# Pauli nonlocality and the nucleon effective mass

Dao T. Khoa1 Doan Thi Loan1 Nguyen Hoang Phuc2 1 Institute for Nuclear Science
and Technology, VINATOM, 179 Hoang Quoc Viet, Hanoi 122772, Vietnam.
2 Phenikaa Institute for Advanced Study (PIAS), Phenikaa University, Hanoi
12116, Vietnam.

###### Abstract

A study of the nucleon mean-field potential in nuclear matter (NM) is done
within an extended Hartree-Fock (HF) formalism, using the CDM3Y6 density
dependent version of the M3Y interaction which is associated with the nuclear
incompressibility $K\simeq 252$ MeV. The momentum dependence of nucleon
optical potential (OP) in NM at the saturation density $\rho_{0}$ is shown to
be due mainly to its exchange term up to $k\approx 2$ fm-1, so that the Pauli
nonlocality is expected to be the main origin of the nucleon effective mass at
low momenta. Because nucleons in neutron-rich NM at $\rho\approx\rho_{0}$ are
either weakly bound or unbound by the in-medium nucleon-nucleon interaction,
the determination of the effective mass of nucleon scattered on targets with
neutron excess at low energies should be of interest for the mean-field
studies of neutron star matter. For this purpose, the folding model is used to
calculate the nonlocal nucleon OP for the optical model analysis of elastic
nucleon scattering on 40,48Ca, 90Zr, and 208Pb targets at energies $E<50$ MeV,
to probe the model reliability and validate the WKB local approximation to
obtain the local folded nucleon OP. The nucleon effective mass $m^{*}$ is then
carefully deduced from the momentum dependence of the local folded nucleon OP
which is resulted from the Pauli nonlocality of the exchange term. The
obtained $m^{*}$ values agree well with the nucleon effective mass given by
the extended HF calculation of the single-particle potential in asymmetric NM.
The neutron-proton effective mass splitting determined at
$\rho\approx\rho_{0}$ from the central strength of the real folded nucleon OP
for 48Ca, 90Zr, and 208Pb targets has been found to depend linearly on the
neutron-proton asymmetry parameter as $m^{*}_{n-p}\approx(0.167\pm
0.018)\delta$, in a good agreement with the recent empirical constraints.

##  I Introduction

Over a wide range of the single-particle energies, the nucleon motion in
medium is overwhelmingly governed by the nuclear mean field, known as the
shell-model potential for bound states and optical potential (OP) for
scattering states. The single-particle potential is also a key quantity in the
mean-field studies of the equation of state (EOS) of nuclear matter (NM) as
well as the structure of finite nuclei [1, 2]. The nucleon OP was widely
studied in the Brueckner-Hartree-Fock (BHF) calculations of NM using free
nucleon-nucleon (NN) interaction [2, 3, 4, 5], and in the mean-field
calculations of NM on the Hartree-Fock (HF) level using different choices of
the effective NN interaction [6, 7, 8, 9, 10, 11]. The mean-field description
of nucleon OP in the NM limit provide an important physics input for the
microscopic models of nucleon OP of finite nuclei, in particular, different
versions of the folding model [11, 12, 13, 14, 15].

The microscopic studies of NM have repeatedly shown the impact by the Pauli
blocking and increasing strength of higher-order NN correlations at high
densities [2]. These in-medium effects are considered as the physics origin of
the density dependence introduced into various effective NN interactions used
in the nuclear structure and reaction studies. In the present work, we employ
the CDM3Y6 density dependent version [16] of the M3Y interaction [20] which
was used successfully in the HF studies of NM [6, 7, 21, 22, 23], and in the
folding model calculation of the nucleon and nucleus-nucleus OP at low and
medium energies [16, 13, 14, 24, 25, 26, 11, 17]. The CDM3Yn density
dependence (n=1-6) of the original M3Y interaction [20] was first parametrized
in Ref. [16] to reproduce the saturation of symmetric NM on the HF level. The
CDM3Y3 and CDM3Y6 versions were later modified to take into account the
rearrangement term (RT) of the single-particle potential in NM into the
folding calculation of the nucleon-nucleus and nucleus-nucleus OP [11, 17],
based on the Hugenholtz and van Hove (HvH) theorem [27, 28] which is exact for
all interacting Fermi systems independently of the interaction between
fermions.

When the antisymmetrization of the nucleon-nucleus system is taken explicitly
into account, the exchange term of the folded nucleon-nucleus potential
becomes nonlocal in coordinate space [12, 15], and a nonlocal folding model of
nucleon OP using the CDM3Yn interaction was suggested in Ref. [29], with the
RT properly taken into account. The calculable $R$-matrix method [30, 31] is
used to solve the OM equation with the nonlocal kernel of the folded nucleon
OP. This method was well tested in the OM analysis of elastic nucleon
scattering at energies up to 40 MeV [32], using the phenomenological nonlocal
nucleon OP [33, 34, 35, 36].

The momentum dependence of nucleon OP at the NM saturation density $\rho_{0}$
is due mainly to the exchange term up to $k\approx 2$ fm-1, and the
antisymmetrization of the nucleon-nucleus system is, therefore, the main
origin of the momentum dependence of nucleon OP at low momenta. As a result,
it becomes possible to determine the nucleon effective mass from the momentum
dependence of the folded nucleon OP of finite nuclei, based on the WKB local
approximation, in the OM analysis of low-energy elastic nucleon scattering on
target nuclei with neutron excess. The present work is our first attempt to
determine the radial- and isospin dependence of the nucleon effective mass
near the saturation density ($\rho\lesssim\rho_{0}$) from the radial strengths
of the real folded nucleon OP. The obtained results for $m^{*}$ allowed us to
determine the neutron-proton effective mass splitting $\Delta m^{*}$ at
$\rho\approx\rho_{0}$ for 48Ca, 90Zr, and 208Pb targets, which is
complementary to the $\Delta m^{*}$ values obtained in different mean-field
calculations of asymmetric NM.

The knowledge about the nucleon effective mass is of wide importance for
different nuclear physics and nuclear astrophysics studies [37, 38, 39, 40,
41]. In particular, the direct link of the nucleon effective mass to the
density dependence of nuclear symmetry energy [5, 42, 43], the liquid-gas
phase transition in the neutron-rich NM, and the temperature profile of hot
protoneutron stars [44] and neutrino emission therefrom [45].

##  II Single-particle potential in nuclear matter

An effective density-dependent NN interaction is the essential input for the
mean-field calculation of NM, and we recall briefly the CDM3Yn density
dependent versions of the M3Y interaction used in our model. Originally,
parameters of the CDM3Yn density dependence were parametrized [16] to
reproduce the NM saturation properties on the HF level. These parameter sets
were extended later to include a realistic isovector part [23, 11, 14] as well
as the rearrangement term of the single-particle potential [11]. The HF
results for the energy per nucleon $E/A$ of NM obtained with the CDM3Yn
interaction are compared in Fig. 1 with results of the _ab initio_ variational
calculation using the Argonne V18 interaction [46]. One can see a nice
agreement of the HF results with those of the ab initio calculation over a
wide range of densities.

Figure 1: Energy per nucleon of the symmetric NM (a) and pure neutron matter
(b) given by the HF calculation using the density- and isospin dependent
CDM3Yn interaction. $K$ is the nuclear incompressibility obtained at the
saturation density $\rho_{0}\approx 0.16$ fm-3. The circles are results of the
ab initio variational calculation by Akmal, Pandharipande and Ravenhall (APR)
[46]. Arrow marks twice the saturation density, indicating the validity of the
CDM3Yn interaction for the low-energy domain of nuclear EOS.

We note that the functional CDM3Yn density dependence of the original M3Y
interaction [20] was parametrized [16] to reproduce on the HF level the
saturation of symmetric NM at $\rho_{0}$, as well as to give a realistic
(real) nucleus-nucleus OP within the double-folding model that properly
describes the nuclear rainbow pattern observed in elastic $\alpha$-nucleus and
light heavy-ion refractive scattering. The rainbow scattering data were proven
to be sensitive to the real OP at small distances where the dinuclear overlap
density is reaching up to twice the saturation density $\rho_{0}$ [24], so
that the use of the CDM3Yn interaction is well validated for the low-energy
domain of nuclear EOS (see Fig. 1).

According to Landau theory for an infinite system of interacting fermions
[47], the single-particle energy is determined [11] as derivative of the
energy per nucleon $\varepsilon\equiv E/A$ of NM with respect to the nucleon
momentum distribution $n_{\tau}(k)$

$E_{\tau}(\rho,k)=\frac{\partial\varepsilon}{\partial
n_{\tau}(k)}=\frac{\hbar^{2}k^{2}}{2m_{\tau}}+U_{\tau}(\rho,k),\ {\rm where}\
\tau=n,p.$ (1)

$E_{\tau}(\rho,k)$ is, thus, the change of the NM energy at density $\rho$
caused by the removal or addition of a nucleon with the momentum $k$. The
single-particle potential $U_{\tau}(\rho,k)$ consists of both the HF and
rearrangement terms

$U_{\tau}(\rho,k)=U^{\rm(HF)}_{\tau}(\rho,k)+U^{\rm(RT)}_{\tau}(\rho,k),$ (2)

with the explicit expressions of $U_{\tau}(k)$ given in Ref. [11]. At the
Fermi momentum $(k\to k_{F\tau}),\ E_{\tau}(k_{F\tau})$ determined from Eqs.
(1)-(2) is exactly the Fermi energy given by the Hugenholtz-van Hove (HvH)
theorem [27], which is satisfied on the HF level only when the effective NN
interaction is density independent, with the RT equal zero [11, 48]. In the
mean-field calculation (1)-(2), the RT originates naturally from the density
dependence of the in-medium NN interaction that implicitly accounts for the
higher-order NN correlations as well as three-body force [1, 4, 5].

For the spin-saturated NM, the (spin-independent) direct (D) and exchange (EX)
parts of the CDM3Yn interaction [16, 14] are used in the HF calculation of NM

$v_{\rm D(EX)}(\rho,s)=F_{0}(\rho)v^{\rm D(EX)}_{00}(s)+F_{1}(\rho)v^{\rm
D(EX)}_{01}(s)(\mbox{\boldmath$\tau$}_{1}\cdot\mbox{\boldmath$\tau$}_{2}).$
(3)

The radial dependence of the isoscalar (IS) and isovector (IV) terms $v^{\rm
D(EX)}_{00(01)}(s)$ is kept unchanged as that used for the original M3Y
interaction [20]. The IS density dependence $F_{0}(\rho)$ was determined [16]
to reproduce the saturation of NM on the HF level. The IV density dependence
$F_{1}(\rho)$ was determined [11] based on the isospin dependence of nucleon
OP given by the BHF calculation of NM by Jeukenne, Lejeune and Mahaux (JLM)
[49], and and fine tuned by the folding model description of the charge
exchange reaction to the isobar analog states in medium-mass nuclei [26].

Using the exact expression of the RT given by the HvH theorem, a compact
method was suggested [11] to account for the RT on the HF level by adding a
correction term to the CDM3Yn density dependence, $F_{0(1)}(\rho)\to
F_{0(1)}(\rho)+\Delta F_{0(1)}(\rho)$, used in the HF-type calculation of the
single-particle potential

$U_{\tau}(\rho,k)=\sum_{k^{\prime}\sigma^{\prime}\tau^{\prime}}\langle\mbox{\boldmath$k$}\sigma\tau,\mbox{\boldmath$k$}^{\prime}\sigma^{\prime}\tau^{\prime}|v_{\rm
D}|\mbox{\boldmath$k$}\sigma\tau,\mbox{\boldmath$k$}^{\prime}\sigma^{\prime}\tau^{\prime}\rangle+\langle\mbox{\boldmath$k$}\sigma\tau,\mbox{\boldmath$k$}^{\prime}\sigma^{\prime}\tau^{\prime}|v_{\rm
EX}|\mbox{\boldmath$k$}^{\prime}\sigma\tau,\mbox{\boldmath$k$}\sigma^{\prime}\tau^{\prime}\rangle,$
(4)

where $|\mbox{\boldmath$k$}\sigma\tau\rangle$ are the ordinary plane waves.
Treating explicitly the isospin dependence, the single-particle potential (4)
is expressed [11] in terms of the IS and IV parts as

$\displaystyle U_{\tau}(\rho,k)$ $\displaystyle=$ $\displaystyle
U_{0}^{\rm(HF)}(\rho,k)+U_{0}^{\rm(RT)}(\rho,k)\pm[U_{1}^{\rm(HF)}(\rho,k)+U_{1}^{\rm(RT)}(\rho,k)]$
(5) $\displaystyle=$ $\displaystyle[F_{0}(\rho)+\Delta
F_{0}(\rho)]U^{\rm(M3Y)}_{0}(\rho,k)\pm[F_{1}(\rho)\pm\Delta
F_{1}(\rho)]U^{\rm(M3Y)}_{1}(\rho,k),$

where (-) sign pertains to proton and (+) sign to neutron. $U^{\rm(M3Y)}_{0}$
and $U^{\rm(M3Y)}_{1}$ are, respectively, the IS and IV parts of the single-
particle potential given by the density independent M3Y interaction

$U^{\rm(M3Y)}_{0(1)}(\rho,k)=\left[J^{D}_{0(1)}+\int\hat{j}_{1}(k_{F}r)j_{0}(kr)v^{\rm
EX}_{00(01)}(r)d^{3}r\right],$ (6) ${\rm where}\ J^{\rm D}_{0(1)}=\int
v_{00(01)}^{\rm D}(r)d^{3}r,\ \hat{j_{1}}(x)=3j_{1}(x)/x\ =\ 3(\sin x-x\cos
x)/x^{3}.$ (7)

Because the original M3Y interaction is momentum independent, the momentum
dependence of the single-particle potential (5) is due to the exchange term of
$U^{\rm(M3Y)}_{0(1)}(\rho,k)$, with the nucleon momentum $k$ determined self-
consistently as

$k=\sqrt{\frac{2m_{\tau}}{\hbar^{2}}[E_{\tau}(\rho,k)-U_{\tau}(\rho,k)]}.$ (8)

Figure 2: Single-particle potential in symmetric NM (8) determined at
$\rho\approx\rho_{0}$ with and without the RT using the CDM3Y6 interaction, in
comparison with the empirical data for nucleon OP taken from Refs. [51]
(circles), [52] (squares), and [53] (triangles).

The extended HF approach (4)-(8) provides a consistent description of both the
single-particle potential for bound nucleons, $U_{\tau}(\rho,k)$ with
$k<k_{F\tau}$, and nucleon optical potential, $U_{\tau}(\rho,k)$ with
$k>k_{F\tau}$. Such an approach is the well-known continuous approximation for
the single-particle potential [1, 50], where nucleon OP in NM is determined as
the mean-field potential felt by nucleon being scattered on NM at energy
$E>0$. The momentum of scattered nucleon is determined by the same relation
(8) but with $E_{\tau}$ replaced by the nucleon incident energy $E$. In this
way, the energy- and momentum dependence of nucleon OP are treated on the same
footing as illustrated in Fig. 2. Therefore, an important constraint for the
present approach is that at $E>0$ the energy dependence of the potential (5)
should agree reasonably with the observed energy dependence of nucleon OP. The
single-particle potential (5) evaluated for symmetric NM at the saturation
density $\rho_{0}$ using the CDM3Y6 interaction is compared with the empirical
data [51, 52, 53] for nucleon OP at $E>0$ in Fig. 2. One can see that the
inclusion of the RT significantly improves the agreement with the empirical
data at energies $E<50$ MeV. Moreover, the momentum dependence of nucleon OP
in this low-energy region is found nearly linear and due mainly to the Pauli
exchange term in Eq. (4).

It should be noted here that the Pauli exchange term in Eq. (4) gives a good
agreement of the energy (or momentum) dependence of nucleon OP with the
empirical data at low energies only. At higher energies ($E>50$ MeV or $k>2$
fm-1) the agreement becomes worse, and nucleon OP (5) is more attractive
compared to the empirical data (see, e.g., Fig. 3 in Ref. [17]). Such a
behavior of nucleon OP given by the HF calculation is well expected in light
of the effective G-matrix interaction derived from the solution of
Brueckner–Bethe–Goldstone equation [12], where both the _direct_ and
_exchange_ parts of the G matrix interaction are energy dependent. This is
also the reason why a slight linear energy dependence was added to the CDM3Y6
interaction, in terms of $g(E)$ factor [16], for the double-folding
calculation of nucleus-nucleus OP at medium energies. Consequently, the
nucleon effective mass $m^{*}$ at high momenta originates not only from the
Pauli nonlocality, but also from the higher-order NN correlations in nuclear
medium. The present work is focused, however, on the determination of $m^{*}$
at low energies ($E<50$ MeV) and corresponding momenta $k<2$ fm-1 from the
momentum dependence of nucleon OP, which is due entirely to the Pauli
nonlocality as shown above.

##  III Folding model of nucleon optical potential

###  III.1 Nonlocal folded nucleon OP

The folding model of nucleon OP [12, 13, 15] is known to generate the first-
order term of the microscopic nucleon OP defined in Feshbach’s formalism of
nuclear reactions [54]. The success of the folding approach in the OM
description of elastic nucleon-nucleus scattering at low and medium energies
confirms that the folded nucleon OP is the dominant part of the microscopic
nucleon OP. In a consistent mean-field consideration, the central OP for
elastic nucleon scattering on a target nucleus $A$ is evaluated using the same
Eq. (4), but with plane waves
$|\mbox{\boldmath$k$}^{\prime}\sigma^{\prime}\tau^{\prime}\rangle$ replaced by
the single-particle wave functions $|j\rangle$ of target nucleons

$U(k)=\sum_{j\in A}[\langle\mbox{\boldmath$k$},j|v_{\rm
D}|\mbox{\boldmath$k$},j\rangle+\langle\mbox{\boldmath$k$},j|v_{\rm
EX}|j,\mbox{\boldmath$k$}\rangle].$ (9)

The antisymmetrization of the nucleon-nucleus system is done in the HF manner,
taking into account explicitly the nucleon knock-on exchange. As a result, the
exchange term of the folded nucleon OP (9) becomes _nonlocal_ in coordinate
space [12], and the OM equation for elastic nucleon scattering at energy $E$
becomes an integro-differential equation

$\displaystyle\left[-\frac{\hbar^{2}}{2\mu}\nabla^{2}+U_{\rm D}(R)+V_{\rm
C}(R)+V_{\rm
s.o.}(R)(\mbox{\boldmath$L$}\cdot\mbox{\boldmath$\sigma$})\right]\Psi(\mbox{\boldmath$R$})$
$\displaystyle\hskip 73.97733pt+\int
K(\rho,\mbox{\boldmath$R$},\mbox{\boldmath$r$})\Psi(\mbox{\boldmath$r$})d^{3}r=E~{}\Psi(\mbox{\boldmath$R$}),$
(10)

where $V_{\rm s.o.}(R)$ is the spin-orbit potential and $V_{\rm C}(R)$ is the
Coulomb potential used for elastic proton scattering only. The scattering wave
function $\Psi(\mbox{\boldmath$R$})$ is obtained from the solution of the OM
equation (10) at each nucleon-nucleus distance $R$. Based on the CDM3Yn
interaction (3) with the RT included, the mean-field part of nucleon OP
consists of the local direct potential $U_{\rm D}(R)$ and the exchange
integral involving a nonlocal density-dependent kernel
$K(\rho,\mbox{\boldmath$R$},\mbox{\boldmath$r$})$. Making explicit the isospin
degrees of freedom, the mean-field part of nucleon OP can be expressed (in the
Lane manner) in terms of the IS and IV components as

$\displaystyle U_{\rm D}(R)=U^{\rm D}_{\rm IS}(R)\pm U^{\rm D}_{\rm IV}(R),$
$\displaystyle K(\rho,\mbox{\boldmath$R$},\mbox{\boldmath$r$})=K_{\rm
IS}(\rho,\mbox{\boldmath$R$},\mbox{\boldmath$r$})\pm K_{\rm
IV}(\rho,\mbox{\boldmath$R$},\mbox{\boldmath$r$}),$ (11)

where (-) sign pertains to proton OP and (+) sign to neutron OP. The IS and IV
parts of the direct folded potential and nonlocal exchange kernel (11) are
given through the IS and IV nucleon density matrices, respectively, as

$\displaystyle U^{\rm D}_{\rm
IS(IV)}(R)=\int\big{[}\rho_{n}(\mbox{\boldmath$r$})\pm\rho_{p}(\mbox{\boldmath$r$})\big{]}v^{\rm
D}_{00(01)}(\rho,s)d^{3}r,$ $\displaystyle K_{\rm
IS(IV)}(\rho,\mbox{\boldmath$R$},\mbox{\boldmath$r$})=\big{[}\rho_{n}(\mbox{\boldmath$R$},\mbox{\boldmath$r$})\pm\rho_{p}(\mbox{\boldmath$R$},\mbox{\boldmath$r$})\big{]}v^{\rm
EX}_{00(01)}(\rho,s),$ (12)

where $s=|\mbox{\boldmath$R$}-\mbox{\boldmath$r$}|$. The nucleon density
matrix is determined from the single-particle wave functions
$\varphi^{(\tau)}_{j}$ of target nucleons as

$\rho_{\tau}(\mbox{\boldmath$R$},\mbox{\boldmath$r$})=\sum_{j\in
A}\varphi^{(\tau)*}_{j}(\mbox{\boldmath$R$})\varphi^{(\tau)}_{j}(\mbox{\boldmath$r$}),\
{\rm with}\
\rho_{\tau}(\mbox{\boldmath$r$})\equiv\rho_{\tau}(\mbox{\boldmath$r$},\mbox{\boldmath$r$})\
{\rm and}\ j\equiv nlj.$ (13)

The direct potential $U_{\rm D}(R)$ is readily obtained by folding the nucleon
densities with the direct part $v^{\rm D}_{00(01)}(\rho,s)$ of the density
dependent CDM3Yn interaction (3), including the contribution of the RT, as

$U^{\rm D}_{\rm
IS(IV)}(R)=\int\big{[}\rho_{n}(\mbox{\boldmath$r$})\pm\rho_{p}(\mbox{\boldmath$r$})\big{]}\big{[}F_{0(1)}(\rho(\mbox{\boldmath$r$}))\pm\Delta
F_{0(1)}(\rho(\mbox{\boldmath$r$}))\big{]}v^{\rm D}_{00(01)}(s)d^{3}r,$ (14)

where the $(\pm)$ signs are used in the same way as in Eqs. (5) and (11), and
the contribution of the RT to the IV part of the direct potential $U^{\rm
D}_{\rm IV}$ via $\Delta F_{1}(\rho)$ is the same for both the proton and
neutron OP. The exact treatment of the nonlocal exchange term is cumbersome
and involves the explicit angular-momentum dependence of the exchange kernel.
Using the multipole decomposition of the radial Yukawa function of the
exchange part of the CDM3Yn interaction (3)

$v^{\rm
EX}_{00(01)}(s)=\sum_{\lambda\mu}\frac{4\pi}{2\lambda+1}X^{(\lambda)}_{00(01)}(R,r)Y^{*}_{\lambda\mu}(\hat{\mbox{\boldmath$R$}})Y_{\lambda\mu}(\hat{\mbox{\boldmath$r$}}),$
(15)

we obtain, after integrating out the angular dependence, the radial equation
for each partial wave

$\displaystyle-\frac{\hbar^{2}}{2\mu}\left[\frac{d^{2}}{dR^{2}}-\frac{L(L+1)}{R^{2}}\right]\chi_{LJ}(R)+\big{[}U_{\rm
D}(R)+V_{\rm C}(R)$ $\displaystyle+A_{LJ}V_{\rm
s.o.}(R)\big{]}\chi_{LJ}(R)+\int_{0}^{\infty}K_{LJ}(\rho,R,r)\chi_{LJ}(r)dr=E~{}\chi_{LJ}(R),$
(16)

where $\chi_{LJ}(R)/R$ is the radial part of nucleon scattering wave function
$\Psi(\mbox{\boldmath$R$})$, $A_{LJ}$ is the s.o. coupling coefficient
determined as $A_{LJ}=L$ if $J=L+1/2$, and $A_{LJ}=-L-1$ if $J=L-1/2$. The
nonlocal density-dependent exchange kernel is then obtained at each partial
wave as

$\displaystyle K_{LJ}(\rho,R,r)$ $\displaystyle=$ $\displaystyle\big{[}K^{\rm
IS}_{LJ}(\rho,R,r)\pm K^{\rm IV}_{LJ}(\rho,R,r)\big{]},$ (17) $\displaystyle
K^{\rm IS}_{LJ}(\rho,R,r)$ $\displaystyle=$
$\displaystyle\big{[}F_{0}(\rho(r))+\Delta
F_{0}(\rho(r))\big{]}\sum_{nlj,\lambda}\left[u^{(n)}_{nlj}(R)u^{(n)}_{nlj}(r)+u^{(p)}_{nlj}(R)u^{(p)}_{nlj}(r)\right]$
(24)
$\displaystyle\times(2j+1)X^{(\lambda)}_{00}(R,r)\left(\begin{array}[]{l}L\\\
0\end{array}\begin{array}[]{l}l\\\ 0\end{array}\begin{array}[]{l}\lambda\\\
0\end{array}\right)^{2},$ $\displaystyle K^{\rm IV}_{LJ}(\rho,R,r)$
$\displaystyle=$ $\displaystyle\big{[}F_{1}(\rho(r))\pm\Delta
F_{1}(\rho(r))\big{]}\sum_{nlj,\lambda}\left[u^{(n)}_{nlj}(R)u^{(n)}_{nlj}(r)-u^{(p)}_{nlj}(R)u^{(p)}_{nlj}(r)\right]$
(31) $\displaystyle\
\times(2j+1)X^{(\lambda)}_{01}(R,r)\left(\begin{array}[]{l}L\\\
0\end{array}\begin{array}[]{l}l\\\ 0\end{array}\begin{array}[]{l}\lambda\\\
0\end{array}\right)^{2}.$

Here $u^{(\tau)}_{nlj}(r)/r$ is the radial part of the single-particle wave
function $\varphi^{(\tau)}_{nlj}(\mbox{\boldmath$r$})$ of target nucleon. The
(-) sign is used with proton OP and (+) sign with neutron OP, and the
contribution of the RT to the IV part of the exchange kernel is also the same
for both proton and neutron OP as found for the IV part of the direct
potential (14). The explicit representation of nucleon OP in terms of the IS
and IV parts should be helpful in revealing the contribution of valence
neutrons to the total OP. Furthermore, the charge exchange form factor (FF) of
the $(p,n)$ reaction to the isobar analog state (IAS) is determined, in the
Lane isospin coupling scheme, entirely by the IV part of nucleon OP [26, 55].
Therefore, the present nonlocal folding model can also be used to calculate
the nonlocal charge exchange FF of the Fermi transition to IAS in the folding
model study of $(p,n)$IAS reaction.

###  III.2 Local approximation for the folded nucleon OP

Although the nucleon OP is well established to be nonlocal in the coordinate
space due to the Pauli blocking (as discussed above) and nonelastic channel
coupling, the nucleon OP of some local form is widely used in numerous OM
analyses of elastic nucleon-nucleus scattering. The obtained elastic
scattering wave function (dubbed as the distorted wave) is then used as a key
input in different DWBA or coupled-channel calculations of direct nuclear
reactions. We briefly discuss here the validity of the local folding model
[13, 14] of nucleon OP, which has been successfully used to study elastic
nucleon scattering.

Applying a local WKB approximation [18, 19] for the change in the scattering
wave function in the OM equation (10) induced by the exchange of spatial
coordinates of scattered nucleon and target nucleon, we obtain

$\Psi(\mbox{\boldmath$r$})=\Psi(\mbox{\boldmath$R+s$})\simeq\Psi(\mbox{\boldmath$R$})\exp\big{(}i{\mbox{\boldmath$k$}}(E,\mbox{\boldmath$R$})\cdot{\mbox{\boldmath$s$}}\big{)},$
(32)

where the local momentum $k(E,R)$ of scattered nucleon is energy- and radial
dependent. Inserting Eq. (32) into Eq. (10), the nonlocal exchange integral
becomes _localized_ and _energy (momentum) dependent_. The local exchange term
of the folded nucleon OP can then be evaluated separately using the nonlocal
nucleon density matrix

$\displaystyle U_{\rm EX}(E,R,k)$ $\displaystyle=$ $\displaystyle U^{\rm
EX}_{\rm IS}(E,R,k)\pm U^{\rm EX}_{\rm IV}(E,R,k),$ $\displaystyle U^{\rm
EX}_{\rm IS(IV)}(E,R,k)$ $\displaystyle=$
$\displaystyle\int\big{[}\rho_{n}(\mbox{\boldmath$R$},\mbox{\boldmath$r$})\pm\rho_{p}(\mbox{\boldmath$R$},\mbox{\boldmath$r$})\big{]}j_{0}\big{(}k(E,R)s\big{)}v^{\rm
EX}_{00(01)}(\rho,s)d^{3}r$ (33) $\displaystyle=$
$\displaystyle\int\big{[}\rho_{n}(\mbox{\boldmath$R$},\mbox{\boldmath$r$})\pm\rho_{p}(\mbox{\boldmath$R$},\mbox{\boldmath$r$})\big{]}\big{[}F_{0(1)}(\rho(\mbox{\boldmath$r$}))\pm\Delta
F_{0(1)}(\rho(\mbox{\boldmath$r$}))\big{]}$ $\displaystyle\ \ \times
j_{0}\big{(}k(E,R)s\big{)}v^{\rm EX}_{00(01)}(s)d^{3}r,$

where the $(\pm)$ signs are used in the same way as in Eq. (11). The local
momentum of scattered nucleon is determined self-consistently from the real
folded nucleon OP as

$k^{2}(E,R)=\frac{2\mu}{\hbar^{2}}\left\\{E-{\rm Re}\left[U_{\rm D}(R)+U_{\rm
EX}(E,R,k)\right]-V_{\rm C}(R)\right\\}.$ (34)

Thus, the momentum- and energy dependences of the local folded nucleon OP of
finite nuclei are directly interrelated and also determined on the same
footing as in the NM limit (see Fig. 2). The method to evaluate the direct
(14) and exchange (33) folded potentials is given in more details in, e.g.,
Ref. [13]. Using a realistic local approximation for the nonlocal density
matrix in Eq. (33), the nuclear density $\rho(\mbox{\boldmath$r$})$ obtained
in any structure model or directly deduced from electron scattering data can
be used in the folding model calculation of the local nucleon OP.

###  III.3 Elastic nucleon scattering on 40,48Ca, 90Zr, and 208Pb targets

To validate the use of the local folded nucleon OP to deduce the nucleon
effective mass, we show here results of the folding model analysis of low-
energy elastic nucleon scattering using both the nonlocal and local folded
nucleon OP. Namely, the nonlocal and local folding models of nucleon OP
discussed above are used to calculate nucleon OP for the OM study of elastic
neutron and proton scattering on 40,48Ca, 90Zr, and 208Pb targets. Because the
nonlocal folding calculation (12)-(13) requires the explicit single-particle
wave functions of target nucleons, we have used the single-particle wave
functions given by the HF calculation of finite nuclei based upon a complete
basis of spherical Bessel functions [56] and finite-range D1S Gogny
interaction [57].

Figure 3: OM description of elastic $n+^{208}$Pb scattering data measured at
26, 30, and 40 MeV [58, 59, 60] given by the nonlocal and local folded neutron
OP’s obtained with the complex CDM3Y6 interaction [29].

To obtain a _complex_ folded nucleon OP, it is necessary to have a realistic
complex parametrization of the density dependent CDM3Yn interaction. For this
purpose, the imaginary density dependence of the CDM3Y6 interaction was
determined using the same density dependent functional $F_{0(1)}(\rho)$ as
that used for the real interaction (3), with the parameters adjusted to
reproduce, at each energy, the imaginary nucleon OP in NM given by the JLM
parametrization of the BHF results [49] on the HF level [14]. The complex
nonlocal and local folded OP’s were then used as input for the OM calculation
of elastic nucleon scattering using the calculable $R$-matrix method [30, 31].
In the present OM calculation, both the nonlocal and local folded OP’s are
supplemented by the local Coulomb and spin-orbit potentials taken from the
global systematics CH89 of nucleon OP [52].

Figure 4: The same as Fig. 3 but for elastic neutron scattering data measured
at 17 and 24 MeV [61, 62, 63] for 40,48Ca and 90Zr targets, respectively.
Figure 5: The same as Fig. 3 but for elastic $p+^{208}$Pb scattering data
measured at 30, 35, and 45 MeV [64, 65].

The reliability of the folded OP is best to be probed in the OM analysis of
elastic neutron scattering from a heavy target at low energies, where the
Coulomb interaction is absent and the mean-field dynamics is well established.
For this purpose, elastic $n+^{208}$Pb scattering data measured accurately at
energies of 26, 30.4, and 40 MeV [58, 59, 60] turned out to be a very good
test ground. Given parameters of the real CDM3Yn interaction fine-tuned by the
HF description of NM shown in Fig. 1, no renormalization of the _real_ part of
both the nonlocal and local folded OP was allowed in the present OM study to
test its proximity to the real nucleon OP. The imaginary folded OP based on
the JLM parametrization delivers a good OM description of elastic proton
scattering, but it gives a stronger absorption of neutron OP, and an overall
renormalization of the _imaginary_ part of both nonlocal and local folded
neutron OP by a factor $\sim 0.8$ was found necessary for a good OM
description of the considered neutron scattering data. From the OM results
obtained for elastic $n+^{208}$Pb scattering shown in Fig. 3 one can see that
both the nonlocal and local folded OP deliver the same good OM description of
data over the whole angular range, which validates the local approximation
(33)-(34) for the exchange term of the folded neutron OP. We note that, in the
absence of the Coulomb interaction, the (diffractive) oscillation of elastic
neutron scattering cross section at small angles can be properly reproduced
only with the inclusion of the RT into the folding calculation [29]. A good
accuracy of the local approximation for the folded OP can also be seen in the
OM results obtained for elastic neutron scattering on the medium-mass 40,48Ca
and 90Zr targets shown in Fig. 4.

Figure 6: The same as Fig. 3 but for elastic $p+^{40}$Ca scattering data
measured at 30, 35, and 45 MeV [66].

The elastic $p+^{208}$Pb scattering data measured at 30.4, 35, and 45 MeV [64,
65] are compared in Fig. 5 with the OM results given by the folded proton OP.
Both the nonlocal and local folded OP were found to give the same good OM
description of elastic $p+^{208}$Pb data at the forward and medium angles,
while the data points at backward angles are better reproduced by the nonlocal
folded OP, especially, at 45 MeV. We note that the OM results shown in Figs. 5
and 6 were obtained without renormalizing the (complex) strength of the folded
proton OP. Like the OM result obtained for elastic $p+^{208}$Pb scattering at
$E=45$ MeV, one can also see a more pronounced difference given by the
nonlocal folded proton OP at medium and large scattering angles from the
results obtained for elastic $p+^{40}$Ca scattering at energies $E\gtrsim 30$
MeV shown in Fig. 6, which indicates the need of taking into account exactly
the Pauli nonlocality of the folded proton OP at these medium energies.

##  IV Nucleon effective mass

As discussed in Sec. II, the momentum dependence of the single-particle
potential in NM at low momenta is determined mainly by the exchange term which
results on a _nonlocal_ single-nucleon potential in the coordinate space. At
positive energy ($E>0$) the single-nucleon potential (4)-(5) can be treated as
nucleon OP in NM, with the associated nucleon effective mass $m^{*}_{\tau}$
determined [1, 37] as

$\frac{m^{*}_{\tau}(\rho,\delta,k)}{m}=\Biggl{[}1+\frac{m}{\hbar^{2}k}\frac{\partial
U_{\tau}(\rho,\delta,k)}{\partial k}\Biggr{]}^{-1},\ \mbox{with}\ \tau=n,p$
(35)

where $m$ is the free nucleon mass. In general, the nucleon effective mass
arises from both the momentum- and energy dependence of the single-particle
potential [1], known as the $k$\- and $E$-effective masses, which characterize
the spacetime nonlocality. Within the time independent HF formalism [11], the
total nucleon effective mass is determined by the same relation (35), and it
is associated with the spatial nonlocality of the nucleon mean-field
potential. At the Fermi momentum ($k\to k_{F\tau}$) , the nucleon effective
mass (35) is obtained naturally from the Fermi energy (1) by the Hugenholtz-
van Hove theorem [27]. The knowledge about $m^{*}_{\tau}$ at the saturation
density $\rho_{0}$ and different neutron-proton asymmetries
$\delta=(\rho_{n}-\rho_{p})/\rho$ is essential for the determination of the
nuclear symmetry energy $E_{\rm sym}$ and its slope parameter $L$, the two key
ingredients of the EOS of neutron rich NM [41, 42].

Figure 7: Neutron- and proton effective masses (35) given by the extended HF
calculation of asymmetric NM using the CDM3Y6 interaction (dashed and dotted
lines, respectively) at $\rho=\rho_{0},\ k=k_{F\tau}$, and different neutron-
proton asymmetries $\delta$. The symbols are those obtained from the real
folded nucleon OP for finite nuclei at $\rho\approx\rho_{0}$, $k\gtrsim
k_{F\tau}$ (see Table 1).

The neutron and proton effective masses (35) obtained from the single-particle
potential (4)-(5) in asymmetric NM at the saturation density $\rho_{0}$ with
$k\to k_{F\tau}$ are shown in Fig. 7 as dashed and dotted lines, respectively.
Although the $m^{*}_{\tau}$ values are still poorly known at high NM densities
and/or large $\delta$ values, the empirical nucleon effective mass in
symmetric NM ($\delta=0$) at the saturation density $\rho_{0}$ is well
established to be around $m^{*}/m\approx 0.73$ [37]. Our extended HF
calculation of NM using the CDM3Yn interaction gives $m^{*}/m\approx 0.737$ at
$\rho\approx\rho_{0}$ and $\delta=0$ (see Fig. 9), in a good agreement with
the empirical data. The isospin dependence of the nucleon effective mass is
governed by the neutron-proton effective mass splitting

$m^{*}_{n-p}=(m^{*}_{n}-m^{*}_{p})/m,$ (36)

which is associated directly with the nuclear symmetry energy and its slope
parameter. The knowledge of $m^{*}_{n-p}$ is also of importance for the
determination of the neutron/proton ratio during stellar evolution or cooling
of protoneutron star [38, 40, 41]. Depending on the isospin dependence of the
in-medium NN interaction, one obtains very different results for the neutron-
proton effective mass splitting. A survey of different mean-field studies in
Ref. [41] shows that $m^{*}_{n-p}$ values depend linearly on the neutron-
proton asymmetry parameter $\delta$, and are ranging widely from negative- to
positive values, up to $m^{*}_{n-p}\approx 0.637~{}\delta$. An analysis of the
empirical constraints for the density dependence of nuclear symmetry energy
[38] from both the nuclear physics experiments and astrophysical observations
has led to the empirical constraint
$m^{*}_{n-p}(\rho_{0},\delta)\approx(0.27\pm 0.25)\delta$. Our extended HF
calculation of asymmetric NM using the CDM3Y6 interaction gives
$m^{*}_{n-p}(\rho_{0},\delta)\approx(0.20\pm 0.02)\delta$ at the Fermi
momentum, which is well inside the empirical boundary.

For many years, the nucleon effective mass in finite nuclei has been a focus
of different nonrelativistic and relativistic nuclear structure studies (see,
e.g., Refs. [68, 69, 70]) which provided accurate estimates of the
$m^{*}_{\tau}$ values for the single-particle states of bound nucleons (with
$k\lesssim k_{F\tau}$). Although the neutron- and proton effective masses, and
the corresponding single-particle energies and spectroscopic factors are well
described by the structure studies, it remains difficult to deduce therefrom
an explicit isospin dependence of $m^{*}_{\tau}$ or the neutron-proton
effective mass splitting (36). The method often used so far for this purpose
is to deduce the isospin dependence of the nucleon effective mass from the
isospin dependence of the phenomenological nucleon OP given by the best OM fit
to elastic nucleon scattering data at different energies [39]. In the present
work, we aim to determine the nucleon effective mass from the semi-microscopic
nucleon OP predicted by the local folding model presented in Sec. III.2. Given
a good OM description of elastic nucleon scattering at low energies by the
local folded nucleon OP shown Sec. III.3, it is reasonable to determine the
effective mass of nucleon scattered by the mean field of target from the
momentum dependence of the real folded OP as

$\displaystyle\frac{m^{*}_{\tau}(E,R,k)}{m}$ $\displaystyle=$
$\displaystyle\left\\{1+\frac{m}{\hbar^{2}k}\left[\frac{\partial
V_{\tau}(E,R,k)}{\partial k}\right]\right\\}^{-1},$ (37)
$\displaystyle\mbox{where}\ V_{\tau}(E,R,k)$ $\displaystyle=$
$\displaystyle{\rm Re}\left[U^{(\tau)}_{\rm D}(R)+U^{(\tau)}_{\rm
EX}(E,R,k)\right].$ (38)

Here $U^{(\tau)}_{\rm D(EX)}$ are the direct and exchange parts of the local
neutron (or proton) folded OP determined from Eqs. (14) and (33),
respectively. The energy or momentum dependence of the folded nucleon OP (38)
is embedded in the exchange term only, which was shown in Sec. III.2 to be
resulted from the Pauli nonlocality of the folded nucleon OP. The in-medium
momentum $k$ of scattered nucleon is determined self-consistently from the
real folded nucleon OP by Eq. (34).

Figure 8: (a) In-medium momentum (34) of scattered neutron and the Fermi
momentum $k_{Fn}$ extracted from the neutron ground-state density of 208Pb.
(b) The radial dependence of the neutron effective mass (38) obtained from the
real folded OP at energies of 1 to 45 MeV for 208Pb target.

It is obvious from Eq. (34) that the momentum $k$ of scattered nucleon depends
explicitly on the nucleon-nucleus distance $R$, and the nucleon effective mass
(38) is, therefore, also radial dependent. The in-medium momentum of scattered
neutron determined from the real folded $n+^{208}$Pb OP at energies of
$E=1\sim 45$ MeV is shown in panel (a) of Fig. 8, and one can see that at each
energy the neutron momentum $k$ changes gradually from its maximum of about
$1.6\sim 2$ fm-1 in the center to $0.2\sim 1.5$ fm-1 at the surface, lying
above the corresponding Fermi momentum. Over the same radial range, the
neutron effective mass $m^{*}_{n}/m$ is changing from about $0.75\sim 0.78$ to
unity at the surface. Such a radial dependence of $m^{*}$ is similar to that
found in the nuclear structure studies [68, 69, 70]. However, the latter is
usually enhanced to above unity at the surface for the bound single-particle
states lying close to the Fermi level.

Figure 9: (a) Neutron and total ground-state densities of 208Pb given by the
HF calculation [56] using the finite-range D1S Gogny interaction [57]. (b) The
radial shape of the real (local) folded $n+^{208}$Pb OP obtained at energies
of 1 to 45 MeV.

From the radial dependence of the neutron effective mass and the neutron- and
total ground-state densities of 208Pb shown in panel (a) of Fig. 9, it is
straightforward to infer the density dependence of the effective mass of
neutron scattered off 208Pb target at different energies, over the density
range $0\lesssim\rho\lesssim\rho_{0}$. One can see in panel (a) of Fig. 9 that
the average total density in the center of 208Pb target is
$\bar{\rho}\approx\rho_{0}$, and the neutron- and proton effective masses
determined at distances $0\lesssim R\lesssim 3$ fm (shown in panels (b) of
Figs. 9 and 10) can represent, therefore, the corresponding $m^{*}_{\tau}/m$
values in asymmetric NM at $\rho\approx\rho_{0}$, $k\gtrsim k_{F\tau}$ and
neutron-proton asymmetry
$\bar{\delta}=(\bar{\rho_{n}}-\bar{\rho_{p}})/\bar{\rho}\approx 0.185$.
Because of the target valence neutrons, the deduced $\bar{\delta}$ value in
the target center is smaller than the total difference in neutron- and proton
numbers $(N-Z)/A$. Since our local folding model (14) and (33) predicts the
real nucleon OP at $E>0$ or $k\gtrsim k_{F\tau}$, approaching the Fermi
momentum from above, we have computed the real folded nucleon OP for 48Ca,
90Zr, and 208Pb targets at $E=0.05$ MeV (i.e., 50 keV above the Fermi level),
and deduced the neutron- and proton effective masses using Eq. (38). The
obtained results are presented in Table 1 and shown in Figs. 7 as rhombuses
and circles. One can see that the $m^{*}_{\tau}/m$ values obtained at
$\bar{\rho}\approx\rho_{0}$ and $k\gtrsim k_{F\tau}$ for these targets follow
approximately the trend given by the extended HF calculation of the single-
particle potential in NM.

Figure 10: The same as Fig. 8 but for the in-medium momentum and effective mass of scattered proton. Table 1: Neutron- and proton effective masses (38) at the average nuclear density $\bar{\rho}\approx\rho_{0}$, deduced from the local folded nucleon OP of 48Ca, 90Zr, and 208Pb targets, at distances $0\lesssim R\lesssim 3$ fm. Nucleus |  48Ca |  90Zr |  208Pb
---|---|---|---
$\bar{\rho}$ | $0.159\pm 0.003$ | $0.160\pm 0.002$ | $0.160\pm 0.001$
$\bar{\delta}$ | $0.0966\pm 0.0069$ | $0.0691\pm 0.0021$ | $0.1853\pm 0.0060$
$m^{*}_{n}/m$ | $0.7490\pm 0.0015$ | $0.7436\pm 0.0003$ | $0.7553\pm 0.0003$
$m^{*}_{p}/m$ | $0.7329\pm 0.0007$ | $0.7322\pm 0.0001$ | $0.7241\pm 0.0003$
$m^{*}_{n-p}/\bar{\delta}$ | $0.167\pm 0.035$ | $0.165\pm 0.011$ | $0.168\pm 0.009$

It is interesting that the neutron-proton effective mass splitting (36)
obtained from the $m^{*}_{\tau}/m$ values given in Table 1 also depends
linearly on the asymmetry parameter,
$m^{*}_{n-p}(\rho_{0},\delta)\approx(0.167\pm 0.018)\delta$, which is within
the empirical boundary of $m^{*}_{n-p}(\rho_{0},\delta)\approx(0.27\pm
0.25)\delta$ deduced from the terrestrial nuclear physics experiments and
astrophysical observations [38]. In neutron-rich NM, nucleons at $\rho_{0}$
are _not_ bound by the in-medium NN interaction (see, e.g., Fig 1 in Ref.
[11]), and the results obtained above for $m^{*}_{\tau}$ at $E>0$ should be
appropriate for nucleons in the outer core of neutron star which are bound by
gravitation only. We note further that the $m^{*}_{n-p}$ value estimated from
the folded nucleon OP of finite nuclei is slightly lower than that given by
nucleon OP in NM using the same interaction, $m^{*}_{n-p}\approx(0.20\pm
0.02)\delta$, and the value $m^{*}_{n-p}\approx(0.41\pm 0.15)\delta$ estimated
from the phenomenological nucleon OP [40], based on the extensive OM analysis
of elastic nucleon scattering.

We note finally that the nucleon effective mass is given entirely by the
momentum dependence of the exchange term of the folded nucleon OP, so that the
nucleon effective masses presented in Table 1 and shown in Fig. 7 originated
solely from the spacial Pauli nonlocality of nucleon OP at low energies. We
have considered the energy region $0<E<50$ MeV, and found that the nucleon
effective mass (38) depends weakly on the energy. For example,
$m^{*}_{n}/m\approx 0.7553+0.0004E$ and $m^{*}_{p}/m\approx 0.7241+0.0005E$
for 208Pb target, and this result agrees fairly with the empirical energy
dependence of about $0.0007E$ established for the isoscalar $k$ effective mass
of nucleons lying above the Fermi level [37].

##  V Summary

The generalized folding model of the nonlocal nucleon OP, with the exchange
potential calculated exactly in the HF manner and rearrangement term properly
included, has been used for the OM analysis of elastic neutron and proton
scattering on 40,48Ca, 90Zr, and 208Pb targets at different energies, where
the WKB local approximation for the exchange term of the nonlocal folded
nucleon OP is validated by a consistently good OM description of the
considered elastic data.

Given the accurate local approximation for the nonlocal folded nucleon OP, a
compact method is proposed to determine the nucleon effective mass at low
momenta from the in-medium momentum dependence of the local folded nucleon OP,
which originates mainly from the Pauli nonlocality. The results obtained for
the effective mass of nucleon being scattered by the target mean-field
potential at $0\lesssim E\lesssim 45$ MeV and $k_{F\tau}\lesssim k\lesssim 2$
fm-1 seem to follow closely the isospin dependence of $m^{*}_{\tau}$ predicted
by the extended HF calculation of single-particle potential in asymmetric NM,
using the same density dependent CDM3Y6 interaction.

The neutron-proton effective mass splitting (36) given by the $m^{*}_{\tau}/m$
values obtained from the real folded nucleon OP of finite nuclei at low
energies was found to depend also linearly on the asymmetry parameter
$\delta$. At positive energies lying slightly above the Fermi level, the
$m^{*}_{n-p}$ value determined at the center of finite nuclei (with the
average density $\bar{\rho}\approx\rho_{0}$) is slightly smaller than that
obtained from the extended HF calculation of the single-particle potential of
nucleons in asymmetric NM at $\rho_{0}$, where the Pauli exchange has been
shown as the main origin of nucleon effective mass.

Because nucleons at density $\rho\approx\rho_{0}$ in the outer core of neutron
star are mainly bound by gravitation, not by the (in-medium) density dependent
NN interaction, the $m^{*}_{\tau}$ values obtained in this work from the
folded nucleon OP of finite nuclei with neutron excess could be of
complementary interest for the mean-field studies of the EOS of neutron star
matter.

## Acknowledgments

The authors are indebted to Pierre Descouvemont for his helpful discussions
and comments on the calculable $R$-matrix method [30, 31]. We also thank Bao-
An Li for his communication and suggestion on the nucleon effective mass. The
present research has been supported, in part, by the National Foundation for
Scientific and Technological Development of Vietnam (NAFOSTED Project No.
103.04-2021.74).

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