Reduction of One-loop Tensor Form-Factors to Scalar Integrals: A General Scheme

TL;DR

This paper tackles the problem of reducing one-loop tensor integrals to scalar integrals in all regions of parameter space, including points where Gram determinants vanish. It generalizes the Passarino–Veltman scheme by introducing new relations that hold at ${\cal D}=0$, and develops explicit reductions for $C_{ij}$ and $D_{ij}$ form factors to scalar integrals like $C_0$ and $B_0$, often via linear systems and cofactor techniques. The authors also derive derivatives of $C_0$ with respect to masses, provide an example with the electron $g-2$ to illustrate the method, and compare with prior approaches, clarifying why spurious divergences cancel in physical results. The work enables analytic, covariant reductions across parameter space and sets the stage for automation, with potential extensions to higher-point functions and two-loop reductions.

Abstract

A general method for reducing tensor form factors, that appear in one-loop calculations in dimensional regularization, to scalar integrals is presented. The method is an extension of the reduction scheme introduced by Passarino and Veltman and is applicable in all regions of parameter space including those where kinematic Gram determinant vanishes. New relations between the the form factors that valid for vanishing Gram determinant play a key role in the extended scheme.