$ud\overline{b}\overline{b}$ tetraquark resonances with lattice QCD potentials and the Born-Oppenheimer approximation

We study tetraquark resonances with lattice QCD potentials computed for a static $\overline{b}\overline{b}$ pair in the presence of two lighter quarks $ud$, the Born-Oppenheimer approximation and the emergent wave method. As a proof of concept we focus on the system with isospin $I=0$, but consider different relative angular momenta $l$ of the heavy quarks $\overline{b}\overline{b}$. For $l=0$ a bound state has already been predicted with quantum numbers $I(J^P)=0(1^{+})$. Exploring various angular momenta we now compute the phase shifts and search for S and T matrix poles in the second Riemann sheet. We predict a tetraquark resonance for $l=1$, decaying into two $B$ mesons, with quantum numbers $I(J^P)=0(1^{-})$, mass $m=10576_{-4}^{+4}\mathrm{MeV}$ and decay width $\mathrm{\Gamma }=112_{-103}^{+90}\mathrm{MeV}$.

Figure 1

(a) $(I=0,j=0)$ potential. (b) $(I=1,j=1)$ potential.

Figure 2

(a) At small separations the static antiquarks $\overline{Q}\overline{Q}$ interact by perturbative one-gluon exchange. (b) At large separations the light quarks $qq$ screen the interaction and the four quarks form two rather weakly interacting $B$ mesons.

Figure 3

Phase shift ${\delta }_l$ as a function of the energy $E$ for different angular momenta $l=0$, 1, 2, 3, 4 for the $(I=0,j=0)$ potential ($\alpha =0.34$, $d=0.45\mathrm{fm}$).

Figure 4

Phase shift ${\delta }_1$ as a function of the energy $E$ for different parameters for the potential. For illustration, we vary parameter $\alpha$ only while fixing $d=0.45\mathrm{fm}$ at the value of the $(I=0,j=0)$ potential. Fixing $d$ and varying $\alpha$ produces comparable results.

Figure 5

T matrix eigenvalue $t_1$ as a function of the complex energy $E$ for the $(I=0,j=0)$ potential ($\alpha =0.34$, $d=0.45\mathrm{fm}$). Along the vertical axis we show the norm $|t_1|$, while the phase $\arg (t_l)$ corresponds to different colors.

Figure 6

Locations for the pole of the eigenvalue $t_1$ of the $\mathrm{T}$ matrix in the complex plane $(\mathrm{Re}(E),\mathrm{Im}(E))$. We illustrate with a cloud of diamond points the computation of the systematic error of the $\alpha$ and $d$ parameters of the $(I=0,j=0)$ potential, utilizing the technique of Ref. [24]. We also depict (solid line) the trajectory of the pole corresponding to a variation of the potential parameters, varying $\alpha$ for $d=0.45\mathrm{fm}$.