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Algebraic-Numerical Evaluation of Feynman Diagrams: Two-Loop Self-Energies

G. Passarino, S. Uccirati

TL;DR

The paper advances a numerically oriented, algebraic framework for evaluating two-loop self-energy diagrams by extending the Bernstein–Tkachov method to all two-loop two-point functions with arbitrary masses. It develops several BT-based strategies and special relations to smooth the integrand, handles Landau singularities, and introduces a novel infrared treatment via pole-residue factorization compatible with dimensional regularization. The authors provide explicit formulations for the S^{121}, S^{131}, and S^{221} topologies, detail contour-management and logarithmic structures, and demonstrate high-precision numerical results that agree with known analytic benchmarks across thresholds. This work underpins automatic, reliable numerical evaluation of multi-loop diagrams in realistic models, including infrared-sensitive cases relevant to electroweak and QCD computations.

Abstract

A recently proposed scheme for numerical evaluation of Feynman diagrams is extended to cover all two-loop two-point functions with arbitrary internal and external masses. The adopted algorithm is a modification of the one proposed by F. V. Tkachov and it is based on the so-called generalized Bernstein functional relation. On-shell derivatives of self-energies are also considered and their infrared properties analyzed to prove that the method which is aimed to a numerical evaluation of massive diagrams can handle the infrared problem within the scheme of dimensional regularization. Particular care is devoted to study the general massive diagrams around their leading and non-leading Landau singularities.

Algebraic-Numerical Evaluation of Feynman Diagrams: Two-Loop Self-Energies

TL;DR

The paper advances a numerically oriented, algebraic framework for evaluating two-loop self-energy diagrams by extending the Bernstein–Tkachov method to all two-loop two-point functions with arbitrary masses. It develops several BT-based strategies and special relations to smooth the integrand, handles Landau singularities, and introduces a novel infrared treatment via pole-residue factorization compatible with dimensional regularization. The authors provide explicit formulations for the S^{121}, S^{131}, and S^{221} topologies, detail contour-management and logarithmic structures, and demonstrate high-precision numerical results that agree with known analytic benchmarks across thresholds. This work underpins automatic, reliable numerical evaluation of multi-loop diagrams in realistic models, including infrared-sensitive cases relevant to electroweak and QCD computations.

Abstract

A recently proposed scheme for numerical evaluation of Feynman diagrams is extended to cover all two-loop two-point functions with arbitrary internal and external masses. The adopted algorithm is a modification of the one proposed by F. V. Tkachov and it is based on the so-called generalized Bernstein functional relation. On-shell derivatives of self-energies are also considered and their infrared properties analyzed to prove that the method which is aimed to a numerical evaluation of massive diagrams can handle the infrared problem within the scheme of dimensional regularization. Particular care is devoted to study the general massive diagrams around their leading and non-leading Landau singularities.

Paper Structure

This paper contains 52 sections, 443 equations, 23 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Arbitrary two-loop diagrams
  3. Reducible two-loop diagrams
  4. Minimal BT algorithm
  5. Special BT relations
  6. The $S_{4}$ family
  7. The case $p^2= -t$ with $t < 0$
  8. Evaluation of $S^{121}$: method I
  9. Single-pole and finite part of $S^{121}$
  10. Landau equations for $S^{121}$
  11. One iteration for $S^{121}$
  12. Distorting the integration contour for $S^{121}$
  13. Around the leading Landau singularity: method II
  14. Two iterations for $S^{121}$
  15. Zeros of $b$-functions
  16. ...and 37 more sections

Figures (23)

  • Figure 1: The arbitrary two-loop diagram $G_{{{L}}}^{\alpha\beta\gamma}$ of Eq.(\ref{['Gdiag']}) and one of the associated subtraction sub-diagrams.
  • Figure 2: The two-loop diagram $S^{121}$ of Eq.(\ref{['defsaba']}). Arrows indicate the momentum flow.
  • Figure 3: The second diagram of the $S_4$-family, Eq.(\ref{['interms']}), which is evaluated in terms of $S^{111}$ (ref. Passarino:2001wv) or $S_{33}$ diagrams (Eq.(\ref{['defstt']})) at zero external momentum.
  • Figure 4: The subtraction diagram, containing a one-loop counter-term (represented by a $\times$) associated with the two-loop diagram $S^{121}$ of Fig. \ref{['tops4']}.
  • Figure 5: The two-particle cut of diagram $S^{121}$ of Fig. \ref{['tops4']}
  • ...and 18 more figures