Anomalous Dimensions in the (Formula) Model Field Theory in 1/N Expansion

Abstract

We analyzed the field-theory model with interaction Lagrangian (<f;{p)<f;¢ where <f; and ¢ are the fermion and boson N-component fields. The theory is solved by applying the 1/N expansion technique. In 3 dimensions {3(g) vanishes identically and so we obtained the scale invariant theory. Anomalous dimensions of the <f; and ¢ fields are calculated. The results are given in Eqs. (3 ·13) and (3 · 26). Anomalous dimensions of the relevant composite tensor operators are also evaluated as shown. in Eq s. (3 · 48) and (3 · 63). All anomalous dimensions are coupling dependent. It is shown that for DoT3 dimensions the theory is UV and IR free and the effective coupling of the theory is confined to be very small. § 1. Introduction and summary In recent years a lot of works1' have been done in order to understand, within the framework of the renormalization group, Bjorken's prediction and the experimental indications of The study of asymptotic behavior of Fourier transform of the coefficient functions in Wilson's operator-product expansion,3l and thus the behavior of leptoproduction structure functions, was possible by means of the Callan-Symanzik equations.4l It has been found1' that non-Abelian gauge field theories are asymptotically free and hence, although they violate logarithmi­ cally Bjorken scaling, they are nearest to the canonical scaling. Meanwhile, recent experimental data with larger 2 seem to show violation of Bjorken scaling,5l which gave a ground for the study of theories which violate canonical scaling. If we have a theory with nontrivial ultraviolet (UV) fixed point then the relevant composite tensor operators in Wilson's expansion acquire anomalous dimensions and hence Bjorken scaling is violated. Once the theory with UV-fixed point is found, we can determine the 2-dependence of the structure functions for large q given the anomalous dimensions of the composite tensor operators. 6l ( 2 being the momentum transfer squared in deep inelastic lepton-hadron scattering.) In some field theories the anomalous dimensions of composite tensor operators have been calculated, e.g., in the cfi theory in 6 + c space-time dimensions'' in the ¢4 and Yukawa theories in 4- c space-time dimensions8l and in the Gross-Neveu model in 2 + c space-time dimensions. 9l Although unrealistic, they can give us better insight and may generate some new ideas about more realistic theories.