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Cuts of Feynman Integrals in Baikov representation

Hjalte Frellesvig, Costas G. Papadopoulos

TL;DR

The paper develops a systematic Baikov-representation framework for computing cuts of Feynman integrals in $d$ dimensions, integrating a loop-by-loop reduction to minimize variables and a differential-equation approach to obtain relations with external kinematics. It defines cuts in Baikov variables such that cut integrals obey the same DE as uncut ones, enabling analysis of the function space (Goncharov polylogarithms, elliptic integrals) governing the full integrals and providing a potential criterion for canonical form via maximally cut cases. Through explicit examples (e.g., double-box, sunset, box-triangle, elliptic double-box), the work demonstrates that maximally cut integrals reflect the underlying analytic structure and can indicate when canonical forms exist or when more complex function classes arise. The findings offer a practical, dimensionally consistent toolkit for classifying and solving multi-loop Feynman integrals and point to future work on ε-singularity algorithms and non-planar topologies.

Abstract

Based on the Baikov representation, we present a systematic approach to compute cuts of Feynman Integrals, appropriately defined in $d$ dimensions. The information provided by these computations may be used to determine the class of functions needed to analytically express the full integrals.

Cuts of Feynman Integrals in Baikov representation

TL;DR

The paper develops a systematic Baikov-representation framework for computing cuts of Feynman integrals in dimensions, integrating a loop-by-loop reduction to minimize variables and a differential-equation approach to obtain relations with external kinematics. It defines cuts in Baikov variables such that cut integrals obey the same DE as uncut ones, enabling analysis of the function space (Goncharov polylogarithms, elliptic integrals) governing the full integrals and providing a potential criterion for canonical form via maximally cut cases. Through explicit examples (e.g., double-box, sunset, box-triangle, elliptic double-box), the work demonstrates that maximally cut integrals reflect the underlying analytic structure and can indicate when canonical forms exist or when more complex function classes arise. The findings offer a practical, dimensionally consistent toolkit for classifying and solving multi-loop Feynman integrals and point to future work on ε-singularity algorithms and non-planar topologies.

Abstract

Based on the Baikov representation, we present a systematic approach to compute cuts of Feynman Integrals, appropriately defined in dimensions. The information provided by these computations may be used to determine the class of functions needed to analytically express the full integrals.

Paper Structure

This paper contains 13 sections, 76 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Baikov representation
  3. The loop-by-loop approach
  4. Deriving differential equations
  5. Cutting Feynman Integrals
  6. Discussion and Outloook
  7. Alternative derivation of the Baikov representation
  8. Examples of cuts performed in Baikov variables
  9. Double-box with massive loop
  10. The sunset
  11. The bubble-triangle
  12. The elliptic box-triangle
  13. The elliptic double-box

Figures (1)

  • Figure 1: The two-loop three-point graph considered in the text.