Lattice $({\mathrm{\Phi }}^4{)}_4$ Effective Potential Giving Spontaneous Symmetry Breaking and the Role of the Higgs Mass

We present a critical reappraisal of the available results on the broken phase of $\lambda ({\mathrm{\Phi }}^4{)}_4$ theory, as obtained from rigorous formal analyses and from lattice calculations. All the existing evidence is compatible with Spontaneous Symmetry Breaking but dictates a trivially free shifted field that becomes controlled by a quadratic hamiltonian in the continuum limit. As recently pointed out, this implies that the simple one-loop effective potential should become effectively exact. Moreover, the usual naive assumption that the Higgs mass-squared $m_h^2$ is proportional to its renormalized'' self-coupling $\lambda_R$ is not valid outside perturbation theory: the appropriate continuum limit has $m_h$ finite and vanishing $\lambda_R$. A Monte Carlo lattice computation of the $\lambda(\Phi^4)_4$ effective potential, both in the single-component and in the O(2)-symmetric cases, is shown to agree very well with the one-loop prediction. Moreover, its perturbative leading-log improvement (based on the concept of $\lambda_R$) fails to reproduce the Monte Carlo data. These results, while supporting in a new fashion the peculiar triviality’’ of the $\lambda ({\mathrm{\Phi }}^4{)}_4$ theory, also imply that, outside perturbation theory, the magnitude of the Higgs mass does not give a measure of the observable interactions in the scalar sector of the standard model.