Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Golem95C: A library for one-loop integrals with complex masses

G. Cullen, J. -Ph. Guillet, G. Heinrich, T. Kleinschmidt, E. Pilon, T. Reiter, M. Rodgers

TL;DR

golem95C extends one-loop integral technology to complex masses, enabling stable evaluation of scalar and tensor integrals in multi-leg amplitudes. The approach combines a robust form-factor reduction with a tensorial reconstruction interface that supports integrand-based unitarity methods, while carefully handling spurious Gram-determinant instabilities and Landau singularities via a dynamic switch to numerical evaluation and the complex-mass regulator. Key contributions include a comprehensive library up to rank six for N-point functions, a modular Fortran95 implementation, and explicit support for complex masses in both algebraic and numerical frameworks, with improved caching and a flexible interface for tensor reconstruction. The work provides a practical, high-precision tool for NLO calculations in processes with unstable particles, accessible to the community via the project repository.

Abstract

We present a program for the numerical evaluation of scalar integrals and tensor form factors entering the calculation of one-loop amplitudes which supports the use of complex masses in the loop integrals. The program is built on an earlier version of the golem95 library, which performs the reduction to a certain set of basis integrals using a formalism where inverse Gram determinants can be avoided. It can be used to calculate one-loop amplitudes with arbitrary masses in an algebraic approach as well as in the context of a unitarity-inspired numerical reconstruction of the integrand.

Golem95C: A library for one-loop integrals with complex masses

TL;DR

golem95C extends one-loop integral technology to complex masses, enabling stable evaluation of scalar and tensor integrals in multi-leg amplitudes. The approach combines a robust form-factor reduction with a tensorial reconstruction interface that supports integrand-based unitarity methods, while carefully handling spurious Gram-determinant instabilities and Landau singularities via a dynamic switch to numerical evaluation and the complex-mass regulator. Key contributions include a comprehensive library up to rank six for N-point functions, a modular Fortran95 implementation, and explicit support for complex masses in both algebraic and numerical frameworks, with improved caching and a flexible interface for tensor reconstruction. The work provides a practical, high-precision tool for NLO calculations in processes with unstable particles, accessible to the community via the project repository.

Abstract

We present a program for the numerical evaluation of scalar integrals and tensor form factors entering the calculation of one-loop amplitudes which supports the use of complex masses in the loop integrals. The program is built on an earlier version of the golem95 library, which performs the reduction to a certain set of basis integrals using a formalism where inverse Gram determinants can be avoided. It can be used to calculate one-loop amplitudes with arbitrary masses in an algebraic approach as well as in the context of a unitarity-inspired numerical reconstruction of the integrand.

Paper Structure

This paper contains 24 sections, 18 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Theoretical background
  3. Form Factors
  4. Integrals
  5. Treatment of potential numerical instabilities
  6. Spurious singularities due to inverse Gram determinants
  7. Landau Singularities
  8. Complex masses
  9. Tensorial Reconstruction of the Integrand
  10. Overview of the software structure
  11. Description of the individual software components
  12. Form factor evaluation
  13. The module matrice_s
  14. Internal structure
  15. Interface for tensorial reconstruction
  16. ...and 9 more sections

Figures (3)

  • Figure 1: A diagram where all propagators can go simultaneously on-shell to develop a leading Landau singularity which can be regulated by introducing complex masses.
  • Figure 2: Singularity structure of the scalar four-point function $A^{4,0}$ (real masses) contained in the diagram of Fig. \ref{['fig:Hbb']} for $m_H=450$ GeV, $m_{\tilde{q}}=800$ GeV, $m_\chi=200$ GeV, $\sqrt{s}=1700$ GeV, $900\,{\rm{GeV}}\leq s_{45}\leq 1200$ GeV.
  • Figure 3: Singularity structure of the scalar four-point function (real masses) contained in the diagram of Fig. \ref{['fig:Hbb']} for $m_H=450$ GeV, $m_{\tilde{q}}^2\to m_{\tilde{q}}^2-i\,m_{\tilde{q}}\Gamma_{\tilde{q}}$, $m_\chi^2\to m_\chi^2-i\,m_\chi\Gamma_\chi$, $\Gamma_{\tilde{q}}=3.5\,{\rm{GeV}}, \Gamma_\chi=1.5$ GeV