a regularization scheme of an appropriate cut-off $\Lambda$ is needed to make the theory finite. To this end we will work in Euclidean space-time (the conventions used are given in the Appendix).

After Wick rotation (transformation to the Euclidean space-time) integrating out the quarks in () one gets the effective action split in fermionic and mesonic parts: $$S_{eff} = S^F_{eff} + S^M_{eff} + S_{eff}^{br} \ ,\eqno(\ACTION)$$ The fermionic part of the action includes the quark loop contribution in the presence of the external meson fields ($\sigma, \vec\pi$): $$S^F_{eff} = -  {\rm Sp}\, \ln (\ {\slash D}_E\ ).
\eqno(\SPLOG)$$ The Dirac operator $\slash D_E$ (in Euclidean space-time) has the form: $${\slash D}_E =  \gamma_4 \ \left[\partial_4 - h(U)\right] .
\eqno(\DIRAC)$$ with the one particle Hamiltonian operator $h_E$ defined as $$h_E(U) \equiv i \, \left[ -i\, \gamma_4 \gamma_k \partial_k +
\gamma_4 \, M U \right] \ .
\eqno(\HAMIL)$$ The constituent quark mass $M=gf_{\pi}$ and $$U = {1\over f_{\pi}} \, (\sigma + i \gamma_5 \vec\tau \vec\pi)
\   \eqno(\DEFU)$$ represents the mesonic fields. It can be checked by inspection that the Hamiltonian is Hermitian: $ (h_E)^{\dagger} = h_E $.

The mesonic part $$S^M_{eff} = {\mu^2 \over 2} \int d^4 x_{E}\,( \sigma^2 +
\vec\pi^2 )
