Signature decay modes of the compact doubly-heavy tetraquarks $T_{bb\bar{u}\bar{d}}$ and $T_{bc\bar{u}\bar{d}}$

Based on the expectations that the lowest-lying doubly-bottom tetraquark $T_{bb\bar{u}\bar{d}}$ ($J^P=1^{+}$) and the bottom-charm tetraquark $T_{bc\bar{u}\bar{d}}$ ($J^P=0^{+}$) are stable against strong and electromagnetic decays, we work out a number of semileptonic and non-leptonic weak decays of these hadrons, making use of the heavy quark symmetry. In doing this, we concentrate on the exclusive decays involving also tetraquarks in the final states, i.e., transitions such as $T_{bb\bar{u}\bar{d}}\to T_{bc\bar{u}\bar{d}}({\mathrm{\ell }}^{-}{\nu }_{\mathrm{\ell }},h^{-})$ and $T_{bc\bar{u}\bar{d}}\to T_{cc\bar{u}\bar{d}}({\mathrm{\ell }}^{-}{\nu }_{\mathrm{\ell }},h^{-})$, where $h^{-}={\pi }^{-},{\rho }^{-},a_1^{-},D_s^{-},D_s^{\ast -}$. So far, only the $J^P=1^{+}$ tetraquark $T_{cc\bar{u}\bar{d}}$ has been discovered, which we identify with the $I=0$ $T_{cc}^{+}$ object, compatible with $J^P=1^{+}$ and having the pole mass relative to the $D^{\ast +}{D}^0$ mass threshold and decay widths $\delta m=M(T_{cc}^{+})-(M(D^{\ast +})+M(D^0))=-360\pm 40_{-0}^{+4}$ keV and $\mathrm{\Gamma }(T_{cc}^{+})=48_{-14}^{+2}$ keV. Experimental discoveries of the transitions worked out here, and related ones involving doubly-heavy baryons, will quantify the diquark-antidiquark component of these tetraquarks.