Yangian Ward Identities for Fishnet Four-Point Integrals

TL;DR

<3-5 sentences high-level summary> This work establishes Yangian Ward identities for the infinite family of four-point ladder integrals and their Basso--Dixon generalisations by viewing fishnet Feynman graphs as correlation functions in a bi-scalar theory and as momentum-space conformal anomalies. The authors derive inhomogeneous PDEs in the conformal cross ratios, with inhomogeneities expressed as dimension- and propagator-shifted linear combinations of BD-type integrals, and they show how these extend to a D-dimensional fishnet generalisation. In two dimensions the Ward identities decouple and can be solved by separation of variables, yielding explicit constructions in terms of Legendre functions and elliptic integrals for the isotropic case, while in higher dimensions they relate vector BD integrals to higher-dimensional scalar counterparts via Tarasov-type decompositions. The results illuminate the role of Yangian symmetry in constraining BD-type integrals, suggest routes to solving the inhomogeneous equations, and point to connections with determinant structures underlying the BD formula and potential extensions to massive theories and N=4 SYM. These insights deepen the interplay between integrability, conformal symmetry, and explicit Feynman integral evaluations in the fishnet framework.

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.