J Steinheimer

S Schramm

H Stöcker

An effective chiral Hadron-Quark Equation of State

Keywords: numbers: 12, 39, -x, 12, 38, Aw

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Introduction

The recent experimental results at the Relativistic Heavy Ion Collider (RHIC), suggesting the creation of a nearly perfect fluid [1,2,3,4], have fueled interest in the study of bulk properties of strongly interacting matter (QCD). Heavy ion experiments at different beam energies try to map out the QCD phase diagram, especially the region where one expects a phase transition from a confined gas of hadrons to a deconfined state of quarks and gluons (QGP) [5,6,7,8,9,10,11,12,13,14,15,16]. To relate any experimental observables to the properties of the matter produced in heavy ion collisions, a deeper understanding of the thermodynamics of QCD has to be obtained and integrated in model simulations of these collisions. One can define two different phase transitions, the first being the chiral phase transition associated with chiral symmetry restoration in the vanishing quark mass limit, where the chiral condensate serves as a well defined order parameter. In the limit of heavy quarks, a deconfinement phase transition with the Polyakov loop as order parameter, is assumed. Physical quarks however have intermediate masses and one would expect that at least the deconfinement order parameter is not so well defined anymore. Both transitions seem to be correlated, accounting for (some) lattice QCD observation that deconfinement and chiral restoration occur at the same temperature (at least at μ B = 0) [17,18,19,20,21], while newest results indicate at least a small shift in the critical temperatures [28,27]. The lattice calculations at finite temperature are an important tool for the investigation of the QCD phase diagram. For the thermodynamics of the pure gauge theory high accuracy data is available [22], and the equation of state (EoS) of strongly interacting matter at vanishing chemical potential is reasonably well understood [23,24]. Here lattice predicts a rapid crossover for the deconfining and chiral phase transitions. At finite baryo-chemical potential, lattice calculations suffer from the so-called sign problem. There are several different approaches to obtain results at finite μ B [25,21,26,29,30,31,32,33], but yet no clear picture, especially about the existence and location of a possible critical end point, has emerged. Recent considerations based on connecting the large N c limit with real-word QCD draw an even more exotic picture of the phase diagram, where the critical temperatures of the deconfinement and chiral phase transitions disconnect and depart in the region of high net baryon densities [34].

PNJL-type models have recently been used to successfully describe lattice results on bulk properties of a strongly interacting matter [35,36] (see also [37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57]). These constituent quark models have the correct degrees of freedom in the asymptotic regime of free quarks and gluons but lack the rich hadronic spectrum.

In our approach we combine, in a single model, a well-established flavor-SU(3) hadronic model with a PNJL-type quark-gluon description of the highly excited matter. This allows us to study the chiral-symmetry and confinement-deconfinement phase structure of the strongly interacting matter at high temperatures and densities. In addition we obtain an equation of state of hadronic and quark matter that is applicable over a wide range of thermodynamical conditions and that can therefore be used in heavy-ion simulations with very different beam energies.

Model description

In our approach we derive the EoS of hot and dense nuclear matter using a single model for the hadronic and quark phase (later also referred to as the hadron quark model HQM). The model includes the correct asymptotic degrees of freedom, namely a free gas of quarks and gluons at infinite temperature, and a gas of hadrons having the correct vacuum properties at vanishing temperature. The model also predicts the structure of finite nuclei, nuclear and neutron matter properties and a first order liquid-vapor phase transition. The two phase transitions that are expected from QCD, the chiral and deconfinement transitions, are included in a consistent manner. In the following we will show how we describe the different phases of QCD and how we combine them in a unified approach.

We describe the hadronic part of the EoS, using a flavor-SU(3) model which is an extension of a non-linear representation of a sigma-omega model including the pseudoscalar and vector octets of mesons and the baryonic octet and decuplet (for a detailed discussion see [58,59,60]).

The Lagrangian density of the model in mean field approximation reads:

L = L kin + L int + L meson , (1)
where besides the kinetic energy term for hadrons, the terms:

L int = - i ψi [γ 0 (g iω ω + g iφ φ) + m * i ]ψ i , (2)
L meson = - 1 2 (m 2 ω ω 2 + m 2 φ φ 2 ) -g 4 ω 4 + φ 4 4 + 3ω 2 φ 2 + 4ω 3 φ √ 2 + 2ωφ 3 √ 2 + 1 2 k 0 (σ 2 + ζ 2 ) -k 1 (σ 2 + ζ 2 ) 2 -k 2 σ 4 2 + ζ 4 -k 3 σ 2 ζ + m 2 π f π σ + √ 2m 2 k f k - 1 √ 2 m 2 π f π ζ , + χ 4 -χ 4 0 + ln χ 4 χ 4 0 -k 4 χ 4 χ 4 0 ln σ 2 ζ σ 2 0 ζ 0 . ( 3
)
represent the interactions between baryons and vector and scalar mesons, the selfinteractions of scalar and vector mesons, and an explicitly chiral symmetry breaking term. The index i denotes the baryon octet and decuplet. Here, the mesonic condensates (determined in mean-field approximation) included are the vector-isoscalars ω and φ, and the scalar-isoscalars σ and ζ (strange quark-antiquark state). Assuming isospin symmetric matter, we neglected the ρ-meson contribution in Eq. 3.

The concept of a broken scale invariance leads to the trace anomaly in (massless) QCD. The last four terms of (3) were introduced to mimick this anomaly on tree level [61,59]. The effect of the logarithmic term χ 4 ln χ is to break the scale invariance which leads to the proportionality χ 4 ∝ θ μ μ . The comparison of the trace anomaly of QCD with that of the effective theory allows for the identification of the χ field with the gluon condensate θ μ μ . This holds only, if the meson-meson potential is scale invariant and can be achieved by multiplying the invariants of scale dimension less then four with an appropriate power of the dilaton field χ.

The effective masses of the baryons (of the octet) are generated by the scalar mesons except for an explicit mass term (δm N = 120 MeV):

m * b = g bσ σ + g bζ ζ + δm b , (4)
while, for simplicity and in order to reduce the number of free parameters, the masses of the decuplet baryons are kept at their vacuum expectation values. With the increase of temperature/density, the σ field (non-strange chiral condensate) decreases in value, causing the effective masses of the particles to decrease towards chiral symmetry restoration. The coupling constants for the baryons [62] are chosen to reproduce the vacuum masses of the baryons, nuclear saturation properties and asymmetry energy as well as the Λ-hyperon potentials. The vacuum expectation values of the scalar mesons are constrained by reproducing the pion and kaon decay constants.

The extension of the hadronic SU(3) model to quark degrees of freedom is constructed in analogy to the PNJL model. The sigma model uses the Polyakov loop Φ as the order parameter for deconfinement. Φ is defined via Φ = 1 3 Tr[exp (i dτ A 4 )], where A 4 = iA 0 is the temporal component of the SU(3) gauge field. One should note that one must distinguish Φ, and its conjugate Φ * at finite baryon densities [63,21,64], as they couple differently to quarks and antiquarks respectively. In our approach the effective masses of the quarks are generated by the scalar mesons except for a small explicit mass term (δm q = 5 MeV and δm s = 105 MeV for the strange quark):

m * q = g qσ σ + δm q , m * s = g sζ ζ + δm s , ( 5
)
with values of g qσ = g sζ = 4.0. Vector type interactions introduce an effective chemical potential for the quarks and baryons, generated by the coupling to the vector mesons:

μ * i = μ i -g iω ω -g iφ φ.
A coupling of the quarks to the Polyakov loop is introduced in the thermal energy of the quarks. Their thermal contribution to the grand canonical potential Ω, can then be written as:

Ω q = -T i∈Q γ i (2π) 3 d 3 k ln 1 + Φ exp E * i -μ * i T (6)
and

Ω q = -T i∈Q γ i (2π) 3 d 3 k ln 1 + Φ * exp E * i + μ * i T ( 7
)
The sums run over all quark flavors, where γ i is the corresponding degeneracy factor, E * i the energy and μ * i the chemical potential of the quark. All thermodynamical quantities, energy density e, entropy density s as well as the densities of the different particle species ρ i , can be derived from the grand canonical potential. In our model it has the form:

Ω V = -L int -L meson + Ω th V -U (8)
Here Ω th includes the heat bath of hadronic and quark quasi particles. The effective potential U (Φ, Φ * , T ), which controls the dynamics of the Polyakov-loop, will be discussed in the following. In our approach we adopt the ansatz proposed in [36]:

U = - 1 2 a(T )ΦΦ * + b(T )ln[1 -6ΦΦ * + 4(Φ 3 Φ * 3 ) -3(ΦΦ * ) 2 ] (9)
with a(T

) = a 0 T 4 + a 1 T 0 T 3 + a 2 T 2 0 T 2 , b(T ) = b 3 T 3 0 T .
This choice of effective potential satisfies the Z(3) center symmetry of the pure gauge Lagrangian. In the confined phase, U has an absolute minimum at Φ = 0, while above the critical Temperature T 0 (for pure gauge T 0 = 270 MeV) its minimum is shifted to finite values of Φ. The logarithmic term originates from the Haar measure of the group integration with respect to the SU(3) Polyakov loop matrix. The parameters a 0 , a 1 , a 2 and b 3 are fixed, as in [36], by demanding a first order phase transition in the pure gauge sector at T 0 = 270 MeV, and that the Stefan-Boltzmann limit of a gas of glouns is reached for T → ∞. Note that T 0 remains a free parameter to adjust the actual critical temperature, of both phase transitions, when both, quarks and hadrons, couple to the scalar fields.

As has been mentioned above, the Lagrangian of the chiral model contains dilaton terms to model the scale anomaly. These terms constrain the chiral condensate, if the dilaton is frozen at its ground state value χ 0 . On the other hand, as deconfinement is realized, the expectation value of the chiral condensate should vanish at some point. On account of this we will couple the Polyakov loop to the dilaton in the following way:

χ = χ 0 (1 -0.5(ΦΦ * )) (10)
Assuming a hard part for the dilaton field which essentially stays unchanged and a soft part, which vanishes when deconfienemt is realized. Hence, allowing the chiral condensate to also approach zero.

Until now all hadrons are still present in the deconfined and chirally restored phase. Since we expect them to disappear, at least at some point above T c , we have to include a mechanism that effectively suppresses the hadronic degrees of freedom, when deconfinement is achieved. In previous calculations baryons were suppressed by introducing a large baryon mass shift for non-vanishing Φ [62].

In the following the suppression mechanism will be provided by excluded volume effects. It is well known that hadrons are no point-like particles, but have a finite volume. Including effects of finite-volume particles, in a thermodynamic model for hadronic matter, was proposed some time ago [65,66,67,68]. We will use an ansatz similar to that used in [69,70], but modify it to also treat the point like quark degrees of freedom consistently.

If one introduces a particle of radius r into a gas of the same particles, then the volume excluded is not just the simple spherical volume, but one-half times the volume of a sphere with radius 2r:

v = 1 2 • 4 3 π(2r) 3 (11)
It is easy to understand that if all other particles also have a radius r then the excluded volume is much bigger than just the volume of a single particle. We expect the volume of a meson to be smaller than that of a baryon. We introduce the quantity v i which is the volume excluded of a particle of species i where we only distinguish between hadronic baryons, mesons and quarks. Consequently v i can assume three values:

v Quark = 0 v Baryon = v v M eson = v/a ( 12
)
where a is a number larger than one. In our calculations we assumed it to be a = 8, which would mean that the radius r of a meson is half of the radius of a baryon. Note that at this point we neglect any possible Lorentz contraction effects on the excluded volumes as introduced in [71,72]. The modified chemical potential μ i , which is connected to the real chemical potential μ i of the i-th particle species, is obtained by the following relation:

μ i = μ i -v i P ( 13
)
where P is the sum over all partial pressures. All thermodynamic quantities can then be calculated with respect to the temperature T and the new chemical potentials μ i . To be thermodynamically consistent, all densities ( e i , ρ i and s i ) have to be multiplied by a volume correction factor f , which is the ratio of the total volume V and the reduced volume V , not being occupied:

f = V V = (1 + i v i ρ i ) -1 (14)
Then the actual densities are:

e = i f e i ( 15
)
ρ i = f ρ i ( 16
)
s = i f s i ( 17
)
Note that in this configuration the chemical potentials of the hadrons are decreased by the quarks, but not vice versa. In other words as the quarks start appearing they effectively suppress the hadrons by changing their chemical potential, while the quarks are only affected through the volume correction factor f . Our implementation of finite-volume corrections as outlined above is a simple approach with as few parameters as possible and can be improved upon in various ways. For one, hadrons differ in size. The size of a hadron could even be density or temperature dependent [74]. In addition, the excluded-volume parameter of a particle does also depend on the density of the system (at dense packing a particle excludes effectively less volume). Furthermore there might be dynamic effects due to interaction which increase with temperature and/or density. Such effects could be introduced to some extent, making the model even more depend on parameters. Such a finetuning could help to better describe e.g. lattice data but requires extensive work on the model. At least some aspects of a density dependend excluded volume can be considered in future works. However, in the following, one should regard the variables v and a as effective parameters for capturing the qualitative effect of an excluded volume correction, which suppresses the hadrons in the quark phase. We would like to stress that these volume corrections enable us to describe a phase transition from hadronic to quark degrees of freedom, having only one single partition function for both phases, in a thermodynamic consistent manner. Furthermore the volume corrections we apply are physically well motivated and are thoroughly discussed in older and recent literature [65,66,67,68,69,70,71,72,75]. They model the fact that hadrons generate a repulsive hard-core interaction for the other particles in the system. This is not necessarily related to confinement. Therefore, a volume correction from the remaining mesons beyond T c that also affect the quarks is not a contradiction to the fact that quarks can propagate freely, it is part of the residual interaction in the system, quarks feel repulsion interacting with the mesons. Our description of the excluded volume effects is at this stage simplified and parameter dependent (yet thermodynamically consistent).

Results at vanishing net baryon density

In this section we concentrate on the properties of the model at vanishing chemical potential. Here lattice calculations suggest a crossover from the hadronic to the quark phase. Different lattice groups obtain different results for the phase transition temperature ranging from T c = 160 MeV to 200 MeV [76,77]. For all following results we set T 0 , the free parameter of the Polyakov-potential, to T 0 = 235 MeV and the excluded volume parameter v = 1fm 3 . This leads to a critical temperature of T c ≈ 183 MeV (T c is defined as the temperature with the largest change in the order parameters as a function of the temperature). In this section we will also distinguish results obtained when the Polyakov loop is coupled to the dilaton in the above described manner (solid lines), and those where the dilaton is not coupled to the Polyakov loop (dashed lines). The lattice data referred to in the following sections are taken from the HotQCD collaboration [73]. Here different actions (p4, asqtad) and lattice spacings (N τ = 6, 8) were compared. Note that the transition region extracted from the lattice data lies between 185 and 195 MeV. The reader should keep in mind that different lattice groups get significantly differing results on all observables. This indicates that the systematic uncertainties on lattice data are still very large (much larger than the statistical errors which are usually plotted). In fact, recent lattice results of the HotQCD group [78] with new actions in order to improve the description of hadrons on the lattice, point to the importance of hadrons to describe the phase transition as well as explicitly state the rather slow and smooth transition from confinement to deconfinement with a wide intermediate region. There are even attempts to combine results on thermodynamics from the lattice with those from a hadronic resonance gas which is expected to be the correct description of matter below T c [79]. We therefore do not expect to get a good agreement of our results on thermodynamics with the lattice data below T c .

Figure (1 left) shows the temperature dependence for the order parameters of both, the deconfinement (Φ), and chiral (σ) phase transition, extracted from our model and compared to lattice data. Both order parameters change smoothly with temperature. The critical temperature is found to be equal for both phase transitions. The lattice results represent a quantity which is called the subtracted chiral condensate (Δ l,s ) and which is defined in the following way:

Δ l,s (T ) = σ(T ) -m q /m s ζ(T ) σ(0) -m q /m s ζ(0) ( 18
)
Here m q and m s refer to the bare mass of up, down and strange quarks. Note that the value of the chiral condensate σ approaches zero only slowly. This originates from the dilaton contribution to the scalar potential in this model (the logarithmic term in eqn.(3) prevents the value of sigma to drop as fast as expected), which generates a repulsive term for small values of σ. Therefore the temperature dependence of the chiral condensate compares less favorably to lattice results than PNJL type models. A simple ansatz to solve this problem would be to simply remove the dilaton contribution from the model, but in the current parametrization of the hadronic model the dilaton contribution is essential for the correct description of the ground state of nuclear matter. To resolve this problem we would have to introduce a different coupling of the dilaton to the hadrons (and quarks) and fields (as for example outlined in [80], here part of the baryon mass is generated through coupling to the dilaton and part through coupling to the chiral condensate). This way one may achieve a satisfactory description of nuclear ground state and a satisfactory behavior of the chiral condensate above T c . Such a major change of the chiral model is out of the scope of the current work and will be pursued in the future.

Like in the PNJL model the parameters of the Polyakov potential are fixed by a fit to pure glue lattice data. Hence, the value of the Polyakov Loop increases somewhat faster as a function of temperature than in recent lattice calculations including quarks. The same behavior can be observed, when a PNJL type of model [57] is compared against the newest lattice results as a function of temperature (and not T/T c which can be misleading at some point).

Figure (1 right) shows the total densities of quarks plus antiquarks (black lines), mesons (green lines) and baryons plus antibaryons (red lines). Below the critical temperature hadrons are the dominant degree of freedom. When the quark number Right : Three times the pressure (red lines) and energy density (black lines) over T 4 as a function of T at μ B = 0. The green dotted line indicates the Boltzmann limit for an ideal gas of three massless quarks and gluons. The symbols denote lattice data from [73], using different lattice actions (asqdat and p4) and lattice spacings N τ . The dashed lines depict results without the dilaton-Polyakov loop coupling. increases around T c , they begin to suppress the hadrons. It is remarkable that the hadrons are still present, and not negligible, up to about 2.0 T c [81]. Especially the mesons contribute strongly to all thermodynamic quantities, since they are quite less suppressed than the baryons (v M < v B ).

Above 2 T c the hadrons are effectively squeezed out of the system by the presence of the quarks. To emphasize this change in degrees of freedom, Figure (2 left) shows the fraction of the total energy density which stems from the quarks and gluons (more precisely the Polyakov potential). As expected for a crossover both degrees of freedom (hadrons and quarks) are present in the temperature range from 0.75 -2 T c . However, around T c the fraction of the energy density, due to quarks and gluons increases rapidly. It converges to unity at around 2 times T c .

Let us now take a closer look at different thermodynamic quantities. Figure (2 right) displays the energy density (black curve) and three times the pressure (red dashed curve), both over T 4 compared to lattice data [73]. In the limit of infinite temperature, both quantities should converge to the Stefan-Boltzmann limit of an ideal gas of quarks and gluons. This limit is indicated as a green dashed line. The strong increase in energy density around T c reflects the rapid change of the relevant degrees of freedom. At three times the critical temperature the energy density is slowly converging to the Stefan-Boltzmann limit, while the pressure is converging even slower as it was also observed in PNJL calculations [38]. At temperatures below T c our calculation gives larger values for the pressure and the energy density.

Around 1.5 T c one can observe a slight 'dip' in the energy density. This 'dip' is connected to the correction factor f of the excluded volume corrections. As has been shown above, the hadronic contribution to the densities disappears only at two time T c and therefore they still exclude some portion of the volume for the quarks. The 'dip' therefore indicates the disappearance of volume correction factors for the quark phase.

In the high temperature limit, where only the quarks (and gluons) remain in the system the energy density and pressure both slightly exceed the data from lattice calculations.

Figure (3 left) displays the difference of the energy density and three times the pressure over T 4 (black solid line). This quantity is also referred to as the 'interaction measure' in lattice calculations. In the Stefan-Boltzmann limit it is 0, while it shows a peak slightly above T c . The height of the peak in our model is comparable to the lower bound from lattice studies [73], while its value at large T is a little bit above that from lattice calculations, because chiral restoration is not fully achieved in our model. We want to point out that, even though our model results below T c agree well with the lattice results for the interaction measure, the corresponding values for the energy density and pressure differ considerably, when we compared to lattice data (Figure 2 right). This points out that the newest lattice results still do not have the resolution to describe the hadronic part of the heat bath sufficiently well [73]. Thus, interaction effects of hadronic states in the hot system are most likely not correctly taken into account in the current lattice data.

Note that our model gives a much better description of the interaction measure as it does for the order parameters when compared to lattice. This might indicate that the conversion from the behavior of the Polyakov loop to the bahavior of thermodynamic quantities might not be as proposed in a PNJL type approach.

An important property of a hot and dense nuclear medium is the speed of sound (c s ):

c 2 s = dp de μ=0 ( 19
)
It is not only closely related to expansion dynamics but also controls the way perturbations (sound-and shock-waves) travel through the fireball [82]. Figure (3 right) shows the square of the speed of sound as a function of temperature. As the temperature increases towards T c one can clearly observe a softening of the EoS due to the crossover. At very high temperature the speed of sound again converges toward its ideal gas limit of c 2 s → 1/3. The dip above T c is again related to the excluded volume corrections. Note that even though the change of degrees of freedom from hadrons to quarks proceeds as a crossover, there is still a substantial softening (i.e. c 2 s goes down to 0.07). This behavior is comparable to results obtained with different versions of the PNJL model [57,41]. It is a result of the fit of the Polyakov loop potential to pure glue data, resulting in a steeper increase of the order parameter when compared to lattice results including quarks.

Overall, the results are not very sensitive on the exact values used in the excludedvolume part of the model. Varying the volume parameter v by a factor of 2 does not alter the temperature dependence of the Polyakov loop as it is controlled mainly by the Polyakov potential (and therefore the parameter T 0 ). Even the phase transition temperature T c of the chiral phase transition is not affected by the volume parameter. For smaller values of the volume the chiral phase transition becomes slightly steeper (a faster increase with temperature). The thermodynamic quantities change at maximum about 10% around T c , but the qualitative behavior of the hadrons beeing suppressed by the quarks stays unchanged. It is rather a question of how much the hadrons get suppressed at a given temperature and therefore how much they still contribute in the region of phase coexistence. At some point above T c the hadrons are removed for any value of the volume parameter. One could probably do some fine tuning to whatever digit desired to get a maybe slightly better description of the lattice data, but that was not the intent of the manuscript. The value of 1 fm 3 for the volume parameter is a rough estimate of a baryons volume which was widely used in the literature.

Results at zero temperature

Most of the physical observables we have accurate knowledge of are measured in the vacuum and, concerning the scales used in this work, at almost zero temperature. Im-ρ κ ρ Figure 4. Left : Densities of the different particle species a a function of net baryon density at T = 0. The grey dashed line is the result from the purely hadronic model and serves as a baseline comparison. The red lines show the results for the quark densities while the proton densities are depicted in black. Results with a quark vector interaction strength of g qω = 3.0 are shown as solid lines while the dashed lines are results with g qω = 0 Right : Binding energy per nucleon as a function of net baryon density at T = 0 for g qω = 3.0. The minimum is located at ≈ 0.16fm -3 and the corresponding binding energy is E/Am N ≈ -16MeV . The resulting incompressibility is κ ≈ 380MeV portant observables for our work are for example the vacuum masses of the hadrons and the properties of nuclear ground state matter. So before extending our work to the realm of finite baryon density and finite temperature we need to make sure that our model gives reasonable results at zero temperature. First we want to investigate the behavior of the different particle densities, at T = 0, as a function of net baryon density.

Here the repulsive vector interaction, transmitted by the vector field ω, starts to play a more important role than the attractive interaction originating from the σ field. In our model the vector interaction strength g nω of the nucleons can easily be constrained by demanding reasonable values for the nuclear binding energy and saturation density. This is not the case for the quark vector-interaction strength. The only reasonable constraint on their part would be to demand that there are no free quarks present in the nuclear ground state. Figure (4 left) shows the densities of quarks (red lines) and protons (black lines) as a function of net baryon density compared to the nucleon density from the purely hadronic model (grey dashed line). In the case of no repulsive quark vector interaction (dotted lines) the free quarks appear already before the ground state density. This would mean that there is no nuclear liquid-gas phase transition and no physical nuclear ground state. If we introduce a finite quark vector interaction strength of g qω = 3.0 ≈ g nω /3 the quarks appear only at larger densities.

In this scenario we can obtain reasonable values for the nuclear ground state saturation density (ρ 0 ≈ 0.16 fm -3 ) as well as the binding energy (E/Am N ≈ -16 MeV) as can be seen in Figure (4 right). We can also calculate the incompressibility modulus at ground state density, defined as κ = 9(dp/dρ b ) T =0,ρ b =ρ 0 and obtain a result of κ ≈ 380MeV . This value is somewhat larger as expected [83] and is due to the hard core repulsive interactions which make the system rather incompressible. We have checked that in fact the speed of sound does stay below c s = 1 for any density or temperature.

Results at finite net baryon density and temperature

Since our model does not sustain the difficulties that lattice calculations have, when going to finite densities, we can simply extend our investigations to finite chemical potentials (μ B = 3μ q = 0) and temperature. As for the case at T = 0 any repulsive vector interaction for the quarks may change the picture of the phase diagram. The value of the repulsive interaction strength g qω is not constrained by any first principle-calculation. It could even be a function of temperature and density. For simplicity we will compare two cases, g qω = 0 and g qω = 3, to show the qualitative changes of the phase diagram when different values of the vector coupling strength are assumed. For calculations at finite μ B , the strange quark chemical potential μ s also plays an important role. If one assumes, that the total net strangeness is globally conserved, hadronic chemistry induces a non zero chemical potential for the strange quark, while it vanishes in the case of a free quark gas . In the present work we always constrain the net strangeness to be zero, but one should also investigate the phase structure of a system where this is not fulfilled. Work along this line is in progress.

Figures (5 left) and (5 right) display a contour plot of the chiral condensate σ (normalized to its ground state value σ 0 ) in the Tμ q phase diagram for the two values of the repulsive quark interaction strength. It is apparent to see that in both cases the chiral phase transition is a smooth crossover for all chemical potentials. Only at very low temperatures the iso-σ lines converge to a first order phase transition. In the case of g qω = 3 this can be identified with the liquid-gas phase transition, which is first order at zero temperature (displayed in the plot as the solid black line with a critical endpoint at T c ≈ 16MeV ). As mentioned above, if the repulse vector interaction strength of the quarks is smaller, they already appear at or before the nuclear ground state and cause the first order jump in the order parameter. At high temperature and high chemical potentials the lines of constant σ are even further apart than at vanishing net baryon density, indicating an even smoother crossover than at μ q = 0. We can still calculate the derivative of the order parameter with respect to the temperature and chemical potential. The grey dashed lines indicate where this gradient, with respect to the temperature and chemical potential, has his maximum in the phase diagram. The black dashed lines indicate a second, very small maximum in this derivative, which is μ caused be a rapid change in the deconfinement order parameter, the Polyakov loop. This suggests that the largest change in the chiral condensate is governed by hadronic interactions and not by the appearance of the quarks.

The hadrons dominate the change in the chiral condensate because they dominate the scalar baryon density in the region where the change in the condensate appears. At low temperatures the quarks are suppressed due to the Polyakov potential and at intermediate temperature there are simply much more hadronic degrees of freedom that couple to the chiral fields as there are quarks degrees of freedom. If one would introduce hadrons as bound states of quarks then of course the change would be dominated by those bound states (again because there are just many more colour neutral states possible then quark states). But then it would again be the hadrons that dominate the chiral dynamics and not the free quarks.

The observation of a smooth crossover can also be found in the contour plot of the Polyakov loop as a function of T and μ q , in Figures ( 6 left) and (6 right). Here again results for two different values of the repulsive vector interaction are shown. The black dashed lines indicate the maximum of the derivative of the Polyakov loop with respect to temperature and chemical potential. There is a visible difference in the chiral (grey dashed) and deconfinement (black dashed) phase transitions at high chemical potentials. While the chiral condensate already drops at the liquid gas phase transition μ q ≈ 313 MeV, the Polyakov loop remains small until much higher chemical potentials.

This behavior is very interesting as it means that, in some parts of the phase diagram, chiral symmetry is partially restored while quarks are still confined. This can be made more clear if one looks at the fraction of the total energy density which can be contributed to quarks and gluons (more precise quarks and the Polyakov potential). This fraction (e QGP /e TOT ) is plotted in Figure (7), using g qω = 3, in the phase diagram where we again indicated the largest change in the chiral condensate (grey dashed line) and the Polyakov loop (black dashed line). Again one observes, at high chemical potentials and intermediate temperatures, a phase which consists quarks and hadrons, while chiral symmetry is partially restored.

The transition which is often called the 'quarkyonic phase transition' would correspond to the usual nuclear liquid gas phase transition in our model. The dominant degrees of freedom are the hadrons, dominated by very light baryonic states. There where attempts to modify the Polyakov potential in a way to make it depend explicitly on the chemical potential [62]. This way one can strongly couple the chiral phase transition and the change in the Polyakov loop. This approach however would introduce a contribution to the net baryon number from the Polyakov potential as well as a contribution to the quark number susceptibility, nothing of which is seen in lattice calculations.

Conclusion

We presented an approach for modeling an EoS that respects the symmetries underlying QCD, and includes the correct asymptotic degrees of freedom, i.e. quarks and gluons at high temperature and hadrons in the low-temperature limit. We achieve this by including quarks degrees of freedom and the thermal contribution of the Polyakov loop in a hadronic chiral sigma-omega model. The hadrons are suppressed at high densities by excluded volume corrections. Nevertheless, we observe a substantial hadronic contribution to the EoS up to temperatures of 2 times T c . We can show that the properties of the EoS are in qualitative agreement with lattice data at μ B = 0. Various quantities, like the pressure and energy density, are in good agreement with lattice data. Deviations from lattice results can be explained by the hadronic contributions and volume corrections. In spite of a continuous phase transition, we obtain a considerably smaller value for the speed of sound around T c (c 2 s ≈ 0.072) when compared to lattice calculations [73]. At finite baryon density, the transition from deconfined to confined matter proceeds as a smooth crossover for all values of μ B . The same is true for the chiral phase transition (except the liquid gas phase transition, which is of first order at very low temperatures). At high chemical potentials and low temperatures we find a very interesting phase structure. In this region chiral symmetry is partially restored, while deconfinement is not yet realized, thus creating an exotic form of matter.

Figure 1 .

 1 Figure 1. Left:The normalized order parameters for the chiral (red lines), and deconfinement (black line) phase transition as a function of T at μ B = 0. Also indicated is the subtracted chiral condensate as defined in the text (orange dash-dotted line). The dashed lines depict results without the dilaton-Polyakov loop coupling. The symbols denote lattice data for the subtracted chiral condensate from[73], using different lattice actions (asqdat and p4) and lattice spacings N τ . Right: Total particle number densities for the different particle species devided by T 3 as a function of T at μ B = 0. The black line shows the total number of quarks+antiquarks per volume while the green (dotted) line refers to the total meson density and the red (dashed) line to the number density of hadronic baryons+antibaryons. The dashed lines depict results without the dilaton-Polyakov loop coupling.

Figure 2 .

 2 Figure2. Left : The fraction of the total energy density that can be assigned to the quark-gluon phase (e QGP contains the energy of the quarks and the Polyakov potential) as a function of T at μ B = 0. The dashed lines depict results without the dilaton-Polyakov loop coupling. Right : Three times the pressure (red lines) and energy density (black lines) over T 4 as a function of T at μ B = 0. The green dotted line indicates the Boltzmann limit for an ideal gas of three massless quarks and gluons. The symbols denote lattice data from[73], using different lattice actions (asqdat and p4) and lattice spacings N τ . The dashed lines depict results without the dilaton-Polyakov loop coupling.

Figure 3 .

 3 Figure3. Left : Energy density minus three times the Pressure over T 4 as a function of T at μ B = 0. Also referred to as the interaction measure. The symbols denote lattice data from[73], using different lattice actions (asqdat and p4) and lattice spacings N τ . The dashed lines depict results without the dilaton-Polyakov loop coupling. Right : The speed of sound squared, as a function of T/T c at μ B = 0. 1/3 is the ideal gas limit. The dashed lines depict results without the dilaton-Polyakov loop coupling.

Figure 5 .

 5 Figure5. Left : Contour plot of the chiral condensate devided by its ground state value as a function of temperature and quark chemical potential. The value for the quark repulsive interaction is g qω = 0. The dashed grey line indicates where the change of the chiral condensate with respect to T and μ q has a maximum. The solid black line shows the same for the change of the Polyakov loop. Right : Contour plot of the chiral condensate divided by its ground state value as a function of temperature and quark chemical potential. The value for the quark repulsive interaction is g qω = 3. The dashed grey line indicates where the change of the chiral condensate with respect to T and μ q has a maximum. The dashed black line shows the same for the change of the Polyakov loop. The liquid gas phase transition, which is first order at T = 0 is shown as the black solid line with a critical endpoint at T cep ≈ 16MeV .

Figure 6 .

 6 Figure 6. Left : Contour plot of the Polyakov loop as a function of temperature and quark chemical potential. The value for the quark repulsive interaction is g qω = 0. The dashed grey line indicates where the change of the chiral condensate with respect to T and μ q has a maximum. The solid black line shows the same for the change of the Polyakov loop. Right : Contour plot of the Polyakov loop as a function of temperature and quark chemical potential. The value for the quark repulsive interaction is g qω = 3. The dashed black line indicates where the change of the Polyakov loop with respect to T and μ q has a maximum.

Figure 7 .

 7 Figure 7. Contour plot of the fraction of the total energy density which comes from the quark contribution and the Polyakov loop potential. The value for the quark repulsive interaction is g qω = 3. The dashed grey line indicates where the change of the chiral condensate with respect to T and μ q has a maximum while the solid black line shows the same for the change of the Polyakov loop. e F symbolizes the energy density of the mean fields.

Acknowledgments

This work was supported by BMBF, HGS-hire and the Hessian LOEWE initiative through the Helmholtz International center for FAIR (HIC for FAIR). The authors thank D. Rischke, C. Greiner, M. Bleicher, J. Noronha, V. Dexheimer and P. Koch-Steinheimer for fruitful discussions. The computational resources were provided by the Frankfurt Center for Scientific Computing (CSC).


M Majewski
email: m.majewski@merlin.phys.uni.lodz.pl

Where is the pseudoscalar glueball?

The pseudoscalar mesons π(1300), K(1460), η(1295), η(1405) and η(1475) are assumed to form the meson decuplet which includes the glueball as the basis state supplementing the standard SU (3)F nonet of light q q states (q = u, d, s). The decuplet is investigated by using the algebraic approach based on the hypothesis of vanishing exotic commutators (VEC) of SU (3)F "charges" and their time derivatives. This leads to a system of master equations (ME) determining: (a) octet contents of the physical isoscalar mesons, (b) the mass formula relating all masses of the decuplet and (c) the mass ordering rule. The states of the physical isoscalar mesons η(1295), η(1405), η(1475) are expressed as superpositions of the "ideal" q q (N and S) states and the glueball G one. The "mixing matrix" realizing transformation from the unphysical states to the physical ones follows from the octet contents and is expressed totaly by the decuplet meson masses. Among four one-parameter families of the resulting mixing matrices (multitude of the solutions arising from bad quality of data on the π(1300) and K(1460) meson masses) there is a family attributing the glueball-dominated composition to the η(1405) meson. The pseudoscalar decuplet is similar in some respects to the scalar one: both are composed of the excited q q states and G; the mass ordering of their N, S, G -dominated isoscalars is the same. Contrary to the Lattice QCD and other predictions, the mass m G -+ of the pseudoscalar pure glueball state is smaller than the scalar m G ++ one.

Introduction

•• The pseudoscalar glueball investigation has been initiated soon after it was realized that bound states of the gluons may play important role in the strong interactions [START_REF] Fritzsch | Proc. of the XVI Int. Conf. on High Energy Physics[END_REF][START_REF] Fritzsch | [END_REF]. From the very beginning the glueball state was traced within structure of the η and η mesons [3,4]. At present, this is not the main purpose of the investigation, but is still continued, and not only within the meson structures [5,6] but also within the baryon ones [7].

The discovery of the ι meson [8,9] rous hopes for existence of the glueball. ι meson has been detected in the gluon rich process of the J/ψ radiative decay and was immediately claimed to be a glueball. However, the glueball may exist as separate particle only if it has exotic quantum numbers; otherwise it should be mixed with the isoscalar q q states having the same signatures J P C . The mass of the ι meson belongs to the region of higher-lying 0 -+ multiplet and the states of this multiplet were (and still are) poorly known. That posed the question of how to certify such assignment. To this end several criteria have been invented which could be used in the cases of deficient multiplets. Some of them, concerning production, are pure qualitative like "creation in the gluon-rich environment", other ones, more regarding decay products, are semi-quantitative (big value of "stickness" [START_REF] Chanowitz | Proc. of the VI Int. Workshop on Photon-Photon Collisions[END_REF] and "gluiness" [START_REF] Close | [END_REF]). At the same time, the question has been risen whether the glueball is necessary for understanding data concerning the pseudoscalar mesons known at that time [12]. This question is still alive [13].

The trend of discussion has changed since the results of the Lattice QCD (LQCD) calculations became available [14,15,16]. They supported the very existence of the pseudoscalar glueball, but the mass attributed was about 2.3GeVmuch above the ι. An attempt to lower the lattice prediction by including quark loops was not very successful [17]. Although the doubts were not dispelled (see e. g. [18]), this became a serious obstacle for ι to be recognized as the glueball candidate, because the results of the Lattice calculations are generally accepted. On the other hand, there is no candidate having the mass predicted by Lattice. Perhaps that induced several attempts to interpret the meson X(1835) as the pseudoscalar glueball [19,20,21], although its mass might be regarded as too low also.

At the same time, starting from late 80's there was growing conviction that ι(1440) signal should be attributed to two different isoscalar mesons [22]. Much experimental effort was devoted to understanding the structure of the signal [START_REF] Bai | [END_REF]24,25,26,27]. As a result, it has been split into η(1405) and η(1475). Hence, since 2004 three isoscalar pseudoscalar mesons has been listed in RPP within the narrow interval of mass [28]:

η 1 = η(1295), η 2 = η(1405), η 3 = η(1475). (1)
Such three isoscalar mesons with similar masses in the vicinity of the isotriplet and isodublet suggest overpopulation of a nonet and possible existence of a glueball which is hidden within the structures of three isoscalar states. The decuplet findings are very important because investigation of its properties is the most promising way of glueball search unless the glueball with exotic quantum numbers will be detected.

•• Information about the structures of the isoscalar mesons η 1 , η 2 , η 3 has been extracted from data on the reactions of their production and from branching ratios of their decays. Data suggest that the meson η 2 is a particle dominated by the glueball state [START_REF] Close | [END_REF]28,29,31].

An unexpected objection has been risen against such a picture: the η 1 has been claimed to be not the q q state [13]. Even its very existence was considered uncertain. That implies non-existence of the decuplet and requires much more complicated spectroscopy of the pseudoscalar mesons. Therefore, we discuss this question in more detail.

The q q structure is put in doubt due to not occurrence of the η 1 in the reactions pp, J/ψ, γγ → ,

"at least not with the expected yields". The base of such expectation is not indicated. However, this is not the only point of view concerning these reactions. The authors of recently published, very careful analysis of the experimental data on η 1 , came to the following conclusions [29]: (i) the charge exchange experiments π --p → nηππ, nK Kπ 0 definitively establish evidence of the η 1 ; (ii) a clear signal of η 1 is seen in the J/ψ radiative decay; (iii) there is an indication for the existence η 1 in pp annihilation; (iv) the LEP data on γγ reaction are compatible with existence of the η 1 signal.

So the η 1 is seen or not in the reactions (2). The three-body decays η 1 → ηππ , K Kπ are strongly suppressed by small phase space (cf ω → πππ) and that may be the reason why it is difficult to observe the η 1 . It is explicitly seen in the reaction π --p → η 1 n, for which high statistics is available, but the number of events observed in reactions ( 2) is many times smaller [28]. Obviously, more measurements are needed to elucidate the situation. But this question has no relevance to the problem of the η 1 internal structure. As in the reaction (2) the η 1 is observed throughout the products of decay, the frequency of its registration depends on width; the subsequent measurements would verify the magnitude of the width. However, the definition of the multiplet does not depend on the widths. Therefore, the widths of the particles cannot be the basis for any conclusion about the structure of the multiplet. Also the width of the η 1 cannot be the base for conclusion about its q q structure. An attempt to call in question this structure resembles confusion which arose after denying the q q structure of the f 0 (980) meson motivated by its small width [30].

In the present paper we admit the η 1 meson to be a q q state and assume that the examined pseudoscalar mesons form a decuplet. We thus focus glueball search again in the region of ι meson -this time being fully aware of the conflict with Lattice prediction.

The glueball assignment of the η 2 meson is also motivated on theoretical ground [32]. It is argued that η 2 meson is a natural pseudoscalar glueball candidate if the f 0 (1500) is the scalar glueball and the glueballs are described as the closed gluonic flux-tubes. Then the f 0 (1500) and the η(1405) would be two parity related glueballs with equal masses. This description deserves attention in view of failure of the Lattice prediction, particularly if it can be treated not too literally.

It is thus interesting to make sure that this assignment can be confirmed by an argument based on the properties of the flavor multiplet as a whole.

•• We conclude the introduction with few comments concerning credibility of the approach we use in this paper. The credibility is especially important in evaluating the glueball contents of the decuplet isoscalar states.

It is currently known that broken SU (3) F symmetry predicts the existence of octets and nonets of light mesons. The multiplets are usually testified by the mass formula relating their masses. The Gell-Mann-Okubo (GMO) and Schwinger (S) mass formulae have been obtained by inclusion into the lagrangean the non-invariant mass term with regard to mixing of the octet isoscalar with the unitary singlet.

Our model unifies and generalizes these mass relations. The model has been introduced at the University of Lodz in the middle of '80s [33,34] and is based on requirement of vanishing the exotic commutators (VEC) of the "charges" and their time derivatives. Apart from the GMO and S mass formulae it gives additional insight into the properties of the multiplet. For the S nonet the model VEC determines the mixing angle and establishes the mass ordering rule which ensures the mixing angle to be a real number. There are two possible orderings. For one of them the mixing angle ϑ is smaller than ideal ϑ < ϑ id , while for the other one it is bigger ϑ > ϑ id (ϑ id ≈ 35 o ).

The model predicts also the ideally mixed (ideal) nonet (I). This nonet has not been derived, as yet, from any other mixing description. In the quark model, where it is the basic object, it is postulated.

The S and I mass formulae are well obeyed by many nonets with various signatures J P C comprising low mass mesons [35]. In general, the S nonets better describe data, although differences between I and S descriptions are small.

For the glueball quest the most important is the prediction of a decuplet [34] -a multiplet comprising three isoscalar mesons. The mass formula, mass ordering rule and the octet contents of the physical isoscalar states follow from the same constraints. The octet contents l 2 i play a key role in determining mixing matrix of the isoscalar states, i.e. the contributions of glueball state to their structures. The orthogonal 3 x 3 mixing matrix can be parametrized by Euler angles. Absolute values of the trigonometric functions of these angles are expressed by the particle masses, i.e. by the masses.

The VEC model does not use additional assumptions nor introduces free parameters to describe multiplets. Its predictions are definite and applicable for decuplets of any signature J P C . If fitted with required experimental input, it offers complete description of the decuplet states. Thus, it bestows quantitative meaning to the most obvious qualitative signature of the glueball presenceoverpopulation of a nonet.

The model has appeared very effective in describing the 0 ++ mesons. It makes possible to sort out twenty scalar mesons among multiplets and attribute the glueball dominating structure to the f 0 (1500) meson [30]. To analyze the 0 -+ decuplet we use essentially the same model. Our present analysis is proceeded in a different way because sample of the input data is different.

The decuplet of pseudoscalar mesons

The model of vanishing exotic commutators (VEC)

The following sequence of exotic commutators is assumed to vanish1 T a , d j T b dt j = 0, (j = 1, 2, 3, ...)

where T is SU (3) F generator, t is the time and (a, b) is an exotic combination of indices, i.e. such that the operator [T a , T b ] does not belong to the octet representation. Substituting dT dt = i[H, T ], and using the infinite momentum approximation for one-particle hamiltonian H = m 2 + p 2 [36], we transform eqs.

(3) into the system: where m2 is the squared-mass operator.

[T a , [ m2 , T b ]] = 0, [T a , [ m2 , [ m2 , T b ]]] = 0, [T a , [ m2 , [ m2 , [ m2 , T b ]]]] = 0, (4)
For the matrix elements of the commutators (4) between one-particle states (we assume one-particle initial, final and intermediate states) we obtain the sequence of equations involving expressions x 8 |(m 2 ) j |x 8 with different powers j = 1, 2, 3, .., where x 8 is the isoscalar state belonging to the octet. Solving these equations, we obtain the sequence of formulae for a multiplet of the light mesons. We find

x 8 | (m 2 ) j | x 8 = 1 3 a j + 2 3 b j (j = 1, 2, 3, ...). ( 5
)
where a is the mass squared of the isovector meson π; b is the mass squared of the subsidiary

ss state, b = 2K -a, (6)
and K, in turn, is the mass squared of the isospinor K meson. The isoscalar octet state | x 8 can be represented as the linear combination of the physical isoscalar states

| x 8 = l i | x i . (7)
The coefficients l 1 , l 2 , l 3 ,.. determine octet contents of the physical isoscalar states |x 1 |x 2 , |x 3 ,... Substituting ( 7) into ( 5) we obtain master equations (ME) of the multiplet.

l 2 i x j i = 1 3 a j + 2 3 b j , (j = 0, 1, 2, 3, ...) ( 8 )
where the x 1 , x 2 , x 3 ,... are isoscalar meson masses squared. Normalization condition of the l i coefficients is included into (8) as equation for j = 0.

Master equations for the decuplet

The states of the decuplet belong to a reducible representation of the SU (3)

F 8 ⊕ 1 ⊕ 1,
where the octet and one of the singlets are considered as q q states while the second singlet is supposed to be a glueball G.

For the decuplet we have following system of the master equations [34,33] which determines masses and mixings of the decuplet states [30]:

l 2 1 x j 1 + l 2 2 x j 2 + l 2 3 x j 3 = 1 3 a j + 2 3 b j , (j = 0, 1, 2, 3) (9)
Table 1: Pseudoscalar mesons merged into decuplet. Status of the particles and their masses (in MeV) are quoted after RPP [28] •π(1300

) ± 100 K(1460) •η(1295) •η(1405) •η(1475) 1300 ± 100 1294 ± 4 1410.3 ± 2.6 1476 ± 4
where x 1 , x 2 , x 3 are the masses squared of the isoscalar mesons η 1 , η 2 , η 3 . The coefficients l 1 , l 2 , l 3 , are real, as all isoscalar mesons are neutral particles. The ME ( 9) are considered as a system of linear equations with respect to unknown coefficients l 2 i . The solution is given by three kinds of relations [34,30]: a) the octet contents (OC) of the isoscalar states

l 2 1 = 1 3 (x 2 -a)(x 3 -a) + 2(x 2 -b)(x 3 -b) (x 1 -x 2 )(x 1 -x 3 ) , ( 10a
)
l 2 2 = 1 3 (x 1 -a)(x 3 -a) + 2(x 1 -b)(x 3 -b) (x 2 -x 1 )(x 2 -x 3 ) , ( 10b
)
l 2 3 = 1 3 (x 1 -a)(x 2 -a) + 2(x 1 -b)(x 2 -b) (x 3 -x 1 )(x 3 -x 2 ) ; ( 1 0 c ) b) the mass formula (MF) f (a) + 2f (b) = 0, (11)
where

f (x) = (x 1 -x)(x 2 -x)(x 3 -x) ( 1 2 )
is characteristic polynomial of the m 2 operator; the numbering of its eigenvalues is chosen such as to satisfy the inequality

x 1 < x 2 < x 3 ; ( 1 3 )
c) the mass ordering rule (MOR)

x 1 < a < x 2 < b < x 3 . (14)
The MF ( 11) is a linear equation with respect to each of the x i , but it is a cubic one with respect to a and b.

The masses and experimental status of the mesons assigned to the decuplet are quoted in the Tab. 1. The table shows that masses of the isoscalar mesons η 1 , η 2 , η 3 are determined with good accuracy; the π(1300) meson mass has large error; the K-meson is not yet established -its mass is unknown. Therefore, these two masses should be considered unknown. It is natural in this model to choose a and b (6) as unknown variables of the ME.

For solving the ME and constructing the mixing matrix of the decuplet the solution of the MF is needed. However, MF is a single equation and its solution cannot be unique. Yet, high precision of the data on isoscalar meson masses provides correct form of the characteristic polynomial f(x) of the m 2 operator as well as precise values of the a and b bands which are required by MOR (14). We hope that restrictions of the model on decuplet states will reduce ambiguity of the solution.

The restrictions imposed by MOR ( 14) are obvious. The unknown variables a and b have to satisfy MOR which requires a ∈ (x 1 , x 2 ) and b ∈ (x 2 , x 3 ). That restricts also the K-meson mass. It follows from ( 6) and ( 14) that

x 1 + x 2 < 2K < x 2 + x 3 , (15)
or 1353MeV < m K < 1443MeV. ( 16
)
From a ∈ (x 1 , x 2 ) we have

1294MeV < m π < 1410MeV. ( 17
)
Comparing the bands (17) with the range of error of the π(1300) meson mass we find that the MOR cuts off lower part of the error range and that the MOR-allowed region covers the upper part of it. This is consistent with treating a as an unknown quantity of the ME.

Families of solutions of the ME

Combining MOR with MF we restrict the unknown masses much stronger. Moreover, as will be seen below, the allowed masses can be attributed to the solutions of ME with explicit flavor properties.

Fig. 1 displays f (x) and -2f (x) (11) (c.f. [37]). The pair of the unknown variables (a, b) provides a solution of the MF if they are such that f (a) = -2f (b). It can be seen from the figure that beside the MOR restrictions a ∈ (x 1 , x 2 ), b ∈ (x 2 , x 3 ) there also appears the MF restriction forbidding a ∈ (x P , x Q ). Hence, the allowed values of a belong to two narrow intervals: a ∈ (x 1 , x P ) and a ∈ (x Q , x 2 ). To each allowed value of a there correspond two values of b (obeying the mass formula) placed on the opposite sides of the point x R . If we wish to have unique solution, we should divide the interval (x 2 , x 3 ) into two parts: (x 2 , x R ) and (x R , x 3 ). Then, we get four domains including unique pairs of values (a, b) making solutions of the MF:

A : a ∈ (x 1 , x P ), b∈ (x R , x 3 ), (18a) B : a ∈ (x 1 , x P ), b∈ (x 2 , x R ), (18b) C : a ∈ (x Q , x 2 ), b∈ (x R , x 3 ), (18c) D : a ∈ (x Q , x 2 ), b∈ (x 2 , x R ), (18d)
where

x P (1.320GeV ) 2 , x Q (1.365GeV ) 2 , x R (1.447GeV ) 2 . ( 19
)
These values correspond to

x 1 = (1.294GeV ) 2 , x 2 = (1.410GeV ) 2 , x 3 = (1.475GeV ) 2 . ( 20
)
The domains A, B, C, D are shown on the Fig. 2. We solve the ME (9) in each of them separately and express b as functions of a. The details of solving the MF as well as properties of the solutions are described in Appendix. Next we calculate the octet contents of the physical isoscalar decuplet states l 2 1 , l 2 2 , l 2 3 [START_REF] Chanowitz | Proc. of the VI Int. Workshop on Photon-Photon Collisions[END_REF] and construct the mixing matrix of the states η 1 , η 2 , η 3 [30]. The mixing matrix V is chosen such that

  η 1 η 2 η 3   = V   N S G   , (21)
where

N = 1 √ 2 (uū + d d), S = ss, G-glueball. (22)
So V expresses the states of the physical isoscalar mesons η 1 , η 2 , η 3 in terms of the decuplet ideal states N , S and G. The V is an orthogonal matrix. Its elements are defined by the masses.

In each of the domains A, B, C, D there is one point where the solution of the ME ( 9) is degenerate. The points are placed at the corners of domains A, B, C, D as is shown in the Fig. 2. In the first three domains we find three different ideally mixed q q nonets and a detached glueball; in the domain D we obtain degenerate decuplet composed of the octet states and two singlet states detached from the octet:

A : x 1 = a, x 3 = b, l 2 1 = 1 3 , l 2 2 = 0, l 2 3 = 2 3 , (23a)
B :

x 1 = a, x 2 = b, l 2 1 = 1 3 , l 2 2 = 2 3 , l 2 3 = 0, (23b)
C :

x 2 = a, x 3 = b, l 2 1 = 0, l 2 2 = 1 3 , l 2 3 = 2 3 , (23c)
D :

x 2 = b = a = x 8 , l 2 1 = 0, l 2 2 = 1, l 2 3 = 0. ( 23d
)
For the wave functions one obtains:

A : η 1 = ±N, η 2 = ±G, η 3 = ±S, (24a) B : η 1 = ±N, η 2 = ±S, η 3 = ±G, (24b) C : η 1 = ±G, η 2 = ±N, η 3 = ±S, (24c) D : η 1 = γ 1 , η 2 = ±x 8 , η 3 = γ 2 . ( 24d
)
Each of the degenerate solutions A, B, C points out its own candidate from among η 1 , η 2 , η 3 as a pure glueball. The solution D describes degenerate decuplet where η 2 is the octet isoscalar η 8 state and η 1 , η 3 are scalar states built as superpositions of the (q q) singlet and G.

The intervals (x 1 , x P ) and (x Q , x 2 ) of the variable a allowed by MF and MOR are small. Also the intervals (x 2 , x R ) and (x R , x 3 ) of the variable b are small. Therefore, the domains A, B, C, D are also small and across any domain the solutions are not much different from the degenerate ones. The solutions of ME in any given domain constitute one-parameter family. To each of the domains there corresponds such a family. The solutions belonging to the same family are dominated by the same structure (N, S, G, η 8 ) which is pure in the degenerate solutions. That can be seen from the Tab. 2. Hence, the dominant structures of the η 1 , η 2 , η 3 in the domains A, B, C, D preserve the patterns of degenerate decuplet (24):

A : η 1 ∼ N, η 2 ∼ G, η 3 ∼ S, (25a)
B : η 1 ∼ N, η 2 ∼ S, η 3 ∼ G, (25b)
C : η 1 ∼ G, η 2 ∼ N, η 3 ∼ S, (25c)
D : η 1 ∼ γ 1 , η 2 ∼ η 8 , η 3 ∼ γ 2 , (25d)
where η 8 is the octet isoscalar state and γ 1 , γ 2 are superpositions of the q q singlet and G. Their contribution to the γ 1 and γ 2 states can be expressed by masses of the physical isoscalar mesons and are slowly varying functions inside the domain D.

Let us give the examples of the mixing matrix of the A, B, C, D solutions near degeneracy.

In each example the value of parameter ∆a is chosen such that the deviation of the π(1300) meson mass m a from its ideal value (i. e. from the η 1 or from the η 2 meson mass) is equal to 6MeV . The choice of this number is to some extend arbitrary. We want to have a decuplet which is deviated both not too little and not too much from the degenerate one; 6 MeV is the difference between the mean RPP values of π(1300) and η 1 masses.

Table 2:

The range of changes of the glueball contents under variation of a within the domains A, B, C and the octet content within the domain D. The intervals of the K-meson mass allowed over these domains are also shown. All masses are in GeV.

A a ∈ ((1.294

) 2 , (1.318) 2 ) 1 ≥ (V 23 ) 2 ≥ (0.764) 2 1.388 ≤ m K ≤ 1.390 B a ∈ ((1.294) 2 , (1.318) 2 ) 1 ≥ (V 33 ) 2 ≥ (0.842) 2 1.353 ≤ m K ≤ 1.374 C a ∈ ((1.377) 2 , (1.410) 2 ) 1 ≥ (V 13 ) 2 ≥ (0.810) 2 1.420 ≤ m K ≤ 1.443 D a ∈ ((1.377) 2 , (1.410) 2 ) 1 ≥ l 2 2 ≥ (0.909) 2 1.403 ≤ m K ≤ 1.410 A. a = (1.300GeV ) 2 , b = (1.471652GeV ) 2 , K = (1.388GeV ) 2 , l 2 1 = 0.
Tab. 2 exhibits also intervals of the admissible K meson mass corresponding to these solutions. Its changes under variations of ∆a are in all domains relatively small and the ranges of admissible values in different domains are strictly separated.

It follows that the states of the pseudoscalar mesons π(1300), K(1460), η 1 , η 2 , η 3 may constitute solution of the ME. The price to pay for ignorance concerning both π(1300) and K(1460) meson masses is the ambiguity of the solution: instead of unique solution we have four qualitatively different one-parameter families of solutions. These families are defined within four separated domains A, B, C, D of the (a, b) plane and can be distinguished due to the fact that the isoscalar physical states are dominated by one of the N , S, G or η 8 component. Hence the domination pattern enables us to distinguish between the families of the solutions A, B, C, D of the ME. With the present data on masses of the π(1300) and K(1460) mesons we can only distinguish between the families. If one of these masses was known then there would be only two solutions (not two families of solutions !) as can be seen from the Fig. 1.

The data on flavor properties of the isoscalar mesons indicate the family A as the one which points out the meson η 2 as a particle dominated by the glueball state.

Comments on solutions of the ME

•• A decuplet of mesons is a multiplet such that the octet isoscalar state η 8 contributes to three isoscalar physical states η i . The contributions of the isoscalar octet state to the physical states η i , given by l 2 i s (10), constitute solution of the ME. The coefficients l i are real numbers, therefore, the following conditions should be satisfied

l 2 i > 0 /i = 1, 2, 3/. ( 30
)
This property is not guaranteed by solution [START_REF] Chanowitz | Proc. of the VI Int. Workshop on Photon-Photon Collisions[END_REF]. Requiring it, we put constraints on the masses of the decuplet. The knowledge of the octet contents l 2 i s provides very convenient way for constructing the mixing matrix. This 3 x 3 orthogonal matrix can be parametrized by Euler angles. The absolute values of trigonometric functions of two of the angles can be expressed explicitly by l 2 i s, i.e. by the masses. The requirement of the glueball flavor independence

< G|m 2 |uū >=< G|m 2 |d d >=< G|m 2 |ss >, (31)
relates them to the trigonometric functions of the third angle. However, some signs of the trigonometric functions cannot be determined if only masses are known. To find them we use available information on domination of the η i states by one of the N , S, G, η 8 states. As a result, all the Euler angles, and consequently, all elements of the mixing matrix are uniquely determined [30].

•• The VEC description of the decuplet depends on the number of ME (9) which are taken into consideration. There are two kinds of decuplets [30]. i) A decuplet which is based on assumption that three exotic commutators vanish. Then four ME arise. If they were applied to the nonet, they would define the ideal (I) one. We may imagine the isoscalar states as superpositions of a glueball and the I nonet states. We say that this decuplet is of the kind I. It complies with one mass formula [START_REF] Close | [END_REF] which, together with conditions (30), defines explicit MOR restrictions for the masses (14). To admit also the degenerate solutions of the ME one must allow l 2 i ≥ 0 for some "i" and in the MOR " ≤ " instead of the " < ". The solutions of the ME do not include free parametersall predicted quantities and relations are expressed by physical masses. ii) A decuplet arising under assumption that two exotic commutators vanish. Then there are three ME. If applied to the nonet, they give the S one. The decuplet is of the kind S if it is formed as superposition of the glueball and the S nonet. In this case the mass formula and ordering rule do not arise. The restrictions on the masses are not so strict and follow from the conditions (30).

If the masses of ten mesons with proper quantum numbers are known and satisfy (30), but do not satisfy the MF, the decuplet is of the kind S. If one of the masses is unknown then we can determine it from the MF and the decuplet becomes of the kind I, provided the masses satisfy (30). However, the states constituting these two distinct decuplets may be not very different.

•• The solutions A, B, C, D described above concern the decuplet I. We now summarize the main features of the solution.

The analysis of the pseudoscalar decuplet presented here does not give the unique result due to the fact that for determining two unknown masses we have only one MF. Moreover, the MF is represented by polynomial of the third degree with respect to each of these variables.

The masses of π(1300) and K(1460) mesons are unknown. So the values a and b which are natural variables in the VEC model are also unknown, but they are bounded by MOR from the below and above. These restrictions are helpful in choosing proper solution of the nonlinear equation [START_REF] Close | [END_REF]. Further restriction is provided by the mass formula which cuts out the central part of the MORallowed interval of a and thus reduce it to two narrow disconnected subintervals (see Fig. 1). To each a belonging to them there correspond two values of b. We divide the interval (x 2 , x 3 ) of the values b to two parts. As a result, the whole domain of the values of a and b is reduced to four small domains A, B, C, D which are shown on the Fig. 2. In these domains solution is unique if one of the variables, a or b is known.

Hence, due to the restrictions of the ME on a and b, the solution is split into four one-parameter, qualitatively different families. The partition allows us to look for solution in each domain separately. Still we have two unknown masses and only one MF equation relating them, but the domains are small and the solutions are only slightly changing across them. The ranges of the masses of the a and K mesons over the domains can be found out from the Tab. 2.

The partition of the whole domain of variables (a, b) is especially helpful for describing properties of mixing matrix. To each of the domains A, B, C, D there is attributed a separate one-parameter family of solutions of the MF determining the decuplet -among them the degenerate one. A family of the MF solutions, in turn, induce one-parameter family of mixing matrices. All the matrices of the family preserve common dominance pattern. This pattern is determined by dominance of one of the N, S, G, η 8 amplitudes in the η 1 , η 2 , η 3 states and can be read out from the degenerate solution. In each domain the degenerate decuplet corresponds to a point at the outer corner of the domain (see Fig. 2). Across the domain the pure state of the degenerate decuplet is transformed into dominating one and all isoscalar states become mixed. Within the domains A, B, C the glueball dominates η 2 , η 3 , η 1 states, respectively.

In the domain D the dominance pattern is different. The degenerate decuplet consists of the octet of exact symmetry and two separate singlets being mixed states of the (q q) singlet and G. The η 2 is pure octet η 8 , while η 1 and η 3 are pure singlets. The rates of the (q q) singlet and G states in the structures of η 1 and η 3 mesons are comparable, slowly changing functions of the parameter ∆a within the domain. In spite of identical flavor properties of the constituents, the properties of the η 1 and η 3 mesons should be different and the difference is changing across the domain. This is mainly due to the fact that they are opposite superpositions of the G and (q q) singlet amplitudes. In this family of solutions the glueball state is not apparent.

On account of so different properties of the families A, B, C, D the qualitative information on the isoscalar mesons is sufficient to make the choice. The proper family can be chosen on the basis of flavor properties of the isoscalar mesons η 1 , η 2 , η 3 . As it has been pointed out, even if all masses are known and satisfy MF and MOR, such an extra information is necessary for constructing the mixing matrix. An exact solution of the ME corresponding to definite values of the π(1300) and K(1460) masses would be determined by suitable value of ∆a.

•• The restrictions following from the Fig. 1 do not hold for decuplet of the type S. There is no mass formula in this case; therefore, there is no connection between the variables a and b and there is no mass gap (x P , x Q ) of the a meson. Also the MOR ( 14) does not exist. Such situation can arise for the decuplet we discuss if the measured masses of the π(1300) and K(1460) mesons will not satisfy MF. However, to define a decuplet of any type we always need mesons having such masses that conditions (30) are satisfied. These conditions give weaker constraints on a and b. We find

x 1 < x 8 < x 3 , (32)
where x 8 is given by the GMO mass formula,

x 8 = 1 3 a + 2 3 b. (33)
The glueball contents of the isoscalar mesons η 1 , η 2 , η 3 can be always calculated from [START_REF] Chanowitz | Proc. of the VI Int. Workshop on Photon-Photon Collisions[END_REF] if the masses m π and m K satisfy l 2 i > 0 /i=1,2,3/. Having known the l 2 i s we can construct the mixing matrix. If the state of η 2 is predicted to be dominated by G the solution of ME should be similar to the one describing the states of the family A.

Pseudoscalar vs scalar meson multiplets

Parity related spectra of the spin 0 mesons

Having described the multiplets of pseudoscalar mesons we get the opportunity to confront its properties with the properties of the corresponding multiplets of scalar mesons [30]. The comparison may reveal some new features of the meson spectroscopy.

Let us compare the 0 -+ and 0 ++ multiplets. The ground states form the nonets:

π, K, η, η , (34)
a 0 (980), K 0 (1430), f 0 (980), f 0 (1710). ( 35
)
which are followed by the decuplets:

π(1300), K(1460), η(1295), η(1405), η(1475), (36)
a 0 (1450), K 0 (1950), f 0 (1370), f 0 (1500), f 0 (2200)/f 0 (2330).

In both cases we have the same sequence of the multiplets. Some of the masses are not exactly known, but this does not spoil the general picture. Let us observe that not only the sequences of the multiplets are similaralso the inner structures of the decuplets are; namely: -the physical mesons f 0 (1500) and η(1405) which are dominated by glueball states are settled just between the remaining isoscalars which are expected to be mostly the N and S quark states, -both decuplets involve excited q q states, hence both glueballs mix with the excited (q q) isoscalar states.

The latter property suggests affinity of the glueball with the excited states. This is especially prompted by the mixing of the 0 ++ glueball. Its mass belongs to the region where the nonet ground states and the decuplet excited states are overlapping, but the glueball prefers mixing just with the excited q q statesthere is no trace of mixing with the ground q q states [30].

The 0 -+ and 0 ++ mesons form the parity related spectra of multiplets (nonets and decuplets). The sequences of these multiplets differ only due to existence of the scalar meson σ(600) which has no adequate pseudoscalar partner. But the nature of this meson is still a matter of discussion. Several authors suggest that its nature is different from the nature of other mesons [START_REF] Pennington | Frascati 1999, Hadron Spectroscopy[END_REF][START_REF] Minkowski | [END_REF][START_REF] Anisovich | [END_REF]. By ignoring the σ(600) we find that 0 -+ and 0 ++ mesons form parity related spectra of multiplets.

The transparency of this picture confirms not only the opinion about the distinct nature of the σ(600), but also supports correctness of sorting out the scalar mesons between the overlapping multiplets [30].

However, there is also a difference between these spectra. The mass spread of the 0 -+ multiplets is shrinking for consecutive multiplets (nonet 137÷958 MeV, decuplet 1295 ÷ 1475 MeV, perhaps degenerate octet at 1800 MeV). The tendency of shrinking the mass spread of the higher lying multiplets is even more clearly expressed in the spectrum of 1 --mesons where all known multiplets above 1400 MeV (at 1400 MeV and 1800 MeV) are degenerate octets [35]. But this tendency is not seen in the spectrum of the 0 ++ multiplets -at least below 2300 MeV.

The masses of the spin 0 glueballs

The Lattice QCD (LQCD) calculations predict for the lower bound of the lightest 0 ++ glueball the mass [14,15,16]

m G ++ ≈ 1.500GeV. ( 38
)
Such mass allows to attribute the glueball nature to several mesons -among them to favored f 0 (1500).

For the lightest 0 -+ glueball these calculations predict the lower bound at

m G -+ ≈ 2.300GeV. ( 39
)
With this value no isoscalar meson discussed here can be assigned to be the glueball. Attempts to diminish this bound were unsuccessful.

The LQCD calculations predict also the lower bounds for the masses of many other glueballs with different J P C . The result is that the mass (38) marks the minimum of these lower bounds. However, this result can be obtained also in other approaches (see, e.g. West's Theorem [41]). Therefore, it is considered more general and independent of particular approach.

The LQCD predicts masses of pure glueball states. Also the flux tube (FT) concerns such states.

The FT approach predicts, however, that masses of the 0 -+ and 0 ++ glueballs should be equal. Since f 0 (1500) is the favored 0 ++ glueball candidate, we should expect the 0 -+ glueball mass at about 1.500GeV . Hence, the LQCD and FT predictions on 0 -+ glueball mass are contradictory.

The VEC prediction of the glueball mass has different source. It refers to broken unitary symmetry which collects the mesons in multiplets -the octets and the nonets. Several of them are well established in the low mass region at various J P C . We assume that at higher masses the mesons are collected in multiplets as well. In the case where the glueball appears we expect the decuplet. The three isoscalar components of the decuplet are superpositions of the q q and G states. There is no pure glueball state but such state may dominate one of the isoscalars.

The mass formula for the decuplet relates physical masses. Also the mixing matrix is explicitly determined by physical masses. There is no ambiguity and only physical masses enter. Therefore, predictions are definite and, in favorable case, may help to perceive something new.

Using the mixing matrix we can calculate the mass of the pure glueball state. We can do this for the decuplets 0 -+ and 0 ++ separately (not assuming any relation between them).

From the decuplet 0 -+ , assuming Solution A and the mass input appropriate to the mixing matrix (26) we find

m G -+ = 1.369GeV. (40)
From the decuplet 0 ++ for Solution 1 [30] we get

m G ++ = 1.497GeV. (41)
The difference between these predictions is approximately equal to the π meson mass.

m G ++ -m G -+ = m π . (42)
Also observe that the inequality

m G -+ < m G ++ (43)
holds for all families A, B, C, D despite of LQCD calculations and West's Theorem predictions. Let us comment.

The VEC search of the glueball is carried on within the isoscalar sector of the decuplet. Prediction of the glueball mass consists in setting all masses of the decuplet and fitting the mixings of the isoscalar components to their flavor properties. Hence, the mixings play important role in glueball determination.

The FT prediction of equality of the 0 -+ and 0 ++ glueball masses is approximately obeyed. This may follow from the fact that G contributions to their structure are high and almost equal: for 0 -+ decuplet (solution A (26))

V A =   xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx -0.94154 xxxxxx xxxxxx xxxxxx   .; ( 4 4 )
for the 0 ++ decuplet [30]

V 1 =   xxxxxx xxxxxx xxxxxx xxxxxx xxxxxx -0.88466 xxxxxx xxxxxx xxxxxx   . (45)
Really the masses of the 0 -+ and 0 ++ glueballs are not identical. Difference between them, although small in the scale of the mass of these glueballs, is not negligible. This difference is a result of parallel but independent examination of the relations between physical masses of the decuplet particles. Perhaps, different masses of these glueballs suggest that different decuplets affect the glueball component not identically. That may concern not only the cases of different J P C but also decuplets of the same J P C in different mass regions (if such ones exist). The West's Theorem is not fulfilled: the observed mass difference (left part of the (42)) has opposite sign. The absolute value of the difference is not predicted. Also the difference between the masses predicted by LQCD has wrong sign. Beside, the value of this difference is probably too large to be explained as the mixing effect.

Conclusions

1. Owing to the unknown masses of π(1300) and K(1460) mesons the solution of ME for decuplet π(1300), K(1460), η(1295), η(1405), η(1475) is not unique. In spite of that due to the restrictions of the VEC model all solutions can be classified into four separate families; one of the families points out η(1405) as the particle dominated by glueball state. 2. The scalar and pseudoscalar glueballs belong to the decuplets formed by mixing G with excited q q isoscalar states. 3. The spectra of the known multiplets of the 0 -+ and 0 ++ mesons are parity related provided the σ(600) is ignored. 4. The mass of the pure pseudoscalar glueball state m G -+ is smaller than the mass m G ++ of the scalar glueball one. and b, together with the functions l 2 i (∆a, x 1 , x 2 , x 3 ), as an approximate solution of the ME (9) in the appropriate domain. This procedure is to be performed for all the domains A, B, C, D.

In the domains A and B we may neglect all terms of (11) containing higher degrees of ∆a or ∆b and restrict ourselves to the linear dependence between them. The approximation is plausible for ∆a covering all the interval (x 1 , x P ). We obtain

A : ∆ b = ∆a 2 x 2 -x 1 x 3 -x 1 , (48a)
B : ∆ b = ∆a 2

x 3 -x 1 x 3 -x 2 . ( 48b
)
In the domains C and D we take into account also the term quadratic in ∆b and all powers of ∆a. This is to avoid l 2 1 < 0 in the domain C and to extend applicability of this approximation towards the largest values of ∆a in the domain D. In these cases the expressions for ∆b are the solutions of the quadratic equation, so they are simple but long and we do not write them out. Two solutions of the quadratic equation for ∆b do not cause confusion, as only one of them complies with the condition l 2 i > 0 for all i=1,2,3. In both regions the approximation is plausible everywhere, except the small surroundings of the point b = x R .

In all the solutions A, B, C, D the value ∆a = 0 implies ∆b = 0. The degeneracy of the decuplet is destroyed if ∆a > 0. An isoscalar state η i having pure G or pure η 8 structure becomes mixed. However, it is still dominated by the same state provided ∆a is sufficiently small. The mixing is intensified and the dominance is getting weaker as ∆a is increasing. By examining the mixing matrix we can check whether the dominance is kept inside all domains.

Tab. 2 shows the range of change of the squared matrix elements V 23 , V 33 , V 13 expressing contribution of the glueball to the η 2 , η 3 , η 1 respectively and the octet content l 2 2 under change of ∆a in the solutions A, B, C, D. It can be seen that dominance of the G and η 8 states is kept over the whole domain of these solutions. N and S dominance of the other η i states belonging to the same solution (not shown in the table) is preserved across the domains A, B, C as well; however, there is no dominance of η 1 , η 2 , η 3 by N, S, G in the case of the solution D. We thus find that within each domain the solution has specific dominance pattern which does not change under variation of ∆a. (Obviously, the degree of the dominance does depend on the ∆a). The dominance patterns of the solutions A, B, etc., are identical with the patterns of ideal structures of the degenerate decuplet (24). These structures correspond to the points at the outer corners of the appropriate domain.

To conclude, all solutions of the ME are split into four separate one-parameter families. The solutions belonging to the same family are slightly different. The solutions belonging to different families have different dominance patterns.

.

  .......................................

Figure 1 :

 1 Figure 1: The allowed values of the unknown quantities a and b. The function f (x) (12) is the characteristic polynomial of the m 2 operator. The eigenvalues x 1 , x 2 , x 3 are squared masses of the physical isoscalar mesons η(1295), η(1405), η(1475). The function -2f (x) is also shown. The a and b are restricted by ordering rule (14) and related by the mass formula (11): f (a) = -2f (b). The horizontal line t which is tangent to the curve -2f (x) at the point R of the local maximum (x R ∈ (x 2 , x 3 )) crosses the curve f (x) at the points P and Q. The figure indicates that the MF cannot be satisfied for a ∈ (x P , x Q ).

  a = (1.300GeV ) 2 , b = (1.414537GeV ) 2 , K = (1.358GeV ) 2 , l 2 1 = 0.297415, l 2 2 = 0.689645, l 2 3 = 0.012941, = (1.404GeV ) 2 , b = (474088GeV ) 2 , K = (1.439GeV ) 2 , a = (1.404GeV ) 2 , b = (1.413137GeV ) 2 , K = (1.409GeV 2 ,

Figure 2 :

 2 Figure 2: The domains A, B, C, D of the solutions of MF. The double circles at the outer corners of all domains correspond to degenerate decuplets. In each domain the pairs of values (a, b), being the solutions of the MF, lie on a curve connecting this corner with the opposite one. The curves are close to diagonals of the domains. The arrows beginning at the points of degeneracy indicate directions towards the growing mixing of the states.

formerly the model was called exotic commutator model (ECM)

Acknowledgments

The authors thank Prof., Prof. P. Maslanka, J. Rembielinski, W. Tybor and management of BLTF JINR for promoting our cooperation as well as dr K. Smolinski for help in computer operations. Especially we thank Prof. S. B. Gerasimov for many valuable discussions in early stage of this work and Prof. P. Kosinski for many interesting comments. This work was financially supported by JINR B-I Fond and by grants N o 690 and N o 795 of University of Lodz.

Appendix. Solving the MF

Introduce in each of the domains A, B, C, D shown on the Fig. 2. the small variables ∆a and ∆b. They can be chosen to be nonnegative (∆a, ∆b ≥ 0). Therefore, in the individual domains A, B, C, D, we have:

Further procedure is the following. Substituting a and b into (10), we express the coefficients l 2 i as functions of the ∆a and ∆b. Putting a and b into MF (46) we get the relation between the ∆b and ∆a. This relation is a cubic equation with respect to any of these variables. For our purposes it is sufficient to find the approximate solution. We have ∆b = ∆b(∆a, x 1 , x 2 , x 3 ).

(47)

Substituting this function into [START_REF] Chanowitz | Proc. of the VI Int. Workshop on Photon-Photon Collisions[END_REF] we obtain the functions l 2 i (∆a, x 1 , x 2 , x 3 ). If all these functions are positive we may consider the corresponding values of a