Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Notes on Fourier transform and its application to three-point momentum-space integrals

Xuhang Jiang

TL;DR

The paper develops a generalized Fourier identity that links massless momentum-space Feynman integrals with three off-shell legs to corresponding position-space graphical functions, extending the classic glue-and-cut framework. It formulates a graph-rule approach and proves the identity via Fourier transform, Mellin transform, and star-triangle relations, with a dimensional extension and edge-weight constraints $\nu_1+\nu_2=d/2$. The generalized identity is then applied to a non-planar family relevant to the off-shell Sudakov form factor, where many integrals factorize into products of ladder-type conformal integrals $F_L(z,\bar{z})$, while some members exhibit elliptic or higher-polylogarithmic structures. The work also clarifies the connection to planar duality through Feynman parameterization and outlines potential extensions to deformed edge weights, higher-point conformal integrals, and non-conformal cases, broadening the toolkit for evaluating challenging Feynman integrals.

Abstract

The Fourier transform of two-point momentum-space Feynman integrals with massless propagators and two off-shell legs can be used to prove identities between their periods, exemplified by the glue-and-cut identity. We generalize this framework to massless momentum-space Feynman integrals with three off-shell legs and obtain a similar family of identities that can be used to calculate these integrals, especially for a non-planar subset of them, which naturally arise in the off-shell Sudakov form factors.

Notes on Fourier transform and its application to three-point momentum-space integrals

TL;DR

The paper develops a generalized Fourier identity that links massless momentum-space Feynman integrals with three off-shell legs to corresponding position-space graphical functions, extending the classic glue-and-cut framework. It formulates a graph-rule approach and proves the identity via Fourier transform, Mellin transform, and star-triangle relations, with a dimensional extension and edge-weight constraints . The generalized identity is then applied to a non-planar family relevant to the off-shell Sudakov form factor, where many integrals factorize into products of ladder-type conformal integrals , while some members exhibit elliptic or higher-polylogarithmic structures. The work also clarifies the connection to planar duality through Feynman parameterization and outlines potential extensions to deformed edge weights, higher-point conformal integrals, and non-conformal cases, broadening the toolkit for evaluating challenging Feynman integrals.

Abstract

The Fourier transform of two-point momentum-space Feynman integrals with massless propagators and two off-shell legs can be used to prove identities between their periods, exemplified by the glue-and-cut identity. We generalize this framework to massless momentum-space Feynman integrals with three off-shell legs and obtain a similar family of identities that can be used to calculate these integrals, especially for a non-planar subset of them, which naturally arise in the off-shell Sudakov form factors.

Paper Structure

This paper contains 7 sections, 1 theorem, 56 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Generalized Fourier identity and its application
  3. Generalized Fourier identity
  4. proof of the generalized Fourier identities
  5. The application to the off-shell form factor
  6. Relation to planar duality by Feynman parameterization
  7. Conclusion and Outlook

Key Result

Proposition 1

If $G^{\prime}_{\mathrm{ms}}(\frac{p_{1}^{2}}{q^{2}},\frac{p_{2}^{2}}{q^{2}})$ is finite for generic kinematics $p_{1}^{2},p_{2}^{2},q^{2}\ne 0$ (no sub-divergence), then $G^{\prime}_{\mathrm{ms}}(\frac{p_{1}^{2}}{q^{2}},\frac{p_{2}^{2}}{q^{2}})=\frac{1}{(q^{2})^{2}}G^{\prime}_{\mathrm{ps}}(z,\bar{z

Figures (2)

  • Figure 1: Two examples of integrals in different representations. The left diagram is in momentum-space with three external legs, while the right diagram is in position space with three external vertices colored pink.
  • Figure 2: The three-loop non-planar integral which will contribute to the off-shell Sudakov form factor (left) and its counterpart in position space (right).

Theorems & Definitions (7)

  • Definition 1
  • Definition 2
  • Proposition 1
  • Example 1
  • Example 2
  • Example 3
  • proof