Tetraquark systems b¯bd¯u in the static limit and lattice QCD

Abstract

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