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Photon Structure Function in Supersymmetric QCD Revisited RyoSaharaa,TsuneoUematsua,YoshioKitadonob aDepartmentofPhysics,GraduateSchoolofScience,KyotoUniversity, Kitashirakawa,Kyoto606-8502,Japan bInstituteofPhysics,AcademiaSinica,Taipei,Taiwan Abstract We investigatethevirtualphotonstructurefunctioninthesupersymmetricQCD (SQCD),wherewe havesquarksandgluinosin 2additiontothequarksandgluons. TakingintoaccounttheheavyparticlemasseffectstotheleadingorderinQCDandSQCDwe 1evaluatethephotonstructurefunctionandnumericallystudyitsbehaviorfortheQCDandSQCDcases. 0 2Keywords: QCD,PhotonStructure,SUSY,LinearCollider n a J Since the experimentsat the Large Hadron Collider (LHC) 0 [1] started there has been much anticipation for the signals of e−(e+) 2theHiggsbosonaswellasforanevidenceofthenewphysics q2=-Q2<0 beyondStandardModelsuchassupersymmetry(SUSY).Once ] hthesesignalsareobservedmoreprecisemeasurementneedsto pbecarriedoutatthefuturee+e−collider,socalledInternational - LinearCollider(ILC)[2].Insuchacase,itisimportanttoknow p hetheItthiseowreeltlickanlopwrendtihcatito,ninsea+teh−igcholelniseirognieesxbpaesreimdeonntsQ,tChDe.cross e+(e−) p2=-P2<0 [section for the two-photonprocesses e+e− → e+e− +hadrons 2 dominates at high energies over the one-photon annihilation Figure 1: e+ e− two-photon processes in supersymmetric QCD. The solid vprocess[3]. Weconsiderherethetwo-photonprocessesinthe (dashed)linedenotesthequark(squark),whilethespiral(spiral-straight)line 5double-tageventswherebothoftheoutgoinge+ ande−arede- impliesthegluon(gluino). 6 tected. Especially, the case in which one of the virtual pho- 7 ton is far off-shell (large Q2 ≡ −q2), while the other is close 4 .to the mass-shell (small P2 = −p2), with Λ2 ≪ P2 ≪ Q2 masssquared,P2,inSQCDwherewehavesquarksandgluinos 1 (Λ: QCD scale parameter), can be viewed as a deep-inelastic inadditiontotheordinaryquarksandgluons. Evolutionequa- 1 1scattering wherethe targetis a virtualphotonand we can cal- tiontotheleadingorder(LO)inSQCDreadsasinQCD[19]: 1culate the photon structure functions in perturbation theories dqγ(t) α :[4,5,6,7,8,9,10,11,12]. = qγ(t)⊗P(0)+ k(0) , (1) v dt α (t) s i Sometimeagotheeffectsofsupersymmetryontwo-photon Xprocess were studied in the literature [13, 14, 15, 16]. In this whereP(0) and k(0)are1-loopparton-partonandphoton-parton rpaper based on the framework of treating heavy parton distri- splittingfunctions,respectively(seeAppendix). Thesymbol⊗ a butions [17, 18] we reexamine the effects of the squarks and denotestheconvolutionbetweenthesplittingfunctionandthe gluinosappearinginSUSYQCD(SQCD)onthephotonstruc- partondistributionfunction. Thevariablet isdefinedinterms turefunctionstotheleadingorderinSQCDwhichcanbemea- oftherunningcouplingαsas[20]: sured in thetwo-photonprocessesof e+e− collision illustrated 2 α (P2) dα (Q2) α (Q2)2 inFig.1. t= β lnαs(Q2), dlsnQ2 =−β0 s4π +O(αs(Q2)3) (2) 0 s withthepartondistributionsprobedbythevirtualphotonwith 1. EvolutionequationsfortheSUSYQCD masssquaredQ2as We consider the DGLAP type evolution equations for the qγ(t)=(G,λ,q1,···qnf,s1,··· ,snf), (3) partondistributionfunctionsinsidethevirtualphotonwiththe wheren isthenumberofactiveflavors. Ineq.(2),β =9−n f 0 f forSQCD.Wedenotethedistributionfunctionofthei-thflavor quark, squark by q(x,Q2,P2), s(x,Q2,P2), (i = 1,··· ,n ), Emailaddresses:[email protected](RyoSahara), i i f and the gluon, gluino by G(x,Q2,P2), λ(x,Q2,P2), respec- [email protected](TsuneoUematsu), [email protected](YoshioKitadono) tively.The1-loopsplittingfunctionswereobtainedin[21,22]. PreprintsubmittedtoPhysicsLettersB January23,2012 We firstconsiderthecase wherealltheparticlesaremassless. wherethesplittingfunctionsare Althoughthisisanunrealisticcase,itisinstructivetoconsider P P the massless case for the later treatment of the realistic case P(0) = qq sq , k(NS) =(K(NS),K(NS)), (14) NS Pqs Pss ! q s withtheheavymasseffects. ForthemasslesspartonstheevolutionstartsatQ2 = P2 and nf nf K(NS) = (e2−he2i)k , K(NS) = (e2−he2i)k . (15) hencewehavetheinitialconditionqγ(t=0)=0[11]. q i qi s i si Xi=1 Xi=1 The1-loopsplittingfunctionisgivenby(seeAppendixA) Fortheflavor-singletpartondistribution PGG PλG PqG PsG qγ =(G,λ,Σ,S), (16) S P P P P P(0) = Gλ λλ qλ sλ  , (4) wehave  PPGGqs PPλλqs PPqqqs PPssqs  ddqtγS = qγS ⊗P(S0)+ αsα(t)k(S) , (17) where P is a splitting functionof Bpartonto A partonwith AB where A,B=G,λ,qands. Whilethesplittingfunctionsofthephoton P P P P intothepartonsG,λ,qands,aredenotedas(seeAppendixB) GG λG qG sG P P P P P(0) = Gλ λλ qλ sλ  , (18) Weki(0n)tr=od(kuGce,ktλh,ekflq,akvso)r.-nonsinglet(NS)combinationsoft(h5e) S  PPGGqs PPλλqs PPqqqs nPPf ssqs  nf quarkandsquarkdistributionfunctionsas k(S) =(K(S),K(S)), K(S) = k , K(S) = k . (19) q s q qi s si nf Xi=1 Xi=1 q (x,Q2,P2)= (e2−he2i)q(x,Q2,P2), (6) Now we should notice that there exist the following super- NS i i Xi=1 symmetricrelationsforthesplittingfunctions[21]: nf P +P = P +P ≡P , (20) s (x,Q2,P2)= (e2−he2i)s(x,Q2,P2), (7) qq sq qs ss φφ NS i i P +P = P +P ≡ P , (21) Xi=1 qG sG qλ sλ φV P +P = P +P ≡ P , (22) Gq λq Gs λs Vφ where e is the i-th flavor charge and he2i = e2/n is the i i i f P +P = P +P ≡ P . (23) average charge squared. We also define the flaPvor-singlet (S) GG λG Gλ λλ VV combinationsforquarksandsquarks Henceifweintroducethefollowingcombinations nf φ≡Σ+S, V ≡G+λ, (24) Σ(x,Q2,P2)= qi(x,Q2,P2), (8) then we obtain the following compact form for the mixing of Xi=1 theflavor-singletpart: nf S(x,Q2,P2)=Xi=1 si(x,Q2,P2). (9) ddφt =Pφφ⊗φ+PφV ⊗V+ αsα(t)Kφ(S), (25) dV We now rearrange the parton components of qγ(t) using the = PVφ⊗φ+PVV ⊗V, (26) dt aboveflavornon-singletandsingletcombinationsas: wherewedenote K(S) = K(S) +K(S) andtheflavor-nonsinglet φ q s qγ(t)=(G,λ,Σ,S,q ,s ). (10) partφ =q +s becomes NS NS NS NS NS dφ α Thenwehavethefollowingsplittingfunction NS =P ⊗φ + K(NS), (27) dt φφ NS α (t) φ s PGG PλG PqG PsG 0 where we introduced K(NS) = K(NS) + K(NS). In terms of the φ q s P P P P  Gλ λλ qλ sλ  flavor singlet and non-singletparton distribution functionswe ThusPfo(0r)t=heflaPPvGGoqsr-noPP0nλλsqsinglPPeqqtqspartPPossnqsdisPPtrqqiqsbutiPPossnqss . (11) caneFx2γp(=rxe,sQsxt2h,ePv2()ier=2itu−axlhXpei2hioe)t(2ioq(nqi(isx(t,xrQ,uQc2t,2uP,rPe2)2f)+u+nscist(iix(o,xnQ,QF2,2γ2P,aP2s)2))) Xi qγNS =(qNS,sNS), (12) +xhe2i (qi(x,Q2,P2)+si(x,Q2,P2)) Xi satisfythefollowingevolutionequation: = x(q (x,Q2,P2)+s (x,Q2,P2)) NS NS dqγ α +xhe2i(Σ(x,Q2,P2)+S(x,Q2,P2)) dNtS = qγNS ⊗P(N0S) + α (t)k(NS) , (13) = xφNS(x,Q2,P2)+xhe2iφ(x,Q2,P2) (28) s 2 InFig.2wehaveplottedthevirtualphotonstructurefunction Theevolutionequationsandthesplittingfunctionsread Fγ(x,Q2,P2)intheSQCD aswellasintheordinaryQCDfor 2 dqγ(t) α PLS 0 Q2 = (1000)2GeV2 andP2 = (10)2GeV2. Wehavealsoshown = qγ(t)⊗P(0)+ k(0) , P(0) = , dt α (t) 0 PLNS ! thequarkaswellasthesquarkcomponentsofthevirtualphoton s structure function Fγ in the case of the SQCD. In contrast to P P nf−1P nf−1P 1 P 1 P 2 GG λG nf qG nf sG nf qG nf sG the QCD, the momentumfractioncarriedbythe quarksin the  P P nf−1P nf−1P 1 P 1 P  fmASginulQdoucndrCricneeitDnaiooflsgsenc.asattfhHsoaceeertodntmtswehcmcpeeoraaetSlrchaleQo-esdxmeCxstapD-odndouditnwsheetdherntieoitbcQcstuhrhCtetoesiaDohgsenoeemcwtsohaifssasesetrtaiholwabeansreeqgohsuegfeaaevtexhrtni,koetsrhifs.reqefqoo.uumFraiit2rγttekthhsbsedeteiarSffcunFQoecdimrtgCute.henr2Deest. PPLLSNS≡≡ PPPPPGGGGGqλqqssq PPPPPλλλλλsλqqqss .nPPf00qqqsqλ nPPf00ssqssλ nPPf00qqqsqλ nPPf00ssqssλ(32,) fromthatoftheQCD. Pqs Pss ! Q2=(1000)2 GeV2, P2=(10)2 GeV2 Whilethephoton-partonsplittingfunctionsare 3.5 Massless QCD k(0) =(k ,k ,k ,k ,k ,k ,k ,k ). (33) 3 Massless SQCD G λ qLS sLS qH sH qLNS sLNS Massless SQCD (quark) Massless SQCD (squark) Nowwetakeintoaccounttheheavymasseffectsbysettingthe 2.5 initialconditionsfortheheavypartondistributionfunctionsas α 2 discussedin[18,27,28]. /2 We note here that the structure function Fγ can be written γF 1.5 asaconvolutionofthepartondistributionqγ(x2,Q2,P2)andthe WilsoncoefficientfunctionC(x,Q2): 1 Fγ(x,Q2,P2)/x= qγ⊗C. (34) 0.5 2 0 Themomentsofthepartondistributionsaredefinedas 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Bjorken x 1 qγ(n,t)≡ dxxn−1qγ(x,Q2,P2), (35) Z Figure2: ThevirtualphotonstructurefunctionFγ(x,Q2,P2)dividedbythe 0 2 QEDcoupling constant αformasslessQCD(solidline) andSQCD(dashed whereweputtheinitialconditions: line)withnf =6,Q2 =(1000)2GeV2andP2 =(10)2GeV2. Alsoshownare thequark(dash-dottedline)andthesquark(double-dottedline)components. qγ(n,t=0)= 0,λˆ(n),0,sˆ (n),qˆ (n),sˆ (n),0,sˆ (n) , (36) LS H H LNS 2. Heavypartonmasseffects andrequirethatthe(cid:16)followingboundaryconditionsaresatisfie(cid:17)d: Many authors have studied heavy quark mass effects in the λ(n,Q2 =m2)=0, s (n,Q2 =m2 )=0, q (n,Q2 =m2)=0, λ LS sq H H nucleon [23] and the photon structure functions [24, 25, 26]. s (n,Q2 =m2 )=0, s (n,Q2 =m2 )=0, (37) Nowweconsidertheheavypartonmasseffects,andwedecom- H sq LNS sq posethepartondistributionsinthecasewherewehaven −1 wherem ,m andm arethemassofthegluino,squarksand f λ sq H lightquarksandoneheavyquarkflavorwhichwetakethen -th theheavy(herewetaketop)quark,respectively.Notethathere f quarkandallthesquarkshavethesameheavymass,whilethe wetakeallthesquarkshavethesamemassm . sq gluinohasanotherheavymass[17,18]: By solving the evolution equation taking into account the qγ(t)=(G,λ,q ,··· ,q ,s ,··· ,s ,q ,s ). (29) aboveboundaryconditionwegetforthemomentofqγ: 1 nf−1 1 nf−1 H H We denote the i-th light flavor quark, squark by qi(x,Q2,P2), qγ(n,t)= α 4π K(0) Pn 1 1− αs(t) 1+din si(x,Q2,P2), (i = 1,··· ,nf − 1), one heavy quark and its 8πβ0αs(t) n Xi i 1+din  "αs(0)#  superpartner (squark) by q , s and the gluon, gluino by G(x,Q2,P2),λ(x,Q2,P2),reHspectHively. +qγ(n,0) Pn αs(t) din , (38) Wenowdefinelightflavor-nonsinglet(LNS)andsinglet(LS) Xi i "αs(0)# combinationofthequarkandthesquarkasfollows: wherethePnistheprojectionoperatorontotheeigenstateλ of i i nf−1 nf−1 theanomalousdimensionmatricesγˆn: q = e2−he2i q, s = e2−he2i s , LNS i L i LNS i L i Xi=1 (cid:16) (cid:17) Xi=1 (cid:16) (cid:17) γˆn = Pniλni , (39) nf−1 nf−1 1 nf−1 Xi qLS = qi, sLS = si, he2iL = n −1 e2i . (30) where the anomalous dimension matrices γˆn is related to the Xi=1 Xi=1 f Xi=1 splittingfunctionP(x)as Thenwerearrangethepartondistributionsas 1 qγ(t)=(G,λ,q ,s ,q ,s ,q ,s ). (31) γˆn ≡−2 dxxn−1P(x), (40) LS LS H H LNS LNS Z 0 3 anddn ≡λn/2β . K(0) istheanomalousdimensioncorrespond- Q2=(1000)2 GeV2, P2=(10)2 GeV2, msquark=300 GeV, mgluino=700 GeV i i 0 n 3.5 ingtothephoton-partonsplittingfunction: Massless QCD Massless SQCD 3 Massive QCD with Threshold 1 Massive SQCD with Threshold (quark) K(0) =2 dxxn−1k0(x). (41) Massive SQCD with Threshold (squark) n Z 2.5 Massive SQCD with Threshold (total) 0 Theinitialvalueqγ(n,0)isdeterminedsothatwehave α 2 /2 2 α (P2) γ qγ(t=t )=0, t = ln s , (42) F 1.5 j mj mj β α (m2) 0 s j 1 or 0= αs4(tπmj)Xi+(cid:16)K(n0)Pqγni(cid:17)(nj,10+)1/dinα1P−nααssα((mPs(22mj))2j)1+ddinin, (43) 0 .05 0 0.1 0.2 0.3 0B.4jor 0k.5en 0x.6 0.7 0.8 0.9 1 for j = λ,s ,qXi,s and s 8πβ.0 Bi!yjsαosl(vPin2)gthe above c(FQliugCdueDrde).3ca:TsheFe.2γTd(xha,sehQde2od,uPb(22led)/-odαto-dtwtaeisdthhecduS)rUvceSuYsrhvoepwacrsotirtchrleeessmpoaanssdswsiveetlolQatChseDtomcpaasstehs.lreeTssshheoSldQdaCsihnD-- LS H H LNS coupled equations we get the initial condition: qγ(n,0) = dotted(dotted)curvecorrespondstothequark(squark)componentofthemas- siveQCD.ThesolidcurvemeanstheFγ/αforthemassiveSQCD.Thekinkat (0,λˆ(n),0,sˆLS(n),qˆH(n),sˆH(n),0,sˆLNS(n)). x=0.89(0.74)correspondstothetop(sq2uark)threshold. Now we write down the moments of the structure function intermsofthepartondistributionfunctionsandthecoefficient functions,whichareO(α0)atLO.Wetake s dotted curve means the squark component for the same case. C(0)(1,0)T =(0,0,he2i ,he2i ,e2,e2,1,1). (44) The sum of these leads to the solid curve which corresponds n L L H H toFγ/αforthemassiveSQCDwithmassivetopandthreshold Thenthen-thmomentofthestructurefunctionFγ tothelead- 2 2 effectsincluded.Here,weadopttheprescriptionfortakinginto ingorderinSQCDisgivenby accountthe threshold effects by rescaling the argumentof the 1 distributionfunction f(x)as[29]: Mγ = dxxn−1Fγ/x= qγ(n)·Cγ(1,0) n Z 2 n 1 0 f(x)→ f(x/x ), x = (46) =he2iLqLS +he2iLsLS +e2Hq2H +e2HsH +qLNS +sLNS . (45) max max 1+ QP22 + 4Qm22 wherex isthemaximalvaluefortheBjorkenvariable.After max 3. Numericalanalysis thissubstitutiontherangeofxbecomes0≤ x≤ x . Atsmall max x,thereisnosignificantdifferencebetweenmasslessandmas- We have solvedthe equations(43) for qγ(n,0) numerically, siveQCD, while thereexistsalargedifferencebetweenmass- andplugthemintothemasterformula(38)forthepartondistri- lessandmassiveSQCD.Atlargex,thesignificantmass-effects butionfunctionsandthenevaluatethemomentsofthestructure functionFγbasedontheformula(45). ByinvertingtheMellin existbothfornon-SUSYandSUSY QCD. TheSQCD case is momentw2egettheFγ asafunctionofBjorkenx. seen to be much suppressed at large x comparedto the QCD. InFig.3,wehavep2lottedournumericalresultsfortheFγ/α. The squarkcontributionto the totalstructurefunctionin mas- The 2dot-dashed and dashed curves correspond to the F2γ/α sive SQCD appears as a broad bump for x < xmax. Here of 2 coursewecouldsetthesquarkmasslargerthan300GeV,e.g. for the massless QCD and SQCD, respectively, where all the around1TeV,asrecentlyreportedbytheATLAS/CMSgroup quarksandsquarksaretakentobemassless. Ofcoursethisis atLHC,forhighervaluesofQ2. the unrealistic case we discussed in the previoussection. For the more realistic case, we take n = 6 and treat the u, d, s, f c and b to be massless and take the top quark t massive. We 4. Conclusion assume that all the squarks possess the same heavy mass and thegluinohasanotherheavymass. Intheseanalyses,wehave In this paper we have studied the virtual photon structure taken Q2 = (1000)2GeV2 and P2 = (10)2GeV2. Forthe mass functionintheframeworkofthepartonevolutionequationsfor values we took the top mass m = 175 GeV, the common thesupersymmetricQCD,wherewehavePDFsforthesquarks top squark mass, m = 300 GeV and the gluino mass m = 700 andgluinosinadditiontothoseforthequarksandgluons. s λ GeV. We considered the heavy parton mass effects for the top The double-dottedcurveshows Fγ/αforthe QCD with the quark, squarks and gluinos by imposing the boundary condi- 2 massofthetopquarkaswellasthethresholdeffectstakeninto tions for their PDFs in the framework treating heavy particle account. The dash-dotted curve shows the quark component distributionfunctions[18]. ThePDFfortheheavyparticlewith forthe massiveSQCD case withmassivetop quark,whilethe masssquared,m2arerequiredtovanishatQ2 =m2.Thiscanbe 4 translatedintotheinitialconditionfortheheavypartonPDFs, where C = 4/3 and C = 3 for SQCD. Hence we have the F A qγ(t = 0). Dueto theinitialconditionthesolutiontothe evo- followinganomalousdimensionsforthesupersymmetriccase: lution equation is altered as given by (38). This change leads 2 to the heavy mass effects for the PDFs. As we have shown γn =γn +γn =γn +γn =2C −2− +4S (n) , in Fig.3, there is no significant difference in the small-x re- φφ qq sq qs ss F" n 1 # gion between QCD and SQCD, while at large x, it turns out 1 thatthereexistsasizabledifferencebetweenthemassiveQCD γn =γn +γn =γn +γn =−4n , φV qG sG qλ sλ fn andSQCD.WhencomparedtothesquarkcontributiontoFγin 2 2 1 thepartonmodelcalculation[30],thesquarkcomponentinthe γn =γn +γn =γn +γn =−4C − , Vφ Gq λq Gs λs F"n−1 n# SQCD is suppressedatlarge x due to the radiativecorrection. We expectthatthefuturelinearcolliderwouldenablesuchan γVnV =γGnG+γλnG =γGnλ+γλnλ analysistobecarriedoutonphotonstructurefunctions. 4 2 =2C −3− + +4S (n) +2n , (A.3) A" n−1 n 1 # f Appendix A. AnomalousDimensionsforSUSYQCD where we have the following replacement: n γn → γn . In f φV φV Notethattheourconventionfortheanomalousdimensionis the case of non-supersymmetric QCD we have the following relatedtotheabovesplittingfunctionas anomalousdimensions: 1 8 2 γinj =−2Z dxxn−1Pij(x). (A.1) γψ0,ψn =γ0N,Sn = 3"−3− n(n+1) +4S1(n)#, 0 The1-loopanomalousdimensionsforSUSYQCDaregivenby γ0,n =−4n n2+n+2 , ψG fn(n+1)(n+2) 2 γqnq =2CF"−2− n(n+1) +4S1(n)#, γ0,n =−16n2+n+2, Gψ 3 n(n2−1) −2 γn =2C , 11 4 4 sq F"n+1# γ0,n =6 − − − +4S (n) GG " 3 n(n−1) (n+1)(n+2) 1 # −2 γn =2C , 4 qs F" n # + nf. (A.4) 3 γn =2C [−2+4S (n)], ss F 1 n2+n+2 γn =−4n , Appendix B. Photon-parton mixing anomalous dimen- qG fn(n+1)(n+2) sions 2 1 γn =−4n =−8n , sG f(n+1)(n+2) f(n+1)(n+2) Thephoton-partonsplittingfunctioncanbeconnectedtothe 1 1 photon-partonmixinganomalousdimensionsgivenby γn =−4n − , qλ f"n n+1# K(0) = K0,n,K0,n,K0,n,K0,n,K0,n,K0,n,K0,n ,K0,n , 1 n G λ qLS sLS qH sH qLNS sLNS γnsλ =−4nf n+1!, (cid:16) (cid:17) (B.1) n2+n+2 where γn =−4C , Gq F n(n2−1) K0,n = K0,n =0, 1 G λ γn =−4C , λq Fn(n+1) K0,n =24(n −1)he2i n2+n+2 , 2 2 qLS f Ln(n+1)(n+2) γn =−4C − , Gs F"n−1 n# K0,n =24(n −1)he2i 1 − n2+n+2 , 1 sLS f L"n n(n+1)(n+2)# γn =−4C , λs Fn n2+n+2 K0,n =24e2 , γn =2C −3− 4 − 4 +4S (n) +2n , qH Hn(n+1)(n+2) GG A" n(n−1) (n+1)(n+2) 1 # f 1 n2+n+2 K0,n =24e2 − , n2+n+2 n2+n+2 sH H"n n(n+1)(n+2)# γλG =−4CAn(n+1)(n+2) =−12n(n+1)(n+2), K0,n =24(n −1) he4i −he2i2 n2+n+2 , γGnλ =−4CA"n−21 − 2n + n+11#, Kq0L,nNS =24(nf−1)(cid:16)he4iL−he2i2L(cid:17)n1(n−+1)n(n2++n2)+2 . sLNS f L L "n n(n+1)(n+2)# 2 2 (cid:16) (cid:17) γλλ =2CA"−3− n + n+1 +4S1(n)#, (A.2) (B.2) 5 References [1] http://lhc.web.cern.ch/lhc. 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