Conformal Four-Point Integrals: Recursive Structure, Toda Equations and Double Copy

TL;DR

This paper explores the mathematical structure of conformal four-point integrals across dimensions, revealing recursive relations, connections to Toda equations, and a double copy formulation, with implications for higher-dimensional and higher-point integrals.

Contribution

It introduces recursive structures and Toda equation relations for conformal integrals in higher dimensions, extending 2D results and establishing a double copy framework.

Findings

Derived new expressions for conformal ladder integrals in all even dimensions

Linked integrability properties to Toda equations and tau-functions

Presented a double copy construction for higher-dimensional integrals

Abstract

We consider conformal four-point Feynman integrals to investigate how much of their mathematical structure in two spacetime dimensions carries over to higher dimensions. In particular, we discuss recursions in the loop order and spacetime dimension. This results e.g. in new expressions for conformal ladder integrals with generic propagator powers in all even dimensions and allows us to lift results on 2d Feynman integrals with underlying Calabi-Yau geometry to higher dimensions. Moreover, we demonstrate that the Basso-Dixon generalizations of these integrals obey different variants of the Toda equations of motion, thus establishing a connection to classical integrability and the family of so-called tau-functions. We then show that all of these integrals can be written in a double copy form that combines holomorphic and anti-holomorphic building blocks. Here integrals in higher dimensions are constructed from an intersection pairing of two-dimensional "periods" together with their derivatives. Finally, we comment on extensions to higher-point integrals which provide a richer kinematical setup.