Electric Dipole Polarizability in ${}^{208}$Pb: insights from the Droplet Model

We study the electric dipole polarizability $\alpha_D$ in ${}^{208}$Pb based on the predictions of a large and representative set of relativistic and non-relativistic nuclear mean field models. We adopt the droplet model as a guide to better understand the correlations between $\alpha_D$ and other isovector observables. Insights from the droplet model suggest that the product of $\alpha_D$ and the nuclear symmetry energy at saturation density $J$ is much better correlated with the neutron skin thickness $\Delta r_{np}$ of ${}^{208}$Pb than the polarizability alone. Correlations of $\alpha_D J$ with $\Delta r_{np}$ and with the symmetry energy slope parameter $L$ suggest that $\alpha_D J$ is a strong isovector indicator. Hence, we explore the possibility of constraining the isovector sector of thenuclear energy density functional by comparing our theoretical predictions against measurements of both $\alpha_D$ and the parity-violating asymmetry in ${}^{208}$Pb. We find that the recent experimental determination of $\alpha_D$ in ${}^{208}$Pb in combination with the range for the symmetry energy at saturation density $J=[31\pm (2)_{\rm est.}]$\,MeV suggests $\Delta r_{np}({}^{208}{\rm Pb}) = 0.165 \pm (0.009)_{\rm exp.} \pm (0.013)_{\rm theo.} \pm (0.021)_{\rm est.} {\rm fm}$ and $L= 43 \pm(6)_{\rm exp.} \pm (8)_{\rm theo.}\pm(12)_{\rm est.}$ MeV.


I. INTRODUCTION
Experimental and theoretical studies of isospin sensitive observables, such as the electric dipole polarizability, the neutron skin thickness, and the parity violating asymmetry, are crucial for a better understanding of the isovector sector of the nucleon-nucleon effective interaction and for constraining present and future nuclear energy density functionals (EDFs) [1][2][3]. The isovector properties of the nuclear Equation of State (EoS) are governed by the nuclear symmetry energy. The symmetry energy S(ρ) encodes the energy cost per nucleon in converting all the protons into neutrons in symmetric nuclear matter. Knowledge of the symmetry energy and of its density dependence is critical for understanding many properties of a variety of nuclear and astrophysical systems, such as the ground and excited state properties of nuclei [4], many aspects of heavy-ion collisions at different projectile-target asymmetries [5], and the structure, composition, and dynamics of neutron stars [6].
The electric dipole polarizability α D in 208 Pb has been recently measured at the Research Center for Nuclear * xavier.roca.maza@mi.infn.it Physics (RCNP) [1] via polarized proton inelastic scattering at forward angles. This experimental technique allows the extraction of the electric dipole response in 208 Pb over a wide energy range with high resolution [1]. By taking the average of all available data on the electric dipole polarizability in 208 Pb [7,8], a value of α D = 20.1±0.6 fm 3 was reported [1]. This value, in combination with the covariance analysis performed for a given Skyrme functional [9] constrained the neutron skin thickness in 208 Pb to be ∆r np = 0.156 +0.025 −0.021 fm [1]. A subsequent systematic study based on a large class of EDFs was able to confirm the correlation between α D and ∆r np [3]. This study extracted a neutron skin thickness ∆r np = 0.168 ± 0.022 fm using the same experimental value of α D .
The purpose of this manuscript is threefold. First, we resort to a macroscopic approach for describing the dipole polarizability, which enables one to qualitatively understand, in a simple and transparent way, the correlation between the electric dipole polarizability and the parameters that characterize the nuclear symmetry energy. Second, through a comprehensive ensemble of microscopic calculations performed with different types of EDFs [10][11][12][13][14][15][16][17] we provide a quantitative analysis which allows to define the regions where the experiment and the adopted microscopic approaches are compatible. Finally, the isospin properties of the considered EDFs are further investigated by the analysis of the dipole polarizability in combination with the parity violating asymmetry measured in polarized elastic electron scattering.
The manuscript has been organized as follows. In Sec. II we introduce the microscopic and macroscopic models used in this work. In particular, we discuss some of the critical insights provided by the macroscopic droplet model. In the next section results are presented for the correlations between the electric dipole polarizability and both the neutron skin thickness and the parity violating asymmetry. Finally, we offer our conclusions in Sec. IV.

II. THEORETICAL FRAMEWORK
In the present section we introduce the theoretical formalism that will be used to compute the various observables discussed in this work. In particular, we briefly review the mean-field plus Random Phase Approximation (RPA) techniques used to compute the distribution of isovector dipole strength. Moreover, we make connection to the macroscopic droplet model (DM) and discuss the critical insights that emerge from such a simplified, yet powerful, description.

A. Microscopic models
For the theoretical calculations presented in this work we use a set of non-relativistic and relativistic selfconsistent mean field models to predict ground-state properties of finite nuclei at either the Hartree-Fock or Hartree levels, respectively. These mean field models have been accurately calibrated to certain groundstate data, such as binding energies and charge radii of selected nuclei (including 208 Pb) as well as to a few empirical properties of infinite nuclear matter at, or around, saturation density. To deal with dynamic properties of the system, such as the electric dipole polarizability, the models adopt the linearization of the timedependent Hartree or Hartree-Fock equations in a fully self-consistent manner. That is, the residual interaction employed in the calculation of the linear response is consistent with the one used to generate the meanfield ground state. This technique is widely known as the Random Phase Approximation [18]. From the RPA calculations we obtain the distribution of the electric dipole strength R(ω; E1) by considering the dipole operator where N , Z, and A are the neutron, proton, and mass numbers, respectively, r n(p) indicates the radial coordinate for neutrons (protons), and Y 1M (r) is the corre-sponding spherical harmonic. Using this definition of the dipole operator allows one to eliminate any contamination to the physical response from the spurious state [18,19]. Further details about these RPA calculations may be found in [9,15,17,20] and references therein. Once the electric dipole strength R(ω; E1) is determined as a function of the excitation energy ω, the dipole polarizability α D can be computed as where m −1 (E1) is the sum of inverse energy weighted strength.

B. Macroscopic model
The RPA formalism described above suggests that the extraction of the inverse energy weighted sum requires the evaluation of the full distribution of dipole strength R(ω; E1). However, given that only the m −1 moment is required-as opposed to the full distribution of strength-a significantly more efficient computation of the dipole polarizability relies on the so-called dielectric theorem [21,22]. In this case, one solves the groundstate problem associated with the model Hamiltonian H under the constraint of a weak one-body term of the form λD, where D is the dipole operator. That is, one searches for the constrained wave function |λ solution of H ′ = H + λD. The dielectric theorem establishes that the m −1 moment may be computed from the expectation value of the Hamiltonian in the constrained ground state as Note that this represents an enormous simplification, as the constrained energy may be obtained from a meanfield calculation, without recourse to the RPA. Applying the same type of procedure but solving the constrained problem classically by using the DM approach of Myers and Swiatecki [23] one obtains the following result: which was first derived by Meyer, Quentin, and Jennings [24]. In this equation r 2 is the mean-square radius of the nucleus, J is the nuclear symmetry energy at saturation density, and Q is the so-called surface stiffness coefficient-which measures the resistance of neutrons against being separated from protons [23]. It was shown in Ref. [25] using a large set of EDFs that the ratio J/Q appearing in Eq. (4) is linearly related to the slope of the symmetry energy at saturation density L. Moreover, the DM gives the symmetry energy coefficient a sym (A) of a finite nucleus of mass number A as follows [23,26]: Expanding Eq. (5) to first order in the "small" parameter JA −1/3 /Q [as was done in deriving Eq. (4)] we can write Eq. (4) as Given that the difference between J and a sym (A) is directly related to the surface symmetry energy, the above result reveals that the electric dipole polarizability is sensitive to the ratio of the surface and bulk nuclear symmetry energies [27]. The DM may also be used to provide an expression for the neutron skin thickness in terms of a few bulk nuclear properties [25,26,28]. That is, is a correction caused by the electrostatic repulsion, and ∆r surf ] is a correction caused by the difference between the surface widths b n and b p of the neutron and proton density profiles [25,28].
In this manner, one may use the DM to relate the dipole polarizability to the neutron skin thickness. For this purpose, one expands Eq. (7) to first-order in JA −1/3 /Q and finds after some algebra the following relation: Adopting a value of J = 31 ± 2 MeV as a reasonable estimate compatible with recent compilations [4,29], one finds for 208 Pb that I C ≈ 0.028 ± 0.002 and ∆r coul np ≈ −0.042 ± 0.003 fm. Moreover, in Ref. [30] it was shown that ∆r surf np ≈ 0.09 ± 0.01 fm for 208 Pb according to the predictions of a large sample of EDFs. Consequently, as a first reasonable approximation, one can neglect the small variations of I C , ∆r coul np , and ∆r surf np in Eq. (8) and explicitly show that for 208 Pb the product α DM D J is linearly correlated with ∆r DM np -in agreement with Ref. [27]. Given the well-known correlation between the neutron skin thickness of a heavy nucleus and the slope of the symmetry energy at saturation density L ≡ 3ρ 0 (dS/dρ) ρ=ρ0 implied by a large set of EDFs [31,32], one can also anticipate the emergence of a linear correlation between α D J and L. To do so we rely on the findings of Ref. [26] that suggest that the symmetry energy coefficient of a finite nucleus is very close to that of the infinite system at an appropriate sub-saturation density ρ A [i.e., a sym (A) ≈ S(ρ A )]. Note that the density ρ A approximately obeys the following simple formula: where c can be chosen so that ρ 208 = 0.1 fm −3 . Using these results in Eq. (6) after expanding S(ρ A ) around saturation density, namely, one arrives at Note that we have defined ǫ A = (ρ 0 − ρ A )/3ρ 0 , which is approximately equal to ǫ A = 1/8 for ρ 0 = 0.16 fm −3 for the case of 208 Pb. This formula suggests how J and L can be related if the dipole polarizability is known.

III. RESULTS
In this section we study correlations between the electric dipole polarizability (mostly in the form of α D J) and both the neutron skin thickness and parity-violating asymmetry in 208 Pb. Our microscopic analysis involves a large and representative body of EDFs. We employ nonrelativistic Skyrme EDFs widely used in the literature (labeled as Skyrme in the figures [10]) and six different families of systematically varied interactions produced, respectively, by a variation of the parameters around an optimal value (without significantly compromising the quality of the merit function). Two of the families are based on non-relativistic Skyrme EDFs (labeled in the figures as SAMi [11] and SV [12]), while three families are based on meson-exchange covariant EDFs (labeled as NL3/FSU [13][14][15], and TF [16]). The last family is based on a meson-exchange covariant EDF but assuming density-dependent coupling constants (labeled as DD-ME [17]).
A. The dipole polarizability and the neutron skin thickness in 208 Pb We start by displaying in Fig. 1(a) the dipole polarizability α D as a function of the neutron skin thickness in 208 Pb as predicted by the large set of EDFs employed in this work. This figure is reminiscent of the corresponding Fig. 1 of Ref. [3] where a significant amount of scatter between the different calculations was observed, although a linear behavior was seen within each family  Fig. 1(b) support the correlation between α D J and ∆r np as suggested by the DM approach, and clearly demonstrate-by comparing the two panels of Fig. 1-that α D J is far better correlated to the neutron skin thickness of 208 Pb than the polarizability alone; note that the correlation coefficient has increased all the way to r = 0.97. The strength of the correlation shown in Fig.1(b) allows one to reliably estimate, within the validity of our theoretical framework, the value of the neutron skin thickness of 208 Pb as a function of J-or viceversaonce the experimental value of α D =20.1±0.6 fm 3 [1] is assumed: ∆r np =−0.157 ± (0.002) theo.
where ∆r np is expressed in fm and J in MeV. The "exp." uncertainties refer to the propagation of the experimental uncertainty of α D , whereas the "theo." uncertainties are associated to the confidence bands resulting from the linear fit shown in Fig. 1. The theoretical uncertainties are meant to indicate the region allowed by the employed EDFs. Moreover, adopting J = [31 ± (2) est. ] MeV as a realistic range of values for the symmetry energy [4,29], and combining this estimate with the measured value of the dipole polarizability [1], we extract from Fig. 1(b) the following constraint on the neutron skin thickness of 208 Pb: ∆r np = 0.165 ± (0.009) exp. ± (0.013) theo. ± (0.021) est. fm .
We have labeled the uncertainty derived from the different estimates on J as "est." because it contains uncertainties coming from both experimental and theoretical analyses which are often not easy to separate. In addition, we use a different label to keep track of the magnitude of the various uncertainties. Finally, we note that the above result for the neutron skin thickness of 208 Pb is in agreement with previous estimates [1-4, 11, 33]. Given the strong correlation between the neutron skinthickness of 208 Pb and the slope of the symmetry energy L, one expects that the strong correlation between α D J and ∆r np will extend also to L. Moreover, based on the DM insights summarized in Eq.(11), we display in Fig. 2 the microscopic predictions for α D J as a function of L for the same models depicted in Fig. 1. The correlation between α D J and L is of particular interest since it provides a direct relation between J and L via the high-precision measurement of the electric dipole polarizability. Specifically, we obtain L = −146±(1) theo. + 6.11±(0.18) exp. ±(0.26) theo. J, (14) where both J and L are expressed in MeV. In particular, adopting as before a value of J = [31 ± (2) est. ] MeV, the above equation translates into the follow constraint on L: Our results show that the analytical formulas (8) and (11) reproduce the trends of the employed microscopic models. For completeness, we now evaluate the quantitative accuracy of these macroscopic formulas in reproducing the present self-consistent results. In doing so, we use the microscopic predictions for the different quantities appearing in the r.h.s. of Eqs. (8) and (11)  of α D , we find that Eqs. (8) and (11) are accurate within a 10% and 12% on average, respectively. We conclude this section noting that the analysis presented here may be systematically extended to other heavy nuclei if α D is experimentally known. This could help tighten the constraint between J and L. The parity violating asymmetry in the elastic scattering of high-energy polarized electrons from 208 Pb has been recently measured at low momentum transfer at the Jefferson Laboratory by the Lead Radius Experiment (PREX) collaboration [2]. The parity violating asymmetry is defined as the relative difference between the differential cross sections of ultra-relativistic elastically scattered electrons with positive and negative helicity [34]: This landmark experiment by the PREX collaboration constitutes the first purely electro-weak measurement of the neutron skin thickness of a heavy nucleus [2]. In a plane-wave Born approximation the parity violating asymmetry is directly proportional to the weakcharge form factor of the nucleus-itself closely related to the neutron form factor. In exact calculations where Coulomb distortions are taken into account a highly linear relation has been found between A PV and ∆r np in 208 Pb within the realm of nuclear EDFs (see Fig. 2 of Ref. [32]). The measured value of the parity violating asymmetry at an average momentum transfer of Q 2 = 0.0088 ± 0.0001 GeV 2 reported by the PREX col- laboration is given by The experimental uncertainty of 9% (dominated by the statistical error) is about three times as large as originally anticipated. By invoking some mild theoretical assumptions, the measurement of A PV was used to extract the following value of the neutron skin thickness in 208 Pb [2,35]: ∆r np = 0.302±(0.175) exp. ±(0.026) theo. ±(0.005) strange fm .
The last contribution to the uncertainty is associated with the experimental uncertainty in the determination of the electric strange quark form factor. The result is consistent with previous estimates-although the central value is larger than the one extracted from the predictions of a large set of EDFs as well as from previous measurements of ∆r np in 208 Pb using hadronic probes [4]. We note, however, that one of the main virtues of an electro-weak extraction of ∆r np is that it is free from most strong-interaction uncertainties. As mentioned, the main source of the experimental uncertainty in PREX arose from the limited statistics, and a new run PREX-II aiming at the original 3% accuracy in the determination of A PV has been scheduled at the Jefferson Laboratory [36]. Moreover, parity violating scattering experiments in 208 Pb with an even higher accuracy may be possible in the near future at the new MESA facility in Mainz [37].
Given the strong correlation displayed by both α D J and A PV with the neutron skin thickness of 208 Pb, it is natural to expect a close relation between α D J and A PV . Note that the parity violating asymmetry A PV is the physical observable directly measured in the experiment. We display in Fig. 3 the predictions for A PV at the PREX kinematics against α D J for the same set of EDFs used in this work. Note that the nuclear physics input for A PV involves both (point) neutron and proton densities -as opposed to only their respective rms radii-properly folded with the proton and neutron electromagnetic form factors. We underscore, however, that such densities are at the core of all nuclear density functionals, so the comparison against experiment may always be done directly in terms of A PV . Also shown in Fig. 3 Although there is some spread in the theoretical predictions, the large value of r = 0.94 suggests that the correlation between α D J and A PV remains strong. It is interesting to note that a more precise measurement of A PV in 208 Pb with a central value lower than 0.7 ppm might rule out most (if not all!) of the state-of-the-art EDFs available in the literature. We stress that such a thought-provoking conclusion was reached by assuming a realistic range for the symmetry energy at saturation (29 ≤ J ≤ 33 MeV) compatible with different estimates [4,29]. Moreover, one may further constrain A PV through its correlation with α D J. Indeed, invoking the experimental value for α D with the alluded value for J leads to: A PV = 0.724 ± (0.003) exp. ± (0.006) theo. ± (0.008) est. ppm . (20) This would correspond to an accuracy of about 1.5%.

IV. CONCLUSIONS
In summary, we have used insights from the droplet model to understand correlations between the electric dipole polarizability, the neutron skin thickness, and the properties of the symmetry energy around saturation density. The correlations suggested by the macroscopic droplet model were verified in a microscopic study using a comprehensive set of EDFs. In particular, we found that the product of the electric dipole polarizability α D and the symmetry energy at saturation density J is a far better isovector indicator than α D alone. We have shown that high-precision measurements of the dipole response of heavy nuclei (such as 208 Pb) can significantly improve our knowledge of the density dependence of the symmetry energy. Indeed, the strong correlation that we found between α D J and the slope of the symmetry energy L was used to establish a tight relation between L and J [see Eq. (14)]. Moreover, by adopting the well accepted range for the symmetry energy of J = [31±(2) est. ] MeV [4,29], the correlation between α D J and the neutron skin thickness displayed in Fig. 1 suggests ∆r np ≈ 0.168 fm for 208 Pb, with properly computed experimental, theoretical, and "estimated" uncertainties [see Eq. (13)]. Given the strong correlation between ∆r np and L, we were also able to constrain the slope of the symmetry energy at saturation density to L ≈ 43 MeV. These values are consistent with the predictions for the neutron skin thickness in 208 Pb and L extracted from different experiments including heavy-ion collisions, giant resonances, antiprotonic atoms, hadronic probes, and spin polarized electron scattering (see Refs. [4,29] and references therein). They also agree nicely with the constraints on L and ∆r np of 208 Pb derived from recent astrophysical observations supplemented with microscopic calculations of neutron matter [38,39].
Further, we found that the parity violating asymmetry A PV measured by the PREX collaboration not only is strongly correlated with the neutron skin thickness but also with α D J. This has the advantage that theoretical calculations of A PV may be directly compared against experiment-without the need to invoke (albeit mild) model-dependent assumptions. Ultimately, we have combined both observables (A PV and α D ) to derive an experimentally allowed region for the theoretical models. The estimated uncertainties derived for L and ∆r np from the high-precision measurement of α D are appreciably smaller than the ones expected from the 3% measurement of A PV at PREX-II. Ideally, a 1% accuracy on A PV may be required to improve the constraint already imposed from α D [32]. However, we underscore that A PV and α D form a critical set of independent isovector indicators that could provide valuable insights into the nature of the nuclear density functional.
Finally, we highlighted the importance of performing high-precision measurements of α D , and when possible A PV [40], in other medium and heavy nuclei (e.g., 48 Ca, 120 Sn, and 208 Pb). We note that whereas a large number of studies-including this one-suggest that the neutron skin thickness of 208 Pb is fairly thin, ruling out a thick neutron skin as suggested by the central value of the PREX experiment may be premature [16]. However, we are confident that systematic studies involving measurements of α D and A PV will help in constraining the density dependence of the symmetry energy. Thus, we encourage systematic studies of these observables as such a program will be of enormous value in constraining the isovector sector of the nuclear EDF.

ACKNOWLEDGMENTS
We are indebted to Prof. W. Nazarewicz and Prof. P.-G. Reinhard for valuable discussions. We also thank