Despite combing through 260 million events that should be able to produce a type of glueball resonance that can't be confused with quarkonium, the researchers once again after about four decades of fruitless searching, have come up empty. The quality of this particular search at Belle, and the relentless failure of searches over the decades to find any trace of glueballs despite increasingly sophisticated efforts to find them has me wondering:

**Does some, as yet unarticulated missing principle of quantum chromodynamics (QCD) forbids glueballs?**

The particularly confounding aspect of this is that glueballs are relatively easy to describe mathematically, since they implicate just one of the Standard Model physical constants, the strong force coupling constant. Unlike other hadrons, no physical constants related to quark masses, the weak force coupling constant, or the CKM matrix needs to be known to describe them.

The masses they are predicted to have are not very different from all sorts of other known hadrons (this search focused on glueballs predicted to have masses from 2.8 to 4.59 GeV (at two sigma extremes from the predicted masses), around the predicted mass of D mesons and B mesons, and well defined, distinctive quantum numbers. These are essentially completely defined from theory.

A search with 260 million events shouldn't be able to miss them if they are produced at all in the studied process at any meaningful rate, but the collaborators found nothing. Branching fractions of as much as 1 per 5,000 decays to glueballs from the Upsilon mesons whose decays were studied, which were promising candidates for giving rise to decay to glueballs at a detectable branching fraction, have been ruled out.

What do we not know about QCD that causes this? If they do exist, why can't we find traces of any of them?

The fact that these are "oddballs" can mix with quarkonium states is particularly notable, because the usual excuse for not being able to see glueball resonances doesn't apply here.

In QCD, the usual rule is that everything that is permitted is mandatory, so finding a decay that isn't prohibited by any of the rules of QCD that can't be detected is a big signal that we're missing something, although it is notable that no QCD theoretical estimate of the branching fraction was provided in this study, as some resonances are simply very rare.

The fact that these are "oddballs" can mix with quarkonium states is particularly notable, because the usual excuse for not being able to see glueball resonances doesn't apply here.

In QCD, the usual rule is that everything that is permitted is mandatory, so finding a decay that isn't prohibited by any of the rules of QCD that can't be detected is a big signal that we're missing something, although it is notable that no QCD theoretical estimate of the branching fraction was provided in this study, as some resonances are simply very rare.

The existence of bound states of gluons (so-called “glueballs”), with a rich spectroscopy and a complex phenomenology, is one of the early predictions of the non-abelian nature of strong interactions described by quantum chromodynamics (QCD). However,despite many years of experimental efforts, none of these gluonic states have been established unambiguously. Possible reasons for this include the mixing between glueballs and conventional mesons, the lack of solid information on the glueball production mechanism, and the lack of knowledge about glueball decay properties. Of these difficulties, from the experimental point of view, the most outstanding obstacle is the isolation of glueballs from various quarkonium states.

Full pdf here.Fortunately, there is a class of glueballs with three gluons and quantum numbers incompatible with quark-antiquark bound states, called oddballs, that are free of this conundrum.The quantum numbers of such glueballs include J P C = 0 −−, 0 +−, 1 −+, 2 +−, 3 −+, and so on.Among oddballs, special attention should be paid to the 0 −− state (G0−− ), since it is relatively light and can be produced in the decays of vector quarkonium or quarkoniumlike states.Two 0 −− oddballs are predicted using QCD sum rules with masses of (3.81 ± 0.12) GeV/c 2 and (4.33 ± 0.13) GeV/c 2 , while the lowest-lying state calculated using distinct bottom-up holographic models of QCD [3] has a mass of 2.80 GeV/c 2 . Although the masses have been calculated, the width and hadronic couplings to any final states remain unknown.Possible G0−− production modes from bottomonium decays are suggested in Ref. [2] including Υ(1S, 2S) → χc1+G0−− , Υ(1S, 2S) → f1(1285)+G0−−, χb1 → J/ψ+G0−− , and χb1 → ω + G0−− . In this paper, we search for 0 −− glueballs in the production modes proposed above and define G(2800), G(3810), and G(4330) as the glueballs with masses fixed at 2.800, 3.810, and 4.330 GeV/c 2 , respectively. All the parent particles in the above processes are copiously produced in the Belle experiment, and may decay to the oddballs with modest rates.

The abstract and paper are as follows:

We report the first search for the J P C = 0−− glueball in Υ(1S) and Υ(2S) decays with data samples of (102 ± 2) million and (158 ± 4) million events, respectively, collected with the Belle detector.Belle Collaboration, "Search for the 0−− Glueball in Υ(1S) and Υ(2S) decays" (November 22, 2016).No significant signals are observed in any of the proposed production modes, and the 90% credibility level upper limits on their branching fractions in Υ(1S) and Υ(2S) decays are obtained.The inclusive branching fractions of the Υ(1S) and Υ(2S) decays into final states with a χc1 are measured to be B(Υ(1S) → χc1 + anything) = (1.90 ± 0.43(stat.) ± 0.14(syst.)) × 10^{−4}with an improved precision over prior measurements and B(Υ(2S) → χc1 + anything) = (2.24 ± 0.44(stat.) ± 0.20(syst.)) × 10^{−4}for the first time.

UPDATE November 29, 2016:

A preprint of a back to the drawing board paper has been posted and notes these results while estimating a best fit for an oddball mass two GeV heavier than the mass used by the Belle Collaboration that is still consistent to within the large two sigma error bars with the heavier Belle Collaboration values.

Alexandr Pimikov, Hee-Jung Lee, Nikolai Kochelev, Pengming Zhang, "Revision of exotic 0−− glueball" (November 26, 2016).We present the new results for the exotic glueball state 0−− in the framework of the QCD sum rules. It is shown that previously used three-gluon current does not couple to any glueball bound state. We suggest considering a new current which couples to this exotic state. The resulting values for mass and decay constant of the 0−− glueball state are MG = 6.3 +0.8 −1.1 GeV and FG = 67 ± 6 keV, respectively.

Their predicted mass at two sigma error bars is 4.1 GeV to 7.9 GeV which isn't too impressive for a pure theoretical calculation that is basically a function of just one experimentally measured Standard Model constant (the strong force coupling constant) that is known to a precision of about 1%. The Belle estimate has a mere 3% uncertainty.

The pseudo-scalar bottom eta meson which is a form of bottomonium has a mass of 9.398 +/- 0.0032 GeV. The measured mass of the parent mesons Y(1S) which is called an upsilon meson, is also a form of bottomonium and is 9.46030 +/- 0.00026 GeV. The measured mass of the Y(2S) which is an excited upsilon meson, which is another form of bottomonium isn't well established in sources I've found off the bat, but would be expected to be heavier than 9 .46 GeV. Measured masses of two kinds of Y bosons with the right quantum numbers whose exact excitation has not been determined are 10.81 GeV and 11.02 GeV. So, the parent would not be barred from decaying into an oddball of this type even if it is on the heavy side of the estimated range, by mass-energy conservation.

The introduction to this paper notes that:

The glueballs are composite particles that contain gluons and no valence quarks. The glueballs carry very important information about the gluonic sector of QCD and their study is one of the fundamental tasks for the strong interaction. While the glueballs are expected to exist in QCD theoretically, there was no clear experimental evidence and so the glueballs remain yet undiscovered (see reviews [1, 2]). This is the reason why the investigation of the possible glueball’s candidates are included in the programs of the running and projected experiments such as Belle (Japan), BaBar (SCAC, USA), BESIII (Beijing, China), RHIC (Brookhaven, USA), LHC (CERN), GlueX (JLAB, USA), NICA (Dubna, Russia), HIAF (China) and FAIR (GSI, Germany).

One of the main problems of the glueball spectroscopy is the possible large mixing of the glueballs with ordinary meson states, which leads to the difficulties in disentangling the glueballs in the experiment. In this connection, the discovery of the exotic 0−− glueball, which can not be mixed with the qq¯ states, is one of the fundamental tasks of the glueball spectroscopy. Therefore, it is very important to investigate the properties of this glueball within the QCD’s based approach. One of such approaches is the QCD Sum Rules (SR). The first study of the 0−− glueball by the QCD SR method has been performed recently in [3] where the authors introduced a very specific interpolating current for this three-gluon state. Unfortunately, they only considered SR for the mass of the glueball and did not check the SR for the decay constant. Below we show that their current has pathology, which leads to the negative sign of the imaginary part of the corresponding correlator and, as the result, SR become inconsistent. Considering the fact the study of the glueball is a very hot topic nowadays and the prediction of the value of the exotic glueball mass is crucial for the experimental observation, the revision of the exotic glueball properties within QCD SR is required.

In this Letter, a new interpolating current, which couples to the 0−− exotic glueball state, is suggested. We calculate the Operator Product Expansion (OPE) for the correlator with this current up to dimension-8 and show that there is a good stability of SR for both mass and decay constant of this state.The paper then reaches the result stated in the abstract and goes on to conclude that:

Our final result is: MG = 6.3 +0.8 −1.1 GeV, FG = 67 ± 6 keV. (11)

The SR analysis in full QCD (Nf = 3) and nonzero quark condensate hJ 2 i) leads to a reduction of the glueball mass by 0.2 GeV. The mass of the exotic glueball in Eq.(11) is not far away from the recent unquenched lattice result MG = 5.166 ± 1.0 GeV [12] obtained with a rather large pion mass mπ = 360 MeV.

Here we would like to note that there are three sources of uncertainties in the above analysis for the mass and decay constant: the variation of gluon condensate, stability of SR triggering Borel parameter M2 dependence in terms of criteria δ min k , and roughly estimated SR uncertainty coming from the OPE truncation. The latter uncertainty for the decay constant comes from the definition of the fiducial interval, Eq. (9), in the standard assumption that the contribution from missing terms is of the order of the last included nonperturbative term squared: (1/3)2 ∼ 10%. The same error for mass can be expected to be suppressed since the related errors for R (SR) k+1 and R (SR) k are correlated. The best threshold value is s bf 0 = 52.4 +12.6% −16.2% GeV2 when only uncertainty of the gluon condensate is included. Note that the fiducial interval for the central value of the gluon condensate is M2 ∈ [3.7, 7.3] GeV2 . We also mention that here we present the results from the k = 0 case for SR (see Eqs.(8,10)). The mass estimation for higher values of k = 1, 2, 3 are in agreement with the considered k = 0 case within the error bars.

Summarizing, we present the revision of the QCD SR result for the exotic three-gluon glueball state with quantum numbers J P C = 0−−. A new interpolating current for this glueball has been constructed. By using this current, we have analyzed the QCD SR consisting of contributions up to the operators of dimension-8 and obtained the estimation of the mass and decay constant of the exotic glueball.

After the paper was completed we were informed about the negative result of the searching of the low mass exotic 0 −− glueball by the Belle Collaboration.UPDATE (December 4, 2016): Marco Frasca has some interesting comments on the subject here.

## 4 comments:

f0(1710) is widely regarded as a glueball.

It is a hypothesis, explored, e.g. at https://arxiv.org/abs/1511.02449 but I'd hardly say that it is "widely regarded" as a glueball.

See, e.g. https://arxiv.org/abs/1503.02463 considering both glueball and non-glueball explanation of f0(1710).

"Similarly, in Refs. [10, 11], the former work of Ref. [12] on the ρρ interaction was extended to SU(3) using the

local hidden gauge formalism for vector-vector interaction and a unitary approach in coupled channels. This

interaction generates resonances, some of which can be associated to known resonances, namely the f2(1270),

f′2 (1525), and K¯ ∗2(1430), as well as the f0(1370) and f0(1710). The results obtained in those former works gave support to the idea of the f2(1270), f′2(1525), K¯ ∗2(1430), f0(1370) and f0(1710) as being quasimolecular states of two vector mesons. In reactions producing these resonances, a pair of vector mesons are primary produced and these two vector mesons rescatter after the production, giving rise to the resonances that can be observed in the invariant mass distributions.

In Ref. [13], the important role of the f2(1270), f′2(1525), and K¯ ∗2(1430) in the J/ψ → φ(ω)V V decays was studied based on the vector-vector molecular structure of those resonances. Related work was also done in Ref. [14] interpreting the J/ψ radiative decay into these resonances. Those latter works were then extended in Ref. [15] to study the decay of ψ(2S) into ω(φ) and f2(1270), f′2(1525) and ψ(2S) into K∗ 0 and K¯ ∗ 02(1430). At the same time, in this latter work the ideas of Ref. [14] in the radiative decay were extended to the decay of Υ(1S), Υ(2S) and ψ(2S). These hadronic and radiative decays for J/ψ have also been addressed within a scheme where the f0(1370), f0(1500) and f0(1710) states emerge as a result of glueball quarkonia mixing [16]. With the steady accumulation of experimental data, new results are now available that can test these theoretical ideas and an update of the theoretical predictions has become timely."

The bottom line in the 2015 article that I quote above is that:

"The overall agreement found with the data on different experiments provides extra support for the picture in

which the tensor states f2(1270), f′2(1525), K¯∗2(1430), as well as the scalar ones f0(1370) and f0(1710) are dynamically generated states from the vector-vector interaction."

Thus, these resonances can be explained without glueballs.

Again, my emerging Baysean prior on all of this is that glueballs probably don't exist and that the problem is to determine what about QCD needs to be tweaked to capture this reality.

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