Inconsistency of the 4 Field Theory in High-Dimensions Constructed from the Broken-Symmetry Phase

Publisher Website
Google Scholar
Abstract
We discuss the construction of the broken-symmetry λφ4 theory from the lattice-regularized model defined on εZd (d-dimensional hypercubic lattice with lattice spacing ε) in the sense that the bare parameter (e.g., λ= λ(ε)) as a function of ε is adjusted approach the critical manifold Γc as ε↓0 such that limε→0 ≪φren(x) >(ε) ≠0. In the case of 0 ≪ limε→0 ≪φren(x)>(ε) ≪ ∞, it is shown that the 3- and 4-point renormalized coupling constant vanish in the continuum limit ε↓0, as long as 0 ≪ λ(ε) ≪ λ0 for sufficiently small λ0, in dimensions d > dc (upper critical dimension) where the critical exponent δ of the magnetization on the critical isotherm line takes its classical (mean-field) value δ= 3. Even for the renormalization which allows ≪φren(x) >(ε) →∞ as ε→0, the same type of triviality is derived for d > 4. These results suggest that the consistency requirement of the λφ4 theory obtained in this scheme leads to the result: ≪φren(x)>(ε) →0 as ε→0, i.e., no breakdown of the symmetry φ→-φ in all d > 4 dimensions.

This publication has 8 references indexed in Scilit: