Twist-2 matching of transverse momentum dependent distributions

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Physics Letters B

www.elsevier.com/locate/physletb

Twist-2 matching of transverse momentum dependent distributions

Daniel Gutiérrez-Reyes^a,∗, Ignazio Scimemi^a, Alexey A. Vladimirov^b

aDepartamentodeFísicaTeóricaII,UniversidadComplutensedeMadrid(UCM),28040Madrid,Spain bInstitutfürTheoretischePhysik,UniversitätRegensburg,D-93040Regensburg,Germany

a r t i c l e i n f o a b s t ra c t

Articlehistory:

Received28February2017 Accepted16March2017 Availableonline21March2017 Editor:B.Grinstein

Wesystematicallystudythelarge-qT (orsmall-b)matchingoftransversemomentumdependent(TMD) distributionstothetwist-2integratedpartondistributions.Performingoperatorproductexpansionfora genericTMDoperatoratthenext-to-leadingorder(NLO)wefoundthecompletesetofTMDdistributions thatmatchtwist-2.Theseareunpolarized,helicity,transversity,pretzelosityandlinearlypolarizedgluon distributions. The NLO matching coefficients for these distributions are presented. The pretzelosity matching coefficientis zero at the presented order,however, it is evident that it is non-zeroin the following orders. This result offers a naturalexplanation of the small value of pretzelosityfound in phenomenologicalfits.Wealsodemonstratethatthecancellationofrapiditydivergencesbytheleading ordersoftfactorimposesthenecessaryrequirementontheLorentzstructureofTMDoperators,whichis supportedonlybytheTMDdistributionsofleadingdynamicaltwist.Additionally,thisrequirementputs restrictionsontheγ5-definitioninthedimensionalregularization.

©^{2017TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

1. Introduction

Thetransversemomentumdependent(TMD)factorizationthe- orems for semi-inclusive deep inelastic scattering (SIDIS) and Drell–Yan type processes formulated in [1–4] allow a consistent treatmentofrapidity divergencesinthe definitionofspin(in)de- pendent TMD distributions. They also provide a self-contained definitionofTMDoperators whichcanbeconsidered individually bystandardmethodsofquantumfield theorywithoutreferringto ascatteringprocess.Inparticular,thelarge-qT (orsmall-b)match- ingofTMDdistributionsonthecorrespondingintegratedfunctions canbeevaluated. Suchconsiderationispracticallyvery important becausetheresultingmatchingcoefficientsserveasaninitialinput tomanymodelsandphenomenologicalansatzesforTMDdistribu- tions. Theunpolarized TMD distributionis themoststudied case and it has been treated using different regularization schemes atthe next-to-leading order (NLO) [1,5,2,4,6–8] andthe next-to- next-to-leading order (NNLO) [9–12]. For polarized distributions suchaprogram hasbeenperformedonlyforhelicity,transversity andlinearlypolarizeddistributionsatNLO[13,14].However,these worksmiss a systematicdiscussion on the relevant renormaliza- tionschemes,whicharefundamentaltoestablishtheircalculation and to provide a spring to higher order analysis. By this article

* Correspondingauthor.

E-mailaddresses:dangut01@ucm.es(D. Gutiérrez-Reyes),ignazios@fis.ucm.es (I. Scimemi),vladimirov.aleksey@gmail.com(A.A. Vladimirov).

we open a seriesof articles devoted to the studyof thesmall-b matchingofpolarizedTMDdistributions.Theprimary goalofthis letteristoprovideadedicatedandconsistentstudyoftheleading twist(twist-2)matchingoftheTMDoperators.

ThequarkandgluoncomponentsofthegenericTMDoperators are

i j(x,b)= dλ

2π^e

−^ixp+λq¯i(λn+b)W(λ,b)qj(0) , (1)

_μν(x,b)= ¹ xp⁺

dλ 2π^e

−^ixp+λF_+μ(λn+^b)W(λ,b)F_+ν(0) , (2) where n is the lightlike vector and we use the standard nota- tionforthelightconecomponentsofvector vμ=^nμv−+ ¯^nμv++ gμν

T vν (withn²= ¯ⁿ²=^0,n· ¯ⁿ=^1,andgμνT =^gμν−^nμn¯ν− ¯^nμnν).

TheoperatorW^is

W(λ,b)= ˜^Wn^T(λn+^b)

X

|^XX| ˜^Wn^T†(0) , (3) with Wilson lines W taken in the appropriate representation of gauge group. The staple contourof the gauge link results in the rapidity divergences, the unique feature of TMD operators. The rapiditydivergencesareremovedbytheproperrapidityrenormal- izationfactorR,whichisbuiltfromtheTMDsoftfactor,

S(b)=^Trcolor Nc ⁰|

S_n^T†˜S_n^T_¯

(b) ˜S^T_n¯^†S_n^T

(0)|⁰, (4)

http://dx.doi.org/10.1016/j.physletb.2017.03.031

0370-2693/©^{2017TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense}(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP³.

where Sn and ˜S_n¯ stand for soft Wilsonlines along n andⁿ¯ ^(for the precise definition of W^T and ˜S^T see e.g. [10]). The struc- tureof factor R follows from the TMD factorization theorem [1, 2,4,10,3]anddependsontherapidityregularizationscheme.How- ever,the expressions forrapidity-divergence-free quantities, such asevolutionkernelsandmatchingcoefficientsareindependenton thescheme.In thefollowingwe use theδ-regularizationscheme formulatedin[15,10].Thisschemeusestheinfinitesimalparame- terδ asa regulator forrapidity divergences incombination with theusual dimensionalregularization (withd=⁴−², >0) for ultravioletandcollineardivergences.Suchcombinationappearsto be very visual and practically convenient. The central statement ofthe TMD factorization theoremis the complete eliminationof rapidity divergences by the rapidity renormalization factor R. In theδ-regularizationschemewhere R=¹/√

S(b),the rapidity di- vergencestakethe formoflnδ anddo notmix withdivergences in , which yield the exact cancellation of lnδ at finite . This non-trivialdemandis necessaryfora consistenthigher-then-NLO evaluationandrequiresthematchingofregularizationsfordiffer- entfieldmodes(see[10]).Italsoresultsintothecorrespondence betweenTMDprocessesandthejetproduction[16].

Thehadron matrix elements of theTMD operators with open vector and spinor indices (1), (2) are to be decomposed over all possible Lorentz variants, which define TMD parton distribu- tionfunctions(TMDPDFs).Intheliterature,thisdecompositionhas beenmadein themomentum space(for spin-1/2 hadronsit can befoundin[17,18](forquarkoperators)andin[19](forgluonop- erators)).However, itisconvenientto considerTMD distributions intheimpactparameter space, whereitisnaturally defined. The correspondencebetweendecompositioninmomentumandimpact parameter spaces can be found in e.g. [20,14]. In this work we needonlyapartofthecompletedecomposition,

_{q←h,i j}(x,b)= ^h|_{i j}(x,b)|^h

=¹ 2

f₁γ_{i j}⁻+^g1LS_L(γ5γ⁻)i j+(S^μ_Tiγ5σ^+μ)i jh₁ +(iγ₅σ^+μ)_{i j}

g^μν_T 2 +^bμbν

b² S^ν_T

2 h^⊥_1T+... , (5)

g←^h,μν(x,b)= ^h|μν(x,b)|^h

=¹ 2

−g^μν_T f₁^g−i_T^μνSLg_1L^g

+^2h⊥₁^g g^μν_T

2 +^bμbν b²

+... ,

(6)

wherethevector bμ isa 4-dimensionalvector oftheimpact pa- rameter(b⁺=^b−=^{0 and}−^b2≡^b2>0),andST,Larecomponents of the hadron spin vector defined in Eq. (11). On the r.h.s. of Eqs. (5), (6) and in the rest of the letter we omit arguments of TMD distributions (x,b), unless they are necessary. Note that in Eq.(5)weusethenormalizationforthedistributionh^⊥_1T different fromthetraditionalone [17].Thetraditionaldefinitioncanbere- covered substituting h^⊥_1T →^h⊥_1T^{b2M2,with M being the massof} hadron.Inthesection4wearguethatsuchnormalizationisnatu- ral.

InEqs.(5),(6)wewriteonlytheTMDdistributionsthatmatch thetwist-2integrateddistributions.ThedotsincludetheTMDdis- tributions that matchthe twist-3 andhigher parton distribution functions(PDF). The reported distributions are usually addressed ashelicity(g_1L and g_1L^g), transversity (h₁), pretzelosity (h^⊥_1T) and linearlypolarizedgluon(h^⊥₁^g)distributions. Thesmall-b matching of these distributions has been performed separately for quarks [13]andgluons[14]indifferentrenormalizationschemes.Further- more,the pretzelosity distribution has beenoverlooked by these

groups. In this letter, we present a uniform and consistent NLO matchingoftheseTMDdistributions.

2. Small-boperatorproductexpansion

The small-b operator product expansion (OPE) is the relation betweenTMDoperatorsandlightconeoperators.Itsleading order canbewrittenas

_{i j}(x,b)= ⁽⁷⁾

C_q←q(b)ab i j ⊗φab

(x)+ ^Cq←g(b)αβ i j ⊗φαβ

(x)+...,

_μν(x,b)= (8)

C_g←q(b)ab μν⊗φ_ab

(x)+ ^Cg←g(b)αβ μν⊗φ_αβ

(x)+...,

wheresymbol⊗^{denotestheMellin}convolutioninthevariablex.

The functionsC(b) aredimensionless, i.e.they dependon b only logarithmically. The dots represent the power suppressed contri- butions, whichpresently havebeenstudied onlyforthe unpolar- ized case(see discussion in[21]). At thisorderofOPE,the func- tions φ (x) aretheformallimit oftheTMD operators(x,0).The hadronicmatrixelementsofφarethePDFs

φ_{q←h,i j}(x)= ^h|φi j(x)|^{h (9)}

=¹ 2

f_q(x)γ_{i j}⁻+f_q(x)S_L(γ5γ⁻)i j

+(S^μ_Tiγ5σ^+μ)i jδf_q(x) +O M

p⁺

,

φg←^h,μν(x)= ^h|φμν(x)|^{h (10)}

=¹ 2

−^gμν_T ^fg(x)−^iμν_T S_Lf_g +O

M p⁺

,

whereMisthemassofhadron,SL andST arethecomponentsof thehadronspinvector

S^μ=^SL

p⁺

Mn¯^μ− ^M 2p⁺n^μ

+^Sμ_T, (11) and μν

T =^+−μν=ⁿαⁿ¯β^αβμν.Forfutureconvenience weintro- ducetheuniversalnotation

^[]_q =^Tr()

2 , ^[]_g =^μν_μν. (12)

Both sidesofEqs.(7),(8)shouldbesupplementedby theultravi- oletrenormalizationconstants. Additionally,the TMDoperator on the l.h.s.is to be multiplied by the rapidity renormalization fac- torR.TherenormalizedTMDoperatorhastheform

^ren(x,b;μ, ζ )=^Z(μ, ζ|)R(b,μ, ζ|, δ)(x,b|, δ), (13) where we explicitly show the dependence on regularization pa- rameters on the r.h.s. The dependence on and δ cancels in the product. The renormalization factors are independent on the Lorentzstructure butdependentonpartonflavor.Theexplicitex- pressionsforthesefactorsuptoNNLOcanbefoundin[10,15].

Thecancellationofrapiditydivergencesforthespin-dependent distributionsisanon-trivialstatement.Letusconsiderthesmall-b OPEforagenericTMDquarkoperator.Atoneloopwefind

^[]_q =^abφab+asCFBBB(−)

−(γ⁺γ⁻+γ⁻γ⁺)^ab (14)

+ ¯^x g^α_T^β

2 −^bαbβ 4BBB

(γ^μγ_αγ_βγ_μ)^ab

+ 1

(1−^x)₊ −^ln δ

p⁺

×

γ⁺γ⁻+γ⁻γ⁺+ⁱγ⁺/b

2BBB +ⁱb/γ⁺

2BBB ab

−ⁱπ

2

γ⁺γ⁻−γ⁻γ⁺+ⁱγ⁺b/

2BBB −ⁱb/γ⁺

2BBB ab

⊗φ_ab+O(a²_s),

where BBB=^b2/4>0, a_s=^g2/(4π)^d/2, and we use the standard PDFnotation,[^f(x)]+=^f(x)−δ(¯^x)

dy f(y)and^x¯=¹−^x.Inthis expression, we omit thegluon operator contribution forsimplic- ity. The complex term in the last line of Eq. (14) is the artifact ofδ-regularization. Thelogarithm ofδ representstherapidity di- vergencewhich istobe eliminatedby thefactor R whichatthis perturbativeorderreads

R=¹+^2asC_FBBB(−)

×

L^√_ζ+^{2 ln} δ

p⁺

−ψ (−)−γE +O(a²_s), (15) where LX =^ln

BBB X²e²γE

. The rapidity divergence cancels in the productRifandonlyif

γ⁺=γ⁺=0, (16) yielding

R^[]_q =^abφ_ab+^asCFBBB(−) (17)

×

−⁴+ ⁴

(1−^x)₊ +²δ(¯x)(L^√_ζ−ψ (−)−γE)

^ab + ¯^x

g^α_T^β 2 −^bαbβ

4BBB

(γ^μγαγβγμ)^ab

⊗φab+O(a²_s) .

The cancellation of rapidity divergences is the fundamental pre- requisite to obtain the matchingcoefficients of the renormalized operatorandφ.

TheconditionsanaloguetoEq.(16)forthegluonoperatorare

^+μ=^−μ=^μ+=^μ−=⁰. (18) TheyfollowfromOPEforagenericgluonTMDoperator^μν sim- ilar to Eq.(14),which we do not presenthere, since it is rather lengthy and not instructive.The conditions inEqs. (16), (18)are satisfiedonlyforthefollowingLorentzstructures

^q= {γ⁺,γ⁺γ⁵,σ^+μ}, ^g= {g^μν_T ,^μν_T ,b^μb^ν/b²}, (19) which exactly correspond to the Lorentz structures for the so called“leadingdynamicaltwist”TMDdistributions.Inthisway,the relationsEqs.(16),(18)provideadefinitionoftheleadingdynam- ical twistfor TMD operators that can be used withno reference to a particular cross-section. On the other hand, our considera- tion shows that TMD operators of non-leading dynamical twist have rapidity singularities that are not canceled by the soft fac- torinEq.(4).Whilewehavenoknowledgeofacalculationofthe correction to the leading order of TMD factorization, ourfinding demonstrates that it has a different structure of rapidity diver- gences(whichcanspoilthefactorization).TherelationinEq.(16) willbe usedinthenext sectionto fixthedefinitionof γ5 inthe dimensionalregularization.

Inordertocalculatethematchingcoefficients,weconsiderthe quark andgluon matrixelements withthe momentumof parton settopμ=^p+n¯μ.Thischoiceofkinematicisallowedforconsider- ationoftwist-2contributiononly(whichisthecaseofthisarticle).

Then,thecalculationsaregreatlysimplified.Inparticular,theper- turbativecorrectionstothepartonmatrixelementofφ’sarezero, due to the absence of a scale in the dimensional regularization.

Therefore,suchmatrixelementsareequaltotheirrenormalization constant, i.e.has notfinitein -terms.Inpractice,it impliesthat the matchingcoefficient is the -finitepart ofthe partonmatrix element of the renormalized TMD operator (13). The evaluation ofOPE fora generalLorentzstructure (asinEq.(17)) isnot very representativebecauseoneneedsonlythecomponentsassociated withtheTMDPDFs.Therefore,weprojectouttherequiredcompo- nentsandpresenttheexpressionsforeachparticulardistribution.

3. Helicitydistribution

In the case ofhelicity distributions the Lorentz structuresfor quarkandgluonoperatorsare

=γ⁺γ⁵, ^μν=^iμν_T . (20) Thecorresponding“orthogonal”projectorsare

=Nsch.

γ⁻γ⁵

2 , ^μν=ⁱNsch.

_T^μν

2 , (21)

where thefactorNsch. dependsonthe definitionofγ5 matrixin dimensionalregularization.Historicallythemostpopularschemes (forQCDcalculations)are’t Hooft–Veltman–Breitenlohner–Maison (HVBM) [22,23], and Larin scheme [24,25]. In both schemes the combination γ⁺γ⁵canbepresentedas

γ⁺γ⁵= ⁱ

3!^+ναβγνγαγβ, (22)

where ^μναβ is the antisymmetric Levi-Civita tensor. The differ- ence between schemes is hidden in the definition of Levi-Civita tensor.InHVBMthe μναβisdefinedonlyfor4-dimensionalsetof indices.I.e. μναβ=^{1 if}{μναβ}^isevenpermutationof{⁰,1,2,3}^{, μναβ}= −^{1 if the} permutation is odd, and ^μναβ=^{0 for any} another case. In Larin scheme the -tensor is non-zero for all set ofd-dimensionalindices. Thevalue of individual components are undefined, however,the product oftwo -tensors isdefined,

^{μ1ν1α1β1μ2ν2α2β2}= −^{gμ1μ2gν1ν2gα1α2gβ1β2}+^{gμ1ν2gν1μ2gα1α2}× g^β1β2−^{.., wherethedotsmeanall 4}!permutationsofindiceswith alternatingsigns.

ThedrawbackofbothschemesistheviolationofAdler–Bardeen theorem for the non-renormalization of the axial anomaly. This mustbefixedbyan extrafiniterenormalizationconstant Z_qq⁵,de- rived from an external condition, see detaileddiscussion in [24, 26,27].TheNNLOcalculationofpolarizeddeep-inelastic-scattering andDrell–Yanprocessinrefs.[26,27]madein(HVBM)haveshown that the finiterenormalization is required only forthe quark-to- quark part (both singlet and non-singlet cases). The same finite renormalization constant can be used for Larin scheme up to

-singulartermsatNNLO [28].However, itseems thatforhigher orderterms(in orinthecouplingconstant)theconstantshould bemodified[28].

Needlesstosay,thatLarinschemeisfarmoreconvenientthen HVBM, because it does not violate Lorentz invariance. However, Larinscheme,asitisoriginallyformulatedandusedinthemodern applications[28],isinapplicableforTMDcalculations.Thepointis that it doesviolate the definitionof the leading dynamical twist Eq.(16).Indeed,intheLarinschemewehave

γ⁺=γ⁺

γ⁺γ⁵

Larin= ⁱ

3!^+ναβγ⁺γνγαγβ=⁰, (23) becausethereisa contributionwhenallindices{ναβ} ^aretrans- verse.Note,thatinHVBMschemethereisnotsuchproblem,since