Yangian Ward Identities for Fishnet Four-Point Integrals

TL;DR

This paper derives Yangian Ward identities for fishnet four-point integrals, revealing new symmetry structures and providing explicit solutions in two dimensions, advancing understanding of conformal integrals and their symmetries.

Contribution

It introduces Yangian Ward identities for fishnet integrals, connecting them to hypergeometric functions and providing explicit solutions in 2D using separation of variables.

Findings

Yangian Ward identities for ladder integrals are derived.

Explicit bootstrap solutions for 2D conformal box integrals are obtained.

Transcendentality patterns differ between 2D and 4D cases.

Abstract

We derive and study Yangian Ward identities for the infinite class of four-point ladder integrals and their Basso-Dixon generalisations. These symmetry equations follow from interpreting the respective Feynman integrals as correlation functions in the bi-scalar fishnet theory. Alternatively, the presented identities can be understood as anomaly equations for a momentum space conformal symmetry. The Ward identities take the form of inhomogeneous extensions of the partial differential equations defining the Appell hypergeometric functions. We employ a manifestly conformal tensor reduction in order to express these inhomogeneities in compact form, which are given by linear combinations of Basso-Dixon integrals with shifted dimensions and propagator powers. The Ward identities naturally generalise to a one-parameter family of D-dimensional integrals representing correlators in the generalised fishnet theory of Kazakov and Olivucci. When specified to two spacetime dimensions, the Yangian Ward identities decouple. Using separation of variables, we explicitly bootstrap the solution for the conformal 2D box integral. The result is a linear combination of Yangian invariant products of Legendre functions, which reduce to elliptic K integrals for an isotropic choice of propagator powers. We comment on differences in the transcendentality patterns in two and four dimensions and their relations to discontinuities.