Sigma meson in pole-dominated QCD sum rules

The properties of $\sigma$(600) meson are studied using the QCD sum rules (QSR) for the tetraquark operators. In the SU(3) chiral limit, we investigate separately SU(3) singlet and octet tetraquark states as constituents of the $\sigma$ meson, and discuss their roles for the classification of the light scalar nonets, $\sigma, f_0, a_0$, and $\kappa$, as candidates of tetraquark states. All our analyses are performed in the the suitable Borel window which is indispensable to avoid the {\it pseudo peak} artifacts outside of the Borel window. The acceptably wide Borel window originates after preparing the favorable set up of a linear combination of operators and the inclusion of the dimension 12 terms in the OPE. Taking into account for the possible large width, we estimate masses for singlet and octet states as $700\sim 850$ MeV, $600\sim 750$ MeV, respectively, although octet states have smaller overlap with the pole than singlet state and may be strongly affected by low energy scattering states. This splitting of singlet and octet states emerges from the number of the $\bar{q}q$ annihilation diagrams, which include both color singlet annihilation processes, $qq\bar{q}\bar{q}\to (q\bar{q})_1$ and color octet annihilation processes, $qq\bar{q}\bar{q}\to G (q\bar{q})_8$. The mass evaluation for the $\sigma$ meson gives the value around $600\sim800$ MeV which is much smaller than the mass obtained by 2-quark correlators, $1.0\sim1.2$ GeV. This indicates $\sigma$ state has the large overlap with the tetraquark states.