Scrutinizing the Light Scalar Quarkonia from LSR at Higher Orders

Abstract

We scrutinize, improve some determinations of the masses and couplings of light scalar quarkonia ($\bar{q}q$ and four-quark states) and present new results for the ${\pi }^{+}{\pi }^{-},K^{+}{K}^{-},\dots$ molecules using QCD Laplace Sum Rule (LSR) truncated at the $D=6$ dimension vacuum condensates. We pay a special attention on the higher order perturbative (PT) corrections up to the (estimated) $\mathcal{O}({\alpha }_s^5)$ which improve the quality of the analysis. We request that the optimal results obey the rigorous constraint: {\it Resonance $\ge$ QCD continuum contributions ($R_{P/C}\ge 1$)} in the LSR which excludes a Breit-Wigner / on-shell (not to be confused with a complex pole) scalar meson mass around (500-600) MeV obtained for values [$t_c\le (1\sim 1.5)$ GeV${}^2$] of the QCD continuum threshold. Mass-splittings due to $SU3$ breakings are small. We discuss the different assignements of the observed scalar mesons below 2 GeV in the Conclusions where the $I=0$ states are compared with the scalar gluonia. The results are compiled in Table 3 to 6.

Comments: Latex file sources, 55 PDF pages, 32 figures, 6 Tables. Version to be published in Nuclear Physics A