Seesaw parametrization for $n$ right-handed neutrinos

Introducing $n$ right-handed neutrinos to the Standard Model yields, in general, massive active neutrinos. We give explicit parametrizations for the involved mixing and coupling matrices in terms of physical parameters for both the top-down and the bottom-up approach for arbitrary $n$. Bounds on the complex mixing angles in the bottom-up approach from perturbativity of the Yukawa couplings to charged lepton flavor violation are discussed. As a novel illustration of possible effects from $n\ne 3$, we extend the neutrino anarchy framework to arbitrary $n$; we show that while the anarchic mixing angles are insensitive to the number of singlets, the observed ratios of neutrino masses prefer small $n$ for the simplest linear measure.

Figure 1

Distribution of the ${\mathcal{M}}_R$ eigenvalues $M_k$, according to the linear measure from Eq. (23) with the boundary constraint $\sum _{k=1}^n{M}_k^2\le M_0^2$.

Figure 2

Distribution of the singular values $y_k$ of the Yukawa matrix, according to the linear measure from Eq. (24) for $n=3$ (black) and $n=20$ (red/dashed) with the boundary constraint $\sum _{k=1}^3{y}_k^2\le y_0^2$.

Figure 3

${log}_{10}[R_{\nu }]\equiv {log}_{10}[(m_2^2-m_1^2)/(m_3^2-m_1^2)]$ for various $n$. The purple/dashed vertical lines indicate the $3\sigma$ range for $R_{\nu }\simeq 0.03$ [30] (for NH). The black vertical line divides the NH and IH solutions.