Landau equation and QCD sum rules for the tetraquark molecular states

Abstract

The quarks and gluons are confined objects, they cannot be put on the mass-shell, it is questionable to apply the Landau equation to study the Feynman diagrams in the QCD sum rules. Furthermore, we carry out the operator product expansion in the deep Euclidean region $p^2\to -\infty$, where the Landau singularities cannot exist. The Landau equation servers as a kinematical equation in the momentum space, and is independent on the factorizable and nonfactorizable properties of the Feynman diagrams in the color space. The meson-meson scattering state and tetraquark molecular state both have four valence quarks, which form two color-neutral clusters, we cannot distinguish the contributions based on the two color-neutral clusters in the factorizable Feynman diagrams. Lucha, Melikhov and Sazdjian assert that the contributions at the order $\mathcal{O}(\alpha_s^k)$ with $k\leq1$ in the operator product expansion, which are factorizable in the color space, are exactly canceled out by the meson-meson scattering states at the hadron side, the tetraquark molecular states begin to receive contributions at the order $\mathcal{O}(\alpha_s^2)$. Such an assertion is questionable, we refute the assertion in details, and choose an axialvector current and a tensor current to examine the outcome of the assertion. After detailed analysis, we observe that the meson-meson scattering states cannot saturate the QCD sum rules, while the tetraquark molecular states can saturate the QCD sum rules. The Landau equation is of no use to study the Feynman diagrams in the QCD sum rules for the tetraquark molecular states, the tetraquark molecular states begin to receive contributions at the order $\mathcal{O}(\alpha_s^0/\alpha_s^1)$ rather than at the order $\mathcal{O}(\alpha_s^2)$.