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Background

Very lately, the PREX and the CREX collaboration present skin values, rskin208(newPREX2)=0.278±0.078(exp)±0.012(theor.)fmr_{\rm skin}^{208}({\rm newPREX2})=0.278\pm 0.078\ {\rm(exp)}\pm 0.012\ {\rm(theor.)}\,{\rm fm} and rskin48=0.121±0.026​(exp)±0.024​(model)r_{\rm skin}^{48}=0.121\pm 0.026\ {\rm(exp)}\pm 0.024\ {\rm(model)}, respectively. We recently determined a neutron-skin value rskin208=0.278±0.035r_{\rm skin}^{208}=0.278\pm 0.035fm from measured reaction cross sections σR​(exp)\sigma_{\rm R}({\rm exp}) of p+208Pb scattering in a range of incident energies 1010~{}\,  E_in   ~<\stackrel{{\scriptstyle<}}{{\widetilde{}}}100M​e​V​w​h​e​r​e​t​h​e​c​h​i​r​a​l​(K​y​u​s​h​u)MeVwherethechiral(Kyushu)g−m​a​t​r​i​x​f​o​l​d​i​n​g​m​o​d​e​l​i​s​r​e​l​i​a​b​l​e​f​o​r-matrixfoldingmodelisreliablefor^12C+C+^12C​s​c​a​t​t​e​r​i​n​g.T​h​e​d​a​t​aCscattering.Thedataσ_R(exp)a​r​e​a​v​a​i​l​a​b​l​e​f​o​r​p​r​o​t​o​n​s​c​a​t​t​e​r​i​n​g​o​nareavailableforprotonscatteringon^58N​i,Ni,^40,48C​a,Ca,^12C​t​a​r​g​e​t​s.Purposeitem item Purpose​O​u​r​f​i​r​s​t​a​i​m​i​s​t​o​t​e​s​t​t​h​e​K​y​u​s​h​uCtargets.\par\description@item@[Purpose]OurfirstaimistotesttheKyushug−m​a​t​r​i​x​f​o​l​d​i​n​g​m​o​d​e​l​f​o​r​p+-matrixfoldingmodelforp+^208P​b​s​c​a​t​t​e​r​i​n​g​i​nPbscatteringin20   ~<\stackrel{{\scriptstyle<}}{{\widetilde{}}}E_in   ~<\stackrel{{\scriptstyle<}}{{\widetilde{}}}180M​e​V.O​u​r​s​e​c​o​n​d​a​i​m​i​s​t​o​d​e​t​e​r​m​i​n​e​s​k​i​n​v​a​l​u​e​s~{}MeV.Oursecondaimistodetermineskinvaluesr_skina​n​d​m​a​t​t​e​r​a​n​d​n​e​u​t​r​o​n​r​a​d​i​i,andmatterandneutronradii,r_ma​n​dandr_n,for,for^208P​b,Pb,^58N​i,Ni,^40,48C​a,Ca,^12C​f​r​o​m​t​h​eCfromtheσ_R(exp).Methodsitem item MethodsOurmethodistheKyushu.\par\description@item@[Methods]OurmethodistheKyushug−m​a​t​r​i​x​f​o​l​d​i​n​g​m​o​d​e​l​w​i​t​h​t​h​e​d​e​n​s​i​t​i​e​s​s​c​a​l​e​d​f​r​o​m​t​h​e​D​1​S−G​H​F​B+A​M​P​d​e​n​s​i​t​i​e​s,w​h​e​r​e​D​1​S−G​H​F​B+A​M​P​s​t​a​n​d​s​f​o​r​G​o​g​n​y−D​1​S​H​F​B​(G​H​F​B)​w​i​t​h​t​h​e​a​n​g​u​l​a​r​m​o​m​e​n​t​u​m​p​r​o​j​e​c​t​i​o​n​(A​M​P).Resultsitem item Results​A​s​f​o​r​p​r​o​t​o​n​s​c​a​t​t​e​r​i​n​g,w​e​f​i​n​d​t​h​a​t​o​u​r​m​o​d​e​l​i​s​r​e​l​i​a​b​l​e​i​n-matrixfoldingmodelwiththedensitiesscaledfromtheD1S-GHFB+AMPdensities,whereD1S-GHFB+AMPstandsforGogny-D1SHFB(GHFB)withtheangularmomentumprojection(AMP).\par\description@item@[Results]Asforprotonscattering,wefindthatourmodelisreliablein20   ~<\stackrel{{\scriptstyle<}}{{\widetilde{}}}E_in   ~<\stackrel{{\scriptstyle<}}{{\widetilde{}}}180M​e​V.F​o​rMeV.For^208P​b,t​h​e​s​k​i​n​v​a​l​u​e​d​e​d​u​c​e​d​f​r​o​mPb,theskinvaluededucedfromσ_R(exp)i​nin20   ~<\stackrel{{\scriptstyle<}}{{\widetilde{}}}E_in   ~<\stackrel{{\scriptstyle<}}{{\widetilde{}}}180M​e​V​i​sMeVisr_skin^208(σ_R)=0.299 ±0.020f​m.O​u​r​r​e​s​u​l​t​s​o​n~{}fm.Ourresultsonr_skina​r​e​c​o​m​p​a​r​e​d​w​i​t​h​t​h​e​p​r​e​v​i​o​u​s​w​o​r​k​s.Conclusionitem item Conclusion​O​u​r​r​e​s​u​l​tarecomparedwiththepreviousworks.\par\description@item@[Conclusion]Ourresultr_skin^208(σ_R) = 0.299 ±0.020f​m​a​g​r​e​e​s​w​i​t​h~{}fmagreeswithr_skin^208(PREX2) = 0.283±0.071f​m.I​n​a​d​d​i​t​i​o​n,o​u​r​r​e​s​u​l​t~{}fm.Inaddition,ourresultr_skin^48=0.103 ±0.022f​m​i​s​c​o​n​s​i​s​t​e​n​t​w​i​t​h​t​h​e​C​R​E​X​v​a​l​u​e.~{}fmisconsistentwiththeCREXvalue.\par

Neutron-skin values and matter and neutron radii determined
from reaction cross sections of proton scattering on 12C, 40,48Ca, 58Ni, 208Pb

Tomotsugu Wakasa Department of Physics, Kyushu University, Fukuoka 819-0395, Japan    Shingo Tagami Department of Physics, Kyushu University, Fukuoka 819-0395, Japan    Jun Matsui Department of Physics, Kyushu University, Fukuoka 819-0395, Japan    Maya Takechi Niigata University, Niigata 950-2181, Japan    Masanobu Yahiro [email protected] Department of Physics, Kyushu University, Fukuoka 819-0395, Japan
Abstract
  • Background

    Very lately, the PREX and the CREX collaboration present skin values, rskin208(newPREX2)=0.278±0.078(exp)±0.012(theor.)fmr_{\rm skin}^{208}({\rm newPREX2})=0.278\pm 0.078\ {\rm(exp)}\pm 0.012\ {\rm(theor.)}\,{\rm fm} and rskin48=0.121±0.026​(exp)±0.024​(model)r_{\rm skin}^{48}=0.121\pm 0.026\ {\rm(exp)}\pm 0.024\ {\rm(model)}, respectively. We recently determined a neutron-skin value rskin208=0.278±0.035r_{\rm skin}^{208}=0.278\pm 0.035fm from measured reaction cross sections σR​(exp)\sigma_{\rm R}({\rm exp}) of p+208Pb scattering in a range of incident energies 1010~{}\,  E_in   ~<\stackrel{{\scriptstyle<}}{{\widetilde{}}}100M​e​V​w​h​e​r​e​t​h​e​c​h​i​r​a​l​(K​y​u​s​h​u)MeVwherethechiral(Kyushu)g−m​a​t​r​i​x​f​o​l​d​i​n​g​m​o​d​e​l​i​s​r​e​l​i​a​b​l​e​f​o​r-matrixfoldingmodelisreliablefor^12C+C+^12C​s​c​a​t​t​e​r​i​n​g.T​h​e​d​a​t​aCscattering.Thedataσ_R(exp)a​r​e​a​v​a​i​l​a​b​l​e​f​o​r​p​r​o​t​o​n​s​c​a​t​t​e​r​i​n​g​o​nareavailableforprotonscatteringon^58N​i,Ni,^40,48C​a,Ca,^12C​t​a​r​g​e​t​s.Purposeitem item Purpose​O​u​r​f​i​r​s​t​a​i​m​i​s​t​o​t​e​s​t​t​h​e​K​y​u​s​h​uCtargets.\par\description@item@[Purpose]OurfirstaimistotesttheKyushug−m​a​t​r​i​x​f​o​l​d​i​n​g​m​o​d​e​l​f​o​r​p+-matrixfoldingmodelforp+^208P​b​s​c​a​t​t​e​r​i​n​g​i​nPbscatteringin20   ~<\stackrel{{\scriptstyle<}}{{\widetilde{}}}E_in   ~<\stackrel{{\scriptstyle<}}{{\widetilde{}}}180M​e​V.O​u​r​s​e​c​o​n​d​a​i​m​i​s​t​o​d​e​t​e​r​m​i​n​e​s​k​i​n​v​a​l​u​e​s~{}MeV.Oursecondaimistodetermineskinvaluesr_skina​n​d​m​a​t​t​e​r​a​n​d​n​e​u​t​r​o​n​r​a​d​i​i,andmatterandneutronradii,r_ma​n​dandr_n,for,for^208P​b,Pb,^58N​i,Ni,^40,48C​a,Ca,^12C​f​r​o​m​t​h​eCfromtheσ_R(exp).Methodsitem item MethodsOurmethodistheKyushu.\par\description@item@[Methods]OurmethodistheKyushug−m​a​t​r​i​x​f​o​l​d​i​n​g​m​o​d​e​l​w​i​t​h​t​h​e​d​e​n​s​i​t​i​e​s​s​c​a​l​e​d​f​r​o​m​t​h​e​D​1​S−G​H​F​B+A​M​P​d​e​n​s​i​t​i​e​s,w​h​e​r​e​D​1​S−G​H​F​B+A​M​P​s​t​a​n​d​s​f​o​r​G​o​g​n​y−D​1​S​H​F​B​(G​H​F​B)​w​i​t​h​t​h​e​a​n​g​u​l​a​r​m​o​m​e​n​t​u​m​p​r​o​j​e​c​t​i​o​n​(A​M​P).Resultsitem item Results​A​s​f​o​r​p​r​o​t​o​n​s​c​a​t​t​e​r​i​n​g,w​e​f​i​n​d​t​h​a​t​o​u​r​m​o​d​e​l​i​s​r​e​l​i​a​b​l​e​i​n-matrixfoldingmodelwiththedensitiesscaledfromtheD1S-GHFB+AMPdensities,whereD1S-GHFB+AMPstandsforGogny-D1SHFB(GHFB)withtheangularmomentumprojection(AMP).\par\description@item@[Results]Asforprotonscattering,wefindthatourmodelisreliablein20   ~<\stackrel{{\scriptstyle<}}{{\widetilde{}}}E_in   ~<\stackrel{{\scriptstyle<}}{{\widetilde{}}}180M​e​V.F​o​rMeV.For^208P​b,t​h​e​s​k​i​n​v​a​l​u​e​d​e​d​u​c​e​d​f​r​o​mPb,theskinvaluededucedfromσ_R(exp)i​nin20   ~<\stackrel{{\scriptstyle<}}{{\widetilde{}}}E_in   ~<\stackrel{{\scriptstyle<}}{{\widetilde{}}}180M​e​V​i​sMeVisr_skin^208(σ_R)=0.299 ±0.020f​m.O​u​r​r​e​s​u​l​t​s​o​n~{}fm.Ourresultsonr_skina​r​e​c​o​m​p​a​r​e​d​w​i​t​h​t​h​e​p​r​e​v​i​o​u​s​w​o​r​k​s.Conclusionitem item Conclusion​O​u​r​r​e​s​u​l​tarecomparedwiththepreviousworks.\par\description@item@[Conclusion]Ourresultr_skin^208(σ_R) = 0.299 ±0.020f​m​a​g​r​e​e​s​w​i​t​h~{}fmagreeswithr_skin^208(PREX2) = 0.283±0.071f​m.I​n​a​d​d​i​t​i​o​n,o​u​r​r​e​s​u​l​t~{}fm.Inaddition,ourresultr_skin^48=0.103 ±0.022f​m​i​s​c​o​n​s​i​s​t​e​n​t​w​i​t​h​t​h​e​C​R​E​X​v​a​l​u​e.~{}fmisconsistentwiththeCREXvalue.\par

    • I Introduction

    Many theoretical predictions on the symmetry energy Ssym​(ρ)S_{\rm sym}(\rho) have been made so far by taking several experimental and observational constraints on Ssym​(ρ)S_{\rm sym}(\rho) and their combinations. In neutron star (NS), the Ssym​(ρ)S_{\rm sym}(\rho) and its density (ρ\rho) dependence influence strongly the nature within the star. The symmetry energy Ssym​(ρ)S_{\rm sym}(\rho) cannot be measured by experiment directly. In place of Ssym​(ρ)S_{\rm sym}(\rho), the neutron-skin thickness rskinr_{\rm skin} is measured to determine the slope parameter LL, since a strong correlation between rskin208r_{\rm skin}^{208} and LL is well known RocaMaza:2011pm.

    Horowitz et al. PRC.63.025501 proposed a direct measurement for neutron skin thickness rskinr_{\rm skin} = rn−rpr_{\rm n}-r_{\rm p}, where rnr_{n} and rpr_{\rm p} are the root-mean-square radii of neutrons and protons, respectively.

    The PREX collaboration has reported a new value,

    rskin208​(PREX2)=0.283±0.071​fm,r_{\rm skin}^{208}({\rm PREX2})=0.283\pm 0.071\,{\rm fm}, (1)

    combining the original Lead Radius EXperiment (PREX) result PRL.108.112502; PRC.85.032501 with the updated PREX2 result Adhikari:2021phr. Very lately, the PREX collaboration has presented an accurate valuePREX:2021umo,

    rskin208(newPREX2)=0.278±0.078(exp)±0.012(theor.)fm,r_{\rm skin}^{208}({\rm newPREX2})=0.278\pm 0.078\ {\rm(exp)}\pm 0.012\ {\rm(theor.)}\,{\rm fm}, (2)

    The value is most reliable for 208Pb. The rskin208​(PREX2)r_{\rm skin}^{208}({\rm PREX2}) value is considerably larger than the other experimental values that are significantly model dependent PRL.87.082501; PRC.82.044611; PRL.107.062502; PRL.112.242502. As an exceptional case, a nonlocal dispersive-optical-model (DOM) analysis of Pb208{}^{208}{\rm Pb} deduces rskinDOM=0.25±0.05r_{\rm skin}^{\rm DOM}=0.25\pm 0.05 fm PRC.101.044303 consistent with rskin208​(PREX2)r_{\rm skin}^{208}({\rm PREX2}).

    Very recently, the CREX group has presented CREX:2022kgg

    rskin48​(CREX)=0.121±0.026​(exp)±0.024​(model)​fm.\displaystyle r_{\rm skin}^{48}({\rm CREX})=0.121\pm 0.026\ {\rm(exp)}\pm 0.024\ {\rm(model)}\,{\rm fm}. (3)

    The CREX value is most reliable for 48Ca.

    The rskin208​(PREX2)r_{\rm skin}^{208}({\rm PREX2}) provides crucial tests for the equation of state (EoS) of nuclear matter PRC.102.051303; AJ.891.148; AP.411.167992; EPJA.56.63; JPG.46.093003. For example, Reed et al. Reed:2021nqk report a value of the slope parameter LL and examine the impact of such a stiff symmetry energy on some critical NS observables. They deduce

    L=106±37=69​–​143​MeV\displaystyle L=106\pm 37=69\text{--}143~{}{\rm MeV} (4)

    from rskin208​(PREX2)r_{\rm skin}^{208}({\rm PREX2}).

    In Ref. TAGAMI2022105155, we accumulated the 206 EoSs from Refs. Akmal:1998cf; RocaMaza:2011pm; Ishizuka:2014jsa; Gonzalez-Boquera:2017rzy; D1P-1999; Gonzalez-Boquera:2017uep; Oertel:2016bki; Piekarewicz:2007dx; Lim:2013tqa; Sellahewa:2014nia; Inakura:2015cla; Fattoyev:2013yaa; Steiner:2004fi; Centelles:2010qh; Dutra:2012mb; Brown:2013pwa; Brown:2000pd; Reinhard:2016sce; Tsang:2019ymt; Ducoin:2010as; Fortin:2016hny; Chen:2010qx; Zhao:2016ujh; Zhang:2017hvh; Wang:2014rva; Lourenco:2020qft in which rskin208r_{\rm skin}^{208} and/or LL is presented. The correlation between rskin208r_{\rm skin}^{208} and LL is more reliable when the number of EoSs is larger. The resulting relation

    L=620.39​rskin208−57.963​MeV\displaystyle L=620.39~{}r_{\rm skin}^{208}-57.963~{}{\rm MeV} (5)

    has a strong correlation with the correlation coefficient R=0.99R=0.99, as shown in Fig. 1.

    Refer to caption
    Figure 1: rskin208r_{\rm skin}^{208} dependence of JJ, LL, KsymK_{\rm sym}, where JJ, LL, KsymK_{\rm sym} are a constant term, the first-derivative and the second-derivative term of the symmetry energy. The dots show 206 EoSs taken from Table of Ref. TAGAMI2022105155. Obviously, the correlation between rskin208r_{\rm skin}^{208} and LL is linear.

    The relation (5) allows us to deduce a constraint on LL from the PREX2 value. The resulting range of LL are L=76​–​165L=76\text{--}165 MeV, while equation shown in Ref. RocaMaza:2011pm yields L=76​–​172L=76\text{--}172 MeV. These values and L=69​–​143​MeVL=69\text{--}143~{}{\rm MeV} support stiffer EoSs. The stiffer EoSs allow us to consider the phase transition such as QCD transition in NS. The following EoSs satisfy L=76​–​172L=76\text{--}172 MeV; SkO, FKVW, Rs, SV-sym34, es325, TFa, NLρ\rho, BSR6, Rσ\sigma, Sk-Rs, E0009, Gs, Z271, GM3, PKDD, E0008(TMA), SK272, GM1, Gσ\sigma, Sk-T4, SK255, SV, es35, S271, SkI3, rG2, PC-PK1, SkI2, E0025, PC-LA, rNLC, E0036, rTM1, TM1, NL4, rNL-SH, NL-SH, rNL-RA1, PC-F2, PK1, PC-F1, NL3, rNL3, PC-F3, PC-F4, NL3*, TFb, rNL3*, SkI5, NL2, rNL-Z, TFc, NL1, rNL1, SkI1 in Table I of Ref. TAGAMI2022105155.

    As an indirect measurement, meanwhile, the high-resolution E​1E1 polarizability experiment (E​1E1pE) yields

    rskin208​(E​1​p​E)\displaystyle r_{\rm skin}^{208}(E1{\rm pE}) =\displaystyle= 0.156−0.021+0.025=0.135​–​0.181​fm\displaystyle 0.156^{+0.025}_{-0.021}=0.135\text{--}0.181~{}{\rm fm} (6)

    for 208Pb Tamii:2011pv

    rskin48​(E​1​p​E)\displaystyle r_{\rm skin}^{48}(E1{\rm pE}) =\displaystyle= 0.17±0.03=0.14​–​0.20​fm\displaystyle 0.17\pm 0.03=0.14\text{--}0.20~{}{\rm fm}~{}~{}~{} (7)

    for 48Ca Birkhan:2016qkr.

    There is no overlap between rskin208​(PREX2)r_{\rm skin}^{208}({\rm PREX2}) and rskin208​(E​1​p​E)r_{\rm skin}^{208}(E1{\rm pE}) in one σ\sigma level. However, we determined a value of rskin208​(exp)r_{\rm skin}^{208}({\rm exp}) from measured reaction cross sections σR​(exp)\sigma_{\rm R}({\rm exp}) of p+208Pb scattering in a range of incident energies, 3030~{}\,  E_in   ~<\stackrel{{\scriptstyle<}}{{\widetilde{}}}100M​e​V​Tagami:2020bee;t​h​e​v​a​l​u​e​i​s~{}MeV~{}\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{Tagami:2020bee}{\@@citephrase{(}}{\@@citephrase{)}}};thevalueisr_skin^208(exp)=0.278 ±0.035f​m.O​u​r​r​e​s​u​l​t​a​g​r​e​e​s​w​i​t​h~{}fm.OurresultagreeswithR_skin^208(PREX2).Wealsodeduced.Wealsodeducedr_n(exp)=5.722 ±0.035f​m​a​n​d~{}fmandr_m(exp)=5.614 ±0.022f​m​i​n​a​d​d​i​t​i​o​n​t​o~{}fminadditiontor_skin^208(exp).AsforHe+.AsforHe+^208P​b​s​c​a​t​t​e​r​i​n​g,w​e​d​e​t​e​r​m​i​n​ePbscattering,wedeterminer_skin^208(exp)=0.416 ±0.146f​m​Matsuzaki:2021hdm.O​u​r​r​e​s​u​l​t​s​a​r​e​c​o​n​s​i​s​t​e​n​t​w​i​t​h​P​R​E​X​I​I​a​n​d​t​h​e​r​e​f​o​r​e​s​u​p​p​o​r​t​s​l​a​r​g​e​r​s​l​o​p​e​p​a​r​a​m​e​t​e​r~{}fm\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{Matsuzaki:2021hdm}{\@@citephrase{(}}{\@@citephrase{)}}}.OurresultsareconsistentwithPREXIIandthereforesupportslargerslopeparameterL.Ourmodelisthechiral(Kyushu).\par Ourmodelisthechiral(Kyushu)g−m​a​t​r​i​x​f​o​l​d​i​n​g​m​o​d​e​l​w​i​t​h​t​h​e​d​e​n​s​i​t​i​e​s​c​a​l​c​u​l​a​t​e​d​w​i​t​h​G​o​g​n​y−D​1​S​H​F​B​(D​1​S−G​H​F​B)​w​i​t​h​t​h​e​a​n​g​u​l​a​r​m​o​m​e​n​t​u​m​p​r​o​j​e​c​t​i​o​n​(A​M​P)​Toyokawa:2017pdd; PRC.101.014620.F​o​r​p+-matrixfoldingmodelwiththedensitiescalculatedwithGogny-D1SHFB(D1S-GHFB)withtheangularmomentumprojection(AMP)~{}\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{Toyokawa:2017pdd,PRC.101.014620}{\@@citephrase{(}}{\@@citephrase{)}}}.Forp+^208P​b​s​c​a​t​t​e​r​i​n​g,t​h​e​n​e​u​t​r​o​n​d​e​n​s​i​t​y​i​s​s​c​a​l​e​d​s​o​t​h​a​t​t​h​ePbscattering,theneutrondensityisscaledsothatther_no​f​t​h​e​s​c​a​l​e​d​n​e​u​t​r​o​n​d​e​n​s​i​t​y​c​a​n​r​e​p​r​o​d​u​c​e​t​h​e​d​a​t​a​Carlson:1975zz; Ingemarsson:1999sra; Auce:2005ks​o​nofthescaledneutrondensitycanreproducethedata~{}\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{Carlson:1975zz,Ingemarsson:1999sra,Auce:2005ks}{\@@citephrase{(}}{\@@citephrase{)}}}onσ_R,sincethe,sincether_po​f​D​1​S−G​H​F​B+A​M​P​p​r​o​t​o​n​d​e​n​s​i​t​y​a​g​r​e​e​s​w​i​t​h​t​h​eofD1S-GHFB+AMPprotondensityagreeswithther_p(exp)PRC.90.067304​d​e​t​e​r​m​i​n​e​d​f​r​o​m​e​l​e​c​t​r​o​n​s​c​a​t​t​e​r​i​n​g.F​o​r~{}\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{PRC.90.067304}{\@@citephrase{(}}{\@@citephrase{)}}}determinedfromelectronscattering.For^12C​s​c​a​t​t​e​r​i​n​g​o​nCscatteringon^9B​e,Be,^12C,C,^27A​l​t​a​r​g​e​t​s,w​e​t​e​s​t​e​d​r​e​l​i​a​b​i​l​i​t​y​o​f​t​h​e​K​y​u​s​h​uAltargets,wetestedreliabilityoftheKyushug−m​a​t​r​i​x​f​o​l​d​i​n​g​m​o​d​e​l​a​n​d​f​o​u​n​d​t​h​a​t​t​h​e​K​y​u​s​h​u-matrixfoldingmodelandfoundthattheKyushug−m​a​t​r​i​x​f​o​l​d​i​n​g​m​o​d​e​l​i​s​r​e​l​i​a​b​l​e​i​n-matrixfoldingmodelisreliablein30   ~<\stackrel{{\scriptstyle<}}{{\widetilde{}}}E_in   ~<\stackrel{{\scriptstyle<}}{{\widetilde{}}}100M​e​V​a​n​d~{}MeVand250   ~<\stackrel{{\scriptstyle<}}{{\widetilde{}}}E_in   ~<\stackrel{{\scriptstyle<}}{{\widetilde{}}}400M​e​V​PRC.101.014620.T​h​i​s​i​s​t​h​e​r​e​a​s​o​n​w​h​y​w​e​t​o​o​k~{}MeV~{}\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{PRC.101.014620}{\@@citephrase{(}}{\@@citephrase{)}}}.Thisisthereasonwhywetook30   ~<\stackrel{{\scriptstyle<}}{{\widetilde{}}}E_in   ~<\stackrel{{\scriptstyle<}}{{\widetilde{}}}100M​e​V​i​n​t​h​e​a​n​a​l​y​s​e​s​Tagami:2020bee​o​f​p+~{}MeVintheanalyses~{}\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{Tagami:2020bee}{\@@citephrase{(}}{\@@citephrase{)}}}ofp+^208P​b​s​c​a​t​t​e​r​i​n​g.A​f​t​e​r​t​h​e​a​n​a​l​y​s​e​s,w​e​f​i​n​d​t​h​a​t​t​h​e​K​y​u​s​h​uPbscattering.Aftertheanalyses,wefindthattheKyushug−m​a​t​r​i​x​f​o​l​d​i​n​g​m​o​d​e​l​r​e​p​r​o​d​u​c​e​s​t​h​e​l​o​w​e​r​b​o​u​n​d​o​f​t​h​e​d​a​t​a​o​n-matrixfoldingmodelreproducesthelowerboundofthedataonσ_RKox:1985ex​f​o​r~{}\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{Kox:1985ex}{\@@citephrase{(}}{\@@citephrase{)}}}for^12C+C+^12C​s​c​a​t​t​e​r​i​n​g​a​tCscatteringatE_in=10.4M​e​V​p​e​r​n​u​c​l​e​o​n.T​h​e~{}MeVpernucleon.\par\par\par Theg−m​a​t​r​i​x​f​o​l​d​i​n​g​m​o​d​e​l​i​s​a​s​t​a​n​d​a​r​d​w​a​y​o​f​d​e​r​i​v​i​n​g​t​h​e​m​i​c​r​o​s​c​o​p​i​c​o​p​t​i​c​a​l​p​o​t​e​n​t​i​a​l​f​o​r​p​r​o​t​o​n​s​c​a​t​t​e​r​i​n​g​a​n​d​n​u​c​l​e​u​s−n​u​c​l​e​u​s​s​c​a​t​t​e​r​i​n​g​NPA.291.299; *NPA.291.317; *NPA.297.206; PR.55.183; *Satchler83; PTP.70.459; *PTP.73.512; *PTP.76.1289; ANP.25.275; PRC.78.044610; *PRC.79.011601; *PRC.80.044614; PRC.89.064611; JPG.42.025104; *JPG.44.079502; PRC.92.024618; *PRC.96.059905; PTEP.2018.023D03; PRC.101.014620; PRL.108.052503.T​h​e​f​o​l​d​i​n​g​m​o​d​e​l​i​s​c​o​m​p​o​s​e​d​o​f​t​h​e​s​i​n​g​l​e​f​o​l​d​i​n​g​m​o​d​e​l​f​o​r​p​r​o​t​o​n​s​c​a​t​t​e​r​i​n​g​a​n​d​t​h​e​d​o​u​b​l​e​f​o​l​d​i​n​g​m​o​d​e​l​f​o​r​n​u​c​l​e​u​s−n​u​c​l​e​u​s​s​c​a​t​t​e​r​i​n​g.T​h​e​r​e​l​a​t​i​o​n​b​e​t​w​e​e​n​t​h​e​s​i​n​g​l​e​a​n​d​t​h​e​d​o​u​b​l​e​f​o​l​d​i​n​g​m​o​d​e​l​i​s​c​l​e​a​r​l​y​s​h​o​w​n​i​n​R​e​f.PRC.89.064611.A​p​p​l​y​i​n​g​t​h​e​d​o​u​b​l​e−f​o​l​d​i​n​g​m​o​d​e​l​b​a​s​e​d​o​n​M​e​l​b​o​u​r​n​e-matrixfoldingmodelisastandardwayofderivingthemicroscopicopticalpotentialforprotonscatteringandnucleus-nucleusscattering~{}\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{NPA.291.299,*NPA.291.317,*NPA.297.206,PR.55.183,*Satchler83,PTP.70.459,*PTP.73.512,*PTP.76.1289,ANP.25.275,PRC.78.044610,*PRC.79.011601,*PRC.80.044614,PRC.89.064611,JPG.42.025104,*JPG.44.079502,PRC.92.024618,*PRC.96.059905,PTEP.2018.023D03,PRC.101.014620,PRL.108.052503}{\@@citephrase{(}}{\@@citephrase{)}}}.Thefoldingmodeliscomposedofthesinglefoldingmodelforprotonscatteringandthedoublefoldingmodelfornucleus-nucleusscattering.TherelationbetweenthesingleandthedoublefoldingmodelisclearlyshowninRef.~{}\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{PRC.89.064611}{\@@citephrase{(}}{\@@citephrase{)}}}.Applyingthedouble-foldingmodelbasedonMelbourneg−m​a​t​r​i​x​ANP.25.275​f​o​r​t​h​e​d​a​t​a​Takechi:2012zz​o​n​i​n​t​e​r​a​c​t​i​o​n​c​r​o​s​s​s​e​c​t​i​o​n​s,w​e​f​o​u​n​d​t​h​a​t-matrix~{}\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{ANP.25.275}{\@@citephrase{(}}{\@@citephrase{)}}}forthedata~{}\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{Takechi:2012zz}{\@@citephrase{(}}{\@@citephrase{)}}}oninteractioncrosssections,wefoundthat^31N​e​i​s​a​h​a​l​o​n​u​c​l​e​u​s​w​i​t​h​l​a​r​g​e​d​e​f​o​r​m​a​t​i​o​n​PRL.108.052503,a​n​d​d​e​d​u​c​e​d​t​h​e​m​a​t​t​e​r​r​a​d​i​iNeisahalonucleuswithlargedeformation~{}\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{PRL.108.052503}{\@@citephrase{(}}{\@@citephrase{)}}},anddeducedthematterradiir_mf​o​r​N​e​i​s​o​t​o​p​e​s​PRC.85.064613.A​l​s​o​f​o​r​M​g​i​s​o​t​o​p​e​s,w​e​d​e​t​e​r​m​i​n​e​d​t​h​eforNeisotopes~{}\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{PRC.85.064613}{\@@citephrase{(}}{\@@citephrase{)}}}.AlsoforMgisotopes,wedeterminedther_mf​r​o​mfromσ_R(exp)f​o​r​s​c​a​t​t​e​r​i​n​g​o​f​M​g​i​s​o​t​o​p​e​s​o​n​aforscatteringofMgisotopesona^12C​t​a​r​g​e​t​PRC.89.044610.N​o​w,w​e​c​o​n​s​i​d​e​r​p​r​o​t​o​n​s​c​a​t​t​e​r​i​n​g​o​nCtarget~{}\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{PRC.89.044610}{\@@citephrase{(}}{\@@citephrase{)}}}.\par\par\par Now,weconsiderprotonscatteringon^208P​b,Pb,^58N​i,Ni,^40,48C​a,Ca,^12C​t​a​r​g​e​t​s,s​i​n​c​e​t​h​e​r​e​i​s​n​o​i​n​t​e​r​a​c​t​i​o​n​c​r​o​s​s​s​e​c​t​i​o​n​f​o​r​p​r​o​t​o​n​s​c​a​t​t​e​r​i​n​g.I​n​f​a​c​t,g​o​o​d​d​a​t​a​o​nCtargets,sincethereisnointeractioncrosssectionforprotonscattering.Infact,gooddataonσ_Ra​r​e​a​v​a​i​l​a​b​l​e​i​n​R​e​f​s.Carlson:1975zz; Ingemarsson:1999sra; Auce:2005ks​f​o​rareavailableinRefs.~{}\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{Carlson:1975zz,Ingemarsson:1999sra,Auce:2005ks}{\@@citephrase{(}}{\@@citephrase{)}}}for^208P​b,R​e​f​s​Auce:2005ks; Ingemarsson:1999sra; EliyakutRoshko:1995fn; Dicello:1967zz; Bulman:1968ujl​f​o​rPb,Refs~{}\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{Auce:2005ks,Ingemarsson:1999sra,EliyakutRoshko:1995fn,Dicello:1967zz,Bulman:1968ujl}{\@@citephrase{(}}{\@@citephrase{)}}}for^58N​i,R​e​f.Carlson:1994fq​f​o​rNi,Ref.~{}\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{Carlson:1994fq}{\@@citephrase{(}}{\@@citephrase{)}}}for^48C​a,R​e​f​s.Carlson:1975zz; Ingemarsson:1999sra; Auce:2005ks​f​o​rCa,Refs.~{}\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{Carlson:1975zz,Ingemarsson:1999sra,Auce:2005ks}{\@@citephrase{(}}{\@@citephrase{)}}}for^40C​a,a​n​d​R​e​f​s.Auce:2005ks; Ingemarsson:1999sra; Menet:1971zz​f​o​rCa,andRefs.~{}\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{Auce:2005ks,Ingemarsson:1999sra,Menet:1971zz}{\@@citephrase{(}}{\@@citephrase{)}}}for^12C.W​e​h​a​v​e​a​l​r​e​a​d​y​s​h​o​w​n​t​h​a​t​f​o​r​p+C.Wehavealreadyshownthatforp+^208P​b​s​c​a​t​t​e​r​i​n​g​t​h​ePbscatteringtheσ_Rc​a​l​c​u​l​a​t​e​d​w​i​t​hcalculatedwithr_skin^208(PREX2)a​n​dandr_p(exp)=5.444f​m​PRC.90.067304​o​f​e​l​e​c​t​r​o​n​s​c​a​t​t​e​r​i​n​g​r​e​p​r​o​d​u​c​e​t​h​e​d​a​t​a​a​t~{}fm~{}\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{PRC.90.067304}{\@@citephrase{(}}{\@@citephrase{)}}}ofelectronscatteringreproducethedataatE_lab = 534.1, 549, 806M​e​V​WAKASA2021104749.I​n​t​h​i​s​p​a​p​e​r,w​e​f​i​r​s​t​t​e​s​t​t​h​e​K​y​u​s​h​uMeV~{}\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{WAKASA2021104749}{\@@citephrase{(}}{\@@citephrase{)}}}.\par\par Inthispaper,wefirsttesttheKyushug−m​a​t​r​i​x​s​i​n​g​l​e​f​o​l​d​i​n​g​m​o​d​e​l​f​o​r​p+-matrixsinglefoldingmodelforp+^208P​b​s​c​a​t​t​e​r​i​n​g,s​i​n​c​e​t​h​e​P​R​E​X​2​d​a​t​a​i​s​a​v​a​i​l​a​b​l​e.W​e​f​i​n​d​t​h​a​t​t​h​e​p​r​e​s​e​n​t​m​o​d​e​l​i​s​r​e​l​i​a​b​l​e​i​nPbscattering,sincethePREX2dataisavailable.Wefindthatthepresentmodelisreliablein20   ~<\stackrel{{\scriptstyle<}}{{\widetilde{}}}E_in   ~<\stackrel{{\scriptstyle<}}{{\widetilde{}}}180M​e​V,a​s​s​h​o​w​n​i​n​S​e​c.LABEL:$^208$Pb.A​f​t​e​r​t​h​e​t​e​s​t​i​n​g,w​e​d​e​t​e​r​m​i​n​e~{}MeV,asshowninSec.~{}\ref{$^{208}$Pb}.Afterthetesting,wedeterminer_m(exp),r_n(exp),r_skin(exp)f​o​rfor^208P​b,Pb,^58N​i,Ni,^40,48C​a,Ca,^12C​f​r​o​m​t​h​eCfromtheσ_R(exp)i​nin20   ~<\stackrel{{\scriptstyle<}}{{\widetilde{}}}E_in   ~<\stackrel{{\scriptstyle<}}{{\widetilde{}}}180M​e​V,a​s​s​h​o​w​n​i​n​S​e​c.LABEL:$^58$Ni,$^48,40$Ca,$^12$C.F​o​r​e​a​c​h​n​u​c​l​e​u​s,t​h​e​D​1​S−G​H​F​B+A​M​P​p​r​o​t​o​n​a​n​d​n​e​u​t​r​o​n​d​e​n​s​i​t​i​e​s​a​r​e​s​c​a​l​e​d​s​o​a​s​t​o​r​e​p​r​o​d​u​c​e~{}MeV,asshowninSec.~{}\ref{$^{58}$Ni,$^{48,40}$Ca,$^{12}$C}.Foreachnucleus,theD1S-GHFB+AMPprotonandneutrondensitiesarescaledsoastoreproduceσ_R(exp)u​n​d​e​r​t​h​a​t​c​o​n​d​i​t​i​o​n​t​h​a​t​t​h​eunderthatconditionthatther_p(scaling)o​f​t​h​e​s​c​a​l​e​d​p​r​o​t​o​n​d​e​n​s​i​t​y​a​g​r​e​e​s​w​i​t​hofthescaledprotondensityagreeswithr_p(exp)o​f​e​l​e​c​t​r​o​n​s​c​a​t​t​e​r​i​n​g.W​e​e​x​p​l​a​i​n​o​u​r​m​o​d​e​l​i​n​S​e​c.II​a​n​d​o​u​r​r​e​s​u​l​t​s​i​n​S​e​c.LABEL:Results.S​e​c​t​i​o​n​LABEL:Summary​i​s​d​e​v​o​t​e​d​t​o​a​s​u​m​m​a​r​y.ofelectronscattering.\par WeexplainourmodelinSec.~{}\ref{Sec-Framework}andourresultsinSec.~{}\ref{Results}.Section\ref{Summary}isdevotedtoasummary.\par

    II Model

    Our model is the Kyushu gg-matrix folding model Toyokawa:2017pdd; PRC.101.014620 with the proton and neutron densities scaled from the D1S-GHFB+AMP densities.

    II.1 The Kyushu gg-matrix folding model

    Kohno calculated the gg matrix for the symmetric nuclear matter, using the Brueckner-Hartree-Fock method with chiral N3LO 2NFs and NNLO 3NFs PRC.88.064005; *PRC.96.059903, where N3LO 3NF is abbreviation of next-to-next-to-next-to-leading-order three-body force and NNLO 2NFs is of next-to-next-to-leading-order two-body force. He set cD=−2.5c_{D}=-2.5 and cE=0.25c_{E}=0.25 so that the energy per nucleon can become minimum at ρ=ρ0\rho=\rho_{0}; see Fig. 2 for the definition of cDc_{D} and cEc_{E}. Toyokawa et al. localized the non-local chiral gg matrix into three-range Gaussian forms Toyokawa:2017pdd, using the localization method proposed by the Melbourne group ANP.25.275; PRC.44.73; PRC.49.1309. The resulting local gg matrix is referred to as “Kyushu gg-matrix”. The Kyushu gg-matrix is constructed from chiral interaction with the cutoff 550 MeV. fig-chiral.epsfig-chiral

    Refer to caption
    Figure 2: 3NNFs in NNLO (next-to-next-to-leading-order). Diagram (a) corresponds to the Fujita-Miyazawa 2π\pi-exchange 3NNF (next-to-next-to-next-to-leading-order) PTP.17.360; *PTP.17.366, and diagrams (b) and (c) correspond to 1π\pi-exchange and contact 3NNFs. The solid and dashed lines denote nucleon and pion propagations, respectively, and filled circles and squares stand for vertices. The strength of the filled-square vertex is often called cDc_{D} in diagram (b) and cEc_{E} in diagram (c).

    The Kyushu gg-matrix folding model is successful in reproducing σR\sigma_{\rm R}, differential cross sections d​σ/d​Ωd\sigma/d\Omega, vector analyzing powers AyA_{y} for 4He scattering in Ein=30​–​200E_{\rm in}=30\text{--}200 MeV per nucleon Toyokawa:2017pdd. The success is true for proton scattering at Ein=65E_{\rm in}=65 MeV Toyokawa:2014yma.

    In Ref. Tagami:2019svt, we tested the Kyushu gg-matrix folding model Toyokawa:2017pdd for 12C scattering on 9Be, 12C, 27Al targets in 3030~{}\,  E_in   ~<\stackrel{{\scriptstyle<}}{{\widetilde{}}}400 M​e​V.W​e​f​o​u​n​d​t​h​a​t​t​h​e​K​y​u​s​h​u~{}MeV.WefoundthattheKyushug−m​a​t​r​i​x​f​o​l​d​i​n​g​m​o​d​e​l​i​s​r​e​l​i​a​b​l​e​f​o​r-matrixfoldingmodelisreliableforσ_Ri​nin30   ~<\stackrel{{\scriptstyle<}}{{\widetilde{}}}E_in   ~<\stackrel{{\scriptstyle<}}{{\widetilde{}}}100 M​e​V​a​n​d~{}MeVand250   ~<\stackrel{{\scriptstyle<}}{{\widetilde{}}}E