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Integration by parts: An introduction

A. G. Grozin

TL;DR

This work surveys the IBP-based reduction framework for multi-loop Feynman integrals, redefining the reduction problem in terms of $L$ loops and $E$ external legs and introducing a unified set of tools to express integrals in terms of a finite master basis. It covers the derivation of IBP using infinitesimal momentum shifts, the generation of recurrence relations, and the recursive sector-based reduction strategy, while also explaining how homogeneity and Lorentz-invariance relations relate to IBP. The paper then surveys and exemplifies leading reduction techniques—Gröbner-bases, Baikov’s method, and the Laporta algorithm—across massless, HQET, and multi-loop diagrams, illustrating both practical implementations (e.g., FIRE, Grinder) and theoretical insights (equivalence between external-leg and vacuum IBP). It highlights the strengths and limitations of each approach, including irreducibility proofs for non-planar diagrams, and points to ongoing challenges in achieving a universal reduction method for extremely high-loop calculations. Collectively, the work provides a comprehensive map of current IBP strategies, their interrelations, and their implications for automating perturbative quantum field theory computations.

Abstract

Integration by parts is used to reduce scalar Feynman integrals to master integrals.

Integration by parts: An introduction

TL;DR

This work surveys the IBP-based reduction framework for multi-loop Feynman integrals, redefining the reduction problem in terms of loops and external legs and introducing a unified set of tools to express integrals in terms of a finite master basis. It covers the derivation of IBP using infinitesimal momentum shifts, the generation of recurrence relations, and the recursive sector-based reduction strategy, while also explaining how homogeneity and Lorentz-invariance relations relate to IBP. The paper then surveys and exemplifies leading reduction techniques—Gröbner-bases, Baikov’s method, and the Laporta algorithm—across massless, HQET, and multi-loop diagrams, illustrating both practical implementations (e.g., FIRE, Grinder) and theoretical insights (equivalence between external-leg and vacuum IBP). It highlights the strengths and limitations of each approach, including irreducibility proofs for non-planar diagrams, and points to ongoing challenges in achieving a universal reduction method for extremely high-loop calculations. Collectively, the work provides a comprehensive map of current IBP strategies, their interrelations, and their implications for automating perturbative quantum field theory computations.

Abstract

Integration by parts is used to reduce scalar Feynman integrals to master integrals.

Paper Structure

This paper contains 18 sections, 141 equations, 16 figures, 1 table.

Table of Contents

  1. Introduction
  2. Feynman graphs and Feynman integrals
  3. Integration by parts
  4. Homogeneity and Lorentz-invariance relations
  5. Massless self-energy diagrams
  6. HQET self-energy diagrams
  7. Equivalence of IBP relations for integrals with the same total number of loop and external momenta
  8. Gröbner-bases methods
  9. Baikov's method
  10. Conclusion
  11. Dimensional regularization
  12. Gröbner bases
  13. Statement of the problem
  14. Monomial orders
  15. Reduction of polynomials
  16. ...and 3 more sections

Figures (16)

  • Figure 1:
  • Figure 2: Symmetric diagrams
  • Figure 3: Sectors in the $N$-dimensional integer space
  • Figure 4: The 1-loop vacuum diagram and its sectors
  • Figure 5: The 1-loop vacuum diagram with masses $m$, 0 and its sectors
  • ...and 11 more figures