Reaction cross section of proton scattering consistent with PREX-II

Background: The neutron skin thickness $R_{\rm skin}^{\rm PV}$ of PREX-II is presented in Phys. Rev. Lett. {\bf 126}, 172502 (2021). The reaction cross section $\sigma_R$ is useful to determine the matter radius $R_m$ and $R_{\rm skin}$. For proton scattering, the reaction cross section $\sigma_R$ are available for $E_{\rm in}>400$ MeV. Method and results: We determine $R_n^{\rm exp}=5.727 \pm 0.071$ fm and $R_m^{\rm exp}=5.617 \pm 0.044$ fm from $R_p^{\rm exp}$ = 5.444 fm and $R_{\rm skin}^{\rm PV}$. The $R_p^{\rm GHFB}$ calculated with D1S-GHFB with the angular momentum projection (AMP). agrees with $R_p^{\rm exp}$. The neutron density calculated with GHFB+AMP is scaled so as to $R_n^{\rm scaling}=5.727$ fm. The Love-Franey $t$-matrix model with the scaled densities reproduces the data on $\sigma_R$. Aim: Our aim is to find the $\sigma_R$ of proton scattering consistent with $R_{\rm skin}^{\rm PV}$. Conclusion: The $\sigma_R$ of proton scattering consistent with $R_{\rm skin}^{\rm PV}$ are $\sigma_R^{\rm exp}$ at $E_{\rm in} = 534.1, 549, 806$ MeV.


I. INTRODUCTION AND CONCLUSION
Background: Horowitz et al. [1] proposed a direct measurement for neutron skin R skin .The measurement is composed of parity-violating (P V ) weak scattering and elastic electron scattering.The neutron radius R n is determined from the former experiment, whereas the proton radius R p is from the latter.
Very recently, the PREX collaboration presented the PREX-II value [2]: combining the original Lead Radius EXperiment (PREX) result [3,4] with the updated PREX-II result.The R P V skin value is most reliable at the present stage, and provides crucial tests for the equation of state (EoS) of nuclear matter [5][6][7][8][9] as well as nuclear structure and reaction.In particular, Reed et al. [10] report a value of the slope parameter of the EoS and examine the impact of such a stiff symmetry energy on some critical neutron-star observables.The R P V skin value is considerably larger than the other experimental values which are significantly model dependent [11][12][13][14].The nonlocal dispersive-optical-model (DOM) analysis of 208 Pb deduces r DOM skin = 0.25 ± 0.05 fm [15], The chiral (Kyushu) gmatrix folding model determines r 208 skin = 0.27 ± 0.03 fm from reaction cross section σ R in 30 ≤ E lab ≤ 100 MeV [16].These values are consistent with R P V skin .Aim: The aim is to find the σ R of p + 208 Pb scattering that supports R skin (PREXII).
Method and results: The reaction cross section σ R is a powerful tool of evaluating the matter radius R m .We first determine R exp n = 5.727 ± 0.071 fm and R exp m = 5.617 ± 0.044 fm from R exp p = 5.444 fm [17] and R PV skin .The R GHFB p calculated with Gongny-D1S HFB (GHFB) with the angular mo- * orion093g@gmail.commentum projection (AMP) agrees with R exp p of electron scaling.The neutron density calculated with GHFB+AMP is scaled so as to R scaling n = 5.727 fm.The Love-Franey tmatrix [18] model with the scaled densities reproduces the data on σ R at E lab = 534.1,549, 806 MeV.Our calculation has no free parameter.

Conclusion:
The σ R of proton scattering consistent with R PV skin are σ exp R at E lab = 534.1,549, 806 MeV.

II. MODEL
Our model is the folding model based on Love-Franey (LF) t-matrix [18].
The formulation of the folding model is shown below.For proton-nucleus scattering, the potential U (R) between an incident proton and a target (T) has the direct and exchange parts, U DR and U EX , as where R is the relative coordinate between an incident proton and T, s = −r T + R, and r T is the coordinate of the interacting nucleon from T. Each of µ and ν denotes the zcomponent of isospin; 1/2 means neutron and −1/2 does proton.The nonlocal U EX has been localized in Eq. (2b) with the local semi-classical approximation [19], where K(R) is the local momentum between an incident proton and T, and M = A/(1 + A) for the target mass number A; see Ref. [20] for the validity of the localization.
The relative wave function ψ is decomposed into partial waves χ L , each with different orbital angular momentum L. The elastic S-matrix elements S L are obtained from the asymptotic form of the χ L .The total reaction cross section σ R is calculable from the S L as where K is an incident momentum.
As proton and neutron densities, , we use the densities calculated with GHFB+AMP [21].As a way of taking the center-of-mass correction to the densities, we use the method of Ref. [22], since the procedure is quite simple.The R GHFB p calculated with GHFB+AMP agrees with R exp p = 5.444 fm [17].The neutron density calculated with GHFB+AMP is scaled so as to R scaling n = 5.727 fm (the central value of R exp n = 5.727 ± 0.071 fm determined in Sec.I).The scaled densities based on R P V skin and R exp p are used for analyses of p+ 208 Pb scattering.Now we explain the scaling of density ρ(r).We can obtain the scaled density ρ scaling (r) from the original density ρ(r) as with a scaling factor

III. RESULTS
The LF t-matrix folding model with the GHFB+AMP densities underestimates the σ R data in 400 ≤ E lab ≤ 900 MeV only by a factor of 0.96, as shown in Fig. 1.The LF t-matrix folding model with the scaled densities reproduces the data in E lab = 534.1,549, 806 MeV.This indicates that the LF t-matrix folding model with the scaled densities is useful in 400 ≤ E lab ≤ 900 MeV.FIG. 1. E lab dependence of reaction cross sections σR for p+ 208 Pb scattering.Open circles stand for the results of the LF t-matrix folding model with GHFB+AMP densities, whereas closed circles correspond to those of the LF t-matrix folding model with the scaled densities.The data are taken from Refs.[23,24].

IV. DISCUSSIONS
Now we discuss how good the LF t-matrix folding model with the scaled densities is for p+ 12     Closed circles stand for the results of the LF t-matrix folding model with the scaled densities.The data are taken from Refs.[27,28].
C scattering at E lab = 800 MeV and p+ 40 Ca scattering at E lab = 700 MeV.For 40 Ca, Zenihiro et al. determined neutron radius R n (RCNP) = 3.375 fm, R p (RCNP) = 3.385 fm and R skin (RCNP) = −0.01 ± 0.049 fm from the differential cross section and the analyzing powers for p+ 40 Ca scattering [25].The GHFB+AMP densities are scaled so as to R p (scaling) = R p (RCNP) and R n (scaling) = R n (RCNP).

For 12 C
, Tanihata et al. determined matter radius R m (σ I ) = 2.35(2) fm from interaction cross sections σ I [26].We deduce neutron radius R n (σ I ) = 2.37 fm from the R m (σ I ) and the R exp p = 2.33 fm of electron scattering.The GHFB+AMP densities are scaled so as to R p (scaling) = R exp p and R n (scaling) = R n (σ I ).

Figure 2
Figure2shows σ R for p+40 Ca scattering at 700 MeV and p+12 C scattering at 800 MeV.The LF t-matrix folding model with the scaled densities is good for p+ 40 Ca scattering at E lab = 700 MeV, and almost reproduces the data for p+12 C scattering at E lab = 800 MeV.

FIG. 2 .
FIG. 2. Mass-numberA dependence of reaction cross sections σR for p+ 40 Ca scattering at 700 MeV and p+ 12 C scattering at 800 MeV.Closed circles stand for the results of the LF t-matrix folding model with the scaled densities.The data are taken from Refs.[27,28].