Computation of the b-quark Mass with Perturbative Matching at the Next-to-Next-to-Leading Order

We compute the two-loop term in the perturbation series for the quark-mass in the lattice Heavy Quark Effective Theory. This is an ingredient in the matching factor required to obtain the $b$-quark mass from lattice simulations of the HQET. Combining our calculations with numerical results from the APE collaboration, we find, at two-loop order, $\bar m_b \equiv m_b^{\bar{MS}}(m_b^{\bar{MS}}) =4.41\pm 0.05\pm 0.10 $GeV. It was expected that the two-loop term would have a significant effect and this is indeed what we find. Depending on the choice of "reasonable" coupling constant in the one-loop estimates the result for $\bar m_b$ can change by several hundred MeV when the two-loop terms are included.