An algebraic/numerical formalism for one-loop multi-leg amplitudes

TL;DR

The paper introduces a universal algebraic/numerical framework for computing one-loop N-point amplitudes with arbitrary external legs and masses. It develops a tensor reduction by subtraction that isolates IR divergences into simple triangle integrals, proves that higher-dimensional integrals are unnecessary for N≥5 in 4D, and builds a Gram-determinant–free form-factor basis using Feynman parameter integrals with nontrivial numerators. Two complementary evaluation routes are provided: an analytic/basis approach fast in most of phase space and a robust numerical contour-deformation method that remains stable near exceptional kinematics, enabling largely automated NLO calculations. A complete set of form factors for N=3,4,5 is given, along with a practical recipe for implementation and numerous consistency relations to aid algebraic simplification and numerical stability. The framework is designed to be readily integrated into automated tools for multi-leg, massive and massless processes, significantly advancing NLO precision in complex collider phenomenology.

Abstract

We present a formalism for the calculation of multi-particle one-loop amplitudes, valid for an arbitrary number N of external legs, and for massive as well as massless particles. A new method for the tensor reduction is suggested which naturally isolates infrared divergences by construction. We prove that for N>4, higher dimensional integrals can be avoided. We derive many useful relations which allow for algebraic simplifications of one-loop amplitudes. We introduce a form factor representation of tensor integrals which contains no inverse Gram determinants by choosing a convenient set of basis integrals. For the evaluation of these basis integrals we propose two methods: An evaluation based on the analytical representation, which is fast and accurate away from exceptional kinematical configurations, and a robust numerical one, based on multi-dimensional contour deformation. The formalism can be implemented straightforwardly into a computer program to calculate next-to-leading order corrections to multi-particle processes in a largely automated way.

Figures (7)

• Figure 1: General $N$-point one-loop graph with momentum and propagator labelling.
• Figure 2: Graphical representation of pinch integrals. Each topology defines an ordered set $S$. The two diagrams correspond to $N=8$, $M=4$, $S=\{1,3,6,8\}$ (left), and $N=6$, $M=4$, $S=\{1,3,5,6\}$ (right).
• Figure 3: Real and imaginary parts of the basis integrals $I_4^6(1)$ and $I_4^6(z_4)$, plotted versus the parameter $x$ which interpolates between exceptional and non-exceptional kinematics, as explained in the text. The solid line stems from the numerical implementation, the dashed curves show the numerical behaviour of the algebraic representation.
• Figure 4: Same as Fig. \ref{['fignum3']} but for the basis integrals $I_4^6(z_3z_4)$ and $I_4^6(z_3z_4^2)$.
• Figure 5: Reduction of $N$-point tensor integrals to basis integrals. The indicated equations can be implemented directly into an algebraic computer program. $I_N^n(1|j_1|j_1,j_2|j_1,j_2,j_3)$ denotes the integral $I^n_N$ with zero, one, two or three Feynman parameters in the numerator.