On the normalization of the partition function of Bethe Ansatz systems

In this note I revisit the calculation of partition function of simple one-dimensional systems solvable by Bethe Ansatz. Particularly I show that by the precise definition and treatment of the partition function the nontrivial normalization factor proposed in a recent work to give the correct O(1)O(1) corrections to the free energy can be derived in a straightforward manner.