One-loop renormalization of quark TMD in the light-cone gauge: CSS evolution

Abstract

Abstract We calculate the one-loop corrections to the quark TMD (focusing on the divergences) in the light-cone gauge using the background field formalism, testing different prescriptions for the extra singularity present in the light-cone gauge propagator. We use the pure rapidity regulator for rapidity divergences. The Collins-Soper-Sterman (CSS) evolution equations are indeed obtained from the one loop renormalization of the quark TMD with all the considered prescriptions. In this setup, in the case of the Mandelstam-Leibbrandt (ML) prescription, the double log contribution to the CSS resummation is found to come from the ghost-like zero-mode brought by the ML prescription, in the diagrams with a gluon propagator ending on the transverse part of the gauge link at infinity. By contrast, when using any prescription defined in terms of a single light-cone vector, such as the Principal Value prescription or retarded or advanced prescriptions, the CSS double log contribution is found to come entirely from the quark-to-quark ladder diagram.