Duality in semileptonic inclusive B-decays in potential models: regular versus singular potentials

Making use of the nonrelativistic potential model for the description of mesons, and working in the Shifman–Voloshin limit, we compare the integrated rate Γ(B→Xclν) calculated as a sum of the individual decay rates to the quantum-mechanical analog of the OPE. In the case of a potential regular at the origin, we find a well-defined duality violation, which is, however, exponentially small. It corresponds to the charm resonances kinematically forbidden in the decay process, but apparently picked up by the OPE. For singular potentials, we do not obtain a full OPE series, but only a limited Taylor expansion, since the coefficients become infinite beyond some order. In this case, we do not find an indication of duality violation: the difference is smaller than the last term of the limited expansion. This emphasizes that the case of singular potentials, which may be relevant for QCD, deserves further study.