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: $[M_{σ_B},f_{σ_B}]=[1.07(13),0.46(16)],~[M_{G_1},f_{G_1}][1.55(12),0.37(11)]$ GeV and the corresponding masses of the radial excitations : $M_{σ'_B}$= 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 $σ$-like ($σ_B,σ'_B$) which decay copiously to $ππ$ from OZI-violating process and the $σ'_B$ to $2(ππ)S$ through $σσ$. -- The $G$-like $(G_1,~G'_1$ and eventually $G_2$) which decay into $η'η, ~ηη$ through the $U(1)_A$ gluonic vertex. Besides some eventual mixings with quarkonia states, we may expect that the observed $σ/f_0(500)$ and $f_0(137)$ are $σ$-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 $ψ_G(0)=2.09(29)$ GeV$^4$ and its slope $10^2ψ'_G(0)=-22(29)$ GeV$^2$. Our results are summarized in Table 1.
Non-Abelian and Type-A Conformal Anomalies from Euler Descent
Racah Institute of Physics, Hebrew University of Jerusalem（雷哈物理学研究所，希伯来大学）；Department of Physics, LEPP, Cornell University（康奈尔大学物理系）；The Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University（特拉维夫大学雷蒙德和贝弗利萨克勒物理与天文学学院）
Adler function and Bjorken polarized sum rule: Perturbation expansions in powers of $SU(N_c)$ conformal anomaly and studies of the conformal symmetry limit
Anomaly-mediated Scalar Gravitational Interactions and the Coupling of Conformal Sectors
Dipartimento di Matematica e Fisica, Università del Salento（萨伦托大学数学与物理系）；INFN Sezione di Lecce（意大利国家核物理研究所莱切分部）；National Center for HPC, Big Data and Quantum Computing（高性能计算、大数据与量子计算国家中心）；CNR-nanotec, Lecce（国家研究委员会纳米技术研究所莱切分部）
