TeV). Using NNLO PDFs,the gluon fusion contribution provides between 58% and 62% of the full NNLO correction. Thetheoretical predictions can be compared to the LHC measurements [46–49] carried out at √ s = 7TeV and √ s = 8 TeV, which are also shown in the plot. We see that the experimental uncertaintiesare still relatively large and that the ATLAS and CMS results are compatible with both the NLOand NNLO predictions. The only exception turns out to be the ATLAS measurement at √ s = 8TeV [48], which seems to point to a lower cross section.We have presented a selection of numerical results on Zγ , W γ and ZZ production at the LHCup to NNLO in QCD perturbation theory. The results for ZZ production were limited to theinclusive cross section for on-shell ZZ pairs. A computation of the two-loop helicity amplitude3igure 1: The transverse energy distribution of the photon in W γ production, computed at NLO(dashes) and NNLO (solid) compared to the ATLAS data. The middle panel shows the ratioDATA/THEORY. The lower panel shows the ratio NNLO/NLO.for q ¯ q → ZZ → l will open the possibility of detailed phenomenological studies at NNLO. Acknowledgements.
I would like to thank Fabio Cascioli, Thomas Gehrmann, Stefan Kallweit,Philipp Maierh¨ofer, Andreas von Manteuffel, Stefano Pozzorini, Dirk Rathlev, Lorenzo Tancredi,Alessandro Torre and Erich Weihs for the nice collaboration on the topics presented in this con-tribution. My research was supported in part by the Swiss National Science Foundation (SNF)under contracts CRSII2-141847, 200021-144352, and by the Research Executive Agency (REA)of the European Union under the Grant Agreements PITN–GA—2010-264564 (
LHCPhenoNet ),PITN–GA–2012–316704 (
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