Numerical NLO QCD calculations

TL;DR

This work develops a fully numerical approach to next-to-leading order QCD calculations for multi-jet final states by combining a comprehensive set of subtraction terms with a robust contour deformation of the loop integration. The method operates at the amplitude level, uses color-Flow and primitive amplitudes, and leverages recurrence relations to avoid explicit Feynman diagrams, enabling efficient computation for both massless and massive partons. Infrared and ultraviolet divergences are handled by explicit subtraction terms whose integrated pieces form an insertion operator that cancels poles against the real-emission contributions, while a carefully designed contour deformation ensures stable four-dimensional Monte Carlo integration. The framework is validated through checks on simple processes and stability studies of contour integration, demonstrating accurate, finite results and highlighting its potential for tackling complex NLO QCD calculations in LHC phenomenology and beyond.

Abstract

We present an algorithm for the numerical calculation of one-loop QCD amplitudes. The algorithm consists of subtraction terms, approximating the soft, collinear and ultraviolet divergences of one-loop amplitudes and a method to deform the integration contour for the loop integration into the complex space. The algorithm is formulated at the amplitude level and does not rely on Feynman graphs. Therefore all required ingredients can be calculated efficiently using recurrence relations. The algorithm applies to massless partons as well as to massive partons.

Figures (16)

• Figure 1: Examples of diagrams regarding left-moving and right-moving primitive amplitudes: Diagrams (a) contributes to the left-moving primitive amplitude, while diagram (b) contributes to the right-moving amplitude.
• Figure 2: The labelling of the momenta for a primitive one-loop amplitude. The arrows denote the momentum flow.
• Figure 3: The recurrence relation for the gluon current. In an off-shell current particle $n+1$ is kept off-shell. This allows to express an off-shell current with $n$ on-shell legs in terms of currents with fewer legs.
• Figure 4: The recurrence relation for the one-loop current in a toy model with a single field and a single three-valent vertex. The one loop currents are represented by an oval with a hole, tree-level currents are represented by an oval without a hole.
• Figure 5: The recurrence relation for the ultraviolet subtraction terms in a toy model with a single field and a single three-valent vertex. Objects with a cross represent ultraviolet subtraction terms.