Finite Mass Corrections to Leptonic Decay Constants in the Heavy Quark Effective Theory

We calculate finite mass corrections to leptonic decay constants in HQET using QCD sum rules. The results are given to two--loop accuracy including renormalization group analysis. The contribution of unknown two--loop anomalous dimension matrix--elements is estimated. We find that the $1/m_Q$ corrections to $f_P$, the decay constant of a pseudoscalar meson, amount $-(23\pm4\pm2)\%$ for the B meson. The first error comes from the sum rule uncertainty, the second one from unknown or not accurately known matrix elements in HQET. We obtain a ratio of vector to pseudoscalar decay constant of $1.00\pm0.07+O(1/m_Q^2)$ for B mesons.