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