$P$-wave states $T_{bb}^{-}$ from diquarks

We investigate the $P$-wave states $T_{bb}^{-}$ in the isospin singlet and three excited modes [excitation occurring in the diquark $[bb{]}_{c_1}^{s_1}$ ($\rho$-mode), antidiquark $[\overline{u}\overline{d}{]}_{c_2}^{s_2}$ ($r$-mode), or between them ($\lambda$-mode)] from diquarks in a quark model. We analyze the dynamical behaviors of the diquark $[bb{]}_{c_1}^{s_1}$, antidiquark $[\overline{u}\overline{d}{]}_{c_2}^{s_2}$, and their correlations in the states $T_{bb}^{-}$ by decomposing the interactions from various sources in the model. The absolute dominant color-spin configuration, more than 99%, in the $\rho$-mode with $1{{}_{1}P}_1$ is $\left[bb{]}_{\overline{3}}^0\right[\overline{u}\overline{d}{]}_3^0$. Its energy is lower by about 18 MeV than the threshold $\overline{B}\overline{B}$ so that it can establish a compact bound state. The chromomagnetic and meson-exchange interactions in the antidiquark $[\overline{u}\overline{d}{]}_3^0$ are responsible for its binding mechanism. Two other excited modes are higher than their respective threshold. The color configuration $6\otimes \overline{6}$ need to be handled discreetly in the tetraquark states.