Probing negative-dimensional integration: two-loop covariant vertex and one-loop light-cone integrals

Abstract

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)

Loop Integrals in Three Outstanding Gauges: Feynman, Light-Cone, and Coulomb

“…(The few exceptions as contemplated by theorems, such as Saalschutzian's, when one of the numerator parameters is negative, generally do not apply to our cases.) Despite this technical problem, NDIM has proven to be an excellent method [6][7][8][9] for computing loop integrals.…”

“…The important point to note [8] is that the dual light-like 4-vector n * µ is necessary in order to span the needed four-dimensional space [18,22]. Without this dual light-like 4-vector, that is, a theory with only onedegree violation of covariance, the calculations lead to unphysical results (i.e., they break causality) [6].…”

Feynman integrals with tensorial structure in the negative dimensional integration scheme

“…Observe that in the process of analytic continuation to our physical world (D > 0), exponents i, j are analytically continued to allow for negative values, whereas the exponent l must be left untouched, since, by definition l ≥ 0 in the original Feynman integral [7].…”

A Possible Way to Relate the "Covariantization" and the Negative Dimensional Integration Method in the Light-Cone Gauge

“…Now we are going to obtain the right answer as it was done in Ref. [6]. To do this, first note that when we defined the light-cone coordinates in Eq.1, there exists the dual of n µ , which we called m µ , and in light-cone computations, this vector always appears in the numerator of the integrands.…”

“…In the late 90's it was tested in the case of working in the light-cone gauge [6]. The best way to illustrate this method is through an example.…”