Composite Fields. Singularities of the Product of Currents at Short Distances and on the Light Cone

Abstract

So far we have considered the R-operation only as applied to individual diagrams. From this chapter on, we shall operate rather with overall renormalized quantities of a theory. For example, we shall try to obtain information about the S-matrix out of the mass shell

$$ {\rm{S}}\left( {\rm{f}} \right)\;\;{\rm{ = }}\;\;{\rm{1}}\;\;{\rm{ + }}\;\;\sum\limits_{\rm{1}} {\frac{{\rm{1}}}{{{\rm{11}}}}} \int {} {{\rm{S}}_{\rm{1}}}\left( {{{\rm{y}}_{\rm{1}}}\;{\rm{,}}\;\;...{\rm{,}}\;\;{{\rm{y}}_{\rm{1}}}} \right)\;\;{\rm{:f}}\left( {{{\rm{y}}_{\rm{1}}}} \right)\;\;...\;\;{\rm{f}}\left( {{{\rm{y}}_{\rm{1}}}} \right)\;{\rm{:}}\;\;{\rm{d}}{{\rm{y}}_{\rm{1}}}\;\;...\;\;{\rm{d}}{{\rm{y}}_{\rm{1}}}\;\;\;\left( {\rm{1}} \right) $$

(1)

using an assumption that the coefficient functions S₁ (Y₁, ..., y₁) admit an asymptotic expansion into power series of the coupling constant and that in every order of the perturbation theory they are represented by a certain sum of renormalized Feynman diagrams. Our main problem will be to search for different relations between the corresponding formal power series. In other words, we shall state that coefficient functions possess some property if for any N this property holds for the sum of the first N terms of the perturbation theory with natural accuracy; that is, if the term which violates the property in question is at least of the (N + 1)th order.