Proton skins, neutron skins, and proton radii of mirror nuclei

We present predictions for proton skins based on isospin-asymmetric equations of state derived microscopically from high-precision chiral few-nucleon interactions. Moreover, we investigate the relation between the neutron skin of a nucleus and the difference between the proton radii of the corresponding mirror nuclei.

In recent years, chiral EFT has evolved into the authoritative approach to construct nuclear two-and many-body forces in a systematic and essentially model-independent manner [7,8]. Nucleon-nucleon (NN) potentials are available from leading order (LO, zeroth order) to N 3 LO (fourth order) [7][8][9][10][11], with the latter reproducing NN data at the high precision level. More recently, NN chiral potentials at N 4 LO have also been developed [12,13].
A large number of applications of chiral NN potentials (usually up to N 3 LO) together with chiral three-nucleon forces (3NF) (generally just at N 2 LO) have been conducted. A fairly extensive, although not exhaustive list is given in Refs. .
We apply the microscopic equations of state (EoS) of symmetric nuclear matter and the ones of pure neutron matter as derived in Ref. [36]. The derivation is based on high-precision chiral NN potentials at next-to-next-tonext-to-leading order (N 3 LO) of chiral perturbation theory [7,10]. The leading 3NF, which is treated as an effective density-dependent force [37], is included.

B. Additional tools
This section provides a very brief summary of previously developed tools to obtain nuclear properties from the infinite-matter EoS [38]. Within the spirit of a liquid droplet model, the energy of a nucleus is written in terms of a volume, a surface, and a Coulomb term as In the above equation, ρ is the total nucleon density, given by ρ n +ρ p , with ρ n and ρ p the neutron and proton densities, respectively. α is the neutron asymmetry parameter, α = ρ I /ρ, where the isovector density ρ I is given by (ρ n − ρ p ). e(ρ, α) is the energy per particle in isospin-asymmetric nuclear matter, written as with e sym (ρ) the symmetry energy. The density functions for protons and neutrons are obtained by minimizing the value of the energy, Eq. (1), with respect to the paramaters of Thomas-Fermi distributions, with i = n, p. The radius and the diffuseness, a i and c i , respectively, are extracted by minimization of the energy while ρ 0 is obtained by normalizing the proton(neutron) distribution to Z(N ). The neutron and proton skins are defined in the usual way, and respectively, where R n and R p are the r.m.s. radii of the neutron and proton density distributions, and T = N or Z. We stress that the above method has the advantage of allowing for a very direct connection between the EoS and the properties of finite nuclei. It was used in Ref. [38] in conjunction with meson-theoretic potentials and found to yield realistic predictions for binding energies and charge radii. The constant f 0 in the surface term is typically obtained from fits to β-stable nuclei and determined to be about 60-70 MeV fm 5 [39]. How this uncertainty impacts the corresponding predictions was discussed in Ref.
[1] and will be taken into account in the present calculations.

III. PREDICTIONS FOR PROTON SKINS
In Table I, we display proton skin predictions for some isotopic chains. The EoS used for these predictions is based upon N 3 LO two-nucleon forces (2NF) plus the leading 3NF. The estimated theoretical errors include uncertainties due to variations of the cutoff in the range 450-500 MeV as well as an error (added in quadrature) to account for the uncertainty originating from the method we use to calculate the skins [1]. The latter error is in the order of ± 0.01 fm, but varies with the size of the skin.
As a general feature, we observe that the proton skins can be quite large. In fact, the neutron skins of the corresponding (neutron-rich) mirror nuclei are smaller. This fact is demonstrated in Table II, where we show, for the most neutron-deficient isotope in each chain, the proton skin together with the neutron skin of the corresponding mirror nucleus. Some data on proton skins can be found in Refs. [48][49][50][51]. In Ref. [49], the existence of neutron and proton skins in β-unstable neutron-or proton-rich Na and Mg isotopes is discussed based on measurements of the interaction cross sections of these isotopes incident on a carbon target around 950A MeV. In Ref. [50], proton skin thickness for isotopes 32−40 Ar were deduced from the interaction cross sections of 31−40 Ar and 31−37 Cl on carbon targets. The obtained matter radii were combined with measured charge radii for Argon isotopes to obtain skin thicknesses.
In Fig. 1, we show our predictions for the proton skins of Argon isotopes in comparison with data deduced from experiments as described in Ref. [50]. Keeping in mind the large experimental errors, the trend of the empirical information is described reasonably well by our predictions, where the proton skin decreases essentially monotonically with increasing number of neutrons in a given isotopic chain.

A. Symmetry of mirror nuclei
Assuming perfect charge symmetry, one has, in mirror nuclei, a relation which we have verified to be exactly satisfied when Coulomb contributions and other charge-dependent effects are turned off. Applying the definition of the neutron skin, we can then immediately conclude from Eq. (7) that  Namely, the neutron skin of nucleus (Z, N ) would be equal to the difference between the proton radii of the mirror pair in the presence of perfect charge symmetry. If charge radii could be measured accurately for mirror pairs in the desired mass range, then the neutron skin of the (Z, N ) nucleus could be obtained from Eq. (9) after theoretical considerations to account for charge effects. Thus, this could be an alternative, although perhaps equally challenging from the experimental side, to the anticipated parity-violating experiments [4].

B. Radii and skins of mirror nuclei for A ≈50
We now move to a specific range within medium mass nuclei, namely A ≈ 48 − 54. This choice can be motivated by the vicinity to 48 Ca, whose neutron skin has already been and is likely to be in the future the object of several investigations, both theoretical and experimental. At the same time, the need to consider mirror pairs limits the spectrum of realistic possibilities. Table IV displays the neutron skin of the neutron-rich isotones from Table III in relation to ∆R p as defined in Eq. (9), with and without Coulomb effects. (Note that the latter case will not be addressed again and is shown here only for numerical verification, since the two items appearing in parentheses in Table IV are expected to be exactly equal to each other on grounds of elementary nuclear physics.) Increasing ∆R p implies increasing the neutron skin, as one might reasonably expect unless Coulomb effects were to reverse the relation in Eq. (9). Note, though, that quantitatively speaking Coulomb effects are significant.
Next we wish to explore the relation between ∆R p and S n (Z, N ) for other chains. In particular, we wish to investigate if and how such relation differs, quantitatively, among chains with different masses. For that purpose, we consider in Table V and VI two isotopic chains, one of them in a mass range considerably different than the one studied in Table IV. A visual representation of Tables IV, V, and VI is provided in Figs. 2, 3, and 4.
The first observation is that, for similar values of ∆R p , the corresponding values of S n (Z, N ) are approximately the same, regardless Z and N . Also, in all three cases the relation is clearly linear. We stress again that the results shown in Figs. 2, 3, and 4 are fundamentally distinct from the correlations discussed in Ref. [4]. The latter are obtained varying the parameters of Skyrme models (each model constrained to produce a chosen value of the neutron skin in 208 Pb) for a fixed mirror pair. Here, we explore to which extent our microscopic EoS yields, within theoretical uncertainties, a unique relation between S n and ∆R p .
The parameters of our predicted linear relation, By means of Eqs. (10-11), a measurement of ∆R p can then be promptly related to the neutron skin of the neutron-rich nucleus in the mirror pair. Microscopic predictions do, of course, differ from one another. Although EFT should, in principle, be a modelindependent approach, even EFT-based predictions can differ between them, depending, for instance, on the details of the input forces (e.g. cutoff) and the chosen many-body method. Moreover, the microscopically-predicted relations between two quantities or observables are not necessarily located on one of the Skyrme models correlations. Here, we suggest that analyses such as the present one, combined with other microscopic predictions, are the best way to provide a global relation between the "observables" being studied (as well as their relation to the density dependence of the symmetry energy), accompanied by a meaningful theoretical uncertainty.

V. SUMMARY AND CONCLUSIONS
Microscopic predictions of the EoS for isospin-asymmetric nuclear matter have been applied to obtain proton and neutron skins of selected chains of nuclei. The calculations of the EoS are based on high-precision chiral forces.
First, we presented proton skin predictions for a few isotopic chains to observe some of their general fetures, particularly in comparison with neutron skins. We find that they are generally large, larger than neutron skins for comparable values of proton-neutron asymmetry. Our predictions compare well with available empirical information.
We then moved the focus on to mirror nuclei in a specific mass range (A ≈ 48-54). At this point we took the opportunity to make some comments about and highlight differences with recent studies [4,6] which have addressed those nuclei.
Using our microscopic predictions and their uncertainties, we constructed a correlation between the skin of a neutron-rich nucleus and the difference between the proton radii of the corresponding mirror pair. We discussed the meaning and significance of such correlation in contrast to those characteristic of phenomenological studies. Given the ab initio nature of the EoS, we are in the position of exploring, for instance, the contribution of 3NF to the predictions, the impact of higher chiral orders, and the order-by-order pattern of the chiral perturbation series.
We conclude by highlighting the importance of taking into account microscopic predictions as a guide towards the planning of future measurements.

Acknowledgments
This work was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Award Number DE-FG02-03ER41270.