Negative dimensional integration method (NDIM) seems to be a very promising technique for evaluating massless and/or massive Feynman diagrams. It is unique in the sense that the method gives solutions in different regions of external momenta simultaneously. Moreover, it is a technique whereby the difficulties associated with performing parametric integrals in the standard approach are transferred to a simpler solving of a system of linear algebraic equations, thanks to the polynomial character of the relevant integrands. We employ this method to evaluate a scalar integral for a massless two-loop threepoint vertex with all the external legs off-shell, and consider several special cases for it, yielding results, even for distinct simpler diagrams. We also consider the possibility of NDIM in non-covariant gauges such as the light-cone gauge and do some illustrative calculations, showing that for one-degree violation of covariance (i.e., one external, gauge-breaking, light-like vector n µ ) the ensuing results are concordant with the ones obtained via either the usual dimensional regularization technique, or the use of principal value prescription for the gauge dependent pole, while for two-degree violation of covariancei.e., two external, light-like vectors n µ , the gauge-breaking one, and (its dual)

We will pursue a way of building up an algebraic structure that involves, in a mathematical abstract way, the well known Grassmann variables. The problem arises when we tried to understand the grassmannian polynomial expansion on the scope of ring theory. The formalization of this kind of variables and its properties will help us to have a better idea of some algebraic structures and the way they are implemented in models concerning theoretical physics .

In this work we present a possible way to relate the method of covariantizing the gauge dependent pole and the negative dimensional integration method for computing Feynman integrals pertinent to the light-cone gauge fields. Both techniques are applicable to the algebraic light-cone gauge and dispense with prescriptions to treat the characteristic poles.

In the usual and current understanding of planar gauge choices for Abelian and non Abelian gauge fields, the external defining vector n µ can either be space-like (n 2 < 0) or time-like (n 2 > 0) but not light-like (n 2 = 0). In this work we propose a light-like planar gauge that consists in defining a modified gauge-fixing term, L GF , whose main characteristic is a two-degree violation of Lorentz covariance arising from the fact that four-dimensional space-time spanned entirely by null vectors as basis necessitates two light-like vectors, namely n µ and its dual m µ , with n 2 = m 2 = 0, n·m = 0, say, e.g. normalized to n · m = 2.

Since the very beginning of it, perhaps the subtlest of all gauges is the light-cone gauge, for its implementation leads to characteristic singularities that require some kind of special prescription to handle them in a proper and consistent manner. The best known of these prescriptions is the Mandelstam–Leibbrandt one. In this work we revisit it showing that its status as a mere prescription is not appropriate but rather that its origin can be traced back to fundamental physical properties such as causality and covariantization methods.