Functional equations for one-loop master integrals for heavy-quark production and Bhabha scattering

TL;DR

This paper applies a recently proposed method of functional equations to one-loop box integrals in heavy-quark production and Bhabha scattering, providing new representations and relationships useful for calculations.

Contribution

It introduces new functional equations for one-loop integrals, deriving hypergeometric representations and relationships between different integral configurations.

Findings

Functional equations relate integrals with different arguments.

New hypergeometric integral representations are derived.

Functional equations aid in analytic continuation and imaginary part calculations.

Abstract

The method for obtaining functional equations, recently proposed by one of the authors, is applied to one-loop box integrals needed in calculations of radiative corrections to heavy-quark production and Bhabha scattering. We present relationships between these integrals with different arguments and box integrals with all propagators being massless. It turns out that functional equations are rather useful for finding imaginary parts and performing analytic continuations of Feynman integrals. For the box master integral needed in Bhabha scattering, a new representation in terms of hypergeometric functions admitting one-fold integral representation is derived. The hypergeometric representation of a master integral for heavy-quark production follows from the functional equation.