Heavy glueballs: status and large- width’s estimate
Glueballs, an old and firm prediction of various QCD approaches (lattice QCD, bag models, AdS/QCD, effective models, etc.), have not yet been experimentally confirmed. While for glueballs below GeV some candidates exist, the situation for heavy glueballs (above GeV) is cloudy. Here, after a brief review of scalar, tensor, and pseudoscalar glueballs, we present predictions for the decays of a putative pseudotensor glueball with a lattice predicted mass of GeV and a putative vector glueball with a lattice predicted mass of GeV. Moreover, we discuss in general the width of heavy glueballs by using large- arguments: we obtain a rough estimate according to which the width of a glueball (such as the vector one) is about MeV. Such a width would be narrow enough to enable measurement at the future PANDA experiment.
Gluons, the force carriers of strong interaction between quarks, carry themselves color charge, hence they are expected to form white bound state, called glueballs. The search for glueballs in the mesonic spectrum of the PDG , is a long-standing activity, see the reviews in Refs. .
Theoretically, bag models  were the first to predict a spectrum of glueballs. Later on, various other approaches have followed, e.g. QCD sum rules, flux-tube model, Hamiltonian QCD, anti De-Sitter/QCD methods . A reliable approach is lattice QCD [5, 6] (both quenched and unquenched), in which a spectrum of glueballs has been evaluated by a numerical simulation of QCD, see Table 1.
In conclusion, there is nowadays a (theoretical) consensus about the existence of glueballs and the qualitative form of the spectrum, but up to now no resonance could be unambiguously classified as a predominantly gluonic. In these proceedings, after a brief review of some glueball’s candidates, we present some recent developments for the decays of the heavy vector and pseudotensor glueballs. Moreover, we present a new, “heuristic”, estimate of the glueball’s width by using large- arguments.
Table 1: Central values of glueball masses from lattice QCD .
|Value [GeV]||Value [GeV]|
2 From light to heavy glueballs
First, we review the status of the three lightest glueballs of Table 1, for which some candidates exists.
Scalar glueball: The resonances and were investigated as glueball’s candidates in various works [7, 8, 9, 10, 11]. In Ref.  the glueball (as a dilaton) was studied within the so-called extended Linear Sigma Model (eLSM) . Quite remarkably, there is only an acceptable scenario: is mostly gluonic. This is in agreement with the original lattice work of Ref. , with the recent lattice study of in Ref. , and also with the AdS/QCD study in Ref. . In conclusion, there is mounting evidence that is predominantly the scalar glueball.
Tensor glueball: In Ref.  it was shown that does not lie on the Regge trajectories. Its mass fits well with lattice (see Tab. 1), it is narrow, the ratio agrees with flavour blindness , and no decay was seen, hence it is a good candidate to be the tensor glueball. Yet, the experimental assessment of this resonance is necessary.
Pseudoscalar glueball: The pseudoscalar glueball has been investigated in a variety of scenarios, see the review . In some works, e.g. Ref. , the pseudoscalar glueball was assigned to the resonance but it is not clear if and are two independent states (see the recent discussion in Ref.  and refs. therein). Moreover, the lattice mass is about GeV, i.e. 1 GeV heavier. In Ref.  the decays of an hypothetical pseudoscalar glueball (linked to the chiral anomaly ) with a mass of about GeV were studied within the eLSM: the decay channels into and are dominant. (For an independent AdS/QCD calculation, see ). A possible experimental candidate is the state measured by BES , yet future measurements are needed.
For the other glueballs listed in Table 1, no candidate is presently known. Very recently, two theoretical studies have been performed with the aim of helping future experimental search.
Pseudotensor glueball: in Ref.  the decays of a pseudotensor glueball, a putative resonance has been studied in a flavour-invariant hadronic model: sizable decay into and are predicted (they are enhanced by isospin factors). Moreover, decays into a vector and a pseudoscalar mesons vanish at leading order, hence .
Vector glueball: The vector glueball is interesting since it can be directly formed in scattering. Up to now, the search for candidates was not successful . The decays of a vector glueball (called ) using the eLSM have been studied in Ref.  (for previous theoretical works, see Ref. ). Three interaction terms have been considered. While the intensity of the corresponding coupling constants cannot be determined, some decay ratios are predicted. In the first two interaction terms (which are also dilatation invariant, then should dominate) the main decay modes are (first term) as well as and (second term). The third interaction terms, which breaks dilatation invariance, predicts decays into vector-pseudoscalar pairs, in particular and .
Both the pseudotensor and the vector glueballs (as well as other heavy glueballs) can be experimentally produced in formation processes at the future PANDA experiment .
3 Glueballs’ widths via large- considerations
The widths of glueballs are presently unknown. The scalar and the pseudoscalar glueball are somewhat special, since they are linked to the trace and axial anomalies. However, for the other glueballs, a (rough!) estimate using large- considerations may be helpful (for reviews on the large- approach see ).
First, we recall that the dominant decay of a meson into two states scales as A typical example is that of the meson ( decays trough creation of a or pair from the vacuum that recombine in ):
Similarly, the vector kaonic state decays into is regulated by a similar strength MeV (it is smaller only due to phase space). From the tensor sector: MeV. Other examples are in the PDG. Interestingly, when going to higher masses, the qualitative picture does not change. For instance, for the state : MeV. Summarizing, whenever a strong (OZI allowed) decay of a conventional meson into two conventional mesons and is kinematically allowed, one has:
Clearly, there are modifications when threshold effects are important and/or certain quantum numbers are considered, but the whole picture is rather stable.
Next, let us consider OZI suppressed decay , which occur through annihilation of the original pair into gluons, which then reconvert into mesons. In the large- language, the scaling holds. A nice example is provided by the decay of the tensor state into . This state is predominantly , hence this transition goes as . Experimentally:
which is a factor (!) smaller than (even if the phase space into is large). A very well known example in the heavy quark sector is given by the decay of the meson into hadrons. The full hadronic decay (also suppressed by ) reads MeV. As shown in , the large- approach naturally explains the validity of the OZI rule  (and is actually the only theoretical framework to derive it). Also in this case, various other examples exist, such as MeV and MeV. In conclusion, one has the following estimate:
Let us now turns to a heavy glueballs above GeV. The large- scaling of a glueball’s decay into two conventional mesons is given by thus one has the relation
Hence, we “guess” that the expected width of a glueball lies between and MeV:
Such a width is relatively small to allow experimental detection of such states (in particular at PANDA ). For the special case of the vector glueball , we expect that its width is in between that of OZI-suppressed and OZI-allowed charm-anticharm decays of vector states: MeV MeV. Namely, the vector glueball is at an intermediate stage between three gluons and quarks (in order to decay, gluons have to completely annihilate into quarks). Moreover, the fact that at least three valence gluons are contained in the vector glueball, is a further hint that its decay should not be large.
Glueballs are expected to exist but were not yet found in experiments. In this work, we have briefly reviewed the status of some candidates and presented predictions for the heavy pseudotensor ad vector glueballs. Moreover, we have discussed the possible width of an heavy glueball, obtaining the heuristic, rough estimate of about MeV. Experimental searches at low energies in the experiments GlueX  and CLAS12  at Jefferson Lab (see also ) and at high energy at the ongoing BESIII [21, 32] and at the future PANDA  experiments are expected to improve our understanding.
Acknowledgments: The author acknowledges support from the Polish National Science Centre NCN through the OPUS project no. 2015/17/B/ST2/01625.
- K. A. Olive et al. (Particle Data Group), Chin. Phys. C38, 090001 (2014).
- W. Ochs, J. Phys. G 40 (2013) 043001; V. Mathieu, N. Kochelev and V. Vento, Int. J. Mod. Phys. E 18 (2009) 1; V. Crede and C. A. Meyer, Prog. Part. Nucl. Phys. 63 (2009) 74; F. Giacosa, EPJ Web Conf. 130 (2016) 01009.
- A. Chodos, R. L. Jaffe, K. Johnson, C. B. Thorn and V. F. Weisskopf, Phys. Rev. D 9 (1974) 3471; R. L. Jaffe and K. Johnson, Phys. Lett. B 60 (1976) 201; R. L. Jaffe, K. Johnson and Z. Ryzak, Annals Phys. 168 (1986) 344.
- H. G. Dosch and S. Narison, Nucl. Phys. Proc. Suppl. 121 (2003) 114 [hep-ph/0208271]; H. Forkel, Phys. Rev. D 71 (2005) 054008; A. P. Szczepaniak and E. S. Swanson, Phys. Lett. B 577 (2003) 61.
- Y. Chen et al., Phys. Rev. D 73, 014516 (2006).
- E. Gregory et al., JHEP 1210, 170 (2012).
- W. J. Lee and D. Weingarten, Phys. Rev. D 61, 014015 (2000).
- C. Amsler and F. E. Close, Phys. Rev. D 53 (1996) 295; F. Giacosa, T. Gutsche, V. E. Lyubovitskij and A. Faessler, Phys. Rev. D 72, 094006 (2005); H. Y. Cheng, C. K. Chua and K. F. Liu, Phys. Rev. D 74, 094005 (2006); P. Chatzis, A. Faessler, T. Gutsche and V. E. Lyubovitskij, Phys. Rev. D 84, 034027 (2011).
- S. Janowski, F. Giacosa and D. H. Rischke, Phys. Rev. D 90 (2014) 11, 114005.
- L. -C. Gui et al, Phys. Rev. Lett. 110 (2013) 021601.
- F. Brünner, D. Parganlija and A. Rebhan, Phys. Rev. D 91 (2015) 10, 106002.
- D. Parganlija, P. Kovacs, G. Wolf, F. Giacosa and D. H. Rischke, Phys. Rev. D 87 (2013) 1, 014011.
- V. V. Anisovich, JETP Lett. 80 (2004) 715; V. V. Anisovich, M. A. Matveev, J. Nyiri and A. V. Sarantsev, Int. J. Mod. Phys. A 20 (2005) 6327.
- F. Giacosa, T. Gutsche, V. E. Lyubovitskij and A. Faessler, Phys. Rev. D 72 (2005) 114021.
- A. Masoni, C. Cicalo and G. L. Usai, J. Phys. G 32 (2006) R293.
- T. Gutsche, V. E. Lyubovitskij and M. C. Tichy, Phys. Rev. D 80 (2009) 014014.
- D. Parganlija and F. Giacosa, Eur. Phys. J. C 77 (2017) no.7, 450.
- W. I. Eshraim, S. Janowski, F. Giacosa and D. H. Rischke, Phys. Rev. D 87 (2013) 5, 054036; W. I. Eshraim et al, Acta Phys. Polon. Supp. 5 (2012) 1101.
- C. Rosenzweig, A. Salomone and J. Schechter, Phys. Rev. D 24, 2545 (1981).
- F. Brünner and A. Rebhan, Phys. Lett. B 770 (2017) 124.
- M. Ablikim et al. [BESIII Collaboration], Phys. Rev. Lett. 106 (2011) 072002.
- A. Koenigstein and F. Giacosa, Eur. Phys. J. A 52 (2016) no.12, 356.
- J. Z. Bai et al. [BES Collaboration], Phys. Rev. D 54 (1996) 1221 Erratum: [Phys. Rev. D 57 (1998) 3187].
- F. Giacosa, J. Sammet and S. Janowski, Phys. Rev. D 95 (2017) no.11, 114004.
- D. Robson, Phys. Lett. B 66 (1977) 267; G. W. S. Hou, In *Minneapolis 1996, Particles and fields, vol. 1* 399-401, [hep-ph/9609363]; G. W. S. Hou, hep-ph/9707526. C. T. Chan and W. S. Hou, Nucl. Phys. A 675 (2000) 367C; M. Suzuki, Phys. Rev. D 65 (2002) 097507.
- M. F. M. Lutz et al. [ PANDA Collaboration ], arXiv:0903.3905 [hep-ex]].
- G. t Hooft, Nucl. Phys. B 72 (1974) 461; E. Witten, Nucl. Phys. B 160 (1979) 57; R. F. Lebed, Czech. J. Phys. 49 (1999) 1273.
- S. Okubo, Phys. Lett. 5, 165 (1963); G. Zweig, CERN Report No.8419/TH412 (1964); J. Iizuka, Prog. Theor. Phys. Suppl. 37 (1966) 21.
- H. Al Ghoul et al. [GlueX Collaboration], AIP Conf. Proc. 1735 (2016) 020001.
- A. Rizzo [CLAS Collaboration], PoS CD 15 (2016) 060.
- T. Gutsche, S. Kuleshov, V. E. Lyubovitskij and I. T. Obukhovsky, Phys. Rev. D 94 (2016) no.3, 034010.
- S. Marcello [BESIII Collaboration], JPS Conf. Proc. 10 (2016) 010009.