"Optical" sum rule for form factors of heavy mesons

A sum rule for the transitions due to the vector current of heavy quarks between the ground negativeparity states and the positive-parity excited states is derived, which is a direct analogue of the sum rule in the scattering of light on atoms, known as the Thomas-Reiche-Kuhn sum rule or the "oscillator strength" sum rule. This sum rule in conjunction with one suggested by Bjorken can be used to place an upper bound on the slope parameter of the Isgur-Wise function $\xi$. The numerical estimates suggest the possibility of a significant differnece of the slope parameter in the $B\to D$ transitions from that in the $B\to D^{*}$ due to an insufficiently heavy mass of the charmed quark. Also the sum rule shows that the slope parameter should be a growing function of the mass of the spectator quark.