S. M. Gerasyuta, M. A. Durnev
The electromagnetic form factors of the exotic baryons are calculated in the framework of the relativistic quark model at small and intermediate momentum transfer. The charge radii of the E+++ baryons are determined.
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The relativistic five-quark equations are found in the framework of the dispersion relation technique.The five-quark amplitudes for the low-lying exotic baryons including the up and down quarks are calculated.The poles of the five-quark amplitudes determine the masses and the widths of nonstrange exotic baryons(isospin I=5/2). The mass spectrum of the E+++ exotic baryons with JP=1/2+,3/2+, 5/2+ is calculated.
We discuss the mass differences for isospin multiplets of the charmed and b-flavored baryons. The mass of the neutral b-flavored sigma baryon, which is not measured, is calculated. We point out, that the measurements of the mass differences between the charmed sigma and chi baryons might be wrong.
The color anti-triplet triquark $q\bar{q}\bar{q}$ is considered as a compact component in the tetraquark structure $cq\bar{q}\bar{q}$ of exotic $D_{s}$ mesons. We discuss the mass spectrum and the flavor mixing of the triquarks by using the instanton induced interaction and the one-gluon exchange potentials. As a characteristic property of the triquark, we investigate the isospin violation. It is shown that the flavor $\bar{\bf 3}$ (isosinglet) and $\bf 6$ (isotriplet) states may be strongly mixed and then are identified with $D_{s}(2632)$.
The relativistic interacting quark-diquark model of baryons, recently developed, is here extended to introduce a spin-isospin transition interaction into the mass operator. The refined version of the model is used to calculate the non strange baryon spectrum. The results are compared to the present experimental data.
We study the phases of a two-flavor Nambu-Jona-Lasinio model at finite temperature $T$, baryon and isospin chemical potentials: $μ_{B}=(μ_{u}+μ_{d})/2$, $μ_{I}=(μ_{u}-μ_{d})/2$. This study completes a previous analysis where only small isospin chemical potentials $μ_{I}$ were considered
Isospin sum rules in charmed baryon weak decays
本文利用粲夸克弱衰变有效哈密顿量在同位旋降算符下为零的性质，推导单粲与双粲重子两体、三体非轻子衰变的数百条同位旋求和规则，用于检验对称性、寻找新衰变模式及研究奇特强子。
We estimate the isospin breaking contributions to the baryon masses which we analyzed recently using a loop expansion in the heavy baryon approximation to chiral effective field theory. To one loop, the isospin breaking corrections come from the effects of the $d, u$ quark mass difference, the Coulomb and magnetic moment interactions, and effective point interactions attributable to color-magnetic effects. The addition of the first meson loop corrections introduces new structure. We estimate the resulting low-energy, long-range contributions to the mass splittings by regularizing the loop inte…
Baryon mass splittings are analyzed in terms of a simple model with general pairwise interactions. At present, the $Δ$ masses are poorly known from experiments. Improvement of these data would provide an opportunity to make a significant test of our understanding of electromagnetic and quark-mass contributions to hadronic masses. The problem of determining resonance masses from scattering and production data is discussed.
Within the QCD sum-rule approach in an external field, we calculate the baryon matrix element of isovector-scalar current, $H_{\rm B}=\langle B|\overline{u}u- \overline{d}d|B\rangle/2M_{\rm B}$, for octet baryons, which appears in the response of the correlator of baryon interpolating fields to a constant isovector-scalar external field. The sum rules are obtained for a general baryon interpolating field with an appropriate form for the phenomenological ansatz of the spectral density. The key phenomenological input is the response of the quark condensates to the external field. To first order …
Data driven isospin analysis of timelike octet baryons electromagnetic form factors and charmonium decay into baryon-anti-baryon
本文提出一种模型无关的数据驱动同位旋分析方法，利用不同同位旋通道的振荡相位差约束同位旋标量与矢量分量的相对相位，并应用于类时八重态重子电磁形状因子及粲偶素衰变至重子-反重子过程。
The current concepts of isospin and baryon mass groups are only well-adapted to deal with baryon multiplets involving both the u and d quarks, and some other quark k. In this paper, we generalize isospin and mass groups to accommodate baryon multiplets involving quarks of any flavor, and the Gell-Mann--Okubo (GMO) formalism is generalized accordingly. Generalized isospin proves to be a simple and valuable framework when working in non-udk baryon multiplets, and provides new quantum numbers that allows us to distinguish $Λ$-like baryons from $Σ$-like baryons in the non-udk multiplets. The gener…
The properties of the two-flavored Gross-Neveu model in the (1+1)-dimensional spacetime are investigated in the presence of the isospin $μ_I$ as well as quark number $μ$ chemical potentials both at zero and nonzero temperatures. The consideration is performed in the limit $N_c\to\infty$, i.e. in the case with an infinite number of colored quarks. In the plane of parameters $μ_I,μ$ a rather rich phase structure is found, which contains phases with and without pion condensation. Moreover, we have found a great variety of one-quark excitations of these phases, including gapless and gapped quasipa…
We construct bi-local interpolating field operators for baryons consisting of three quarks with two flavors, assuming good isospin symmetry. We use the restrictions following from the Pauli principle to derive relations/identities among the baryon operators with identical quantum numbers. Such relations that follow from the combined spatial, Dirac, color, and isospin Fierz transformations may be called the (total/complete) Fierz identities. These relations reduce the number of independent baryon operators with any given spin and isospin. We also study the Abelian and non-Abelian chiral transfo…
Magnetic Moment of Octet Baryons in Isospin Asymmetric Magnetized Strange Matter
结合手征SU(3)夸克平均场模型和手征组分夸克模型，研究强磁场下同位旋不对称奇异强子物质中八重态重子的磁矩，发现狄拉克海效应导致磁催化，重子有效质量随磁场单调增加，真空极化效应在确定重子电磁性质中起关键作用。
In this paper, the electromagnetic mass differences of heavy hadrons are discussed, while ignoring the relevant hyperfine interactions. The effects of one-photon exchange interaction and up-down quark mass difference are parameterized. Two mass difference equations 2Σ_c^+ - (Σ_c^{++} + Σ_c^0) = 2Σ_b^0 - (Σ_b^+ + Σ_b^-) and (Ξ_{cc}^+ - Ξ_{cc}^{++}) + (Ξ_{bb}^- - Ξ_{bb}^{0}) =2(Ξ_{bc}^0 - Ξ_{bc}^{+}) for the heavy baryons are obtained. In addition, the masses of Σ_b^0, Ξ_b^0, and Ξ_{cc}^{++} are predicted based on the known experimental data.
A study of the meson-baryon interaction in the $S=-1$ sector is performed, employing a chiral SU(3) lagrangian up to next-to-leading order (NLO) and implementing unitarization in coupled channels. The model is constrained by a large set of experimental data, paying a especial attention to processes that are sensitive to the NLO contributions, such as the $K^- p\to K^+Ξ^-, K^0Ξ^0$ reactions. The consideration of additional cross sections in single isospin channels, $K^-p\to ηΛ, ηΣ$, has been found to provide more homogeneous and reliable values of the low-energy constants, the stability of whic…
We identify the quantum numbers of baryon exotics in the Quark Model, the Skyrme Model and QCD, and show that they agree for arbitrary colors and flavors. We define exoticness, E, which can be used to classify the states. The exotic baryons include the recently discovered qqqq qbar pentaquarks (E=1), as well as exotic baryons with additional q qbar pairs (E >=1). The mass formula for non-exotic and exotic baryons is given as an expansion in 1/N, and allows one to relate the moment of inertia of the Skyrme soliton to the mass of a constituent quark.
The properties of the two-flavored Gross-Neveu model with nonzero current quark mass are investigated in the (1+1)-dimensional spacetime at finite isospin $μ_I$ as well as quark number $μ$ chemical potentials and zero temperature. The consideration is performed in the limit $N_c\to\infty$, i.e. in the case with an infinite number of colored quarks. In the plane of parameters $μ_I,μ$ a rather rich phase structure is found, which contains phases with and without pion condensation. We have found a great variety of one-quark excitations of these phases, including gapless and gapped quasiparticles.…
Current measurements of isospin splittings in mesons and baryons are sufficiently precise that they allow estimates of the mass difference between constituent up and down quarks. Some previous results are updated in the light of these new measurements, and the importance of better measurements of some observables such as $M(K^{*\pm})$, $M(B^{*0})-M(B^0)$, and isospin splittings in bottom baryons is noted.
In the present paper, the rotational states of the baryons, dibaryons and baryon octets are shown. A comparison of the S = +1 theta baryons is made to the spectrum of the narrow width dibaryons. From this comparison, the rotational spectra of the exotic theta baryons are proposed. We analyze the importance of the 70 MeV quantum proposed by Mac Gregor and its relationship in the location of exotic baryons belonging to the antidecuplet. Antidecuplet mass separations are suggested on the order of the pion mass or 120 - 150 MeV from symmetry to the spectrums of the baryon decuplet and charmed bary…
Magnetic moments of decuplet baryons in isospin asymmetric magnetized strange matter
在统一手征有效框架下，研究有限温度同位旋不对称磁化奇异物质中十重态重子的介质质量和磁矩，采用手征SU(3)夸克平均场模型和手征组分夸克模型，考虑狄拉克海效应和反常磁矩。
The status of exotic baryons is analyzed in the limit of a large number of colors Nc. Several toy models reproducing the main features of the large-Nc QCD are studied in order to clarify the recent controversy concerning the consistency of various approaches to the 1/Nc-expansion for exotic baryons. The analysis reveals several difficulties of the rigid-rotator approximation including the problems of this approximation in the description of exotic baryons.
Basis light-front quantization approach to $Λ$ and $Λ_c$ and their isospin triplet baryons
本研究采用基矢光前量子化方法，通过含三维禁闭势和单胶子交换的有效哈密顿量，计算了$Λ$、$Λ_c$及其同位旋三重态重子的质量、电磁性质和部分子分布函数，结果与实验、格点QCD等相符。
We study the isospin symmetry breaking and mass splittings of the eight lowest-lying baryons. We consider three kinds of baryon mass terms, including the bare mass term, the electromagnetic terms and the spontaneous chiral symmetry breaking terms. We include the mixing term between flavor-octet and flavor-decuplet baryons. This assumes that the lowest-lying Sigma and Xi baryons contain a few decuplet components and so are not purely flavor-octet. We achieve a good fitting that the difference between every fitted mass and its experimental value is less than 0.2 MeV.
Medium modification of the magnetic moments of octet baryons in the isospin asymmetric nuclear medium at finite temperature has been calculated using medium modified quark and baryon masses derived in chiral SU(3) quark mean field model. The magnetic moments of baryons are found to vary significantly as a function of density of nuclear medium as well as with the increase in isospin asymmetry of the medium. The rise of temperature is found to decrease the effect of isospin asymmetry on masses and magnetic moments of baryons. The results of present investigation may be helpful to understand the …
The novel observation of an exotic strangeness S=+1 baryon state at 1.54 GeV will trigger an intensified search for this and other baryons with exotic quantum numbers. This state was predicted long ago in topological soliton models. We use this approach together with the new datum in order to investigate its implications for the baryon spectrum. In particular we estimate the positions of other pentaquark and septuquark states with exotic and with non-exotic quantum numbers.
We discuss the isospin-breaking mass differences among baryons, with particular attention in the charm sector to the $Σ_c^{+}-Σ_c^0$, $Σ_c^{++}-Σ_c^0$, and $Ξ_c^+-Ξ_c^0$ splittings. Simple potential models cannot accommodate the trend of the available data on charm baryons. More precise measurements would offer the possibility of testing how well potential models describe the non-perturbative limit of QCD.
Isospin sum rules for bottom-baryon weak decays
本研究利用底夸克弱衰变有效哈密顿量在同位旋降算符下为零的性质，无需 Wigner-Eckart 不变量即可推导数百条底重子两体及三体非轻子衰变的同位旋求和规则，为新衰变模式和五夸克态同位旋伙伴提供线索。
A recent general analysis of light-baryon isospin splittings is updated and extended to charmed baryons. The measured $Σ_c$ and $Ξ_c$ splittings stand out as being difficult to understand in terms of two-body forces alone. We also discuss heavy-light mesons; though the framework here is necessarily less general, we nevertheless obtain some predictions that are not strongly model-dependent.
We present an extensive study of the static properties of light exotic and non-exotic baryons, based on the formulation of chiral quark-soliton model in terms of light-cone wave functions. We discuss in particular vector, axial and tensor charges as well as an estimation of the electromagnetic moments for the baryon octet, decuplet and hypothetical antidecuplet. Contributions from Fock components up to the seven-quark sector are considered and appear to be sizeable, reflecting the importance of the pion cloud. All these observables are further decomposed into flavor, valence quark, sea quark a…
We discuss the isospin-breaking mass differences among baryons, with particular attention in the charm sector to the $Σ_c^{+}-Σ_c^0$, $Σ_c^{++}-Σ_c^0$, and $Ξ_c^+-Ξ_c^0$ splittings. Simple potential models cannot accommodate the trend of the available data on charmed baryons. More precise measurements would offer the possibility of testing how well potential models describe the non-perturbative limit of QCD.
In this paper we investigate the phase structure of a (1+1)-dimensional schematic quark model with four-quark interaction and in the presence of baryon ($μ_B$), isospin ($μ_I$) and chiral isospin ($μ_{I5}$) chemical potentials. It is established that in the large-$N_c$ limit ($N_c$ is the number of colored quarks) there exists a duality correspondence between the chiral symmetry breaking phase and the charged pion condensation (PC) one. The role and influence of this property on the phase structure of the model are studied. Moreover, it is shown that the chemical potential $μ_{I5}$ promotes th…
We calculate total cross section for production of the $Ξ_{3/2}(1862)$ exotic baryon in $\bar{K}N \to KΞ_{3/2}$ reaction assuming the following spin-parity values of the $Ξ_{3/2}$ baryon 1/2+, 1/2-, 3/2+, 3/2-. We demonstrate that the reaction total cross section strongly depends on the spin of the $Ξ_{3/2}$ baryon.
We investigate the QCD phase diagram of isospin asymmetric matter using the Polyakov loop extended quark meson (PQM) model with vector interaction. The critical point temperature is found to decrease in isospin asymmetric matter and disappear at large isospin chemical potential. We also discuss the QCD phase transition in the neutron star core. From comparison of the QCD phase diagram in PQM and corresponding baryon and isospin chemical potentials of neutron star matter in relativistic mean field models, we show that the order of the chiral phase transition in the neutron star core could be cr…
We investigate the pentaquark($P$) exotic baryons as soliton-antiflavored heavy mesons bound states in the limit of infinitely heavy meson mass. Our approach respects the chiral symmetry as well as the heavy quark symmetry. The results reveal a possibility for the loosely bound non-strange $P$-baryon(s).
In this work we extend our formalism to study meson-baryon interactions by including $s$- and $u$-channel diagrams for pseudoscalar-baryon systems. We study the coupled systems with strangeness $-1$ and focus on studying the isospin-1 resonance(s), especially in the energy region around 1400 MeV. By constraining the model parameters to fit the cross section data available on several processes involving relevant channels, we find resonances in the isoscalar as well as the isovector sector in the energy region around 1400 MeV.
We calculate the N_c dependence of the decay widths of exotic eikosiheptaplet within the framework of Chral Quark Soliton Model. We also discuss generalizations of regular baryon representations for arbitrary N_c.
We present a study of the 1+1 flavor system of strongly interacting matter in terms of the Polyakov$-$Nambu$-$Jona-Lasinio model. We find that though the small isospin symmetry breaking brought in through unequal light quark masses is too small to affect the thermodynamics of the system in general, it may have significant effect in baryon-isospin correlations and have a measurable impact in heavy-ion collision experiments.
Recently observed exotic baryon resonance with positive strangeness is discussed. The chiral soliton model, which allowed to predict the mass and width of this state, predicts also a number of other exotic states, strange and non-strange, some of them are, probably, observed in experiments. The existence of exotic multibaryons is expected as well, with positive strangeness or beauty, and negative charm. The possibility of binding of anticharm and antibeauty is pointed out.
We calculate the mass splittings within the heavy baryon isospin multiplets Σ_{c(b)} and Ξ_{c(b)}' in chiral perturbation theory to leading one-loop order. The pattern of the mass splittings in the Σ_c iso-triplet, which is different from that of any other known isospin multiplet, can be explained. We predict m_{Ξ_c'^+} - m_{Ξ_c'^0}=-0.2\pm0.6 MeV, m_{Ξ_b'^0} - m_{Ξ_b'^-}=-4.0\pm1.9 MeV and the mass of the Σ_b^0 to be 5810.3\pm1.9 MeV.
The extraction of meson-baryon scattering parameters from lattice QCD is complicated by the necessity, in some channels, of including annihilation diagrams. We consider a strategy to avoid the need for these extremely costly and noisy contributions. The strategy is based on simulations with an isospin chemical potential which, contrary to a baryon chemical potential, has no sign problem. When the isospin chemical potential is larger than a critical value, a charged pion condensate forms. Baryons propagating in the pion condensate will have their masses modified by pion-baryon scattering parame…
Properties of a hypothetical baryonium with the quark content ($uds\ov{u}\ov{d}\ov{s}$) are discussed. The MIT bag model predicts its mass to be unexpectedly low, approximately 1210 MeV. Possible hadronic decay modes of this state are analyzed. Ultrarelativistic heavy-ion collisions provide favorable conditions for the formation of such particles from the baryon-free quark-gluon plasma. We estimate multiplicities of such exotic baryonia on the basis of a simple thermal model.
I briefly review the history of exotic meson (4-quark) and baryon (5-quark) states, which is rooted in the formalism of Regge pole and duality. There are robust model-independent predictions for the exchange of 4-quark (Baryonium) Regge trajectories in several processes, which are strongly supported by experiment. On the other hand the predictions for the spectroscopy of 4-quark resonances are based on specific QCD inspired models, with some experimental support. The corresponding predictions for the recently discovered exotic baryon (Pentaquark) state are briefly discussed.
In the large $N_c$ limit both heavy and light baryons are described by the universal mean field, which allows us to relate the properties of heavy baryons to light ones. With the only input from the decays of light octet baryons (due to the universality of the chiral mean field), excellent description of strong decays of both charm and bottom sextets is obtained. The parameter-free prediction for the widths of exotic anti-decapentaplet ($\overline{\boldsymbol{15}}$) baryons is also made. The exotic heavy baryons should be anomalously narrow despite of large phase space available. In particular…
We explore the phase diagram of two-flavor QCD at imaginary values of baryon and isospin chemical potentials analyzing the thermodynamic potential of QCD analytically and that of the Polyakov-loop extended Nambu-Jona-Lasinio (PNJL) model numerically. QCD has no pion condensation at imaginary baryon and isospin chemical potentials, and therefore has discrete symmetries that are not present at real baryon and isospin chemical potentials. The PNJL model possesses all the discrete symmetries. The PNJL model can reproduce qualitatively lattice QCD data presented very lately.
We emphasize the importance of identifying non-exotic SU(3) partners of the Theta^+ pentaquark, and indicate possible ways how to do it. We also use the soliton picture of baryons to relate Reggeon couplings of various baryons. These relations are used to estimate the Theta^+ production cross section in high energy processes. We show that the corresponding cross sections are significantly suppressed relative to the production cross sections of usual baryons. Finally, we present spin non-flip form factors of the anti-decuplet baryons in the framework of the chiral quark soliton model.
We use the Nambu--Jona-Lasinio model to study the effects of the isospin chemical potential on the QCD phase diagram at nonzero temperature and baryon chemical potential. We find that the phase diagram is qualitatively altered by a small isospin chemical potential. There are two first order phase transitions that end in two critical endpoints, and there are two crossovers at low baryon chemical potential. These results have important consequences for systems where both baryon and isospin chemical potentials are nonzero, such as heavy ion collision experiments. Our results are in complete agree…
We incorporate fine-structure corrections into the dynamical diquark model of multiquark exotic hadrons. These improvements include effects due to finite diquark size, spin-spin couplings within the diquarks, and most significantly, isospin-dependent couplings in the form of pionlike exchanges assumed to occur between the light quarks within the diquarks. Using a simplified two-parameter interaction Hamiltonian, we obtain fits in which the isoscalar $J^{PC} = 1^{++}$ state---identified as the $X(3872)$---appears naturally as the lightest exotic (including all states that are predicted by the m…
We generalize the usual octet, decuplet and exotic antidecuplet and higher baryon multiplets to any number of colours Nc. We show that the multiplets fall into a sequence of bands with O(1/Nc) splittings inside the band and O(1)splittings between the bands characterized by "exoticness", that is the number of extra quark-antiquark pairs needed to compose the multiplet. Each time one adds a pair the baryon mass is increased by the same constant which can be interpreted as a mass of a quark-antiquark pair. At the same time, we prove that masses of exotic rotational multiplets are reliably determi…
In this talk, we present our recent investigation on the magnetic moments of the exotic pentaquark states, based on the chiral quark-soliton model, all relevant intrinsic parameters being fixed by using empirical data.
The pion nucleon sigma-term is extracted on the basis of the soliton picture of the nucleon from the mass spectrum of usual and the recently observed exotic baryons, assuming that they have positive parity. The value found is consistent with that inferred by means of conventional methods from pion nucleon scattering data. The study can also be considered as a phenomenological consistency check of the soliton picture of baryons.
This paper has been withdrawn by the author since it is too simple and naive.
We briefly review the formulation of chiral quark soliton model and explain the difference and similarities with the Skyrme model. Next, we apply the model to calculate non-exotic and exotic mass spectra. We concentrate on large $N_{c}$ counting both for mass splittings and decay widths. It is shown that pure large $N_{c}$ arguments do not explain the small width of exotic pentaquark states.
We extend a previous study of photoproduction of exotic baryon resonances to the reaction $γ+ d \to J/ψ+ n + p$, which permits simultaneous investigation of the reactions $γ+ p \to P_c^+ \to J/ψ~p$ \hbox{($n$ spectator)} and $γ+ n \to P_c^0 \to J/ψ~n$ \hbox{($p$ spectator).} Here $P_c^+$ is an exotic baryon with quark content $c \bar c uud$, and $P_c^0$ is its hypothetical isospin partner with quark content $c \bar c ddu$. We find: \hbox{(1) The} cross section for $J/ψ~n$ photoproduction should be equal to that for $J/ψ~p$ photoproduction if these processes are dominated by the photon coupling…
The 1/N expansion for exotic baryons is developed, and applied to the masses, meson couplings and decay widths. Masses and widths of the 27 and 35 pentaquark states in the same tower as the Theta+ are related by spin-flavor symmetry. The 27 and 35 states can decay within the pentaquark tower, as well as to normal baryons, and so have larger decay widths than the lightest pentaquark Theta. The 1/N expansion also is applied to baryon exotics containing a single heavy antiquark. The decay widths of heavy pentaquarks via pion emission, and to normal baryons plus heavy D^(*),B^(*) mesons are studie…
We consider charge-symmetry violations in the nucleon-nucleon force which result from isospin-violating meson-baryon coupling constants. The vector mesons are assumed to couple to the nucleon's electromagnetic current, which we decompose into isoscalar and isovector quark components. We compute these currents in the context of a constituent quark model. The isospin violations in the meson-baryon couplings arise from the difference in the up and down constituent quark masses. We show that class IV charge-symmetry-breaking potentials arise in the resulting $ω$ and $ρ$ exchange contributions to t…
The octet, decuplet and antidecuplet baryon mass spectrum as a function of the Skyrme charge e and the SU(3)_f symmetry breaking parameters is given in tabular form. We also estimate the decuplet-octet and the antidecuplet-octet mass difference. Comparison with existing literature is given.
Isospin splittings in baryons with two heavy quarks and a $u$ or $d$ quark are calculated using simple methods proposed previously by the authors. The results are $M(Ξ_{cc}^{++})-M(Ξ_{cc}^+) = 1.41 \pm 0.12^{+0.76}$ MeV, $M(Ξ_{bb}^0) - M(Ξ_{bb}^-) = - 4.78 \pm 0.06^{+0.03}$ MeV, and $M(Ξ_{bc}^+)-M(Ξ_{bc}^0)=-1.69 \pm0.07 ^{+0.39}$ MeV, where the statistical errors reflect uncertainties in input mass splittings, and the systematic errors are associated with the choice of constituent-quark masses.
The significantly screened heavy-quark potential in hot medium provides the possibility to study exotic quantum states of three-heavy-quark systems. By solving the Schrödinger equation for a three-charm-quark system at finite temperature, we found that, there exist Borromean states which might be realized in high energy nuclear collisions, and the binding energies of the system satisfy precisely the scaling law for Efimov states in the resonance limit.
In this talk I look into three different topics, addressing first a method to determine the quantum numbers of the $Θ^+$, then exploiting the possibility that the $Θ^+$ is a bound state of $K πN$ and in the third place I present results on a new resonant exotic baryonic state which appears as dynamically generated by the Weinberg Tomozawa $ΔK$ interaction.
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