# Separation of variables and scalar products at any rank

## TL;DR

This paper develops a method to determine scalar product measures in higher-rank integrable models using separation of variables, enabling explicit calculations of form factors and potentially broadening applications in integrable quantum field theories.

## Contribution

It introduces a novel approach to find scalar product measures in higher-rank models without explicit SoV construction, revealing non-symmetric couplings of Q-functions and expressing form factors as determinants.

## Key findings

- Scalar product measure for higher-rank models derived without explicit SoV
- Q-functions at different nesting levels are coupled non-symmetrically
- Form factors expressed as ratios of determinants

## Abstract

Separation of variables (SoV) is a special property of integrable models which ensures that the wavefunction has a very simple factorised form in a specially designed basis. Even though the factorisation of the wavefunction was recently established for higher rank models by two of the authors and G. Sizov, the measure for the scalar product was not known beyond the case of rank one symmetry. In this paper we show how this measure can be found, bypassing an explicit SoV construction. A key new observation is that the measure for spin chains in a highest-weight infinite dimensional representation of sl(N) couples Q-functions at different nesting levels in a non-symmetric fashion. We also managed to express a large number of form factors as ratios of determinants in our new approach. We expect our method to be applicable in a much wider setup including the problem of computing correlators in integrable CFTs such as the fishnet theory, N=4 SYM and the ABJM model.