Renormalization, Wilson lines, and transverse-momentum-dependent parton-distribution functions

Abstract

We perform an analysis of transverse-momentum dependent parton distribution functions, making use of their renormalization properties in terms of their leading-order anomalous dimensions. We show that the appropriate Wilson line in the light-cone gauge, associated with such quantities, is a cusped one at light-cone infinity. To cancel the ensuing cusp anomalous dimension, we include in the definition of the transverse-momentum dependent parton distribution functions an additional soft counter term (gauge link) along that cusped transverse contour. We demonstrate that this is tantamount to an "intrinsic (Coulomb) phase", which accumulates the full gauge history of the color-charged particle.

Evolution of the helicity and transversity. Transverse-momentum-dependent parton distributions

“…[17,18] (see also the closely-related work of Cherednikov and Stefanis Refs. [19,20]). The relation between the approach of Collins and that of Echevarria, Idilbi, and Scimemi has been analyzed in Ref.…”

Non-dipolar Wilson links for transverse-momentum-dependent wave functions

“…The vertical Wilson lines connecting the longitudinal Wilson lines in eq. (2.1) at infinity do not contribute in covariant gauge [28]. In contrast to a space-like gauge vector for defining a TMD parton density in ref.…”

“…The point resides in the design of the associated Wilson links and the introduction of soft subtraction, so that rapidity divergences [24] and Wilson-line self-energy divergences are avoided [25]. As light-like Wilson lines are adopted in the un-subtracted TMD definition, rapidity divergences from radiative gluons collimated to the Wilson lines are produced [26][27][28][29][30]. As these rapidity divergences are regularized by rotating the Wilson lines away from the light cone [26] (a non-light-like axial gauge n · A = 0 with n 2 = 0 was chosen actually), the self-energy divergences attributed to the infinitely long dipolar Wilson lines [25] appear.…”

Transverse-Momentum-Dependent Parton Distributions at the Edge of the Lightcone

“…This jeopardizes the renormalizability of TMDs and calls for a certain generalized renormalization procedure to augment the insufficient dimensional regularization. In our works 19 , we worked out such a procedure that enabled us to obtain a well-defined and fully gaugeinvariant TMD PDF, free of undesirable divergences. This TMD PDF has calculable renormalization-group properties and obeys an evolution equation with respect to rapidity in the impact parameter space.…”

“…In this work, we present and discuss an operator definition of the quark TMD that embodies gauge invariance in terms of lightlike longitudinal and transverse gauge links with the appropriate behavior at lightcone infinity 11,19 .…”