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

Abstract

Recent lattice simulations of $(\hat{I}»\hat{I}{¦}^4{)}_4$ theories in the broken phase show that : a) the shifted field propagator is well reproduced by the simple 2-parameter form $\frac{Z_{\mathrm{prop}}}{p^2+M_h^2}$ at finite momenta but strongly differs for $p\to 0$ b) the bare zero-momentum two-point function $\hat{I}{“}_2(0)=\frac{d^2{V}_{\mathrm{eff}}}{d\ddot{I}{•}_B^2}{|}_{\ddot{I}{•}_B=\pm v_B}$ gives a value of $Z_{\ddot{I}}•\equiv \frac{M_h^2}{\hat{I}{“}_2(0)}$ 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 $\frac{d^2{V}_{\mathrm{eff}}}{d\ddot{I}{•}_R^2}{|}_{\ddot{I}{•}_R=\pm v_R}=M_h^2$. New lattice data collected around the phase transition confirm this scenario. By denoting $M_{\mathrm{SB}}\equiv M_h=\mathcal{O}(v_R)$ the scale of the broken phase, our results suggest the existence of a `hierarchy' of scales $\hat{I}{“}_2(0)\ll M_{\mathrm{SB}}^2\ll v_B^2$ 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.

15 pages, 5 figures