Criticality in the scale invariant standard model (squared)

Understand

We consider first the standard model Lagrangian with $\mu_h^2$ Higgs potential term set to zero.

• We point out that this clasically scale invariant theory potentially exhibits radiative electroweak/scale symmetry breaking with very high vacuum expectation value (VEV) for the Higgs field, $< \phi > \approx 10^{17-18}$ GeV.
• Furthermore, if such a vacuum were realized then cancellation of vacuum energy automatically implies that this nontrivial vacuum is degenerate with the trivial unbroken vacuum.
• Such a theory would therefore be critical with the Higgs self-coupling and its beta function nearly vanishing at the symmetry breaking minimum, $\lambda (\mu=< \phi >)\approx \beta_{\lambda} (\mu=< \phi >)\approx 0$.