  Effective Polyakov line actions are a powerful tool to study the finite
temperature behaviour of lattice gauge theories. They are much simpler to
simulate than the original (3+1) dimensional LGTs and are affected by a milder
sign problem. However it is not clear to which extent they really capture the
rich spectrum of the original theories, a feature which is instead of great
importance if one aims to address the sign problem. We propose here a simple
way to address this issue based on the so called second moment correlation
length $\xi_{2nd}$. The ratio $\xi/\xi_{2nd}$ between the exponential
correlation length and the second moment one is equal to 1 if only a single
mass is present in the spectrum, and becomes larger and larger as the
complexity of the spectrum increases. Since both $\xi_{exp}$ and $\xi_{2nd}$
are easy to measure on the lattice, this is an economic and effective way to
keep track of the spectrum of the theory. In this respect we show using both
numerical simulation and effective string calculations that this ratio
increases dramatically as the temperature decreases. This non-trivial behaviour
should be reproduced by the Polyakov loop effective action.
