Cobimaximal mixing pattern from a $\Delta(27)$ inverse seesaw model

This paper proposes a $\Delta(27)$-based inverse seesaw model that successfully explains neutrino mass hierarchies and cobimaximal mixing patterns while accounting for the observed baryon asymmetry of the Universe via leptogenesis in the normal mass hierarchy scenario.

The Gist

Imagine the universe as a giant, complex orchestra. For decades, physicists have been trying to write the sheet music for this orchestra, specifically focusing on a group of tiny, ghostly musicians called neutrinos. These particles are everywhere, but they are incredibly hard to catch and even harder to understand because they have almost no mass and they "mix" (change identities) as they travel.

This paper proposes a new, elegant piece of sheet music to explain how these neutrinos get their mass and why they mix the way they do. Here is the breakdown of their idea, using simple analogies.

1. The Problem: The "Flavor" Mystery

In the Standard Model (our current best theory of physics), there is a puzzle.

• Quarks (the building blocks of protons and neutrons) are like a choir where everyone sings slightly different notes, but they stay mostly in their own lanes. Their mixing is small.
• Leptons (which include electrons and neutrinos) are like a jazz band where the musicians are constantly swapping seats and changing instruments. Two of their mixing angles are huge, and one is small.

Why is the neutrino "jazz" so different from the quark "choir"? The authors suggest that there is a hidden conductor or a set of rules (symmetries) that we haven't fully seen yet.

2. The Solution: A New Rulebook (The $\Delta(27)$ Symmetry)

The authors introduce a new set of rules based on a mathematical group called $\Delta(27)$.

• The Analogy: Imagine a dance floor with 27 specific spots. The rules of $\Delta(27)$ dictate exactly how the dancers (particles) can move from one spot to another.
• The Twist: To make the dance work perfectly, they add a few extra dancers (new particles) and a few extra dance instructors (new scalar fields).
• The Goal: They want to achieve a specific dance pattern called "Cobimaximal Mixing." Think of this as a perfect, balanced choreography where the neutrinos mix in a very specific, predictable way that matches what we see in experiments.

3. How Neutrinos Get Their Tiny Mass: The "Inverse Seesaw"

We know neutrinos have mass, but it's incredibly tiny—like a feather compared to an elephant.

• The Classic Seesaw: Usually, physicists imagine a seesaw where a heavy kid on one side pushes a light kid (the neutrino) up very high. The heavier the kid, the lighter the neutrino.
• The Inverse Seesaw: The authors use a clever variation. Imagine a seesaw where the "heavy" side is actually a pair of twins holding hands very tightly. Because they are so tightly linked, they act almost like a single unit, but with a tiny "wiggle room" (a small breaking of symmetry). This wiggle room allows the neutrino to have a tiny, non-zero mass without needing the "heavy kid" to be impossibly massive. It's a more natural way to get those feather-light masses.

4. The Secret Sauce: Breaking the Rules

For the dance to look right, the perfect symmetry has to be broken in a very specific way.

• The Analogy: Imagine a perfectly round table where everyone is sitting equidistant. To get the right pattern, some people have to stand up and move to specific chairs.
• The Mechanism: The authors use a series of "vacuum expectation values" (VEVs). Think of these as the particles "settling down" into specific positions. They use a hierarchy of these positions (some are big, some are small) to explain why the electron is light, the muon is heavier, and the tau is heaviest. It's like a staircase where each step is a different size, creating the mass differences we observe.

5. The Big Payoff: Explaining the Universe's Imbalance

One of the biggest mysteries in cosmology is: Why is there more matter than antimatter? (If they were created equally, they would have annihilated each other, and we wouldn't be here).

• Leptogenesis: The paper shows that their model can explain this. As the heavy "twin" neutrinos decayed in the early universe, they created a slight imbalance between matter and antimatter.
• The Catch: This only works if the neutrinos follow the Normal Ordering (where the lightest neutrino is the lightest, and the heaviest is the heaviest). If they followed the "Inverted Ordering" (a different arrangement), the math doesn't add up, and the universe would have been empty.
• The Result: Their model successfully reproduces the exact amount of matter we see in the universe today, but only for the Normal Ordering scenario.

6. The Verdict: A Winning Model

The authors ran the numbers and compared their "sheet music" against real-world data:

• Neutrino Mixing: It fits the observed angles perfectly.
• Neutrinoless Double Beta Decay: This is a rare experiment looking for proof that neutrinos are their own antiparticles. Their model predicts a signal that is currently allowed by experiments (but only for the Normal Ordering).
• Cosmology: The total mass of neutrinos predicted by their model fits within the limits set by the Planck satellite (which maps the cosmic microwave background).

Summary

In short, this paper proposes a symmetry-based dance floor ($\Delta(27)$) with a special seesaw mechanism to explain why neutrinos are so light and mix so wildly. It successfully explains the mass of the electron, the mixing of neutrinos, and the existence of our matter-filled universe—but it strongly suggests that the neutrinos must be arranged in a Normal Hierarchy (lightest to heaviest) for everything to work. It's a tidy, predictive package that solves several puzzles at once.

Technical Summary

Here is a detailed technical summary of the paper "Cobimaximal mixing pattern from a $\Delta(27)$ inverse seesaw model" by A. E. Cárcamo Hernández, I. de Medeiros Varzielas, and N. A. Pérez-Julve.

1. Problem Statement

The Standard Model (SM) successfully describes particle physics but fails to explain several phenomena, most notably:

• Neutrino Masses: The origin of the tiny masses of active neutrinos.
• Flavor Problem: The hierarchical structure of fermion masses and the distinct mixing patterns between quarks (small mixing angles) and leptons (two large, one small mixing angle).
• Baryon Asymmetry: The observed matter-antimatter asymmetry in the Universe.

While the SM accommodates neutrino oscillations via non-zero masses, it does not predict the specific mixing angles or the CP-violating phase required to match experimental data. Specifically, the paper aims to explain the cobimaximal mixing pattern (where the atmospheric mixing angle $\theta_{23}$ is maximal or close to maximal, and the Dirac CP phase $\delta_{CP}$ is close to $\pm \pi/2$) within a theoretically consistent framework that also generates neutrino masses and explains the baryon asymmetry.

2. Methodology and Model Construction

The authors propose a supersymmetric (SUSY) low-scale inverse seesaw model extended by specific discrete symmetries.

A. Symmetry Group

The model extends the SM gauge symmetry with a family symmetry group:
$$G = SU(3)_C \times SU(2)_L \times U(1)_Y \times \Delta(27) \times Z_2 \times Z_3 \times Z_9$$

• $\Delta(27)$: A non-Abelian discrete group with triplet and anti-triplet representations, crucial for generating specific mixing patterns.
• $Z_3$: Separates scalar triplets involved in charged lepton interactions from those generating Lepton Number Violating (LNV) terms.
• $Z_9$: Responsible for generating the hierarchical mass pattern of charged leptons (specifically the electron mass suppression).
• $Z_2$: Prevents unwanted dimension-5 operators that would require unnatural fine-tuning for the electron mass.

B. Particle Content

• Fermions: Standard Model leptons ($l_L, l_R$) plus six heavy right-handed Majorana neutrinos ($\nu_R, N_R$) which are singlets under the SM gauge group.
• Scalars: The MSSM Higgs doublets ($H_u, H_d$) plus four gauge-singlet scalars ($\sigma, \rho, \phi, \eta$) transforming under the discrete groups.

C. Mechanism

• Inverse Seesaw: The light neutrino masses are generated via the inverse seesaw mechanism, involving the mass matrix structure:
  $$M_\nu \approx \begin{pmatrix} 0 & m_D & 0 \\ m_D^T & 0 & M \\ 0 & M^T & \mu \end{pmatrix}$$
  where $\mu$ is a small LNV parameter. The light neutrino mass is approximately $m_\nu \approx m_D M^{-1} \mu (M^T)^{-1} m_D^T$.
• Symmetry Breaking: The discrete symmetries are broken at an intermediate scale $\Lambda_{int}$ via specific Vacuum Expectation Value (VEV) alignments:
  • $\langle \eta \rangle \propto (1, 0, 0)$
  • $\langle \phi \rangle \propto (1, \omega, \omega^2)$ where $\omega = e^{2\pi i/3}$.
  • These alignments are critical for enforcing the cobimaximal mixing pattern.

3. Key Contributions

1. Predictive Cobimaximal Mixing: The model successfully derives the cobimaximal mixing pattern (large $\theta_{23}$, $\delta_{CP} \approx \pm \pi/2$) directly from the $\Delta(27)$ symmetry breaking pattern, without ad-hoc assumptions.
2. Charged Lepton Hierarchy: The model naturally reproduces the charged lepton mass hierarchy ($m_e \ll m_\mu \ll m_\tau$) through the spontaneous breaking of the $Z_9$ symmetry, utilizing higher-dimensional operators suppressed by powers of a small parameter $\lambda \approx 0.225$.
3. Leptogenesis Viability: The paper analyzes the generation of the baryon asymmetry via leptogenesis from the decay of the lightest pseudo-Dirac sterile neutrinos, accounting for destructive interference effects characteristic of the inverse seesaw.

4. Results

A. Neutrino Oscillation Parameters

The model was tested against global fit data for both Normal Ordering (NO) and Inverted Ordering (IO) of neutrino masses.

• Fit Quality: The model reproduces the observed neutrino mass squared differences ($\Delta m^2_{21}, \Delta m^2_{31}$), mixing angles ($\theta_{12}, \theta_{23}, \theta_{13}$), and the Dirac CP phase $\delta_{CP}$ within the $3\sigma$ experimental ranges for both NO and IO scenarios.
• JUNO Compatibility: The predicted value for $\sin^2 \theta_{12}$ is compatible with recent JUNO experiment projections.

B. Neutrinoless Double Beta Decay ($0\nu\beta\beta$)

The effective Majorana mass $m_{ee}$ was calculated:

• NO Scenario: Predicted range $4.1 \text{ meV} \lesssim m_{ee} \lesssim 540 \text{ meV}$.
• IO Scenario: Predicted range $36 \text{ meV} \lesssim m_{ee} \lesssim 550 \text{ meV}$.
• Constraint: The model is consistent with current KamLAND-Zen limits ($m_{ee} \lesssim 122 \text{ meV}$) only in the Normal Ordering (NO) scenario. The Inverted Ordering scenario is largely excluded by current limits in this specific model configuration.

C. Baryon Asymmetry of the Universe (BAU)

The authors analyzed leptogenesis considering the interference between the two pseudo-Dirac states ($N^\pm$).

• Result: The model successfully reproduces the observed baryon asymmetry ($Y_{\Delta B} \approx 0.87 \times 10^{-10}$) only for the Normal Ordering (NO) scenario.
• Condition: This success requires the trace of the LNV submatrix $\mu$ to satisfy $10^{-3} \text{ eV}^2 \lesssim \text{Tr}[\mu\mu^\dagger] \lesssim 10^{-1} \text{ eV}^2$.
• IO Exclusion: The Inverted Ordering scenario fails to generate the correct baryon asymmetry within the allowed parameter space.

5. Significance and Conclusion

This paper presents a minimal, predictive supersymmetric model that unifies the explanation of neutrino masses, mixing patterns, and the matter-antimatter asymmetry of the Universe.

• Theoretical Achievement: It demonstrates that the $\Delta(27)$ family symmetry, combined with auxiliary cyclic groups, can naturally enforce the cobimaximal mixing pattern and the charged lepton mass hierarchy.
• Phenomenological Impact: The model makes a strong prediction: Nature likely favors the Normal Ordering of neutrino masses. Both the neutrinoless double beta decay constraints and the successful generation of the baryon asymmetry via leptogenesis are only satisfied in the NO scenario within this framework.
• Testability: The model predicts specific ranges for the effective Majorana mass ($m_{ee}$) and the sum of neutrino masses ($\sum m_\nu$), which are within the reach of next-generation experiments like KATRIN, LEGEND, and cosmological surveys (DESI, Planck).

In summary, the work provides a robust theoretical framework where the observed flavor structure and cosmological asymmetry are direct consequences of discrete symmetry breaking, strongly favoring the Normal Ordering of neutrino masses.