Now, using the ansatz () the generating functional () can be re-expressed in terms of the rotational matrix $R(t)$ in Euclidean space-time as $$Z'_{NJL}[A_{\mu}] = \int
{\cal D}R(t) \ \exp \big\lbrack - S_{eff}[A_{\mu}, R(t), \Omega] \,
\big\rbrack \ ,
\eqno(\NEWGEN)$$ where $\Omega$ is a function of $R\equiv R(t)$ (see eq. ()). Having the explicit form of the Dirac operator () we can go back to the action (). Since we treat the meson fields classically ([*i.e.*]{} at the 0–loop level) the only non-trivial part of the action is the fermionic part (). By construction (see \[,\]) it is splitted in a natural way into valence and Dirac sea parts. The valence part is finite and it will not be regularized. In this case the calculation is straightforward and the detailed derivation can be found [*e.g*]{} in \[\]. Hence, from now on we will concentrate on the sea contribution and we will restrict our discussion to that part of the full action.

As we can see from (), (), the operator () is non-Hermitian. Hence, in general, the action () will have the real and imaginary part. To make clear distinction we write down both parts explicitly:

$$\openup 2\jot
\eqalignno{
S_{eff}^F 