Treating thebquark distribution function with reliable uncertainties

The parton distribution function for a $b$ quark in the $B$ meson (called the shape function) plays an important role in the analysis of the $B\to X_s\gamma$ and $B\to X_u\ell \overline{\nu }$ data, and gives one of the dominant uncertainties in the determination of $|V_{ub}|$. We introduce a new framework to treat the shape function, which consistently incorporates its renormalization group evolution and all constraints on its shape and moments in any short distance mass scheme. At the same time it allows a reliable treatment of the uncertainties. We develop an expansion in a suitable complete set of orthonormal basis functions, which provides a procedure for systematically controlling the uncertainties due to the unknown functional form of the shape function. This is a significant improvement over fits to model functions. Given any model for the shape function, our construction gives an orthonormal basis in which the model occurs as the first term, and corrections to it can be studied. We introduce a new short distance scheme, the “invisible scheme,” for the kinetic energy matrix element ${\lambda }_1$. We obtain closed form results for the differential rates that incorporate perturbative corrections and a summation of logarithms at any order in perturbation theory, and present results using known next-to-next-to-leading order expressions. The experimental implementation of our framework is straightforward.