Vacuum alignment with and without elementary scalars

We systematically elucidate differences and similarities of the vacuum alignment issue in composite and renormalizable elementary extensions of the Standard Model featuring a pseudo-Goldstone Higgs. We also provide general conditions for the stability of the vacuum in the elementary framework, thereby extending previous studies of the vacuum alignment.

Figure 1

The figure shows the parameter space of the theory allowing for a pseudo-Goldstone Higgs vacuum solution (shaded region) occurring for $0<\theta <\pi /2$. The plot is obtained in the minimal scenario with only the $\sigma$ field and the one-loop contributions coming from the gauge and the fermion sectors. The star represents the SM embedding in this scenario, with the three massive gauge bosons and the top mass; see Sec. 4 for details.

Figure 2

The red curve gives the minimum with respect to $\theta$, and the yellow (blue) area is excluded by tree-level vacuum stability (perturbativity). We fix $M_S=0.5M_{\sigma }$ ($\beta =0.5$) and $M_S=2M_{\sigma }$ ($\beta =2$) in the upper and lower panels, respectively.

Figure 3

The plot shows the stationary points of the potential with respect to $\theta$ (blue solid curve) when the explicit symmetry-breaking term, $V_B$, is introduced. The colored region corresponds to positive $C_B$, and the color gradient indicates the value of ${\log }_{10}{C}_B$.

Figure 4

The blue contour shows the stationary points with respect to $\theta$, and on the blue regions the second derivative with respect to $\theta$ is positive, so the blue contour on a blue region represents a minimum. The red contour represents the points giving the correct EW gauge boson and top quark masses. This benchmark scenario was considered in Ref. [12], and we follow the analysis and definitions therein. The figure corresponds to Fig. 1 in Ref. [12].