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NNLO phenomenology using N-jettiness subtraction 6 1 0 2 n Radja Boughezal∗† a J ArgonneNationalLaboratory 1 E-mail: [email protected] 2 ] h Wediscussnext-to-next-to-leadingorderQCDresultsforHiggs,W-bosonandZ-bosonproduc- p tion in association with a jet in hadronic collisions, obtained using the recently developed N- - p jettinesssubtractionmethod. e h [ 2 v 8 2 9 4 0 . 1 0 6 1 : v i X r a 12thInternationalSymposiumonRadiativeCorrections(Radcor2015)andLoopFestXIV(Radiative CorrectionsfortheLHCandFutureColliders) 15-19June,2015 UCLADepartmentofPhysics&AstronomyLosAngeles,USA ∗Speaker. †Wethanktheorganizersforthekindinvitationandfortheverystimulatingconference. (cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ NNLOphenomenologyusingN-jettinesssubtraction RadjaBoughezal 1. Introduction RunIoftheLargeHadronCollider(LHC)wasmarkedbythediscoveryandinitialcharacter- ization of the Higgs boson. The comparison of Standard Model (SM) predictions with data from Run I of the LHC was limited by the statistical precision of the experimental data. This will no longerbethecaseduringRunII,andsystematicerrorswilldominate. Thelargestsystematicerror currently hindering our understanding of Higgs properties is the theoretical precision of the SM predictions. Thisisthecaseforthewell-measureddi-bosonmodes[1]whichdominatetheoverall signal-strength determination. The theoretical uncertainties must be reduced in order to sharpen our understanding of the mechanism of electroweak symmetry breaking in Nature. Calculations throughnext-to-next-to-leadingorder(NNLO)inperturbativeQCDhavebecomeincreasinglynec- essarytomatchtheprecisionofLHCmeasurements. Inparticularimprovementsinboththeoverall productionrateoftheHiggsbosonandinthemodelingofitskinematicdistributionsareneededto matchtheexpectedexperimentalprecisionofRunII. NNLO calculations for scattering processes with ﬁnal-state jets at hadron colliders possess a complex singularity structure. Partial results for 2→2 processes such as inclusive jet produc- tion [2], Higgs+jet production [3, 4] and Z+jet production [5] are available (full results for Z+j usingantennasubtractionwillbeavailablesoon[6]). MorerecentlycompleteresultsforHiggs+jet production [7], W++jet production [8] and Z+jet production [9] were achieved using N-jettiness subtraction method [8, 10, 11, 12]. The purpose of this contribution is to summarize the method andtheresultsachievedfortheseprocesses. 2. DescriptionofN-jettinesssubtraction We review here the salient features of the N-jettiness subtraction scheme for NNLO calcu- lations, which was recently introduced in the context of the NNLO computation of W+ boson and Higgs boson production in association with a jet [7, 8, 9]. We begin with the deﬁnition of N-jettiness,T ,aglobaleventshapevariabledesignedtovetoﬁnal-statejets[11]: N (cid:26) (cid:27) 2p ·q T =∑min i k . (2.1) N i Q k i ThesubscriptN denotesthenumberofjetsdesiredintheﬁnalstate,andisaninputtothemeasure- ment. FortheHiggs+jet,W++jetandZ+jetprocessesconsideredhere,wehaveN=1. Valuesof T nearzeroindicateaﬁnalstatecontainingasinglenarrowenergydeposition,whilelargervalues 1 denoteaﬁnalstatecontainingtwoormorewell-separatedenergydepositions. The p arelight-like i vectors for each of the initial-state beams and ﬁnal-state jets in the problem, while the q denote k the four-momenta of any ﬁnal-state radiation. TheQ are dimensionful variables that characterize i the hardness of the beam-jets and ﬁnal-state jets. We set Q =2E, twice the energy of each jet. i i ThecrosssectionforT lessthansomevalueT cut canbeexpressedintheform[13,14] N N (cid:34) (cid:35) (cid:90) N σ(T <T cut)= H⊗B⊗B⊗S⊗ ∏J +···. (2.2) N N n n The function H contains the virtual corrections to the process. The beam function B encodes the effect of radiation collinear to one of the two initial beams. It can be written as a perturbative 2 NNLOphenomenologyusingN-jettinesssubtraction RadjaBoughezal matchingcoefﬁcientconvolutedwithapartondistributionfunction. Sdescribesthesoftradiation, whileJ containstheradiationcollineartoaﬁnal-statejet. Theellipsisdenotespower-suppressed n terms which become negligible for T (cid:28)Q. Each of these functions obeys a renormalization- N i groupequationthatallowslogarithmsofT toberesummed. Ifthisexpressionisinsteadexpanded N to ﬁxed-order in the strong coupling constant, it reproduces the cross section for low T . The N derivation of this factorization theorem in the small-T limit relies upon the machinery of Soft- N CollinearEffectiveTheory[15]. ThebasicideabehindN-jettinesssubtractionisthatT fullycapturesthesingularitystructure N of QCD amplitudes with ﬁnal-state partons. This allows us to calculate the NNLO corrections to processessuchasHiggs+jet,W++jetandZ+jetinthefollowingway. Wedividethephasespace according to whether T is greater than or less than T cut. For T >T cut there are at least two N N N N hard partons in the ﬁnal state, since all singularities are controlled by N-jettiness. This region of phase space can therefore be obtained from, for example, a NLO calculation of Higgs production inassociationwithtwojetsinthecasewherethebornprocessisHiggs+jet. BelowT cut,thecross N section is given by the factorization theorem of Eq. (2.2) expanded to second order in the strong coupling constant. As long as T cut is smaller than any other kinematic invariant in the problem, N powercorrectionsbelowthecutoffareunimportant. AllingredientsofEq.(2.2)areknowntotheappropriateordertodescribethelowT region N through second order in the strong coupling constant. The two-loop virtual corrections for the processes discussed here are known [16, 17]. The beam functions are known through NNLO [18, 19],asarethejetfunctions[20,21]andsoftfunction[10]. Itisthereforepossibletocombinethis informationtoprovidethefullNNLOcalculationofHiggs+jetandW+/Z+jet. A full NNLO calculation requires as well the high T region above T cut. However, a ﬁnite N N value of T implies that there are actually N+1 resolved partons in the ﬁnal state. This is the N crucial observation; T completely describes the singularity structure of QCD amplitudes that N contain N ﬁnal-state partons at leading order. The high T region of phase space is therefore N described by a NLO calculation with N+1 jets. We must choose T cut much smaller than any N other kinematical invariant in the problem in order to a void power corrections to Eq. (2.2) below thecutoff. 3. NumericalResults WenowpresentsomenumericalresultsforHiggs,W+ andZproductioninassociationwitha jet. Forvalidationchecksoftheresultspresentedherewereferthereadertothedetaileddescription in [7, 8, 9]. We focus on 8 TeV proton-proton collisions. Jets are reconstructed using the anti-k T algorithm [22] with R=0.5. For the Higgs+jet process we show results using the NNPDF [23] parton distribution functions (PDFs), for the W++jet we use CT10 PDFs [24] while for Z + jet we use CT14 PDFs [25]. We use the perturbative order of the PDFs that is consistent with the partoniccrosssectionunderconsideration: LOPDFswithLOpartoniccrosssections,NLOPDFs with NLO partonic cross sections, and NNLO PDFs with NNLO partonic cross sections. We set therenormalizationandfactorizationscalesequaltothemassoftheHiggsboson, µ =µ =m R F H (cid:113) forHiggs+jet, µ =MW forW++jetand µ =µ0= m2ll+∑(pTjet)2 forZ+jet. Forthelatterscale choice the sum is over the transverse momenta of all ﬁnal-state jets, and m is the invariant mass ll 3 NNLOphenomenologyusingN-jettinesssubtraction RadjaBoughezal 0.16 LO 0.10 LO 0.14 NLO NLO V]0.12 NNLO V]0.08 NNLO Ge0.10 Ge [pb/0.08 [pb/0.06 jetσ/dpT0.06 Hσ/dpT0.04 d0.04 d 0.02 0.02 2.2 NLO 2.2 1.8 LO 1.8 NLO K 1.4 NNLO K 1.4 LO NLO NNLO 1.0 1.0 NLO 0.6 0.6 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180 pTjet[GeV] pTH[GeV] Figure 1: The transverse momentum of the lead- Figure2: ThetransversemomentumoftheHiggs ingjetfortheHiggs+jetprocessatLO,NLO,and boson at LO, NLO, and NNLO in the strong cou- NNLOinthestrongcouplingconstant. Thelower pling constant. The lower inset shows the ratios inset shows the ratios of NLO over LO cross sec- of NLO over LO cross sections, and NNLO over tions, and NNLO over NLO cross sections. Both NLO cross sections. Both shaded regions in the shadedregionsintheupperpanelandthelowerin- upperpanelandthelowerinsetindicatethescale- setindicatethescale-variationerrors. variationerrors. ofthedi-leptonpairarisingfromtheZ-bosondecay. Toestimatetheresidualtheoreticalerror,we varythesescalessimultaneouslyaroundthecentralvaluebyafactoroftwo. Wesetthemassofthe jet Higgsbosonasm =125GeV.Weimposethefollowingcutsontheﬁnal-statejet: p >30GeV, H T |η |<2.4 for Higgs+jet and |η |<2.5 forW++jet. For the Z + jet process we show a plot for jet jet the dependence of the ratio σ /σ on the power corrections as a function of τcut and refer NNLO NLO 1 thereadertothecorrespondingpaperformorephenomenologicalstudies[9]. We begin by showing few distributions in the Higgs plus jet production case. In Fig. 1 we showthetransversemomentumdistributionoftheleadingjet. Thereisashapedependencetothe jet corrections, with the K-factor decreasing as p is increased. This trend is visible when going T from LO to NLO in perturbation theory, and also when going from NLO to NNLO. We note that theNNLOresultisentirelycontainedwithintheNLOscale-variationband. Theshapedependence jet andmagnitudeoftheNNLOcorrectionsforthe p distributionareinagreementwiththeresults T ofRef.[4],obtainedusingsector-improvedresiduesubtractionscheme[27,28]. InFig.2weshow the transverse momentum of the Higgs boson. The NLO corrections range from 40% to 120% near pH =60 GeV, depending on the scale choice. The magnitude of this correction decreases as T thetransversemomentumoftheHiggsincreases. TheNNLOcorrectionsaremoremild,reaching only20%atmostforthecentralscalechoiceµ =m . Theyalsodecreaseslightlyasthetransverse H momentumoftheHiggsincreases. TheshapedependenceandmagnitudeoftheNNLOcorrections forthe pH distributionareinagreementwiththeresultsofRef.[4]. Wenotethatwehavecombined T the two bins closest to the boundary pH = 30 GeV to avoid the well-known Sudakov shoulder T effect[26]. In Fig. 3 we show the transverse momentum spectrum of the leading jet for W+ + jet at LO, NLO and NNLO in perturbation theory. The ratios of the NLO cross section over the LO result, as well as the NNLO cross section over the NLO one, are shown in the lower inset. The 4 NNLOphenomenologyusingN-jettinesssubtraction RadjaBoughezal LO LO NLO NLO ] 101 NNLO ] 101 NNLO V V Ge Ge pb/ pb/ [ [ jetσ/dpT100 Wσ/dpT100 d d 10-1 10-1 2.2 K 11..48 NNLLNOOLO K 11..04 NNLLNOOLO 1.0 NLO NLO 0.6 0.6 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180 pTjet[GeV] pTW[GeV] Figure 3: The transverse momentum spectrum of Figure 4: The transverse momentum spectrum of theleadingjetforW++jetatLO,NLOandNNLO theW+-bosonatLO,NLOandNNLOinperturba- in perturbation theory. The bands indicate the es- tion theory. The bands indicate the estimated the- timated theoretical error. The lower inset shows oretical error. The lower inset shows the ratios of the ratios of the NLO over the LO cross section, theNLOovertheLOcrosssection,andtheNNLO and the NNLO over the NLO cross section. Both over the NLO cross section. Both shaded regions shadedregionsintheupperpanelandthelowerin- in the upper panel and the lower inset indicate the setindicatethescale-variationerrors. Thedashed scale-variation errors. The dashed and solid black andsolidblacklinesinthelowerinsetrespectively linesinthelowerinsetrespectivelyshowthedistri- show the distribution for Tcut = 0.05 GeV and butionforTcut =0.05GeVandTcut =0.07GeV, 1 1 1 T1cut =0.07GeV,forthescalechoiceµ =2MW. forthescalechoiceµ =2MW. shaded bands in the upper inset indicate the theoretical errors at each order estimated by varying therenormalizationandfactorizationscalesbyafactoroftwoaroundtheircentralvalue,asdothe shadedregionsinthelowerinset. InthelowerinsetwehaveshowntheresultsforbothT cut =0.05 N GeVandT cut =0.07GeV,forthescalechoice µ =2M ,todemonstratetheT cut independence N W N in every bin studied. The NLO corrections are large and positive for this scale choice, increasing jet jet the cross section by 40% at p =40 GeV and by nearly a factor of two at p =180 GeV. The T T jet scale variation at NLO reaches approximately ±20% for p =180 GeV. The shift when going T fromNLOtoNNLOismuchmoremild, givingonlyapercent-leveldecreaseofthecrosssection jet thatvariesonlyslightlyas p isincreased. ThescalevariationatNNLOisatthepercentleveland T isnearlyinvisibleonthisplot. ThetransversemomentumspectrumoftheW-bosonisshowninFig.4. TheNLOcorrections areagain40%for pW ≥50GeVwithasizablescaledependence,whiletheNNLOcorrectionsare T ﬂatinthisregionanddecreasethecrosssectionbyasmallamount. Thephase-spaceregion pW < T 30 GeV only opens up at NLO, leading to a different pattern of corrections for these transverse momentum values. The instability of the perturbative series in the bins closest to the boundary pW =30GeViscausedbythewell-knownSudakov-shouldereffect[26]. T Finally, in Fig. 5 we show the dependence of the sum of the cross sections above and below theN-jettinesscutoffT cut andtheeffectofpowercorrections. Thevalidationisdonefortheratio N σ /σ in 13 TeV proton-proton collisions. We have checked that the NLO cross section NNLO NLO obtained with N-jettiness subtraction agrees with the result obtained with standard techniques. 5 NNLOphenomenologyusingN-jettinesssubtraction RadjaBoughezal ThesecrosssectionsareobtainedusingCT14PDFsatthesameorderinperturbationtheoryasthe partoniccrosssection,andcontainthefollowingﬁducialcutsontheleadingﬁnal-statejetandthe twoleptonsfromCMS[31]: pjet >30GeV,|η |<2.4, pl >20GeV,|η |<2.4and71GeV < T jet T l m <111GeV. TheATLASanalysisissimilarbutwithslightlydifferentcuts. Wereconstructjets ll (cid:113) using the anti-kT algorithm [22] with R=0.5. A dynamical scale µ0 = m2ll+∑pTjet,2 is chosen to describe this process, where the sum is over the transverse momenta of all ﬁnal-state jets, and m the invariant mass of the di-lepton pair arising from the Z-boson decay. In this validation plot ll wehavesettherenormalizationandfactorizationscalestoµ =µ =2×µ ;sincethecorrections R F 0 arelargerforthisscalechoice,itiseasiertoillustratetheimportantaspectsoftheT cut variation. 1 1.10 1.09 1.08 O 1.07 L N /σ 1.06 O L N 1.05 N σ 1.04 1.03 1.02 0.04 0.10 0.25 0.50 1.00 2.00 5.00 cut[GeV] T1 Figure5: PlotoftheNNLOcrosssectionovertheNLOresult,σ /σ ,asafunctionofTcut,forthe NNLO NLO 1 scalechoiceµ =2×µ . Theverticalbarsaccompanyingeachpointindicatetheintegrationerrors. 0 A few features can be seen in Fig. 5. First, in the region T cut <0.2 GeV the result becomes 1 independent of the particular value of the cut chosen within the numerical errors. The NNLO correction for µ =2×µ corresponds to a +3% shift in the cross section. The plot makes clear 0 that we have numerical control over the NNLO cross section to the per-mille level, completely sufﬁcient for phenomenological predictions. We observe an approximately linear dependence of σ onln(T cut)intheregion0.2GeV<T cut <0.5GeV,indicatingtheonsetoftheneglected NNLO 1 1 power corrections. These power corrections have the form (T /Q)lnn(T /Q), where n ≤ 3 at N N jet NNLO[12]andQisahardscalesuchas p . T 4. Conclusions We have presented in this proceedings the complete NNLO calculation of W+/Z + jet and Higgs boson production in association with a jet in hadronic collisions. To perform this compu- tation we have used a new subtraction scheme based on the N-jettiness event-shape variable T . N WewillfurtherstudythephenomenologicalimpactofourNNLOresultinfuturework, including the prediction for the exclusive one-jet bin, where an intricate interplay between various sources of higher-order corrections was recently pointed out [29]. 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Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.