Zb tetraquark channel from lattice QCD and Born-Oppenheimer approximation

Abstract

Two $Z_b$ hadrons with exotic quark structure $\bar{b}b\bar{d}u$ were discovered by Belle experiment. We present a lattice QCD study of the $\bar{b}b\bar{d}u$ system in the approximation of static $b$ quarks, where the total spin of heavy quarks is fixed to one. The energies of eigenstates are determined as a function of the separation $r$ between $b$ and $\bar{b}$. The lower eigenstates are related to a bottomonium and a pion. The eigenstate dominated by $B{\bar{B}}^{\ast }$ has energy significantly below $m_B+m_{B^{\ast }}$, which points to a sizable attraction for small $r$. The attractive potential $V(r)$ between $B$ and ${\bar{B}}^{\ast }$ is extracted assuming that this eigenstate is related exclusively to $B{\bar{B}}^{\ast }$. The Schrödinger equation for $B{\bar{B}}^{\ast }$ within the extracted potential leads to one bound state below $B{\bar{B}}^{\ast }$ threshold, whose mass depends on the parametrization of the lattice potential. For certain parametrizations, the bound state is very close to the $B{\bar{B}}^{\ast }$ threshold and renders a narrow peak in the $B{\bar{B}}^{\ast }$ rate above threshold - these features could be related to $Z_b(10610)$ in the experiment.

arXiv comment5 pages and supplement. The submatrix of the correlation matrix was multiplied by a wrong overall constant, which affected the eigen-energies. All figures expect Fig. 1 are replaced in v4. The main conclusion does not change - Zb is related to attraction between B and B*. The correct potential is less attractive and it results only in one bound state (see Note added)

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