# The charge and mass symmetry breaking in the $KK\bar{K}$ system

I. Filikhin1, R. Ya. Kezerashvili2,3,4, and B. Vlahovic1 1North Carolina
Central University, Durham, NC, USA
2New York City College of Technology, The City University of New York,
Brooklyn, NY, USA
3The Graduate School and University Center, The City University of New York,
New York, NY, USA
4Long Island University, Brooklyn, NY, USA

###### Abstract

In the framework of the Faddeev equations in configuration space, we
investigate the $K$(1460) meson as a resonant state of the $KK\bar{K}$ kaonic
system. We perform calculations for the particle configurations
$K^{0}K^{+}K^{-}$ and $K^{0}K^{+}\overline{{K}^{0}}$ within two models: the
$ABC$ model, in which all three particles are distinguishable, and the $AAC$
model when two particles are identical. The models differ in their treatment
of the kaon mass difference and the attractive Coulomb force between the
$K^{+}K^{-}$ pair. We found that the Coulomb shift adds over 1 MeV to the
three-body binding energy. The expected correction to the binding energy due
to mass redistribution from $AA$ to $AB$ is found to be negligible, up to a
maximum of 6% of the relative mass correction. At the same time, the symmetry
of the wave function is distorted depending on the mass ratio value. We found
that the repulsive $KK$ interaction plays essential role in the binding energy
of the $KK\bar{K}$ system and report the mass of 1461.8 or 1464.1 MeV for the
neutral $K^{0}$(1460) and 1466.5 or 1468.8 MeV for the charged $K^{+}$(1460)
resonances, respectively, depending on the parameter sets for $KK$ and
$K\bar{K}$ interactions.

##  I Introduction

Few-body physics has received interest for decades. Since 1961, when the
Faddeev equations Fad  in the momentum representation were formulated and a
few years later the Faddeev-Noyes equations in configuration space were
suggested Noyes1968 , special attention has been given to three-body systems
constituted by nucleons, mesons, two nucleons and a meson, two mesons and a
nucleon, quarks, and three-particle cluster systems. At low energies, the
general approach for solving the three-body problem is based on the use of
methods for studying the dynamics of three particles in discrete and continuum
spectra. Among the most powerful approaches are the method of Faddeev
equations in momentum Fad ; Fad1  or coordinate Noyes1968 ; Noyes1969 ;
Gignoux1974 ; FM  spaces. However, the method of hyperspherical harmonics, the
variational method in the harmonic-oscillator basis, and the variational
method complemented with the use of explicitly correlated Gaussian basis
functions has been successfully employed for the solution of a few-body
problem in atomic, nuclear, high energy physics, and even in condensed matter
physics RKez2019 .

Three-body systems can be composed of three identical particles ($AAA$ model),
two identical and the third one ($AAC$ model), and three non-identical
particles ($ABC$ model). The $nnn$, $nns$, and $ssn$ baryons are examples of
the quantum system with three and two identical quarks, while 3$\alpha$
(cluster model for 12C), $ppn$(3He), and $nnp$(3H) nucleon systems are
composed of three and two identical particles FSV19 , respectively. One can
also have systems with a meson and two baryons or two mesons and a baryon,
such as kaonic clusters $NN{\bar{K}}$, ${\bar{K}\bar{K}}N$, $K{\bar{K}}N$, and
$KK{\bar{K}}$, which are considered as systems with two identical or three
non-identical particle, depending on the configuration of the particles or
theoretical approach for the description of kaonic clusters. When two
identical particles are fermions or bosons the wave functions of the three-
body kaonic clusters are antisymmetric or symmetric, correspondingly, with
respect to the two identical particles exchange.

The kaonic clusters $NN{\bar{K}}$, ${\bar{K}\bar{K}}N$, and $K{\bar{K}}N$ were
intensively studied in the framework of the Faddeev equations in momentum and
coordinate representation Ikeda2007A ; Shevchenko2007A ; Ikeda2007 ;
Shevchenko2007B ; Ikeda2009 ; Ikeda2010 ; Torres2010 ; 5 ; K2015 ; FV ; FVK20
. The Faddeev equations were used to study four-body kaonic clusters (see
review RKezNNNK ). It is a very challenging task to solve the Faddeev
equations exactly and usually introduce some reasonable approximations of the
Faddeev equations, such as the use of separable potentials, energy-independent
kernels, on-shell two-body scattering amplitudes, the Faddeev-type Alt-
Grassberger-Sandhas equations’ AGS . In the framework of the fixed-center
approximation for the Faddeev equations, dynamically generated three-body
resonances formed via meson-meson and meson-baryon interactions were studied
(see 13 ; 12 ; 922 ; Oset2020  and references herein).

The $K$(1460) pseudoscalar was a subject of interest since the middle of the
seventieths. In 1976, the first high statistics study of the
$K^{\pm}p\rightarrow K^{\pm}\pi^{+}\pi^{-}p$ process was carried out at SLAC,
using a 13 GeV incident $K^{\pm}$ beam Brandenburg . The $J^{\pi}=0^{-}$
partial-wave analysis of the $\pi\pi K$ system in this reaction led to the
report of the first evidence for a strangeness-one pseudoscalar meson with a
mass of $\sim$ 1400 MeV and a width of $\sim$ 250 MeV. A few years later the
ACCORD collaboration DAUM1981  using SLAC $K^{-}$ beam at 63 GeV investigated
the diffractive process $K^{-}p\rightarrow K^{-}\pi^{+}\pi^{-}p$ and the data
sufficient for partial-wave analysis extending up to a mass of 2.1 GeV were
collected. The data analysis confirmed the existence of a broad $0^{-}$
resonance with a mass of $\sim$ 1460 MeV. However, even the 2018 PDG PDG2018
did not list the $K$(1460) as an ”established particle”. In the most recent
studies of the resonance structure in $D^{0}\rightarrow
K^{\mp}\pi^{\pm}\pi^{\pm}\pi^{\mp}$ decays using $pp$ collision, data
collected at 7 and 8 TeV with the LHCb experiment LHCb , and the precise
measurements of the $a_{1}^{+}$(1260), $K_{1}^{-}$(1270) and $K$(1460)
resonances are made. Within a model-independent partial-wave analysis
performed for the $K$(1460) resonance, it is found that the mass is roughly
consistent with previous studies Brandenburg ; DAUM1981 . They showed the
evidence for the K(1460) in ${\bar{K}}^{\ast}(892)\pi^{-}$ and
$[\pi^{+}\pi^{-}]^{L=0}K^{-}$ channels and confirmed the resonant nature of
the $K$(1460) with the mass 1482.40$\pm$3.58$\pm$15.22 MeV and width
335.60$\pm$6.20$\pm$8.65 MeV. Thus, the $K$(1460) is listed in the 2020 PDG
PDG2020  and in the most recent PDG PDG2022 .

Several theoretical models Izgur85 ; Longacre90 ; Albaladejo ; Torres2011 ;
RKSh.Ts2014 ; KezerasVail2015 ; Parganlija2017 ; SHY19 ; KezerasPRD2020 ;
Torrez2021 ; Meisner2022  considered the dynamic generation of the
pseudoscalar $K$(1460) resonance as three $K$ mesons: $KK{\bar{K}}$. The first
time the resonance $K$(1460) was considered as a $2^{1}S_{0}$ excitation of
the kaon in a unified quark model in Ref. Izgur85  and the mass 1450 MeV for
this resonance was obtained. In Ref. Longacre90  within the isobar assumption
shown that the final-state interaction leads to the formation of an exotic
($I(J^{\pi})=\frac{3}{2}(0^{-})$) $K^{+}K^{+}\overline{K^{0}}$ threshold
enhancement with a mass around 1500 MeV and a width of approximately 200 MeV.
Assuming isospin symmetry in the effective kaon-kaon interactions that are
attractive for the $K{\bar{K}}$ pair and repulsive for the $KK$ pair, in Ref.
Torres2011  the $K$(1460) pseudoscalar resonance was studied as the
$KK{\bar{K}}$ system. The authors have performed the investigation of the
$KK{\bar{K}}$ system using two methods: the single-channel variational
approach in the framework of the model KanadaJido ; JidoKanada , and the
Faddeev equations in momentum representation. In the latter case, the authors
determined the two-body on-shell $t$ matrices for $KK$ and $K{\bar{K}}$
interactions using the Bethe-Salpeter equation in a couple-channel approach
and considered the on-shell factorization method. Dynamic generation of the
pseudoscalar $K$(1460) resonance was considered in Ref. Albaladejo , by
studying interactions between the $f_{0}(980)$ and $a_{0}(980)$ scalar
resonances and the lightest pseudoscalar mesons. In Torres2011  have included
$\pi\pi K$ and $\pi\eta K$ channels. The same channels have been considered in
an effective way in Ref. Albaladejo . In Ref. RKSh.Ts2014  using the single-
channel description of the $KK{\bar{K}}$ system in the framework of the
hyperspherical harmonics method was obtained a reasonable agreement with
experimental data for the mass of the $K$(1460) resonance. Authors
Parganlija2017 , based on the extended version of the linear sigma model
GellMann60 ; Gasiorowicz69 ; Ko94 , discussed to what extent it is possible to
interpret the pseudoscalar meson $K(1460)$ ($I(J^{\pi})=\frac{1}{2}(0^{-})$)
as excited $\bar{q}q$ states. In a framework of the Faddeev equations in
configuration space using the $AAC$ model $KK{\bar{K}}$ system was
investigated in Ref. KezerasPRD2020 . In Ref. SHY19  the $KK{\bar{K}}$ system
was considered based on the coupled-channel complex-scaling method by
introducing three channels $KK{\bar{K}}$, $\pi\pi K$, and $\pi\eta K$. The
resonance energy and width were determined using two-body potentials that fit
two-body scattering properties. The model potentials having the form of one-
range Gaussians were proposed based on the experimental information related to
$a_{0}$ and $f_{0}$ resonances. In this model, the $K\bar{K}$ interaction
depends on the pair isospin. In particular, the isospin triplet
$K{\bar{K}}(I=1)$ interaction is essentially weaker than the isospin singlet
$K{\bar{K}}-K{\bar{K}}$ interaction in the channel $\pi K-K{\bar{K}}(I=0)$.
The most recently in Ref. Meisner2022  the $KK{\bar{K}}$ three-body system in
$\frac{1}{2}(0^{-})$ channel is studied in a formalism that satisfies two-body
and three-body unitarity using the isobar approach, where the two-body
$K{\bar{K}}$ interaction is parametrised via the $f_{0}(980)$ and $a_{0}(980)$
poles.

There are the following physical particle configurations of the $KK{\bar{K}}$
system: $K^{0}K^{0}{\bar{K}}$, $K^{0}K^{+}K^{-}$,
$K^{0}K^{+}\overline{{K}^{0}}$, $K^{+}K^{+}{\bar{K}}$. In these
configurations, there are particles and antiparticles, that have different
masses, isospins, and electric charges. The $K^{0}K^{0}\overline{{K}^{0}}$ and
$K^{+}K^{+}{\bar{K}}$ configurations can be considered in the framework of
Faddeev equations as an $AAC$ model with two identical particles. The
configurations $K^{0}K^{+}K^{-}$ and $K^{0}K^{+}\overline{{K}^{0}}$ present
the system with three non-identical particles due to the different masses and
isospins of mesons. The latter two configurations must be considered within
the Faddeev formalism as the $ABC$ system. Theoretical approaches for these
systems utilizing average masses of particles can lead to the loss of
important information about the nature of the systems. In particular, the
formalization of the kaons as identical particles allows us to discuss an
exchange effect as was discussed for the $NN{\bar{K}}$ system YA07 .

We focus on the particle configurations $K^{0}K^{+}K^{-}$ and
$K^{0}K^{+}\overline{{K}^{0}}$ for the bound three-body $KK{\bar{K}}$ system.
Previously, the kaonic $KK{\bar{K}}$ resonance is considered within the $AAC$
model using average kaon masses and disregarding the Coulomb force.
Considering the Coulomb potential or difference of the masses of kaons, we
propose the $ABC$ model for the description of the $K^{0}K^{+}K^{-}$ and
$K^{0}K^{+}\overline{{K}^{0}}$ particle configurations rigorously within the
Faddeev equation in configuration space. One of the objectives of this paper
is to compare the binding energies of the $KK{\bar{K}}$ system obtained within
the $AAC$ and $ABC$ models using the same interaction potentials. This allows
us to draw a conclusion on the $AAC$ model’s validity. We show that the proper
averaging procedure within the $ABC$ model leads to the $AAC$ model with a
negligible correction of the binding energy - up to a maximum of 6% of the
relative mass correction. For the higher relative mass correction, the $AAC$
model is unacceptable, and one must use the $ABC$ model.

In this paper we focus on the single-channel description of the $KK{\bar{K}}$
system in the absence of the $\pi\pi K$ and $\pi\eta K$ inelastic channels. It
is worth noticing that despite its simplicity, the single-channel model can
reproduce the mass of the $K(1460)$ resonance. The smallness of the kinetic
energy of the kaons in a bound system in comparison with the kaon mass, makes
possible the study of the single channel $KK{\bar{K}}$ using a non-
relativistic potential model. While the single-channel and non-relativistic
potential model description clearly do not fully represent reality, it still
allows us to address the issues related to the charge and mass symmetry
breaking in the $KK\bar{K}$ system and the validity of the $AAC$ model. In the
$AAC$ and $ABC$ models, we are using $s$-wave $K{\bar{K}}$ and $KK$ two-body
potentials Torres2011  and in the $ABC$ model experimental kaon masses, as the
inputs.

This paper is organized as follows. In Secs. II and III we present the Faddeev
equations in configuration space for $ABC$ and $AAC$ models and their
application for the $K^{0}K^{+}K^{-}$ and ${K}^{0}K^{+}\overline{K^{0}}$
configurations. Averaged mass and potential approaches which lead to the
reduction of the $ABC$ model to the $AAC$ one are given in Secs. IV and V,
respectively. Results of numerical calculations for $AAC$ and $ABC$ models are
presented in Sec. VI. The concluding remarks follow in Sec. VII.

##  II Faddeev equations in configuration space

The three-body problem can be solved in the framework of the Schrödinger
equation or using the Faddeev approach in the momentum Fad ; Fad1  or
configuration Noyes1968 ; FM ; K86  spaces. The Faddeev equations in the
configuration space have different forms depending on the type of particles
and can be written for: i. three non-identical particles; ii. three particles
when two are identical; iii. three identical particles. The identical
particles have the same masses and quantum numbers. In the particle
configurations $K^{0}K^{+}K^{-}$ and $K^{0}K^{+}\overline{K^{0}}$ the $K^{+}$
and $K^{-}$, and $K^{0}$ and $\overline{K^{0}}$ have equal masses,
respectively. However, these particle configurations cannot be considered in
the framework of the $AAC\ $model because $K^{-}$ and $\overline{K^{0}}$ are
antiparticles of $K^{+}$ and $K^{0}$, respectively, hence, are non-identical
particles. Moreover, in the $K^{0}K^{+}K^{-}$ particle configuration the
$K^{+}$ and $K^{-}$ are a particle and antiparticle, respectively, which have
the same masses but different charges. The non-identical particles are un-
exchangeable bosons. Thus, the $K^{0}K^{+}K^{-}$ and
$K^{0}K^{+}\overline{K^{0}}$ particle configurations each consist of three
non-identical particles and must be treated within the ABC model.

###  II.1 Faddeev equations for the ABC model

In the Faddeev method in configuration space, alternatively, to the finding
the wave function of the three-body system using the Schrödinger equation, the
total wave function is decomposed into three components Noyes1968 ; FM ; K86 :

$\Psi(\mathbf{x}_{1},\mathbf{y}_{1})=U(\mathbf{x}_{1},\mathbf{y}_{1})+W(\mathbf{x}_{2},\mathbf{y}_{2})+Y(\mathbf{x}_{3},\mathbf{y}_{3}).$
(1)

Each Faddeev component corresponds to a separation of particles into
configurations $(kl)+i$, $i\neq k\neq l=1,2,3$. The Faddeev components are
related to their own set of the Jacobi coordinates $\mathbf{x}_{i}$ and
$\mathbf{y}_{i}$, $i=1,2,3$. There are three sets of Jacobi coordinates. The
total wave function is presented by the coordinates of one of the sets shown
in Eq. (1) for $i=1$. The mass-scaled Jacobi coordinates $\mathbf{x}_{i}$ and
$\mathbf{y}_{i}$ are expressed via particle coordinates $\mathbf{r}_{i}$ and
masses $m_{i}$ in the following form:

$\mathbf{x}_{i}=\sqrt{\frac{2m_{k}m_{l}}{m_{k}+m_{l}}}(\mathbf{r}_{k}-\mathbf{r}_{l}),\qquad\mathbf{y}_{i}=\sqrt{\frac{2m_{i}(m_{k}+m_{l})}{m_{i}+m_{k}+m_{l}}}(\mathbf{r}_{i}-\frac{m_{k}\mathbf{r}_{k}+m_{l}\mathbf{r}_{l})}{m_{k}+m_{l}}).$
(2)

In Eq. (1), each component depends on the corresponding coordinate set which
is expressed in terms of the chosen set of mass-scaled Jacobi coordinates. The
orthogonal transformation between three different sets of the Jacobi
coordinates has the form:

$\left(\begin{array}[]{c}\mathbf{x}_{i}\\\
\mathbf{y}_{i}\end{array}\right)=\left(\begin{array}[]{cc}C_{ik}&S_{ik}\\\
-S_{ik}&C_{ik}\end{array}\right)\left(\begin{array}[]{c}\mathbf{x}_{k}\\\
\mathbf{y}_{k}\end{array}\right),\ \ C_{ik}^{2}+S_{ik}^{2}=1,$ (3)

where

$C_{ik}=-\sqrt{\frac{m_{i}m_{k}}{(M-m_{i})(M-m_{k})}},\quad
S_{ik}=(-1)^{k-i}\mathrm{sign}(k-i)\sqrt{1-C_{ik}^{2}}.$

Here, $M$ is the total mass of the system. The components
$U(\mathbf{x_{1}},\mathbf{y_{1}})$, $W(\mathbf{x}_{2},\mathbf{y}_{2})$, and
$Y(\mathbf{\ x}_{3},\mathbf{y}_{3})$ satisfy the Faddeev equations in the
coordinate representation written in the form FM :

$\begin{array}[]{l}(H_{0}+V_{23}(\mathbf{x_{1}})-E)U(\mathbf{x},\mathbf{y})=-V_{23}(\mathbf{x_{1}})(W(\mathbf{x}_{2},\mathbf{y}_{2})+Y(\mathbf{\
x}_{3},\mathbf{y}_{3}),\\\
(H_{0}+V_{13}(\mathbf{x_{2}})-E)W(\mathbf{x},\mathbf{y})=-V_{13}(\mathbf{x_{2}})(U(\mathbf{x}_{1},\mathbf{y}_{1})+Y(\mathbf{\
x}_{3},\mathbf{y}_{3}),\\\
(H_{0}+v_{12}(\mathbf{x_{3}})-E)Y(\mathbf{x},\mathbf{y})=-V_{12}(\mathbf{x_{3}})(U(\mathbf{x}_{1},\mathbf{y}_{1})+W(\mathbf{\
x}_{2},\mathbf{y}_{2}).\end{array}$ (4)

Here, $H_{0}=-(\Delta_{\mathbf{x}}+\Delta_{\mathbf{y}})$ is the kinetic energy
operator with $\hbar^{2}=1$ and $V_{kl}$ is the interaction potential between
the pair of particles $(kl)$, $i\neq k\neq l$. In the system of equations (4),
the independent variables are $\mathbf{x}$ and $\mathbf{y}$ and can be chosen
to be $\mathbf{x_{i}}$ and $\mathbf{y_{i}}$, where $i$ is $1$ or $2$, or $3$.
After that, the remaining coordinates are expressed by the chosen coordinates
according to the Jacobi coordinates transformation (3).

###  II.2 Faddeev equations for the AAC model

The system of Eqs. (4) can be reduced to a simpler form for a case of two
identical particles when a particle $B$ in the $ABC$ model is replaced by a
particle $A$. In this case, for the Bose particles, the total wave function of
the system is decomposed into the sum of the Faddeev components $U$ and $W$
corresponding to the $(AA)C$ and $(AC)A$ types of rearrangements:

$\Psi=U+W+PW,$

where $P$ is the permutation operator for two identical particles.
Consequently, Eqs. (4) can be rewritten as follows K86 :

$\begin{array}[]{l}{(H_{0}+V_{AA}-E)U=-V_{AA}(W+PW)},\\\
{(H_{0}+V_{AC}-E)W=-V_{AC}(U+PW)}.\end{array}$ (5)

In Eqs. (5) ${V_{AA}}$ and ${V_{AC}}$ are the interaction potentials between
identical and non-identical particles, respectively.

##  III  The $ABC$ model versus $AAC$ for $K^{0}K^{+}K^{-}$ and
${K}^{0}K^{+}\overline{K^{0}}$ particle configurations

We study the $KK{\bar{K}}$ system using the available $s$-wave effective
phenomenological $KK$ and $K{\bar{K}}$ potentials Torres2011  and considering
the $K$ and ${\bar{K}}$ kaon’s experimental masses and charges based on Ref.
PDG2022 . This leads to the consideration of the $K(1460)$ resonance according
to the following neutral or charged particle configurations:
$K^{0}K^{0}{\bar{K}}$, $K^{0}K^{+}K^{-}$, $K^{+}K^{+}{\bar{K}}$,
$K^{0}K^{+}\overline{K^{0}}$. For the description of the $KK{\bar{K}}$ system
within the $ABC$ and $AAC$ models, we focus, as on the representative, on the
following two particle configurations: $K^{0}K^{+}K^{-}$ and
$K^{0}K^{+}{\bar{K}}^{0}$. The configuration $K^{0}K^{+}K^{-}$ includes the
Coulomb interaction and $K^{+}$ and $K^{-}$ have the same masses but are non-
identical that make three particles distinguishable. The configuration
$K^{0}K^{+}\overline{{K}^{0}}$ does not include the Coulomb interaction and
two particles $K^{0}$ and $\overline{K^{0}}$ have the same masses but are non-
identical. Therefore, the exact treatment of the $K^{0}K^{+}K^{-}$ and
$K^{0}K^{+}\overline{K^{0}}$ particle configurations must be done within the
$ABC$ model. We consider $s$-wave pair potentials. Hence, the bound state
problem for the $KK{\bar{K}}$ system should be formulated using the Faddeev
equations in the $s$-wave approach K2015 . In the $s$-wave approach for the
$K^{0}K^{+}K^{-}$ particle configuration Eqs. (4) reads:

$\begin{array}[]{l}(H_{0}+v_{K^{0}K^{+}}+v_{C}^{\mathcal{U}}-E)\mathcal{U}=-v_{K^{0}K^{+}}(\mathcal{W}+\mathcal{Y}),\\\
(H_{0}+v_{K^{+}K^{-}}+v_{C}^{\mathcal{W}}-E)\mathcal{W}=-v_{K^{+}K^{-}}(\mathcal{U}+\mathcal{Y}),\\\
(H_{0}+v_{K^{0}K^{-}}+v_{C}^{\mathcal{Y}}-E)\mathcal{Y}=-v_{K^{0}K^{-}}(\mathcal{U}+\mathcal{W}).\end{array}$
(6)

In Eqs. (6) $v_{K^{0}K^{+}}$, $v_{K^{+}K^{-}}$ and $v_{K^{0}K^{-}}$ are the
$s$-wave $KK$ and $K{\bar{K}}$ potentials, while $v_{C}^{U}$,
$v_{C}^{\mathcal{W}}$ and $v_{C}^{\mathcal{Y}}$ are the components of the
Coulomb potential related to the $K^{+}K^{-}$ electrostatic interaction
depending on the mass-scaled Jacobi coordinate of the corresponding Faddeev
components $\mathcal{U}$, $\mathcal{W}$, and $\mathcal{Y}$. A consideration of
the Coulomb interaction in the framework of the Faddeev formalism is a
challenging problem Mer80 . Following FSV19 , we consider the Coulomb
$K^{+}K^{-}$ interaction included on the left-hand side of the Faddeev
equations (6) as a perturbation.

The Coulomb attraction in the $K^{0}K^{+}K^{-}$ violates the $AAC$ symmetry
and makes three kaons distinguishable. If one neglects the Coulomb attraction
in the $K^{0}K^{+}K^{-}$ in Eqs. (6), even the equality of $K^{+}$ and $K^{-}$
masses does not allow to treat the $K^{0}K^{+}K^{-}$ configuration in the
framework of the $AAC$ model. However, if one considers $K^{0}$ and $K^{+}$ as
identical particles with masses equal to the average of their masses and
neglects the Coulomb attraction, $K^{0}K^{+}K^{-}$ can be considered within
the $AAC$ model. A schematic for the $K^{0}K^{+}K^{-}$ is presented in Fig. 1
when it is treated as $ABC$ and $AAC$ models. Within the $AAC$ model the
Faddeev equations (5) for two identical particles in the three-body
$K^{0}K^{+}K^{-}$ system for the $s$-wave interparticle interactions can be
written in the following form:

$\begin{array}[]{l}(H_{0}+v_{K^{0}K^{+}}-E)\mathcal{U}=-v_{K^{0}K^{+}}(1+P)\mathcal{W}\;,\\\
(H_{0}+v_{K^{+}K^{-}}-E)\mathcal{W}=-v_{K^{+}K^{-}}(U+P\mathcal{W})\;.\end{array}$
(7)

In Eqs. (7) the $\mathcal{U}$ and $\mathcal{W}$ are the $s$-wave Faddeev
components of the wave function, and the exchange operator $P$ acts on the
particles’ coordinates only.

The $K^{0}K^{+}\overline{K^{0}}$ particle configuration also can be considered
within the $AAC$ model, if one considers $K^{0}$ and $K^{+}$ as identical
particles with masses equal to the average of their masses. In this case, Eqs.
(7) describe $K^{0}K^{+}\overline{K^{0}}$ configuration with only difference
that $v_{K^{+}K^{-}}$ interaction should be replaced by the
$v_{K^{0}\overline{K^{0}}}$ potential.

Figure 1:  The schematic represents the reduction of the $ABC$ model ($a$) to
$AAC$ model ($b$). The kaons and antikaon of the $K^{0}K^{+}K^{-}$ particle
configuration are shown in blue and red colors, respectively, together with
the experimental masses. The kaon pair with the Coulomb interaction is
encircled by the oval. The crosses indicate the position of the middle point
between $A$ and $B$ particles and two $A$ particles in the $ABC$ and $AAC$
models, respectively. The Jacobi coordinates related to configurations $(AB)C$
and $(AA)C$ are shown by arrows.

##  IV The mass difference of kaons: from the $AAC$ to $ABC$ model

In the previous studies $KK{\bar{K}}$ system was considered within the $AAC$
model. Such consideration is valid if we ignore the difference between $K^{+}$
and $K^{0}$ masses and the Coulomb interaction between charged kaons. As the
first step for a realistic consideration, we are using the experimental kaon
masses instead of the average kaon mass used in the $AAC$ model. After that,
we consider that the $AB$ pair is the kaonic pair and antikaon is considered
as the particle $C$. Within this $ABC$ model, the masses of $A$ and $B$ kaons
are varied around the average value of the kaon mass ${m}=(m_{A}+m_{B})/2$.
These variations have different signs for the $A$ and $B$ kaons. In other
words, we consider the $ABC$ model with variable masses of the $A$ and $B$
kaons but keeping the sum of masses of the $AB$ kaon pair constant. This
approach allows us to understand how the binding energy of the $ABC$ is
sensitive to the variation of the $A$ and $B$ masses.

Consider the particles in the $ABC$ model, where $C$ is the antiparticle, have
masses $m_{A}$, $m_{C}$, and $m_{B}$, and can be numbered manually as

$\begin{array}[]{l}m_{1}=(m_{A}+m_{B})/2+m_{A}/2-m_{B}/2=m+\Delta m,\\\
m_{2}=(m_{A}+m_{B})/2+m_{B}/2-m_{A}/2=m-\Delta m,\\\ m_{3}=m_{C},\end{array}$
(8)

where $\Delta m=(m_{A}-m_{B})/2$ is small and kaons are particles 1 and 2 and
the antikaon is particle 3. Following Friar at al. F90 , we write the kinetic
energy operator $\hat{H_{0}}$ in terms of the individual momenta of the
particles in the center-of-mass as:

$\hat{H_{0}}=\sum_{i=1,2,3}\frac{\pi_{i}^{2}}{2m_{i}}\approx\frac{\pi_{3}^{2}}{2m_{3}}+\frac{\pi_{1}^{2}}{2m}+\frac{\pi_{2}^{2}}{2m}-\frac{\Delta
m}{m}\frac{\pi_{1}^{2}}{2m}+\frac{\Delta m}{m}\frac{\pi_{2}^{2}}{2m}.$ (9)

This expression follows from the Taylor series for the $1/m$. In the first
order perturbation theory, the correction for the energy can be presented as
$\langle{\hat{H_{0}}}\rangle\approx\langle\hat{H_{0}}^{\Delta
m=0}\rangle+\langle\Delta H_{0}\rangle$, where $\langle\hat{H_{0}}^{\Delta
m=0}\rangle=\langle\Psi^{R}|\frac{\pi_{3}^{2}}{2m_{3}}+\frac{\pi_{1}^{2}}{2m}+\frac{\pi_{2}^{2}}{2m}|\Psi^{R}\rangle$
and

$\langle\Delta H_{0}\rangle=\langle\Psi^{R}|\frac{\Delta
m}{m}\frac{\pi_{2}^{2}}{2m}-\frac{\Delta
m}{m}\frac{\pi_{1}^{2}}{2m}|\Psi^{R}\rangle.$ (10)

Here, the $\Psi^{R}$ is the coordinate part of the wave function
$\Psi=\eta_{isospin}\otimes\Psi^{R}$.

Within the $AAC$ model, $\langle\Delta H_{0}\rangle=0$, due to $\Delta m=0$.
In the $ABC$ model, this relation is approximately satisfied

$\langle\Delta H_{0}\rangle\approx 0.$ (11)

The possible value for the $\Delta m$ is restricted so that $|\Delta m/m|\ll
1$. The linear approximation Eq. (9) is applicable when we consider only the
first two terms of the Taylor series for the function $1/{m}$ near the point
$m=1$. The next quadratic terms of the expansion cannot be compensated
similarly to Eq. (11) due to the alternating series of the Tailor expansion:
$m/m_{i}=\sum_{n}(-1)^{n}(\Delta m/m)^{n}$, $i=1,2$.

Therefore, we can assume that the symmetrical variation of kaons’ masses
described by Eq. (8) for the $ABC$ model does not lead to a significant change
in the $AAC$ binding energy when $|\Delta m/m|\ll 1$. This assumption follows
from the compencation effect for the three-body Hamiltonian expressed by Eqs.
(10)-(11).

##  V Coulomb interaction: from the $ABC$ to $AAC$ model

Let us ignore the repulsive potential acting between the $K^{0}$ and $K^{+}$
kaons. Such truncation shifts the three-body energy to a fixed value. After
that the equation for the Faddeev component $\mathcal{U}$ in Eqs. (6) is
eliminated. Also, we can neglect the Coulomb interaction and consider only the
nuclear interaction between kaons and antikaon. The corresponding system of
equations reads:

$\begin{array}[]{l}(H_{0}+v_{2}-E)\mathcal{W}=-v_{2}\mathcal{Y},\\\
(H_{0}+v_{3}-E)\mathcal{Y}=-v_{3}\mathcal{W},\end{array}$ (12)

where, for simplicity, we denoted $v_{K^{+}K^{-}}=v_{2}$ and
$v_{K^{0}K^{-}}=v_{3}$. Assuming that the difference $\Delta
v=|(v_{3}-v_{2})/2|$ of the potentials $v_{2}$ and $v_{3}$ is small one can
introduce the average potential

$\overline{v}=\frac{(v_{2}+v_{3})}{2},$ (13)

so that

$v_{2}=\frac{(v_{2}+v_{3})}{2}+\frac{(v_{2}-v_{3})}{2},\quad
v_{3}=\frac{(v_{2}+v_{3})}{2}+\frac{(v_{3}-v_{2})}{2},$ (14)

and rewrite Eqs. (12) in the form

$\begin{array}[]{l}(H_{0}+\overline{v}+\Delta
v-E)\mathcal{W}=-v_{2}\mathcal{Y},\\\ (H_{0}+\overline{v}-\Delta
v-E)\mathcal{Y}=-v_{3}\mathcal{W}.\end{array}$ (15)

In the first order of the perturbation theory, by averaging Eqs. (15), one
obtains

$\begin{array}[]{l}\langle\mathcal{W}_{0}|(H_{0}+\overline{v}+\Delta
v-E)|\mathcal{W}_{0}\rangle=-\langle\mathcal{W}_{0}|(\overline{v}+\Delta
v)|\mathcal{Y}_{0}\rangle,\\\
\langle\mathcal{Y}_{0}|(H_{0}+\overline{v}-\Delta
v-E)|\mathcal{Y}_{0}\rangle=-\langle\mathcal{Y}_{0}|(\overline{v}-\Delta
v)|\mathcal{W}_{0}\rangle,\end{array}$ (16)

where $\mathcal{W}_{0}$ and $\mathcal{Y}_{0}$ are the solutions of Eqs. (15)
with the potential $\overline{v}$, when $\Delta v$ is omitted, that gives the
energy of $E_{0}$. Therefore, from (16) we obtain

$\begin{array}[]{l}\langle\mathcal{W}_{0}|E_{0}+\Delta
v-E)|\mathcal{W}_{0}\rangle=-\langle\mathcal{W}_{0}|\Delta
v|\mathcal{Y}_{0}\rangle,\\\ \langle\mathcal{Y}_{0}|(E_{0}-\Delta
v-E)|\mathcal{Y}_{0}\rangle=-\langle\mathcal{Y}_{0}|(-\Delta
v)|\mathcal{W}_{0}\rangle.\end{array}$ (17)

For equal kaon masses the functions $\mathcal{W}_{0}$ and $\mathcal{Y}_{0}$
are the same and, therefore, $\langle\mathcal{W}_{0}|\Delta
v|\mathcal{W}_{0}\rangle=\langle\mathcal{Y}_{0}|\Delta
v|\mathcal{Y}_{0}\rangle$ and $\langle\mathcal{W}_{0}|\Delta
v|\mathcal{Y}_{0}\rangle=\langle\mathcal{Y}_{0}|\Delta
v|\mathcal{W}_{0}\rangle$. By adding the algebraic equations in (17), we
obtain $E=E_{0}$. In other words, in the first order of perturbation theory,
the average potential gives $E=E_{0}$.

Now let’s assume that masses of $K^{0}$ and $K^{+}$ are equal to their average
mass. The Coulomb potential is a perturbation with respect to the strong kaon-
kaon interaction. The Coulomb potential can be treated within the given above
scheme. One can denote the Coulomb part of the $K^{+}K^{-}$ potential as
$v_{C}^{\mathcal{W}}$. The $v_{C}^{\mathcal{W}}$ is proportional to the
$\frac{1}{x}$ and $v_{C}^{\mathcal{Y}}$ ($v_{C}^{\mathcal{U}}$) is
proportional to the $\frac{1}{x^{\prime\prime}}$ ($\frac{1}{x^{\prime}}$).
Here, the mass-scaled Jacobi coordinate $x^{\prime}=|\mathbf{x_{1}}|$
corresponds to the $\mathcal{U}$ channel and is expressed by coordinates
$x=|\mathbf{x_{2}}|$ and $y=|\mathbf{y_{2}}|$ of the channel $\mathcal{W}$.
The $x^{\prime\prime}=|\mathbf{x_{3}}|$ is the coordinate in the channel
$\mathcal{Y}$ conjugated to the $\mathcal{W}$ channel and is expressed via
coordinates $x$ and $y$ of the channel $\mathcal{W}$ (see Eq. (4)).

The average potential is
$\overline{v}_{C}=(v_{C}^{\mathcal{W}}+v_{C}^{\mathcal{Y}})/2$. The $\Delta
v_{C}=(v_{C}^{\mathcal{W}}-v_{C}^{\mathcal{Y}})/2$ defines the difference
between the channels’ potentials $v_{C}^{\mathcal{W}}$ and
$v_{C}^{\mathcal{Y}}$. One can repeat the procedure presented by Eqs.
(16)-(17) and finally for the $K^{0}K^{+}K^{-}$ with the Coulomb interaction
obtains the following equation that corresponds to the $AAC$ model:

$\begin{array}[]{l}(H_{0}+v_{KK}+\overline{v}_{C}^{\mathcal{U}}-E)\mathcal{U}=-v_{KK}(1+P)\mathcal{W},\\\
(H_{0}+v_{KK^{-}}+\overline{v}_{C}^{\mathcal{W}}-E)\mathcal{W}=-v_{KK^{-}}(U+P\mathcal{W}),\end{array}$
(18)

where

$\begin{array}[]{l}v_{KK}=v_{K^{0}K^{+}},\quad\overline{v}_{C}^{\mathcal{U}}=-n/x^{\prime},\\\
v_{KK^{-}}=v_{K^{0}K^{-}},\quad\overline{v}_{C}^{\mathcal{W}}=-n(1/x^{\prime\prime}+1/x)/2,\end{array}$
(19)

and the masses of the kaons are equal to their average mass. The $n$ is the
Coulomb charge parameter.

##  VI Numerical Results

###  VI.1 Interaction potentials

In the present work, we use the $s$-wave effective potentials for ${K}{K}$ and
$K{\bar{K}}$ interactions from Ref. Torres2011 . The $K{\bar{K}}$ is an
attractive interaction that makes the $KK{\bar{K}}$ system bound, while the
$KK$ interaction is described by a repulsive potential. The $K{\bar{K}}$ and
$KK$ potentials are written in the form of one range Gaussian:

$v_{{\bar{K}}K}(r)=v_{0}exp((-r/b)^{2}),$ (20)

where $v_{0}$ and $b$ are the strength (depth) and the range of the potential,
respectively. In calculations are used two model potentials A and B with the
set of parameters listed in Table 1.

Table 1: The parameter sets A and B for $KK$ and $K\bar{K}$ interactions.
Parameters of potential
---
| Set A ($b=0.66$ fm) | Set B ($b=0.47$ fm)
Interaction |  $v_{0}$ ($I=0$), MeV |  $v_{0}$ ($I=1$), MeV |  $v_{0}$ ($I=0$), MeV |  $v_{0}$ ($I=1$), MeV
$K\overline{K}$ | $-630.0-210i$ | $-630.0-210i$ | $-1155.0-283i$ | $-1155.0-283i$
$KK$ | 0 | 104 | 0 | 313

###  VI.2 From the $AAC$ to $ABC$ model: Effects of kaons mass difference and
Coulomb force

Within the theoretical formalism presented in the previous sections, we
calculate the binding energies $E_{3}$ of the $K^{0}K^{+}K^{-}$ and
${K}^{0}K^{+}\overline{K^{0}}$ and $E_{2}^{K{\bar{K}}}$ of the bound
$K{\bar{K}}$ pairs. The three-particle energy $E_{3}(V_{KK}=0)$, due to the
kaon-antikaon interactions, but with the omitted interaction between two
kaons, is another significant characteristic of the three-particle kaonic
system. An analysis of the $E_{3}(V_{KK}=0)$ shows that the repulsive $KK$
interaction plays essential role in the resonance energy.

The results of calculations of these energies for the $K^{0}K^{+}K^{-}$ and
${K}^{0}K^{+}\overline{K^{0}}$ particle configurations in the $AAC$ and $ABC$
models are presented in Table 2.

The analysis of the results leads to the following conclusions:

i. the binding energies $E_{3}$ and $E_{3}(V_{KK}=0)$ of $K^{0}K^{+}K^{-}$ and
${K}^{0}K^{+}\overline{K^{0}}$ calculated in the $AAC$ model, with the average
kaon masses $\overline{m}_{K}=495.7$ MeV, and $ABC$ model are the same. Thus,
the difference of the kaon masses does not affect $E_{3}$ and
$E_{3}(V_{KK}=0)$: the mass distinguishability is not important for $E_{3}$
and $E_{3}(V_{KK}=0)$ energies when the Coulomb interaction is neglected. The
difference of 2.5 MeV for $E_{3}$ and 2.3 MeV for $E_{3}(V_{KK}=0)$ is related
to the parameter sets A and B for $KK$ and $K{\bar{K}}$ interactions,
respectively;

ii. the mass distinguishability has a small effect on the binding energy of
$K{\bar{K}}$ pairs;

iii. the consideration of the Coulomb interaction in the framework of the
$ABC$ model leads to an increase of the binding energy $E_{3}$ and the energy
$E_{2}^{K^{+}K^{-}}$ of the bound $K^{+}K^{-}$ pair in the $K^{0}K^{+}K^{-}$
particle configuration for the parameter sets A and B for $KK$ and
$K{\bar{K}}$ interactions, respectively;

iv. the repulsive $KK$ interaction plays an essential role in the binding
energy of the $KK\bar{K}$ system. The comparison of the $E_{3}$ and
$E_{3}(V_{KK}=0)$ of $K^{0}K^{+}K^{-}$ and ${K}^{0}K^{+}\overline{K^{0}}$
energies shows that contribution of the repulsive $KK$ interaction decreases
the three-particle binding energy by about $38\%$ and $25\%$ for the set of
parameters A and B, respectively;

v. finally, the mass of neutral resonance $K(1460)$ in the $ABC$
($K^{0}K^{+}K^{-}$) model is 1464.1 Mev and 1461.8 MeV for the parameter sets
A and B for $KK$ and $K{\bar{K}}$ interactions, respectively. The mass of the
charged resonance $K^{+}(1460)$ in the $ABC$ (${K}^{0}K^{+}\overline{K^{0}}$)
model is 1468.8 MeV and 1466.5 MeV for the parameter sets A and B for $KK$ and
$K{\bar{K}}$ interactions, respectively. Our results for the mass of neutral
and charged $K$(1460) resonance obtained within the $ABC$ model are in
reasonable agreement with the reported experimental value of the $K$(1460)
mass PDG2022 .

Note, in contrast to the mass of the $K(1460)$ resonance, calculations of the
width using the Faddeev equations for the $AAC$ model in momentum
representation ($\Gamma=50$ MeV) Torres2011  and configuration space
($\Gamma=104$ MeV) KezerasPRD2020 , variational method ($\Gamma=110$ MeV)
Torres2011 , hyperspherical harmonics method ($\Gamma=49$ MeV) RKSh.Ts2014
did not reproduce the quite sizeable experimental width,
335.60$\pm$6.20$\pm$8.65 MeV LHCb ; PDG2022 . In Ref. Longacre90  reported the
width of approximately 200 MeV for the ${K}^{+}K^{+}\overline{K^{0}}$
resonance and Albaladejo  presented the estimation for the width of
$\Gamma\geq 100$ MeV for $K(1460)$ resonance. The study of the $KK\bar{K}$
within the non-perturbative three–body dynamics did not calculate the width
for this system.

We calculate the width follow KezerasPRD2020  using A and B sets for the
potential parameters listed in Table 1. The comparison of the widths obtained
in the $ABC$ and $AAC$ models shows that they are close enough with a
negligible difference about 1 - 2 MeV: $\Gamma_{KK\bar{K}}=104-106$ MeV and
$\Gamma_{KK\bar{K}}=117-119$ MeV for the potential with A and B parameter
sets, respectively.

In this work we study the resonance $K(1460)$ considering only the channel
$KK\bar{K}$. One could also have channels like $\pi\pi K$ and $\pi\eta K$.
These channels are included, in an effective way, in Albaladejo , using
Faddeev equations in momentum space by Martinez Torres et al. Torres2011 , and
employing the complex-scaling method Dote2015  in the semi-relativistic
framework in Ref. SHY19 . While the formation of the resonance $K(1460)$ could
be due to the $KK\bar{K}$ channel mainly, the inclusion of other channels may
have a relevant role in its mass and, more importantly, in its width. However,
consideration of these channels in Torres2011  gives a significantly smaller
width ($\Gamma=50$ MeV) than our single-channel result. Resonance positions in
three-channel $KK\bar{K}-\pi\pi K-\pi\eta K$ and two-channel $KK\bar{K}-\pi\pi
K$ and $KK\bar{K}-\pi\eta K$ calculations presented in SHY19  demonstrate that
the coupling to the $\pi\pi K$ channel is significant to reproduce the large
width of the resonance and the coupling to $\pi\eta K$ channel makes a large
contribution to the mass of the resonance. However, the consideration of the
coupled-channel $\pi\pi K$ and $\pi\eta K$ and variation of the interaction
range and strength of the one-range Gaussian potentials did not reproduce the
experimental width. In Ref. Parganlija2017  the channels $K^{*}_{0}(1430)\pi$,
$K\rho$, and $K^{*}_{0}(892)\pi$ are quoted as “decaying channels”. These
channels require two-body dynamics either beyond $s-$wave (to form
$K^{*}(892)$ or $\rho$) or well above 1 GeV (to form $K^{*}(1430)$), and these
effects are not included in Albaladejo ; Torres2011 ; SHY19 . It is then
natural that the width reported in those and present works is much smaller
than the one quoted indicated in Brandenburg ; DAUM1981  ($\Gamma\sim 250$
MeV) or in the recent LHCb analysis LHCb  ($\Gamma\sim 335$ MeV).

Table 2:  The binding energy of the $K^{0}K^{+}K^{-}$ and
$K^{+}K^{0}\bar{K}^{0}$ particle configurations in the $AAC$ and $ABC$ model.
Calculations are performed for the potentials with the A and B parameter sets,
respectively. In the $AAC$ model, the masses of the $K^{+}$ and $K^{0}$ kaons
are equal to the average value of their masses ($\overline{m}_{K}=495.7$ MeV)
and the Coulomb attraction is omitted. AC and BC correspond to the
calculations in the $ABC$ model when the Coulomb attraction is included. The
$E_{3}$ is the binding energy of the $KK\bar{K}$ system and
$E_{3}(V_{K^{0}K^{+}}=0)$ is the three-body energy, when the repulsive
interaction between $K^{+}$ and $K^{0}$ is omitted. The $E_{2}^{K\bar{K}}$ and
$E_{2}^{K^{+}\bar{K}}$, $E_{2}^{K^{0}\bar{K}}$ are the energy of the bound
$K\bar{K}$ and $K\bar{K}$, $K^{0}\bar{K}$ pairs, in $AAC$ and $ABC$ models,
respectively. The energies are given in MeV. $AAC$ model
---
| Resonance | System |  Mass, MeV | Potentials | $\ \ E_{3}$ | $E_{3}(V_{K^{0}K^{+}}=0)$ | $\ E_{2}^{K\bar{K}}$
---|---|---|---|---|---|---
${K^{0}}(1460)$ | $K^{0}K^{+}K^{-}$ |  | $m_{K^{-}}=493.7$ |  A | $-19.4$ | $\ \ \ \ \ -31.2\ \ \ \ \ $ | $-10.95$
---|---|---|---|---
$\overline{m}_{K}=495.7$ | B | $-21.9$ | $-28.9$ | $-11.03$
${K^{+}}(1460)$ | $K^{0}K^{+}\overline{{K}^{0}}$ |  | $\overline{m}_{K}=495.7$ |  A | $-20.1$ | $\ \ \ \ \ -32.1\ \ \ \ \ $ | $-11.40$
---|---|---|---|---
$m_{\overline{{K}^{0}}}=497.6$ |  B | $-22.4$ | $-29.6$ | $-11.34$
$ABC$ model
Resonance | System |  Mass, MeV |  Potentials | $\ \ \ E_{3}$ |  $\ E_{3}(V_{K^{0}K^{+}}=0)$ | $\ \ \ E_{2}^{K^{+}\bar{K}}$ | $E_{2}^{K^{0}\bar{K}}$
${K^{0}}(1460)$ | $K^{0}K^{+}K^{-}$ |  | $m_{K^{-}}=493.7$
---
$m_{K^{+}}=493.7$
$m_{K^{0}}=497.6$
A | $\ -19.4$ | $\ \ \ \ \ \ \ \ -31.2\ \ \ $ | $-10.74$ | $-11.17$
---|---|---|---|---
B | $-21.9$ | $\ \ \ \ -28.9$ | $-10.86$ | $-11.17$
AC | $-20.9$ | $-$ | $-12.37$ | $-11.17$
BC | $-23.2$ | $-$ | $-12.27$ | $-11.17$
${K^{+}}(1460)$ | $K^{0}K^{+}\overline{{K}^{0}}$ |  | $\ \ m_{K^{+}}=493.7$
---
$m_{K^{0}}=497.6$
$\ m_{\overline{{K}^{0}}}=497.6$
A | $\ -20.1$ | $\ \ \ \ \ \ \ -32.1\ \ \ $ | $-11.17$ | $-11.62$
---|---|---|---|---
B | $\ -22.4$ | $\ \ -29.6$ | $-11.17$ | $-11.49$
Figure 2:  The bound state energy $E_{3}(\zeta)$ of the $K^{0}K^{+}K^{-}$
system (thin line) and the binding energies of bound $\ E_{2}^{K^{0}K^{-}}$
($E_{2}^{+}(\zeta)$) and $E_{2}^{K^{+}K^{-}}$($E_{2}^{-}(\zeta)$) kaon pairs
(thin dashed lines) as functions of the mass scaling parameter $\zeta$. The
dependencies of bound state energy $E_{3}(\xi)$ (thick line) and the binding
energies of bound $\ E_{2}^{K^{0}K^{-}}$ ($E_{2}^{+}(\xi)$) and
$E_{2}^{K^{+}K^{-}}$($E_{2}^{-}(\xi)$) kaon pairs (thick dashed lines) in the
$K^{0}K^{+}K^{-}$ system on the potential scaling parameter $\xi$.
Calculations are performed with the set B for $KK$ and $K{\bar{K}}$
interactions.

Figure 3: (Left panel) The mass redistribution effect on the root mean squared
distances between kaons in the $K^{0}K^{+}K^{-}$ system. The results of
calculations for the $AAC$ model ($a$), the $ABC$ model using the experimental
kaon masses ($b$), the $ABC$ model with the different unrealistic masses for
the $K^{0}$ and $K^{+}$ kaons ($c$). The total mass of the $K^{0}K^{+}$ pair
is constant and equals to the sum of experimental masses. In ($a$)-($c$) cases
the total binding energy $E_{3}(\zeta)=-21.9$ MeV is the same, while the
$r.m.s.$ distances between kaons are different. (Right panel) The $r.m.s.$
distances between $K^{+}K^{-}$ (solid curve) and $K^{0}K^{-}$ (dashed curve)
kaons in three-body systems as functions of the $\zeta$ parameter. The
vertical line corresponds to the parameter $\zeta$ related to experimental
masses of $K^{+}$ and $K^{0}$ kaons. Calculations are performed with the set B
for $KK$ and $K{\bar{K}}$ interactions.  Figure 4:  The two-body widths
$\Gamma_{2}$ of $K^{+}K^{-}$ (solid line) and $K^{0}K^{-}$ (dashed line)
versus the $\zeta$ parameter. Calculations are performed with the set B for
$K{\bar{K}}$ interaction. The vertical line shows the parameter $\zeta$
related to experimental values for masses of $K^{+}$ and $K^{0}$ kaons.

###  VI.3 The mass-symmetry breaking in the $AA$ pair: from the $AAC$ to $ABC$
model

Let us start with the bosonic $AAC$ model with two identical particles with
the average mass of two kaons and violate the mass-symmetry of this model by
changing the masses of two identical particles but keeping the total mass of
the $AA$ pair constant. Such mass redistribution leads to the transformation
of the $AAC$ model with the symmetric wave function with respect to the
exchange of $AA$ particles to the $ABC$ model with a lack of this symmetry.
Now, in the $AA$ pair of ”identical” particles, we have the masses

$m_{K^{0}}(\zeta)=\overline{m}_{K}(1-\zeta),\qquad
m_{K^{+}}(\zeta)=\overline{m}_{K}(1+\zeta),$

where $\overline{m}_{K}=\left(m_{K^{0}}+m_{K^{+}}\right)/2$ the average mass
of the $AB$ pair and $\zeta$ is a mass scaling parameter that can be varied.
The total mass of this pair is constant:
$m_{K^{0}}(\zeta)+m_{K^{+}}(\zeta)=m_{K^{0}}+m_{K^{+}}$. The kaonic system
with the mass $m_{K^{-}}$ and variable masses $m(K^{0})$ and $m(K^{+})$, must
be considered within the $ABC$ model. The cases when $\zeta=0$ and
$\zeta=0.004$ correspond to the $AAC$ model with average masses of $K^{0}$ and
$K^{+}$ kaons and the $ABC$ model that describes $K^{0}K^{+}K^{-}$ with the
experimental masses of kaons, respectively.

Results of calculations of the binding energy within the $ABC$ model with
variable masses of two kaons and the energy of the bound $AC$ and $BC$ kaon
pairs as functions of the mass scaling parameter $\zeta$ are shown in Fig. 2.
The total energy $E_{3}(\zeta)$ does not depend on the mass redistribution
between $A$ and $B$ kaons up to $\zeta\leq$ 0.06. The later value shows when
the limit of the approximation (11) is reached. However, the bound $AC$ and
$BC$ kaon pairs energies are sensitive to the variation of the parameter
$\zeta$. The $K^{+}K^{-}$ and $K^{0}K^{-}$ kaon pairs energies
$E_{2}^{+}(\zeta)$ and $E_{2}^{-}(\zeta)$, which correspond to the increase of
the $K^{0}$ mass and the decrease of the $K^{+}$ mass from the average value
to the experimental mass, increases and decreases, respectively, with the
$\zeta$ increase. Thus, the redistribution of the mass between two kaons
violates the exchange symmetry of the wave function in the $AAC$ model and
leads to the $ABC$ model that gives the same total energy as the $AAC$ model,
but increases $E_{2}^{+}(\zeta)$ and decreases $E_{2}^{-}(\zeta)$ energies of
the bound kaon pairs. The violation of the wave function symmetry from the
symmetric with respect to the exchange of $AA$ particles to a wave function
without such symmetry should affect the average distance between kaons. To
demonstrate that we calculated the root mean squared ($r.m.s.$) distances
between kaons. The latter is illustrated in Fig. 3, the left panel, by
presenting the $r.m.s.$ distances between kaon pairs for the $K^{0}K^{+}K^{-}$
particle configuration for the different values of $\zeta$. In the $AAC$ model
$\zeta=0$ and one gets the isosceles triangle. Consideration of experimental
masses in the $ABC$ model leads to the different root mean squared distances
between particles. In Fig. 3, the right panel, is shown the dependence of the
$r.m.s.$ distances on $\zeta$. One can conclude that the mass flow affects the
$r.m.s.$ distances between kaons up to $\zeta\sim 0.06$. This effect is non-
linear and tends to saturate.

According to Eq. (2) and Eq. (4), the mass-scaled coordinates in the $AAC$
model are $\mathbf{x}_{1}=\sqrt{\overline{m}_{K}}\mathbf{r}_{23}$ and
$\mathbf{x}_{2}=\sqrt{\frac{2\overline{m}_{K}m_{\bar{K}}}{\overline{m}_{K}+m_{\bar{K}}}}\mathbf{r}_{13}$.
These coordinates in the $ABC$ model are
$\mathbf{x}_{1}=\sqrt{\frac{\overline{m}_{K}^{2}-(\Delta
m)^{2}}{\overline{m}_{K}}}\mathbf{r}_{23}$,
$\mathbf{x}_{2}=\sqrt{\frac{2(\overline{m}_{K}+\Delta
m)m_{\bar{K}}}{\overline{m}_{K}+\Delta m+m_{\bar{K}}}}\mathbf{r}_{13}$,
$\mathbf{x}_{3}=\sqrt{\frac{2(\overline{m}_{K}-\Delta
m)m_{\bar{K}}}{\overline{m}_{K}-\Delta m+m_{\bar{K}}}}\mathbf{r}_{12}$ and
have a $\Delta m$-dependence. Thus, the dependence of pair potentials as a
function of the $\mathbf{x}$ coordinate in Eq. (4) is expressed as a $\Delta
m$-dependence.

One would assume that three-body kinetic energy operator is affected by the
mass scaling parameter $\zeta$. However, in the first order perturbation
theory for $\zeta\leq 0.064$, as we have shown in Section IV, the kinetic
energy matrix element $\langle{\hat{H_{0}}}\rangle$ does not depend on
$\zeta$. The binding energy
$E=\langle{\hat{H_{0}}}\rangle+\langle(v_{K^{0}\bar{K}}+v_{K^{+}\bar{K}}+v_{KK})\rangle$
is also a constant. This means the matrix element of the total potential
energy does not changed with the $\zeta$ increasing. Because the width
$\Gamma$ is evaluated from the imaginary part of the complex potentials, the
three-body width is also independent on the parameter $\zeta$ and is the same
in the $AAC$ and $ABC$ models. The two-body $K^{+}\bar{K}$ and $K^{0}\bar{K}$
subsystems do not have the compensation mechanism due to the absence of the
third particle that has the three-body $KK\bar{K}$ system as expressed by Eqs.
(10)-(11). Two-body widths depend on the mass scaling parameter and repeat the
$E_{2}(\zeta)$ dependence as depicted in Fig. 4. The widths of the
$K^{+}\bar{K}$ and $K^{0}\bar{K}$ pairs are the same, $\Gamma_{2}=59.2$ MeV,
in the $AAC$ model ($\zeta=0$). In the $ABC$ model, $\zeta=0.004$, the widths
are $\Gamma_{2}=59.8$ MeV and $\Gamma_{2}=58.6$ MeV for the $K^{+}\bar{K}$ and
$K^{0}\bar{K}$, respectively, for the set of the parameters B.

###  VI.4 From the $AAC$ to $ABC$ model through asymmetry of $AC$ and $BC$
potentials

Above we considered the bosonic $AAC$ model and its transformation to the
$ABC$ model due to changing the masses of two identical particles. However,
the bosonic $AAC$ model can be also transformed to the $ABC$ model in the case
when interactions in the $AC$ and $BC$ pairs are different. For example, in
Ref. SHY19  to interpret the three-body resonance as K(1460) were used two-
body potentials with the different strength and range parameters. In our
consideration of the $K^{0}K^{+}K^{-}$ particle configuration the kaon-
antikaon interaction we use the same strong interaction in the $K^{0}K^{-}$
and $K^{+}K^{-}$ pairs. However, interactions in the $K^{0}K^{-}$ and
$K^{+}K^{-}$ pairs are different due to the Coulomb attraction between $K^{+}$
and $K^{-}$. The latter means that even if one considers the average mass for
$K^{0}$ and $K^{+}$ kaons, the $K^{0}K^{+}K^{-}$ configuration should be
described within the $ABC$ model. Let us demonstrate this in general via a
model where a strong interaction in the $AC$ and $BC$ pairs is different by
introducing scaled potentials

$v_{AC}\to v_{K^{0}K^{-}}(\xi)=\overline{v}(1-\xi),\qquad v_{BC}\to
v_{K^{+}K^{-}}(\xi)=\overline{v}(1+\xi).$

In the last expressions $\overline{v}$ is the average potential of the $AC$
and $BC$ pairs, $\xi$ is the potential scaling parameter and $v_{AC}=v_{BC}$
when $\xi=0$. Within this model we calculate the binding energies of the
$K^{0}K^{+}K^{-}$ and the bound pairs $K^{0}K^{-}$ and $K^{+}K^{-}$ by keeping
the average potential constant. In Fig. 2 are shown dependencies of the
binding energy $E_{3}(\xi)$ of the $K^{0}K^{+}K^{-}$ and energies $\
E_{2}^{K^{0}K^{-}}$ ($E_{2}^{+}(\xi)$) and
$E_{2}^{K^{+}K^{-}}$($E_{2}^{-}(\xi)$) of the $K^{0}K^{-}$ and $K^{+}K^{-}$
pairs, respectively, on the potential scaling factor $\xi$. Results when
$\xi=0$ correspond to the $AAC$ model, while the increment of $\xi$ leads to
the $ABC$ model. The increase of $\xi$ leads to the increase or decrease of
the two-body energies of the $K^{0}K^{-}$ and $K^{+}K^{-}$ pair, respectively.
At the same time, the binding energy of the $K^{0}K^{+}K^{-}$ system remains
unchanged up to a value of $\xi=0.03$, as is demonstrated in Fig. 2 and can be
seen from Eq. (17). A similar situation was demonstrated in the previous
subsection when the mass scaling parameter $\zeta$ increases up to the value
of $\zeta=0.06$.

The Coulomb attraction acting in the single kaon-antikaon pair leads also to
the $ABC$ model and increases three-body energy by 1.5 MeV and 1.3 MeV for the
parameter sets A and B for $KK$ and $K{\bar{K}}$ interactions, respectively.
It can be noted that the averaging procedures presented in sections IV and V
allow us to consider the Coulomb interaction in the $AAC$ model.

##  VII Conclusions

In the framework of the Faddeev equations in configuration space, we
investigated the $K$(1460) resonance dynamically generated via the $KK\bar{K}$
system. We considered the $KK{\bar{K}}$ system as the $K^{0}K^{+}K^{-}$ and
$K^{0}K^{+}\overline{{K}^{0}}$ particle configurations that are analyzed using
$AAC$ and $ABC$ models. We demonstrated that the $ABC$ model can be reduced to
the $AAC$ one, where the wave function is symmetric with respect to the
exchange of identical particles. The reduction is possible by averaging the
masses of the $AB$ pair or averaging $AC$ and $BC$ potentials, if they are
different.

It is shown that the repulsive $KK$ interaction plays essential role in the
binding energy of the $KK\bar{K}$ system: contribution of the repulsive $KK$
interaction decreases the three-particle binding energy by about 38% and 25%
for the A and B parameter sets, respectively.

Our three-body non-relativistic single-channel model predicts a quasi-bound
state for the $KK\bar{K}$ system. The mass of neutral $K(1460)$ resonance
calculated in the $ABC$ model for the $K^{0}K^{+}K^{-}$ particle configuration
is 1464.1 MeV or 1461.8 MeV, while the mass of the charged $K^{+}(1460)$
resonance for the ${K}^{0}K^{+}\overline{K^{0}}$ particle configuration is
1468.8 MeV or 1466.5 MeV. These values are obtained for the parameter sets A
and B for $KK$ and $K{\bar{K}}$ interactions, respectively. The results are in
fair agreement with the experimental value of the $K$(1460) mass,
1482.40$\pm$3.58$\pm$15.22 MeV PDG2022 .

Due to the Coulomb attraction of three-body binding energy $E_{3}$ increases,
and the energy shift is 1.5 and 1.3 MeV for the parameter sets A and B for
$KK$ and $K{\bar{K}}$ interactions, respectively. Let us note that within the
$AAC$ model, consideration of the Coulomb attraction is also possible using
the averaging procedure for the Coulomb potential. The binding energy $E_{3}$
of $K^{0}K^{+}K^{-}$ and ${K}^{0}K^{+}\overline{K^{0}}$ calculated in the
$AAC$ model with the average kaon masses and in the $ABC$ model with the
experimental kaon masses are the same. Effectively, the $AAC$ model can
reproduce the binding energy obtained in the $ABC$ model if the corresponding
relative correction of masses is not larger than 6%. Thus, the small
difference of the kaon masses does not affect the binding energy of the
$K^{0}K^{+}K^{-}$ kaonic system when the Coulomb interaction is neglected.

An increase in the mass difference of the kaons leads to the mass-symmetry
violation of the system with the same binding energy. However, when the
relative mass correction exceeds 6% the binding energies calculated using the
$AAC$ and $ABC$ models differ. It should be noted that this effect has not
been reported before. The three-body kaonic system allows us to demonstrate
this effect clearly. One can consider similar nuclear systems, for example,
$np\Lambda$ or $np\alpha$, instead of the $KK\bar{K}$ kaonic system but the
effect will be small due to a small difference between the proton and neutron
masses. We have found that mass correction does not change three-body energy,
however, violates the exchange symmetry that affects the $ABC$ model wave
function symmetry.

In the $AAC$ model, there is the exchange symmetry, related to the symmetric
localization of identical particles. In contrast to the $AAC$ model,
consideration of experimental masses in the $ABC$ model leads to the violation
of this symmetry. This indicated by different root mean squared distances
between kaons. The different $r.m.s.$ distances between kaons are due to
different kaon masses and/or potentials in the $AAC$ and $ABC$ models.

We considered the $KK\bar{K}$ system using the single-channel description with
effective $s$-wave potentials. Some refinements can be done, such as using
more realistic two-body potentials, including $p$-wave components, and/or
considering the coupled-channel approach. It is important to note that the
choice of the model and assumptions made in our analysis can always have an
impact on the results. Therefore, it is essential to carefully consider the
limitations and uncertainties for description $KK\bar{K}$ system relating to
the mass and charge symmetry braking.

## Acknowledgments

This work is supported by the National Science Foundation grant HRD-1345219
and DMR-1523617 awards Department of Energy$/$National Nuclear Security
Administration under Award Number NA0003979 DOD-ARO grant #W911NF-13-0165.

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