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All-Purpose Numerical Evaluation of One-Loop Multi-Leg Feynman Diagrams

A. Ferroglia, G. Passarino, M. Passera, S. Uccirati

TL;DR

This paper delivers a comprehensive numerical framework for evaluating one-loop multi-leg Feynman diagrams up to six external legs across arbitrary kinematics, including complex masses. It combines the Bernstein–Tkachov (BT) theorem with Mellin–Barnes (MB) representations and sector decomposition to produce stable, analytic-compatible integral representations for all N-point functions (C, D, E, F families) and their infrared divergent parts. By classifying diagrams according to the leading BT factor $B_G$ and employing multiple representations (BT, MB, and square/comp representations) near Landau thresholds, the authors derive explicit formulas and expansions up to ${\cal O}(\epsilon)$, along with new integral representations for cross-checks. The framework enables automatic, robust numerical evaluation and provides detailed treatments of regular and singular configurations, including infrared poles and Landau singularities, facilitating coupling to higher-loop calculations and complex-mass scenarios with precise pole structure control.

Abstract

A detailed investigation is presented of a set of algorithms which form the basis for a fast and reliable numerical integration of one-loop multi-leg (up to six) Feynman diagrams, with special attention to the behavior around (possibly) singular points in phase space. No particular restriction is imposed on kinematics, and complex masses (poles) are allowed.

All-Purpose Numerical Evaluation of One-Loop Multi-Leg Feynman Diagrams

TL;DR

This paper delivers a comprehensive numerical framework for evaluating one-loop multi-leg Feynman diagrams up to six external legs across arbitrary kinematics, including complex masses. It combines the Bernstein–Tkachov (BT) theorem with Mellin–Barnes (MB) representations and sector decomposition to produce stable, analytic-compatible integral representations for all N-point functions (C, D, E, F families) and their infrared divergent parts. By classifying diagrams according to the leading BT factor and employing multiple representations (BT, MB, and square/comp representations) near Landau thresholds, the authors derive explicit formulas and expansions up to , along with new integral representations for cross-checks. The framework enables automatic, robust numerical evaluation and provides detailed treatments of regular and singular configurations, including infrared poles and Landau singularities, facilitating coupling to higher-loop calculations and complex-mass scenarios with precise pole structure control.

Abstract

A detailed investigation is presented of a set of algorithms which form the basis for a fast and reliable numerical integration of one-loop multi-leg (up to six) Feynman diagrams, with special attention to the behavior around (possibly) singular points in phase space. No particular restriction is imposed on kinematics, and complex masses (poles) are allowed.

Paper Structure

This paper contains 41 sections, 356 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Prolegomena
  3. Generalized two-point function
  4. Three-point functions (C-family)
  5. Evaluation of $C_0$ for $B_3 \ne 0$
  6. Form factors in the C-family
  7. $C_0$ at ${\cal O}\left(\epsilon\right)$
  8. Evaluation of $C_0$ when $B_3 \approx 0$
  9. The regular case: method I
  10. The regular case: method II
  11. The singular case
  12. A new integral representation for $C_0$
  13. Infrared divergent $C_0$
  14. $B_{\rm th} \ne 0$
  15. Infrared configuration and $B_{\rm th} \approx 0$
  16. ...and 26 more sections

Figures (5)

  • Figure 1: The two-point Green function.
  • Figure 2: The one-loop, three-point Green function. The second diagram, although having $4$ external lines, is included in the $C$-family of $3$ internal lines.
  • Figure 3: The one-loop, four-point Green function of Eq.(\ref{['origD0']}). All momenta are flowing inwards.
  • Figure 4: The one-loop, five-point Green function of Eq.(\ref{['EqDefE']}). Propagators are $q^2+m^2_1\cdots\,$$(q+p_1+\cdots + p_4)^2+m^2_5$.
  • Figure 5: Diagrammatical representation of the BT algorithm of Eq.(\ref{['pdec']}) for the pentagon. The symbol $[i]$ denotes multiplication of the corresponding box by a factor $w_i/(4\,B_5)$.