A Probabilistic Angle on One Loop Scalar Integrals

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.