our operator exponent ($-\tau W$), where:

$$W \equiv \  \tilde{\slash D}_E\,^{\dagger} \
\tilde{\slash D}_E
\ , \eqno(\WDEF)$$

that does not contain $A_{\mu}$ and $\Omega$. This part of $W$ we will call $W_0$. On the other hand, for the operator ${\cal B}$ we take the remaining terms of the operator $W$ and we will call them $W_1, W_2,...,W_5$. The first type of terms ($W_1$) is defined as the sum of all terms in the exponent that are linear in $A_{\mu}$,  $W_2 \sim A_{\mu} \times \Omega$,  $W_3 \sim
\Omega$,  $W_4 \sim \Omega^2$ and $W_5 \sim A_{\mu}^2$. The $W_5$ contribution to () is exactly zero, while the $W_4$ contribution we neglect as being small (quadratic in $\Omega$). Hence, the operator ${\cal B}$ will include the terms $W_1$–$W_3$ only. Finally, this gives us the following expressions for the terms $W_0$–$W_3$ of the operator $W$ ():

$$\openup 2\jot \eqalignno{
W_0 =  &+\omega^2 + h^2 \ , &(\WIa) \cr
W_1 = &- (A_4 + A_4^{\dagger}) \  R^{\dagger} \hat Q R \, \omega +
(A_k - A_k^{\dagger}) \  \gamma_4 \gamma_k \, R^{\dagger} \hat Q R
\, \omega \  + \cr
&+ i\,  ( h \, A_4\,  R^{\dagger} \hat Q R -
A_4^{\dagger}\,  R^{\dagger} \hat Q R \, 