# New approach to scalar products of Bethe vectors

## TL;DR

This paper introduces a new method for calculating scalar products of Bethe vectors in quantum integrable models with $rak{gl}(2)$ symmetry, providing determinant formulas for both periodic and boundary conditions.

## Contribution

The authors develop a novel approach based on transfer matrix action to derive determinant representations of scalar products in Bethe ansatz models.

## Key findings

- Determinant formulas for scalar products in periodic boundary conditions
- Determinant formulas for scalar products with reflection algebra boundary conditions
- New method simplifies calculations of scalar products in integrable models

## Abstract

We consider quantum integrable models solvable by the algebraic Bethe ansatz and possessing $\mathfrak{gl}(2)$-invariant $R$-matrix. We study the models of both periodic boundary conditions and boundary conditions based on reflection algebra. We present a new method to calculate the scalar products based on formula of an action of transfer matrix of a model onto Bethe vector. Using this method we identify some coefficient in the multiple action of the transfer matrix with the scalar product between on-shell and off-shell Bethe vectors. This allows us to find determinant representation of the scalar products in both types of boundary conditions.