Tetraquark systems $\bar{b}b\bar{d}u$ in the static limit and lattice QCD

Abstract

Two hadrons with exotic quark content $Z_b^{+}\simeq \bar{b}b\bar{d}u$ were discovered by Belle. We present a lattice study of the $\bar{b}b\bar{d}u$ systems with various quantum numbers using static bottom quarks. Only one set of quantum numbers that couples to $Z_b$ and ${\rm Y}\pi$ was explored on the lattice before; these studies found an attractive potential between $B$ and ${\bar{B}}^{\ast }$ resulting in a bound state below the threshold. The present study considers the other three sets of quantum numbers. Eigenenergies of the $\bar{b}b\bar{d}u$ system are extracted as a function of separation between $b$ and $\bar{b}$. The resulting eigenenergies do not show any sizable deviation from noninteracting energies of the systems $\bar{b}b+\bar{d}u$ and $\bar{b}u+\bar{d}b$, so no significant attraction or repulsion is found. A slight exception is a small attraction between $B$ and ${\bar{B}}^{\ast }$ at small distance for the quantum number that couples to $Z_b$ and ${\eta }_b\rho$.

Comments: 8 pages, 9 figures, one sign in operator O1 of equation (4) corrected, the sign in the calculation was correct