Analytic results for one-loop integrals in dimensional regularisation

TL;DR

This work develops a comprehensive framework to obtain analytic $\varepsilon$-expanded results for one-loop triangle, box, and pentagon integrals with arbitrary scales in a $D= d-2\varepsilon$ dimensional setting. Central to the method is the mapping of $\varepsilon=0$ integrals to volumes of hyperbolic simplices, which are computable via multiple polylogarithms, followed by a one-fold auxiliary-mass integration to generate higher-order terms in $\varepsilon$. The authors demonstrate algorithmic expressions in terms of MPLs for up to five external legs and provide explicit constructions for triangle, box, and pentagon cases, including detailed analytic continuation across kinematic regions. The approach unifies geometric (hyperbolic-volume) and analytic (MPL) techniques, enabling precise, cross-checked results relevant for NLO and beyond in dimensional regularisation. The results have potential for broad applicability to higher-point one-loop integrals and inform future extensions to more legs and complex mass configurations.

Abstract

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.

Figures (6)

• Figure 1: Dissecting a three-simplex into six orthoschemes in the Poincaré ball model.
• Figure 2: Depiction of the projective dissection with point $P'_\infty$ in the centre of the simplex. First the original simplex $\mathcal{S}$ is dissected into four simplices $\mathcal{S}_i$ (blue lines). We illustrate on the example of the simplex at the bottom how to further dissect this simplex into orthoschemes.
• Figure 3: The different kinematic regions in $(\tilde{\mathcal{Q}}_{2,2},\tilde{\mathcal{Q}}_{3,3})$-space relevant for the triangle integral.
• Figure 4: (left) Contour $\gamma(t)=(\tilde{\mathcal{Q}}_{2,2}(t),\tilde{\mathcal{Q}}_{3,3}(t))$ in the normalized $\tilde{\mathcal{Q}}_{2,2}$--$\tilde{\mathcal{Q}}_{3,3}$ plane used for analytic continuation. (right) Value of $I_3^4(\mathcal{Q}_{1,1},\mathcal{Q}_{1,1}\tilde{\mathcal{Q}}_{2,2}(t),\mathcal{Q}_{1,1}\tilde{\mathcal{Q}}_{3,3}(t))$ along the contour, plotted versus the curve parameter $t$. The solid lines are obtained from our analytic result (real: blue, imaginary: red), while the crosses represent numerical results from pySecDec.
• Figure 5: Comparison of the $\mathcal{O}(\varepsilon^1)$ contribution from $I_3^{4-2\varepsilon}(\mathcal{Q}(t))$ with the numerical results from pySecDec using the same conventions as in figure \ref{['fig:analytic_continuation_tri_eps=0']}.