  We consider the interactions in a mesonic system, referred here to as
`tetron', consisting of two heavy quarks and two lighter antiquarks (which may
still be heavy in the scale of QCD), i.e. generally $Q_a Q_b \bar q_c \bar
q_d$, and study the existence of bound states below the threshold for decay
into heavy meson pairs. At a small ratio of the lighter to heavier quark masses
an expansion parameter arises for treatment of the binding in such systems. We
find that in the limit where all the quarks and antiquarks are so heavy that a
Coulomb-like approximation can be applied to the gluon exchange between all of
them, such bound states arise when this parameter is below a certain critical
value. We find the parametric dependence of the critical mass ratio on the
number of colors $N_c$, and confirm this dependence by numerical calculations.
In particular there are no stable tetrons when all constituents have the same
mass. We discuss an application of a similar expansion in the large $N_c$ limit
to realistic systems where the antiquarks are light and their interactions are
nonperturbative. In this case our findings are in agreement with the recent
claims from a phenomenological analysis that a stable $b b \bar u \bar d$
tetron is likely to exist, unlike those where one or both bottom quarks are
replaced by the charmed quark.
