Non-universality of transverse momentum dependent parton distribution functions

In the field theoretical description of hadronic scattering processes, single transverse-spin asymmetries arise due to gluon initial and final state interactions. These interactions lead to process dependent Wilson lines in the operator definitions of transverse momentum dependent parton distribution functions. In particular for hadron-hadron scattering processes with hadronic final states this has important ramifications for possible factorization formulas in terms of (non)universal TMD parton distribution functions. In this paper we will systematically separate the universality-breaking parts of the TMD parton correlators from the universal T-even and T-odd parts. This might play an important role in future factorization studies for these processes. We also show that such factorization theorems will (amongst others) involve the gluonic pole cross sections, which have previously been shown to describe the hard partonic scattering in weighted spin asymmetries.

In the field theoretical description of hadronic scattering processes, single transverse-spin asymmetries arise due to gluon initial and final state interactions. These interactions lead to process dependent Wilson lines in the operator definitions of transverse momentum dependent parton distribution functions. In particular for hadron-hadron scattering processes with hadronic final states this has important ramifications for possible factorization formulas in terms of (non)universal TMD parton distribution functions. In this paper we will systematically separate the universality-breaking parts of the TMD parton correlators from the universal T -even and T -odd parts. This might play an important role in future factorization studies for these processes. We also show that such factorization theorems will (amongst others) involve the gluonic pole cross sections, which have previously been shown to describe the hard partonic scattering in weighted spin asymmetries.

I. INTRODUCTION
Many theoretical as well as experimental studies in recent years have been aimed at better understanding the processes that cause spin asymmetries in hadronic scattering. A mechanism to generate single-spin asymmetries (SSA) through soft gluon interactions between the target remnants and the initial and final state partons was first proposed in the context of collinear factorization [1,2,3,4,5,6,7,8]. This collinear factorization formalism involves, apart from the usual twist-two quark correlators, also twist-three collinear quark-gluon matrix elements. Since they contain the field operator of a zero-momentum gluon, they are referred to as gluonic pole matrix elements. An important example is the Qiu-Sterman matrix element T F (x,x) [1,2,3,4,5].
Several other mechanisms to generate SSA's through the effects of the intrinsic transverse momenta of the partons have also been proposed. For instance, in the Sivers effect the asymmetry arises in the initial state due to a correlation between the intrinsic transverse motion of an unpolarized quark and the transverse spin of its parent hadron [9,10]. The effect can be described by a transverse momentum dependent (TMD) distribution function f ⊥ 1T (x,p 2 T ) [11]. Such a function can exist by the grace of soft gluon interactions between the target remnants and the active partons [12,13]. These interactions give rise to process dependent Wilson lines, also called gauge links, in the definitions of TMD parton distribution and fragmentation functions. The Wilson lines secure the gauge invariance of these definitions. At the same time they prevent the use of time-reversal to argue that the Sivers effect should vanish. Instead, timereversal can be used to derive non-trivial 'universality' relations between the Sivers functions in different processes. For instance, it was shown that the Sivers function in SIDIS, which contains a future pointing Wilson line, has opposite sign [12,13,14] as the TMD function in Drell-Yan scattering, which involves a past pointing Wilson line. Moreover, the Wilson lines are also crucial ingredients in the derivation [15] of the relation between the Sivers function and the Qiu-Sterman matrix element, 2M f ⊥(1) 1T (x) = −gT F (x,x), demonstrating that the first transverse moment of the Sivers function is a gluonic pole matrix element.
The process dependence of the Wilson lines in TMD parton correlators makes the study of the (non)universality of these functions particularly important. In the basic electroweak processes, SIDIS, Drell-Yan scattering and e + e −annihilation, the hard partonic parts of the process are just simple electroweak vertices (at tree-level). Depending on the particular process only initial or final-state gluon interactions contribute and, correspondingly, only future and past pointing Wilson lines occur. However, when going to hadronic processes that involve hard parts with more colored external legs, such as in hadronic dijet or photon-jet production, there can be both initial and final state gluon interactions. As a result, the Wilson lines resulting from a resummation of all exchanged collinear gluons will also be more complicated than just the simple future and past pointing Wilson lines [16,17,18]. In particular, for each of the Feynman diagrams that contribute to the hard partonic part of the hadronic scattering process there is, in principle, a different gauge link structure.
For the TMD distribution functions this at first sight seems to complicate things considerably. However, for the collinear distribution functions remarkable simplifications occur. Upon integration over intrinsic transverse momenta all the effects of the complicated gauge link structures in the TMD correlators disappear, while for the transverse moment they contribute a gluonic pole matrix element with multiplicative prefactors, referred to as gluonic pole strengths. These are color-fractions that, in principle, differ for each Feynman diagram that contributes to the partonic subprocess. Therefore, for a given subprocess one can multiply the color factors with the contribution of each partonic diagram and collect them in modified (but manifestly gauge invariant) hard cross sections [17,19]. These modified hard functions, called gluonic pole cross sections, appear whenever gluonic pole matrix elements (such as the first moments of the Sivers and Boer-Mulders functions) contribute. This is typically the case in weighted azimuthal spin asymmetries.
The effects of the gluon initial and final-state interactions for the fully TMD treatment of these processes is less clear-cut. In Refs [20,21] a TMD factorization formula based on one-gluon radiation was proposed for the quark-Sivers contribution to the SSA in dijet production in proton-proton scattering. This result involves the gluonic pole cross sections found in Refs [17,19] as hard parts, folded with the TMD distribution functions as measured in SIDIS (i.e. with a future pointing Wilson line in their definitions). On the other hand, in Refs [16,17,18] it was observed that complicated Wilson line structures occur in the TMD distribution (and fragmentation) functions in such processes. Those results, in concurrence with a model calculation, led the authors of Ref. [22] to conclude that a TMD factorization formula for spin asymmetries in processes such as dijet production in proton-proton scattering cannot be written down with universal distribution functions. It is also asserted that a proof of TMD factorization for such processes will be essentially different from the existing proofs for SIDIS and Drell-Yan scattering and that it will probably involve 'effective' TMD parton distribution functions [23]. Recent extensions [24,25] of the work in [20,21,22] also include the contributions of two collinear gluons (as was previously discussed for Drell-Yan [26]). These indicate that the Feynman graph calculations in Refs [16,17,18,19], [20,21] and [22] are "mutually consistent" up to two-gluon contributions [24].
By using the gluonic pole strengths we will in this paper systematically separate the universality-breaking parts of the TMD parton correlators from the universal T -even and T -odd matrix elements. It is a non-trivial observation that this is possible and we believe that it constitutes another important ingredient in trying to relate the results of Refs [16,17,18,19] and Refs [20,21]. We demonstrate that the gluonic pole cross sections are also encountered in unintegrated, unweighted processes. In particular, we will argue that the gluonic pole cross sections can also emerge in unweighted spin-averaged processes and that ordinary partonic cross sections can also arise in unweighted single-spin asymmetries, though they appear in such a way that they will vanish for the integrated and weighted processes [17,19,27,28], respectively. We will start by recapitulating the collinear case in section II and the appearance of universal collinear correlators in hadronic cross sections in section III. The study of the non-universality of the TMD parton correlators will be presented in sections IV and V, followed by a discussion on how the non-universal TMD correlators affect hadronic cross sections (section VI). After summarizing in section VII we list all universalitybreaking matrix elements that are encountered at tree-level in 2→2 hadronic scattering processes (appendix A).

II. COLLINEAR CORRELATORS
For a twist analysis of hadronic variables in high-energy physics it is useful to make a Sudakov decomposition p µ = xP µ +σn µ +p µ T of the momentum p µ of each active parton. The Sudakov vector n is an arbitrary light-like four-vector n 2 ≡ 0 that has non-zero overlap P ·n with the hadron's momentum P µ . We will choose the Sudakov vector such that this overlap is positive and of the order of the hard scale. Up to subleading twist its coefficient σ = (p·P −xM 2 )/(P ·n) is always integrated over. The vector p T is called the intrinsic transverse momentum of the parton. It is orthogonal to both P and n, i.e. p T ·P = p T ·n = 0, and will appear suppressed by one power of the hard scale with respect to the collinear term. Vectors in the transverse plane can be obtained by using the transverse projectors g µν T ≡ g µν −P {µ n ν} /(P ·n) and ǫ µν T ≡ ǫ µνρσ P ρ n σ /(P ·n). Note that each observed hadron can have a different transverse plane.
We consider collinear quark distribution functions as being obtained from transverse momentum dependent (TMD) quark distribution functions. Those are projections of the TMD quark correlator defined on the light-front (LF: ij (x,p T ;P ,S) = The Wilson line or gauge link U [η;ξ] = Pexp −ig C ds·A a (s) t a is a path-ordered exponential along the integration path C with endpoints at η and ξ. Its presence in the hadronic matrix element is required by gauge-invariance. In the TMD correlator (1) the integration path C in the gauge link is process-dependent. In the diagrammatic approach the Wilson lines arise by resumming all gluon interactions between the soft and the hard partonic parts of the hadronic process [15,26,29,30]. Consequently, the integration path C is fixed by the (color-flow structure of) the hard partonic scattering [18]. Going beyond the simplest electroweak processes such as SIDIS, Drell-Yan scattering and e + e − -annihilation, the competing effects of the gluonic initial and final-state interactions lead to gauge link structures that can be quite more complicated than the future or past pointing Wilson lines [16,17,18]. The situation becomes particularly notorious when considering processes which have several Feynman diagrams that contribute to the partonic scattering. In that case each cut Feynman diagram D can, in principle, lead to a different gauge-invariant correlator [16,17,18]. This observation leads to a broad spectrum of different TMD parton correlators that appear in hadronic scattering processes.
For collinear correlators the situation is simpler. For instance, in the p T -integrated correlator defined on the lightcone (LC: all process-dependence of the gauge link disappears, leaving just a straight Wilson line U n [0;ξ] in the lightcone ndirection, where n is the lightlike vector in the Sudakov decomposition of the quark momentum p (we will use the non-calligraphed letter U to indicate straight line segments). Another situation is encountered in the transverse momentum weighted correlators (the transverse moments). In the transverse moments a (sub)process-dependence remains as a direct consequence of the presence of the gauge links in the TMD correlators. Nevertheless, a simple decomposition can still be made (omitting the Dirac indices) [15,17,19]: with collinear correlators The only process dependence due to the Wilson lines in the TMD correlators resides in the multiplicative factors C G . They are color-fractions that are fixed by the color-flow structure of the hard partonic function of the scattering process [17,19]. We will refer to them as gluonic pole strengths. Important examples are the transverse moments of the correlators Φ [+] in SIDIS and Φ [−] in Drell-Yan scattering, for which one has C [15]. Transverse momentum dependent gluon distribution functions are projections of the TMD correlator Here Tr indicates a trace over color-triplet indices. Writing the field-operators in the color-triplet representation requires the inclusion of two Wilson lines U [0;ξ] and U ′ [ξ;0] [18]. They again arise from the resummation of gluon initial and final-state interactions. In general this will lead to two unrelated Wilson lines U and U ′ . In the particular case that U ′ = U † , the gluon correlator can also be written as the product F a U ab F b of two gluon fields with the Wilson line U in the adjoint representation of SU (N ). This is for instance the case for the gluon correlators in Figs 1c and d, but not for the gluon correlators in Figs 1e and f.
In the p T -integrated correlator on the lightcone the process dependence of the TMD gluon correlator disappears, However, as for the quark correlator, a subprocess-dependence due to the Wilson lines in the TMD gluon correlators remains in the transverse moments. The analogue of the decomposition (3) in the case of the gluon correlator is (with and omitting the gluon field indices µ and ν) [19]: The matrix elements Γ G f and Γ G d are the two gluonic pole matrix elements that correspond to the two possible ways to construct color-singlets from three gluon fields [19,31]. They involve the antisymmetric f and symmetric d structure constants of SU (3), respectively. The only process dependence coming from the Wilson lines in the TMD correlators is contained in the gluonic pole strengths C . The collinear correlators are and The collinear (anti)quark and gluon fragmentation correlators can be analyzed in the same way. The matrix elements in (3) and (8) contain the collinear T -even and T -odd parton distribution functions, see e.g. [19].

III. COLLINEAR FUNCTIONS IN HADRONIC CROSS SECTIONS
In the diagrammatic approach, the calculation of the hadronic cross sections starts off with the transverse momentum dependent parton correlators (TMD distribution and fragmentation functions), which will appear in combination with squares of hard partonic amplitudes. In general the hard amplitude contains more terms, that is H = i H i . In the squared amplitude one therefore has terms like H * i H j ≡Σ [D] , where D refers to the cut Feynman diagram that is the pictorial representation of the product of the amplitude H j and conjugate hard amplitude H * i . The hadronic cross section dσ of a hadronic scattering process mediated by a two-to-two partonic subprocess a(p 1 )b(p 2 )→c(k 1 )d(k 2 ) and where the outgoing hadrons and/or jets are produced with large perpendicular component with respect to the beam will contain the following structure in the integrand: where the parton momenta are approximately (compared to the hard scale) on-shell. The convolutions '⊗' represent the appropriate Dirac and color traces for the hard functionΣ [D] . To get to the hadronic cross section one has to multiply by the flux factor and integrate over the final-state phase-space and parton momenta including a delta function for momentum conservation on the partonic level. Since our aim in this paper is to display some general features that a k T -factorization formula (if it exists) will have due to the process-dependence of the Wilson lines that arise in the diagrammatic TMD gauge link approach, we focus our discussion on the Wilson lines and neglect soft factors. In the full k T -factorization formula such factors will most likely also be present to account for soft-gluon effects.
The TMD correlators in (11) are the gauge invariant non-universal (anti)quark/gluon correlators that contain the appropriate Wilson lines for the particular color-flow diagram D that contributes to the partonic subprocess ab→cd. This is the reason why in expression (11) the summation over cut diagrams D is displayed explicitly. The hard functions, i.e. the expressionsΣ [D] of the individual Feynman diagrams are, themselves, not gauge invariant. For azimuthal dependence originating from only one of the partons one can effectively use the correlators calculated in Ref. [18]. Furthermore, in the tree-level discussion employed here the Wilson lines are along the lightlike n-direction, though a non-lightlike n 2 = 0 direction may be required when higher-order corrections are taken into account [32,33].
From momentum conservation on the partonic level it will follow that, depending on the process, some components of the partonic momenta can be measured (e.g. in a way similar to the identification of the incoming parton momentum fraction x with the Bjorken scaling variable x B in deep inelastic scattering). This also works for the transverse momenta. For instance, for a hadronic scattering process with a two-to-two hard subprocess the structure in (11) will appear with a delta function for momentum conservation enforcing the relation p 1 +p 2 −k 1 −k 2 ≡ 0. There are ways to measure one or several components of which is not required to vanish by momentum conservation since the directions of the intrinsic transverse momenta of the partons can be different for each observed hadron (in back-to-back jet production in hadron-hadron scattering it is the component along the outgoing jet direction in the plane perpendicular to the beam axis that is experimentally accessible through the relation q T ·K ⊥ jet ∝ sin(δφ), where δφ is the azimuthal imbalance of the two jets in the perpendicular plane [17,19,27,34]). This quantity defines a scale much smaller than the hard scale of the process. Using these components one can construct integrated and weighted hadronic cross sections. Integrated cross sections will involve the structure while weighted cross sections will involve (making use of the decomposition in Eq. (3)) and similar expressions Σ 2 ∂ , Σ 1 ′ ∂ and Σ 2 ′ ∂ which are obtained by weighting with p 2T , k 1T and k 2T , respectively. In these expressions only universal collinear correlators appear. In Eqs (12) and (13) we have defined the hard functionŝ The factors C G (a) are the gluonic pole strengths that appear in the decomposition of the transverse moment of the TMD correlator of parton a. In expression (13) this parton was implicitly taken to be a quark. If it were a gluon there would have been twoΣ [g]b→cd terms, one corresponding to each of the gluonic pole matrix elements Γ G f and Γ G d , cf. Eq. (8) or Ref. [19]. The hard functions in Eqs (14a) and (14b) no longer depend on the individual (diagrammatic) contributions D, but only on the hard process ab→cd. Moreover, in contrast to the hard functionsΣ [D] that appear in (11), they are gauge invariant expressions. After performing the traces theΣ ab→cd reduce to the partonic cross sections dσ ab→cd and theΣ [a]b→cd reduce to the gluonic pole cross sections dσ [a]b→cd calculated in Refs [17,19,28].

IV. TRANSVERSE MOMENTUM DEPENDENT CORRELATORS
To study azimuthal asymmetries arising from one of the partons in hadronic processes mediated by 2→2 partonic subprocesses at tree-level it is possible, as we will show explicitly, to organize the TMD correlators in a decomposition analogous to (3) containing TMD correlators with special properties: Here D refers to a particular cut Feynman diagram that contributes to the cross section of the partonic process ab→cd. The gluonic pole factors C

[D]
G are the same as those in the decomposition of the collinear correlators in (3). An important difference between the decomposition of the collinear correlator in (3) and the decomposition of the TMD correlator in (15) is that the matrix elements in the latter decomposition are not universal in general. They depend on the partonic process ab→cd but, in contrast to the TMD quark correlators Φ [D] on the l.h.s. of the decomposition, they do not depend on the individual cut Feynman diagram D. The only diagram dependence resides in the gluonic pole factors. The matrix elements on the r.h.s. of (15) have been chosen such that they reduce to the familiar universal (process independent) collinear matrix elements when integrating over or weighting with the intrinsic transverse momenta: The most straightforward illustration of the decomposition (15) with the T -even and T -odd quark correlators The factors C [±] G = ±1 are the same as those for the transverse moments (3) in those processes. In contrast to (17), we observe that the TMD matrix elements Φ (ab→cd) (x,p T ) and πΦ (ab→cd) G (x,p T ) on the r.h.s. of (15) in general do not have definite behavior under time-reversal. However, the process dependent universalitybreaking parts of the TMD correlators can be separated from the universal T -even and T -odd parts in (18a) and (18b): In these expressions all process dependence due to Wilson lines on the light-front is now contained in process-dependent universality-breaking matrix elements δΦ (ab→cd) (x,p T ) and πδΦ (ab→cd) G (x,p T ), which we will refer to as junk-TMD. Also these in general have no definite behavior under time-reversal, but they do have the special properties that they vanish after p T -integration and weighting: This is consistent with the properties expressed in (16). The expressions in Eq. (20) will be used in section VI to show that the universality-breaking matrix elements vanish in integrated and weighted hadronic cross sections. Moreover, as can be seen from their explicit expressions in appendix A the (anti)quark/gluon universality-breaking matrix elements vanish in an order g expansion of the Wilson lines (i.e. the one-gluon radiation contribution). All universality-breaking matrix elements that occur at tree-level in proton-proton scattering with hadronic final states are listed in appendix A. The TMD correlators Φ [D] (x,p T ) in these processes have already been derived in Ref. [18] and are given in the tables of that reference (the TMD correlators and gluonic pole factors that appear at tree-level in direct photon-jet production in proton-proton scattering can be found in Ref. [35]). It is straightforward to verify that these results are reproduced with the matrix elements given in appendix A and through the decompositions (15) and (19). This should not come as a surprise, since the matrix elements in the appendix were defined that way. It is a remarkable and non-trivial observation that with the gluonic pole strengths all the quark correlators encountered in a certain partonic process ab→cd can be decomposed in terms of only the two matrix elements Φ (ab→cd) and πΦ  16)). It should be mentioned, though, that this decomposition is not unique. For instance, one could also have made a decomposition in terms of more matrix elements. That is, by including matrix elements that do not contribute to the zeroth (p T -integration) and first (p T -weighting) transverse moments in p T , but do contribute to the second moment, third moment, etc. It is conceivable that the inclusion of these additional matrix elements will allow one to summarize the TMD quark correlators encountered in different partonic processes, in the same way as it was possible to summarize all quark correlators associated to the different Feynman diagrams D that contribute to one specific partonic process ab→cd by the two matrix elements Φ (ab→cd) and πΦ (ab→cd) G . At present this is just speculation, though, and the verification or falsification will require more insight into the way that the different Wilson line structures contribute to higher transverse moments. However, regardless of all these cautionary remarks we believe that the notational advantage, the points concerning gauge invariance of the hard functions that will be addressed in section VI and the possible role that it could play in relating the gauge link formalism to the results of Refs [20,21] provide more than enough justification for the decomposition in Eq. (15).
In the case of gluon distributions we start by defining T -even and T -odd gluon correlators (cf. Figs 1c-f), where as in Ref. [15] a gluon correlator is called T -odd if it vanishes when identifying the future and past-pointing Wilson lines. Note that in contrast to Γ (T -even) and Γ (T -odd) (f ) , the correlator Γ cannot be written as a matrix element of two gluon fields with a single Wilson line in the adjoint representation. After p T -integration the T -even and T -odd correlators reduce to the universal collinear gluon matrix elements in the expressions in Eqs (7) and (8): Since there are two distinct ways to construct T -odd gluon correlators, it will follow in the next section that there are also two distinct TMD gluon-Sivers distribution functions (cf. Eq. (28b)). There is actually also a second way to construct a T -even gluon correlator: Γ ′ (T -even) = 1 2 (Γ [+,− † ] + Γ [−,+ † ] ). However, this correlator is not needed in the decomposition (23) of the TMD gluon correlators, since the difference between Γ ′ (T -even) and Γ (T -even) is a matrix element that vanishes upon p T -integration and p T -weighting. This difference may, therefore, be absorbed in the universality-breaking matrix elements δΓ (ab→cd) to be defined in (24a).
A decomposition resembling the one in (8) for TMD gluon correlators can almost be made: This leaves only a specific type of matrix elements with colorless intermediate states unaccounted for (we will return to this point in a moment). The TMD matrix elements on the r.h.s. of (23) only depend on the process and not on the particular Feynman diagram D that contributes to that process. The multiplicative factors C (f ) G are the gluonic pole factors calculated in Ref. [19], the same that also appear in the decomposition (8) of the collinear correlator. Under p T -integration and weighting the matrix elements Γ (ab→cd) (x,p T ) and Γ (ab→cd) G f /d (x,p T ) have the same behavior as Γ (T -even) (x,p T ) and Γ (T -odd) (f /d) (x,p T ), respectively. Therefore, one can make the further separation in which all process-dependence due to Wilson lines on the light-front has been gathered in the universality-breaking matrix elements δΓ (ab→cd) (x,p T ) and πδΓ , which have the special properties that they vanish after a p T -integration and p T -weighting.
It is also straightforward to check that through the decompositions in (23)- (24) and with the universality-breaking matrix elements given in appendix A the TMD gluon correlators in the tables 4, 5 and 8 of Ref. [18] corresponding to qg→qg,qg→qg and gg→gg scattering are reproduced. However, in the TMD correlators in the tables 6 and 7 for the processes qq→gg and gg→qq one does not recover terms of the form P ,S| Tr ∂ (x). They can be included in (23) by adding diagram-dependent universality-breaking matrix elements which will not appear in integrated and weighted hadronic cross sections. However, it could be that they contribute to the second or higher transverse moments.

V. PARAMETRIZATIONS OF PARTON CORRELATORS
At leading twist the parametrizations of the different TMD quark correlators are given by and with similar parametrizations for the matrix elements πδΦ , etc. The quark distribution functions that appear in the parametrizations (25a) and (25b) are the familiar T -even and T -odd (respectively) quark distribution functions as measured in SIDIS. On the other hand, the quark distribution functions in (25c) and in the parametrization of δΦ G are process dependent. From the properties in (20) one finds that the functions δf 1 , δh 1T and δg 1L vanish upon p T -integration, for instance illustrated in Figure 2. Also the functions δh 1T , δh vanish, e.g., this assumption is sufficiently generic for our conclusions to be applicable for many hadronic processes, in particular to back-to-back dijet or photon-jet production in proton-proton scattering. By inserting the decompositions (15) and (23) of the TMD parton correlators into the expression for the unintegrated hadronic cross section in Eq. (11) the parton contribution to a hard 2→2 process becomes which forms the central result of this paper. Again it is implicitly implied that parton a is an (anti)quark, since if it were a gluon there would be two gluonic pole terms Γ G fΣ (f ) [g]b→cd and Γ G dΣ (d) [g]b→cd . In (29) both terms in the decomposition of the TMD correlator of (anti)quark a have been given explicitly, while only the first terms of the decompositions (15) and (23) were used for the other partons. The "+ · · · " contains the other possible combinations of the terms in those decompositions and also the contributions where parton a is a gluon.
The hard functions in expression (29) are the partonic and gluonic pole cross sections in (14a) and (14b) (in the collinear expansions of the hard functions, which have corrections at order O(1/s)). Hence, the expression in (29) demonstrates that by using the decompositions in (15) and (23) and by introducing the gluonic pole cross sections, also the unintegrated, unweighted hadronic cross section can be written as a product of soft TMD parton correlators and hard partonic functions that are separately and manifestly gauge invariant. After performing the traces these hard functions reduce to the partonic and gluonic pole cross sections calculated in Refs [17,19,28]. Hence, it follows that the gluonic pole cross sections that have been seen [17,19,27,28] to represent the hard partonic functions in weighted spin asymmetries already appear in the fully TMD cross sections. This conclusion is consistent with the work in Refs [20,21], where a TMD factorization formula for the quark-Sivers contribution to single transverse-spin asymmetries in dijet production is proposed with the gluonic pole cross sections of Refs [17,19] as hard functions. However, the work in Refs [20,21] limits to one-gluon exchange and as a result it obtains the distribution functions measured in SIDIS (i.e., transverse momentum dependent distribution functions with a future pointing Wilson line in the hadronic matrix elements defining them). In contrast, our expression (29) involves correlators with process dependent Wilson lines in the operator definitions and a factorized form with universal distribution functions would only be achieved if, regardless of the appearance of process dependent Wilson lines in their definitions, the TMD matrix elements in (29) are identical for all partonic channels and equal those in semi-inclusive deep inelastic scattering. In the present context this translates into a vanishing of all universality-breaking matrix elements, for which we see no reason. Indeed, universality has recently also been disputed in Ref. [22], where it is argued that a TMD factorization theorem for this process with universal distribution functions is not possible. This is confirmed by an explicit calculation including the exchange of two collinear gluons [25]. A recent extension [24] of the work in Refs [20,21] including two-gluon exchange also points to non-universality. Moreover, it shows that Refs [16,17,18,19], [20,21] and [22] are consistent to (at least) that order. We want to emphasize that for a full connection the role of soft factors, which have been neglected in the present study, should also be investigated.
With the parametrizations (25) and (28) inserted in (29), an expression for the hadronic cross section in terms of TMD parton distribution and fragmentation functions is obtained. After performing the traces these hard functions reduce to the partonic and gluonic pole cross sections encountered in Refs [17,19,27,28]. As an illustration we consider the contribution to (29) of an unpolarized quark in an unpolarized hadron, important for unintegrated spin averaged hadronic cross sections (summations over parton types are understood): Taking as a second example the contribution of an unpolarized quark in a transversally polarized hadron, important for unintegrated single-spin asymmetries, one finds The terms with only universal T -even functions will appear folded with partonic cross sections and the universal T -odd functions with gluonic pole cross sections. In addition, there are various universality-breaking functions that appear with partonic cross sections or gluonic pole cross sections. A shorter notation for some of the terms in these expressions could have been obtained by not extracting the universal T -even and T -odd parts of the distribution functions f 1 and f ⊥ 1T , as indicated by the underbraces in (30). In particular, due to these universality-breaking functions the gluonic pole cross sections also appear in the unweighted spin-averaged cross sections (30a) and the usual partonic cross sections also appear in the unweighted single-spin asymmetries (30b). However, in the light of the properties in (26)- (27) it is seen that terms with universality-breaking matrix elements do not contribute to the integrated and q T -weighted (where q T is as defined in section III) hadronic cross sections which are expressed in terms of the structures in (12) and (13): weighted: For back-to-back jet production in polarized proton-proton scattering (p ↑ p→jjX) this (in essence) reproduces the results of Refs [17,19,27], while for photon-jet production (p ↑ p→γjX) it reproduces the results in Ref. [28].
Only if the universality-breaking matrix elements vanish in the unintegrated, unweighted cross sections (29) and (30) does one also in the TMD case arrive at the situation with universal functions only. Otherwise, the non-universality of the unweighted processes is important. It will affect the results of experiments that try to look at explicit p Tdependence or that construct weighted cross sections involving convolutions of TMD functions which upon integration do not factorize into transverse moments, e.g. when looking at sin(φ h ±φ S ) , rather than P π⊥ sin(φ h ±φ S ) asymmetries in SIDIS to extract transversity or Sivers functions.

VII. SUMMARY
We have argued that the gluonic pole cross sections, the hard partonic scattering functions that are folded with the collinear parton distribution functions in weighted spin-asymmetries, also appear in unintegrated, unweighted hadronic cross sections. Assuming as a starting point that the hadronic cross section factorizes at the diagrammatic level in a hard partonic function and for each of the observed hadrons a soft parton correlator, we have shown that the transverse momentum dependent cross section can be written in terms of soft and hard functions that are separately manifestly gauge-invariant. The hard functions are the partonic cross sections or the gluonic pole cross sections. The latter do not show up in the integrated cross sections. The soft functions are the TMD parton distribution (and fragmentation) functions with Wilson lines on the light-front in their field theoretical operator definitions. These Wilson lines can be considered as the collective effect of gluon initial and final state interactions. Since they are process dependent, the TMD parton distributions are in general non-universal. By systematically separating the universality-breaking parts from the universal T -even and T -odd parts, we arrived at an expression that has soft parts multiplying the partonic and gluonic pole cross sections, as was also found in Refs [20,21]. However, in the gauge link approach taken here the gluonic pole cross section can also emerge in TMD spin-averaged processes and that ordinary partonic cross sections can also arise in TMD single-spin asymmetries. They appear with universalitybreaking functions that will vanish for the integrated and weighted processes considered in Refs [17,19,27,28]. They also vanish at the level of one-gluon radiation contributions, which corresponds to the order g term of the Wilson lines.
The non-universal terms are well-defined matrix elements. All universality-breaking matrix elements that are encountered at tree-level in proton-proton scattering with 2→2 partonic processes have been calculated and are given in the appendix of this paper. In particular, the process-dependent universality-breaking matrix elements δΦ, etc. disappear in the simple electroweak processes with underlying hard parts like qγ * →q and qq→γ * . We believe that the explicit identification of universality-breaking matrix elements is an important contribution to arrive at a unified picture of TMD factorization of hadronic scattering processes.