More conservation laws and sum rules in the heavy quark limit

Abstract

This is the continuation of a previous article in which the Bjorken and Voloshin sum rules were interpreted as statements of conservation of probability and energy. Here the formalism is extended to higher moments of the Hamiltonian operator. From the conservation of the second moment of the Hamiltonian operator one can derive a sum rule which, in the small velocity limit, reduces to the Bigi-Grozin-Shifman-Uraltsev-Vainshtein sum rule. On the other hand, the conservation of the third moment of the Hamiltonian operator gives a new sum rule, which is related to the matrix element of the heavy quark counterpart of the Darwin term in atomic physics. This sum rule allows a model-independent estimate of this matrix element, with results in good agreement with those obtained from the factorization approximation. The general case of the higher order moments is also discussed.