Bounds on the derivatives of the Isgur-Wise function

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 ξ(w) is an alternate series in powers of (w−1). Moreover, one gets that the n-th derivative of ξ(w) at w = 1 can be bounded by the (n−1)-th one, and an absolute lower bound for the n-th derivative (−1)nξ(n)(1) ≥ (2n+1)!! 22n . Moreover, for the curvature we find ξ′′(1) ≥ 1 5 [4ρ + 3(ρ)] where ρ = −ξ′(1). We show that the quadratic term 3 5 (ρ) 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 ξ(w) used to extract |Vcb|. These results are consistent with the dispersive bounds, and they strongly reduce the allowed region of the latter for ξ(w).