Lattice measurement of the scalar propagator near the symmetry breaking phase transition

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Recent lattice simulations of (λΦ⁴)₄ theories in the broken phase show that: a) the shifted field propagator is well reproduced by the simple 2-parameter form Z propp² + M²ₕ at finite momenta but strongly differs for p → 0 b) the bare zero-momentum two-point function Γ₂(0)= d² V effd ϕ²B|ϕB₌ ± ᵥB gives a value of Zϕ≡ M²ₕΓ₂(0) that increases when approaching the continuum limit. This supports theoretical expectations where vB is related by an infinite re-scaling to the `physical Higgs condensate' vᵣ defined through d² V effd ϕ²ᵣ|ϕᵣ₌ ± ᵥᵣ=M²ₕ. New lattice data collected around the phase transition confirm this scenario. By denoting M SB ≡ Mₕ = O (vᵣ) the scale of the broken phase, our results suggest the existence of a `hierarchy' of scales Γ₂(0) ≪ M² SB ≪ v²B 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 Mₕ and accomodate very different mass scales in the framework of a spontaneously broken theory.