On the finiteness of scaling sum rules

The convergence of sum rules relating the matrix elements of local operators to integrals over deep-inelastic structure functions is studied critically. It is found that the matrix elements may always be written as the q² → ? ∞ limit of finite expressions, regardless of the (Regge) asymptotic behavior of the structure functions or the possible occurrence of J = 0 fixed singularities. The correct form of the sum rule for the operator Schwinger term is taken as a paradigm case. It is derived from the Bjorken-Johnson-Low theorem and agrees with the results of parton model and light-cone analyses. It readily encompases the results of second order φ³ theory (where the Schwinger term diverges logarithmically) and second order vector gluon theory (where it vanishes). Sufficient conditions for the finiteness of the operator Schwinger term are the scaling of the longitudinal structure function and the absence of J = 0 fixed singularities with nonpolynomial residues. The treatment is readily applicable to other scaling and fixed q² sum rules needing regulation. A compendium of these is given.