Paul Mork | Searcharxiv

arXiv2026

Recursive construction of scalar one-loop integrals in dimensional regularisation

We derive a novel recursive structure for dimensionally regularised scalar one-loop Feynman integrals based on Schl\"afli's differential formula for hyperbolic simplices. The recursion relates the Laurent coefficients in the dimensional regulator $\varepsilon$ of an $N$-point integral to lower-order coefficients of integrals with additional external legs. The construction is seeded by the $\varepsilon=0$ contributions, which admit a geometric interpretation as volumes of simplices in hyperbolic space and are known in terms of multiple polylogarithms (MPLs). Iterating the recursion therefore provides a constructive algorithm for computing arbitrary orders in the $\varepsilon$-expansion of scalar one-loop integrals with arbitrary masses and kinematics, while remaining entirely within the class of MPLs. In particular, this establishes that all coefficients in the Laurent expansion of dimensionally regularised scalar one-loop integrals can be expressed in terms of MPLs. As a first application beyond existing results, we obtain an explicit closed MPL expression for the $\mathcal{O}(\varepsilon)$ coefficient of the scalar hexagon with arbitrary masses and off-shell Euclidean kinematics.

hep-th↗

arXiv2025

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.

hep-ph↗