# New effective interactions for hypernuclei in density dependent relativistic
mean field model

Yu-Ting Rong  Shan-Gui Zhou<EMAIL_ADDRESS>
###### Abstract

New effective $\Lambda N$ interactions are proposed for the density dependent
relativistic mean field model. The multidimensionally constrained relativistic
mean field model is used to calculate ground state properties of eleven known
$\Lambda$ hypernuclei with $A\geq 12$ and the corresponding core nuclei. Based
on effective $NN$ interactions DD-ME2 and PKDD, the ratios $R_{\sigma}$ and
$R_{\omega}$ of scalar and vector coupling constants between $\Lambda N$ and
$NN$ interactions are determined by fitting calculated $\Lambda$ separation
energies to experimental values. We propose six new effective interactions for
$\Lambda$ hypernuclei: DD-ME2-Y1, DD-ME2-Y2, DD-ME2-Y3, PKDD-Y1, PKDD-Y2 and
PKDD-Y3 with three ways of grouping and including these eleven hypernuclei in
the fitting. It is found that the two ratios $R_{\sigma}$ and $R_{\omega}$ are
correlated well and there holds a good linear relation between them.

###### keywords:

Effective $\Lambda N$ interaction , hypernuclei , density dependent
relativistic mean field model , in-medium effect

[ITP]organization=CAS Key Laboratory of Theoretical Physics, Institute of
Theoretical Physics, Chinese Academy of
Sciences,city=Beijing,postcode=100190,country=China

[UCAS]organization=School of Physical Sciences, University of Chinese Academy
of Sciences,city=Beijing,postcode=100049,country=China

[CTNP]organization=Center of Theoretical Nuclear Physics, National Laboratory
of Heavy Ion Accelerator,city=Lanzhou,postcode=730000,country=China

[HNU]organization=Synergetic Innovation Center for Quantum Effects and
Application, Hunan Normal
University,city=Changsha,postcode=410081,country=China

##  1 Introduction

Hyperon-nucleon ($YN$) interaction is an important part of baryon-baryon
interactions [1, 2, 3, 4, 5, 6]. It is crucial for understanding the structure
of hypernuclei and properties of compact stars and vice versa, i.e., the
studies of hypernuclear structure and compact stars are crucial for
constraining the $YN$ interaction [7, 8]. Many experiments have been carried
out to study hypernuclear structure at LHC, DA$\Phi$NE, JLab, J-PARC, MAMI,
RHIC and GSI [9, 10, 11, 12, 13, 14]. The investigations of single-$\Lambda$,
double-$\Lambda$ [15], $\Xi$ hypernuclei [16, 17] and anti-hypernuclei [18]
have provided valuable information on $YN$ and $YY$ interactions. From the
theory side, the SU(3) flavor symmetry, which reveals the relation between
baryon and meson octet states, is generally used for determining the $YN$
interaction. With the constraint of SU(3) flavor symmetry or experimental
observables, many approaches, such as Lattice QCD [19, 20, 21, 22], the chiral
effective field theory [23, 24, 25], the Nijmegen soft-core model [26, 2, 27],
the Jülich hyperon-nucleon model [28, 29, 30], the Skyrme Hartree-Fock model
[31, 32, 33] and the relativistic mean-field (RMF) model [34, 35, 36, 37],
have been used to study the structure of hypernuclei based on various
(effective) $YN$ and $YY$ interactions.

In the RMF model proposed by Walecka [38], there are only linear coupling
terms in the interaction Lagrangian, i.e., mesons do not interact among
themselves. Such linear couplings lead to an improper description of the
incompressibility of nuclear matter and surface properties in finite nuclei.
To describe adequately these essential nuclear properties, either nonlinear
(NL) self-couplings of meson fields [39, 40, 41, 42, 43] or density dependent
(DD) nucleon-meson couplings [44, 45, 46, 47, 48, 49, 50] have been introduced
in the RMF model. The NL-RMF model has been extended and widely applied to the
study of hypernuclei and many effective interactions have been proposed.
However, there are much less DD-RMF effective interactions for hypernuclei
[51, 52, 53, 54, 55, 56, 57, 58]. The effective $\Lambda N$ interactions were
usually obtained by fitting the calculated $\Lambda$ separation energies to
experimental values of known hypernuclei and in the fitting procedure the RMF
calculations were carried out with the restriction of spherical symmetry. Yet
most observed hypernuclei are deformed and there are only few spherical
hypernuclei. In this letter, we propose new DD effective interactions for
hypernuclei by using a deformed RMF model—the multidimensionally constrained
(MDC) RMF model.

The paper is organized as follows. In Sec. 2, we introduce the MDC-RMF model
for $\Lambda$ hypernuclei with density dependent couplings. In Sec. 3, we
present our results and discuss $\Lambda$ separation energies and deformation
effects. Finally a summary is given in Sec. 4.

##  2 Theoretical framework

The RMF model has been very successful in describing properties of nuclear
matter and finite nuclei in the whole nuclear chart [59, 60, 61, 62, 63, 64,
65, 66, 67, 68, 69]. Both the NL-RMF and DD-RMF models have been extended to
the study of hypernuclei [70, 71, 34, 35, 72, 73, 36, 51, 52, 53, 74, 75, 76,
54, 56, 77, 55, 78, 57, 79, 58, 37]. In the DD-RMF model, the Lagrangian for a
$\Lambda$ hypernucleus is written as

$\begin{split}{\cal
L}=&\sum_{B}\bar{\psi}_{B}\left(i\gamma_{\mu}\partial^{\mu}-M_{B}-g_{\sigma
B}\sigma-g_{\sigma^{*}B}\sigma^{*}-g_{\omega
B}\gamma_{\mu}\omega^{\mu}-g_{\phi B}\gamma_{\mu}\phi^{\mu}-g_{\rho
B}\gamma_{\mu}\vec{\tau}\cdot\vec{\rho}^{\mu}-e\gamma_{\mu}\dfrac{1-\tau_{3}}{2}A^{\mu}\right)\psi_{B}\\\
&+\psi_{\Lambda}\dfrac{f_{\omega\Lambda\Lambda}}{4M_{\Lambda}}\sigma_{\mu\nu}\Omega^{\mu\nu}\psi_{\Lambda}+\dfrac{1}{2}\partial^{\mu}\sigma\partial_{\mu}\sigma-\dfrac{1}{2}m_{\sigma}^{2}\sigma^{2}+\dfrac{1}{2}\partial_{\mu}\sigma^{*}\partial^{\mu}\sigma^{*}-\dfrac{1}{2}m_{\sigma^{*}}\sigma^{*2}-\dfrac{1}{4}\Omega^{\mu\nu}\Omega_{\mu\nu}+\dfrac{1}{2}m_{\omega}^{2}\omega^{\mu}\omega_{\mu}\\\
&-\dfrac{1}{4}S^{\mu\nu}S_{\mu\nu}+\dfrac{1}{2}m_{\phi}^{2}\phi^{\mu}\phi_{\mu}-\dfrac{1}{4}\vec{R}^{\mu\nu}\vec{R}_{\mu\nu}+\dfrac{1}{2}m_{\rho}^{2}\vec{\rho}^{\mu}\vec{\rho}_{\mu}-\dfrac{1}{4}F^{\mu\nu}F_{\mu\nu},\end{split}$
(1)

where $B$ represents neutron, proton or $\Lambda$ and $M_{B}$ is the
corresponding mass. $\sigma$ and $\sigma^{*}$ are scalar-isoscalar meson
fields coupled to baryons, $\omega^{\mu}$ and $\phi^{\mu}$ are vector-
isoscalar meson fields coupled to baryons, $\vec{\rho}^{\mu}$ is the vector-
isovector meson field coupled to nucleons and $A^{\mu}$ is the photon field.
$\Omega_{\mu\nu}$, $S_{\mu\nu}$, $\vec{R}_{\mu\nu}$ and $F_{\mu\nu}$ are field
tensors of the vector mesons $\omega^{\mu}$, $\phi^{\mu}$ and
$\vec{\rho}^{\mu}$ and photons $A^{\mu}$. $m_{\sigma}$ ($g_{\sigma B}$),
$m_{\sigma^{*}}$ ($g_{\sigma^{*}B}$), $m_{\omega}$ ($g_{\omega B}$),
$m_{\phi}$ ($g_{\phi B}$) and $m_{\rho}$ ($g_{\rho B}$) are the masses
(coupling constants) for meson fields. The coupling constants are dependent on
the total baryonic density $\rho^{\upsilon}$,

$g_{mB}(\rho^{\upsilon})=g_{mB}(\rho_{\text{sat}})f_{mB}(x),\ \ \
x=\rho^{\upsilon}/\rho_{\text{sat}},$ (2)

where $m$ represents mesons and $\rho_{\text{sat}}$ is the saturation density
of nuclear matter.

Starting for the Lagrangian (1), the equations of motion can be derived via
the variational principle. The Dirac equation for baryons reads

$\displaystyle\left[\alpha\cdot
p+V_{B}+T_{B}+\Sigma_{R}+\beta(M_{B}+S_{B})\right]\psi_{iB}=\varepsilon_{i}\psi_{iB},$
(3)

and the Klein-Gordon equations for mesons and photon are

$\begin{split}(-\Delta+m_{\sigma}^{2})\sigma&=-g_{\sigma
N}\rho_{N}^{s}-g_{\sigma\Lambda}\rho_{\Lambda}^{s},\\\
(-\Delta+m_{\sigma^{*}}^{2})\sigma^{*}&=-g_{\sigma^{*}\Lambda}\rho_{\Lambda}^{s},\\\
(-\Delta+m_{\omega}^{2})\omega_{0}&=g_{\omega
N}\rho_{N}^{\upsilon}+g_{\omega\Lambda}\rho_{\Lambda}^{\upsilon}-\dfrac{f_{\omega\Lambda\Lambda}}{2M_{\Lambda}}\rho_{\Lambda}^{T},\\\
(-\Delta\phi+m_{\phi}^{2})\phi&=g_{\phi\Lambda}\rho_{\Lambda}^{\upsilon},\\\
(-\Delta+m_{\rho}^{2})\rho_{0}&=g_{\rho
N}(\rho_{n}^{\upsilon}-\rho_{p}^{\upsilon}),\\\ -\Delta
A_{0}&=e\rho_{p}^{\upsilon}.\end{split}$ (4)

The Dirac equation (3) and Klein-Gordon equations (4) are coupled via the
scalar, vector and tensor densities

$\begin{split}\rho_{B}^{s}&=\sum_{i}\bar{\psi}_{iB}\psi_{iB},\\\
\rho_{B}^{\upsilon}&=\sum_{i}\bar{\psi}_{iB}\gamma^{0}\psi_{iB},\\\
\rho_{\Lambda}^{T}&=i\partial_{k}\left(\sum_{i}\psi_{i\Lambda}^{\dagger}\gamma^{k}\psi_{i\Lambda}\right),\end{split}$
(5)

and various potentials

$\begin{split}V_{B}&=g_{\omega B}\omega_{0}+g_{\phi B}\phi_{0}+g_{\rho
B}\tau_{3}\rho_{0}+e\dfrac{1-\tau_{3}}{2}A_{0},\\\ S_{B}&=g_{\sigma
B}\sigma+g_{\sigma^{*}B}\sigma^{*},\\\
T_{\Lambda}&=-\dfrac{f_{\omega\Lambda\Lambda}}{2M_{\Lambda}}\beta(\alpha\cdot
p)\omega_{0},\\\ \Sigma_{R}&=\dfrac{\partial g_{\sigma
N}}{\partial\rho^{\upsilon}}\rho_{N}^{s}\sigma+\dfrac{\partial g_{\omega
N}}{\partial\rho^{\upsilon}}\rho_{N}^{\upsilon}\omega+\dfrac{\partial g_{\rho
N}}{\partial\rho^{\upsilon}}(\rho_{n}^{\upsilon}-\rho_{p}^{\upsilon})\rho_{0}+\dfrac{\partial
g_{\sigma^{*}\Lambda}}{\partial\rho^{\upsilon}}\rho_{\Lambda}^{s}\sigma^{*}+\dfrac{\partial
g_{\phi\Lambda}}{\partial\rho^{\upsilon}}\rho_{\Lambda}^{\upsilon}\phi_{0}+\dfrac{1}{2M_{\Lambda}}\dfrac{\partial
f_{\omega\Lambda\Lambda}}{\partial\rho^{\upsilon}}\rho_{\Lambda}^{T}\omega_{0}.\end{split}$
(6)

The rearrangement term $\Sigma_{R}$ is present in the DD-RMF model to ensure
the energy-momentum conservation and thermodynamic consistency [48].

Under the mean field and no-sea approximations, the Dirac equation (3) and
Klein-Gordon equations (4) have been solved in different bases, including the
coordinate space [80, 81, 82, 83], the harmonic oscillator basis [84, 85, 86],
the Woods-Saxon basis [87] and the Lagrange mesh [88]. To simplify the solving
procedure of these equations, most RMF models were developed with certain
spatial symmetries imposed on nuclei. Note that the solution of the Dirac
equation in a 3D lattice was achieved recently and in such a RMF model, the
nuclei in question are not restricted by any spatial symmetry [89, 90]. In the
present work, we use the MDC-RMF model [91, 92, 68] in which an axially
deformed harmonic oscillator (ADHO) basis [84] is used, the pairing
correlations are treated by the BCS approach, and the $V_{4}$ symmetry is
assumed for nuclear shapes, i.e., all deformations characterized by
$\beta_{\lambda\mu}$ with even $\mu$, e.g., $\beta_{20}$, $\beta_{22}$,
$\beta_{30}$, $\beta_{32}$, $\beta_{40}$, $\cdots$, are included self-
consistently. The MDC-RMF model has been used to study the potential energy
surfaces and fission barriers of heavy and superheavy nuclei [91, 92, 93, 94,
95], non-axial octupole $Y_{32}$ correlations [96, 97] and the octupole
correlations in chiral doublet bands [98, 99], etc.

Both NL and DD couplings have been implemented in the MDC-RMF model for normal
nuclei. The MDC-RMF model with NL self-couplings of meson fields has been also
extended to $\Lambda$ hypernuclei [75, 77, 37]. In the present work, we have
included the DD couplings in the MDC-RMF model for $\Lambda$ hypernuclei and,
under the $V_{4}$ symmetry, solved Eqs. (3) and (4) in the ADHO basis. To keep
the time reversal symmetry, when dealing with the unpaired baryon, we adopt
the equal filling approximation which has been widely used in mean-field
calculations [100]. The spurious motion due to the breaking of the
translational invariance is treated by including the center of mass correction
$E_{\text{c.m.}}=-\langle P^{2}\rangle/2M$ with $M=(A-1)M_{N}+M_{\Lambda}$ in
the binding energy.

##  3 Results and discussions

In this section we will determine the parameters of coupling constants
$g_{mB}(\rho^{\upsilon})$ in Eq. (2) for hypernuclei based on available DD-RMF
effective interactions for normal nuclei. For the mesons in Eq. (1), one
usually considers $\sigma$, $\omega$ and $\rho$ for normal nuclei. In this
work, we focus on single-$\Lambda$ hypernuclei in which $\sigma^{*}$ and
$\phi$ can be omitted because they only couple to stranges quarks according to
the OZI rule. Furthermore, the electromagnetic field and $\rho$ mesons do not
couple to $\Lambda$ because $\Lambda$ hyperon is charge neutral with isospin
$\tau=0$. Therefore we are left with $g_{\sigma\Lambda}$ and
$g_{\omega\Lambda}$ to be fixed. Many DD-RMF effective interactions have been
proposed for normal nuclei, e.g., TW99 [49], DD-ME1 [50], PKDD [43], DD-ME2
[101], DD [102], D3C [102], PKO1 and PKO2 [103], PKO3 [104], PKA1 [105], DD2
[106], DDME$\delta$ [107], DDME-X [108], DD-LZ1 [109] and DDV, DDS, DDVT,
DDST, DDVTD and DDSTD [110]. Most of them can provide a good description for
the properties not only of nuclear matter but also of finite nuclei around and
far from the valley of $\beta$ stability. In this work we choose two typical
ones, PKDD [43] and DD-ME2 [101], in which the density dependence in Eq. (2)
is taken as

$f_{mN}(x)=\left\\{\begin{array}[]{l}a_{m}\displaystyle\frac{1+b_{m}\left(x+d_{m}\right)^{2}}{1+c_{m}\left(x+d_{m}\right)^{2}},\
\ m=\sigma~{}\text{or}~{}\omega,\\\ e^{\displaystyle-a_{\rho}(x-1)},\ \
m=\rho,\\\ \end{array}\right.$ (7)

with nine parameters. Under five constraints,
$f_{\sigma}(1)=1,f_{\omega}(1)=1,f_{\sigma}^{\prime\prime}(0)=0,f_{\omega}^{\prime\prime}(0)=0$
and $f_{\sigma}^{\prime\prime}(1)=f_{\omega}^{\prime\prime}(1)$, there are
four free parameters which have been adjusted to properties of nuclear matter
and selected finite nuclei [43, 101]. We assume the same density dependence of
$g_{m\Lambda}(\rho^{\upsilon})$ as that of $g_{mN}(\rho^{\upsilon})$
($m=\sigma$ or $\omega$). So there are two parameters
$g_{\sigma\Lambda}(\rho_{\text{sat}})$ and
$g_{\omega\Lambda}(\rho_{\text{sat}})$ to be determined.

In $\Lambda$ hypernuclei, the single particle potential depth for $\Lambda$ is
about 30 MeV [111, 112] and the energy splitting between spin partners in
single-$\Lambda$ states is very small compared to that for nucleons [113, 114,
115]. The shallow potential and small spin-orbit splittings for $\Lambda$ have
been understood under the following mechanisms: (i) The effective scalar and
vector boson exchange interactions with $\Sigma$ and $\Delta$-isobar
intermediate states [116]; (ii) the combined quark-gluon exchange between the
valence baryon and the nucleons of the core [117]; (iii) a weak SU(3) symmetry
breaking and a tensor $\omega\Lambda\Lambda$ coupling [118, 119]. In the RMF
model, the last mechanism has been used mostly for the study of $\Lambda$
hypernuclei [120, 72, 75, 121, 78, 79]. According to the OZI rule, the tensor
coupling constant $f_{\omega\Lambda\Lambda}$ is the same as that of
$g_{\omega\Lambda}$ and
$R_{\omega\Lambda\Lambda}=f_{\omega\Lambda\Lambda}/g_{\omega\Lambda}=-1$ [119,
122]. In the present work we follow this convention concerning the tensor
coupling between $\omega$ and $\Lambda$ and assume
$f_{\omega\Lambda\Lambda}(\rho^{\upsilon})=-g_{\omega\Lambda}(\rho^{\upsilon})$.
According to the quark model, the ratio
$R_{m}=g_{m\Lambda}(\rho_{\text{sat}})/g_{mN}(\rho_{\text{sat}})=2/3$ with
$m=\sigma$ or $\omega$ [123, 34, 124]. These two ratios, together with the
parameters of DD-ME2 or PKDD, define completely the DD-RMF functionals for
$\Lambda$ hypernuclei. We use DD-ME2-Y0 and PKDD-Y0 to label these two
effective interactions. Since they can not give even a decent description for
$\Lambda$ separation energies of hypernuclei with $A\geq 12$, as seen in Table
1, we will adjust the ratios $R_{\sigma}$ and $R_{\omega}$ to experimental
values of $\Lambda$ separation energies of selected hypernuclei.

Until now $\Lambda$ separation energies of 33 single-$\Lambda$ hypernuclei
have been measured. Most of these hypernuclei are very light and there are
fifteen with $A\geq 12$: ${}^{12}_{~{}\Lambda}$B,
${}^{12-14}_{~{}~{}~{}~{}\Lambda}$C, ${}^{15,16}_{~{}~{}~{}~{}\Lambda}$N,
${}^{16}_{~{}\Lambda}$O, ${}^{28}_{~{}\Lambda}$Si, ${}^{32}_{~{}\Lambda}$S,
${}^{40}_{~{}\Lambda}$Ca, ${}^{51,52}_{~{}~{}~{}~{}\Lambda}$V,
${}^{89}_{~{}\Lambda}$Y, ${}^{139}_{~{}~{}\Lambda}$La and
${}^{208}_{~{}~{}\Lambda}$Pb. In the present work, the difference of $\Lambda$
separation energies of mirror hypernuclei, (${}^{12}_{~{}\Lambda}$B,
${}^{12}_{~{}\Lambda}$C) and (${}^{16}_{~{}\Lambda}$N,
${}^{16}_{~{}\Lambda}$O), can not be reproduced because $\Lambda$ interacts
with both protons and neutrons via the exchange of the same mesons and the
charge symmetry breaking is not enough. Therefore we select only one
hypernucleus in each pair, namely, ${}^{12}_{~{}\Lambda}$C and
${}^{16}_{~{}\Lambda}$O. ${}^{14}_{~{}\Lambda}$C and ${}^{15}_{\Lambda}$N are
not included because their separation energies were measured only by using the
photographic emulsion technique which should not be trusted for hypernuclei
with $A\geq 12$ [7]. Up to date, all effective $YN$ interactions used in the
RMF model were obtained with the restriction of spherical symmetry though many
of the observed hypernuclei are deformed. In this work, we will use the MDC-
RMF model and consider axial and reflection symmetric deformations when
adjusting the parameters of DD-RMF effective interactions. For simplicity, the
pairing correlations are ignored.

We first calculate the ground state properties of the core nuclei of these
eleven selected hypernuclei: 11,12C, 15O, 27Si, 31S, 39Ca, 50,51V, 88Y, 138La
and 207Pb. The relative deviations of the calculated binding energies from the
experimental values are shown in figure 1(a). The experimental binding
energies are taken from AME2016 [125]. From figure 1(a), one can see that
binding energies calculated with both DD-ME2 and PKDD are close to the
experimental values for all of these nuclei. The relative deviations from
AME2016 are within 5% and the largest deviation occurs for 12C. Except 15O and
207Pb, all the other nuclei are deformed though ${}^{39}_{~{}\Lambda}$Ca is
almost spherical, as seen in figure 1(b). From MDC-RMF calculations with PKDD
and DD-ME2, similar deformations are obtained for ten of these eleven nuclei
with 11C as an exception: 11C is oblate with DD-ME2 but prolate with PKDD.

Figure 1: (Color online) (a) The relative deviations of the calculated binding energies of 11,12C, 15O, 27Si, 31S, 39Ca, 50,51V, 88Y, 138La and 207Pb from the experimental values [125]. (b) Quadrupole deformation parameters of these nuclei. The MDC-RMF calculations are carried out with the density dependent interactions DD-ME2 and PKDD, respectively.  Table 1:  The calculated $\Lambda$ separation energies $B_{\Lambda}$ (in MeV) for selected hypernuclei with DD-ME2-Y$i$ and PKDD-Y$i$ ($i=0,1,2$ and $3$) in comparison with experimental values. $\bar{\chi}^{2}$ ($\bar{\chi}^{2}_{\rm all}$) represents the average least-square deviation of the calculated $\Lambda$ separation energies of hypernuclei in each group (all of the eleven hypernuclei) from the experimental values. The bold-faced $B_{\Lambda}$ values denote that the experimental values of the corresponding hypernuclei are used in the parametrization fitting. See text for the grouping of hypernuclei used in the fitting. The root mean square (rms) deviation $\Delta$ is given in MeV and the root of relative square (rrs) deviation $\delta$ is in percentage. The corresponding ratios $R_{\sigma}$ and $R_{\omega}$ of the effective $\Lambda N$ interactions are listed at the bottom of the table. The tensor coupling constant between $\Lambda$ and $\omega$ is fixed to $R_{\omega\Lambda\Lambda}=-1.0$.  |  | DD-ME-Y$i$ | PKDD-Y$i$
---|---|---|---
Hypernuclei | Exp. | $i=0$ | $i=1$ | $i=2$ | $i=3$ | $i=0$ | $i=1$ | $i=2$ | $i=3$
${}^{12}_{~{}\Lambda}$C | $11.36\pm 0.20$ | 25.514 | 10.789 | 10.588 | 10.120 | 25.566 | 10.854 | 10.514 | 10.013
${}^{13}_{~{}\Lambda}$C | $12.0\pm 0.2$ | 28.080 | 12.262 | 12.023 | 11.422 | 27.814 | 11.932 | 11.534 | 10.944
${}^{16}_{~{}\Lambda}$O | $13.0\pm 0.2$ | 27.516 | 12.849 | 12.643 | 12.174 | 27.771 | 13.089 | 12.684 | 12.105
${}^{28}_{~{}\Lambda}$Si | $17.2\pm 0.2$ | 34.724 | 17.643 | 17.560 | 17.450 | 34.857 | 17.731 | 17.578 | 17.469
${}^{32}_{~{}\Lambda}$S | $17.5\pm 0.5$ | 36.814 | 18.865 | 18.895 | 18.827 | 36.341 | 18.824 | 18.695 | 18.640
${}^{40}_{~{}\Lambda}$Ca | $18.7\pm 1.1$ | 36.600 | 19.566 | 19.448 | 19.265 | 36.730 | 19.756 | 19.493 | 19.235
${}^{51}_{~{}\Lambda}$V | $21.5\pm 0.6$ | 39.126 | 21.227 | 21.228 | 21.401 | 39.095 | 21.251 | 21.221 | 21.407
${}^{52}_{~{}\Lambda}$V | $21.8\pm 0.3$ | 39.429 | 21.422 | 21.440 | 21.662 | 39.348 | 21.400 | 21.402 | 21.649
${}^{89}_{~{}\Lambda}$Y | $23.6\pm 0.5$ | 41.882 | 23.511 | 23.576 | 23.974 | 41.788 | 23.501 | 23.578 | 24.018
${}^{139}_{~{}~{}\Lambda}$La | $25.1\pm 1.2$ | 42.691 | 24.306 | 24.479 | 25.215 | 42.226 | 23.987 | 24.295 | 25.210
${}^{208}_{~{}~{}\Lambda}$Pb | $26.9\pm 0.8$ | 44.489 | 25.687 | 25.893 | 26.746 | 44.029 | 25.337 | 25.694 | 26.729
$\bar{\chi}^{2}$ |  |  | 2.543 | 1.867 | 0.185 |  | 2.580 | 1.889 | 0.211
$\bar{\chi}^{2}_{\text{all}}$ |  | 2956.907 | 2.543 | 3.009 | 6.690 | 2954.595 | 2.580 | 3.666 | 9.233
$\Delta$ |  | 17.175 | 0.711 | 0.672 | 0.668 | 17.048 | 0.816 | 0.712 | 0.716
$\delta$ |  | 98.221 | 3.759 | 3.840 | 4.817 | 97.776 | 4.018 | 4.060 | 5.419
$R_{\sigma}$ |  | 0.667 | 0.366 | 0.417 | 0.577 | 0.667 | 0.367 | 0.464 | 0.659
$R_{\omega}$ |  | 0.667 | 0.352 | 0.415 | 0.611 | 0.667 | 0.353 | 0.472 | 0.712

Based on effective $NN$ interactions DD-ME2 and PKDD, we take
$m_{\Lambda}=1115.6$ MeV and determine the two ratios $R_{\sigma}$ and
$R_{\omega}$ by minimizing the average least-square deviation

$\bar{\chi}^{2}(\bm{a})=\frac{1}{N}\sum_{i}^{N}\left(\dfrac{B_{\Lambda,i}^{\text{exp.}}-B_{\Lambda,i}^{\text{cal.}}(x_{i};\bm{a})}{\Delta
B_{\Lambda,i}^{\text{exp.}}}\right)^{2},$ (8)

where $\bm{a}$ is the ensemble of parameters to be determined ($R_{\sigma}$
and $R_{\omega}$), $i$ numbers each hypernucleus, $B_{\Lambda}$ is the
separation energy and $\Delta B_{\Lambda}^{\text{exp.}}$ represents the
experimental uncertainty. The experimental values of
$B_{\Lambda,i}^{\text{exp.}}$ and $\Delta B_{\Lambda,i}^{\text{exp.}}$ are
taken from Ref. [7] and references therein, except for
${}^{40}_{~{}\Lambda}$Ca of which $B_{\Lambda,i}^{\text{exp.}}$ and $\Delta
B_{\Lambda,i}^{\text{exp.}}$ are taken from Ref. [126]. The eleven hypernuclei
cover a large mass interval with $A=12$–208. When we made the fitting, it was
found that the two ratios $R_{\sigma}$ and $R_{\omega}$ both deviate from 2/3:
The more light hypernuclei are included in the fitting, the smaller the two
ratios are. To show such dependence of the deviations on the mass interval, we
arrange these eleven hypernuclei into three groups: (1) All of them
($A=12$–208), (2) nine of them (${}^{16}_{~{}\Lambda}$O and heavier, i.e.,
$A=16$–208) and (3) only six of them (${}^{40}_{~{}\Lambda}$Ca and heavier
ones, i.e., $A=40$–208). Based on either of the two effective $NN$
interactions DD-ME2 and PKDD, three new parameter sets are obtained. These six
new effective interactions are labelled as DD-ME2-Y$i$ and PKDD-Y$i$ with
$i=1$, 2 and 3 and listed in Table 1.

From Table 1, one can see that if only six medium-heavy and heavy hypernuclei
(those in Group 3) are used in the fitting, the average deviation
$\bar{\chi}^{2}$ is the smallest which is around 0.2 for both DD-ME2-Y3 and
PKDD-Y3. When more light hypernuclei are included, $\bar{\chi}^{2}$ becomes
larger and the two ratios $R_{\sigma}$ and $R_{\omega}$ become smaller. To
check the overall description of the new effective interactions for all of
these eleven hypernuclei, we calculate and list in Table 1 the average
deviation $\bar{\chi}^{2}_{\rm all}$ as defined in Eq. (8), the root mean
square (rms) deviation $\Delta$ and the root of relative square (rrs)
deviation $\delta$

$\Delta=\sqrt{\frac{1}{N}\sum_{i}^{N}\left({B_{\Lambda,i}^{\text{exp.}}-B_{\Lambda,i}^{\text{cal.}}}\right)^{2},}\
\ \
\delta=\sqrt{\frac{1}{N}\sum_{i}^{N}\left(\dfrac{B_{\Lambda,i}^{\text{exp.}}-B_{\Lambda,i}^{\text{cal.}}}{B_{\Lambda,i}^{\text{exp.}}}\right)^{2},}$
(9)

with $N=11$, regardless how many hypernuclei are used in the fitting.
Therefore $\bar{\chi}^{2}_{\rm all}$ is larger than $\bar{\chi}^{2}$ for DD-
ME2-Y$i$ and PKDD-Y$i$ with $i>1$. For example, the average deviation
$\bar{\chi}^{2}_{\rm all}$ is 6.690 for DD-ME2-Y3 and 9.233 for PKDD-Y3. Such
large $\bar{\chi}^{2}_{\rm all}$ values are mainly due to the fact that the
uncertainties of $\Lambda$ separation energies of light hypernuclei are quite
small, $\Delta B_{\Lambda}^{\text{exp.}}=0.2$ MeV for
${}^{12,13}_{~{}~{}~{}~{}\Lambda}$C, ${}^{16}_{~{}\Lambda}$O and
${}^{28}_{~{}\Lambda}$Si. Compared with $\bar{\chi}^{2}_{\rm all}$, the rms
and the rrs deviations are more adequate in describing the agreement between
the calculation and experiment: $\Delta=0.67$–0.82 MeV and $\delta=3.8$–5.4%
which are fairly small, meaning a reasonably good agreement.

Figure 2: (Color online) (a) The calculated $\Lambda$ separation energies
compared with the experimental values and (b) the calculated quadrupole
deformation parameters with DD-ME2-Y1, DD-ME2-Y2 and DD-ME2-Y3, respectively.
The experimental values of $B_{\Lambda}$ are taken from [7] except
${}^{40}_{\Lambda}$Ca [126]. The inset in (a) shows the results for
${}^{51,52}_{~{}~{}~{}~{}\Lambda}$V. The “Core” in (b) represents the
quadrupole deformation parameters of the corresponding core nuclei calculated
with DD-ME2.

The $\Lambda$ separation energies $B_{\Lambda}$ calculated with DD-ME2-Y$i$
($i=1,2$ and 3) are compared with experimental values in figure 2(a). It can
be seen that each of these three new effective interactions can give a good
description of $B_{\Lambda}$ for the selected hypernuclei with an exception of
${}^{32}_{~{}\Lambda}$S. Although all of these eleven hypernuclei are used in
the parametrization fitting for DD-ME2-Y1, the light ones weigh more due to
the smaller experimental errors $\Delta B_{\Lambda,i}^{\text{exp.}}$ (0.2
MeV). Therefore, with DD-ME2-Y1, the calculated separation energies of the
four light hypernuclei are very much close to the experimental values while
those of heavier ones deviate more from the experiment. With DD-ME2-Y3, the
opposite is true: The calculated $B_{\Lambda}$ for heavy and medium heavy
hypernuclei, which are used in the fitting, are very much close to the
experiment while noticeble discrepancies can be seen for
${}^{12,13}_{~{}~{}~{}~{}\Lambda}$C and ${}^{16}_{~{}\Lambda}$O. As for
${}^{32}_{~{}\Lambda}$S, the calculated $B_{\Lambda}$ with these three
parameter sets are very similar (18.865, 18.895 and 18.827 MeV) and all larger
than the experimental value ($17.5\pm 0.5$ MeV) considerably. The reason for
this discrepancy is not clear to us yet.

Figure 2(b) shows the quadrupole deformation parameters of the eleven selected
hypernuclei calculated with DD-ME2-Y$i$ ($i=1,2$ and $3$) and of the
corresponding core nuclei calculated with DD-ME2. From figure 2(b) one can
find that the deformation parameter of a deformed hypernucleus is always
smaller than that of its normal nuclear core. This is particularly true for
light hypernuclei, e.g., ${}^{12}_{~{}\Lambda}$C. Such shape polarization
effects of $\Lambda$ have been discussed in Refs. [127, 128, 129, 75, 130,
131, 77]. Although 12C is obviously oblate with $\beta=-0.347$,
${}^{13}_{~{}\Lambda}$C is spherical with DD-ME2-Y$i$ ($i=1,2$ and 3); such a
shape change is similar to that discussed in Ref. [75] where NL-RMF
functionals were used. ${}^{32}_{~{}\Lambda}$S is spherical with DD-ME2-Y1
though its core, 31S, is moderately deformed with $\beta=0.217$. DD-ME-Y2 and
DD-ME-Y3 both predict a prolate ${}^{32}_{~{}\Lambda}$S with $\beta$ slightly
smaller than that of 31S. Similar discussions hold for $\Lambda$ separation
energies and deformation parameters calculated with PKDD-Y$i$ ($i=1,2$ and 3)
and will not be repeated.

Among the six parameter sets DD-ME2-Y$i$ and PKDD-Y$i$ ($i=1,2$ and 3), the
two ratios $R_{\sigma}$ and $R_{\omega}$ change a lot. However, they are
correlated linearly with each other as shown in figure 3. We made a linear fit
of these two parameters and the relation

$R_{\omega}=1.228R_{\sigma}-0.097,$ (10)

is obtained and shown as the blue line in figure 3. This linear relation can
be explained as follows. In the RMF model, the central potentials $U_{B}$
($B=N$ or $\Lambda$) for $\Lambda$ hypernuclei can be calculated from scalar
and vector potentials approximately, $-U_{B}\approx g_{\sigma
B}\sigma+g_{\omega B}\omega_{0}$. With the restriction
$R_{m}=g_{m\Lambda}/g_{mN}$ ($m=\sigma$ or $\omega$), one obtains

$R_{\omega}\approx\dfrac{-U_{\Lambda}-R_{\sigma}g_{\sigma
N}\sigma}{-U_{N}-g_{\sigma N}\sigma}.$ (11)

Generally speaking, the potential depths, represented by $U_{B}(0)\equiv
U_{B}(r)|_{r=0}$, are about 70 MeV for nucleons and 30 MeV for $\Lambda$. The
scalar potential depth $g_{\sigma N}\sigma(0)$ for nucleons is about $-400$
MeV. With these values, Eq. (11) becomes $R_{\omega}\approx
1.212R_{\sigma}-0.091$, which is very close to Eq. (10). Similar linear
behaviors between $R_{\sigma}$ and $R_{\omega}$ in nonlinear parameter sets
have been discussed in Refs. [51, 121]. Several NL-RMF effective interactions
NLSH-B [72], PK1-Y1 [121] and TM1-A [120] and DD-RMF effective interactions
DDME2D-a [57] and DDME2-a [57], are also shown in figure 3. They all fall well
on the line defined in Eq. (10). The ratios $R_{\sigma}=R_{\omega}=2/3$
predicted from the quark model deviate from the blue line but slightly, as
seen in figure 3. However, the $\Lambda$ separation energies calculated with
the corresponding parameter sets (DD-ME2-Y0 and PKDD-Y0) are much larger than
the experimental values as listed in Table 1. This means that these two ratios
are connected strongly and correlated closely through the linear relation
(10).

Figure 3: (Color online) The correlation between $R_{\sigma}$ and $R_{\omega}$
in DD-ME2-Y$i$ and PKDD-Y$i$ ($i=1,2$ and 3). The blue line shows the linear
relation (10) obtained by a linear fitting of $R_{\sigma}$ and $R_{\omega}$ in
these six parameter sets. Predictions from the quark model (QM), NL effective
interactions NLSH-B [72], PK1-Y1 [121] and TM1-A [120] and DD effective
interactions DDME2D-a [57] and DDME2-a [57] are also shown for comparison.

##  4 Summary

We investigate the effective interactions for $\Lambda$ hypernuclei in the
density dependent relativistic mean field model and propose new parameter
sets. Based on effective $NN$ interactions DD-ME2 and PKDD, the two ratios of
scalar and vector coupling constants between effective $\Lambda N$ and $NN$
interactions, namely, $R_{\sigma}$ and $R_{\omega}$, are obtained by fitting
calculated $\Lambda$ separation energies to experimental values of eleven
single-$\Lambda$ hypernuclei with $A\geq 12$. The calculations were carried
out by using the MDC-RMF model in which deformations are allowed for these
hypernuclei and their normal core nuclei. With three ways of grouping and
including these eleven selected $\Lambda$ hypernuclei in the fitting, six new
effective interactions DD-ME2-Y$i$ and PKDD-Y$i$ ($i=1,2$ and $3$) are
obtained.

Ground state properties of the eleven selected hypernuclei and the
corresponding core nuclei are described well by the MDC-RMF model. For core
nuclei, the calculated binding energies agree satisfactorily with the
experiment and most of them are deformed. For hypernuclei, the calculated
$\Lambda$ separation energies are close to the experimental values with a
small average least-square deviation weighted by experimental uncertainties.
Shape polarization effects of $\Lambda$ and the shape change of a hypernucleus
compared to its core nucleus are obvious in the $A\leq 40$ mass region. The
two ratios $R_{\sigma}$ and $R_{\omega}$ in these six new effective
interactions vary largely. But they are correlated well with each other and
follow closely a linear relation.

## Acnowledgements

Helpful discussions with Johann Haidenbauer, Hoai Le, Andreas Nogga, Xiang-
Xiang Sun and Kun Wang are gratefully acknowledged. This work has been
supported by the National Key R&D Program of China (Grant No. 2018YFA0404402),
the National Natural Science Foundation of China (Grants No. 11525524, No.
12070131001, No. 12047503, and No. 11961141004), the Key Research Program of
Frontier Sciences of Chinese Academy of Sciences (Grant No. QYZDB-SSWSYS013),
the Strategic Priority Research Program of Chinese Academy of Sciences (Grant
No. XDB34010000), the Inter-Governmental S&T Cooperation Project between China
and Croatia, and the IAEA Coordinated Research Project “F41033”. The results
described in this paper are obtained on the High-performance Computing Cluster
of ITP-CAS and the ScGrid of the Supercomputing Center, Computer Network
Information Center of Chinese Academy of Sciences.

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