Scalar One-Loop Integrals using the Negative-Dimension Approach

TL;DR

This paper develops a negative-dimension approach to evaluate massive one-loop integrals, providing a unified hypergeometric function representation for all kinematic regions, including new explicit formulas for complex vertex integrals.

Contribution

It introduces a general framework for negative-dimension analysis of one-loop integrals, enabling simultaneous solutions across kinematic regions and deriving new explicit hypergeometric representations.

Findings

Unified hypergeometric representation for all kinematic regions

Recovery of known results for standard integrals

New explicit formulas for complex vertex integrals

Abstract

We study massive one-loop integrals by analytically continuing the Feynman integral to negative dimensions as advocated by Halliday and Ricotta and developed by Suzuki and Schmidt. We consider n-point one-loop integrals with arbitrary powers of propagators in general dimension D. For integrals with m mass scales and q external momentum scales, we construct a template solution valid for all n which allows us to obtain a representation of the graph in terms of a finite sum of generalised hypergeometric functions with m+q-1 variables. All solutions for all possible kinematic regions are given simultaneously, allowing the investigation of different ranges of variation of mass and momentum scales. As a first step, we develop the general framework and apply it to massive bubble and vertex integrals. Of course many of these integrals are well known and we show that the known results are recovered. To give a concrete new result, we present expressions for the general vertex integral with one off-shell leg and two internal masses in terms of hypergeometric functions of two variables that converge in the appropriate kinematic regions. The kinematic singularity structure of this graph is sufficiently complex to give insight into how the negative-dimension method operates and gives some hope that more complicated graphs can also be evaluated.