Comment on the Dispersion Relations used to Calculate $\Delta\rho$
  We use the operator product expansion (OPE) to show that non-perturbative QCD corrections to $\Delta\rho$ can be calculated using unsubtracted dispersion relations for either the transverse or the longitudinal vacuum polarization functions. Recent calculations of the non-perturbative contribution to $\Delta\rho$ based on a non-relativistic calculation of corrections to the $\ttbar$ threshold are inconsistent with this result. 
Tatsu Takeuchi, Aaron K. Grant and Mihir P. Worah