Perturbative Corrections to a Sum Rule for the Heavy Quark Kinetic Energy

We calculate the perturbative corrections to order \alpha_s^2\beta_0 to the sum rule derived from the second moment of the time-ordered product of b \to c currents near zero recoil. This sum rule yields a bound on \lambda_1, the expectation value of the b quark kinetic energy operator inside the B meson. The perturbative corrections significantly weaken the bound relative to the tree level result, yielding \lambda_1 < -0.15 GeV^2.