Lepton mixing and charged lepton flavour violation from inverse seesaw with non-degenerate heavy states

This paper analyzes an inverse seesaw model with non-degenerate pseudo-Dirac heavy states governed by specific discrete flavor symmetries, finding that while current charged lepton flavor violation limits do not strongly constrain the parameter space compatible with observed lepton mixing, upcoming experiments like Mu3E, COMET, and Mu2e are expected to provide significant tests.

The Gist

The Ghostly Dance of Particles: A New Theory of Neutrinos

The Big Mystery
Imagine the Standard Model of physics as a giant, mostly complete puzzle. It explains almost everything we see in the universe, from atoms to stars. But there is one piece missing: Neutrinos.

Neutrinos are like cosmic ghosts. They zip through everything without touching it. For a long time, physicists thought they had no mass. But then, scientists discovered they actually do have mass, and they can change their identity (or "flavor") as they travel—like a chameleon changing colors. The Standard Model puzzle doesn't have a piece that explains how these ghosts get mass or why they change colors.

The Proposed Fix: The "Inverse Seesaw"
To fix the puzzle, the authors of this paper suggest adding new pieces. They use a mechanism called the Inverse Seesaw.

Think of a playground seesaw. Usually, if you put a heavy kid on one side, the light kid on the other side goes flying up. In the "Inverse" version, imagine the heavy kid is actually helping keep the light kid grounded. In physics terms, they propose adding new, very heavy particles (called "sterile neutrinos") that interact with our regular neutrinos. These heavy particles are so massive that they force the regular neutrinos to remain incredibly light.

The New Twist: "Option 3"
Previous versions of this theory (called Option 1 and Option 2) assumed these new heavy particles were identical twins—they all weighed exactly the same.

This paper introduces Option 3. Imagine a family of triplets. In the old theories, the triplets were clones. In this new theory, the triplets are siblings with different heights. They come in pairs, but each pair has a different weight than the others. This "non-degenerate" (different mass) setup changes how the particles behave mathematically.

The Secret Code: Flavor Symmetry
To make sure these particles don't just do whatever they want, the authors use a "Flavor Symmetry." Think of this as a strict dance choreography.

In a dance troupe, everyone knows their specific steps. The authors use complex mathematical patterns (groups like $\Delta(3n^2)$) to dictate how the particles dance together. This ensures the theory fits the data we already have from experiments. It's like a rulebook that says, "Electrons dance with electrons, Muons with Muons, but sometimes they can switch partners."

The Smoking Gun: The Forbidden Switch
The most exciting part of this paper is a prediction about Charged Lepton Flavour Violation (cLFV).

In our everyday world, a red ball stays a red ball. In the Standard Model, a Muon (a heavy cousin of the electron) should stay a Muon. But this theory predicts that, very rarely, a Muon could spontaneously turn into an Electron and a flash of light (a photon).

• Analogy: Imagine you are watching a magician. You expect a red card to stay red. But in this theory, the magician might occasionally turn the red card into a blue card.

What the Experiments Say
The authors ran the numbers to see if this "magic trick" is possible.

1. Current Limits: Right now, our detectors aren't sensitive enough to catch this switch happening. The "magic" is too subtle.
2. Future Hope: However, new experiments coming online soon (like Mu3E, COMET, and Mu2e) are building super-sensitive detectors. They are like high-powered microscopes looking for that red-to-blue card trick.
3. The Prediction: The paper says that if these new experiments find the switch, it would support this specific "Option 3" theory.

The "Noise-Canceling" Effect
One of the coolest findings is something called cancellation.
Imagine two speakers playing sound waves. If they play the exact opposite wave, the sound disappears (noise-canceling headphones).
In this theory, the different heavy particles can sometimes "cancel each other out" regarding the Muon-to-Electron switch. Depending on the exact weights of the heavy particles, the signal might vanish completely or become very weak. This is a unique fingerprint of this theory that helps distinguish it from the other versions (Option 1 and 2).

The Bottom Line
This paper is a blueprint for a specific version of a new physics theory. It says:

1. Neutrinos get their mass from heavy, invisible partners.
2. These partners aren't identical; they have different weights.
3. We might be able to prove this is true by watching Muons turn into Electrons in the next few years.

If the upcoming experiments catch this rare event, it will be a huge step forward in understanding the fundamental rules of the universe. If they don't, this specific version of the theory might need to be rewritten.

Technical Summary

Here is a detailed technical summary of the paper "Lepton mixing and charged lepton flavour violation from inverse seesaw with non-degenerate heavy states" by F. P. Di Meglio and C. Hagedorn.

1. Problem Statement

The Standard Model (SM) cannot explain the observed neutrino masses and lepton mixing pattern (encoded in the PMNS matrix). While various Seesaw mechanisms (Type-I, II, III) exist, they typically predict new particles at mass scales far beyond current experimental reach. The Inverse Seesaw (ISS) mechanism offers a solution by allowing neutrino masses to be small due to small lepton number violation, keeping new particle masses within the reach of current or near-future experiments.

Previous studies of ISS scenarios often assumed degenerate heavy sterile states or specific symmetry breaking patterns. This paper addresses a specific variant of the ISS model (referred to as "Option 3") where the Dirac mass matrix of the gauge singlets carries the non-trivial flavour structure. The authors aim to analyze the phenomenology of this setup, specifically focusing on:

1. The mass spectrum and mixing of light and heavy neutral states.
2. Charged Lepton Flavour Violation (cLFV) processes, such as $\mu \to e\gamma$, $\mu \to 3e$, and $\mu-e$ conversion in nuclei.
3. How this specific variant compares to other ISS options (Option 1 and Option 2) regarding experimental constraints and predictions.

2. Methodology

The authors employ a theoretical framework combining the Inverse Seesaw mechanism with non-abelian discrete flavour symmetries.

• Model Setup: The model extends the SM with 3+3 gauge singlets ($N_i$ and $S_j$). The flavour symmetry group is chosen from the series $\Delta(3n^2)$ or $\Delta(6n^2)$, combined with CP symmetry.
• Symmetry Breaking:
  • Charged Leptons: The mass matrix is diagonal, determined by a residual $Z_3$ symmetry.
  • Neutral States: The symmetry is broken to a residual group $G_\nu = Z_2 \times CP$. Crucially, in "Option 3," the Dirac mass matrix $M_{NS}$ (coupling $N_i$ and $S_j$) is the source of symmetry breaking, while the neutrino Yukawa coupling ($m_D$) and the Majorana mass matrix ($\mu_S$) remain invariant under the full flavour group.
• Analytical Derivation: The authors derive the mass matrices for light and heavy neutrinos. They diagonalize the $9 \times 9$mass matrix to obtain the light neutrino masses and the heavy sterile state spectrum. They derive analytical expressions for the PMNS mixing matrix and the non-unitarity parameter$\eta$, which quantifies mixing between light and heavy states.
• Numerical Scans: The parameter space is scanned numerically. Parameters varied include the Yukawa coupling $y_0$, the lepton number violation scale $\mu_0$, the heavy mass parameters $M_f$, and mixing angles $\theta_N$ and $\theta_S$.
• Constraints: The scans are constrained by:
  • Neutrino oscillation data (mass squared differences and mixing angles at 3$\sigma$ level).
  • Cosmological bounds on the sum of neutrino masses.
  • Current experimental limits on cLFV processes (e.g., MEG, SINDRUM).
  • Projected future limits from experiments like Mu3E, COMET, and Mu2e.

3. Key Contributions

• Non-Degenerate Heavy States: Unlike previous ISS studies (Options 1 and 2) where heavy sterile states were often assumed to be (nearly) degenerate, this work demonstrates that in Option 3, the heavy sterile states form three pseudo-Dirac pairs with generally distinct masses.
• Flavour Structure: The paper explicitly details how the flavour structure is encoded in the Dirac mass matrix $M_{NS}$, leading to specific predictions for lepton mixing angles dependent on the residual symmetry parameters.
• cLFV Cancellation Mechanisms: The authors identify conditions under which the $\mu-e$ conversion rate can undergo exact cancellation. They show that unlike Option 2 (where cancellation is universal and depends only on the nucleus), in Option 3, the cancellation depends on the specific group theory case, parameters, and the lightest neutrino mass.
• Phenomenological Comparison: A comprehensive comparison is made between Option 3 and Options 1 & 2, highlighting distinct experimental signatures that could differentiate these models.

4. Results

• Mass Spectrum: The heavy sterile states form three pseudo-Dirac pairs. Their masses are split by terms of order $\mu_0$. Depending on the lightest neutrino mass ($m_0$), the heavy masses range from 150 GeV to 700 TeV.
• Lepton Mixing: The PMNS mixing matrix is determined by the parameter $\theta_N$. The model accommodates experimental mixing angles at the 3$\sigma$ level or better.
• cLFV Signals:
  • Current Bounds: Current limits on $\mu \to e\gamma$, $\mu \to 3e$, and $\mu-e$ conversion do not strongly constrain the parameter space.
  • Future Sensitivity: Upcoming experiments (Mu3E, COMET, Mu2e) will have a relevant impact. For viable data points, the branching ratio (BR) for $\mu \to e\gamma$ is typically $< 6 \times 10^{-16}$, and $\mu \to 3e$ is suppressed but potentially within reach of Mu3E Phase 2.
  • $\mu-e$ Conversion: The conversion rate (CR) can be significantly suppressed or even vanish (exact cancellation) for specific parameter values. The minimum CR can reach values as low as $10^{-24}$.
  • Tau Decays: cLFV tau decays ($\tau \to \ell\gamma$, etc.) are generally too small ($< 10^{-14}$) to be observed at Belle II.
• Parameter Dependence: The signal strength depends on the angle $\theta_S$. In some cases (e.g., Case 1), the signal is enhanced when $\cos 2\theta_S \approx 0$ and suppressed when $|\cos 2\theta_S| \approx 1$.
• Comparison with Other Options:
  • Option 1: Predicts very suppressed cLFV signals (similar to SM with light neutrinos) and degenerate heavy states.
  • Option 2 & 3: Both predict observable cLFV signals. However, Option 3 allows for non-degenerate heavy masses and case-dependent cancellation in $\mu-e$ conversion, whereas Option 2 has universal cancellation.

5. Significance

This work is significant for the following reasons:

1. Experimental Discrimination: It provides specific predictions that allow experimentalists to distinguish between different realizations of the Inverse Seesaw mechanism. The non-degenerate nature of the heavy states in Option 3 is a key differentiator.
2. Future Experiment Targets: The paper identifies specific regions of the parameter space that will be probed by next-generation cLFV experiments (Mu3E, COMET, Mu2e), guiding the design and interpretation of these searches.
3. Theoretical Consistency: It demonstrates that viable lepton mixing patterns can be achieved in ISS models with non-degenerate heavy states without violating current experimental bounds, expanding the viable landscape of neutrino mass models.
4. Cancellation Phenomena: The discovery that $\mu-e$ conversion cancellation is not universal in this model (unlike in Option 2) implies that a null result in $\mu-e$ conversion experiments does not rule out the model but rather constrains specific combinations of parameters and group theory choices.

In conclusion, the paper establishes that the "Option 3" ISS scenario with non-degenerate heavy states is a phenomenologically viable framework that predicts observable charged lepton flavour violation, offering distinct signatures for future high-precision experiments.