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Tools for NLO automation: extension of the golem95C integral library

J. Ph. Guillet, G. Heinrich, J. F. von Soden-Fraunhofen

TL;DR

The paper tackles automating NLO calculations for multi-leg one-loop amplitudes when tensor ranks exceed the number of propagators, as encountered in effective theories and spin-2 loop scenarios, by extending the golem95C integral library to higher ranks $r$ expressed relative to $N$.The approach combines an extended tensor reduction framework with a basis of form factors, stabilized via a parameter-space approach that leverages $D$-dimensional and $D+2m$-dimensional integrals and numerically robust representations, while supporting complex masses and unitarity-based tensor reconstruction.Key contributions include enabling $r\le 10$ for $N\le 4$ and $r\le N+1$ for $N\ge 5$, providing higher-rank form factors, and delivering practical usage through demos and installation guidance, making the tool applicable within and beyond diagrammatic methods.The work delivers an openly available, versatile library for automated one-loop calculations in BSM and effective theories, facilitating reliable tensor and scalar integral computations in both traditional and unitarity-inspired workflows.

Abstract

We present an extension of the program golem95C for the numerical evaluation of scalar integrals and tensor form factors entering the calculation of one-loop amplitudes, which supports tensor ranks exceeding the number of propagators. This extension allows various applications in Beyond the Standard Model physics and effective theories, for example higher ranks due to propagators of spin two particles, or due to effective vertices. Complex masses are also supported. The program is not restricted to the Feynman diagrammatic approach, as it also contains routines to interface to unitarity-inspired numerical reconstruction of the integrand at the tensorial level. Therefore it can serve as a general integral library in automated programs to calculate one-loop amplitudes.

Tools for NLO automation: extension of the golem95C integral library

TL;DR

The paper tackles automating NLO calculations for multi-leg one-loop amplitudes when tensor ranks exceed the number of propagators, as encountered in effective theories and spin-2 loop scenarios, by extending the golem95C integral library to higher ranks $r$ expressed relative to $N$.The approach combines an extended tensor reduction framework with a basis of form factors, stabilized via a parameter-space approach that leverages $D$-dimensional and $D+2m$-dimensional integrals and numerically robust representations, while supporting complex masses and unitarity-based tensor reconstruction.Key contributions include enabling $r\le 10$ for $N\le 4$ and $r\le N+1$ for $N\ge 5$, providing higher-rank form factors, and delivering practical usage through demos and installation guidance, making the tool applicable within and beyond diagrammatic methods.The work delivers an openly available, versatile library for automated one-loop calculations in BSM and effective theories, facilitating reliable tensor and scalar integral computations in both traditional and unitarity-inspired workflows.

Abstract

We present an extension of the program golem95C for the numerical evaluation of scalar integrals and tensor form factors entering the calculation of one-loop amplitudes, which supports tensor ranks exceeding the number of propagators. This extension allows various applications in Beyond the Standard Model physics and effective theories, for example higher ranks due to propagators of spin two particles, or due to effective vertices. Complex masses are also supported. The program is not restricted to the Feynman diagrammatic approach, as it also contains routines to interface to unitarity-inspired numerical reconstruction of the integrand at the tensorial level. Therefore it can serve as a general integral library in automated programs to calculate one-loop amplitudes.

Paper Structure

This paper contains 12 sections, 19 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Theoretical background
  3. Form Factors
  4. Integrals
  5. Extension of the program to higher rank
  6. Installation
  7. Usage and examples
  8. Three-point rank 4 example
  9. Four-point rank 5 example
  10. Five-point rank 6 example
  11. Conclusions
  12. Appendix: $\tilde{k}$ - integrals

Figures (1)

  • Figure 1: Examples of diagrams with graviton exchange, where rank four 3-point and rank five 4-point integrals are needed. The graviton is denoted by $G$.