June, 2015
Revisiting Scalar Glueballs
HaiYang Cheng, ChunKhiang Chua and KehFei Liu
Institute of Physics, Academia Sinica
Taipei, Taiwan 115, Republic of China
Department of Physics and Center for High Energy Physics
Chung Yuan Christian University
ChungLi, Taiwan 320, Republic of China
Department of Physics and Astronomy, University of Kentucky
Lexington, KY 40506
Abstract
It is commonly believed that the lowestlying scalar glueball lies somewhere in the isosinglet scalar mesons and denoted generically by . In this work we consider lattice calculations and experimental data to infer the glue and components of . These include the calculations of the scalar glueball masses in quenched and unquenched lattice QCD, measurements of the radiative decays , the ratio of decays to , and , the ratio of decays to and , the contributions to , and the near mass degeneracy of and . All analyses suggest the prominent glueball nature of and the flavor octet structure of .
I Introduction
The existence of glueballs is an archetypal prediction of QCD as a confining theory. It is generally believed that the lowestlying scalar glueball lies somewhere in the isosinglet scalar mesons with masses above 1 GeV. To see this we first give a short review on scalar mesons (see e.g. Close:2002 ; Klempt ; Amsler ; Liu:2007 ). Many scalar mesons with masses lower than 2 GeV have been observed and they can be classified into two nonets: one nonet with mass below or close to 1 GeV, such as (or ), (or ), and and the other nonet with mass above 1 GeV such as , and two isosinglet scalar mesons. Of course, the two nonets cannot be both lowlying states simultaneously. If the light scalar nonet is identified with the Pwave states, one will encounter two major difficulties: First, why are and degenerate in their masses? In the two quark model, the latter is dominated by the component, whereas the former cannot have the content since it is an state. Second, why are and so broad compared to the narrow widths of and even though they are all in the same nonet? These difficulties with mass degeneracy and the hierarchy of widths can be easily overcome in the tetraquark model Jaffe . Therefore, this suggests that the heavy scalar nonet is composed of Pwave states, while the light nonet is made of Swave tetraquark states.
Finalstate interactions of , , etc., are known to be very important in the region below 2 GeV. Such interactions can be described in unitarized chiral perturbation theory (ChPT) or unitarized quark models with coupled channels. It follows that the light scalar mesons , , and can be dynamically generated through pseudoscalar mesonpseudoscalar meson scattering within the framework of unitarized ChPT valid up to 1.2 GeV (see Pelaez and references therein). ^{1}^{1}1A coupled channel study of the mesonmeson wave in terms of 13 coupled channels in Albaladejo:2008qa indicates that all the resonances with masses below 2 GeV and and 1/2 can be dynamically generated. This implies that these light scalars may have nonnegligible contents of hadronic molecules. The dynamically generated bound state or resonance is characterized by a strong coupling to the coupled channel. For example, both and have been advocated to be molecular states Weinstein ; Locher , while a resonance. By the same token, it has been shown that and can be dynamically generated from the interaction in a hidden gauge unitary approach Molina ; Geng . That is, they have molecular components in addition to the content.
Although the light scalar nonet is composed of tetraquark and/or molecular states, it is allowed to have a small amount of the component for several reasons: (i) A mixing of the heavy scalar nonet with the light nonet will enable us to understand the near degeneracy of and Black . (ii) The large dependence of unitarized twoloop ChPT partial waves for the description of pionpion scattering suggests a subdominant component of the possibly originates around 1 GeV Pelaez:2006 . (iii) If is a loosely bound state of , it will be hard to understand its prompt production in decays. This will require an core component in . Likewise, the heavy scalar nonet dominated by can have molecular and tetraquark components.
In principle, twoquark, fourquark and molecular components of light and heavy scalar mesons can be studied in lattice QCD with the corresponding interpolating fields. So far, the lattice calculation with all the interpolating fields available at the same time is not yet practical (for a review of previous works for light scalar mesons in full lattice QCD, see Wakayama ). For heavier scalar mesons, the masses of and have been calculated using the twoquark interpolation field Mathur:2006 . The chirally extrapolated masses GeV for and GeV for suggest that the mesons and are predominantly states.
Taking the lattice result as a cue, we shall assume in this work that the scalar meson nonet above 1 GeV is primarily a state in nature. To the lowest order approximation we will not consider the possible tetraquark and molecular contributions. Experimentally, there exist three isosinglet scalars , , above 1 GeV. They cannot be all accommodated in the nonet picture. One of them could be primarily a scalar glueball. It has been suggested that is predominately a scalar glueball in Close1 . Lattice calculations indicate that the mass of the lowlying scalar glueball lies in the range of GeV (see Table 1 below). This suggests that does not have a sizable glue content. Among the two remaining isoscalar mesons, and , it has been quite controversial as to which of the two is the dominant scalar glueball. Since the glueball is hidden somewhere in the quark sector, this is the main reason why the glueball is so elusive.
It is worthy mentioning that the very existence of has long been considered to be questionable (see e.g. Klempt and Ochs for detailed discussions). Its mass and width are quoted by PDG (Particle Data Group) PDG to be MeV and MeV, respectively. It appears that the decays into two pion isobar can be described by the two poles and , while four pion isobar can be also described by the two poles and . However, there is no any single publication showing the need of three states simultaneously. Hence, the hypothesis of three distinct poles , and is not a general consensus and there is (probably) not a single experiment favoring this hypothesis.
In spite of the controversies on the identification of the scalar glueball, the 2006 version of PDG PDG2006 attempted to conclude the status as “Experimental evidence is mounting that has considerable affinity for glue and that the and have large and components, respectively”. This has been toned down to “The or, alternatively, the have been proposed as candidates for the scalar glueball” in the latest version of PDG.
Using the CLEO data, Dobbs et al. Dobbs:2015dwa have recently analyzed the radiative decays of and into , and . They have determined the product branching fractions for the radiative decays of and to scalar resonances such as and found (see also Table 2)
(1) 
For a pure, unmixed glueball, its decays to pseudoscalar pairs are expected to be flavor blind. Hence, decays to , , , and should have branching fractions proportional to apart from the phase space factor. ^{2}^{2}2For a pure, unmixed glueball, the ratio defined in Eq. (14) below approaches to 3/4 in the SU(3) limit. Taking into account of phase space corrections, we find for MeV and 0.98 for MeV. Therefore, Dobbs et al. concluded that is not a pure scalar glueball. By the same token, the large deviation of the experimental measurement PDG
(2) 
from the value of 3/4 also implies that cannot be a pure glueball either.
Denoting and , we write
(3) 
with being , respectively, for . At first sight, it appears that Eq. (1) implies while Eq. (2) leads to . However, this may be misleading because the component contributes to both and , while contributes only to . Therefore, it is possible to accommodate even with .
The abovementioned flavor blindness of glueball decays is valid for glueballs. For a scalar glueball, it cannot decay into a quarkantiquark pair in the chiral limit (see Sec. III.C below for discussion). Consequently, a large suppression of the production relative to is expected in the spin0 glueball decay, though it is difficult to quantify the effect of chiral suppression. Therefore, the ratio to be defined in Eq. (14) below will be naturally small. Comparison of this with Eqs. (1) and (2) suggests that is likely to have a large glueball component.
In the literature, there exist two different types of models for the mixing between the scalar glueball and the scalar quarkonia and (see Mathieu ; Crede ; Ochs for reviews). In the first type of models, is composed primarily of a glueball with large mixing with states, is predominately a state and is dominated by the content. In contrast, in the second type of models, is primarily a glueball state and is dominated by the component, while is still governed by the .
In our previous work CCL , we have employed two simple and robust results as inputs for the mass matrix which is essentially the starting point for the mixing model between scalar quarkonia and the glueball. We have shown that is composed primarily of a scalar glueball. In this work, we shall point out that new results from the unquenched lattice QCD calculation of the glueball spectrum, new measurements of radiative decays of , a new lattice calculation of and new experimental results on the scalar meson contribution to all support the prominent glueball nature of .
This work is organized as follows. In Sec. II, we first outline the general expected features of a pure glueball and then discuss two different types of models for the mixing between the glueball and quarkoina states. We proceed to discuss various signals for the existence of a scalar glueball, such as the lattice calculations of the glueball spectrum, the radiative decays of to isosinglet scalar mesons, , etc. In the vicinity of there exist several possible other states. Their mixing effects are briefly discussed in Sec. IV. Discussion and conclusions are presented in Sec. V.
Ii Model for Scalar glueballquarkonia mixing
A pure glueball state is expected to exhibit the following signatures (see e.g. Amsler:2006 )

It is produced copiously in the gluerich environment such as radiative decays (or Nussinov ) as the glueball couples strongly to the colorsinglet digluon.

It is suppressed in reactions.

Its width is commonly believed to be narrow, say, of order 100 MeV, as inferred from the large argument that the glueball decay width scales as , while the width of the state is . Hence, the very broad does not appear to be a good scalar glueball candidate.

The decay amplitude for glueballs is flavor symmetric, namely, its coupling is flavor independent Close1 . A scalar glueball cannot decay into a massless quark pair or a photon pair to leading order. Hence, its decay amplitude is subject to chiral suppression (see Sec. III.C below for detailed discussions and references). However, this feature does not hold for pseudoscalar glueballs owing to the axial anomaly Frere . Consequently, the scalar glueball decay to mesons is sensitive to flavor or SU(3) breaking.
The above features provide qualitative criteria for distinguishing glueballs from states with the same quantum numbers. The suppression in reactions is usually not a good criterion because the quark mixing can be adjusted in such a way that the state has a weak or even vanishing coupling to two photons.
A physical glueball state is an admixture of the glueball with the state or even the tetraquark state with the same quantum numbers so that a pure glueball is not likely to exist in nature. In the following we shall consider two different types of models for the mixing of the scalar glueball with the scalar quarkonia:
(i) Model I: as primarily a scalar glueball Amsler and Close Close1 claimed discovered at LEAR as an evidence for a scalar glueball because its decay to is not compatible with a simple picture. This is best illustrated in the argument given by Amsler Amsler02 . Let . The suppression of the production relative to (cf. Eq. (2)) indicates that is dominated. This is also well established in and collisions. By contrast, the nonobservation of in reactions implies that is dominated. This is because (see Eq. (29) below), and hence a small rate implies that is close to . Obviously, the above two conclusions are in contradiction. This led Amsler to argue that is not a state but rather something else and suggested that it is primarily a glueball. This can explain why its coupling is weak and why it is produced abundantly in and collisions. However, this interpretation has a difficulty with the large suppression of production relative to .
A typical result of the mixing matrices obtained by Amsler, Close and Kirk Close1 , Close and Zhao Close2 , He et al. He:2006 and Yuan et al. Yuan is the following
(4) 
taken from Close2 . Eq. (4) will be referred as Model I. A common feature of these analyses is that, before mixing, the quarkonium mass is larger than the glueball mass which, in turn, is larger than the quarkonium mass , with close to 1500 MeV and of the order of MeV. In this model, is considered mainly as a state, while is dominated by the content and is composed primarily of a glueball with possible large mixing with states.
(ii) Model II: as primarily a scalar glueball Based on the lattice calculations, Lee and Weingarten Lee found that to be composed mainly of the scalar glueball, is dominated by the quark content, and is mainly governed by the component, but it also has a glueball content of 25%. Their mixing matrix is
(5) 
In this scheme, MeV, MeV and MeV.
To improve this model, it is noted in CCL that two crucial facts need to be incorporated as the starting point for the mixing calculation. First of all, it is known empirically that flavor SU(3) is an approximate symmetry in the scalar meson sector above 1 GeV. The multiplets of the light scalar mesons , and are nearly degenerate. In the scalar charmed meson sector, and ^{3}^{3}3In spite of its notation, the mass of , MeV PDG , is almost identical to the mass of , MeV. have very similar masses even though the former contains a strange quark. It is most likely that the same phenomenon also holds in the scalar bottom meson sector Cheng:2014 . This unusual behavior is not understood as far as we know and it serves as a challenge to the existing hadronic models, but the degeneracy of and is confirmed in the quenched lattice calculation Mathur:2006 . This requires that there not be a 200 MeV difference between the state and the in the diagonal matrix elements in the mixing matrix as have been done in all the previous calculations. Second, a latest quenched lattice calculation of the glueball spectrum at the infinite volume and continuum limits based on much larger and finer lattices have been carried out Chen:2005mg . The mass of the scalar glueball is calculated to be MeV. This suggests that should be close to 1700 MeV rather than 1550 MeV from the earlier lattice calculations Bali .
We begin by considering exact SU(3) symmetry as a first approximation for the mass matrix, namely, with being the masses of the scalar quarkonia , and , respectively, before mixing. In this case, two of the mass eigenstates are to be identified with and which are degenerate with the mass before mixing. Taking to be the experimental mass of MeV of , it is a good approximation for the mass of at MeV PDG . Thus, in the limit of exact SU(3) symmetry, is an SU(3) isosinglet octet state and is degenerate with . In the absence of glueballquarkonium mixing, would be a pure glueball and a pure SU(3) singlet and its mass is shifted down by 3 times the coupling between the and states which is MeV lower than . When the glueballquarkonium mixing is turned on, there will be additional mixing between the glueball and the SU(3)singlet . As a result, the mass shift of and due to this mixing is only of order 10 MeV. Since the SU(3) breaking effect is expected to be weak, it can be treated perturbatively. The obtained mixing matrix is ^{4}^{4}4We have updated the fit results in CCL by taking into account the experimental uncertainties of the isosinglet scalar meson masses and branching fractions. The other updated parameters in fit (ii) are , and .
(6) 
with MeV, MeV and MeV will be referred as Model II. It is evident that is composed primarily of the scalar glueball, is close to an SU(3) octet, and consists of an approximated SU(3) singlet with some glueball component (). Unlike , the glueball content of is very tiny because an SU(3) octet does not mix with the scalar glueball.
For other glueballquarkonium mixing models in this category, namely, is predominantly a glueball, see Janowski .
Iii Signal for Scalar glueball and its mixing with quarkonium
In this section we shall consider the calculations of the scalar glueball mass in quenched and unquenched lattice QCD, the radiative decay , the ratio of decays to , and , the ratio of decays to and , the scalar contributions to , and the near mass degeneracy of and . They will provide clues on the coefficients and in Eq. (3) for isosinglet scalar mesons . For example, the radiative decay is sensitive to the glue content of , while the study of scalar contributions to can be used to explore the component of . For the study of the scalar glueball production in hadronic decays, see He:2015 .
iii.1 Masses from lattice calculations
Lattice calculations of the scalar glueball mass in quenched and unquenched QCD are summarized in Table 1. Except for the earlier calculation by Bali et al. Bali , the mass of a pure gauge scalar glueball falls in the range of 16501750 MeV. The latest quenched lattice calculation of the glueball spectroscopy by Chen et al. Chen:2005mg shows that the lightest scalar glueballs has a mass of order 1710 MeV. The predicted masses in quenched lattice QCD are for pure glueballs in the YangMills gauge theory. The question is what happens to the glueballs in the presence of quark degrees of freedom? Is the QCD glueball heavier or lighter than the one in YangMills theory? In full QCD lattice calculations, glueballs will mix with fermions, so pure glueballs does not exist. The unquenched calculation carried out in Gregory gives MeV for the lowestlying scalar glueball. ^{5}^{5}5An earlier full QCD lattice study in UKQCD did not give numerical results on glueball masses except in the last figure of the paper. In unquenched lattice QCD, the glueball is not the lowest state. There are other mesons below it. This makes it harder to isolate and identify the glueball. Hence, there are not many unquenched calculations. It suggests that the unquenching effect is small; the mass of the scalar glueball is not significantly affected by the quark degree of freedom.
It is clear that both quenched and unquenched lattice calculations indicate that should have a large content of the scalar glueball. In principle, the percentage of the glue component in can be calculated in full lattice QCD by considering the overlap of with the glue and operators. ^{6}^{6}6Notice that quenched lattice QCD has been used in Lee to estimate the mixing between the glue and states.
In the glueballquarkonia mixing models considered in Sec. II, the parameter is the mass of the scalar glueball in the pure gauge sector. In Model I, MeV in fit 1 and MeV in fit 2 Close2 , while it is of order 1665 MeV in Model II CCL . Obviously, the latter lies in the range of quenched lattice results for a pure scalar glueball.
Bali et al. (1993) Bali  

H. Chen et al. (1994) Chen:1994  
Morningstar, Peardon (1999) Morningstar  
Vaccarino, Weingarten (1999) Vaccarino  
Loan et al. (2005) Loan  
Y. Chen et al. (2006) Chen:2005mg  
Gregory et al. (2012) Gregory 
iii.2 Radiative decays
The radiative decay is an ideal place to test the scalar glueball content of since the leading shortdistance mechanism for the inclusive decay is . If is composed mainly of the scalar glueball, it should be the most prominent scalar produced in radiative decays. Hence, it is expected that
(7) 
Branching fractions of radiative decays of to and measured by BES and CLEO are listed in Table 2. When summing over various channels in the table, we obtain
(8) 
and
(9) 
where we have used the average of BES and CLEO measurements whenever both available. It is clear that the lower limit for the radiative decay of is one order of magnitude larger than . Using the measured branching fractions and PDG , we find
(10) 
Likewise, we have
(11) 
where the branching fractions and Albaladejo:2008qa have been used. ^{7}^{7}7For the sake of consistency, we use the results of Albaladejo:2008qa for both and obtained from the same data analysis. Therefore, we conclude that
(12) 
The radiative decay of to a scalar glueball has been studied by the CLQCD Collaboration within the framework of quenched lattice QCD CLQCD . The result is
(13) 
Comparing this with Eqs. (10) and (11), it is edvident that has a larger overlap with the pure glueball than other scalar mesons as expected in Model II.
Decay Mode  BES  CLEO Dobbs:2015dwa 

PDG  
BES:etaeta  
PDG  
PDG  
BES:omegaomega  
BES:etaeta 
In Model I, one may argue that the constructive interference between the and glueball components can lead to a large radiative rate for . On the other hand, since in this model, it is clear that the radiative decay to is mainly governed by its glueball content as the constructive and destructive interferences between the and glueball components tend to cancel each other. Therefore, it will be difficult to understand why is largely suppressed relative to if is primarily a glueball.
iii.3 Ratio of decays to , and
Since glueballs are flavor singlets, their decays are naively expected to be flavor symmetric. For example, considering a pure glueball decay into and , we have
(14) 
where the glueball couplings to two pseudoscalar mesons are expected to be flavor independent, namely, . In the SU(3) limit, . Taking into account of phase space corrections, we find and 0.98 for MeV and 1500 MeV, respectively.
However, the above argument is no longer true for scalar glueballs due to chiral suppression. It was noticed long time ago by Carlson et al. Carlson , by Cornwall and Soni Cornwall and revitalized recently by Chanowitz Chanowitz that a scalar glueball cannot decay into a quarkantiquark pair in the chiral limit, i.e., . Consequently, scalar glueballs should have larger coupling to than to . Nevertheless, chiral suppression for the ratio at the hadron level should not be so strong as the current quark mass ratio . It has been suggested Chao that should be interpreted as the scale of chiral symmetry breaking. A precise estimate of the chiral suppression effect is a difficult issue because of the hadronization process from to and the possible competing mechanism is not wellknown Carlson ; Chao ; Jin ; Chanowitz:reply . The only reliable method for tackling with the nonperturbative effects is lattice QCD. An earlier lattice calculation Sexton did support the chiralsuppression effect with the result
(15) 
which are in sharp contrast to the flavorsymmetry limit with . Although the errors are large, the lattice result did show a sizable deviation from the flavorsymmetry limit. Therefore, .
The experimental results
(16) 
clearly indicate that the production in decays is largely suppressed relative to . Theoretically, the ratio of and productions in decays is given by CCL
(17) 
where and are the coefficients of the wave function defined in Eq. (3), is the c.m. momentum of the hadron and the parameter denotes a possible SU(3) breaking effect in the OZI allowed decays when the pair is created relative to the and pairs. In Model II, has the smallest content of (see Eq. (6)) even though it decays dominantly to ; the smallness of arises from the chiral suppression of scalar glueball decay. Specifically, the parameters , and were chosen in CCL . The ratio is consistent with the lattice calculation (15). Substituting Eq. (6) into Eq. (17) leads to .
Note that in the absence of chiral suppression the smallness of can be naturally explained in terms of the large component of in Model I. For example, we found for and . However, the presence of chiral suppression will render the ratio even smaller. If we apply the same parameters , and as in Model II, we will obtain which is too small compared to experiment. Hence, if the chiral suppression effect is confirmed in the future, this will favor Model II over Model I.
Although in Model II has the largest content of , the production is largely suppressed relative to due to the destructive interference between and components
(18) 
The experimental value of for PDG can be fitted with two possible solutions
(19) 
Setting for the moment, we are led to or . The second solution is nothing but a flavor octet as advocated in Model II before. With a small SU(3) breaking in the parameter , namely, , we obtain in excellent agreement with experiment. ^{8}^{8}8After taking into account the contribution from the glueball content, we obtain . The above discussion explains why the measurement of favors the flavor octet nature of .
In Model I, is dominated by the glueball content. Since is of order unity for flavorindependent couplings, one needs a large mixing with the glueball component in order to accommodate the experimental result of in this model. The destructive interference between the and components have to be adjusted in such a way that the production of the pair is severely suppressed so that the quark component alone will lead to a very huge to compensate for the smallness of produced by the glueball component. From Eq. (17) with and and the wave function , we find which is slightly smaller than the value of 2.4 obtained in Close2 . At any rate, the predicted ratio is still smaller than experiment.
Can the experimental ratio be accommodated in Model I ? To see this, we notice that
(20) 
Taking and , the experimental measurement can be accommodated by having either or . Neither of the relations can be satisfied in Model I with , and . In principle, one can introduce chiral suppression to accommodate . For example, , and will lead to . However, the same set of parameters also leads to a too small ratio . In other words, it is difficult to explain the ratios of and productions in and decays simultaneously in Model I.
We next turn to the modes and consider two ratios that have been measured: and . Their theoretical expressions are given by CCL
(21) 
where
(22) 
with being the mixing angle defined by
(23) 
In Eq. (III.3), the coupling is the ratio of the doubly OZI suppressed coupling to that of the OZI allowed one CCL .