Numerical evaluation of virtual corrections to multi-jet production in massless QCD

TL;DR

This work presents NJet, a C++ library for numerical evaluation of one-loop virtual corrections in massless QCD for multi-jet production. It extends NGluon with generalized unitarity to handle arbitrary fermion–gluon primitives, constructs full colour-decomposed amplitudes, and provides squared matrix elements up to 7 external partons, enabling five-jet NLO predictions. The paper details implementation, including a cache for tree amplitudes, FDH scheme pole handling with CD R conversion, and a BLHA interface for integration with external Monte Carlo tools, complemented by the NParton multi-precision primitives. Validation against analytic results and other codes, plus extensive accuracy and speed studies, demonstrate reliability and competitive performance, with successful applications to 3- and 4-jet NLO QCD and potential for heavy-quark extensions.

Abstract

We present a C++ library for the numerical evaluation of one-loop virtual corrections to multi-jet production in massless QCD. The pure gluon primitive amplitudes are evaluated using NGluon. A generalized unitarity reduction algorithm is used to construct arbitrary multiplicity fermion-gluon primitive amplitudes. From these basic building blocks the one-loop contribution to the squared matrix element, summed over colour and helicities, is calculated. No approximation in colour is performed. While the primitive amplitudes are given for arbitrary multiplicities we provide the squared matrix elements only for up to 7 external partons allowing the evaluation of the five jet cross section at next-to-leading order accuracy. The library has been recently successfully applied to four jet production at next-to-leading order in QCD.

Figures (6)

• Figure 1: Constructing parent diagrams for the primitive amplitudes. In case that no internal propagator can be connected since the corresponding vertices do not exist a blank propagator is inserted, represented graphically as a shaded blob.
• Figure 2: Matching Feynman diagrams to the primitive amplitudes $A_5^{[m]}(1_{\overline{d}},2_u,3_{\overline{u}},4_g,5_d)$ and $A_5^{[m]}(1_{\overline{d}},4_g,3_{\overline{u}},2_u,5_d)$. Each vertex is either ordered (red) or unordered (blue) with respect to the primitive in question. A diagram contributes with a coefficient $-1$ when it contains an odd number of unordered vertices and $+1$ otherwise. Any diagram which cannot be obtained from the original primitive by "pinching" and "pulling" operations will have a $0$ coefficient.
• Figure 3: Triple cut for two primitives with different helicity and ordering of external legs. The partial trees $A(\ell_{1;g}^{\pm},1_q^-,2_g^+,3_g^-,\ell_{4;\bar{q}}^{\pm})$ are the same for both primitives.
• Figure 4: Accuracy for 4-jet amplitudes: (a) shows the six gluon process, (b) the $d{\overline{d}} \to 4g$ process, (c) $d {\overline{d}} \to {\overline{u}} u + 2g$ and (d) the six unlike quark pair process. The thicker histograms show computations in double precision whereas the thinner curves show the distribution in quadruple precision for points which did not pass the relative accuracy of $10^{-4}$ when calculated in double precision (region marked by the shaded area). Red histograms show the $\tfrac{1}{\epsilon^2}$ poles, green histograms the $\tfrac{1}{\epsilon}$ and blue histograms the finite part of the amplitudes.
• Figure 5: Accuracy for 5-jet amplitudes: (a) shows the seven gluon process and (b) the $d {\overline{d}} \to {\overline{d}} d + 3g$ process. The thicker histograms show computations in double precision whereas the thinner curves show the distribution in quadruple precision for points which did not pass the relative accuracy of $10^{-4}$ when calculated in double precision. Red histograms show the $\tfrac{1}{\epsilon^2}$ poles, green histograms the $\tfrac{1}{\epsilon}$ and blue histograms the finite part of the amplitudes.