Cut vertices and their renormalization: A generalization of the Wilson expansion

Cut vertices, a generalization of matrix elements of composite operators, are introduced. Their renormalization is discussed. The Bogolubov-Parasiuk-Hepp-Zimmermann method of renormalization of cut vertices allows one to obtain a generalization of the Wilson expansion where cut vertices multiplied by singular functions appear rather than local operators times singular functions. A Callan-Symanzik equation for the moments of the structure function in $e^{+}+e^{-}\to \mathrm{hadron} \mathrm{}\left(p\right)+\mathrm{anything}$ is derived. This equation is valid to all orders of perturbation theory in both gauge and nongauge theories. Examples of renormalization through the two-loop level are given.