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\title{$p$ and $sd$shell $\Lambda$hypernuclei with shell model approach}
\author{A.~Umeya\thanks{Email: aumeya@nit.ac.jp}}
\instlist{\inst{}
Liberal Arts and Sciences, Nippon Institute of Technology,
Miyashiro, Saitama 3458501, Japan}
\begin{document}
\maketitle
\begin{abstract}
As a higher stage of hypernuclear studies,
new experiments with high intensity and high resolution
are being planned at
the Japan Proton Accelerator Research Complex
and the Jefferson Laboratory.
We focus our attention on the interplay
between the hyperon motion and the nuclear core states.
Taking two typical examples
of the $p$shell hypernucleus ${}_{\;\Lambda}^{10}\mathrm{Be}$
and the $sd$shell hypernucleus ${}_{\;\Lambda}^{27}\mathrm{Mg}$,
we discuss novel coupling features of the $p_{}^{\Lambda}$ orbital
and the core deformation
within the extended shellmodel framework.
\end{abstract}
\section{Introduction}
Hypernuclear structure studies have been progressing steadily
through the $K$ and $\pi$induced production reaction experiments,
especially by the $\gamma$ray coincidence measurements
with the large volume Ge detector~\cite{has06}.
Moreover a series of recent $(e, e_{}^{\prime} K_{}^{+})$ reaction experiments
from the Jefferson Laboratory (JLab) provide highresolution data
of the lowlying energy levels for $p$shell hypernuclei~\cite{gog21,urc15}.
These data are quite helpful in better understanding
of hyperonnucleon interactions,
though the data are still limited to about ten hypernuclear species.
\par
As a higher stage of hypernuclear studies,
new projects of highintensity and highresolution
$(K_{}^{}, \pi_{}^{} \gamma)$ and $(\pi_{}^{+}, K_{}^{+} \gamma)$
reaction experiments are being scheduled at the
Japan Proton Accelerator Research Complex (JPARC) facility.
New experiments are also planned at JLab.
In order to meet these experimental projects,
updated theoretical studies are needed
for prediction and/or comparison with the coming quality data.
\par
The recent
${}_{}^{10}\mathrm{B}\, (e, e_{}^{\prime} K_{}^{+})\,
{}_{\;\Lambda}^{10}\mathrm{Be}$
reaction experiment done at JLab~\cite{gog16}
provides us with an interesting new aspect
that is not seen in ordinary nuclear structures.
This high resolution experiment
has confirmed the four major peaks predicted
by the distortedwave impulse approximation (DWIA) calculations~\cite{mot94}
based on the normalparity nuclear core wave functions
coupled with a $\Lambda$ hyperon in $s$ orbit.
At the same time,
the data also show an extra subpeak (bump)
which seems difficult to be explained
within the $p$shell nuclear normalparity configurations employed so far.
\par
In this article, we focus our attention on the interplay
between the hyperon motion and the nuclear core states
in typical $p$ and $sd$shell hypernuclei.
The extended shellmodel calculation~\cite{ume19}
proved to be successful for the first time
in explaining the extra subpeak observed in the
${}_{}^{10}\mathrm{B}\, (e, e_{}^{\prime} K_{}^{+})\,
{}_{\;\Lambda}^{10}\mathrm{Be}$
experiment~\cite{gog16}.
It is attributed to the lowering of $p_{/\!/}^{\Lambda}$ state
(See Fig.\,\ref{f01} for this notation)
due to the strong coupling with $\alpha$$\alpha$ like nuclear core deformation
as already known
in the case of ${}_{\Lambda}^{9}\mathrm{Be}$~\cite{mot85,dal76}.
In the $sd$shell region,
we will show the results of new calculations
for a hypernuclear structure of ${}_{\;\Lambda}^{27}\mathrm{Mg}$,
which is expected to be well produced by the
${}_{}^{27}\mathrm{Al}\, (e, e_{}^{\prime} K_{}^{+})$
and
${}_{}^{27}\mathrm{Al}\, (\pi_{}^{}, K_{}^{0})$
reactions.
It is noted that the eveneven core nucleus ${}_{}^{26}\mathrm{Mg}$
is shown to have rotational bands.
Thus we see interesting interplay
in the coupling of the $p_{}^{\Lambda}$ orbital and the core deformation.
\section{Multiconfiguration shellmodel framework}
In the extended shellmodel calculation~\cite{ume19},
each hypernuclear state of $J_{}^{\pm}$
is described by taking four types configurations,
(A) $[ J_{\mathrm{core}(i)}^{} \otimes s_{}^{\Lambda} ]_{J_{}^{}}$,
(B) $[ J_{\mathrm{core}(i)}^{} \otimes p_{}^{\Lambda} ]_{J_{}^{+}}$,
(C) $[ J_{\mathrm{core}(i)}^{+} \otimes s_{}^{\Lambda} ]_{J_{}^{+}}$, and
(D) $[ J_{\mathrm{core}(i)}^{+} \otimes p_{}^{\Lambda} ]_{J_{}^{}}$,
where
$s_{}^{\Lambda}$ and $p_{}^{\Lambda}$
denote the $\Lambda$ singleparticle states.
$J_{\mathrm{core}(i)}^{\pm}$
denotes all the possible spinparity states of core nucleus,
which are labeled with $i$.
In the traditional treatments appeared so far,
only (A) and (B) configurations are taken into account.
However, we found it important to extend the model space
to include (C) and (D) configurations as well.
For the case of ${}_{\;\Lambda}^{10}\mathrm{Be}$,
the naturalparity (unnaturalparity) core state
$J_{\mathrm{core}(i)}^{}$ ($J_{\mathrm{core}(i)}^{+}$)
is constructed in the $0\hbar\omega$ ($1\hbar\omega$) space,
and nuclear core configurations with natural and unnaturalparity,
(B) and (C),
can be mixed easily by the $\Lambda N$ interaction
at appropriate excitation energy
where the lowering of $p_{/\!/}^{\Lambda}$ favors as mentioned above.
\par
Secondly, in order to study coupling features of $\Lambda$
and typical rotational motion,
we further apply the extended shell model to $sd$shell hypernuclei.
For the core nucleus ${}_{}^{26}\mathrm{Mg}$,
naturalparity nuclear core states
are described in the conventional model space as
\mbox{$(0s)_{}^{4}(0p)_{}^{12}(sd)_{}^{10};J_{\mathrm{core}(i)}^{+}\rangle$},
and unnaturalparity nuclear core states
are described by the $1p$$1h$ $1\hbar\omega$ excitations as
$(0s)_{}^{4}(0p)_{}^{11}(sd)_{}^{11};J_{\mathrm{core}(i)}^{}\rangle$.
For the singleparticle states of the $\Lambda$ hyperon,
$0s$, $0p$, and $sd$shell orbitals are taken into account,
so that new configurations such as
$[ {}_{}^{26}\mathrm{Mg}(J_{\mathrm{core}}^{\pm}) \otimes (sd)_{}^{\Lambda} ]$
should be added.
Thus each hypernuclear state of $J_{}^{\pm}$
is described by taking six types configurations.
For $sd$shell hypernucleus ${}_{\;\Lambda}^{27}\mathrm{Mg}$,
we will show interesting results from the first but limited calculation
done within each of the configurationdiagonal spaces
for the positiveparity core states.
\section{Results and discussion}
In the ${}_{\Lambda}^{9}\mathrm{Be}$ hypernucleus,
it is well known that
the $p_{}^{\Lambda}$ state splits into two orbital states
expressed by $p_{\perp}^{\Lambda}$ and $p_{/\!/}^{\Lambda}$
as shown in Fig.\,\ref{f01}.
The splitting is due to the strong coupling with nuclear core deformation
having the $\alpha$$\alpha$ structure~\cite{mot85,dal76}.
For the ${}_{}^{9}\mathrm{Be}\, (K_{}^{}, \pi_{}^{})$ reaction,
the DWIA cross sections obtained
by using the extended shellmodel wave functions
are in good agreement
with those by using the clustermodel wave functions~\cite{mot85}.
For the other $p$shell hypernuclei,
${}_{\;\Lambda}^{10}\mathrm{Be}$, ${}_{\;\Lambda}^{10}\mathrm{B}$,
and ${}_{\;\Lambda}^{11}\mathrm{B}$,
the extended shellmodel calculation also
shows the $p_{}^{\Lambda}$state splitting~\cite{ume19,ume22}.
Especially,
in the ${}_{\;\Lambda}^{10}\mathrm{Be}$ hypernucleus,
the lowenergy $p_{/\!/}^{\Lambda}$ states
can mix with $s_{}^{\Lambda}$ states.
These paritymixed wave functions can explain,
for the first time,
the extra subpeak observed in the
${}_{}^{10}\mathrm{B}\, (e, e_{}^{\prime} K_{}^{+})\,
{}_{\;\Lambda}^{10}\mathrm{Be}$
reaction experiment.
\par
Figure \ref{f02} shows
the calculated energy levels of ${}_{\;\Lambda}^{27}\mathrm{Mg}$,
together with the lowlying energy levels of ${}_{}^{26}\mathrm{Mg}$.
For ${}_{}^{26}\mathrm{Mg}$,
the calculated energy levels are in good agreement with the experimental levels.
For ${}_{\;\Lambda}^{27}\mathrm{Mg}$,
the lowlying hypernuclear states
consist of the $0s_{}^{\Lambda}$ singleparticle state.
We note that the energy spacings of
the $5/2_{1}^{+}$$3/2_{1}^{+}$ ($5/2_{2}^{+}$$3/2_{2}^{+}$) doublet
with the $2_{1}^{+}$ ($2_{2}^{+}$) core state
is narrow.
This is attributed to the small spinorbit component
of the $\Lambda N$ interaction adopted reasonably in this calculations.
One notes also that
the contribution from $\Lambda N$ spinspin component is small.
\begin{figure}
\centerline{\includegraphics[scale=0.6]{f01.eps}}
\caption{The $p_{\perp}^{\Lambda}$ (left) and $p_{/\!/}^{\Lambda}$ (right)
states realized in ${}_{\Lambda}^{9}\mathrm{Be}$
due to the $\alpha$$\alpha$ core deformation.}
\label{f01}
\end{figure}
\begin{figure}
\centerline{\includegraphics[width=0.9\textwidth]{f02.eps}}
\caption{Calculated energy levels
of ${}_{\;\Lambda}^{27}\mathrm{Mg}$ (right column),
together with the experimental and calculated levels of ${}_{}^{26}\mathrm{Mg}$
(central and left columns).}
\label{f02}
\end{figure}
\begin{figure}
\centerline{\includegraphics[width=0.7\textwidth]{f03.eps}}
\caption{Rotational bands in the energy levels of ${}_{}^{26}\mathrm{Mg}$,
together with strengths of $E2$ transitions.
The arrows denote $E2$ transitions.}
\label{f03}
\end{figure}
\begin{figure}
\centerline{\includegraphics[width=0.86\textwidth]{f04.eps}}
\caption{The DWIA cross sections of the
${}_{}^{27}\mathrm{Al}\, (\gamma, K_{}^{+})$
${}_{\;\Lambda}^{27}\mathrm{Mg}$ reaction without quasifree (QF) contribution.
The solid, dashed, and dotted bars denote
$s_{}^{\Lambda}$, $p_{}^{\Lambda}$, and $(sd)_{}^{\Lambda}$ states,
respectively.}
\label{f04}
\end{figure}
The Antisymmetrized Molecular Dynamics (AMD) calculation
provides the detailed analyses of the rotational bands
in the ${}_{}^{26}\mathrm{Mg}$ nucleus~\cite{kan20}.
In the present shellmodel calculation,
we obtain four rotational bands
with the help of $B(E2)$ estimates for possible transitions.
One notices that in the ${}_{\Lambda}^{9}\mathrm{Be}$ case
the nuclear core deformation is based on the dumbbell structure
of the $\alpha$$\alpha$ system,
while in ${}_{}^{26}\mathrm{Mg}$ the rotational motion
is mainly based on the quadrupole deformation.
Here we remark that
the large scale shell model within certain limitation works
in describing effectively the various kinds of rotational states
as far as the amount of deformation is tractable.
Figure \ref{f03} shows
these bands in the energy levels of ${}_{}^{26}\mathrm{Mg}$,
together with strengths of $E2$ transitions.
The first column shows the ground band.
In the theoretical $B(E2)$ estimates,
the effective charges are adjusted to reproduce
the experimental $B(E2; 2_{1}^{+}{\to}0_{\mathrm{g.s.}}^{+})$ value of
$61.3\,e_{}^{2}\mathrm{fm_{}^{4}}$.
The numerical results
help us to classify the calculated energy levels
into the members belonging to four rotational bands, respectively,
since intraband $B(E2)$ cascades
are obtained in the present shellmodel calculation.
The $2_{1}^{+}$ state is in the ground band,
and the $2_{2}^{+}$ state is the band head of the rotational band with $K=2$.
The property of these rotational level structures
has been pointed out in the AMD calculation~\cite{kan20}.
\par
In hypernuclei,
a $\Lambda$ hyperon in the $p_{}^{\Lambda}$ orbits
induces a mixing of nuclear core states by $\Lambda N$ interaction,
while that in the $s_{}^{\Lambda}$ orbit does not.
In the lowlying negativeparity states
of heavier hypernuclei ${}_{\hspace*{2.5em}\Lambda}^{145155}\mathrm{Sm}$,
a covariant density functional theory suggested that
the admixture of the $p_{}^{\Lambda}$ configurations
coupled with nuclear core states
having $J_{\mathrm{core}}^{}$ and $J_{\mathrm{core}}^{} \pm 2$~\cite{mei17}.
This mechanism
generally applies to the $\Lambda$particlerotor coupling systems.
In fact we found that the similar mixing occurs
in the $sd$shell hypernucleus ${}_{\;\Lambda}^{27}\mathrm{Mg}$.
For example, in the present shellmodel calculation,
configurations of the ${}_{\;\Lambda}^{27}\mathrm{Mg}$ negativeparity states
$1/2_{}^{}$ ($E_{x}^{\mathrm{cal}}=10.615\,\mathrm{MeV}$) and
$3/2_{}^{}$ ($E_{x}^{\mathrm{cal}}=10.685\,\mathrm{MeV}$) are
\begin{eqnarray}
\label{e2}
1/2_{}^{}\rangle
&=&
\sqrt{0.70}\,
 0_{\mathrm{g.s.}}^{+} \otimes p_{1/2}^{\Lambda} \rangle
+
\sqrt{0.28}\,
 2_{1}^{+} \otimes p_{3/2}^{\Lambda} \rangle
+
\cdots,
\\
\label{e3}
3/2_{}^{}\rangle
&=&
\sqrt{0.68}\,
 0_{\mathrm{g.s.}}^{+} \otimes p_{3/2}^{\Lambda} \rangle
+
\sqrt{0.15}\,
 2_{1}^{+} \otimes p_{3/2}^{\Lambda} \rangle

\sqrt{0.15}\,
 2_{1}^{+} \otimes p_{1/2}^{\Lambda} \rangle
+
\cdots,
\end{eqnarray}
respectively.
The mixing amplitudes are quite large in these negativeparity states
which consist of the $p$state $\Lambda$ particle and the deformed core.
On the other hand,
configurations of the ${}_{\;\Lambda}^{27}\mathrm{Mg}$ positiveparity states
$1/2_{\mathrm{g.s.}}^{+}$,
$5/2_{1}^{+}$ ($E_{x}^{\mathrm{cal}}=1.932\,\mathrm{MeV}$), and
$3/2_{1}^{+}$ ($E_{x}^{\mathrm{cal}}=1.935\,\mathrm{MeV}$) are
$1/2_{}^{+}\rangle
=
\sqrt{0.99}\,
 0_{\mathrm{g.s.}}^{+} \otimes s_{1/2}^{\Lambda} \rangle
+
\cdots$,
$5/2_{}^{+}\rangle
=
\sqrt{0.99}\,
 2_{1}^{+} \otimes s_{1/2}^{\Lambda} \rangle
+
\cdots$, and
$3/2_{}^{+}\rangle
=
\sqrt{0.99}\,
 2_{1}^{+} \otimes s_{1/2}^{\Lambda} \rangle
+
\cdots$,
respectively.
Due to the weak coupling
between the nuclear core and the $\Lambda$ hyperon in the $s$ orbit
which is completely spherical,
the $s_{1/2}^{\Lambda}$ hyperon does not induce a mixing of nuclear core states.
\par
Figure \ref{f04} shows the DWIA calculation of the
${}_{}^{27}\mathrm{Al}\, (\gamma, K_{}^{+})$
${}_{\;\Lambda}^{27}\mathrm{Mg}$ reaction cross sections.
The ${}_{\;\Lambda}^{27}\mathrm{Mg}$ hypernucleus
can be obtained by the
${}_{}^{27}\mathrm{Al}\, (e, e_{}^{\prime} K_{}^{+})$
reaction experiment at JLab and the
${}_{}^{27}\mathrm{Al}\, (\pi_{}^{}, K_{}^{0})$
reaction experiment at JPARC.
In the future work,
comparing the DWIA cross sections of these hypernuclear productions
within the extended shellmodel framework
is valuable
to understand the structure of the ${}_{\;\Lambda}^{27}\mathrm{Mg}$ hypernucleus.
\section{Concluding remarks}
We have investigated the structures of the
${}_{\;\Lambda}^{10}\mathrm{Be}$ and ${}_{\;\Lambda}^{27}\mathrm{Mg}$
hypernuclei by using the extended shellmodel framework.
Strong coupling between $p_{}^{\Lambda}$ state and core deformation
is realized in these hypernuclei.
In ${}_{\;\Lambda}^{10}\mathrm{Be}$,
the $p_{}^{\Lambda}$ state
splits into $p_{\perp}^{\Lambda}$ and $p_{/\!/}^{\Lambda}$ states
under the strong effect of the nuclear core deformation,
and the lower $p_{/\!/}^{\Lambda}$ state comes down in energy
and couples easily with $s_{}^{\Lambda}$ state
coupled with different parity coreexcited states.
In ${}_{\;\Lambda}^{27}\mathrm{Mg}$,
the lowlying negativeparity states
show large admixture of the $p_{}^{\Lambda}$ configurations
coupled with nuclear core states having $0_{}^{+}$ and $2_{}^{+}$,
which manifests one of the general properties for the
$\Lambda$ particle plus nuclear rotor systems.
It should be remarked that the present large scale shell model
is effective
also in describing certain amount of deformed hypernuclear systems.
The wave functions are used to predict the $(\gamma, K_{}^{+})$ spectrum
for future experiments.
\acknowledgments
The author thanks Dr. T. Motoba and Dr. K. Itonaga
for collaboration and discussions.
This work was supported by JSPS KAKENHI Grant Number JP20K03950.
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\end{thebibliography}
\end{document}
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