Neutrino mass and mixing, resonant leptogenesis and charged lepton flavor violation in a minimal inverse seesaw model with $S_4$ symmetry

This paper proposes a minimal inverse seesaw model with $S_4$ symmetry, characterized by only three parameters, which successfully explains neutrino oscillation data, predicts a normal mass ordering with specific CP and mixing angles, and accounts for the observed baryon asymmetry and charged lepton flavor violation within experimental limits.

The Gist

Imagine the universe as a giant, complex orchestra. For a long time, scientists thought they knew the score: the Standard Model of physics. But recently, they noticed a few instruments playing notes that didn't quite fit the sheet music. Specifically, tiny particles called neutrinos seem to have mass, and the universe has way more matter than antimatter. Also, there are hints that particles like muons and taus might be able to "switch places" in forbidden ways.

This paper proposes a new "sheet music" to fix these issues. The authors, V. V. Vien and Mayengbam Kishan Singh, suggest a specific mathematical framework called a Minimal Inverse Seesaw Model enhanced with a symmetry rule called $S_4$.

Here is a breakdown of their proposal using simple analogies:

1. The Puzzle: Why Neutrinos are Weird

In the old story, neutrinos were supposed to be weightless ghosts. But experiments show they have a tiny bit of weight and can change their "flavor" (like a chameleon changing colors) as they travel.

• The Analogy: Imagine you have three identical twins (neutrinos). In the old model, they were all perfectly weightless. In reality, they have tiny, different weights, and they constantly swap identities. The authors built a machine (the model) to explain exactly how heavy they are and how they swap.

2. The Machine: The "Inverse Seesaw" with a Secret Rule

To explain the tiny weights, the authors use a mechanism called the Inverse Seesaw.

• The Analogy: Think of a playground seesaw. Usually, if one side goes up, the other goes down. In this "Inverse" version, the authors set up a system where heavy weights (heavy particles) are balanced in a way that forces the light weights (our neutrinos) to be incredibly tiny.
• The $S_4$ Symmetry: To make the math work without getting messy, they added a "traffic rule" called $S_4$ symmetry.
  • The Analogy: Imagine a dance floor with specific rules about who can hold hands with whom. The $S_4$ rule is like a strict choreographer that says, "Only these specific dancers can pair up." This rule forces the particles to arrange themselves in a very specific, neat pattern, preventing the math from becoming a chaotic mess.

3. The Ingredients: Simplicity is Key

The authors pride themselves on using the fewest possible ingredients.

• The Analogy: Instead of a recipe requiring 50 spices, they claim to make the perfect soup with just three main ingredients: one real number (a simple weight) and two complex numbers (numbers that have a "direction" or angle to them).
• They added a few new "heavy" particles to the mix (like adding heavy anchors to the seesaw), but they kept the number of new rules to a minimum.

4. The Results: What the Model Predicts

When the authors ran their "simulation" (a complex calculation) using real-world data, their model made several specific predictions:

• The Order of Neutrinos: The model predicts that the neutrinos are arranged in a "Normal Hierarchy."
  • The Analogy: Think of three runners. The model says the lightest runner is almost weightless, the middle one is slightly heavier, and the heaviest is significantly heavier. It rules out the idea that the heaviest runner is actually the lightest.
• The "Octant" of the Mix: It predicts that the mixing angle $\theta_{23}$ is in the "higher octant."
  • The Analogy: Imagine a clock face. The model says the hand is pointing past the halfway mark (towards the 6 o'clock position), rather than before it.
• The CP Violation (The "Time Travel" Effect): It predicts a specific value for the "Dirac CP phase," which relates to why the universe prefers matter over antimatter.
  • The Analogy: This is the "twist" in the dance. The model predicts the dancers are turning in a specific direction (a "lower half-plane" of angles), which helps explain why we exist instead of being annihilated by antimatter.
• The Total Weight: The model predicts the sum of all three neutrino masses is about 59 milli-electron-volts.
  • The Analogy: If you put all three neutrinos on a super-sensitive scale, they would weigh about 0.00000000000000000006 grams. This fits perfectly with what astronomers see when looking at the cosmic microwave background (the afterglow of the Big Bang).

5. The "Heavy" Side: Resonant Leptogenesis

The model also explains how the universe got its matter.

• The Analogy: Imagine two heavy twins (heavy neutrinos) who are almost identical in weight, but one is slightly heavier. Because they are so close in weight, they can "resonate" like two tuning forks hitting the same note. This resonance amplifies a tiny difference, creating a huge imbalance between matter and antimatter in the early universe. The authors show their model creates just the right amount of this imbalance to match what we see today.

6. The Safety Check: Forbidden Transitions

Finally, they checked if their model breaks any known laws. One specific law is that a muon (a heavy cousin of an electron) shouldn't turn into an electron and a photon (light) easily.

• The Analogy: It's like checking if a car can drive through a wall. The authors calculated that in their model, the car can drive through the wall, but only so slowly that current detectors (like the MEG II experiment) won't see it yet, but future, more sensitive detectors might. Their model stays within the "speed limits" set by current experiments.

Summary

In short, this paper says: "We found a simple, elegant set of rules (using $S_4$ symmetry) that explains why neutrinos are light, why they mix the way they do, why the universe is made of matter, and why we haven't seen forbidden particle switches yet. It fits all the current data perfectly and gives us a clear target for what future experiments should look for."

Technical Summary

Technical Summary: Neutrino Mass and Mixing, Resonant Leptogenesis, and cLFV in a Minimal Inverse Seesaw Model with $S_4$ Symmetry

Problem Statement
The Standard Model (SM) faces significant challenges regarding neutrino oscillation data, the observed baryon asymmetry of the Universe (BAU), and the potential for charged lepton flavor violation (cLFV). While canonical seesaw mechanisms explain small neutrino masses, they require heavy scales difficult to test experimentally. Low-scale alternatives, such as the inverse seesaw (ISS) mechanism, offer testable predictions at the TeV scale. However, constructing a minimal ISS model that simultaneously accommodates neutrino oscillation parameters, generates the BAU via resonant leptogenesis, and satisfies cLFV constraints often requires complex symmetry structures or numerous free parameters. Previous works, such as Ref. [55], utilized $S_4$ symmetry but required multiple singlet scalars and specific assignments of singlet fermions to different $S_4$ representations, leading to distinct mass matrix structures.

Methodology
The authors propose a minimal extension of the SM incorporating $S_4$ discrete symmetry augmented by Abelian symmetries ($Z_5, Z_3, Z_2$). The particle content includes two right-handed neutrinos ($\nu_R$) and two singlet neutral fermions ($S$), alongside specific singlet scalars ($\phi_l, \phi_\nu, \chi$).

• Symmetry Assignment: Unlike previous studies where singlet fermions $S_1$ and $S_2$ were assigned to different $S_4$ singlets ($1_1$ and $1_2$), this model assigns both $S_1$ and $S_2$ to a single $S_4$ doublet ($2$). Consequently, the mass matrices $M_R$ and $\mu$ are generated by a single $SU(2)_L$ singlet scalar $\chi$ in the trivial singlet $1_1$.
• Mass Generation: The model yields a specific structure for the Dirac mass matrix $M_D$ and off-diagonal Majorana mass matrices $M_R$ and $\mu$. The light neutrino mass matrix is derived via the ISS mechanism ($m_\nu \approx M_D M_R^{-1} \mu M_R^{-1} M_D^T$).
• Analytical Framework: The model reduces the neutrino sector to three parameters: one real mass scale ($m_0$) and two complex dimensionless parameters ($\alpha, \beta$). Analytical expressions are derived for mixing angles, CP phases, and the sum of neutrino masses.
• Numerical Analysis: A $\chi^2$ minimization using the Multinest sampling package is performed against global neutrino oscillation data (Ref. [1]). The analysis scans parameter spaces to fit oscillation data, calculate the baryon asymmetry via Boltzmann equations for resonant leptogenesis, and evaluate cLFV branching ratios (specifically $\mu \to e\gamma$).

Key Contributions

1. Minimal Model Construction: The paper presents a simplified ISS(2,2) framework where the $S_4$ symmetry naturally enforces zero diagonal elements in $M_R$ and $\mu$ without fine-tuning Yukawa couplings, differing structurally from previous $S_4$-based ISS models.
2. Parameter Reduction: The model successfully describes neutrino oscillations using only three effective parameters ($m_0, \alpha, \beta$) in the neutrino sector.
3. Unified Phenomenology: The study simultaneously addresses neutrino mass ordering, the BAU via resonant leptogenesis, and cLFV constraints within a single consistent framework.

Results

• Neutrino Oscillations: The model strongly favors Normal Ordering (NO). The best-fit parameters yield:
  • $\sin^2 \theta_{23} \approx 0.560$, indicating a higher octant for $\theta_{23}$ ($\theta_{23} \approx 48.4^\circ$).
  • A Dirac CP-violating phase $\delta_{CP}$ in the lower half-plane, with a best-fit value of $\approx 339.8^\circ$.
  • The sum of neutrino masses is predicted at $\sum m_\nu \approx 58.98$ meV, consistent with Planck cosmological bounds ($< 72$ meV).
  • The effective Majorana mass for neutrinoless double beta decay is predicted at $m_{\beta\beta} \approx 6.2$ meV, within the range $(5.92 - 7.46)$ meV.
• Resonant Leptogenesis: Successful generation of the observed BAU ($\eta_B \approx 6.12 \times 10^{-10}$) is achieved for a narrow range of the lepton-number-violating parameter $a_\mu \sim (10^{-3} - 10^{-2})$ GeV and heavy neutrino masses $M_R \sim (1 - 3)$ TeV. The analysis indicates that resonant enhancement compensates for washout effects, with CP asymmetries $\epsilon_{max} \sim 10^{-5} - 10^{-3}$.
• Charged Lepton Flavor Violation: The predicted branching ratios for $\mu \to e\gamma$ and $\tau \to \ell\gamma$ processes satisfy current experimental limits from MEG II and Belle. The Yukawa couplings required for successful leptogenesis remain within perturbative limits ($|h| \sim 10^{-5} - 10^{-2}$).
• Inverted Ordering: The model is found to be incompatible with Inverted Ordering (IO), yielding a $\chi^2_{min} > 100$ which falls outside the experimental $3\sigma$ range.

Significance
The paper claims that this minimal $S_4$-symmetric inverse seesaw model provides a robust and testable explanation for current neutrino data while naturally accommodating the matter-antimatter asymmetry of the Universe. By predicting a higher octant for $\theta_{23}$ and a specific lower-half-plane value for $\delta_{CP}$, the model offers clear targets for future experiments like T2K and NO$\nu$A. Furthermore, the prediction of heavy neutrinos at the TeV scale ($M_R \sim 10^4$ GeV) suggests that the mechanism could be directly tested by future colliders, distinguishing it from high-scale canonical seesaw scenarios. The consistency of the model with both cosmological constraints on neutrino mass sums and experimental bounds on cLFV underscores its viability as a minimal extension of the Standard Model.