Mirror channel eigenvectors of the $d$-dimensional fishnets

TL;DR

This paper develops a basis of eigenvectors for graph operators in planar fishnet Feynman integrals across dimensions, enabling spectral decomposition useful for computing higher-dimensional Basso-Dixon integrals.

Contribution

It introduces a novel eigenvector basis for fishnet lattice operators using Zamolodchikovs-Faddeev algebra and integral representations of fused R-matrices with O(d) symmetry.

Findings

Eigenvectors depend on quantum numbers (u_k, l_k) for lattice excitations.

Wave-functions constructed with creation/annihilation operators satisfying Zamolodchikovs-Faddeev algebra.

Spectral decomposition applicable to higher-dimensional Basso-Dixon integrals.

Abstract

We present a basis of eigenvectors for the graph building operators acting along the mirror channel of planar fishnet Feynman integrals in $d$-dimensions. The eigenvectors of a fishnet lattice of length $L$ depend on a set of $L$ quantum numbers $(u_k,l_k)$, each associated with the rapidity and bound-state index of a lattice excitation. Each excitation is a particle in $(1+1)$-dimensions with $O(d)$ internal symmetry, and the wave-functions are formally constructed with a set of creation/annihilation operators that satisfy the corresponding Zamolodchikovs-Faddeev algebra. These properties are proved via the representation - new to our knowledge - of the matrix elements of the fused R-matrix with $O(d)$ symmetry as integral operators on the functions of two spacetime points. The spectral decomposition of a fishnet integral we achieved can be applied to the computation of Basso-Dixon integrals in higher dimensions.