Tetraquark systems $\bar bb\bar du$ in the static limit and lattice QCD

Two hadrons with exotic quark content $Z_b^+\simeq \bar bb \bar du$ were discovered by Belle. We present a lattice study of the $\bar bb\bar du$ systems with various quantum numbers using static bottom quarks. Only one set of quantum numbers that couples to $Z_b$ and $\Upsilon\;\pi$ was explored on the lattice before; these studies found an attractive potential between $B$ and $\bar B^*$ resulting in a bound state below the threshold. The present study considers the other three sets of quantum numbers. Eigenenergies of the $\bar bb \bar du$ 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 $\bar bb+\bar du$ and $\bar bu+\bar db$, so no significant attraction or repulsion is found. A slight exception is a small attraction between $B$ and $\bar B^*$ at small distance for the quantum number that couples to $Z_b$ and $\eta_b\;\rho$.