We calculate the mass of the vector meson in the chiral symmetry restored vacuum. This is accomplished by separating the four quark operators appearing in the vector and axial vector meson sum rules into chiral symmetric and symmetry breaking parts depending on the contribution of the fermion zero modes. We then identify each part from the fit to the vector and axial vector meson masses. By taking the chiral symmetry breaking part to be zero while keeping the symmetric operator to the vacuum value, we find that the chiral symmetric part of the vector and axial vector meson mass to be between 550 and 600 MeV. This demonstrates that chiral symmetry breaking, while responsible for the mass difference between chiral partner, accounts only for a small fraction of the symmetric part of the mass.
We calculate the masses of the vector and axial-vector mesons as well as the nucleon and the delta resonance in the chiral symmetry restored vacuum. This is accomplished by separating the quark operators appearing in the QCD sum rules for these hadrons into the chiral symmetric and symmetry breaking parts depending on the contributions of the fermion zero modes. We then extract the vacuum expectation values of all the separated parts of the quark operators using the QCD sum rule relations for these hadrons with their vacuum masses and widths. By taking the chiral symmetry breaking parts to be …
We probe effects of the partial chiral symmetry restoration to the mass of heavy-light mesons in a constituent quark model by changing the constituent quark mass of the light quark. Due to the competing effect between the quark mass and the linearly rising potential, whose contribution to the energy increases as the quark mass decreases, the heavy-light meson mass has a minimum value near the constituent quark mass typically used in the vacuum. Hence, the meson mass increases as one decreases the constituent quark mass consistent with recent QCD sum rule analyses, which show an increasing $D$ …
In the hadronic sector of relativistic heavy ion physics, the $ρ<=> 2π$ reaction is the strongest one, strong enough to equilibrate the $ρ$ with the pions throughout the region from chemical freezeout to thermal freezeout when free-particle interactions (with no medium-dependent effects) are employed. Above the chiral restoration temperature, only $ρ$'s and $π$'s are present, in that the chirally restored $A_1$ is equivalent to the $ρ$ and the mesons have an SU(4) symmetry, with no dependence on isospin and negligible dependence on spin. In the same sense the $σ$ and $π$ are "equivalent" scala…
The masses of hadrons in the vacuum, where the chiral symmetry is restored, and in the medium are in general different even when the changes in the order parameters of chiral symmetry are the same. Here, we first discuss the relation between the hadron masses and the chiral symmetry breaking in approaches based on operator product expansion (OPE). We then discuss what additional changes occur to the hadron masses when going from the chiral symmetry restored vacuum to nuclear medium and/or finite temperature. The work will highlight how we can identify the effects of chiral symmetry restoration…
Extending previous work we study the constraints of QCD sum rules on mass and width of light vector and axial-vector mesons in vacuum and in a medium with finite nuclear density. For the latter case especially the effect of nuclear pions leading to vector-axial-vector mixing is included in the analysis.
Moments of the $ϕ$ meson spectral function in vacuum and in nuclear matter are analyzed, combining a model based on chiral SU(3) effective field theory (with kaonic degrees of freedom) and finite-energy QCD sum rules. For the vacuum we show that the spectral density is strongly constrained by a recent accurate measurement of the $e^+ e^- \to K^+ K^-$ cross section. In nuclear matter the $ϕ$ spectrum is modified by interactions of the decay kaons with the surrounding nuclear medium, leading to a significant broadening and an asymmetric deformation of the $ϕ$ meson peak. We demonstrate that both…
We present a two-flavour linear sigma model with global chiral symmetry and vector and axial-vector mesons. We calculate pion-pion scattering lengths and the decay widths of scalar, vector, and axial-vector mesons. It is demonstrated that vector and axial-vector meson degrees of freedom play an important role in these low-energy processes and that a reasonable theoretical description requires globally chirally invariant terms other than the vector meson mass term. An important question for meson vacuum phenomenology is the quark content of the physical scalar f0(600) and a0(980) mesons. We inv…
We study the chiral dynamics of vector mesons in two-flavor QCD in vacuum by utilizing a functional renormalization group approach. This allows us to capture the dynamical transition from the quark-gluon phase at high energies to the hadronic phase at low energies without the necessity of model parameter tuning. We use this to analyze the scaling of vector meson masses towards the chiral symmetry breaking scale, the decoupling of the mesons at high energies and the validity of vector meson dominance.
The properties of open strange meson $K_1^\pm$ in nuclear matter are estimated in the QCD sum rule approach. We obtain a relation between the in-medium mass and width of $K_1^-$ ($K_1^+$) in nuclear matter, and show that the upper limit of the mass shift is as large as -249 (-35) MeV. The spectral modification of the $K_1$ meson is possible to be probed by using kaon beams at J-PARC. Such measurement together with that of $K^*$ will shed light on how chiral symmetry is partially restored in nuclear matter.
Unusual constituent quark axial current couplings to light vector mesons, $ρ$ and $ω$, are derived in the vacuum and under weak magnetic field by considering a quark-antiquark interaction mediated by a non perturbative gluon exchange. Similarly, light axial mesons are found to couple anomalously with the constituent quark vector current. These interactions are of the type of the Wess-Zumino-Witten terms, being strongly anisotropic and dependent on the vector (or axial) meson polarization. They also provide axial (vector) form factors for the vector (axial) mesons and are quite small, suppresse…
We study scalar, pseudoscalar, vector, and axial-vector mesons with non-strange and strange quantum numbers in the framework of a linear sigma model with global chiral $U(N_f)_L \times U(N_f)_R$ symmetry. We perform a global fit of meson masses, decay widths, as well as decay amplitudes. The quality of the fit is, for a hadronic model that does not consider isospin-breaking effects, surprisingly good. We also investigate the question whether the scalar $\bar{q}q$ states lie below or above 1 GeV and find the scalar states above 1 GeV to be preferred as $\bar{q}q$ states. Additionally, we also d…
The properties of the light vector mesons in the presence of local parity breaking medium with a chiral imbalance are considered in the vector-meson dominance model. Applying the finite lowest-order radiatively induced local effective Lagrangian, initially developed for the QED, for the vector $ρ$ and $ω$ mesons, we obtained the mass spectrum as a function of momentum and chiral chemical potential $μ_5$. We showed that in addition to the Chern-Simons term, splitting the transverse polarisations of the mesons, there is another radiatively induced contribution that becomes important at momentum …
We propose ``vector manifestation (VM)'' of the Wigner realization of the chiral symmetry in which the symmetry is restored at the critical point by the massless degenerate pion (and its flavor partners) and rho meson (and its flavor partners) as the chiral partner, in sharp contrast to the traditional manifestation á la linear sigma model where the symmetry is restored by the degenerate pion and scalar meson. The application to the chiral phase transition of the large Nf QCD is performed using the hidden local symmetry Lagrangian. Combined with the Wilsonian matching proposed recently, the VM…
The infinite tower of vector mesons encoded in holographic dual QCD bring drastic changes to the soliton structure of the nucleon. The nucleon is given by a point-like instanton in 5D surrounded by a vector meson cloud with the vector dominance restored by the infinite tower. I discuss the possible relevance of this structure in hot and dense hadronic matter.
At zero-density, meson-nucleon vertex corrections to vacuum polarization are studied in the $σ-ω$ model with the cutoff. It is shown that the properties of the vertex corrections to vacuum polarization are somewhat different from those described by the ordinary renormalization procedures when the cutoff is small (< 5 GeV). The low-energy effective Lagrangian is constructed in the framework of the renormalization group method. The weak tensor and derivative couplings of meson-nucleon interactions may be needed in the low-energy effective theory of mesons and nucleons.
We describe in-medium properties of hadrons in dense matter near chiral restoration using a Wilsonian matching to QCD of an effective field theory with hidden local symmetry at the chiral cutoff $Λ$. We find that chiral symmetry is restored in vector manifestation à la Harada and Yamawaki at a critical matter density $n_c$. We express the critical density in terms of QCD correlators in dense matter at the matching scale. In a manner completely analogous to what happens at the critical $N_f^c$ and at the critical temperature $T^c$, the vector meson mass is found to vanish (in the chiral limit) …
We study the impact of the fermion vacuum term in the SU(3) quark meson model on the equation of state and determine the vacuum parameters for various sigma meson masses. We examine its influence on the equation of state and on the resulting mass radius relations for compact stars. The tidal deformability $Λ$ of the stars is studied and compared to the results of the mean field approximation. Parameter sets which fulfill the tidal deformability bounds of GW170817 together with the observed two solar mass limit turn out to be restricted to a quite small parameter range in the mean field approxi…
We revisit the renormalization prescription for the quark-meson model in an extended mean-field approximation, where vacuum quark fluctuations are included. At a given cutoff scale the model parameters are fixed by fitting vacuum quantities, typically including the sigma-meson mass $m_σ$ and the pion decay constant $f_π$. In most publications the latter is identified with the expectation value of the sigma field, while for $m_σ$ the curvature mass is taken. When quark loops are included, this prescription is however inconsistent, and the correct identification involves the renormalized pion de…
We explore the effect of including quantum fluctuations in the two flavor Nambu$-$Jona-Lasinio model at finite temperature. This is accomplished, in a symmetry preserving way, by including collective and noncollective modes in the one-meson-loop gap equation which originate from poles and branch cuts in the complex plane, respectively. The inclusion of a boson cutoff, $Λ_b$, is necessary to regularize the meson-loop momenta. This new parameter is used to study the influence of going beyond the usual mean field approximation in the quark condensate. As the temperature increases, chiral symmetry…
We derive the equation of state for hot nuclear matter using Walecka model in a nonperturbative formalism. We include here the vacuum polarisation effects arising from the nucleon and scalar mesons through a realignment of the vacuum. A ground state structure with baryon-antibaryon condensates yields the results obtained through the relativistic Hartree approximation (RHA) of summing baryonic tadpole diagrams. Generalization of such a state to include the quantum effects for the scalar meson fields through the $σ$-meson condensates amounts to summing over a class of multiloop diagrams. The tec…
A symmetry-preserving truncation of the two-body bound-state problem in relativistic quantum field theory is used to compute the leading-twist parton distribution amplitudes (PDAs) for the first radial excitations of the $π$- and $K$-mesons. In common with ground states in these channels, the PDAs are found to be dilated with respect to the relevant conformal-limit form and skewed toward the heavier valence-quark in asymmetric systems. In addition, the PDAs of radially-excited pseudoscalar mesons are not positive definite, owing to the fact that dynamical chiral symmetry breaking (DCSB) forces…
Inhomogeneous chiral-symmetry breaking phases at non-vanishing chemical potential and temperature are studied within a two-flavor quark-meson model in the chiral limit. The analysis is performed beyond the standard mean-field approximation by taking into account the Dirac-sea contributions of the quarks. Compared with the case where the Dirac sea is neglected, we find that the inhomogeneous phase shrinks, but in general does not disappear. It is shown within a Ginzburg-Landau analysis that the Lifshitz point of the inhomogeneous phase coincides with the tricritical point if the ratio between s…
Recent topics on mesons in nuclei are discussed by especially emphasizing the role of the partial restoration of chiral symmetry in the nuclear medium. The spontaneously broken chiral symmetry in vacuum is considered to be incompletely restored in finite nuclear density systems with moderate reduction of the magnitude of the quark condensate. On the partial restoration of chiral symmetry, the wave function renormalization is important to be taken into account for the Nambu-Goldstone bosons. We also discuss the possible change of the meson properties in the nuclear medium and meson-nucleus syst…
A scaled SU(3) Nambu - Jona-Lasinio Lagrangian is used to compute the mass of the nine scalar mesons in the vacuum and the mass of the gluball. It is shown that a suitable choice of the vacuum gluon condensate allows to reproduce the experimental masses of the scalar mesons except for the {$K^*_{0}$}(1430). This choice corresponds to a weak coupling between the gluon and quark condensates, giving a {$f_{0}$}(1500) or a {$f_{J}$}(1710) which is nearly a pure glueball.
We investigate a linear sigma model with global chiral $U(2)_{R} \times U(2)_{L}$ symmetry. The mesonic degrees of freedom are the standard scalar and pseudoscalar mesons and the vector and axial-vector mesons. The baryonic degrees of freedom are the nucleon, $N$, and its chiral partner, $N^{*}$, which is usually identified with N(1535). The chiral partner is incorporated in the so-called mirror assignment, where the nucleon mass is not solely generated by the chiral condensate but also by a chirally invariant mass term, $m_{0}$. The presence of (axial-) vector fields modifies the expressions …
The vacuum fluctuation (VF) effects on the properties of the hyperonic neutron star matter are investigated in the framework of the relativistic mean field (RMF) theory. The VF corrections result in the density dependence of in-medium baryon and meson masses. We compare our results obtained by adopting three kinds of meson-hyperon couplings. The introduction of both hyperons and VF corrections soften the equation of state (EoS) for the hyperonic neutron star matter and hence reduce hyperonic neutron star masses. The presence of the $δ$ field enlarges the masses and radii of hyperonic neutron s…
We derive the effective Lagrangian for the low-energy massive meson excitations of the color-flavor-locking (CFL) phase of QCD with 3 flavors of light quarks. We compute the decay constants, the maximum velocities, and the masses of the mesons at large baryon chemical potential mu. The decay constants are linear in mu. The meson maximum velocities are close to that of sound. The meson masses in the CFL phase are significantly smaller than in the normal QCD vacuum and depend only on bare quark masses. The order of the meson masses is, to some extent, reversed compared to that in the QCD vacuum.…
An updated investigation of QCD sum rules for the first two moments of rho meson spectral functions, both in vacuum and in-medium, is performed with emphasis on the role of the scale related to spontaneous chiral symmetry breaking in QCD. It is demonstrated that these lowest moments of vector current spectral distributions do permit an accurate sum rule analysis with controlled input including QCD condensates of the lowest dimensions, whereas higher moments are subject to uncertainties from higher dimensional condensates. Possible connections with Brown-Rho scaling are discussed. The factoriza…
Correlation functions provide information on the properties of mesons in vacuum and of hot nuclear matter. In this Letter, we present a new method to derive a well-defined spectral representation for correlation functions. Combining this method with the quark gap equation and the inhomogeneous Bethe-Salpeter equation in the rainbow-ladder approximation, we calculate in-vacuum masses of light mesons and the electrical conductivity of the quark-gluon plasma. The analysis can be extended to other observables of strong-interaction systems.
QCD sum rules for $D$ and $B$ mesons in a strongly interacting environment require the decomposition of Gibbs averaged operators related to tensor structures. We present a procedure to decompose these operators into vacuum and medium-specific parts, thus defining plain medium-specific condensates with coefficients vanishing in vacuum. Our decomposition allows for an identification of potential elements of order parameters for chiral restoration, in particular for higher mass-dimension quark-condensates which would be masked otherwise if operators with uncontracted Lorentz indices would be link…
In this work the neutral meson properties have been investigated in the presence of thermo-magnetic background using two-flavor Nambu--Jona-Lasinio model. Mass, spectral function and dispersion relations are obtained in the scalar ($σ$) and pseudo-scalar ($π^0$) channels as well as in the vector ($ρ^0$) and axial vector ($a^0_1$) channels. The general Lorentz structures for the vector and axial-vector meson polarization functions have been considered in detail. The ultra-violet divergences appearing in this work have been regularized using a mixed regularization technique where the gamma funct…
By comparing the two- and three-flavor Nambu--Jona-Lasinio (NJL) models, we demonstrate that the naively expected vacuum superconductivity (VSC) in constant magnetic field ${\bf B}=B\hat z$ is disfavored due to the splitting magnetic catalysis effect (MCE) to chiral condensates with different quark flavors. Based on the simple two-flavor NJL model, we illuminate, in the lowest Landau level approximation, the similar origins of $π^0$ and $\barρ^+_1$ ($ρ^+$ meson with spin $S_z=1$) mass reductions with smaller $B$ and their different features at larger $B$. With the full Landau levels, the two-f…
We study the effect of the chiral phase transition on pion production in hot hadronic matter. The phase of restored chiral symmetry is characterized by the appearance of the scalar $σ$-meson as a chiral partner of the pion as well as by the degeneracy of the vector and axial-vector mesons. We find rapid thermal and chemical equilibration of these degrees of freedom in the symmetric phase. Provided that the chiral transition temperature is not considerably high, the presence of a chirally symmetric phase will result in $\sim 1.6$ times more thermal pions in the final state.
A detailed analysis of the lowest two moments of the $ϕ$ meson spectral function in vacuum and nuclear matter is performed. The consistency is examined between the constraints derived from finite energy QCD sum rules and the spectra computed within an improved vector dominance model, incorporating the coupling of kanonic degrees of freedom with the bare $ϕ$ meson. In the vacuum, recent accurate measurements of the $e^+ e^- \to K^+ K^-$ cross section allow us to determine the spectral function with high precision. In nuclear matter, the modification of the spectral function can be described by …
Spontaneous chiral symmetry breaking is one of the most important features in low-energy QCD. The chiral symmetry is expected to be restored at very high temperature and/or density. Accompanied by the chiral phase transition, properties of hadrons will be changed especially near the critical point. The study of the phenomena associated with the chiral phase transition will give us some clues on the connection between the chiral symmetry and the low-energy hadron dynamics. We develop the theory based on the hidden local symmetry (HLS) at finite temperature, which is an effective field theory of…
It has been suggested recently that an arbitrary induced theta-vacuum state could be created in heavy ion collisions. If such a state can be created, it would decay by various mechanisms to the fundamental theta=0 state which is the true ground state of our world. In the following we will discuss the possibility of studying this unusual state through the emission of pions, eta-mesons, and eta'-mesons. We will also present the spectrum of the produced particles in this non-zero theta background. We use the instantaneous perturbation theory for our estimates.
The vacuum fluctuation (VF) effects on asymmetric nuclear matter are investigated. Masses of nucleons and mesons are modified in the nuclear medium by calculating the loop-diagram corrections and the density dependence of hadron masses is obtained. The relativistic Lagrangian density with the isovector scalar $δ$ meson is used to calculate the nuclear equation of state (EOS) in the framework of the relativistic mean-field (RMF) approach, the effects of the in-medium hadron masses on the properties of neutron stars are finally studied. With the inclusion of the VF corrections, the nuclear EOS b…
The problem of propagation of photons and massive vector mesons in the presence of Lorenz and CPT invariance violating medium is studied when the parity-odd medium is bounded by a hyperplane separating it from the vacuum. The solutions in both half-spaces are carefully discussed and in the case of space-like boundary stitched on the boundary with help of the Bogolubov transformations provided by the space-like Chern-Simons vector. The presence of two different Fock vacua is shown and the probability amplitude for transmission of particles from vacuum to parity breaking medium is calculated. We…
We revisit the question of whether the U_A(1) symmetry is effectively restored in hot and dense medium. In particular, by generalizing the Witten-Veneziano formula to finite temperature, we investigate whether the mass of eta'-meson will change in medium due to the restoration of chiral symmetry.
We discuss results of numerical analyses of the 3+1 dimensional Nambu--Jona-Lasinio model at non-zero baryon density. By studying relevant order parameters, it is shown that the model goes through a transition between a vacuum with broken chiral symmetry, and a chirally restored phase at high baryon chemical potential (mu) with a diquark condensate roughly proportional to the area of the Fermi surface and a broken U(1) baryon number symmetry. The condensation of diquark pairs from the fermionic spectrum is shown to lead to an energy gap (Delta) about the Fermi energy (E_F), which is found to b…
In this work, we construct a phenomenological effective model for the heavy-light quark systems, which consist of (u,d,c,b) quarks, i.e. extended nonlocal chiral-quark model (ExNLChQM) to compute the heavy-meson weak-decay constants f_D and f_B. ExNLChQM is based on the heavy-quark effective field theory as well as the dilute instanton-vacuum configuration. In ExNLChQM, a certain portion of the heavy-meson mass is considered to be generated from the nontrivial QCD vacuum contribution, being similar to that for the light quarks in usual instanton approaches. Hence, the effective heavy- and ligh…
We study the structural change of the vector spectral function and integrated production rates of dileptons in the presence of the chiral mixing induced exclusively at finite density. The mixing produces multiple bumps and peaks around the vacuum masses of the $ρ, ω$ and $ϕ$ resonances in the spectral function. The arising modification becomes pronounced when the mass difference between parity partners decreases. In particular, the emergent enhancement around the vacuum $ϕ$ meson in the production rates serves as an excellent signature of the partially-restored chiral symmetry in heavy-ion col…
We investigate the in-medium masses of a $\bar{D} (0^-)$ meson and a $\bar{D}_0^* (0^+)$ meson and spectral functions for $\bar{D}$ and $\bar{D}_0^*$ meson channels in nuclear matter. These mesons are introduced as chiral partner in the chiral symmetry broken vacuum, hence they are useful to explore the partial restoration of the broken chiral symmetry in nuclear matter. We consider the linear sigma model to describe the chiral symmetry breaking. Our study shows that the loop corrections to $\bar{D}$ and $\bar{D}_0^*$ meson masses provide a smaller mass splitting at finite density than that in…
We study the neutron star core-crust transition density $ρ_t$ with the inclusion of the vacuum polarization in the dielectric function in the nonlinear relativistic Hartree approach (RHAn). It is found that the strong correlation between the $ρ_{t}$ and the scalar meson mass $m_σ$ strikingly overwhelms the uncertainty of the nuclear equation of state in the RHAn models, in contrast to the usual awareness that $ρ_{t}$ is predominantly sensitive to the isovector nuclear potential and symmetry energy. The accurate extraction of $ρ_{t}$ through the future gravitational wave measurements can thus p…
Cherenkov gluons may be responsible for the asymmetry of dilepton mass spectra near rho-meson observed in experiment. They can be produced only in the low-mass wing of the resonance. Therefore the dilepton mass spectra are flattened there and their peak is slightly shifted to lower masses compared with the in-vacuum rho-meson mass. This feature must be common for all resonances.
We discuss QCD sum rule constraints based on moments of vector meson spectral distributions in the vacuum and in a nuclear medium. Sum rules for the two lowest moments of these spectral distributions do not suffer from uncertainties related to QCD condensates of dimension higher than four. We exemplify these relations for the case of the omega meson and discuss the issue of in-medium mass shifts from this viewpoint.
The sigma meson mass and width are studied at finite baryonic density in the framework of a chiral unitary approach which successfully reproduces the meson meson phase shifts and generates the f0 and sigma resonances in vacuum.
We study the reaction $e^+e^-\toπ^0γ$ based on a dispersive representation of the underlying $π^0\toγγ^*$ transition form factor. As a first application, we evaluate the contribution of the $π^0γ$ channel to the hadronic-vacuum-polarization correction to the anomalous magnetic moment of the muon. We find $a_μ^{π^0γ}\big|_{\leq 1.35\,\text{GeV}}=43.8(6)\times 10^{-11}$, in line with evaluations from the direct integration of the data. Second, our fit determines the resonance parameters of $ω$ and $ϕ$. We observe good agreement with the $e^+e^-\to3π$ channel, explaining a previous tension in the…
In this paper, the equation of state (EOS) of deconfined quark stars is studied in the framework of the two-flavor NJL model, and the self-consistent mean field approximation is employed by introducing a parameter $α$ combining the original Lagrangian and the Fierz-transformed Lagrangian, $\mathcal{L}_R= (1-α)\mathcal{L}+α\mathcal{L}_F$, to measure the weights of different interaction channels. It is believed that the deconfinement of phase transition happens along with the chiral phase transition. Thus, due to the lack of description of confinement in the NJL model, the vacuum pressure is set…
The quantum vacuum in a many-body system of finite nuclei has been investigated within the relativistic Hartree approach which describes the bound states of nucleons and anti-nucleons consistently. The contributions of the Dirac sea to the source terms of the meson-field equations are taken into account up to the one-nucleon loop and one-meson loop. The tensor couplings for the $ω$- and $ρ$-meson are included in the model. After adjusting the parameters of the model to the properties of spherical nuclei, a large effective nucleon mass $m^{*}/M_{N} \approx 0.78$ is obtained. The overall nucleon…
Monte Carlo simulations of the 3+1 dimensional NJL model are performed with baryon chemical potential mu>0. For mu>Sigma_0, the constituent quark mass in vacuum, chiral symmetry is restored and a diquark condensate <qq_+> forms. We analyse the fermion propagator and find evidence for particle-hole mixing in the vicinity of the Fermi surface and an energy gap Delta>0, both of which provide evidence for superfluidity at high baryon density induced by a BCS mechanism. At (mu a)=0.8 the ratio between the BCS gap and the vacuum quark mass is Delta/Sigma_0=0.15(2).
Using the chirally invariant overlap Dirac operator we remove its lowest-lying quasizero modes from the valence quark propagators and study evolution of isovector mesons with J=1. At the truncation level about 50 MeV SU(2)_L \times SU(2)_R and U(1)_A symmetries get restored. However, we observe a degeneracy not only within the chiral and U(1)_A multiplets, but also a degeneracy of all possible chiral multiplets, i.e., the observed quantum levels have a symmetry larger than U(2)_L \times U(2)_R and their energy does not depend on the spin orientation of quarks and their parities. We offer a pos…
