Higgs vacua with potential barriers

Scenarios in which the Higgs vacuum arises radiatively and is separated from the origin by a potential barrier at zero temperature are known to be attainable in models with extra singlet scalars, which in the limit of zero barrier height give rise to Coleman-Weinberg realizations of electroweak-symmetry breaking. However, this requires large values of Higgs-portal couplings or a large number $N$ of singlets. This is quantified in detail by considering, for varying $N$, the full two-loop effective potential at zero temperature, as well as finite-temperature effects including the dominant two-loop corrections due to the singlets. Despite the large couplings, two-loop effects near the electroweak scale are under control, and actually better behaved in models with larger couplings yet fewer singlets. Strong first-order phase transitions are guaranteed even in the Coleman-Weinberg scenarios. Cubic Higgs couplings and Higgs associated-production cross sections exhibit deviations from the standard model predictions which could be probed at a linear collider.

Figure 1

Effective potentials in GeV units at one (left) and two loops (right), for $N=1$ (top), $N=6$ (middle row), $N=12$ (bottom), for different values of ${\lambda }_{SH}$ yielding a large barrier. ${\lambda }_S$ and $m_S^2$ were fixed at 0.1 and ${10}^4{\mathrm{GeV}}^2$, respectively. The behavior under rescalings is illustrated by choosing $\mu =175\mathrm{GeV}$ (solid red line), $\mu =246\mathrm{GeV}$ (dashed blue line), and a field-dependent choice $\mu =\exp (-h^2/3\times {10}^4)+h$ (dotted black line). Note the improved scaling behavior in the two-loop case.

Figure 2

Effective potentials in GeV units at one (solid blue line) and two loops (dashed red line), for $N=1,{\lambda }_S=0.1,{\lambda }_{SH}=7.8,m_S^2={10}^4{\mathrm{GeV}}^2,\mu =\exp (-h^2/3\times {10}^4)+h$.

Figure 3

Two-loop effective potentials in the Coleman-Weinberg scenarios for $N=1$ (solid line), $N=3$ (dashed line), and $N=6$ (dotted line).

Figure 4

Quartic root of the barrier height in GeV units at one (left) and two loops (right), for $N=1$ (top), $N=3$ (middle row), and $N=6$ (bottom), in terms of ${\lambda }_{SH}$ and $m_S$. ${\lambda }_S$ was fixed at 0.1. Points to the right of the red line have an electroweak vacuum with energy greater than $V(h=0)$.

Figure 5

Two-loop diagrams involving the Higgs-portal coupling and a nonzero background for the Higgs field. Higgs propagators are denoted with single lines, and singlet propagators with double lines. The dots denote one-loop counterterms.

Figure 6

Effective potential for $N=1$ (top) and $N=6$ (bottom), at one (left) and two loops (right), for the same choices of parameters as the corresponding graphs in Fig. 1. The choices of temperatures are zero (solid blue line), critical temperature (dashed red line; from top to bottom and left to right, 74.5,67.2,63.2, and 64.5 GeV), 100 GeV (dotted green line), and 140 GeV (dash-dotted black line).

Figure 7

Strength of the first-order phase transition at one (left) and two loops (right), for $N=1$ (top), $N=3$ (middle row), and $N=6$ (bottom), in terms of ${\lambda }_{SH}$ and $m_S$. ${\lambda }_S$ was fixed at 0.1. The red lines correspond to $\frac{v_c}{T_c}=\infty$, and to their right the electroweak vacuum lies above the origin at zero temperature.

Figure 8

Enhancement of the cubic Higgs coupling with respect to its standard model value ($a_h/a_h^{\mathrm{SM}}$), in models $N=1$ (top), $N=3$ (middle row), and $N=6$ (bottom), at one loop (left) and two loops (right). ${\lambda }_S$ was fixed at 0.1.