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Multi-loop techniques for massless Feynman diagram calculations

A. V. Kotikov, S. Teber

Abstract

We review several multi-loop techniques for analytical massless Feynman diagram calculations in relativistic quantum field theories: integration by parts, the method of uniqueness, functional equations and the Gegenbauer polynomial technique. A brief, historically oriented, overview of some of the results obtained over the decades for the massless 2-loop propagator-type diagram is given. Concrete examples of up to $5$-loop diagram calculations are also provided.

Multi-loop techniques for massless Feynman diagram calculations

Abstract

We review several multi-loop techniques for analytical massless Feynman diagram calculations in relativistic quantum field theories: integration by parts, the method of uniqueness, functional equations and the Gegenbauer polynomial technique. A brief, historically oriented, overview of some of the results obtained over the decades for the massless 2-loop propagator-type diagram is given. Concrete examples of up to -loop diagram calculations are also provided.

Paper Structure

This paper contains 37 sections, 1 theorem, 209 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Basics of Feynman diagrams
  3. Notations
  4. Massless vacuum diagrams (and Mellin-Barnes transformation)
  5. Massless one-loop propagator-type diagram
  6. Massless two-loop propagator-type diagram
  7. Basic definition
  8. Symmetries
  9. Brief historical overview of some results
  10. Methods of calculations
  11. Parametric integration
  12. Integration by parts
  13. Presentation of the method
  14. Example of an important application
  15. The method of uniqueness
  16. ...and 22 more sections

Key Result

Theorem 1

Multiple zeta values are sufficient for the Laurent expansion of the two-loop integral, $G(D,\alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_5)$, with $D=2m-2\varepsilon$ ($m \in \mathbb{N}$) if all powers of the propagators are of the form $\alpha_i = n_i + a_i \varepsilon$ where the $n_i$ are positive

Figures (8)

  • Figure 1: Two-loop scalar massless propagator-type diagram.
  • Figure 2: Two-loop primitive, or recursively one-loop, diagrams. Diagram a) corresponds to $\alpha_5=0$. Diagram b) corresponds to $\alpha_4=0$. Diagram c) corresponds to $\alpha_1=\alpha_3=0$.
  • Figure 3: Some simple two-loop massless propagator diagrams which satisfy the rule of triangle and can be computed exactly using IBP identities.
  • Figure 4: Examples of two-loop massless p-type diagrams which are beyond IBP identities and uniqueness.
  • Figure 5: Examples of more complicated two-loop massless p-type diagrams.
  • ...and 3 more figures

Theorems & Definitions (1)

  • Theorem 1: Bierenbaum and Weinzierl (2003)