Transverse momentum dependent (TMD) distribution and fragmentation functions
are described as Fourier transforms of matrix elementscontaining nonlocal
combinations of quark and gluon fields. These matrix elements also contain a
gauge link operator with a process dependent path, of which the process
dependence that can be traced back to the color flow in the process. Expanding
into irreducible tensors built from the transverse momenta $p_\st$, we can
define a universal set of TMD correlators of definite rank with a well-defined
operator structure.Comment: 6 pages, to be published in proceedings of the Third Worshop on the
  QCD Structure of the Nucleon (QCD N'12), Bilbao, Spain, 22-26 October 201

Buffing, M. G. A.

Mukherjee, A.

Mulders, P. J.

English

arXiv

Transverse momentum dependent (TMD) distribution and fragmentation functions are described as Fourier transforms of matrix elements containing nonlocal combinations of quark and gluon fields. These matrix elements also contain

a gauge link operator with a process dependent path, of which the process dependence that can be traced back to the color flow in the process. Expanding into irreducible tensors built from the transverse momenta pT , we can define a universal set of TMD correlators of definite rank with a well-defined operator structure

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DOI 10.1393/ncc/i2013-11590-6Colloquia: QCDN12IL NUOVO CIMENTO Vol. 36 C, N. 5 Settembre-Ottobre 2013Universality of TMD distribution functions of definite rankP. J. Mulders(1), M. G. A. Buffing(1) and A. Mukherjee(2)(1) Nikhef and Department of Physics, Faculty of Sciences, VU UniversityAmsterdam, The Netherlands(2) Department of Physics, Indian Institute of TechnologyBombay, Powai, Mumbai 400076, Indiaricevuto il 18 Aprile 2013Summary. — Transverse momentum dependent (TMD) distribution and fragmen-tation functions are described as Fourier transforms of matrix elements containingnonlocal combinations of quark and gluon fields. These matrix elements also containa gauge link operator with a process dependent path, of which the process depen-dence that can be traced back to the color flow in the process. Expanding intoirreducible tensors built from the transverse momenta pT , we can define a universalset of TMD correlators of definite rank with a well-defined operator structure.PACS 12.38.Aw – General properties of QCD (dynamics, confinement, etc.).PACS 12.38.-t – Quantum chromodynamics.PACS 13.88.+e – Polarization in interactions and scattering.1. – IntroductionTransverse momentum dependent (TMD) distribution functions (PDF) and fragmen-tation functions (PFF), also simply referred to as TMDs, basically are forward matrixelements of parton fields, for instance for quarksΦij(p|p) =∫d4ξ(2π)4ei p·ξ 〈P |ψj(0)ψi(ξ)|P 〉,(1)including a summation over color indices. For a single incoming fermion one would justhave Φ ∝ (/p + m). In a diagrammatic approach in principle all sorts of correlators areneeded, among them quark-quark-gluon correlators definedΦμA ij(p− p1, p1|p) =∫d4ξ d4η(2π)8ei (p−p1)·ξ ei p1·η 〈P |ψj(0)Aμ(η)ψi(ξ)|P 〉.(2)c© Societa` Italiana di Fisica 8384 P. J. MULDERS, M. G. A. BUFFING and A. MUKHERJEEThe basic idea is to factorize these hadronic (soft) parts in a full diagrammatic approachand parametrize them in terms of PDFs. This will not work for the unintegrated correla-tors above. At high energies, however, the hard scale provides for each hadron light-likevectors P and n such that P · n = 1. For instance n = P ′/P · P ′, where P ′ is themomentum of another hadron obeying P ·P ′ ∝ s. Using the light-like vectors, one writesdown a Sudakov expansion of the parton momenta, p = xP + pT + (p·P − xM2)n withx = p+ = p · n. In any contraction with vectors not appearing in the correlator, thecomponent xP contributes at order√s, the transverse component at order M and theremaining component contributes at order M2/√s. This allows consecutive integrationof the components. Starting from the fully unintegrated result in eq. (1) one obtains theTMD light-front (LF) correlatorΦij(x, pT ;n) =∫dξ·P d2ξT(2π)3ei p·ξ 〈P |ψj(0)ψi(ξ)|P 〉∣∣∣∣ξ·n=0,(3)and the collinear light-cone (LC) correlatorΦij(x) =∫dξ · P2πei p·ξ 〈P |ψj(0)ψi(ξ)|P 〉∣∣∣∣ξ·n=ξT=0 or ξ2=0,(4)or the local matrix elementΦij = 〈P |ψj(0)ψi(ξ)|P 〉∣∣ξ=0.(5)The importance of integrating at least the light-cone (minus) component p− = p · Pis that the nonlocality is at equal (light-cone) time, i.e. time-ordering is not relevantanymore for TMD or collinear PDFs [1]. For local matrix elements one can calculatethe anomalous dimensions, which show up as the Mellin moments of the splitting func-tions that govern the scaling behavior of the collinear correlator Φ(x). We note that thecollinear correlator is not simply an integrated TMD. The dependence on upper limitΦ(x;Q2) =∫ Q d2pT Φ(x, pT ) is governed by the anomalous dimensions (splitting func-tions). One has a αs/p2T behavior of TMDs that is calculable using collinear TMDs andwhich matches to the intrinsic non perturbative pT -behavior [2]. We note that in oper-ator product expansion language, the collinear correlators involve operators of definitetwist, while TMD correlators involve operators of various twist.In order to determine the importance of a particular correlator in a hard process,one can do a dimensional analysis to find out when they contribute in an expansion inthe inverse hard scale. Dominant are the ones with lowest canonical dimension obtainedby maximizing contractions with n, for instance for quark or gluon fields the minimalcanonical dimensions dim[ψ(0)/nψ(ξ)] = dim[Fnα(0)Fnβ(ξ)] = 2, while an example fora multi-parton combination gives dim[ψ(0)/nAαT (η)ψ(ξ)] = 3. Equivalently, one canmaximize the number of P ’s in the parametrization of Φij . Of course one immediatelysees that any number of collinear n·A(η) = An(η) fields doesn’t matter. Furthermore onemust take care of color gauge-invariance, for instance when dealing with the gluon fieldsand one must include derivatives in color gauge-invariant combinations. With dimensionzero there is iDn = i∂n + gAn and with dimension one there is iDαT = i∂αT + gAαT . Thecolor gauge-invariant expressions for quark and gluon distribution functions actuallyUNIVERSALITY OF TMD DISTRIBUTION FUNCTIONS OF DEFINITE RANK 85include gauge link operators,U[0,ξ] = P exp(−i∫ ξ0dζμAμ(ζ))(6)connecting the nonlocal fields,Φ[U ]q ij(x, pT ;n) =∫dξ · P d2ξT(2π)3ei p·ξ 〈P |ψj(0)U[0,ξ] ψi(ξ)|P 〉∣∣∣∣LF,(7)Φ[U,U′]μνg (x, pT ) =∫d(ξ·P ) d2ξT(2π)3eip·ξ Tr〈P, S|Fnμ(0)U[0,ξ] Fnν(ξ)U ′[ξ,0] |P, S〉∣∣∣∣LF.(8)For transverse separations, the gauge links involve gauge links running along the mi-nus direction to ±∞ (dimensionally preferred), which are closed with one or moretransverse pieces at light-cone infinity [3, 4]. The two simplest possibilities are U [±] =Un[0,±∞] UT[0T ,ξT ]Un[±∞,ξ], leading to gauge link dependent quark TMDs Φ[±]q (x, pT ). Whichcorrelator is relevant in which process is a matter of doing the (diagrammatic) calculationand resummation [5, 6]. For gluons, the correlator involves color gauge-invariant tracesof field operators Fnα, which are written in the color-triplet representation, requiring theinclusion of two gauge links U[0,ξ] and U ′[ξ,0]. Thus even with the simplest links, one hasalready four gluon TMDs Φ[±,±]g (x, pT ).2. – Parametrization of TMDsIn principle quark and gluon TMDs including a gauge link need to be parametrizedwith a set of PDFs (or in the case of fragmentation PFFs), which just as the correlatordepend on the gauge link U . For quarks [7, 8] one finds the following set of functionsdepending on x and p2T :Φ[U ](x, pT ;n) ={f[U ]1 (x, p2T )− f⊥[U ]1T (x, p2T )	pT STTM+ g[U ]1s (x, pT )γ5(9)+h[U ]1T (x, p2T ) γ5 /ST + h⊥[U ]1s (x, pT )γ5 /pTM+ ih⊥[U ]1 (x, p2T )/pTM}/P2,with the spin vector parametrized as Sμ = SLPμ+SμT +M2 SLnμ and shorthand notationsfor g[U ]1s and h⊥[U ]1s ,g[U ]1s (x, pT ) = SLg[U ]1L (x, p2T )−pT · STMg[U ]1T (x, p2T ).(10)For quarks, these include not only the functions that survive upon pT -integration, fq1 (x) =q(x), gq1(x) = Δq(x) and hq1(x) = δq(x), which are the well-known collinear spin-spindensities (involving quark and nucleon spin) but also momentum-spin densities such asthe Sivers function f⊥q1T (x, p2T ) (unpolarized quarks in a transversely polarized nucleon)and spin-spin-momentum densities such as g1T (x, p2T ) (longitudinally polarized quarks in86 P. J. MULDERS, M. G. A. BUFFING and A. MUKHERJEEa transversely polarized nucleon). The parametrization for gluons, following the namingconvention in ref. [9], reads2xΓμν[U ](x, pT ) = −gμνT fg[U ]1 (x, p2T ) + gμνT	pT STTMf⊥g[U ]1T (x, p2T )(11)+i	μνT gg[U ]1s (x, pT ) +(pμT pνTM2− gμνTp2T2M2)h⊥g[U ]1 (x, p2T )−	pT {μT pν}T2M2h⊥g[U ]1s (x, pT )−	pT {μT Sν}T + 	ST {μT pν}T4Mhg[U ]1T (x, p2T ).For TMD correlators time-reversal does not provide constraints as future- and past-pointing is interchanged. Depending on the behavior of the Dirac structure, one candistinguish T-even and T-odd functions satisfying f [U ] = ±f [Ut], where U t is the time-reversed link of U . In the quark-quark correlator f⊥q1T and h⊥q1 are T-odd, in the gluon-gluon correlator f⊥g1T , hg1T , h⊥g1L and h⊥g1T are T-odd. Since time reversal is a good sym-metry of QCD, the appearance of T-even or T-odd functions in the parametrization ofthe correlators is linked to specific observables with this same character. In particularsingle spin asymmetries are T-odd observables.3. – Moments of TMDsIn order to study the gauge link dependence, it is convenient to construct moments ofTMDs. The procedure of moments is well-known for the moments of collinear functions.For Φ(x) in eq. (4) one constructs momentsxNΦ(x) =∫dξ · P2πei p·ξ 〈P |ψ(0) (i∂n)N Un[0,ξ] ψ(ξ)|P 〉∣∣∣∣LC(12)=∫dξ · P2πei p·ξ 〈P |ψ(0)Un[0,ξ] (iDn)N ψ(ξ)|P 〉∣∣∣∣LC.Integrating over x one finds the connection of the Mellin moments of PDFs with lo-cal matrix elements having specific anomalous dimensions, which via an inverse Mellintransform define the splitting functions. In the same way one can consider transversemoments [4] starting with the light-front TMD in eq. (3),pαT Φ[±](x, pT ;n) =(13) ∫dξ · P d2ξT(2π)3ei p·ξ 〈P |ψ(0)Un[0,±∞] UT[0T ,ξT ] iDαT (±∞)Un[±∞,ξ]ψ(ξ)|P 〉∣∣∣∣LF.Integrating over pT gives the lowest transverse moment. This moment involves twist-3(or higher) collinear multi-parton correlators, in particular light-cone quark-quark-gluoncorrelator ΦnαF (x − x1, x1|x) starting from eq. (2), and the similarly defined correlatorΦαD(x− x1, x1|x). The particular combinations that are needed in the moments areΦ˜α∂ (x) = ΦαD(x)− ΦαA(x)(14)=∫dx1 ΦαD(x− x1, x1|x)−∫dx1 PV1x1ΦnαF (x− x1, x1|x),ΦαG(x) = πΦnαF (x, 0|x).(15)UNIVERSALITY OF TMD DISTRIBUTION FUNCTIONS OF DEFINITE RANK 87The latter is referred to as a gluonic pole or ETQS-matrix element [10-12,4]. The collinearcorrelators have trivial gauge links bridging the light-like separation and thus they haveno gauge link dependence. The functions are T-even or T-odd under time reversal. Whilethe collinear function Φ(x) is T-even, the functions contributing to the first momentinvolve the T-even correlator Φ˜α∂ (x) and the T-odd correlator ΦαG(x). For the highermoments one finds that the relevant correlators involve similar multi-partonic correlators,e.g. for the double weighting one needs the T-even correlator Φ˜αβ∂∂ (x), the symmetrized T-odd correlator Φ˜αβ{∂G}(x) and the T-even double gluonic pole correlator ΦαβGG(x). In termsof these functions we get for the moments Φα1...αn[U ]∂...∂ ≡∫d2pT pα1T . . . pαnT Φ[U ](x, pT )Φ[U ](x) = Φ(x), Φ[U ]α∂ (x) = Φ˜α∂ (x) + C[U ]G ΦαG(x),(16)Φ[U ]αβ∂∂ (x) = Φ˜αβ∂∂ (x) + C[U ]G Φαβ{∂G}(x) +2∑c=1C[U ]GG,c ΦαβGG,c(x),(17)etc. The gauge link dependence is contained in calculable gluonic pole factors C [U ]G ,etc. For instance the factors C [±]G = ±1. An additional summation is needed if thereare multiple color arrangements possible for the fields, e.g. the summation over c in thedouble gluonic pole contribution of the quark-quark correlator is needed because one canhave color structures Tr(GGψψ) (c = 1) or Tr(ψψ)Tr(GG)/Nc (c = 2). For the simplestgauge links in quark correlators one has C [±]GG,1 = 1 and C[±]GG,2 = 0, but if Wilson loopsappear in U the latter coefficient is nonzero.4. – Universal TMDs of definite rankThe parametrization of quark-quark correlators in eq. (9) shows that all weightingsbeyond double pT -weighting are zero. From it we deduce for a nucleon [13]Φ[U ](x, pT ) = Φ(x, p2T ) +pT iMΦ˜i∂(x, p2T ) +pT ijM2Φ˜ij∂∂(x, p2T )(18)+ C [U ]G⎧⎩pT iMΦiG(x, p2T ) +pT ijM2Φ˜ ij{∂G}(x, p2T )⎫⎭+2∑c=1C[U ]GG,cpT ijM2ΦijGG,c(x, p2T ),where pijT is the second rank traceless tensor pijT = piTpjT − 12 p2T gijT . We refer to thecorrelators in eq. (18) as TMD correlators of definite rank. Each of these definite rankcorrelators is parametrized in terms of universal TMD PDFs. Using the T-behavior ofthe functions and their tensorial structure the identification is straightforward, with forinstance f⊥1T and h⊥1 appearing in ΦG. Furthermore, one obtainsf[U ]1 (x, p2T ) = f1(x, p2T ), g[U ]1T (x, p2T ) = g1T (x, p2T ), . . . ,(19)f⊥[U ]1T (x, p2T ) = C[U ]G f⊥1T (x, p2T ), h⊥[U ]1 (x, p2T ) = C[U ]G h⊥1 (x, p2T ),(20)h⊥[U ]1T (x, p2T ) = h⊥(A)1T (x, p2T ) + C[U ]GG,1 h⊥(B1)1T (x, p2T ) + C[U ]GG,2 h⊥(B2)1T (x, p2T ).(21)Functions like f1 and g1T are universal. The T-odd rank-1 TMD PDFs like h⊥1 also areuniversal but appear in cross sections with calculable process dependent gluonic pole88 P. J. MULDERS, M. G. A. BUFFING and A. MUKHERJEEfactors. At rank-2, however, one has three universal T-even pretzelocity functions h⊥...1Tappearing in different, but calculable,combinations in hard cross sections. For a spin-1/2 target the rank-2 quark-quark correlator Φ{∂G} = 0, but such a rank-2 correlatoris relevant for a spin one hadron [13] or for gluon-gluon TMD correlators [14]. Theanalogous treatment for fragmentation functions is simpler. In that case there is noprocess dependence [15]. Phrased in terms of operators, the gluonic pole matrix elementsvanish in that case [16-18]. Nevertheless, there exist T-odd fragmentation functions, buttheir QCD operator structure is T-even, similar to the structure of Φ˜α∂ .The universal p2T -dependent TMD functions and correlators show up in azimuthalasymmetries, with the azimuthal dependence contained in the tensors in eq. (18). Fora given rank m they just are |pT |m e±imϕ. For the study of the p2T -dependence, it isconvenient to use Bessel transforms [19] of the (−p2T/2M2)-moments,f (m/2)... (x, p2T ) =(−p2T2M2)m/2f...(x, p2T ) =∫ ∞0db√|pT |b Jm(|pT |b) f (m/2)... (x, b).(22)For quarks one gets multiple color possibilities at rank two, leading to three universalpretzelocities. For gluons [14] there are more situations in which multiple color configu-rations show up, among them in the study of the gluon Boer-Mulders function h⊥g1 (x, pT )corresponding to linear gluon polarization in an unpolarized nucleon.∗ ∗ ∗This research is part of the research program of the Foundation for FundamentalResearch of Matter (FOM) and the Netherlands Organization for Scientific Research(NWO), and the FP7 EU-programme HadronPhysics3 (contract no 283286).REFERENCES[1] Diehl M. and Gousset T., Phys. Lett. B, 428 (1998) 359.[2] Collins J. C., Soper D. E. and Sterman G. F., Nucl. Phys. B, 250 (1985) 199.[3] Belitsky A. V., Ji X. and Yuan F., Nucl. Phys. B, 656 (2003) 165.[4] Boer D., Mulders P. J. and Pijlman F., Nucl. Phys. B, 667 (2003) 201.[5] Bacchetta A., Bomhof C. J., Mulders P. J. and Pijlman F., Phys. Rev. D, 72 (2005)034030.[6] Bomhof C. J. and Mulders P. J., JHEP, 02 (2007) 029.[7] Mulders P. J. and Tangerman R. D., Nucl. Phys. B, 461 (1996) 197 (Erratum 484,(1997) 538).[8] Bacchetta A. et al., JHEP, 02 (2007) 093.[9] Meissner S., Metz A. and Goeke K., Phys. Rev. D, 76 (2007) 034002.[10] Efremov A. V. and Teryaev O. V., Phys. Lett. B, 150 (1985) 383.[11] Qiu J.-W. and Sterman G. F., Phys. Rev. Lett., 67 (1991) 2264.[12] Boer D., Mulders P. J. and Teryaev O. V., Phys. Rev. D, 57 (1998) 3057.[13] Buffing M. G. A., Mukherjee A. and Mulders P. J., Phys. Rev. D, 86 (2012) 074030.[14] Buffing M. G. A., Mukherjee A. and Mulders P. J., arXiv:1306.5897 [hep-ph].[15] Collins J. C. and Metz A., Phys. Rev. Lett., 93 (2004) 252001.[16] Meissner S. and Metz A., Phys. Rev. Lett., 102 (2009) 172003.[17] Gamberg L. P., Mukherjee A. and Mulders P. J., Phys. Rev. D, 77 (2008) 114026.[18] Gamberg L. P., Mukherjee A. and Mulders P. J., Phys. Rev. D, 83 (2011) 071503.[19] Boer D., Gamberg L., Musch B. and Prokudin A., JHEP, 10 (2011) 021.

Universality of TMD distribution functions of definite rank

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