The Shifman-Vaĭnshteĭn-Zakharov method: Why it works, why it fails, and ways to improve it

Shifman, Vaĭnshteĭn, and Zakharov (SVZ) have proposed a procedure for calculating hadronic masses and determining nonperturbative parameters in QCD using the operator-product expansion for two-point functions and (exponential) moments of the corresponding spectral functions. In this paper we present a detailed theoretical analysis of the SVZ procedure in the context of nonrelativistic potential theory. We find that the phenomenological success of the usual first-order SVZ method in relating hadronic energies (masses) is due to a hidden variational principle and a semiclassical structure which gives correct JWKB-type relations between energies. The first-order method fails theoretically: it does not reproduce the correct potential-model or field-theoretic parameters, e.g., the gluon-condensate parameter of QCD. We show why it breaks down in this application, and that its reliability can be greatly improved in all applications by using higher-order approximations for the moment function. Our results are directly relevant for the SVZ analysis of charmonium and $b$-quarkonium. The general conclusions should also hold for light-quark systems.