Measurements of the induced polarization in the quasi-elastic $A(e,e'\vec p\,)$ process in non-coplanar kinematics

We report measurements of the induced polarization $\vec P$ of protons knocked out from $^2$H and $^{12}$C via the $A(e,e'\vec p\,)$ reaction. We have studied the dependence of $\vec P$ on two kinematic variables: the missing momentum $p_{\rm miss}$ and the "off-coplanarity" angle $\phi_{pq}$ between the scattering and reaction planes. For the full 360$\degree$ range in $\phi_{pq}$, both the normal ($P_y$) and, for the first time, the transverse ($P_x$) components of the induced polarization were measured with respect to the coordinate system associated with the scattering plane. $P_x$ vanishes in coplanar kinematics, however in non-coplanar kinematics, it is on the same scale as $P_y$. We find that the dependence on $\phi_{pq}$ is sine-like for $P_x$ and cosine-like for $P_y$. For carbon, the magnitude of the induced polarization is especially large when protons are knocked out from the $p_{3/2}$ shell at very small $p_{\rm miss}$. For the deuteron, the induced polarization is near zero at small $|p_{\rm miss}|$, and its magnitude increases with $|p_{\rm miss}|$. For both nuclei such behavior is reproduced qualitatively by theoretical results, driven largely by the spin-orbit part of the final-state interactions. However, for both nuclei, sizeable discrepancies exist between experiment and theory.