XXXX-XXXX

2]Department of Physics, Tokyo Institute of Technology, 2-12-1 Ookayama,
Megro, Tokyo 152-8551, Japan

3]Department of Life and Environmental Agricultural Sciences, Tottori
University, Tottori 680-8551, Japan

4]RIKEN Nishina Center for Accelerator-Based Science, RIKEN, Wako 351-0198,
Japan 5]RIKEN Cluster for Pioneering Research, RIKEN, Wako 351-0198, Japan

6]Department of Mathematical and Physical Sciences, Chubu University, Kasugai,
Aichi 487-8501, Japan

7]Department of Physics, Nara Women’s University, Nara 630-8506, Japan

# Investigation of kaonic atom optical potential by the high precision data of
kaonic 3He and 4He atoms

J. Yamagata-Sekihara<EMAIL_ADDRESS>Department of Physics, Kyoto
Sangyo University, Kyoto 603-8555, Japan  Y. Iizawa  [  D. Jido  N. Ikeno  [
T. Hashimoto  [  [  S. Okada  [  S. Hirenzaki  [

###### Abstract

We theoretically investigate kaonic atom states based on the latest high
precision data of the $2p$ states of the kaonic atoms in 3He and 4He (J-PARC
E62). We consider the phenomenological form for the optical potential
proportional to the nuclear density distributions and determine the potential
parameters by the latest data. We find two sets of potential parameters for
each of $K^{-}-^{3}$He and $K^{-}-^{4}$He that reproduce the data. We then
extract the potential parameters consistent with the data of 3He and 4He atoms
simultaneously. Using the potential, we investigate the structure of the
kaonic atoms and kaonic nuclei for various nuclei up to Sn. We also report the
optical potential parameters suited for the global description of the atomic
data, the possible strong isospin dependence of the potential, and the
different strength of the data of $K^{-}-^{3}$He and $K^{-}-^{4}$He in the
restrictions on the potential parameters. In addition, we find that the study
of the $1s$ states of the kaonic helium atom is very interesting as a next
step, and the determination of the orbital angular momenta of the kaonic
nuclear states is helpful for the unified understandings of the atomic and
nuclear bound states of kaon because of the mutual influence between the
atomic and the nuclear states of kaon with the same orbital angular momentum.

xxxx, xxx

##  1 Introduction

Meson-nucleus systems are interesting research targets to investigate the in-
medium meson properties and the aspect of the symmetry of the strong
interaction at finite density Batty:1997zp ; Friedman:2007zza ;
Yamazaki:2012zza ; Hayano:2008vn ; Metag:2017yuh ; Hirenzaki:2022dpt ;
Itahashi:2023boi ; piAF:2022gvw . Especially, the ${\bar{K}}$-nucleus bound
states, that is, kaonic atoms and kaonic nuclei, are promissing for the
studies of the kaon properties at finite nuclear density and the kaon-nucleon
interaction in the nuclear medium. Since kaon is the lightest meson with
strangeness and is believed to be one of the Nambu-Goldstone bosons of the
SU(3) chiral symmetry breaking, the behavior of kaon at finite density is
expected to provide quite relevant information on the studies of the strong
interaction symmetry. In addition, the quantitative knowledge of kaon
properties at finite density is considered to be crucial to study the high
density nuclear matter and to clarify the role of kaon there, for example, to
show the clear criteria for the kaon condensation occurrence.

Recently, there have been important developments in experimental studies of
the kaon-nucleus bound states. The experimental evidence of the kaonic nuclear
states could open a new era of the meson-nuclear physics J-PARCE15:2018zys .
And we are also interested in the recent experimental achievement of the high
precision measurement of the kaonic 3He and 4He atoms (J-PARC E62)
J-PARCE62:2022qnt . The accuracy of the data of the kaonic-helium atoms is
almost two orders of magnitude better than the previous data and clearly opens
up the precision frontier of the kaon physics. The experimental results
clearly denied the existence of the possible large shifts and widths of the
kaonic atoms suggested theoretically in connection with the studies of the
kaonic nuclear states in Ref. Akaishi:2005mw .

Theoretically, the kaon-nucleus interaction has been studied using the data of
kaonic atoms by the $\chi^{2}$ fitting procedure Batty:1997zp ;
Friedman:2007zza ; Mares:2006vk . The developments of the chiral unitary
approach Ramos:1999ku ; Kaiser:1995eg ; Oset:1997it  enable us to deepen our
understandings of the hadron resonances in the ${\bar{K}}N$ channel
Jido:2003cb  and kaon-nucleus bound systems Hirenzaki:2000da . A global
analysis of kaonic atom spectra Iizawa:2019uiu  suggested that repulsive
shifts of kaonic atomic states stem from large nuclear absorption of kaon
instead of the existence of nuclear bound states of kaon. Recently, the
progress has been also made in the studies of the kaonic nuclear states by the
microscopic few body calculations Akaishi:2002bg ; Yamazaki:2002uh ;
Shevchenko:2007zz ; Wycech:2008wf ; Dote:2008hw ; Ikeda:2008ub ; Barnea:2012qa
; Sekihara:2016vyd ; Ohnishi:2017uni ; Dote:2017wkk .

In this article, we theoretically investigate the features and the
implications of the kaon-nucleus optical potential determined by the latest
high precision data of the $2p$ states of the $K^{-}-^{3,4}$He atoms. In Sect.
2, we explain our theoretical model used in the present analyses and introduce
the optical potential in a phenomenological form proportional to the nuclear
density distributions. In Sect. 3, we determine the potential parameters and
summarize the features of the obtained potential. We mention the restrictions
on the potential parameters required by the data of $K^{-}-^{3}$He and
$K^{-}-^{4}$He systems, the possible strong isospin dependence of the optical
potential, and the comparison of the data with the calculated results by the
theoretically proposed potentials Mares:2006vk ; Ramos:1999ku . In Sect. 4, we
apply the potential obtained in Sect. 3 to the various kaonic atoms and kaonic
nuclei, and discuss the implications of the potential to find out the
important observables to develop the unified understandings of the kaon-
nucleus bound states including atomic and nuclear states. Section 5 is devoted
to the summary and the conclusion of this article.

##  2 Theoretical model

We adopt the standard optical potential description of the kaon-nucleus bound
systems, and we solve the relativistic Klein-Gordon (KG) equation with the
Lorentz scalar type optical potential $U(r)$ and the electromagnetic potential
$V_{\rm em}(r)$, which is the time component of the Lorentz vector potential,
to obtain the theoretical binding energies and widths of the kaonic atom
states. The KG equation is written as,

$[-\nabla^{2}+\mu^{2}+2\mu U(r)]\phi(\bm{r})=[\omega-V_{\rm
em}(r)]^{2}\phi(\bm{r})~{},$ (1)

where $\mu$ is the kaon-nucleus reduced mass, $\omega$ the complex eigen
energy, and $\phi(\bm{r})$ the kaon wave function. We write the complex energy
with two real parameters as $\displaystyle\omega=E+\mu-i\frac{\Gamma}{2}$,
where $E$ is the binding energy and $\Gamma$ is the width of the bound state.
As the electromagnetic potential $V_{\rm em}(r)$, we consider the Coulomb
potential between kaon and nucleus including the effects of the vacuum
polarization. The finite size charge distribution of the nucleus is also taken
into account and $V_{\rm em}(r)$ is expressed as Ikeno:2015ioa ,

$V_{\rm em}(r)=-\alpha\int\frac{\rho_{\rm
ch}(r^{\prime})Q(|\bm{r}-\bm{r^{\prime}}|)}{|\bm{r}-\bm{r^{\prime}}|}d\bm{r^{\prime}}~{},$
(2)

where $\alpha$ is the fine structure constant and $\rho_{\rm ch}(r)$ the
charge distribution of the nucleus. The function $Q$ is defined as,

$Q(r)=1+\frac{2\alpha}{3\pi}\int^{\infty}_{1}~{}du~{}e^{-2mru}\left(1+\frac{1}{2u^{2}}\right)\frac{(u^{2}-1)^{1/2}}{u^{2}}~{},$
(3)

where $m$ indicates the electron mass. The second term on the right-hand side
indicates the short range vacuum polarization effects. We confirm that the
electromagnetic potential adopted here is accurate enough to study the energy
shifts due to the strong interaction reported in Ref. J-PARCE62:2022qnt  by
comparing the calculated results with those in Ref. Santos:2004bw .

We consider a phenomenological optical potential $U(r)$, which is assumed to
be proportional to the nuclear density distribution, defined as,

$U(r)=(V_{0}+iW_{0})\frac{\rho(r)}{\rho_{0}}~{}~{},$ (4)

where $\rho_{0}$ is the normal nuclear density $\rho_{0}=0.17~{}{\rm
fm}^{-3}$. The parameters $V_{0}$ and $W_{0}$ show the potential strength at
the normal nuclear density, and are used as the parameters to study the kaon-
nucleus bound systems in this article. The nuclear density distribution
$\rho(r)$ used in the optical potential is the distribution of the center of
the nucleon. The systematic analyses of the kaonic atoms using the similar
potential have been reported in Ref. Iizawa:2019uiu .

We use realistic nuclear density distributions for $U(r)$ and realistic charge
distributions for $V_{\rm em}(r)$, which are obtained by a precise few-body
calculation Hiyama  for 3He and 4He. The nuclear density distributions
$\rho(r)$ are shown in Fig. 1. As can be seen in the figure, the central
densities of these nuclei are higher than the normal nuclear density
$\rho_{0}$, especially for 4He. They are $\rho(0)\approx 1.32\rho_{0}$ for 3He
and $\rho(0)\approx 2.02\rho_{0}$ for 4He. To use the realistic densities is
important for the theoretical analyses here since the calculated results of
the binding energies and the widths of the kaonic $2p$ atoms in He can be
improved by a few eV from those obtained with the single Gaussian densities.

The potential picture for the kaonic atoms explained above is considered to be
standard for the studies of the atomic states, however this picture would be
modified for the systems having so-called mesonic nuclear states where the
degrees of freedom of each nucleon are expected to be more important and the
few-body calculations of the whole systems would be required.

Figure 1: The nuclear density distributions $\rho(r)$ of 3He and 4He in Eq.
(4) obtained by a microscopic calculation Hiyama .

##  3 Optical potential from $K^{-}-^{3}$He and $K^{-}-^{4}$He atomic $2p$
states

###  3.1 Experimental data and calculated results with theoretical potentials

First, we briefly summarize the latest experimental data J-PARCE62:2022qnt
and the calculated results by the theoretical potentials Mares:2006vk ;
Ramos:1999ku  for $K^{-}-^{3}$He and $K^{-}-^{4}$He atoms.

The latest high precision experimental data reported in Ref. J-PARCE62:2022qnt
, which we mainly study in this article, are summarized in Table 1. We mention
here that the data have two interesting features. As far as we look at the
central values of the data, we find (i) the opposite sign of the shifts of
$K^{-}-^{3}$He and $K^{-}-^{4}$He, and (ii) the larger width of $K^{-}-^{3}$He
than $K^{-}-^{4}$He. These features could change in future because of the
improvements of the errors of the present data. The square root of the
quadratic sum of the statistical and systematical errors is also shown in the
parenthesis for each datum in Table 1, and is considered to represent the
experimental error of each datum in the following analyses of this article.
The transition energy shift $\Delta E$ is defined as $\Delta E=E^{\rm
exp}_{3d\to 2p}-E^{\rm em}_{3d\to 2p}$ by the transition energies from $3d$ to
$2p$ states observed in the experiment, $E^{\rm exp}_{3d\to 2p}$, and
calculated with the electromagnetic potential, $E^{\rm em}_{3d\to 2p}$. The
positive shift $\Delta E>0$ means $E^{\rm exp}_{3d\to 2p}>E^{\rm em}_{3d\to
2p}$ and thus the larger experimental transition energy. Since the observed
energy $E^{\rm exp}_{3d\to 2p}$ is considered to correspond to the transition
energy calculated with the full potential, $E^{\rm opt+em}_{3d\to 2p}$, the
positive shift $\Delta E>0$ indicates that the effects of the optical
potential makes the $2p$ state deeper than the corresponding pure
electromagnetic state and that the attractive shift of the $2p$ state.

Table 1: Experimental $3d\to 2p$ transition energy shifts and $2p$ widths of the kaonic 3He and 4He atoms J-PARCE62:2022qnt  in units of eV. The errors shown in the parentheses, which are the square root of the quadratic sum of the statistical and systematical errors in Ref. J-PARCE62:2022qnt , are considered to represent the experimental errors in this article. It should be noted that the shift of $K^{-}-^{3}$He is repulsive, while that of $K^{-}-^{4}$He is attractive. eV | Shift | Width
---|---|---
$K^{-}-^{3}$He |  $-0.2\pm 0.4\,$(stat)$\pm 0.3\,$(syst) |  $2.5\pm 1.0$ (stat)$\pm 0.4\,$(syst)
| ($\pm 0.5$) | ($\pm 1.1$)
$K^{-}-^{4}$He |  $0.2\pm 0.3\,$(stat)$\pm 0.2\,$(syst) |  $1.0\pm 0.6\,$(stat)$\pm 0.3\,$(syst)
| $(\pm 0.4)$ | $(\pm 0.7)$

In order to check how well existing theoretical potentials Ramos:1999ku ;
Mares:2006vk  work for the new experimental data, we calculate the energies of
the atomic $2p$ and $3d$ states for $K^{-}-^{3}$He and $K^{-}-^{4}$He atoms
with these potentials and compare the results with the latest data
J-PARCE62:2022qnt . The potential in Ref. Mares:2006vk  was obtained by the
$\chi^{2}$ fitting to the previous kaonic atom data and its strength of the
real and imaginary parts is ($-226,-91$) MeV at $\rho_{0}$. The potential in
Ref. Ramos:1999ku  was obtained by the chiral unitary model and show the
strength ($-45,-61$) MeV at $\rho_{0}$. The results are shown in Fig. 2 (Left)
for $K^{-}-^{3}$He and Fig. 2 (Right) for $K^{-}-^{4}$He with the experimental
data J-PARCE62:2022qnt . We find that both the potentials reproduce the data
reasonably well. However, the calculated width of $K^{-}-^{3}$He by the chiral
unitary potential and the energy shift of $K^{-}-^{4}$He by the $\chi^{2}$
fitting potential are found to show small discrepancies with the experimental
results.

|
---|---
Figure 2: The calculated energy shifts of the $3d\to 2p$ transitions and the
widths of the kaonic atom $2p$ states in 3He (Left) and 4He (Right) with the
latest experimental data J-PARCE62:2022qnt . The theoretical potentials
obtained by the analyses of the $\chi^{2}$ fitting to the previous kaonic atom
data Mares:2006vk  and by the chiral unitary model Ramos:1999ku  are used.

###  3.2 Optical potential parameters fixed by the latest data

We extract here the kaon-nucleus optical potential parameters that are
consistent with the latest data of the transition energy shifts and the level
widths of the kaonic helium atoms in Ref. J-PARCE62:2022qnt . We calculate the
energy shift and width of kaonic 3He with given $V_{0}$ and $W_{0}$. In Fig.
3, we show the contour plots of the energy shifts and the widths of kaonic 3He
in the $V_{0}-W_{0}$ plane of the potential parameters defined in Eq. (4). In
Fig. 3 (a) and (b), the solid lines show the $V_{0}$ and $W_{0}$ values that
reproduce the central values of the experimental data, and the dashed and
dotted lines show the $V_{0}$ and $W_{0}$ values reproducing the upper and
lower bounds of the experimental data, respectively. We find that two sets of
the potential parameters $(V_{0},W_{0})=(-178,-274)$ MeV (HE3-A) and
$(-300,-95)$ MeV (HE3-B) reproduce the both data of the shift and the width of
the kaonic 3He atomic $2p$ state. These solutions are listed in Table 2 and
shown by the solid diamonds in Fig. 3. The obtained two parameter sets are
summarized as a potential with a shallower real part and stronger absorption
and one with a deeper real part and weaker absorption, respectively. We
present in Fig. 3 (c) the overlaid contour plot of the shift and the width
shown in Figs. 3 (a) and (b). The hatched area indicates the potential
parameters that are consistent with the experimental data of the shift and
width of 3He. The relatively wide region of parameters are found to be
acceptable for the kaonic 3He atomic data.

|
---|---
Figure 3: The contour plots of the shift of the $3d\to 2p$ transition energy (a) and the $2p$ level width of the kaonic 3He atom (b) in the plane of the potential parameters $V_{0}$ and $W_{0}$ defined in Eq. (4). The $V_{0}$ and $W_{0}$ values that reproduce the central values of the experimental data are shown by the solid lines. The $V_{0}$ and $W_{0}$ values reproducing the upper and lower bounds of the experimental data are shown by the dashed and the dotted lines. The experimental shift and width of the $K^{-}-^{3}$He atom are reproduced simultaneously by the potential parameters $(V_{0},W_{0})=(-178,-274)$ MeV (HE3-A) and $(-300,-95)$ MeV (HE3-B) shown by the solid diamonds in the figures. Panel (c) shows the overlaid contour plot of the transition energy shift (a) and the width of the $K^{-}-^{3}$He atom (b) in the same $V_{0}-W_{0}$ plane. The hatched region indicates the potential parameters that are consistent with the experimental data of the shift and width of 3He. Table 2: The strengths of the optical potential defined in Eq. (4) that reproduce the $2p$ atomic data in the $K^{-}-^{3}$He system J-PARCE62:2022qnt  listed in Table 1. The potential depths at normal nuclear density ($V_{0}$ and $W_{0}$ parameters) and those at the center of the 3He nucleus are shown for two sets of parameters HE3-A and HE3-B. These potentials are shown as the solid diamonds in Figs. 3, 5, 10, 12. 3He | HE3-A | HE3-B
---|---|---
[MeV] | Real | Imag | Real | Imag
$U$ at $\rho=\rho_{0}$ | $-178$ | $-274$ | $-300$ | $-95$
($V_{0}$ and $W_{0}$)
$U(r=0)$ | $-235$ | $-362$ | $-396$ | $-126$

In Fig. 4, we show the results for the $K^{-}-^{4}$He $2p$ atomic state
obtained by the optical potential as in Fig. 3. For the $K^{-}-^{4}$He system,
we also find that two sets of the potential parameters
$(V_{0},W_{0})=(-58,-56)$ MeV (HE4-A) and $(-287,-43)$ MeV (HE4-B) reproduce
the central values of the experimental data both for shift and width of $2p$
state in $K^{-}-^{4}$He. These two sets of the potential parameters are listed
in Table 3 and shown by the solid circles in Fig. 4. In this $K^{-}-^{4}$He
case, both parameter sets indicate the small values of the $W_{0}$ parameter
for the imaginary potential, while the corresponding real potential parameters
$V_{0}$ take small and large values. It should be noted here that the central
density $\rho(0)$ of 4He is close to $2\rho_{0}$ and 1.53 times larger than
that of 3He. In Fig. 4 (c), the overlaid contour plot of the shift and width
of the $K^{-}-^{4}$He atom are shown. We find that the hatched areas, where
the potential parameters are consistent with the data, exist in two regions in
the $V_{0}-W_{0}$ plane separately. The data of the $K^{-}-^{4}$He atom
provide the stronger constraints to the potential parameters than those of the
$K^{-}-^{3}$He atom.

|
---|---
Figure 4: Same as Fig. 3 except for the kaonic 4He atomic state. The experimental shift and width of the $K^{-}-^{4}$He atom are reproduced simultaneously by the potential parameters $(V_{0},W_{0})=(-58,-56)$ MeV (HE4-A) and $(-287,-43)$ MeV (HE4-B) shown by the solid circles in the figures. Table 3: Same as Table 2 except for the $K^{-}-^{4}$He system. These potential parameters HE4-A and HE4-B are shown as the solid circles in Figs. 4, 5, 11, 12. 4He | HE4-A | HE4-B
---|---|---
[MeV] | Real | Imag | Real | Imag
$U$ at $\rho=\rho_{0}$ | $-58$ | $-56$ | $-287$ | $-43$
($V_{0}$ and $W_{0}$)
$U(r=0)$ | $-117$ | $-113$ | $-581$ | $-87$

###  3.3 Combined study of kaonic 3He and 4He atoms

We consider combined analyses of both data of 3He and 4He in this section.
First, we assume the isoscalar form (IS) for the optical potential, which does
not distinguish the proton and neutron densities, and extract the potential
parameters consistent with the data of 3He and 4He atoms simultaneously. We
show in Fig. 5 the overlaid contour plot of Fig. 3 (c) and Fig. 4 (c). The
potential parameters indicated by the solid triangles reproduce atomic states
consistent with both $K^{-}-^{3}$He and $K^{-}-^{4}$He atom data. These
parameter sets can be categorized as weak-attraction strong-absorption
potential ($V_{0},W_{0})=(-90,-120)$ MeV (IS-A) and strong-attraction weak-
absorption potential ($V_{0},W_{0})=(-280,-70)$ MeV (IS-B). We summarize the
strengths of the potential IS-A and IS-B in Table 4. Note that because of the
larger central densities of 3He and 4He than the normal nuclear density
$\rho_{0}=0.17$ fm-3, the depth of the potentials at the nuclear centers are
larger than the potential strengths at $\rho_{0}$ expressed by the
$(V_{0},W_{0})$ parameters. These potential parameters are considered to
reflect average behavior of the kaon interaction with nucleus and are
candidates of the potential parameters suited for global description of kaonic
atoms. We think, hence, the parameters obtained here are one of the important
results deduced from the latest data J-PARCE62:2022qnt  in this article. These
potential parameters are further investigated by heavier kaonic atoms in Sect.
4.2.

Figure 5: The overlaid plot of the figures Fig. 3 (c) and Fig. 4 (c) is shown. The solid triangles show the two sets of potential parameters IS-A ($V_{0},W_{0})=(-90,-120)$ MeV and IS-B $(V_{0},W_{0})=(-280,-70)$ MeV that are consistent with both $K^{-}-^{3}$He and $K^{-}-^{4}$He atomic data in Ref. J-PARCE62:2022qnt . Table 4: The strengths of the isoscalar (IS) kaon-nucleus optical potential IS-A and IS-B, that reproduce all data in Ref. J-PARCE62:2022qnt  compiled in Table 1 within the errors for both 3He and 4He, are summarized for the phenomenological potential form defined in Eq. (4). The potential depths at normal nuclear density ($V_{0}$ and $W_{0}$ parameters) and those at the center of the 3He and 4He nuclei are shown for the potentials. These potentials are corresponding to the solid triangles in Figs. 5, 10, 11, 12. 3He and 4He | IS-A | IS-B
---|---|---
[MeV] | Real | Imag | Real | Imag
$U$ at $\rho=\rho_{0}$ | $-90$ | $-120$ | $-280$ | $-70$
($V_{0}$ and $W_{0}$)
$U(r=0)$ | 3He | $-119$ | $-159$ | $-370$ | $-93$
4He | $-182$ | $-243$ | $-566$ | $-142$

We, then, consider the potential with the isospin dependence for the
$K^{-}-$He systems. Actually, the potential strengths $V_{0}$ and $W_{0}$ for
the kaonic 3He and 4He atoms listed in Table 2 and 3 may suggest the different
contributions from $K^{-}p$ and $K^{-}n$ interaction. The isospin dependence
of the ${\bar{K}}N$ interaction is also anticipated theoretically
Miyahara:2015bya . Though the accuracy of the data seems not enough to
determine the isospin dependence, we consider, as an estimation, a
phenomenological potential form with the isospin dependence as,

$\tilde{U}(r)=[V_{p}+iW_{p}]\frac{\rho_{p}(r)}{\rho_{0}}+[V_{n}+iW_{n}]\frac{\rho_{n}(r)}{\rho_{0}}~{}~{},$
(5)

where $\rho_{p}$ and $\rho_{n}$ are the proton and neutron distributions.
Assuming the same form for the proton and neutron distributions for each
nucleus, we have $\rho_{p}(r)=\displaystyle\frac{2}{3}\rho(r)$ and
$\displaystyle\rho_{n}(r)=\frac{1}{3}\rho(r)$ for 3He and
$\displaystyle\rho_{p}(r)=\rho_{n}(r)=\frac{1}{2}\rho(r)$ for 4He and, thus,
the potential takes forms as,

$\tilde{U}(r)=\frac{2V_{p}+V_{n}}{3}\frac{\rho(r)}{\rho_{0}}+i\frac{2W_{p}+W_{n}}{3}\frac{\rho(r)}{\rho_{0}}~{}~{}~{}~{}(\rm{for}~{}^{3}{\rm
He})~{},$ (6)

and

$\tilde{U}(r)=\frac{V_{p}+V_{n}}{2}\frac{\rho(r)}{\rho_{0}}+i\frac{W_{p}+W_{n}}{2}\frac{\rho(r)}{\rho_{0}}~{}~{}~{}~{}(\rm{for}~{}^{4}{\rm
He})~{}.$ (7)

If we adopt these potential forms to the parameter sets
$(V_{0},W_{0})=(-300,-95)$ MeV for $K^{-}-^{3}$He and $(-287,-43)$ MeV for
$K^{-}-^{4}$He in Tables 2 and 3, we have $(V_{p},W_{p})=(-326,-199)$ MeV, and
($V_{n},W_{n})=(-248,113)$ MeV. The values for proton and neutron terms are
largely different and they could indicate the existence of the relatively
large isospin dependence in the kaon-nucleus optical potential. At the same
time the sign of $W_{n}$ is found to be positive, which is unnatural. This
might indicate that effects beyond linear density contributions, such as two
nucleon absorptions of kaon, could be significant in such higher densities as
reported in Ref. Sekihara:2012wj . For further studies, we need more accurate
data to determine the isospin dependence.

##  4 Discussions on kaonic atom optical potential

In this section, we investigate the features and the implications of the kaon-
nucleus optical potential with the parameters which are determined in the last
section and are listed in Tables 2, 3, and 4.

###  4.1 Kaonic $1s$ atoms in 3He and 4He

In order to study further implications and explore the validity of the
potentials HE3-A, -B and HE4-A, -B, we investigate the atomic $1s$ states in
3He and 4He. The atomic $1s$ states are more sensitive to the strong
interaction than the $2p$ state, because the $s$ states are free from the
centrifugal barrier. Thus, we expect to obtain further information on the
optical potential by experimental studies of the atomic $1s$ states. Here we
show our theoretical predictions of the binding energies and widths for the
atomic $1s$ state in 3He and 4He calculated with the potentials determined by
the experimental data of the $2p$ states. With comparison of these
calculations with future experiments we will be able to distinguish better
potentials.

We adopt the potential parameters HE3-A, -B, and HE4-A, -B determined in Sect.
3.2 to the theoretical study of the $1s$ states. The calculated results are
shown in Table 5. The difference of the energy of the electromagnetic states
in 3He and 4He mainly stems from the difference of the reduced mass of the
systems. Since the width of the $1s$ state is much smaller than the level
spacing of the kaonic atom states, the $1s$ states will be able to find
experimentally as a discrete state. We also find clear differences in the
predictions of the $1s$ states for two parameter sets both of 3He and 4He,
although these parameters provide almost same energies and widths of the $2p$
states. In the $K^{-}-^{3}$He system, the calculation shows around $0.5$ keV
difference in the shift of the $2p\to 1s$ transition energy between the
results by HE3-A and HE3-B. The difference of the widths of the $1s$ state is
so large that the width by HE3-B is more than twice of that by HE3-A. In
$K^{-}-^{4}$He, the difference of the energy shift is larger than $3$ keV,
while the difference in the $1s$ width is as small as around $0.6$ keV.

As for possible experiments for the $1s$ state observation, we could think of
the X-ray spectroscopy as in Ref. J-PARCE62:2022qnt  first as a realistic
method. As another possible candidate of the experiments, we could think of
the $(K^{-},p)$ reaction with nuclear targets that was studied theoretically
in Ref. Yamagata:2007cp . In this reaction one observes the energy of the
emitted protons and obtains the energy spectrum of kaonic atoms by the missing
mass technique. We hope that our research will encourage to conduct
experiments in the near future.

Table 5: The calculated energies ($E$) and widths ($\Gamma$) of the kaonic
$1s$ and $2p$ atomic states for 3He and 4He in units of keV. The energies
calculated only with the electromagnetic potential are shown in the column of
‘Electromagnetic state’. The results obtained with the full potential for
parameters HE3-A, -B and HE4-A, -B are shown in the columns in the table as
indicated. The $2p\to 1s$ transition energies ($E_{2p\to 1s}$) and their
shifts ($\Delta E$) due to the optical potential are also shown.

3He [keV] | Electromagnetic state | HE3-A | HE3-B
---|---|---|---
Atomic state | $E$ | $E$ | $\Gamma$ | $E$ | $\Gamma$
$1s$ | $-44.7484$ | $-38.1513$ | $5.6302$ | $-38.6419$ | $11.5738$
$2p$ | $-11.1951$ | $-11.1949$ | $2.50\times 10^{-3}$ | $-11.1949$ | $2.49\times 10^{-3}$
$E_{2p\to 1s}$ | $33.5533$ | $26.9564$ | $27.4470$
$\Delta E$ |  | $-6.5969$ | $-6.1063$

4He [keV] | Electromagnetic state | HE4-A | HE4-B
---|---|---|---
Atomic state | $E$ | $E$ | $\Gamma$ | $E$ | $\Gamma$
$1s$ | $-46.5017$ | $-40.1397$ | $5.2594$ | $-37.0823$ | $4.6759$
$2p$ | $-11.6242$ | $-11.6244$ | $1.01\times 10^{-3}$ | $-11.6244$ | $9.99\times 10^{-4}$
$E_{2p\to 1s}$ | $34.8775$ | $28.5153$ | $25.4579$
$\Delta E$ |  | $-6.0564$ | $-9.4196$

###  4.2 Kaonic atoms in heavy nuclei

We apply the isoscalar form of the optical potentials with the parameter sets
IS-A and IS-B in Table 4 to the global study of heavier kaonic atoms. We
compare our calculated energy shifts and widths with the data in Refs.
Batty:1976ue ; Backenstoss:1972yg ; Wiegand:1973ac ; Barnes:1974iu ;
Batty:1979zr ; Kunselman:1971ib , which are summarized in Table 6, and we find
out the potential parameters better suited for the global description of the
kaonic atoms in the periodic table. The potential suited for the global fit is
expected to describe well the average behavior of the kaon-nucleus
interaction. In the following calculations, we consider isotope nuclei having
the largest natural abundance for each atomic number. We use the charge
distributions of the modified harmonic oscillator form for the nuclei with the
atomic number $3\leq Z\leq 8$ and the Woods-Saxon form for those with larger
$Z$. The parameters of the charge distributions are taken from Refs.
DeJager:1974liz ; DeVries:1987atn . The nuclear density distributions
$\rho(r)$ in Eq. (4) are deduced from the charge distributions as explained in
Ref. Nieves:1993ev .

The calculated results are shown and compared with the experimental data in
Figs. 6-9.

Table 6: Observed data of the shift of the transition energies $\Delta E$ and the level widths $\Gamma$ for the heavier kaonic atoms considered in this article. The data are taken from Refs. Batty:1976ue ; Backenstoss:1972yg ; Wiegand:1973ac ; Barnes:1974iu ; Batty:1979zr ; Kunselman:1971ib . $Z$ | Nucleus | Transition |  $\Delta E$ [keV] |  $\Gamma$ [keV] | Ref.
---|---|---|---|---|---
3 | Li | $3d\to 2p$ | $0.002\pm 0.026$ | $0.055\pm 0.029$ | Batty:1976ue
4 | Be | $3d\to 2p$ | $-0.079\pm 0.021$ | $0.172\pm 0.58$ | Batty:1976ue
5 | B | $3d\to 2p$ | $-0.167\pm 0.035$ | $0.700\pm 0.080$ | Backenstoss:1972yg
6 | C | $3d\to 2p$ | $-0.590\pm 0.080$ | $1.730\pm 0.150$ | Backenstoss:1972yg
8 | O | $4f\to 3d$ | $-0.025\pm 0.018$ | $0.017\pm 0.014$ | Batty:1979zr
12 | Mg | $4f\to 3d$ | $-0.027\pm 0.015$ | $0.214\pm 0.015$ | Batty:1979zr
13 | Al | $4f\to 3d$ | $-0.130\pm 0.050$ | $0.490\pm 0.160$ | Barnes:1974iu
|  |  | $-0.076\pm 0.014$ | $0.442\pm 0.022$ | Batty:1979zr
14 | Si | $4f\to 3d$ | $-0.240\pm 0.015$ | $0.810\pm 0.120$ | Barnes:1974iu
|  |  | $-0.130\pm 0.015$ | $0.800\pm 0.033$ | Batty:1979zr
15 | P | $4f\to 3d$ | $-0.330\pm 0.08$ | $1.440\pm 0.120$ | Backenstoss:1972yg
16 | S | $4f\to 3d$ | $-0.550\pm 0.06$ | $2.330\pm 0.200$ | Backenstoss:1972yg
|  |  | $-0.43\pm 0.12$ | $2.310\pm 0.170$ | Wiegand:1973ac
|  |  | $-0.462\pm 0.054$ | $1.96\pm 0.17$ | Batty:1979zr
17 | Cl | $4f\to 3d$ | $-0.770\pm 0.40$ | $3.80\pm 1.0$ | Backenstoss:1972yg
|  |  | $-0.94\pm 0.40$ | $3.92\pm 0.99$ | Kunselman:1971ib
|  |  | $-1.08\pm 0.22$ | $2.79\pm 0.25$ | Wiegand:1973ac
27 | Co | $5g\to 4f$ | $-0.099\pm 0.106$ | $0.64\pm 0.25$ | Batty:1979zr
28 | Ni | $5g\to 4f$ | $-0.180\pm 0.070$ | $0.59\pm 0.21$ | Barnes:1974iu
|  |  | $-0.246\pm 0.052$ | $1.23\pm 0.14$ | Batty:1979zr
29 | Cu | $5g\to 4f$ | $-0.240\pm 0.220$ | $1.650\pm 0.72$ | Barnes:1974iu
|  |  | $-0.377\pm 0.048$ | $1.35\pm 0.17$ | Batty:1979zr
47 | Ag | $6h\to 5g$ | $-0.18\pm 0.12$ | $1.54\pm 0.58$ | Batty:1979zr
48 | Cd | $6h\to 5g$ | $-0.40\pm 0.10$ | $2.01\pm 0.44$ | Batty:1979zr
49 | In | $6h\to 5g$ | $-0.53\pm 0.15$ | $2.38\pm 0.57$ | Batty:1979zr
50 | Sn | $6h\to 5g$ | $-0.41\pm 0.18$ | $3.18\pm 0.64$ | Batty:1979zr

As can be seen clearly in the figures, the parameter IS-A
$(V_{0},W_{0})=(-90,-120)$ MeV is significantly better suited to describe the
kaonic atom data in the wide range of the periodic table than IS-B. The
potential with the parameter IS-B has a so large real part and provides the
nuclear state with the same angular momenta as the atomic state as explained
in the following sections. Consequently, the level repulsion between atomic
and nuclear states makes the shift of the atomic states repulsive, while the
potential with the parameter IS-A has the large imaginary part which makes the
shift repulsive. Thus, the results obtained in this article, that the
parameters IS-A is better suited for the global description of the kaonic atom
data, support the conclusion in Ref. Iizawa:2019uiu  and show the same
features of the kaon-nucleus optical potential which can reproduce the kaonic
atom data in the wide range of the periodic table.

|
---|---
Figure 6: The calculated results of the energy shifts of the $3d\to 2p$ transition (Left) and the $2p$ level widths of the kaonic atoms (Right) with the experimental data compiled in Table 6 Batty:1976ue ; Backenstoss:1972yg ; Wiegand:1973ac ; Barnes:1974iu ; Batty:1979zr ; Kunselman:1971ib  . The horizontal axis shows the atomic number $Z$ of the nucleus. We consider an isotope nucleus with the largest natural abundance for each $Z$ for the theoretical calculations. For helium, we show the data of 4He in Ref. J-PARCE62:2022qnt . The parameters of the optical potential (IS-A and IS-B) used for the calculations are indicated in the figures.  |
---|---
Figure 7: Same as Fig. 6 except for the $4f\to 3d$ transitions and the $3d$ level widths of the kaonic atoms. |
---|---
Figure 8: Same as Fig. 6 except for the $5g\to 4f$ transitions and the $4f$ level widths of the kaonic atoms. |
---|---
Figure 9: Same as Fig. 6 except for the $6h\to 5g$ transitions and the $5g$
level widths of the kaonic atoms.

###  4.3 Structure of kaonic atoms and effects form possible kaonic nuclei in
${}^{3,~{}4}$He

If the optical potential is attractively strong enough, kaonic nuclear states,
which are compact bound states induced by strong interaction, can be formed.
If atomic states have the same quantum numbers with such nuclear states, they
may receive certain influence from the nuclear states, such as level repulsion
induced by state mixing. Here we consider the effects of possible nuclear
states on the kaonic atom structure in 3He and 4He. In the studies of the
kaonic nuclear structure, the hadron degrees of freedom and the energy
dependence of the interaction are expected to be more important. Nevertheless,
because we focus on the nature of the atomic states, our study on kaonic
nuclear states here should be considered to be rather qualitative. The mutual
influence between the atomic and nuclear states of kaon with the same angular
momentum are interesting to be mentioned. Here we just use the same optical
potential also for the nuclear states.

We show in Fig. 10 (Lower) the contour plots of the shift and width of the
$2p$ atomic state of the $K^{-}-^{3}$He system in the $V_{0}-W_{0}$ plane,
which are essentially the same plots as Fig. 3 but contain more detailed
information. In Fig. 10, we show the boundary of the potential parameters
which allow the existence of the kaonic nuclear $p$ states by the white line
and the nuclear $p$ state(s) exists in the parameter region left to the white
line. The shifts and widths obtained with smaller $W_{0}$ have strong $V_{0}$
dependence. In particular, closed to the white line the shift suddenly changes
its sign and the width has a maximum. This feature is clearly shown in the
upper figures of Fig. 10 where the shift and width are plotted as functions of
$V_{0}$ for a fixed $W_{0}=-50$ MeV. These plots indicate typical influence of
the existence of a nuclear state on the shift and width of the atomic state
with the same angular momentum. Closed to the white line which is the boundary
for the existence of the nuclear state, the binding energy of the nuclear
state is so small that large mixing of the atomic and nuclear states with the
same quantum numbers take place. These characteristic behavior of the shift
and width stem from the occurrence of the level crossing between the nuclear
state and atomic states. In Fig. 11, we show the same contour plots as Fig. 10
(Lower) for the atomic $2p$ state in $K^{-}-^{4}$He. The boundary of the
existence of the nuclear $p$ state is again represented by the white line. As
we can see from Figs. 10 and 11, two kinds of the potential parameter sets in
Tables 2, 3, and 4 determined by the data J-PARCE62:2022qnt  are distinguished
by the boundaries of the existence of the kaonic nuclear $p$ states.

Figure 10: (Upper) The $V_{0}$ dependence of the shift of the $3d\to 2p$
transition energy and the $2p$ level width of the kaonic 3He atom calculated
with $W_{0}=-50$ MeV. (Lower) the contour plots of the shift and the width of
the kaonic 3He atom in the plane of the parameters $V_{0}$ and $W_{0}$. The
potential parameters in the region to the left of the white line allow the
existence of the kaonic nuclear $p$ state(s). The solid diamonds in the
figures indicate the potential parameter sets of HE3-A and HE3-B in Table 2 by
which we can reproduce the experimental shift and width of the $K^{-}-^{3}$He
atom simultaneously. The solid triangles indicate the parameter sets of IS-A
and IS-B in Table 4. Figure 11: Same contour plots as Fig. 10 (Lower) except
for the kaonic 4He atomic state. The solid circles indicate the potential
parameter sets of HE4-A and HE4-B in Table 3 by which we can reproduce the
experimental shift and width of the $K^{-}-^{4}$He atom simultaneously. Figure
12: The contour plots of the shifts of the $2p\to 1s$ transition energies
(Left) and the $1s$ level widths (Right) of kaonic 3He (Upper) and 4He (Lower)
atoms. The solid triangles show the two sets of the potential parameters IS-A
and IS-B in Table 4. The solid diamonds and circles indicate the potential
parameters listed in Table 2 for 3He (HE3-A and HE3-B) and Table 3 for 4He
(HE4-A and HE4-B), respectively.

We show in Fig. 12 the contour plots of the shifts of the $2p\to 1s$
transition energies and the widths of the $1s$ states of kaonic atoms in 3He
and 4He. The parameter sets, IS-A and IS-B, are also indicated in the figures.
We can see the similar behavior of the shift and width for smaller $|W_{0}|$
region even with small $|V_{0}|$ to those in the contour plots of Fig. 10 and
11. This indicates the existence of the nuclear $s$ states because this
characteristic behavior of the shift and width stems from the mixing of the
atomic and nuclear states. Considering also the results by E15
J-PARCE15:2018zys  and the tendency of the mass number dependence of the
binding energy of the kaonic nuclear $s$ states found theoretically in Refs.
Akaishi:2002bg ; Yamazaki:2002uh ; Shevchenko:2007zz ; Wycech:2008wf ;
Dote:2008hw ; Ikeda:2008ub ; Barnea:2012qa ; Sekihara:2016vyd ;
Ohnishi:2017uni ; Dote:2017wkk , we conclude that the existence of the nuclear
$s$ state(s) in 3He and 4He is highly plausible. All the potential parameter
sets reported in this article do not contradict this expectation as we can see
in Fig. 12. The nuclear $s$ states can strongly affect the atomic $s$ states
and change the binding energy and the width of the atomic $s$ states largely.
Thus, the origin of the discrepancies in the calculated results for the atomic
$1s$ states by HE3-A, -B for 3He and HE4-A, -B for 4He in Table 5 is naturally
expected to be related to the structure of the kaonic nuclear $s$ states.
Hence, the observation of the atomic $1s$ state in 3,4He is expected to
provide the information on the potential parameters and the kaonic nuclear $s$
states, too. At the same time, it will be interesting to fix the angular
momentum of the kaonic nulcear state(s) in 3He and 4He, and clarify the (non-)
existence of the kaonic nuclear $p$ state to develop the unified
understandings of the structure of kaonic atoms and kaonic nuclei in He.

###  4.4 Kaonic nuclear states in heavy nuclei

We also examine the possibility of the existence of kaonic nuclear states by
using our isoscalar optical potential parameters IS-A and IS-B given in Table
4 for heavier nuclei listed in Table 6, for which the energy and width of the
kaonic atomic states are observed. We study the kaonic nuclear states that
have the same angular momentum $\ell$ with the atomic state for each nucleus
shown in Table 6, because these nuclear states can give influence to the
atomic states observed in experiments due to the state mixing as discussed in
the previous section. We find that, the optical potential IS-B, which has a
larger attractive potential, provides kaonic nuclear states with the same
angular momentum as the atomic state for all nucleus in Table 6, while the
potential IS-A, which has a shallower attractive potential, does not give the
corresponding nuclear states for nuclei with $Z\leq 28$ and provides bound
states in Cu with $\ell=3$ and heavier nuclei. For the shallower IS-A
potential, kaonic nuclear $p$ states exist for nuclei with $Z\geq 29$, which
are much heavier than C, which is the heaviest nucleus in which the kaonic
$2p$ atomic state is observed.

Finally, we mention here a past work Ichikawa:2020doq  in which a set of the
potential parameters was extracted by investigating observed 12C$(K^{-},p)$
spectrum using a phenomenological potential similar with Eq. (4). The spectrum
was reproduced by their best potential with the strength $(-80,-40)$ MeV at
the nuclear center of 11B for a wide range of $K^{-}$ kinetic energy
$(E-M_{K})$ from -300 MeV to $+40$ MeV Ichikawa:2020doq . It should be noted
that their nuclear density at the center of 11B used in Ref. Ichikawa:2020doq
is slightly different from the normal nuclear density and the potential
strength of Ref. Ichikawa:2020doq  is corresponding to the value of
$(V_{0},W_{0})=(-74,-37)$ MeV in our formulation. These potential parameters
are different from our IS-A and IS-B that we find with the kaonic He atoms,
while they provide $0.46$ eV for the shift and $1.01$ eV for the width for the
$2p$ state of the $K^{-}-^{4}$He atom, which are consistent with the recent
4He data reported in Ref. J-PARCE62:2022qnt  within the errors. The comparison
of the results obtained in this article and those in Ref. Ichikawa:2020doq
may provide the information on the energy dependence of the potential and/or
the effects of possible nuclear deformation of the kaonic nuclear states.

##  5 Summary

We study the features and the implications of the kaon-nucleus optical
potential determined based on the latest high precision kaonic atom data in
Ref. J-PARCE62:2022qnt  in this article. We adopt the phenomenological form of
the optical potential and find the complex potential parameters by the
experimental data in Ref. J-PARCE62:2022qnt . The energy shift and width of
the $K^{-}-^{3}$He data and the $K^{-}-^{4}$He data are reproduced by the
potential parameters shown in Table 2 (HE3-A and HE3-B) and Table 3 (HE4-A and
HE4-B), respectively. The parameters in Table 4 (IS-A and IS-B) provide the
energy shifts and widths consistent with the both data of $K^{-}-^{3}$He and
$K^{-}-^{4}$He. We find that the obtained parameters in Table 2 and Table 3
may indicate the possible strong isospin dependence of the kaon-nucleus
optical potential, and we also find that the constraints to the potential
parameters by the data of 3He are weaker than those by the data of 4He. The
comparison of the data with the results by the theoretical potentials
Mares:2006vk ; Ramos:1999ku  are also shown.

We apply the potential parameters to the studies of other kaonic atom states.
We calculate the atomic $1s$ states in 3He and 4He using the potential
parameters HE3-A, -B and HE4-A, -B. We find that the $1s$ states exist as
discrete states and they will provide the further information on the optical
potential. We also study the kaonic atoms in heavier nuclei using the IS-A and
IS-B potential parameters and find that the IS-A parameters are better suited
for the global description of the kaonic atoms in the wide range of the
periodic table. The IS-A parameter set has the relatively weak real part and
strong imaginary part as studied in Ref. Iizawa:2019uiu . We then consider the
kaonic nuclear states in 3He and 4He by the same theoretical framework with
the optical potential and investigate the effects of the nuclear states on the
observables of the kaonic atoms. We find that the existence of the kaonic
nuclear states with the same angular momentum as the atomic states affects the
atomic state structure. Thus, we think that the determination of the quantum
numbers of the kaonic nuclear states is important not only for the study of
the kaonic nuclear states but also for the studies of the kaonic atoms and the
optical potential. We also study the existence of the kaonic nuclear states
with the same angular momentum as the existing atomic data for the heavier
nuclei.

Based on the present study of the kaon-nucleus bound systems by the optical
potential framework, we conclude that the further studies of the kaonic atoms
especially $1s$ states in 3,4He and the determination of the orbital angular
momentum of the kaonic nuclear states are interesting and important to
determine the kaon-nucleus interaction decisively and to develop the unified
understandings of the kaonic atoms and kaonic nuclei. The widths of the kaonic
$1s$ atoms in He are significantly smaller than the level spacing and the $1s$
states exist as the discrete states as deeply bound pionic atoms
Yamazaki:2012zza ; Hirenzaki:2022dpt ; Itahashi:2023boi ; piAF:2022gvw  which
can be observed if we can find the appropriate methods. Experimental methods
based on hadronic reaction Yamagata:2007cp  and/or X-ray spectroscopy
J-PARCE62:2022qnt  would be candidates. The improvements of the data of kaonic
$2p$ atoms in He is also important for the better determination of the kaon-
nucleus interaction as discussed in this article. The determination of the
quantum numbers of the kaonic nuclear states is one of the important next
challenge after the confirmation of the existence of the state
J-PARCE15:2018zys , and it has the important meanings for the unified study of
the kaon-nuclear systems because of the mutual influence between the atomic
and nuclear states with the same angular momentum.

In addition, we could even study the possible differences of the $K^{-}-$
nucleus interaction between the $1s$ and $2p$ atomic states since all nucleons
in a helium nucleus are in the $s$ state in a good approximation and thus, the
angular momenta between $K^{-}-$ nucleus and $K^{-}-$ nucleon are expected to
be same. Hence, the atomic $1s$ state is expected to be more sensitive to
${\bar{K}}N$ $s$ wave amplitude than the atomic $2p$ state.

## Acknowledgement

This work was partly supported by JSPS KAKENHI Grant Numbers JP22K03607,
JP23K03417. The work of D.J. was partly supported by Grants-in-Aid for
Scientific Research from JSPS (21K03530, 22H04917, 23K03427).

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