Sphalerons in composite and nonstandard Higgs models

After the discovery of the Higgs boson and the rather precise measurement of all electroweak boson’s masses the local structure of the electroweak symmetry breaking potential is already quite well established. However, despite being a key ingredient to a fundamental understanding of the underlying mechanism of electroweak symmetry breaking, the global structure of the electroweak potential remains entirely unknown. The existence of sphalerons, unstable solutions of the classical action of motion that are interpolating between topologically distinct vacua, is a direct consequence of the Standard Model’s $\mathrm{SU}(2{)}_L$ gauge group. Nevertheless, the sphaleron energy depends on the shape of the Higgs potential away from the minimum and can therefore be a litmus test for its global structure. Focusing on two scenarios, the minimal composite Higgs model $\mathrm{SO}(5)/\mathrm{SO}(4)$ or an elementary Higgs with a deformed electroweak potential, we calculate the change of the sphaleron energy compared to the Standard Model prediction. We find that the sphaleron energy would have to be measured to $\mathcal{O}(10)\%$ accuracy to exclude sizeable global deviations from the Standard Model Higgs potential. We further find that because of the periodicity of the scalar potential in composite Higgs models a second sphaleron branch with larger energy arises.

Figure 1

Left: Schematic representation of $V_{\mathrm{bos}}^{\text{saddle}}[N_{\mathrm{CS}}]$ in the presence of a single-valued branch of extremal solutions. Note the translation and reflection symmetries of the graph. Right: Illustration of $V_{\mathrm{bos}}[N_{\mathrm{CS}}]$ evaluated at nonextremal paths between vacua when bisphalerons are present.

Figure 2

Left: Profiles for $R$, $S$, in units of $m_W$, in the SM sphaleron configuration obtained by solving the reduced system of 2 differential equations. The vertical line marks the scale at which the low $r$ solution (red) was matched with the high $r$ solution (blue). Right: Contributions to the dimensionless integrand in ${\tilde{V}}_{\mathrm{bos}}^{\mathrm{SM}}$, evaluated on the sphaleron solution, due to the gauge fields (solid blue), derivatives of the scalar field (dashed orange) and the potential energy density of the Higgs (dotted green).

Figure 3

Deformed Higgs potential for $\gamma =0.1$ and varying values of $\beta$. The dots represent the Higgs minimum, with VEV and curvature fixed by the $W$ and Higgs masses. On the left hand, the values of $\beta$ ensure absolute stability of the Higgs vacuum. The right-hand plot shows the extremal values of $\beta$ for which the metastable Higgs vacuum is still sufficiently long lived at zero temperature. The shaded areas reflect the region of the potential probed by sphaleron configurations.

Figure 4

In red, sphaleron energy as a function of $\beta$ in models with a deformed Higgs potential, for $\beta$ in the allowed stability window. The dash-dotted gray line represents the SM result. $\gamma$ was fixed to 0.1, and hardly influences the results. The shaded band corresponds to absolutely stable Higgs vacua, as in the left plot in Fig. 3.

Figure 5

Properties of sphaleron configurations with $\gamma =0.1$ and $\beta =0.495$. The left plot shows the profiles for $R$, $S$, in units of $m_W$, with the vertical lines marking the scale at which the low $r$ solution (red) was matched with the high $r$ solution (blue). The plot on the right shows the contributions to the dimensionless integrand in ${\tilde{V}}_{\mathrm{bos}}$, evaluated on the sphaleron solution, due to the gauge fields (solid blue), derivatives of the scalar field (dashed orange) and the potential energy density of the Higgs (dotted green). In all plots, the gray dash-dotted lines correspond to the SM results.

Figure 6

Composite Higgs potential for $f_{\pi }=260\mathrm{GeV}$, for the physically equivalent realizations with $\alpha <0$ (left) and $\alpha >0$ (right). Both cases yield the correct Higgs and $W$ masses at tree level. The darker shade shows the region of the potential probed by the sphaleron branch in common with the SM, while the lighter shade corresponds to the region probed by the new, higher-energy sphaleron branch present in composite models.

Figure 7

Energy barrier between topological vacua as a function of $f_{\pi }$, in the $S(0)=0$ (left) and $S(0)\ne 0$ (right) branches. The horizontal line marks the SM limit.

Figure 8

Profile functions for $R$, $S$ in the sphaleron configurations obtained by solving the system of 2 differential equations. The vertical line marks the scale at which the low $r$ solution (red) was matched with the high $r$ solution (blue). Top: solutions in the $S(0)=0$ branch, with $f_{\pi }=250\mathrm{GeV}$ (left) and $f_{\pi }=1\mathrm{TeV}$ (right). Bottom: solutions in the $S(0)\ne 0$ branch, with $f_{\pi }=250\mathrm{GeV}$ (left) and $f_{\pi }=1\mathrm{TeV}$ (right).

Figure 9

Contributions to the integrand of the dimensionless bosonic energy functional for ${\widehat{f}}_{\pi }=260\mathrm{GeV}$ in the $S(0)=0$ branch (left) and for $S(0)\ne 0\mathrm{GeV}$ (right). The contributions in solid blue are due to the gauge fields; those coming from derivatives of the scalar fields are shown with dashed orange lines, while those coming from the scalar potential energy density are shown with dotted green curves.