Application of the negative-dimension approach to massless scalar box integrals

TL;DR

This paper introduces a novel negative-dimension approach to evaluate massless one-loop box integrals, providing finite hypergeometric function representations that facilitate calculations of complex multi-loop Feynman diagrams.

Contribution

It develops a negative-dimension method for massless box integrals, yielding explicit hypergeometric and polylogarithmic representations for complex multi-loop integrals.

Findings

Finite hypergeometric function representations for box integrals with up to three scales.

Explicit formulas for two-loop box graphs with one off-shell leg.

Polylogarithmic expressions in the on-shell case.

Abstract

We study massless one-loop box integrals by treating the number of space-time dimensions D as a negative integer. We consider integrals with up to three kinematic scales (s, t and either zero or one off-shell legs) and with arbitrary powers of propagators. For box integrals with q kinematic scales (where q=2 or 3) we immediately obtain a representation of the graph in terms of a finite sum of generalised hypergeometric functions with q-1 variables, valid for general D. Because the power each propagator is raised to is treated as a parameter, these general expressions are useful in evaluating certain types of two-loop box integrals which are one-loop insertions to one-loop box graphs. We present general expressions for this particular class of two-loop graphs with one off-shell leg, and give explicit representations in terms of polylogarithms in the on-shell case.