A Probabilistic Angle on One Loop Scalar Integrals

TL;DR

This paper introduces a probabilistic approach to evaluate one-loop scalar integrals, deriving recurrence relations and analytical expressions, simplifying calculations without hypergeometric functions, and demonstrating its application to tensor integral reduction.

Contribution

It presents a novel probabilistic framework for scalar integrals, enabling recurrence relations and analytical solutions in common cases, including tensor integral reduction.

Findings

Derived integral recurrence relations and exact expressions.

Established epsilon expansions relating different dimensions.

Reduced tensor integrals of rank 2 with N=5.

Abstract

Recasting the $N$-point one loop scalar integral as a probabilistic problem, allows the derivation of integral recurrence relations as well as exact analytical expressions in the most common cases. $\epsilon$ expansions are derived by writing a formula that relates an $N$-point function in decimal dimension to an $N$-point function in integer dimension. As an example, we give relations for the massive 5-point function in dimension $n=4-2\epsilon$, $n=6-2\epsilon$. The reduction of tensor integrals of rank 2 with $N=5$ is achieved showing the method's potential. Hypergeometric functions are not needed but only integration of arcsine function whose analytical continuation is well known.