Analytical Results for Dimensionally Regularized Massless On-shell Double Boxes with Arbitrary Indices and Numerators

TL;DR

This paper develops an algorithm for the analytical evaluation of complex massless on-shell double box Feynman diagrams with arbitrary numerators and propagator powers, expressing results in terms of polylogarithms.

Contribution

It introduces a method to reduce double box diagrams to master integrals and explicitly evaluates them using polylogarithms, advancing analytical techniques in quantum field theory.

Findings

Explicit analytical expressions for master double box integrals in terms of polylogarithms.

Reduction of complex diagrams to a linear combination of simpler master diagrams.

Application of differential relations to express complex master integrals through simpler ones.

Abstract

We present an algorithm for the analytical evaluation of dimensionally regularized massless on-shell double box Feynman diagrams with arbitrary polynomials in numerators and general integer powers of propagators. Recurrence relations following from integration by parts are solved explicitly and any given double box diagram is expressed as a linear combination of two master double boxes and a family of simpler diagrams. The first master double box corresponds to all powers of the propagators equal to one and no numerators, and the second master double box differs from the first one by the second power of the middle propagator. By use of differential relations, the second master double box is expressed through the first one up to a similar linear combination of simpler double boxes so that the analytical evaluation of the first master double box provides explicit analytical results, in terms of polylogarithms $\Li{a}{-t/s}$, up to $a=4$, and generalized polylogarithms $S_{a,b}(-t/s)$, with $a=1,2$ and $b=2$, dependent on the Mandelstam variables $s$ and $t$, for an arbitrary diagram under consideration.