  This paper focuses on tetraquarks containing two heavy quarks in the formal
limit where the heavy quark masses are taken to be arbitrarily large. There are
well-established model-independent arguments for the existence of deeply bound
exotic $\bar{q}\bar{q}'QQ$ tetraquark in the formal limit of arbitrarily large
heavy quark masses. However, these previous arguments did not address the
question whether tetraquark states exist close to the threshold for breaking up
into two heavy mesons. Such states are not stable under strong
interactions---they can emit pions and decay to lower-lying tetraquark states.
This raises the issue of whether the nature of the heavy quark limit requires
such states to exist as resonances that are narrow. Here we present a
model-independent argument that establishes the existence of parametrically
narrow tetraquark states. The argument here is based on Born-Oppenheimer and
semi-classical considerations. The results derived here are only valid in a
formal limit that is well outside the regime occurring for charm quarks in
nature. However the analysis may give some insight in the experimentally
observed narrow near-threshold tetraquark states containing a heavy quark and
antiquark.
