Difficulties of an Infrared Extension of Differential Renormalization

We investigate the possibility of generalizing differential renormalization of D.Z.Freedman, K.Johnson and J.I.Latorre in an invariant fashion to theories with infrared divergencies via an infrared $\tilde{R}$ operation. Two-dimensional $\sigma$ models and the four-dimensional $\phi^4$ theory diagrams with exceptional momenta are used as examples, while dimensional renormalization serves as a test scheme for comparison. We write the basic differential identities of the method simultaneously in co-ordinate and momentum space, introducing two scales which remove ultraviolet and infrared singularities. The consistent set of Fourier-transformation formulae is derived. However, the values for tadpole-type Feynman integrals in higher orders of perturbation theory prove to be ambiguous, depending on the order of evaluation of the subgraphs. In two dimensions, even earlier than this ambiguity manifests itself, renormalization-group calculations based on infrared extension of differential renormalization lead to incorrect results. We conclude that the extended differential renormalization procedure does not perform the infrared $\tilde{R}$ operation in a self-consistent way, as the original recipe does the ultraviolet $R$ operation.