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

Using the OPE and the trace formalism, we formulate new sum rules in the heavy quark limit of QCD that include the sum over all excited states for any value jP of the light cloud. We show that these sum rules imply that the elastic Isgur–Wise function ξ(w) is an alternate series in powers of (w−1). Moreover, one gets that any nth derivative of ξ(w) at w=1 can be bounded by the (n−1)th one. For the curvature σ2=ξ″(1), this proves rigorously our already conjectured bound σ2⩾54ρ2. Moreover, we obtain the absolute lower bound for the nth derivative (−1)nξ(n)(1)⩾(2n+1)!!22n, that generalizes the results ρ2⩾3/4 and σ2⩾15/16. These bounds should be taken into account in the empirical parametrizations of ξ(w) used to extract |Vcb|.