Sum Rules in the Heavy Quark Limit of QCD and Isgur-Wise Functions

Using the OPE, we formulate new sum rules in the heavy quark limit of QCD. These sum rules imply that the elastic Isgur-Wise function $\xi (w)$ is an alternate series in powers of $(w-1)$. Moreover, one gets that the $n$-th derivative of $\xi (w)$ at $ w=1$ can be bounded by the $(n-1)$-th one, and an absolute lower bound for the $n$-th derivative $(-1)^n \xi^{(n)}(1) \geq {(2n+1)!! \over 2^{2n}}$. Moreover, for the curvature we find $\xi ''(1) \geq {1 \over 5} [4 \rho^2 + 3(\rho^2)^2]$ where $\rho^2 = - \xi '(1)$. We show that the quadratic term ${3 \over 5} (\rho^2)^2$ has a transparent physical interpretation, as it is leading in a non-relativistic expansion in the mass of the light quark. These bounds should be taken into account in the parametrizations of $\xi (w)$ used to extract $|V_{cb}|$. These results are consistent with the dispersive bounds, and they strongly reduce the allowed region of the latter for $\xi (w)$. The method is extended to the subleading quantities in $1/m_Q$, namely $\xi_3(w)$ and $\bar{\Lambda}\xi (w)$.}]