Search for the tensor glueball

Quantum chromodynamics (QCD), the fundamental theory of strong interactions, predicts the existence of a full spectrum of glueballs, of composite particles containing gluons but no valence quarks. Their existence is a direct consequence of the nonabelian nature of QCD and of confinement. The properties of glueballs have been studied in many models since their prediction in the 1970s [1, 2] but experimentally, no generally accepted view had emerged. Recent reviews of glueballs and of light-quark mesons can be found elsewhere [3, 4, 5, 6, 7]. The scalar glueball is expected in the 1500 - 2000 MeV mass range [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. Based on lattice calculations, we have to expect the tensor glueball mass 600 to 800 MeV above the scalar glueball. However, a small mass gap is also possible. The authors of Ref. [15] use QCD sum rules and predict 1780 MeV for the scalar, and 1860 MeV for the tensor glueball.

The radiative decay branching ratios for producing glueballs were predicted by lattice gauge calculations [24, 25]:

	$\displaystyle\Gamma_{J/\psi\to\gamma/G_{0^{++}}}/\Gamma_{\rm tot}$	$\displaystyle=$	$\displaystyle(3.8\pm 0.9)10^{-3}\,,$		(1)
	$\displaystyle\Gamma_{J/\psi\to\gamma/G_{2^{++}}}/\Gamma_{\rm tot}$	$\displaystyle=$	$\displaystyle(11\pm 2)10^{-3}\,.$		(2)

These are large numbers. The yield of $f_{2}(1270)$ in radiative $J/\psi$ decays is $(1.64\pm 0.12)10^{-3}$, about six times weaker than the predicted rate for the tensor glueball!

Little is known about the glueball width. Arguments based on the $1/N_{c}$ expansion (see, e.g., Ref. [19]) suggest that glueballs might be narrow, 100  MeV or less. Narison [20] gave a $2\pi$ partial decay width of the scalar glueball of $(119\pm 36)$ MeV, not incompatible with a large total width. Minkowski and Ochs assume a width exceeding 1 GeV [21]. The authors of Refs. [22, 23] reproduce the decay rates of $f_{0}(1710)$ assuming that this is the scalar glueball and predict the tensor glueball to be very wide.

Recently we have presented a coupled-channel analysis [26] of the $S$-wave amplitude from BESIII data on radiative $J/\psi$ decays. The data on radiative $J/\psi$ into $\pi^{0}\pi^{0}$ and $K_{s}K_{s}$ were published including a bin-wise partial-wave decomposition into $S$-wave and $D$-wave  [27, 28], the data on decays into for $\eta\eta$ and $\phi\omega$ were published only in an energy-dependent amplitude analysis [29, 30]. A large number of further data were included in the coupled-channel analysis, references to the additional data can be found in Ref. [26]. Ten scalar isoscalar resonances were required to fit the data. Five of them were interpreted as mainly singlet, five as mainly octet resonances in SU(3). The yield of resonances showed a striking peak at $(1865\pm 25^{\,+10}_{\,-30})-i(185\pm 25^{\,+15}_{\,-10})$ MeV called $G_{0}(1865)$. In a subsequent paper we studied the decays of the scalar mesons into pairs of pseudoscalar mesons. We found that the decays can be understood only when these mesons contain an additional flavor singlet fraction beyond the one expected for any mixing angle.

This peak at 1865 MeV showed properties expected from a scalar glueball:

• 1.

  $G_{0}(1865)$ is produced abundantly in radiative $J/\psi$ decays above a very low background. Its mass is $1\sigma$ compatible with the mass calculated in unquenched lattice QCD [10] and the yield is $1.6\sigma$ compatible with the yield calculated in lattice QCD.
  

• 2.

  The decay analysis of the scalar isoscalar mesons shows that the assignment of mesons to mainly-octet and mainly-singlet states is correct. Even the production of mainly-octet scalar mesons - which should be forbidden in radiative $J/\psi$ decays - peaks at 1865 MeV.
  

• 3.

  The decay analysis requires a small glueball content in the flavor wave function of several scalar resonances. The glueball content as a function of the mass shows a peak compatible with the peak in the yield of scalar isoscalar mesons. The sum of the fractional glueball contributions is compatible with one [31].
  

• 4.

  In the reaction $B_{s}\to J/\psi+K^{+}K^{-}$, a primary $s\bar{s}$ couples to mesons having a strong coupling to $K^{+}K^{-}$ [32]. Two peaks in the $K^{+}K^{-}$ mass spectrum are seen due to $\phi(1020)$ and $f_{2}^{\prime}(1525)$ (see Fig. 7 in Ref. [32], Fig. 2c below shows a fit to the tensor wave) but there is no sign of higher-mass scalar mesons. In particular $f_{0}(1710)$ with its prominent $K\bar{K}$ decay mode is not seen in $s\bar{s}\to f_{0}(1710)\to K\bar{K}$. It is, however, produced very strongly in the process gluon-gluon$\to f_{0}(1710)\to K\bar{K}$: Obviously, $f_{0}(1710)$ is produced by two initial-state gluons but not by an $s\bar{s}$ pair in the initial state. $f_{0}(1710)$ must have a sizable glueball fraction!
  

For these reasons, we are convinced that the scalar glueball is distributed among the agglomeration of scalar isoscalar mesons in the range from 1500 to 2300 MeV. Based on lattice calculations, we have to expect the tensor glueball above 2500 MeV even though smaller masses are possible as well [15].

In this Letter we search for the tensor glueball expected to be produced in radiative $J/\psi$ decays into $\pi^{0}\pi^{0}$ and $K_{s}K_{s}$. In Section 2 we compare the intensities of the $\pi\pi$ and $K\bar{K}$ invariant masses in the scalar and tensor wave. The tensor wave reveals a wide high-mass resonance. Subsequently, in Section 3, we discuss if the high-mass enhancement is split into several states and if these contain a small fraction of the tensor glueball. The results are discussed and summarized in Section 4.

The $\pi^{0}\pi^{0}$ or $K_{s}K_{s}$ systems produced in radiative $J/\psi$ decays are limited to even angular momenta due to Bose symmetry. Practically, only $S$ and $D$-waves are relevant. These two partial waves can be written in the multipole basis [34, 33]. The scalar intensity originates from the electric dipole transition $E0$. Three electromagnetic amplitudes, $E1,M2$, and $E3$, lead to the production of tensor mesons where the $E1$ amplitude is the most significant one. These three amplitudes and relative phases are discussed below.

The $E0$ and $E1$ squared amplitudes lead to strikingly different mass distributions (see Fig. 1). The distributions were derived in Refs. [27, 28] by exploiting the statistical precision provided by $(1.311\pm 0.011)\times 10^{9}J/\psi$ decays collected by the BESIII collaboration. The $\pi\pi$ mass distribution in the scalar partial wave (Fig. 1a) is characterized by a mountainous landscape starting with a steady rise growing up to 1450 MeV and a rapid fall-off. After a minimum, the intensity increases due to the $f_{0}(1710)/f_{0}(1770)$ complex. After a further deep minimum, a wide and asymmetric structure due to $f_{0}(2020)/f_{0}(2100)$ follows. At the highest mass, at about 2300 MeV, a small dip-peak structure is seen. In the $K\bar{K}$ distribution (Fig. 1c), the intensity rises slightly up to 1450 MeV, followed by a very significant interference pattern and then rises steeply to an asymmetric peak at about 1700 MeV that dominates the mass distribution. After a fast drop on its right side, a second peak at about 2100 MeV appears with a high-mass shoulder.

The squared amplitudes that describe production of tensor mesons do not show such rich structures. Below 1 GeV, the $\pi\pi-D$-wave (Fig. 1b) shows only a tail of the $f_{2}(1270)$ and no entries in $K\bar{K}$ mass distribution (Fig. 1d). Above 1 GeV the $\pi\pi$ $2^{++}$ intensity exhibits only one strong peak due to $f_{2}(1270)$ production and a wide enhancement that reaches a maximum at about 2200 MeV. The $K\bar{K}$ intensity exhibits only one peak due to $f_{2}^{\prime}(1525)$. Above, only little intensity is seen.

In a first fit, we describe the high-mass region by one additional resonance. Neither the mass distribution nor the phase difference are well reproduced. The $\chi^{2}/N_{\rm data}=1088/765$ for the mass distributions and $2584/677$ for the phase differences. The mass distribution is reasonably described, the phase differences qualitatively only. However, apparent discrepancies are often enforced by adjacent high-statistics points, and some structures are limited to a narrow mass window.

Alternatively, we allow for one further resonance $f_{2}(1640)$, the fit improves only marginally. The fit with or without $f_{2}(1640)$ gives a narrower or wider high-mass tensor resonance. Since we do not know if $f_{2}(1640)$ participates in the reaction, we increase the errors correspondingly:

$\displaystyle M=(2210\pm 60)\,{\rm MeV};\ \ \Gamma=(360\pm 120)\,{\rm MeV}\,.$

(3)

The error does not contain the possibility that the production amplitude of tensor mesons may be reduced dynamically with decreasing photon energy. Only the phase space is taken into account. Tentatively, we call these resonances $X_{2}(2210)$ (not $f_{2}(2210)$ since it might be a cluster of resonances).

An energy-dependent partial-wave analysis of the same data on was reported by the JPAC Collaboration [34]. Four scalar and three tensor mesons were identified. The tensor amplitude was described by $f_{2}(1270)$, $f_{2}^{\prime}(1525)$, and a further state at about $f_{2}(1950)$ and a width of 700 MeV. A possible tensor glueball was not discussed. The difference in mass might be due a different choice of the ambiguous solutions of the energy-independent partial wave analysis. With our choice, the $|E1|^{2}$ distribution shows a clear peak at about 2200 MeV.

In our fit, $X_{2}(2210)$ was parameterized by a three-channel relativistic Breit-Wigner amplitude with $\pi\pi$, $K\bar{K}$, and $\rho\rho$ as decay channels. The ratio of the frequencies of $X_{2^{++}}(2210)$ decays into $K\bar{K}$ and $\pi\pi$ is

$\displaystyle BR_{K\bar{K}/\pi\pi}=0.23\pm 0.05\,.$

(4)

In Table 1 the properties of $f_{2}(1270)$ and $f_{2}(1525)$ are compared with values given in the Review of Particle Physics (RPP) [37] and with other determinations using radiative $J/\psi$ decay.

The $f_{2}(1270)$ mass found here is incompatible with the RPP value. We note that in an analysis of BESII data on $J/\psi\to\gamma\pi\pi$, the $f_{2}(1270)$ mass was determined to $(1262^{+1}_{-2}\pm 8)$ MeV [38], and from CLEO data on this reaction, (1259$\pm$4$\pm$4) MeV was deduced [36]. JPAC finds masses between 1262 and 1282 MeV [34].

The $K\bar{K}/\pi\pi$ ratio for the $f_{2}(1270)$ could be determined with a large uncertainty from the faint peak at about 1270 MeV in the $K\bar{K}$ mass distribution. Here, we fix the ratio to the RPP value. Also, the small $\pi\pi$ decay mode of $f_{2}^{\prime}(1525)$ is fixed to the RPP value. Our radiative yields of $f_{2}(1270)$ and $f_{2}^{\prime}(1525)$ yields are fully compatible with RPP values.

In the reaction $J/\psi\to\gamma$ plus a tensor meson, the production process couples to the mesonic flavor-singlet component only. The ratio $R$ of $f_{2}^{\prime}(1525)/f_{2}(1270)$ production is related to the mixing angle via

$\displaystyle\tan^{2}\theta_{\rm tens}=\frac{1}{\lambda}\cdot R\cdot\frac{q_{f_{2}}}{q_{f_{2}^{\prime}}}$

(5)

from which we find a tensor mixing angle $\theta^{\rm tens}=(35$$\pm$$2)^{\circ}$. The mixing angle identifies the tensor meson nonet as ideally mixed but is inconsistent with $29.8(28.0)^{\circ}$ derived from the quadratic (linear) GMO formula.

We now need to ask: Is it plausible that just one tensor resonance above 1700 MeV is produced in radiative $J/\psi$ decays? There is at most marginal evidence for $f_{2}(1640)$, and no evidence at all for $f_{2}(1910/1950)$. Both states are seen in several experiments [37]. But these states are at most very weakly produced in radiative $J/\psi$ decays. Above these two states, a tensor meson at 2210 MeV suddenly appears. Could $X_{2}(2210)$ contain a fraction of the tensor glueball? And why is the fit to the phases bad with a single-resonance fit? Are several tensor resonances hidden in $X_{2}(2210)$?

The total observed yield in $\pi\pi$ and $K\bar{K}$ is

$\displaystyle\sum_{M=1.9\,{\rm GeV}}^{M=2.5\,{\rm GeV}}Y_{J/\psi\to\gamma f_{2},f_{2}\to\pi\pi,K\bar{K}}=(0.35\pm 0.15)\,10^{-3}\,.$

(6)

Data on $\pi\pi$ elastic $D$-wave scattering in this mass range do not exist. The missing intensity cannot be determined from the data included in our fits. An estimate can be obtained from tensor states reported to be seen in radiative $J/\psi$ decays. The reactions and their contributions to the high-mass region are listed in Table 2. Summation yields

$\displaystyle\sum_{M=1.9\,{\rm GeV}}^{M=2.5\,{\rm GeV}}Y_{J/\psi\to\gamma f_{2}}=(3.1\pm 0.6)\,10^{-3}\,.$

(7)

This is a substantial yield even though still smaller than the observed yield of the scalar glueball. We note that the $4\pi$ tensor contribution is rather small when compared to the $K^{*}(892)\bar{K}^{*}(892)$ and $\phi\phi$ contributions. The $6\pi$ tensor contribution is completely unknown. Hence there may still be missing intensity. Here we emphasize the importance of further studies of these channels with the much larger statistics taken by the BESIII Collaboration.

Summarizing, we have presented a coupled-channel analysis of BESIII data on $J/\psi$ decays into $\pi^{0}\pi^{0}$ and $K_{s}K_{s}$. The data are dominated by $S$-wave and $D$-wave contributions. This fit is important since it does not only provide the tensor wave but also shows that $S$ and $D$-waves both are consistently described.

In the tensor wave we find an enhancement at $M=(2210\pm 60)$ MeV, $\Gamma=360\pm 120$ MeV and a yield (in $\pi\pi$ and $K\bar{K}$) of $(0.35\pm 0.10)\,10^{-3}$ called $X_{2}(2210)$. There are arguments speaking in favor and against a glueball interpretation.

Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 196253076 – TRR 110T and the Russian Science Foundation (RSF 22-22-00722).