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Two-Loop Tensor Integrals in Quantum Field Theory

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

TL;DR

The work develops a unified, numerically stable framework for evaluating general massive tensor two-loop diagrams with two- and three-point functions by reducing them to generalized scalar integrals. It combines tensor decompositions into form factors with gauge-consistent projectors, integration-by-parts identities, and Tarasov-style recurrences, together with smooth integral representations inspired by Bernstein–Tkachov methods. The authors treat multiple vertex families (V$^{E,I,M,G,K,H}$), providing explicit vector and rank-two tensor reductions, and extend the representation to rank-three tensors, including diagrammatic interpretations. The approach aims to enable two-loop renormalization and precision predictions in the Standard Model by avoiding Gram-determinant instabilities and keeping a close link between algebraic reductions and numerical integration.

Abstract

A comprehensive study is performed of general massive, tensor, two-loop Feynman diagrams with two and three external legs. Reduction to generalized scalar functions is discussed. Integral representations, supporting the same class of smoothness algorithms already employed for the numerical evaluation of ordinary scalar functions, are introduced for each family of diagrams.

Two-Loop Tensor Integrals in Quantum Field Theory

TL;DR

The work develops a unified, numerically stable framework for evaluating general massive tensor two-loop diagrams with two- and three-point functions by reducing them to generalized scalar integrals. It combines tensor decompositions into form factors with gauge-consistent projectors, integration-by-parts identities, and Tarasov-style recurrences, together with smooth integral representations inspired by Bernstein–Tkachov methods. The authors treat multiple vertex families (V), providing explicit vector and rank-two tensor reductions, and extend the representation to rank-three tensors, including diagrammatic interpretations. The approach aims to enable two-loop renormalization and precision predictions in the Standard Model by avoiding Gram-determinant instabilities and keeping a close link between algebraic reductions and numerical integration.

Abstract

A comprehensive study is performed of general massive, tensor, two-loop Feynman diagrams with two and three external legs. Reduction to generalized scalar functions is discussed. Integral representations, supporting the same class of smoothness algorithms already employed for the numerical evaluation of ordinary scalar functions, are introduced for each family of diagrams.

Paper Structure

This paper contains 58 sections, 479 equations, 18 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Conventions and notation
  3. Definition of one-loop generalized functions
  4. Alphameric classification of graphs
  5. Tensor integrals and gauge cancellations
  6. Projector techniques
  7. Techniques for the reduction of two-loop two-point functions
  8. Reduction to sub-loops
  9. Integration by parts identities and generalized recurrence relations
  10. Reduction for tensor two-point functions
  11. Strategies for the evaluation of two-loop self-energies
  12. Reduction of tensor two-loop three-point functions
  13. The $V^{{{E}}}$-family ($\alpha = 1, \beta = 2, \gamma = 1$)
  14. Vector integrals in the $V^{{{E}}}$ family
  15. Rank two tensor integrals in the $V^{{{E}}}$ family
  16. ...and 43 more sections

Figures (18)

  • Figure 1: $I$-family contribution to the decay $H \to \gamma\gamma$. Internal dotted lines represent a Higgs-Kibble $\phi$-field, while solid ones indicate a $W$-field.
  • Figure 2: Scalar diagram of the $S^{{{A}}}$-family, the so-called sunset (or sunrise) configuration.
  • Figure 3: Irreducible classes for two-loop two-point functions.
  • Figure 4: Example of a diagram belonging to the $S^{{{C}}}$-family and contributing to the $Z$ self-energy. Dashed lines represent a $H$-field.
  • Figure 5: A contribution of the $V^{{{K}}}$ family to $H \to Z^* Z \to Z \overline f f$. External momenta flow inwards.
  • ...and 13 more figures