Power of two: Assessing the impact of a second measurement of the weak-charge form factor of ${}^{208}\mathrm{Pb}$

Background: Besides its intrinsic value as a fundamental nuclear-structure observable, the weak-charge density of ${}^{208}\mathrm{Pb}$—a quantity that is closely related to its neutron distribution—is of fundamental importance in constraining the equation of state of neutron-rich matter.

Purpose: To assess the impact that a second electroweak measurement of the weak-charge form factor of ${}^{208}\mathrm{Pb}$ may have on the determination of its overall weak-charge density.

Methods: Using the two putative experimental values of the form factor, together with a simple implementation of Bayes' theorem, we calibrate a theoretically sound—yet surprisingly little known—symmetrized Fermi function, that is characterized by a density and form factor that are both known exactly in closed form.

Results: Using the charge form factor of ${}^{208}\mathrm{Pb}$ as a proxy for its weak-charge form factor, we demonstrate that using only two experimental points to calibrate the symmetrized Fermi function is sufficient to accurately reproduce the experimental charge form factor over a significant range of momentum transfers.

Conclusions: It is demonstrated that a second measurement of the weak-charge form factor of ${}^{208}\mathrm{Pb}$ supplemented by a robust theoretical input in the form of the symmetrized Fermi function would place significant constraints on the neutron distribution of ${}^{208}\mathrm{Pb}$. In turn, such constraints will become vital in the interpretation of hadronic experiments that will probe the neutron-rich skin of exotic nuclei at future radioactive beam facilities.

Figure 1

(a) Correlation plot between the half-density radius $c$ and the surface diffuseness $a$ that define the symmetrized Fermi function. The number of points represent the raw data obtained from the Monte Carlo simulation. Also shown are the respective averages and standard deviations. The solid red line represents the functional form obtained from solving the equation $F_{\mathrm{ch}}(q_1;a,c)=0.210$. (b) Probability distribution function for the charge radius of ${}^{208}\mathrm{Pb}$ obtained from the Monte Carlo simulation.

Figure 2

Variability in the charge form factor of ${}^{208}\mathrm{Pb}$ generated by the distribution of parameters displayed in Fig. 1. Also shown with the green solid line is the relative uncertainty in the model predictions as a function of the momentum transfer.

Figure 3

(a) Correlation plot between the half-density radius $c$ and the surface diffuseness $a$ that define the symmetrized Fermi function. The number of points represent the raw data obtained from the Monte Carlo simulation. Also shown are the 39% and 95% confidence ellipses. The solid red and purple lines represents the functional form obtained from solving the equations: $F_{\mathrm{ch}}(q_1;a,c)=0.210F_{\mathrm{ch}}(q_2;a,c)=-0.061$. (b) Probability distribution function for the charge radius of ${}^{208}\mathrm{Pb}$ obtained from the Monte Carlo simulation. The black solid line represents a fit to a Gaussian probability distribution.

Figure 4

(a) Charge form factor and (b) the corresponding charge density of ${}^{208}\mathrm{Pb}$. The two red points on the left-hand panel represent the sole input used in the calibration of the symmetrized Fermi function. The theoretical predictions are displayed by an uncertainty band (in cyan) and the experimental data are from Ref. [2].

Figure 5

(a) Charge form factor and (b) the corresponding charge density of ${}^{208}\mathrm{Pb}$. The two red points represent the sole input used in the calibration of the Helm function. The theoretical predictions are displayed by an uncertainty band (in cyan) and the experimental data are from Ref. [2].

Figure 6

Charge and weak-charge density of ${}^{208}\mathrm{Pb}$ as predicted by a collection of accurately calibrated mean-field models. The labels indicate the predicted neutron-skin thickness of ${}^{208}\mathrm{Pb}$. The inset displays the fastest Gaussian falloff of the Helm form factor relative to the exponential falloff of symmetrized Fermi function. The experimental data are from Ref. [2].