The mean-square radius of the neutron distribution in the
relativistic and non-relativistic mean-field models

Haruki<EMAIL_ADDRESS>and Toshio
<EMAIL_ADDRESS>
1 Department of Physics, Graduate School of Science, Chiba University, Chiba
263-8522, Japan

2 Research Center for Electron Photon Science, Tohoku University, Sendai
982-0826, Japan

It is investigated why the root-mean- square radius of the point neutron
distribution is smaller by about 0.1 fm in non-relativistic mean-field models
than in relativistic ones. The difference is shown to stem from the different
values of the product of the effective mass and the strength of the one-body
potential in the two frameworks. The values of those quantities are
constrained by the Hugenholtz-Van Hove theorem. The neutron skin is not a
simple function of the symmetry potential, but depends on the nucleon
effective mass.

##  1 Introduction

Recently much has been written on the neutron distribution in nuclei[1, 2, 3,
4, 5]. It is one of the most fundamental problems in nuclear physics together
with the proton distribution[1, 6]. The neutron distribution, however, has not
been well determined experimentally so far. This fact is because the neutron
density has been studied through hadron probes, where the ambiguity as to the
interaction and the reaction mechanism is not avoidable yet[3].

In contrast to the neutron distribution, the proton distribution is widely
investigated throughout the periodic table of the stable nuclei
theoretically[6] and experimentally[7]. The relationship between the point
proton and charge density distributions is defined unambiguously[6, 8]. The
latter is deduced from electron scattering cross sections rather model-
independently[7], compared with the strong interaction, since the
electromagnetic interaction is well understood, and is so weak that the
density distribution of the nuclear ground state is not disturbed[6, 9].

It has been believed for a long time that electron scattering is useless in
the study of the neutron distribution in nuclei[6, 10]. Recently, the present
authors have proposed a new way to deduce the neutron distribution from
electron scattering data[8]. They have derived the exact expression of the
$n$th-order moment of the nuclear charge distribution, and shown that the
mean-square radius(msr) of the charge distribution($R^{2}_{c}$) is dominated
by the msr of the point proton distribution($R^{2}_{p}$) and is independent of
the neutron’s msr($R^{2}_{n}$), but that the $n$th-order ($n\geq 4$) moment of
the charge density depends on the ($n\\!-\\!2$)th-order moment of the neutron
distribution[8]. For example, the fourth-order moment of the charge
density($Q^{4}_{c}$) depends on $R^{2}_{n}$. Their relationship is uniquely
defined, and the value of $Q^{4}_{c}$ is well determined in electron
scattering experiment[7, 11]. The value of $R^{2}_{n}$, however, is not
separated from $Q^{4}_{c}$ experimentally. In order to deduce the value of
$R^{2}_{n}$ from the experimental value of $Q^{4}_{c}$, it is necessary to
rely on a model-dependent analysis. An advantage to use $Q^{4}_{c}$ for
deducing the value of $R^{2}_{n}$ is that we need not take care of assumptions
on the interaction and reaction mechanism in electron scattering, but are able
to focus discussions of the model-dependence on nuclear structure.

At present, the nuclear structure is not investigated without invoking
phenomenological models. Moreover, most of the models are constructed for
different purposes independently. Hence, it is not appropriate for the
separation of $R^{2}_{n}$ from $Q^{4}_{c}$ to choose one model among many
existing models. The present authors[4] have proposed the least squares
analysis(LSA) for the separation by employing as many previous models as
possible together. Through the LSA, they explore the constraints which are
inherent in the framework of the nuclear models. The procedure of the
separation is as follows. First $R^{2}_{n}$ and $Q^{4}_{c}$ are calculated
using several models in the same framework, and then the least-square
line(LSL) for those values is obtained in the $R^{2}_{n}-Q^{4}_{c}$ plane.
Next, the value of $R^{2}_{n}$ in the framework is determined by the cross
point of the LSL and the line of $Q^{4}_{c}$ corresponding to its experimental
value. In order to confirm the obtained result, the LSA of $R^{2}_{n}$ against
the other moments also has been performed. The estimated values of $R^{2}_{n}$
are not model-independent, but are derived on the basis of the data from the
well-known electromagnetic probe, with utilizing the knowledge on the
phenomenological models accumulated for a long time in nuclear physics. A
similar method has been proposed for analyzing the parity violating electron
scattering[2], and employed actually in the analysis of the recent JLab
experiment[5, 12].

In the Ref.[4], the values of $R^{2}_{n}$ in 40Ca, 48Ca and 208Pb have been
estimated, for which the experimental values of $Q^{4}_{c}$ are available at
present. They have arbitrary chosen 11 relativistic and 9 non-relativistic
models among more than 100 versions accumulated for the last 50 years[13, 14,
15, 16]. Those models well reproduce fundamental nuclear properties within the
mean-field framework, assuming some nuclei to be a doubly closed shell
nucleus. The LSL is obtained with a small standard deviation and the values of
$R^{2}_{n}$ are determined within the 1$\%$ error including experimental
one[4]. In this analysis, it has been shown that the relativistic and non-
relativistic frameworks yield different values of $R^{2}_{n}$ from each other
in 48Ca and 208Pb. The value of $R_{n}$ in the non-relativistic models is
smaller by about 0.1 fm than that in the relativistic models in both nuclei.
Since in those mean-field models, the values of $R_{p}$ are fixed so as to
reproduce the experimental values, the neutron skin defined by $\delta
R=R_{n}-R_{p}$ differs by about 0.1 fm in the two frameworks. The difference
is not between each models, but between the two frameworks, so that the result
is apparently understood to reflect an essential difference between the
structures of the two mean-field approximations.

It should be noted that the 0.1 fm is not small for the neutron skin itself.
As seen later, for example, in 208Pb, $\delta R$ is 0.275 and 0.162 fm in the
relativistic and non-relativistic models, respectively. Understanding the 0.1
fm difference may be important for the study of nuclear fission and fusion
phenomena which are sensitive to the structure of nuclear surface[1, 17].
Recent detailed calculations[18] may not neglect order of 0.1 fm difference in
describing asymmetric nuclei. The 0.1 fm difference has been pointed out to be
crucial for the neutron star physics also[3, 18].

The purpose of the present paper is to investigate why the values of
$R^{2}_{n}$ in the non-relativistic mean-field models is smaller than in the
relativistic ones. The difference will be shown to stem mainly from the
difference between the products of the effective mass and the strength of the
one-body potential in the two frameworks. These two quantities are constrained
in each framework by Hugenholtz-Van Hove(HVH) theorem[19, 20, 21]. The theorem
has been proved in the mean-field approximation in the non-relativistic
framework for the symmetric nuclear matter. As will be shown in the present
paper, the theorem also holds in the asymmetric nuclear matter in both
relativistic and non-relativistic models, and is numerically maintained in the
mean-field approximation for finite nuclei also.

In §2, the root msr $R$ in the one-body potential will be discussed in order
to derive an analytical expression of $R$ in terms of the strength of the
potential and the nucleon effective mass, using the Woods-Saxon and harmonic
potentials. In §3, the equations of motion in the relativistic models will be
shown to have the same structure as the Schrödinger equation in the non-
relativistic models. In §4, the HVH theorem will be extended to asymmetric
nuclear matter. In §5, the complexity of the mean-field models due to a large
variety of interaction parameters will be simplified by using the Woods-Saxon
type function, aiming to make clear the difference between the relativistic
and non-relativistic model. In §6, the difference between $\delta R$’s in the
two frameworks will be investigated in detail, according to the HVH theorem.
The final section will be devoted to a brief summary of the present paper.

##  2 The nuclear radius in the one-body potential

Many phenomenological models have been proposed with various interaction
parameters[13, 16]. Whether the nuclear radius($R$) is $R_{n}$ or $R_{p}$, it
may be a complicated function of their parameters, and the function would be
different from one model to another. The radius, however, is one of the most
fundamental quantities which determine the structure of the nucleus, and,
hence, $R_{c}$ is used as an input to fix the free parameters of the models.
This fact implies that the relationship of $R$ to other key quantities of
nuclei like those in the one-body potential must be almost the same in the
mean-field models, although those key quantities may also depend on the
parameters in complicated ways.

Such relationships of $R$ to other key quantities should hold even in
simplified one-body potential models, if they describe well the gross
properties of nuclei[1]. As a simple one, Woods-Saxon(WS) potential is most
widely used in the literature[1]. It may become a guide for the present
purpose also, if we have an analytical formula for relationship between $R$
and the parameters of the one-body Hamiltonian with WS potential,

$H=\frac{\boldsymbol{p}^{2}}{2M}+V_{\rm ws}(r),\quad V_{\rm
ws}(r)=\frac{V_{\rm ws}}{1+e^{(r-R_{\rm ws})/a_{\rm ws}}}.$ (1)

Aiming to have an analytical expression of the relationship, we require the
help of the harmonic oscillator(HO) potential,

$V_{\rm H}(r)=\frac{k}{2}\left(r^{2}-R^{2}_{\rm H}\right),\qquad
k=M\omega^{2},$ (2)

$R_{\rm H}$ being a constant which determines the value of $V_{\rm H}(0)$.
Bohr and Mottelson have shown that the single-particle wave functions in WS
potential, which determine the value of $R$, are well reproduced by those of
HO potential[1]. In HO potential, the dimension analysis yields the expression
of the radius $R_{\rm ho}$ as

$R_{\rm ho}=\frac{C}{(Mk)^{1/4}}$ (3)

with $C$ denoting a constant. For the above exact formula, let us search for
the expression of $R_{\rm ho}$ in terms of the WS parameters by minimizing the
following quantity with respect to the variables, $k$ and $R_{\rm H}$,

$F_{n}(k,R_{\rm H})=\int^{R_{\rm H}}_{0}dr\,r^{n}\Bigl{(}V_{\rm H}(r)-V_{\rm
ws}(r)\Bigr{)}^{2},\quad$ (4)

where $n$ is considered to be 0 for the surface integral and 2 for the volume
integral. The value of $n$ is chosen by referring to Ref.[1], which shows a
similarity of the wave functions in the two potentials with $\omega=8.6$ MeV
and $V_{\rm H}(0)=-55$ MeV and with $V_{\rm ws}=-50$ MeV, $R_{\rm ws}=5.8$ fm
and $a_{\rm ws}=0.65$ fm. The numerical method yields the minimum values of
$F_{n}$ for the same WS parameters at $\omega=8.63$ MeV and $V_{\rm
H}(0)=-55.20$ MeV for $n=0$, and at $\omega=9.66$ MeV and $V_{\rm
H}(0)=-62.65$ MeV for $n=2$. Comparing these values with those in Ref.[1], it
may be reasonable to employ $n=0$, rather than $n=2$, for reproducing the wave
functions in WS potential.

Once we determine the value of $n$, it is possible to derive the analytical
formula for the approximate relationship between $R_{\rm ho}$ and the WS
parameters. Eq.(4) for $n=0$ is written as

$F_{0}(k,R_{\rm H})=\int^{R_{\rm H}}_{0}dr\Bigl{(}V_{\rm H}(r)-V_{\rm
ws}(r)\Bigr{)}^{2}=\frac{2}{15}k^{2}R_{\rm H}^{5}+F_{\rm H}+F_{\rm V},$

where we have defined

$\displaystyle F_{\rm H}$ $\displaystyle=-2\int_{0}^{R_{\rm H}}dr\,V_{\rm
H}(r)V_{\rm ws}(r)=-kV_{\rm ws}\int_{0}^{\infty}dr\,\frac{r^{2}-R^{2}_{\rm
H}}{1+e^{(r-R_{\rm ws})/a_{\rm ws}}}+\delta F_{\rm H},$ (5) $\displaystyle
F_{\rm V}$ $\displaystyle=\int_{0}^{R_{\rm H}}dr\,V^{2}_{\rm ws}(r)=V^{2}_{\rm
ws}\int_{0}^{\infty}\frac{dr}{\left(1+e^{(r-R_{\rm ws})/a_{\rm
ws}}\right)^{2}}+\delta F_{\rm V},$ (6)

with

$\delta F_{\rm H}=kV_{\rm ws}\int_{R_{\rm
H}}^{\infty}dr\,\frac{r^{2}-R^{2}_{\rm H}}{1+e^{(r-R_{\rm ws})/a_{\rm
ws}}},\qquad\delta F_{\rm V}=-V^{2}_{\rm ws}\int_{R_{\rm
H}}^{\infty}\frac{dr}{\left(1+e^{(r-R_{\rm ws})/a_{\rm ws}}\right)^{2}}.$ (7)

Using the identity for a general function $g(r)$,

$\int_{R_{\rm H}}^{\infty}dr\,\frac{g(r)}{\left(1+e^{(r-R_{\rm ws})/a_{\rm
ws}}\right)^{n}}=\varDelta^{n}a_{\rm ws}\int_{0}^{\infty}dx\,\frac{g(a_{\rm
ws}x+R_{\rm H})e^{-nx}}{(1+\varDelta e^{-x})^{n}},\quad\varDelta=e^{-(R_{\rm
H}-R_{\rm ws})/a_{\rm ws}},$

we can neglect $\delta F_{\rm H}$ and $\delta F_{\rm V}$ in Eq.(5) and (6),
assuming $\varDelta\ll 1$. Then, $F_{\rm V}$ in Eq.(6) is independent of $k$
and $R_{\rm H}$, and it is enough to minimize the only first term of the most
right-hand side of Eq.(5). The integral of the first term is performed with
the use of Sommerfeld expansion. In neglecting contributions of relative order
$e^{-R_{\rm ws}/a_{\rm ws}}$[1], it is written as

$\int_{0}^{\infty}dr\,\frac{g(r)}{1+e^{(r-R_{\rm ws})/a_{\rm
ws}}}=\int_{0}^{R_{\rm ws}}dr\,g(r)+\frac{\pi^{2}a^{2}_{\rm
ws}}{6}g^{\prime}(R_{\rm ws})+\frac{7\pi^{4}a^{4}_{\rm
ws}}{360}g^{\prime\prime\prime}(R_{\rm ws})+\cdots.$

Since $g^{\prime\prime\prime}(r)=0$ for Eq.(5), we have

$F_{0}(k,R_{\rm H})=\frac{2}{15}k^{2}R^{5}_{\rm H}+kV_{\rm ws}\left(R^{2}_{\rm
H}R_{\rm ws}-\frac{1+b_{\rm ws}}{3}R^{3}_{\rm ws}\right),\quad b_{\rm
ws}=\left(\frac{\pi a_{\rm ws}}{R_{\rm ws}}\right)^{2}.$

It should be noticed that there is no higher-order contribution from the
diffuseness parameter. The partial differentials of the above equation $F_{0}$
with respect to $k$ and $R_{\rm H}$ yield its minimum value at

$\displaystyle k$ $\displaystyle=-3\left(\frac{3}{5}\right)^{3/2}\frac{V_{\rm
ws}}{R^{2}_{\rm ws}(1+b_{\rm ws})^{3/2}},$ (8) $\displaystyle R^{2}_{\rm H}$
$\displaystyle=\frac{5}{3}(1+b_{\rm ws})R^{2}_{\rm ws},\qquad V_{\rm
H}(0)=\frac{3}{2}\sqrt{\frac{3}{5}}\frac{V_{\rm ws}}{\sqrt{1+b_{\rm ws}}}.$

When employing the values, $V_{\rm ws}=-50$ MeV, $R_{\rm ws}=5.8$ fm and
$a_{\rm ws}=0.65$ fm in Ref.[1], the above equations provide $\omega=8.49$ MeV
and $V_{\rm H}(0)=-54.80$ MeV, which reproduce almost the same values obtained
by the numerical method mentioned above.

Finally, inserting Eq.(8) into Eq.(3), $R_{\rm ho}$ is described approximately
in terms of the WS parameters as

$R_{\rm ho}\approx B\left(-\frac{R^{2}_{\rm ws}}{m^{\ast}V_{\rm
ws}}\right)^{1/4}\left(1+b_{\rm ws}\right)^{3/8},$ (9)

$B$ being a constant. In the above equation, the nucleon mass has been
replaced by the effective mass, $M^{\ast}=Mm^{\ast}$. Eq.(9) expresses well
our expectation such that the value of $R$ increases with $R_{\rm ws}$, and
decreases with increasing $(-V_{\rm ws})$ and $m^{\ast}$. Indeed, the first
parenthesis of the right-hand side may be derived in the square-well potential
with the depth $V_{\rm ws}$ and the width $R_{\rm ws}$. Eq(9) shows that the
diffuseness parameter contributes to the radius in the form of $(a_{\rm
ws}/R_{\rm ws})^{2}$.

If the neutron potential, $V_{n}$, and effective mass, $m_{n}^{\ast}$, are
different from $V_{p}$ and $m^{\ast}_{p}$ of the proton, the value of $R_{n}$
may be different from that of $R_{p}$. In the same way, if $V_{n}$ and
$m_{n}^{\ast}$ in the one model are different from those in another model,
their $R_{n}$’s are different from each other. When comparing the nuclear
radius, $R_{1}$ in the one framework with $R_{2}$ in another one, the
following expression is useful,

$\frac{R_{1}}{R_{2}}=\left(\frac{m_{2}^{\ast}V_{\rm ws,2}}{m^{\ast}_{1}V_{\rm
ws,1}}\right)^{1/4}\left(\frac{R_{\rm ws,1}}{R_{\rm
ws,2}}\right)^{1/2}\left(\frac{1+b_{\rm ws,1}}{1+b_{\rm ws,2}}\right)^{3/8}.$
(10)

##  3 Equations of motion of the mean-field models

Eq.(9) and (10) are simple enough to understand the relationship between
$R_{\rm ho}$ and the key quantities of the one-body potential. The effective
mass and the one-body potential are well-defined quantities in the mean-field
models. Expecting that such a simple relationship holds approximately in those
phenomenological models also, let us investigate how they appear in the
equations of motion in the relativistic and non-relativistic models.

In the relativistic nonlinear $\sigma-\omega-\rho$ model, the nuclear
Lagrangian is given, using the notations in the literature[14, 16, 22], by

$\displaystyle\mathcal{L}$
$\displaystyle=\overline{\psi}\left(i\gamma_{\mu}\partial^{\mu}-M-g_{\sigma}\sigma-
g_{\omega}\gamma_{\mu}\omega^{\mu}-g_{\rho}\gamma_{\mu}\boldsymbol{\tau}\\!\cdot\\!\boldsymbol{b}^{\mu}-e\gamma_{\mu}A^{\mu}\frac{1+\tau_{3}}{2}\right)\psi$
$\displaystyle\hphantom{=}+\frac{1}{2}(\partial_{\mu}\sigma)^{2}-\frac{m^{2}_{\sigma}}{2}\sigma^{2}-\frac{g_{3}}{3}\sigma^{3}-\frac{g_{4}}{4}\sigma^{4}-\frac{1}{4}\omega_{\mu\nu}\omega^{\mu\nu}+\frac{m^{2}_{\omega}}{2}\omega_{\mu}\omega^{\mu}+\frac{c_{4}}{4}(\omega_{\mu}\omega^{\mu})^{2}$
$\displaystyle\hphantom{=}-\frac{1}{4}\boldsymbol{b}_{\mu\nu}\\!\cdot\\!\boldsymbol{b}^{\mu\nu}+\frac{m^{2}_{\rho}}{2}\boldsymbol{b}_{\mu}\\!\cdot\\!\boldsymbol{b}^{\mu}+\lambda
g^{2}_{\rho}\boldsymbol{b}_{\mu}\\!\cdot\\!\boldsymbol{b}^{\mu}g^{2}_{\omega}\omega_{\nu}\omega^{\nu}-\frac{1}{4}A_{\mu\nu}A^{\mu\nu}.$
(11)

Then, Euler-Lagrange equation provides us with the equations of motion for the
static mean-field,

$\displaystyle\Bigl{(}-i\boldsymbol{\alpha}\cdot\boldsymbol{\nabla}+\gamma_{0}(M+V_{\sigma})+V_{0}\Bigr{)}\psi=(E+M)\psi,$
(12)
$\displaystyle\left(-\boldsymbol{\nabla}^{2}+m^{2}_{\sigma}\right)V_{\sigma}=-g^{2}_{\sigma}\left(\rho_{S}+\frac{g_{3}}{g^{3}_{\sigma}}V^{2}_{\sigma}+\frac{g_{4}}{g^{4}_{\sigma}}V^{3}_{\sigma}\right),$
(13)
$\displaystyle\left(-\boldsymbol{\nabla}^{2}+m^{2}_{\omega}\right)V_{\omega}=g^{2}_{\omega}\left(\rho-\frac{c_{4}}{g^{4}_{\omega}}V^{3}_{\omega}-2\lambda
V_{\omega}V^{2}_{\rho}\right),$ (14)
$\displaystyle\left(-\boldsymbol{\nabla}^{2}+m^{2}_{\rho}\right)V_{\rho}=g^{2}_{\rho}\bigl{(}\rho_{p}-\rho_{n}-2\lambda
V^{2}_{\omega}V_{\rho}\bigr{)},$ (15)
$\displaystyle-\boldsymbol{\nabla}^{2}V_{c}=e^{2}\rho_{p}.$ (16)

In the above equations from Eq.(12) to (16), we have defined $\psi$ as a
single-particle wave function, and used following notations,
$V_{\sigma}=g_{\sigma}\sigma,V_{\omega}=g_{\omega}\omega^{0},V_{\rho}=g_{\rho}b_{3}^{0}$
and $V_{c}$ for the Coulomb potential, $V_{c}=eA^{0}$. Moreover, $V_{0}$ is
given by

$V_{0}(\boldsymbol{r})=V_{\omega}(\boldsymbol{r})+V_{\rho}(\boldsymbol{r})\tau_{3}+V_{c}(\boldsymbol{r})\frac{1+\tau_{3}}{2}$
(17)

with $\tau_{3}=+1(-1)$ for protons(neutrons), and the nucleon densities are

$\rho_{S}(\boldsymbol{r})=\sum_{\alpha}\overline{\psi}_{\alpha}(\boldsymbol{r})\psi_{\alpha}(\boldsymbol{r}),\quad\rho_{\tau}(\boldsymbol{r})=\sum_{\alpha\in\tau}\psi_{\alpha}^{\dagger}(\boldsymbol{r})\psi_{\alpha}(\boldsymbol{r}),\quad\rho(\boldsymbol{r})=\rho_{n}(\boldsymbol{r})+\rho_{p}(\boldsymbol{r}),$

with $\tau=p$ for protons and $\tau=n$ for neutrons.

Eq.(12) represents the two coupled equations for the upper component,
$\psi_{u}(\boldsymbol{r})$, and the lower two component,
$\psi_{d}(\boldsymbol{r})$, of $\psi(\boldsymbol{r})$. One of them gives

$\psi_{d}(\boldsymbol{r})=-\,\frac{1}{2M^{\ast}_{\tau}(\boldsymbol{r})}i\boldsymbol{\sigma}\\!\cdot\\!\boldsymbol{\nabla}\psi_{u}(\boldsymbol{r})\,,$
(18)

writing the effective nucleon mass, $M^{\ast}_{\tau}(\boldsymbol{r})$, as

$M^{\ast}_{\tau}(\boldsymbol{r})=\frac{2M+E+V_{\sigma}(\boldsymbol{r})-V_{0}(\boldsymbol{r})}{2}.$
(19)

In inserting Eq.(18) into the other equation of Eq.(12), we obtain the
Schrödinger-like equation as

$\left(-\boldsymbol{\nabla}\frac{1}{2M^{\ast}_{\tau}(\boldsymbol{r})}\\!\cdot\\!\boldsymbol{\nabla}+V_{\tau}(\boldsymbol{r})+V_{c}(\boldsymbol{r})\frac{1+\tau_{3}}{2}-i\left(\boldsymbol{\nabla}\frac{1}{2M^{\ast}_{\tau}(\boldsymbol{r})}\right)\\!\cdot\\!(\boldsymbol{\nabla}\\!\times\\!\boldsymbol{\sigma})\right)\psi_{u}(\boldsymbol{r})=E\psi_{u}(\boldsymbol{r}).$
(20)

In the above equation, the nuclear potential, $V_{\tau}(\boldsymbol{r})$, is
defined by

$V_{\tau}(\boldsymbol{r})=V_{\sigma}(\boldsymbol{r})+V_{\omega}(\boldsymbol{r})+V_{\rho}(\boldsymbol{r})\tau_{3}.$
(21)

We note that the effective mass, $M^{\ast}_{\tau}(\boldsymbol{r})$, is written
approximately as

$M^{\ast}_{\tau}(\boldsymbol{r})\approx
M+\frac{1}{2}\left(V_{\sigma}(\boldsymbol{r})-V_{\omega}(\boldsymbol{r})-V_{\rho}(\boldsymbol{r})\tau_{3}-V_{c}(\boldsymbol{r})\frac{1+\tau_{3}}{2}\right),$
(22)

using the fact that $2M+E\approx 2M$. For 208Pb, the values of the potentials
around the center of the nuclear density are about $V_{\sigma}\approx-380$
MeV, $V_{\omega}\approx 306$ MeV and $V_{\rho}\approx-6$ MeV[22]. It should be
noted that the effective mass in the relativistic models is almost isoscalar,
and is dominated by $V_{\sigma}$ and $V_{\omega}$ in the same way as the spin-
orbit potential in the last term of the left-hand side in Eq.(20).

The root-msr’s of the point proton and neutron distributions calculated with
NL3[22] are listed in Table 1. They are defined as

$\displaystyle R^{2}_{\tau}$
$\displaystyle=\frac{1}{N_{\tau}}\sum_{\alpha\in\tau}\int^{\infty}_{0}dr\,r^{2}\Bigl{(}G^{2}_{\alpha}(r)+F^{2}_{\alpha}(r)\Bigr{)}\,,\qquad(R_{\tau})^{2}_{G}=\frac{1}{N_{\tau}}\sum_{\alpha\in\tau}\int^{\infty}_{0}dr\,r^{2}G^{2}_{\alpha}(r)\,,$
$\displaystyle(R_{\tau})^{2}_{N}$
$\displaystyle=\frac{1}{N_{\tau}}\sum_{\alpha\in\tau}\int^{\infty}_{0}dr\,r^{2}\frac{G^{2}_{\alpha}(r)}{n_{G}}\,,\qquad
n_{G}=\int^{\infty}_{0}dr\,G^{2}_{\alpha}(r)\,,$

where $G_{\alpha}(r)/r$ and $F_{\alpha}(r)/r$ denote the radial part of the
large and the small component of $\psi_{\alpha}(\boldsymbol{r})$,
respectively, with the normalization,
$\int^{\infty}_{0}dr\bigl{(}G^{2}_{\alpha}(r)+F^{2}_{\alpha}(r)\bigr{)}=1$.
Moreover, we have defined $N_{\tau}=N(Z)$ for $\tau=n(p)$, and $n_{G}$ for the
normalization of the upper component used in $(R_{\tau})^{2}_{N}$. Table 1
also shows the ratios of $(R_{\tau})_{G}$ and $(R_{\tau})_{N}$ to $R_{\tau}$
in the parentheses. As seen from $(R_{\tau})_{G}$ in Table 1, the contribution
of the lower component to $R_{\tau}$ is about $1\%$, and it is absorbed into
$(R_{\tau})_{N}$ which is calculated with the renormalized large component
$G_{\alpha}(r)/\sqrt{n_{G}}$. Similar results are obtained in other
relativistic models. According to these results, we will use the renormalized
large component, ignoring the small component, when comparing the relativistic
models with the non-relativistic ones below.

We note that in principle, the two-component framework equivalent to the four-
component one should be derived by the Foldy-Wouthuysen unitary
transformation[9]. In order to obtain the normalized two component wave
functions, Eq.(20) will be used only in the present paper for comparison with
non-relativistic models, for simplicity and transparency. In Ref.[4], the
calculations of the msr in the relativistic models have been performed within
the four-component framework.

| $R_{n}$ | $(R_{n})_{G}$ | $(R_{n})_{N}$ | $R_{p}$ | $(R_{p})_{G}$ | $(R_{p})_{N}$
---|---|---|---|---|---|---
48Ca | $3.6050$ | $3.5736(0.991)$ | $3.6082(1.001)$ | $3.3789$ | $3.3522(0.992)$ | $3.3846(1.002)$
208Pb | $5.7405$ | $5.6888(0.991)$ | $5.7522(1.002)$ | $5.4600$ | $5.4135(0.991)$ | $5.4656(1.001)$
Table 1:  The root msr of the point neutron($R_{n}$) and proton($R_{p}$)
distribution calculated with NL3 for 48Ca and 208Pb. The number is given in
units of fm, except for the one in the parenthesis which denotes the ratio to
the $R_{\tau}$. For details, see the text.

In the Skyrme Hartree-Fock approximation in the non-relativistic models, the
Schrödinger equation is written as [23, 24],

$\left(-\boldsymbol{\nabla}\frac{1}{2M^{\ast}_{\tau}(\boldsymbol{r})}\\!\cdot\\!\boldsymbol{\nabla}+V_{\tau}(\boldsymbol{r})+V_{c}(\boldsymbol{r})\frac{1+\tau_{3}}{2}-i\boldsymbol{W}_{\tau}(\boldsymbol{r})\\!\cdot(\boldsymbol{\nabla}\\!\times\\!\boldsymbol{\sigma})\right)\varphi(\boldsymbol{r})=E\varphi(\boldsymbol{r}),$
(23)

where using the same notations as in Ref.[24],
$M^{\ast}_{\tau}(\boldsymbol{r})$, $V_{\tau}(\boldsymbol{r})$ and
$\boldsymbol{W}_{\tau}(\boldsymbol{r})$ are given as,

$\displaystyle\frac{1}{M^{\ast}_{\tau}(\boldsymbol{r})}$
$\displaystyle=\frac{1}{M}+\frac{t_{1}(2+x_{1})+t_{2}(2+x_{2})}{4}\rho(\boldsymbol{r})+\frac{t_{2}(1+2x_{2})-t_{1}(1+2x_{1})}{4}\rho_{\tau}(\boldsymbol{r})\,,$
(24) $\displaystyle V_{\tau}(\boldsymbol{r})$
$\displaystyle=\frac{t_{0}}{2}\Bigl{(}(2+x_{0})\rho(\boldsymbol{r})-(1+2x_{0})\rho_{\tau}(\boldsymbol{r})\Bigr{)}+\frac{t_{3}}{24}(2+x_{3})(2+\alpha)\rho^{\alpha+1}(\boldsymbol{r})$
$\displaystyle\hphantom{=}-\frac{t_{3}}{24}(2x_{3}+1)\Bigl{[}2\rho^{\alpha}(\boldsymbol{r})\rho_{\tau}(\boldsymbol{r})+\alpha\rho^{\alpha-1}(\boldsymbol{r})\Bigl{(}\rho^{2}_{p}(\boldsymbol{r})+\rho^{2}_{n}(\boldsymbol{r})\Bigr{)}\Bigr{]}$
$\displaystyle\hphantom{=}+\frac{t_{1}(2+x_{1})+t_{2}(2+x_{2})}{8}K(\boldsymbol{r})+\frac{t_{2}(1+2x_{2})-t_{1}(1+2x_{1})}{8}K_{\tau}(\boldsymbol{r})$
$\displaystyle\hphantom{=}+\frac{t_{2}(2+x_{2})-3t_{1}(2+x_{1})}{16}\boldsymbol{\nabla}^{2}\rho(\boldsymbol{r})+\frac{3t_{1}(1+2x_{1})+t_{2}(1+2x_{2})}{16}\boldsymbol{\nabla}^{2}\rho_{\tau}(\boldsymbol{r})$
$\displaystyle\hphantom{=}-\frac{W_{0}}{2}\boldsymbol{\nabla}\\!\cdot\\!\bigl{(}\boldsymbol{J}(\boldsymbol{r})+\boldsymbol{J}_{\tau}(\boldsymbol{r})\bigr{)}\,,$
(25) $\displaystyle\boldsymbol{W}_{\tau}(\boldsymbol{r})$
$\displaystyle=\frac{W_{0}}{2}\boldsymbol{\nabla}\bigl{(}\rho(\boldsymbol{r})+\rho_{\tau}(\boldsymbol{r})\bigr{)}+\frac{t_{1}-t_{2}}{8}\boldsymbol{J}_{\tau}(\boldsymbol{r})-\frac{t_{1}x_{1}+t_{2}x_{2}}{8}\boldsymbol{J}(\boldsymbol{r})$
$\displaystyle\approx\frac{W_{0}}{2}\boldsymbol{\nabla}\bigl{(}\rho(\boldsymbol{r})+\rho_{\tau}(\boldsymbol{r})\bigr{)}.$
(26)

In Eq.(25), $K(\boldsymbol{r})=K_{n}(\boldsymbol{r})+K_{p}(\boldsymbol{r})$
has been defined with
$K_{\tau}(\boldsymbol{r})=\sum_{\alpha\in\tau}|\boldsymbol{\nabla}\varphi_{\alpha}(\boldsymbol{r})|^{2}$,
and
$\boldsymbol{J}(\boldsymbol{r})=\boldsymbol{J}_{n}(\boldsymbol{r})+\boldsymbol{J}_{p}(\boldsymbol{r})$,
where $\boldsymbol{J}_{\tau}(\boldsymbol{r})$ denotes the spin density given
in Ref.[24].

It is seen that Eq.(23) in the non-relativistic models has the same structure
as Eq.(20) in the relativistic models. They are composed of the four parts,
$M^{\ast}_{\tau}(\boldsymbol{r})$, $V_{\tau}(\boldsymbol{r})$,
$V_{c}(\boldsymbol{r})$ and the spin-orbit potential. If the strengths and the
coordinate-dependences of these parts were the same in the two frameworks, one
could not distinguish one framework from another, in spite of their
complicated parameter sets. Among the four parts, the last two ones are
expected to play a minor role in the present purpose to explore the difference
between $\delta R$’s in the two frameworks. The Coulomb potential is almost
the same, and the strengths of the spin-orbit potentials reproduce
experimental values of the single-particle energy levels in both
frameworks[14, 25]. In contrast to these, the first two parts are strongly
model-dependent. The values of the effective masses are spread out over a wide
range [13]. Similarly, there is no reason why the one-body potentials are
almost the same in all the mean-field models. Hence, the 0.1 fm difference
between $\delta R$’s may be related to $M^{\ast}_{\tau}(\boldsymbol{r})$ and
$V_{\tau}(\boldsymbol{r})$ depending on the different interaction parameters.

This observation is consistent with Eq.(9) and (10) which clearly indicate
that the difference problem is related to the effective mass and one-body
potential. It is also apparent that they are not independent of each other. On
the one hand, the product of the $M^{\ast}_{p}(\boldsymbol{r})$ and
$V_{p}(\boldsymbol{r})$ is constrained by hand so as to reproduce the
experimental value of $R_{c}$ in both relativistic and non-relativistic
models. On the other hand, there is not a similar constraint on the neutron
distribution, but both frameworks predict the values of $R_{n}$ which are
distributed within a narrow range around each average value[4]. If the
difference between $\delta R$’s is actually related to the effective mass and
the one-body potential, there should be another constraint on the variations
of these two quantities, which works differently in the relativistic and non-
relativistic models.

As one of such candidates, it may be natural to expect the symmetry energy[2].
The symmetry energy coefficient, $a_{4}$[26], is composed of the potential and
kinetic parts[1], which are given in the present relativistic and non-
relativistic mean-field models, respectively, as[13]

$\displaystyle a_{4,{\rm rel}}$ $\displaystyle=\frac{k_{\rm
F}^{2}}{6\sqrt{k^{2}_{\rm
F}+M^{2}_{\sigma}}}+\frac{\rho}{2}\frac{g^{2}_{\rho}}{m^{2}_{\rho}+2\lambda
g^{2}_{\rho}V^{2}_{\omega}},\,\qquad M_{\sigma}=M+V_{\sigma},$ (27)
$\displaystyle a_{4,{\rm non}}$
$\displaystyle=\left(\frac{3\pi^{2}}{2}\right)^{2/3}\left(\frac{\rho^{2/3}}{6M}+\frac{-3t_{1}x_{1}+t_{2}(5x_{2}+4)}{24}\rho^{5/3}\right)$
$\displaystyle\hphantom{=}-\frac{2x_{0}+1}{8}t_{0}\rho-\frac{2x_{3}+1}{48}t_{3}\rho^{\alpha+1},$
(28)

where $k_{\rm F}$ denotes the Fermi momentum, and $\rho$ the nucleon density
in the nuclear matter. Actually, they are related to the difference between
the neutron and proton potentials in Eq.(21) and (25), and the effective mass
in Eq.(19) and (24). The relationship of $a_{4}$ to $\delta R$, however, does
not seem to be described explicitly. In fact, there is more fundamental
restriction on the relationship between the potential and the effective mass.
It is known as the Hugenholtz-Van Hove(HVH) theorem[19, 20, 21], which holds
in any mean-field model for symmetric nuclear matter.

##  4 Hugenholtz-Van Hove theorem

According to the HVH theorem, the binding energy per nucleon is equal to the
Fermi energy in symmetric nuclear matter. Both relativistic and non-
relativistic models have been constructed so as to satisfy the theorem at the
values of the binding energy of the nucleon to be about $-16$MeV and of the
Fermi momentum to be about 1.3 fm-1. These values are used as inputs in order
to fix their free parameters in the nuclear interactions. The Fermi energy is
given by the sum of the kinetic and potential energy, so that the strength of
the potential and the value of the effective mass are constrained by these
inputs. Since the HVH theorem has been proved for the only symmetric nuclear
matter[19, 20, 21], however, we will extend the theorem to the relativistic
and non-relativistic asymmetric nuclear matter, and utilize the theorem as a
guide of the analysis of $\delta R$ in neutron-rich finite nuclei.

###  4.1  HVH theorem in the symmetric nuclear matter

Hugenholtz and Van Hove have shown that the following equation holds in the
non-relativistic mean-field model for symmetric nuclear matter[19, 20, 21],

$\frac{\mbox{\Large$\varepsilon$}}{\rho}=E_{\rm F},\quad{\rm
when}\quad\frac{d}{d\rho}\frac{\mbox{\Large$\varepsilon$}}{\rho}=0,$ (29)

where $\varepsilon$ stands for the total energy density of the system, and
$E_{\rm F}$ the Fermi energy. The value of $\mbox{\Large$\varepsilon$}/\rho$
represents the binding energy per nucleon, $E_{\rm B}$, to be written in the
non-relativistic models, as

$E_{\rm B}=E_{\rm F}=\frac{k^{2}_{\rm F}}{2m^{\ast}M}+V.$ (30)

In the relativistic models, $\mbox{\Large$\varepsilon$}/\rho$ and $E_{\rm F}$
contain the nucleon rest mass. Hence, $E_{\rm B}$ and $E_{\rm F}$ are given by

$E_{\rm B}=\frac{\mbox{\Large$\varepsilon$}}{\rho}-M=E_{\rm F}-M\,.$

In the present relativistic models, $E_{\rm F}$ in the symmetric nuclear
matter is written as[14]

$E_{\rm F}=\sqrt{k^{2}_{\rm F}+M^{2}_{\sigma}}+V_{\omega},\qquad
M_{\sigma}=M+V_{\sigma}.$

In setting

$E_{\rm B}=K+V,\quad K=\sqrt{k^{2}_{\rm F}+M_{\sigma}^{2}}-M_{\sigma},\quad
V=V_{\sigma}+V_{\omega},$

$K$ is described as

$K=\frac{k^{2}_{\rm F}}{2M^{\ast}}(1-\delta)\approx\frac{k^{2}_{\rm
F}}{2M^{\ast}},$ (31)

with $M^{\ast}=M+\bigl{(}V_{\sigma}-V_{\omega}\bigr{)}/2$ from Eq.(22). We
have defined

$\delta=1-\frac{2M^{\ast}}{k^{2}_{\rm F}}\left(\sqrt{k^{2}_{\rm
F}+M^{2}_{\sigma}}-M_{\sigma}\right),$ (32)

and used the fact that $\delta\ll 1$ in taking the values of Ref.[22] for the
right-hand side. Thus, in the relativistic models also, $E_{\rm B}$ is
expressed in the form as Eq.(30). Finally, in both relativistic and non-
relativistic models, the nuclear potential is inversely proportional to the
effective mass, according to the HVH theorem. In the case of Eq.(30), we have

$V=\frac{a}{m^{\ast}}+b,$ (33)

where $a\approx-35$ MeV and $b\approx-16$ MeV for $k_{\rm F}\approx 1.3$ fm-1
and $E_{\rm B}\approx-16$ MeV.

Indeed, it is verified that all the relativistic and non-relativistic models
employed in the present paper satisfy Eq(29) explicitly. In the non-
relativistic models, we have for the protons and neutrons, separately,

$\frac{\partial\mbox{\Large$\varepsilon$}}{\partial\rho_{\tau}}=\frac{k^{2}_{{\rm
F}\tau}}{2M^{\ast}_{\tau}}+V_{\tau}=E_{{\rm F}\tau},$ (34)

while in the relativistic $\sigma-\omega-\rho$ models,

$\frac{\partial\mbox{\Large$\varepsilon$}}{\partial\rho_{\tau}}=\sqrt{k^{2}_{{\rm
F}\tau}+M_{\sigma}^{2}}+V_{\omega}+\tau V_{\rho}=E_{{\rm F}\tau}.$ (35)

In the above equations, the total energy density in the non-relativistic
models is written as[23, 24]

$\varepsilon$
$\displaystyle=\frac{K_{p}}{2M^{\ast}_{p}}+\frac{K_{n}}{2M^{\ast}_{n}}+\frac{t_{0}}{4}\Bigl{(}(2+x_{0})\rho^{2}-(2x_{0}+1)(\rho^{2}_{p}+\rho^{2}_{n})\Bigr{)}$
$\displaystyle\hphantom{\,=\,\frac{K_{p}}{2M^{\ast}_{p}}+\frac{K_{n}}{2M^{\ast}_{n}}}+\frac{t_{3}}{24}\Bigl{(}(2+x_{3})\rho^{\alpha+2}-(2x_{3}+1)\rho^{\alpha}(\rho^{2}_{p}+\rho^{2}_{n})\Bigr{)},$

where we have defined $K_{\tau}=3k^{2}_{{\rm F}\tau}\rho_{\tau}/5$ with
$k_{{\rm F}\tau}=(3\pi^{2}\rho_{\tau})^{1/3}$. In the relativistic models, it
is given by

$\mbox{\Large$\varepsilon$}=\mbox{\Large$\varepsilon$}_{K}+V_{\omega}\rho+V_{\rho}(\rho_{p}-\rho_{n})+U_{\sigma}-U_{0},$

using the abbreviations,

$\displaystyle\mbox{\Large$\varepsilon$}_{K}$
$\displaystyle=\frac{2}{(2\pi)^{3}}\int^{k_{{\rm
F}p}}_{0}d^{3}k\sqrt{k^{2}+M^{2}_{\sigma}}+\frac{2}{(2\pi)^{3}}\int^{k_{{\rm
F}n}}_{0}d^{3}k\sqrt{k^{2}+M^{2}_{\sigma}},$ $\displaystyle U_{\sigma}$
$\displaystyle=\frac{m^{2}_{\sigma}}{2g^{2}_{\sigma}}V^{2}_{\sigma}+\frac{g_{3}}{3g^{3}_{\sigma}}V^{3}_{\sigma}+\frac{g_{4}}{4g^{4}_{\sigma}}V^{4}_{\sigma},\qquad
U_{0}=\frac{m^{2}_{\omega}}{2g^{2}_{\omega}}V^{2}_{\omega}+\frac{c_{4}}{4g^{4}_{\omega}}V^{4}_{\omega}+\frac{m^{2}_{\rho}}{2g^{2}_{\rho}}V^{2}_{\rho}+\lambda
V^{2}_{\omega}V^{2}_{\rho},$ (36)

which satisfy the equations of motion for the mesons,

$\displaystyle\frac{\partial U_{\sigma}}{\partial V_{\sigma}}$
$\displaystyle=\frac{m^{2}_{\sigma}}{g^{2}_{\sigma}}V_{\sigma}+\frac{g_{3}}{g^{3}_{\sigma}}V^{2}_{\sigma}+\frac{g_{4}}{g^{4}_{\sigma}}V^{3}_{\sigma}=-\rho_{S},\qquad\rho_{S}=\frac{\partial\mbox{\Large$\varepsilon$}_{K}}{\partial
M_{\sigma}}\,,$ $\displaystyle\frac{\partial U_{0}}{\partial V_{\omega}}$
$\displaystyle=\frac{m^{2}_{\omega}}{g^{2}_{\omega}}V_{\omega}+\frac{c_{4}}{g^{4}_{\omega}}V^{3}_{\omega}+2\lambda
V_{\omega}V^{2}_{\rho}=\rho\,,\qquad\frac{\partial U_{0}}{\partial
V_{\rho}}=\frac{m^{2}_{\rho}}{g^{2}_{\rho}}V_{\rho}+2\lambda
V^{2}_{\omega}V_{\rho}=\rho_{p}-\rho_{n}.$

In both relativistic and non-relativistic models,
$\partial\mbox{\Large$\varepsilon$}/\partial\rho_{\tau}=E_{{\rm F}\tau}$ holds
at any value of $\rho$ or $k_{\rm F}$, so that we have for
$\rho_{p}=\rho_{n}=\rho/2$,

$\frac{d\mbox{\Large$\varepsilon$}}{d\rho}=\frac{1}{2}\left(\frac{\partial\mbox{\Large$\varepsilon$}}{\partial\rho_{p}}+\frac{\partial\mbox{\Large$\varepsilon$}}{\partial\rho_{n}}\right)=\frac{E_{{\rm
F}n}+E_{{\rm F}n}}{2}=E_{\rm
F},\qquad\rho\frac{d}{d\rho}\frac{\mbox{\Large$\varepsilon$}}{\rho}=\frac{d\mbox{\Large$\varepsilon$}}{d\rho}-\frac{\mbox{\Large$\varepsilon$}}{\rho}=E_{\rm
F}-\frac{\mbox{\Large$\varepsilon$}}{\rho},$

as should be. The last equation yields Eq.(29) for
$(d/d\rho)(\mbox{\Large$\varepsilon$}/\rho)=0$. Thus, in the mean-field
models, Eq.(34) and (35), which hold for protons and neutrons separately, are
essential for the HVH theorem to be valid.

###  4.2  HVH theorem in the asymmetric nuclear matter

In order to discuss neutron-rich nuclei using the HVH theorem as a guide, we
have to extend the theorem so as to be applicable to the relativistic and non-
relativistic asymmetric nuclear matter.

One of the naive ways to the extension may be to minimize the total energy per
nucleon, assuming $\rho_{n}=\nu\rho$ and $\rho_{p}=(1-\nu)\rho$ for a fixed
value of $\nu$[1, 13, 24]. This choice is not, however, appropriate for the
present purpose, since $E_{{\rm F}n}$ and $E_{{\rm F}p}$ remain as in Eq.(34)
and (35) without the Coulomb energy. Moreover, if $\rho_{n}(r)=\nu\rho(r)$ and
$\rho_{p}(r)=(1-\nu)\rho(r)$ were realized in finite nuclei, one would have
$\delta R=0$ even for $N\neq Z$ nuclei. In order to extend the HVH theorem for
asymmetric nuclear matter, it is be better to avoid these defects. For this
purpose, without using the parameter $\nu$, we make a model for the neutron
and proton system in taking account effects of the “Coulomb potential”
explicitly as below.

We require for asymmetric nuclear matter

$\frac{\partial}{\partial\rho_{\tau}}\frac{\mbox{\Large$\varepsilon$}_{\rm
asym}}{\rho}=0,$ (37)

adding $v_{c}\rho_{p}$ as the “Coulomb term” to the total energy density[27],

$\mbox{\Large$\varepsilon$}_{\rm
asym}=\mbox{\Large$\varepsilon$}+v_{c}\rho_{p},$ (38)

where $v_{c}$ is a constant. The above equation is assumed in order to make a
model which may be used just as a guide for the following discussions on the
stable finite nucleus where Fermi energies of neutrons and protons are the
same and the Coulomb potential is necessary. We will see later that the final
results of this paper listed in Table 12 do not depend on the above form of
the “Coulomb term” and its strength $v_{c}$. Then, since Eq.(34) and (35)
still hold, we have the expression of the binding energy,

$E_{\rm B}=E_{{\rm F}\tau}=\frac{k^{2}_{{\rm
F}\tau}}{2M^{\ast}_{\tau}}+V_{\tau}+\frac{1+\tau_{3}}{2}v_{c},$ (39)

in the non-relativistic models, while in the relativistic models,

$\displaystyle E_{\rm B}=E_{{\rm F}\tau}-M$ $\displaystyle=\sqrt{k^{2}_{{\rm
F}\tau}+M^{2}_{\sigma}}-M_{\sigma}+V_{\tau}+\frac{1+\tau_{3}}{2}v_{c}$
$\displaystyle=(1-\delta_{\tau})\frac{k^{2}_{{\rm
F}\tau}}{2M^{\ast}_{\tau}}+V_{\tau}+\frac{1+\tau_{3}}{2}v_{c},$ (40)

where $\delta_{\tau}$ is given by Eq.(32) with $k_{{\rm F}\tau}$ and
$M^{\ast}_{\tau}$ instead of $k_{\rm F}$ and $M^{\ast}$, respectively, while
$V_{\tau}$ and $M^{\ast}_{\tau}$ are given by Eq.(21) and (22). The Coulomb
potential in Eq.(22) is neglected here, since its role is expected to be
small, compared to that from
$\left(V_{\sigma}(\boldsymbol{r})-V_{\omega}(\boldsymbol{r})-V_{\rho}(\boldsymbol{r})\right)$
in Eq.(22). The value of
$\left(V_{\sigma}(\boldsymbol{r})-V_{\omega}(\boldsymbol{r})-V_{\rho}(\boldsymbol{r})\right)$
at $r=0$ is about $-680$ MeV, as noted below Eq.(22). Eq.(39) and (40) are
accepted as the HVH theorem in asymmetric nuclear matter, and imply the
relationship between $V_{\tau}$ and $m^{\ast}_{\tau}$ as in Eq.(33),

$V_{\tau}=\frac{a_{\tau}}{m^{\ast}_{\tau}}+b_{\tau},\qquad
a_{\tau}=-(1-\delta_{\tau})\frac{k^{2}_{{\rm F}\tau}}{2M},\qquad
b_{\tau}=\left\\{\begin{array}[]{ll}E_{\rm B},&\tau=n\\\\[2.0pt] E_{\rm
B}-v_{c},&\tau=p,\end{array}\right.$ (41)

where $\delta_{\tau}=0$ in the non-relativistic models, while in the
relativistic models, $|\delta_{\tau}|\ll 1$ being almost constant. The values
of $k^{2}_{{\rm F}\tau}$, which provide the values of $\rho_{\tau}$ in the
relativistic and non-relativistic models, are determined by the two equations
in Eq.(37), once $v_{c}$ is given by hand.

Figure 1:  The relationship between the effective mass and the one-body
potential for neutrons($\tau=n$) and protons ($\tau=p$) in the mean-field
models for asymmetric nuclear matter. The black circles show the values
calculated by the 11 relativistic models, while the red ones show those by the
9 non-relativistic models. Each circles are accompanied by the number which
indicates the used model specified in the text. The least-square lines are
shown for the four groups. The two black lines are obtained from the black
circles for neutrons and protons, respectively, and the red lines from the red
circles.

For simplicity, for both relativistic and non-relativistic models, we take
$v_{c}$ from the strength at $r=0$ of the Coulomb potential for a uniformly
charged sphere of radius $r_{c}A^{1/3}$,

$v_{c}=\frac{3}{2}\frac{Z\alpha}{r_{c}A^{1/3}}.$ (42)

It yields $v_{c}=22.144$ MeV for 208Pb with $r_{c}=1.350$ fm. In employing
this value, we obtain Fig.1 for the $1/m^{\ast}_{\tau}-V_{\tau}$ relationship
corresponding to Eq.(41). The black circles indicate the values from the 11
relativistic models, while the red ones from the 9 non-relativistic models.
These models have been employed in Ref.[4]. Each circle is accompanied by the
number which shows the used model. The numbering is according to Ref.[4], as
1.L2[15], 2.NLB[15], 3.NLC[15], 4.NL1[28], 5.NL3[22], 6.NL-SH[29], 7.NL-Z[30],
8.NL-S[31], 9.NL3II[22], 10.TM1[32] and 11.FSU[16] for the relativistic
nuclear models, and 1.SKI[25], 2.SKII[25], 3.SKIII[33], 4.SKIV[33],
5.SkM∗[34], 6.SLy4[24], 7.T6[35], 8.SGII[36] and 9.Ska[37] for the non-
relativistic models. The above numbering of the models will be used throughout
the present paper.

In Fig.1, the least-square lines(LSL) of these circles are drawn by the black
ones for the relativistic models, while by the red ones for the non-
relativistic models. The LSL’s for neutrons and protons are well separated
from each other both in the relativistic models and in the non-relativistic
ones. The effective mass and the one-body potential are complicated functions
of the interaction parameters whose values are different from one another
between the mean-field models. Nevertheless, as seen in Fig.1, all the circles
are almost on their own LSL’s.

On the one hand, the circle at $(m^{\ast}_{\tau,i},V_{\tau,i})$ of the model
$i$ is given by

$V_{\tau,i}=a_{\tau,i}/m^{\ast}_{\tau,i}+b_{\tau,i},$ (43)

according to Eq.( 41) by the HVH theorem. On the other hand, LSL satisfies

$V^{\rm L}_{\tau,i}=a^{\rm L}_{\tau}/m^{\ast}_{\tau,i}+b^{\rm L}_{\tau},$ (44)

where $a^{\rm L}_{\tau}$ and $b^{\rm L}_{\tau}$ denote the slope and intercept
of LSL, respectively. In writing the average value of Eq.(43) as $\langle
V_{\tau,i}\rangle$ and that of Eq.(44) as $\langle V^{\rm L}_{\tau,i}\rangle$,
they are equal to each other by the definition of the LSL, $\langle
V_{\tau,i}\rangle=\langle V^{\rm L}_{\tau,i}\rangle$, yielding

$\langle a_{\tau,i}/m^{\ast}_{\tau,i}\rangle+\langle b_{\tau,i}\rangle=a^{\rm
L}_{\tau}\langle 1/m^{\ast}_{\tau,i}\rangle+b^{\rm L}_{\tau}.$ (45)

Hence, if the following approximation is valid,

$\langle a_{\tau,i}/m^{\ast}_{\tau,i}\rangle\approx\langle
a_{\tau,i}\rangle/\langle m^{\ast}_{\tau,i}\rangle,\quad\langle
1/m^{\ast}_{\tau,i}\rangle\approx 1/\langle m^{\ast}_{\tau,i}\rangle,$ (46)

then we have

$\langle m^{\ast}_{\tau}\rangle\langle V_{\tau}\rangle\approx\langle
a_{\tau}\rangle+\langle b_{\tau}\rangle\langle m^{\ast}_{\tau}\rangle\approx
a^{\rm L}_{\tau}+b^{\rm L}_{\tau}\langle m^{\ast}_{\tau}\rangle$ (47)

by writing $\langle V_{\tau,i}\rangle=\langle V_{\tau}\rangle$, $\langle
a_{\tau,i}\rangle=\langle a_{\tau}\rangle$, $\langle b_{\tau,i}\rangle=\langle
b_{\tau}\rangle$ and $\langle m^{\ast}_{\tau,i}\rangle=\langle
m^{\ast}_{\tau}\rangle$.

| $a^{\rm L}_{\tau}$ | $b^{\rm L}_{\tau}$ | $\langle a_{\tau}\rangle$ | $\langle b_{\tau}\rangle$ | $\langle m^{\ast}_{\tau}\rangle$ | $\langle V_{\tau}\rangle$ | $\langle\rho_{\tau}\rangle$
---|---|---|---|---|---|---|---
${\rm Rel}$ | $n$ | $-41.059$ |  2$-1.176$ | $-38.011$ |  2$-5.918$ | $0.6426$ | $-65.160$ | $0.0832$
| $p$ | $-36.489$ | $-20.421$ | $-31.530$ | $-28.063$ | $0.6488$ | $-76.742$ | $0.0611$
${\rm Non}$ | $n$ | $-36.743$ | $-10.069$ | $-39.927$ |  2$-5.840$ | $0.7524$ | $-61.219$ | $0.0903$
| $p$ | $-29.960$ | $-29.656$ | $-31.168$ | $-27.984$ | $0.7217$ | $-73.256$ | $0.0623$
Table 2: The values of the gradient($a^{\rm L}_{\tau}$) and the
intercept($b^{\rm L}_{\tau}$) of the least-square lines in units of MeV for
the relationship between $1/m^{\ast}_{\tau}$ and $V_{\tau}$ in Fig.1. The
average values of the coefficients in Eq.(43) are also listed as $\langle
a_{\tau}\rangle$ and $\langle b_{\tau}\rangle$, together with those of the
effective masses, the strengths of the one-body potentials (MeV) and the
nuclear matter densities ( fm-3 ) in the relativistic (Rel) and non-
relativistic (Non) models. The notations of $n$ and $p$ indicate that the
values in the corresponding rows are for neutrons and protons, respectively.
For details, see the text.

In Table 2 are listed the values of the slope $a^{\rm L}_{\tau}$ and intercept
$b^{\rm L}_{\tau}$ of the LSL. The average values of $a_{\tau,i}$ and
$b_{\tau,i}$ calculated by each model are tabulated as $\langle
a_{\tau}\rangle$ and $\langle b_{\tau}\rangle$. The average values of the
effective mass $\langle m^{\ast}_{\tau}\rangle$ and of the strengths of the
one-body potentials $\langle V_{\tau}\rangle$ are also listed, together with
the average values of $\rho_{\tau,i}$ as $\langle\rho_{\tau}\rangle$.

The difference between the values of $\langle a_{\tau}\rangle$ in the
relativistic and non-relativistic models is related to those of
$\langle\rho_{\tau}\rangle$ through the Fermi momentum. The values of $\langle
b_{\tau}\rangle$ are almost the same between the two frameworks, since
$\langle b_{\tau}\rangle$ satisfies the relationship as $\langle E_{\rm
B}\rangle=\langle b_{n}\rangle$ and $v_{c}=\langle b_{n}\rangle-\langle
b_{p}\rangle$, where $\langle E_{\rm B}\rangle$ denote the average value of
${E_{\rm B}}_{,i}$, according to Eq.(41).

The values of $a^{\rm L}_{\tau}$ and $b^{\rm L}_{\tau}$ depend on the
distributions of the points ($m^{\ast}_{\tau,i},V_{\tau,i}$) and have no
simple relationship to $\langle\rho_{\tau}\rangle$, $\langle E_{\rm B}\rangle$
and $v_{c}$. They, however, are implicitly constrained by the HVH theorem
through Eq.(47). Since the values of $v_{c}$ and $\langle E_{\rm B}\rangle$
are almost the same in the relativistic and non-relativistic models, Eq.(47)
provides the relationship between the effective mass, the strength of the one-
body potential and the nucleon density. This fact implies that the LSL
coefficients $a^{\rm L}_{\tau}$ and $b^{\rm L}_{\tau}$ are dominated by
$\langle\rho_{\tau}\rangle$ implicitly.

Eq.(47) is rewritten as $a^{\rm L}_{\tau}-\langle
a_{\tau}\rangle\approx\bigl{(}\langle b_{\tau}\rangle-b^{\rm
L}_{\tau}\bigr{)}\langle m^{\ast}_{\tau}\rangle$, which provides the
relationship as $a^{\rm L}_{\tau}\gtrsim\langle a_{\tau}\rangle$ for $b^{\rm
L}_{\tau}\lesssim\langle b_{\tau}\rangle$, and $a^{\rm
L}_{\tau}\lesssim\langle a_{\tau}\rangle$ for $b^{\rm L}_{\tau}\gtrsim\langle
b_{\tau}\rangle$. The non-relativistic models obey the first case, while the
relativistic ones the second case. The value of $|b^{\rm L}_{n}|$ is made much
smaller by the small $\langle m^{\ast}_{n}\rangle$ in the relativistic models,
compared to that in the non-relativistic one.

| $\langle m^{\ast}_{\tau}\rangle\langle V_{\tau}\rangle$ | $a^{\rm L}_{\tau}+b^{\rm L}_{\tau}\langle m^{\ast}_{\tau}\rangle$ | $\langle a_{\tau}\rangle+\langle b_{\tau}\rangle\langle m^{\ast}_{\tau}\rangle$
---|---|---|---
${\rm Rel}$ | $n$ | $-41.872$ | $-41.815$ | $-41.814$
| $p$ | $-49.790$ | $-49.738$ | $-49.737$
${\rm Non}$ | $n$ | $-46.061$ | $-44.319$ | $-44.321$
| $p$ | $-52.869$ | $-51.363$ | $-51.364$
Table 3: The product of the mean values of the effective mass $\langle
m^{\ast}_{\tau}\rangle$ and the one-body potential $\langle V_{\tau}\rangle$.
The numbers are given in units of MeV. For details, see the text.

In Table 3, the value of each term in Eq.(47) is listed. It shows that the
values of $\langle m^{\ast}_{\tau}\rangle\langle V_{\tau}\rangle$ are a little
larger than those of $\langle a_{\tau}\rangle+\langle b_{\tau}\rangle\langle
m^{\ast}_{\tau}\rangle$ and $a^{\rm L}_{\tau}+b^{\rm L}_{\tau}\langle
m^{\ast}_{\tau}\rangle$ in the non-relativistic models, because the
approximations in Eq.(46) are a little worse in the non-relativistic models
than in the relativistic models. This difference, however, is not essential
for the present discussions on $\delta R$.

In finite nuclei, Eq.(9) indicates that the radius depends on
$\bigl{(}-\langle m^{\ast}_{\tau}\rangle\langle
V_{\tau}\rangle\bigr{)}^{-1/4}$. Table 3 implies a possibility that $R_{n}$ is
larger in the relativistic models than in the non-relativistic ones, if the
same tendency maintains in finite nuclei. In the mean-field models for finite
nuclei, however, the effective mass and one-body potential may have
complicated coordinate dependences. In order to confirm the above implications
for finite nuclei, we need a way to extract from them the values of the
effective mass and the strength of the one-body potentials which are
appropriate for the use in Eq.(9). Moreover, it is desirable to explore
whether or not they are constrained by the HVH theorem as in asymmetric
nuclear matter. Although we do not have for finite nuclei an equation like
Eq.(37) to yield the HVH constraint in asymmetric nuclear matter, Eq.(47) may
be helpful for understanding roles of the HVH theorem in finite nuclei.
Bearing these facts in mind, we proceed discussions for finite nuclei from the
next section.

##  5 Simplification of the mean-field models

One of the ways to find a common structure of the models is to simplify them
without loosing their main characteristics. By defining the effective mass and
the one-body potential by such a way, we may find their relationships to $R$
and a restriction between them like the HVH lines which are hidden in the
complexity of the calculated results of the mean-field models for finite
nuclei.

In this section, we will analyze the structure of the relativistic and non-
relativistic models by simplifying their descriptions as much as possible. As
mentioned in §3, $R_{\tau}$ may be a functional of
$V_{\tau}(r),M^{\ast}_{\tau}(r),V_{c}(r)$ and the spin-orbit potential,
$V_{\ell s,\tau}(r)$, but among them, it is expected for $V_{c}(r)$ and
$V_{\ell s,\tau}(r)$ to play a minor role in the difference between $\delta
R$’s in the two frameworks, $\delta R[V_{\tau},M^{\ast}_{\tau},V_{c},V_{\ell
s,\tau}]\approx\delta R[V_{\tau},M^{\ast}_{\tau}]$. Using these facts as a
guide, let us express approximately all the Hamiltonians in the both
frameworks, using the same basis.

###  5.1 Nuclear potential and effective mass

The fundamental properties of nuclei are well described with the WS
potential[1], and its structure is clear for the present purpose to discuss
$\delta R$, as in Eq.(10). Hence, we approximate the mean-field potential,
$V_{\tau}(r)$, and the effective mass, $M^{\ast}_{\tau}(r)$, in both
frameworks by using the WS type function,

$f_{\tau}(r)=f_{\tau}(r,R_{\tau},a_{\tau})=\frac{1}{1+\exp\left((r-R_{\tau})/a_{\tau}\right)},$
(48)

that is,

$\displaystyle V_{\tau}(r)$ $\displaystyle\approx V_{{\rm ws},\tau}f(r,R_{{\rm
ws},\tau},a_{{\rm ws},\tau}),$ (49) $\displaystyle m^{\ast}_{\tau}(r)$
$\displaystyle\approx\left\\{\begin{array}[]{ll}1+(m^{\ast}_{{\rm
ws},\tau}-1)f(r,R^{\ast}_{{\rm ws},\tau},a^{\ast}_{{\rm
ws},\tau})-\dfrac{V_{c}(r)}{2M}\dfrac{1+\tau_{3}}{2},&({\rm Rel}),\\\\[12.0pt]
1+(m^{\ast}_{{\rm ws},\tau}-1)f(r,R^{\ast}_{{\rm ws},\tau},a^{\ast}_{{\rm
ws},\tau}),&({\rm Non}),\end{array}\right.$ (52)

where $m^{\ast}_{\tau}(r)=M^{\ast}_{\tau}(r)/M$ is defined, and Rel and Non
indicate the relativistic and non-relativistic models, respectively. The
three-parameters in the right-hand sides of the above equations are determined
by minimizing, for example, for $V_{\tau}(r)$, the following quantity with
respect to $V_{{\rm ws},\tau},R_{{\rm ws},\tau}$, and $a_{{\rm ws},\tau}$,

$\int^{\infty}_{0}dr\,r^{2}\Bigl{(}V_{\tau}(r)-V_{{\rm ws},\tau}f(r,R_{{\rm
ws},\tau},a_{{\rm ws},\tau})\Bigr{)}^{2}.$ (53)

Here, the volume integral has been chosen in order to minimize the above
deviation, since both $V_{\tau}(r)$ and $V_{\rm ws,\tau}f(r,R_{{\rm
ws},\tau},a_{{\rm ws},\tau})$ are expected to have a similar shape to that of
the nuclear density whose volume integral value is constrained by the nucleon
number. In deriving Eq.(9), we have used $n=0$ in Eq.(4), since there is not
such a constraint on the HO potential, but since it is important to keep a
similarity of the wave functions in the HO and WS potential.

###  5.2 Coulomb potential

In the relativistic models, the Coulomb energy is calculated by taking into
account the only direct term of the interaction in the same way as for other
interactions, while the exchange term is also included in the non-relativistic
models. The Coulomb interaction, however, plays a minor role in the present
purpose on $\delta R$, so that we simply express it by that of the uniform
charge distribution with the radius, $R_{\rm coul}$,

$V_{c}(r)\approx V_{\rm sph}(r),\qquad V_{\rm
sph}(r)=\left\\{\begin{array}[]{ll}\dfrac{Z\alpha}{2R_{\rm
coul}}\left(3-\dfrac{r^{2}}{R^{2}_{\rm coul}}\right)\,,&r<R_{\rm
coul},\\\\[12.0pt] \dfrac{Z\alpha}{r},&r>R_{\rm coul}.\end{array}\right.$ (54)

The radius $R_{\rm coul}$ is determined by minimizing the deviation:

$\int^{\infty}_{0}dr\,r^{2}\Bigl{(}V_{c}(r)-V_{\rm sph}(r)\Bigr{)}^{2}.$

The value of $R_{\rm coul}$ of the each model will be shown later.

###  5.3 Spin-orbit potential

We express the spin-orbit potential in the form:

$V_{\ell s,\tau}(r)=V_{\ell
s,\tau}\frac{1}{r}\frac{df_{\tau}(r)}{dr}\boldsymbol{\ell}\\!\cdot\\!\boldsymbol{\sigma}.$
(55)

In the relativistic models, it is written from Eq.(20) as

$\displaystyle V_{\ell
s,\tau}(r)=\frac{1}{r}\frac{d}{dr}\frac{1}{2M^{\ast}_{\tau}(r)}\boldsymbol{\ell}\\!\cdot\\!\boldsymbol{\sigma}$
$\displaystyle=-\frac{1}{2M}\frac{1}{m^{\ast
2}_{\tau}(r)}\frac{1}{r}\frac{dm^{\ast}_{\tau}(r)}{dr}\boldsymbol{\ell}\\!\cdot\\!\boldsymbol{\sigma}$
$\displaystyle\approx\frac{1-m^{\ast}_{{\rm ws},\tau}}{2M}\frac{1}{m^{\ast
2}_{\tau}(r)}\frac{1}{r}\frac{d}{dr}f(r,R^{\ast}_{{\rm
ws},\tau},a^{\ast}_{{\rm
ws},\tau})\boldsymbol{\ell}\\!\cdot\\!\boldsymbol{\sigma},$ (56)

with neglecting $V_{c}(r)$ in Eq.(52). In the calculations, the further
approximation has been used, as $m^{\ast}_{\tau}(r)\approx
m^{\ast}_{\tau}(R^{\ast}_{{\rm ws},\tau})=(1+m^{\ast}_{{\rm ws},\tau})/2$.

In the non-relativistic models, the spin-orbit potential of Eq.(26) is
approximated as

$V_{\ell
s,\tau}(r)=\frac{W_{0}}{2r}\frac{d}{dr}\Bigl{(}\rho(r)+\rho_{\tau}(r)\Bigr{)}\boldsymbol{\ell}\\!\cdot\\!\boldsymbol{\sigma}\approx\frac{W_{0}\rho_{0}}{2}\left(1+\frac{N_{\tau}}{A}\right)\frac{1}{r}\frac{d}{dr}f(r,R_{{\rm
den}},a_{{\rm den}})\,\boldsymbol{\ell}\\!\cdot\\!\boldsymbol{\sigma},$ (57)

where the value of $W_{0}$ is fixed at 120 MeV fm5 and we have written the
nuclear density as

$\rho(r)=\rho_{p}(r)+\rho_{n}(r)\approx\rho_{0}f(r,R_{{\rm den}},a_{{\rm
den}})\,,\quad$

with

$4\pi\rho_{0}\int^{\infty}_{0}dr\,r^{2}f(r,R_{{\rm den}},a_{{\rm den}})=A.$

The details of the calculation of the nuclear density will be mentioned in
§5.6.

In fact, the spin-orbit potentials are expected not to play an important role
in understanding the difference of $\delta R$ between the relativistic and
non-relativistic models, since their strengths are similar and the isospin-
dependences are small, in addition to the reason mentioned before.

###  5.4 A few examples

Before summarizing the results of the present section, let us compare
$V_{\tau}(r),\,\rho_{\tau}(r)$ and $m^{\ast}_{\tau}(r)$ from the exact mean-
field calculations with those of the corresponding simplified Hamiltonian, by
taking a few examples. After minimizing Eq.(53), the only values of $V_{{\rm
ws},n}$ for the relativistic WS potentials have been multiplied by 0.99, so as
to reproduce well the values of $R_{n}$ in the exact relativistic mean-field
calculations. This factor makes the mean value of $R_{n}$ from the WS
potential smaller by about $0.015$ fm.

Figure 2: The one-body potentials, the effective masses and the densities for
neutrons($\tau=n$) and protons($\tau=p$) in 208Pb. The solid curves are
obtained by the relativistic mean-field model with NL3, while the dashed ones
by its simplified Hamiltonian.

Figure 3: The one-body potentials, the effective masses and the densities for
neutrons($\tau=n$) and protons($\tau=p$) in 208Pb. The solid curves are
obtained by the non-relativistic mean-field model with SkM∗, while the dashed
ones by its simplified Hamiltonian.

Fig.3 shows the results of the one-body potentials, the effective masses and
the neutron and the proton densities for 208Pb calculated with NL3. The solid
curves are obtained by the full calculations and the dashed ones by the
simplified Hamiltonians. All other relativistic models yield similar results.
In non-relativistic models, we show the results for SkM∗ in Fig.3. These
results of SKM∗ are similar to those of other models except for the effective
mass in SLy4. In SLy4, the coordinate dependences of the effective mass are
similar to those in Fig.3, but the relation of the magnitude between the
$m^{\ast}_{{\rm ws},p}$ and $m^{\ast}_{{\rm ws},n}$ is opposite to that in
other non-relativistic models. It is seen that all the results by simplified
versions well reproduce the corresponding ones obtained by the full
calculations.

###  5.5 Results by the simplified models

Table 4 shows the root msr’s of the point neutron distributions in 48Ca and
208Pb determined in Ref.[4]. Those of the point proton distributions obtained
in a similar way are also listed. The errors in the parentheses are given by
taking into account the experimental error and the standard deviation of the
LSL. Since both relativistic and non-relativistic models employ the
experimental values of the msr’s of the nuclear charge distributions as an
input, the values of $R_{p}$ in the two frameworks are almost equal to each
other, while the values of $R_{n}$ are larger by about 0.1 fm in the
relativistic models than in the non-relativistic ones in both 48Ca and 208Pb.
The purpose of the present paper is to understand this difference between
$R_{n}$’s.

| $R_{n}$ | $R_{p}$ | $\delta R$
---|---|---|---
| Rel | $3.597(0.021)$ | $3.378(0.005)$ | $0.220(0.026)$
48Ca | Non | $3.492(0.028)$ | $3.372(0.009)$ | $0.121(0.036)$
| Rel | $5.728(0.057)$ | $5.454(0.013)$ | $0.275(0.070)$
208Pb | Non | $5.609(0.054)$ | $5.447(0.014)$ | $0.162(0.068)$
Table 4: The results of the least squares analysis in Ref.[4]. The numbers in
the parentheses denote the error which is obtained taking account of the
experimental error and the standard deviation of the calculated values from
the least-square line. All the numbers are given in units of fm. For details,
see the text.

We note that the new data from JLab have been reported in Ref.[5], according
to the parity violating electron scattering experiment. It provides $\delta R$
in 208Pb to be $0.283\pm 0.071$ fm. This is almost the same as the value of
$\delta R$ in Table 3 in the relativistic models, and is not incompatible with
the non-relativistic one in taking into account their errors. The analysis of
the JLab data[5] is model-dependent as in Ref.[4], and Ref.[38] has obtained
$\delta R=0.19\pm 0.02$ fm from the JLab data on the basis of the different
model-analysis.

Figure 4: The strength of the one-body potentials for neutrons($\tau=n$) and
protons($\tau=p$) in 208Pb. On the left-hand side, the relativistic and non-
relativistic models are indicated. The black circles represent the strengths
of the neutron potentials, the red ones those of the proton potentials, and
the blue ones the differences between their two strengths. The vertical lines
stand for the average values of the corresponding circles in the relativistic
and non-relativistic models, separately. The scale of the bottom side is for
the strength of the potentials, while that of the top side for the difference
between the strengths of neutron and proton potentials.

Let us summarize the results in the present section for 208Pb. Fig.4 shows the
values of $V_{{\rm ws},\tau}$ in Eq.(49) for the relativistic and non-
relativistic models. The strengths of the neutron potentials are shown by the
black circles, and those of the proton ones by the red circles. It is seen
that the non-relativistic ones are distributed over a wide range, as expected,
in contrast to those of relativistic models. The straight vertical lines show
their average values. The difference between $V_{{\rm ws},n}-V_{{\rm ws},p}$,
however, is almost equal independently of the models, as shown by the blue
circles and the straight lines indicating their average values. Thus, the
difference is only a little larger in the relativistic models than in the non-
relativistic ones. This fact implies that the difference between $\delta R$’s
in the two frameworks may not be due to the symmetry potentials only.

Figure 5: The radius parameters of the Woods-Saxon potentials for
neutrons($\tau=n$) and protons($\tau=p$) in 208Pb obtained from the
relativistic and non-relativistic mean-field models. On the left-hand side,
the used relativistic and non-relativistic models are indicated. The black
circles represent the values of the radius parameter $R_{{\rm ws},n}$, while
the red ones those of $R_{{\rm ws},p}$. The blue ones stand for the values of
the radius parameters of the Coulomb potentials. The vertical lines indicate
the average values of the corresponding circles in the relativistic and non-
relativistic models, separately.

Figure 6: The diffuseness parameters of the Woods-Saxon potentials for
neutrons($\tau=n$) and protons($\tau=p$) in 208Pb obtained from the
relativistic and non-relativistic mean-field models. On the left-hand side,
the used relativistic and non-relativistic models are indicated. The black
circles represent the values of the diffuseness parameter $a_{{\rm ws},n}$,
while the red ones those of $a_{{\rm ws},p}$. The vertical lines indicate the
average values of the corresponding circles, in the relativistic and non-
relativistic models, separately.

Fig.6 shows the values of $R_{{\rm ws},\tau}$ in a similar way as in Fig.4.
The black and red circles for the non-relativistic models are again
distributed over a wide region, compared to those of the relativistic ones,
although their regions are overlapped. The solid lines express the mean values
of the corresponding circles. It is seen that the value of the difference,
$\langle R_{{\rm ws},p}\rangle-\langle R_{{\rm ws},n}\rangle$, in the
relativistic models is rather smaller than that in the non-relativistic models
as in Table 5. Thus, the spread of the values of $R_{{\rm ws},\tau}$ in the
non-relativistic models does not seem to cause the difference between $\delta
R$’s in the two frameworks. The values of $R_{\rm coul}$ are indicated by the
blue circles for reference.

Fig.6 shows the values of $a_{{\rm ws},\tau}$. The straight lines stand for
their average values. Those of the relativistic and non-relativistic models
are distributed similarly over a wide region, but are small, compared to
$R_{{\rm ws},\tau}$, as $(a_{{\rm ws},\tau}/R_{{\rm ws},\tau})^{2}\approx
0.01$. The difference between $\delta R$’s in the relativistic and non-
relativistic models may not be due to these distributions of $a_{{\rm
ws},\tau}$.

Figure 7:  The effective mass of neutrons($\tau=n$) and protons($\tau=p$) in
208Pb obtained from the relativistic and non-relativistic mean field models.
On the left-hand side, the used relativistic and non-relativistic models are
indicated. The black circles represent the values of the effective masses of
neutrons, while the red ones those of protons. In T6, the value for neutrons
is the same as that for protons. The vertical lines indicate the average
values of the corresponding circles, in the relativistic and non-relativistic
models, separately.

In Fig.7 are shown the values of $m^{\ast}_{{\rm ws},\tau}$. The black circles
represent those of the neutrons, while the red circles the protons. The
straight lines indicate their average values, which are in the relativistic
models $\langle m^{\ast}_{{\rm ws},n}\rangle\approx 0.6321$ and $\langle
m^{\ast}_{{\rm ws},p}\rangle\approx 0.6387$, and in the non-relativistic
models $\langle m^{\ast}_{{\rm ws},n}\rangle\approx 0.7508$ and $\langle
m^{\ast}_{{\rm ws},p}\rangle\approx 0.7207$. As seen in the figure, the
circles of the relativistic models are almost at the same value and the ratio,
$\langle m^{\ast}_{{\rm ws},p}\rangle/\langle m^{\ast}_{{\rm ws},n}\rangle$,
is about 1.010, while those of the non-relativistic models are spread out, as
in nuclear matter, but the ratio in each model is almost the same and is on
average $\langle m^{\ast}_{{\rm ws},p}\rangle/\langle m^{\ast}_{{\rm
ws},n}\rangle\approx 0.9601$.

Table 5 lists the mean values of the WS parameters, the strengths of the one-
body potentials and the effective masses in the present simplified models for
the relativistic and non-relativistic mean-field ones, respectively. It should
be noticed that the values of $\langle m^{\ast}_{{\rm ws},\tau}\rangle$ and
$\langle V_{{\rm ws},\tau}\rangle$ are almost the same as those of $\langle
m^{\ast}_{\tau}\rangle$ and $\langle V_{\tau}\rangle$ in Table 2.

| $\langle R_{{\rm ws},n}\rangle$ | $\langle R_{{\rm ws},p}\rangle$ | $\langle a_{{\rm ws},n}\rangle$ | $\langle a_{{\rm ws},p}\rangle$ | $\langle V_{{\rm ws},n}\rangle$ | $\langle V_{{\rm ws},p}\rangle$ | $\langle m^{\ast}_{{\rm ws},n}\rangle$ | $\langle m^{\ast}_{{\rm ws},p}\rangle$
---|---|---|---|---|---|---|---|---
Rel | $7.046$ | $7.153$ | $0.733$ | $0.719$ | $-66.376$ | $-78.888$ | $0.6321$ | $0.6387$
Non | $7.096$ | $7.239$ | $0.630$ | $0.632$ | $-61.368$ | $-72.759$ | $0.7508$ | $0.7207$
Table 5:  The mean values of the Woods-Saxon parameters, the strengths of the
one-body potentials and the effective masses in the relativistic and non-
relativistic models. The numbers of $\langle R_{{\rm ws},\tau}\rangle$ and
$\langle a_{{\rm ws},\tau}\rangle$ are given in units of fm, and those of
$\langle V_{{\rm ws},\tau}\rangle$ in units of MeV. For details, see the text.

###  5.6 The proton and neutron distributions

It may be useful to see directly how the difference between $\delta R$’s is
caused in terms of the neutron and proton densities. We approximate the
neutron and proton distributions, $\rho_{\tau}(r)$, in the mean-field models
by the Fermi-type function which is widely employed[1, 7]. The approximation
is performed for

$\rho_{{\rm ws},\tau}(r)\approx\rho_{0,\tau}f_{\tau}(r,R_{{\rm
den},\tau},a_{{\rm den},\tau}),$ (58)

in the same minimization method as in Eq.(53). Since the obtained function
satisfies the normalization, $\int d^{3}r\rho_{{\rm ws},\tau}(r)=N_{\tau}$
with a small error about $0.5\%$, we slightly correct $R_{{\rm den},\tau}$ to
satisfy the normalization. We note that the correction by $\rho_{0,\tau}$
instead of $R_{{\rm den},\tau}$ yields almost the same values as those which
will be seen in Tables 6 and 7. The minimization under the constraint on the
nucleon-number also yields similar results.

The msr, $R^{2}_{\tau}$, by Eq.(58) is given as[1],

$R^{2}_{\tau}\approx\frac{3}{5}\left(\frac{3N_{\tau}}{4\pi\rho_{0,\tau}}\right)^{2/3}+\pi^{2}a^{2}_{{\rm
den},\tau},$ (59)

which provides the relationship between $R^{2}_{n}$ and $R^{2}_{p}$ as

$R^{2}_{n}=\frac{R^{2}_{p}-\pi^{2}a^{2}_{{\rm
den},p}}{(1-\epsilon)^{2/3}}+\pi^{2}a^{2}_{{\rm
den},n},\quad\epsilon=1-\frac{Z}{N}\frac{\rho_{0,n}}{\rho_{0,p}}.$ (60)

When keeping order up to $O(\epsilon)$, the difference, $\delta
R=R_{n}-R_{p}$, is written as

$\delta R\approx\frac{\epsilon}{3}R_{p}+\frac{\pi^{2}}{2}\frac{a^{2}_{{\rm
den},n}-a^{2}_{{\rm den},p}}{R_{p}}.$ (61)

| $\rho_{0,\tau}$ | $a_{{\rm den},\tau}$ | $R_{{\rm den},\tau}$ | $\epsilon$
---|---|---|---|---
Rel | $n$ | $0.0860$ | $0.553$ | $6.903$ | $0.1044$
| $p$ | $0.0625$ | $0.454$ | $6.692$ |
Non | $n$ | $0.0911$ | $0.554$ | $6.766$ | $0.0556$
| $p$ | $0.0628$ | $0.475$ | $6.672$ |
Table 6: The mean values of the parameters for the Fermi-type neutron(n) and proton(p) distributions. They are obtained by approximating the densities in the relativistic and non-relativistic mean-field models for 208Pb. The values of $\rho_{0,\tau}$ are given in units of fm-3 and those of $a_{{\rm den},\tau}$ and $R_{{\rm den},\tau}$ are in fm. For the definition of $\epsilon$, see the text. | $R_{n}$ | $R_{p}$ | $\delta R$
---|---|---|---
Rel | MF | 5.749 | 5.466 | 0.283
| WS | 5.740 | 5.457 | 0.283
| Eq.(59) | 5.728 | 5.451 | 0.277
Non | MF | 5.617 | 5.455 | 0.161
| WS | 5.621 | 5.462 | 0.159
| Eq.(59) | 5.629 | 5.460 | 0.169
Table 7: The mean values of the root msr’s of the neutron($R_{n}$) and
proton($R_{p}$) distributions and their difference $(\delta R=R_{n}-R_{p})$
obtained by the three approximations for 208Pb. MF indicates the mean values
in the relativistic and non-relativistic mean-field models, while WS the ones
obtained by approximating the mean-field potentials by Woods-Saxon potentials.
Eq.(59) stands for the equation number in the text used for the calculations
of $R_{\tau}$ and $\delta R$ given in its row. All the values are listed in
units of fm. For details, see the text.

Table 6 lists the average values of the parameters for the Fermi-type
densities in Eq.(58) in the present relativistic and non-relativistic models.
Table 7 shows the values of Eq.(59) using the results in Table 6. The average
values of $R_{\tau}$ and $\delta R$ in the mean-field models and their
simplified versions also are listed in the rows named MF and WS, respectively.
It is seen that the values in Eq.(59) and in WS almost reproduce the results
of the mean-field models.

The values of the two terms in the right-hand side of Eq.(61) are given as

$\delta R_{\rm rel}\approx 0.190+0.091=0.280\,\mbox{fm},\qquad\delta R_{\rm
non}\approx 0.101+0.074=0.175\,\mbox{fm}.$ (62)

Thus the difference about 0.1 fm between $\delta R$’s in the relativistic and
non-relativistic models mainly comes from the first terms $\epsilon R_{p}/3$,
and the diffuseness parameters yielding the second terms play a rather minor
role. It should be noticed that the first term proportional to $\epsilon$
disappears, when $\rho_{0,n}=(N/A)\rho$ and $\rho_{0,p}=(Z/A)\rho$.

Since the values of $R_{p}$ and $\rho_{0,p}$ in the relativistic and non-
relativistic models are almost the same, the difference between $\delta R$’s
in Eq.(61) comes from $\rho_{0,n}$ in $\epsilon$. Table 6 provides

$\frac{(\rho_{0,n})_{\rm rel}}{(\rho_{0,n})_{\rm
non}}=0.944,\qquad\frac{(\rho_{0,p})_{\rm rel}}{(\rho_{0,p})_{\rm
non}}=0.995.$

The 5.6$\%$ decrease of $(\rho_{0,n})_{{\rm rel}}$ provides the increase of
$(R_{n})_{\rm rel}$ by $0.944^{-1/3}=1.019$, yielding the 0.1 fm-difference
which we are discussing. Fig.3 shows qualitatively such a broadening of the
neutron density in NL3, in comparing with that in Fig.3.

In Table 2 are listed the mean values of the neutron and proton densities for
the nuclear matter obtained by Eq.(37). Those values are almost the same as
the corresponding ones in Table 6. They provide the values of $\epsilon$ in
Eq.(60) to be 0.1138 and 0.0567 for the relativistic and non-relativistic
models, respectively, which are comparable with the values for 208Pb in Table
6. Thus, the various parameters including $\langle m^{\ast}_{\tau}\rangle$ and
$\langle V_{\tau}\rangle$ at $r=0$ for 208Pb are similar to those for nuclear
matter.

It may be noticed that the values listed in the rows of MF in Table 7 are a
little different from those of LSA in Table 4, since the former is the mean
values of $R_{\tau}$ calculated by the models, while the latter has been
obtained by the least squares analysis of the calculated values comparing with
the experimental data[4].

##  6 The HVH lines in 208Pb

In the previous section, we have shown that the mean values of $\langle
m^{\ast}_{\tau}\rangle$, $\langle V_{\tau}\rangle$ and
$\langle\rho_{\tau}\rangle$ in Table 2 for nuclear matter are almost the same
as the corresponding ones in Tables 5 and 6 for 208Pb. In order to explore the
reason why $\delta R$’s in the relativistic and non-relativistic schemes are
different from each other, the contributions from $R_{{\rm ws},\tau}$ and
$a_{{\rm ws},\tau}$ in Figs.5 and 6 to $R_{\tau}$ should be also examined, in
addition to those from $V_{{\rm ws},\tau}$ and $m^{\ast}_{{\rm ws},\tau}$ in
Figs.4 and 7.

In the present section, first, it will be discussed that the similar equations
to Eqs.(9) and (10) hold for the dependence of $R_{\tau}$ on the WS parameters
and $m^{\ast}_{{\rm ws},\tau}$ with the values in Fig.4 to 7. Second, the
value of $\delta R$ will be shown to be dominated by $m^{\ast}_{{\rm
ws},\tau}$ and $V_{{\rm ws},\tau}$, rather than by $R_{{\rm ws},\tau}$ and
$a_{{\rm ws},\tau}$. Third, it will be investigated whether or not the
constraint on the values of $m^{\ast}_{{\rm ws},\tau}$ and $V_{{\rm ws},\tau}$
by the HVH theorem holds in the same way as in Fig. 1 for the nuclear matter.
Finally, the difference between $\delta R$’s between the two schemes will be
explained in terms of $m^{\ast}_{{\rm ws},\tau}$ and $V_{{\rm ws},\tau}$.

We discuss $R_{\tau}$ of the finite nucleus 208Pb on the basis of Eqs.(9) and
(10). For this purpose, first we examine if it is appropriate for Eqs.(9) and
(10) to employ $m^{\ast}_{{\rm ws},\tau}$ and $V_{{\rm ws},\tau}$ defined in
Eqs.(52) and (49). When $R_{\tau}$ calculated by the simplified models is
expressed in terms of $m^{\ast}_{{\rm ws},\tau}$ and $V_{{\rm ws},\tau}$ as

$R_{\tau}\approx B_{\tau}\left(-\frac{R^{2}_{{\rm ws},\tau}}{m^{\ast}_{{\rm
ws},\tau}V_{{\rm ws},\tau}}\right)^{1/4}(1+b_{{\rm ws},\tau})^{3/8},\quad
b_{{\rm ws},\tau}=\left(\frac{\pi a_{{\rm ws},\tau}}{R_{{\rm
ws},\tau}}\right)^{2},$ (63)

then the value of the coefficient $B_{\tau}$ corresponding to $B$ in Eq.(9)
should be almost constant independently of the various interaction parameters
of the mean-field models. In order to estimate the value of $B_{\tau}$, both
sides except for $B_{\tau}$ of the above equation are calculated for each
model, according to §5. The results are listed in the left-hand side named WS
in Table 8, where the mean values of $B_{\tau}$ are shown as $\langle
B_{\tau}\rangle$ in units of $({\rm fm}^{2}{\rm MeV})^{1/4}$ in the
relativistic and non-relativistic models, separately. The table shows that the
values of the standard deviation($\sigma$) are small enough for our purpose.

The meaning of $B_{\tau}$ may be qualitatively understood according to
Ref.[1], where the values of $C$ in Eq.(3) are estimated by summing a single
particle radii over the occupied orbits in HO potential. Their approximations
yield the values which are in the same order of magnitude as those of
$B_{\tau}$ in Table 8, as $B_{n}\approx 5.44$ for $N=126$ and $B_{p}\approx
5.07$ for $Z=82$ in units of $({\rm fm}^{2}{\rm MeV})^{1/4}$.

|  $\langle B_{\tau}\rangle$ ($\sigma$)
---|---
| WS | MF
Rel |  n | 5.295 (0.0063) | 5.304 (0.0076)
|  p | 5.243 (0.0151) | 5.253 (0.0151)
Non |  n | 5.288 (0.0217) | 5.283 (0.0273)
|  p | 5.268 (0.0801) | 5.261 (0.0685)
Table 8: The mean values of the proportional constant $B_{\tau}$ between the
root msr and the function of the Woods Saxon parameters in Eq.(63). The
standard deviation $\sigma$ is given in the parenthesis. WS lists the mean
values in using $R_{\tau}$ of the simplified models, while MF shows for
reference those in using $R_{\tau}$ of the full mean-field approximation. The
numbers are written in units of $({\rm fm}^{2}{\rm MeV})^{1/4}$. For details,
see the text.

We note that the values of $\sigma$ for protons are larger than those for
neutrons in both models. This fact may be due to the Coulomb potential which
is not explicitly taken into account in the right-hand side in Eq.(63). If
$v_{c}=22$ MeV is added to $V_{{\rm ws},p}$ by hand for reference, the values
of $\sigma$ for protons become comparable with those for neutrons, as 0.0089
and 0.0217 in the relativistic and non-relativistic models, respectively. We
expect, however, that these results do not change the following discussions on
the difference between $\delta R$’s in the relativistic and non-relativistic
models.

It should be also made sure that the 0.1 fm difference between $\delta R$’s is
not due to the enhancement of $R_{n,{\rm rel}}$ by the factor $B_{n,{\rm
rel}}$. The equation corresponding to Eq.(10) is described as

$\frac{R_{n}}{R_{p}}=\frac{B_{n}}{B_{p}}\left(\frac{m^{\ast}_{{\rm
ws},p}V_{{\rm ws},p}}{m^{\ast}_{{\rm ws},n}V_{{\rm
ws},n}}\right)^{1/4}\left(\frac{R_{{\rm ws},n}}{R_{{\rm
ws},p}}\right)^{1/2}\left(\frac{1+b_{{\rm ws},n}}{1+b_{{\rm
ws},p}}\right)^{3/8}$ (64)

Since $\delta R$ is written as $\delta R=R_{p}(R_{n}/R_{p}-1)$ and the value
of $R_{p}$ is almost fixed due to the fitting in both relativistic and non-
relativistic models, the difference between $\delta R$’s in the two frameworks
stems from their values of $R_{n}/R_{p}$. Table 8 provides the ratio of
$(\langle B_{n}\rangle/\langle B_{p}\rangle)_{\rm rel}/(\langle
B_{n}\rangle/\langle B_{p}\rangle)_{\rm non}=1.0061$. In the right-hand side
of Table 8, the values of $\langle B_{\tau}\rangle$ in using $R_{\tau}$ from
the mean-field calculations in Eq.(63) are listed. The table shows that the WS
calculations yield almost the same results as those of the mean-field ones, as
$(\langle B_{n}\rangle/\langle B_{p}\rangle)_{\rm rel}/(\langle
B_{n}\rangle/\langle B_{p}\rangle)_{\rm non}=1.0055$. These values imply that
the factor $B_{\tau}$ is not enough to explain the 0.1 fm difference. Thus, it
is reasonable to use $V_{{\rm ws},\tau}$ and $m^{\ast}_{{\rm ws},\tau}$
defined in Eqs.(49) and (52) in the analysis of $\delta R$ in 208Pb.

Assuming that Eq.(64) holds for the mean values in Table 5, we have

$R_{n}\approx\frac{B_{n}}{B_{p}}\left(\frac{\langle m^{\ast}_{{\rm
ws},p}\rangle\langle V_{{\rm ws},p}\rangle}{\langle m^{\ast}_{{\rm
ws},n}\rangle V_{{\rm ws},n}\rangle}\right)^{1/4}\left(\frac{\langle R_{{\rm
ws},n}\rangle}{\langle R_{{\rm
ws},p}\rangle}\right)^{1/2}\left(\frac{1+\langle b_{{\rm
ws},n}\rangle}{1+\langle b_{{\rm ws},p}\rangle}\right)^{3/8}R_{p}\noindent$
(65)

with $\langle b_{{\rm ws},\tau}\rangle=(\pi\langle a_{{\rm
ws},\tau}\rangle/\langle R_{{\rm ws},\tau}\rangle)^{2}$. Using the values in
Tables 5 and 8, the above equation provides for the relativistic and non-
relativistic models, respectively, as

$R_{n,{\rm rel}}\approx 1.0517\,R_{p,{\rm rel}}\,,\quad R_{n,{\rm non}}\approx
1.0274\,R_{p,{\rm non}}\,.$ (66)

If we put the numbers of $R_{p}$ obtained by the WS approximation in Table 7
into the right-hand sides of Eq.(66), then we have the values of $R_{n,{\rm
rel}}\approx 5.739$ fm and $R_{n,{\rm non}}\approx 5.612$ fm. It is seen that
they are almost the same values as those of the WS approximation in Table 7.
Thus, Eq.(64) holds well also for the mean values in Tables 5 and 8.

Second, the value of $\delta R$ will be shown to be dominated by
$m^{\ast}_{{\rm ws},\tau}$ and $V_{{\rm ws},\tau}$, rather than by $B_{\tau}$,
$R_{{\rm ws},\tau}$ and $a_{{\rm ws},\tau}$, with the use of their mean
values. The one way to show this fact is by taking the numbers in Eq.(66)
which imply that $R_{n,{\rm rel}}>R_{n,{\rm non}}$, when $R_{p,{\rm
rel}}\approx R_{p,{\rm non}}$, indicating the $0.1$ fm difference problem.
Those numbers have been obtained by

$1.0517\approx 1.0468\times 1.0047\,,\quad 1.0274\approx 1.0329\times
0.9947\,,$ (67)

where the first numbers in the right-hand sides of the above equations come
from the factor $(\langle m^{\ast}_{{\rm ws},p}\rangle\langle V_{{\rm
ws},p}\rangle/\langle m^{\ast}_{{\rm ws},n}\rangle V_{{\rm
ws},n}\rangle)^{1/4}$, while the second numbers from the rest of the factors
in Eq.(65). Thus, the difference between $R_{n}$ and $R_{p}$ is mainly due to
the first number coming from the values of $\langle m^{\ast}_{{\rm
ws},\tau}\rangle$ and $\langle V_{{\rm ws},\tau}\rangle$ in both relativistic
and non-relativistic scheme. The second numbers from $B_{\tau}$, $\langle
R_{{\rm ws},\tau}\rangle$ and $\langle a_{{\rm ws},\tau}\rangle$ play a minor
role in their differences. This fact also implies that the distribution of
$R_{{\rm ws},\tau}$ and $a_{{\rm ws},\tau}$ over a wide region in Figs.5 and 6
is not worrisome for the 0.1 fm problem. The minor role of $a_{{\rm ws},\tau}$
is consistent with the results of Eq.(62).

It may be seen in another way qualitatively that, compared to $B_{\tau}$,
$\langle R_{{\rm ws},\tau}\rangle$ and $\langle b_{{\rm ws},\tau}\rangle$,
$\langle m^{\ast}_{{\rm ws},\tau}\rangle$ and $\langle V_{{\rm
ws},\tau}\rangle$ play an important roles in $\delta R$ of the two frameworks.
We write Eq.(63) in terms of the mean values,

$R_{\tau,{\rm rel}}\approx B_{\tau,{\rm rel}}\left(-(\langle m^{\ast}_{{\rm
ws},\tau}\rangle\langle V_{{\rm ws},\tau}\rangle)_{\rm
rel}\right)^{-1/4}(\langle R_{{\rm ws},\tau}\rangle_{\rm rel})^{1/2}(1+\langle
b_{{\rm ws},\tau}\rangle_{\rm rel})^{3/8},$ (68)

for the relativistic scheme. In the above equation, keeping the values of
$B_{\tau,{\rm rel}}$, $\langle R_{{\rm ws},\tau}\rangle_{\rm rel}$ and
$\langle b_{{\rm ws},\tau}\rangle_{\rm rel}$, we replace $(\langle
m^{\ast}_{{\rm ws},\tau}\rangle\langle V_{{\rm ws},\tau}\rangle)_{\rm rel}$ by
that of the non-relativistic one, $(\langle m^{\ast}_{{\rm
ws},\tau}\rangle\langle V_{{\rm ws},\tau}\rangle)_{\rm non}$. Then, the values
of $R_{\tau,{\rm rel}}$ of the WS approximation in Table 7 and the mean values
of Table 5 provide

$R_{\tau,{\rm rel}}\left(\frac{\bigl{(}\langle m^{\ast}_{{\rm
ws},\tau}\rangle\langle V_{{\rm ws},\tau}\rangle\bigr{)}_{\rm
rel}}{\bigl{(}\langle m^{\ast}_{{\rm ws},\tau}\rangle\langle V_{{\rm
ws},\tau}\rangle\bigr{)}_{\rm
non}}\right)^{1/4}=\left\\{\begin{array}[]{ll}5.607\,\mbox{fm},&\tau=n,\\\\[2.0pt]
5.403\,\mbox{fm},&\tau=p.\end{array}\right.$ (69)

The above equation yield $\delta R=5.607-5.403=0.204$ fm, which should be
compared to $\delta R=0.159$ fm in WS for the non-relativistic models in Table
7. The difference between $\delta R$’s in the two frameworks is reduced from
$0.283-0.159=0.124$ fm to $0.204-0.159=0.045$ fm by $64\%$. Thus, it is seen
that the $0.1$ fm problem is deeply related to the difference between
$(\langle m^{\ast}_{{\rm ws},\tau}\rangle\langle V_{{\rm
ws},\tau}\rangle)_{\rm rel}$ and $(\langle m^{\ast}_{{\rm
ws},\tau}\rangle\langle V_{{\rm ws},\tau}\rangle)_{\rm non}$ in $R_{\tau}$ .

Third, let us investigate whether or not there is a constraint on
$m^{\ast}_{{\rm ws},\tau}$ and $V_{{\rm ws},\tau}$ in 208Pb, as in nuclear
matter. In Fig.8 are plotted the values of $V_{{\rm ws},\tau}$ in Fig.4 and
those of $m^{\ast}_{{\rm ws},\tau}$ in Fig.7 in the $1/m^{\ast}_{{\rm
ws},\tau}-V_{{\rm ws},\tau}$ plane. The black circles show the values for
neutrons and protons in the relativistic models, while the red circles in the
non-relativistic models. The numbers attached to each circle indicate the used
model, according to the numbering mentioned in §4.2. The pair of the same
number represent the values for neutrons and protons calculated by the same
model.

The slanting lines are obtained by the least square method for the values of
each group. The upper and lower black lines are drawn for neutrons and protons
in the relativistic models, respectively, while the upper and lower red lines
are in the non-relativistic models. It is remarkable that the values of each
group follow well the corresponding line, and that the four lines are well
separated from one another, as in Fig.1 for nuclear matter. We notice that the
only FSU(11)[16] among the relativistic models yields a point on the neutron
line for the non-relativistic models. This may reflect the fact that FSU has
added two additional parameters to the Lagrangian of, for example, NL3(5)[22],
so as to reduce the value of $R_{n}$. The values of the gradient($a^{\rm
L}_{{\rm ws},\tau}$) and the intercept($b^{\rm L}_{{\rm ws},\tau}$) of the
LSL,

$V_{{\rm ws},\tau}=a^{\rm L}_{{\rm ws},\tau}/m^{\ast}_{{\rm ws},\tau}+b^{\rm
L}_{{\rm ws},\tau},$ (70)

are listed in Table 9 for relativistic(Rel) and non-relativistic(Non) models.
The values of the correlation coefficient, $r$, are also shown, which are
nearly equal to 1.

Figure 8: The relationship between the effective mass and the one-body potential for neutrons($\tau=n$) and protons ($\tau=p$) in the mean-field models for 208Pb. The black circles show the values calculated by the 11 relativistic models, while the red ones those by the 9 non-relativistic models. The least-square lines are shown for the four groups. The two black lines are obtained from the black circles for neutrons and protons, respectively, and the red lines from the red circles.  | Rel | Non
---|---|---
| $a^{\rm L}_{{\rm ws},\tau}$ | $b^{\rm L}_{{\rm ws},\tau}$ | $r$ | $a^{\rm L}_{{\rm ws},\tau}$ | $b^{\rm L}_{{\rm ws},\tau}$ | $r$
$n$ | $-40.120$ |  2$-2.795$ | $0.958$ | $-38.609$ |  2$-7.438$ | $0.997$
$p$ | $-34.933$ | $-24.098$ | $0.937$ | $-32.588$ | $-25.215$ | $0.997$
Table 9: The values of the gradient($a^{\rm L}_{{\rm ws},\tau}$) and the
intercept($b^{\rm L}_{{\rm ws},\tau}$) of the least-square line for the
relationship between the depth of the one-body potential and the nucleon
effective mass in Fig.8. The numbers of $a^{\rm L}_{{\rm ws},\tau}$ and
$b^{\rm L}_{{\rm ws},\tau}$ are given in units of MeV. The values of the
correlation coefficient, $r$, are also listed. For details, see the text.

In Fig.8, it is seen that the variation of the effective mass and the strength
of the one-body potential in finite nuclei is also constrained in a similar
way to that in Fig.1 for nuclear matter. Eq.(70) has the same form as in
Eq.(44) from the HVH theorem. Thus, the HVH theorem seems to be inherent in
the mean field approximation for finite nuclei also. From now on, we will
refer to the LSL of 208 Pb as the HVH line.

We compare the coefficients of the HVH line for 208Pb in Table 9 to those of
Eq.(44) for nuclear matter listed in Table 2. The coefficients of Eq.(44) have
shown to be constrained by the HVH theorem through Eq.(47). It is seen in
Tables 2 and 9 that the values of the corresponding coefficients are not the
same as each other, but the magnitude relation of the corresponding two values
are almost the same as those of other pair. More important values for the
present discussion is those in Eq.(47). In Table 10 is listed one of the
values in Eq.(47), $\langle a_{\tau}\rangle+\langle b_{\tau}\rangle\langle
m^{\ast}_{\tau}\rangle$, in the first column, and the corresponding values
obtained from Tables 5 and 9 in the second and the last column. It is seen
that the values for nuclear matter in the first column are almost the same as
those for 208Pb in other columns. Since the values in the first column is
nothing but the results due to the HVH theorem, it is confirmed that those in
the second and third columns also reflect the constraint by the theorem.

We note that the values of the first column has been obtained by introducing a
model with $v_{c}$ in Eq.(38). It is made so as to provide neutrons and
protons with the same average binding energies by Eq.(37), as in stable
nuclei. The value of $v_{c}$ is employed which approximately corresponds to
the energy of the Coulomb potential for 208Pb in Eq.(42). Although the model
has been used as a guide for discussions of the finite nucleus, Table 10 shows
conversely that such a simple model almost reproduces results for the finite
nucleus and is useful for describing asymmetric nuclear matter.

In Fig.8, it should be noticed that on the one hand, that the distance between
two black lines for the relativistic models is about 12.5 MeV at a fixed value
of $1/m^{\ast}_{{\rm ws},\tau}$, as listed in Table 11. It is almost the same
as the mean value of $V_{3}=V_{{\rm ws},n}-V_{{\rm ws},p}$ in Fig.4. This is
because of $m^{\ast}_{{\rm ws},n}\approx m^{\ast}_{{\rm ws},p}$ in the
relativistic models. On the other hand, in the non-relativistic models, the
distance between the two red lines for the same value of $1/m^{\ast}_{{\rm
ws},\tau}$ is about 7.7 MeV, in spite of the fact that $V_{3}\approx 11.6$ MeV
as in Table 11. This is because, in the non-relativistic models, the value of
the neutron effective mass is larger than that of the proton one, except for
SLy4(6)[24], as in Fig.7. The value of $V_{3}\approx 11.6$ MeV is
approximately kept by providing neutrons and protons with different effective
masses. As seen below, it is essential for understanding 0.1 fm difference
that the values of the effective mass for neutrons are different from the ones
for protons in the non-relativistic models, while those in the relativistic
models are almost the same.

So far, understanding the dependence of $R_{\tau}$ on $m^{\ast}_{{\rm
ws},\tau}$ and $V_{{\rm ws},\tau}$ is simply based on their mean values as in
Eqs.(67) and (69), aiming to emphasize their roles in the $0.1$ fm problem.
Finally, we investigate roles of of $m^{\ast}_{{\rm ws},\tau}$ and $V_{{\rm
ws},\tau}$ in $R_{\tau}$ by using their values themselves, together with the
HVH line in Fig.8.

Eq.(69) has been obtained by replacing the mean values, $\left(\langle
m^{\ast}_{{\rm ws},\tau}\rangle\langle V_{{\rm ws},\tau}\rangle\right)_{\rm
rel}$ by $\left(\langle m^{\ast}_{{\rm ws},\tau}\rangle\langle V_{{\rm
ws},\tau}\rangle\right)_{\rm non}$. In order to explore in more detail how the
value of $R_{\tau}$ in the relativistic scheme approaches to that in the non-
relativistic one by changing $m^{\ast}_{{\rm ws},\tau}$ and $V_{{\rm
ws},\tau}$, we replace the values of $m^{\ast}_{{\rm ws},\tau}$ and $V_{{\rm
ws},\tau}$ in $R_{\tau}$ in each relativistic model by $\langle m^{\ast}_{{\rm
ws},\tau}\rangle_{\rm non}$ and $\langle V_{{\rm ws},\tau}\rangle_{\rm non}$.
The replacement will be made keeping the values of $R_{{\rm ws},\tau}$ and
$b_{{\rm ws},\tau}$ in each model, and using the HVH line in Fig.8. By this
procedure, we will see the roles of $m^{\ast}_{{\rm ws},\tau}$ and $V_{{\rm
ws},\tau}$ in $R_{\tau}$ separately, as follows.

In Fig.9 is shown $R_{\tau}$ as a function of $m^{\ast}_{{\rm ws},\tau}$ in
the case of NL3(5) as an example. The closed and open circles indicate the
values of $R_{\tau}$ in the full mean-field calculation and in the simplified
one in §5, respectively, at the value of $m^{\ast}_{{\rm ws},\tau}$ for NL3.
The solid curves are calculated by keeping the values of $R_{{\rm ws},\tau}$
and $a_{{\rm ws},\tau}$ of NL3 and using $V_{{\rm ws},\tau}$ given by the HVH
line for the relativistic models in Fig.8. The closed and open circles are
seen to be almost on the curves. The dashed curves also show $R_{\tau}$, but
using $V_{{\rm ws},\tau}$ given by the HVH line for the non-relativistic
models in Fig.8.

In Fig.9, we have specified the six points on the curves, where Xn, Yn and Zn
are for the neutrons, and others for the protons. The points Xτ indicates the
position of the open circles. The points Yτ and Zτ are on the dashed curves.
The former indicates the place where NL3 provides $m^{\ast}_{{\rm ws},\tau}$,
and the latter the place of $m^{\ast}_{{\rm ws},\tau}=\langle m^{\ast}_{{\rm
ws},\tau}\rangle_{\rm non}$ as shown by the vertical lines. The replacement of
$m^{\ast}_{{\rm ws},\tau}$ and $V_{{\rm ws},\tau}$ in NL3 by $\langle
m^{\ast}_{{\rm ws},\tau}\rangle_{\rm non}$ and $\langle V_{{\rm
ws},\tau}\rangle_{\rm non}$ is made by using the values at point $Z_{\tau}$.
We make, however, the replacement by two steps according to the curves in
Fig.9. In the first step, the values at Xτ are replaced by those at Yτ, and in
the second step the values at Yτ by those at Zτ. This process is shown in
Fig.9 by the arrows. The blue arrow indicates the first step, while the green
one the second step. In this way, we may see how $R_{\tau}$ in NL3 varies by
$m^{\ast}_{{\rm ws},\tau}$ and $V_{{\rm ws},\tau}$ separately and approaches
to $R_{\tau}$ in the non-relativistic scheme.

Fig.9 shows that the value of $R_{n}$ is decreased in the first step, because
$V_{{\rm ws},n}$ becomes deeper as seen in Fig.8. From Yn to Zn, the potential
becomes shallower, but the value of the effective mass is increased and a role
of the kinetic part as a repulsive potential declines. As a result, the value
of $R_{n}$ further shrinks, as in Fig.9. The decrease of $R_{\tau}$ from Yτ to
Zτ with the increasing $m^{\ast}_{{\rm ws},\tau}$ is understood qualitatively
by Eqs.(63) and(70) which yield

$m^{\ast}_{{\rm ws},\tau}=-\,\frac{1}{b^{\rm L}_{{\rm ws},\tau}}\left(a^{\rm
L}_{{\rm ws},\tau}+\frac{B_{\tau}^{4}}{R_{\tau}^{4}}R^{2}_{{\rm
ws},\tau}\bigl{(}1+b_{{\rm ws},\tau}\bigr{)}^{3/2}\right).$ (71)

Thus, the value of $R_{n}$ in the relativistic models approaches to that in
the non-relativistic models, following the path under the constraint of the
HVH theorem on $V_{{\rm ws},\tau}$ and $m^{\ast}_{{\rm ws},\tau}$.

With respect to $R_{p}$, Fig.9 shows its increase from Xp to Yp, because of
the decreasing strength of $|V_{{\rm ws},p}|$. From Yp to Zp, the value of
$R_{p}$ decreases in the same way as that of $R_{n}$ from Yn to Zn, according
to Eq.(71). The final value of $R_{p}$ at the point Zp returns to the almost
original value at Xp, since the value of $R_{p}$ at Xp for the relativistic
model is fixed by the experimental value of $R_{c}$ as an input, while the
value at Zp is almost equal to the values of $R_{p}$ for the non-relativistic
models which are fixed in the same way.

In the above analysis, it should be noticed that the value of $\langle
m^{\ast}_{{\rm ws},n}\rangle_{\rm non}$ is larger than that of $\langle
m^{\ast}_{{\rm ws},p}\rangle_{\rm non}$ as in Table 5. Owing to the fact, the
path from Yn to Zn is longer than that from Yp to Zp as seen in Fig.9. This
difference also works to make $R_{n}$ smaller in the path from Yn to Zn.

Fig.11 shows the values of $R_{\tau}$ which are obtained by the same procedure
as in Fig.9 for all the relativistic mean-field models taken in the present
paper. The black and red circles show the results of the full mean-field
calculations and the simplified ones in §5, respectively, where the closed
circles are for neutrons and the open circles for protons. The vertical lines
indicate their mean values. It is seen that the simplified calculations
reproduce well the values of $R_{\tau}$ by the full calculations. Those for
the non-relativistic models also are shown in the same way.

| $\langle a_{\tau}\rangle+\langle b_{\tau}\rangle\langle m^{\ast}_{\tau}\rangle$ | $\langle m^{\ast}_{{\rm ws},\tau}\rangle\langle V_{{\rm ws},\tau}\rangle$ | $a^{\rm L}_{{\rm ws},\tau}+b^{\rm L}_{{\rm ws},\tau}\langle m^{\ast}_{{\rm ws},\tau}\rangle$
---|---|---|---
${\rm Rel}$ | $n$ | $-41.814$ | $-41.956$ | $-41.887$
| $p$ | $-49.737$ | $-50.386$ | $-50.324$
${\rm Non}$ | $n$ | $-44.321$ | $-46.075$ | $-44.193$
| $p$ | $-51.364$ | $-52.437$ | $-50.760$
Table 10: The values for the combination composed of the effective mass and the coefficients of the least-square line for asymmetric nuclear matter(the first column) and for 208Pb(the third column) in the relativistic and non-relativistic models. The values of the product of the effective mass and the strength of the one-body potential is also listed in the second column for 208Pb. All the numbers are given in units of MeV. For details, see the text. | $V^{\rm rel}_{{\rm ws},\tau}\bigl{(}\langle m^{\ast}_{\tau}\rangle_{\rm rel}\bigr{)}$ | $V^{\rm non}_{{\rm ws},\tau}\bigl{(}\langle m^{\ast}_{\tau}\rangle_{\rm rel}\bigr{)}$ | $V^{\rm non}_{{\rm ws},\tau}\bigl{(}\langle m^{\ast}_{\tau}\rangle_{\rm non}\bigr{)}$
---|---|---|---
$n$ | $-66.264$ | $-68.517$ | $-58.865$
$p$ | $-78.792$ | $-76.238$ | $-70.434$
$V_{n}-V_{p}$ |  $-$$12.528$ |  $-7$$7.721$ |  $-$$11.569$
Table 11: The neutron(n) and proton(p) potentials at the mean value of the
effective mass. The numbers are given in units of MeV. For details, see the
text. Figure 9: The root msr as a function of the effective mass for $\tau=n$
and $\tau=p$ in NL3. The closed circles are obtained by the full mean-field
approach, and the open ones by the simplified Hamiltonian of NL3. The solid
curves are calculated by using the least-square lines for the relationship
between $m^{\ast}_{{\rm ws},\tau}$ and $V_{{\rm ws},\tau}$ in the relativistic
models in Fig.8, while the dashed ones by using those in the non-relativistic
models. The top two lines are for neutrons, and the bottom two ones for the
protons. In these curves, the values of the Woods-Saxon parameters, $R_{{\rm
ws},\tau}$ and $a_{{\rm ws},\tau}$, are taken from those determined by NL3.
The vertical lines indicate the average values of the effective masses for
neutrons($\langle m^{\ast}_{{\rm ws},n}\rangle_{\rm non}$) and
protons($\langle m^{\ast}_{{\rm ws},p}\rangle_{\rm non}$) in the non-
relativistic models. The point, Xn, indicates the place of the open circle for
the neutrons, while Yn the point on the dashed curve at the same value of
$m^{\ast}_{{\rm ws},n}$ as that for Xn. The point, Zn shows the intersection
point between the dashed curve and the vertical line for $\langle
m^{\ast}_{{\rm ws},n}\rangle_{\rm non}$. The points, Xp, Yp and Zp are given
in a similar way. The blue and green arrows are used for discussions in the
text.

Figure 10: The root msr $R_{\tau}$ of 208Pb calculated in the relativistic and
non-relativistic models. On the left-hand side, the used models are indicated.
The black closed and open circles are obtained by the full mean-field
approximations for the neutrons($\tau=n$) and protons($\tau=p$), respectively.
The red ones are by the simplified Hamiltonian for each model. The blue ones
are obtained through the first step discussed in the text, while the green
ones by the second step. The vertical lines show the average values of the
same color circles, respectively. For details, see the text.

Figure 11: The difference between the root msr’s of the neutron and the proton
distribution in 208Pb calculated in the relativistic and non-relativistic
models. On the left-hand side, the used models are indicated. The black closed
circles are obtained by the full mean-field approximations, and the red ones
by the simplified Hamiltonian for each model. The blue ones are obtained
through the first step, while the green ones through the second step discussed
in the text. The vertical lines show the average values of the same color
circles, respectively. For details, see the text.

The blue circles are obtained by the first step from Xτ to Yτ mentioned in
Fig.9, while the green ones by the second step. All the models show the
similar change of $R_{\tau}$ as in Fig.9 such that the values of $R_{n}$
decrease by the two steps, while those of $R_{p}$ come back to almost the same
values by the second step from Yτ to Zτ.

Fig.11 shows the values of $\delta R$, using the same designating symbols as
in Fig.11. The values from the two steps shown by the green circles are almost
the same as the blue ones obtained by the first step, since the values of
$R_{p}$ return to the original ones by the second step.

The results of $R_{\tau}$ and $\delta R$ in Figs.11 and 11 are summarized in
Table 12 in units of fm. The mean values of $R_{\tau}$ in the relativistic
models are listed in the columns named Red, Blue and Green according to the
colors of those figures. From Red to Green, the value of $R_{n}$ decreases,
while that of $R_{p}$ increases from Red to Blue and decreases from Blue to
Green, up to almost the Red one, as shown in the figures. In changing the
values $V_{\tau}$ and $m^{\ast}_{\tau}$ in the relativistic models following
the HVH lines, the value of $\delta R$ shrinks from 0.283 fm to 0.193 fm,
which should be compared to 0.159 fm of the non-relativistic models. The
difference between $\delta R$’s in the relativistic and non-relativistic
models becomes smaller by $73\%$, changing its value from 0.124 fm to 0.034
fm.

| Rel | Non
---|---|---
| Red | Blue | Green | Red
$R_{n}$ | $5.740$ | 5.691 | 5.645 | $5.621$
$R_{p}$ | 5.457 | 5.493 | 5.452 | 5.462
$\delta R$ | $0.283$ | 0.197 | 0.193 | 0.159
Table 12:  The average values of the root msr of the proton and neutron
distributions in the relativistic(Rel) and non-relativistic(Non) models in
various approximations. Red, Green and Blue indicate the average values
corresponding to those in Figs.11 and 11, respectively. All the numbers are
given in units of fm. For details, see the text.

It is concluded that most of the 0.1 fm difference between $\delta R$’s in the
relativistic and non-relativistic models is attributed to the difference
between the values of their $V_{\tau}$ and $m^{\ast}_{\tau}$, which are
constrained by $\rho_{0,\tau}$ through the HVH theorem. The remaining
difference may be caused by the sum of many small contributions, in addition
to those from $R_{{\rm ws},\tau}$, $a_{{\rm ws},\tau}$ and $B_{\tau}$, from
the used approximations. The exchange term of the Coulomb force, the center of
mass correction, the small component of the wave functions, etc., are also
managed differently in the two frameworks. Discussions on those effects,
however, are beyond the present purpose.

##  7 Summary

Ref.[4] has pointed out that the neutron skin thickness defined by $\delta
R=R_{n}-R_{p}$ in 208Pb is larger by about 0.1 fm in the relativistic mean-
field models than in the non-relativistic ones. Here, $R_{n}$ and $R_{p}$
represent the root msr(mean-square-radius) of the point neutron and proton
distributions in the nucleus, respectively. The value of the charge radius
$R_{c}$ of 208Pb is about 5.503 fm. The 0.1 fm difference is not small for
nuclear physics[1, 10, 17, 18], but also for astrophysics[3, 5, 18]. In this
paper, it has been investigated why the difference is avoidable in the present
mean-field models, even though both relativistic ad non-relativistic models
are constructed phenomenologically with free parameters to be fixed by
experimental values.

The value of $R_{p}$ is one of the most important inputs, together with the
binding energy per nucleon and the Fermi momentum in nuclear matter in all of
the phenomenological models[14, 24]. The relationship between $R_{p}$ and
$R_{c}$ is unambiguously defined theoretically[8], and the latter is observed
experimentally through electromagnetic probes, whose reaction mechanism are
well understood[6, 7, 9]. Hence, the 0.1 fm problem is due to the difference
between the values of $R_{n}$ in the two frameworks.

It is shown that the values of $R_{\tau}$ are dominated by those of
$(-m^{\ast}_{\tau}V_{\tau})^{-1/4}$, as in Eq.(63), where $m^{\ast}_{\tau}$
and $V_{\tau}$ represent the effective mass in units of $M$ and the strength
of the one-body potential near the center of the nucleus($r\approx 0$),
respectively, and the subscript indicates $\tau=p$ for protons and $\tau=n$
for neutrons. Although $m^{\ast}_{\tau}$ and $V_{\tau}$ are complicated
functions of the interaction parameters in the phenomenological models, they
are not independent of each other. Their variations are constrained together
with the nucleon density $\rho_{\tau}$ at $r\approx 0$ by the Hugenholtz-Van
Hove(HVH) theorem[19, 20, 21].

In writing the average values of $m^{\ast}_{\tau}$ and $V_{\tau}$ in each
framework as $\langle m^{\ast}_{\tau}\rangle$ and $\langle V_{\tau}\rangle$,
respectively, their product is approximately expressed by the HVH equation as
$\langle m^{\ast}_{\tau}\rangle\langle V_{\tau}\rangle\approx
a_{\tau}+b_{\tau}\langle m^{\ast}_{\tau}\rangle$, where $a_{\tau}$ and
$b_{\tau}$ are constants. The values of $a_{\tau}$ and $b_{\tau}$ depend on
the average values of $\rho_{\tau}$($\langle\rho_{\tau}\rangle$), the binding
energy per nucleon $E_{\rm B}$ and Coulomb energy $v_{c}$ of the corresponding
asymmetric nuclear matter with $N$ and $Z$. Since the values of $E_{\rm B}$
and $v_{c}$ are almost the same in the relativistic and non-relativistic
models, the difference between the two frameworks in the right-hand side of
the HVH equation is attributed to the difference between the values of
$\langle\rho_{\tau}\rangle$ and $\langle m^{\ast}_{\tau}\rangle$. Indeed, the
values of $\langle\rho_{\tau}\rangle$ and $\langle m^{\ast}_{\tau}\rangle$ in
the nuclear matter in Table 2 are almost the same as those for 208Pb in Table
5 and 6. The difference of the right-hand side of the HVH equation for the two
frameworks is expressed by $\langle m^{\ast}_{\tau}\rangle\langle
V_{\tau}\rangle$ in the left-hand side, which induces the difference of
$R_{n}$ between the relativistic and non-relativistic models, according to
Eq.(63).

Table 10 provides their average values as $(\langle m^{\ast}_{n}\rangle\langle
V_{n}\rangle)_{\rm non}=-46.075$ MeV for the non-relativistic models against
$(\langle m^{\ast}_{n}\rangle\langle V_{n}\rangle)_{\rm rel}=-41.956$ MeV for
the relativistic models. The ratio of these values yields

$(46.075/41.956)^{1/4}=1.0237,$

which is comparable to the value showing the 0.1 fm difference of $R_{n}$ as

$R_{n,{\rm rel}}/R_{n,{\rm non}}=5.740/5.621=1.0212$

in Table7. This comparison assumes the same relationship between the average
values of $R_{n}$ and $(-m^{\ast}_{n}V_{n})^{-1/4}$, as in Eq.(63). The
results of the more detailed analysis without using the average values have
been summarized in Table 12, which shows that about $70\%$ of the 0.1 fm
difference is explained according to the HVH theorem.

We note that the 0.1 fm problem is observed using the limited number of the
Skyrme-type interactions and the relativistic mean-field models in Ref.[4], so
that the problem has been investigated within the same models in the present
paper. It may be interesting to explore in other phenomenological models[2]
whether or not there is a similar difference problem and the HVH theorem is
useful for understanding the difference.

The 0.1 fm difference has been observed in 48Ca also in Ref.[4]. It would be
discussed in a similar way as for 208Pb in the present paper, but a new method
must be devised for comparing the results for 48Ca with those for nuclear
matter in the mean-field models.

## Acknowledgment

The authors would like to thank Professor T. Suda for useful discussions.

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