Variations on Slavnov's scalar product

Abstract

We consider the rational six-vertex model on an L×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 Casoratian, and consequently that tree-level N =4 SYM structure constants that are known to be determinants, remain determinants at 1-loop level. © SISSA 2012.