Generalized Mandelstam-Leibbrandt regularization

TL;DR

This work extends the Mandelstam-Leibbrandt regularization to algebraic non-covariant gauges outside the light cone, using a scale-symmetry plus regularity approach to obtain closed-form, dimensional-regularized loop integrals that preserve naive power counting and gauge invariance. A central role is played by a null vector $F_\mu$ fixed by Lorentz symmetry and a consistency condition $F\cdot F=0$, enabling ML-consistent single- and multi-pole integrals whose light-cone limit reproduces known results. The authors derive a general recurrence framework for higher-order spurious poles and implement a FORM-based method to compute the resulting expressions, providing a practical tool for gauge-theory calculations and potential applications in Very Special Relativity and non-local models. The closed-form results unify pole and finite parts with the generalized ML prescription and offer robust checks against existing literature, while clarifying the mathematical structure of ML regularization beyond the light cone.

Abstract

Algebraic non-covariant gauges are used often in string theory, Chern-Simons theory, gravitation and gauge theories. Loop integrals, however, have spurious singularities that need to be regularized. The most popular and consistent regularization is the Mandelstam- Leibbrandt(ML) prescription. This paper extends the ML prescription outside the light cone. It shares all the properties of light-cone ML regularization: It preserves naive power counting and gauge invariance. Moreover, using dimensional regularization(DR), we get a closed form for the basic integrals, including divergent and finite pieces. These results simplify calculations in gauge theories and open new avenues for applications in non-local models.