operator ${\cal A}$ its proper-time regularized form reads \[\]:

$$\ln {\cal A} = - \lim_{\Lambda\to\infty} \,
\int^{\infty}_{1/\Lambda^2} {d\tau\over\tau}
e^{-\tau{\cal A}}
\ . \eqno(\PROP)$$

The Schwinger method is especially convenient for the logarithmic expressions resulting from the fermionic determinant. Applying () to the action () we obtain:

$$S^F_{eff} = {N_c \over 2}  \sum_{n} \int d^4 x_{E}\,
Tr_{\gamma} Tr_{\tau} \int_{1 \over\Lambda^2}^\infty {d\tau \over\tau}
\,\psi_{n}^{\dagger} (x_{E})\,\, \exp[\,-\tau
\,\tilde{\slash D}\,^{\dagger} \,
\tilde{\slash D} \,] \ \psi_{n} (x_{E})
\ , \eqno(\REGACTIO)$$

where $\psi_n(x)$ is any complete set of eigenfunctions. Actually, in our calculations we shell use for $\psi_n(x)$ eigenfunctions of the Hamiltonian $h$. For the meson fields $\sigma, \vec\pi$ we substitute the static selfconsistent hedgehog solution $\bar U(\vec x)$ of the chiral soliton \[\] form:

$$\sigma(x) = \sigma(r), \quad \quad \vec\pi(x) =
\hat r \, \pi(r)
\ , \eqno(\HEDANZ)$$

restricted to the chiral circle (). Numerically, we will follow the method of Ripka and Kahana \[\] and will use the eigenfunctions of the Hamiltonian $h$ to compute all matrix elements (see also Section 4).

Now, our task is to evaluate the current () with the regularized action (). The difficulty lies in the fact that the expression () contains rather complicated operator exponent in the 