Analytic results for one-loop integrals in dimensional regularisation

We present a method to obtain analytic results in terms of multiple polylogarithms for one-loop triangle, box and pentagon integrals depending on an arbitrary number of scales and to any desired order in the Laurent expansion in the dimensional regulator $\varepsilon$. Our method leverages the fact that for $\varepsilon=0$ one-loop integrals compute volumes of simplices in hyperbolic spaces, which can always be evaluated in terms of polylogarithms using an algorithm recently introduced in pure mathematics. The higher orders in $\varepsilon$ can then be expressed as a one-fold integral involving the result for $\varepsilon=0$. Remarkably, we find that for up to five external legs, all integrals can be evaluated algorithmically in terms of polylogarithms using direct integration techniques, which, in particular, requires us to rationalise all appearing square roots. We also discuss how we can use the connection to hyperbolic geometry to perform the analytic continuation from the Euclidean region to other kinematic regions.

💡 Research Summary

The paper presents a systematic method for obtaining analytic expressions of one‑loop scalar integrals with up to five external legs (triangles, boxes and pentagons) in dimensional regularisation, expressed to arbitrary order in the Laurent expansion around the dimensional regulator ε. The key observation is that for ε = 0 the Feynman‑parameter representation of a one‑loop integral coincides with the volume of a simplex in hyperbolic space H^{N‑1}, where N is the number of propagators. This geometric interpretation allows the authors to use recent pure‑mathematics algorithms that evaluate hyperbolic simplex volumes in terms of multiple polylogarithms (MPLs).

The authors first review the connection between the Gram matrix Q_{ij} built from external momenta and masses and the hyperbolic simplex’s Gram matrix. They then describe an algorithm that recursively decomposes any simplex into (N‑1)! orthoschemes (right‑angled simplices). For each orthoscheme, explicit volume formulas are known: in H^3 the volume reduces to dilogarithms, in H^5 to trilogarithms, and for general even N the volume can be expressed as MPLs of weight ≤ N‑1 using the algorithm of Ref.