#  $\Xi$ hypernuclei ${}^{15}_{\Xi}$C, ${}^{12}_{\Xi}$Be and $\Xi N$ two-body
interaction

Yusuke Tanimura  Department of Physics, Tohoku University, Sendai, 980-8578,
Japan  Graduate Program on Physics for the Universe, Tohoku University,
Sendai, 980-8578, Japan  Hiroyuki Sagawa  Center for Mathematics and Physics,
the Univeristy of Aizu, Aizu-Wakamatsu, Fukushima 965-8580, Japan  RIKEN,
Nishina Center, Wako, Saitama 351-0198, Japan  Tingting Sun  School of Physics
and Microelectronics, Zhengzhou University, Zhengzhou 450001, China  Emiko
Hiyama  Department of Physics, Tohoku University, Sendai, 980-8578, Japan
RIKEN, Nishina Center, Wako, Saitama 351-0198, Japan

###### Abstract

We study the energy spectra of $\Xi$ hypernuclei ${}^{15}_{\Xi}$C and
${}^{12}_{\Xi}$Be with a relativistic mean field (RMF) model with meson
exchange $\Xi N$ interactions. The RMF parameters are optimized to reproduce
the average energy of KINKA and IRRAWADDY events for the ground state and also
the average energy of KISO and IBUKI events for the excited state in
${}^{15}_{\Xi}$C. The potential depth of average $\Xi N$ mean field potential
is found to be about $-12$ MeV in the nuclear matter limit. We further
introduce the two-body $s$\- and $p$-wave interactions between valence
nucleons and $\Xi$ particle. We found that the $s$-wave interaction deduced
from the HAL LQCD results is rather weak to obtain the energy difference
between IRRAWADDY and KINKA events. The $p$-wave interaction is added and
fitted to reproduce the energy difference. The resulting interaction together
with the $s$-wave one gives a reasonable energy simultanously for the IBUKI
event as an excited $\Xi_{p}$ state. The model is further applied to predict
the energy spectrum of ${}^{12}_{\Xi}$Be.

$Xi$-hyper nuclei, RMF model, $N\Xi$ two-body spin-spin interaction

##  I Introduction

The study of interaction of the strangeness $S=-2$ sector is important in
hypernuclear physics. To this end, so far several $YY$ potential models such
as Njimegen group esc  and chiral effective field theory chiral  have been
proposed. However, due to the difficulty of performing
hyperon($Y$)-hyperon($Y$) scattering experiment, the proposed potentials have
a large ambiguity depending on the models. Instead, as an alternative way to
obtain information on $S=-2$ sector, more pragmatic models were proposed to
reproduce empirical data with less ambiguity. For such approaches, it is
crucial to study structure of double strangeness hypernuclei such as double
$\Lambda$ hypernuclei and $\Xi$ hypernuclei. Experimentally, some double
$\Lambda$ hypernuclei, ${}^{6}_{\Lambda\Lambda}$He (NAGARA event) NAGARA ;
DEMACHI , ${}^{10}_{\Lambda\Lambda}$Be (DEMACHI-YANAGI event)DEMACHI , and
${}^{11}_{\Lambda\Lambda}$Be (HIDA DEMACHI  and MIKAGE events ekawa ) have
been observed by KEK-E373 and J-PARC-E07 experiments. In comparisons with the
data and theoretical calculations, it was found that the ${}^{1}S_{0}$ of
$\Lambda\Lambda$ interaction is attractive, but much less attractive than that
of $\Lambda N$ interaction. Further analysis of J-PARC-E07 experiment is still
in progress.

Regarding $\Xi$ hypernuclei, the first evidence of bound $\Xi$ hypernucelus,
${}^{15}_{\Xi}$C (${}^{14}{\rm N}+\Xi^{-}$), was reported in 2015 by KEK-E373
experiment, which is named KISO event kiso . At that time, there were two
possible $\Xi^{-}$ binding energy interpretations from the analyses of the
kinematic formation and decay modes, $\Xi^{-}+^{14}{\rm
N}\rightarrow^{15}_{\Xi}{\rm C}\rightarrow^{10}_{\Lambda}{\rm
Be}+^{5}_{\Lambda}$He. One is $B_{\Xi}\equiv-E(^{14}{\rm N})=4.38\pm 0.25$ MeV
assuming the ground state of ${}^{10}_{\Lambda}$Be after the decay from
${}^{15}_{\Xi}$C. The other is $B_{\Xi}=1.11\pm 0.25$ MeV assuming the decay
from the first excited state of ${}^{10}_{\Lambda}$Be. Afterwards, due to the
revision of the observed binding energy of the ${}^{10}_{\Lambda}$Be, the
$B_{\Xi}$ of KISO event was updated accordingly to be $3.87\pm 0.21$ MeV or
$1.03\pm 0.18$ MeV HiNa18 . Thanks to the observation of KISO event, it was
found that $\Xi$-nucleus interaction should be attractive, and it was
requested to interpret theoretically that KISO event is the observation of
either ${}^{14}{\rm N(g.s.)}+\Xi(1s)$ or ${}^{14}{\rm N(g.s.)}+\Xi(1p)$.
Motivated by the observed data in Ref.kiso , some of the present authors (T.T.
S, H. S and E. H.) studied the structure of ${}^{15}_{\Xi}$C using
relativistic mean field (RMF) and Skyrme-Hartree-Fock (SHF) models, and
interpreted KISO event as ${}^{14}{\rm N(g.s.)}+\Xi(1p)$ Sun16 . Afterwards,
${}^{14}{\rm N(g.s.)}+\Xi(1p)$ has been observed as a IBUKI event with the
binding energy $B_{\Xi}=1.27\pm 0.21$ MeV ibuki . Very recently, two new
events ${}^{14}{\rm N(g.s.)}+\Xi(1s)$ has been reported by J-PARC-E07. One is
called KINKA event and the other is called IRRAWADDY event irra . The observed
$B_{\Xi}$ of the former is $8.00\pm 0.77$ MeV or $4.96\pm 0.77$ MeV, and that
of the latter is $6.27\pm 0.27$ MeV. These observations could give a great
contribution to the information on $s$-wave $\Xi N$ interaction. For the
further study of $\Xi N$ interaction, it is planned to perform a search
experiment of ${}^{12}_{\Xi}$Be at J-PARC by ${}^{12}{\rm C}(K^{-},K^{+})$
reaction.

Considering this situation, it is important to reproduce all of observed
events theoretically. We consider that KINKA and IRRAWADDY events are partner
of the ground-state doublet and that IBUKI and KISO events are also partner of
the excited-state doublet because there should exists $\sigma\cdot\sigma$ and
$\tau\cdot\tau$ dependent terms in the $\Xi N$ interaction, whereas spin-orbit
contribution is negligible for the ground state doublet of ${}^{15}_{\Xi}$C.
In this paper, we adopt the RMF model with the meson-exchange $\Xi N$
interactions, which were also employed in the previous paper Sun16 . In
addition, in this paper, we introduce the $\sigma\cdot\sigma$ and
$\tau\cdot\tau$ dependent two-body $\Xi N$ interactions for valence nucleons
and $\Xi$ particle. Parameters of the two-body $\Xi N$ interaction should be
optimized so as to reproduce the IRRAWADDY event and Kiso event. Moreover, the
$\Xi N$ interaction employed in ${}^{15}_{\Xi}$C, is applied to study of
${}^{12}_{\Xi}$Be. It is planned to produce ${}^{12}_{\Xi}$Be by
$(K^{-},K^{+})$ reaction using 12C target as J-PARC-E70 experiment. To
encourage the experiment, we predict energy spectra of this hypernucleus.

Since the core nuclei, 14N and 11B, have the compact shell structure, it is
hardly compressed by the addition of $\Xi$ hyperon. Thus, the RMF approach is
suited for the present work. In fact, in the previous paper Sun16  by RMF, we
accomplished interpreting KISO event as ${}^{14}{\rm N(g.s.)}+\Xi(1p)$. Thus
it is highly expected that we can interpret the IRRAWADDY event and predict
low-lying level structure of ${}^{12}_{\Xi}$Be in a consistent manner.

This paper is organized as follows. Sec. II is devoted for the introduction of
the model. Results of ${}^{15}_{\Xi}$C and ${}^{12}_{\Xi}$Be are presented in
Sec. III. The average $\Xi N$ mean-field potentials for various $\Xi$ nuclei
is also discussed in Sec. III. Summary is given in Sec. IV.

##  II Model

###  II.1 Relativistic mean-field model for $\Xi$ hypernuclei

To describe the $\Xi$ hypernucleus, we employ a meson-exchange model with non-
linear couplings for RMF theory BB77 . The Lagrangian density is given by

$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle\bar{\psi}_{N}(i\partial\\!\\!\\!/-m_{N})\psi_{N}+\bar{\psi}_{\Xi}(i\partial\\!\\!\\!/-m_{\Xi})\psi_{\Xi}$
(1)
$\displaystyle+\frac{1}{2}(\partial_{\mu}\sigma)(\partial^{\mu}\sigma)-\frac{1}{2}m_{\sigma}^{2}\sigma^{2}-\frac{c_{3}}{3}\sigma^{3}-\frac{c_{4}}{4}\sigma^{4}$
$\displaystyle-\frac{1}{4}G^{\mu\nu}G_{\mu\nu}+\frac{1}{2}m_{\omega}^{2}\omega^{\mu}\omega_{\mu}+\frac{d_{4}}{4}(\omega^{\mu}\omega_{\mu})^{2}$
$\displaystyle-\frac{1}{4}\vec{R}^{\mu\nu}\cdot\vec{R}_{\mu\nu}+\frac{1}{2}m_{\rho}^{2}\vec{\rho}^{\mu}\cdot\vec{\rho}_{\mu}$
$\displaystyle-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}$
$\displaystyle-\bar{\psi}_{N}\left(g_{\sigma N}\sigma+g_{\omega
N}\omega\\!\\!\\!/+g_{\rho
N}\vec{\rho}\\!\\!\\!/\cdot\vec{\tau}_{N}+eA\\!\\!\\!/\frac{1-\tau_{N,3}}{2}\right)\psi_{N}$
$\displaystyle-\bar{\psi}_{\Xi}\left(g_{\sigma\Xi}\sigma+g_{\omega\Xi}\omega\\!\\!\\!/+g_{\rho\Xi}\vec{\rho}\\!\\!\\!/\cdot\vec{\tau}_{\Xi}\right.$
$\displaystyle\hskip
28.45274pt\left.+\frac{f_{\omega\Xi}}{4m_{\Xi}}G_{\mu\nu}\sigma^{\mu\nu}+eA\\!\\!\\!/\frac{-1-\tau_{\Xi,3}}{2}\right)\psi_{\Xi},$

where $\psi_{N}$ and $\psi_{\Xi}$ are nucleon and $\Xi$ hyperon fields, and
$G^{\mu\nu}=\partial^{\mu}\omega^{\nu}-\partial^{\nu}\omega^{\mu}$,
$\vec{R}^{\mu\nu}=\partial^{\mu}\vec{\rho}^{\nu}-\partial^{\nu}\vec{\rho}^{\mu}$,
and $F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}$ are the field
tensors of the vector mesons $\omega$ and $\rho$, and the photon,
respectively. $\vec{\tau}_{N}$ and $\tau_{\Xi}$ are the Pauli matrix in the
isospin space. We adopt PK1 parameter set pk1  for the nucleon-meson
couplings. The $\rho-\Xi$ coupling constant is taken to be
$g_{\rho\Xi}=g_{\rho N}$ as in Refs. MJ94 ; Sun16 . The remaining coupling
constants $g_{\sigma\Xi}$, $g_{\omega\Xi}$, and $f_{\omega\Xi}$ in the hyperon
sector are fitted to experimental data (IRRAWADDY irra  and KINKA events, see
Sec. III). The $\Xi$ hyperon mass $m_{\Xi}$ is taken to be $1321.7$ MeV pdg20
.

The model Lagrangian given above is solved within the mean-field and the no-
sea approximations. We impose the time-reversal invariance and charge
conservation of the mean-field state, i.e., the time-odd or charged vector
fields vanish. The only non-zero components are their time-like and neutral
components, $\omega^{0}$, $\rho^{0}_{3}$, and $A^{0}$, where the subscript $3$
on the $\rho$ meson field means the third component in the isospace.

The scalar-isoscalar density $\rho_{S}$, vector-isoscalar and -isovector
densities $j^{0}$ and $j_{3}^{0}$ of nucleon are defined in terms of the
single-particle wave functions of nucleon $\psi_{k}^{(N)}$ as

$\displaystyle\rho_{S}(\bm{r})$ $\displaystyle=$ $\displaystyle\sum_{k\in{\rm
occ}}\bar{\psi}_{k}^{(N)}(\bm{r})\psi_{k}^{(N)}(\bm{r}),$ (2) $\displaystyle
j^{0}(\bm{r})$ $\displaystyle=$ $\displaystyle\sum_{k\in{\rm
occ}}\bar{\psi}_{k}^{(N)}(\bm{r})\gamma^{0}\psi_{k}^{(N)}(\bm{r}),$ (3)
$\displaystyle j^{0}_{3}(\bm{r})$ $\displaystyle=$
$\displaystyle\sum_{k\in{\rm
occ}}\bar{\psi}_{k}^{(N)}(\bm{r})\gamma^{0}\tau_{3}\psi_{k}^{(N)}(\bm{r}),$
(4)

where $k$ runs over the occupied nucleon states. The scalar, vector, and
tensor densities of $\Xi$ hyperon are defined in terms of the single-particle
wave function of $\Xi$ hyperon $\psi_{k}^{(\Xi)}$ as

$\displaystyle\rho_{S\Xi}(\bm{r})$ $\displaystyle=$
$\displaystyle\bar{\psi}_{k}^{(\Xi)}(\bm{r})\psi_{k}^{(\Xi)}(\bm{r}),$ (5)
$\displaystyle j^{0}_{\Xi}(\bm{r})$ $\displaystyle=$
$\displaystyle\bar{\psi}_{k}^{(\Xi)}(\bm{r})\gamma^{0}\psi_{k}^{(\Xi)}(\bm{r}),$
(6) $\displaystyle j^{0}_{\Xi 3}(\bm{r})$ $\displaystyle=$
$\displaystyle\bar{\psi}_{k}^{(\Xi)}(\bm{r})\gamma^{0}\tau_{3}\psi_{k}^{(\Xi)}(\bm{r}),$
(7) $\displaystyle\bm{V}_{T\Xi}(\bm{r})$ $\displaystyle=$
$\displaystyle\bar{\psi}_{k}^{(\Xi)}(\bm{r})i\bm{\alpha}\psi_{k}^{(\Xi)}(\bm{r}),$
(8)

where $k$ here denotes the occupied $\Xi$ hyperon state.

The equations of motion for the meson and electromagnetic fields read

$\displaystyle(-\bm{\nabla}^{2}+m_{\sigma}^{2})\sigma$ $\displaystyle=$
$\displaystyle-g_{\sigma N}\rho_{S}-c_{3}\sigma^{2}-c_{4}\sigma^{3}$ (9)
$\displaystyle-g_{\sigma\Xi}\rho_{S\Xi},$
$\displaystyle(-\bm{\nabla}^{2}+m_{\omega}^{2})\omega^{0}$ $\displaystyle=$
$\displaystyle g_{\omega N}j^{0}-d_{4}\left(\omega^{0}\right)^{3}$ (10)
$\displaystyle+g_{\omega\Xi}j^{0}_{\Xi}+\frac{f_{\omega\Xi}}{2m_{\Xi}}\bm{\nabla}\cdot\bm{V}_{T\Xi},$
$\displaystyle(-\bm{\nabla}^{2}+m_{\rho}^{2})\rho^{0}_{3}$ $\displaystyle=$
$\displaystyle g_{\rho N}j^{0}_{3}+g_{\rho\Xi}j^{0}_{\Xi 3},$ (11)
$\displaystyle-\bm{\nabla}^{2}A^{0}$ $\displaystyle=$ $\displaystyle
e\left[\frac{1}{2}(j^{0}-j^{0}_{3})-\frac{1}{2}(j^{0}_{\Xi}-j^{0}_{\Xi
3})\right].$ (12)

The Dirac equation for the nucleon single-particle wave function is given by

$\left[-i\bm{\alpha}\cdot\bm{\nabla}+V_{N}+\beta(m_{N}+S_{N})\right]\psi_{k}^{(N)}=\epsilon_{k}\psi_{k}^{(N)},$
(13)

where

$\displaystyle S_{N}$ $\displaystyle=$ $\displaystyle g_{\sigma N}\sigma,$
(14) $\displaystyle V_{N}$ $\displaystyle=$ $\displaystyle g_{\omega
N}\omega^{0}+g_{\rho N}\rho^{0}_{3}\tau_{N,3}+eA^{0}\frac{-1-\tau_{N,3}}{2}.$
(15)

Note that $\tau_{N,3}=+1$ for neutron and $\tau_{N,3}=-1$ for proton. The
Dirac equation for the single-particle wave function of $\Xi$ hyperon is given
by

$\left[-i\bm{\alpha}\cdot\bm{\nabla}+V_{\Xi}+\beta(m_{\Xi}+S_{\Xi})+i\beta\bm{\alpha}\cdot\bm{T}_{\Xi}\right]\psi_{k}^{(\Xi)}=\epsilon_{k}\psi_{k}^{(\Xi)},$
(16)

where

$\displaystyle S_{\Xi}$ $\displaystyle=$ $\displaystyle g_{\sigma\Xi}\sigma,$
(17) $\displaystyle V_{\Xi}$ $\displaystyle=$ $\displaystyle
g_{\omega\Xi}\omega^{0}+g_{\rho\Xi}\rho^{0}_{3}\tau_{\Xi,3}+eA^{0}\frac{1+\tau_{\Xi,3}}{2},$
(18) $\displaystyle\bm{T}_{\Xi}$ $\displaystyle=$
$\displaystyle\frac{f_{\omega\Xi}}{2m_{\Xi}}\left(-\bm{\nabla}\omega^{0}\right).$
(19)

The set of equations, Eqs. (2-19), are solved self-consistently. In the
numerical calculations, we further impose the spherical symmetry, and the
radial coordinate is discritized up to 40 fm with the step size of $0.1$ fm.
The spurious $\Xi-\Xi$ interaction effect via the $\rho$ meson is removed in
the same way as in Ref. MJ94 .

###  II.2 Residual interactions

We consider spin-isospin-dependent residual interaction, which will be
introduced in Sec. III. Once the residual interaction is given, its effect on
the energy spectra of ${}^{15}_{\Xi}$C and ${}^{12}_{\Xi}$Be will be estimated
by the first-order perturbation theory. We construct the unperturbed states
based on the RMF wave functions.

The 14N subsystem of ${}^{15}_{\Xi}$C nucleus is described as the $p_{1/2}$
neutron and proton coupled to the total spin-parity $J^{\pi_{np}}_{np}=1^{+}$
and isospin $T_{np}=0$ on top of the inert 12C core. The $\Xi$ particle is
then coupled to the nucleon pair to make the total quantum numbers $J^{\pi}T$,
where $J^{\pi}$ is the total spin-parity of the system, and $T=1/2$ in the
case of ${}^{15}_{\Xi}$C. The energy shift due to the residual interaction
$V_{N\Xi}$ is given as

$\displaystyle\Delta E$ $\displaystyle=$
$\displaystyle\left\langle\left[\left[\nu 1p_{1/2}~{}\pi
1p_{1/2}\right]^{1^{+}}\Xi
nl_{j}\right]^{J^{\pi}}\right|V_{N\Xi}\left|\left[\left[\nu 1p_{1/2}~{}\pi
1p_{1/2}\right]^{1^{+}}\Xi nl_{j}\right]^{J^{\pi}}\right\rangle,$ (20)

where we take here the proton-neutron formalism, and the obvious isospin
quantum number $T=1/2$ is implicit. $nlj$ denotes the radial quantum number,
orbital and total angular momentum of the $\Xi$ particle, respectively.

The ${}^{12}_{\Xi}$Be nucleus is regarded as a particle-hole state, i.e., a
$p_{3/2}$ nucleon hole and a 1$s_{1/2}$ $\Xi$ particle on top of the 12C core,
which are coupled to good angular momentum and isospin $J^{\pi}T$. In the case
of ${}^{12}_{\Xi}$Be, the energy shift due to the residual interaction is
given as

$\displaystyle\Delta E$ $\displaystyle=$
$\displaystyle\left\langle\left[N1p_{3/2}^{-1}~{}\Xi
nl_{j}\right]^{J^{\pi}T}\right|V_{N\Xi}\left|\left[N1p_{3/2}^{-1}~{}\Xi
nl_{j}\right]^{J^{\pi}T}\right\rangle.$ (21)

##  III Results

###  III.1 $\Xi N$ interaction and energy spectra of ${}^{15}_{\Xi}$C

We adjust the parameters, $g_{\sigma\Xi}$, $g_{\omega\Xi}$, and
$f_{\omega\Xi}$ so as to reproduce the observed $B_{\Xi}=6.27\pm 0.24$ MeV,
IRRAWADDY event for ${}^{14}{\rm N(g.s.)}+\Xi(1s)$, and $B_{\Xi}=1.03\pm 0.18$
MeV, KISO event for ${}^{14}{\rm N(g.s.)}+\Xi(1p)$. It should be noted that we
have other two observed data, IBUKI event, $B_{\Xi}=1.27\pm 0.21$ MeV, KINKA
event $B_{\Xi}=8.00\pm 0.77$ MeV or $4.96\pm 0.77$ MeV, which are expected to
be ${}^{14}{\rm N(g.s.)}+\Xi(1p)$ and ${}^{14}{\rm N(g.s.)}+\Xi(1s)$,
respectively. First, assuming the KINKA and IRRAWADDY events are the ground-
state spin doublet, and KISO and IBUKI events are a part of the excited-state
multiplets, we fit the $\Xi-\omega$ Yukawa couplings $g_{\sigma\Xi}$ and
$g_{\omega\Xi}$ so that $B_{\Xi(1s)}$ is located between IRRAWADDY and KINKA,
and spin-averaged $B_{\Xi(1p)}$ is close to KISO and IBUKI. Furthermore, we
introduce $\Xi-\omega$ tensor coupling $f_{\omega\Xi}$ to produce spin-orbit
force of $\Xi N$ so that $B_{\Xi(1p_{3/2})}$ and $B_{\Xi(1p1/2)}$ are 0.86 MeV
and 0.72 MeV, respectively. The parameters thus determined leads to
$B_{\Xi(1s)}=5.69$ MeV for the RMF ground-state binding energy. The resultant
values of the coupling constants are tabulated in Table 1. In view of the
order of magnitude, our phenomenological values are consistent with the values
with the naive quark counting and the quark model CW91 . Note that
$g_{\omega\Xi}/g_{\omega N}\approx 0.5$ is close to the value adopted in the
recent quark mean field model study Hu21 .

Table 1: $\Xi$-meson coupling constants. The “Quark model” values $1/3$ for $g_{\sigma\Xi}/g_{\sigma N}$ in the first row and for $g_{\omega\Xi}/g_{\omega N}$ in the second row come from the naive quark counting, and $-0.4$ for $f_{\omega\Xi}/g_{\omega\Xi}$ from a quark model in Ref. CW91 .  Coupling | Fitted | Quark model
---|---|---
$g_{\sigma\Xi}/g_{\sigma N}$ | $0.497$ | $1/3$
$g_{\omega\Xi}/g_{\omega N}$ | $0.5377$ | $1/3$
$f_{\omega\Xi}/g_{\omega\Xi}$ | $-0.8$ | $-0.4$

Next, we study spin-isospin-dependent residual $\Xi N$ interactions. The
$s$-wave interaction is defined by

$\displaystyle V_{N\Xi}$ $\displaystyle=$ $\displaystyle\sum_{i\in{\rm
nucleons}}(v_{\sigma}\bm{\Sigma}_{i}\cdot\bm{\Sigma}_{\Xi}+v_{\tau}\vec{\tau}_{i}\cdot\vec{\tau}_{\Xi}$
(22) $\displaystyle+$ $\displaystyle
v_{\sigma\tau}\bm{\Sigma}_{i}\cdot\bm{\Sigma}_{\Xi}\vec{\tau}_{i}\cdot\vec{\tau}_{\Xi})\delta(\bm{r}_{i}-\bm{r}_{\Xi}),$

where $\bm{\Sigma}=\left(\begin{array}[]{cc}\bm{\sigma}&0\\\
0&\bm{\sigma}\end{array}\right)$ is the spin operator acting on a Dirac
spinor. In this work, we use the information on a potential based on first
principle lattice QCD simulations, the HAL QCD $\Xi N$ potential (HAL
potential) hal20  in the following way: First we calculate volumes of HAL
potential defined by

$\displaystyle V_{\beta}=\int d^{3}rf_{\beta}(r)V_{\beta}^{\rm HAL}(r),$ (23)

where $\beta=0,\sigma,\tau,\sigma\tau$, and

$\displaystyle f_{\beta}(r)=\frac{1}{1+{\rm exp}[-(r-R_{\beta})/d]}.$ (24)

Here, we take $d=0.1$ fm and $R_{\beta}=(0.234,0.816,0.548,0.684)$ fm for the
spin-isospin channels $(2S+1,2T+1)=(11,31,13,33)$, respectively, so that
$V_{\beta}^{\rm HAL}(R)=0$. As a result, the factors $f_{\beta}(r)$ suppress
the repulsive cores of the HAL potential, which would induce the short-range
correlation missing in our simple framework. Also, we take zero-range
reduction of RMF meson-exchange interaction TH12 ; BMMR02 :

$\displaystyle\alpha$ $\displaystyle=$ $\displaystyle-\frac{g_{\sigma
N}g_{\sigma\Xi}}{m_{\sigma}^{2}}+\frac{g_{\omega
N}g_{\omega\Xi}}{m_{\omega}^{2}}=-327\ {\rm MeV\ fm}^{3}.$ (25)

Thus, finally, we employ

$\displaystyle v_{\sigma}$ $\displaystyle=$
$\displaystyle(V_{\sigma}/V_{0})\alpha=4.87\ {\rm MeV\ fm}^{3},$ (26)
$\displaystyle v_{\tau}$ $\displaystyle=$
$\displaystyle(V_{\tau}/V_{0})\alpha=26.83\ {\rm MeV\ fm}^{3},$ (27)
$\displaystyle v_{\sigma\tau}$ $\displaystyle=$
$\displaystyle(V_{\sigma\tau}/V_{0})\alpha=-53.56\ {\rm MeV\ fm}^{3}.$ (28)

Among the spin and isospin dependent terms of $\Xi N$ interaction given in Eq.
(22), only $v_{\sigma}$ contributes to the binding energy of ${}^{15}_{\Xi}$C
since the core is the isospin $T=0$ nucleus, 14N. As seen in Eq. (26), the
value $v_{\sigma}$ is rather small so that we cannot reproduce the spin-
doublet state of IRRAWADDY and KINKA events by the $s$-wave interaction. It
should also be noted that the HAL potential provides the $s$-wave $\Xi N$
interaction only, and a missing part of $\Xi N$ interaction, $p$-wave spin and
isospin dependent terms, might be important in ${}^{15}_{\Xi}$C since the core
14N is a $p$-shell nucleus.

Next, we introduce a $p$-wave spin dependent interaction defined by

$\displaystyle V^{p}_{N\Xi}$ $\displaystyle=$ $\displaystyle\sum_{i\in{\rm
nucleons}}v_{\sigma}^{p}\bm{\Sigma}_{i}\cdot\bm{\Sigma}_{\Xi}\overleftarrow{\bm{\nabla}}\cdot\delta(\bm{r}_{i}-\bm{r}_{\Xi})\overrightarrow{\bm{\nabla}},$
(29)

where $v_{\sigma}^{p}$ is parameter to be optimized by the IRRAWADDY and KINKA
events.

Fig. 1 shows the energy spectrum of ${}^{15}_{\Xi}$C as a function of $p$-wave
spin-spin interaction $v_{\sigma}^{p}$. We see that for the negative value of
$v_{\sigma}^{p}\sim-150$ MeV fm5, the energies of $J=1/2^{+}$ and $3/2^{+}$
are in good agreement with IRRAWADDY event and KINKA events, respectively. In
this case, IRRAWADDY event (KINKA event) can be interpret as observation of
the ground (excited) state. On the other hand, for the positive value of
$v_{\sigma}^{p}\sim 100$ MeVfm5 the energies of $J=3/2^{+}$ and $1/2^{+}$ are
in good agreement with IRRAWADDY event and KINKA events, respectively. In this
case, IRRAWADDY event (KINKA event) can be interpret as observation of the
ground (excited) state, too. It should be noticed that the opitimized $p$-wave
$N\Xi$ interaction is more or less the same magnitude of the $p$-wave $NN$
ineteraction in the Skyrme EDF, which was determined by using the empirical
nuclear structure information of several double-closed shell nucleus VB72 .

Figure 1: The estimated energy spectrum of ${}^{15}_{\Xi}$C nucleus as a
function of the $p$-wave spin-spin interaction strength $v_{\sigma}^{p}$.
Shown by lines are the energies of the levels labeled by the orbital occupied
by the $\Xi^{-}$ particle, $\Xi_{lj}$, and the total spin parity, $J^{\pi}$.
The experimental data of $B_{\Xi^{-}}$ are shown by color bands. The measured
binding energy of KISO event was reported in Refs. kiso ; HiNa18 , IRRAWADDY
and KINKA events in Ref. irra , and IBUKI event in Ref. ibuki . For KISO and
KINKA events there are two possibilities each that correspond to the ground
and an excited state of the decay products. Figure 2: The estimated energy
spectra of ${}^{12}_{\Xi}$Be nucleus, in which $\Xi^{-}$ occupies the $1s$
state. (a) shows the spectrum calculated by RMF with the $s$-wave interaction
as given in Eq. (22). (b) and (c) represent the spectra calculated by RMF with
the $p$-wave interaction as given in Eq. (29) together with the $s$-wave. The
spectrum (b) is obtained with a repulsive $p$-wave interaction
$v_{\sigma}^{p}=100$ MeV fm5, while the spectrum (c) is with an attractive
$p$-wave one $v_{\sigma}^{p}=-150$ MeV fm5. The four states obtained with RMF
without the residual interactions are degenerate and the energy is shown by
the straight dotted line. The energies of the levels are shown in unit of MeV.

###  III.2 Energy spectra of ${}^{12}_{\Xi}$Be

Fig. 2 shows the estimated energy spectra of ${}^{12}_{\Xi}$Be nucleus, in
which $\Xi^{-}$ occupies the $1s$ state. Fig. 2(a) shows the spectrum
calculated by RMF with only the $s$-wave interaction as given in Eq. (22).
Fig. 2(b) and 2(c) represent the spectra calculated with the $p$-wave
interaction as given in Eq. (29) as well as the $s$-wave interaction. The
spectrum (b) is obtained with a repulsive $p$-wave interaction
$v_{\sigma}^{p}=100$ MeV fm5, while the spectrum (c) is with an attractive one
$v_{\sigma}^{p}=-150$ MeV fm5. The four states by RMF results without the
residual interactions are degenerate and the energy is shown by the straight
dotted line.

In Fig.2 (a), we illustrate the spectra calculated with the HAL potential. We
see that the ground state of ${}^{12}_{\Xi}$Be becomes $T=0,J^{\pi}=2^{-}$.
Qualitatively, the spectrum in Fig. 2(a) with the $s$-wave interaction can be
understood in the following argument. Since the $\sigma\tau$ term is dominant
in the $s$-wave interaction, we demonstrate the $s$-wave effect to calculate
the two-body $N\Xi$ matrix element taking only the spin-isospin operators,

$\displaystyle\Delta E\propto$
$\displaystyle\left\langle\left[N1p_{3/2}^{-1}~{}\Xi
nl_{j}\right]^{J^{\pi}T}\right|\bm{\sigma}_{N}\cdot\bm{\sigma}_{\Xi}\vec{\tau}_{N}\cdot\vec{\tau}_{\Xi}\left|\left[N1p_{3/2}^{-1}~{}\Xi
nl_{j}\right]^{J^{\pi}T}\right\rangle.$

The isospin part is estimated to be

$\displaystyle\left\langle
T|\vec{\tau}_{N}\cdot\vec{\tau}_{\Xi}|T\right\rangle=\begin{cases}1&\text{for}\,\,\,T=1,\\\
-3&\text{for}\,\,\,T=0.\end{cases}$ (31)

The spin part of Eq. (III.2) is also calculated as the reduced matrix form

$\displaystyle\left\langle\left[\nu 1p_{3/2}^{-1}~{}\Xi
nlj\right]^{J^{\pi}T}\right\|\bm{\sigma}_{N}\cdot\bm{\sigma}_{\Xi}\left\|\left[\nu
1p_{3/2}^{-1}~{}\Xi nlj\right]^{J^{\pi}T}\right\rangle$ (34) $\displaystyle=$
$\displaystyle(-)^{(J+1)}\left\\{\begin{array}[]{lcl}1/2&3/2&J\\\ 3/2&1/2&1\\\
\end{array}\right\\}$ $\displaystyle\times$ $\displaystyle\langle
p_{3/2}\|\bm{\sigma}\|p_{3/2}\rangle\langle
s_{1/2}\|\bm{\sigma}_{\Xi}\|s_{1/2}\rangle,$ (35)

where we use the $6j$ symbol, which is given by

$(-)^{(J+1)}\left\\{\begin{array}[]{lcl}1/2&3/2&J\\\ 3/2&1/2&1\\\
\end{array}\right\\}=\begin{cases}-\left(\frac{1}{40}\right)^{1/2}&\text{for}\,\,\,J=2,\\\
+\left(\frac{5}{24}\right)^{1/2}&\text{for}\,\,\,J=1.\end{cases}$ (36)

For the spin-isospin channel, the strength $v_{\sigma\tau}$ is attractive so
that $J=2,T=0$ state obtains the most attractive energy, while $J=1,T=0$
receives the most repulsive effect in Fig. 2(a). For the $T=1$ state, the
spin-isospin interaction works in opposite directions because of Eq. (31),
which makes smaller contributions for $J=2,T=1$ and $J=1,T=1$ states,
respectively, compared with $T=0$ states. Mathematical structures of $p$-wave
interactions are more complicated, but essentially follows the features of
$s$-wave interactions for the attractive $p$-wave in Fig. 2(c), where both the
$s$\- and $p$-wave interactions work coherently. For the repulsive $p$-wave
case in Fig. 2(b), two interactions cancel partially for $T=0$ states, but the
net effect gives more energy shift for $T=1$ states than $T=0$ states.

###  III.3 $\Xi$ mean-field potential and $\Xi$ single-particle energy

With the parameters thus obtained, we discuss the isoscalar part of the $\Xi$
mean field potential,

$\displaystyle V_{\sigma+\omega}=g_{\sigma\Xi}\sigma+g_{\omega\Xi}\omega^{0},$
(37)

for various single-$\Xi$ hypernuclei with mass $A=12$ to $209$ shown in Fig.
3. The potential increases smoothly from the center to the surface without a
plateau in the interior part of $\Xi$ potential for the mass $A\leq 20$. The
central depth is $-20$ MeV for 11B core and gradually become shallower for 14N
and 16O cores. For heavier $\Xi$ hypernuclei with $A\geq 40$, the central part
of the potential becomes flat and the depth converges to be around $-12$ MeV
in heavier systems, which can be considered as an infinite matter limit. The
width of the potential increases to follow $A^{1/3}$ law.

The single-particle states of $\Xi$ hypernuclei with mass $A=12$ to 209 are
shown in Fig. 4: (a) for $\Xi^{-}$ with the Coulomb interaction between a
$\Xi^{-}$ particle and core protons; (b) for $\Xi^{0}$ without the Coulomb
interaction. The Coulomb interaction between $\Xi^{-}$ and protons are
attractive and makes the potential depth deeper about $-17$ MeV for 208Pb core
and $-2$ MeV in 11B core. In Fig. 4(a), the $s$-orbit bounds $B_{\Xi}=27$ MeV
for ${}^{209}_{\Xi}$Tl, and a light $\Xi$ hypernucleus ${}^{12}_{\Xi}$Be has a
small $B_{\Xi}=3$ MeV. In ${}^{209}_{\Xi}$Tl, the orbits only up to $l=8$ are
shown although there a lot more Coulomb-assisted bound states. Only the
$s$-orbit is bound in the $\Xi$ mean field potential of ${}^{12}_{\Xi}$Be. Due
to the contribution of Coulomb interaction, the behaviour of the single
particle energies of $\Xi^{-}$ hypernuclei is similar to that of $\Lambda$
hypernuclei TH12 .

In Fig. 4(b), the $s$-orbit is bound in all $\Xi^{0}$ nuclei starting from 11B
core to 208Pb core. The binding energy of the $s$-orbit of 208Pb core is about
10 MeV, which is almost the same as the potential depth. For $A=11,\ 14$ and
$16$ cores, only the $s$-orbit is bound, but the $p$-orbit of $\Xi^{0}$
hypernucleus becomes bound for $A\geq 40$ core. For 208Pb core, the states
with the angular momentum $l\leq 3$ are bound.

Figure 3: The isoscalar part $V_{\sigma+\omega}$ of the mean field potential
for $\Xi$ particle in light to heavy hypernuclei with $A=11-208$ cores.

Figure 4: Binding energies of single $\Xi$ states: (a) for $\Xi^{-}$ particle
and (b) for $\Xi^{0}$ particle in light to heavy hypernuclei with the mass
$A=11-208$ core. The difference between (a) and (b) is with and without
Coulomb interaction between $\Xi$ particle and protons.

##  IV Summary

We studied the energy spectra of $\Xi$ hypernuclei ${}^{15}_{\Xi}$C and
${}^{12}_{\Xi}$Be with an RMF model taking into account meson exchange $N\Xi$
interactions. The mean field potential parameters are optimized to reproduce
the average energy of KINKA and IRRAWADDY events for possible ground states
and also the average energy of KISO and IBUKI events for possible excited
states in ${}^{15}_{\Xi}$C. We further introduce the two-body $s$-wave
interaction between valence nucleons and $\Xi$ particle. The strength of
$s$-wave interaction is extracted from HAL Lattice QCD calculations, which
gives a strong attraction in the $(T,S)=(0,0)$ channel. We demonstrate that
the HAL $s$-wave interaction has rather small effect on the energy of
${}^{15}_{\Xi}$C since the core nucleus 14N has the isospin $T$=0 and the
isoscalar spin-spin interaction is rather small. We then introduce $p$-wave
$N\Xi$ interaction to fit the energy difference between IRRAWADDY and KINKA
event, and found that both repulsive and attractive spin-spin $p$-wave $N\Xi$
interaction gives a reasonable energy splitting with the coupling strength
$v_{\sigma}^{p}=-$150 MeV fm5 or 100 MeV fm5, respectively. The attractive
$p$-wave gives $J^{\pi}=1/2^{+},3/2^{+}$ states for the ground state and the
first excited state, respectively. On the other hand, the repulsive one give a
reversed spectrum, i.e., $J^{\pi}=3/2^{+},1/2^{+}$ states for the ground state
and the first excited state, respectively. These $p$-wave coupling strength
gives also a reasonable energy for the IBUKI event as an excited $\Xi
p$-state.

The energy spectrum of ${}^{12}_{\Xi}$Be is also studied in the same
theoretical model. The HAL $s$-wave interaction in the spin-isospin channel is
strongly attractive so that the $(J^{\pi},T)=(2^{-},0)$ state becomes the
lowest ground state and the states with $(J^{\pi},T)=(1^{-},1),(2^{-},1)$ and
$(1^{-},0)$ become the excited states in order. When the $p$-wave interaction
is introduced, the attractive $p$-wave one gives more attractive energy for
$(J^{\pi},T)=(2^{-},0)$ state, while the repulsive interaction cancels largely
with the $s$-wave contribution each other and makes $(J^{\pi},T)=(1^{-},1)$
state to be the ground state.

The mean field $N\Xi$ potential of the present RMF model gives the potential
depth $-12$ MeV in the nuclear matter limit. This potential depth is one-
fourth of nuclear mean field and a half of $N\Lambda$ mean field. These
features might give a critical influence on the structure, especially the mass
radius relation and the inert core of neutron star.

###### Acknowledgements.

We would like to thank to Prof. Nakazawa for helpful discussion of the present
work. This work was supported in part by JSPS KAKENHI of Grant Numbers
JP19K03858, JP19K03861, JP20H00155, Grand-In-Aid for scientific Research on
Innovative Areas, 18H05407 and and the National Natural Science Foundation of
China (Grant No. U2032141).

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