Scotogenic A4 Neutrino Mass Model

• The scotogenic neutrino mass model is a radiative framework where neutrino masses are induced at one loop via A4 flavor symmetry and an exact Z2 symmetry that also stabilizes dark matter.
• It employs inert scalar doublets and heavy Majorana singlets with specific A4 representations to generate predictive mass textures and nonzero mixing angles such as θ13.
• Phenomenological implications include testable signals in lepton flavor violation, neutrinoless double beta decay, and dark matter searches via collider and direct-detection experiments.

The scotogenic neutrino mass model encompasses a broad class of extensions to the Standard Model (SM) in which tiny neutrino masses are generated radiatively, typically at one loop, with the same new physics sector providing a viable dark matter candidate. A distinctive feature of this class is the interplay of a discrete or gauge symmetry (most commonly an exact $\mathbb{Z}_2$) that both forbids tree-level neutrino masses and ensures dark-matter stability. Within this landscape, models employing non-Abelian discrete flavor groups, especially $A_4$, have realized predictive structures for masses and mixing, naturally linking phenomena such as nonzero $\theta_{13}$, large leptonic CP violation, and both dark matter and lepton-flavor violating signals. Below is a comprehensive exposition of the foundational principles, typical realizations, flavor symmetries, radiative mechanisms, phenomenology, and theoretical extensions characterizing the scotogenic $A_4$ model and its broader context.

1. Foundations: Field Content and Symmetries

The archetypal scotogenic $A_4$ model extends the SM by

• Three left-handed lepton doublets $L_i=(\nu_i, \ell_i)_L$, transforming as a triplet of $A_4$.
• Three right-handed charged-lepton singlets: $\ell_1^c \sim 1$, $\ell_2^c \sim 1'$, $\ell_3^c \sim 1''$ under $A_4$.
• Three heavy Majorana singlets $N_k\sim 3$ under $A_4$.
• Three SM Higgs doublets $\Phi_i \sim 3$ under $A_4$, all even under a discrete $\mathbb{Z}_2$ ("dark parity").
• One inert scalar doublet $\eta$ (odd under $\mathbb{Z}_2$ and $A_4$-singlet).
• Three scalar singlets $\sigma_i \sim 3$ under $A_4$, $\mathbb{Z}_2$-even.

The symmetry-breaking pattern is as follows:

• $A_4$ is spontaneously broken by $\langle \sigma_i\rangle$, with $\langle\phi_i^0\rangle$ breaking $A_4\to \mathbb{Z}_3$ (lepton triality).
• $\mathbb{Z}_2$ remains exact, forbidding a tree-level Dirac mass for neutrinos and stabilizing the lightest $\mathbb{Z}_2$-odd particle.
• A global or gauged $B-L$ can be imposed to forbid an explicit Majorana mass for $N_k$.

Charged-lepton masses arise at tree level via vacuum expectation values (VEVs) $\langle\phi_i^0\rangle$, while neutrino masses are generated radiatively through loops containing $\mathbb{Z}_2$-odd fields.

2. Radiative Neutrino Mass Generation

Neutrinos acquire Majorana masses at one loop via the exchange of inert scalars and singlet fermions, with the key Lagrangian terms

$$-\mathcal{L}_\text{Yuk} \supset h_{ik} \overline{L_i} \widetilde\eta N_k + \frac{1}{2}(M_N)_{k\ell}\overline{N^c_k}N_\ell + \lambda_5(\Phi^\dagger\eta)^2 + \text{h.c.}$$

where $(M_N)_{k\ell}$ is an $A_4$-structured Majorana mass induced from $\langle\sigma_i\rangle$ and the $\lambda_5$ term controls the splitting between the real and imaginary components of $\eta^0$.

The resulting one-loop neutrino mass matrix is

$$(M_\nu)_{ij} = \sum_k \frac{h_{ik} h_{jk} M_k}{16\pi^2} \left[ \frac{m_R^2}{m_R^2 - M_k^2} \ln\frac{m_R^2}{M_k^2} - \frac{m_I^2}{m_I^2 - M_k^2} \ln\frac{m_I^2}{M_k^2} \right]$$

where $M_k$ are the eigenvalues of $M_N$ and $m_{R,I}$ are the masses of $\text{Re}\,\eta^0$ and $\text{Im}\,\eta^0$. In the limit of small scalar mass splitting, the dominant term reduces to

$$(M_\nu)_{ij} \simeq \frac{\lambda_5 v^2}{8\pi^2} \sum_k \frac{h_{ik} h_{jk} M_k}{m_0^2 - M_k^2}\left[1 - \frac{M_k^2}{m_0^2 - M_k^2} \ln\frac{m_0^2}{M_k^2}\right]$$

with $m_0^2 = (m_R^2 + m_I^2)/2$.

The $A_4$ symmetry structures the heavy Majorana sector, typically yielding a $1\oplus 2$ block (tribimaximal basis), leading to predictive textures and mixing-angle relations.

3. Structure of the Neutrino Mass Matrix and PMNS Correlations

Following the $A_4$ flavor structure, after rotation to the tribimaximal basis, the Majorana mass matrix takes the form

$$M_N^{(1,2,3)} = \begin{pmatrix} A+D & 0 & 0 \ 0 & A & C \ 0 & C & A-D \end{pmatrix},\quad C = \sqrt{2}E,\quad F = -E$$

Thus, the light-neutrino mass matrix inherits a nontrivial $2\times2$ substructure in the $(2,3)$ block, where the off-diagonal entry $C$ induces nonzero $\theta_{13}$ and a leptonic Dirac CP phase $\delta_{CP}$.

Diagonalization proceeds via a rotation in the $(2,3)$ sector: $$U_\epsilon = \begin{pmatrix} 1 & 0 & 0 \ 0 & \cos\theta & -\sin\theta e^{i\varphi} \ 0 & \sin\theta e^{-i\varphi} & \cos\theta \end{pmatrix}$$ The PMNS matrix is then $U = U_\text{TB}\, U_\epsilon^\dagger$ up to unphysical phases. Key mixing parameters are extracted as

• $\theta_{13}$ from $U_{e3}\simeq -(\sin\theta/\sqrt{3}) e^{-i\varphi}$,
• $\theta_{12}$ from $\tan^2\theta_{12} = (1 - 3\sin^2\theta_{13})/2$,
• $\theta_{23}$ and $\delta_{CP}$ from off-diagonal structure and complex phases.

Numerical studies yield $0.05\leq \sin^2 2\theta_{13} \leq 0.15$ and enforce necessarily large $|\tan\delta_{CP}| \gtrsim 1.2$ (i.e., $|\delta_{CP}| \sim 45^\circ-90^\circ$).

Mass eigenvalue patterns (NO, IO, QD) are realized depending on input parameters, with the oscillation data fixing the differences.

4. Dark Matter Stability and Phenomenology

The imposed $\mathbb{Z}_2$ ("dark parity") guarantees that the lighter among $\text{Re}\,\eta^0$ or $\text{Im}\,\eta^0$ is stable, making it a WIMP candidate. Its relic density is controlled by SU(2) and Higgs-portal annihilation channels, and direct-detection rates are set by the effective coupling $$\lambda_L \simeq (\lambda_3 + \lambda_4 + \lambda_5)/2$$. The $\lambda_5$-induced mass splitting $m_R - m_I \sim \lambda_5 v^2 / m_0$ is critical to evade strong inelastic direct-detection constraints.

Typical successful regions have $m_\eta \sim$ TeV scale or lower, consistent with relic-abundance and XENON/LUX-type bounds.

5. Phenomenological Consequences: Lepton Flavor Violation, $0\nu\beta\beta$, and Collider Probes

Lepton Flavor Violation

One-loop exchange of $N_k$ and $\eta^\pm$ naturally induces charged LFV processes: $$\text{Br}(\mu\to e\gamma) \simeq \frac{3\alpha}{32\pi} \left| \sum_k h_{\mu k} h_{e k}^* F\left(\frac{M_k^2}{m_\eta^2}\right) \right|^2$$ where $F(x)$ is the loop function displayed in (Ma et al., 2012). Current and future LFV searches (e.g., $\mu\to e\gamma$) stringently constrain the parameter space, especially the size of $h_{ik}$.

Neutrinoless Double Beta Decay

The model predicts the effective $0\nu\beta\beta$ mass parameter

$$m_{ee} = \left| \sum_j U_{ej}^2 m_j' e^{i\alpha_j} \right|$$

Typically, $m_{ee}$ falls in the $0.002$–$0.05$ eV range, depending on the mass hierarchy, testable in current or next-generation $0\nu\beta\beta$ experiments.

Collider and Other Signatures

Charged components $\eta^\pm$ are accessible via Drell-Yan pair production, with signatures such as $\eta^+\eta^- \to \ell^+\ell^- + E_T^{\text{miss}}$, and potentially exotic decays of $\sigma_i$ if kinematically allowed.

6. Variations and Extensions: Modular and Non-Abelian Flavor

Alternative scotogenic frameworks exploit other flavor symmetries. For instance, modular $A_4$ models replace explicit flavons with modular forms, generating all flavor structure from modular weights and VEV of the modulus $\tau$, achieving neutrino mass matrices and mixing predictions with fewer parameters and dynamically ensuring dark matter stability (Behera et al., 2020).

Similarly, $\Delta(27)$ and other non-Abelian groups yield related structures for $M_\nu$, with specific predictions for CP violation and $0\nu\beta\beta$ (Ma et al., 2014). The core mechanism remains analogous: a combination of radiative mass generation and symmetry-induced flavor textures.

7. Theoretical Constraints and Ultraviolet Completions

The stability and naturalness of small parameters, such as $\lambda_5$, are subject to theoretical scrutiny. UV completions have been constructed in which $\mathbb{Z}_2$ and small $\lambda_5$ arise as low-energy remnants of a spontaneously broken global symmetry (e.g., U(1)$_L$), supplemented by new fields such as scalar triplets and singlets (Escribano et al., 2021). In these completions, dark parity and lepton number violation are dynamically generated, resolving the origin of apparent ad hoc ingredients in the low-energy effective theory.

The high-scale behavior, including renormalization group evolution of all parameters, can lead to vacuum instability or spontaneous breaking of dark parity at high scales unless quartic couplings and masses are judiciously chosen (Bouchand et al., 2012, Escribano et al., 2020).

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In summary, the scotogenic $A_4$ neutrino mass model realizes a scenario with radiative Majorana neutrino masses linked to a $\mathbb{Z}_2$-stabilized dark sector, predictive flavor correlations, dark matter candidates, and a suite of associated phenomenological signatures (LFV, $0\nu\beta\beta$, direct and indirect DM detection, colliders), all rooted in and constrained by the underlying flavor symmetry and scalar-fermion dynamics (Ma et al., 2012, Behera et al., 2020, Ma et al., 2014, Escribano et al., 2021). The interplay between radiative suppression, discrete symmetry, and flavor structure positions this framework as a technically natural and phenomenologically rich alternative to traditional seesaw mechanisms.