11institutetext: Tetsuo Hyodo 22institutetext: Tokyo Metropolitan University,
Hachioji 192-0397, Tokyo, Japan
22email<EMAIL_ADDRESS>33institutetext: Wolfram Weise 44institutetext:
Physics Department,Technical University of Munich, D-85748 Garching, Germany
44email<EMAIL_ADDRESS>
# Theory of kaon-nuclear systems

Tetsuo Hyodo and Wolfram Weise corresponding author

###### Abstract

The strong interaction between an antikaon and a nucleon is at the origin of
various interesting phenomena in kaon-nuclear systems. In particular, the
interaction in the isospin $I=0$ channel is sufficiently attractive to
generate a quasi-bound state, the $\Lambda(1405)$ resonance, below the
$\bar{K}N$ threshold. Based on this picture, it may be expected that the
$\bar{K}N$ interaction also generates quasi-bound states in kaon-nuclear
systems, sometimes called kaonic nuclei. At the same time, the $\bar{K}N$
quasi-bound picture of the $\Lambda(1405)$ is also related to the discussion
of hadronic molecules in hadron spectroscopy. Here an overview is presented of
the theoretical studies developed for kaon-nucleon and kaon-nuclear systems.
We start from the modern understanding of the $\Lambda(1405)$ resonance. We
then discuss the $\bar{K}N$ interaction and various aspects of few-body kaonic
nuclei. Heavier kaon-nuclear systems are examined from the viewpoint of
nuclear many-body physics, with focus on the properties of antikaons in
nuclear matter. Related topics, such as the $K^{-}p$ momentum correlation
functions in high-energy collisions and the studies of kaonic atoms, are also
discussed.

##  1 Introduction

The kaons ($K=K^{+},K^{0}$) and antikaons ($\bar{K}=K^{-},\bar{K}^{0}$) are
the lightest pseudoscalar mesons with strangeness ParticleDataGroup:2020ssz .
Their nature is closely related to the symmetry breaking pattern in low-energy
QCD. In the $SU(3)$ octet of the light pseudoscalar mesons, kaons figure as
flavor partners of the pions, the Nambu-Goldstone (NG) bosons associated with
the spontaneous breaking of chiral symmetry in QCD. While the smallness of the
pion mass has its origin in the almost vanishing up- and down-quark masses
($m_{u}\simeq 2$ MeV, $m_{d}\simeq 5$ MeV at a renormalization scale of 2
GeV), kaons are relatively massive because of the more sizable strange quark
mass ($m_{s}\simeq 0.1$ GeV) reflecting the pronounced explicit chiral
symmetry breaking in the strangeness sector. These features are illustrated in
Fig. 1 where the masses of the lowest lying hadrons are plotted: the kaons are
not as light as the pions, but at the same time not as heavy as the ordinary
hadrons other than the NG bosons. This intermediate nature of the kaons leads
to various interesting phenomena of nonperturbative QCD at low energies.

Figure 1: Masses of the lowest-lying hadrons in the strangeness $S=0$ and
$S=-1$ sectors.

One such aspect is the possible formation of bound states of nuclei with
antikaons ($\bar{K}$). Because the $\bar{K}$ has isospin $I=1/2$, there are
two independent components of the $\bar{K}N$ interaction, $I=0$ and $I=1$, as
in the nuclear force. Moreover, the $NN$ and $\bar{K}N$ interactions share a
common feature. The $NN$ interaction in the $I=0$ (${}^{3}S_{1}$) channel is
sufficiently attractive to form the deuteron bound state, and the $I=1$
(${}^{1}S_{0}$) channel is also known to have an attractive scattering length.
In a similar way, despite the otherwise obviously different physics of the
$NN$ and $\bar{K}N$ systems, the $\bar{K}N$ interaction in the $I=0$ channel
also has a quasi-bound state below the threshold, the $\Lambda(1405)$, and the
$I=1$ interaction is likewise considered to be attractive (see Table 1). A
qualitative difference, though, is that the deuteron is stable, whereas the
$\bar{K}N$ system couples strongly to the lower-energy $\pi\Sigma$ and
$\pi\Lambda$ channels. Consequently, the $\Lambda(1405)$ as a quasibound state
imbedded in the $\pi\Sigma$ continuum has a short lifetime, i.e. a large decay
width, $\Gamma\simeq 50$ MeV. Nonetheless the question was raised whether a
self-bound system can be obtained by replacing one of the nucleons in nuclei
by an antikaon. In fact, shortly after the prediction Dalitz:1959dn ;
Dalitz:1960du  and the experimental observation of the $\Lambda(1405)$
Alston:1961zzd , possible bound kaon-nuclear systems have been discussed
PL7.288 . Recent interest in the kaon-nuclear systems was triggered by the
work of Akaishi and Yamazaki Akaishi:2002bg  who proposed the existence of
possible deeply bound kaonic nuclei with a narrow width. Since then, many
theoretical and experimental investigations have been devoted kaon-nuclear
systems with these issues in mind.

Table 1: Qualitative features of the $NN$ and $\bar{K}N$ $s$-wave interactions. | $I=0$ | $I=1$
---|---|---
$NN$ |  $d$ (bound state) | attractive
$\bar{K}N$ |  $\Lambda(1405)$ (quasi-bound state) | attractive

This chapter introduces the theoretical framework for kaon-nuclear systems,
starting with the $\Lambda(1405)$ as a prototype example. We then describe
few-body and many-body kaon-nuclear systems, including a discussion of kaon
properties in nuclear matter. We also refer here to several review articles
covering related topics Ramos2001 ; Friedman:2007zza ; Hyodo:2011ur ;
Gal:2016boi ; Tolos:2020aln ; Meissner:2020khl ; Mai:2020ltx ; Hyodo:2020czb .

##  2 The $\Lambda(1405)$ resonance

The $\Lambda(1405)$ resonance is nominally located just 27 MeV below the
$K^{-}p$ threshold ($M_{p}+m_{K^{-}}=1432$ MeV). Its properties are therefore
closely linked to the low-energy $\bar{K}N$ interaction. The $\Lambda(1405)$
is the lowest lying excited baryon in the strangeness $S=-1$ and isospin $I=0$
sector, rated as a four-star resonance in PDG ParticleDataGroup:2020ssz .
Studies of the $\Lambda(1405)$ go back to 1959 when Dalitz and Tuan
theoretically predicted a resonance below the $\bar{K}N$ threshold
Dalitz:1959dn ; Dalitz:1960du . Based on the coupled-channels unitarity of the
$\bar{K}N$-$\pi\Sigma$ system and the empirical $\bar{K}N$ scattering length,
they deduced the existence of a resonance in the $\pi\Sigma$ scattering
amplitude. Shortly after the prediction, experimental evidence for the
$\Lambda(1405)$ was reported in the $\pi\Sigma$ invariant mass distribution of
the $K^{-}p\to\pi\pi\pi\Sigma$ reaction at 1.15 GeV Alston:1961zzd . Since
then a large amount of theoretical and experimental investigations have been
devoted to clarify the nature of the $\Lambda(1405)$ (see Refs. Hyodo:2011ur ;
Meissner:2020khl ; Mai:2020ltx ; Hyodo:2020czb ). In particular, recent
developments have firmly established the following basic properties of the
$\Lambda(1405)$:

  * •

Eigenenergy: The generalized eigenenergy of an unstable state is expressed by
the position of the resonance pole of the scattering amplitude in the complex
energy plane (the real and imaginary parts respectively correspond to the mass
and half width of the resonance). The pole positions of the $\Lambda(1405)$
have been pinned down Ikeda:2011pi ; Ikeda:2012au ; Guo:2012vv ; Mai:2014xna ,
thanks to accurate constraints from the precise measurement of kaonic hydrogen
by the SIDDHARTA collaboration SIDDHARTA:2011dsy ; Bazzi:2012eq .

  * •

Spin and parity: Experimental determination of the spin and parity has been
carried out by the CLAS collaboration in the photoproduction $\gamma p\to
K^{+}\Lambda(1405)$ CLAS:2014tbc . The result confirms the expected quantum
numbers of $J^{P}=1/2^{-}$.

Current interest on the $\Lambda(1405)$ is focused on its internal structure.
In conventional constituent quark models, the negative parity excited baryons
are described by the internal excitation of three quarks in the confining
potential. In a systematic study of negative parity baryons using such a quark
model Isgur:1978xj , it is shown that the mass of the $\Lambda(1405)$ deviates
significantly (by $\sim 100$ MeV) from the standard quark model prediction, in
contrast to other negative parity states which are well described by the quark
model. This suggests that the $\Lambda(1405)$ has a more exotic internal
structure, unlike that of a simple three-quark state. In particular, the
static quark model picture lacks the dynamical aspect of the excited baryons
which couple to the meson-baryon continuum channels. An alternative picture of
the $\Lambda(1405)$, as a weakly bound $\bar{K}N$ molecule induced by the
attractive $\bar{K}N$ interaction, is in line with Ref. Dalitz:1967fp . This
picture is also supported by a recent analysis of the compositeness
Kamiya:2015aea ; Kamiya:2016oao  and emphasizes once again the attractive
nature of the $\bar{K}N$ interaction. In this sense, the internal structure of
the $\Lambda(1405)$ is important, not only in hadron spectroscopy to search
for non-conventional configurations of hadrons, but also for the deeper
understanding of the $\bar{K}N$ interaction and its implications for kaon-
nuclear physics.

As a resonance in $\pi\Sigma$ scattering located close to the $\bar{K}N$
threshold, the $\Lambda(1405)$ requires a dynamical treatment in terms of
coupled-channel meson-baryon scattering. The pseudoscalar $\pi$ and $\bar{K}$
are combined with the respective $1/2^{+}$ baryons in $s$ wave to form the
$J^{P}=1/2^{-}$ baryonic system with strangeness $S=-1$. The lowest lying
pseudoscalar mesons are regarded as Nambu-Goldstone bosons associated with the
spontaneous breaking of $N_{f}=3$ chiral symmetry. Therefore their low-energy
$s$-wave interactions with the octet baryons are constrained by the Weinberg-
Tomozawa theorem Weinberg:1966kf ; Tomozawa:1966jm . Based on this
observation, a series of works Kaiser:1995eg ; Oset:1997it ; Oller:2000fj
developed a successful theoretical framework, called chiral $SU(3)$ dynamics,
in which the coupled-channel meson-baryon scattering amplitude satisfying the
unitarity condition is constructed in a systematic manner.

Figure 2: Feynman diagrams for the scattering equation. Solid (dashed) lines
represent the baryon (meson) propagator. Open (shaded) blobs correspond to the
interaction kernel $V$ (scattering amplitude $T$) and the meson-baryon
intermediate loop stands for the loop function $G$.

Denoting the meson-baryon channels by the indices $i,j$, the scattering
amplitude takes a matrix form, $T_{ij}$, in channel space. In the s-wave
($\ell=0$) channels, $T_{ij}$ is a function of the total meson-baryon energy
$W$. In chiral $SU(3)$ dynamics, $T_{ij}$ is then given by the solution of a
coupled-channel Lippmann-Schwinger scattering equation;

$\displaystyle T_{ij}=V_{ij}+V_{ik}G_{k}T_{kj},$ (1)

where the summation over the repeated indices is implicit. In Fig. 2, Feynman
diagram representation of the scattering equation is shown. The kernel
$V_{ij}$ represents the meson-baryon interaction, constructed so that it
systematically satisfies the chiral symmetry constraints Ecker:1994gg ;
Bernard:1995dp ; Pich:1995bw ; Bernard:2007zu ; Scherer:2012xha . Furthermore
$G_{k}$ is the loop function representing the propagation of the meson and
baryon in the intermediate state with index $k$. Substituting $T_{kj}$ on the
right-hand-side of Eq. (1) iteratively, one finds

$\displaystyle
T_{ij}=V_{ij}+V_{ik}G_{k}V_{kj}+V_{ik}G_{k}V_{kl}G_{l}V_{lj}+\dotsb,$ (2)

where the multiple scatterings in coupled channels are iterated to all orders
in $T_{ij}$. This guarantees the unitarity of the scattering amplitude. The
same strategy is applied in the study of the nuclear force within the
framework of chiral effective field theory Epelbaum:2008ga ; Machleidt:2011zz
. In the following, we present the properties of $V_{ij}$ and $G_{i}$.

Chiral perturbation theory Ecker:1994gg ; Bernard:1995dp ; Pich:1995bw ;
Bernard:2007zu ; Scherer:2012xha  classifies the interaction kernel $V$ by the
chiral order in terms of a low-momentum scale $p$. The chiral order, denoted
by $\mathcal{O}(p^{n})$, refers to an expansion in powers of
$p/\Lambda_{\chi}$ where $\Lambda_{\chi}\sim 1$ GeV is the characteristic
scale of the spontaneous breaking of chiral symmetry. The terms with smaller
chiral orders dominate in the low-energy region, $p\ll\Lambda_{\chi}$, thus
low-energy phenomena can be described by a finite number of terms.

In meson-baryon scattering the leading order (LO) terms start at
$\mathcal{O}(p)$ and involve couplings of the $SU(3)$ pseudoscalar meson and
$J^{P}=1/2^{+}$ baryon octets through their vector and axial vector currents.
For example, the dominant Weinberg-Tomozawa (WT) term in the $K^{-}$ proton
elastic scattering channel derives from the following piece of the chiral
interaction Hamiltonian:

$\displaystyle\delta H_{\rm WT}(K^{-}p\rightarrow K^{-}p)=-{i\over 2f^{2}}\int
d^{3}x\,\bar{\psi}_{p}(x)\gamma^{\mu}\psi_{p}(x)\,K^{+}(x)\partial_{\mu}K^{-}(x)~{}~{},$
(3)

where $\psi_{p}$ and $K^{-}$ are the proton and antikaon fields, respectively.
Note that the WT term is completely determined by the pseudoscalar meson decay
constant, $f\simeq 0.1$ GeV. The interaction vanishes in the limit of zero
meson four-momentum, $q^{\mu}=(E,\vec{q})$, a feature characteristic of Nambu-
Goldstone bosons. The s-wave interaction is governed by the time component of
the vector current. In the limit of a pointlike, static nucleon, the leading
WT term is therefore a contact interaction proportional to $E/f^{2}$, with the
meson energy $E=\sqrt{m^{2}+\vec{q}^{2}}$. At threshold the WT interaction
kernels in $K^{-}p$ and $K^{-}n$ channels have the form

$\displaystyle V_{\rm WT}(K^{-}p)=2\,V_{\rm WT}(K^{-}n)\propto-{m_{K}\over
f^{2}}~{}~{}.$ (4)

For $K^{+}p$ and $K^{+}n$ scattering the corresponding terms are repulsive, of
equal magnitude as in Eq. (4) but with opposite sign. The important point to
note here is that the leading $\bar{K}N$ interaction close to threshold is
much stronger than the s-wave $\pi N$ interaction. It scales with the kaon
mass $m_{K}$ (rather than the small pion mass $m_{\pi}$) and reflects the
stronger explicit chiral symmetry breaking characteristic of the strange
quark.

The meson-baryon interaction up to and including the next-to-leading order
(NLO) terms of $\mathcal{O}(p^{2})$ is schematically written as

$\displaystyle V=\underbrace{V_{\rm WT}+V_{\rm
Born}}_{\mathcal{O}(p^{1})}+\underbrace{V_{\rm
NLO}}_{\mathcal{O}(p^{2})}+\dotsb~{}~{}.$ (5)

The corresponding Feynman diagrams are shown in Fig. 3. The ellipsis stands
for the $\mathcal{O}(p^{3})$ and higher order contributions. In each
interaction term there are low-energy constants (LECs), whose strength cannot
be determined from symmetry arguments alone. The precision of the calculation
can be increased by introducing higher order terms, but this requires a
sufficient amount of experimental data to determine all the LECs.

The dominant contribution in the leading order $\mathcal{O}(p^{1})$ terms is
the Weinberg-Tomozawa term $V_{\rm WT}$ (Fig. 3, first term in the right hand
side). In the chiral dynamics framework the pertinent low-energy theorems are
automatically built in, and the low-energy limit of $V$ reduces to $V_{\rm
WT}$ satisfying the Weinberg-Tomozawa theorem. As mentioned, $V_{\rm WT}$ has
no LEC apart from the pseudoscalar meson decay constant. The sign and strength
of the interaction are determined by flavor $SU(3)$ symmetry.

The Born terms $V_{\rm Born}$ are constructed by the $s$\- and $u$-channel
baryon exchange diagrams (Fig. 3, second and third terms). The strength
parameters of the meson-baryon-baryon three-point vertices in these diagrams
are given by the axial vector coupling constants of the participating baryons,
satisfying Goldberger-Treiman relations Goldberger:1958vp . Their values
reflect the internal structure of the baryons. While $V_{\rm Born}$ is counted
as $\mathcal{O}(p)$, the main contribution of this term is in $p$-wave, and
the $s$-wave projected part appears at higher order than $V_{\rm WT}$ in the
non-relativistic expansion Weinberg:1996kr . Thus, in the low-energy limit,
$V_{\rm Born}$ is much smaller than $V_{\rm WT}$, and the meson-baryon
interaction is model-independently given by the leading Weinberg-Tomozawa
term, thanks to chiral symmetry. When confronted with high-precision data,
such as the measurement of kaonic hydrogen by the SIDDHARTA collaboration
SIDDHARTA:2011dsy ; Bazzi:2012eq , the precision of the theoretical framework
is increased by introducing the higher order terms. This is achieved by
including the NLO terms $V_{\rm NLO}$ at $\mathcal{O}(p^{2})$ Ikeda:2011pi ;
Ikeda:2012au ; Guo:2012vv ; Mai:2014xna  (fourth terms in Fig. 3).

Figure 3: Feynman diagrams for the interaction kernel $V$. Black dots
represent the $\mathcal{O}(p)$ vertices, and black square stands for the
$\mathcal{O}(p^{2})$ vertex. The first term in the right hand side is $V_{\rm
WT}$, the second and third terms are $V_{\rm Born}$, the forth term is $V_{\rm
NLO}$, and ellipsis shows the $\mathcal{O}(p^{3})$ contributions.

The scattering equation (1) is in general an integral equation, with the off-
shell structure of the interaction kernel $V_{ij}$ being integrated over. In
practical applications the equation is usually reduced into algebraic form by
using the on-shell factorization. This formulation preserves the unitarity
condition. It is also justified by the N/D method (see Refs. Oset:1997it ;
Oller:2000fj ; Hyodo:2011ur ; Mai:2020ltx ). Because the leading order $V_{\rm
WT}$ is a contact interaction, the momentum integration in the loop function
involves an ultraviolet divergence. This divergence is usually tamed by
dimensional regularization, and the finite part of the loop function $G_{i}$
is specified by a subtraction constant which plays the role of an ultraviolet
cutoff. The one-loop diagram is counted as $\mathcal{O}(p^{3})$, and therefore
the renormalization procedure of the meson-baryon scattering is completed at
$\mathcal{O}(p^{3})$. In the unitarized amplitude in Eq. (1) with the
$\mathcal{O}(p^{2})$ interaction kernel ($V=V_{\rm WT}+V_{\rm Born}+V_{\rm
NLO}$), the subtraction constants need to be determined by fitting
experimental data. As mentioned, $V_{\rm WT}$ is completely determined by
chiral symmetry. The Born terms contain the axial vector coupling constants of
the octet baryons, conventionally denoted by $D$ and $F$, which are determined
by the semi-leptonic decays of hyperons. Thus, when the $\mathcal{O}(p^{1})$
interactions $V_{\rm WT}$ and $V_{\rm Born}$ are used, the only free
parameters are the subtraction constants. In the strangeness $S=-1$ coupled-
channel scattering matrix there are six isospin states
($\bar{K}N,\pi\Lambda,\pi\Sigma,\eta\Lambda,\eta\Sigma,K\Xi$), and six
subtraction constants can be used to fit the experimental data. In $V_{\rm
NLO}$, there are seven LECs, which should be determined also by the
experimental data in the $\mathcal{O}(p^{2})$ calculations, in addition to the
subtraction constants. In the current situation the LECs up to
$\mathcal{O}(p^{2})$ terms can be determined by experimental data, but in
order to perform $\mathcal{O}(p^{3})$ calculations, the data base is not (yet)
sufficient.

Figure 4 shows the result of a chiral $SU(3)$ dynamics calculation of the
$K^{-}p$ forward scattering amplitude Ikeda:2012au , featuring the prominent
emergence of the $\Lambda(1405)$ as a $\bar{K}N$ quasibound state imbedded in
the $\pi\Sigma$ continuum. This amplitude has served as input to many
investigations of $\bar{K}$ interactions in more complex systems.

Figure 4: Real and imaginary parts of the $K^{-}p\rightarrow K^{-}p$ forward
scattering amplitude, $F=T/8\pi\sqrt{s}$, from chiral $SU(3)$ dynamics at NLO
(solid curve) and constrained by the SIDDHARTA kaonic hydrogen measurements.
The dot at threshold marks the corresponding deduced scattering length.
Uncertainties are represented by the grey band. Also shown (dashed curve) for
comparison is an effective Weinberg-Tomozawa amplitude in a reduced three-
channel ($\bar{K}N-\pi\Sigma-\pi\Lambda$) scheme using values $f_{\pi}=92.4$
MeV, $f_{K}=109$ MeV of the meson decay constants and three subtraction
constants as fit parameters. Adapted from Ref. Ikeda:2012au .

Poles of the scattering amplitude $T_{ij}$ in the complex energy plane
represent the generalized eigenenergies of unstable resonances Hyodo:2020czb .
Usually there is a single pole for a resonance state, whose position $z$ in
the complex energy plane is related to the mass $M_{R}$ and the width
$\Gamma_{R}$ as

$\displaystyle M_{R}=\text{Re }z,\quad\Gamma_{R}=-2\ \text{Im }z.$ (6)

It was first pointed out in Ref. Oller:2000fj  that there are actually two
poles with $I=0$ and $S=-1$ in the $\Lambda(1405)$ energy region. This fact
has been confirmed in many subsequent studies Meissner:2020khl ; Mai:2020ltx ;
Hyodo:2020czb , including the cloudy bag model Fink:1989uk , meson-exchange
model Haidenbauer:2010ch , dynamical coupled-channel model Kamano:2014zba ;
Kamano:2015hxa , and Hamiltonian effective field theory Liu:2016wxq . One pole
is located near the $\bar{K}N$ threshold with a narrow decay width. A second
pole is found around the $\pi\Sigma$ threshold with relatively broad width.
Namely, the nominal “$\Lambda(1405)$” is not a single resonance. It is
realized as a superposition of two eigenstates in the same energy region. In
fact, in the latest version of PDG ParticleDataGroup:2020ssz , the lower
energy pole is called $\Lambda(1380)$ as a new two-star resonance, and
$\Lambda(1405)$ is reserved for the higher energy pole around 1420 MeV. Here
we note that the two resonance poles in the complex energy plane do not
necessarily exhibit two peaks in the projected spectrum on the real axis. The
shape of the spectrum depends on the relative phase of the residues of the
poles. In the $\Lambda(1405)$ case there is only one broad and asymmetric peak
structure on the real axis (see Fig. 5). In other words, the $\Lambda(1405)$
spectrum is produced by the contributions from both poles.

Figure 5: The absolute value of the scattering amplitude in the complex energy
$z$ plane. Adapted from Ref. Hyodo:2011ur .

The origin of the two poles can be understood by the properties of the
Weinberg-Tomozawa interaction Jido:2003cb ; Hyodo:2007jq . The meson-baryon
channel can be expressed either in the physical basis (such as $K^{-}p$,
$\bar{K}^{0}n$, …), in the isospin basis ($\bar{K}N(I=0)$, $\bar{K}N(I=1)$,
…), or in the $SU(3)$ basis ($\bm{1}(I=0)$, $\bm{8}(I=0)$, …). The basis
transformation can be done through the $SU(2)$ Clebsch-Gordan coefficients and
the $SU(3)$ isoscalar factors. Representing the interaction in the isospin
basis and in the $SU(3)$ basis is useful in order to study the symmetry
underlying the interaction. The coupling strength of $V_{\rm WT}$ is $SU(3)$
symmetric, hence $V_{\rm WT}$ becomes a diagonal matrix in the $SU(3)$ basis
where the mixing of different representations is absent. Therefore, in the
$SU(3)$ symmetric limit of hadron masses, the coupled-channel equation reduces
to a set of independent single-channel problems Hyodo:2006yk ; Hyodo:2006kg .
There are four channels with $I=0$ ($\bm{1}$, $\bm{8}$, $\bm{8}^{\prime}$, and
$\bm{27}$). Three of those are shown to be sufficiently attractive to generate
bound states in the $SU(3)$ limit Jido:2003cb . It is found that one of the
poles evolves into the $\Lambda(1670)$ resonance. The other two poles move
towards the $\Lambda(1405)$ energy region, along with the $SU(3)$ breaking
towards physical masses. In the same way, there are four channels in the
isospin basis ($\pi\Sigma$, $\bar{K}N$, $\eta\Lambda$, and $K\Xi$), and
$V_{\rm WT}$ is attractive in the diagonal $\pi\Sigma$, $\bar{K}N$ and $K\Xi$
channels. In the absence of off-diagonal couplings, the $\bar{K}N$ attraction
generates a bound state below the $\bar{K}N$ threshold, and the $\pi\Sigma$
attraction creates a resonance above the $\pi\Sigma$ threshold Hyodo:2007jq .
By introducing the channel couplings gradually, these eigenstates evolve into
the two poles between the $\bar{K}N$ and $\pi\Sigma$ thresholds. In this way,
the origin of the two poles can be understood by the two attractive components
in the Weinberg-Tomozawa term, which are model-independently constrained by
the low-energy theorem of chiral symmetry.

Now we explain how the pole positions in PDG are determined, based on Refs.
Ikeda:2011pi ; Ikeda:2012au ; Guo:2012vv ; Mai:2014xna . In these works of NLO
chiral $SU(3)$ dynamics, a systematic uncertainty analysis was performed with
inclusion of the SIDDHARTA kaonic hydrogen measurement. The experimental data
base for this analysis can be classified as follows:111The $K^{-}p$
correlation function data was not available when Refs. Ikeda:2011pi ;
Ikeda:2012au ; Guo:2012vv ; Mai:2014xna  were published, but the data in Ref.
ALICE:2019gcn  is shown to be consistent with the scattering amplitude of Ref.
Ikeda:2011pi ; Ikeda:2012au , as we describe below Kamiya:2019uiw .

  * (i)

elastic and inelastic total cross sections of $K^{-}p$ scattering,

  * (ii)

threshold branching ratios Tovee:1971ga ; Nowak:1978au ,

  * (iii)

energy level shift and width of the kaonic hydrogen SIDDHARTA:2011dsy ;
Bazzi:2012eq ,

  * (iv)

$\pi\Sigma$ invariant mass distributions in various reactions Ahn:2003mv ;
Niiyama:2008rt ; CrystallBall:2004ovf ; Zychor:2007gf ; HADES:2012csk ;
CLAS:2013rjt ; CLAS:2013zie .

The total cross section $\sigma_{ij}$ from channel $j$ to $i$ at the energy
$W=\sqrt{s}$ is derived from $T_{ij}$ as (here we follow the convention in
Refs. Ikeda:2011pi ; Ikeda:2012au ):

$\displaystyle\sigma_{ij}(W)=\frac{q_{i}}{q_{j}}\frac{|T_{ij}(W)|^{2}}{16\pi
W^{2}},$ (7)

where $q_{i}$ is the magnitude of the three-momentum in channel $i$. The
threshold branching ratios are calculated by the combination of the cross
section at the $K^{-}p$ threshold, $\sigma_{ij}(W=m_{K^{-}}+M_{p})$. The
energy shift $\Delta E$ and width $\Gamma$ of kaonic hydrogen are related to
the $K^{-}p$ scattering length $a_{K^{-}p}$ through the improved Deser formula
Meissner:2004jr , and $a_{K^{-}p}$ can be calculated from the scattering
amplitude as

$\displaystyle a_{K^{-}p}=\left.\frac{T_{K^{-}p,K^{-}p}(W)}{8\pi
W}\right|_{W=m_{K^{-}}+M_{p}}.$ (8)

In this way, data sets (i)-(iii) are related to the two-body scattering
amplitude $T_{ij}$ and can be used as direct constraints on $T_{ij}$. In
contrast, the $\pi\Sigma$ spectra cannot be calculated in terms of the
corresponding scattering amplitude $T_{ij}$ only. Elastic $\pi\Sigma$
scattering experiments are practically not possible, so the $\pi\Sigma$
spectra can only be measured in production reactions, where particles other
than $\pi\Sigma$ exist in the final state. Consider for example the simplest
case of a three-body final state, such as it appears in the photoproduction
reaction, $\gamma p\to K^{+}\pi\Sigma$. Let channel $i$ denote the $\pi\Sigma$
subsystem in the final state. The distribution of the invariant $\pi\Sigma$
mass $M_{I}$ can be written as

$\displaystyle\frac{d\sigma_{i}(M_{I})}{dM_{I}}\propto\left|\sum_{j}T_{ij}(M_{I})G_{j}(M_{I})C_{j}\right|^{2}~{}~{}.$
(9)

The coefficients $C_{j}$ represent the weight of the initial channel $j$,
which reflect various aspects of the reaction, such as the kinematics (initial
energy, scattering angle of $K^{+}$, etc.). In order to extract the
information of the two-body scattering amplitude $T_{ij}$, one needs to
estimate the $C_{j}$s in some way. In addition, Eq. (9) does not include the
final state interactions of other hadron pairs ($\pi K$ and $K\Sigma$), so the
full three-body dynamics is missing. Therefore, in comparison with the direct
constraints (i)-(iii), using the $\pi\Sigma$ spectra (iv) requires special
care.

In Refs. Ikeda:2011pi ; Ikeda:2012au , the meson-baryon scattering amplitude
was constructed using NLO chiral $SU(3)$ dynamics together with the
constraints (i)-(iii). A systematic uncertainty analysis was performed to
determine the subtraction constants and the LECs in the NLO term, achieving an
accuracy $\chi^{2}/{\rm d.o.f}=0.96$. This indicates that the “puzzle” caused
by the DEAR experiment DEAR:2005fdl , a possible inconsistency of the
scattering data and an earlier kaonic hydrogen measurement Borasoy:2004kk ;
Borasoy:2005ie , has been resolved. Even when the interaction is restricted to
$V_{\rm WT}$ only, the overall description is reasonable ($\chi^{2}/{\rm
d.o.f}=1.12$). This means that the scattering amplitude is essentially
determined by $V_{\rm WT}$, while NLO corrections are required as a further
improvement to deal with the accurate SIDDHARTA data. It also turns out that
the SIDDHARTA result reduces the uncertainty of the subthreshold extrapolation
of the scattering amplitude, in comparison to a similar analysis that only
uses the data (i) and (ii) Borasoy:2006sr . In Ref. Guo:2012vv , in addition
to (i)-(iii), the cross sections of $K^{-}p\to\eta\Lambda$ CrystalBall:2001uhc
, $\pi\Lambda$ phase shift at $\Xi^{-}$ mass FNALE756:2003kkj ;
HyperCP:2004not , and the $\pi\Sigma$ mass distributions in the
$\Sigma^{+}(1660)\to\pi^{+}\pi^{-}\Sigma^{+}$ reaction Hemingway:1984pz  and
in the $K^{-}p\to\pi^{0}\pi^{0}\Sigma^{0}$ reaction CrystallBall:2004ovf  were
included in the analysis. The effect of the $SU(3)$ breaking in the meson
decay constants was discussed in detail. In Ref. Mai:2014xna , the $\pi\Sigma$
spectra in the photoproduction data by CLAS CLAS:2013rjt  was used in addition
to (i)-(iii). It was shown that some solutions allowed by (i)-(iii) were
rejected by the CLAS data, and two solutions were finally obtained. The
results of the pole positions in Refs. Ikeda:2011pi ; Ikeda:2012au ;
Guo:2012vv ; Mai:2014xna  are shown in Table 2, and plotted in Fig. 6. In all
those analyses, two poles are found in the energy region of the coupled
$\bar{K}N$-$\pi\Sigma$ systems. The position of the $\Lambda(1405)$ pole (near
the $\bar{K}N$ threshold) has small ambiguity and converges to the region
around 1420 MeV. This reflects the strong constraint from the SIDDHARTA data
at the $K^{-}p$ threshold. On the other hand, the pole position of the
$\Lambda(1380)$ (near the $\pi\Sigma$ threshold) shows sizable ambiguities in
different analyses. It is therefore desirable to determine that pole position
quantitatively in future studies.

Table 2: Resonance poles of $\Lambda(1405)$ and $\Lambda(1380)$ in PDG ParticleDataGroup:2020ssz . |  $\Lambda(1405)$ [MeV] |  $\Lambda(1380)$ [MeV]
---|---|---
Refs. Ikeda:2011pi ; Ikeda:2012au  NLO | $1424^{+7}_{-23}-i26^{+3}_{-14}$ | $1381^{+18}_{-6}-i81^{+19}_{-8}$
Ref. Guo:2012vv  Fit II | $1421^{+3}_{-2}-i19^{+8}_{-5}$ | $1388^{+9}_{-9}-i114^{+24}_{-25}$
Ref. Mai:2014xna  solution #2 | $1434^{+2}_{-2}-i\,10^{+2}_{-1}$ | $1330^{+4~{}}_{-5~{}}-i\,56^{+17}_{-11}$
Ref. Mai:2014xna  solution #4 | $1429^{+8}_{-7}-i\,12^{+2}_{-3}$ | $1325^{+15}_{-15}-i\,90^{+12}_{-18}$
Figure 6: Resonance poles of $\Lambda(1405)$ and $\Lambda(1380)$ in the
complex energy ($z$) plane. Triangles, squares, crosses, and circles show the
results in Refs. Ikeda:2011pi ; Ikeda:2012au  (NLO), Ref. Guo:2012vv  (Fit
II), Ref. Mai:2014xna  (Solution #2), and Ref. Mai:2014xna  (Solution #4).
Dotted lines represent the threshold energies of meson-baryon channels:
$\pi^{0}\Sigma^{0}$, $\pi^{-}\Sigma^{+}$, $\pi^{+}\Sigma^{-}$, $K^{-}p$,
$\bar{K}^{0}n$ from left to right. Adapted from Ref. Hyodo:2020czb .

The spin of the $\Lambda(1405)$ was considered to be consistent with $1/2$ in
the past experiments Engler:1965zz ; Thomas:1973uh ; Hemingway:1984pz , but
its parity had not been determined. In 2014, the direct experimental
determination of the parity of the $\Lambda(1405)$ was performed in Ref.
CLAS:2014tbc  with the CLAS photoproduction data of $\gamma p\to
K^{+}\Lambda(1405)$. Consider the decay angular distribution of the
$\Lambda(1405)\rightarrow\pi^{-}\Sigma^{+}$ decay. In the unpolarized two-body
decay via the strong interaction, the angular dependence is determined only by
the spin, independent of the parity. If the polarization of the $\Sigma^{+}$
($\vec{Q}$) and that of the $\Lambda(1405)$ ($\vec{P}$) are specified, then
the difference arises in the direction of $\vec{Q}$ with respect to $\vec{P}$,
as shown in the left panel of Fig. 7. For instance, when the initial state is
$J^{P}=1/2^{-}$ with an $s$-wave decay, the direction of $\vec{Q}$ is
independent of the decay angle as in panel (a). On the other hand, for a
$1/2^{+}$ state with a $p$-wave decay, the direction of $\vec{Q}$ rotates
around the $\vec{P}$ vector as in panel (b). Thus, taking the $z$ axis in the
direction of $\vec{P}$, the $z$ component of the $\Sigma$ polarization $Q_{z}$
is constant for $1/2^{-}$, while it changes sign depending on the decay angle
for $1/2^{+}$. Experimentally, the $\Lambda(1405)$ polarization $\vec{P}$ is
determined by the reaction plane specified by the photon and the $K^{+}$. The
$\Sigma^{+}$ polarization $\vec{Q}$ is given by the weak decay of the
$\Sigma^{+}$. The experimental result for the total energy $2.65<W<2.75$ GeV
and the scattering angle of $K^{+}$ in the center-of-mass system,
$0.70<\cos\theta_{K^{+}}^{\mathrm{cm}.}<0.80$, is shown in the right panel of
Fig. 7. The observed $Q_{z}$ does not depend on the decay angle
$\cos\theta_{\Sigma^{+}}$ with respect to $\vec{P}$, and hence the
$J^{P}=1/2^{-}$ quantum numbers are confirmed for the $\Lambda(1405)$.

Figure 7: Upper figure: schematic illustration of the polarization transfer of
$Y^{*}\rightarrow Y\pi$ decay with $Y^{*}$ having $J^{P}=1/2^{-}$ (a) or
$J^{P}=1/2^{+}$ (b). Lower figure: $z$-component of the $\Sigma^{+}$
polarization $Q_{z}$ as a function of the decay angle
$\cos\theta_{\Sigma^{+}}$. Solid line: average of the experimental data;
dotted line: prediction for $p$-wave decay; dashed line marks the case without
polarization. Adapted from Ref. CLAS:2014tbc .

##  3 $K^{-}p$ correlation functions

In recent years, two-particle momentum correlation functions in the high-
energy collisions have been of particular interest, as a new tool to extract
properties of hadronic interactions that are not easily accessible in direct
scattering experiments. In high-energy heavy-ion and proton-proton collisions
with high multiplicities in the TeV region, hadrons are statistically emitted
from the high-energy source created by the collisions Bauer:1992ffu ;
Lisa:2005dd ; ExHIC:2017smd . The produced hadrons can interact pairwise
before they reach the detector. The momentum correlation of such a pair of
hadrons reflects the effect of their two-body interaction.

The correlation function of two hadrons (with momenta $\bm{p}_{1}$ and
$\bm{p}_{2}$) is given by

$\displaystyle C(\bm{q})$
$\displaystyle=\frac{N_{12}(\bm{p}_{1},\bm{p}_{2})}{N_{1}(\bm{p}_{1})N_{2}(\bm{p}_{2})},$
(10)

where $\bm{q}$ is the relative momentum of the hadron pair. The numerator
represents the production yield with simultaneous detection of the hadron
pair. The denominator corresponds to the product of the yields of the
individually detected hadrons, with $N_{i}$ normalized to unity when
integrated over the momentum. If the hadronic final state interactions were
completely absent, the correlation function of a distinguishable hadron pair
would be $C(\bm{q})=1$, and that of a pair of identical hadrons would be the
correlation induced by quantum statistics. Information about the hadron
interactions can therefore be extracted by precise measurements of the
deviation from these baseline correlations.

Figure 8: $(K^{-}p\oplus K^{+}\bar{p})$ correlation functions in proton-proton
collisions at $\sqrt{s}=5,7,13$ TeV (from left to right) by ALICE at LHC. The
rightmost panel shows the result of the combined analysis of all collision
energies. Adapted from Ref. ALICE:2019gcn .

Theoretical calculation of the correlation function of a pair of hadrons
without channel couplings can be performed using the Koonin-Pratt formula
Koonin:1977fh ; Pratt:1986cc :

$\displaystyle C(\bm{q})$ $\displaystyle=\int
d\bm{r}\;S(\bm{r})|\Psi^{(-)}(\bm{q};\bm{r})|^{2},$ (11)

where $S(\bm{r})$ is the source function characterizing the shape and size of
the hadron emitting source. The scattering wave function
$\Psi^{(-)}(\bm{q};\bm{r})$ of the hadron pair with relative coordinate
$\bm{r}$ is determined by the hadronic final state interactions. It follows
from Eq. (11) that the measurement of the correlation function can serve two
complementary purposes. First, using a hadron pair with a well specified
interaction, the measured correlation function permits to extract information
about the emission source $S(\bm{r})$ in the collision systems
Goldhaber:1960sf ; ALICE:2020ibs . On the other hand, if the source function
can be estimated through other considerations, one can extract the information
of the two-body interaction in $\Psi^{(-)}(\bm{q};\bm{r})$ from the
correlation function measurement. Utilizing the latter strategy, the STAR
collaboration at RHIC STAR:2014dcy  and the ALICE collaboration at LHC
ALICE:2018ysd ; ALICE:2020mfd  have studied several baryon-baryon
interactions, which are difficult or impossible to measure in standard
scattering experiments.

Measurements of the $K^{-}p$ correlation functions in proton-proton collisions
at $\sqrt{s}=5,7,13$ TeV were reported in Ref. ALICE:2019gcn . In the
analysis, the data of the $K^{-}p$ pair and its charge conjugation
$K^{+}\bar{p}$ are combined (Fig. 8). A remarkable fact is the non-monotonic
behavior of the correlation function at the relative momentum $\sim 58$ MeV.
Because this momentum corresponds to the threshold energy of the
$\bar{K}^{0}n$ pair, this behavior can be interpreted as a threshold cusp
effect. The threshold energy difference between $K^{-}p$ and $\bar{K}^{0}n$ is
about 5 MeV and caused by isospin symmetry breaking. A $K^{-}p$ scattering
experiment at such low momentum is technically difficult. The non-optimal
resolution of the total cross section data at such low energy prohibited an
experimental signal of the $\bar{K}^{0}n$ threshold cusp. In contrast, the
high precision of the correlation function data in Fig. 8 clearly show the
$\bar{K}^{0}n$ cusp effect. In addition, there are data points even below the
$\bar{K}^{0}n$ threshold, corresponding to $K^{-}p$ scattering at extremely
low energy. The $K^{-}p$ correlation function can therefore provide an
additional new constraint on the properties of the $\Lambda(1405)$.

Figure 9: The $K^{-}p$ correlation function deduced from $pp$ collisions at
$\sqrt{s}=13$ TeV (data) in comparison with theoretical calculation based on
chiral $SU(3)$ dynamics with coupled channels (solid line). Dotted line
includes the contribution of the $\Lambda(1520)$ with $S=-1$ and
$J^{P}=3/2^{-}$. Dashed line represents the prediction of the correlation
function with larger source size. Adapted from Ref. Kamiya:2019uiw .

An improved calculation of the $K^{-}p$ correlation function requires
generalizing Eq. (11) by taking into account the following items:

  * •

Coupled-channel effects: the $K^{-}p$ system couples to the $\pi\Sigma$ and
$\pi\Lambda$ channels at lower energies, so a generalization of the Koonin-
Pratt formula for the coupled-channel case is needed, as discussed in Refs.
Lednicky:1998r ; Haidenbauer:2018jvl .

  * •

Isospin symmetry breaking: to describe the $\bar{K}^{0}n$ cusp, the isospin
symmetric formulation is not suitable. Physical hadron masses with inclusion
of isospin symmetry breaking must be used as input.

  * •

Coulomb interaction: in the diagonal $K^{-}p$, $\pi^{+}\Sigma^{-}$, and
$\pi^{-}\Sigma^{+}$ potentials, the Coulomb force is active in addition to the
strong interaction, and the wave function components must be calculated for
these channels with the appropriate asymptotic boundary conditions for Coulomb
wave functions.

A theoretical framework including all these effects has been completed in Ref.
Kamiya:2019uiw , and applied to the $K^{-}p$ correlation function. In this
work, the meson-baryon coupled-channel potential based on chiral $SU(3)$
dynamics Miyahara:2018onh  was used for the strong interaction to calculate
the wave function. The source function $S(\bm{r})$ is assumed to be of
spherical gaussian shape with a size parameter $R$. In Ref. ALICE:2019gcn ,
the source size is estimated as $R\sim 1$ fm from the $K^{+}p$ correlation
function where the interaction is relatively well understood. By fixing
$R=0.9$ fm and varying the weight of the $\pi\Sigma$ source,
$\omega_{\pi\Sigma}$, it is shown that the ALICE data can be well described
(Fig. 9). Once the theoretical framework is set, it is also possible to study
the source size dependence of the correlation functions by changing the value
of $R$. The prediction for a larger source is shown by the dashed line in Fig.
9. The correlation function in the small-momentum region is seen to be
suppressed when the source size is increased. This tendency is consistent with
the recent experimental result of Pb-Pb collisions reported by the ALICE
collaboration in Ref. ALICE:2021szj .

##  4 $\bar{K}N$ interaction and few-body kaonic nuclei

Let us now turn to kaon-nuclear few-body physics. Several methods have been
established to solve the few-body Schrödinger equation rigorously
Suzuki:1998bn ; Kamada:2001tv ; Hiyama:2003cu . These methods permit to
calculate few-body systems accurately for given interaction potentials.
Realistic nuclear forces have been constructed by phenomenological potentials
Wiringa:1994wb ; Machleidt:2000ge  and more recently based on chiral effective
field theory Epelbaum:2008ga ; Machleidt:2011zz . These potentials achieve an
accuracy of $\chi^{2}/{\rm d.o.f}\sim 1$. At the same time recent developments
in lattice QCD Ishii:2006ec ; Aoki:2009ji  are promising in providing yet
another approach to the $NN$ interaction by generating these potentials from
the underlying QCD.

Studies of few-body kaonic nuclei rely on realistic $\bar{K}N$ interactions in
combination with advanced few-body techniques. Such calculations have been
performed using variational methods Yamazaki:2007cs ; Dote:2008in ;
Dote:2008hw ; Barnea:2012qa ; Ohnishi:2017uni  or solving Faddeev equations
Shevchenko:2006xy ; Shevchenko:2007ke ; Ikeda:2007nz ; Ikeda:2010tk ;
Shevchenko:2014uva ; Revai:2014twa . The most advanced computations
Ohnishi:2017uni ; Shevchenko:2014uva ; Revai:2014twa  use $\bar{K}N$
interactions constrained by experimental data including the SIDDHARTA
measurement at the level of $\chi^{2}/{\rm d.o.f}\sim 1$.

Such potentials (called Kyoto $\bar{K}N$ potentials) have been presented in
Refs. Miyahara:2015bya ; Miyahara:2018onh , following the prescription
developed in Ref. Hyodo:2007jq  to construct an equivalent local potential
that produces the scattering amplitude resulting from NLO chiral $SU(3)$
dynamics. Based on this scattering amplitude Ikeda:2011pi ; Ikeda:2012au , an
effective single-channel $\bar{K}N$ potential has been constructed
Miyahara:2015bya , in which the coupled-channel decays of $\bar{K}N$ to
$\pi\Sigma$ and $\pi\Lambda$ are incorporated by the imaginary part of the
$\bar{K}N$ potential. The energy dependence of the potential reflects the
derivative coupling of the NG bosons in chiral perturbation theory and the
elimination of the $\pi$-hyperon channels. Alternatively, coupled-channel
$\bar{K}N$-$\pi\Sigma$-$\pi\Lambda$ potentials have also been constructed
Miyahara:2018onh . In this case, all coupling strengths are real and the
energy dependence is milder than that of the effective single-channel
$\bar{K}N$ potential, as the $\pi Y$ channels are explicitly active in this
framework. As far as the $\bar{K}N$ component is concerned, both the single-
channel and coupled-channel potentials give equivalent results, while the
coupled-channel potential is suitable for the computation of quantities for
which the $\pi\Sigma$ and $\pi\Lambda$ channels become important, such as the
$K^{-}p$ correlation functions. In addition, the coordinate space potential is
useful to extract the spatial structure of the system by displaying the wave
function. In Fig. 10, we plot the density distribution
$\rho(r)=r^{2}|\Psi(r)|^{2}$ at the $\Lambda(1405)$ pole together with the
real and imaginary parts of the $\bar{K}N$ potential in the $I=0$ channel at
the same energy. It is seen that the two-body density spreads over a large
$\bar{K}N$ distance, even beyond the range of the potential. This indicates
again the $\bar{K}N$ molecular nature of the $\Lambda(1405)$.

Figure 10: The $\bar{K}N$ density distribution $\rho(r)=r^{2}|\Psi(r)|^{2}$
(thick solid line) at the $\Lambda(1405)$ pole energy, $(1424-26i)$ MeV. Also
shown are the real part (thin solid line) and the imaginary part (dotted line)
of the Kyoto $\bar{K}N$ potential $(I=0)$ at that energy. Adapted from Ref.
Miyahara:2015bya .

The simplest prototye kaonic nucleus is the three-body $\bar{K}NN$ system.
Before proceeding to the actual calculations, let us first classify the
possible configurations. Both $\bar{K}$ and $N$ have isospin $I=1/2$, so we
can construct the $\bar{K}NN$ system in three different isospin configurations
PL7.288 :

$\displaystyle{\rm(i)}:|\bar{K}[NN]_{I=0}\rangle_{I=1/2},\quad{\rm(ii)}:|\bar{K}[NN]_{I=1}\rangle_{I=1/2},\quad{\rm(iii)}:|\bar{K}[NN]_{I=1}\rangle_{I=3/2}.$
(12)

Assuming that all the pairs are combined in $s$ waves in the ground state, the
$[NN]_{I=0}$ ($[NN]_{I=1}$) pair is in a spin $J=1$ ($J=0$) state. Then the
states (i), (ii), and (iii) have different quantum numbers:
$I(J^{P})=1/2(1^{-})$, $1/2(0^{-})$, and $3/2(0^{-})$, respectively.222The
state (ii) is sometimes called $K^{-}pp$, but this assignment is not
appropriate because the $K^{-}pp$ configuration mixes with $\bar{K}^{0}pn$.
The physical state is the superposition of these two configurations.
Furthermore (ii) is an isospin doublet. The isospin partner of the $K^{-}pp$
\- $\bar{K}^{0}pn$ state is a superposition of $K^{-}pn$ and $\bar{K}^{0}nn$.
Let us now consider the lowest energy state among (i)-(ii). The $[NN]_{I=0}$
($[NN]_{I=1}$) configuration corresponds to the ${}^{3}S_{1}$ (${}^{1}S_{0}$)
channel, so only (i) is bound in the absence of $\bar{K}$. However, by
performing the isospin recombination, one can extract the fractions of the
$\bar{K}N$ pairs in different isospin states:

$\displaystyle{\rm(i)}:|\bar{K}[NN]_{I=0}\rangle_{I=1/2}$
$\displaystyle=-\frac{1}{2}|[\bar{K}N]_{I=0}N\rangle_{I=1/2}+\frac{\sqrt{3}}{2}|[\bar{K}N]_{I=1}N\rangle_{I=1/2},$
(13) $\displaystyle{\rm(ii)}:|\bar{K}[NN]_{I=1}\rangle_{I=1/2}$
$\displaystyle=\frac{\sqrt{3}}{2}|[\bar{K}N]_{I=0}N\rangle_{I=1/2}+\frac{1}{2}|[\bar{K}N]_{I=1}N\rangle_{I=1/2},$
(14) $\displaystyle{\rm(iii)}:|\bar{K}[NN]_{I=1}\rangle_{I=3/2}$
$\displaystyle=|[\bar{K}N]_{I=1}N\rangle_{I=3/2}.$ (15)

This means that the fraction of the $\bar{K}N$ pair in the $I=0$ channel is
$1/4$, $3/4$, and $0$ for (i), (ii), and (iii), respectively. Now we recall
that the $\bar{K}N$ pair in $I=0$ has a quasi-bound state $\Lambda(1405)$
below the threshold, while the $I=1$ component is unbound, though attractive
(see Table 1). Thus, from the viewpoint of the $\bar{K}N$ interaction, the
configuration (ii) can earn the largest attraction where the fraction of the
$I=0$ $\bar{K}N$ pair is maximal. In this way, the $NN$ correlation favors the
state (i), while the $\bar{K}N$ correlation favors the state (ii) as the
ground state. The numerical calculation shows that there exists a quasi-bound
state in channel (ii), while (i) is unbound with respect to the $K^{-}d$
threshold Barnea:2012qa . Thus, in the two-baryon systems, the $\bar{K}N$
correlation is so strong that the ground state spin is changed when a
$\bar{K}$ is added to the nucleons.

Now we present numerical results of a comprehensive study of few-body kaonic
nuclei in Ref. Ohnishi:2017uni , where the ground states of up to the seven-
body system ($\bar{K}NNNNNN$) are computed using the stochastic variational
method with a correlated gaussian basis Suzuki:1998bn . The AV4’ potential
Wiringa:2002ja  is adopted for the nuclear force which reproduces the few-body
nuclei in the relevant mass number region $A=2$-6 reasonably well. The
$\bar{K}N$ potential is the single-channel Kyoto potential from Ref.
Miyahara:2015bya  where the effects of the $\pi\Sigma$ and $\pi\Lambda$
channels are renormalized into the imaginary part of the potential.

The results of the few-body calculations Ohnishi:2017uni  are summarized in
Table 3. Note that the absorption processes of the antikaon by multi-nucleons,
such as $\bar{K}NN\to YN$, are not included (for estimates of such effects,
see Refs. Dote:2008hw ; Sekihara:2009yk ; Sekihara:2012wj ; Bayar:2013lxa ).
Up to $A=4$ systems, $\bar{K}NNNN$, we find a quasi-bound state in each mass
number below the lowest threshold. With $A=5$, no state is found below the
threshold of $(\bar{K}NNNN)+N$. In the $A=6$ system, both the $J^{P}=0^{-}$
and $1^{-}$ system have a quasibound state at almost degenerated energy. The
ground state of the $\bar{K}NN$ system is confirmed to be $J^{P}=0^{-}$ (the
state (ii) in Eq. (12)), and the other states are found to be unbound. The
uncertainties of the binding energy $B$ and the mesonic decay width
$\Gamma_{\pi YN}$ come primarily from the treatment of the energy dependence
of the $\bar{K}N$ system in the few-body kaonic nuclei.

Table 3: Isospin $I$, spin-parity $J^{P}$, binding energy $B$, and the mesonic decay width $\Gamma_{\pi YN}$ of the ground states of the few-body kaonic nuclei Ohnishi:2017uni . Uncertainties stem mainly from the treatment of the energy dependence of the $\bar{K}N$ interaction and also from the difference in the isospin multiplet. The $0^{-}$ and $1^{-}$ states of $\bar{K}NNNNNN$ are almost degenerate. | $\bar{K}NN$ | $\bar{K}NNN$ | $\bar{K}NNNN$ | $\bar{K}NNNNNN$
---|---|---|---|---
$I(J^{P})$ | $1/2(0^{-})$ | $0(1/2^{-})$ | $1/2(0^{-})$ | $1/2(0^{-},1^{-})$
$B$ [MeV] | 25-28 | 45-50 | 68-76 | 70-81
$\Gamma_{\pi YN}$ [MeV] | 31-59 | 26-70 | 28-75 | 24-76

As mentioned above, in the three-body system, the $\bar{K}N$ correlation
overcomes the $NN$ correlation and the spin-parity of the ground state is
changed by the injection of the antikaon. Consider next the interplay between
the $\bar{K}N$ and $NN$ correlations in the four-nucleon systems, the
$\bar{K}NNNN$ state. The wavefunction of the $\bar{K}NNNN$ system with
$J^{P}=0^{-}$, $I=1/2$, $I_{3}=+1/2$ can be written:

$\displaystyle|\bar{K}NNNN\rangle$
$\displaystyle=C_{1}|K^{-}pppn\rangle+C_{2}|\bar{K}^{0}ppnn\rangle,$ (16)

where $C_{i}$ is the weight of each component. The $\bar{K}N$ correlation
tends to increase the fraction of the $I=0$ pair ($K^{-}p$ or $\bar{K}^{0}n$),
and therefore the first term in Eq. (16) is favored. The $NN$ correlation, on
the other hand, favors the second component, because $ppnn$ can form an
$\alpha$ particle cluster which has a large binding energy. In this way, the
$C_{i}$ weight should be determined by the competition between the $\bar{K}N$
and $NN$ correlations which act in opposite directions. The results of the
dynamical few-body calculation shows that $|C_{1}|^{2}=0.08$ and
$|C_{2}|^{2}=0.92$: the $NN$ correlation (in particular the $\alpha$ particle
correlation) wins over $\bar{K}N$ correlation in the $\bar{K}NNNN$ system.

From the relatively large binding energies of the kaonic nuclei in Table 3,
one might expect that the density in the center of the system is also
increased. In fact early studies Akaishi:2002bg  considered kaonic nuclei to
be candidates of possible high-density matter. Here we comment on the density
of the system from the viewpoint of the few-body calculation. In Fig. 11, the
density distributions of the nucleons in the $\bar{K}NNNN$ system measured
from the center-of-mass of the nucleons are shown by the solid and dashed
lines (the difference arises from the treatment of the energy dependence of
the $\bar{K}N$ interaction). Compared with the density of 4He without the
antikaon (dash-dotted line), the central density increases slightly, due to
the attraction of the $\bar{K}N$ interaction. For comparison, the
$\bar{K}NNNN$ system is also calculated with the Akaishi-Yamazaki potential
Yamazaki:2007cs . The binding energy of the $\bar{K}NNNN$ system is obtained
as $B\sim 85$ MeV, larger than $B=68-76$ MeV with the Kyoto potential which is
less attractive. In Fig. 11, however, the central density (dotted line) is
smaller than the result of the Kyoto potential. Thus, the larger binding
energy does not necessarily imply a higher central density of the system. This
is because the binding energy mainly reflects the tail of the wave function at
large $r$. In addition, one should be cautious about the notion of the
“density” in the few-body systems, where in any case only a few nucleons are
present.

Figure 11: Nucleon density distribution in the $\bar{K}NNNN$ system measured
from the center-of-mass of the nucleons by the Kyoto $\bar{K}N$ potential
(solid and dashed lines) and by the Akaishi-Yamazaki potential (dotted line).
For comparison, the density distribution of 4He is shown by the dash-dotted
line. Adapted from Ref. Ohnishi:2017uni .

Obviously, remaining uncertainties concern the widths of $\bar{K}$-nuclear
clusters with three and more nucleons. The widths listed in Table 3 are lower
limits. They do not include multinucleon absorption processes such as
$\bar{K}NNN\rightarrow\Lambda NN$ and $\Sigma NN$ which become more likely
with increasing number of nucleons, as the momentum mismatch characteristic of
antikaon absorption on a two-nucleon pair can then be shared between those
additional nucleons.

Faddeev calculations of $\bar{K}NN$ systems have been performed
Shevchenko:2014uva ; Revai:2014twa  using as input several $\bar{K}N$
potentials, both phenomenological and chiral. In search for $K^{-}pp$ bound
clusters, binding energies and widths are found which are slightly larger than
those listed in Table 3. A study comparing two-pole and and one-pole effective
potential subject to SIDDHARTA constraints Shevchenko:2012np  pointed out
almost equivalent results with pole positions of the $\Lambda(1405)$ at
$1426-i48$ MeV in the one-pole model, and $1414-i58$ MeV and $1386-i104$ MeV
in the two-pole model.

In search for a $\bar{K}NN$ quasibound cluster, the J-PARC E15 collaboration
measured the $\Lambda p$ invariant mass spectrum in the reaction ${}^{3}{\rm
He}(K^{-},\Lambda p)n$ reaction E15a:2019 ; E15b:2020 . A binding energy
$B=42\pm 3(stat.)^{+3}_{-4}(sys.)$ MeV was reported for the proposed
$``K^{-}pp"$ cluster, accompanied by a very large width, $\Gamma=100\pm
7(stat.)^{+19}_{-9}(sys.)$ MeV, more than twice the binding energy. This large
width includes primary direct two- and three-nucleon absorption mechanisms,
$K^{-}(pp)n\rightarrow(Yp)n$ and $K^{-}(ppn)\rightarrow YNN$. It raises,
however, the issue whether the extremely short lifetime of such an object
still justifies the notion of a bound system.

##  5 Kaonic atoms

Kaonic atoms are systems of a negatively charged $K^{-}$ with positively
charged ordinary nuclei, bound by the Coulomb interaction. Although the
$K^{-}$ interacts with nucleons also through the strong interaction, most of
the energy levels of kaonic atoms follow the pattern of a hydrogen-like atom,
replacing the reduced mass of the system correspondingly. This is because the
typical length scale of kaonic atoms is much larger than the range of the
strong interaction. Consider the simplest example of kaonic hydrogen, the
$K^{-}p$ system. The Bohr radius of kaonic hydrogen is $r_{B}\sim 84$ fm,
which is much larger than the $K^{-}p$ interaction range of $\lesssim 1$ fm.
Nonetheless, the wave function of the ground state (the $1s$ level) of kaonic
hydrogen has non-zero probability of the $K^{-}p$ pair to overlap with the
strong-interaction range at short distance. This induces an energy shift,
$\Delta E$, for the 1s level expected from the electromagnetic interaction
alone, and also a finite width $\Gamma$ of the energy level, reflecting the
transitions $K^{-}p\to\pi\Sigma$ and $\pi\Lambda$.

In the study of pionic atoms, the Deser-Trueman formula Deser:1954vq ;
Trueman:1961zz  was developed to relate the shift $\Delta E$ and the width
$\Gamma$ to the two-body scattering length of the strong interaction. This
formula has been further improved by including the isospin symmetry breaking
effect by the effective Lagrangian approach as Meissner:2004jr ;
Meissner:2006gx

$\displaystyle\Delta E-\frac{i\Gamma}{2}$
$\displaystyle=-2\mu^{2}\alpha^{3}a_{K^{-}p}\left[1-2\mu\alpha(\ln\alpha-1)a_{K^{-}p}\right],+\dotsb,$
(17)

with the reduced mass $\mu=m_{K^{-}}M_{p}/(m_{K^{-}}+M_{p})$, and the
electromagnetic fine structure constant $\alpha\simeq 1/137$. The scattering
length of the $K^{-}p$ system, $a_{K^{-}p}$, is related to the $\bar{K}N$
scattering lengths in the isospin basis as

$\displaystyle a_{K^{-}p}$
$\displaystyle=\frac{1}{2}(a_{I=0}+a_{I=1})+\dotsb,$ (18)

where the ellipsis stands for the isospin symmetry breaking effects. In Ref.
Baru:2009tx , a further improvement was suggested by a resummation of
logarithmically enhanced correction terms as

$\displaystyle\Delta
E-\frac{i\Gamma}{2}=-\frac{2\mu^{2}\alpha^{3}a_{K^{-}p}}{1+2\mu\alpha(\ln\alpha-1)a_{K^{-}p}}+\dotsb.$
(19)

As already mentioned, accurate kaonic hydrogen measurements were performed by
the SIDDHARTA collaboration SIDDHARTA:2011dsy ; Bazzi:2012eq . The 1s energy
shift and width deduced from this experiment, $\Delta E=-283\pm 36\pm 6$ eV
and $\Gamma=541\pm 89\pm 22$ eV, translate through the Deser formula (17) into
the complex $K^{-}p$ scattering length shown in Fig. 4.

Experiments measuring the 1s energy shift and width of kaonic deuterium are
the next important step in order to further constrain the $\bar{K}N$
interactions, as they provide additional sensitive access to the $I=1$
component through the $K^{-}n$ interaction. Such measurements are in
preparation (SIDDHARTA-2 at DA$\Phi$NE Curceanu:2019  and J-PARC E57
Zmeskal:2015 ). The theoretical description of kaonic deuterium is challenging
because there is no principal separation of scales to start from. The $K^{-}d$
Bohr radius of about 55 fm and the strong-interaction distance scale of about
1 fm have to be dealt with at the same level of high computational precision
in order to achieve accurate results for the ground state energy shift and
width. Calculations of kaonic deuterium have been performed Hoshino:2017mty
solving the three-body Schrödinger equation using a variational method. The
Kyoto $\bar{K}N$ potential Miyahara:2015bya  has been used as input. The
predicted energy shift and width of kaonic deuterium is

$\displaystyle\Delta E-\frac{i\Gamma}{2}=(670-i508)\ {\rm eV}~{}~{},$ (20)

with an uncertainty of about 10% implied by the so far unconstrained $K^{-}n$
$(I=1)$ interaction. A kaonic deuterium measurement with 25% accuracy is
expected to improve the determination of this $I=1$ component significantly.

A further observation concerns the applicability of the modified Deser-Trueman
formulae, Eqs. (17) and (19). While these relations (especially the resummed
version) work well for kaonic hydrogen, they are not reliable for kaonic
deuterium. A benchmark comparison can be made with the $K^{-}d$ scattering
length,

$\displaystyle a_{K^{-}d}=(-1.42+i1.60)\ {\rm fm}~{}~{},$ (21)

calculated in fixed-scatterer approximation using the full $K^{-}d$
Schrödinger wave function. With this value of $a_{K^{-}d}$ the estimated
$\Delta E$ resulting from (19) deviates by more than 20% from Eq. (20). The
deviation even exceeds 35% for the improved Deser formula (17). Hence the
Deser formulae are not reliable for kaonic deuterium. Faddeev calculations
Revai:2016muw  have arrived at similar conclusions.

Heavier kaonic atoms include nuclear mass numbers from A = 4 to A = 63,
systematically covering energy shifts and widths of kaonic 2p, 3d and 4f
levels. This systematics has been useful in establishing a phenomenological
$K^{-}$-nuclear optical potential Friedman:2007zza . More recent studies
connect this optical potential to microscopic theoretical input for the
$\bar{K}N$ interaction and address at the same time the role of nuclear
$K^{-}$ absorption Friedman:2016rfd .

The complex $K^{-}$\- nuclear potential $U$ enters the Klein-Gordon equation
describing the kaonic atom:

$\displaystyle\big{[}\vec{\nabla}^{2}+\big{(}E-V_{c}(\vec{r})\big{)}^{2}-m_{K}^{2}-2\mu_{K}\,U(\vec{r})\big{]}\phi(\vec{r})=0~{}~{},$
(22)

where $\mu_{K}=m_{K}M_{A}/(m_{K}+M_{A})$ is the $K^{-}$-nucleus reduced mass
and $V_{c}$ is the Coulomb potential. In the limit of low proton and neutron
densities, $\rho_{p}$ and $\rho_{n}$, the leading order s-wave optical
potential reduces to

$\displaystyle U(\vec{r})=-{2\pi\over\mu_{K}}\left[{\cal
F}_{K^{-}p}\,\rho_{p}(\vec{r})+{\cal
F}_{K^{-}n}\,\rho_{n}(\vec{r})\right]~{}~{},$ (23)

with the $K^{-}N$ forward scattering amplitudes in the kaon-nucleus c.m.
frame, ${\cal F}_{K^{-}N}=\left(1+{A-1\over A}{\mu_{K}\over
M_{N}}\right)F_{K^{-}N}$ in terms of the threshold $K^{-}$nucleon c.m.
amplitudes. This leading-order single-nucleon potential is now commonly
constructed using realistic theoretical input from chiral $SU(3)$ dynamics,
but it still needs to be extended by additional phenomenological pieces to
account for correlations in the nucleus and for kaon absorption on two and
more nucleons.

Figure 12: Absorptive interaction densities
$r^{2}|\phi(r)|^{2}\,\textrm{Im}\,U(r)$ for several kaonic atoms for lower
(solid curves) and upper (dashed curves) atomic levels in those selected
examples. Also shown are the corresponding density distributions (in units of
$\rho_{0}=0.17$ fm-3) of several atomic nuclei. Adapted from Ref.
Friedman:2016rfd .

In Ref. Friedman:2016rfd  a systematic investigation has been performed using
an optical potential ansatz $U=U^{(1)}+U^{(2)}$, where $U^{(1)}$ is based on
Eq. (23) using realistic chiral $SU(3)$ coupled-channel amplitudes as input,
but with additional corrections for subthreshold kinematics and Pauli
principle effects Waas1997b  taken into account. A phenomenological second
term $U^{(2)}$ is introduced to represent multinucleon processes, in
particular $K^{-}$ absorption:

$\displaystyle
U^{(2)}(\vec{r})=-{2\pi\over\mu_{K}}B\,\rho(\vec{r})\left({\rho(\vec{r})\over\rho_{0}}\right)^{\alpha}~{}~{},$
(24)

with a complex strength $B$ and exponent $\alpha\geq 1$, and the density
$\rho=\rho_{p}+\rho_{n}$ scaled in units of the ground state density of
nuclear matter, $\rho_{0}=0.17$ fm-3. A quantity of interest in this context
is the fraction of single- and multi-nucleon absorption rates. Single nucleon
absorption refers to the $K^{-}N\rightarrow\pi\Lambda,\pi\Sigma$ reactions
encoded in the imaginary parts of the amplitudes $F_{K^{-}N}$, for which a
fraction of $0.75\pm 0.05$ is quoted in Ref. Friedman:2016rfd . The remaining
25% are empirically assigned to multinucleon absorptive processes, with no
significant dependence on nuclear mass number. Given this constraint, an
instructive outcome of a precision fit to a large kaonic atom data base is
that, out of a variety of $\bar{K}N$ potentials, only two which are derived
from chiral $SU(3)$ dynamics Ikeda:2012au ; Cieply:2012  actually qualify as
being compatible with the data.

When combined with the single-nucleon potential $U^{(1)}$ constructed with
chiral $SU(3)$ amplitudes Ikeda:2012au , a best fit for the strength of the
phenomenological potential (24), with fixed $\alpha=1$, arrives at an
absorption length, $B=(-0.9+i\,1.4)$ fm, with an uncertainty of about 20%. The
$K^{-}$ absorption processes are localized at the nuclear surface, as shown in
Fig. 12. One can distinguish between absorption from a lower and an upper
level of a given kaonic atom as indicated. The maximum overlaps of the
absorption profiles with the nuclear density distributions are at relatively
low densities, around $0.15-0.2\,\rho_{0}$ for the lower state and
$0.1-0.15\,\rho_{0}$ for the upper state Friedman:2016rfd .

##  6 Many-body physics and kaons in baryonic matter

Kaon physics in the context of the nuclear many-body problem includes a broad
range of topics, from kaon- and antikaon scattering and production on complex
nuclei to more exotic phenomena such as kaon condensation in dense baryonic
matter and the possible role of strangeness in neutron stars. These themes
have been addressed in several review articles Ramos2001 ; Tolos:2020aln . In
this section, after some short introductory digressions, we shall primarily
report on theoretical explorations in quest for possible quasibound antikaon-
nuclear systems with more complex nuclei, following the previous assessment of
few-body kaonic nuclei.

Historically, the attractive nature of the $\bar{K}N$ interaction as it
follows from the leading Weinberg-Tomozawa amplitude provided a strong
motivation for investigating the phenomenon of kaon condensation in dense
baryonic matter Kaplan1986 . The bulk of neutron star matter is traditionally
viewed as being composed of neutrons plus a small fraction of protons and
electrons subject to beta equilibrium. At some critical density, a rapidly
dropping in-medium $K^{-}$ mass might cross the continuously increasing
electron chemical potential. At that point it begins to be energetically
favorable for the $K^{-}$ to replace the electrons, and a kaon condensate is
formed Brown:1994 . However, such a process would soften the equation-of-state
of neutron star matter, lowering its maximum mass to typically less than 1.5
solar masses. Within the last decade, new stringent conditions for the
stiffness of the neutron star equation-of-state have been established by the
observation of several heavy neutron stars with masses $M\simeq 2\,M_{\odot}$
and even beyond. As a consequence kaon condensation in neutron stars can be
ruled out. At the same time the discussion about the role of strangeness in
neutron stars has turned to a new direction under the heading of the “hyperon
puzzle”, the fact that the present maximum mass constraints can also not
easily be reconciled with possible admixtures of hyperons in the neutron star
core Djapo:2010 ; Logoteta:2019 ; Gerstung:2020 .

The starting point for studying interactions of antikaons in a nuclear medium
with proton and neutron densities $\rho_{p}$ and $\rho_{n}$ (baryon density
$\rho=\rho_{p}+\rho_{n}$) is the dispersion equation (energy $E$ and momentum
$\vec{q}$ ) in homogeneous nuclear matter:

$E^{2}-\vec{q}^{2}-m_{K}^{2}-\Pi(E,\vec{q};\rho_{p},\rho_{n})=0~{}~{},$ (25)

where $m_{K}$ is the $\bar{K}$ mass in vacuum and the self-energy $\Pi$
summarizes all interactions of the antikaon with nucleons in the medium. A
corresponding optical potential $U$ is defined as

$U(E,\vec{q};\rho_{p},\rho_{n})={1\over
2E}\Pi(E,\vec{q};\rho_{p},\rho_{n})~{}~{}.$ (26)

The energy and momentum dependent potential $U$ for antikaons is in general
complex. Its imaginary part quantifies the absorption of the meson in the
medium, as already encountered in the case of kaonic atoms. At zero momentum,
$\vec{q}=0$, the self-consistent solution of Eq. (25) determines an effective
in-medium mass of the meson:

$m_{K}^{*}(\rho)=\textrm{Re}\,E(\vec{q}=0;\rho)~{}~{}.$ (27)

Correspondingly, the in-medium meson decay width is given by

$\Gamma(\rho)=-2\,\textrm{Im}\,E(\vec{q}=0;\rho)\simeq-2\,\textrm{Im}\,U(m_{K}^{*},\vec{q}=0;\rho)~{}~{},$
(28)

where the approximation on the r.h.s. holds if the width is small compared to
the effective mass.

Consider now the self-energies of $K^{-}$ and $K^{+}$ in a nuclear medium. In
the low-density limit,

$\displaystyle\Pi^{\pm}=-4\pi{\sqrt{s}\over
M_{N}}\left(F_{K^{\pm}p}\,\rho_{p}+F_{K^{\pm}n}\,\rho_{n}\right)~{}~{},$ (29)

in terms of the $K^{\pm}N$ forward scattering c.m. amplitudes. At low energies
close to threshold these amplitudes are governed by chiral symmetry and the
Weinberg-Tomozawa theorem as in Eq. (4):

$\displaystyle F^{WT}_{K^{-}p}=-F^{WT}_{K^{+}p}={E\over
4\pi\,f^{2}}~{},~{}~{}~{}~{}~{}~{}F^{WT}_{K^{-}n}=-F^{WT}_{K^{+}n}={E\over
8\pi\,f^{2}}~{}~{},$ (30)

implying a characteristic downward (upward) shift of the effective $K^{-}$
($K^{+}$) masses by their attractive (repulsive) s-wave interactions,
respectively, in the nuclear medium. Such a splitting is actually observable
in the rapidity distributions of $K^{-}$ and $K^{+}$ produced in $1-2$ AGeV
heavy-ion collisions Song:2021 .

It is instructive to compare the s-wave optical potentials for pions and kaons
from the point of view of the underlying chiral symmetry breaking pattern. In
the chiral limit of massless $u,d$ and $s$ quarks, all interactions of both
pions and kaons as Nambu-Goldstone bosons would strictly vanish at
$E=\vec{q}=0$. For the pion, the leading s-wave potential at threshold,
$U=-{2\pi\over m_{\pi}}b_{0}\rho$, is determined by the isospin-even $\pi N$
scattering length $b_{0}$ for which a chiral low-energy theorem as well as
pion-nucleon data give a very small value. Next-to-leading order nuclear
correlation effects are therefore important. For kaons and antikaons the
driving s-wave terms (30) are much stronger at threshold ($E=m_{K}$), where
the comparatively large kaon mass reflects the explicit chiral symmetry
breaking by the mass of the strange quark. For example and orientation, the
Weinberg-Tomozawa potential for a $K^{-}$ in symmetric nuclear matter becomes

$\displaystyle U^{WT}_{K^{-}}=-{3(1+m_{K}/M_{N})\over
8\,f_{K}^{2}}\,\rho\simeq-60\,{\rho\over\rho_{0}}\,\textrm{MeV}~{}~{},$ (31)

using the empirical kaon decay constant $f_{K}\simeq 110$ MeV and
$\rho_{0}=0.17$ fm-3.

Figure 13: In-medium mass $m^{*}_{K}(\rho)$ and width of a $K^{-}$ in
symmetric nuclear matter as function of baryon density $\rho$ in units of
normal nuclear matter density, $\rho_{0}=0.17$ fm-3. Calculations based on
chiral $SU(3)$ dynamics with coupled channels and including effects of Pauli
blocking and Fermi motion (dashed lines). Full lines: two-nucleon correlations
additionally included. Also shown is the in-medium $K^{+}$ mass for
comparison. Adapted from Ref. Waas1997b .

Theoretical studies of kaons and antikaons in nuclear matter using amplitudes
from chiral $SU(3)$ coupled-channel dynamics were reported in Refs. Waas1996 ;
Waas1997a ; Waas1997b . Also included in these works were effects nonlinear in
density, i.e. beyond the non-interacting nuclear Fermi gas, in particular
Pauli blocking of nucleons and short-range $NN$ correlations. A typical result
for the effective $K^{-}$ and $K^{+}$ masses is shown in Fig. 13. The primary
Weinberg-Tomozawa mass splitting pattern between antikaon and kaon in matter
is still prominent. Pauli blocking effects are next in importance in shifting
this pattern, whereas short-range correlations turn out to have moderate
influence and enter significantly at densities above twice the density of
normal nuclear matter. The large width shown in the lower part of Fig. 13
reflects the disappearance of the $K^{-}$ in the processes
$\bar{K}N\rightarrow\pi Y$ with $Y=\Lambda,\Sigma$. This width should be taken
as a lower limit because multinucleon absorption mechanisms such as
$\bar{K}NN\rightarrow YN,~{}\bar{K}NNN\rightarrow YNN$, … have not been
considered. At very low densities the formation of the $\Lambda(1405)$ and its
in-medium subthreshold dynamics induces strong non-linear density dependence.
In Ref. Lutz2002  this behaviour has been further elaborated by a self-
consistent treatment of the $\Lambda(1405)$ and other hyperons in the nuclear
medium.

Figure 14: Binding energy $B_{K}$ and width $\Gamma$ of a $K^{-}$ bound in
${}^{16}{\rm O}$ as function of the central nuclear density $\rho_{0}$. Solid
curves (LO): leading order $s$\- and $p$-wave interactions ($T\rho$
approximation). Short-dashed and dash-dotted curves: sequentially adding Pauli
and short-range correlations. Long-dashed curve with grey uncertainty band:
estimated additional width from $\bar{K}NN$ two-body absorption. Adapted from
Ref. Weise:2008aj .

The strong in-medium attraction experienced by the $\bar{K}$ has motivated
several systematic studies of possible quasibound states in complex nuclei
Weise:2008aj ; Cieply:2011 ; Gazda:2012zz ; Hrtankova:2017wrg . The
investigation Weise:2008aj  includes $s$\- and $p$-wave $\bar{K}N$
interactions together with Pauli and short-range correlations, plus an
estimate of $\bar{K}NN$ two-nucleon absorption, in a calculation solving Eq.
(25) with local nuclear densities, $\rho(\vec{r})$. Binding energies $B_{K}$
and widths $\Gamma$ of the lowest antikaon quasibound states are computed for
prepared pseudo-nuclei, varying the central density $\rho_{0}$ from its
nominal value in order to explore the systematics of a possibly compressed
$\bar{K}$-nuclear system. An example is shown in Fig. 14 for the case of
${}^{16}{\rm O}+K^{-}$. The strong binding in the leading ($T\rho$)
approximation is significantly reduced by Pauli effects and, to a lesser
extent, by additional short-range correlations. The estimated two-body
absorption width competes with the binding energy and begins to overwhelm
$B_{K}$ as the central density increases, even though this width is presumably
still to be considered a lower limit since absorption processes on more than
two nucleons add further contributions to the overall width.

Figure 15: $\Lambda p$ invariant mass spectrum (left) and proton momentum
distribution (right) following $K^{-}$ absorption on ${}^{12}C$ in the KLOE
detector (LNF). Color codes show contributions to total absorption from
various processes involving two, three and four baryons. Adapted from Ref.
DelGrande:2019 .

In this context, a more detailed understanding of $\bar{K}$-nuclear systems
clearly requires as much information as possible about the multi-nucleon
absorption rates of antikaons through $\bar{K}NN\rightarrow
YN,~{}\bar{K}NNN\rightarrow YNN,...$ multi-nucleon processes, with hyperons
$Y=\Lambda,\Sigma$ in the final state. Experimental data have been reported in
Refs. VazquezDoce:2015szw ; HADES:2018qkj ; DelGrande:2019 . Figure 15 shows
the $\Lambda p$ spectrum following low-momentum $K^{-}$ absorption on
${}^{12}\rm{C}$ layers in the KLOE detector at LNF. These $\Lambda p$ spectra
also incorporate conversions from $\Sigma^{0}p$ final states. The hyperon-
nucleon pairs are produced in various multi-nucleon absorption processes that
are simulated to reconstruct the overall rate with an envelope extending over
the region $2.1$-$2.3$ GeV, covering as well the potential range of possible
bound “$K^{-}pp$” clusters. Also shown in Fig. 15 is the momentum spectrum of
protons emerging from the absorption reactions. One notes large proton recoil
momenta characteristic of two-body absorption kinematics, while three and more
particles in the final state can share the momentum more evenly between them.

Figure 16: Real and imaginary parts of the $K^{-}$-nucleus potential for the
${}^{208}{\rm Pb}+K^{-}$ system. The chiral $SU(3)$ dynamics based potential
Ikeda:2012au  has been used in combination with the phenomenological
absorptive potential of Eq. (24). with $\alpha=1$. Different contributions
from the $K^{-}N$ single-nucleon potential ($KN$, including subthreshold
kinematics) and the phenomenological multi-nucleon potential ($KNN$) are
displayed, together with the total potential including uncertainty estimates.
The chiral $K^{-}N$ potential (solid blue lines - in the absence of the multi-
nucleon potential) is also shown for comparison. Adapted from Ref.
Hrtankova:2017wrg .

The competition between binding and absorption is obviously the key to the
possible existence of quasibound $\bar{K}$-nuclear states. This issue has been
further elaborated Hrtankova:2017wrg  in an investigation that follows closely
the constraints implied by the systematics of pionic atoms discussed in the
previous section. The input is now again the single-nucleon chiral
$SU(3)$-based coupled-channel potential plus the complex phenomenological
multi-nucleon potential (24). The real and imaginary parts of the resulting
potential that could, in principle, support a quasibound ${}^{208}{\rm
Pb}+K^{-}$ system is shown in Fig. 16. While the one-body potential derived
from chiral $SU(3)$ dynamics would indeed be able to generate such a bound
state, the large imaginary part from multi-nucleon absorption, together with
the accompanying repulsive real part, both work against binding. The total
real part of the potential adds up to a central strength, Re $U_{K}\simeq-40$
MeV, whereas the imaginary part turns out to be about four times as large, Im
$U_{K}\simeq-160\pm 30$ MeV, though with considerable uncertainties.
Nonetheless, multinucleon absorption of antikaons is expected to reduce the
lifetime of possible $\bar{K}$-nuclear quasibound states to a low level that
will make such states difficult to observe. The same kind of absorptive
mechanisms would also work against kaon condensation in dense baryonic matter.

##  7 Summary

The interest in strangeness nuclear physics is in large part motivated by the
special role that the strange quark plays in low-energy QCD, with its mass
intermediate between light and heavy, featuring a subtle combination of
spontaneous and explicit chiral symmetry breaking. This symmetry breaking
pattern implies that the interaction of antikaons with nucleons close to
threshold is quite strongly attractive. As a consequence, the question has
been raised whether quasibound states of $\bar{K}$ with nucleons and nuclei
can exist. For the isospin $I=0$ channels of the $\bar{K}$-nucleon system,
this question found a convincing answer in the form of the $\Lambda(1405)$ as
a $\bar{K}N$ quasibound state embedded in the $\pi\Sigma$ continuum with a
nontrivial, quasimolecular intrinsic structure. Viewed in a simple constituent
quark picture, the $\Lambda(1405)$ can indeed be considered as historically
the first example of an “exotic” baryon with an underlying five-quark
($udu\bar{u}s$ and $udd\bar{d}s$) content.

The framework for the theoretical description of kaon- and antikaon-nuclear
systems close to threshold is an effective field theory (EFT) based on
spontaneously broken chiral $SU(3)_{L}\times SU(3)_{R}$ symmetry, as a
realization of low-energy QCD with $N_{f}=3$ quark flavors. Unlike the two-
flavor case, however, chiral perturbation theory is not an option to proceed
because of the significant explicit chiral symmetry breaking by the strange
quark mass and the appearance of the $\Lambda(1405)$ just below $\bar{K}N$
threshold. Instead, a combination of this EFT with non-perturbative coupled-
channel methods (chiral $SU(3)$ coupled-channel dynamics) proves to be
successful in comparison with all known empirical $\bar{K}N$ data, including
accurate measurements of the atomic energy shift and width of kaonic hydrogen
and the $K^{-}p$ correlation function from high-energy proton-proton
collisions. Important steps in the near future for further improvements
include measurements of kaonic deuterium that are expected to provide more
detailed information on the isospin $I=1$ part of the $\bar{K}N$ interaction
through the $K^{-}n$ channel.

The systematics of strong interaction energy shifts and widths in kaonic atoms
with heavier nuclei have always been an important source of constraints on the
$K^{-}$-nuclear optical potential. With the leading two-body $\bar{K}N$
interaction from chiral $SU(3)$ dynamics as a reliable input, the focus on
many-body effects such as $K^{-}$ absorption in kaonic atoms with heavier
nuclei can now be better controlled.

Given the strong attraction in the $\bar{K}N$ interaction that is manifest
already in its leading-order (Weinberg-Tomozawa) term, much scientific
intensity has been devoted to theoretical and experimental investigations of
possible bound $\bar{K}$-nuclear systems. Several calculations have been
addressed to the prototype $\bar{K}NN$ bound cluster and to light antikaonic
nuclei with up to six nucleons, using sophisticated few-body techniques. While
such calculations produce binding energies that increase with the number of
nucleons, ranging from 25 MeV for $\bar{K}NN$ to about 80 MeV for
$\bar{K}NNNNNN$, the widths of these states are suggested to be of similar
magnitudes but with large uncertainties, in particular because $\bar{K}$
multi-nucleon absorption processes are not taken into account. For heavier
antikaonic nuclei, such $\bar{K}$ absorption reactions, when constrained by
kaonic atom data and translated into a phenomenological potential with non-
linear density dependence, produce a large imaginary part of the potential
that competes with the binding energy. Thus the key issue in the physics of
antikaonic nuclei that remains to be further clarified is the balance between
nuclear $\bar{K}$ binding and absorption.

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