Sigma meson in pole-dominated QCD sum rules

Abstract

The properties of the $\ensuremath{\sigma}$ meson are studied using the QCD sum rules for tetraquark operators. In the SU(3) chiral limit, we separately investigate SU(3) singlet and octet tetraquark states as constituents of the $\ensuremath{\sigma}$ meson and discuss their roles for the classification of the light scalar nonets $\ensuremath{\sigma}$, ${f}_{0}$, ${a}_{0}$, and $\ensuremath{\kappa}$ as candidates of tetraquark states. All of our analyses are performed in the suitable Borel window which is indispensable to avoid the pseudopeak artifacts outside of the Borel window. We can set up acceptably wide Borel windows after preparing favorable linear combinations of operators and including the dimension 12 terms in the operator-product expansion. Taking into account the possible large widths, we evaluate masses for singlet and octet states as 700--850 and 600--750 MeV, respectively, although the octet operators have a smaller overlap with the tetraquark states than the singlet case, which requires careful interpretations. The splitting of the singlet and octet states emerges from the number of the $\overline{q}q$ annihilation diagrams, which include the color singlet annihilation processes $qq\overline{q}\overline{q}\ensuremath{\rightarrow}(q\overline{q}{)}_{1}$ and the color octet annihilation processes $qq\overline{q}\overline{q}\ensuremath{\rightarrow}G(q\overline{q}{)}_{8}$. The mass of the $\ensuremath{\sigma}$ meson is evaluated as 600--800 MeV, which is much closer to the experimental value $\ensuremath{\sim}500\text{ }\text{ }\mathrm{MeV}$ than the mass evaluated by 2-quark correlator analyses, $\ensuremath{\sim}1.0\text{ }\text{ }\mathrm{GeV}$. This indicates that the tetraquark state shares a larger fraction in the $\ensuremath{\sigma}$ meson than ordinary two quark meson states.