Di-gluonium sum rules, I = 0 scalar mesons and conformal anomaly

We revisit, scrutinize, improve, confirm and complete our previous results [1-3] from the scalar di-gluonium sum rules within standard SVZ-expansion at N2LO without instantons beyond minimal duality ansatz : "one resonance + QCD continuum" parametrization of spectral function which is necessary for a better understanding complex spectra $I=0$ mesons. select different (un)subtracted (USR) moments degree $\leq$ 4 extracting two lowest gluonia masses couplings. obtain: $[M_{\sigma_B},f_{\sigma_B}]=[1.07(13),0.46(16)],~[M_{G_1},f_{G_1}][1.55(12),0.37(11)]$ GeV corresponding radial excitations $M_{\sigma'_B}$= 1.11(12) $M_{G'_1}=1.56(14)$ are (unexpectedly) almost degenerated with ground states. The 2nd excitation found to have much heavier mass: $M_{G_2}\simeq$ 2.99(22) GeV. Combining these some Low-Energy Vertex Sum Rules (LEV-SR), we predict hadronic widths classify them into groups -- $\sigma$-like ($\sigma_B,\sigma'_B$) decay copiously $\pi\pi$ OZI-violating process $\sigma'_B$ $2(\pi\pi)S$ through $\sigma\sigma$. $G$-like $(G_1,~G'_1$ eventually $G_2$) $\eta'\eta, ~\eta\eta$ $U(1)_A$ gluonic vertex. Besides eventual mixings quarkonia states, may expect that observed $\sigma/f_0(500)$ $f_0(137)$ while $f_0(1.5)$ $f_0(1.7)$ gluonia. high mass $G_2(2.99)$ can also mix $G_1,~G'_1$ bring gluon component candidates above 2 estimate conformal charge $\psi_G(0)=2.09(29)$ GeV$^4$ its slope $10^2\psi'_G(0)=-22(29)$ GeV$^2$. Our summarized in Table 1.