$A\left(\vec{e},e^{\prime }\vec{p}\right)B$ responses: From bare nucleons to complex nuclei

We study the occurrence of factorization in polarized and unpolarized observables in coincidence quasielastic electron scattering. Starting with the relativistic distorted wave impulse approximation, we reformulate the effective momentum approximation and show that the latter leads to observables which factorize under some specific conditions. Within this framework, the role played by final state interactions and, in particular, by the spin-orbit term is explored. Connection with the nonrelativistic formalism is studied in depth. Numerical results are presented to illustrate the analytical derivations and to quantify the differences between factorized and unfactorized approaches.

Figure 1

Reduced cross section for proton knockout from $1p_{1∕2}$ (upper panel) and $1p_{3∕2}$ (middle panel) in ${}^{16}\mathrm{O}$ and from $2s_{1∕2}$ in ${}^{40}\mathrm{Ca}$ (lower panel). RDWIA calculations (solid line) are compared to EMA (short-dashed line) and EMA-noLS (dotted line) results. The corresponding relativistic distorted wave momentum distribution is also plotted (thin solid line). Negative (positive) $p_m$ values correspond to ${\varphi }_F=0°$ $\left(180°\right)$, respectively.

Figure 2

$A_{TL}$ asymmetry for proton knockout from $1p_{1∕2}$ (left panel) and $1p_{3∕2}$ (middle panel) in ${}^{16}\mathrm{O}$ and from $2s_{1∕2}$ in ${}^{40}\mathrm{Ca}$ (right panel). RDWIA calculations (solid line) are compared to RDWIA-noLS (dashed line), EMA (short-dashed line), EMA-noLS (dotted line) and factorized (dash-dotted line) results. The EMA-noLS calculation coincides in all panels with the factorized $\left(A_{TL}^{bare}\right)$ result. In the right hand panel EMA, as well as EMA-noLS, coincides with the factorized $\left(A_{TL}^{bare}\right)$ result.

Figure 3

Electron analyzing power $A$ at $(q,\omega )$ constant kinematics and azimuthal angle ${\varphi }_F=225°$. The labeling of the curves is as in Fig. 2. For this observable factorization is only achieved in the EMA-noLS curve on the right hand panel. See text for details.

Figure 4

Induced polarization $P_n$ at coplanar kinematics with ${\varphi }_F=0°$. Kinematics and labeling as in Fig. 2. Only the EMA-noLS calculation for a $s_{1∕2}$ shell factorizes.

Figure 5

Longitudinal transferred polarization $P_l^{\prime }$ (top panels) and sideways transferred polarization $P_s^{\prime }$ (bottom panels) at coplanar kinematics $\left({\varphi }_F=0°\right)$. In this case, factorization is obtained within the EMA approach when there is no spin-orbit coupling in the final state (EMA-noLS, dotted line).