  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 lattice model and are affected by a milder sign
problem, but it is not clear to which extent they really capture the rich
spectrum of the original theories. 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
it becomes larger and larger as the complexity of the spectrum increases. Since
both $\xi$ and $\xi_{2nd}$ are easy to measure on the lattice, this is a cheap
and efficient way to keep track of the spectrum of the theory. As an example of
the information one can obtain with this tool we study the behaviour of
$\xi/\xi_{2nd}$ in the confining phase of the ($D=3+1$) $\mathrm{SU}(2)$ gauge
theory and show that it is compatible with 1 near the deconfinement transition,
but it increases dramatically as the temperature decreases. We also show that
this increase can be well understood in the framework of an effective string
description of the Polyakov loop correlator. This non-trivial behaviour should
be reproduced by the Polyakov loop effective action; thus, it represents a
stringent and challenging test of existing proposals and it may be used to
fine-tune the couplings and to identify the range of validity of the
approximations involved in their construction.
