[Paper Review] Lepton Mixing and Cancellation of the Dirac Mass Hierarchy in SO(10) GUTs with Flavor Symmetries T7 and Sigma(81)

Claudia Hagedorn, Michael A. Schmidt+1 more|Max Planck Institute for Plasma Physics|Nov 18, 2008

Particle physics theoretical and experimental studies3 citations

TL;DR

This paper proposes a mechanism to cancel the hierarchical structure of the Dirac mass matrix in SO(10) Grand Unified Theories using additional SO(10) singlet fermions and discrete flavor symmetries $T_7$ and $\Sigma(81)$. By balancing the hierarchy in the Dirac mass $m_D$ with an opposite hierarchy in the $M_{NS}$ matrix, the effective neutrino mass matrix becomes non-hierarchical, enabling natural realization of tri-bimaximal mixing and small $\theta_{13}$, while preserving quark mass hierarchies.

ABSTRACT

In SO(10) grand unified theories (GUTs) the hierarchy which is present in the Dirac mass term of the neutrinos is generically as strong as the one in the up-type quark mass term. We propose a mechanism to partially or completely cancel this hierarchy in the light neutrino mass matrix in the seesaw context. The two main ingredients of the cancellation mechanism are the existence of three fermionic gauge singlets and of a discrete flavor symmetry G_f which is broken at a higher scale than SO(10). Two realizations of the cancellation mechanism are presented. The realization based on the Frobenius group T7 = Z7 x Z3 leads to a partial cancellation of the hierarchy and relates maximal 2-3 lepton mixing with the geometric hierarchy of the up-quark masses. In the realization with the group Sigma(81) the cancellation is complete and tri-bimaximal lepton mixing is reproduced at the lowest order. In both cases, to fully accommodate the leptonic data we take into account additional effects such as effects of higher-dimensional operators involving more than one flavon. The heavy neutral fermion mass spectra are considered. For both realizations we analyze the flavon potential at the renormalizable level as well as ways to generate the Cabibbo angle.

Motivation & Objective

• To resolve the conflict between the strong hierarchy in charged fermion masses and the mild hierarchy in neutrino masses in SO(10) GUTs.
• To explain the observed lepton mixing patterns—particularly tri-bimaximal mixing (TBM) and large $\theta_{12}$—within a unified GUT framework.
• To decouple neutrino mass and mixing parameters from the up-quark mass hierarchy by introducing additional SO(10) singlet fermions and flavor symmetries.
• To maintain diagonal quark mass matrices at leading order while generating the Cabibbo angle via higher-dimensional operators.

Proposed method

• Introduce additional SO(10) singlet fermions $S_i$ that mix only with neutrinos, leading to a three-component neutral fermion mass matrix involving $\nu_L$, $N$, and $S$.
• Use the double seesaw (DS) and linear seesaw (LS) mechanisms to derive the effective light neutrino mass matrix $m_\nu \approx m_\nu^{DS} + m_\nu^{LS}$.
• Implement the cancellation mechanism by ensuring the product $F = m_D M_{NS}^{-T}$ has $\mathcal{O}(1)$ entries, thereby suppressing the Dirac mass hierarchy in the effective $m_\nu$.
• Realize the cancellation using discrete flavor symmetries $T_7$ and $\Sigma(81)$, which constrain the vacuum alignment and coupling structures of flavon fields.
• Include higher-dimensional operators involving multiple flavon fields to generate corrections, particularly for $\theta_{12}$ and the Cabibbo angle.
• Analyze the flavon potential at the renormalizable level to ensure consistent vacuum alignment and symmetry breaking.

Experimental results

Research questions

• RQ1 Can the strong hierarchy in the Dirac mass matrix $m_D$ be canceled in SO(10) GUTs while preserving the observed neutrino mass and mixing patterns?
• RQ2 How can the $T_7$ and $\Sigma(81)$ flavor symmetries be used to achieve partial or complete cancellation of the Dirac mass hierarchy?
• RQ3 Can the resulting neutrino mass matrix reproduce tri-bimaximal mixing and small $\theta_{13}$ without fine-tuning?
• RQ4 How can the Cabibbo angle be generated in this framework without disrupting the neutrino mixing structure?
• RQ5 What is the mass spectrum of the heavy neutral fermions (including $N$ and $S$) in the model?

Key findings

• In the $T_7$ model, partial cancellation of the Dirac mass hierarchy leads to maximal atmospheric mixing and a small $\theta_{13} \sim \mathcal{O}(0.05)$, consistent with observations.
• The $T_7$ model requires an additional Higgs field to generate the large solar mixing angle $\theta_{12}$, which contributes only to the linear seesaw term and does not disrupt the DS cancellation.
• In the $\Sigma(81)$ model, complete cancellation of the Dirac mass hierarchy results in a neutrino mass matrix compatible with tri-bimaximal mixing, though the atmospheric mass squared difference vanishes at leading order.
• The inclusion of higher-dimensional operators in the $\Sigma(81)$ model restores the atmospheric mass splitting, resolving the vanishing $\Delta m_{32}^2$ issue.
• The quark sector maintains diagonal mass matrices at leading order, with the Cabibbo angle generated via higher-dimensional operators involving $\underline{\mathbf{16}}_H$ fields.
• The heavy neutral fermion spectrum includes states with masses near the GUT scale and the Planck scale, depending on the $M_{NS}$ and $M_{SS}$ matrices.