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HERWIRI2: CEEX Electroweak Corrections in a Hadronic MC 2 1 0 2 ScottYost† ∗ n TheCitadel a J Charleston,SC29409,USA 7 E-mail: [email protected] 2 ValerieHalyo‡ ] h PrincetonUniversity p Princeton,NJ08544,USA - E-mail: [email protected] p e h MiroslavHejna [ PrincetonUniversity 1 Princeton,NJ08544,USA v E-mail: [email protected] 6 0 B.F.L.Ward§ 9 BaylorUniversityWaco,TX76798,USA 5 . E-mail: [email protected] 1 0 2 Reachingthe1%precisionlevelforW andZ productioncalculationsfortheLHCwillrequirea 1 mixtureofhigherorderQCD andelectroweakcorrections. Asafirststep towardimplementing : v the combined QED QCD exponentiation proposed in previous work, we have implemented Xi the O(a ) electrowea⊗kcorrectionsand YFS exponentiationstructure of the KK Monte Carlo in r a HERWIG. We discuss the current status of this program and sketch the further developments neededtoreachthedesiredprecisionlevel. 10thInternationalSymposiumonRadiativeCorrections(ApplicationsofQuantumFieldTheoryto Phenomenology) September26-30,2011 Mamallapuram,India BU-HEPP-11-06 Speaker. ∗ †ThisworkanditspresentationweresupportedinpartbyD.o.E.grantDE-PS02-09ER09-01andgrantsfromThe CitadelFoundation. ‡WorksupportedinpartbyD.o.EgrantDE-FG02-91ER40671. §WorksupportedinpartbyD.o.E.grantDE-FG02-09ER41600. (cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ HERWIRI2:CEEXElectroweakCorrectionsinaHadronicMC ScottYost Vector boson production is one of the most important Standard Model processes observed at the LHC, and electroweak radiative corrections will be needed for analysis at the percent level. Previous studies [1] by some of the authors have found that electroweak corrections alone can exceed 1% for some cuts of interest. These studies were based on HORACE,[2] which provides state-of-the-art O(a ) radiative corrections with a final-state photon shower, and PHOTOS, [3] whichaddsfinalstatephotonicradiation. Otherprogramsdevelopedforimplementingelectroweak corrections for hadronic collisions include WINHAC [4] and ZINHAC[5]. Initial state photonic radiation has been considered only in certain MRST parton distribution functions. [6] However, noneofthemostrecentPDFsinclude QEDeffects. For electron-positron colliders, precision electroweak corrections have been implemented in theprogram KKMC[7]usingYFS[8]exponentiated multiple-photon radiation forboththeinitial and final state, together with O(a ) electroweak corrections in the DIZET [9] package developed for ZFITTER[10]. The DIZET corrections can be applied to any parton-level process, so there is noobstacletoextendingthemtohadroniccollisions. YFSexponentiation providesthebasisforan efficient representation of multi-photon phase space, which can also be implemented for hadronic initial and final states. Some of the authors have proposed using a non-abelian extension of YFS exponentiation as a basis for an integrated event generator implementing both multi-photon and multi-gluon corrections inaunifiedframework,calledQCD QEDexponentiation. [11] ⊗ The collection of programs implementing QCD QEDexponentiation has been called HER- ⊗ WIRI, for High Energy Radiation with Infra-Red Improvements, with a version number distin- guishing the class of corrections included. The name acknowledges that the initial versions build upontheHERWIG[12]partonshowergenerator. Thefirsttobereleased, HERWIRI1,[13]imple- mentedIR-improvedsplitting kernels[14]obtainedusingtheQCDanalogofYFSexponentiation. This program is publicly available, and tests are in progress. The IR-improved kernels have also been implemented [15] in MC@NLO[16] and POWHEG[17]. The structure of HERWIRIis not tiedtoaparticular shower, andourultimategoalisacomplete showergenerator basedentirely on QCD QEDexponentiation withexactO(a 2,a a ,a 2)residuals. [18] s s ⊗ Thesecondversion, HERWIRI2,implementstheelectroweak radiativecorrections ofKKMC in a hadronic shower generator. This note describes a version of HERWIRI2 which is presently nearingcompletion.1 HERWIRI2ismotivatedbythesuccessfulapplicationofYFSexponentiation in BHLUMI[19], BHWIDE[20], KoralZ[21], KoralW[22], YFSWW3[23], KKMC, and related programsforLEPphysics. Alloftheseprogramsbenefitfromaveryefficientrepresentation ofN- photon phasespace, withcomplete control overthesoftandcollinear singularities foranarbitrary number ofphotons. Realand virtual IR singularities cancel exactly toall orders. Thenon-abelian extension toQCD QEDshouldhavesimilaradvantages. ⊗ WhilebasedonHERWIG,HERWIRI2islargelyindependentoftheunderlyingshower. HER- WIGgeneratesthepartonmomentaandshower,andHERWIRI2passesthegeneratedhardprocess momenta to KKMC routines to add photons and electroweak corrections. Although KKMC was developed for e+e collisions, it was designed to be extended to more general processes, so the − ability to select quarks as the incoming state already exists in all but the lowest-level generation routines. 1HERWIRI2doesnotincorporateHERWIRI1,althoughthetwoprogramscanbeusedincombination. 2 HERWIRI2:CEEXElectroweakCorrectionsinaHadronicMC ScottYost KKMC is a precision generator for e+e f f +ng , f = m ,t ,d,u,s,c,b for CMS energies − → from 2mt to 1 TeV. The precision tag for LEP2was 0.2%. ISR and FSR g emission is calculated uptoO(a 2),including interference. TheMCstructure isbased onYFSexponentiation, including residuals calculated perturbatively to the relevant orders in a kLl. (L=ln(s/m2)). Exact collinear e bremsstrahlung is implemented for up to three photons. Electroweak corrections [24, 25, 26] are included viaDIZET6.21[9],andbeamstrahlung canbemodeledoverawiderangeofenergies via abuilt-inoruser-defined distribution. Therearetwomodesofoperation: exclusiveexponentiation(EEX)andcoherentexclusiveex- ponentiation (CEEX).EEXappliesYFSexponentiation todifferentialcross-sections, whileCEEX applies it at the amplitude level. The CEEX mode is written in a manner that is most readily extended to quark scattering, so it is taken as the basis for HERWIRI2. The orders of residuals included inCEEXmodearea ,a L,a 2L2,anda 2L. CEEX was introduced for pragmatic reasons, because the traditional (EEX) exponentiation of spin-summed differential cross sections suffered from a proliferation of interference terms in processes with multiple diagrams, limiting its utility in reaching the desired 0.2% precision tag for LEP2.[27] CEEX works at the level of spinor helicity amplitudes, greatly facilitating the cal- culation of effects such as ISR-FSR interference, which are included in KKMC, and therefore in HERWIRI2. CEEXis maximally inclusive: all real photons radiated are kept in the event record, no matter how soft or collinear. There is no need to integrate out a region of soft phase space, because theexponentiated amplitudes arewell-behaved atk=0. TheCEEXcrosssectionforqq f f hastheform[7] → ¥ s = 1 (cid:229) dPSr (n) (~p,~k) (1) flux CEEX n=0Z where 2 r (n) = 1 eY(~p,Emin)1 (cid:229) M ~p ~k (2) CEEX n! 4~l ,~m (cid:12)(cid:12) ~l ~m !(cid:12)(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) TheYFSformfactoris (cid:12) (cid:12) Y(~p,E ) = Q2Y(p ,p ,E )+Q2Y(p ,p ,E )+QQ Y(p ,p ,E ) min i 1 2 min f 3 4 min i f 1 3 min +QQ Y(p ,p ,E ) QQ Y(p ,p ,E ) QQ Y(p ,p ,E ), (3) i f 2 4 min i f 1 4 min i f 2 3 min − − whereQ,Q areinitialandfinalpartoncharges, and i f Y(p,p ,E )=2a B(p,p ,E )+2a ReB(p,p ), (4) i j min i j min i j withrealandvirtualformfactors definedreespectively by d3k p p 2 i j B = , (5) −Zk0<Emin 8p 2k0(cid:18)pi·k− pj·k(cid:19) i d4k 2p +k 2p k e i j B = − . (6) (2p )3 k2 2p k+k2 −2p k k2 Z (cid:18) i· j· − (cid:19) 3 HERWIRI2:CEEXElectroweakCorrectionsinaHadronicMC ScottYost Then-photon helicity-spinor amplitudecanbeexpandedintermsofordera r havingtheform Mn(r)=(cid:229)P (cid:213)i=n1Si(Pj)b 0(r) ~~lp; XP!+(cid:229)j=n1b 1(r)(cid:18)S~l~pmj(Pk ;j)XP(cid:19)+···+1<j1(cid:229)<···<jnSb n(jr(1)P(cid:18)j1~l~)p··~~mk·S; Xj(nPPj(cid:19)n)      (7)  withresidualspinoramplitudes b (r) andcomplexsoftphotonfactorsS withtheproperty i j 2 S(Pj) = 2pa Q2 pa pb (8) j − p k − p k a j b j (cid:12) (cid:12) (cid:18) · · (cid:19) (cid:12) (cid:12) whereQ,p ,p belong tothe(cid:12)initial(cid:12)orfinalfermionsdepending onthepartition P . a b j KKMCincorporates the DIZETlibrary (version 6.21) from the program ZFITTER.[10] The g andZ propagators aremultiplied byvacuum polarization factors: Hg = 2 1P g , HZ =4sin2(2q W)r E8Wpa G√m M2Z2. (9) − Vertex corrections are incorporated into the coupling of Z to f via form factors in the vector coupling: (f) T g(Z,f) = 3 Q F(f)(s)tanq . (10) V sin(2q )− f v W W Box diagrams contain these plus a new angle-dependent form-factor in the doubly-vector compo- nent: (i) (f) (i) (f) (f) (i) (i,f) T T 2T Q F (s) 2QT F (s)+4QQ F (s,t) g(Z,i)g(Z,f) = 3 3 − 3 f v − i 3 v i f box . (11) V V sin2(2q ) W Thecorrection factorsarecalculated atthebeginning ofarunandstoredintables. The Drell-Yan cross section with multiple-photon emission can be expressed as an integral overtheparton-level processq(p )q(p ) f(p )f(p )+ng (k),integratedoverphasespaceand i 1 i 2 3 4 → summedoverphotons. Thepartonmomenta p ,p aregeneratedusingpartondistributionfunctions 1 2 givingaprocess atCMSenergyqandmomentumfractions x ,x suchthatq2=x x s: 1 2 1 2 s = dx1dx2(cid:229) f(q,x )f(q,x )s (q2)d (q2 x x s), (12) DY x x i 1 i 2 i − 1 2 Z 1 2 i wherethefinalstatephase spaceincludes p ,p andk,i=1, ,nandmultiplegluon radiation + 3 4 i ··· hadronization isincluded through ashower. The parton-level cross section s (q2) can be calculated by KKMC, which integrates over a i finalspacephasespacewithtwofermionsandanarbitrary numberofphotons: ¥ s (q2)= (cid:229) dPS s (~p,~k) (13) i 2+n i n=0Z HERWIRI2 uses HERWIG 6.5 as the shower generator, which creates the hard process first, at Born level, in subroutine HWEPRO (HWHDYP), and then passes it to the cascade generator HWBGEN.HERWIRI2findstheZ/g andthepartons interacting withitintheeventrecord. The ∗ 4 HERWIRI2:CEEXElectroweakCorrectionsinaHadronicMC ScottYost initial partons define p , p , which are transformed to the CM frame and projected on-shell to 1 2 create a starting point for KKMC,which generates the final fermion momenta p ,p and photons 3 4 k (bothISRandFSR.)Thegeneratedparticlesaretransformedbacktothelabframeandplacedin i theeventrecord. InadditiontothebasicDYprocess,HERWIGgenerates“Compton”eventsg+q q+Z/g . ∗ → About 10% of the events have this form. This is factorized into gluon emission times a hard EW process at a shifted value of q2. These have a different profile in the event record, but can be processedbyKKMCaswell. Thereisalsoathirdclassofeventswiththeemissionofanadditional hardgluon. About1%oftheeventshavethisform,andalsohaveasignificantshiftoftheZenergy fromitsgeneration scale. Withachangeofvariables, theDrell-Yancrosssection inHERWIGcanbeexpressed as s = dx1dx2(cid:229) f(q,x )f(q,x )s (q2)d (q2 x x s) DY x x i 1 i 2 i − 1 2 Z 1 2 i = qmaxdqP(q) 1 dx1(cid:229) P W(i) (q2,x )= W (14) Zqmin Zq2/s x1 i i HW 1 h HWi where P(q) is a normalized, integrable, crude probability distribution for q, P is the crude proba- i bilityofgenerating partoni,andW istheHERWIGeventweight. Thisweightdepends onlyon HW thehardBorncrosssectionandisnotalteredbytheshower. Thecrudeprobability distributions usedbyHERWIGare 1 Ng N2q P(q)= 2[Pg(q)+PZ(q)], Pg(q)= q4, PZ(q)= (q2 M2)+G 2M2 (15) − Z Z Z TheHERWIGeventweightis W =(cid:229) W(i) , W(i) = 1 f(q,x )f(q,x )ln s s (i) (q2) (16) HW HW HW P(q) i 1 i 2 q2 HW i (cid:18) (cid:19) andthecorresponding probability forselectingpartoniis (i) P =W /W (17) i HW HW Wehavechosentointroduceelectroweakcorrectionsinaminimallyinvasiveway,incorporat- ingtheminaformfactor s (q2) F(i)(q2)= i (18) EW s (i) (q2) Born KKMCwillcalculate theEWformfactor, andmultiplyitbytheHERWIGBorncrosssection. s (i)(q2) s = W , W =F(i)(q2)W =W KK . (19) HW+EW h Toti Tot EW HW HWs (i) (q2) Born TheKKMCcrosssectioniscalculated usingaprimarydistribution ds P(dir)i(vs,v) =s B(io)rn(s(1−v))21 1+√11 v g ivgi−1vmgi−ingi (20) (cid:18) − (cid:19) 5 HERWIRI2:CEEXElectroweakCorrectionsinaHadronicMC ScottYost with 2a s 2a s g = Q2 ln 1 , g = Q2ln (21) i p i m2 − i p i m2 (cid:20) (cid:18) i (cid:19) (cid:21) (cid:18) i (cid:19) togeneratethefactorvgivingthefractionofsremainingafterISRphotonemission,s =s(1 v). X − TheKKMCcrosssectionis ds ds s (q2)= ds Cru Mod =s W W . (22) Prids ds Prih Cru Modi Z Pri Cru W iscalculated duringISRgeneration andW isgenerated afters isavailable. Cru Mod X TheHERWIGandKKMCweightsarecombinedtocalculatethetotalHERWIRI2weight, s (q2) W(i)W(i) s = W i = W s (i)(q2) Cru Mod , (23) Tot * HWs (i)⋆(q2)+ * HW Pri s (i)⋆(q2)+ Born Born This average could be calculated using a joint probability distribution for q and v, D(q,v)= P(q)ds /dv,withP(q)fromHERWIG.AnadaptiveMC(S.Jadach’sFOAM[28])couldcalculate Pri thenormalizationofthedistributionatthebeginningoftherun,inasimilarmannertohowKKMC presently integrates the one-dimensional primary distribution. To account for beamsstrahlung, KKMC already permits such a user-defined distribution, in up to three variables. However, as a first step, we have tried to run HERWIRI2 using KKMC’s one-dimensional primary distribution. Thisrequiresfixinganoverallscaleq toinitialize KKMC(e.g.,q =M ). 0 0 Z Thebuilt-inprimarydistribution forelectronsatscaleq canbeusedforthelow-levelgenera- 0 tion ofv. Thetransformation from thisdistribution toadistribution atHERWIG’sgenerated scale qforquarkiisthenobtained byachangeofvariables: ds (i)(q2,v) W(i) W(i) s =s (e) W Pri Crud Mod (24) Tot Pri * HW ds (e)(q2,v)! s (i)⋆(q2) !+ Pri 0 Born with ds (i)(q2,v) s (i) (q2(1 v)) Pri =Wg(i) Born − , (25) ds (e)(q2,v) s (e) (q2(1 v)) Pri 0 Born 0 − where Wg = gg i vv gi−gevmgi−inge. (26) e(cid:18) min(cid:19) Theg factorsarecalculated usingq2/m2 forpartoniandq2/m2 fortheelectron. i 0 e Shufflingthenumeratorsanddenominators aboutgivestheexpression usedinHERWIRI2: s Tot= WHWWModWKarlWFFWg (27) (cid:10) (cid:11) withtwonewweights s (e)W(i) s (i) (q2(1 v)) W = Pri Crud , W = Born − . (28) Karl FF s (e) (q2(1 v)) s (i)⋆(q2) Born 0 − Born HERWIRI2 is still under development, so any numerical results must be treated as preliminary. Arunfor ppcollisions at10TeVwiththeZ/g invariant massbounded by40GeVand140 ∗ 6 HERWIRI2:CEEXElectroweakCorrectionsinaHadronicMC ScottYost GeV, using HERWIG 6.520 default parameters and PDFs, yields a cross-section of 1183.7 1.3 ± pb, compared to1098.8 1.0 pbforHERWIGalone, givinganelectroweak contribution of7.7%. ± Turning on ISR gives a much wider weight distribution and consequently, greatly reduced effi- ciency. Preliminary results give a cross-section of 1212 109 pb, showing an additional 2.4% ± contribution fromISR. Work is in progress to optimize MC generation in the presence of ISR. Once HERWIRI2 is complete and thoroughly tested, it will be compared to other available hadronic/electroweak generators. Inparticular, itwillbeinterestingtoseetheeffectofinitialstateradiation,whichisnot presentintheotherprograms,butappearstoenteratthe2–3%level,makingitcrucialtoprecision calculations. 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