Information content of the weak-charge form factor

Background: Parity-violating electron scattering provides a model-independent determination of the nuclear weak-charge form factor that has widespread implications across such diverse areas as fundamental symmetries, nuclear structure, heavy-ion collisions, and neutron-star structure.

Purpose: We assess the impact of precise measurements of the weak-charge form factor of ${}^{48}$Ca and ${}^{208}$Pb on a variety of nuclear observables, such as the neutron skin and the electric-dipole polarizability.

Methods: We use the nuclear density functional theory with several accurately calibrated nonrelativistic and relativistic energy density functionals. To assess the degree of correlation between nuclear observables and to explore systematic and statistical uncertainties on theoretical predictions, we employ the chi-square statistical covariance technique.

Results: We find a strong correlation between the weak-charge form factor and the neutron radius, that allows for an accurate determination of the neutron skin of neutron-rich nuclei. We determine the optimal range of the momentum transfer $q$ that maximizes the information content of the measured weak-charge form factor and quantify the uncertainties associated with the strange quark contribution. Moreover, we confirm the role of the electric-dipole polarizability as a strong isovector indicator.

Conclusions: Accurate measurements of the weak-charge form factor of ${}^{48}$Ca and ${}^{208}$Pb will have a profound impact on many aspects of nuclear theory and hadronic measurements of neutron skins of exotic nuclei at radioactive-beam facilities.

Figure 1

Systematic trends as predicted by the various families of models for the following isovector observables: the neutron skin, electric-dipole polarizability, weak-charge form factor $F_W(q=0.475{\mathrm{fm}}^{-1})$ of ${}^{208}$Pb, and the weak-charge form factor $F_W(q=0.778{\mathrm{fm}}^{-1})$ of ${}^{48}$Ca. Experimental values are indicated by black squares with error bars ($r_{\mathrm{skin}}^{208}$ and $F_W^{208}$ from Refs. [1, 2] and ${\alpha }_D^{208}$ from Ref. [41]).

Figure 2

Correlation coefficients (9) derived from a covariance analysis between the weak-charge form factors of ${}^{208}$Pb ($F_W^{208}$) and ${}^{48}$Ca ($F_W^{48}$), and a variety of nuclear observables, including the strong isovector indicators such as neutron skin, electric dipole polarizability, symmetry energy $J$, slope of the symmetry energy $L$, and a strong isoscalar indicator: incompressibility $K$ at saturation density. In all cases correlation coefficients are obtained from the covariance matrix associated with each model.

Figure 3

Weak-charge form factors with corresponding theoretical errors for ${}^{48}$Ca and ${}^{208}$Pb as predicted by SV-min and FSUGold. Note that the theoretical error bars have been artificially increased by a factor of 10. Indicated in the figure are the values of the momentum transfer appropriate for PREX-II ($q=0.475{\mathrm{fm}}^{-1}$) and CREX ($q=0.778{\mathrm{fm}}^{-1}$).

Figure 4

Correlation coefficient (9) between $r_n^{208}$ and $F_W^{208}\left(q\right)$ as a function of the momentum transfer $q$. Panel (a) shows the absolute value of the correlation coefficient predicted by SV-min and FSUGold assuming no strange-quark contribution to the nucleon form factor. Panel (b) shows the impact of including the experimental uncertainty in the strange-quark contribution to the nucleon form factor. The arrow marks the PREX-II momentum transfer of $q=0.475{\mathrm{fm}}^{-1}$. The first dashed vertical line indicates the position of the first zero of $F_W^{208}\left(q\right)$, the second one marks the position of the first maximum of $|F_W^{208}\left(q\right)|$ (from which the surface thickness can be deduced).