Lepton flavor at the electroweak scale: A complete $A_4$ model

Apparent regularities in fermion masses and mixings are often associated with physics at a high flavor scale, especially in the context of discrete flavor symmetries. One of the main reasons for that is that the correct vacuum alignment requires usually some high scale mechanism to be phenomenologically acceptable. Contrary to this expectation, we present in this paper a renormalizable radiative neutrino mass model with an $A_4$ flavor symmetry in the lepton sector, which is broken at the electroweak scale. For that we use a novel way to achieve the vacuum expectation value alignment via an extended symmetry in the flavon potential proposed before by two of the authors. We discuss various phenomenological consequences for the lepton sector and show how the remnants of the flavor symmetry suppress large lepton flavor violating processes. The model naturally includes a dark matter candidate, whose phenomenology we outline. Finally, we sketch possible extensions to the quark sector and discuss its implications for the LHC, especially how an enhanced diphoton rate for the resonance at 125 GeV can be explained within this model.

Figure 1

Neutrino mass generation at one loop.

Figure 2

The deviations from tribimaximal mixing of the form $U=U_{\mathrm{HPS}}{U}_{12}$ and $U=U_{\mathrm{HPS}}{U}_{13}$ generated by the angle $\theta$ defined in Eq. (3.18). The yellow point represents TBM, the continuous lines give the deviations from TBM with the angle $\theta$ given by the color codes in the top right corner for $\delta =\frac{n}{5}\frac{\pi }{2}$ for $n=0,\dots ,5$, where $n=0$ is the outermost parabola etc. The 1, 2, and 3 sigma regions of a recent global fit [7] are indicated by dotted, dashed, and continuous contours, respectively.

Figure 3

Dependence of the reactor angle ${\theta }_{13}$ on the atmospheric mixing angle ${\theta }_{23}$. The various color codings are given next to each scatter plot. Top left: For ${sin}^2{\theta }_{23}<1/2({sin}^2{\theta }_{23}>1/2)$ the model predicts ${\delta }_{CP}=0$, $2\pi ({\delta }_{CP}=\pi )$. Top right: The scatterplot shows a band structure in ${sin}^2{\theta }_{12}$. Bottom left: For the points in the experimentally allowed region, $\widehat{b}$ has to be of similar size as $\widehat{a}$, $\widehat{d}$. Bottom right: For the points in the experimentally allowed region, $\widehat{e}$ has to be of approximately 1 order of magnitude smaller than $\widehat{a}$, $\widehat{d}$. The 1, 2, and 3 sigma regions of Ref. 7 are again indicated by dotted, dashed, and continuous contours, respectively.

Figure 4

Numerical evaluation of the approximate atmospheric sum rule (3.25). The numerical evaluation shows that the sum rule holds to a good degree of approximation.

Figure 5

Expectation for the effective mass of neutrinoless double beta decay. The pink points lie within the 3 sigma region for all oscillation parameters. The points with color coding lie within the 3 sigma range for all observables except ${\theta }_{13}$.

Figure 6

Lowest order $\mu \to e\gamma$ processes mediated by $\widehat{\eta }$ (left) and ${\phi }^{\prime (\prime )}$ (right). There has to be a coupling to the VEVs $⟨{\varphi }_1{\varphi }_2⟩$ of the neutrino sector, which suppresses the amplitudes.

Figure 7

The functions $F_2$ (red dashed) and $G_2$ (blue solid).

Figure 8

Lepton flavor violating rare decay ${\tau }^{-}\to {\mu }^{-}{\mu }^{-}{e}^{+}$. (a) Tree level contribution of ${\phi }^{\prime \prime }$. (b) One-loop contribution of $S\mathrm{\text{−}}\eta$.

Figure 9

Contour lines for different values of $h_1=h_2$ with the correct DM abundance ${\Omega }_S{h}^2=0.12$.

Figure 10

Plot of function $h$ of Eq. (7.8).

Figure 11

$R_{\gamma \gamma }$ in the case where all charged scalars have the same common mass $M$ as a function of $\frac{1}{3}\sum _{J=1}^3{c}_{\eta J}$.