Lower bounds on the curvature of the Isgur-Wise function

Using the OPE, we obtain new sum rules in the heavy quark limit of QCD, in addition to those previously formulated. Key elements in their derivation are the consideration of the non-forward amplitude, plus the systematic use of boundary conditions that ensure that only a finite number of $j^P$ intermediate states (with their tower of radial excitations) contribute. A study of these sum rules shows that it is possible to bound the curvature $\sigma^2 = \xi''(1)$ of the elastic Isgur-Wise function $\xi (w)$ in terms of its slope $\rho^2 = - \xi '(1)$. Besides the bound $\sigma^2 \geq {5 \over 4} \rho^2$, previously demonstrated, we find the better bound $\sigma^2 \geq {1 \over 5} [4 \rho^2 + 3(\rho^2)^2]$. 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. At the lowest possible value for the slope $\rho^2 = {3 \over 4}$, both bounds imply the same bound for the curvature, $\sigma^2 \geq {15 \over 16}$. We point out that these results are consistent with the dispersive bounds, and, furthermore, that they strongly reduce the allowed region by the latter for $\xi (w)$.