Photon Structure Function in Supersymmetric QCD Revisited

0.11 MB·English

by  Ryo Sahara

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.
[2] http://www.linearcollider.org/cms.
[3] T.F.Walsh,Phys.Lett.36B(1971)121;S.J.Brodsky,T.Kinoshitaand
H.Terazawa,Phys.Rev.Lett.27(1971)280.
[4] M.Krawczyk, A.Zembrzuski and M. Staszel, Phys.Rept. 345 (2001)
265;R.Nisius,Phys.Rept.332(2000)165;M.Klasen,Rev.Mod.Phys.
74(2002)1221;I.Schienbein,Ann.Phys.301(2002)128;R.M.God-
bole,Nucl.Phys.Proc.Suppl.126(2004)414.
[5] T.F.WalshandP.M.Zerwas,Phys.Lett.44B(1973)195;R.L.Kingsley,
Nucl.Phys.B60(1973)45.
[6] E.Witten,Nucl.Phys.B120(1977)189.
[7] W.A.BardeenandA.J.Buras,Phys.Rev.D20(1979)166;21(1980)
2041(E).
[8] G.Altarelli,Phys.Rep.81(1982)1.
[9] R.J.DeWitt,L.M.Jones,J.D.Sullivan,D.E.WillenandH.W.Wyld,
Jr.,Phys.Rev.D19(1979)2046;D20(1979)1751(E).
[10] M.Glu¨ckandE.Reya,Phys.Rev.D28(1983)2749.
[11] T.UematsuandT.F.Walsh,Phys.Lett.101B(1981)263;Nucl.Phys.B
199(1982)93.
[12] G.Rossi,Phys.Rev.D29(1984)852;
F.M.BorzumatiandG.A.Schuler,Z.Phys.C58(1993)139;
M.DreesandR.M.Godbole,Phys.Rev.D50(1994)3124;
P.MathewsandV.Ravindran,Int.J.Mod.Phys.A11,(1996)2783;
J.Chy´la,Phys.Lett.B488(2000)289.
[13] E.Reya,Phys.Lett.B124(1983)424.
[14] D.A.RossandL.J.Weston,Eur.Phys.JC18(2001)593.
[15] M.Drees,M.Glu¨ckandE.Reya,Phys.Rev.D30(1984)2316.
[16] I.Antoniadis,C.KounnasandR.Lacaze,Nucl.Phys.B211(1983)216.
[17] Y.Kitadono,K.Sasaki,T.UedaandT.Uematsu,Prog.Theor.Phys.121
(2009)495;Phys.Rev.D81(2010)074029;
Y.Kitadono,Phys.Lett.B702(2011)135.
[18] Y.Kitadono,R.Sahara,T.UedaandT.Uematsu,Eur.Phys.J.C70(2010)
999.
[19] K.SasakiandT.Uematsu,Phys.Rev.D59(1999)114011.
[20] W.FurmanskiandR.Petronzio,Z.Phys.C11(1982)293.
[21] C.KounnasandD.A.Ross,Nucl.Phys.B214(1983)317.
[22] S.K.JonesandC.H.LlewellynSmith,Nucl.Phys.B217(1983)145.
[23] M.Buza,Y.Matiounine,J.Smith,R.MigneronandW.L.vanNeerven,
Nucl.Phys.B472(1996)611;
I.Birenbaum,J.Blu¨mleinandS.Klein,Nucl.Phys.B780(2007)40;820
(2009)417.
[24] M.Glu¨ck,E.ReyaandM.Stratmann,Phys.Rev.D51(1995)3220;D54
(1996)5515;
M.Glu¨ck,E.ReyaandI.Schienbein,Phys.Rev.D60(1999)054019;D
62(2000)019902(E);Phys.Rev.D63(2001)074008.
[25] K.Sasaki,J.SofferandT.Uematsu,Phys.Rev.D66(2002)034014.
[26] F.Cornet, P.Jankowski, M.Krawczyk andA.Lorca, Phys.Rev. D68
(2003)014010;F.Cornet,P.JankowskiandM.Krawczyk,Phys.Rev.D
70(2004)093004.
[27] M.Fontannaz,Eur.Phys.J.C38(2004)297;
[28] P.Aurenche, M.Fontannaz andJ.P.Guillet, Z.Phys.C64(1994) 621;
Eur.Phys.J.C44(2005)395;
[29] M. Aivazis, J. C. Collins, F. Olness and W. K. Tung, Phys. Rev. D50
(1994)3102.
[30] Y. Kitadono, Y. Yoshida, R. Sahara and T. Uematsu, Phys. Rev. D84
(2011)074031.
6