Next-to-next-to-leading order vacuum polarization function of heavy quark near threshold and sum rules for bb̄ system

A correlator of the vector current of a heavy quark is computed analytically near threshold in the next-to-next-to-leading order in perturbative and relativistic expansion that includes αs2, αsv and v2 corrections in the coupling constant and velocity of the heavy quark to the nonrelativistic Coulomb approximation. Based on this result, the numerical values of the b-quark pole mass and the strong coupling constant are determined from the analysis of sum rules for the ϒ system. The next-to-next-to-leading corrections are found to be of the order of the next-to-leading ones.