Slope parameters determined from CREX and PREX2

[Background] Very lately, the CREX group presents a skin value $\Delta R_{\rm skin}^{48}({\rm CREX}) =0.121 \pm 0.026\ {\rm (exp)} \pm 0.024\ {\rm (model)}=0.071\sim 0.171$~fm. Meanwhile, the PREX group reported a skin value $\Delta R_{\rm skin}^{208}({\rm PREX2}) = 0.283\pm 0.071=0.212 \sim 0.354$~fm. In our previous paper, we determined both the $L$--$\Delta R_{\rm skin}^{48}$ relation and the $L$--$\Delta R_{\rm skin}^{208}$ one, using 206 EoSs, where $L$ is a slope parameter. [Purpose] We determine $L$ from $\Delta R_{\rm skin}^{48}({\rm CREX})$ and $\Delta R_{\rm skin}^{208}({\rm PREX2}) $, using 207 EoSs. [Results] The $\Delta R_{\rm skin}^{48}({\rm CREX})$ yields $L({\rm CREX})=0 \sim 51$~MeV and the $\Delta R_{\rm skin}^{208}({\rm PREX2}) $ does $L({\rm PREX2})=76 \sim 165$~MeV. [Conclusion] There is no overlap between $L({\rm CREX})$ and $L({\rm PREX2})$. This is a big problem to be solved.


Matter:
Among the basic physical quantities that determine the equation of state (EoS) of nuclear systems, the symmetry energy (S sym ) and its dependence on the nucleon density (ρ) are receiving a lot of attention, because of their critical role in shaping the structure of nuclei and neutron stars (NSs) [17][18][19][20][21][22][23]. Many predictions on the symmetry energyS sym (ρ) have been made so far by taking several experimental and observational constraints on S sym (ρ) and their combinations. The ∆R 208 skin (E1pE) and the ∆R 48 skin (CREX) are the most important experimental constraint on the slope parameter L, since a strong correlation between r 208 skin and L is well known [24,25]. In this paper, we will show a strong correlation between r 48 skin and L and between r 208 skin and L from 207 EoSs. As an essential constraint on the EoS from astrophysics, one may take M = 1.97 ± 0.04M sun [26], where M is the mass of NS. For a pulsar in a binary system, detection of the general relativistic Shapiro delay allows us to determine M .
Usually, the S sym is expanded into in terms of the nuclear density ρ around the saturation density ρ 0 . For the S sym (ρ), at the present stage, a major aim is to determine L at ρ = ρ 0 . The symmetry energy S sym (ρ) cannot be measured by experiment directly. In place of S sym (ρ), the arXiv:2210.03313v1 [nucl-th] 7 Oct 2022 neutron-skin thickness r skin is measured to determine L. This subject is currently under experimental investigation for 208 Pb and 48 Ca nuclei at Jefferson Lab [27][28][29]. Figure 1 shows the binding energy per nucleon E/A as a function of ρ and δ = (N − Z)/A. The E/A is expanded into in terms of δ; note that J = S 0 (ρ 0 ). This equation shows the relation between E/A(ρ, δ) and S sym (ρ). Two curves correspond to the symmetric-nuclear matter with δ = 0 and pureneutron matter with δ = 1, respectively. This figure shows that L is a slope for pure-neutron matter at ρ = ρ 0 , because dS 0 (ρ)/dρ is zero at ρ = ρ 0 . We accumulate the 205 EoSs from Refs. [20,24, in which ∆R 208 skin and/or L is presented, since a strong correlation between ∆R 208 skin and a slope parameter L is shown. In the 205 EoSs of Table I, the number of Gogny EoSs is much smaller than that of Skyrme EoSs. In Ref. [56], we constructed D1MK and D1PK. Eventually, we get the 207 EoSs, as shown in Table I. The correlation is more reliable when the number of EoSs is larger. For this reason, we take the 207 EoSs.
Among the 207 EoSs, APR of Ref. [30] is the most reliable EoS, since they calculated properties of dense symettric -nuclear and pure-nuclear matter and the structure of neutron stars with the variational chain summation methods for the Argonne v18 two-nucleon interaction plus the Urbana three nucleon interaction. The methods are hard to calculate. In fact, Steiner el.al. have fit bulk properties of APR for symmetric-nuclear and neutron matter with a Skyrme-like Hamiltonian [40]. The EoS is called NRAPR. Brown and Schwenk modified NRAPR slightly [43]. The EoS is called NRAPR-B in this paper. Tsang el.al. made further modification for NRAPR-B [46]. The EoS is called NRAPR-T in this paper.
The ∆R 208 skin (PREX2) and ∆R 48 skin (CREX) is most reliable, and provides L of nuclear matter. In fact, using 207 EoSs of Table I, we found the L-∆R 48 skin relation [56] as ∆R 48 skin = 0.0009L + 0.125 > 0.125 fm with a high correlation coefficient R = 0.98, because of L > 0. Equation (7)  Results: Figure 3 shows the relation between ∆R 208 skin and ∆R 48 skin . The 207 EoSs do not satisfy both ∆R 208 skin (PREX2) and ∆R 48 skin (CREX) in the one-σ level. If one considers the two-σ level of ∆R 208 skin (PREX2) and ∆R 48 skin (CREX), one may find that some EoSs satisfy ∆R 208 skin (PREX2) and There is no overlap between them. Note that the value of L should be positive. The later value is consistent with L = 76 ∼ 172 MeV of Ref. [24]. Reed et al. [57] report a value of the slope parameter L = 106 ± 37 = 69 ∼ 143 MeV.
Conclusion:There is no overlap between L(CREX) and L(PREX2) in the one-σ level, as shown in Fig. 3. This is a big problem to be solved. As another problem, we can say that APR [30,31,44] as the most reliable EoS is ruled out.     [54,55]