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The Art of Computing Loop Integrals

Stefan Weinzierl

TL;DR

The article surveys a comprehensive toolkit for computing loop integrals in perturbative quantum field theory, starting from basic Feynman rules and dimensional regularisation to advanced analytic and numerical methods. It highlights how loop calculations are organised via parameterisations, tensor reductions, and master integrals, and how the analytic structure is governed by graph polynomials ${\mathcal U}$ and ${\mathcal F}$ and by the class of multiple polylogarithms. The work systematically connects algebraic frameworks (Z-/S-sums, Hopf algebras) with practical techniques (IBP, Mellin–Barnes, differential equations, sector decomposition) to enable both exact and numerical evaluations, including one- and two-loop cases and beyond. It also points to current frontiers, such as elliptic integrals and unitarity-based methods, illustrating the ongoing development of the field and the utility of these methods for precise predictions in high-energy physics.

Abstract

A perturbative approach to quantum field theory involves the computation of loop integrals, as soon as one goes beyond the leading term in the perturbative expansion. First I review standard techniques for the computation of loop integrals. In a second part I discuss more advanced algorithms. For these algorithms algebraic methods play an important role. A special section is devoted to multiple polylogarithms. I tried to make these notes self-contained and accessible both to physicists and mathematicians.

The Art of Computing Loop Integrals

TL;DR

The article surveys a comprehensive toolkit for computing loop integrals in perturbative quantum field theory, starting from basic Feynman rules and dimensional regularisation to advanced analytic and numerical methods. It highlights how loop calculations are organised via parameterisations, tensor reductions, and master integrals, and how the analytic structure is governed by graph polynomials and and by the class of multiple polylogarithms. The work systematically connects algebraic frameworks (Z-/S-sums, Hopf algebras) with practical techniques (IBP, Mellin–Barnes, differential equations, sector decomposition) to enable both exact and numerical evaluations, including one- and two-loop cases and beyond. It also points to current frontiers, such as elliptic integrals and unitarity-based methods, illustrating the ongoing development of the field and the utility of these methods for precise predictions in high-energy physics.

Abstract

A perturbative approach to quantum field theory involves the computation of loop integrals, as soon as one goes beyond the leading term in the perturbative expansion. First I review standard techniques for the computation of loop integrals. In a second part I discuss more advanced algorithms. For these algorithms algebraic methods play an important role. A special section is devoted to multiple polylogarithms. I tried to make these notes self-contained and accessible both to physicists and mathematicians.

Paper Structure

This paper contains 44 sections, 249 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Basic techniques
  3. Feynman rules
  4. Regularisation
  5. Dimensional regularisation schemes
  6. Loop integration in $D$ dimensions
  7. Feynman and Schwinger parameterisation
  8. Shift of the integration variable
  9. Wick rotation
  10. Generalised spherical coordinates
  11. Euler's Gamma and Beta function
  12. Result for the momentum integration
  13. Tensor integrals
  14. Passarino - Veltman reduction
  15. General reduction method
  16. ...and 29 more sections

Figures (6)

  • Figure 1: A one-loop Feynman diagram contributing to the process $e^+ e^- \rightarrow q g \bar{q}$.
  • Figure 2: Integration contour for the Wick rotation. The little circles along the real axis exclude the poles.
  • Figure 3: An example for an irreducible scalar product in the numerator: The scalar product $2p_1 k_2$ cannot be expressed in terms of inverse propagators.
  • Figure 4: The triangle trick: Integration by parts eliminates the propagators $1$, $2$, $3$ or $4$.
  • Figure 5: Sketch of the proof for the multiplication of $Z$-sums. The sum over the square is replaced by the sum over the three regions on the r.h.s.
  • ...and 1 more figures