Exotic Tetraquark states with two $\bar{b}$-quarks and $J^P=0^+$ and $1^+$ $B_s$ states in a nonperturbatively-tuned Lattice NRQCD setup

Abstract: We use $n_f=2+1$ Wilson-clover gauge-field ensembles from the CLS consortium in a Lattice NRQCD setup to predict the binding energy of a $I(J^P)=0(1^+)$ $ud\bar{b}\bar{b}$ tetraquark and a $\frac{1}{2}(1^+)$ $\ell s\bar{b}\bar{b}$ tetraquark. We determine the binding energies with respect to the relevant $BB^*$ and $B_sB^*$ thresholds respectively to be $112.0(13.2)$ MeV for the $ud\bar{b}\bar{b}$, and $46.4(12.3)$ MeV for the $\ell s\bar{b}\bar{b}$. We also determine the ground-state $J^P=0^+$ $B_{s0}^*$ and $1^+$ $B_{s1}$ mesons to lie $75.4(14.0)$ and $78.7(13.9)$ MeV below the $BK$ and $B^*K$ thresholds respectively. Our errors are entirely dominated by systematics due to discretisation effects. To achieve these measurements, we performed a neural network based nonperturbative tuning of the Lattice NRQCD Hamiltonian's parameters against the basic bottomonium spectrum. For all lattice spacings considered we can reproduce the continuum splittings of low-lying bottomonia. It is worth remarking that our nonperturbative tuning parameters deviate from 1 by significant amounts, particularly the term $c_2$.