  We compute the free energy in the presence of a chemical potential coupled to
a conserved charge in effective O($n$) scalar field theory (without explicit
symmetry breaking terms) to NNL order for asymmetric volumes in general
$d$--dimensions, using dimensional (DR) and lattice regularizations. This
yields relations between the 4-derivative couplings appearing in the effective
actions for the two regularizations, which in turn allows us to translate
results, e.g. the mass gap in a finite periodic box in $d=3+1$ dimensions, from
one regularization to the other. Consistency is found with a new direct
computation of the mass gap using DR. For the case $n=4, d=4$ the model is the
low-energy effective theory of QCD with $N_{\rm f}=2$ massless quarks. The
results can thus be used to obtain estimates of low energy constants in the
effective chiral Lagrangian from measurements of the low energy observables,
including the low lying spectrum of $N_{\rm f}=2$ QCD in the $\delta$--regime
using lattice simulations, as proposed by Peter Hasenfratz, or from the
susceptibility corresponding to the chemical potential used.
