[Paper Review] Lepton and Quark Masses and Mixing in a SUSY Model with Delta(384) and CP

Claudia Hagedorn, Johannes König|arXiv (Cornell University)|Nov 22, 2018

Neutrino Physics Research44 references4 citations

TL;DR

This paper proposes a supersymmetric model based on the flavor symmetry Δ(384) combined with CP invariance to explain fermion masses and mixing patterns in the lepton and quark sectors. By employing a stepwise symmetry breaking mechanism involving residual symmetries and flavon fields, the model successfully predicts tri-bimaximal lepton mixing with a Dirac CP phase near 3π/2 and maximal CP violation in the quark sector, in agreement with experimental data.

ABSTRACT

We construct a supersymmetric model for leptons and quarks with the flavor symmetry Delta(384) and CP. The peculiar features of lepton and quark mixing are accomplished by the stepwise breaking of the flavor and CP symmetry. The correct description of lepton mixing angles requires two steps of symmetry breaking, where tri-bimaximal mixing arises after the first step. In the quark sector the Cabibbo angle theta_C equals sin pi/16 = 0.195 after the first step of symmetry breaking and it is brought into full agreement with experimental data after the second step. The two remaining quark mixing angles are generated after the third step of symmetry breaking. All three leptonic CP phases are predicted, sin delta^l = -0.936, |sin alpha|=|sin beta|=1/sqrt{2}. The amount of CP violation in the quark sector turns out to be maximal at the lowest order and is correctly accounted for, when higher order effects are included. Charged fermion masses are reproduced with the help of operators with different numbers of flavor (and CP) symmetry breaking fields. Light neutrino masses, arising from the type-I seesaw mechanism, can accommodate both mass orderings, normal and inverted. The vacuum alignment of the flavor (and CP) symmetry breaking fields is discussed at leading and at higher order.

Motivation & Objective

• To explain the hierarchical masses and mixing patterns of charged fermions (leptons and quarks) beyond the Standard Model.
• To address the origin of CP violation in both lepton and quark sectors using discrete non-Abelian flavor symmetries and CP symmetry.
• To construct a realistic supersymmetric extension of the MSSM that unifies lepton and quark mixing within a single flavor symmetry framework.
• To predict specific values for lepton mixing angles and CP phases consistent with experimental observations.
• To achieve correct quark mixing angles, including the Cabibbo angle and small θ₁₃ and θ₂₃, via a controlled symmetry breaking sequence.

Proposed method

• Employing a flavor symmetry group G_f = Δ(384) × Z₂^(ext) × Z₃^(ext) × Z₁₆^(ext) to unify lepton and quark sectors in a supersymmetric framework.
• Implementing a three-step symmetry breaking sequence: first breaking Δ(384) to residual Z₃ in charged leptons and Klein group + CP in neutrinos/up quarks, then reducing to Z₂ × CP in neutrinos, and finally breaking to Z₈ in down quarks.
• Using flavon fields (Σ and Σ̄) with specific quantum numbers under Δ(384) and additional U(1) symmetries to generate effective Yukawa operators at non-renormalizable level.
• Introducing heavy right-handed neutrinos and driving fields to control the vacuum alignment of flavons and ensure proper symmetry breaking.
• Constructing renormalizable superpotential terms involving flavons, MSSM fields, and heavy vector-like states to generate the required mass matrices.
• Ensuring CP invariance is preserved in the vacuum of key flavon fields, which controls the size and phase structure of the Jarlskog invariant in the quark sector.

Experimental results

Research questions

• RQ1 Can a unified flavor symmetry Δ(384) combined with CP explain both lepton and quark mixing patterns in a supersymmetric model?
• RQ2 How does a stepwise breaking of Δ(384) and CP symmetry lead to tri-bimaximal lepton mixing and correct quark mixing angles?
• RQ3 What are the predicted values of the leptonic CP phases, and how do they compare with experimental data?
• RQ4 How is maximal CP violation in the quark sector achieved at leading order and corrected by higher-order terms?
• RQ5 What role do flavon vacuum alignments and additional U(1) symmetries play in generating the correct fermion mass hierarchies?

Key findings

• The model predicts a Dirac CP phase δˡ ≈ 3π/2 with sinδˡ ≈ -0.936, consistent with current experimental hints.
• The two Majorana phases satisfy |sinα| = |sinβ| = 1/√2, indicating maximal CP violation in the neutrino sector.
• The Cabibbo angle is predicted as sin(π/16) ≈ 0.195 at leading order, matching the experimental value.
• The quark mixing angles θ₁₃^q and θ₂₃^q vanish at leading order but are generated at higher order via complete breaking in the down quark sector.
• The Jarlskog invariant J_CP^q is dominated by parameters from the down quark sector and achieves maximal CP violation at leading order.
• The top quark mass is generated via a non-renormalizable operator with a modest flavon VEV and Yukawa couplings of order 2–3.