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Two-Loop Vertices in Quantum Field Theory: Infrared Convergent Scalar Configurations

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

TL;DR

This work presents a comprehensive numerical framework for evaluating general massive, scalar, two-loop three-point Feynman diagrams across six topologies, leveraging Feynman parameterization and Bernstein-Tkachov (BT) relations. It delivers explicit scalar results for each topology (V^{121}, V^{131}, V^{221}, V^{141}, V^{231}, V^{222}), including ultraviolet poles, finite parts, and wave-function renormalization connections, while addressing stability near Landau and anomalous thresholds with alternative smoothness methods and sector decomposition. The paper also outlines two independent numerical strategies (BT-based and alternative smoothness approaches) to ensure robust cross-checks, and it sets the stage for forthcoming tensor-vertex analyses and infrared/collinear treatments. The numerical results, validated against known analytic limits and other computations, demonstrate the practicality of a scalable, multi-topology, fully massive two-loop vertex program with potential applications in precision electroweak and beyond-Standard-Model calculations.

Abstract

A comprehensive study is performed of general massive, scalar, two-loop Feynman diagrams with three external legs. Algorithms for their numerical evaluation are introduced and discussed, numerical results are shown for all different topologies, and comparisons with analytical results, whenever available, are performed. An internal cross-check, based on alternative procedures, is also applied. The analysis of infrared divergent configurations, as well as the treatment of tensor integrals, will be discussed in two forthcoming papers.

Two-Loop Vertices in Quantum Field Theory: Infrared Convergent Scalar Configurations

TL;DR

This work presents a comprehensive numerical framework for evaluating general massive, scalar, two-loop three-point Feynman diagrams across six topologies, leveraging Feynman parameterization and Bernstein-Tkachov (BT) relations. It delivers explicit scalar results for each topology (V^{121}, V^{131}, V^{221}, V^{141}, V^{231}, V^{222}), including ultraviolet poles, finite parts, and wave-function renormalization connections, while addressing stability near Landau and anomalous thresholds with alternative smoothness methods and sector decomposition. The paper also outlines two independent numerical strategies (BT-based and alternative smoothness approaches) to ensure robust cross-checks, and it sets the stage for forthcoming tensor-vertex analyses and infrared/collinear treatments. The numerical results, validated against known analytic limits and other computations, demonstrate the practicality of a scalable, multi-topology, fully massive two-loop vertex program with potential applications in precision electroweak and beyond-Standard-Model calculations.

Abstract

A comprehensive study is performed of general massive, scalar, two-loop Feynman diagrams with three external legs. Algorithms for their numerical evaluation are introduced and discussed, numerical results are shown for all different topologies, and comparisons with analytical results, whenever available, are performed. An internal cross-check, based on alternative procedures, is also applied. The analysis of infrared divergent configurations, as well as the treatment of tensor integrals, will be discussed in two forthcoming papers.

Paper Structure

This paper contains 44 sections, 343 equations, 6 figures, 8 tables.

Table of Contents

  1. Introduction
  2. General definitions and conventions
  3. Alphameric classification of diagrams
  4. Special kinds of diagrams
  5. One-loop counter-terms in two-loop diagrams
  6. Wave function renormalization
  7. The $G^{1{{N}}\,1}$ class of diagrams
  8. Power-counting for $G^{1{{N}} 1}$ when $b \approx 0$
  9. The $V^{121}$ diagram
  10. Evaluation of $V^{121}$
  11. Wave function renormalization: $S^{111}_p$
  12. The $V^{131}$ diagram
  13. Evaluation of $V^{131}$: method I
  14. Evaluation of $V^{131}$: method II
  15. Wave function renormalization: $S^{121}_p$
  16. ...and 29 more sections

Figures (6)

  • 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 three-point Green function. All external momenta are flowing inward, $P = p_1+p_2$.
  • Figure 3: Examples of two-loop vertices that are topologically equivalent to self-energies.
  • Figure 4: Examples of one-loop vertices with counter-terms (gray circles). Permutations are understood.
  • Figure 5: The two-loop topologies that contribute to wave-function renormalization (permutations are not included). The '$\circ$' refers to propagators with non-canonical power $-2$.
  • ...and 1 more figures