Radiative corrections to the parity-violating spin asymmetry

Abstract

The parity-violating spin asymmetry Apv for elastic electron scattering from spin-zero nuclei, together with its QED corrections, is evaluated non-perturbatively within the phase-shift analysis. Dispersion corrections, taking into account low-lying transient nuclear excitations, are estimated with the help of the lowest-order $γZ$ box diagrams. Collision energies between 5 $-$ 500 MeV are considered, and results are provided for the $^{12}$C and $^{208}$Pb target nuclei. In addition, we have evaluated Apv at GeV energies and small scattering angles -- relevant for the Pb Radius Experiment (PREx) -- revealing that the low-lying nuclear excited states give no measurable dispersive contribution. However, they are important at lower energies and backward angles.

Figures (14)

• Figure 1: Feynman box diagrams for dispersion. Diagram (a) describes the electron (e) nucleus (N) coupling by a photon ($\gamma$) with momentum $q_1$ and subsequently by a $Z$ boson with momentum $q_2$. $N^*$ indicates the excitation of the nucleus $N$ while $k_i$ and $k_f$ are the momenta of the incoming and scattered electron. In diagram (b), the order is reversed.
• Figure 2: Ground-state charge density $\varrho_0$ of $^{12}$C as function of the distance $R_N$ from the nuclear center. Shown are the parametrization Gauss (-------), the Fourier-Bessel fit $(\cdots\cdots$) and the result from the nuclear model $(-\cdot -\cdot -)$. Included is the negative value of the weak density $\varrho_{\rm w}\;(----)$.
• Figure 3: Differential cross section $\frac{d\sigma_{\rm coul}}{d\Omega}$ for unpolarized electrons (a) scattering from $^{12}$C at 55 MeV ($----$) and at 155 MeV (----------), as well as from $^{208}$Pb at 155 MeV ($-\cdot -\cdot -)$ as function of the scattering angle $\theta$, using the charge distribution Gauss. Included are the results from the numerical $\varrho_0$ for 155 MeV ($\cdots\cdots$). (b) scattering from $^{12}$C at $\theta = 60^\circ \;(-\cdot -\cdot -)$ and $100^\circ$ (----------) as function of collision energy $E_{\rm i,kin}$, using the Gauss $\varrho_0$. Included are the results from the numerical $\varrho_0$ for $60^\circ \; (\cdots\cdots)$ and $100^\circ \; (----)$.
• Figure 4: Parity-violating asymmetry $A_{\rm pv}$ for electron scattering from $^{12}$C (a) at 55 MeV ($-\cdot -\cdot -$, multiplied by a factor of 10) and 155 MeV (----------) as function of scattering angle $\theta$ and (b) at $60^\circ \; (-\cdot -\cdot -)$ and $100^\circ$ (-------------) as function of collision energy, using the charge distribution Gauss. The results from the numerical charge density are also shown, at 55 MeV $(\cdots\cdots$, multiplied by 10) and 155 MeV $(----)$ in (a) and at $60^\circ \; (\cdots\cdots)$ and $100^\circ \;(-----)$ in (b).
• Figure 5: $^{12}$C transition form factors as function of momentum transfer (a) for the isovector $1^-$ state at 19.6 MeV and (b) for the isoscalar $2^+$ state at 4.439 MeV. Shown are the proton form factor $F_L^{\rm c}$ (-----------), the weak form factor $F_L^{\rm c,w}$ (--$\circ$--), the transverse electric form factors $F_{L,L+1}^{\rm te}\; (-\cdot - \cdot -)$ and $F_{L,L-1}^{\rm te}\;(----)$ as well as their weak counterparts $F_{L,L+1}^{\rm te,w}\; (-\cdot -\circ - \cdot -)$ and $F_{L,L-1}^{\rm te,w}\; (--\circ --)$ for (a) $L=1$ and (b) $L=2$.