Parity violation and dynamical relativistic effects in $(\vec{e},e^{\prime }N)$ reactions

It is well known that coincidence quasielastic $(\vec{e},e^{\prime }N)$ reactions are not appropriate to analyze effects linked to parity violation due the presence of the fifth electromagnetic response $R^{T{L}^{\prime }}$. Nevertheless, in this work we develop a fully relativistic approach to be applied to parity-violating (PV) quasielastic $(\vec{e},e^{\prime }N)$ processes. This is of importance as a preliminary step in the subsequent study of inclusive quasielastic PV $(\vec{e},e^{\prime })$ reactions. Moreover, our present analysis allows us to disentangle effects associated with the off-shell character of nucleons in nuclei, gauge ambiguities, and the role played by the lower components in the nucleon wave functions, i.e., dynamical relativistic effects. This study can help in getting clear information on PV effects. Particular attention is paid to the relativistic plane-wave impulse approximation where the explicit expressions for the PV single-nucleon responses are shown for the first time.

Figure 1

General scheme of the scattering process $A(e,e^{\prime }N)B$. The scattering frame is defined by $\{\widehat{\mathit{x}},\widehat{\mathit{y}},\widehat{\mathit{z}}\}$, whereas the hadronic reference system is given by $\{\widehat{\mathbf{1}},\widehat{\mathbf{2}},\widehat{\mathbf{3}}\}$. Also shown are the four-momenta and angular variables that enter into the description of the process.

Figure 2

Transverse single-nucleon responses: $T$ (upper panels), $TT$ (middle), and $T^{\prime }$ (bottom). The separate projection components are shown: $uu$ (left panels), $uv$ (central), and $vv$ (right). Results are presented for the CC1 (thin lines) and CC2 (thick) current operators.

Figure 3

As for Fig. 2, except that now the results are for the longitudinal responses: $L$ (upper panels), interference $TL$ (middle), and $T{L}^{\prime }$ (bottom). Again the two currents have been selected, CC1 (thin lines) and CC2 (thick); and the three gauges: Landau (solid lines), Coulomb (dashed), and Weyl (dot-dashed).

Figure 4

Hadronic response functions for a proton in the $1p_{1/2}$ shell in ${}^{16}\mathrm{O}$. The labels of the various curves are as in Fig. 3.

Figure 5

Energy projection contributions to the hadronic response functions for the $1p_{1/2}$ shell in ${}^{16}\mathrm{O}$: $uu$ (blue dotted line), $uv$ (red dashed), and $vv$ (green dot-dashed). The total responses are represented by the solid black lines. The six top panels correspond to results obtained with the Landau gauge and CC1 current operator, whereas the bottom ones refer to the CC2 current.

Figure 6

As for Fig. 5 but now for a proton in the $1p_{3/2}$ shell in ${}^{16}\mathrm{O}$.

Figure 7

PV hadronic responses for a proton in the $1p_{1/2}$ shell in ${}^{16}\mathrm{O}$. In the upper panels results in RPWIA (green lines) are compared with the RDWIA ones using the EDAI-O optical potential (blue). The bottom panels present RDWIA-EDAI-O results (green) compared with the ones evaluated with the effective momentum approximation (EMA) (red). See text for details.

Figure 8

Asymmetry ${\mathcal{A}}_{\mathrm{excl}}$ for the case of a proton in the $1p_{1/2}$ shell in ${}^{16}\mathrm{O}$. The kinematics are fixed to $q=0.5$ GeV/$c$ and $\omega =0.132$ GeV, and two values for the scattering angle have been selected: ${\theta }_e={15}^{\circ }$ (forward scattering, left panels) and ${\theta }_e={140}^{\circ }$ (backward, right panels). Coplanar kinematics are used, i.e., ${\varphi }_N=0^{\circ }$. The top (bottom) panels refer to results evaluated with the NCC1 (NCC2) prescriptions. In each case we compare the asymmetry corresponding to the RPWIA, RPWIA-uu, RDWIA, and EMA approaches. FSI have been evaluated using the EDAI-O potential.

Figure 9

Exclusive helicity asymmetry for the $p_{1/2}$ shell (protons) in ${}^{16}\mathrm{O}$. Results correspond to RDWIA and NCC2 prescription. The azimuthal angle has been fixed to ${\varphi }_N=0.{01}^{\circ }$ (top panels) and ${\varphi }_N=0.{001}^{\circ }$ (bottom). Forward and backward kinematics have been explored in the left and right panels, respectively.