Negative dimensional approach for scalar two-loop three-point and three-loop two-point integrals

The study of quantum field theories in the perturbative regime can sometimes become a very challenging and arduous issue to deal with, especially when one needs to go on calculating multi-loop Feynman integrals. There are several techniques that have been developed and refined in the course of time, but usually the calculations are rather involved and analytic results are difficult to obtain. Depending on the approach one uses to tackle the problem, the obstacles to surmount can be more or less demanding. For example, if one uses the standard Feynman parametrization, the integral over momentum space is straightforward — in the sense that nowadays one can easily fint them out even tabulated in textbooks on quantum field theories — but the resulting parametric integrals that are left over become quite complicated and cumbersome to tackle, especially with increasing number of internal momenta in the loops. Of course, if one is clever enough, he/she may be able to overcome some of these hurdles and even perform really hard calculations [1,2]. There are, of course, other techniques which have been considered in the literature, e.g., integration by parts and Gegenbauer polynomial method in configuration space [9]; the Mellin-Barnes representation for massive propagators [10] to cite just a few of them [11], each one of those with its own strengths and weaknesses. On the other hand, there is this novel technique known as negative dimensional integration method (NDIM), first devised by Halliday and Ricotta [3]. It has as its starting point the principle of analytic continuation, it has the remarkable property of being equivalent to Grassmannian integration in (positive) D-dimensional space [4], it requires only the integration of Gaussian-like integrals and solving of linear algebraic equations and is suitable to deal with massless [5] or massive [6] diagrams in onand off-shell regimes [7] and even to carry out light-cone gauge integrals, with added troublesome gauge-dependent poles [8]. Our aim in this work is to demonstrate the simplicity of NDIM calculating some scalar three-point integrals, with five and six massless propagators, at two-loop level and some two-point three-loop integrals also in the massless case. The outline for our work is as follows: in section II we present a detailed calculation of a two-loop integral in the NDIM approach; in section III we write down the results for the remaining twoand three-loop scalar integrals, while in section IV we give our concluding remarks about this work.