Z-boson production in association with a jet at next-to-next-to-leading order in perturbative QCD
Radja Boughezal, John M. Campbell, R. Keith Ellis, Christfried Focke, Walter T. Giele, Xiaohui Liu, Frank Petriello
FFERMILAB-PUB-15-519-T Z -boson production in association with a jet at next-to-next-to-leading order inperturbative QCD Radja Boughezal, ∗ John Campbell, † R. Keith Ellis, ‡ ChristfriedFocke, § Walter Giele, ¶ Xiaohui Liu, ∗∗ and Frank Petriello
1, 4, †† High Energy Physics Division, Argonne National Laboratory, Argonne, IL 60439, USA Fermilab, P.O.Box 500, Batavia, IL 60510, USA Institute for Particle Physics Phenomenology, Department of Physics, University of Durham, Durham, DH1 3LE, UK Department of Physics & Astronomy, Northwestern University, Evanston, IL 60208, USA Maryland Center for Fundamental Physics, University of Maryland, College Park, Maryland 20742, USA (Dated: June 3, 2016)We present the first complete calculation of Z -boson production in association with a jet inhadronic collisions through next-to-next-to-leading order in perturbative QCD. Our computationuses the recently-proposed N -jettiness subtraction scheme to regulate the infrared divergences thatappear in the real-emission contributions. We present phenomenological results for 13 TeV proton-proton collisions with fully realistic fiducial cuts on the final-state particles. The remaining theoret-ical uncertainties after the inclusion of our calculations are at the percent-level, making the Z +jetchannel ready for precision studies at the LHC Run II. INTRODUCTION
The production of a Z -boson in association with a jetis an important process for the physics program of theLarge Hadron Collider (LHC). It serves as a backgroundto searches for supersymmetry and for dark matter in themono-jet channel, and in measurements of properties ofthe Higgs boson. The measurement of the Z +jet processcan also be used to improve the determination of thegluon distribution function. For all of these purposes aprecision Standard Model prediction of this process ishighly desirable.The next-to-leading order (NLO) corrections in thestrong coupling constant for Z +jet production have beenknown for some time [1]. The NLO electroweak correc-tions were considered in Ref. [2]. First partial results fornext-to-next-to-leading order (NNLO) corrections to the qg , q ¯ q , and gg partonic channels in the leading-color ap-proximation were recently presented [3]. However, noneof these results are suitable for precision phenomenologyat the LHC. The scale uncertainty of the NLO calcula-tion is comparable to the combination of all other exper-imental systematic errors at high transverse momenta ofthe leading jet [4, 5]. Inclusion of partial NNLO correc-tions is a first step to improve this situation, but evensmall partonic channels can shift the distribution shapesin non-trivial ways, in particular at high transverse mo-mentum [6]. A complete calculation is highly desirable.In this paper we report on a complete calculation of Z -boson production in association with a jet at NNLO inperturbative QCD, including all partonic channels andmaintaining the full color dependence. We investigatethe effects of higher-order QCD corrections on the kine-matics of the Z -boson, the leading jet, and the leptonsarising from the Z -boson decay in 13 TeV LHC colli-sions. Fully realistic acceptance cuts are imposed on the final-state particles. We find that the NNLO correctionsare at the percent-level over most of the studied phasespace, and have minimal kinematic dependence. Afterour calculation the Z +jet channel is ready for a preci-sion comparison with the upcoming data from the LHCRun II.To derive these predictions we use the recently-proposed N -jettiness subtraction technique [7, 8], whichhas been used to provide the first complete predictionsfor both W -boson and Higgs boson production in asso-ciation with a jet at NNLO [7, 9]. We incorporate ourresults into a new version of the MCFM program [13]that supports NNLO calculations using the N -jettinessframework. An interesting feature of the computationalalgorithm used in this new version is that it exhibitsstrong scaling to many thousands of nodes, and can runon modern supercomputing platforms. This makes possi-ble calculations and phenomenological studies that werepreviously intractable. We will discuss the details of ourapproach for color-singlet production in a recent publi-cation [14]. THEORETICAL FRAMEWORK
We sketch here the N -jettiness subtraction scheme.The implementation of this scheme used in obtaining ourresults was presented in Ref. [7]. Another description ofthe method is also given in Ref. [8].We begin with the definition of the N -jettiness vari-able T N , a global event shape designed to veto final-state The Higgs plus jet process has also been calculated using otherNNLO subtraction methods [10–12]. a r X i v : . [ h e p - ph ] J un jets [15]: T N = (cid:88) k min i (cid:26) p i · q k Q i (cid:27) . (1)The subscript N denotes the number of jets desired in thefinal state; for the Z +jet process considered here, N = 1.Values of T near zero indicate a final state containing asingle narrow energy deposition, while larger values de-note a final state containing two or more well-separatedenergy depositions. The p i are light-like reference vec-tors for each of the initial beams and final-state jets inthe problem. The reference vectors for the final-state jetscan be determined by using a jet algorithm, as discussedin Refs. [15, 16]. The determination of the p i is insen-sitive to the choice of jet algorithm in the small- T cutN limit [15]. The q k denote the four-momenta of any final-state radiation. The Q i characterize the hardness of thebeam-jets and final-state jets. We set Q i = 2 E i , twicethe lab-frame energy of each jet.We briefly outline the procedure through which we use T N to obtain the complete NNLO correction to the Z +jetprocess. The NNLO cross section consists of contribu-tions with Born-level kinematics, and processes with ei-ther one or two additional partons radiated. We partitionthe phase space for each of these terms into regions aboveand below a cutoff on T N , which we label T cutN : σ NNLO = (cid:90) dΦ N |M N | + (cid:90) dΦ N +1 |M N +1 | θ
We now discuss how we obtain the various componentsof Eq. (2) needed to obtain the complete cross sectionat NNLO. Above T cutN we need a NLO calculation of Z +2-jets. We use an improved version of MCFM [13]optimized to handle the N -jettiness subtraction schemeto obtain this contribution efficiently. Upon integrationover the phase space of the final-state leptons, we cancheck our implementation of the hard function againstPeTeR [26]; we have done so and have found perfectagreement. The calculation and validation of the nec-essary N -jettiness soft function has been detailed in aseparate publication [25]. The necessary two-loop beamand jet functions for this process are also known [23, 24]. We clarify here an unclear point in Ref. [21]: to obtain the virtualcorrections for all helicity amplitudes, only the spinor productsshould be conjugated in Eqs. (2.24-2.25), not the α , β and γ coefficients. The primary check of the N -jettiness formalism is thatthe logarithmic dependence on T cutN that occurs sepa-rately in the low and high T N regions cancels when theyare summed. This requires that almost all parts of thecalculation are implemented correctly and consistently;the beam, soft, and jet functions, as well as the NLOcorrections to Z +2-jets, are probed by this check. Weshow in Fig. 1 the results of this validation for the ratio σ NNLO /σ NLO in 13 TeV proton-proton collisions (we notethat NNLO PDFs are used in the numerator, while NLOPDFs are used in the denominator). We have checkedthat the NLO cross section obtained with N -jettinesssubtraction agrees with the result obtained with stan-dard techniques. These cross sections are obtained us-ing CT14 parton distribution functions [27] at the sameorder in perturbation theory as the partonic cross sec-tion, and contain the following fiducial cuts on the lead-ing final-state jet and the two leptons from CMS [5]: p jetT >
30 GeV, | η jet | < . p lT >
20 GeV, | η l | < . < m ll <
111 GeV. The ATLAS analysis issimilar but with slightly different cuts [4]. We reconstructjets using the anti- k T algorithm [28] with R = 0 .
5. A dy-namical scale µ = (cid:113) m ll + (cid:80) p jet, T is chosen to describethis process, where the sum is over the transverse mo-menta of all final-state jets, and m ll the invariant massof the di-lepton pair arising from the Z -boson decay. Inthis validation plot we have set the renormalization andfactorization scales to µ R = µ F = 2 × µ ; since the cor-rections are larger for this scale choice, it is easier toillustrate the important aspects of the T cut variation. T cut [GeV] σ NN L O / σ N L O Figure 1: Plot of the NNLO cross section over the NLO result, σ NNLO /σ NLO , as a function of T cut , for the scale choice µ =2 × µ . The vertical bars accompanying each point indicatethe integration errors. A few features can be seen in Fig. 1. First, in the re-gion T cut < .
08 GeV the result becomes independentof the particular value of the cut chosen within the nu-merical errors. The NNLO correction for µ = 2 × µ corresponds to an almost +5% shift in the cross sec-tion. The plot makes clear that we have numerical controlover the NNLO cross section to the per-mille level, com-pletely sufficient for phenomenological predictions. Weobserve an approximately linear dependence of σ NNLO on ln ( T cut ) in the region 0 . < T cut < . T N /Q ) ln n ( T N /Q ), where n ≤ Q is a hard scale such as p jetT .The other possible checks of the N -jettiness formalisminvolve comparison with other NNLO results obtained us-ing different techniques. We have previously checked thatthe agreement between Higgs+jet production as com-puted with N -jettiness and with other techniques [10]agree at the per-mille level [9]. A selection of processeswithout final-state jets have also been computed withboth N -jettiness subtraction and other techniques, andshow a similar level of agreement [8, 14]. NUMERICAL RESULTS
We present here numerical results for Z -boson produc-tion in association with a jet at NNLO. Our central scalechoice is the dynamical scale µ = µ , as described in theprevious section. To obtain an estimate of the theoret-ical errors we vary µ away from this choice by a factorof two. We use the same cuts on the jets and leptons asdescribed in the previous section. We include the con-tributions from both the Z -boson and a virtual photondecaying to leptons in our numerical results. p ZT [GeV] -4 -3 -2 -1 d σ / d p Z T [ pb / G e V ] LONLONNLO K NLOLONNLONLO
Figure 2: Plot of the Z -boson p T distribution at LO, NLOand NNLO in QCD perturbation theory, for 13 TeV collisionswith the central scale µ = (cid:113) m ll + (cid:80) p jet, T . The K -factorsare shown in the lower inset. We note that the cross sections at each order in per-turbation theory for the cuts described above are: σ LO = 97 . +3 . − . pb ,σ NLO = 133 . +5 . − . pb ,σ NNLO = 135 . +0 . − . pb . (4)The NNLO correction results in a +1% increase in thefiducial cross section. The scale dependence is greatlyreduced with respect to the NLO result. We note thatthe full NNLO corrections are in good agreement with analternative calculation found in Ref. [3] that has recentlybecome available. We next show the Z -boson transversemomentum distribution in Fig. 2, focusing on the range p ZT <
500 GeV. The distributions at LO, NLO and NNLOin QCD perturbation theory are shown, as are the usual K -factors: the ratio of the NLO over the LO cross sec-tion, and the NNLO over the NLO result. To producethis distribution and all other ones, we average the re-sults from T cut = 0 . , . , . , .
06 GeV. A reducedscale dependence is obtained when the NNLO correctionsare included, and a significantly smaller correction is ob-served when going from NLO to NNLO than when goingfrom LO to NLO, indicating stability of the perturba-tive expansion. A slight increase of the NNLO correctionoccurs as p ZT is increased. The analogous transverse mo-mentum distribution for the leading jet is shown in Fig. 3.In this case the NLO corrections grow with p jetT , reach-ing a K -factor of 2.5 for p jetT = 500 GeV. The NNLOcorrections are far more mild, but they grow with p jetT ,increasing the NLO result by 10% at p jetT = 500 GeV.It is essential to account for these corrections when com-paring with measurements, as the experimental errors areonly at the few-percent level in this region. p jetT [GeV] -4 -3 -2 -1 d σ / d p j e t T [ pb / G e V ] LONLONNLO
100 200 300 400 5001.01.52.02.5 K NLOLONNLONLO
Figure 3: Plot of the leading-jet p T distribution at LO, NLOand NNLO in QCD perturbation theory, for 13 TeV collisionswith the central scale µ = (cid:113) m ll + (cid:80) p jet, T . The K -factorsare shown in the lower inset. We now study distributions of the lepton that comes from the Z → l + l − decay; the anti-lepton distributionsare similar. The lepton transverse momentum distribu-tion at LO, NLO and NNLO in QCD perturbation the-ory is shown in Fig. 4. We focus on the range p l − T ≤ p l − T is increased. The variation of the K -factors that appearsfor low- p l − T arises from the leading-order kinematic re-striction that p ZT >
30 GeV, which occurs because of the p jetT >
30 GeV cut. This in turn restricts the allowedvalues of p l − T that can occur. This restriction is lifted atNLO when additional radiation is present, but leads tolarge corrections near the LO kinematic boundary. Fi-nally, we show in Fig. 5 the rapidity distribution of thelepton. The kinematic variation of the K -factor is smallat both NLO and NNLO, with the corrections being aconstant +40% shift at NLO and nearly zero at NNLO.Although not shown explicitly here, we find a similarpattern of corrections for the jet and Z -boson rapiditydistributions. p l − T [GeV] -2 -1 d σ / d p l − T [ pb / G e V ] LONLONNLO
20 40 60 80 100 120 140 160 1801.01.41.8 K NLOLONNLONLO
Figure 4: Plot of the lepton p T distribution at LO, NLOand NNLO in QCD perturbation theory, for 13 TeV collisionswith the central scale µ = (cid:113) m ll + (cid:80) p jet, T . The K -factorsare shown in the lower inset. Before concluding we comment briefly on some compu-tational aspects of our calculation. It was recently shownthat a multi-threaded version of the Vegas integration al-gorithm [29] could significantly reduce the time neededto obtain NLO cross sections [30]. We have extendedthis parallelization to use the MPI protocol in order toallow communication between the separate nodes presenton modern computing clusters. Numerical tests on theMira supercomputer at the Argonne Leadership Com-puting Facility and at the NERSC facility at Berkeleyshow that our code exhibits strong scaling to the several-thousand node level. We anticipate that the techniques η l − d σ / d η l − [ pb ] LONLONNLO K NLOLONNLONLO
Figure 5: Plot of the lepton rapidity distribution at LO, NLOand NNLO in QCD perturbation theory, for 13 TeV collisionswith the central scale µ = (cid:113) m ll + (cid:80) p jet, T . The K -factorsare shown in the lower inset. we have developed will become increasingly importantfor theoretical predictions to match the ever-improvingquality and precision of high energy collider data. CONCLUSIONS
In this manuscript we have presented the completeNNLO corrections to the Z +jet process in hadronic col-lisions. Our calculation utilizes the N -jettiness subtrac-tion scheme, which has proven to be a powerful tool forobtaining higher-order QCD cross sections. We havegiven phenomenological results for 13 TeV LHC colli-sions. The NNLO corrections are small throughout moststudied regions of phase space, and are at or below thepercent level for p T values up to 100 GeV. However, theyreach up to 10% in the tails of the jet and Z -boson trans-verse momentum distributions, and must be included inany comparison of theory with experiment in this region.The corrections to the rapidity distributions of the jet, Z -boson and leptons are flat, and are at or below thefew-percent level for all scale choices. The Z +jet predic-tion exhibits an extremely stable perturbative expansion,and upon inclusion of the complete NNLO corrections isready for a precision comparison with LHC Run II data.The N -jettiness subtraction scheme has now been ap-plied to obtain the complete NNLO results for severalimportant LHC processes. One great virtue of this ap-proach is its simplicity: all complications associated withthe double-unresolved singular limit of QCD are handledby the factorization theorem of Eq. (3). Another advan-tage of this approach is the ease with which the necessarynumerical integrations can be efficiently run on massivelyparallel computing platforms. Only the real-radiation in- tegration in the region T N > T cutN is computationally ex-pensive. The calculational method scales to the largestavailable computing platforms. The conceptual appeal,simplicity and computational advantages of N -jettinesssubtraction will make it a powerful tool whenever preci-sion predictions for scattering processes are required. Acknowledgements .We thank T. LeCompte for many helpful discussions.R. B. is supported by the DOE contract DE-AC02-06CH11357. J. C., K. E. and W. G. are supported bythe DOE contract DE-AC02-07CH11359. C. F. is sup-ported by the NSF grant PHY-1520916. X. L. is sup-ported by the DOE grant DE-FG02-93ER-40762. F. P.is supported by the DOE grants DE-FG02-91ER40684and DE-AC02-06CH11357. This research used resourcesof the National Energy Research Scientific ComputingCenter, a DOE Office of Science User Facility supportedby the Office of Science of the U.S. Department of Energyunder Contract No. DE-AC02-05CH11231. 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