Integrand reduction of one-loop scattering amplitudes through Laurent series expansion

TL;DR

This paper introduces a semi-analytic method for one-loop integrand reduction in dimensional regularization that leverages Laurent expansions of the integrand decomposition to determine master integral coefficients through a diagonal, corrected system. By performing the subtraction at the coefficient level rather than at the integrand level, the authors remove the need for 5-point coefficients and exclude spurious 4-point contributions, while enabling independent reconstruction of 3-, 2-, and 1-point functions. The approach relies on systematic polynomial division and one-dimensional asymptotic expansions across quadruple, triple, double, and single cuts, with universal correction terms computable a priori. The method is implemented in $\texttt{C++}$ and $\texttt{Mathematica}$, validated on higher-rank examples, and extended to cases where the numerator rank exceeds the number of denominators, signaling potential applicability beyond standard one-loop reductions.

Abstract

We present a semi-analytic method for the integrand reduction of one-loop amplitudes, based on the systematic application of the Laurent expansions to the integrand-decomposition. In the asymptotic limit, the coefficients of the master integrals are the solutions of a diagonal system of equations, properly corrected by counterterms whose parametric form is konwn a priori. The Laurent expansion of the integrand is implemented through polynomial division. The extension of the integrand-reduction to the case of numerators with rank larger than the number of propagators is discussed as well.