Multiple integral formulae for the scalar product of on-shell and off-shell Bethe vectors in SU(3)-invariant models

TL;DR

This paper derives new multiple integral formulas for the scalar product between on-shell and off-shell Bethe vectors in SU(3)-invariant models, extending known results from SU(2) cases and providing recursive relations.

Contribution

It introduces recursive relations and multiple integral formulas for scalar products in SU(3) models, generalizing the Slavnov determinant approach from SU(2).

Findings

Derived recursion relations for scalar products.

Obtained multiple integral expressions involving determinants.

Connected results to known SU(2) scalar product formulas.

Abstract

We study the scalar product S_{l,m} between an on-shell and an off-shell Bethe state in models with SU(3)-invariance, where l and m denote the cardinalities of the two sets of Bethe roots. We construct recursion relations relating S_{l,m} to scalar products of smaller dimension, namely S_{l-1,m} and S_{l,m-1}. Solving these recursion relations we obtain new multiple integral expressions for S_{l,m}, whose integrands are (l+m) \times (l+m) determinants, and closely related to the Slavnov determinant expression for the SU(2) scalar product.