Heavy Quark Effective Theory at Large Orders in 1/m

The existing derivations of a heavy quark effective theory (HQET) are analyzed beyond the next-to-leading order in $1/m$. With one exception they are found to be incorrect. The problem is a wrong normalization of the heavy quark field in the effective theory. We argue that the correct effective theory should be given by a Foldy--Wouthuysen type field transformation to all orders in $1/m$. The renormalization of the resulting Lagrangian to order $1/m^2$ is performed allowing for inclusion of effects arising through vacuum polarization. Our results for the anomalous dimensions disagree with the existing ones. Some applications are considered.