Di-Gluonium Sum Rules, Conformal Charge and I = 0 Scalar Mesons

We revisit, improve and confirm our previous results [1-3] from the scalar digluonium sum rules within the standard SVZ-expansion at N2LO {\it without instantons} and {\it beyond the minimal duality ansatz} : "one resonance $\oplus$ QCD continuum" parametrization of the spectral function. We select different unsubtracted sum rules (USR) moments of degree $\leq$ 4 for extracting the two lowest gluonia masses and couplings. We obtain in units of GeV: $(M_{G},f_G)=[1.04(12),0.53(17)]$ and $[1.52(12),0.57(16)]$. We attempt to predict the masses of their first radial excitations to be $M_{\sigma'} \simeq 1.28(9)$ GeV and $M_{G_2}\simeq 2.32(18)$ GeV. Using a combination of the USR with the subtracted sum rule (SSR), we estimate the conformal charge (subtraction constant $\psi_G(0)$ of the scalar gluonium two-point correlator at zero momentum) which agrees completely with the Low Energy Theorem (LET) estimate. Combined with some low-energy vertex sum rules (LEV-SR), we confront our predictions for the widths with the observed $I=0$ scalar mesons spectra. We confirm that the $\sigma$ and $f_0(980)$ meson can emerge from a maximal (destructive) ($\bar uu+\bar dd$) meson - $(\sigma_B$) gluonium mixing [10]. The $f_0(1.37)$ and $f_0(1.5)$ indicate that they are (almost) pure gluonia states (copious decay into $4\pi$) through $\sigma\sigma$, decays into $\eta\eta$ and $\eta'\eta$ from the vertex $U(1)_A$ anomaly with a ratio $÷$ to the square of the pseudoscalar mixing angle sin$^2\theta_P$.