Revisiting the topic of determining fraction of glueball component in $f_0$ mesons via radiative decays of $J/\psi$

The QCD theory predicts existence of glueballs, but so far all experimental endeavor fails to identify any of such states. To remedy the obvious discrepancy between the QCD which is proved to be a successful theory for strong interaction and the failure of experimentally searching for glueballs, one is tempted to accept the most favorable interpretation that the glueballs mix with regular $q\bar q$ states of the same quantum numbers. The lattice estimate on the masses of the pure $0^{++}$ glueballs ranges from 1 to 2 GeV which is the region of the $f_0$ family. Thus many authors suggest that the $f_0$ mesonic series is an ideal place to study possible mixtures of glueballs and $q\bar q$. In this paper following the strategy proposed by Close, Farrar and Li, we try to determine the fraction of glueball components in $f_0$ mesons using the measured branching ratios of $J/\psi$ radiative decays into $f_0$ mesons. Since the pioneer papers by Close et al. more than 20 years elapsed and more accurate measurements have been done by several experimental collaborations, so that based on the new data, it is time to revisit this interesting topic. Our numerical results show that $f_0(500)$, $f_0(980)$, are almost pure quark anti-quark bound states, while for $f_0(1370)$, $f_0(1500)$ and $f_0(1710)$, to fit both the experimental data of $J/\psi$ radiative decay and their mass spectra, glueball components are needed. Moreover, the mass of the pure $0^{++}$ glueball is phenomenologically determined.


I. INTRODUCTION
The QCD theory demands existence of glueballs because of interactions among gluons.
The glueballs behave differently from the q q systems, for example they do not directly couple to photons, so that their special characteristic helps to identify glueball states.Generally, the glueballs of J P C = 0 ++ should be the most low-lying hadronic states.They have the same quantum numbers as the iso-singlet scalar meson f 0 family (i.e the so far observed series of f 0 (500), f 0 (980), f 0 (1370), f 0 (1500), f 0 (1710), f 0 (2020) and f 0 (2100)).
Even though the theoretical estimate on the mass of the 0 ++ glueball is so diverse, they all suggest the mass to be within a range of 1.2 GeV∼ 1.7 GeV.Because of the situation, a study on the mass of 0 ++ glueball based on analysis of the data might be welcome.
On other aspects, due to the failure of experimental search for glueballs, we are tempted to consider that the QCD interaction would cause glueballs to mix with the q q states of the same quantum numbers, so that the possibility that pure glueballs exist as individuals in the nature seems to be much slim, even though not completely ruled out.Namely glueballs would mix with q q state to make hadrons.Definitely, this scenario can reconcile the discrepancy between the QCD prediction and the experimental observation.In fact the mass of a pure glueball is indeed only a parameter which does not have a definite physical meaning.
In Refs.[8,[14][15][16][17][18][19], the authors considered f 0 (1370), f 0 (1500) and f 0 (1710) as mixtures of glueball and J P C = 0 ++ q q bound states.They preferred f 0 (1500) as a hadron predominated by a glueball component.Furthermore in Ref. [20] the authors further extended the scenario by involving possible components of hybrid state q qg which may provide better fits to the available data.By contrary to the above consideration, in Ref. [17,18], f 0 (1710) was supposed to be dominated by the glueball component, but not a pure glueball.
As the first step, in this work, we restrict ourselves to the scenario where only mixtures of glueballs and q q are considered while a possible contribution of hybrids to the mass spectra of the f 0 family is ignored .
We first would calculate the masses of the q q bound states by solving the Schrödinger equations.Some authors extended the equation into relativistic form for estimating the mass spectra of heavy-light mesons and the results seem to be closer to the data.Following Ref. [21] we calculate the light quark-antiquark system in the relativistic Schrödinger equations.
The results indicate that the experimentally measured masses of f 0 (500), f 0 (980) can correspond to the q q states (ground states of uū+d d √ 2 and ss), so can be considered as pure bound states of quark-antiquark.Whereas there are no values corresponding to the masses of f 0 (1370), f 0 (1500) and f 0 (1710).It signifies that they cannot be pure q q states and extra components should be involved.To evaluate the fractions of glueballs in those states, diagonalizing the mass matrix whose eigen-values correspond to the masses of the physical states and the transformation unitary matrix determine the fractions of q q and glueball in the mixtures.But this manipulation is not sufficient to fix all the four parameters: and ss states and mass of a pure glueball.Following the strategy provided by Close, Farrar and Li [7], by analyzing the radiative decays of J/ψ → γ + f 0 , we may determine all of those parameters.
After this introduction, in section II we calculate the mass spectra of q q states of 0 ++ by solving the relativistic Schrödinger equations.Then in the following section, we present the scheme of Close, Farrar and Li about J/ψ → γ + f 0 where f 0 refers f 0 (1370), f 0 (1500) and f 0 (1710) and extract the useful information about the fraction of glueball components in those mesons.In section IV, via a full analysis we confirm three physical states (f 0 (1370), f 0 (1500) and f 0 (1710)) as mixtures of q q and glueballs.A brief discussion and conclusion are presented in the last section.
II. 0 ++ q q SYSTEMS First, in terms of the relativistic Schrödinger equation let us calculate the mass spectra of f 0 by assuming them to be made of a light quark and an anti-quark.Since f 0 mesons are 0 ++ states, the relative orbital angular momentum l = 1.Following Ref. [21], the effective Hamiltonian is where m 1 and m 2 are the masses of the light quark and anti-quark respectively, ∇ 2 1 and ∇ 2 2 acting on the fields of q 1 and q2 , V 0 (r) is a combination of the QCD-Coulomb term and a linear confining term [22][23][24] V 0 (r) = −4 3 αs(r) r Here α s (r) is the coupling constant, for the concerned energy scale of Λ QCD ∼ 300 MeV the non-perturbative QCD effect dominates and so far α s (r) cannot be determined by a general principle, thus mostly one needs to invoke concrete models where the model dependent parameters are adopted by fitting data.Indeed, theoretical uncertainties are not avoidable.
In this work, we take the value given in Refs.[21,24] and the phenomenological constants b and c are fixed by fitting data.
Since we are dealing with the P-wave structure of q q, the spin-spin hyperfine interaction and spin-orbit interaction are concerned and an extra hamiltonian H ′ can be written as The spin-spin hyperfine interaction is with [21] where σ is a phenomenological parameter and The spin-orbit interaction is where L is the orbital angular momentum between the quark and anti-quark.For 0 ++ state, we have In our numerical computations we set m = 0.3 GeV for u and d quark, and m = 0.5 GeV for s quark, It has been supposed that f 0 (500) ( m 0 (500) is a rather wide resonance as m f 0 (500) = 400 ∼ 550 MeV, so sometimes it is named as m 0 (400) or σ particle) and f 0 (980) are mixtures of ground states of uū+d d √ 2 and ss By fitting data of m f 0 (500) = 400 ∼ 550MeV, m f 0 (980) = 990 ± 20MeV, we obtained b = 0.31 ± 0.02 GeV 2 and c = −1.65 ± 0.07 GeV, λ = 201 ∼ 263MeV and θ = 30.7 • ∼ 33.9 • .
The numerical results about the 0 ++ q q states are listed in Tab.I, and for a clear comparison, we also include some corresponding experimental data in the table.
Once all the relevant parameters are fixed, we go on estimating the mass spectra of the excited states of uū+d d √ 2 and ss with higher principal quantum numbers n=2 and 3.
It is noted that if one only considers the q q structure, the range from a few hundreds of MeV to 2 GeV can only accommodate six P-wave 0 ++ eigenstates, namely the masses of those excited eigen-states of n ≥ 3 or l ≥ 3 would be beyond this range.On other aspect, there indeed exist seven 0 ++ physical mesons which are experimentally observed within the aforementioned range.This fact signifies that there should exist something else beside the pure q q structures, obviously, the most favorable candidate is mixtures of glueballs and q q.This observation inspires all researchers to explore the possible fractions of glueball components in the observed meson states.
Our next job is to estimate the fractions of glueball components in the observed mesons.
Fortunately, the work of Close, Farrar and Li offers a possibility to make such an estimate via radiative decays of J/ψ.  and ss states with principal quantum numbers n=1, 2 and 3. part (b): The masses of f 0 family which are experimentally measured [28].
In this section let us briefly introduce the results of Close, Farrar and Li without bothering details of the derivations.In their pioneer work, it was proposed to determine the fraction of glueball components in a meson via J/ψ → γ + f 0 decay.Especially, we focus on mixtures of f 0 (1370), f 0 (1500) and f 0 (1710) states with glueball because if they are pure quark-antiquark states the theoretically estimated values of their masse spectra obviously deviate from the data (see above section).In Ref. [7,30], for searching glueball fraction, an ideal reaction is the radiative decays of J/ψ.Close, Farrar and Li formulated the decay branching ratios as where and c R = 2/3 for 0 ++ state, H 0 ++ (x) is a loop integral and its numerical result is given in Ref. [7].Taking experimental data [28] BR(J/ψ → γ + gg) = 8.8%, and by Eq.8 we can obtain all results which are listed in Tab.II.As indicated in Ref. [7], the width of f 0 is determined by the inclusive processes of f 0 → gg and f 0 → q q.It is noted that the contribution of the glueball component to gg final state is In Fig. 1 we show that the value of b(R[q q] → gg) would be different if f 0 is a glueball or q q bound state where readers can ignore the irrelevant hadronization processes.
In next section we will further determine the fractions of , |S = |ss and |G = |gg which are no longer on-mass shell inside the physical f 0 states.

IV. STUDY ON THE MIXING OF QUARKONIUM AND GLUEBALL
Our work is a phenomenological study and fully based on the available data.As discussed in the introduction, we find that the energy region of 1 ∼ 2 GeV cannot accommodate seven pure 0 ++ q q states in contrary to the experimental observation.Thus a scenario about mixture of glueball and q q within this energy region is favored.The decay rates and ss, (b) J/ψ → γ + f 0 with f 0 believed as q q bound state, (c) J/ψ → γ + f 0 with f 0 believed as glueball.
The lattice estimate suggests that the mass of 0 ++ pure glueball is about 1.5 GeV, so that one can naturally conjecture that f 0 (1370), f 0 (1500) and f (1710) are mixtures of q q and glueball with certain fractions.The rest 0 ++ q q states would have negligible probability to mix with glueballs because their masses are relatively apart from that of pure glueball.
and U is a unitary matrix with the compact form as By imposing the unitary condition on U, we should determine all the elements of U up to an arbitrary phase.Furthermore we will enforce a few additional conditions to the shape the matrix: (1) let the determinant of the matrix be unity and (2) deliberately choose the solutions where the diagonal elements are larger than the off-diagonal ones, (3) we restrict the matrix elements to be real.Those requirements serve as a convention for fixing the unitary matrix.In fact, the condition ( 2) is consistent with the shape of the transformation which is established based on a perturbation conjecture as suggested in literature.
The unitary matrix U transforms the unphysical states |N , |S and |G into the physical eigenstates |f 0 (1370) , |f 0 (1500) and |f 0 (1710) , and at the same time diagonalizes the mass matrix M as with Namely, m f 0 (1370) , m f 0 (1370) , and m f 0 (1710) are the three roots of equation Generally, we have four unknowns in the hermitian matrix M: λ n−s , λ s−g , λ n−g and m g , meanwhile there are three independent equations, so we need an extra relation to fix all of the four unknowns.Fortunately, the work of Close, Farrar and Li offers such an opportunity.
Let us employ the relations b 1370 : b 1500 : b 1710 which are extracted from Eqs.( 8) and ( 9), to determine the fourth unknown parameter.
By the quantum field theory, |c i1 gg|N +c i2 gg|S +c i3 gg|G | 2 ×A |c i1 q q|N +c i2 q q|S +c i3 q q|G | 2 ×B+|c i1 gg|N +c i2 gg|S +c i3 gg|G | 2 ×A (18) where A and B are the phase factors for the two final states q q and gg respectively.The integration over the final states is proportional to dΠ 2 = |p| 4π2m f 0 , if neglecting the mass difference between final states gg and q q, we have A ≈ B. Considering the follow relations: one has The solutions show that for f 0 (1370) and f 0 (1500), the main components are q q bound states, whereas the glueball component in f 0 (1710) is overwhelmingly dominant.It also suggests the mass of a pure glueball of 0 ++ to be 1637 ∼ 1698 MeV.

V. DISCUSSION AND CONCLUSION
The main purpose of this work is to explore probability of mixing between 0 ++ q q states and glueball.To serve this goal, we first calculate the mass spectra of six 0 ++ light q q bound states by solving the relativistic Schrödinger equation.
The numerical estimates indicate that in order to fit the observed experimentally measured spectra of f 0 (500), f 0 (980), f 0 (1370), f 0 (1500), f 0 (1710), f 0 (2020) and f 0 (2100), an extra hadronic structure is needed to accommodate the seventh member of the f 0 family existing in the energy range from a few hundreds of MeV to 2 GeV.As suggested in literature, the most favorable scenario is the mixing between q q and glueball of the same quantum numbers.Instead of calculating the mixing based on complete theoretical frameworks, we investigate the mixing in terms via analyzing experimental data.Besides properly diagonalizing the mass matrix, supplementary information about the fractions of the glueball components in the f 0 mesons can be extracted from the data of J/ψ radiative decays to f 0 .It is found that in f 0 (1370) and f 0 (1500) mainly there are q q bound states whereas in f 0 (1710) a glueball component dominates.
In this work, we obtain the mixing parameters by a phenomenological study, while some authors tried to directly calculate them in terms of certain models.Within this energy range, the dominant dynamics is the non-perturbative QCD which induces the mixing.Since the solid knowledge about the non-perturbative QCD is still lacking, the theoretical calculation heavily relies on the adopted models where some model-dependent parameters have to be input and cause uncertainties of the theoretical estimates.Among those calculations, the results of the lattice calculations [1][2][3] and that based on the QCD sum rules [4-6, 12, 13] may make more sense even though still are not completely trustworthy.Combining the phenomenological studies by analyzing the experimental data and those estimates based on theoretical framework may shed light on this intriguing field.Now let us briefly discuss another two 0 ++ states f 0 (2020) and f 0 (2100).Since their masses are heavier than the assumed glueball mass, according to the principle of quantum mechanics, the probability of their mixing with glueball is small.However, since the theoretically estimated masses do not accommodate them, it is suggested that another hadronic structure needs to be invoked, namely the mixing between q q and hybrids q qg, or even more complicated mixtures among q q, glueball and hybrids should also be taken into account, [20,31].We will come to that subjects in our coming works.
The reason that we are able to carry out this exploration is that much more data in this energy range has been accumulated and the measurements are obviously more accurate than 25 years ago.However, as one can see, the precision is still far from accurately determining the mixing parameters yet, therefore we lay hope on the experimental progresses which will be realized by the BESIII, BELLE, LHCb and probably the future charm-tau factory.
Definitely, to verify this mixing scenario one needs to do more theoretical works, including estimating the production (not only via the radiative decays of J/ψ) and decay rates of f 0 families.Further works, both experimental and theoretical are needed badly.
λ s−g , λ n−s and m g which respectively are the mixing parameters between uū+d d √ 2 and glueball; ss and glueball; uū+d d √ 2

2 and
Now we propose that the physical states f 0 (1370), f 0 (1500) and f (1710) are mixtures of the second excited states of |N = | uū+d d √ |S = |ss with glueball state |G .A unitary matrix U transforms them into the physical states as

TABLE I :
part (a): the theoretically predicted mass spectra of uū+d d √ 2