A model independent analysis of gluonic pole matrix elements and universality of TMD fragmentation functions
aa r X i v : . [ h e p - ph ] M a r A model independent analysis of gluonic pole matrix elements and universality ofTMD fragmentation functions
L. P. Gamberg, ∗ A. Mukherjee, † and P.J. Mulders ‡ Division of Science, Penn State University-Berks, Reading, Pennsylvania 19610, USA Physics Department, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India Department of Physics and Astronomy, VU UniversityNL-1081 HV Amsterdam, the Netherlands (Dated: May 29, 2018)Gluonic pole matrix elements explain the appearance of single spin asymmetries (SSA) in high-energy scattering processes. They involve a combination of operators which are odd under timereversal (T-odd). Such matrix elements appear in principle both for parton distribution functionsand parton fragmentation functions. We show that for parton fragmentation functions these gluonicpole matrix elements vanish as a consequence of the analytic structure of scattering amplitudes inQuantum Chromodynamics. This result is important in the study of the universality of transversemomentum dependent (TMD) fragmentation functions.
PACS numbers: 12.38.-t; 13.85.Ni; 13.88.+e
Cross sections for high energy scattering processes aregiven by a convolution of partonic cross sections withparton distribution functions (PDF) and parton frag-mentation functions (PFF). These functions interpretedas momentum distributions and parton decay functions,respectively, are given as matrix elements of quark andgluon operators [1–4]. Among such observables, gluonicpole or Qiu-Sterman matrix elements involve a combi-nation of operators which is odd under time reversal(T-odd). These matrix elements have been extensivelystudied [5–12]. They explain the appearance of singlespin asymmetries (SSA) in high-energy scattering pro-cesses. These correlation functions show up in combi-nation with calculable hard parts that may differ fromthe partonic cross sections through specific calculable fac-tors and signs [9, 13, 14]. In this paper we use generalproperties of scattering amplitudes in Quantum Chro-modynamics (QCD) to study the support properties ofthese parton correlation functions. Specifically, assumingunitarity and analyticity properties to hold for forwardparton-hadron scattering amplitudes, we uncover theirsingularity structure which then determines the corre-sponding properties of the PDFs and PFFs. Using thisanalysis for fragmentation functions, we show that sin-gle gluon and multi-gluon pole matrix elements vanishin the limit when the momenta of these gluons becomezero. Since these multi-gluonic pole matrix elements ap-pear in integrated and weighted transverse momentumdependent (TMD) fragmentation functions, as a conse-quence of them being zero, all leading T-odd effects inthe matrix elements are all part of the final state interac-tions among the fragmentation remnant and final statehadron [4, 15] rather than from T-odd partonic operatorcombinations. Thus, when transverse momentum depen-dent (TMD) T-odd fragmentation functions appear inobservables like SSA they are convoluted with the stan-dard partonic cross sections. The vanishing of gluonic pole matrix elements for any number of gluons, providesa general proof of the universality of these TMD frag-mentation functions [16].Spectral studies of the gluonic pole matrix elements forfragmentation, specifically for quark-quark-gluon matrixelements with just one gluon field [17, 18] already in-dicated that they vanish. Our arguments presented hereare more general and can be applied to multi-gluonic polematrix elements. They depend only on the analytic struc-ture of scattering amplitudes, yet they do not depend onthe details of partonic or hadronic masses, and are insen-sitive to integrations over transverse momentum.We begin our analysis by considering high-energy scat-tering processes where the structure of hadrons is ac-counted for using quark and gluon correlators, which areFourier transforms of forward matrix elements of non-local quark and gluon operators between hadronic states.For instance the quark-quark correlatorΦ [ U ] ij ( x,k T ) = Z d ( ξ · P ) d ξ T (2 π ) e ik · ξ ×h P | ψ j (0) U [0; ξ ] ψ i ( ξ ) | P i (cid:5) LF , (1)where U [ η ; ξ ] = P exp (cid:2) − ig R C ds · A a ( s ) t a (cid:3) is the gauge linkthat ensures gauge invariance [19–22]. The non-localityof the matrix elements needed to describe the distributionfunctions is restricted to the light-front (LF) and it isconvenient to use the Sudakov decomposition k = x P + σ n + k T , (2)in terms of a generic light-like four-vector n satisfying n = 0 and P · n = 1. In a particular hard pro-cess, its role is played by other momenta that are hardwith respect to the hadron under consideration, e.g. n ≈ P ′ /P · P ′ . We can then also work with light-cone co-ordinates. Including mass effects one would have n − = n and n + = P − M n ; with k ± ≡ k · n ∓ . These are P P Φ (k;P,S)k k (a) PPk k ∆ (k; P,S) (b) P P Φ k−k k k(k,k−k ;P,S) (c)FIG. 1: The quark-quark correlators that establish thenon-perturbative connection between partons and hadrons fordistribution functions (a) or fragmentation functions (b) anda quark-quark-gluon (multi-parton) correlator (c). k + = k · n = x and k − = k · P − xM = σ + xM .The transverse momentum is orthogonal to n and P . In ahard process the dependence on k − of a particular corre-lator is not important and it is integrated over, leaving uswith the restricted light-front non-locality ξ + = 0 (LF).The expansion of these correlators (in Dirac space) con-tains the TMD distribution functions depending on mo-mentum fraction x and transverse momentum k T . Uponintegration over k T one obtains the collinear correlatorsΦ( x ) = Z d ( ξ · P )2 π e i x ξ · P h P | ψ (0) U n [0; ξ ] ψ ( ξ ) | P i (cid:5) LC , (3)where non-locality is restricted to the light-cone (LC: ξ · n = ξ T = 0) and the gauge link is unique, being thestraight-line path along n . These collinear correlators areexpanded in the ’standard’ parton distribution functions,depending solely on the momentum fraction x . Here wewill not discuss the scale dependence [23–26].The quark-quark light-front correlator that plays a rolein the fragmentation of partons is∆ [ U ] ij ( z, k T ) = X X Z d ( ξ · P ) d ξ T (2 π ) e i k · ξ h |U [0 ,ξ ] ψ i ( ξ ) ×| P, X ih P, X | ¯ ψ j (0) | i| LF , (4)with the quark momentum, k = z P + k T + σ n , i.e. aSudakov expansion with x = 1 /z >
1. In this case oneoften refers to the hadron transverse momentum P ⊥ = − z k T (in a frame in which the parton does not havea transverse momentum ( k ⊥ = 0)). Diagrammaticallythese correlators are represented in Fig. 1. In this paper,we start the investigation of multi-parton correlators bylooking at the case with one additional gluon, as given x < −1P P−k u −k x > 1sP Pkk0 < x < 1kP PksP P−ku−k −1 < x < 0 k = xPkkP Pus−(k+P) k−PP−kP+k FIG. 2: Integrating parton correlators over k − allows connect-ing them to a single anti-parton - hadron scattering four-pointfunction A ( k ; s, u ) (middle). Depending on the value of x ,the imaginary part of this amplitude represents the (anti)-parton distribution or fragmentation correlators. in Fig. 1 (c). They appear in azimuthal asymmetriesinvolving the k T -weighted correlator,Φ α [ U ] ∂ ( x ) = Z d k T k α T Φ [ U ] ( x,k T )= ˜Φ α∂ ( x ) + C [ U ] G π Φ αG ( x, x ) , (5)which is decomposed in pieces ˜Φ ∂ and Φ G , that con-tain T-even and T-odd operator combinations, respec-tively [22]. The T-odd parts come with calculable glu-onic pole factors C [ U ] G that depend on the gauge link. Inthis paper we focus on the connection of this part to thezero momentum ( x →
0) limit of a quark-quark-gluoncorrelatorΦ αG ( x, x − x ) = Z d ( ξ · P )2 π d ( η · P )2 π e ix ( η · P ) e i ( x − x )( ξ · P ) × h P | ψ (0) U n [0; η ] gG nα ( η ) U n [ η ; ξ ] ψ ( ξ ) | P i (cid:5) LC , (6)where G nα = n µ G µα represent specific components ofthe color field strength tensor ( α being transverse). Thezero momentum limit of this correlator is the gluonic polematrix element mentioned above. It is the support ofΦ G ( x, x − x ) that we are after and specifically the modelindependent proof that it vanishes in the limit x → | x | > k − and k − -integrations in the quark-quark andquark-quark-gluon correlators lead to light-front correla-tors, for which time-ordering is irrelevant. Therefore thematrix elements can be considered as forward matrix ele-ments of time-ordered products of operators. These rep-resent scattering amplitudes and their analytic structureenables one to make statements about the support of theassociated parton correlation functions. This can be donefor quark-quark [27] and multi-parton correlators [28] andfor TMD as well as collinear correlators [29]. In thislanguage the diagrams of Figs. 1 are just hadron-partonamplitudes, e.g. the quark-quark correlator Eq. (3) re-lated to the forward antiquark-hadron scattering ampli-tude A ( k ; s, u ) (see Fig. 2). Depending on the precisestructure these are untruncated Green functions (time-ordered) or related to such Green functions via the LSZformalism [30].The second step is the study of the analytic structureof an amplitude and in particular the singularities arisingfrom cuts in the forward amplitudes. These are cuts inthe (untruncated) legs, in particular the parton virtual-ity k , and the Mandelstam invariants. The virtualitiesare conventionally placed just below the real axis, or theinvariants are replaced by p + iǫ , s + iǫ . The integra-tions over k − and k − imply integrations over some of theinvariants. At this point one must make the standard as-sumption that it is possible to use analyticity for QCD-amplitudes. We illustrate this step first for the standardquark-quark correlators. In that case one works withthe four-point Green functions, shown in the middle ofFig. 2, where the cut amplitude depending on the particu-lar value of x gives the quark distributions (0 < x <
1) orthe quark fragmentation functions (0 < z < x > x describes theanti-quark distributions ( − < x <
0) and anti-quarkfragmentation ( − < z < x < − k and in the Mandelstamvariables, for which we choose s = ( P − k ) and u =( P + k ) . For the forward amplitude, the invariants areconstrained to s + u = 2 k + 2 M , where we will neglectthe hadron masses (they don’t play any essential role in −k < x < k > s < (0 u > u < (0 s > (a) quarks −k −1 < x < s < (0 u > u < (0 s > k > (b) antiquarksFIG. 3: The integration contours for the k − integration withrespect to the kinematic singularities in the (forward) anti-parton - hadron scattering amplitude for the case of (non-vanishing) distribution functions for quarks (a) and anti-quarks (b). su k−kkP Pks − channel
11 11
P−k (a) su k−kkP Pku − channel
11 11
P+k (b)FIG. 4: The additional invariants for the amplitude A ( k ; s, u ; s , u ; k , ( k − k ) ) relevant for gluonic pole matrixelements, (a) for the case s > u > our proof). These singularities then constitute the s -cut(for s > u <
0) and the u -cut (for u > s < k in Eq. (2), one has k = 2 xk − + k T or s = 2( x − k − + k T and a similarexpression for u . The transverse momenta, just as any ofthe parton or hadron masses, have little bearing on ourresults so we omit them in the expressions for k − , k − = s + iǫ x −
1) = u + iǫ x + 1) = k + iǫ x , (7)and one sees that for distribution functions the k − inte-gration with respect to s , u and k singularities followsthe (dashed) contour in Fig. 3. The integration contourscan be wrapped around the s and u -cuts for positive andnegative x -values respectively if | x | <
1, cuts that (in k − ) smoothly vanish when | x | →
1. Neither masses, nortransverse momenta matter and the support propertiesare valid for collinear and TMD PDFs. We getΦ( x ) = θ ( x ) θ (1 − x ) Disc [ s ] A + θ ( − x ) θ (1 + x ) Disc [ u ] A . (8)As discussed for instance in Ref. [31], the case for frag-mentation is different since one in essence discusses theparton propagator for positive k (sitting on the cut).For | x | >
1, one simply has∆( x ) = θ ( x −
1) Disc [ s ] A + θ ( − − x ) Disc [ u ] A . (9)In wrapping the integration around the s - or u -cut wehave to assume convergence in the variable k − (or k ),or use subtracted relations.It is important to mention here that the integration ofEq. (1) and A ( k ; s, u ) over k T leads to ultraviolet diver-gences [26, 28, 29]. However, this does not invalidate the (k−k ) > < x < −k k > u < )( s < )(0 u > s > (a) Φ G ( x, x ) −k x > (k−k ) > k > (b) ∆ G ( x, x )FIG. 5: The integration contours for the k − integra-tion with respect to the singularities in the amplitude A ( k ; s, u ; s , u ; k , ( k − k ) ) relevant for gluonic pole con-tributions. The figure shows for a given value of positive s (relevant for x >
0) how the k − integration bypasses the cutsin s , u and the parton virtualities in the limit x → +0.The cases 0 < x < x > assumption that the integral over k − alone is sufficientlywell behaved when k + and k T are fixed. Integration over k T as well as weighting with k T is anyway intimatelylinked with QCD evolution of TMDs [32].We can extend this analyticity analysis to the multi-parton distribution and fragmentation functions inEq. (6), by looking at the multi-parton amplitude A ( k ; s, u ; s , u ; k , ( k − k ) ) shown in Fig. 4, by study-ing the contours for the additional integrations. Definingthe momentum of the additional parton as k as shownin Fig. 1, one retains the definitions and relations for s and u . For given positive s ( s -channel, x >
0) or posi-tive u ( u -channel, x <
0) one gets additional invariants s = ( P ∓ k ± k ) and u = ( P ∓ k ) (cf. Figs 4 (a)and (b), respectively). Note that t = k in both cases.Furthermore one has parton virtualities. Depending onif one is dealing with the s - or u -cut discontinuities, onehas slightly different constraints for s + u , but for givenvalues of s and u in the two cases of Fig. 4, one has cutsalong s > u <
0) and u > s <
0) as well as forpositive parton virtualities. The relevant singularities for k − are found from k − = s + iǫ x − ( x ∓ k − = u + iǫ x ∓ k + iǫ x = ( k − k ) + iǫ x − x ) + k − (10)(with ∓ referring to s - and u -channel cuts, respectively).Again, parton or hadron masses as well as transversemomenta have little bearing as all they do is move theendpoints of the cuts. However, it is important to notethat k T in the complete expression for the numerator of Eq. (10) protects against the cut starting at zero inthe zero mass limit. Depending on the value of x , theintegration contour in k − bypasses the singularities en-countered in the complex plane in a particular way, whichdictates the support properties of the quark-gluon-quarkcorrelation functions. The denominators in Eq. (10) inthe expressions relating k − to s and u tell us that onlywhen x ∈ [ x − ,
1] (for positive x ) or x ∈ [ − , x +1] (fornegative x ) the singularities in s and u are relevant. Westudy the case of the s -channel ( x > u -channelis analogous. Looking at gluonic poles, we consider thelimit x →
0. For 0 < x <
1, the value x = 0 lies inthe interval for which the s and u discontinuities cancontribute. These are shown in Fig. 5(a), now togetherwith the singularities arising from the parton virtualities k and ( k − k ) . For the case x > G also the u -channel contribution) in thelimit x → G ( x, x ) = θ ( x ) θ (1 − x ) Disc [ s,s ] A + θ ( − x ) θ (1 + x ) Disc [ u,u ] A , (11)∆ G ( x, x ) = 0 , (12)where for ∆ G ( x, x ) the k − integration can be wrappedaround the k cut, which smoothly vanishes for x → +0.This is described by the arrow inside the branch cut inFigs. 5(a) and (b), indicating that it harmlessly recedesto infinity. Moreover it matches continuously to the casethat x <
0. Starting from x → − k cutis then along the negative k − axis. This establishes theproof that gluonic pole matrix elements for fragmenta-tion correlators vanish. Similar results were obtained inthe spectator model field theory in Ref. [16], the spec-tator approach in Ref. [17], and the general spectral ap-proach of Ref. [18]. But now it has been demonstratedin a completely general way by assuming unitarity andanalyticity properties to hold for QCD. Similar to theearlier discussion, we note that the integration of Eq. (6)and A ( k ; s, u ; s , u ; k , ( k − k ) ) over k T is ultravioletdivergent [26, 28, 29]. Again, the integral over k − aloneis well behaved when k +1 and k T are fixed with again fullintegration over k T corresponding to appropriate regu-larisation of A ( k ; s, u ; s , u ; k , ( k − k ) ) [33] and studyof the QCD evolution [32].In our last step, we show that our arguments forvanishing gluonic pole matrix elements hold for generalmulti-gluonic and even multi-partonic pole matrix ele-ments. Considering the analytic properties of generalmulti-gluonic pole matrix elements we can proceed in-ductively. For two gluons one simply extends the nestingof momenta k − k and k by a nesting k − k − k , k − k and k , which adds to the set ( s , u , s , u ) two new invari-ants ( s , u ), without changing the behavior in the oth-ers. The gluonic pole matrix element ∆ GG ( x, x, x ) thusdisappears as do all higher pole matrix elements. Sincethese higher pole matrix elements appear in the higher k T -moments of the correlator ∆ [ U ] ij ( z, k T ) in Eq. (4), weconclude based on our very general assumptions of ana-lyticity for QCD amplitudes that this TMD correlator isuniversal and will be convoluted with the standard par-tonic cross sections. There is no proliferation of functionsoriginating from the structure of the gauge links [14].This universality thus applies to all TMD fragmentationfunctions, T-even or T-odd, and for quark as well as forgluon PFFs. Most well-known are the T-odd ones forquarks such as the Collins function H ⊥ [4, 15] and thepolarization fragmentation function D ⊥ T [4, 34]. Thesefunctions are simply allowed T-odd parts in the fragmen-tation correlator ∆( z, k T ) being a decay function. Thecorresponding T-odd TMD distribution functions, theBoer-Mulders function h ⊥ and the Sivers function f ⊥ T originate from the difference Φ [ U + ] ( x, p T ) − Φ [ U − ] ( x, p T )of correlators with different gauge links and as a con-sequence will be convoluted with non-standard gluonicpole cross sections [13, 14, 35]. Our result also impliesuniversality for the TMD fragmentation functions of glu-ons [36]; including for instance the T-even TMD frag-mentation functions H ⊥ ( g )1 , which just as the correspond-ing distribution function h ⊥ ( g )1 has a non-trivial gaugelink dependence [35, 37, 38]. The T-even fragmentationfunctions H ⊥ ( g )1 , however, is universal. In the case ofthese T-even functions, the non-trivial gauge link depen-dence only becomes visible in even k T -moments involvingcontributions from T-even multi-gluonic pole matrix el-ements with an even number of gluons, all of which forfragmentation functions, however, will vanish.This research is part of the Integrated InfrastructureInitiative Hadron Physics 2 (Grant 227431). LG acknowl-edges support from U.S. Department of Energy undercontract DE-FG02-07ER41460. AM thanks BRNS (sanc-tion no. 2007/37/60/BRNS/2913) for support. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected][1] J. C. Collins and D. E. Soper, Nucl. Phys.
B194 , 445(1982).[2] R. L. Jaffe and X.-D. Ji, Phys. Rev. Lett. , 552 (1991).[3] P. J. Mulders and R. D. Tangerman, Nucl. Phys. B461 ,197 (1996), hep-ph/9510301.[4] D. Boer and P. J. Mulders, Phys. Rev.
D57 , 5780 (1998),hep-ph/9711485.[5] A. V. Efremov and O. V. Teryaev, Sov. J. Nucl. Phys. , 140 (1982).[6] A. V. Efremov and O. V. Teryaev, Phys. Lett. B150 ,383 (1985). [7] J.-W. Qiu and G. Sterman, Phys. Rev. Lett. , 2264(1991).[8] J.-W. Qiu and G. Sterman, Nucl. Phys. B378 , 52 (1992).[9] J.-W. Qiu and G. Sterman, Phys. Rev.
D59 , 014004(1999), hep-ph/9806356.[10] Y. Kanazawa and Y. Koike, Phys. Lett.
B478 , 121(2000), hep-ph/0001021.[11] H. Eguchi, Y. Koike, and K. Tanaka, Nucl. Phys.
B763 ,198 (2007), hep-ph/0610314.[12] Y. Koike and K. Tanaka, Phys. Lett.
B646 , 232 (2007),hep-ph/0612117.[13] A. Bacchetta, C. J. Bomhof, P. J. Mulders, and F. Pijl-man, Phys. Rev.
D72 , 034030 (2005), hep-ph/0505268.[14] C. J. Bomhof and P. J. Mulders, JHEP , 029 (2007),hep-ph/0609206.[15] J. C. Collins, Nucl. Phys. B396 , 161 (1993), hep-ph/9208213.[16] J. C. Collins and A. Metz, Phys. Rev. Lett. , 252001(2004), hep-ph/0408249.[17] L. P. Gamberg, A. Mukherjee, and P. J. Mulders, Phys.Rev. D77 , 114026 (2008), 0803.2632.[18] S. Meissner and A. Metz, Phys. Rev. Lett. , 172003(2009), 0812.3783.[19] A. V. Efremov and A. V. Radyushkin, Theor. Math.Phys. , 774 (1981).[20] D. Boer and P. J. Mulders, Nucl. Phys. B569 , 505 (2000),hep-ph/9906223.[21] A. V. Belitsky, X. Ji, and F. Yuan, Nucl. Phys.
B656 ,165 (2003), hep-ph/0208038.[22] D. Boer, P. J. Mulders, and F. Pijlman, Nucl. Phys.
B667 , 201 (2003), hep-ph/0303034.[23] J. C. Collins and D. E. Soper, Nucl. Phys.
B193 , 381(1981).[24] X.-d. Ji, J.-p. Ma, and F. Yuan, Phys. Rev.
D71 , 034005(2005), hep-ph/0404183.[25] J. C. Collins, T. C. Rogers, and A. M. Stasto, (2007),arXiv:0708.2833 [hep-ph].[26] A. Bacchetta, D. Boer, M. Diehl, and P. J. Mulders,JHEP , 023 (2008), 0803.0227.[27] P. V. Landshoff and J. C. Polkinghorne, Phys. Rept. ,1 (1972).[28] R. L. Jaffe, Nucl. Phys. B229 , 205 (1983).[29] M. Diehl and T. Gousset, Phys. Lett.
B428 , 359 (1998),hep-ph/9801233.[30] C. Itzykson and J. B. Zuber, QUANTUM FIELD THE-ORY, 1980, New York, Usa: Mcgraw-hill (1980) 705P.(International Series In Pure and Applied Physics).[31] J. C. Polkinghorne, MODELS OF HIGH-ENERGYPROCESSES, 1980, Cambridge, Uk: Univ.Pr.(1980)131p.[32] S. M. Aybat and T. C. Rogers, (2011), 1101.5057.[33] Z.-B. Kang, (2010), 1012.3419.[34] D. Boer, Z.-B. Kang, W. Vogelsang, and F. Yuan, Phys.Rev. Lett. , 202001 (2010), 1008.3543.[35] C. J. Bomhof and P. J. Mulders, (2007), arXiv:0709.1390[hep-ph].[36] P. J. Mulders and J. Rodrigues, Phys. Rev.
D63 , 094021(2001), hep-ph/0009343.[37] F. Dominguez, C. Marquet, B.-W. Xiao, and F. Yuan,(2011), 1101.0715.[38] D. Boer, S. J. Brodsky, P. J. Mulders, and C. Pisano,(2010), 1011.4225., 094021(2001), hep-ph/0009343.[37] F. Dominguez, C. Marquet, B.-W. Xiao, and F. Yuan,(2011), 1101.0715.[38] D. Boer, S. J. Brodsky, P. J. Mulders, and C. Pisano,(2010), 1011.4225.