Massless limits, anomalous dimensions from the renormalization group, and the Callan-Symanzik equations for gφ4 + fφ6 theory

The renormalization-group and Callan-Symanzik equations for gφ⁴ + fφ⁶ theory are obtained by using the technique of differential vertex operations in a normal-product algorithm. The Lagrangian constructed by the addition of all possible counterterms obtained from power-counting arguments contains two over-subtracted terms, in contrast to the usual Lagrangians containing only mass-type over-subtractions. The solutions of partial-differential equations are discussed in detail, which contain as a particular case the saddle-point solution implying the nonexistence of an anomalous dimension, while the usual fixed-point solution mimics the situation of the coupling-constant-dependent engineering dimension of fields. The massless limits are also considered, which involve the strong-coupling limit g-->∞ of the theory not encountered in other scalar field theories. By some means or other one can ascertain the m-->0 limit of Γ^N (x₁,x₂,x₃,...,x_n), yet the massless limits of Γ (N₁[φ²],x₁,...,x_n) and Γ(N₂[φ⁴],x₁,...,x_n) require higher-order partial-differential equations. This peculiarity regarding the anomalous dimension and composite operators all seems to stem from the masslike dimension of g. In this connection the functional forms of the group equations are also derived, which are nothing but the generalizations of the equations obtained by Migdal.