Wilson coefficients of dimension 6 gluon operators with spin indices in the heavy quark current correlation functions

We calculate the Wilson coefficients of dimension 6 gluon operators with spin indices in the two point correlation function of the heavy scalar, pseudoscalar, and axialvector currents. Our result completes the list of all Wilson coefficients of gluon operators up to dimension 6 for the correlation functions between heavy quark currents without derivatives. We then use the result to investigate the stability of the QCD sum rule results for the $\eta_c$, $J/\Psi$, $\chi_{c0}$, and $\chi_{c1}$ mesons near $T_c$. While the inclusion of the dimension 6 operators increases the stability of the sum rules for all currents, all of them break down slightly above $T_c$ with that for the $J/\psi$ persisting to relatively highest temperature.