Asymptotically freeφ4theory

Exact properties of asymptotically free $g{\phi }^4$ theory with negative (renormalized) $g$ are deduced by renormalization-group and other methods. It has been argued that the effective potential $\mathcal{U}\left(\chi \right)$ for the model approaches -$\infty$. for $\chi \to \infty$, 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 $\mathcal{U}\left(\chi \right)=\mathrm{const}\times {\chi }^2$. 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 $\phi \mathrm{}\left(x\right)\mathrm{}\to \phi \mathrm{}\left(x\right)+\mathrm{const}$, 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 or an indication that the model really is unstable after all.