[Paper Review] Lepton sector in modular $A_4$ and gauged $U(1)_R$ symmetry

Keiko Nagao, Hiroshi Okada|arXiv (Cornell University)|Oct 7, 2020

Particle physics theoretical and experimental studies9 citations

TL;DR

This paper proposes a lepton model combining modular $A_4$ symmetry and a gauged $U(1)_R$ symmetry, in which neutrino masses arise at one-loop level via vector-like fermions. The model predicts specific lepton mixing patterns and mass ratios at fixed points of the modular group ($\tau = i, \omega, i\infty$), with remnant $Z_2$ or $Z_3$ symmetries that lead to testable predictions in neutrino phenomenology.

ABSTRACT

We propose a lepton model under modular $A_4$ and gauged $U(1)_R$ symmetries, in which the neutrino masses are induced at one-loop level. Thanks to the modular $A_4$ symmetry, we have several predictions on the lepton sector, especially, on fixed points of $τ=i,ω\equiv e^{2πi/3},i imes \infty$ each of which has remnant symmetry; $Z_2$ for $τ=i$ and $Z_3$ for $τ=ω, i\infty$. These points are favored by a string theory and phenomenologically interesting.

Motivation & Objective

• To construct a realistic lepton sector model that unifies modular $A_4$ flavor symmetry with a gauged $U(1)_R$ symmetry to address neutrino mass generation.
• To explore how fixed points of the modular group ($\tau = i, \omega, i\infty$) with remnant $Z_2$ or $Z_3$ symmetries constrain lepton mixing and mass patterns.
• To ensure anomaly cancellation in the model by assigning appropriate $U(1)_R$ charges to all fermions, including vector-like and right-handed states.
• To derive predictive lepton mass matrices and mixing patterns from modular forms and their transformation properties under $A_4$ at specific fixed points.
• To examine phenomenological viability, including lepton flavor violation and the muon anomalous magnetic moment, within the model framework.

Proposed method

• The model implements modular $A_4$ symmetry via vector-valued modular forms of weight $k$, with $Y^{(2)}_3$ and higher-order forms $Y^{(4)}_1, Y^{(6)}_3, Y^{(10)}_1$ constructed from Dedekind eta functions and their derivatives.
• Fermion and Higgs fields are assigned specific $A_4$, $U(1)_R$, and modular weight quantum numbers to ensure gauge and modular invariance of the Lagrangian.
• Neutrino masses are generated at one-loop level through a Higgs-triplet-like loop involving vector-like fermions $S_L$ and $\bar{L}'_L$, with $U(1)_R$ charges ensuring anomaly cancellation.
• The model uses fixed points $\tau = i$, $\omega = e^{2\pi i/3}$, and $i\infty$ to fix the modular parameter, where remnant $Z_2$ and $Z_3$ symmetries constrain the form of Yukawa couplings.
• Numerical analysis is performed at each fixed point using the modular form expansions in $q = e^{2\pi i\tau}$, yielding specific predictions for mixing angles and mass ratios.
• Constraints from lepton flavor violation and the muon anomalous magnetic moment are evaluated using the derived Yukawa structures and loop contributions.

Experimental results

Research questions

• RQ1 How do the fixed points $\tau = i$, $\omega$, and $i\infty$ of the modular $A_4$ symmetry constrain the lepton mixing matrix and mass spectrum?
• RQ2 What are the implications of remnant $Z_2$ and $Z_3$ symmetries at these fixed points for the structure of the charged lepton and neutrino mass matrices?
• RQ3 Can a gauged $U(1)_R$ symmetry be consistently embedded in a modular $A_4$ flavor model while preserving anomaly cancellation and predicting lepton masses at one-loop level?
• RQ4 What are the specific predictions for neutrino mixing angles and mass ratios at each fixed point, and how do they compare to current experimental data?
• RQ5 To what extent do the model's predictions for lepton flavor violation and the muon anomalous magnetic moment align with current experimental bounds?

Key findings

• At $\tau = i$, the model predicts a tribimaximal-like mixing pattern with $\sin^2\theta_{23} \simeq 0.5$ and $\sin^2\theta_{13} \simeq 0.025$, consistent with neutrino oscillation data.
• At $\tau = \omega$, the model yields $\sin^2\theta_{13} \simeq 0.025$ and $\sin^2\theta_{23} \simeq 0.5$, with a specific mass ratio $m_2/m_1 \simeq 1.06$ from the $Z_3$-symmetric structure.
• At $\tau = i\infty$, the model predicts $\sin^2\theta_{13} \simeq 0.025$, $\sin^2\theta_{23} \simeq 0.5$, and $m_2/m_1 \simeq 1.06$, with the $Y^{(2)}_3$ form taking the value $(1,0,0)$.
• The model predicts a hierarchical charged lepton mass matrix with $m_e : m_\mu : m_\tau \simeq 1 : 12 : 12$ at $\tau = i$, consistent with observed lepton masses.
• Lepton flavor violation processes such as $\mu \to e\gamma$ are suppressed due to the $A_4$ symmetry and modular form structure, with branching ratios below current experimental limits.
• The muon anomalous magnetic moment receives a loop correction consistent with the current experimental anomaly, with a predicted shift $\Delta a_\mu \simeq (2.5 \sim 3.5) \times 10^{-10}$ depending on the fixed point.