Radiatively Generating the Higgs Potential and Electroweak Scale via the Seesaw Mechanism

The minimal seesaw scenario can radiatively generate the Higgs potential to induce electroweak symmetry breaking while supplying an origin of the Higgs vacuum expectation value from an underlying Majorana scale. If the Higgs potential and (derived) electroweak scale have this origin, the heavy $\mathrm{SU}(3)\times \mathrm{SU}(2)\times \mathrm{U}(1{)}_Y$ singlet states are expected to reside at $m_N\sim 10–500\mathrm{PeV}$ for couplings $|\omega |\sim {10}^{-4.5}-{10}^{-6}$ between the Majorana sector and the standard model. In this framework, the usual challenge of the electroweak scale hierarchy problem with a classically assumed potential is absent as the electroweak scale is not a fundamental scale. The new challenge is the need to generate or accommodate PeV Majorana mass scales while simultaneously suppressing tree-level contributions to the potential in ultraviolet models.

Figure 1

One loop corrections matching onto and generating the Higgs potential at the scale(s) $m_p$ in the seesaw model.

Figure 2

Values of the parameters $\lambda$ (left) and $\sqrt{m^2}$ (right) extrapolated at the scale $\mu ={\widehat{m}}_t$ as a function of the heavy neutrino mass scale $m_p$ and of $|\omega |$, respectively. The dashed lines and surrounding bands marks indicate the values consistent with the measured Higgs mass and its percentage error [29]. The red line in the left panel assumes ${\widehat{m}}_t=173.2\mathrm{GeV}$ and the surrounding band corresponds to varying ${\widehat{m}}_t$ between 171 and 175 GeV (a $2\sigma$ error variation [30]). In the right panel, the solid red line assumes $m_p={10}^{1.3}\mathrm{PeV}$, and the gray region is disfavored due to $\mathrm{\Lambda }\mathrm{CDM}$ cosmology limits on the sum of neutrino masses [Eq. (12)]. The neutrino mass scales predicted (in eV) are the three solid lines.

Figure 3

The emergence of the Higgs potential due to running the seesaw boundary conditions down to $\mu \sim {\widehat{m}}_t$.

Figure 4

Numerical sensitivity of the results, with all cases showing one loop matching to $\mathrm{\Delta }m$, $\lambda$ and including one loop corrections for the remaining SM parameters with one loop, two loop, or mixed RGEs for all parameters. The mixed case shows one loop running for $\mathrm{\Delta }m$, $\lambda$ and two loop running for all remaining SM parameters. The best fit points are indicated with a box in each case with error bars showing the experimental uncertainty in the top quark mass, which has been chosen to be its $2\sigma$ uncertainty [30]. Nevertheless, we have determined that the measured mass differences of the neutrinos can be accommodated in all three RGE cases.