\ ,\eqno(\ACTa)$$ and $$S_{eff}^{br} = - {m_o \mu^2 \over g} \int d^4x_E \ \sigma \
,\eqno(\ACTb)$$ includes an explicit symmetry breaking term ().

In order to compute the electromagnetic current matrix elements we include the electromagnetic coupling in a minimal way and we take into account the rotational zero modes. We start with the Dirac operator ${\slash D}_E$ with electromagnetic potential $A_\mu$: $${\slash D}_E =  \gamma_4 \ \left[
\partial_4 - h(U) - i A_4 \hat Q +
i \gamma_4 \gamma_k A_k \hat Q \right] \ ,
\eqno(\DIRACEL)$$ The quark charge matrix $\hat Q \equiv {1\over 6} \hat1 + {1\over2} \tau^3$.

In the next step, we introduce the rotating soliton: $$\tilde U(\vec x,x_4) \equiv R(x_4) \, \bar U(\vec x) \,
R^{\dagger}(x_4), \qquad \ R\in SU_2,\qquad R^{\dagger} R = \hat 1.
\eqno(\MESF)$$ where the meson field $\bar U(\vec x)$ corresponds to the stationary point (minimum) of the effective action (). In the Euclidean space-time the rotation matrix $R(x_4)$, can be represented as:

$$R(x_4) = \exp(-i x_4 \, \Omega) \ , \quad\quad
\Omega \equiv {1\over 2} \tau^A \, \Omega^A
\quad \hbox{and}\quad
\Omega_E = - i \ R^{\dagger} \, \dot{R}
\ ,  \eqno(\RDEF)$$

where as usually the summation convention over repeated indices is assumed.

