K &=&  (\partial p_{scat}/\partial \theta)(1/p_{scat}),\\
C &=&  (\partial p_{scat}/\partial p_{beam})(p_{beam}/p_{scat}),\\
D &=& (\partial p_{scat}/\partial Q)(1/p_{scat}).
\end{eqnarray}

Here, $T, \theta$ and $Q$ are a cosine of an angle between a beam direction and a normal to a target plane, a scattering angle,
and an excitation energy, respectively. Values of $K$ and $C$  are respectively a derivative of the scattered momentum ($p_{scat}$)
to the scattering angle ($\theta$) and that to the beam momentum ($p_{beam}$). $D$ is a derivative of $p_{scat}$ to $Q$.
The position $x_f$ at the final focal plane of kaon spectrometer can be expressed as:

\begin{equation}
x_f = (\partial x_f/\partial x_0)x_0 + (\partial x_f/\partial \theta_0)\theta_0
+(\partial x_f/\partial \delta_0) \delta_0 + (\partial x_f/\partial \theta)\theta
+s_{16}DQ. \label{Eq:HIHR6}
\end{equation}

The matching conditions are obtained as follows.

\begin{eqnarray}
(\partial x_f/\partial x_0) &=& s_{11} b_{11} T + s_{12} b_{21} \rightarrow {\rm minimize},\\
(\partial x_f/\partial \theta_0) &=& s_{11} b_{12} T + s_{12} b_{22} \rightarrow 0,\\
(\partial x_f/\partial \delta_0) &=& s_{11} b_{16} T + s_{12} b_{26} + s_{16}C \rightarrow 0,\\
(\partial x_f/\partial \theta) &=& s_{12} + s_{16} K \rightarrow 0.
\end{eqnarray}

Fixing reaction kinematics and a scattering angle, magnetic component of beam line and spectrometers are tuned
to satisfy the above conditions. Degrees of freedom to be tuned in beam line and spectrometer systems are 6 and 3, respectively.
Once the matching conditions are satisfied, the excitation energy $Q$ can be given as $x_f/(s_{16} D$ with an energy
resolution of $(\partial x_f/\partial x_0 ) x_0$. In the real operation of the beam line, higher order terms of matrices
should be taken into account and the momentum matching parameter optimization will be performed with
an established parameter minimization algorism on the computer.

§.§ High Precision Spectroscopy of $\Lambda$ Hypernuclei with the $(\pi^+, K^+)$ Reaction at HIHR

§.§.§ Introduction

The major goal of nuclear physics is understanding the nature of many-body system
whose dynamics is dictated by the strong interaction. There is a hierachy of three such systems in our universe:
1) baryons/mesons, the bound system of quarks, 2) nuclei, the self-bound system of baryons and
3) neutron stars, isospin-asymmetric nuclear matter bound by the gravitational force.
The size scale of 1) and 2) is femtometers and that of 3) is  10 km. Though they differ by 10$^{19}$ order of magnitudes,
it is common that the strong interaction plays a key role to govern their structure and properties.
The strong interaction in the low energy region where the QCD is not perturbative has been investigated in the framework of
baryon potential models. The construction of NN potentials has taken advantage upon the availability of a substantial number
of scattering data. High-precision potential models fit all these data with extreme accuracy.
However, when only two-body forces are accounted for, light nuclei turn out to be under-bound and the saturation
properties of infinite isospin-symmetric nuclear matter are not correctly reproduced.
It indicates the need for a three-body interaction. The same theoretical framework could be extended to the strange sector.
Though the strange quark is heavier than the u and d quarks, it is still lighter than the QCD cut-off ($\sim$1 GeV)
unlike the heavy quarks $(c, t, b)$. Hence, the strange quark can be treated in the framework of the flavor SU(3) symmetry
which is a natural extension of the isospin symmetry for ordinary nucleons.
To devise a unified description of the baryonic interaction within the flavor SU(3) basis, one must then quantitatively
understand the hyperon-nucleon (YN) and the hyperon-hyperon (YY) interactions.
Investigation on double-$\Lambda$ and $\Xi$, $S=-2$ hypernuclei and direct measurement of YN scattering experiments
made great progresses at the existing J-PARC hadron hall.
However, the bare YN interaction and effective YN interaction in nuclei are different
due to quantum many-body effects; spectroscopic investigation of $\Lambda$ hypernuclei, nuclear many-body systems
containing one $\Lambda$ particle, provides a unique and, currently, the only practical tool to provide
effective YN interaction in various nuclei with an excellent precision.

While several models of YN forces were proposed in the past, recent observation of neutron stars with masses
of  $2M_\odot$ (two solar mass) poses several important questions.
There is a very simple argument to justify the appearance of strange degrees of freedom in the inner core of a neutron star.
In the pure neutron matter case, whenever the chemical potential becomes sufficiently large to match the chemical potential
of a hyperon in the same matter, the hyperon becomes stable since it is a distinguishable particle, and creates its own Fermi sea,
thereby lowering the kinetic energy of the system. This results in a so-called "softening" of the equation of state (EOS),
due to a decrease of the Fermi pressure. In turn, a soft EOS predicts a lower sustainable mass for a neutron star.
So far, existing YN interactions typically predict a maximum mass no larger than $1.5~M_\odot$,
in strong contrast with the astrophysical observations (the "hyperon puzzle").
The key for solving this apparent contradiction is a more repulsive YN interaction,
which increases the hyperon chemical potential, moving the onset of hyperons at higher densities.
Most models nowadays agree on this aspect, but additional constraints are needed,
and they can only be inferred from accurate spectroscopic data of hypernuclei.
The relation between hypernuclei and matter inside a neutron star is not straightforward.
It is clear that the hyperon chemical potential at high density cannot be effectively constrained
from the existing hypernuclear data with a few MeV resolution, unless very accurate measurements on hypernuclei
are performed in wide mass range. Tiny changes in the binding energy, and consequently on the determination of
the YN/YNN force, can have dramatic consequences on the EOS of matter at supra-saturation conditions.
Another interesting aspect of the YN interaction is the charge symmetry breaking (CSB).
Recently decay-pion spectroscopy of electro-produced $^4_\Lambda$H, which was originally proposed at JLab <cit.>,
was successfully carried out at Mainz <cit.> and the excitation energy of $^4_\Lambda$He $1^+$ state was also
successfully measured by the gamma-spectroscopy at J-PARC <cit.>.
These new experimental data strongly support the fact that the A=4 hypernuclear isospin-doublet has a large CSB
for the ground states ($0^+$) and small CSB for the $1^+$ excited states.
Although the origin of such a large CSB is not fully understood, it is clear that the discussion based on
two-body YN/NN interaction only does not suffice and that the inclusion of $\Lambda$N-$\Sigma$N coupling
and three-body force are essential.
Behavior of $\Lambda$ in symmetric nuclear matter and neutron-rich environments would be quite different
and  $\Lambda$N-$\Sigma$N coupling and 3-body force are important for the discussion of nuclear matter property <cit.>.
It is quite important to investigate these effects for the heavier and neutron richer hypernuclear systems,
but these effects tend to be smaller <cit.> and thus high resolution is an important key.

Though a systematic study of $A \leq 208~~\Lambda$ hypernuclei with a few MeV accuracy has already been carried out
with the conventional $(\pi^+, K^+)$ reaction spectroscopy at BNL-AGS and KEK-PS <cit.>,
the accuracy of existing hypernuclear data is not adequate to the features of the baryonic force models needed to properly address
the hyperon puzzle, and thus higher precision data are necessary.
We propose to measure $\Lambda$ binding energies of $\Lambda$ hypernuclei in wide mass region via
the $(\pi^+, K^+)$ reaction at the HIHR beam line which will be realized in the extended hadron experimental hall of J-PARC.
Precise hypernuclear spectroscopy in wide mass region is crucially important to constrain or provide necessary information
to construct a reliable interaction which can be used to calculate EOS of the neutron stars.
Such high precision hypernuclear spectroscopy is only possible at momentum dispersion match $\pi$ beam line, HIHR and
cannot be performed at other facilities.

§.§.§ Experiment of the $(\pi^+, K^+)$ hypernuclear spectroscopy at HIHR

The proposed experiment investigates $\Lambda$ hypernuclei in wide mass range
with the $^A Z(\pi^+,K^+) _\Lambda^A~Z$ reaction (Fig. <ref>).

The elementary process of the $n(\pi^+, K^+)\Lambda$.
A neutron in a nucleus is converted by this reaction to a $\Lambda$ hyperon.
Some fraction of produced $\Lambda$ is sticked to the nucleus to produce
a $\Lambda$ hypernucleus.

There are various methods for hypernuclear spectroscopy. It can be categorized in the reaction spectroscopy
such as the missing mass spectroscopy with the  $(\pi^+,K^+)$ reaction and others which include decay product
studies such as gamma and decay-pion spectroscopies. The production information is complementary
to the information obtained by decay product studies.
Reaction spectroscopy with the missing mass technique is quite powerful way of investigation of hypernuclei.
It provides reaction cross sections and detailed information of energy levels of hypernuclear states.

The hypernuclear reaction spectroscopy started with the $(K^-,\pi^-)$ reaction. An s-quark in the beam $K^-$ is
exchanged with a d-quark in a neutron to produce a $\Lambda$ in the $(K^-,\pi^-)$ reaction.
On the other hand, the $(\pi^+,K^+)$ reaction produces $s$ and $\overline{s}$ quark pair associatively and the a $\Lambda$ and
a $K^+$ are produced simultaneously. These reactions convert a neutron to a $\Lambda$, and thus both $(K^-,\pi^-)$ and
$(\pi^+,K^+)$ reactions produced the same hypernuclei with the same targets while they are exothermic and endothermic reactions.
Typical cross sections of $\Lambda$ hypernuclei for these reactions are 1 mb/sr and 10 $\mu$b/sr for
$(K^-,\pi^- )$ and  $(\pi^+,K^+)$ reactions, respectively.

Beam momentum dependence of hypernuclear
recoil momentum with a $^{12}$C target for various reactions <cit.>.

Figure <ref> shows recoil momentum transfer to the produced hypernuclei
as a function of beam momentum.
The $(K^-,\pi^- )$ reaction gives small momentum transfer, and thus a neutron is converted to a $\Lambda$
without changing angular momentum. It populates predominantly substitutional states; the produced $\Lambda$ occupies
the orbit with the same angular momentum as the converted neutron. Contrary to the  $(K^-,\pi^- )$ reaction,
$(\pi^+,K^+)$ and $(\gamma,K^+) / (e,e'K^+)$ give large momentum transfer to hypernculei and
they can populate high-spin states with a neutron hole in large angular momentum orbits and a $\Lambda$
in small angular momentum states. This feature allows $(\pi^+,K^+)$ reaction to have a large cross section to populate
deeply bound $\Lambda$ states like a ground state by converting a neutron with a high angular momentum to a
$\Lambda$ in $s$ orbit while the cross section of such a state is highly suppressed for the $(K^-,\pi^- )$ reaction
due to small angular momentum transfer.
The  $(e,e'K^+)$  reaction has similar features to populate deep bound $\Lambda$ hypernuclei as the $(\pi^+,K^+)$ reaction,
because both are endothermic reactions and momentum transfer is large. However, its hypernuclear production cross section is
extremely small, 0.1 $\mu$b/sr which is 100 times less than it of the $(\pi^+,K^+)$ reaction.
In order to compensate small production cross section, very high intensity primary electron
beam (100 $\mu$A=$6.3 \times 10^{14}~e^-/sec$) is used.
High quality electron beam allows to avoid measuring momentum and position of electron beams and combination of high-resolution spectrometers for kaon and scattered electrons make it possible to measure hypernuclear mass with a resolution of 0.5 MeV (FWHM) , however accidental coincidence background originating from the high intensity primary electron beam causes a serious problem about signal-to-noise ratio <cit.>.
So far, the $(\pi^+,K^+)$ reaction spectroscopy has limitations for resolution as well as hypernuclear yields.
Since $\pi^+$ beam is produced as the secondary particles and its large momentum spread makes analysis of
beam momentum inevitable. Intensity of $\pi^+$ beam is limited by operational rate limit of detector system on the beam line.
Due to these reasons, the energy resolution and beam intensity for the $(\pi^+,K^+)$ hypernuclear spectroscopy have
been limited to be a few MeV and several million $\pi^+$s per a second, respectively.
Introduction of HIHR will remove those limitations from the  $(\pi^+,K^+)$ $\Lambda$ hypernuclear spectroscopy and
it will realize the energy resolution of $<0.4$ MeV (FWHM) which is equivalent or better than it of $(e,e'K^+)$ spectroscopy
and such a good energy resolution enables us to determine binding energies of various $\Lambda$ hypernuclei with
an accuracy of $<100$ keV. Furthermore, there is no limitation for $\pi^+$ beam intensity.
Detail of HIHR will be given in following sections.
It should be noted that the $(\pi^+,K^+)$ reaction populates spin non-flip states in forward angles while the
$(\gamma,K^+) /(e,e'K^+)$ reaction populates both spin-flip and non-flip states because photon has spin 1.
This spin states selectivity of the $(\pi^+,K^+)$ reaction spectroscopy may simplify the analysis of heavy hypernuclei of
which density of energy levels becomes high. Since the $(\pi^+,K^+)$ reaction converts a neutron to a $\Lambda$ and
the $(e,e'K^+)$ reaction converts a proton, different hypernuclei which belong to the same iso-spin multiplet will
be studied with the same target. They are highly complementary, and it is important to have the same level of accuracy
for these reaction spectroscopic techniques in order to discuss subtle effect of charge symmetry breaking effects
of the $\Lambda$N interaction.
Table <ref> summarizes the features of $(K^-,\pi^- ), (\pi^+,K^+)$ and $(e,e'K^+)$ reaction spectroscopy of $\Lambda$ hypernuclei.
After the first proposal of the $(\pi^+,K^+)$ reaction spectroscopy by H.Thiessem <cit.>, pioneering experiments were
performed at BNL AGS <cit.> and then intensive research was carried out at KEK-PS with
SKS spectrometer <cit.>.

Reactions for $\Lambda$ hypernuclear spectroscopy

Reaction     Resolution     Cross section     Momentum     Elementary     Signal to

(MeV, FWHM)     ($\mu$b/sr) for     Transfer     process     Noise ratio

$(K^-, \pi^-)$     $>2$     1000     Small     $n (K^-, \pi^-) \Lambda$     $\bigcirc$

$(\pi^+, K^+)$     $1.5$     10     Large     $n (\pi^+,K^+) \Lambda$     $\bigcirc$

$\mathbf \rightarrow 0.4$ (HIHR)

$(e,e^\prime K^+)$     $0.5$     0.1     Large     $p (e,e^\prime K^+) \Lambda$     $\bigcirc$

Figure <ref> shows $\Lambda$ binding energy spectra of $^{12}_\Lambda$C measured with the
$(\pi^+,K^+)$ reaction at BNL-AGS, KEK-SKS and J-PARC-HIHR (expectation).
The energy resolutions are respectively
3 MeV, 1.5 MeV and 0.4 MeV (expectation) for BNL, KEK and HIHR. Comparing the spectra from BNL and KEK,
one can easily see that a factor of two improvement of the energy resolution provides much richer information.
Further improvement of energy resolution from KEK to HIHR by a factor of more than three, determination
precision of energies for major peaks which correspond to $\Lambda$ in $s$, $p$-orbits is improved drastically as well as it will
enable us to separate sub-peaks originating from core nucleus excited states. Since heavier hypernuclei have more complex
nuclear structure, better energy resolution contributes to improve sensitivity of small cross section peaks.

Figure <ref> shows $^{208}_\Lambda$Pb energy spectrum measured with SKS at KEK <cit.> and
peaks corresponding to the states with $\Lambda$ in $s,p,d,f,g,h$-orbits are barely observed for a major neutron hole series
of  $(i_(13/2) )^{-1}$ but a clear observation of these peaks was hampered by overlapping with sub-major neutron series
from a neutron hole in $(h_(9/2) )^{-1}$ where a neutron was converted to a $\Lambda$.
Due to small cross section of $^{208}_\Lambda$Pb, a thicker target (3.418 g/cm$^2$)  was used at SKS and
it results in energy resolution of 2.3 MeV (FWHM), though a thinner carbon target (0.859 g/cm$^2$) was used
to achieve so far best energy resolution of 1.45 MeV as the $(\pi^+,K^+)$ reaction.
Sub-MeV resolution is necessary to observe clear peaks for heavy hypernuclei and a high intensity $\pi^+$ beam
enables us to use a thin target enough not to deteriorate resolution; these requirements can be met only at HIHR.

In the following section, the physics justification of the measurement will be given.

$\Lambda$ binding energy spectra of $^{12}_\Lambda$C measured at BNL-AGS, KEK-SKS and HIHR(expectation).
Simulation for HIHR was carried out based on theoretical calculation for major peaks <cit.> and quasi-free $\Lambda$
production events, background events were generated to be consistent with the experimental data <cit.>.

$\Lambda$ binding energy spectra of $^{208}_\Lambda$Pb measured at KEK-SKS <cit.>.
Major neutron hole series of $(i_{13/2})^{-1}$ with a $\Lambda$ in $s, p, d, f, g, h$ orbits are barely
observed as $s_\Lambda, p_\Lambda, d_\Lambda, f_\Lambda, g_\Lambda$ and $h_\Lambda$
peaks with an energy resolution of 2.3 MeV.

§.§.§ Neutron stars and the hyperon puzzle

Neutron stars (NS) are the most compact and dense stars in the universe, with typical masses
$M \sim 1.4~M_\odot$ and radii $R \sim 10$ km. Their central densities can be several times larger
than the nuclear saturation density, $\rho_0 = 0.16$ fm$^{-3}$. Since the Fermi energy of fermions at such
densities is in excess of tens of MeV, thermal effects have little influence on the structure of NS.
Therefore, they exhibit the properties of cold matter at extremely high densities, very far from being realized
in present terrestrial experiments. In the era of multi-messenger astronomical observations, NS offers a
unique opportunity to test a broad class of theories, from nuclear physics to general relativity, including the
recent observation of gravitational waves <cit.> and X-ray hotspot measurement by NICER
which constraints on mass-radius ratio <cit.>.
These recent progresses in astronomical observations have deepened our understanding of macroscopic
features of NS, such as mass, radius, and stiffness. Therefore, microscopic understanding becomes more
important than ever; we should answer what kind of physics determines the size and mass of NS and why NS is so stiff.

Chemical potentials $\mu$, and concentrations $Y$ of the stellar constituents
in hyperonic matter as a function of the baryon density <cit.>.

From the surface to the interior of a NS, stellar matter undergoes a number of transitions. From electron and neutron-rich ions
in the outer envelopes, the composition is believed to change into a degenerated gas of neutrons, protons, electrons, and
muons in the outer core. At densities larger than $\sim 2\rho_0$, new hadronic degrees of freedom or exotic phases are
likely to appear. Figure <ref> shows the chemical potentials and concentrations of stellar constituents in beta-stable
hyperonic matter as a function of baryon density, obtained from a recent theoretical calculation employing modern baryonic
potentials <cit.>.
The appearance of hyperons in the core of a NS was already advocated in 1960 <cit.>. In the degenerate dense matter
forming the inner core of a NS, Pauli blocking would prevent hyperons from decaying by limiting the phase space
available to nucleons. When the nucleon chemical potential is large enough, the conversion of nucleons into hyperons
becomes energetically favorable. This results in a reduction of the Fermi pressure exerted by the baryons and a softening
of the equation of state (EOS). As a consequence, the maximum mass determined by the equilibrium condition between
gravitational and nuclear forces is reduced. The value of about $1.5~M_\odot$ for the maximum mass of a NS, inferred
from neutron star mass determinations <cit.>, was considered the canonical limit, and it was compatible with most
EOS of matter containing strangeness. However, the recent measurements of the large mass values of the millisecond pulsars
J1614-2230 ($1.97(4)~M_\odot$)

<cit.> PSR J0348+0432 ($2.01(4)~M_\odot$) <cit.>

MSP J0740+6620 ($2.14(20)~M_\odot$)

require a much stiffer equation of state.

This seems to contradict the appearance of strange baryons in high-density matter given what is known at present about
the hyperon-nucleon interaction. This apparent inconsistency between NS mass observations and theoretical calculations
is a long-standing problem known as “hyperon puzzle”. Its solution requires better understanding of the
hyperon-nucleon (YN) interaction in a wide range of systems from light to medium and heavy hypernuclei as well
as more accurate theoretical calculation frameworks.

Currently there is no general agreement among the predicted results for the EOS and the maximum mass of NS including
hyperons. This is due to incomplete knowledge of the interactions governing the system (both two- and three-body forces
for hypernuclei) as well as difficulties of theoretical treatment of many-body systems. There are theoretical calculations
which were applied to the hyperonic sector and resulted in the appearance of hyperons at around $2-3~\rho_0$, and
a strong softening of EOS, implying a sizable reduction of the maximum mass <cit.>. On the other hand,
other approaches suggest much weaker effects arising from the presence of strange baryons in the core of the
star <cit.>.

Rich nucleon-nucleon (NN) scattering data allow one to derive satisfactory models of two-body nuclear forces,
either purely phenomenological <cit.> or built on the basis of an effective field theory <cit.>.
In contract to NN scattering, quite limited scattering data are available for hyperon-nucleon (YN) scattering, though a
$\Sigma$N scattering experiment was recently performed at J-PARC <cit.> and a new $\Lambda$p scattering experiment
is planned. Furthermore, there is no scattering data for $\Lambda$-neutron an additional information on the three-nucleon interaction is inferred from saturation properties of
symmetric nuclear matter (Urbana IX force <cit.>), the resulting PNM EOS turns out to be stiff enough to be compatible
with astrophysical observations <cit.>.

Recent analysis of $^{16}$O+$^{16}$O scattering data shows that the established meson exchange
potential model (Nijmegen ESC08c <cit.>) cannot reproduce the cross section at large scattering angles where
density of the system becomes large and inclusion of 3-body force (TBF) solves the problem as shown in Fig. <ref> <cit.>.

Differential cross sections for $^{16}$O+$^{16}$O elastic scattering at $E/A = 70$ MeV <cit.>.

Energy as a function of baryon density <cit.>. The vertical dotted lines indicate the
$\Lambda$ threshold densities. In the inset, neutron
and $\Lambda$ fractions correspond to the two hyper-neutron matter EOSs.

It is a quite promising scenario to solve the hyperon puzzle that inclusion of such 3-body repulsive force in the $\Lambda$NN
interaction makes EOS harder or prevents cross over of chemical potentials of neutron and $\Lambda$ to suppress the
appearance of $\Lambda$ in NS.

Figure <ref> shows one example of EOS of neutron star calculated by quantum Monte Carlo method <cit.>.
The EOS of NS without hyperons (Pure Neutron Matter; green) is hard but inclusion of $\Lambda$ makes the EOS soft ($\Lambda$N; red)
where density of system becomes higher ($\rho >1.5~\rho_0 \sim 0.24$ fm$^{-3}$),  and inclusion of 3-body repulsive force (blue),
stiffness of EOS recovers. Inset plot shows neutron and $\Lambda$ fractions. One can see that the 3-body repulsive
force pushes the $\Lambda$ appearance threshold up.

Thus, there is a general indication that 3-body repulsive forces become quite significant at high density, and it harden
the EOF of neutron stars. The binding energies of light hypernuclei with high statics YN scattering data are necessary
to construct realistic YN interaction models, but they are not enough to solve the hyperon puzzle.
Additional information must necessarily be inferred from the properties of medium and heavy hypernuclei
in order to extrapolate to the infinite-mass limit for discussion of highly massive asymmetric nuclear matter such as
neutron stars and strange hadronic matters ($n_u \sim n_d \sim n_s$).

Such information is essential to extrapolate the hyperon behavior to the infinite limit and thus to reliably predict neutron star

§.§.§ Theoretical models of structure calculation of $\Lambda$ hypernuclei

Except for very light $\Lambda$ hypernuclei which ab-inito or detailed cluster calculation can handle,
the structure of $\Lambda$ hypernuclei is generally studied by employing the weak-coupling
approximation that assumes the wave function of a $\Lambda$ hypernucleus to be decomposed into a core nucleus and
a $\Lambda$ hyperon. In this picture, the hypernuclear Hamiltonian consists of the Hamiltonian for the core-nucleus,
the $\Lambda$ kinetic energy and the sum of $\Lambda$N interaction terms that can be derived with various theoretical frameworks.

Realistic nuclear interaction models have been constructed based on rich nucleon-nucleon (NN) scattering data
but scarce hyperon-nucleon (YN) and completely no YY scattering data limits construction of reliable baryonic interaction models.
Therefore, the analysis of hypernuclear binding energies with the limited number of available hyperon-nucleon (YN) scattering
data play key role to formulate various baryon-baryon interaction models. Traditionally, interaction models based on meson exchange
picture such as Nijmegen ESC <cit.> and Bonn-Jülich <cit.> are widely used.
Recently SU(3) chiral effective field theory (ChEFT) <cit.> attracts attention since it allows improvement of precision
of the calculation in systematical way by power counting and two- and three-body forces are derived in a consistent way.
Recent progresses of lattice QCD calculation with massive CPU power computers enable us also to derive the baryonic
interaction potential models <cit.> and coming closer to a level where they give additional constraints
on the baryonic interaction, however, direct calculation of light baryonic systems by lattice QCD is still challenging task.

Ab-initio calculations and microscopic cluster calculations enable us to use these bare interactions to calculate the binding energies
of light hypernuclear systems <cit.>.
However, analysis of heavier hypernuclei needs effective interactions which have been derived from the bare $\Lambda$N force
by various calculation techniques such as G-matrix methods based on Bruckner theory <cit.> because heavier nuclear systems
are too complicated for those microscopic calculations, and it is practically impossible to take all quantum many-body effects
into account.

In order to perform theoretical analysis of binding energy information of hypernuclei in wide mass range, following information
are necessary: 1) reliable two-body baryonic interaction, 2) systematic method to convert bare two-body interaction to the
effective interaction which takes quantum many body effects adequately into account, and 3) nuclear structure calculation
with the derived effective interaction model.

Let us give one example of such calculations: a meson exchange baryon-baryon interaction, ESC08c <cit.>, which takes
two-meson and meson pair exchanges as well as various one boson exchanges into account with broken SU(3) symmetry,
is selected as the bare baryon-baryon interaction. As already recognized in the non-strange sector by the study of $^{16}$O+$^{16}$O
scattering data <cit.>, the ESC08c model needs to be supported by a 3/4-body repulsive force originating from multi-Pomeron
exchange (MPa) that can be universally extended to the strange sector. Finally, effective YN and YY interaction models
were constructed from ESC08c+MPa by Gaussian parametrization of the G-matrices theory in nuclear matter and
EOS of neutron star was constructed. Then, Tolmann-Oppenheimer-Volkoff (TOV) equation was solved for the hydrostatic
structure and mass-radius relations of neutron stars was obtained <cit.>.
Figure <ref> shows neutron star masses as a function of the radius. Dotted black curve is for ESC only without
three body forces and it was shown that inclusion of hyperon makes EOS very soft. Inclusion of three-body force makes
EOS stiffer and maximum mass of NS becomes larger; the case for MPa, two solar-mass NS can be supported.

Neutron star masses as a function of the radius <cit.>. Dotted black curve is for ESC
only without three body forces and it was shown that inclusion of hyperon makes EOS very soft.
Red solid line shows the case with multi-body forces, MPa. Blue
and green lines are different parameter set of multi-body forces <cit.>.

Energy spectra of various $\Lambda$ hypernuclei. Experimental data were marked by
open circles and theoretical calculation results with 3-body force (MPa) is red solid line and without
3-body force (ESC) is black dots <cit.>.

Obtaining $k_f$-dependent local potential derived from the G-matrices calculation for the interaction which recovered
the stiffness of NS, hypernuclear energy spectra were calculated with Skyrme Hatree-Fock wave functions (Fig. <ref>).
Figure <ref> shows the effects of three-body force which were obtained from the $\Lambda$ binding energies
in $s$-orbit with multi-body force (ESC+MPa) and without it (ESC only). Energy differences of the experimental data
obtained by previous $(\pi^+,K^+)$  spectroscopy and calculated values without multi-body force were plotted, too.
Though 3-body force plays a significant role for NS, the effects for hypernuclear energies are small, typically less than 1 MeV.
It is clear that the experimental data have too large errors as well as ambiguities to constraint theoretical models.
Experiments at HIHR will provide $\Lambda$ binding energies in wide mass region with an accuracy of $< 100$ keV.
It should be noted that similar discussions for $\Lambda$ binding energies for $p,d,f,...$-orbits are possible and
interaction models are further constrained with those information when reliable $\Lambda$ binding energies are obtained at HIHR.

Expected effects of 3-body repulsive $\Lambda$NN force for various $\Lambda$ hypernuclei <cit.>.
Open black circles are previous data obtained by previous  $(\pi^+, K^+)$ spectroscopy of $\Lambda$
hypernuclei. Recently possible ambiguity of 0.5 MeV was pointed out <cit.> for all
these data and blue circles shows the experimental data with this ambiguity.
Expected precision of experiments at HIHR was shown by error bars of closed circles.

There are variety of theoretical calculations depending on choice of bare interactions and treatment of quantum many-body problem.
The effective G-matrix potential from the Nijmegen potential including many-body force was combined
with a microscopic calculation, Hyper-AMD to calculate the binding energies of  $^{12}_\Lambda$B, $^{13}_\Lambda$C,
$^{16}_\Lambda$O, $^{28}_\Lambda$Si, and $^{51}_\Lambda$V <cit.>.
The different version of Nijmegen potential with Pomeron and Odderon exchanges (ESC16) with the G-matrix calculation <cit.> was also carried out.

Recently, the auxiliary field diffusion Monte Carlo (AFDMC) technique for strange systems has made substantial progresses.
By using this microscopic ab-initio approach, an accurate analysis of the $\Lambda$ separation energy of light- and
medium-heavy hypernuclei has been carried out <cit.> using a phenomenological interaction <cit.>
in which the two-body potential has been fitted on the existing $\Lambda$p scattering data.
As shown in Fig. <ref>(a), when only the two-body $\Lambda$N force is considered (red curve), the calculated hyperon separation energies tend to
disagree with the experimental data (green curve) as the density increases. This has a sizable effect on the predicted NS structure, Fig. <ref>(b).
The inclusion of the three-body $\Lambda$NN force in this scheme leads to a satisfactory description of the hyperon separation energies in a wide mass range
and for the $\Lambda$ occupying different single particle state orbitals ($s, p$ and $d$ wave), as shown in Figs. <ref>(b).
The resulting EOS spans the whole regime extending from the appearance of a substantial fraction of hyperons at $\sim 2\rho_0 \simeq 0.32$ fm$^{-3}$
to the absence of $\Lambda$ particles in the entire density range of the star, as shown in Fig. <ref>(a).

Recently SU(3) Chiral Effective Field Theory (ChEFT), which based on chiral symmetry of QCD Lagrangian, attracts wide
attention <cit.>. ChEFT established a prescription to extend the interaction model to include many-body forces.
Performed next-to-leading order (NLO) calculations already included some components of the 3-body force (the ones that can be reduced to 2-body terms),
but next-to-next-to-leading order (NNLO) calculations are necessary for the inclusion of a genuine 3-body interaction, and more experimental inputs
such as high statistics YN scattering data to be available at K1.1 beam line of J-PARC extended hadron hall and high precision binding energy information
of light hypernuclei, are required to constraint the Low-Energy Constants which are necessary parameters to make ChEFT predictable.

(a) Equations of state. The vertical dotted lines indicate the $\Lambda$ threshold densities.
In the inset, neutron and $\Lambda$ fractions correspond to the two hyper-neutron matter EOSs.

(b) Mass-radius relations given by AFDMC. Closed circles represent the predicted
maximum masses. Horizontal bands at $2 M_\odot$ are
the observed masses of the heavy neutron stars

EOS and neutron star mass-radius relations calculated by AFDMC <cit.>.

(a) Experimental $B_\Lambda$ values in s wave and AFDMC calculation results with
2-body $\Lambda$N interaction alone, and two different
parametrizations of the 3-body YNN interaction (updated from <cit.>).

(b) Experimental results for $\Lambda$ in $s, p, d, f$ and $g$ orbits.
Red open circles are the AFDMC results obtained including the most recent 2-body plus 3-body hyperon-nucleon
phenomenological interaction model (updated from <cit.>).

$\Lambda$ separation energies as a function of $A^{-2/3}$ calculated by AFDMC <cit.>.

Although the above calculations are based on different bare potentials: meson exchange Nijmegen potential, phenomenological one or ChEFT,
and they adopt different approaches how to apply these potentials to hypernuclei: G-matrix calculation, microscopic ab-initio quantum Monte Carlo
calculation, the general tendencies of results are consistent.
They predict that inclusion of 3-body repulsive force makes relatively small differences in the $\Lambda$ separation energies of hypernuclei but it gives
dramatically different results for the properties of the infinite medium like a NS: hardening of EOS or even suppression of the hyperon
appearance <cit.>.

However, while 2-body baryonic force models based on different theoretical frameworks are expected to be reasonably accurate
from new data of YN scattering at K1.1 J-PARC and information of light hypernuclei, detailed information on the 3-body hyperon-nucleon interaction
is still missing. This lack of knowledge is to be attributed to a poor experimental information for medium to heavy hypernuclei,
which are the key to infer properties of the infinite hyper-nuclear matter. Therefore, in order to properly assess the role of hyperons in
NSs and reconcile theoretical predictions with astrophysical observations, $i.e.$ solve the hyperon puzzle, precise experimental investigation on light,
medium and heavy targets is of paramount importance. In order to provide such accurate experimental data efficiently, realization of HIHR beam line is essential.

§.§.§ Choice of targets, additional physics outputs

Other proposals for wide variety of targets will follow after the success of the proposed experiment but let us limit the choice targets for
the first campaign of hypernuclear research at HIHR for now.
As shown in Fig. <ref>, 3-body force effect depends on mass number of hypernuclei, choice of targets should cover wide mass range from
light to heavy targets. As discussed in section <ref>, thin targets are necessary to realize a high-resolution experiment.
Therefore, self-supporting solid targets with a thickness of about millimeters were chosen for the first campaign of hypernuclear experiments at HIHR.
Surely, the main goal of the proposed experiment is to solve the hyperon puzzle by systematic study of $\Lambda$ hypernuclei in wide mass range,
but physics outputs of high resolution spectroscopy of $\Lambda$ hypernuclei are not limited to that.
So far $^{12}$C target was used as a reference for the $(\pi^+,K^+)$ spectroscopy of hypernuclei due to its easy handling.
Because the proposed experiment would be the first experiment at HIHR, beam line commissioning including dispersion matching parameters’ optimization
will be carried out.

Li, Be, B targets can be prepared as self-supporting targets and they are theoretically within scope of precise few body calculation techniques,
such as microscopic cluster model <cit.> and no-core shell model calculation which was extended to $^7_\Lambda$Li  for hypernuclei <cit.>
and to $^{12}$C for normal nuclei <cit.>.
Importance of $\Lambda$N-$\Sigma$N coupling is widely recognized for discussion of Charge Symmetry Breaking (CSB)
of $\Lambda$ hypernuclei  <cit.>. For heavier and neutron richer hypernuclear systems, such $\Lambda$N-$\Sigma$N coupling and
3-body force become more important, implying that the behavior of $\Lambda$ in symmetric nuclear matter and neutron-rich environments would
be quite different. It was pointed out that the $\Lambda$N-$\Sigma$N coupling is quite important also for the discussion of NS <cit.>.
Since there is high resolution ($< 1$ MeV resolution) spectroscopic experiments with electron beams for $^7_\Lambda$He <cit.>, $^9_\Lambda$Li <cit.>
and $^{10}_\Lambda$Be <cit.> at JLab, and thus measurement of their isospin multiplet partners, $^7_\Lambda$Li, $^9_\Lambda$Be and
$^{10}_\Lambda$B are important to study with same or better energy resolution at HIHR; such precise data will enable us to discuss CSB and
the $\Lambda$N-$\Sigma$N coupling in detail.
The region of medium mass hypernuclei is also interesting. Precision measurement of them will clarify the single-particle behavior of a $\Lambda$ hyperon
in the nuclear system, investigation of baryonic many-body system with strangeness degree of freedom, and the effective $\Lambda$-N interaction.
Dynamics of a core nucleus, which is a many-body system of protons and neutrons, couples to a $\Lambda$ hyperon motion which is free
from Pauli effects of nucleons, and then new symmetric states, “genuine hypernuclear states”, will appear <cit.>.
Deformation of nuclei can be studied by measuring these states.

Though recent progresses of microscopic calculations such as Hyper-AMD <cit.> and AFDMC <cit.> make it possible to discuss medium-heavy
hypernuclei, the mean-field picture is also important.

$^{40}$Ca <cit.> and $^{51}$V <cit.> spectra obtained by the $(\pi^+, K^+)$ reaction.

In the investigation of hadronic many-body systems with strangeness, there is a fundamental question, “to what extent does a $\Lambda$ hyperon keep its
identity as a baryon inside a nucleus?” <cit.>. Spectroscopic data in heavier hypernuclei can help to answer this question. Indeed, the relevance of
the mean-field approximation in nuclear physics is one of the prime questions related to the role that the sub-structure of nucleons plays in the nucleus.
The mean-field dynamics dominates the structure of medium to heavy nuclei and $\Lambda$ hypernuclei prove the existence of single-particle motion
from the deepest $s$-orbit up to large $\Lambda$ valence orbits. The existing data from $(\pi^+, K^+)$ reactions obtained at KEK, however, do not
resolve the fine structure in the missing mass spectra due to limited energy resolution, and theoretical analyses suffer from those uncertainties
as shown in Fig. <ref>. The improved energy resolution at HIHR, which is comparable to the spreading widths of the excited hypernuclear states,
will provide wide variety of information. For example: 1) differentiation between the effects of the static spin-orbit potential and dynamical self-energies
due to core polarization; 2) access to collective motion of the core nucleus, namely deformation of the core nucleus, utilizing the $\Lambda$ as a probe;
3) modify energy levels of a core nucleus by adding a $\Lambda$ as an impurity and so on.
Effective masses of a $\Lambda$ hyperon in the nuclear potential will be obtained, which appear to be closer to that of the free value in contrast
to the case of ordinary nuclei. Therefore, the proposed precision measurement of the single particle levels can address the degree of
non-locality of the
effective $\Lambda$-Nucleus potential and can be compared, for example, with the advanced mean field calculations based on the quark-meson
coupling (QMC) model <cit.> and on DDRH <cit.>. This can be related to the nature of the $\Lambda$N and $\Lambda$NN interactions,
and to the $\Lambda$N short range interactions <cit.>. In a more exotic way, the binding energies were discussed in terms of the
distinguishability of a $\Lambda$ hyperon in the nuclear medium, which will result in a different $A$ dependence of the binding energy
as suggested by Dover <cit.>.
Lead target is a holy grail for the $\Lambda$ hypernuclear spectroscopy. The $^{208}$Pb nucleus is doubly magic nucleus and heaviest
target which can be easily available as a self-supporting plate. Its charge density distribution is nearly constant for a very large fraction ( 70%)
of the nuclear volume as shown in Fig. <ref> <cit.>, and its properties were expected to reflect those of uniform nuclear matter.
List of targets for the proposed experiment will be compiled in section <ref> with the requesting beamtime.

Charge density distribution of doubly closed-shell nuclei <cit.>.

§.§.§ Experimental setup

Details of HIHR beam line were already given in section <ref>.
Detection of 0.7 GeV/$c$ $K^+$ is possible with the well-established experimental techniques.
Detector package based on it of SKS in K1.8 <cit.> can be adopted, namely a plastic scintillation counter for
time-of-flight measurement, aerogel and lucite Cherenkov counters for kaon identification at the trigger level and
a drift chamber system to measure positions and angles of $K^+$s at the focal plane of the spectrometer.
As discussed in section <ref>, position resolution of 0.2 mm (rms) which was achieved by the SKS detector
is good enough for the tracking devices at HIHR.

A set of plastic scintillation walls are placed just downstream of the focal plane
in GEANT4 simulation model (detailed discussion about GEANT4 simulation will be given
in section <ref>).

Figure <ref> shows a set of plastic scintillation walls placed just downstream of the kaon spectrometer’s focal plane
with a distance of 1 meter. Particle hit position distribution was estimated with GEANT4 simulation. From the hit position distribution
shown in Fig. <ref>, TOF walls should cover at least 1200 mm$\times$200 mm area.
Scatter plot for time of flight between TOF1 and TOF2, and energy deposit in TOF1 for $0.69-0.73$ GeV/$c$
$\pi^+, K+$ and $p$ are given in Fig.<ref>. Central values of TOF for $\pi^+, K^+$ and $p$ are respectively 3.39, 4.09 and 5.66 ns.
The momentum acceptances of them make their widths as 0.11, 0.11 and 0.19 ns (rms).
Assuming a standard resolution of 0.15 ns for TOF counters, more than 3.7$\sigma$ separation of $K^+$ from $\pi^+$ and $p$ is possible.
GENAT4 simulation shows a conventional counter technique can separate them clearly.
At the trigger level of the experiment, aerogel Cherenkov counters are used for pion rejection and
lucite Cherenkov counters will separate kaons from protons.

Hit position distribution at the front TOF wall.

Time of flight between two TOF counters vs. energy deposit in TOF1nt TOF wall.

§.§.§ Expected Yield of Hypernuclei

The hypernuclear spectroscopy with the $(\pi^+, K^+)$ reaction at the novel dispersion match beam line HIHR aims
to investigate hypernuclei in wide mass range with a sub MeV resolution while maintaining good hypernuclear yield as well as signal/accidental ratio.
In this section, yield of hypernuclei is estimated based on the current conceptual design of HIHR.

§.§.§ Primary target and $\pi^+$ beam extraction

A beam spot size of the primary target (T1, 66 mm length gold target) in the existing the hadron hall
is $2.5^H \times 1.0^V$ mm$^2$.
It is known that optimization of the primary beam line magnetic components, the beam spot size on the beam optics study
shows that beam size on the T2 target can be maintained as it on T1 even taking multiple scattering effect into account (Fig. <ref>).

Optics study of the beam transport between T1 and T2 primary targets.
Taking multiple scattering into account, beam spot size of $2.5^H \times 1.0^V$ mm$^2$ can be maintained.

The horizontal beam image $f(x)$ is expressed by the convolution of the horizontal primary beam distribution assumed as
a gaussian on the target and a flat distribution with a target image length $T \cos \theta_{ex}$. Here, $\theta_{ex}$ is a beam extraction angle.

\[
f(x)= \int^{(T \sin \theta_{ex})/2}_{(-T \sin \theta_{ex})/2} \exp\left( \frac{-(x-u)^2}{2(x_{ptarg})^2} \right) \cos \theta_{ex}~du.
\]

Let us calculate root mean square of $f(x)$ assuming a platinum target of 60 mm length ($T=60$ mm) of which material thickness
is equivalent to it of 66 mm gold.
Figure <ref> shows the $\theta_{ex}$ dependence of root mean square of horizontal beam image distribution $f(x)$.
Though smaller extraction angle $\theta_{ex}$ and beam size ($x_{ptarg}$) on the target are favored in terms of beam image size
which affects directly the energy resolution of hypernuclear spectroscopy, careful optimization is necessary to have a reasonable $\pi^+$
yield, beam spot size and not to disturb other beam lines in the extended hadron hall.
It should be noted that vertical beam image size is almost independent of target length and beam extraction angle.

Beam extraction angle ($\theta_{ex}$) dependence of horizontal beam image size.
Energy resolution of HIHR is highly affected by horizontal beam image size but not by vertical one.
Platinum target of 60 mm length is assumed as the primary target.
Blue line is for the same beam spot size on existing T1 target ($\sigma_x=2.5$ mm, $\sigma_y=1$ mm)
and orange line is for ($\sigma_x= 1.0$ mm, $\sigma_y=2.5$ mm) option.

Figure <ref> shows expected beam intensities in the case that a 30-GeV, 50-kW primary proton beam is irradiated on
a 60-mm thick platinum target. About $2.5 \times 10^8$ $\pi^+$s per beam spill for 1.1 GeV/$c$ $\pi^+$ with the beam extraction
angle of $\theta_{ex}= 6$ degrees.
Figure <ref> shows the extraction angle dependence of $\pi^+$ intensity.
The beam extraction angle should be optimized in a range of $3-6$ degrees taking other beam lines conditions into account.
For now, extraction angle of 3 degrees for 50-kW primary beam and $\pi^+$ intensity of $2 \times 10^8$  $\pi^+$/spill
at the experimental target are assumed for following hypernuclear yield estimation.
It should be noted that no rate limitation exists for detector system of HIHR and beam with a higher pion rate is desirable
if future improvement of accelerator makes it possible.

Expected beam intensity as a function of beam momentum for 30-GeV, 50-kW primary
proton beam is irradiated on a 60-mm thick platinum target and pion beam is extracted with an angle of 6 degrees.
The calculation was performed with the Sanford-Wang formula <cit.>.

Expected beam intensity as a function of beam extraction angle for 30-GeV, 50-kW
primary proton beam is irradiated on a 60-mm thick platinum target,
calculated by using the Sanford-Wang formula <cit.>.

§.§.§ Solid angle estimation of kaon spectrometer

Angle distribution of $K^+$ with a momentum of 0.725 GeV/$c$ at the experimental target.
Events are selected for the particles reached
to the focal plane. $x_{tgt}'$ and $y_{tgt}'$ are dispersive and non-dispersive angles, respectively.

Solid angle of the kaon spectrometer of HIHR was estimated based on GEANT4 Monte Carlo simulation.
Effective width and gap of dipole magnets (D1S, D2S) of the kaon spectrometer were assumed as 100 cm $\times$ 20 cm, respectively.

Figure <ref> shows angular distribution of 0.725 GeV/$c$ $K^+$s passed through the kaon spectrometer.
Solid angle of the spectrometer was estimated as $\Omega = 4 \pi N_{fp}/N_{gen}$, where $N_{fp}$ is the number of $K^+$s
which reach to the focal plane and $N_{gen}$ is the number of uniformly generated $K^+$s in the simulation.

Figure <ref> shows momentum dependence of the solid angles of the kaon spectrometer with and without
vertical angular selection at the experimental target. There would be a room for optimization, but resolution for the events
with larger $|y_{target}' |$  tends to be worse. Therefore $|y_{target}' | < 0.1$ rad is now assumed to make the solid angle
flat over kaon momentum acceptance. Applying this cut, solid angle is about 20 msr over entire momentum acceptance.

Figure 4-6: Solid angle of the kaon spectrometer as a function of kaon momentum.
Red dots show the solid angle with a selection of $|y_{target}' |<0.1$ rad and block ones without the selection.
On the experimental target, uniform distribution of reaction points
for $|x_{target} |<50$ mm was assumed for this simulation.

§.§.§ Yield estimation of $^{12}_\Lambda$C

Reaction spectroscopy of $\Lambda$ hypernuclei normally utilize a carbon target as a reference and commissioning of setup
because easy handling of solid target and nuclear structure of $^{12}_\Lambda$C has been well studied.

It is assumed that primary proton of 30 GeV, 50 kW bombarded the target of platinum 60 mm length with a beam spot size of
$2.5 \times 1.0$ mm$^2$. Beam repetition cycle of 5.2 s and the meson extraction angle of 3 degrees was assumed.
The experimental target of $^{12}$C of 400 mg/cm$^2$ is assumed to achieve about 400 keV energy resolution
as discussed in the next section.
Cross section of $^{12}$C$(\pi^+, K^+)^{12}_\Lambda$C is assumed as 8.1 $\mu$b/sr which is an average of $2-14$ degrees
for kaon emission angles <cit.>. In the proposed experiment, kaon emission angle of 0 degrees will be measured and
thus cross section is expected to be slightly larger than the assumed value.
Key parameters for $^{12}_\Lambda$C yield estimation are summarized in Table <ref>.

Yield estimation of $^{12}_\Lambda$C hypernucleus at HIHR

Primary proton beam     30 GeV, 50 kW

repetition cycle 5.2 s

Central momentum of $\pi^+$ beam     1.1 GeV/$c$

$\pi^+$ beam rate at target     $3.85 \times 10^7$ /s

(200 M $\pi^+$/spill

Reaction     $^{12}$C$(\pi^+, K^+)^{12}_\Lambda$C

Cross section of ground state of $^{12}_\Lambda$C     8.1 $\mu$b/sr

Central momentum of $K^+$     0.71 GeV/$c$

Solid angle of kaon spectrometer     $>20$ msr

Kaon survival ratio     0.12

11.4 m for QSQDMD

Estimated count rate of $^{12}_\Lambda$C$_{gs}$     53.1 counts/hr

§.§.§ Resolution estimation

Detailed design of HIHR and target will be optimized in collaboration with J-PARC beam line group.
The energy resolution estimation was carried out with an optics code TRANSPORT and Monte Carlo package GEANT4.

§.§.§ TRANSPORT, optics code study

The following factors contribute to the total energy resolution of the $(\pi^+, K^+)$ reaction spectroscopy experiment.

* Beam momentum resolution.

Though a separate discussion of beam momentum resolution is not necessary to evaluate the total energy resolution of
$\Lambda$ hypernuclei for a dispersion matching experiment, the beam momentum resolution plays an important role
in calibration as well as a possible option of non-dispersion matching experiment. Therefore, it is useful to estimate beam
momentum resolution based on the first order beam optics. Assuming beam image size at IF2 as $\sigma_x = 1$ mm,
magnification $(x_{tgt} |x_0)=1.3$ and dispersion $(x_{tgt} |\delta)=11.28$ cm/%, momentum resolution is estimated
as $dp/p = 2.4 \times 10^{-4}$ for a 1.1 GeV/$c$ $\pi^+$ beam. It corresponds to 250 keV/$c$ (FWHM).

* Kaon spectrometer momentum resolution.

Similar to the beam momentum resolution, separate discussion of kaon spectrometer is not necessary to estimate
the total energy resolution of HIHR. Position resolution of 0.2 mm (rms) and tracking efficiency of 99.8% is assumed
as they are achieved for SKS tracking chamber at K1.8 beam line of J-PARC <cit.>.
Dispersion $(x_{fp} |\delta)=10.72$ cm/% and a point source are assumed, momentum resolution of
$dp/p = 1.9 \times 10^{-4}$ for a 0.71 GeV/$c$ $K^+$ beam. It corresponds to 32 keV/$c$ (FWHM).
Beam size on the target will contribute in addition to this value. With magnification of $(x_{fp} |x_{tgt})=1.84$ and reaction point uncertainty
of 1.3 mm, momentum resolution including this contribution is estimated as 370 keV/$c$ (FWHM).
Momentum dispersion match operation needs a thin solid target but additional tracking device after the target will make it
possible to optically separate beam and kaon spectrometers for non-dispersion matching operation.
It should be noted that high resolution (sub-MeV) spectroscopy is also possible for a thick gaseous target with
this non-dispersion matching operation of HIHR at the cost of maximum $\pi^+$ intensity.

* Energy loss and straggling in the target.

Since we have no information of reaction point in a solid target (typically the thickness of the target is less than
a millimeter for a material thickness of $100-400$ mg/cm$^2$), energy loss of charged particles can be corrected only
as an average. Its distribution including straggling will contribute the final mass resolution.
Most probable energy loss ($\Delta_p$) and straggling ($w$) were calculated with Landau-Vavilov formula and compiled in
Tab. <ref>. Angular distribution affects the effective thickness of the target, however, horizontal angular acceptance
of the kaon spectrometer is $\pm50$ mr and this effect is estimated as $\{1/\cos(5 \times 10^{-2})\}-1|\simeq 1.3 \times 10^{-3}$
which can be safely neglected.

With the above estimation into account, the second order ion optics calculation with TRANSPORT was performed.

Most probable energy loss and straggling for typical solid targets with a thickness of 50/100/200 mg/cm$^2$ .
They correspond to the cases for reaction at the center of 100/200/400 mg/cm$^2$ thick target. Unit is keV.

Target     Thickness    2|c|1.1 GeV/$c$ $\pi^+$     2|c|0.71 GeV/$c$ $K^+$

(mg/cm$^2$)     $\Delta_p$ (MPV)     $w$ (FWHM)     $\Delta_p$ (MPV)     $w$ (FWHM)

$^{10}$B     200     136     31     124     36

$^{12}$C     100     65     16     59     18

$^{12}$C     200     136     31     124     36

$^{12}$C     400     283     62     260     72

$^{28}$Si     400     258     62     232     72

$^{51}$V     400     221     56     196     65

$^{89}$Y     400     203     55     176     63

$^{139}$La     400     182     51     156     59

$^{208}$Pb     400     162     49     135     57

Horizontal position distribution at the focal plane calculated by the second order optical calculation.

Figure <ref> shows horizontal position distribution of $K^+$s emitted at the focal plane calculated
by the second order optical calculation. Kaon emission angle at the experimental target was assumed as $\theta_K=0$ and
energy loss and straggling effects in the target were not included in this calculation. Therefore, this result reflects purely beam line
and spectrometer optical features. The first peak corresponds to the ground state of $^{12}_\Lambda$C and
the second peak is an artificially created excited state with $E_x= 5$ MeV.
Peak width was fitted as $\sigma = 0.161$ cm and it can be converted to the energy resolution (optics) as:

\[\Delta E_{opt} = 2\sqrt{2 \ln 2} \times 0.161~{\rm cm} \times \frac{5~\rm MeV}{6.02~\rm cm} = 315~{\rm keV (FWHM)}.\]

This resolution shows ideally achievable value for a very thin target with complete dispersion matching conditions are satisfied.

Let us summarize results of energy resolution studies with the optics code TRANSPORT in Table <ref>.
Taking  energy loss and straggling effects into account, energy resolution is estimated as better than 0.4 MeV
for the ground state of $^{12}_\Lambda$C.

Summary of energy resolution study with TRANSPORT

Optical code

Reaction      $^{12}$C$(\pi^+, K^+)^{12}_\Lambda$C

$\pi^+$ beam line momentum resolution (FWHM)     0.25 MeV/$c$

$K^+$ spectrometer momentum resolution (FWHM)     0.37 MeV/$c$

Energy resolution of      0.32 MeV

Energy loss and straggling     0.09 MeV

in target (100 mg/cm$^2$ $^{12}_\Lambda$C)

Mass resolution of $^{12}_\Lambda$C     0.33 MeV

§.§.§ GEANT4, Monte Carlo simulation study

As shown in Fig. <ref>, the HIHR dispersion matching beam line after IF2,
an experimental target and the kaon spectrometer were modeled in GEANT4 simulation code.
Optics code is useful for quick optimization of magnet parameters, and it is suitable to examine
the rough behavior of the system. On the contrary, Monte Carlo simulation takes longer time and
tunes of beam element parameters are not straightforward as the optics code does, but it is necessary to
investigate the behavior of a complex system in which various factors are intertwined.
Ideal magnetic fields for dipole, quadrupole and sextupole magnets were now assumed in the model and
they should be replaced by more realistic fields map calculated by 3D finite element method such as TOSCA
or measured magnetic field after detailed design or fabrication of magnets.

Example of beam trajectories simulated by GEANT4. Beam color (red, yellow, black, magenta and blue)
corresponds to $\pi^+$
beam momenta of 1.045, 1.0475, 1.050, 1.0525 and 1.055 GeV/$c$, respectively.

Figure <ref> shows GEANT4 simulated beam trajectories. It can be seen that momentum spread of beam
$\pi^+$ was converted to spatial distribution at the experimental target ($z=0$) as expected.

In the simulation, following processes were taken into account. Passing through the experimental target, beam
$\pi^+$ loses its energy, experiences multiple scattering and finally react with a target nucleus through
the $^{12}$C$(\pi^+, K^+)^{12}_\Lambda$C reaction. In this section, excitation energies of 0, 5, 10, 15 MeV were
considered (they are nothing to do with real $^{12}_\Lambda$C structure) and $K^+$ emission angles were uniformly
distributed in the angular acceptance.

Example of $K^+$ trajectories simulated by GEANT4. Beam color (green, black yellow and blue)
corresponds to the excitation energy of 15, 10, 5, 0 MeV for $^{12}$C$(\pi^+, K^+)^{12}_\Lambda$C reaction.
Measurement of beam position gives excitation energy.

Lambda binding energy spectrum for $p_\pi=1.05$ GeV/$c$ (left) and
$1.045< p_\pi<1.055$ GeV/$c$ (right).

Figure <ref> shows that $K^+$s with the same excitation energy converge at the focal plane and
it is different from the case that $K^+$s with the same momentum from a point source converge at the focal plane
of a normal spectrometer.

The $K^+$ horizontal position distribution was converted to Lambda binding energy ($B_\Lambda$).
Figure <ref>(left) shows $-B_\Lambda$ for beam pion momentum was selected as $p_\pi =1.05$ GeV/$c$,
and (right) shows it for $1.045< p_\pi <1.055$ GeV/$c$. Energy resolution for $p_\pi =1.05$ GeV/$c$, was 0.48 MeV (FWHM)
and it for $1.045 < p_\pi <1.055$ GeV/$c$ was 0.86 MeV (FWHM).  It is known that deterioration of energy resolution
in this simulation was caused by an imperfect alignment of focal planes for different beam momenta.
Ideally intrinsic resolution of HIHR beam optics system is expected to approach the ideal value, 315 keV (FWHM) as shown
by TRANSPORT calculation in the previous section. Further GEANT4 simulation study on beam elements tunes with
higher order optical elements for better dispersion matching condition is in progress.

It should be noted that the resolution was discussed here assuming use of the kaon spectrometer as a “hardware spectrometer” which
has a clean focal plane and gives energy or momentum information as a dispersive position information.
Recent progresses of computer power enable us to obtain resolution of $\Delta p/p = 2 \times 10^{-4}$  for a so-called
“software spectrometer”, which has no clear focal plane, by using 6 orders polynomial function of position and angle information of
$x, x', y, y'$ at reference plane <cit.>. Introduction of such techniques would enable us to compensate higher order aberration
which cannot be compensated perfectly by hardware and contribute to improve energy resolution.

By using GEANT4 simulation, how target thickness affects energy resolution was studied.
Target thickness affects energy resolution in following two ways and these effects are automatically included
in the result of GEANT4 simulation.

* Effective thickness changes due to $K^+$ emission angle. This effect is already discussed in the previous section.  and it is safely neglected.

* Energy loss depends on the reaction position. If the reaction happens near to front surface of the target,
$p(\pi^+)$ is larger than it is expected for the case the reaction happens at the center of the target,
since energy loss in the target is smaller. The same thing happens for emitted $K^+$. If the reaction happens near to front,
actual $p(K^+)$ is also larger than it measured by the kaon spectrometer. If reaction happens near to back surface of the target,
the opposite effects will be observed. In addition to energy loss effects, energy straggling occurs even if the reaction point does not move.
These effects can be easily estimated with GEANT4 simulation. Preparing relatively thick target in the model and reaction points were
uniformly distributed along the incoming beam direction, then spatial distribution of $K^+$s at the focal plane was measured and
converted to the energy scale. This study shows that 1 mm change of reaction point along the beam direction
in the $^{12}$C target corresponds to 78.5 keV change of missing mass of $^{12}_\Lambda$C.
Since 200 mg/cm$^2$ of $^{12}$C (density 1.8 g/cm$^3$) is 1.1 mm thick, energy loss and straggling give 87.2 keV deterioration
to the energy resolution.

Final result of target thickness dependence of energy resolution for $^{12}$C (density 1.8 g/cm$^3$) with
$p_\pi=1.05$ GeV/$c$ was shown in Fig. <ref>.  Black points show the resolution for all generated events
and red points for the events with $K^+$ angular acceptance cuts of $|x_{target}'|<20$ mr and $|y_{target}' |<20$ mr.
Red points give better resolution because momentum matching conditions are less satisfied for the events with larger
emission angles. It indicates that further study of optics tuning is essentially important as well as limitation of
angular acceptance results in improvement of resolution (250 keV (FWHM) for the thin target limit) at the cost of yield.
Looking black points, it can be seen that up to 400 mg/cm$^2$ which corresponds to 2 mm-thick $^{12}$C ,
intrinsic resolution of optics system dominates under currently achieved matching condition.
For target thickness over 400 mg/cm$^2$ energy straggling and other factors in target begin to contribute
to the energy resolution. Limitation of $K^+$ angular acceptance makes intrinsic optical resolution good enough
to see almost pure target thickness dependence as shown by red points.

Target thickness dependence of the energy resolution for $^{12}$C target.
Black points and red points are respectively without and with angular acceptance
selection of $|x_{target}' |<20$ mr and $|y_{target}' |<20$ mr.

§.§.§ Summary of energy resolution study

The energy resolution estimation was carried out with an optics code TRANSPORT and a Monte Carlo package GEANT4.
Optics code has advantages of quick matrix tuning and easier investigation of optical features of the beam line and
spectrometer systems. However, detailed study with reactions in the target cannot be carried out.
On the contrary to the optics code, Monte Carlo simulation can easily consider various reactions in materials on the
beam line but tuning of beam elements is not easy as the optics code. During conceptual design works, both optics code
and simulation studies will be complementarily performed.
When a reliable design of magnets is finalized and detailed magnetic field maps are implemented in the simulation code,
a thorough optimization of optical parameters will be performed. After such an optimization, adjustments for minor
change of experimental condition will be quickly done with a dedicated GEANT4 simulation code to be developed for
this purpose.
Though further study and optimization of beam line components are necessary, current study shows that the
dispersion match technique enables an energy resolution of better than 400 keV (FWHM) for less than 400 mg/cm2 targets at HIHR. Limitation of acceptance with a thin target might enable further improvement of the resolution at the cost of hypernuclear yield. Target thickness and acceptance limitation will be optimized depending on the experimental requirements, such as resolution or yield.
Figure <ref> shows expected $\Lambda$ binding energy spectra for $^{12}_\Lambda$C (left) and $^{208}_\Lambda$Pb (right).
They should be compared with previously measured spectra at KEK (Fig. <ref>).

Expected $\Lambda$ binding energy spectra for $^{12}_\Lambda$C (left) and
$^{208}_\Lambda$Pb (right). Resolution of 0.4 MeV (FWHM) and number of the events
for the ground states of 1000 and 100 were assumed for $^{12}_\Lambda$C
and $^{208}_\Lambda$Pb, respectively.

Previously measured $\Lambda$ binding energy spectra for $^{12}_\Lambda$C (left) <cit.> and
$^{208}_\Lambda$Pb (right) <cit.>. Resolutions were 1.45 MeV (FWHM) and
2.3 MeV (FWHM) for $^{12}_\Lambda$C and $^{208}_\Lambda$Pb, respectively.

§.§.§ Beamtime request

The targets listed in Table <ref> are required to achieve the original goal of this proposal,
but choice of targets is not limited, because HIHR aims to be the hypernuclear factory for wide variety of experiments.
Since HIHR is a brand-new beam line, commission time for hardware development will be requested separately.
Assuming the energy resolution of HIHR is 400 keV (FWHM) as discussed in section <ref>,
more than 100 events of a peak can determine the peak centroid with a statistical error better than
$400~{\rm keV} / \sqrt{100} \simeq 40~{\rm keV}$. Typically, cross sections of excited bound states of $\Lambda$
binding energy of heavy hypernuclei are several times larger than they for the ground states, for example, cross sections of
$\Lambda$ in $s$-orbit (ground state) and $p$-orbit for the $^{208}$Pb$(\pi^+, K^+)^{208}_\Lambda$Pb reaction are
respectively 0.2 and 0.8 $\mu$b/sr. Therefore, statistical uncertainty of binding energies for excited bound states
would be a few 10 keV.
Background shape ambiguity, accuracy of energy scale calibration would contribute to systematic errors,
but overall uncertainties of binding energies of hypernuclei can be controlled to be less than 100 keV.
Carbon target is normally used as a reference for reaction spectroscopy of hypernuclei.
Optimized parameters of magnetic elements for dispersion matching condition will be investigated with the carbon runs.
During beam line tuning, limitation of acceptance will be applied to have clean events with the best energy resolution
which is optically achievable and thus higher statistics is essentially important. Reference data for various target
thickness (100, 200, 400 mg/cm$^2$) will be accumulated.
It should be noted that $(\pi^+, K^+)$ reaction hypernuclear spectroscopy cannot use
hyperon ($\Lambda$ and $\Sigma^0$) masses as the absolute mass calibration sources due to unavailability of
neutron target, and thus $^{12}_\Lambda$C mass will serve as one of energy references.
All previous $(\pi^+, K^+)$ experiments used the $^{12}_\Lambda$C mass as the mass calibration,
but it was pointed out that old $^{12}_\Lambda$C mass measured by nuclear emulsion almost half a century ago
would have a large error <cit.>. Recent state-of-art hybrid emulsion experiment, J-PARC E07,
has lots of  $^{12}_\Lambda$C events and analysis of these events will provide a new $^{12}_\Lambda$C mass reference
in near future. Decay pion spectroscopy experiment at HIHR is now planned <cit.> and it will serve as
an independent calibration source.

For light to medium heavy targets, hypernuclear production cross section for ground state is relatively large
and thus beamtime for 100 events accumulations with 200 mg/cm$^2$ thick targets are requested.
For heavy targets, cross sections of ground states are small and energy level density is large. Therefore luminosity-oriented
beamtime to measure ground state which needs high statistics and resolution-oriented beamtime to separate complicate
excitation energy levels which have larger cross sections than the ground state, are separately requested.
Beamtimes for light to medium-heavy hypernuclei need 724 hours (30 days) and heavy targets need 1764 hours (73 days),
ground total of 2488 hours (104 days) for the first campaign of experiments at HIHR.
Divided beamtime allocation is desirable rather than a continuous one, because time for analysis of accumulated data
is important to feed back its result to the next beamtime.

Summary of requesting beamtime for 50 kW proton beam power. Differential cross sections at
$\theta_K \sim 0$ were estimated by using data of prior $(\pi^+, K^+)$  experiments <cit.>.

Hypernuclei     Assumed g.s.      Target thickness     Expected     Requested      Beam

cross section     (mg/cm$^2$)                         Yield (/h)     number of       time (h)

($\mu$b/sr)          events for g.s.

$^{12}_\Lambda$C     8.1     100     13.3     1000     79

$^{12}_\Lambda$C     8.1     200     26.6     2000     79

$^{12}_\Lambda$C     8.1     400     53.1     2000     39

$^{6}_\Lambda$Li     1.9     200     12.7     100     8

$^{7}_\Lambda$Li     1.9     200     10.9     100     10

$^{9}_\Lambda$Be     0.2     200     1.1     100     98

$^{10}_\Lambda$B     0.9     200     3.5     100     30

$^{11}_\Lambda$B     0.9     200     3.2     100     33

$^{28}_\Lambda$Si     0.5     400     1.4     100     75

$^{40}_\Lambda$Ca     0.5     400     0.94     100     112

$^{51}_\Lambda$V     1.2     400     1.8     100     59

$^{89}_\Lambda$Y     0.6     200     0.53     100     199

Sub-total                     724

(light to                      (30 days)

midium heavy

$^{139}_\Lambda$La     0.3     200     0.085     20     236

$^{139}_\Lambda$La     0.3     400     0.17     80     471

$^{208}_\Lambda$Pb     0.3     200     0.057     20     352

$^{208}_\Lambda$Pb     0.3     400     0.11     80     705

Sub-total                     1764

(heavy)                      (73 days)

Grand                 2488

total                  (104 days)

§.§.§ Summary of spectroscopic study of $\Lambda$ hypernuclei with the ($\pi^+, K^+)$ reaction at HIHR

Based on a newly designed momentum matching pion beam line with a specially designed
kaon spectrometer system, HIHR at the J-PARC hadron extension hall, a campaign of $\Lambda$
hypernuclear $(\pi^+, K^+)$ reaction spectroscopy experiments was proposed.
Unprecedent energy resolution of $<400$ keV (FWHM) as a hypernuclear reaction spectroscopy
with high statistics $>100$ events for ground states of various $\Lambda$ hypernuclei will
enable us to determine $\Lambda$ binding energies with an accuracy better than 100 keV.
Precise determination of hypernuclear binding energies in wide mass region enables us to clarify
the existence of the $\Lambda$NN three-body repulsive force which is a key to solve
the puzzle of heavy neutron stars (hyperon puzzle). HIHR will be a unique hypernuclear
factory which provides various high precision data to construct a reliable baryonic interaction model
and such experiments cannot be performed at other facilities.

§.§.§ Spectroscopic study of neutron-rich $\Lambda$ hypernuclei with the double charge exchange (DCX) reaction

HIHR is designed primarily for the study of $\Lambda$ hypernulei with the $(\pi^+, K^+)$ reaction, but the dispersion match
technique can be applied to the $\pi^-$ beam.  The double charge exchange reaction, $^A Z(\pi^-, K^+)^A_\Lambda (Z-2)$, enables us to investigate
neutron rich hypernuclei. Isospin asymmetric systems serve as an ideal test ground for study of $\Lambda N - \Sigma N$ coupling effect
which is one of key issues to solve the hyperon puzzle.
It should be noted that inclusion of the coherent $\Lambda - \Sigma$ coupling effect which can be treated as the effective three-body force with a hyperon
is essentially important to theoretically reproduce experimentally-measured energies of the $s$-shell hypernuclei <cit.>.

As it is well known, some of neutron rich nuclei near the neutron drip line exhibit interesting properties such
as neutron halos and skins and they are characterized by having loosely bound neutrons <cit.>.
Adding a $\Lambda$ hyperon to such systems,  the glue like role of $\Lambda$ makes the system tighter bound and
changes their basic properties such as radius and surface diffuseness <cit.>.

Figure <ref> shows isospin asymmetry of $\Lambda$ hypernuclei produced by the $(\pi^\pm, K^+)$ reactions.
It tells us that the $(\pi^-, K^+)$ (DCX) reaction produces neutron richer hypernuclei with light targets.
Though cross sections of DCX are small, it enables us to access the new region of neutron-rich hypernuclei which are beyond neutron drip line for normal nuclei.

The isospin asymmetric parameter, $|N-Z|/(N+Z)$, for various $\Lambda$ hypernuclei produced
via the $(\pi^-, K^+)$ and $(\pi^+, K^+)$ reactions. Closed circle, closed square, and open triangle show the asymmetric
parameters for the target nuclei and hypernuclei produced with the $(\pi^-, K^+)$ and $(\pi^+, K^+)$ reactions, respectively.

Though clear peaks of neutron rich hypernuclei has not been observed yet with the DCX reactions,
experiments with $(\pi^-, K^+)$ at KEK-PS <cit.> and $(K^-_{stop},  \pi^+)$ at DA$\Phi$NE <cit.> were already carried out.
For the $^{10}{\rm B}(\pi^-, K^+)^{10}_\Lambda$Li reaction, events in the bound region were observed (Fig. <ref>) and
the number of the events was converted to the cross section, $11.3 \pm 1.9$ nb/sr <cit.>.
Recently heavy hyper-hydrogen ($^6_\Lambda$H) search experiment was carried out with the $^6$Li$(\pi^-, K^+) X$ reaction and
an upper limit of cross section, 0.56 nb/sr, was obtained<cit.>.

Assuming 1.20 GeV/$c$ momentum for $\pi^-$ and 0.88 GeV/$c$ for $K^+$ which were the same setting as KEK-PS-E521 <cit.>,
$\pi^-$ intensity at HIHR is $2 \times 10^8 \pi$/spill (Fig. <ref>) which is 50 times stronger than KEK-PS K6 beam line
and almost the same as $\pi^+$ intensity of 1.05 GeV/$c$ at HIHR.
With an adequate adjustment of momentum dispersion match conditions for $\pi^-$, the same energy resolution of 0.4 MeV is expected
for the $(\pi^-, K^+)$ reaction at HIHR. Therefore, yield estimation needs to take cross section difference between $(\pi^+, K^+)$
and $(\pi^-, K^+)$ DCX reactions into account.

Table <ref> compares important parameters between prior DCX experiment at KEK-PS, ($\pi^+, K^+)$ and DCX at HIHR.
Assuming a cross section of 10 nb/sr for both $^{9}{\rm Be}(\pi^-, K^+)^{9}_\Lambda$He and $^{10}{\rm B}(\pi^-, K^+)^{10}_\Lambda$Li reactions,
$^{9}_\Lambda$He yield of 0.092 h$^{-1}$ and
$^{10}_\Lambda$Li yield of 0.078 h$^{-1}$ are expected. In order to achieve similar precision to $(\pi^+, K^+)$ reaction spectroscopy by accumulating 100 counts,
1091 hours (= 45 days) and 1282 hours (= 53 days) of beamtimes are necessary
respectively for $^{9}_\Lambda$He and $^{10}_\Lambda$Li
with 50 kW proton beam on the primary target.
In order to study wider variety of neutron rich hypernuclei in reasonable beamtime with DCX reaction at HIHR, usage of higher beam intensity or introduction
of thicker target at the cost of resolution could be considered.
Further optimization on the experimental conditions is in progress.

Missing mass spectrum of the $^{10}{\rm B}(\pi^-, K^+)$ reaction. Inset shows an expanded view near the threshold <cit.>.

Important parameters for $(\pi^\mp, K^+)$ reaction spectroscopy of $\Lambda$ hypernuclei.

Reaction     Target     $p_\pi$      $p_K$      Resolution
Cross section    Yield

thickness     (GeV/$c$)     (GeV/$c$)     (MeV, FWHM)
(nb/sr)      (/h)

7cKEK PS K6 <cit.>

$^{10}{\rm B}(\pi^-, K^+)^{10}_\Lambda$Li      3.5     1.20     0.88     2.5
$11.3 \pm 1.9$

$^{9}{\rm Be}(\pi^+, K^+)^{9}_\Lambda$Be      0.4     1.05     0.72     0.4
$2.4 \times 10^2$     2.2

$^{10}{\rm B}(\pi^+, K^+)^{10}_\Lambda$B      0.4     1.05     0.72     0.4
$9 \times 10^2$     7.0

$^{9}{\rm Be}(\pi^-, K^+)^{9}_\Lambda$He      0.4     1.20     0.88     0.4
10$^*$     0.092

$^{10}{\rm B}(\pi^-, K^+)^{10}_\Lambda$Li      0.4     1.20     0.88     0.4
10$^*$    0.078

7r$^*$assumed value.

§.§.§  Cusp spectroscopy for $\Sigma$$N$ interaction

The $\Sigma$$N$ interaction has recently studied via high-statistics
$\Sigma^+$$p$ and $\Sigma^-$$p$ scattering experiment (J-PARC E40) <cit.>.
This experiment will make a great contribution to
construction of the realistic $Y$$N$ interaction models.
However, in this type of experiment,  $\Sigma$$N$ scattering cross sections
at very low energy ($p_Y^{lab} <$ 400 MeV/$c$)
is difficult to measure.
In order to study low-energy $\Sigma$$N$ interaction,
another type of  experiment is proposed
to measure a high-resolution
missing-mass spectrum
for the $d(\pi^+,K^+)$ reaction
around the $\Sigma$$N$ threshold.

A sharp enhancement (“cusp”) in the $d(\pi^+,K^+)$
and $d(K^-, \pi^-)$ missing mass spectra at the $\Sigma$$N$ threshold
was previously observed in several experiments, but their limited resolution
did not allow extraction of the scattering length.
In the proposed experiment, a $d(\pi^+,K^+)$ missing mass spectrum will be
measured with a pion beam momentum of 1.69 GeV/c
with the unprecedented high resolution ($< 400$ keV(FWHM))
by the HIHR beam line  <cit.>.
By tagging the $\Lambda$$p$ final state,
events in the isospin $T=1/2$ cannel is separated from those in the $T=3/2$ channel, and
the real and imaginary parts of the $\Sigma$$N$ scattering length for $T=1/2$
will be accurately obtained for the first time.

§.§.§  Precise decay pion spectroscopy for hypernuclear ground state mass

In the precise measurement of $\Lambda$ binding energies
in hypernuclei proposed at the HIHR beam line,
precise and reliable energy calibration of the hypernuclear excitation spectrum is
of great importance.
In the $(e,e'K^+)$ spectroscopy at JLab, $p(e,e'K^+)$$\Lambda$,$\Sigma^0$ reactions
with a hydrogen target are used for energy calibration of the spectrometer system.
This method cannot be used for the $(\pi^+,K^+)$ reaction.
In the $(\pi^+,K^+)$ experiments, the ground-state peak of $^{12}_\Lambda$C were long used
for energy calibration. However, the reference $B_\Lambda$
value of the $^{12}_\Lambda$C ground state measured in old emulsion experiments
was found to be incorrect <cit.>, and thus a precise and reliable
calibration method is awaited.

The $B_\Lambda$ values of hypernuclear ground states can be reliably determined
by measuring the pion momentum in the two-body mesonic decays,
$^A_\Lambda$Z$_{gs}$ $\to$  $^A$(Z+1) $+ \pi^-$, as was successfully performed
for $^4_\Lambda$H <cit.>.
At the HIHR beam line, ground states of light hypernuclei are produced
by intense pion beams on a thin target, and the momentum spectrum
of decay pions are measured with an additional magnetic spectrometer
around the target. Here, a compact superconducting solenoid magnet is planned.

For the calibration purpose, $^9_\Lambda$Be or $^{13}_\Lambda$C
hypernuclei are best suited
because of a large branching ratio of the two-body $\pi^-$ decay to the
ground state of the daughter nucleus.
In addition, measured pion spectra can be used to assign the ground-state
spins as is mentioned above for the $(\pi^-,K^0)$ reaction.

This measurement is also essential in studies of charge symmetry breaking (CSB) effects,
because the CSB effect in the $B_\Lambda$ values
for $p$-shell and heavier $\Lambda$ hypernuclei
is expected to be of the order of 100 keV.
By comparing the $B_\Lambda$ values with those of the corresponding mirror hypernuclei measured
in the $(e,e'K^+)$ reaction,
as well as by comparing their level schemes to be measured in $\gamma$-ray
spectroscopy as mentioned above,
the CSB effects will be investigated beyond the $A=4$ hypernuclei
and the origin of this phenomenon will be understood
in terms of the $B$$B$ interactions in nuclear matter.

§.§.§  Search for $\eta$ and $\eta'$ nuclear bound states

The HIHR beam line will be also used in non-strangeness experiments
for high resolution missing mass spectroscopy using intense pions beams below 2 GeV/c.

Meson-nucleus bound states as well as mesonic atoms
will provide us with clear information on hadron-hadron interactions and
hadron properties in nuclear matter related to partial restoration of chiral symmetry.
In J-PARC, $K^-$ nuclear bound states and $K^-$ atoms have been
intensively studied, and are further planned, mainly at the K1.8BR beam line with $K^-$ beams.

In the extended hadron facility at J-PARC,
$\eta$ and $\eta'$ nuclear bound states, which are not observed yet <cit.>
probably due to small cross sections,
will be searched for via the $(\pi,N)$ reaction <cit.>
taking advantage of intense ($> 10^8$/spill) pion beams
and an excellent ($< 400$ keV (FWHM)) energy resolution.
In particular, behavior of $\eta$' meson in nuclear matter has
attracted physicists' attention because
the heavy mass of $\eta$' is believed to be explained by the UA(1) anomaly.
In the $\eta'$ nucleus search experiment,
we employ the $^{12}$C($\pi^+,p$) reaction with a 1.8 GeV/c pion beam and
high energy protons from
$\eta'NN \to NN$ decay will be tagged around the target
to overcome huge background.

§.§.§  Pure neutron and neutron-rich nuclei via $(\pi^-,\pi^+)$ reaction

Another type of non-strangeness experiments suitable for the HIHR beam line
are studies of pure-neutron and very neutron-rich light nuclei via
the double charge exchange $(\pi^-,\pi^+)$ reaction.

A candidate of a tetraneutron resonance state, $^4$n, were
reported at RIBF via a heavy-ion double charge exchange reaction,
$^4$He($^8$He,$^8$Be)$^4$n <cit.>.
In order to confirm its existence and investigate the structure,
a missing mass measurement of
the $^4$He$(\pi^-,\pi^+)$$^4$n reaction at HIHR beam line is proposed <cit.>.
When an intense pion beam ($\sim 1.6\times 10^8$ pions/spill)
at 980 MeV/c is irradiated to a 2.0g/cm$^2$-thick $^4$He target,
$\sim$100 events of $^4$n will be collected in two weeks for a cross section of 1nb/sr.
With a mass resolution ($< 400$ keV (FWHM))
better than the previous RIBF experiment (1 MeV (FWHM)),
the width as well as the accurate energy of the state can be obtained.

In addition, the $(\pi^-,\pi^+)$ reaction can be used to study heavy hydrogens
($^4$H, $^5$H, $^6$H, $^7$H) and other neutron-rich nuclei.
Among the heavy hydrogens, $^6$H is not observed yet and should be searched for at HIHR.

§.§ Other Experiments Planned at HIHR

§.§.§ Cusp spectroscopy for $\Sigma$$N$ interaction

The $\Sigma$$N$ interaction has recently been studied via high-statistics
$\Sigma^+$$p$ and $\Sigma^-$$p$ scattering experiments (J-PARC E40) <cit.>.
This experiment will make a great contribution to
construction of the realistic $Y$$N$ interaction models.
However, in this type of experiment,  $\Sigma$$N$ scattering cross sections
at very low energies ($p_Y^{lab} <$ 400 MeV/$c$)
are difficult to measure.
In order to study low-energy $\Sigma$$N$ interaction,
another type of  experiment is proposed
to measure a high-resolution
missing-mass spectrum
for the $d(\pi^+,K^+)$ reaction
around the $\Sigma$$N$ threshold.

A sharp enhancement (“cusp”) in the $\Lambda p$ invariant mass spectra of
$K^- d \to \pi^- \Lambda p$, $\pi^+ d \to K^+ \Lambda p$, and $pp \to K^+ \Lambda p$
reactions at the $\Sigma$$N$ threshold was previously observed in several experiments, but
the obtained peak position and width were not consistent with each other <cit.>.
It was discussed that the mass spectrum can be expressed in terms of the $\Sigma N$ scattering length of
the spin-triplet $T$(isospin) $= 1/2$ channel <cit.>.
However, the past experiments did not allow extraction of the scattering length
due to their limited mass resolution.
Figure <ref> shows the calculated mass spectra when we choose the scattering length as
$A_{\Sigma} = 2.06 - i4.64$ fm.
The black line shows the original calculated spectrum without smearing due to the mass resolution.
The points with statistical error bars show
the expected yield when we assume the total yield as
$5 \times 10^{4}$ events.
Two prominent peaks are expected to appear due to $\sim$2 MeV difference
of the $\Sigma N$ thresholds (2128.9 MeV for $\Sigma^+ n$ and 2130.9 MeV for $\Sigma^0 p$), because the cusp structure should be seen for both
$\Sigma^+ n$ and $\Sigma^0 p$ thresholds.
The yield ratio should be
related to the amplitude of the elementary processes of $K^- p \to \pi^- \Sigma^{+}$ and $K^- n \to \pi^- \Sigma^{0}$ and their interference.
Figure <ref> shows the case for the yield ratio as $\Sigma^+ n : \Sigma^0 p = 1 : 1$.
The colored lines show the smeared spectra with the experimental resolution
of $\Delta M = 0.4$, 2, and 3 MeV in FWHM.
The resolution of $\Delta M = 2$ MeV corresponds to the best value of the past experiment
achieved by HIRES at COSY <cit.>.
The prominent two peak structure in the original spectrum disappears for $\Delta M \geq 2$ MeV,
and much better resolution is necessary to determine the scattering length from the cusp spectrum.
Moreover, it is difficult to distinguish the mass spectra
with $\Delta M \geq 2$ MeV from
a simple Breit-Wigner distribution.
Therefore, the mass resolution is a key for the $\Sigma N$ cusp measurement.
The inconsistency of the peak positions and widths among the past experiments may originate from the yield ratio between the $\Sigma^+ n$ and $\Sigma^0 p$ channels, which is different depending on the reactions and the beam momenta.

Calculated spectra with the scattering length of $A_{\Sigma} = 2.06 - i4.64$ fm.
The black line shows the original spectrum and
the colored lines show the smeared spectra with
experimental resolutions of $\Delta M = 0.4$ (red, HIHR), 2 (green, COSY HIRES<cit.>),
and 3 (J-PARC E27

<cit.>) MeV in FWHM.
The points with statistical error bars show the expected yield when we assume the total yield as
$5 \times 10^{4}$ events.

In the proposed experiment, a $d(\pi^+,K^+)$ missing mass spectrum will be
measured with a pion beam momentum of 1.7 GeV/$c$
with the unprecedented high resolution ($< 0.4 $ MeV (FWHM))
by the HIHR beam line <cit.>.
By tagging the $\Lambda$$p$ final state,
events in the isospin $T=1/2$ channel is separated from those in the $T=3/2$ channel, and
the real and imaginary parts of the $\Sigma$$N$ scattering length for the
spin-triplet $T=1/2$ channel
will be accurately obtained for the first time.
By using the HIHR beam line, not only the $\Sigma N$ cusp but also
other cusp spectra can be measured with a high resolution.
In particular, measurement of the $\bar{K} N$ (isospin $T = 1$) cusp is promising by using the $p(\pi^{\pm}, K^+)\Lambda \pi^{\mp}$

§.§.§ Precise decay pion spectroscopy for hypernuclear ground state mass

In the precise measurement of $\Lambda$ binding energies
in hypernuclei proposed at the HIHR beam line,
precise and reliable energy calibration of the hypernuclear excitation spectrum is
of great importance.
In the $(e,e'K^+)$ spectroscopy at JLab, $p(e,e'K^+)$$\Lambda$,$\Sigma^0$ reactions
with a hydrogen target are used for energy calibration of the spectrometer system.
This method cannot be used for the $(\pi^+,K^+)$ reaction
because of the absence of a neutron target.
Instead, the ground-state binding energy of $^{12}_\Lambda$C, which was
reported to be $10.76 \pm 0.19\,\mathrm{MeV}$
by old emulsion experiments,
has been used for the energy calibration.
However, it has been pointed out <cit.> that
the ground-state binding energies of other $p$-shell hypernuclei thus obtained are systematically shifted by $\approx 0.54\text{--}0.6\,\mathrm{MeV}$
from those obtained in old emulsion experiments as well as those measured by the FINUDA experiment using the $(K^-_{\rm stop}, \pi^-)$ reaction.
It should be noted that the FINUDA experiment
could make use of the two-body decay of $K^+$ into $\mu^+\nu_\mu$ and $\pi^+\pi^0$ for calibration.
an independent method to measure the hypernuclear ground-state binding energies is called for, in order to obtain precise and reliable standard values of some $p$-shell hypernuclei.

The $B_\Lambda$ values of hypernuclear ground states can be reliably determined
by measuring the pion momentum in the two-body mesonic decays,
$^A_\Lambda$Z$_{gs}$ $\to$  $^A$(Z+1) $+ \pi^-$, as was successfully performed
for $^4_\Lambda$H <cit.>.
At the HIHR beam line, the ground states of light hypernuclei are produced
by intense pion beams on a thin target, and the momentum spectrum
of their decay pions are measured with
an additional magnetic spectrometer around the target.
One of the promising ideas is to use a compact superconducting solenoid magnet, in which a low-momentum decay pion moves along a helical orbit.
The momenta of the decay pions range between $90\,\mathrm{MeV}/c$ and $135\,\mathrm{MeV}/c$.

For the calibration purpose, $^9_\Lambda$Be or $^{13}_\Lambda$C
hypernuclei are best suited.
The first reason is that the ground state is not split into the doublet because the spin of the even-even core nucleus ($^8\mathrm{Be}$ or $^{12}\mathrm{C}$)  is 0.
Under this condition, one
can safely relate the lowest-energy peak in the hypernuclear mass spectrum
to the ground state;
this is not the case when both the upper and lower state in the ground-state doublet can be populated.
The second reason is that the excitation energy of the first excited state
of the daughter nucleus ($^{9}\mathrm{C}$ or $^{13}\mathrm{N}$) is
sufficiently large.
Otherwise, an experimental determination of the absolute pion momentum for the $^A_\Lambda$Z$_{gs}$ $\to$  $^A$(Z+1)$_{gs}+ \pi^-$ decay will be more or less complicated.
In addition, $^{12}_\Lambda\mathrm{C}$ may be used for the energy calibration as well, taking the advantage of a large formation cross section.
The formation of the upper $2^-$ state of the $^{12}\mathrm{C}$ ground-state doublet will be suppressed because of the non-spin-flip nature of the $(\pi^+,K^+)$ reaction. It is noted that the $^{12}\mathrm{C}$ ground state with the spin-parity $1^-$ decays dominantly to the first excited state of $^{12}\mathrm{N}$, but not to the ground state [1].

In Refs. [1, 2], excitation spectra for pionic decay from the lower and upper states of various hypernuclear ground-state doublets are shown.
Since the spectrum is totally different between the lower and upper states, spin-parity assignment of the hypernuclear ground states is feasible. This technique was already applied in the FINUDA experiment [3], resulting in the first spin assignment of $^{15}_\Lambda$N to be $3/2^+$.
One of the interesting objects to be investigated is a neutron-rich  hypernucleus $^{12}\mathrm{Be}$ [4], which can be populated in $^{12}$C$(\pi^-,K^+)$ reaction.
Whereas the core nucleus $^{11}\mathrm{Be}$ has a positive-parity ground state, as is known as parity inversion, the inclusion of a $\Lambda$ hyperon may revert the parity of the ground state.

This measurement is also essential in studies of charge symmetry breaking (CSB) effects,
because the CSB effect in the $B_\Lambda$ values
for $p$-shell and heavier $\Lambda$ hypernuclei
is expected to be of the order of 100 keV.
By comparing the $B_\Lambda$ values measured in the $(\pi^*,K^*)$ reaction at HIHR with those of the corresponding mirror hypernuclei measured
in the $(e,e'K^+)$ reaction,
as well as by comparing their level schemes to be measured in $\gamma$-ray
spectroscopy as mentioned above,
the CSB effects will be investigated beyond the $A=4$ hypernuclei
and the origin of this phenomenon will be understood
in terms of the $B$$B$ interactions in nuclear matter.

§.§.§ Search for $\eta$ and $\eta'$ nuclear bound states

The HIHR beam line will be also used in non-strangeness experiments
for high resolution missing mass spectroscopy using intense pions beams below 2 GeV/c.

Meson-nucleus bound states as well as mesonic atoms
will provide us with clear information on hadron-hadron interactions and
hadron properties in nuclear matter related to partial restoration of chiral symmetry.
In J-PARC, $K^-$ nuclear bound states and $K^-$ atoms have been
intensively studied, and are further planned, mainly at the K1.8BR beam line with $K^-$ beams.

In the extended hadron facility at J-PARC,
$\eta$ and $\eta'$ nuclear bound states, which are not observed yet <cit.>
probably due to small cross sections,
will be searched for via the $(\pi,N)$ reaction <cit.>
taking advantage of intense ($> 10^8$/spill) pion beams
and an excellent ($< 400$ keV (FWHM)) energy resolution.
In particular, behavior of $\eta$' meson in nuclear matter has
attracted physicists' attention because
the heavy mass of $\eta$' is believed to be explained by the UA(1) anomaly.
In the $\eta'$ nucleus search experiment,
we employ the $^{12}$C($\pi^+,p$) reaction with a 1.8 GeV/c pion beam and
high energy protons from
$\eta'NN \to NN$ decay will be tagged around the target
to overcome huge background.

§.§.§ Pure neutron system and neutron-rich nuclei via $(\pi^-,\pi^+)$ reaction

Another type of non-strangeness experiments suitable for the HIHR beam line
is a study of pure-neutron systems and very neutron-rich light nuclei via
the double charge exchange $(\pi^-,\pi^+)$ reaction.

A candidate of a tetraneutron resonance state, $^4$n, were
reported at RIBF via a heavy-ion double charge exchange reaction,
$^4$He($^8$He,$^8$Be)$^4$n <cit.>.
In order to confirm its existence and investigate the structure,
a missing mass measurement of
the $^4$He$(\pi^-,\pi^+)$$^4$n reaction at HIHR beam line is proposed <cit.>.
When an intense pion beam ($\sim 1.6\times 10^8$ pions/spill)
at 980 MeV/c is irradiated to a 2.0 g/cm$^2$-thick $^4$He target,
$\sim$100 events of $^4$n will be collected in two weeks for a cross section of 1nb/sr.
With a mass resolution ($< 400$ keV (FWHM))
better than the previous RIBF experiment (1 MeV (FWHM)),
the width as well as the accurate energy of the state can be obtained.

In addition, the $(\pi^-,\pi^+)$ reaction can be used to study heavy hydrogens
($^4$H, $^5$H, $^6$H, $^7$H) and other neutron-rich nuclei.
Among the heavy hydrogens, $^6$H is not observed yet and should be searched for at HIHR.

§.§.§  Cusp spectroscopy for $\Sigma$$N$ interaction

The $\Sigma$$N$ interaction has recently studied via high-statistics
$\Sigma^+$$p$ and $\Sigma^-$$p$ scattering experiment (J-PARC E40) <cit.>.
This experiment will make a great contribution to
construction of the realistic $Y$$N$ interaction models.
However, in this type of experiment,  $\Sigma$$N$ scattering cross sections
at very low energy ($p_Y^{lab} <$ 400 MeV/$c$)
is difficult to measure.
In order to study low-energy $\Sigma$$N$ interaction,
another type of  experiment is proposed
to measure a high-resolution
missing-mass spectrum
for the $d(\pi^+,K^+)$ reaction
around the $\Sigma$$N$ threshold.

A sharp enhancement (“cusp”) in the $d(\pi^+,K^+)$
and $d(K^-, \pi^-)$ missing mass spectra at the $\Sigma$$N$ threshold
was previously observed in several experiments, but their limited resolution
did not allow extraction of the scattering length.
In the proposed experiment, a $d(\pi^+,K^+)$ missing mass spectrum will be
measured with a pion beam momentum of 1.69 GeV/c
with the unprecedented high resolution ($< 400$ keV(FWHM))
by the HIHR beam line  <cit.>.
By tagging the $\Lambda$$p$ final state,
events in the isospin $T=1/2$ cannel is separated from those in the $T=3/2$ channel, and
the real and imaginary parts of the $\Sigma$$N$ scattering length for $T=1/2$
will be accurately obtained for the first time.

§.§.§  Precise decay pion spectroscopy for hypernuclear ground state mass

In the precise measurement of $\Lambda$ binding energies
in hypernuclei proposed at the HIHR beam line,
precise and reliable energy calibration of the hypernuclear excitation spectrum is
of great importance.
In the $(e,e'K^+)$ spectroscopy at JLab, $p(e,e'K^+)$$\Lambda$,$\Sigma^0$ reactions
with a hydrogen target are used for energy calibration of the spectrometer system.
This method cannot be used for the $(\pi^+,K^+)$ reaction.
In the $(\pi^+,K^+)$ experiments, the ground-state peak of $^{12}_\Lambda$C were long used
for energy calibration. However, the reference $B_\Lambda$
value of the $^{12}_\Lambda$C ground state measured in old emulsion experiments
was found to be incorrect <cit.>, and thus a precise and reliable
calibration method is awaited.

The $B_\Lambda$ values of hypernuclear ground states can be reliably determined
by measuring the pion momentum in the two-body mesonic decays,
$^A_\Lambda$Z$_{gs}$ $\to$  $^A$(Z+1) $+ \pi^-$, as was successfully performed
for $^4_\Lambda$H <cit.>.
At the HIHR beam line, ground states of light hypernuclei are produced
by intense pion beams on a thin target, and the momentum spectrum
of decay pions are measured with an additional magnetic spectrometer
around the target. Here, a compact superconducting solenoid magnet is planned.

For the calibration purpose, $^9_\Lambda$Be or $^{13}_\Lambda$C
hypernuclei are best suited
because of a large branching ratio of the two-body $\pi^-$ decay to the
ground state of the daughter nucleus.
In addition, measured pion spectra can be used to assign the ground-state
spins as is mentioned above for the $(\pi^-,K^0)$ reaction.

This measurement is also essential in studies of charge symmetry breaking (CSB) effects,
because the CSB effect in the $B_\Lambda$ values
for $p$-shell and heavier $\Lambda$ hypernuclei
is expected to be of the order of 100 keV.
By comparing the $B_\Lambda$ values with those of the corresponding mirror hypernuclei measured
in the $(e,e'K^+)$ reaction,
as well as by comparing their level schemes to be measured in $\gamma$-ray
spectroscopy as mentioned above,
the CSB effects will be investigated beyond the $A=4$ hypernuclei
and the origin of this phenomenon will be understood
in terms of the $B$$B$ interactions in nuclear matter.

§.§.§  Search for $\eta$ and $\eta'$ nuclear bound states

The HIHR beam line will be also used in non-strangeness experiments
for high resolution missing mass spectroscopy using intense pions beams below 2 GeV/c.

Meson-nucleus bound states as well as mesonic atoms
will provide us with clear information on hadron-hadron interactions and
hadron properties in nuclear matter related to partial restoration of chiral symmetry.
In J-PARC, $K^-$ nuclear bound states and $K^-$ atoms have been
intensively studied, and are further planned, mainly at the K1.8BR beam line with $K^-$ beams.

In the extended hadron facility at J-PARC,
$\eta$ and $\eta'$ nuclear bound states, which are not observed yet <cit.>
probably due to small cross sections,
will be searched for via the $(\pi,N)$ reaction <cit.>
taking advantage of intense ($> 10^8$/spill) pion beams
and an excellent ($< 400$ keV (FWHM)) energy resolution.
In particular, behavior of $\eta$' meson in nuclear matter has
attracted physicists' attention because
the heavy mass of $\eta$' is believed to be explained by the UA(1) anomaly.
In the $\eta'$ nucleus search experiment,
we employ the $^{12}$C($\pi^+,p$) reaction with a 1.8 GeV/c pion beam and
high energy protons from
$\eta'NN \to NN$ decay will be tagged around the target
to overcome huge background.

§.§.§  Pure neutron and neutron-rich nuclei via $(\pi^-,\pi^+)$ reaction

Another type of non-strangeness experiments suitable for the HIHR beam line
are studies of pure-neutron and very neutron-rich light nuclei via
the double charge exchange $(\pi^-,\pi^+)$ reaction.

A candidate of a tetraneutron resonance state, $^4$n, were
reported at RIBF via a heavy-ion double charge exchange reaction,
$^4$He($^8$He,$^8$Be)$^4$n <cit.>.
In order to confirm its existence and investigate the structure,
a missing mass measurement of
the $^4$He$(\pi^-,\pi^+)$$^4$n reaction at HIHR beam line is proposed <cit.>.
When an intense pion beam ($\sim 1.6\times 10^8$ pions/spill)
at 980 MeV/c is irradiated to a 2.0g/cm$^2$-thick $^4$He target,
$\sim$100 events of $^4$n will be collected in two weeks for a cross section of 1nb/sr.
With a mass resolution ($< 400$ keV (FWHM))
better than the previous RIBF experiment (1 MeV (FWHM)),
the width as well as the accurate energy of the state can be obtained.

In addition, the $(\pi^-,\pi^+)$ reaction can be used to study heavy hydrogens
($^4$H, $^5$H, $^6$H, $^7$H) and other neutron-rich nuclei.
Among the heavy hydrogens, $^6$H is not observed yet and should be searched for at HIHR.

§.§ Design of K1.1 and K1.1BR Beam Lines

The K1.1 and K1.1BR are mass-separated charged beam lines for low-momentum
kaons of $\sim$1.1 and $\sim$0.8 GeV/c, respectively.
As shown in Fig <ref>,
the beam line is extracted at  6$^\circ$ and use the cross fields type separators.
Total length of 28.32 and 20.78 m for K1.1 and K1.1BR, respectively.
Almost all elements such as magnets, electrostatic separators, slits, and
power supplies  already exist for use of the K1.1BR and K1.1 in the current
Hadron Experimental Hall.
The most upstream magnet D1, which will be installed in the T2 vacuum chamber,
will be newly fabricated to conform with the configuration around
the production target (T2).
Then the distance between the T2 to K1.1D1 becomes shorter than that of
the beam line at the current Hadron Hall (2.0 m $\rightarrow$ 1.2m).
This optimization to the K1.1 beam line enhances the beam line acceptance about twice.
Field (at pole ) values are listed in Table <ref>.
The D2 maximum field limits the maximum momentum of $\sim$1.2GeV/c.

Layout of the K1.1 and K1.1BR beamliines.

Field (at pole) values of the K1.1 beam line elements.

Elements	    Max. field [kG]	    K1.1 (1.1GeV/c)	 [kG]	    K1.1BR (0.8GeV/c) [kG]

D1			    (new)				    12.3948						     9.0144

Q1			     7.75				     5.9733						     4.2744

Q2			    11.65				    -9.0733						    -6.5176

D2			    15.512			    14.2068						    10.33216

Q3			    1223				     7.0645						     5.92312

O1			     4.09

Q4			    12.34				    -8.0711						    -5.65928

S1			     5.22				    -1.3024						    -0.62264

S2			     5.22				    -1.3024						    -0.63856

Q5			    12.07				    -9.0749						    -7.4448

Q6			    12.07				     8.047						    8.53512

D3			    21.18				    -14.6772						    14.825512

Q7			    11.87				    -5.104

S3			    2.56				    -1.1848

S4			    2.56				     0.3611

Q8			    11.87				    -9.4507

Q9			    11.4				     9.4533

D4			    21.46				    17.927

Q10		     8.2				    -6.7281

Q11		    11.5				    -6.1168

Q7-BR		   					   								    -7.56032

Q8-BR		   					   								     8.392336

Figure <ref> shows the beam envelope of the K1.1 beam line
calculated by TRANSPORT.
Beam source is assumed to be an ellipse shape with $\pm$7.7 mm (horizontal)
and $\pm$1.7 mm (vertical) and spreads of $\pm$30 mrad in horizontal,
$\pm$15mrad in vertical, and momentum spread of $\pm$3%.
This source size is taken into account of the primary proton beam size of
1.0 mm (H) $\times$ 2.5 mm (V) in rms and the length of the prodction target
of 66 mm and the beam extraction angle of 6.0 degree.
Top half of the figure shows vertical one, while bottom half shows horizontal.
Red and  white solid curves show optics only the 1st order and including the
2nd order, respectively. Dashed one shows one due to the momentum spread.
The beam is focused vertically at the following 4 points, IF (Intermediate Focus),
MS1 (Mass Slit 1), MS2, and FF (Final Focus), for particle-mass separation.
At IF vacuum window is installed to separate accelerator-class vacuum with
the secondary beam line class vacuum.
To minimize the effects of the vacuum window and redefine the vertical beam size,
the beam is vertically focused at IF.
From IF to MS1 is the 1st mass-separator section.
The particles with different mass, namely different velocity, are focused at
the different vertical position on MS1 by using an 1.9m long electrostatic field
and compensation magnetic field.
Mass slit with the tapered shape is installed at MS1 to prevent
the unwanted particles passing through.
Momentum slit is installed the upstream of the mass slit 1, where
momentum dispersion is large, to define momentum bite of the secondary beam.
From MS1 to MS2 is the 2nd mass-separator section similar to the 1st
mass-separator section.
After MS2, beam analyzer is located.
At FF, beam is focused horizontally, vertically, and achromatically.

Beam envelope of the K1.1 beam line optics.
Beam source is assumed to be an ellipse shape with $\pm$7.7 mm (horizontal)
and $\pm$1.7 mm (vertical) and spreads of $\pm$30mrad in horizontal,
$\pm$15mrad in vertical, and momentum spread of $\pm$3%.
Red curves show the 1st order optics, while white one show
the one including the 2nd order.

Figure <ref> shows the K1.1 beam analyzer
with DQQ configuration.
It is necessary to measure at least 5 independent observables
in order to determine particle's momentum.
In the beam analyzer, particle trajectory on both horizontal
and vertical positions and their directions at the downstream
part (VI) with trackers such as two sets of wire chambers,
At the upstream part (VO), it is enough to measure the horizontal
position to determine particle momentum.
Scintillating fiber tracker (FT) is used
Coefficients of  VO $\rightarrow$ VI transport are as follows;
$R_{11}$=1.38544, $R_{33}$=-0.66853 (horizontal and vertical magnifications),
$R_{12}$=0.43615 cm/rad.
and $R_{16}$=2.108 cm/$ (dispersion).
In the 1st order optics calculation, momentum resolution is expressed as follows,

\begin{equation}
\Delta p/p = \frac{\sqrt{1 + R_{11}^2}}{R_{16}} \Delta X,
\end{equation}

where $\Delta X$ is the resolution of the horizontal position measurements.
If we assume $\delta X$=0.2 mm (RMS), $\Delta p/p$= 1.6$\times$10$^{-4}$ (RMS),
namely 3.8$\times$10$^{-4}$ (FWHM), is expected.
However  $R_{12}$ can not be adjusted to small or zero like QQDQQ configuration
The multiple scattering by the FT at VO affects the momentum resolution.
In case of  0.5g/cm$^2$ material at VO, the multiple scattering gives
$\delta\theta \sim$1.2mrad (RMS).
Then the momentum resolution by the multiple scattering effect is estimated
as follows,

\begin{eqnarray}
\Delta p/p 	& = & \frac{\sqrt{1 + R_{11}^2}}{R_{16}} \times R_{12}\Delta\theta \\
& = & 4.2 \times 10^{-4} {\rm (RMS)} \nonumber \\
& \rightarrow & 1.0 \times 10^{-3} {\rm (FWHM)} \nonumber
\end{eqnarray}

The multiple scattering effect significantly contributes the momentum resolution of the beam analyzer.

K1.1 beam analyzer.

Figure <ref> shows the beam envelope of t
he K1.1BR beam line calculated by TRANSPORT.
The same beam source as the K1.1 beam line is assumed.
In the K1.1BR optics, the beam is horizontally focused after
D3 (HFOC).

Beam envelope of the K1.1BR beam line optics
Beam source size and spread are same as those of K1.1 beam line.

Kaon yield and purity were estimated using TURLE .
The assumption of the simulation is as follows;
5.0$\times$10$^{13}$ protons[ 5.0$\times$10$^{13}$
corresponds to 48kW (30GeV) with 5.0 s repetition cycle.
Note that number of proton on the T2 target
depends on not only MR beam power but also the T1 target.]
on the 50% loss production target,
production cross section by Sanford-Wang parameterization with kinematic reflection,

and no cloud-pion effect.
Table <ref> shows 1.1GeV/c $K^-$ yield and purity for each
slit opening at K1.1 beam line, while table <ref> shows
those of 0.8 GeV/c $K^\pm$ at K1.1BR beam line.
IFH and MomS are fully open.

Yield and purity of the 1.1 GeV/c $K^-$ at K1.1 beam line.

2c|slit opening		    1c|acceptance
2c|$E$=70kV/cm		     2c$E$=60kV/cm

IFV 	    MS1/MS2				    1c| [msr %]
yield	    purity		    yield	    purity

$\pm$1.5mm	    $\pm$1.0mm	    1.19	    219k	    98%	    220k	    97%

$\pm$2.0mm      $\pm$1.0mm        1.28      234k      98%      234k      95%

$\pm$3.0mm	    $\pm$1.0mm	    1.31	    242k	    98%	    242k	    95%

$\pm$1.5mm	    $\pm$1.5mm	    1.70	    312k	    71%	    312k	    16%

$\pm$2.0mm      $\pm$1.5mm        1.94      357k      41%      357k      10%

$\pm$3.0mm	    $\pm$1.5mm	    2.14	    393k	    32%	    394k	    9.2%

$\pm$1.5mm	    $\pm$2.0mm	    1.95 	    358k	    14%	    358k	    4.0%

$\pm$2.0mm      $\pm$2.0mm        2.38      437k      6.2%     437k      2.8%

$\pm$3.0mm,	    $\pm$2.0mm	    2.79	    511k	    4.9%	    515k	    1.5%

Yield and purity of the 0.8 GeV/c $K^\pm$ at K1.1BR beam line
with $E$=60 kV/cm.

3c|slit opening		    1c|acceptance
2c|$K^+$		     2c$K^-$

IFV	    MS1	    HFOC				    1c| [msr %]
yield	    purity		    yield	    purity

$\pm$1.5mm	    $\pm$1.0mm	    $\pm$10mm	    1.36	    131k	    91%	     79k	    88%

$\pm$2.0mm      $\pm$1.0mm        $\pm$10mm     1.50      145k      83%       87k      78%

$\pm$3.0mm	    $\pm$1.0mm	    $\pm$10mm	    1.62	    156k	    54%	     94k	    45%

$\pm$1.5mm	    $\pm$1.5mm	    $\pm$10mm	    1.85	    179k	    67%	    108k	    60%

$\pm$2.0mm      $\pm$1.5mm        $\pm$10mm     2.21      213k      59%      128k      51%

$\pm$3.0mm	    $\pm$1.5mm	    $\pm$10mm	    2.55	    245k	    39%	    148k	    32%

$\pm$1.5mm	    $\pm$2.0mm	    $\pm$10mm	    2.09	    201k	    47%	    121k	    38%

$\pm$2.0mm      $\pm$2.0mm        $\pm$10mm     2.60      250k      43%      151k      35%

$\pm$3.0mm	    $\pm$2.0mm	    $\pm$10mm	    3.21	    310k	    31%	    187k	    24%

§.§ $\Lambda$p Scattering Experiment at the K1.1 Beam Line

§.§.§ Background of YN interaction and YN scattering experiment

Nucleus is a many-body system of nucleons which are bound by a nuclear force.
The nuclear force is crucially important interaction which determines the structures from nuclei up to compact stars.
The nuclear force shows the attractive nature in the middle and long ranges and the averaged attractive force binds the nuclei.
In order to make the nuclei and compact stars be a stable bound state,
the balance between the attractive force in the middle-long range and repulsive force in the short range is very important,
because the repulsive force in the short range prevents nuclei and compact stars from collapsing due to its attractive force.
The nuclear force is a strong interaction between the color-less nucleon and can be described theoretically by not taking into account the role of quarks.
For example, boson exchange models try to explain the nuclear force by considering the possible meson exchange including  pseudo scalar, vector, scalar mesons
and its pair or multi-meson exchange diagrams.
Experimentally, the nuclear force has been intensively studied by plenty of proton-proton and neutron-proton scattering experiments.
Very precise scattering data such as differential cross section and spin observables exist and these data were essential to construct so-called "realistic models" <cit.> for nuclear force
which reproduce these experimental observables with a reduced $\chi^{2} \sim 1$.
Such modern interaction models are used to understand the structure of the nuclei and three-body nuclear force.

In order to understand the role of quark flavor  in the nuclear force, we should extend the nuclear force to the baryon-baryon (BB) interaction including hyperon-nucleon (YN) and hyperon-hyperon (YY) interactions,
because new interaction-multiplets are expected to show very different features in the short range region where two baryons overlap with each other.
The quark cluster model predicts the quite repulsive core in the 10 and $8_{s}$ multiplets due to the Pauli effect in quark level
and the attractive core in the flavor singlet channel due to the color-magnetic interaction <cit.>.
These predictions are now reproduced by lattice QCD simulations which become a powerful theoretical tool to derive the YN and YY potentials from the first principle in QCD <cit.>.
The BB interaction is an essential test to describe the meson exchange picture with a uniform treatment assuming the SU(3) flavor symmetry <cit.>.
In order to test these theoretical models of two-body BB interaction, important experimental inputs are the binding energy of few-body bound system including hyperon such as hypertriton
and two-body scattering data between hyperon and proton.
As we mentioned, the $pp$ and $np$ scattering data played an essential role to establish the realistic nuclear force models.
On the other hand, the hyperon-proton scattering experiment was quite difficult experimentally due to the low intensity of the hyperon beam and its short lifetime.
36 cross section data of the $\Lambda p$ and $\Sigma p$ scatterings  <cit.> were actually indispensable to construct the present BB interaction models.
However the quality and quantity of these data are insufficient to impose a strict constraint on the theoretical models.
Therefore, historically, the BB interaction has been investigated from the hypernuclear structure because their binding energies and level structures reflect the YN interaction <cit.>.
The effective two-body interaction potential in the hypernuclei was obtained by the G-matrix calculation using the bare two-body interaction <cit.>.
Then, the calculated energy levels of the hypernulear system were compared with the hypernuclear data and the bare two-body interaction was updated so as to reproduce the experimental data.
This theoretical and experimental strategy seems to work well at least for the many hypernucler phenomena.
However, we know that a neutron star with two-solar mass <cit.> can not be supported by the present two-body YN interaction due to softening of the equation of state by the appearance of hyperons in the high density region in the neutron star.
The many-body repulsive interactions such as YNN three-body force are expected to play an essential role in the such high-density region to support the massive neutron star with hyperon.
Such many-body effect is expected to appear in the heavy $\Lambda$ hypernuclei as the mass difference
in the ground state.
A new project to measure the $\Lambda$ hypernuclear bounding energy with ultra precise resolution are just launching to explore the $\Lambda NN$ three-body interaction by the ($\pi^{+}, K^{+}$) spectroscopy in the extended hadron experimental facility by utilizing a high-intensity high-resolution (HIHR) beam line <cit.>.
In such a situation, it is crucially important to establish the realistic two-body YN interaction from the two-body system, that is, YN scattering data.
That is because the current theoretical treatment to derive YN two-body interaction from the hypernuclear structure is already suffered from uncertainties from the many-body effect in the hypernuclei.
Therefore, we have to really change the strategy for deriving the YN two-body interaction, that is, the realistic two-body YN interaction should be constructed based on the two-body scattering data.

Recently, we have realized a high-statistics $\Sigma p$ scattering in the J-PARC E40 experiment where
$\sim$5,000 scattering events for both $\Sigma^{-}p$ and $\Sigma^{+}p$ channels were identified <cit.>.
We have introduced a new experimental technique to overcome the experimental difficulty of a hyperon-proton scattering experiment.
A liquid hydrogen target was used as both hyperon production and hyperon proton scattering targets to identify the hyperon production and the scattering events kinematically
without any imaging data to identify the scattering topology.
High intensity $\pi$ beams were handled to accumulate $\Sigma$ beams as much as possible and
more than 100 times more $\Sigma$ beams ($\sim$17 M $\Sigma^{-}$ and $\sim$70 M $\Sigma^{+}$) than that in the past KEK experiment were accumulated.
The differential cross sections measured in E40 together with a past measurement (KEK-PS E289 data for 400 $< p$ (MeV/$c$) $<$ 700 <cit.>) and theoretical calculations are shown in
Figure 17 in reference<cit.>.

The statistical error of 10% level was achieved  with a fine angular step of $d\cos \theta = 0.1$ by identifying  the largest statistics of about 4,500 $\Sigma^{-}p$ elastic scattering  events from 1.72 $\times$ $10^{7}$ $\Sigma^{-}$ particles.
The differential cross sections show clear forward peaking structure and the forward and backward ratio is large particularly in the higher momentum regions.
Although the experimental inputs of the two-body hyperon-proton scattering were quite limited up to now, the success of the $\Sigma^{-}p$ scattering is a remarkable step to provide accurate data to improve the $BB$ interaction models and to establish the realistic $BB$ interactions with both theoretical and experimental efforts.

We apply the same experimental method to the $\Lambda p$ scattering experiment to derive the differential cross section and spin observables with good accuracy.
By synthesizing all the experimental information ($\Sigma N$ channel in E40 and $\Lambda N$ channel in the proposed experiment), better understanding of the $BB$ interactions will be achieved in near future.

§.§.§ Formalism of the spin-dependent YN interaction

In this proposed experiment, the important new aspect is to measure the spin observables with a polarized $\Lambda$ beam.
The hyperon-proton scattering is the scattering between two spin 1/2 particles and the spin observables are substantial tool for the study of the spin-dependent YN interaction<cit.>.

The $T$ matrix for the elastic scattering between particle $a$ and $b$ is represented in terms of spin-independent, spin-spin, symmetric $LS$ (SLS), antisymmetric $LS$ (ALS) and tensor components as

\begin{equation}
M = V_{c} + V_{\sigma}(\bm{s_{a}}\cdot \bm{s_{b}})+V_{SLS}(\bm{s_{a}}+\bm{s_{b}})\cdot \bm{L} + V_{ALS}(\bm{s_{a}}-\bm{s_{b}}) \cdot \bm{L} + V_{T}([\bm{s_{a}} \otimes \bm{s_{b}}]^{(2)} \cdot \bm{Y_{2}}(\hat{\bm{r}}))
\end{equation}

where $\bm{L}$ is the $a$-$b$ relative orbital angular momentum, $\bm{r}$ is the $a$-$b$ relative coordinate and $V$'s are form-factor functions
for the spin-independent central interaction $V_{c}$, the spin-spin interaction $V_{\sigma}$, the SLS interaction $V_{SLS}$, the ALS interaction $V_{ALS}$, and the tensor interaction $V_{T}$.
In order to describe the scattering observables, new scalar amplitudes, vector amplitudes and tensor amplitudes are defined as follows,

\begin{equation}
U_{\alpha} \equiv <\bm{k}_{f}|V_{c}|\bm{k}_{i}>,  U_{\beta} \equiv <\bm{k}_{f}|V_{\sigma}|\bm{k}_{i}>
\end{equation}

for the scalar amplitudes,

\begin{equation}
S_{ALS} \equiv <\bm{k}_{f}|V_{ALS}L_{1}|\bm{k}_{i}> , S_{SLS} \equiv <\bm{k}_{f}|V_{SLS}L_{1}|\bm{k}_{i}>
\end{equation}

for the scalar amplitudes and

\begin{equation}
T_{j} = \frac{1}{2} <\bm{k}_{f}|V_{T}Y_{2j-1}|\bm{k}_{i}>
\end{equation}

for the tensor amplitudes for $j=1, 2, 3$. For later convenience, the following $T_{\alpha}$ and $T_{\beta}$ are also used,

\begin{equation}
T_{\alpha} = \frac{1}{\sqrt{6}}T_{1}+T_{3},  T_{\beta} = \frac{1}{\sqrt{6}}T_{1}-T_{3},
\end{equation}

which give

\begin{equation}
T_{2} = -\tan \theta (\frac{1}{2}T_{\alpha}+T_{\beta}).
\end{equation}

The differential cross section is described by the sum of the all contributions as follows,

\begin{equation}
\Bigl(\frac{d\sigma}{d\Omega} \Bigl)=\frac{1}{4}{\rm Tr}(MM^{\dag})=|U_{\alpha}|^{2}+\frac{3}{16}|U_{\beta}|^{2}+\frac{1}{2}(|S_{SLS}|^{2} + |S_{ALS}|^{2})+\frac{1}{4}|T_{1}|^{2}+\frac{1}{2}(|T_{2}|^{2}+|T_{3}|^{2}).
\end{equation}

Therefore, in order to determine the each component separately, a lot of spin observables are necessary.
In the YN scattering case, the measurable spin observables are limited.
However, such spin observables data are quite important to pin down the each spin-dependent component.
The analyzing power $A_{y}(Y)$ for the polarized hyperon beam is described as,

\begin{equation}
A_{y}(Y) = -\frac{1}{\sqrt{2}\sigma ( \theta )} {\rm {Im}} \left \{  (U_{\alpha} + \frac{1}{4}U_{\beta} )^{*} S_{SLS}+ (U_{\alpha} - \frac{1}{4}U_{\beta} )^{*} S_{ALS} -\frac{1}{2}T_{\alpha}^{*} (-S_{ALS}+S_{SLS}) \right\},
\end{equation}

where $\sigma(\theta)$ represents the differential cross section.
If we regard that  the component including the tensor amplitude ($T_{\alpha}$) is the 2nd order value, the analyzing power is sensitive to the LS forces.
In the NN interaction, the ALS is not allowed for the isospin symmetry and the ALS is the purely the new interaction appeared in the YN sector.

By measuring the change of the polarization of hyperon before and after the scattering, the depolarization can be measured.
The depolarization is described as

\begin{dmath}
%D_{y}^{y} \approx \frac{1}{\sigma(\theta)}{\rm Re} \left\{  |U_{\alpha}|^{2}-\frac{1}{16}|U_{\beta}|^{2} + \cdots \right\},
D_{y}^{y}  =  \frac{1}{\sigma(\theta)}{\rm Re} \left\{  \frac{1}{2 \sqrt{3}} \left ( U_{0}+\frac{1}{\sqrt{3}} U_{1}\right)^{*}U_{1} + \frac{1}{2} \left ( U_{0}-\frac{1}{\sqrt{3}} U_{1}\right)^{*} \left (\frac{1}{\sqrt{6}}T_{1}+T_{3}  \right ) -S_{1}^{*}S_{2}   +  \frac{1}{2}|S_{3}|^{2}  - \frac{1}{\sqrt{6}}T_{1}^{*} \left (\frac{1}{\sqrt{6}} T_{1}-T_{3} \right ) - \frac{1}{2}|T_{2}|^{2} \right\}.
\end{dmath}

Although the equation is complicated form of several amplitudes, $D_{y}^{y}$ is expected to be rather sensitive to tensor force <cit.>.
It is still difficult to determine each spin-dependent component separately from these limited observables.
However, accumulation of these differential measurements can contribute to impose strong constraints on the present YN interaction models.

As for the type of the existing YN scattering observables, total cross section data exist even though the accuracy is not sufficient.
Therefore theoretical calculations of all models become similar for the total cross section because the existing cross section data are used to fit parameters in the theoretical models as shown in next section.
However, since there are almost no spin observable data except for the analyzing power measured in KEK <cit.>,
the theoretical prediction are quite different among the different theoretical frameworks.
Therefore the spin observable measurement is quite important to test and improve the theoretical frameworks.

§.§.§ Theoretical studies for the $\Lambda p$ scattering

[]Total cross section of the $\Lambda p$ elastic scattering and theoretical calculation. (1)Calculation by ESC16 model <cit.>. The solid line (a) shows the calculation including the higher wave contribution, whereas dotted lines (b) and (c) are obtained by the effective range approximation with some scattering length and effective range.
(2) Calculation by the chiral EFT model with contributions up to the next leading order <cit.>. The red and cyan bands represent the result for NLO13 and the alternative version NLO19, respectively. The dotted line show the result for the Nijmegen NSC97f potential. (3) Calculation by the quark cluster model, that is, fss2 (solid curve) and FSS (dashed curve) <cit.>.

In Figure <ref>, the total cross sections of the $\Lambda p$ elastic scattering are plotted with theoretical models such as the Nijmegen Extended Soft-Core (ESC) 16 model <cit.>,
the chiral Effective Field Theory (EFT) extended to YN sector <cit.> and the quark cluster model <cit.>.
In the low energy below 400 MeV/$c$, there exist several cross section measurements with hydrogen bubble chambers and these are the essential inputs to determine the S-wave contribution of the $\Lambda p$ channel.
On the other hand, in the higher momentum range higher than 400 MeV/$c$, the cross section data were reported by one bubble chamber experiment
and the total cross sections for each momentum region were determined from only $\sim$20 scattering events.
Due to such limited statistics, there is no differential information such as differential cross section and spin observables.

The relationship between the isospin basis and the flavor SU(3) basis for the $NN$ and $YN$ with $S=-1$ channels

$S$     $BB$ channel (I)     $^{1}E$ or $^{3}O$     $^{3}E$ or $^{1}O$

0     $NN (I=0)$     $--$     $(10^{*})$

$NN (I=1)$     (27)     $--$

$\Lambda N (I=1/2)$     $\frac{1}{\sqrt{10}}[(8_{s})+3(27)]$      $\frac{1}{\sqrt{2}}[-(8_{a})+(10^{*})]$

$-1$        $\Sigma N (I=1/2)$     $\frac{1}{\sqrt{10}}[3(8_{s})-(27)]$      $\frac{1}{\sqrt{2}}[(8_{a})+(10^{*})]$

$\Sigma N (I=3/2)$     (27)      (10)

The $NN$ and $YN$ interactions should be understood in a unified way using (broken) SU(3) symmetry.
Table <ref>  shows the relationship between the isospin basis and the flavor SU(3) basis for the $NN$ and $YN$ channels with strangeness $S=-1$.
These channels are related each other through the SU(3) flavor multiplet and $\Lambda N$ and $\Sigma N$ channels are discussed together theoretically.
Now, the $\Sigma p$ scattering cross section data will be drastically updated by the E40 experiment.

All theories predict the sizable cusp (enhancement) of the cross section at the $\Sigma N$ threshold due to the strong $\Lambda N - \Sigma N$ ($I=1/2$) $^{3}S_{1}-^{3}D_{1}$ coupling caused by the tensor force of the pion exchange potential.
In the quark cluster model, the importance of the ALS which causes the $\Lambda N ^{3}P_{1} - \Lambda N ^{1}P_{1} $  and $\Lambda N - \Sigma N (I=1/2) ^{3}P_{1} - ^{1}P_{1} $ transitions
is discussed because the additional ALS force due to the quark-quark interaction is predicted in the quark picture.
Such ALS effect should appear in the Analyzing power (polarization) in the $\Lambda p$ scattering at this $\Lambda N-\Sigma N$ threshold region.

In the following subsection, we summarize the characteristics of  several theoretical models.

* Nijmegen ESC model

Nijmegen extended-soft-core (ESC) models whose latest version is ESC16 <cit.>, describe $NN$, $YN$ and $YY$ interactions in a unified way using broken SU(3) symmetry.
The potentials consists of local and nonlocal potentials due to one-boson exchange which are the members of nonets of pseudoscalar, vector, scalar and axial-vector mesons,
two psudoscalar exchange, meson-pair exchange and diffractive exchange.
The meson-baryon coupling constants are calculated by imposing the SU(3) symmetry.
In the Nijmegen model, there have been difficulties to reproduce the sufficiently repulsive short-range interaction in $\Sigma N (I=3/2, ^{3}S_{1})$ channel and
$\Sigma N (I=1/2, ^{1}S_{0})$ channel which are explained quite naturally as the Pauli effect in the quark level in the (10) and (8$_{s}$) multiplets in the SU(3) representation  in the quark level in the quark cluster model.
In the latest ESC16 version, such "forbidden state" effect are taken into account phenomenologically by making an effective Pomeron potential as the sum of