Universality of TMD correlators

Transcription

1 EPJ Web of Conferenes 85, DOI: 0.05/ epjonf/ C Owned by the authors, published by EDP Sienes, 205 Universality of MD orrelators M..A. Buffing,a, A. Mukherjee 2,b, and P.J. Mulders, Nikhef and Department of Physis and Astronomy, VU University Amsterdam, De Boelelaan 08, NL-08 HV Amsterdam, the Netherlands 2 Department of Physis, Indian Institute of ehnology Bombay, Powai, Mumbai , India Abstrat. In a high-energy sattering proess with hadrons in the initial state, olor is involved. ransverse momentum dependent distribution funtions MDs desribe the quark and gluon distributions in these hadrons in momentum spae with the inlusion of transverse diretions. Apart from the anti-quarks and gluons that are involved in the hard sattering proess, additional gluon emissions by the hadrons have to be taken into aount as well, giving rise to Wilson lines or gauge links. he MDs involved are sensitive to the proess under onsideration and hene potentially nonuniversal due to these Wilson line interations with the hard proess; different hard proesses give rise to different Wilson line strutures. We will show that in pratie only a finite number of universal MDs have to be onsidered, whih ome in different linear ombinations depending on the hard proess under onsideration, ensuring a generalized universality. For quarks this gives rise to three Pretzeloity funtions, whereas for gluons a riher struture of funtions arises. Introdution In the desription of hadroni sattering proesses, one has to onsider both hard sattering ontributions as well as parton distribution funtions PDFs that desribe the hadrons initiating the interations. We onsider transverse momentum dependent PDFs MDs by inluding transverse diretions in momentum spae in the desription of these objets []. New phenomena appear and manifest themselves for example in the form of angular orrelations between the partiles involved in the proess. Another effet is the sensitivity to polarization modes of the hadron and onstituent partons that would not have been possible without the inlusion of these transverse diretions. It is therefore relevant to study these MDs. In these proeedings, whih are based on the Refs. [2, 3], we fous on the universality properties of these MDs. In a olor gauge invariant desription, gauge links, path ordered exponentials, have to be inluded in the definition of MDs. hese gauge links appear as a result of gluon emissions oupling to the olored partiles in the hard sattering proess. It is this interplay between gauge links and the hard proess that introdues a sensitivity and potential proess dependene of the MDs to the proess in whih it appears, sine the gauge link struture is proess dependent itself. We refer to Ref. [4] for a tabulation of the strutures. he Sivers effet is a onsequene of the presene of different gauge link strutures in different proesses [5]. In turn, this warrants a study to make a lassifiation of all the MD strutures that appear in the various a m.g.a.buffing@vu.nl b asmita@phy.iitb.a.in p.j.g.mulders@vu.nl proesses, investigating the existene of a more generalized form of universality. In Setion 2 we outline the generalized universality for quarks, published in Ref. [2] and in Setion 3 we fous on the generalized universality for gluons, whih has been published in Ref. [3]. In Setion 4 we present some general onlusions and a brief disussion of the results. 2 Quarks For quarks, the matrix element desribing the orrelator is given by d ξ Pd Φ [U] 2 ξ ij x, p ; n = e ip ξ 2π 3 P ψ j 0 U [0,ξ] ψ i ξ P, ξ n=0 whih ontains a biloal ombination of quark fields onneted by a gauge link U [0,ξ]. his gauge link, ensuring olor gauge invariane in the proess, onsists of a path ordered exponential. As will be explained later, the path depends on the proess under onsideration and is onstruted out of staple like piees, running through light one infinity. hey are of the form U [±] [0,ξ] = U[n] [0,± ] U [0,ξ ] U[n] [±,0], with n being the diretion along the light one and the diretion in the transverse plane. he two simplest paths are indiated in Fig., onneting the fields through either plus or minus light one infinity. hese gauge links emerge due to soft gluon emission from the quark orrelator oupling to the partile involved in the hard proess. Initial state interations ISIs give rise to minus gauge links and final state interations FSIs imply plus gauge links. his is an Open Aess artile distributed under the terms of the Creative Commons Attribution Liense 4.0, whih permits unrestrited use, distribution, and reprodution in any medium, provided the original work is properly ited. Artile available at or

2 EPJ Web of Conferenes a Figure. he two simplest gauge links for quark distribution funtions. he dots indiate the positions 0 and ξ of the two quark fields in the orrelator, while the path of the gauge link is indiated by the line onnetion the two positions. In the simplest onfiguration the gauge link runs through either plus or minus light one infinity, illustrated in a and b respetively. Figures taken from Ref. [2]. he seond way to desribe the orrelator is by writing an expansion in terms of transverse momentum dependent parton distribution funtions MDs. he ontributions for an unpolarized hadron are given by { Φ [U] x, p ; n = f [U] x, p 2 ih [U] x, p 2 / p } /P M 2, 2 with h [U] x, p 2 being the Boer-Mulders funtion, the funtion desribing transversely polarized quarks in an unpolarized proton, whereas f x, p 2 desribes the unpolarized quark in an unpolarized proton. By inluding linearly or transversely polarized hadrons more MDs have to be inluded in the parametrization, for whih we refer to Ref. [6]. As of now, we have two desriptions, whih should be related to eah other. In order to do so, we use transverse moments, weightings with transverse momenta, a proedure whih an be applied at the level of both the MDs and the matrix elements. For the matrix elements, the result of a single transverse weighting is given by [7] Φ α[u] x d 2 p p α Φ [U] x, p = Φ α D x Φα A x Φα x = Φ α x C[U] Φα x. 3 he matrix element Φ α x is referred to as gluoni pole or Efremov-eryaev-Qiu-Sterman matrix element [8] and appears multiplied with a gluoni pole prefator. All proess dependene is isolated in these alulable gluoni pole prefators. he matrix elements in Eq. 3 are defined through [9] Φ α D x = dx Φ α D x x, x x, 4 Φ α A x dx PV i Φ nα F x x x, x x, 5 Φ α x = π Φnα F x, 0 x, 6 with dξ Pdη P Φ α Dij x x, x x = e ip ηip p ξ P ψ 2π 2 j 0 U [0,η] id α η U [η,ξ] ψ i ξ P, 7 LC b dξ Pdη P Φ α Fij x x, x x = e ip ηip p ξ P ψ 2π 2 j 0 U [0,η] F nα η U [η,ξ] ψ i ξ P. 8 LC Note a redefinition of the these definitions ompared to the Refs. [2, 7, 9, 0] regarding fators of π, whih was required for synhronization with the onvention used in Ref. [3]. For fragmentation orrelators the gluoni poles vanish []. We therefore do not expet proess dependene for the fragmentation orrelators and fous on the distribution orrelators only. In the above single transverse weighting example, only one additional operator shows up, i.e. one gluoni pole or partial derivative operator ombination. For higher transverse weightings, we get ontributions with multiple of suh operators in their definition. Antiipating results for transverse weightings with more fators of p, we an write down an expansion of the quark orrelator as [2] Φ [U] x, p = Φx, p 2 p i M Φ i x, p2 p ij ij Φ M2 x, p2 p ijk ijk Φ M 3 x, p2... pi M Φi x, p2 p ij ij Φ M2 { } x, p2 p ijk M 3 pij, M 2 Φij, x, p2 p ijk M 3 ijk Φ { } x, p2... { }, x, p2... ijk Φ pijk, M 3 Φijk, x, p , 9 where the index aounts for the possibility to have multiple ways of traing the olor. Note that there is no summation over for the single gluoni pole, sine only one olor struture is allowed in that situation. Contributions like Φ { } indiate the symmetrized ombination Φ { } = Φ Φ. he important realization is that eah of these ontributions has a ertain behavior under time-reversal symmetry. Contributions with an odd number of gluoni poles are -odd, whereas all other ontributions are -even. We define the number of operators in the definition of the matrix elements i.e. the number of gluoni poles and s as the rank of the matrix element, whih equals the number of transverse weightings that is required to obtain the objet. Furthermore, as an be seen in Eq. 3 for the single weighted ase, all proess dependene is identified with gluoni pole ontributions in the form of prefators, alulable numerial fators that depend on the gauge link only. When performing transverse weightings, we basially weight the expression in Eq. 9. Weighting the orrelator Φ [U] x, p with zero fators of p implies that on the r.h.s. of Eq. 9 only the objet Φx, p 2 survives or atually the integrated version of it. All other ontributions have fators of p i, p ij, p ijk, et., whih do not survive the integration over transverse momentum. Here, these transverse 0200-p.2

3 RANSVERSIY 204 momenta are defined as the symmetri and traeless tensors, e.g. p i, p ij = p ip j 2 p2 g ij. 0 For a single transverse weighting, we have to multiply Eq. 9 with p i and integrate over transverse momentum. Due to the definitions of the transverse momentum tensors, on the r.h.s. only the matrix elements with the prefator p i survive, i.e. the integrated versions of Φ i x, p2 and Φi x, p2. his an be generalized for transverse weightings with an arbitrary number of arbitrary rank. Applying the transverse weightings on MDs, we obtain the weighted funtions p f... n[u] x, p 2 2 n = f [U] 2M 2... x, p 2. a e b d f Usually only the integrated funtions f... n[u] x are referred to as transverse moment. We will extend this name to funtions that still depend on p 2, but are azimuthally averaged. he behavior of the MDs under time reversal symmetry is known. E.g. f is -even, while the Boer-Mulders funtion h [U] is -odd. We ould therefore identify at the level of transverse moments whih MD orresponds to whih matrix element in the expansion in Eq. 9. For example, h orresponds to C[U] Φ x, p 2, see e.g. Ref. [7], whereas f orresponds to Φx, p 2. his way, all MDs ould be assoiated with one or more matrix elements. For the rank 2 Pretzeloity funtion a ompliation arises, sine it orresponds to the matrix elements Φ ij x, p2 and, Φij, x, p2, the latter oming in two olor ontributions, see the Refs. [2, 0]. herefore, we have three Pretzeloity funtions, h [U] x, p 2 = h A x, p2, h B x, p 2,2 h B2 x, p 2. 2 Note that we stritly speaking only make the identifiation at the level of transverse moments using our methods. Inidentally, for both Drell-Yan and SIDIS we get the same linear ombination of them, namely h [±] x, p2 = h A x, p2 h B x, p 2. 3 Nevertheless, it still is important to realize the underlying struture of these funtions. 3 luons For gluons, a similar approah an be used and the matrix element for the gluon orrelator is given by [4, 2, 3] Γ [U,U ] d ξ Pd μν 2 ξ x, p ; n = e ip ξ P,S F nμ 0 2π 3 U [0,ξ] F nν ξ U [ξ,0] P,S LF. 4 Note that a olor traing is still required in the above definition. Sine the gluon fields are olor otets rather than olor triplets, two gauge link ontributions are required for a proper gauge invariant desription, indiated by U and g Figure 2. Examples of gauge link strutures for gluons. he dots represent the loations of the two gluon fields in the gluon orrelator, whereas the lines indiate the path of the gauge link. See the main text for an explanation. Figures taken from Ref. [3]. U in the above equation. Both ontributions onsist of staple like gauge links, with optionally additional Wilson loops. he three types of strutures that an be onstruted this way in the relevant 2 2 proesses are given by type : r F nμ 0 U [0,ξ] F nν ξ U [ξ,0] type 2: r F nμ 0 U [0,ξ] F nν ξ U [ξ,0] r U [loop] N type 3: h r F nμ 0 U [loop] r F nν ξ U [loop ] N he first type orresponds to orrelators ontaining a single olor trae only, among them the four simplest gluon gauge link strutures allowed, illustrated in Fig. 2a-d. hese four gauge link strutures onsist of the staple links going through plus or minus light one infinity. Sine there are two possibilities for both of them, it leaves us with four strutures. More involved strutures also allow for e.g. the situation that U and U are a ombination of three staple links, illustrated in more detail in Fig. 2e. Correlators of the seond type have two or more olor traes and are extensions of the first type. Starting from the struture of the first type, one an allow for olor traes ontaining gauge link loops only and multiply the type orrelator with them, see e.g. Fig. 2f. In this, we define the gauge link loops as U [ ] = U [] [0,ξ] U[ ] [ξ,0] or U[ ] = U [ ] [0,ξ] U[] [ξ,0]. ype 3 orrelators are required too, see Fig. 2g-h, but in these proeedings the fous will be on the type and type 2 olor strutures. Just as for quarks, the minus gauge links ome from initial state interations and the plus gauge links from final state interations. A simple illustration for gluons involves 0200-p.3

4 EPJ Web of Conferenes a b Figure 3. hree Feynman diagrams with a different olor flow ontribution eah. In a all olor remains in the initial state, in b all olor flows into the final state and in we have olor splitting, with olor flowing in both the initial and final state. See the main text for the impliations for the orresponding gauge link strutures. the diagrams in Fig. 3. In Fig. 3a there are only ISIs and alulations show the gauge link struture to be the one in Fig. 2b. For Fig. 3b, with only FSIs, we find the gauge link struture of Fig. 2a. he Feynman diagram in Fig. 3 has olor splitting, with olor flowing into both the initial and final state. Due to this olor struture, we find the gauge link struture in Fig. 2d, with one staple link running through plus light one infinity and one staple link running through minus light one infinity. On the other hand, MDs ould be used to parametrize the orrelator as well, giving for the unpolarized hadron ontributions the expression 2x Γ μν[u] x,p = g μν p μ pν M 2 f g[u] x,p 2 g μν p 2 2M 2 x,p 2, 5 where we use the naming onvention of Ref. [4]. We refer to Ref. [2] for the full parametrization. Applying weightings at the level of the matrix elements, one is muh more sensitive to the type of gauge link strutures involved ompared to the quark situation, due to more ompliated gauge link strutures for gluon orrelators. As will be shown later, this results in a muh larger set of olor ombinations of the operator strutures. he olor index that for quarks in Eq. 9 only beame relevant for the Pretzeloity funtion will play a more signifiant role for gluons. Let s start with the ontributions for type orrelators and fous on the matrix elements ontaining gluoni poles only. We then have depending on the gauge link strutures involved the matrix elements Γ α, r [ F0 α ξ, Fξ ], 6a { F0 α Γ α Γ α α 2,2 r, r Γ α α 2 ξ, Fξ }, 6b F0 [ α ξ, [ α 2 F0 { α ξ, { α 2 ξ, Fξ ]], 6,2 r ξ, Fξ }}, 6d Γ α α 2 α 3, r [ F0 α ξ, [ α 2 ξ, [ α 3 ξ, Fξ ]]], 6e Γ α α 2 α 3,2 r { F0 α ξ, { α 2 ξ, { α 3 ξ, Fξ }}}. 6f Note that we omitted the gauge links themselves for the sake of simpliity of this illustration. Eah time, the gluoni poles enter either in a ommutator or antiommutator ombination with the gluon field Fξ. For the single weighted ase, these funtions were introdued in Ref. [3] with the subsripts d and f. For type 2 and type 3 more ompliated strutures are allowed, sine there is an additional olor trae that ould reeive operators due to transverse weighting. Examples of gluoni pole only strutures that arise are Γ α α 2,3 2 { N r α ξ, α 2 ξ } r F0Fξ, 7a Γ {α α 2 },4 2 N r F0 {α ξ { r α 2 } ξ, Fξ }. 7b he parentheses around some indies in some of the above equations indiates a symmetrization over those indies. For ontributions reeiving ontributions only, defined through i α = id α A α, we find that they always ome in the ommutator ombination, i.e. Γ α r [ F0 i α, Fξ ]], 8a Γ α α 2 [ r F0 i α ξ, [ i α 2 ξ, Fξ ]], 8b Γ α α 2 α 3 [ r F0 i α ξ, [ i α 2 ξ, [ i α 3 ξ, Fξ ]]]. 8 On top of this, also a number of mixed terms exists that have both gluoni pole and ontributions. A full list of all these ontributions and the gluoni pole ontributions not shown above an be found in Ref. [3]. Again writing down the expansion of the orrelator in terms of matrix elements ontaining gluoni poles and ontributions, we find [3, 5] Γ [U] x, p = Γx, p 2 p i M Γ i x, p2 p ij M pi, 2 Γ ij x, p2 p ijk ijk Γ M3 x, p2... M Γi, x, p2 p ij ij Γ M2 { }, x, p2 pij, M 2 Γij, x, p2 p ijk ijk Γ M3 { }, x, p2... { }, x, p2... p ijk ijk Γ M3 pijk, M 3 Γijk, x, p , 9 he number of olor strutures, of whih some were illustrated above and of whih a omplete list an be found in Ref. [3], runs to two for =, it runs to four for = 2 and it runs to seven for = 3. Starting from the Eqs. 5 and 9, we an again perform transverse weightings and ompare the results of the two separate approahes. For the approah in terms of matrix elements, we give the double weighted ase as an example. We find that Γ α α 2 [U] x d 2 p p α p α 2 Γ [U] x, p 0200-p.4

5 = Γ α α 2 x, Γ α α 2 { }, x, Γα α 2, x. 20 We ould find this by looking at the r.h.s. of Eq. 9. Only the rank 2 objets on the r.h.s. of that equation survive weighting over two transverse momenta, the reason for whih is analogues to the explanation we gave at the end of Setion 2 for transverse weighting of the quark orrelator. Among the surviving matrix elements are speifi ontributions with zero, one and two gluoni poles whih ome in different olor onfigurations, hene the summation over the index. Applying transverse weightings on the MDs, using the definition in Eq., we an identify whih MD orresponds to whih matrix element in the expansion in Eq. 5. It turns out that h g is the only gluon MD ontributing at rank 2. It is a -even funtion this funtion is multiplied by two fators of p in Eq. 5 and ould therefore orrespond to both Γ x, p 2 and, Γ,x, p 2, with running from to 4, sine there are four possibilities to trae the olor. his implies that there are five h g funtions, whih depending on the proess under onsideration appear in different linear ombinations, sine four of them ome with a proess dependent gluoni pole fator. here is no identifiation with Γ α α 2 x, sine there are no -odd { }, rank 2 ontributions at leading twist that ould be identified with it. Inluding the results for the MDs not expliitly mentioned in Eq. 5, this leads for the gluon MDs to the results f g[u] x, p 2 = h g[u] x, p2 = L x, p 2 = 2 = 2 = 2 =, f ga x, p 2, 2, hga x, p 2, 22, h ga L x, p 2, 23 x, p 2 = h ga x, p 2 4, h gb x, p 2, 24 x, p 2 = 2 = =, h ga x, p 2 7 =, h gb x, p he three MDs not mentioned above in the Eqs. 2-25, namely f g, gg and gg, are proess independent. o illustrate this generalized universality for the h, onsider the situations for the three diagrams illustrated in Fig. 3. We find for both the Higgs prodution through gluon fusion and the sattering of a gluon on a Higgs partile that h g[, ] x, p 2 = h ga x, p 2 h gb x, p 2, 26a h g[,] x, p 2 = h ga x, p 2 h gb x, p 2, 26b RANSVERSIY 204 whereas we find for the olor splitting example in Fig. 3 that h g[,] x, p 2 = h ga x, p 2 h gb2 x, p In order to find the funtions h gb3 x, p 2 and h gb4 x, p 2 more ompliated diagrams have to be onsidered. 4 Conlusions For the quarks, a result of applying the method of generalized universality is the disovery of three Pretzeloity funtions rather than one. In any proess in partiular it is a linear ombination of these funtions that appears. It is the gauge link struture of the diagram under onsideration that determines whih linear ombination appears. For Drell-Yan and SIDIS one does find the same linear ombination of Pretzeloity funtions. Nevertheless it is still important to know the preise operator struture underlying the MDs, sine it is important for studies wherein the operator strutures involved beome relevant, e.g. in lattie alulations. For the gluon MDs f g[u], h g[u], h g[u] L, and multiple funtions appear and linear ombinations of these funtions have to be onsidered. his brings the number of MDs operator-wise at 23, although there still are only 8 observable MD strutures. Nevertheless, it an be alulated how eah of these observable strutures are onstruted out of the 23 objets for any given proess. Aknowledgements his onferene proeeding ontribution is based on the talk given by MAB. his researh is part of the researh program of the Stihting voor Fundamenteel Onderzoek der Materie FOM, whih is finanially supported by the Nederlandse Organisatie voor Wetenshappelijk Onderzoek NWO. We also aknowledge support of the FP7 EU-programme Hadron- Physis3 ontrat no and QWORK ontrat Referenes [] J.P. Ralston and D.E. Soper, Nul. Phys. B 52, ; R.D. angerman and P.J. Mulders, Phys. Rev. D5, ; D. Boer, Phys. Rev. D60, [2] M..A. Buffing, A. Mukherjee and P.J. Mulders, Phys. Rev. D86, [3] M..A. Buffing, A. Mukherjee and P.J. Mulders, Phys. Rev. D88, [4] C.J. Bomhof, P.J. Mulders and F. Pijlman, Eur. Phys. J. C47, [5] D.W. Sivers, Phys. Rev. D 4, ; D.W. Sivers, Phys. Rev. D 43, ; J.C. Collins, Nul. Phys. B 396, ; J.C. Collins, Phys. Lett. B 536, ; S.J. Brodsky, D.S. Hwang and I. Shmidt, Nul. Phys. B 642, p.5

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