Corrections to the Bjorken and Voloshin sum rules

Abstract

We calculate near zero recoil the order ${\alpha }_s$ corrections to the Bjorken and Voloshin sum rules that bound the $B\to D^{(\ast )}\ell \bar{\nu }$ form factors. These bounds are derived by relating the result of inserting a complete set of physical states in a time ordered product of weak currents to the operator product expansion. The sum rules sum over physical states with excitation energies less than a scale $\Delta$. We find that the corrections to the Bjorken bound are moderate, while the Voloshin bound receives sizable corrections enhanced by $\Delta /{\Lambda }_{QCD}$. With some assumptions, we find that the slope parameter for the form factor $h_{A_1}$ in $B\to D^{\ast }\ell \bar{\nu }$ decay satisfies $0.4\lesssim {\rho }_{A_1}^2\lesssim 1.3$.

Comments: 26 pages, revtex, 5 figures included using epsf. Some references added; to appear in Phys. Rev. D