All-Mass $n$-gon Integrals in $n$ Dimensions

TL;DR

This paper establishes a geometric framework linking one-loop Feynman integrals with hyperbolic simplicial geometry in exactly n dimensions, providing explicit dilogarithmic formulas and symbols for all-mass integrals.

Contribution

It introduces a geometric approach to all-mass n-particle integrals in n dimensions, deriving explicit dilogarithmic expressions and symbols using hyperbolic geometry and classical theorems.

Findings

Derived dilogarithmic formulas for all-mass box and pentagon integrals.

Connected Feynman integrals to hyperbolic simplicial geometry and dual conformal symmetry.

Provided a unified geometric perspective for these integrals across different space-time signatures.

Abstract

We explore the correspondence between one-loop Feynman integrals and (hyperbolic) simplicial geometry to describe the "all-mass" case: integrals with generic external and internal masses. Specifically, we focus on $n$-particle integrals in exactly $n$ space-time dimensions, as these integrals have particularly nice geometric properties and respect a dual conformal symmetry. In four dimensions, we leverage this geometric connection to give a concise dilogarithmic expression for the all-mass box in terms of the Murakami-Yano formula. In five dimensions, we use a generalized Gauss-Bonnet theorem to derive a similar dilogarithmic expression for the all-mass pentagon. We also use the Schl\"afli formula to write down the symbol of these integrals for all $n$. Finally, we discuss how the geometry behind these formulas depends on space-time signature, and we gather together many results related to these integrals from the mathematics and physics literature.