Massless limits, anomalous dimensions from the renormalization group, and the CallanSymanzik equations for gφ^{4} + fφ^{6} theory
Abstract
The renormalizationgroup and CallanSymanzik equations for gφ^{4} + fφ^{6} theory are obtained by using the technique of differential vertex operations in a normalproduct algorithm. The Lagrangian constructed by the addition of all possible counterterms obtained from powercounting arguments contains two oversubtracted terms, in contrast to the usual Lagrangians containing only masstype oversubtractions. The solutions of partialdifferential equations are discussed in detail, which contain as a particular case the saddlepoint solution implying the nonexistence of an anomalous dimension, while the usual fixedpoint solution mimics the situation of the couplingconstantdependent engineering dimension of fields. The massless limits are also considered, which involve the strongcoupling 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} (x_{1},x_{2},x_{3},...,x_{n}), yet the massless limits of Γ (N_{1}[φ^{2}],x_{1},...,x_{n}) and Γ(N_{2}[φ^{4}],x_{1},...,x_{n}) require higherorder partialdifferential 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.
 Publication:

Physical Review D
 Pub Date:
 April 1977
 DOI:
 10.1103/PhysRevD.15.2186
 Bibcode:
 1977PhRvD..15.2186C