  In this work, the non-relativistic asymptotic behavior of the transition
$q\overline{q}\rightarrow2g\rightarrow q\overline{q}$ in the $^1S_0$ channel is
discussed. Different with the usual calculation which expands the physical
amplitude around the quark anti-quark threshold, we take the quark anti-quark
pairs as off shell and only expand the expression on the three-dimensional
momenta of the quarks and anti-quarks. We calculate the results to order 6. The
imagine part of the results after applying the on shell conditions can
reproduce the non-relativistic QCD (NRQCD) results in leading order of
$\alpha_s$. The real part of the results can be used to estimate the mass shift
of the $^1S_0$ heavy quark anti-quark system due to the $2g$ annihilation
effect. The results can also be used to estimate the energy shifts of the
positronium system due to the two-photon annihilation.
