Bounds on the derivatives of the Isgur-Wise function from sum rules in the heavy quark limit of QCD
Résumé
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 $\\sigma^2 = \\xi\'\'(1)$, this proves the already proposed bound $\\sigma^2 \\geq {5 \\over 4} \\rho^2$. Moreover, we obtain the absolute bound for the $n$-th derivative $(-1)^n \\xi^{(n)}(1) \\geq {(2n+1)!! \\over 2^{2n}}$, that generalizes the results $\\rho^2 \\geq {3 \\over 4}$ and $\\sigma^2 \\geq {15 \\over 16}$.