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

We study the scalar product Sℓ,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 Sℓ,m to scalar products of smaller dimension, namely Sℓ−1,m and Sℓ,m−1. Solving these recursion relations we obtain new multiple integral expressions for Sℓ,m, whose integrands are (ℓ+m)×(ℓ+m) determinants, and closely related to the Slavnov determinant expression for the SU(2) scalar product.