Recursive construction of scalar one-loop integrals in dimensional regularisation

Abstract

We derive a novel recursive structure for dimensionally regularised scalar one-loop Feynman integrals based on Schläfli'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.