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

TL;DR

This paper refines sum rule analyses of scalar gluonium states, predicting their masses, decay modes, and gluonic content, enhancing understanding of the complex scalar meson spectrum.

Contribution

It introduces improved sum rule calculations for scalar gluonium, including higher-order corrections and spectral function parametrizations, to better identify and classify scalar mesons.

Findings

Predicted scalar gluonium masses around 1.07, 1.55, and 2.99 GeV.

Classified scalar mesons into $\sigma$-like and $G$-like groups based on decay modes.

Estimated the conformal charge and its slope, providing insights into gluonic contributions.

Abstract

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.