[Paper Review] Tri-Bimaximal Neutrino Mixing, A4 and the Modular Symmetry

Guido Altarelli, Ferruccio Feruglio|arXiv (Cornell University)|Dec 8, 2005

Neutrino Physics Research4 citations

TL;DR

This paper presents a 4-dimensional supersymmetric (SUSY) realization of an A₄ flavor symmetry model that naturally produces tri-bimaximal neutrino mixing, with small corrections from higher-dimensional operators. It shows that vacuum alignment is robust under SUSY, and proposes a novel link between A₄ and the modular group, where the charged lepton mass basis coincides with the symmetry basis. The model predicts a diagonal CKM matrix in the quark sector, requiring new A₄-breaking sources to match observations.

ABSTRACT

We formulate and discuss a 4-dimensional SUSY version of an A4 model for tri-bimaximal neutrino mixing which is completely natural. We also study the next-to-the-leading corrections and show that they are small, once the ratios of A4 breaking VEVs to the cutoff are fixed in a specified interval. We also point out an interesting way of presenting the A4 group starting from the modular group. In this approach, which could be interesting in itself as an indication on a possible origin of A4, the lagrangian basis where the symmetry is formulated coincides with the basis where the charged leptons are diagonal. If the same classification structure in A4 is extended from leptons to quarks, the CKM matrix coincides with the unit matrix in leading order and a study of non leading corrections shows that the departures from unity of the CKM matrix are far too small to accomodate the observed mixing angles.

Motivation & Objective

• To construct a 4D SUSY version of an A₄ model that naturally realizes tri-bimaximal neutrino mixing without relying on extra dimensions.
• To analyze the pattern of next-to-leading-order corrections in the neutrino and charged lepton sectors, showing they remain small under specific parameter constraints.
• To explore the dynamical origin of A₄ by relating it to the modular group, where the Lagrangian basis aligns with the charged lepton diagonal basis.
• To investigate the implications of extending the A₄ classification to quarks, revealing a diagonal CKM matrix in leading order.

Proposed method

• Formulate a 4D SUSY model based on A₄ flavor symmetry with flavon fields φS and φT responsible for neutrino and charged lepton masses.
• Use a presentation of A₄ via generators S and T satisfying S² = (ST)³ = T³ = 1, enabling a direct link to the modular group.
• Implement vacuum alignment via F- and D-term minimization in the SUSY framework, ensuring the required VEV directions for φS and φT.
• Perform a perturbative expansion in 1/Λ to compute next-to-leading-order corrections to vacuum expectation values.
• Derive explicit expressions for corrections to φS and φT VEVs in terms of coupling constants and soft-breaking parameters.
• Extend the A₄ classification to quarks and analyze the resulting CKM matrix, showing it is diagonal at leading order.

Experimental results

Research questions

• RQ1 Can tri-bimaximal neutrino mixing be achieved in a 4-dimensional SUSY model with A₄ flavor symmetry, without requiring extra dimensions?
• RQ2 How small are the next-to-leading-order corrections to the vacuum alignment in the SUSY A₄ model, and under what parameter conditions are they suppressed?
• RQ3 Can the A₄ group be naturally embedded into the modular group, and does this provide a dynamical origin for A₄ symmetry?
• RQ4 Why does a direct extension of the A₄ classification to quarks lead to a diagonal CKM matrix, and what new sources of A₄ breaking are needed to generate observed quark mixing?
• RQ5 What is the role of the modular group in connecting A₄ to string-theoretic structures, and how does this affect the physical basis of the Lagrangian?

Key findings

• The 4D SUSY A₄ model successfully reproduces tri-bimaximal neutrino mixing with small corrections, provided the ratios of A₄-breaking VEVs to the cutoff scale lie within a specified interval.
• Next-to-leading-order corrections to the vacuum alignment are suppressed and remain small, as shown by explicit one-loop level calculations in the SUSY framework.
• The A₄ group can be generated from the modular group via generators S and T, with the Lagrangian basis coinciding with the charged lepton diagonal basis, simplifying the model's structure.
• Extending the A₄ classification to quarks leads to a diagonal CKM matrix at leading order, with higher-order corrections insufficient to generate the observed mixing angles.
• The model requires new sources of A₄ breaking in the quark sector to account for the non-zero CKM mixing angles, indicating a need for sector-specific flavor symmetry breaking.