  Recent lattice simulations of $(\lambda \Phi^4)_4$ theories in the broken
phase show that : a) the shifted field propagator is well reproduced by the
simple 2-parameter form ${{Z_{\rm prop}}\over{p^2 + M^2_h}}$ at finite momenta
but strongly differs for $p \to 0$ b) the bare zero-momentum two-point function
$\Gamma_2(0)= \frac{d^2 V_{\rm eff}}{d \phi^2_B}|_{\phi_B= \pm v_B}$ gives a
value of $Z_\phi \equiv {{M^2_h}\over{\Gamma_2(0)}}$ that increases when
approaching the continuum limit. This supports theoretical expectations where
$v_B$ is related by an infinite re-scaling to the `physical Higgs condensate'
$v_R$ defined through $\frac{d^2 V_{\rm eff}}{d \phi^2_R}|_{\phi_R= \pm
v_R}=M^2_h$. New lattice data collected around the phase transition confirm
this scenario. By denoting $M_{\rm SB} \equiv M_h ={\cal O} (v_R)$ the scale of
the broken phase, our results suggest the existence of a `hierarchy' of scales
$\Gamma_2(0) \ll M^2_{\rm SB} \ll v^2_B$ that become infinitely far in the
continuum limit. This may open unexpected possibilities to reconcile an
infinitesimal slope of the effective potential with finite values of $M_h$ and
accomodate very different mass scales in the framework of a spontaneously
broken theory.

High Energy Physics - Phenomenology;High Energy Physics - Lattice;High Energy Physics - Theory