[Paper Review] Zee-Babu model in modular $A_4$ symmetry

Hiroshi Okada, Yonghui Qi|arXiv (Cornell University)|Sep 28, 2021

Neutrino Physics Research4 citations

TL;DR

This paper proposes a Zee-Babu model extended with modular $A_4$ flavor symmetry to predict neutrino masses, mixing angles, CP phases, and the sum of neutrino masses. By assigning modular weights and using a $Δ\chi^2$ analysis, the model predicts that the tau lepton is localized near the $i\infty$ fixed point in both normal and inverted neutrino mass hierarchies, leading to constrained predictions for neutrinoless double beta decay and lepton flavor violation.

ABSTRACT

We study a Zee-Babu model in a modular $A_4$ flavor symmetry, in which we search for several predictions such as phases, sum of neutrino masses, and neutrinoless double beta decay, satisfying neutrino oscillation data in a minimum framework of the charge assignments of modular weight. We perform $Δχ^2$ analysis to get our results and find $τ$ is localized nearby at one of the fixed points of $i imes \infty$ for both of normal and inverted mass hierarchies.

Motivation & Objective

• To extend the Zee-Babu model with modular $A_4$ flavor symmetry to reduce free parameters and enhance predictive power in the lepton sector.
• To constrain the neutrino mass matrix, mixing patterns, CP phases, and sum of neutrino masses using modular symmetry and experimental oscillation data.
• To investigate the localization of the tau lepton in the complex upper half-plane under modular $A_4$ symmetry and its phenomenological consequences.
• To perform a $\Delta\chi^2$ analysis to identify viable parameter regions consistent with neutrino oscillation data.
• To predict observable signatures such as neutrinoless double beta decay and lepton flavor violation within a minimal framework.

Proposed method

• Assign modular weights $[-2, -4, -6]$ to left-handed charged leptons and zero weight to right-handed charged leptons under $A_4$.
• Construct Yukawa interactions using modular forms $Y^{(2)}_3$, $Y^{(4)}_3$, and higher-order forms $Y^{(6)}_3$, $Y^{(10)}_{1'}$, etc., transforming under $A_4$ representations.
• Implement the Higgs potential and charged-lepton mass matrix using $A_4$-invariant combinations of modular forms and fields with assigned modular weights.
• Derive the neutrino mass matrix via two-loop diagrams involving the doubly charged scalar $k^{++}$ and the singly charged scalar $s^{-}$, ensuring $A_4$ and modular invariance.
• Use the $\Delta\chi^2$ analysis to scan parameter space and identify best-fit values consistent with global neutrino oscillation data.
• Analyze the behavior of the modular parameter $\tau$ and determine the fixed point localization of the tau lepton, particularly near $i\times\infty$.

Experimental results

Research questions

• RQ1 Can modular $A_4$ symmetry in the Zee-Babu model reduce the number of free parameters and predict specific values for neutrino masses, mixing angles, and CP phases?
• RQ2 Where is the tau lepton localized in the complex upper half-plane under modular $A_4$ symmetry, and how does this localization affect the neutrino mass spectrum?
• RQ3 What are the predictions for the sum of neutrino masses and neutrinoless double beta decay in this framework?
• RQ4 How do the modular weights and transformation properties of scalar and fermion fields constrain the Yukawa couplings and mass matrices?
• RQ5 What is the role of the $\Delta\chi^2$ analysis in identifying viable parameter regions consistent with neutrino oscillation data?

Key findings

• The tau lepton is localized near the $i\times\infty$ fixed point of the modular $A_4$ group in both normal and inverted neutrino mass hierarchies.
• The model predicts a vanishing lightest neutrino mass due to the anti-symmetric structure of the two-loop neutrino mass matrix, consistent with the Zee-Babu mechanism.
• The $\Delta\chi^2$ analysis identifies a preferred region where the modular parameter $\tau$ for the tau lepton is close to $i\times\infty$, indicating strong symmetry breaking at this point.
• The sum of neutrino masses is predicted to be in the range of $\sim 0.06$ eV to $\sim 0.1$ eV, depending on the hierarchy, with a preference for the inverted hierarchy.
• Neutrinoless double beta decay is suppressed in this model due to the small effective Majorana mass, with predictions below $0.02$ eV in the inverted hierarchy.
• The model predicts suppressed lepton flavor violation processes, consistent with current experimental bounds, due to the constrained flavor structure from modular symmetry.