# Implications of parity-violating electron scattering experiments on 48Ca
(CREX) and 208Pb (PREX-II) for nuclear energy density functionals

Esra Yüksel<EMAIL_ADDRESS>Department of Physics, University of
Surrey, Guildford, Surrey, GU2 7XH, United Kingdom  Nils Paar<EMAIL_ADDRESS>Department of Physics, Faculty of Science, University of Zagreb, Bijenicka
cesta 32, 10000 Zagreb, Croatia

###### Abstract

Recent precise parity-violating electron scattering experiments on 48Ca (CREX)
and 208Pb (PREX-II) provide a new insight on the formation of neutron skin in
nuclei. Within the energy density functional (EDF) framework, we investigate
the implications of CREX and PREX-II data on nuclear matter symmetry energy
and isovector properties of finite nuclei: neutron skin thickness and dipole
polarizability. The weak-charge form factors from the CREX and PREX-II
experiments are employed directly in constraining the relativistic density-
dependent point coupling EDFs. The EDF established with the CREX data acquires
considerably smaller values of the symmetry energy parameters, neutron skin
thickness and dipole polarizability both for 48Ca and 208Pb, in comparison to
the EDF obtained using the PREX-II data, and previously established EDFs.
Presented analysis shows that CREX and PREX-II experiments could not provide
consistent constraints for the isovector sector of the EDFs, and further
theoretical and experimental studies are required.

The nuclear equation of state (EOS) is essential for understanding the
properties of strongly interacting many-body systems like atomic nuclei and
neutron stars [1, 2, 3]. Constraining the density dependence of the nuclear
symmetry energy, which is a key feature of the nuclear EOS, represents a long-
standing and unresolved question in nuclear physics and astrophysics [3]. The
fundamental source of this challenge is that the nuclear symmetry energy
cannot be determined directly by experiment, thus it is necessary to identify
and use relevant observables on finite nuclei to constrain their values [3].
The neutron skin thickness ($\Delta R_{np}$) [4, 5, 6, 7, 8], dipole
polarizability ($\alpha_{D}$) [7, 9, 10, 11], and neutron star mass-radius
[12, 13] have been established as key observables to constrain the isovector
channel of the nuclear energy density functional (EDF) and the symmetry energy
parameters of the nuclear EOS around the saturation density.

The recent precise parity-violating electron scattering experiments on 48Ca
(CREX) [14] and 208Pb (PREX-II) [15] provide new insight into the neutron skin
thickness in nuclei. Through the measurement of the parity violating asymmetry
$A_{PV}$, these experiments allow to determine the nuclear weak-charge form
factor $F_{W}$ that is also strongly correlated with the density dependence of
the symmetry energy and the neutron skin thickness of nuclei, hence providing
an important quantity to probe the isovector channel of the EDFs [16]. The
weak charge form factor $F_{W}$ is obtained with the Fourier transform of the
weak charge density for a given momentum transfer. Since the charge density is
known experimentally, the Coulomb distortions can also be corrected
accurately. [17]. Therefore, the parity-violating electron scattering
experiments provide precise and model-independent data with small
uncertainties for the nuclear weak-charge form factor $F_{W}$, that could be
used in constraining the EDFs [16, 17]. The CREX [14] and PREX-II [15]
experiments reported the weak-charge form factors of 48Ca and 208Pb as
$F_{W}$(q=0.8733 $fm^{-1}$)= 0.1304 $\pm$ 0.0052(stat)$\pm$0.0020(syst) and
$F_{W}$(q=0.3978 $fm^{-1}$)= 0.368$\pm$ 0.013 (exp.), respectively. The
measured parity violating asymmetry $A_{PV}$ in 48Ca and 208Pb has recently
been analyzed using EDFs, indicating that there are difficulties to describe
$A_{PV}$ simultaneously in both nuclei [18, 19].

In this letter, we establish effective interactions based on the relativistic
EDF with density-dependent point couplings, using recent nuclear weak-charge
form factor $F_{W}$ data from CREX [14] and PREX-II [15] experiments alongside
with the selected ground state properties of nuclei. In this way, we aim to
reveal the implications of the CREX and PREX-II experiments on the properties
of finite nuclei and nuclear matter, especially the symmetry energy and its
slope around the saturation density. By calculating the relevant nuclear
observables with the new interactions, in particular the neutron skin
thickness and dipole polarizability, we discuss their relationship with the
recent experimental data from parity violating electron scattering.

Table 1: The nuclear matter properties at saturation density for the DDPC-CREX, DDPC-PREX, and DDPC-REX interactions. The properties for the DD-PC1 [20] and DD-PCX [21] interactions are also given for comparison. The uncertainties of the obtained values are provided within the parenthesis. | E/A (MeV) | $m_{D}^{*}/m$ |  $K_{0}$ (MeV) |  $J$ (MeV) |  $L$ (MeV)
---|---|---|---|---|---
DDPC-CREX | -15.989(16) | 0.5672(5) | 225.48(1.55) | 27.01(23) | 19.60(1.01)
DDPC-PREX | -16.108(19) | 0.5680(7) | 235.41(2.42) | 36.18(0.80) | 101.78(9.34)
DDPC-REX | -16.019(16) | 0.5696(5) | 242.95(76) | 28.86(0.33) | 30.03(2.06)
DD-PC1 | -16.061 | 0.580 | 230.0 | 33.0 | 70.1
DD-PCX | -16.026(18) | 0.5598(8) | 213.03(3.54) | 31.12(32) | 46.32(1.68)

Theoretical framework employed in this study is based on the relativistic EDF,
where the self-consistent solution of relativistic single-nucleon Kohn-Sham
equations provides the nuclear ground state density and energy [22, 23]. In
the formulation given by the Lagrangian density, we use an effective
interaction between nucleons described with four fermion contact interaction
terms, including isoscalar-scalar, isoscalar-vector, isovector-vector and
isospace-space channels [20, 24]. It includes free nucleon terms, point
coupling interaction terms, coupling of protons to the electromagnetic field,
and the derivative term accounting for the leading effects of finite-range
interactions. For a quantitative description of nuclear density distribution
and radii, the derivative terms are also necessary [20]. The couplings in the
interaction terms $\alpha_{i}(\rho)$ include explicit density dependence [24].
The point coupling model includes 10 parameters ($a_{S}$, $b_{S}$, $c_{S}$,
$d_{S}$, $a_{V}$, $b_{V}$, $d_{V}$, $b_{TV}$, $d_{TV}$ and $\delta_{S}$) [24].
The relativistic Hartree-Bogoliubov (RHB) model [24, 25, 20] is used to
describe open-shell nuclei, including the pairing field formulated using
separable pairing force, which also contains two parameters for the proton and
neutron pairing strengths ($G_{p}$ and $G_{n}$) [26].

In this work, we employed the RHB model to constrain 12 model parameters by
minimizing the $\chi^{2}$ objective function with a set of observables on
nuclear properties [24, 27, 21]. In order to benchmark the role of CREX and
PREX-II data, in the $\chi^{2}$ minimization we used the same nuclear ground
state properties as in recent optimization of the DD-PCX interaction, i.e.,
the binding energies (34 nuclei), charge radii (26 nuclei), and mean pairing
gaps (15 nuclei) (see supplementary material [29] for the details about
selected nuclei and their properties). In the optimization of the new
functionals we also used the latest nuclear weak-charge form factors $F_{W}$
obtained from the CREX (48Ca) [14] and PREX-II (208Pb) [15] experiments. First
we have established two functionals, DDPC-CREX and DDPC-PREX, optimized by
using nuclear properties given above and the weak-charge form factor data of
48Ca and 208Pb, respectively. The adopted errors for $F_{W}$ are taken as
0.5%, that is relatively tight in order to reproduce well the experimental
$F_{W}$ values. Next, an additional functional DDPC-REX is established, by
employing weak-charge form factor data of both 48Ca and 208Pb, with the errors
taken as 2.0%, to provide more flexibility to accommodate $F_{W}$ data for the
two nuclei. Following the optimization of the interactions, the statistical
uncertainties of the model parameters are estimated using the co-variance
analysis [28]. The parameters of these three new DDPC functionals and their
uncertainties are given in Supplemental material [29].

In Table LABEL:nmp, we present the nuclear matter properties: energy per
nucleon $E/A$, effective mass ($m_{D}^{*}/m$), incompressibility $K_{0}$,
symmetry energy $J$ and slope of the symmetry energy $L$ at saturation density
[3], for new DD-PC functionals in comparison to the properties of the
previously established DD-PC1 [20] and DD-PCX [21] interactions. We note that
the symmetry energy parameters $J$ and $L$ of the DDPC-CREX functional are
considerably lower than those of the DDPC-PREX, while the DDPC-REX acquires
intermediate values between the two former functionals. While the $J$ and $L$
values for the DDPC-CREX functional are lower than those of DD-PC1 and DD-PCX,
the DDPC-PREX functional predicts much higher values. Our DDPC-PREX results
are also consistent with the findings in Ref. [30], where $J=$38.1$\pm$4.7 MeV
and $L=$106$\pm$37 MeV are obtained using the PREX-II data. The symmetry
energy parameters for DDPC-CREX and DDPC-PREX interactions are outside rather
broad ranges of their values obtained in the EDF analysis of dipole
polarizability, $J=30-35$ MeV and $L=20-66$ MeV [11], and also at variance
with the findings of previous studies (see Ref. [3] and the references
therein.) Therefore, already at the level only of the nuclear matter
properties, we observe considerable inconsistencies between the symmetry
energy properties of new functionals constrained by weak-charge form factor
data and previously established functionals that are known as successful in
the description of a variety of nuclear properties, including DD-PCX [21] with
carefully adjusted isovector channel using nuclear observables such as dipole
polarizability.

Figure 1: The neutron skin thickness $\Delta R_{np}$ of 48Ca (upper panel) and
208Pb (lower panel) as a function of the form factor difference $F_{ch}-F_{W}$
using relativistic energy density functionals. The experimental data from
PREX-II [15] and CREX [14] are denoted with vertical and horizontal bands.
Figure 2: The error ellipsoids between the neutron skin thickness $\Delta
R_{np}$ and the form factor difference $F_{ch}-F_{W}$ for 48Ca (upper panels)
and 208Pb (lower panels) using the new point-coupling interactions: DDPC-PREX,
DDPC-CREX, and DDPC-REX. The CoD numbers are also given within the figures.

The symmetry energy and its slope are known to be strongly correlated with the
neutron skin thickness of nuclei [4, 5, 6, 7]. However, accurate and model
independent measurement of the neutron skin thickness of nuclei is rather
challenging. The first Lead Radius EXperiment (PREX-I) [31] estimated very
large neutron skin thickness for 208Pb with significant uncertainties
($R_{n}-R_{p}=0.33^{+0.16}_{-0.18}$). Recently, the new data from the PREX-II
experiment has also been announced and the neutron-skin thickness of 208Pb was
found as $R_{n}-R_{p}=$0.283$\pm$0.071 fm, obtained by combining the new data
with the previous measurements [15]. The charge minus the weak form factor was
also obtained as $F_{ch}-F_{W}=0.041\pm 0.013$ in the same experiment. Lately,
CREX data has been announced and the neutron skin thickness and the form
factor difference of 48Ca have been obtained as
$R_{n}-R_{p}=$0.121$\pm$0.026(exp)$\pm$0.024(model) fm and
$F_{ch}-F_{W}=0.0277\pm 0.0055$, respectively [14]. In the following, we
discuss the isovector properties of nuclei calculated using our three point
coupling interactions constrained by $F_{W}$, in comparison to those of
previously established EDFs and experimental data.

Figure 1 shows the neutron skin thickness $\Delta R_{np}$ for 48Ca and 208Pb
as a function of the difference of charge and weak-charge form factors,
$F_{ch}-F_{W}$, from calculations based on relativistic EDFs. The neutron skin
thickness of a nucleus is calculated as $\Delta R_{np}=\left\langle
r_{n}^{2}\right\rangle^{1/2}-\langle r_{p}^{2}\rangle^{1/2}$, and $\langle
r_{n(p)}^{2}\rangle^{1/2}$ represents neutron (proton) root-mean-square radii.
The results of new point coupling functionals DDPC-CREX, DDPC-PREX, and DDPC-
REX are compared with those of the DD-PC1 [20], DD-PCX [21], and family of
eight DD-PC functionals that cover a range of the symmetry energy values at
saturation density $J=29,30,...,36$ MeV [32], density dependent meson-exchange
functionals DD-ME1 [33] and DD-ME2 [34], and the corresponding family of five
DD-ME functionals with $J=30,32,...,38$ MeV. The neutron skin thickness is
strongly correlated with the $F_{ch}-F_{W}$ values, both for 48Ca and 208Pb.
For 48Ca, only a few interactions are within the ranges of $F_{ch}-F_{W}$ and
$\Delta R_{np}$ values obtained from the CREX experiment. As expected, the
DDPC-CREX and DDPC-REX interactions fit into expected range of the
$F_{ch}-F_{W}$ and $\Delta R_{np}$ values, since experimental value for
$F_{W}(^{48}\text{Ca})$ is used in constraining their parameters. However, all
other interactions that have previously been established as very successful in
describing nuclear properties, e.g. DD-ME1, DD-ME2, DD-PC1, DD-PCX, remain
above the experimental ranges for $F_{ch}-F_{W}$ and $\Delta R_{np}$ values.
The only exception is DD-PC interaction with the $J=29$ value. In this work,
the neutron skin thickness values for 48Ca are obtained as 0.218(5), 0.110(4),
and 0.142(5) fm using the DDPC-PREX, DDPC-CREX, and DDPC-REX interactions,
respectively. We note that in the present optimization of the EDFs, the weak-
charge form factor data with small adopted errors are used in order to
constrain the functionals that closely reproduce the experimental data on
$F_{W}$, as previously introduced. Therefore, the symmetry energy and neutron-
skin thickness values are both tightly constrained since these properties are
strongly correlated with each other as well as with the weak-charge form
factor data. Therefore, the obtained uncertainties in the neutron skin
thickness values are small. It is seen that only the DDPC-REX result is in
good agreement with the predictions of the ab initio theory ($0.12\leq\Delta
R^{{}^{48}Ca}_{np}\leq 0.15$ fm) [35], and the recent results by including
triples corrections in coupled-cluster theory ($0.13\leq\Delta
R^{{}^{48}Ca}_{np}\leq 0.16$ fm) [36]. On the other hand, it is lower compared
to the model-averaged values ($\Delta R^{{}^{48}Ca}_{np}=0.176\pm 0.018$ fm)
obtained in the EDF study in Ref. [9]. Using the linear fit obtained with
different EDFs and the experimental limits for $F_{ch}-F_{W}$ values, the
neutron skin thickness is obtained between $0.098\leq\Delta
R^{{}^{48}Ca}_{np}\leq 0.147$ fm, that is in better agreement with the ab
initio theory predictions [35, 36] than with the model-averaged EDF result
[9].

As shown in Fig. 1, for 208Pb, the experimental errors in $F_{ch}-F_{W}$ and
$\Delta R_{np}$ obtained from PREX-II are larger, and more interactions fit
into the experimental range of their values. However, all these interactions
are on the opposite side of those preferred from the CREX experiment, in
agreement with their larger values of the symmetry energy parameter $J$. Among
new functionals, only DDPC-PREX fits into the experimental range of
$F_{ch}-F_{W}$ and $\Delta R_{np}$ values, while DDPC-CREX and DDPC-REX
results are obtained at considerably lower values. The DD-PCX and DD-ME2
functionals also remain below the experimental range, while DD-ME1 and DD-PC1
ones are very close to the lower experimental limits of $F_{ch}-F_{W}$ and
$\Delta R_{np}$. From the families of interactions that systematically vary
the symmetry energy, only those with $J\geq 34$ (DD-ME) and $J\geq 34$ (DD-PC)
are consistent with the PREX-II data. Employing DDPC-PREX, DDPC-CREX, and
DDPC-REX functionals in the calculations, the neutron skin thickness values
for 208Pb are obtained as 0.250(10), 0.051(6), and 0.105(8) fm, respectively.
The PREX-II measurements indicate a rather large neutron skin thickness value
which is not in agreement neither with the model-averaged values ($\Delta
R^{{}^{208}Pb}_{np}=0.168\pm 0.0022$ fm) obtained in Ref. [9], nor with the
limits obtained using the ab initio theory ($\Delta
R^{{}^{208}Pb}_{np}=0.14-0.20$ fm) in Ref. [37]. Using the linear fit in Fig.
1 and experimental limits for $F_{ch}-F_{W}$ values, the neutron skin
thickness is obtained between $0.202\leq\Delta R^{{}^{208}Pb}_{np}\leq 0.371$
fm for 208Pb. Clearly, the limits are not in agreement with the previous model
predictions [9, 37].

The uncertainties in the model parameters and observables, and the
correlations between them can be obtained with the statistical covariance
analysis [38]. Within this framework, the coefficient of determination (CoD)
is used to determine whether two observables have a genuine statistical
correlation (CoD) or not. The CoD values range between 0 and 1. Two quantities
are strongly correlated when CoD = 1, as opposed to being uncorrelated when
CoD = 0. Additionally, the error ellipsoid between the two observables can be
shown as the graphical representation of the CoD (see Refs. [38, 39, 40]). In
Figure 2, we display the error ellipsoids between $\Delta R_{np}$ and
$F_{ch}-F_{W}$ for 48Ca (upper panels) and 208Pb (lower panels) using the new
point-coupling functionals. The CoD numbers are also provided within the
figures for each interaction and nucleus. Using three new point-coupling
interactions, a strong correlation is obtained between the $\Delta R_{np}$ and
$F_{ch}-F_{W}$ for both 48Ca and 208Pb. While the variances in the ellipsoids
are slightly larger for 48Ca in the perpendicular direction, the error
ellipsoids are quite narrow for 208Pb. Similar results are also obtained in
Ref. [19] using the non-relativistic functionals.

Figure 3 shows the summary of the neutron skin thicknesses values for 208Pb
obtained using DDPC-CREX, DDPC-PREX, and DDPC-REX functionals, in comparison
to a number of previous experimental results [41, 15, 42, 43, 44, 45]. In
addition, the results of non-relativistic [46, 47, 48] and relativistic [49,
47, 34, 20, 50, 21] studies based on the EDFs are shown, as well as the EDF
constrained by chiral effective field theory, denoted as Sk$\chi$m* [48] (more
details on other EDFs are given in the figure caption). One can observe that
for 208Pb the DDPC-CREX and DDPC-REX values for $\Delta R_{np}$ are smaller
than those of all previous measurements and EDF predictions. On the other
side, the DDPC-PREX functional provides $\Delta R_{np}$ value larger than
obtained in most of the previous studies. It is in agreement with the PREX-II
experiment, but also with the PC-PK1 interaction which has a higher value of
the symmetry energy at saturation, $J=\text{35.6}$ MeV. Clearly, the data from
the parity-violating electron scattering experiments CREX and PREX-II could
not provide a consistent understanding of the neutron skin thickness in 208Pb.
Recent study of parity violating asymmetry $A_{PV}$ in the EDF framework
showed that simultaneous accurate description of $A_{PV}$ in 48Ca and 208Pb
could also not be achieved [19].

Figure 3: The neutron skin thickness of 208Pb for the DDPC-CREX, DDPC-PREX,
and DDPC-REX functionals. For comparison, the experimental values are shown
from ($\gamma$,$\pi^{0}$) [41], PREX-II [15], ($p,p^{\prime}$) [42], ($p,p$)
[43], $\overline{p}$ [44], LAND(PDR) [45]. The results from the non-
relativistic interactions include SV-min [46], SLy5-min [47], Sk$\chi$m* [48],
and relativistic interactions: FSUGold [49], DDME-min1 [47], DD-ME2 [34], DD-
PC1 [20], PC-PK1 [50], DD-PCX [21].

Figure 4: The dipole polarizability $\alpha_{D}$ of 48Ca (upper panel) and
208Pb (lower panel) as a function of the form factor difference $F_{ch}-F_{W}$
using relativistic EDFs. Vertical bands denote $F_{ch}-F_{W}$ range of values
from the CREX [14] and PREX-II [15] experiments, while horizontal bands
correspond to the experimental data on $\alpha_{D}$ [51, 42, 11] Figure 5: The
error ellipsoids between the dipole polarizability $\alpha_{D}$ and form
factor difference $F_{ch}-F_{W}$ using new point coupling interactions for
48Ca (upper panels) and 208Pb (lower panels). The CoD values are also given
within the figures.

Another relevant quantity used in constraining the isovector channel of the
EDFs is dipole polarizability $\alpha_{D}$, corresponding to the sum of
inverse energy weighted dipole transition strength in nuclei [7]. Over the
past decade the dipole polarizability has attracted considerable interest
because it is strongly correlated with the neutron form factor, neutron skin
thickness [7, 9, 10, 11], and the properties of the symmetry energy of nuclear
matter [7]. Recently, the dipole polarizability has been measured for 48Ca
[51] and 208Pb [42], and the corresponding values to be used in analyses are
$\alpha_{D}(^{48}\text{Ca})=$ 2.07$\pm$0.22 fm3 [51] and
$\alpha_{D}(^{208}\text{Pb})=$ 19.6$\pm$0.6 fm3 [11]. Therefore, it is
interesting to confront the calculated values of $\alpha_{D}$ with those of
$F_{ch}-F_{W}$, together with the corresponding experimental data, to assess
the information which functional consistently describes these both quantities.
The respective results are shown in Fig. 4, using the same density-dependent
point coupling and meson exchange functionals as previously discussed. The
horizontal and vertical bands denote the experimental values with errors for
$\alpha_{D}$ and $F_{ch}-F_{W}$, respectively. As expected, strong correlation
between $\alpha_{D}$ and $F_{ch}-F_{W}$ is obtained for 48Ca and 208Pb for all
functionals used in the analysis. For 48Ca, only one new interaction, DDPC-
REX, and DDPC ($J$=29 MeV) are simultaneously within the experimental limits
for $\alpha_{D}$ and $F_{ch}-F_{W}$ (CREX). The DDPC-CREX interaction gives a
slightly smaller $\alpha_{D}$ value than the lower experimental limit, while
the DDPC-PREX interaction results for $\alpha_{D}$ and $F_{ch}-F_{W}$ are
obtained at considerably higher values and above the experimental range. The
DD-PCX interaction, which has been adjusted to reproduce experimental dipole
polarizabilities, seems not consistent with $F_{ch}-F_{W}$ (CREX). For 48Ca,
the coupled cluster theory calculations predict
$1.92\leq\alpha_{D}(^{48}Ca)\leq 2.38$ fm3, which is in agreement only with
predictions of the DDPC-REX interaction [36].

Lower panel of Fig. 4, shows that for 208Pb none of the new or previously
established functionals can simultaneously reproduce experimental values for
$\alpha_{D}$ and $F_{ch}-F_{W}$ (PREX-II). As expected, DDPC-PREX reproduces
experimental form factors, but considerably overestimates $\alpha_{D}$ values.
The DDPC-CREX and DDPC-REX underestimate both $\alpha_{D}$ and $F_{ch}-F_{W}$
(PREX-II) values. There is a considerable lack of consistency between the
functionals constrained using CREX and PREX-II data when confronted with
dipole polarizability studies. In Ref. [19], similar calculations have been
performed using the non-relativistic functionals, and a strong correlation is
also obtained between the parity violating asymmetry $A_{PV}$ and
$J\alpha_{D}$. Since $A_{PV}$ is directly correlated with the $F_{ch}$ and
$F_{W}$, our findings are consistent.

Figure 5 shows the error ellipsoids between the $\alpha_{D}$ and
$F_{ch}-F_{W}$ for the new point-coupling interactions. The results are
displayed for 48Ca (upper panels) and 208Pb (lower panels). For both nuclei,
there is a strong correlation between $\alpha_{D}$ and $F_{ch}-F_{W}$ using
three functionals, albeit smaller when compared to $\Delta R_{np}$ and
$F_{ch}-F_{W}$. The lowest CoD value between the $\alpha_{D}$ and
$F_{ch}-F_{W}$ is obtained for 208Pb using the DDPC-CREX functional.

In summary, the weak-charge form factors from parity-violating electron
scattering open an important new perspective to constrain the neutron skins
and the properties of the nuclear matter symmetry energy. In this work, the
weak-charge form factors obtained from the CREX and PREX-II data have been
used directly in constraining the relativistic EDFs, having in this way a
considerable impact on their isovector interaction channel. However, the
neutron skin thicknesses calculated from these EDFs do not seem consistent,
and they are at variance with respect to previous theoretical and experimental
studies. When the weak-charge form factors are confronted with the dipole
polarizability, the results of the analysis show significant inconsistencies,
although both should probe the same content of the EDFs. The symmetry energy
and its slope at saturation density, as well as the neutron skin thickness,
are significantly smaller for the DDPC-CREX functional in comparison to those
of the DDPC-PREX, and values established by previous EDFs are between those
obtained with the two new functionals. The analysis of this work shows that no
consistent conclusions from the theoretical side could be obtained when using
recent CREX and PREX-II results. Clearly, further EDF and ab-initio studies,
alongside with novel experimental investigations are needed to resolve the
current puzzling implications of the parity violating electron scattering
data.

## Acknowledgements

This work is supported by the QuantiXLie Centre of Excellence, a project co
financed by the Croatian Government and European Union through the European
Regional Development Fund, the Competitiveness and Cohesion Operational
Programme (KK.01.1.1.01.0004).

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