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

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.

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.

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:

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:

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.

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

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:

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

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

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).

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:

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

so that

and rewrite Eqs. (12) in the form

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

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

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:

where

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