Lattice effective potential of $(\lambda\Phi^4)_4$: nature of the phase transition and bounds on the Higgs mass

We present a detailed discussion of Spontaneous Symmetry Breaking (SSB) in $(\lambda\Phi^4)_4$. In the usual approach, inspired by perturbation theory, one predicts a second-order phase transition, the Higgs mass $m_h$, related to the value of the renormalized 4-point coupling, gets smaller when increasing the ultraviolet cutoff and this leads to the generally quoted upper bounds $m_h<$700-900 GeV. On the other hand, by exploring the structure of the effective potential in those approximation consistent with `triviality', where the Higgs mass does not represent a measure of any observable interaction, SSB does not require an ultraviolet cutoff, the phase transition is first-order, such that the massless `Coleman-Weinberg' regime lies in the broken phase, and one gets only $m_h<$3 TeV from vacuum stability. To separate out the two alternatives, we present a precise lattice computation of the slope of the effective potential in the region of bare parameters indicated by the Luscher~\&~Weisz and Brahm's analysis of the critical line. Our lattice data strongly support the latter description of SSB. Indeed, our data cannot be reproduced in perturbation theory, and then they confirm the existence on the lattice of a remarkable phase of $(\lambda\Phi^4)_4$ where SSB is generated through ``dimensional transmutation'', and show no evidence for residual self-interaction effects of the shifted ``Higgs'' field $h(x)=\Phi(x)-\langle\Phi\rangle$, in agreement with ``triviality''.