On the n - 0 limit of gamma_gg(a) in QCD

arXiv:hep-ph/0008067v18Aug2000Onthen→0limitofγgg(a)inQCDJ.F.Bennett&J.A.Gracey,TheoreticalPhysicsDivision,DepartmentofMathematicalSciences,UniversityofLiverpool,PeachStreet,Liverpool,L697ZF,UnitedKingdom.Abstract.Weconsiderthen→0limitoftheDGLAPsplittingfunctionγgg(a)atallordersinthestrongcouplingconstant,a,byanalysingtheleadingorderlargeNfformoftheassociatedd-dimensionalcriticalexponent.Weshowthatforunpolarizedscatteringthepoleatn=0whichappearsinsuccessiveordersinperturbationtheoryisabsentintheresummedexpression.LTH4831TheDGLAPequation,[1],iswidelyusedtoevolvethepartonstructurefunctionsofthenucleonconstituentsoveralargerangeofenergyscales.Centraltotheequationarethesplittingfunctionswhichdependonthevariablexwhichrepresentsthefractionofthemomentumcarriedbythatpartoninthenucleon.Alternativelyonecanworkwiththeanomalousdimensionoftheunderlyingtwist-2operatorsbuiltoutofthequarkandgluonfieldsofQCDwhichdependonthevariablen.TheyarerelatedtothesplittingfunctionsviaaMellintransformrestrictedtotheunitintervalinx.ItisgenerallyacceptedthatthesolutionoftheDGLAPevolutionwhicheffectivelyrepresentsperturbativeQCD,fitsthedataextremelywell.See,forexample,[2].Moreover,itappearstotranscendtheregionwheretheperturbativeapproximationoughtnottobevalid.Therefore,onewouldhopethateitherbyresummingperturbationtheoryinsomefashionordevelopingnon-perturbativemethods,suchastheBFKLformalism[3],itmightbepossibletobegintoexplorephenomenainthemoreextremenon-perturbativeregions.Onerecentissue,[4],hasbeentheproblemofunderstandingthen→0behaviouroftheDGLAPsplittingfunctionsinQCD.Itisknownthatinperturbationtheoryeachtermoftheexpansionoftheoperatoranomalousdimensionsinthestrongcouplingconstanthassuccessivelyhigherorderpolesatn=0,[5,6,7].However,otherconsiderationssuggestthatthefunctionisfiniteatn=0,[4,8],thoughthereissomedisagreementwiththispointofview,[9].Inthisletterweprovidesomemoreinsightintothen→0limitofthegluon-gluonsplittingfunction,γgg(a),wherea=αs/(4π)isthestrongcouplingconstant.ThisisachievedbyexaminingthelargeNfresultforγgg(a)whichhasbeencomputedin[10]whereNfisthenumberofquarkflavours.Essentiallythe1/NfexpansionsumsadifferentsetofFeynmandiagramsfromthosewhichareordinarilycomputedinthelooporcouplingconstantexpansionofperturbationtheory.Moreover,theresummationistoallordersina.Thistechniquehasbeenusedin[11,12,10,13]todeterminetheanomalousdimensionsofallthetwist-2operatorsusedindeepinelasticscatteringforbothunpolarizedandpolarizedprocessesatO(1/Nf).Theallordersresultsareexpressedasafunctionofd,wheredisthedimensionofspacetime,knownascriticalexponents.Thecoefficientsofthecorrespondingrenormalizationgroupfunctionarededucedfromknowledgeofthed-dimensionalfixedpointoftheQCDβ-functionandpropertiesofthecriticalrenormalizationgroupequation.Theresultshavebeenshowntobeinagreementwithallknownexplicittwoandthreeloopperturbativeresults,[5,6,7,14,15],whichputsthevalidityofthelargeNfresultsinrelationtodeepinelasticscatteringonafirmfooting.Hence,inthisletterwewillexaminethen→0limitofthesingletgluonsplittingfunctionatO(1/Nf)andallordersina.First,werecallthebasicformalismoftheproblem.Astheflavoursingletgluonictwist-2operatormixeswiththefermionicoperatorunderrenormalizationinperturbationtheory,onehastodealwithamatrixofanomalousdimensions,γij(a).SincewewillbeconsideringitinthelargeNfexpansionwedefinethecoefficientsofitsperturbativeexpansionformallybyγij(a)=γqq(a)γgq(a)γqg(a)γgg(a)!(1)whereγqq(a)=a1a+(a21Nf+a22)a2+(a31N2f+a32Nf+a33)a3+O(a4)γgq(a)=b1a+(b21Nf+b22)a2+(b31N2f+b32Nf+b33)a3+O(a4)γqg(a)=c1Nfa+c2Nfa2+(c31N2f+c32Nf+c33)a3+O(a4)γgg(a)=(d11Nf+d12)a+(d21Nf+d22)a2+(d31N2f+d32Nf+d33)a3+O(a4).(2)Theexplicitvaluesofthecoefficientstotwoloopsasafunctionofnaregivenin[5,6,7]andexactvaluesforthelowoperatormomentsatthreeloopsarefoundin[14,15].Inparticularwenoted11=43T(R)(3)2whereT(R)isgivenbytr(TaTb)=T(R)δabandTaarethegeneratorsofthecolourgroupwhosestructureconstantsarefabc.In[10]itwasarguedthatthesetofcoefficientsoftheC2(G)sectorofγgg(a)atleadingorderin1/Nf,correspondingtodC2(G)l1atthelthloop,couldbewrittencompactlyinthecriticalexponentformasλC2(G)+,1(ac)=h[32μ5n2+32μ5n+32μ5+8μ4n4+16μ4n3−120μ4n2−128μ4n−160μ4−32μ3n4−64μ3n3+160μ3n2+192μ3n+316μ3+48μ2n4+96μ2n3−78μ2n2−126μ2n−306μ2−31μn4−62μn3+31μn+146μ+7n4+14n3+7n2−28]Γ(n+2−μ)Γ(μ)/[8n(μ−1)3(n+2)(n2−1)Γ(2−μ)Γ(μ+n)]−[32μ5n2+32μ5n+32μ5−144μ4n2−144μ4n−160μ4−4μ3n4−8μ3n3+240μ3n2+244μ3n+316μ3+16μ2n4+32μ2n3−180μ2n2−196μ2n−306μ2−20μn4−40μn3+59μn2+79μn+146μ+8n4+16n3−6n2−14n−28]/[8n(μ−1)3(n+2)(n2−1)]+2(μ−1)S1(n)μC2(G)ηo1(2μ−1)(μ−2)T(R)(4)whereSl(n)=P∞r=11/rl,d=2μ,facdfbcd=C2(G)δabandηo1=(2μ−1)(μ−2)Γ(2μ)4Γ2(μ)Γ(μ+1)Γ(2−μ).(5)WerecallthatafeatureofthelargeNfapproachtocomputinginformationontheperturbativecoefficientsof(2)wasthattheanomalousdimensionsoftheeigen-operatorsof(1)atcriticalityweredeterminedanddenotedbyλ±(ac)=P∞i=1λ±,i(ac)/Nif,[10].Thateigen-operatorwhichwaspredominantlygluonicincontentcorrespondstotheeigen-criticalexponentλ+(ac)whilstλ−(ac)correspondstothedimensionofthemainlyfermioniceigen-operator.Thelocationofthefixedpoint,ac,isgivenbythenon-tr