On the physical part of the factorized correlation functions of the XXZ chain

It was recently shown by Jimbo, Miwa and Smirnov that the correlation functions of a generalized XXZ chain associated with an inhomogeneous six-vertex model with disorder parameter $\alpha$ and with arbitrary inhomogeneities on the horizontal lines factorize and can all be expressed in terms of only two functions $\rho$ and $\omega$. Here we approach the description of the same correlation functions and, in particular, of the function $\omega$ from a different direction. We start from a novel multiple integral representation for the density matrix of a finite chain segment of length $m$ in the presence of a disorder field $\alpha$. We explicitly factorize the integrals for $m=2$. Based on this we present an alternative description of the function $\omega$ in terms of the solutions of certain linear and nonlinear integral equations. We then prove directly that the two definitions of $\omega$ describe the same function. The definition in the work of Jimbo, Miwa and Smirnov was crucial for the proof of the factorization. The definition given here together with the known description of $\rho$ in terms of the solutions of nonlinear integral equations is useful for performing e.g.\ the Trotter limit in the finite temperature case, or for obtaining numerical results for the correlation functions at short distances. We also address the issue of the construction of an exponential form of the density matrix for finite $\alpha$.