Where is the pseudoscalar glueball?
The pseudoscalar mesons , , , and are assumed to form the meson decuplet which includes the glueball as the basis state supplementing the standard nonet of light states . The decuplet is investigated by using the algebraic approach based on the hypothesis of vanishing exotic commutators (VEC) of ”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 , , are expressed as superpositions of the ”ideal” ( and ) states and the glueball 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 and meson masses) there is a family attributing the glueball-dominated composition to the meson. The pseudoscalar decuplet is similar in some respects to the scalar one: both are composed of the excited 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 of the pseudoscalar pure glueball state is smaller than the scalar one.
The pseudoscalar glueball investigation has been initiated soon after it was realized that bound states of the gluons may play an important role in the strong interactions [1, 2]. From the very beginning the glueball state was traced within structure of the and 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 .
The discovery of the meson [8, 9] rous hopes for the existence of the glueball. The meson has been detected in the gluon rich process of the 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 states having the same signatures . The mass of the meson belongs to the region of a higher-lying 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”  and ”gluiness” ). 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 . This question is still alive .
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 - much above the . An attempt to lower the lattice prediction by including quark loops was not very successful . Although the doubts were not dispelled (see e. g. ), this became a serious obstacle for 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 as the pseudoscalar glueball [19, 20, 21], although its mass might also be regarded as too low.
At the same time, starting from late 1980s there was growing conviction that the signal should be attributed to two different isoscalar mesons . Much experimental effort was devoted to understanding the structure of the signal [23, 24, 25, 26, 27]. As a result, it has been split into and . Hence, since 2004 three isoscalar pseudoscalar mesons have been listed in RPP within the narrow interval of mass :
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 , , has been extracted from data on
the reactions of their production and from branching ratios of their
decays. Data suggest that the meson is a particle dominated
by the glueball state [11, 28, 29, 31].
An unexpected objection has been risen against such a picture: the has been claimed not to be the state . Even its very existence was considered uncertain. This implies the nonexistence of the decuplet and requires much more complicated spectroscopy of the pseudoscalar mesons. Therefore, we discuss this question in more detail.
The structure is put into doubt due to not occurrence of the in the reactions
”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 , came to the following
(i) the charge exchange experiments definitively establish evidence of the ;
(ii) a clear signal of is seen in the radiative decay;
(iii) there is an indication for the existence in annihilation;
(iv) the LEP data on reaction are compatible with the existence of the signal.
So the may or may not be seen in the reactions (2). The three-body decays , are strongly suppressed by small phase space (cf and that may be the reason why it is difficult to observe the . It is explicitly seen in the reaction , for which high statistics is available, but the number of events observed in reactions (2) is many times smaller . Obviously, more measurements are needed to elucidate the situation. But this question has no relevance to the problem of the internal structure. As in the reaction (2) the 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 cannot be the base for conclusion about its structure. An attempt to call in question this structure resembles confusion which arose after denying the structure of the meson motivated by its small width .
In the present paper we admit the meson to be a state and assume that the examined pseudoscalar mesons form a decuplet. We thus focus the glueball search again in the region of meson - this time being fully aware of the conflict with the lattice prediction.
The glueball assignment of the meson is also motivated on theoretical ground . It is argued that meson is a natural pseudoscalar glueball candidate if the is the scalar glueball and the glueballs are described as the closed gluonic flux-tubes. Then the and the 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 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 is smaller than ideal , while for the other one it is bigger ().
The model also predicts 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 comprising low mass mesons . 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  - 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 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.
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 . 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 presence – overpopulation of a nonet.
The model has appeared very effective in describing the mesons. It makes possible to sort out 20 scalar mesons among multiplets and attribute the glueball dominating structure to the meson . To analyze the decuplet we use essentially the same model. Our present analysis is proceeded in a different way because sample of the input data is different.
2 The decuplet of pseudoscalar mesons
2.1 The model of vanishing exotic commutators (VEC)11footnotemark: 1
The following sequence of exotic commutators is assumed to vanish 22footnotetext: formerly the model was called exotic commutator model (ECM)
where is generator, is the time and is an exotic combination of indices, i.e. such that the operator does not belong to the octet representation. Substituting , and using the infinite momentum approximation for one-particle hamiltonian , we transform eqs. (3) into the system:
where is the squared-mass operator.
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 with different powers , where 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
where is the mass squared of the isovector meson ; is the mass squared of the subsidiary state,
and , in turn, is the mass squared of the isospinor meson. The isoscalar octet state can be represented as the linear combination of the physical isoscalar states
where the , , ,… are isoscalar meson masses squared. Normalization condition of the coefficients is included into (8) as equation for .
2.2 ME for the decuplet
The states of the decuplet belong to a reducible representation of the
where the octet and one of the singlets are considered as states while the second singlet is supposed to be a glueball .
where are the masses squared of the isoscalar mesons
, , .
The coefficients , , , 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 .
b) the mass formula (MF)
is the characteristic polynomial of the operator; the numbering of its eigenvalues is chosen such as to satisfy the inequality
c) the mass ordering rule (MOR)
The MF (11) is a linear equation with respect to each of the , but it is a cubic one with respect to and .
The masses and experimental status of the mesons assigned to the decuplet are quoted in tab. 1. The table shows that masses of the isoscalar mesons , , are determined with good accuracy; the meson mass has large error; the -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 and (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 operator as well as precise values of the and bands which are required by MOR (14). We hope that restrictions of the model on decuplet states will reduce ambiguity of the solution.
From we have
Comparing the bands (17) with the range of error of the
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 as an unknown
quantity of the ME.
2.3 Families of solutions of the ME
Combining the 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.
It can be seen from the figure that beside the MOR restrictions , there also appears the MF restriction forbidding . Hence, the allowed values of belong to two narrow intervals: and . To each allowed value of there correspond two values of (obeying the mass formula) placed on the opposite sides of the point . If we wish to have unique solution, we should divide the interval into two parts: and . Then, we get four domains including unique pairs of values making solutions of the MF:
These values correspond to
The domains A, B, C, D are shown in figure 2. We solve the ME (9) in each of them separately and express as functions of . The details of solving the MF as well as properties of the solutions are described in the appendix.
So V expresses the states of the physical isoscalar mesons , , in terms of the decuplet ideal states , and . 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 Fig. 2. In the first three domains we find three different ideally mixed 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:
For the wave functions one obtains:
Each of the degenerate solutions A, B, C points out its own candidate from among , , as a pure glueball. The solution D describes degenerate decuplet where is the octet isoscalar state and , are scalar states built as superpositions of the and G.
The intervals and of the variable allowed by MF and MOR are small. Also the intervals and of the variable 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, ) which is pure in the degenerate solutions. That can be seen from the Tab. 2. Hence, the dominant structures of the , , in the domains A, B, C, D preserve the patterns of degenerate decuplet (24):
where is the octet isoscalar state and , are superpositions of the singlet and G. Their contribution to the and 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 is chosen such
that the deviation of the meson mass from its
ideal value (i. e. from the or from the meson
mass) is equal to . 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 and
A. , , ,
, , ,
B. , , ,
, , ,
C. , , ,
, , ,
D. , , ,
, , ,
Table 2 also exhibits intervals of the admissible K meson mass corresponding to these solutions. Its changes under variations of 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 , , , , may constitute solution of the ME. The price to pay for ignorance concerning both and 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 plane and can be distinguished due to the fact that the isoscalar physical states are dominated by one of the , , or 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 and 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 as a
particle dominated by the glueball state.
3 Comments on solutions of the ME
A decuplet of mesons is a multiplet such that the octet isoscalar state contributes to three isoscalar physical states . The contributions of the isoscalar octet state to the physical states , given by s (10), constitute a solution of the ME. The coefficients are real numbers, therefore, the following conditions should be satisfied
This property is not guaranteed by solution (10). Requiring it, we put constraints on the masses of the decuplet.
The knowledge of the octet contents 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 s, i.e. by the masses. The requirement of the glueball flavor independence
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 states by one of the ,
, , states. As a result, all the Euler angles, and
consequently, all elements of the mixing matrix are uniquely
i) A decuplet which is based on the 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 (11) which, together with conditions
(30), defines explicit MOR restrictions for the masses
(14). To also admit the degenerate solutions of the ME one
must allow for some and in the MOR
instead of the . The solutions of the ME do not include free
parameters – all 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 and mesons are unknown. So the values and 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 the proper solution of the nonlinear equation (11). Further restriction is provided by the mass formula which cuts out the central part of the MOR-allowed interval of and thus reduce it to two narrow disconnected subintervals (see figure 1). To each belonging to them there correspond two values of . We divide the interval of the values to two parts. As a result, the whole domain of the values of the and is reduced to four small domains A, B, C, D which are shown in figure 2. In these domains the solution is unique if one of the variables, or is known.
Hence, due to the restrictions of the ME on and , the solution is split into four one-parameter, qualitatively different families. The partition allows us to look for a 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 and mesons over the domains can be found out from the table 2.
The partition of the whole domain of variables 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 a common dominance pattern. This pattern is determined by the dominance of one of the N, S, G, amplitudes in the , , 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 figure 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 , , states, respectively.
In the domain D the dominance pattern is different. The degenerate decuplet consists of the octet of the exact symmetry and two separate singlets being mixed states of the and . is the pure octet , while and are pure singlets. The rates of the and states in the structures of the and mesons are comparable, slowly changing functions of the parameter within the domain. In spite of the identical flavor properties of the constituents, the properties of the and 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 and 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 the flavor properties of the isoscalar mesons , , . 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 and K(1460) masses would be determined by suitable value of .
The restrictions following from figure 1 do not hold for the decuplet of the type S. There is no mass formula in this case; therefore, there is no connection between the variables and and there is no mass gap () of the meson. Also the MOR (14) does not exist. Such a situation can arise for the decuplet we discuss if the measured masses of the 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 and . We find
where is given by the GMO mass formula,
The glueball contents of the isoscalar mesons , , can be always calculated from (10) if the masses and satisfy /i=1,2,3/.
Having known the s we can construct the mixing matrix. If the state of is predicted to be dominated by the solution of the ME should be similar to the one describing the states of the family A.
4 Pseudoscalar versus scalar meson multiplets
4.1 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 . The comparison may reveal some new features of the meson spectroscopy.
Let us compare the and multiplets.
The ground states form the nonets:
which are followed by the decuplets:
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
similar, but also the inner structures of the decuplets are; namely:
- the physical mesons and which are dominated by glueball states are settled just between the remaining isoscalars which are expected to be mostly the and quark states,
- both decuplets involve excited states; hence both glueballs mix with the excited states.
The latter property suggests affinity of the glueball with the excited states. This is especially prompted by the mixing of the 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 states – there is no trace of mixing with the ground states .
The and 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 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 [38, 39, 40]. By ignoring the we find that and mesons form parity related spectra of multiplets.
The transparency of this picture confirms not only the opinion about the distinct nature of the , but also supports correctness of sorting out the scalar mesons between the overlapping multiplets .
However, there is also a difference between these spectra. The mass spread of the multiplets is shrinking for consecutive multiplets (nonet , decuplet , 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 mesons where all known multiplets above 1400 MeV (at 1400 MeV and 1800 MeV) are degenerate octets . But this tendency is not seen in the spectrum of the multiplets - at least below 2300 MeV.
4.2 The masses of the spin 0 glueballs
Such mass allows to attribute the glueball nature to several mesons – among them to favored .
For the lightest glueball these calculations predict the lower bound at
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 also predict the lower bounds for the masses of many other glueballs with different . 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 ). 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 and glueballs should be equal. Since is the favored glueball candidate, we should expect the glueball mass at about . Hence, the LQCD and FT predictions on 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 . 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 and 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 and separately (not assuming any relation between them).
From the decuplet , assuming solution A and the mass input appropriate to the mixing matrix (26) we find
From the decuplet for solution 1  we get
The difference between these predictions is approximately equal to the meson mass.
Also observe that the inequality
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. The 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 an important role in glueball determination.
The FT prediction of the equality of the and
glueball masses is approximately obeyed. This may follow from the
contributions to their structure are high and almost equal:
for decuplet (solution A (26)),
for the decuplet ,
In fact, the masses of the and 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 the 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 nonidentically. That may concern not only the cases of different but also decuplets of the same 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.
1. Owing to the unknown masses of the and
mesons the solution of the ME for decuplet , ,
, , 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 as the particle dominated by the glueball state.
2. The scalar and pseudoscalar glueballs belong to the decuplets formed by mixing with excited isoscalar states.
3. The spectra of the known multiplets of the and mesons are parity related provided the is ignored.
4. The mass of the pure pseudoscalar glueball state is smaller than the mass of the scalar glueball one.
The authors thank Professors 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 Professor S. B. Gerasimov for many valuable discussions in early stage of this work and Professor P. Kosinski for many interesting comments. This work was financially supported by JINR B-I Fond and by grants nos690 and 795 of University of Lodz.
7 Appendix. Solving the MF
Introduce in each of the domains A, B, C, D shown in figure 2 the small variables and . They can be chosen to be nonnegative . Therefore, in the individual domains A, B, C, D, we have:
Further procedure is the following. Substituting and into (10), we express the coefficients as functions of the and . Putting and into MF (46) we get the relation between the and . 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
Substituting this function into (10) we obtain the functions . If all these functions are positive, we may consider the corresponding values of and , together with the functions , 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 or and restrict ourselves to the linear dependence between them. The approximation is plausible for covering all the interval . We obtain
In the domains C and D we also take into account the term quadratic in and all powers of . This is to avoid in the domain C and to extend applicability of this approximation towards the largest values of in the domain D. In these cases the expressions for 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 do not cause confusion, as only one of them complies with the condition for all i=1,2,3. In both regions the approximation is plausible everywhere, except the small surroundings of the point .
In all the solutions A, B, C, D the value implies . The degeneracy of the decuplet is destroyed if . An isoscalar state having pure or pure structure becomes mixed. However, it is still dominated by the same state provided is sufficiently small. The mixing is intensified and the dominance is getting weaker as is increasing. By examining the mixing matrix we can check whether the dominance is kept inside all domains.
Table 2 shows the range of change of the squared matrix elements , , expressing contribution of the glueball to the , , respectively and the octet content under change of in the solutions A, B, C, D. It can be seen that dominance of the and states is kept over the whole domain of these solutions. The and dominance of the other 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 by 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 . (Obviously, the degree of the dominance does depend on the ). 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.
-  H. Fritzsch, M. Gell-Mann, Proc. of the XVI Int. Conf. on High Energy Physics, Chicago-Batavia Ill. 1972 vol. 2, p. 135
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