Lower bounds on the smallest lepton mixing angle

We give minimal values for the smallest lepton mixing parameter $|U_{e3}|$, applying 2-loop renormalization group equations in an effective theory approach. This is relevant in scenarios that predict an inverted neutrino mass spectrum with the smallest mass and $|U_{e3}|$ being zero at tree level, a situation known to be preserved at 1-loop order. At 2-loop, $|U_{e3}|$ is generated at a level of ${10}^{-12}–{10}^{-14}$. Such small values are of interest in supernova physics. Corresponding limits for the normal mass ordering are several orders of magnitude larger. Our results show that $|U_{e3}|$ can in general be expected to be nonzero.

Figure 1

Rank-changing 2-loop diagram in the SM which breaks the 1-loop RG invariance $m_3=|U_{e3}|=0$. ${\ell }_L$ ($e_R$) are left- (right-) handed leptons and $H$ is the Higgs doublet.

Figure 2

Generated value of $|U_{e3}|$ at $\lambda =100\mathrm{GeV}$ assuming a vanishing value at $\Lambda ={10}^{12}\mathrm{GeV}$ in the leading-log approximation. The solid line corresponds to fixing all mixing parameters to the best-fit values and the shaded region shows the $3\sigma$ region, which is taken from 4.