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Motivated by a number of theoretical considerations, predicting the deeplybound double-heavy tetraquarks $T^{\{bb\}}_{[\bar u \bar d]}$,$T^{\{bb\}}_{[\bar u \bar s]}$ and $T^{\{bb\}}_{[\bar d \bar s]}$, we explorethe potential of their discovery at Tera-$Z$ factories. Using the process $Z\to b \bar b b \bar b$, we calculate, employing the Monte Carlo generatorsMadGraph5$\_$aMC@NLO and Pythia6, the phase space configuration in which the~$bb$ pair is likely to fragment as a diquark. In a jet-cone, defined by aninvariant mass interval $m_{bb} < M_{T^{\{bb\}}_{[\bar q \bar q']}} + \DeltaM$, the sought-after tetraquarks $T^{\{bb\}}_{[\bar q \bar q^\prime]}$ as wellas the double-bottom baryons,~$\Xi_{bb}^{0,-}$, and $\Omega_{bb}^-$, can beproduced. Using the heavy quark--diquark symmetry, we estimate $\mathcal{B} (Z\to T^{\{bb\}}_{[\bar u \bar d]} + \; \bar b \bar b) = (1.4^{+1.1}_{-0.5})\times 10^{-6}$, and about a half of this for the$T^{\{bb\}}_{[\bar{u}\bar{s}]}$ and $T^{\{bb\}}_{[\bar d \bar s]}$. We alsopresent an estimate of their lifetimes using the heavy quark expansion,yielding $\tau(T^{\{bb\}}_{[\bar q \bar q^\prime]}) \simeq 800$~fs. Measuringthe tetraquark masses would require decays, such as $T^{\{bb\} -}_{[\bar u \bard]} \to B^- D^- \pi^+$, $T^{\{bb\} -}_{[\bar u \bar d]} \to J/\psi \overlineK^0 B^-$, $T^{\{bb\} -}_{[\bar u \bar d]} \to J/\psi K^- \overline B^0$,$T^{\{bb\} -}_{[\bar u \bar s]} \to \Xi_{bc}^0 \Sigma^-$, and $T^{\{bb\}0}_{[\bar d \bar s]} \to \Xi_{bc}^0 \bar\Sigma^0$, with subsequent decay chainsin exclusive non-leptonic final states. We estimate a couple of the decaywidths and find that the product branching ratios do not exceed~$10^{-5}$.Hence, a good fraction of these modes will be required for a discovery of$T^{\{bb\}}_{[\bar q \bar q']}$ at a Tera-$Z$ factory.

Prospects of discovering stable double-heavy tetraquarks at a Tera-$Z$ factory