Renormalization-group properties of transverse-momentum dependent parton distribution functions in the light-cone gauge with the Mandelstam-Leibbrandt prescription

TL;DR

This paper analyzes the renormalization-group properties of transverse-momentum dependent parton distribution functions (TMD PDFs) in the light-cone gauge using the Mandelstam-Leibbrandt prescription. It derives the transverse gauge field at light-cone infinity, then computes the leading-order anomalous dimension, demonstrating the absence of cusp-induced rapidity divergences and showing the soft factor reduces to unity at this order. The results show the ML-gauge yields the same UV structure as covariant gauges, implying standard TMD evolution without contour-obstruction artifacts. This simplifies the factorized treatment of SIDIS processes in ML-gauge and clarifies the role of soft factors in TMD definitions at one loop.

Abstract

The renormalization-group properties of transverse-momentum dependent parton distribution functions in the light-cone gauge with the Mandelstam-Leibbrandt prescription for the gluon propagator are addressed. An expression for the transverse component of the gauge field at light-cone infinity, which plays a crucial role in the description of the final-/initial-state interactions in the light-cone axial gauge, is obtained. The leading-order anomalous dimension is calculated in this gauge and the relation to the results obtained in other gauges is worked out. It is shown that, using the Mandelstam-Leibbrandt prescription, the ensuing anomalous dimension does not receive contributions from extra rapidity divergences related to a cusped junction point of the Wilson lines.

Figures (4)

• Figure 1: Integration contour and poles in the $( \hbox{\bf Re} \ q^0, {\textbf{Im}{}} \ q^0)$ plane: the poles of the gluon propagator using the ML-prescription (position 1) and those in a covariant gauge (position 2) belong to the same, i.e., second and fourth, quadrants. This is in contrast to the poles pertaining to the principal-value prescription (position 3). The Wick rotation can be performed without changing the position of the poles.
• Figure 2: The integration contour associated with the additional soft counter term.
• Figure 3: Virtual one-loop gluon contributions (curly lines) to the UV-divergences of the TMD PDF in the light-cone gauge---graphs (a) and (b). The graphs (c) and (d) are corresponding contributions originating from the soft factor $R$. Double lines denote gauge links. The vertical ones represent the transverse gauge links. The Hermitian conjugated diagrams are not shown.
• Figure 4: Real gluon contributions (curly lines) to the TMD PDF in the light-cone gauge using the ML pole prescription ("ML gauge"). Double lines denote gauge links. The Hermitian conjugated diagrams are not shown.