[Paper Review] Zee model in a modular $A_4$ symmetry

Takaaki Nomura, Hiroshi Okada+1 more|arXiv (Cornell University)|Nov 22, 2021

Particle physics theoretical and experimental studies4 citations

TL;DR

This paper proposes a minimal Zee model with modular $A_4$ symmetry to generate radiative neutrino masses, predicting a normal neutrino mass hierarchy with $\tau$ localized near $2.3i$, leading to a $B_1$ two-zero texture in the neutrino mass matrix. The model successfully reproduces Majorana CP-violating phases consistent with $B_1$ texture, while avoiding flavor-changing neutral currents via symmetry constraints.

ABSTRACT

We study a Zee model applying a modular $A_4$ symmetry in supersymmetry framework, where we construct our lepton model as minimum modular weight as possible. We show our predictions on phases and neutrino masses through numerical analysis. We find a model that can fit in both normal and inverted hierarchy cases of neutrino mass. The predictions of neutrino observables are shown where we obtain more constrained allowed region in the inverted hierarchy case.

Motivation & Objective

• To construct a minimal Zee model using modular $A_4$ symmetry to generate radiative neutrino masses.
• To reduce free parameters in the lepton sector by leveraging modular symmetry, enabling predictive power.
• To analyze the $\Delta\chi^2$ in the lepton sector to test predictions on neutrino masses, mixing angles, and CP-violating phases.
• To investigate the role of the modular parameter $\tau$ in stabilizing the neutrino mass matrix texture.
• To compare model predictions with the $B_1$ two-zero texture and cosmological constraints.

Proposed method

• Assign minimal modular weights $[-2,-2,-4]$ to left-handed lepton doublets under $A_4$ symmetry to minimize free parameters.
• Construct a renormalizable Lagrangian with $H_1$, $H_2$, $s^-$, and $\varphi$ fields transforming under $SU(2)_L \times U(1)_Y \times A_4$ with specific modular weights.
• Fix Higgs masses and mixing angles numerically to simplify $\Delta\chi^2$ analysis in the lepton sector.
• Derive the neutrino mass matrix from loop diagrams involving charged scalars and the $A_4$-invariant Yukawa couplings.
• Perform $\Delta\chi^2$ analysis to constrain $\tau$, neutrino masses, and CP-violating phases.
• Compare phase predictions with the $B_1$ texture via analytical expressions for $\alpha_{21}$ and $\alpha_{31}$.

Experimental results

Research questions

• RQ1 Can a minimal modular $A_4$ symmetry in the Zee model reproduce the $B_1$ two-zero texture in the neutrino mass matrix?
• RQ2 What is the predicted value of the modular parameter $\tau$ and how does it relate to fixed points in the complex plane?
• RQ3 How do the Majorana CP-violating phases in the model compare with those predicted by the $B_1$ texture?
• RQ4 Is the normal neutrino mass hierarchy favored over the inverted hierarchy in the $\Delta\chi^2$ analysis?
• RQ5 What are the cosmologically allowed ranges for the sum of neutrino masses and the effective electron-neutrino mass $\langle m_{ee} \rangle$?

Key findings

• The normal neutrino mass hierarchy is favored in the $\Delta\chi^2$ analysis, with $\tau$ localized near $2.3i$, close to the fixed point $i\times\infty$.
• The neutrino mass matrix in the $\tau \to i\times\infty$ limit reduces to the $B_1$ two-zero texture, with $(m_\nu)_{22} = (m_\nu)_{13} = 0$.
• The predicted CP-violating phases $\alpha_{21}$ and $\alpha_{31}$ are in good agreement with the analytical predictions of the $B_1$ texture, particularly $\alpha_{21} \simeq -\alpha_{31} \simeq \delta_{\text{CP}} - \pi/2$.
• The sum of neutrino masses is predicted to be in the range $58-61$ meV for the normal hierarchy, consistent with cosmological bounds.
• The effective electron-neutrino mass $\langle m_{ee} \rangle$ is predicted to be $2.6-4.4$ meV, providing a testable signal for future double-beta decay experiments.
• The model avoids flavor-changing neutral currents in the quark sector due to the structure of the modular $A_4$ symmetry, a key advantage over generic Zee models.