Parametrizing Majorana neutrino couplings in the Higgs sector

Nonzero masses for the active neutrinos—regardless of their nature or origin—arise only after electroweak symmetry breaking. We discuss the parameterization of neutrino couplings to a Higgs sector consisting of one $SU(2{)}_L$ scalar doublet and one $SU(2{)}_L$ scalar triplet, and allow for right-handed neutrinos whose Majorana mass parameters arise from the vacuum expectation value of a standard model scalar singlet. If the neutrinos are Majorana fermions, all Yukawa couplings can be expressed as functions of the neutrino mass eigenvalues and a subset of the elements of the neutrino mixing matrix. In the mass basis, the Yukawa couplings are, in general, not diagonal. This is to be contrasted to the case of charged fermions or Dirac neutrinos, where couplings to the Higgs-boson are diagonal in the mass basis and proportional only to the fermion masses. Nonetheless, all physically distinguishable parameters can be reached if all neutrino masses are constrained to be positive, all mixing angles constrained to lie in the first quadrant ($\theta \in [0,\pi /2]$), and all Majorana phases to lie in the first two quadrants ($\varphi \in [0,\pi \}$), as long as all Dirac phases vary within the entire unit circle ($\delta \in [0,2\pi \}$). We discuss several concrete examples and comment on the Casas-Ibarra parameterization for the neutrino Yukawa couplings in the case of the type I seesaw Lagrangian. We also comment on the case where $SU(2{)}_L$ fermion triplets replace the right-handed neutrinos.

Figure 1

Three level diagrams that mediate the decay of a neutrino mass eigenstate ${\nu }_j$ into three neutrino mass eigenstates ${\nu }_i$.

Figure 2

${ϵ}_3$, as defined in Eq. (5.11), as a function of the Majorana phase ${\varphi }_1$ for $m_2=2m_1$, $m_3=100m_1$, $m_4=1000m_1$, ${\theta }_2=\pi /5$, $C=\sqrt{2}{e}^{-i\pi /6}$, and $S=3^{1/4}{e}^{-i3\pi /4}$, in arbitrary units. In the top panel, $\det (R)=+1$ (black curve) and $\det (R)=-1$ (red [grey] curve) clearly lead to different values of ${ϵ}_3$. In the bottom panel we fix $\det (R)=+1$ and double the ${\varphi }_1$ range. The vertical dotted line corresponds to ${\varphi }_1=\pi$. It is clear that the imagine of ${\varphi }_1\in [\pi ,2\pi \}$ is the same as the imagine of the $\det (R)=-1$, ${\varphi }_1\in [0,\pi \}$ points in the top figure.