Large $|U_{e3}|$ and tribimaximal mixing

We investigate in a model-independent way to what extent one can perturb tribimaximal mixing in order to generate a sizable value of $|U_{e3}|$, while at the same time keeping solar neutrino mixing near its measured value, which is close to ${sin}^2{\theta }_{12}=\frac{1}{3}$. Three straightforward breaking mechanisms to generate $|U_{e3}|\simeq 0.1$ are considered. For charged lepton corrections, the suppression of a sizable contribution to ${sin}^2{\theta }_{12}$ can be achieved if $CP$ violation in neutrino oscillations is almost maximal. Generation of the indicated value of $|U_{e3}|\simeq 0.1$ through renormalization group corrections requires the neutrinos to be quasidegenerate in mass. The consistency with the allowed range of ${sin}^2{\theta }_{12}$ together with large running of $|U_{e3}|$ forces one of the Majorana phases to be close to $\pi$. This implies large cancellations in the effective Majorana mass governing neutrinoless double beta ($(\beta \beta {)}_{0\nu }$) decay, constraining it to lie near its minimum allowed value of $m_0\cos 2{\theta }_{12}$, where $m_0\gtrsim 0.1\mathrm{eV}$. Finally, explicit breaking of the neutrino mass matrix in the inverted hierarchical and quasidegenerate neutrino mass spectrum cases is similarly correlated with the $(\beta \beta {)}_{0\nu }$-decay effective Majorana mass, although to a lesser extent. The implied values for the atmospheric neutrino mixing angle ${\theta }_{23}$ are given in all cases.

Figure 1

Charged lepton corrections to tribimaximal mixing. The left panel shows $|U_{e3}|$ against ${sin}^2{\theta }_{23}$ (the axes cover the whole $1\sigma$ range), while the right panel gives $|U_{e3}|$ against $J_{CP}$. The solid blue lines display the maximal value that $|J_{CP}|$ can take.

Figure 2

The running of $|U_{e3}{|}^2={sin}^2{\theta }_{13}$ in SM for both the normal (left panel) and the inverted (right panel) mass orderings. The high scale values of the mixing angles are kept fixed at TBM values, while the masses and phases are varied randomly such that after RG evolution the parameter values are within current experimental ranges.

Figure 3

Scatter plots showing the running of ${sin}^2{\theta }_{13}$ with $m_0$, for MSSM with normal mass ordering and $\tan \beta =5$, 20. The high scale mixing angles are fixed at TBM and the masses and phases are varied randomly such that after RG evolution the parameter values are within current experimental ranges. For a given $\tan \beta$, the allowed regions are the same as above for the inverted mass ordering.

Figure 4

Scatter plots in the $m_0-|{\alpha }_2|$ plane for MSSM ($\tan \beta =5$, 20) and normal mass ordering, both for high (black circles) and low (red squares) energy scales. The high scale mixing angles are fixed at the TBM values and the masses and phases are varied randomly at the high scale so that the low energy parameters are consistent with the current experimental data. The data for the inverted mass ordering show the same variation.

Figure 5

Scatter plots showing the correlation between ${sin}^2{\theta }_{13}$ and ${sin}^2{\theta }_{23}$ for MSSM with $\tan \beta =5$, 20. The left panel shows the case with normal mass ordering, while the right panel is for the inverted mass ordering. The high scale mixing angles are fixed at TBM and the masses and phases are varied randomly such that after RG evolution the parameter values are within current experimental ranges.

Figure 6

Scatter plots for the effective neutrino mass $⟨m⟩$ that contributes to neutrinoless double beta decay as a function of $m_0$, in MSSM with $\tan \beta =5$, 20 and normal mass ordering. The solid (black) lines indicate the maximum and minimum possible values of $⟨m⟩$ for given $m_0$, obtained by varying the oscillation parameters in their current $3\sigma$ range and the phases between 0 to $2\pi$. The cases with inverted ordering show identical characteristics.

Figure 7

Scatter plots for an explicitly broken TBM mass matrix of $|U_{e3}|$ against the smallest neutrino mass for the normal (left panel) and inverted (right panel) mass ordering.

Figure 8

Scatter plots of $|U_{e3}|$ against ${sin}^2{\theta }_{23}$ as well as of $|U_{e3}|$ against $⟨m⟩$ for an explicitly broken TBM mass matrix in case of an inverted hierarchy. The left plots show the cases ${\alpha }_2=0$ and ${\alpha }_2=\pi$; the right plots have free ${\alpha }_2$. Indicated also are the $1\sigma$ ranges of the oscillation parameters, and the upper and lower limits of the effective mass.

Figure 9

Same as Fig. 8 for quasidegenerate neutrinos.