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The LeClair-Mussardo series and nested Bethe Ansatz

We consider correlation functions in one dimensional quantum integrable models related to the algebra symmetries $\mathfrak{gl}(2|1)$ and $\mathfrak{gl}(3)$. Using the algebraic Bethe Ansatz approach we develop an expansion theorem, which leads to an infinite integral series in the thermodynamic limit. The series is the generalization of the LeClair-Mussardo series to nested Bethe Ansatz systems, and it is applicable both to one-point and two-point functions. As an example we consider the ground state density-density correlator in the Gaudin-Yang model of spin-1/2 Fermi particles. Explicit formulas are presented in a special large coupling and large imbalance limit.