The Bethe Ansatz

\[ S_n (\bar{v} | \bar{u}) = \frac{1}{d(\bar{v}) d(\bar{u})} ~\langle 0 | C(\bar{v}) B(\bar{u}) | 0 \rangle \]

Duality relations for Monodromy matrix operators: for \(c=i\), as stated in the text before equation Slavnov2018-9-1.3,

\[ D(u^*) = A^\dagger (u), \hspace{10mm} C(u^*) = -B^\dagger (u) \]

which directly comes from the \(L\) operator and extending to the monodromy matrix, using the self-duality of the Bethe roots \(\bar{u}^* = \bar{u}\) valid for on-shell vectors.

Special products:

\[ \Delta_n (\bar{v}) = \prod_{1 \leq j < k \leq n} g(v_k, v_j), \hspace{10mm} \Delta'_n (\bar{v}) = \prod_{1 \leq j < k \leq n} g(v_j, v_k) = (-1)^{n(n-1)/2} \Delta_n(\bar{v}) \]

Useful functions:

\[ h(u,v) = \frac{f(u,v)}{g(u,v)} = \frac{u-v+c}{c}, \hspace{10mm} t(u,v) = \frac{g(u,v)}{h(u,v)} = \frac{c^2}{(u-v)(u-v+c)} \]

Final forms for the scalar product:

\[ S_n (\bar{v} | \bar{u}) = \Delta'_n (\bar{u}) \Delta_n (\bar{v}) h(\bar{v}, \bar{u}) ~\mbox{det}_n~ {\cal M}_{jk}, \]

\[ {\cal M}_{jk} = t(v_k, u_j) \left( \kappa - r(v_k) \frac{f(\bar{u}_j, v_k)}{f(v_k, \bar{u}_j)} \right) \]

This can alternatively be expressed in terms of (derivatives of) the eigenvalue of the transfer matrix,

\[ \tau(v|\bar{u}) = a(v) f (\bar{u}, v) + d(v) f(v, \bar{u}) \]

as

\[ S_n (\bar{v} | \bar{u}) = \frac{\Delta'_n (\bar{u}) \Delta(\bar{v})}{g(\bar{v}, \bar{u})} ~\mbox{det}~ \left( \frac{c}{d(v_k)} \frac{\partial}{\partial u_j} \tau (v_k | \bar{u}) \right) \]

which corresponds to ssp.