  We consider the spectrum of correlation lengths of the spin-$\frac{1}{2}$ XXZ
chain in the antiferromagnetic massive regime. These are given as ratios of
eigenvalues of the quantum transfer matrix of the model. The eigenvalues are
determined by integrals over certain auxiliary functions and by their zeros.
The auxiliary functions satisfy nonlinear integral equations. We analyse these
nonlinear integral equations in the low-temperature limit. In this limit we can
determine the auxiliary functions and the expressions for the eigenvalues as
functions of a finite number of parameters which satisfy finite sets of
algebraic equations, the so-called higher-level Bethe Ansatz equations. The
behaviour of these equations, if we send the temperature $T$ to zero, is
different for zero and non-zero magnetic field $h$. If $h$ is zero the
situation is much like in the case of the usual transfer matrix. Non-trivial
higher-level Bethe Ansatz equations remain which determine certain complex
excitation parameters as functions of hole parameters which are free on a line
segment in the complex plane. If $h$ is non-zero, on the other hand, a
remarkable restructuring occurs, and all parameters which enter the description
of the quantum transfer matrix eigenvalues can be interpreted entirely in terms
of particles and holes which are freely located on two curves when $T$ goes to
zero.
