Bounds on the derivatives of the Isgur-Wise function from sum rules in the heavy quark limit of QCD

Abstract

Using the OPE and the trace formalism, we have obtained a number of sum rules in the heavy quark limit of QCD that include the sum over all excited states for any value $j^P$ of the light cloud. We show that these sum rules imply that the elastic Isgur-Wise function $\xi (w)$ is an alternate series in powers of $(w-1)$. Moreover, we obtain sum rules involving the derivatives of the elastic Isgur-Wise function $\xi (w)$ at zero recoil, that imply that the $n$-th derivative can be bounded by the $(n-1)$-th one. For the curvature ${\sigma }^2={\xi }^{\prime \prime }(1)$, this proves the already proposed bound ${\sigma }^2\ge \frac{5}{4}{\rho }^2$. Moreover, we obtain the absolute bound for the $n$-th derivative $(-1{)}^n{\xi }^{(n)}(1)\ge \frac{(2n+1)!!}{2^{2n}}$, that generalizes the results ${\rho }^2\ge \frac{3}{4}$ and ${\sigma }^2\ge \frac{15}{16}$.

Comments: 9 pages, Latex