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

TL;DR

This paper explores the negative dimensional approach to evaluate complex multi-loop scalar integrals in quantum field theory, providing new analytical results for specific configurations and comparing them with existing literature.

Contribution

It introduces the application of negative dimensional integration to compute two-loop three-point and three-loop two-point scalar integrals with arbitrary propagator exponents.

Findings

Derived analytical expressions for specific two-loop three-point integrals.

Computed three three-loop two-point scalar integrals.

Validated results against existing literature.

Abstract

The well-known $D$-dimensional Feynman integrals were shown, by Halliday and Ricotta, to be capable of undergoing analytic continuation into the domain of negative values for the dimension of space-time. Furthermore, this could be identified with Grassmannian integration in positive dimensions. From this possibility follows the concept of negative dimensional integration for loop integrals in field theories. Using this technique, we evaluate three two-loop three-point scalar integrals, with five and six massless propagators, with specific external kinematic configurations (two legs on-shell), and four three-loop two-point scalar integrals. These results are given for arbitrary exponents of propagators and dimension, in Euclidean space, and the particular cases compared to results published in the literature.