Mixing angle and phase correlations from $A_5$ with generalized $CP$ and their prospects for discovery

The observed leptonic mixing pattern could be explained by the presence of a discrete flavor symmetry broken into residual subgroups at low energies. In this scenario, a residual generalized charge parity ($CP$) symmetry allows the parameters of the Pontecorvo-Maki-Nakagawa-Sakata matrix, including Majorana phases, to be predicted in terms of a small set of input parameters. In this article, we study the mixing parameter correlations arising from the symmetry group ${\mathrm{A}}_5$ including generalized $CP$ subsequently broken into all of its possible residual symmetries. Focusing on those patterns which satisfy present experimental bounds, we then provide a detailed analysis of the measurable signatures accessible to the planned reactor, superbeam and neutrinoless double-beta decay experiments. We also discuss the role which could be played by high-precision measurements from longer term projects such as the Neutrino Factory. This work provides a concrete example of how the synergies of the upcoming experimental program allow flavor symmetric models to be thoroughly investigated. Indeed, thanks to the rich tapestry of observable correlations, we find that each step of the experimental program can make important contributions to the assessment of such flavor-symmetric patterns, and ultimately all patterns that we have identified can be excluded, or strong evidence found for their continued relevance.

Figure 1

Mixing angles for ${\mathbb{Z}}_3$ as a function of the internal parameter $\theta$. This pattern predicts $|\sin \delta |=1$ and $\sin {\alpha }_{21}=\sin {\alpha }_{31}=0$. The shaded regions show the $3\sigma$ allowed region for the corresponding mixing angle according to current global data [25].

Figure 2

The two patterns of mixing angles for ${\mathbb{Z}}_5$ as a function of the internal parameter $\theta$. Both patterns predict $\sin {\alpha }_{21}=\sin {\alpha }_{31}=0$. The right pattern predicts $|\sin \delta |=0$ while the left pattern predicts $|\sin \delta |=1$. The shaded regions show the $3\sigma$ allowed region for the corresponding mixing angle according to current global data [25].

Figure 3

Allowed mixing angles for $G_e={\mathbb{Z}}_2\times {\mathbb{Z}}_2$ as a function of the unphysical parameter $\theta$. There are two possible sets of predictions of the mixing angles which have the same ${\theta }_{12}$ and ${\theta }_{13}$ predictions but distinct ${\theta }_{23}$ predictions (solid and dotted lines) related by the mapping ${\theta }_{23}\to \frac{\pi }{2}-{\theta }_{23}$. All complex phases are $CP$ conserving for these patterns: $\sin \delta =\sin {\alpha }_{21}=\sin {\alpha }_{31}=0$. The shaded regions show the $3\sigma$ allowed region for the corresponding mixing angle according to current global data [25].

Figure 4

Predictions for ${\theta }_{12}$ as a function of ${\theta }_{13}$. All of the patterns of mixing parameters associated with a given charged-lepton residual symmetry have the same prediction (solid lines). The dashed line close to each prediction shows the linearized predictions in Eqs. (15)–(17). The grey regions show the 1 and $3\sigma$ allowed regions for ${\theta }_{12}$ and the vertical lines show the $3\sigma$ range for ${\theta }_{13}$ from current global data [25].

Figure 5

Red (green) lines show the exclusion regions at 1, 2 and $3\sigma$ for ${\theta }_{23}=\pi /4$ and $\delta =3\pi /2$ ($\delta =\pi /2$) expected at LBNF with a 34 kton LAr detector after $5+5$ years running. In this region outside the curves, the two sets of predictions can be excluded at the given confidence. The side panels show the appropriate marginalized $\mathrm{\Delta }{\chi }^2$ and the 1, 2 and $3\sigma$ confidence levels (one d.o.f).

Figure 6

${\theta }_{23}$ as a function of ${\theta }_{13}$ for the patterns which predict $CP$ conservation (solid lines). The dashed line close to each solid line shows the linearized expression in Eqs. (18)–(19). The grey regions show the 1 and $3\sigma$ allowed regions for ${\theta }_{23}$ while the vertical lines show the $3\sigma$ range for ${\theta }_{13}$ from current global data [25].

Figure 7

The regions of true parameter space for which the upper-octant relations in Eqs. (18)–(19) can be excluded at 1, 2 and $3\sigma$ by a 2000 km, 10 GeV LENF using a magnetized iron neutrino detector (MIND). The two regions start to overlap at low values of ${\theta }_{13}$ at the $5\sigma$ confidence level.

Figure 8

$|m_{ee}|$ versus the lightest neutrino mass for the IO (NO) for the solid (dashed) lines. The predictions in a given panel all have the same phase assignment, shown in the top left of the plot. The red (blue) shaded region shows the most general predictions for $|m_{ee}|$ with IO (NO) obtained by varying the oscillation parameters over their current $3\sigma$ global ranges [25]. The green band indicates the combined limit using data from CUORICINO, EXO-200, GERDA, KamLAND-Zen and NEMO-3 [89] and the light blue band displays the projected limit of future experiments.