  The scale of low energy c-cbar and b-bbar cross sections on light hadrons is
of great importance to searches for the quark gluon plasma using the
heavy-quarkonium suppression signature. Unfortunately, little is known about
these near-threshold cross sections at present, and recent theoretical
estimates span many orders of magnitude. Here we use experimental data on the
four observed closed-flavor heavy quarkonium hadronic decays psi' -> pi pi
J/psi, Upsilon' -> pi pi Upsilon, Upsilon'' -> pi pi Upsilon and Upsilon'' ->
pi pi Upsilon', combined with simple models of the transition amplitudes, to
estimate the pion scattering cross sections of c-cbar and b-bbar mesons near
threshold. Specifically we consider the closed-flavor reactions pi J/psi -> pi
psi', pi Upsilon -> pi Upsilon', pi Upsilon -> pi Upsilon'' and pi Upsilon' ->
pi Upsilon'' and their time-reversed analogues. Our results may be useful in
constraining theoretical models of the strong interactions of heavy quarkonia,
and can be systematically improved through future detailed studies of dipion
decays, notably psi' -> pi pi J/psi and Upsilon'' -> pi pi Upsilon.
