Slope parameters determined from CREX and PREX2 Shingo Tagami Department of Physics Kyushu University Fukuoka 819-0395 Japan

2025-05-02 0 0 465.51KB 8 页 10玖币

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Slope parameters determined from CREX and PREX2

Shingo Tagami

Department of Physics, Kyushu University, Fukuoka 819-0395, Japan

Tomotsugu Wakasa

Department of Physics, Kyushu University, Fukuoka 819-0395, Japan

Masanobu Yahiro∗

Department of Physics, Kyushu University, Fukuoka 819-0395, Japan

Background: Very lately, the CREX group presents a skin value ∆R48

skin(CREX) = 0.121±0.026 (exp)±0.024 (model) =

0.071 ∼0.171 fm. Meanwhile, the PREX group reported a skin value ∆R208

skin(PREX2) = 0.283 ±0.071 = 0.212 ∼

0.354 fm. In our previous paper, we determined both the L–∆R48

skin relation and the L–∆R208

skin one, using 206 EoSs, where L

is a slope parameter.

Purpose: We determine Lfrom ∆R48

skin(CREX) and ∆R208

skin(PREX2), using 207 EoSs.

Results: The ∆R48

skin(CREX) yields L(CREX) = 0 ∼51 MeV and the ∆R208

skin(PREX2) does L(PREX2) = 76 ∼

165 MeV.

Conclusion: There is no overlap between L(CREX) and L(PREX2). This is a big problem to be solved.

I. INTRODUCTION AND CONCLUSION

Background on experiments:

Horowitz, Pollock and Souder proposed a direct measure-

ment for neutron skin thickness ∆Rskin =rn−rp[1], where

rpand rnare proton and neutron radii, respectively. This di-

rect measurement consists of parity-violating and elastic elec-

tron scattering. In fact, the PREX collaboration has reported

a new value,

∆R208

skin(PREX2) = 0.283 ±0.071 = 0.212 ∼0.354 fm,

(1)

combining the original Lead Radius EXperiment (PREX) re-

sult [2, 3] with the updated PREX2 result [4]. For 48Ca, the

CREX group presents [5]

∆R48

skin(CREX) = 0.121 ±0.026 (exp) ±0.024 (model)

= 0.071 ∼0.171 fm.(2)

Note that the direct values are obtained from a single momen-

tum transfer q. As an ab initio method for 48Ca, we should

consider the coupled-cluster (CC) method [6, 7] with chiral in-

teraction. The CC result ∆R48

skin(CC) = 0.12 ∼0.15 fm [7]

is consistent with ∆R48

skin(CREX).

As an indirect measurement on ∆Rskin, the high-resolution

E1polarizability experiment (E1pE) was made for 208Pb [8]

and 48Ca [9] in RCNP. The results are

∆R208

skin(E1pE) = 0.156+0.025

−0.021 = 0.135 ∼0.181 fm,(3)

∆R48

skin(E1pE) = 0.14 ∼0.20 fm.(4)

The ∆R48

skin(CREX) is consistent with ∆R48

skin(E1pE), but

∆R208

skin(PREX2) is not with ∆R208

skin(E1pE).

∗orion093g@gmail.com

Reaction cross section σRis a standard observable to deter-

mine the matter radius rmand the skin value ∆Rskin. High-

accuracy data σR(exp) are available for 42−51Ca + 12C scat-

tering at 280 MeV per nucleon [10]. The chiral (Kyushu) g-

matrix folding model [11] yields ∆R48

skin(exp) = 0.105 ±

0.06 fm from the data [12]. The result is consistent with

∆R48

skin(CREX). High-accuracy data σR(exp) are available

also for p + 208Pb scattering in 21 ≤Elab ≤180 MeV [13–

15]. The chiral (Kyushu) g-matrix folding model yields

∆R208

skin(exp) = 0.278 ±0.035 fm [16]. The value is con-

sistent with ∆R208

skin(PREX2).

Matter:

Among the basic physical quantities that determine the

equation of state (EoS) of nuclear systems, the symmetry en-

ergy (Ssym) 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–

23]. Many predictions on the symmetry energySsym(ρ)have

been made so far by taking several experimental and obser-

vational constraints on Ssym(ρ)and their combinations. The

∆R208

skin(E1pE) and the ∆R48

skin(CREX) are the most impor-

tant experimental constraint on the slope parameter L, since a

strong correlation between r208

skin and Lis well known [24, 25].

In this paper, we will show a strong correlation between r48

skin

and Land between r208

skin and Lfrom 207 EoSs.

As an essential constraint on the EoS from astrophysics,

one may take M= 1.97 ±0.04Msun [26], where Mis 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 Ssym is expanded into

Ssym(ρ) = J+L(ρ−ρ0)

3ρ0

+Ksym(ρ−ρ0)2

18ρ2

+· · · .(5)

in terms of the nuclear density ρaround the saturation density

ρ0. For the Ssym(ρ), at the present stage, a major aim is to

determine Lat ρ=ρ0. The symmetry energy Ssym(ρ)cannot

be measured by experiment directly. In place of Ssym(ρ), the

arXiv:2210.03313v1 [nucl-th] 7 Oct 2022

neutron-skin thickness rskin is measured to determine L. This

subject is currently under experimental investigation for 208Pb

and 48Ca nuclei at Jefferson Lab [27–29].

Figure 1 shows the binding energy per nucleon E/A as a

function of ρand δ= (N−Z)/A. The E/A is expanded into

E(ρ, δ)

A=S0(ρ) + Ssym(ρ)δ2+· · · (6)

in terms of δ; note that J=S0(ρ0). This equation shows the

relation between E/A(ρ, δ)and Ssym(ρ). Two curves corre-

spond to the symmetric-nuclear matter with δ= 0 and pure-

neutron matter with δ= 1, respectively. This ﬁgure shows

that Lis a slope for pure-neutron matter at ρ=ρ0, because

dS0(ρ)/dρ is zero at ρ=ρ0.

FIG. 1. Illustration on the physics meaning of the slope parameter L

and the incompressibility K.

We accumulate the 205 EoSs from Refs. [20, 24, 30–55] in

which ∆R208

skin and/or Lis presented, since a strong correlation

between ∆R208

skin and a slope parameter Lis 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 Ta-

ble 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 re-

liable EoS, since they calculated properties of dense symet-

tric -nuclear and pure-nuclear matter and the structure of neu-

tron 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 calcu-

late. 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 modiﬁca-

tion for NRAPR-B [46]. The EoS is called NRAPR-T in this

paper.

The ∆R208

skin(PREX2) and ∆R48

skin(CREX) is most reli-

able, and provides Lof nuclear matter. In fact, using 207

EoSs of Table I, we found the L-∆R48

skin relation [56] as

∆R48

skin = 0.0009L+ 0.125 >0.125 fm (7)

with a high correlation coefﬁcient R= 0.98, because of

L > 0. Equation (7) indicates that the lower limit of ∆R48

skin

is 0.125 fm. The same derivation is possible the L-∆R208

skin

relation:

L= 620.39 ∆R208

skin −57.963 (8)

has R= 0.99. The two relations are visualized by Fig. 2.

FIG. 2. Skin value ∆Ras a function of Lfor 208Pb and 48Ca. Two

straight line show Eq. (7) and Eq. (8), respectively. Dots denote 207

EoSs for 208Pb and 48Ca.

Results: Figure 3 shows the relation between ∆R208

skin and

∆R48

skin. The 207 EoSs do not satisfy both ∆R208

skin(PREX2)

and ∆R48

skin(CREX) in the one-σlevel. If one considers

the two-σlevel of ∆R208

skin(PREX2) and ∆R48

skin(CREX),

one may ﬁnd that some EoSs satisfy ∆R208

skin(PREX2) and

∆R48

skin(CREX)

FIG. 3. Relation between ∆R208

skin and ∆R48

skin.

The L–∆R48

skin realtion of Eq. (7) yields L(CREX) = 0 ∼

51 MeV and the L–∆R208

skinofEq. (8) does L(PREX2) =

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摘要：

SlopeparametersdeterminedfromCREXandPREX2ShingoTagamiDepartmentofPhysics,KyushuUniversity,Fukuoka819-0395,JapanTomotsuguWakasaDepartmentofPhysics,KyushuUniversity,Fukuoka819-0395,JapanMasanobuYahiroDepartmentofPhysics,KyushuUniversity,Fukuoka819-0395,JapanBackground:Verylately,theCREXgrouppresentsa...

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Slope parameters determined from CREX and PREX2 Shingo Tagami Department of Physics Kyushu University Fukuoka 819-0395 Japan

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