Asymptotically freeφ4theory

Abstract

Exact properties of asymptotically free gphi/sup 4/ theory with negative (renormalized) g are deduced by renormalization-group and other methods. It has been argued that the effective potential V (chi) for the model approaches - infinity for chi ..-->.. infinity, so that the model is inconsistent with positivity. It is shown here how this difficulty may be avoided because of deduced results which imply that actually V (chi) = const x chi/sup 2/. These results are exact zero-momentum theorems which state that the proper vertex functions (except for the inverse two-point function) vanish whenever one of their four-momentum arguments vanish. These theorems are deduced as a consequence of the fact that the exact field equation of the theory is invariant, apart from mass terms and mass counterterms, to the transformation phi (x) ..-->.. phi (x) + const, which only adds a constant (reflection-symmetry breaking) term to the field equation. This partial symmetry and the associated theorems arise as a consequence of renormalization: they are not true order by order in perturbation theory. The perturbation series in g for the vertex functions is therefore not an asymptotic expansion when a momentum vanishes. This is either a remarkable property of the model ormore » an indication that the model really is unstable after all. (AIP)« less