  We analyze the analytic structure of meson propagators in the
Nambu--Jona-Lasinio model with a proper-time regulator. We show that the
regulator produces unphysical complex singularities. As a result the naive use
of the Wick rotation is no longer allowed. Formulas involving integration over
mesonic momenta, such as meson-loop contributions or dispersion relations for
meson Green's functions, cannot be written in usual forms.
