Nuclear transparency and shadowing of the two-step process in the $A(e,e'\pi^+)$ reaction

We investigate nuclear transparency induced by pion electroproduction on nuclei, $A(e,e'\pi^+)$, with our interest in the role of the two-step process in the Glauber multiple scattering theory. Based on the framework in which the quantum diffusion model is incorporated into the Glauber theory, the experimental data in the range of photon virtuality $0.5< Q^2< 10$ GeV$^2/c^2$ are analyzed, including the recent JLab data at the 6 GeV electron beam together with the proposal for the 12 GeV upgrade. Application of the quantum diffusion model during the formation length of the quark/antiquark $(q\bar{q})$ structure of a pion reproduces the increase in the nuclear transparency up to high $Q^2$, showing evidence of color transparency. In contrast, through the coherent length of the $\gamma^*$ fluctuation into $\rho^0$, the two-step process with the $\rho N$ scattering cross section chosen to be $\sigma_{\rho N}=\sigma_{\pi N}$ contributes to a reduction of the nuclear transparency in the whole range of $Q^2$. Combined with the one-step process (direct photon coupling) usually adopted in the original formulation of the Glauber theory, this reaction mechanism provides another source of the $Q^2$ dependence of the transparency at low virtualities through the shadowing in the initial state interaction. By the further absorption due to the shadowing, a better agreement is obtained with the pion transparency measured in the $A(e,e'\pi^+)$ reaction for $^{12}$C, $^{27}$Al, $^{63}$Cu, and $^{197}$Au nuclei. A discussion of the $Q^2$ dependence of the parameter $\alpha$ for the transparency $T_A=A^{\alpha-1}$ is given.