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

TL;DR

This paper re-evaluates the broken phase of the ermi4 theory, showing that the effective potential aligns with one-loop predictions and that the Higgs mass does not directly indicate interaction strength outside perturbation theory.

Contribution

It demonstrates that the ermi4 theory's effective potential is accurately described by one-loop calculations and challenges the assumption that Higgs mass correlates with self-coupling beyond perturbation theory.

Findings

Monte Carlo results agree with one-loop effective potential.

Leading-log perturbative improvements fail to match lattice data.

Higgs mass does not measure observable interactions outside perturbation theory.

Abstract

We present a critical reappraisal of the available results on the broken phase of $\lambda(\Phi^4)_4$ 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 $m^2_h$ 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 $\lambda(\Phi^4)_4$ 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.