Electric Dipole Polarizability of ⁴⁰Ca

Introduction.– The nuclear equation of state (EOS) determines not only basic properties of nuclei [1] but also plays a key role for the properties of neutron stars and the dynamics of core-collapse supernovae and neutron star mergers [2]. New observations from neutron stars and mergers provides constraints for the EOS of neutron-rich matter that can be compared with those derived from nuclear physics (see, e.g., Refs. [3, 4, 5]). However, while the EOS of symmetric nuclear matter is well determined around saturation density, the properties of neutron-rich matter are less explored experimentally. The latter depends on the symmetry energy, whose properties are typically encoded in an expansion around saturation density $n_{0}$, with the symmetry energy at saturation density $J(n_{0})$ and its density dependence $L=3n_{0}\partial J(n_{0})/\partial n$.

Theoretically, a model-dependent correlation between $L$ and the neutron-skin thickness $r_{\rm skin}$ in nuclei with neutron excess has been established [6, 7, 8, 9]. This correlation was also recently confirmed in ab initio computations of the neutron skin in ²⁰⁸Pb [10]. Experimental attempts to determine the neutron skin thickness have been performed with a variety of probes (see, e.g., Ref. [11] and references therein), but many of them suffer from systematic uncertainties entering in the description of the reaction processes. Parity-violating elastic electron scattering (a weak process mediated by the $Z^{0}$ boson) can be used for a nearly model-independent extraction of the neutron distribution in nuclei and, by comparison with accurately measured charge radii, the neutron skin thickness. Recently, results with this technique have been reported by the CREX and PREX collaborations for ⁴⁸Ca [12] and ²⁰⁸Pb [13], respectively. The $r_{\rm skin}$ values inferred with selected nuclear models favor a comparatively small neutron skin in the former and a large skin in the latter case.

Alternatively, the electric dipole polarizability $\alpha_{\rm D}$ has been established as a possible measure of the neutron skin, based on the strong correlation with $r_{\rm skin}$ [8, 14]. Data for $\alpha_{\rm D}$ extracted from proton inelastic scattering experiments at extreme forward angles have been presented for both ⁴⁸Ca [15] and ²⁰⁸Pb [16]. In these papers, two theoretical approaches have been used to describe $\alpha_{\rm D}$: ab initio coupled-cluster (CC) calculations [17, 18] starting from chiral two- and three-nucleon interactions [19, 20] and energy density functional (EDF) theory [21].

Attempts to simultaneously describe $\alpha_{\rm D}$(²⁰⁸Pb) and the parity-violating asymmetry from PREX and CREX with EDF models have shown limited success [22, 23, 24, 25]. The values derived for $r_{\rm skin}$ [13] and $L$ [26] from PREX are in tension with EDFs capable of describing [27] the presently available results on $\alpha_{\rm D}$ in ⁴⁸Ca [15], ⁶⁸Ni [28], ¹²⁰Sn [29, 30], and ²⁰⁸Pb [16]. While the CREX results is in excellent agreement with ab initio predictions [18], the PREX result is in mild tension with the recent ab initio computations of ²⁰⁸Pb [10].

Correlations between experimental observables and symmetry energy properties are well explored in EDF theory [6, 7, 8, 14], but predictions for isovector observables like $\alpha_{\rm D}$ are less well constrained. On the other hand, ab initio calculations provide a direct link to the EOS, as nuclear matter properties can be calculated based on the same chiral interactions [19, 18, 31, 10, 32]. Results presented here are based on the set of two- and three-nucleon interactions from Refs. [19, 20] applied to study $\alpha_{\rm D}$ in ⁴⁸Ca [18, 15]. The calculations of the $E1$ response are based on merging the Lorentz Integral Transform approach with CC theory, as described in Refs. [33, 34]. Recent work has extended the original two-particle–two-hole (2p-2h) CC truncation to include correlations up to three-particle–three-hole (3p-3h), so-called triples corrections, in the computation of $\alpha_{\rm D}$ [35]. Their inclusion leads to a reduction of the predictions for $\alpha_{\rm D}$(⁴⁸Ca) of the order of 10%, allowing an improved simultaneous description of the charge radius [35]. A similar improvement was achieved for ⁶⁸Ni [36].

In this Letter, we present the measurement of the dipole polarizability for ⁴⁰Ca and confront it with CC and EDF calculations. This tests the emerging picture that nuclear theory can describe very well the neutron skin in medium-mass nuclei and related observables.

Experiment.– Cross sections for the ⁴⁰Ca$(p,p^{\prime})$ reaction have been measured at RCNP, Osaka, at an incident proton energy of 295 MeV. Data were taken with the Grand Raiden spectrometer [37] in a laboratory scattering angle range $0.4^{\circ}-14.0^{\circ}$ and for excitation energies in the range $5-25$ MeV. Dispersion matching techniques were applied to achieve an energy resolution of about 30 keV (full width at half maximum). The experimental techniques and the raw data analysis are described in Ref. [38].

In the top panel of Fig. 1 we show representative energy spectra measured at laboratory scattering angles $\Theta_{\rm lab}=0.4^{\circ}$, $1.74^{\circ}$, $3.18^{\circ}$, and $5.15^{\circ}$. The predominant cross sections lie in the energy region above 10 MeV. $M1$ strength in ⁴⁰Ca is known to be concentrated in a single prominent transition at 10.32 MeV [39]. The cross sections above 10 MeV show a broad resonance structure peaking at about 19 MeV increasing towards $0^{\circ}$. The angular dependence is consistent with relativistic Coulomb excitation of $E1$ transitions. We identify this resonance structure as the isovector giant dipole resonance.

The various contributions to the spectra were separated using a multipole decomposition analysis (MDA) as described in Ref. [40]. Results for the most forward angle measured are presented in the bottom part of Fig. 1 as example, where the spectra was rebinned to 200 keV. Theoretical angular distributions for the relevant multipoles were obtained from Distorted Wave Born Approximation calculations with transition amplitudes from quasiparticle-phonon-model calculations similar to the analysis of ⁴⁸Ca [15]. Additionally, a background due to pre-equilibrium multistep scattering was considered. Its angular dependence was taken from experimental systematics [41, 42] while the amplitude was derived by two means. Initially, an unconstrained fit was done at each energy bin of the set of spectra. The resulting cross sections could be well approximated by a simple Fermi function but showed strong fluctuations for certain excitation energy bins due to the similarity to some of the $E1$ theoretical angular distributions. Thus, in the final analysis, the continuum contribution was determined by fitting a Fermi function to the unconstrained excitation energy dependence.

Photoabsorption cross sections and dipole polarizability.– The $E1$ cross sections resulting from the MDA were converted into equivalent photoabsorption cross sections using the virtual photon method [45]. The virtual photon spectrum was calculated in an eikonal approach [46] to Coulomb excitation, integrated over the distribution of scattering angles covered in the solid angle of each angular bin. The photoabsorption spectra derived from scattering data at $0.40^{\circ}$ and $1.00^{\circ}$ were essentially identical, and that at $1.74^{\circ}$ deviated only slightly, consistent with an estimate of the grazing angle ($1.33^{\circ}$) at which Coulomb-nuclear interference becomes relevant. The resulting photoabsorption cross section is displayed as blue histogram in Fig. 2 (a).

The electric dipole polarizability $\alpha_{D}$ was obtained from the photoabsorption cross section over the energy range $10-25$ MeV leading to a contribution 1.60(14) fm³. The integration was extended to 60 MeV, where the cumulative sum plotted in Fig. 2 (b) shows saturation. The data at higher excitation energies were taken for $25-31$ MeV from Ref. [44] and for $31-60$ MeV from Ref. [43] to obtain the total $\alpha_{D}(^{40}{\rm Ca})=1.92(17)$ fm³. The uncertainty considers systematic errors of (i) the absolute cross sections, (ii) the MDA (determined as described, e.g., in Ref. [47]), and (iii) the parameterization of the continuum background (evaluated by varying the amplitude of the Fermi function), added in quadrature. The latter, dominating the total uncertainty budget, was estimated by the variation needed to change the $\chi^{2}$ value of the MDA fit by one. Statistical errors turned out to be negligible. A detailed breakdown of the error contributions is given in Table 1.

Comparison with coupled-cluster calculations.– The extracted value of $\alpha_{\rm D}$ serves as a benchmark for CC theory [33, 18, 35, 34]. Coupled-cluster calculations were recently performed for the dipole polarizability of ⁴⁸Ca [15] and ⁶⁸Ni [36], which led to an improved understanding of the neutron and proton distributions in nuclei, as well as their difference encoded in the neutron skin. We have performed CC computations of $\alpha_{\rm D}$ in ⁴⁰Ca starting from a Hartree-Fock reference state considering a basis of 15 major harmonic oscillator shells. To gauge the convergence of our results we varied the oscillator frequency in the range $\hbar\omega=12-16$ MeV. Three-nucleon contributions had an additional energy cut of $E_{\rm 3max}=16\hbar\omega$.

Figure 3 explores the correlation between $\alpha_{\rm D}$ for ⁴⁰Ca and ⁴⁸Ca as predicted by theory. Panel (a) shows the CC results including triples contributions, not available for ⁴⁰Ca so far. The theoretical uncertainties for the different Hamiltonians stem from the truncation of the CC expansion and the residual dependence on CC convergence parameters, calculated as described in Ref. [34]. Similarly to ⁴⁸Ca, we find that the inclusion of 3p-3h correlations reduces the value of $\alpha_{\rm D}$(⁴⁰Ca) by an amount varying between 10% – 20% for different interactions. While the EM and PWA interactions [19] are not simultaneously compatible with both ⁴⁰Ca and ⁴⁸Ca experimental data, the set of employed interactions shows an approximately linear trend between the two quantities overlapping with both experimental results. A particular improvement in the reproduction of both $\alpha_{\rm D}$(⁴⁸Ca) and $\alpha_{\rm D}$(⁴⁰Ca) is seen for the NNLO_sat interaction [20], which is capable of accurately describing binding energies and radii of nuclei up to ⁴⁰Ca as well the saturation point of symmetric nuclear matter. The different interactions predict a range of symmetry energy parameters $J=27-33$ MeV, $L=41-49$ MeV [18], with the NNLO_sat values at the lower end ($J=27$ MeV, $L=41$ MeV).

Comparison with EDF approaches.– Recently, it was investigated whether the dipole polarizability and the parity-violating asymmetry $A_{\rm PV}$ for ²⁰⁸Pb and ⁴⁸Ca can be simultaneously accounted for with modern EDFs [24]. We use the four representative forms of functionals from that study: non-relativistic Skyrme functionals SV [48] and RD [49], the latter with different forms of density dependence, and relativistic functionals DD [50] with finite-range meson-exchange coupling and PC [51] with point coupling. All four have been calibrated to the same set of ground-state data to determine the model parameters. With these sets, it was shown that PREX and CREX results for $A_{\rm PV}$ (and $r_{\rm skin}$) cannot be consistently explained within the model uncertainties while the $\alpha_{\mathrm{D}}$ were reproduced. Hence, the present result in ⁴⁰Ca provides an important test of the global predictive power of these EDFs.

Figure 3 (b) displays the EDF results for $\alpha_{\rm D}$ with $1\sigma$ error ellipses (for their definition see Refs. [22, 24]). The parametrizations as given from the ground-state fits are shown by filled ellipses. The DD functional performs rather well. The other predictions tend to slightly overestimate the experimental mean values of both ⁴⁰Ca and ⁴⁸Ca, while their $1\sigma$ error ellipses do overlap with the experimental bands, except for PC. In all cases, the two $\alpha_{\mathrm{D}}$ values are highly correlated. We note that the same holds for the description of $\alpha_{D}$(²⁰⁸Pb) [16] after correction for the quasi-deuteron contribution [27]. Thus, all the models are capable to account for the mass dependence of the polarizability.

The dashed ellipses show results from a refit where additionally the experimental $\alpha_{\mathrm{D}}$ value of ²⁰⁸Pb [16] corrected for the quasideuteron part [27] was included yielding the functionals SV-alpha; RD-alpha, PC-alpha, and DD-alpha [22, 24]. This improves the agreement with experiment, particularly for the PC model, and shrinks most error ellipsoids. The uncertainty reduction is especially large for the DD model because this functional has the least isovector freedom. The linear trend shown by the different theoretical approaches in Fig. 3 is similar although the CC calculations tend to underestimate the $\alpha_{D}$ in ⁴⁰Ca and perform nicely for ⁴⁸Ca. The bulk symmetry energies range from $J=30$ MeV for DD to 35 MeV for PC and accordingly from 32 MeV to 82 MeV for $L$. The fits which include also $\alpha_{\mathrm{D}}$ in ²⁰⁸Pb narrow the prediction to $J=30-32$ MeV and $L=35-52$ MeV which correlates nicely to the narrower range of predictions for $\alpha_{\mathrm{D}}$ in ^40,48Ca.

Conclusions.– We have extracted the dipole polarizability of ⁴⁰Ca from a combination of relativistic Coulomb excitation measurement in inelastic proton scattering under very forward angles with total photoabsorption data at high excitation energies. Together with a similar analysis on ⁴⁸Ca the new data serve as a benchmark test of state-of-the art theoretical approaches. A representative set of EDFs can describe these data. An improvement is obtained when the EDFs are optimized by adding the dipole polarizability of ²⁰⁸Pb to the calibration dataset. Coupled-cluster computations for the NNLO_sat interaction simultaneously describe well the dipole polarizability of ⁴⁰Ca and ⁴⁸Ca, as well as the corresponding charge radii and the neutron skin thickness [34]. A nearly linear systematic trend is obtained for other interactions, as in the case of EDF theory. This analysis supports the robustness of current theoretical approaches in the description of $\alpha_{D}$ and their constraints of symmetry energy parameters discussed, e.g., in Refs. [22, 24, 52].