Lattice study on a tetra-quark state $T_{bb}$ in the HAL QCD method

Abstract

We study a doubly-bottomed tetra-quark state $(bb\bar{u}\bar{d})$ with quantum number $I(J^P)=0(1^{+})$, denoted by $T_{bb}$, in lattice QCD with the Non-Relativistic QCD (NRQCD) quark action for $b$ quarks. Employing $(2+1)$-flavor gauge configurations at $a\approx 0.09$ {fm} on $32^3\times 64$ lattices, we have extracted the coupled channel potential between $\bar{B}{\bar{B}}^{\ast }$ and ${\bar{B}}^{\ast }{\bar{B}}^{\ast }$ in the HAL QCD method, which predicts an existence of a bound $T_{bb}$ below the $\bar{B}{\bar{B}}^{\ast }$ threshold. By extrapolating results at $m_{\pi }\approx 410,570,700$ {MeV} to the physical pion mass $m_{\pi }\approx 140$ {MeV}, we obtain a biding energy with its statistical error as $E_{\mathrm{binding}}^{(\mathrm{single})}=155(17)$ MeV and $E_{\mathrm{binding}}^{(\mathrm{coupled})}=83(10)$ MeV, where ``coupled" means that effects due to virtual ${\bar{B}}^{\ast }{\bar{B}}^{\ast }$ states are included through the coupled channel potential, while only a potential for a single $\bar{B}{\bar{B}}^{\ast }$ channel is used in the analysis for ``single". A comparison shows that the effect from virtual ${\bar{B}}^{\ast }{\bar{B}}^{\ast }$ states is quite sizable to the binding energy of $T_{bb}$. We estimate systematic errors to be $\pm 20$ MeV at most, which are mainly caused by the NRQCD approximation for $b$ quarks.

Comments: 27 pages, 8 figures, references added, table 1 and figure 8 modified