Sum rules for heavy flavor transitions in the small velocity limit

We show how sum rules for the weak decays of heavy flavor hadrons can be derived as the moments of spectral distributions in the small velocity limit. Our derivation of the sum rules is based on the operator product expansion. This systematic approach allows us to determine corrections to these sum rules, to obtain new sum rules, and it provides us with a transparent physical interpretation; it also opens a new perspective on the notion of the heavy quark mass. Applying these sum rules we derive a lower bound on the deviation of the exclusive form factor ${\mathit{F}}_{\mathit{B}\to \mathit{D}}^{\mathrm{*}}$ from unity at zero recoil; likewise we give a field-theoretical derivation of a previously formulated inequality between the expectation value for the kinetic energy operator of the heavy quark and for the chromomagnetic operator. We analyze how the known results on nonperturbative corrections must be understood when one takes into account the normalization point dependence of the low scale parameters. The relation between the field-theoretic derivation of the sum rules and the quantum-mechanical approach is elucidated.