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Leading singularities in Baikov representation and Feynman integrals with uniform transcendental weight

Christoph Dlapa, Xiaodi Li, Yang Zhang

TL;DR

This work develops a leading-singularity–in–Baikov framework to identify Feynman integrals with uniform transcendental weight, exploiting rationalization of radicals and syzygy-based IBP to handle multi-scale two-loop configurations. The method is demonstrated by constructing the UT basis for a two-loop double box with three external masses, obtaining a canonical differential equation with a 30-letter alphabet. By combining loop-by-loop Baikov representations, dlog-integrand construction, and shifted-integral techniques, the authors obtain a full UT basis across 47 master integrals in 33 sectors and provide actionable procedures for automating the UT search. The results advance multi-scale UT integral determination and pave the way for automated differential-equation analyses in complex Feynman-integral families.

Abstract

We provide a leading singularity analysis protocol in Baikov representation, for the searching of Feynman integrals with uniform transcendental (UT) weight. This approach is powered by the recent developments in rationalizing square roots and syzygy computations, and is particularly suitable for finding UT integrals with multiple mass scales. We demonstrate the power of our approach by determining the UT basis for a two-loop diagram with three external mass scales.

Leading singularities in Baikov representation and Feynman integrals with uniform transcendental weight

TL;DR

This work develops a leading-singularity–in–Baikov framework to identify Feynman integrals with uniform transcendental weight, exploiting rationalization of radicals and syzygy-based IBP to handle multi-scale two-loop configurations. The method is demonstrated by constructing the UT basis for a two-loop double box with three external masses, obtaining a canonical differential equation with a 30-letter alphabet. By combining loop-by-loop Baikov representations, dlog-integrand construction, and shifted-integral techniques, the authors obtain a full UT basis across 47 master integrals in 33 sectors and provide actionable procedures for automating the UT search. The results advance multi-scale UT integral determination and pave the way for automated differential-equation analyses in complex Feynman-integral families.

Abstract

We provide a leading singularity analysis protocol in Baikov representation, for the searching of Feynman integrals with uniform transcendental (UT) weight. This approach is powered by the recent developments in rationalizing square roots and syzygy computations, and is particularly suitable for finding UT integrals with multiple mass scales. We demonstrate the power of our approach by determining the UT basis for a two-loop diagram with three external mass scales.

Paper Structure

This paper contains 14 sections, 89 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Singularity analysis in Baikov representation
  3. Leading Singularities and dlog integrals
  4. Baikov representation
  5. Example: Double box with three massive external legs
  6. Top sector and six-propagator sector
  7. Five-propagator sectors and lower sectors
  8. Differential equation and the alphabet
  9. Reducing shifted integrals to Feynman integrals through syzygies
  10. Finding dlog integrands
  11. Reducing shifted integrals
  12. Details about the IBP reduction in the loop-by-loop approach
  13. Summary and outlook
  14. The perfect square and Gram-determinant relations

Figures (3)

  • Figure 1: A schematic algorithm for finding UTs in a specific sector of a Feynman integral family. In steps one to three, a LS analysis is done.
  • Figure 2: The two-loop double box Feynman integral with three massive external momenta $k_1, k_2, k_3$ and seven massless propagators. The massive legs are indicate by thick lines.
  • Figure 3: The different topologies of the MIs. The thick external lines represent massive external momenta. The sectors $\mathcal{I}$ are labeled by the lines that are present in the diagram and we also show the number of master integrals in each sector.