General subtraction method for numerical calculation of one-loop QCD matrix elements

TL;DR

This work introduces a subtraction scheme to enable numerical evaluation of arbitrary one-loop QCD amplitudes with many external legs in four dimensions. By constructing local UV, soft, and collinear counterterms at the integrand level and summing infrared pieces analytically over graphs, the method reproduces the known MSbar ultraviolet structure and the standard infrared pole pattern that cancels against real emission. The approach preserves the original loop integrand except for the subtractions, and provides explicit expressions for the infrared operator $\mathbf{I}^V(\epsilon)$ that governs the pole structure. Together with a consistent LSZ-based treatment of external legs, the scheme delivers a practical path to automated NLO calculations for complex multi-leg processes, reducing reliance on fully analytic loop integrals. The framework thus expands the scope of problems accessible to numerical NLO QCD calculations while maintaining theoretical consistency with established renormalization and factorization properties.

Abstract

We present a subtraction scheme for eliminating the ultraviolet, soft, and collinear divergences in the numerical calculation of an arbitrary one-loop QCD amplitude with an arbitrary number of external legs. The subtractions consist of local counter terms in the space of the four-dimensional loop momentum. The ultraviolet subtraction terms reproduce MSbar renormalization. The key point in the method for the soft and collinear subtractions is that, although the subtraction terms are defined graph-by-graph and the matrix element is also calculated graph-by-graph, the sum over graphs of the integral of each the subtraction term can be evaluated analytically and provides the well known simple pole structure that arises from subtractions from real emission graphs, but with the opposite sign.