Transverse-momentum parton densities: gauge links, divergences and soft factor
aa r X i v : . [ h e p - ph ] A ug November 22, 2018 13:43 WSPC - Proceedings Trim Size: 9in x 6in chst˙jlab˙040810 RUB-TPII-06/2010
TRANSVERSE-MOMENTUM PARTON DENSITIES:GAUGE LINKS, DIVERGENCES AND SOFT FACTOR ∗ I. O. CHEREDNIKOV ‡† INFN Cosenza, Universit ` a della CalabriaI-87036 Rende (CS), ItalyandBogoliubov Laboratory of Theoretical PhysicsJINR RU-141980 Dubna, Russia ‡ E-mail: [email protected]
N. G. STEFANIS § Institut f¨ur Theoretische Physik II,Ruhr-Universit¨at Bochum, D-44780 Bochum, Germany § E-mail: [email protected]
We discuss the state-of-the-art of the theory of transverse-momentum depen-dent parton densities (TMD)s, paying special attention to their renormalizationproperties, the structure of the gauge links in the operator definition, and therole of the soft factor in the factorization formula within the TMD approach tothe semi-inclusive processes. We argue that the use of the lightcone axial gaugeoffers certain advantages for a consistent definition of TMDs as compared tothe off-the-light-cone gauges, or covariant gauges with off-the-lightcone gaugelinks.
Keywords : Parton distribution functions, Wilson lines, renormalization.
1. Introduction
The distribution functions of partons (in what follows we consider onlyquark distributions), depending on the longitudinal components x , as wellas on the transverse components k ⊥ , of their momenta (hence TMD)s,accumulate useful information about the intrinsic motion of the hadron’sconstituents and enter as a nonperturbative input in the QCD approach ∗ Talk presented at Workshop on “Exclusive Reactions at High Momentum Transfer”,18-21 May 2010, TJNAF (Newport News, VA, USA) † Also at: ITPM, Moscow State University, Russia ovember 22, 2018 13:43 WSPC - Proceedings Trim Size: 9in x 6in chst˙jlab˙040810 to the semi-inclusive hadronic processes (see, e.g., Refs. [1–4]). The QCDfactorization formula for a semi-inclusive structure function is expected tohave the following symbolic form [5–7] F ( x B , z h , P h ⊥ , Q ) = X i e i · H ⊗ F D ⊗ F F ⊗ S , (1)where z h and P h ⊥ are the longitudinal and transverse fractions of the mo-mentum of the produced hadron, respectively. This expression contains thehard (perturbatively calculable) part H , the (nonperturbative) distributionand fragmentation functions F D and F F , and the soft part S . The latter isabsent in the collinear (fully inclusive) picture, and will be discussed below.However, several problems arise in attempting to formulate the TMD ap-proach in terms of the quantum field operators and their matrix elements:( i ) Extra (rapidity) divergences appear already at the one-loop level, whichinvalidate the standard renormalization procedure [3,8–10]. ( ii ) A muchmore complicated (compared to the collinear case) structure of the gaugelinks leads to the non-universality of distribution or fragmentation func-tions (see, e.g., Refs. [11–13]). ( iii ) Several counter-examples have beengiven showing that the straightforward factorization formula (1) may fail,at least in some specific situations [14,15]. ( iv ) The role and explicit ex-pression of the soft factor S can be different in different schemes. In whatfollows, we basically concentrate on the first and the last problem.
2. Divergences and renormalization properties of TMDs
The operator definition of the quark TMD (without the soft term) reads[5,6,9,10,16–21]˜ F i/h ( x, k ⊥ ) = 12 Z dξ − d ξ ⊥ π (2 π ) e − ik + ξ − + i k ⊥ ξ ⊥ × D h | ¯ ψ i ( ξ − , ξ ⊥ )[ ξ − , ξ ⊥ ; ∞ − , ξ ⊥ ] † [ n ] [ ∞ − , ξ ⊥ ; ∞ − , ∞ ⊥ ] † [ l ] × γ + [ ∞ − , ∞ ⊥ ; ∞ − , ⊥ ] [ l ] [ ∞ − , ⊥ ; 0 − , ⊥ ] [ n ] ψ i (0 − , ⊥ ) | h E . (2)This expression may be given a physical meaning, because it is formallygauge invariant. The gauge invariance is ensured by means of the path-ordered gauge links[ ∞ − , ξ ⊥ ; ξ − , ξ ⊥ ] [ n ] ≡ P exp (cid:20) ig Z ∞ dτ n − µ A µa t a ( ξ + n − τ ) (cid:21) , [ ∞ − , ∞ ⊥ ; ∞ − , ξ ⊥ ] [ l ] ≡ P exp (cid:20) ig Z ∞ dτ l · A a t a ( ξ ⊥ + l τ ) (cid:21) , (3) ovember 22, 2018 13:43 WSPC - Proceedings Trim Size: 9in x 6in chst˙jlab˙040810 where we distinguish between longitudinal (lightlike, n = 0) [ ... ] [ n ] andtransverse [ ... ] [ l ] links. (A generalized definition, which includes into theWilson lines the spin-dependent Pauli term F µν [ γ µ , γ ν ], was recentlyworked out in Ref. [22]).Of course, the above expressions have to be quantized, using, for in-stance, functional-derivative techniques. This means that the gluon poten-tial in the gauge link has to be Wick contracted with corresponding terms inthe interaction Lagrangian, accompanying the Heisenberg fermion (quark)field operators.At the tree-level, the “distribution of a quark in a quark” (here we con-sider only ultraviolet (UV) and rapidity divergences, which are independentof the particular hadronic state) is normalized as˜ F (0) q/q ( x, k ⊥ ) = 12 Z dξ − d ξ ⊥ π (2 π ) e − ik + ξ − + ik ⊥ · ξ ⊥ ×h p | ¯ ψ ( ξ − , ξ ⊥ ) γ + ψ (0 − , ⊥ ) | p i = δ (1 − x ) δ (2) ( k ⊥ ) , (4)and, formally, the integration over k ⊥ yields the usual collinear (integrated)PDF Z dk (2) ⊥ ˜ F (0) q/q ( x, k ⊥ ) = F (0) q/q ( x ) = δ (1 − x ) . (5)However, already in the calculation of the one-gluon contributions, oneencounters—besides the normal UV divergences—certain pathological sin-gularities. Namely, one has at the one-loop level the following singularterms:(1) Standard UV poles ∼ ε in the dimensional regularization: theycan be removed by the usual R − operation and are controlled byrenormalization-group evolution equations (analoguous to the DGLAPequation in the integrated case).(2) Pure rapidity divergences : they give rise to logarithmic and double-logarithmic terms of the form ∼ ln η , ln η . These terms, although theydepend on the additional rapidity parameter η [1,5,6,8], do not affectthe UV renormalization properties and can be safely resummed, e.g.,by means of the Collins-Soper equation.(3) Pathological overlapping divergences : they contain the UV and rapiditypoles simultaneously ∼ ε ln η and are considered to be highly problem-atic . The reason is that they prevent the removal of all UV-singularitiesby the standard R − procedure. Therefore, a special generalized renor-malization procedure is needed in order to take care of those terms andenable the construction of well-defined renormalizable TMDs. ovember 22, 2018 13:43 WSPC - Proceedings Trim Size: 9in x 6in chst˙jlab˙040810 It is interesting to note that working in the lightcone gauge with theMandelstam-Leibbrandt prescription [23,24], one doesn’t get any overlap-ping divergences, at least in the leading loop order [25]. The renormalizationproperties of operators and matrix elements containing Wilson lines andloops with or without obstructions have been extensively studied in varioussituations—see, e.g., Refs. [26–30]. The specifics of the TMD consist in thefact that, though the fermion fields are separated by a spacelike distance,the gauge links lay on pure lightlike rays, or on the 2 D − transverse planeat lightcone infinity.The analysis of the one-loop anomalous dimension of the TMD, given byEq. (2), shows that the contribution of the overlapping singularity is nothingelse, but the cusp anomalous dimension [29]. Therefore, in order to renor-malize expression (2), one can apply, apart from the standard R − operation,an additional renormalization factor, which depends on the cusp angle andcan be written as a vacuum matrix element of the Wilson lines evaluatedalong a special contour with an obstruction (cusp); viz., Z − χ = * (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P exp " ig Z χ dζ µ t a A aµ ( ζ ) + . (6)The UV singularity of this factor cancels the cusp anomalous dimensionfrom the overlapping divergence, thus rendering the re-defined TMD (2)renormalizable [9,10]. Therefore, the generalized renormalization procedurefor the TMD can be formulated as˜ F ren ( x, k ⊥ , χ, ... ) = Z R · Z χ · ˜ F ( x, k ⊥ , χ, ... ) , (7)where Z R is the usual renormalization constant, while Z χ can be includedin the definition of the TMD itself. In that case, it is treated as a “softfactor”: Z χ ≡ [Soft Factor] , (8)which is defined as[Soft Factor] = h | P e ig R C χ dζ µ t a A aµ ( ζ ) · P − e − ig R C′ χ dζ µ t a A aµ ( ξ + ζ ) | i , (9)where the contours C and C ′ are explicitly given in Ref. [10]. Therefore, the ovember 22, 2018 13:43 WSPC - Proceedings Trim Size: 9in x 6in chst˙jlab˙040810 generalized definition of the TMD reads [9,10,25] F i/h ( x, k ⊥ ) = 12 Z dξ − d ξ ⊥ π (2 π ) e − ik + ξ − + i k ⊥ · ξ ⊥ × D h (cid:12)(cid:12)(cid:12) ¯ ψ i ( ξ − , ξ ⊥ )[ ξ − , ξ ⊥ ; ∞ − , ξ ⊥ ] † [ n ] [ ∞ − , ξ ⊥ ; ∞ − , ∞ ⊥ ] † [ l ] × γ + [ ∞ − , ∞ ⊥ ; ∞ − , ⊥ ] [ l ] [ ∞ − , ⊥ ; 0 − , ⊥ ] [ n ] ψ i (0 − , ⊥ ) (cid:12)(cid:12)(cid:12) h E × [Soft Factor] . (10)This function is free (at least, at the one-loop order) of pathologicaldivergences and is a well-defined renormalizable quantity.
3. Factorization and role of the soft factor
The soft factor, introduced above, naturally enters in the factorization for-mula (1): [Soft Factor] = S . However, its interpretation is twofold.On the one hand, it formally looks similar to the “intrinsic” Coulombphase found by Jakob and Stefanis [31] in QED for Mandelstam chargedfields involving a gauge contour which is a timelike straight line. The name“intrinsic” derives from the fact that this phase is different from zero evenin the absence of external charge distributions. Its origin was ascribed in[31] to the long-range interaction of the charged particle with its oppositelycharged counterpart that was removed “behind the moon” after their pri-mordial separation. Note that the existence of a balancing charge “behindthe moon” was postulated before by several authors—see [31] for relatedreferences—in an attempt to restore the Lorentz covariance of the chargedsector of QED. This phase is acquired during the parallel transport of thecharged field along a timelike straight line from infinity to the point of in-teraction with the photon field and is absent in the local approach, i.e., forlocal charged fields joined by a connector. It is different from zero only fora Mandelstam field with its own gauge contour attached to it and keepstrack of its full history since its primordial creation. Keep in mind that theconnector is introduced ad hoc in order to restore gauge invariance and isnot part of the QCD Lagrangian. In contrast, when one associates a distinctcontour with each quark field, one, actually, implies that these Mandelstamfield variables should also enter the QCD Lagrangian (see [31] for more de-tails). However, a consistent formulation of such a theory for QCD is stilllacking and not without complications of its own.The analogy to our case is the following. First, formally adopting adirect contour for the gauge-invariant formulation of the TMD in the light-cone gauge, the connector gauge link does not contribute any anomalous ovember 22, 2018 13:43 WSPC - Proceedings Trim Size: 9in x 6in chst˙jlab˙040810 dimension—except at the endpoints; this anomalous dimension being, how-ever, irrelevant for the issue at stake. Hence, there is no intrinsic Coulombphase in that case. Second, splitting the contour and associating eachbranch to a quark field, transforms it into a Mandelstam field and, as aresult, adding together all gluon radiative corrections at the one-loop or-der, a η -dependent term survives that gives rise to an additional anoma-lous dimension. We have shown that this extra anomalous dimension canbe viewed as originating from a contour with a discontinuity in the four-velocity ˙ x ( σ ) at light-cone infinity—a cusp obstruction.Classically, it is irrelevant how the two distinct contours are joined, i.e.,smoothly or by a sharp bend. But switching on gluon quantum corrections,the renormalization effect on the junction point reveals that the contours arenot smoothly connected, but go instead through a cusp [10]. Here, we have asecond analogy to the QED case discussed above. Similarly to the “particlebehind the moon”, this cusp-like junction point is “hidden” and manifestsitself only through the path-dependent phase after renormalization.In that case, the soft factor looks like an intrinsic property of the gauge-invariant operators containing the fermion fields and must be taken into ac-count in order to construct consistently the gauge-invariant renormalizabletwo-particle matrix elements.On the other hand, the soft factor appears as the result of the separationthe “soft” contributions from the one-loop graphs [7]. In this situation it isneeded in order to avoid the double-counting in the factorization formula(1). In general, these two soft factors might be different. In particular, ithas been shown in Ref. [32] that the anomalous dimensions of the TMDwithin different subtraction schemes of the soft factors are different. Therelationship between these frameworks will be studied elsewhere.
4. Conclusions
The above mentioned results have been obtained by using the lightconeaxial gauge with a proper regularization of the gluon propagator [10,25].The extra rapidity divergences can be treated by different methods, e.g.,one may shift the gauge links off the lightcone, or use, alternatively, theoff-the-lightcone axial gauge [5–7]. In these cases, the additional rapidityvariable parameterizes the deviation of the gauge links from the light raysin terms of the axial-gauge fixing vector. An advantage of the “pure light-cone” frameworks is the more straightforward physical interpretation of thefactorization and the role of the collinear Wilson lines in the definition ofTMDs, as well as the direct relationship between the (unintegrated) TMDs ovember 22, 2018 13:43 WSPC - Proceedings Trim Size: 9in x 6in chst˙jlab˙040810 and the (integrated) collinear PDFs: one can get the collinear PDF, satisfy-ing the DGLAP evolution equation, by simple k ⊥ -integration. In contrast,the “off-the-lightcone” frameworks don’t allow us to perform such a proce-dure, so that more sophisticated methods must be invented. The completeproof of the QCD factorization, within the TMD approach (in particularwithin the “pure lightcone” scheme), as well as the clarification of the roleplayed by the soft factors in different approaches, are still lacking. References
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