The massless non-adjacent double off-shell scalar box integral -- branch cut structure and all-order epsilon expansion

TL;DR

This paper extends the analytic understanding of a specific scalar box integral with two off-shell points, providing explicit formulas for its branch cut structure and all-order epsilon expansion using hypergeometric functions and polylogarithms.

Contribution

It generalizes previous results to the case of two non-adjacent off-shell points, deriving explicit formulas in terms of hypergeometric functions and single-valued polylogarithms.

Findings

Explicit all-order epsilon expansion in terms of single-valued polylogarithms.

Analytic splitting of real and imaginary parts of the integral.

Representation in terms of hypergeometric functions enabling detailed analysis.

Abstract

We generalize the result of our recent paper on the massless single off-shell scalar box integral to the case of two non-adjacent end points off the light cone. An analytic result in $d=4-2\varepsilon$ dimensions is established in terms of four Gauss hypergeometric ${_2}\mathrm{F}_1$ functions respectively their single-valued counterparts. This allows for an explicit splitting of real and imaginary parts, as well as an all-order $\varepsilon$-expansion in terms of single-valued polylogarithms.