Neutron skins of heavy nuclei and tidal deformability of neutron star

In this paper, I discuss the numerical predictions for the neutron-skin thickness (NST) of various finite nuclei starting from $^{40}$Ca to $^{238}$U using recently developed effective relativistic mean-field models G3 and IOPB-I [6, 7]. The calculated results compared with the PREX-II data, and the experiment has done with antiprotons at CERN. Further, I also calculated the dimensional tidal deformability of a canonical neutron star 1.4$M_\odot$ and compared with the recent observation of GW1701817.


Introduction
The accurate description of the matter distribution in nuclei is an important problem in nuclear physics to understand the nuclear structure of the nuclei.The proton distribution has measured with very high accuracy using electron-nucleus elastic scattering.But it is more difficult to accurately determine the neutron density distributions of nuclei by any experimental probe because of the electrically neutral particle.Several experiments of charge distribution of neutron have been carried out worldwide over the last decade by using proton scattering and interaction cross-sections in heavy-ion collisions at relativistic energies.Till now, the determination of neutron radii is very poorly known compared to proton radii.However, the newly designed experiments such as PREX-II at JLab and the Bates Laboratory at MIT with polarized beams and targets have resulted in a significant better clearer picture of the neutron charge distribution [1, 2].However, PREX-II experiment has mainly designed for neutron radius of 208 Pb, and give the first model-independent neutron-skin thickness (NST) of 208 Pb to be 0.33 +0.16  −0.18 fm.The total error is almost half the median value, but the PREX experiment is exciting, and future higher statistics data are required to reduce the uncertainty.Using the tight bound of the tidal deformability of the canonical neutron star, Fattoyev et al. have suggested an upper limit of NST of 208 Pb to be about ∆r np 0.25 fm [3].
Thanks to GW170817, the first detected binary neutron star merger.The first analysis of the tidal phasing has placed the upper bound of Λ ≤ 800 at 90% confidence on the tidal deformability of a 1.4M ⊙ neutron star [4].This analysis did not explicitly consider that both compact objects were neutron stars.Then they have re-analysis again with using an ordinary equation of state for both the stars and placed Λ 1.4 = 190 +390 −120 that translates to a stringent bound of Λ 1.4 = 580 at the 90% confidence level [5].These bounds can, in turn, constrain the neutron star equation of state.
In this contribution, I calculated NST for the finite nuclei starting from 40 Ca to 238 U using the recently developed new energy density functional G3 and IOPB-I.Next, I show the result of the dimensional tidal deformability of the neutron stars, which was observed by the aLIGO-VIRGO collaboration.
Throughout this paper, I use geometric units, c=1=G, where c is the speed of light, and G is the gravitational constant, respectively.

The extended relativistic mean-field (ERMF) model
The relativistic mean-field model is the effective model of QCD formalism.The mean-field formalism approximates the effect of vacuum fluctuation and Fock term in the calculation by fitting the coupling constant with experimental observables.It is the virtue of mean-field that can ignore the basic formalism difficulties like renormalization and divergence of the system and take the simple way just by fitting the coupling constant.In this calculation, I have taken the non-linear coupling in the σ−meson like Boguta Lagrangian, ω − ρ cross-coupling.Additionally, I have included the δ−meson, that is responsible for the mass isospin asymmetry of the nucleon.Here, I start with the energy density functional for the ERMF model [6,7]: where Φ, W, R, D and A are the fields and g σ , g ω , g ρ , g δ and e 2 4π are the coupling constants of the fields.m σ , m ω , m ρ and m δ are the masses of the mesons for σ, ω, ρ, δ and photon field, respectively.The extended energy density functional with δ−meson contains the nucleons and other exchange mesons like σ, ω and ρ−meson and photon A µ .The effects of the δ−meson to the bulk properties of finite nuclei are nominal, but, the effects are significant for highly asymmetric dense nuclear matter.The δ− meson causes the splits the effective masses of proton and neutron which modify the properties of the neutron star and heavy ion reaction.Additionally, from the energy density functional in Eq.1, the terms having g γ , f, β σ and β ω are responsible for the effects related with the electromagnetic structure of the pion and nucleon [8].

Results
I have calibrated the parameters of the energy density functional as given by Eq.(1) [6,7].The optimization of the energy density functional performed for a given set of fit data using the simulated annealing method.This method allows one to search for the best fit parameter in a given region of the parameter space.I have fitted the coupling constants of Eq.(1) to the properties of eight spherical nuclei together with some constraints on the properties of the nuclear matter at the saturation density.Newly generated ERMF models G3 and IOPB-I are very well suitable for the study ground-state properties of finite nuclei, nuclear matter, and neutron star.Here, I examine the NST of 26 stable nuclei starting from 40 Ca and 238 U.The result for the difference between the neutron and proton rms radii gives the NST of the nucleus.The NST ∆r np is defined as ( Where R n and R p are the rms radii for the neutron and proton distribution, respectively.The addition of ω−ρ cross-coupling into the Lagrangian density constrains the NST of finite nuclei.Finally, I will discuss in this section the neutron star properties such as tidal deformability of the binary neutron star.To calculate the tidal deformability, I need to solve the TOV equation together with perturbing matric, where the nuclear equation of state use as an input.Tidal deformability λ is highly sensitive to the fifth power of the radius (R 5 ) or the compactness (C = M/R) of the star.The relation defines the dimensionless version of the tidal deformability [9]: where k 2 is the dimensionless tidal love number which gives the internal information of the neutron star.M is the mass of the star.For the binary neutron star, the individual quantities of the tidal deformability Λ 1 and Λ 2 associated with the phase of the gravitational wave given by It is noticed that Λ = Λ 1 = Λ 2 is for the equal-mass neutron star.Notably, the tidal deformability determined from the first BNS merger is already convincing enough to rule out the various types of viable EoSs.Fig. 1(b) shown the Λ 1 and Λ 2 values for the fixed chirp mass 1.188M ⊙ from GW170818 for the high-mass M 1 and low-mass M 2 components of the binary system.From the GW170817, the values of Λ ≤ 800 (first analysis) and Λ = 190 +390 −120 (re-analysis the GW170817 data ) in the low-spin case are within the 90% credible intervals which are consistent with the 680.79, 622.06, and 461.03 of the 1.4M ⊙ NS binary for the IOPB-I, FSUGarnet, and G3 parameter sets, respectively .Furthermore, NL3 equation of state is ruled out on the basis of the GW170817 data because of the stiffer nature.

Conclusion
In this analysis, I studied the NST of various finite nuclei and tidal deformability of the neutron star using the effective interactions G3 and IOPB-I.Both parameter sets have very well reproduced the finite nuclei, infinite nuclear matter, and neutron star properties.In particular, the calculated value Λ 1.4 of a neutron star is consistent with the recent observation of the GW170817.This work is supported in part by JSPS KAKENHI Grant No.18H01209.

Fig. 1 .
Fig. 1. (a)The NST as a function of the asymmetry parameter.(b) Dimensionless tidal deformability for a 1.88M ⊙ chirp-mass binary neutron star merger consistent with GW170817.The figures are adopted from[7].