Determination of the quarkonium--gluonium content of the isoscalar tensor resonances $f_2(1920)$, $f_2(2020)$, $f_2(2240)$, $f_2(2300)$ and of the broad state $f_2(2000)$ based on decay couplings to $\pi^0\pi^0,\eta\eta,\eta\eta'$

In the reactions $p\bar p\to\pi^0\pi^0,\eta\eta,\eta\eta'$ there are four relatively narrow resonances $f_2(1920)$, $f_2(2020)$, $f_2(2240)$, $f_2(2300)$, and a broad one $f_2(2000)$ in the mass region 1990--2400 MeV. In the framework of quark combinatorics we carry out an analysis of the decay constants for all five resonances. It is shown that the relations for the decay constants corresponding to the broad resonance $f_2(2000)\to\pi^0\pi^0,\eta\eta, \eta\eta'$ are the same as those corresponding to a glueball. An additional argument in favour of the glueball--nature of $f_2(2000)$ is the fact that $f_2(1920)$, $f_2(2020)$, $f_2(2240)$, $f_2(2300)$ fit well the $q\bar q$ trajectories in the $(n,M^2)$-plane (where $n$ is the radial quantum number), while the broad $f_2(2000)$ resonance turns out to be an unnecessary extra state for these trajectories.