Variations on Slavnov's scalar product

TL;DR

This paper extends Slavnov's scalar product in the rational six-vertex model, revealing new determinant formulas and connecting them to discrete KP tau-functions, with implications for calculating structure constants in N=4 SYM.

Contribution

It introduces a new determinant expression for scalar products in the six-vertex model, linking them to discrete KP tau-functions and demonstrating their persistence at 1-loop in N=4 SYM.

Findings

New determinant expression for scalar products

Connection to discrete KP tau-functions

Determinant structure of N=4 SYM constants at 1-loop

Abstract

We consider the rational six-vertex model on an L-by-L lattice with domain wall boundary conditions and restrict N parallel-line rapidities, N < L/2, to satisfy length-L XXX spin-1/2 chain Bethe equations. We show that the partition function is an (L-2N)-parameter extension of Slavnov's scalar product of a Bethe eigenstate and a generic state, with N magnons each, on a length-L XXX spin-1/2 chain. Decoupling the extra parameters, we obtain a third determinant expression for the scalar product, where the first is due to Slavnov [1], and the second is due to Kostov and Matsuo [2]. We show that the new determinant is a discrete KP tau-function in the inhomogeneities, and consequently that tree-level N = 4 SYM structure constants that are known to be determinants, remain determinants at 1-loop level.