$A_4$ realization of the inverse seesaw mechanism: Neutrino masses, ${\theta }_{13}$, and leptonic nonunitarity

We provide an $A_4$-based flavor symmetric scenario to accommodate the inverse seesaw mechanism for explaining light neutrino masses and mixing. We find that the lepton mixing, in particular, the tribimaximal mixing pattern and its deviation through nonzero ${\theta }_{13}$, is originated solely from the flavor structure of the lepton number violating contribution of the neutral lepton mass matrix. Here we discuss in detail how a nonzero value of ${\theta }_{13}$ is correlated with the other parameters in the framework and its impact on the Dirac $CP$ phase $\delta$. We also analyze the nonunitarity effects on the lepton-mixing matrix and its implication in terms of the lepton flavor violating decays, etc.

Figure 1

(Left) Plot for $\sin {\theta }_{13}$ vs $\beta$. Here the $3\sigma$ range for $\sin {\theta }_{13}$ fixes $\beta$ in the range 0.328–0.413. (Right) $r=0.03$ contour in the $\alpha –\beta$ plane.

Figure 2

(Left) Absolute neutrino masses vs $\alpha$ (blue dotted, magenta large dashed, orange dashed, and red continuous lines represent $m_1$, $m_2$, $m_3$, and $\sum m_i$, respectively). (Right) Plot for $|m_{ee}|$ vs $\alpha$ [case A].

Figure 3

Contour plot for $r=0.03$ (dashed line) and $\sin {\theta }_{13}=0.153$ (continuous line) for $\delta =30°$ (left) and $\delta =60°$ (right), respectively. Red dotted lines represent a $3\sigma$ variation of $\sin {\theta }_{13}$, while black dots stand for intersection (solution) points for best fit values of $\sin {\theta }_{13}$ and $r$ in both panels.

Figure 4

Contour plot for $r=0.03$ in the $\alpha -\cos {\varphi }_{ba}$ plane for ${\varphi }_{ba}=0$. The disallowed range of $\alpha$, $\cos {\varphi }_{ba}$ is indicated by the dotted portion.

Figure 5

Absolute neutrino masses vs $\alpha$ (blue dotted, magenta large dashed, orange dashed, and red continuous lines represent $m_1$, $m_2$, $m_3$, and $\sum m_i$, respectively). The left panel is for $\cos {\varphi }_{ba}<0$, and the right panel is for $\cos {\varphi }_{ba}>0$.

Figure 6

Contour plot for $r=0.03$ (dashed line) and $\sin {\theta }_{13}=0.153$ (continuous line) for $\delta =30°$ (left) and $\delta =60°$ (right), respectively. Red dotted lines represent a $3\sigma$ variation $\sin {\theta }_{13}$, and black dots stands for solution points for best fit values of $\sin {\theta }_{13}$ and $r$ in both panels.

Figure 7

Allowed range (represented by the blue patch) of $\alpha$ and $\beta$ in order to satisfy the $3\sigma$ range of $\sin {\theta }_{13}$ and $r$. The phases ${\varphi }_{ba,da}$ are allowed to vary within $0–2\pi$.

Figure 8

Allowed ranges of ${\varphi }_{ba}$ and ${\varphi }_{da}$ in order to produce $\sin {\theta }_{13}$ in the $3\sigma$ range, correct $r$, satisfying $\sum m_i<0.23\mathrm{eV}$.

Figure 9

(Left) $\sum m_i$ vs $\beta$ which satisfy the $3\sigma$ range of $\sin {\theta }_{13}$ and $r$. The horizontal orange patch represents the excluded region from the upper bound on the sum of all three light neutrino masses ($\sum m_i<0.23\mathrm{eV}$). (Right) $|m_{ee}|$ vs $\beta$ satisfying $\sum m_i<0.23\mathrm{eV}$. In both panels, $\alpha$, ${\varphi }_{ba}$, and ${\varphi }_{da}$ vary according to Figs. 7 and 8.

Figure 10

Dirac $CP$ phase $\delta$ vs $\beta$ satisfying constraints from the $3\sigma$ range of $\sin {\theta }_{13}$ and $r$ (with $\sum m_i<0.23\mathrm{eV}$). Here both ${\varphi }_{ba}$ and ${\varphi }_{da}$ vary between 0 and $2\pi$.

Figure 11

Contour plots for $k=0.0183\mathrm{eV}$ in the $v_f–\mathrm{\Lambda }$ plane [using Eq. (5.9)] for ${\varphi }_{ba}={\varphi }_{da}=0$ (and $|{\mu }_1|=1$). The dotted portion in each curve indicates the excluded part in view of Eq. (5.8). Here the orange, magenta, and red lines stand for $\lambda =0.01$, 0.1, and 1, respectively.

Figure 12

(Left) Contour plot for $k=0.0195\mathrm{eV}$ in the $v_f–\mathrm{\Lambda }$ plane for ${\varphi }_{ba}=0$ and $\delta =30°$. (Right) Contour plot for $k=0.0220\mathrm{eV}$ in the $v_f–\mathrm{\Lambda }$ plane for ${\varphi }_{ba}=0$ and $\delta =60°$. In both panels, orange, magenta, and red lines stand for $\lambda =0.01$, 0.1, and 1, respectively.

Figure 13

(Left) Contour plot for $k=0.0544\mathrm{eV}$ in the $v_f–\mathrm{\Lambda }$ plane for ${\varphi }_{da}=0$ and $\cos {\varphi }_{ba}=-0.3$ ($\alpha =0.814$) [IH, inverted hierarchy]. (Right) Contour plot for $k=0.0218\mathrm{GeV}$ in the $v_f–\mathrm{\Lambda }$ plane for ${\varphi }_{ba}=0$ and $\cos {\varphi }_{ba}=0.8$ ($\alpha =1.904$) [NH, normal hierarchy]. In both panels, orange, magenta, and red lines stand for $\lambda =0.01$, 0.1, and 1, respectively.

Figure 14

(Left) Contour plot for $k=0.0274\mathrm{eV}$ in the $v_f–\mathrm{\Lambda }$ plane for ${\varphi }_{ba}={\varphi }_{da}=\varphi$ and $\delta =30°$. (Right) Contour plot for $k=0.0658\mathrm{eV}$ in the $v_f–\mathrm{\Lambda }$ plane for ${\varphi }_{ba}={\varphi }_{da}=\varphi$ and $\delta =60°$. In both panels, orange, magenta, and red lines stand for $\lambda =0.01$, 0.1, and 1, respectively.

Figure 15

Contour plot for $k=0.0147\mathrm{eV}$ in the $v_f–\mathrm{\Lambda }$ plane where $\alpha =2.25$, $\beta =1$, ${\varphi }_{ba}=0.5$, and ${\varphi }_{da}=2$ and with $C_1=7.5\times {10}^{-4}$ (left) and $C_1=4\times {10}^{-4}$ (right). In both panels, orange, magenta, and red lines stand for $\lambda =0.01$, 0.1, and 1, respectively.