Analytic structure of meson propagators in the proper-time regularized Nambu--Jona-Lasinio model

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.