Lower bounds on the curvature of the Isgur-Wise function

Using the operator product expansion, 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 nonforward amplitude, plus the systematic use of boundary conditions that ensure that only a finite number of jP intermediate states (with their tower of radial excitations) contribute. A study of these sum rules shows that it is possible to bound the curvature σ²=ξ^''(1) of the elastic Isgur-Wise function ξ(w) in terms of its slope ρ²=-ξ^'(1). In addition to the bound σ²>~5/4ρ², previously demonstrated, we find the better bound σ²>~1/5[4ρ²+3(ρ²)²]. We show that the quadratic term 3/5(ρ²)² has a transparent physical interpretation, as it is leading in a nonrelativistic expansion in the mass of the light quark. At the lowest possible value for the slope ρ²=3/4, both bounds imply the same bound for the curvature σ²>~15/16. 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 ξ(w).