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We consider the spectrum of correlation lengths of the spin-$\frac{1}{2}$ XXZchain in the antiferromagnetic massive regime. These are given as ratios ofeigenvalues of the quantum transfer matrix of the model. The eigenvalues aredetermined by integrals over certain auxiliary functions and by their zeros.The auxiliary functions satisfy nonlinear integral equations. We analyse thesenonlinear integral equations in the low-temperature limit. In this limit we candetermine the auxiliary functions and the expressions for the eigenvalues asfunctions of a finite number of parameters which satisfy finite sets ofalgebraic equations, the so-called higher-level Bethe Ansatz equations. Thebehaviour of these equations, if we send the temperature $T$ to zero, isdifferent for zero and non-zero magnetic field $h$. If $h$ is zero thesituation is much like in the case of the usual transfer matrix. Non-trivialhigher-level Bethe Ansatz equations remain which determine certain complexexcitation parameters as functions of hole parameters which are free on a linesegment in the complex plane. If $h$ is non-zero, on the other hand, aremarkable restructuring occurs, and all parameters which enter the descriptionof the quantum transfer matrix eigenvalues can be interpreted entirely in termsof particles and holes which are freely located on two curves when $T$ goes tozero.

Low-temperature spectrum of correlation lengths of the XXZ chain in the antiferromagnetic massive regime