Lattice QCD Evidence for Exotic Tetraquark Resonance

We study the manifestly exotic tetraquark D$_{\rm s0}^{++}(cu\bar s \bar d)$ and the scalar tetraquark $f_0(ud \bar u \bar d)$ in SU(3)$_c$ anisotropic quenched lattice QCD with the $O(a)$-improved Wilson (clover) fermion with various quark masses including the idealized SU(4)$_f$ case. For $f_0(ud\bar u \bar d)$ etc., we only consider connected diagrams at the quenched level, i.e., the tetraquark $f_0(ud\bar u\bar d)$ is identical with D$_{\rm s0}^{++} (cu\bar s \bar d)$ in the idealized SU(4)$_f$ case. First, for comparison, we study the lowest $q\bar q$ scalar meson, and find that it has a large mass of about 1.37GeV after chiral extrapolation, which corresponds to $f_0(1370)$. Second, we investigate the lowest 4Q state in the spatial periodic boundary condition, and find that it is just a scattering state of two pseudoscalar mesons, as is expected. Third, to extract spatially-localized 4Q resonances, we use the Hybrid Boundary Condition (HBC) method, where anti-periodic and periodic boundary conditions are imposed on quarks ($c$,$u$) and antiquarks ($\bar s$,$\bar d$), respectively. By applying the HBC on a finite-volume lattice, the threshold of the two-meson scattering state is raised up, while the mass of a compact 4Q resonance is almost unchanged. In HBC, we find a nontrivial 4Q resonance state about 100 MeV below the two-meson threshold in some quark-mass region. Its chiral behavior largely differs from a two-meson scattering state. The scalar tetraquark $f_0(ud\bar u\bar d)$ is found to have the mass of about 1.1GeV after chiral extrapolation, and seems to correspond to $f_0(980)$. Then, the manifestly exotic tetraquark D$_{\rm s0}^{++}(cu\bar s \bar d)$ would exist around 1GeV in the idealized SU(4)$_f$ chiral limit. Finally, MEM analysis is applied to obtain the spectral function of the 4Q system.