Probing $0νββ$ and $μ\to eγ$ via Fully Determined Dirac Mass Terms in LRSM with Double Seesaw

This paper investigates the phenomenological implications of a Left-Right Symmetric Model extended with a double type-I seesaw mechanism, demonstrating how fully determined Dirac mass textures can enhance right-handed neutrino contributions to neutrinoless double beta decay and the charged lepton flavor violation process $\mu \to e\gamma$ within parameter spaces accessible to current and future experiments.

The Gist

Imagine the universe as a giant, complex machine. For decades, physicists have been trying to figure out how a specific part of this machine—the neutrino—works. Neutrinos are ghostly, tiny particles that zip through everything, including you, without leaving a trace. The Standard Model (the current "instruction manual" for physics) says these ghosts should have no weight. But experiments have proven they do have a tiny bit of mass. This is a glitch in the manual, suggesting there are hidden gears and levers we haven't seen yet.

This paper is like a team of mechanics (the authors) proposing a new blueprint to fix the manual. They are testing a specific theory called the Left-Right Symmetric Model (LRSM) with a "Double Seesaw" mechanism.

Here is a breakdown of their work using everyday analogies:

1. The "Double Seesaw" Mechanism

Imagine a playground seesaw. Usually, if you put a heavy kid on one end, the light kid on the other flies up high. In physics, this explains why neutrinos are so light: they are "light kids" being balanced by "heavy kids" (heavy, invisible particles) on the other side.

The authors propose a Double Seesaw. Imagine a seesaw sitting on top of another seesaw.

• The First Seesaw: Heavy, invisible particles (called "sterile neutrinos") push down on a second set of heavy particles (called "right-handed neutrinos").
• The Second Seesaw: These right-handed particles then push down on the tiny, visible neutrinos we know.
• The Result: Because there are two layers of heavy weights, the tiny neutrinos end up incredibly light, which matches what we observe.

2. The Two Blueprints (Case I and Case II)

To make their math work, the team had to decide how the "gears" (masses) of these invisible particles connect. They tested two different designs:

• Case I (The "Uniform" Design): They assumed the connections between the particles are perfectly symmetrical, like a set of identical gears. It's a simple, clean starting point, like assuming all the wheels on a car are exactly the same size.
• Case II (The "Custom" Design): They didn't just guess; they built the gears based on the specific rules of their machine. This design is more complex and "fully determined" by the theory itself. It's like building a custom engine where every bolt is placed according to a strict, pre-written recipe. This makes the theory very predictive—it leaves less room for guessing.

3. The Two Tests: The "Flash" and the "Double-Click"

The team wanted to know: "If our blueprint is right, what strange things should we see in experiments?" They focused on two specific events:

• The "Flash" (µ → eγ): Imagine a muon (a heavy cousin of an electron) suddenly deciding to turn into an electron and flash a photon (light) in the process. In our current manual, this is so rare it's practically impossible. But in the authors' new blueprint, the heavy invisible particles act like a shortcut, making this "flash" happen much more often. They calculated exactly how often this should happen based on their two designs.
• The "Double-Click" (Neutrinoless Double Beta Decay): Imagine two atoms in a nucleus trying to change their identity. Normally, they spit out two electrons and two invisible neutrinos to balance the books. But in the authors' theory, the invisible neutrinos cancel each other out inside the machine, so the atoms only spit out the two electrons. This is a "double-click" with no neutrinos. If we hear this click, it proves the neutrinos are their own anti-particles (like a coin that is heads on both sides).

4. The Findings: What the Team Discovered

The authors ran simulations to see if their blueprints could explain these events without breaking the rules of the universe.

• The "Flash" Results:

  • In Case I (Uniform), they found that if the heavy particles are very massive (thousands of times heavier than a proton), the "flash" could happen often enough to be seen by upcoming experiments like MEG-II.
  • In Case II (Custom), the results depended heavily on how the heavy particles were arranged (their "hierarchy"). They found specific arrangements where the "flash" would be visible, but only if the particles were heavy enough and arranged in a specific way. Interestingly, if all the heavy particles were the exact same weight, the "flash" would disappear entirely (a phenomenon called GIM suppression), making it a great test to rule out that specific scenario.

• The "Double-Click" Results:

  • They checked if their theory would make the "double-click" happen fast enough to be detected by experiments like LEGEND-200 or KamLAND-Zen.
  • They found that in the regions where the "flash" is likely to be seen, the "double-click" is also boosted, but often not enough to be seen immediately unless the heavy particles are very specific.
  • However, in a "sweet spot" where the heavy particles are lighter (around 300 GeV), the "double-click" rate gets a massive boost, potentially making it detectable soon.

5. The Bottom Line

The paper concludes that their "Double Seesaw" blueprint is a strong candidate for explaining the universe's mysteries.

• It offers a way to see new physics in the near future.
• Case II is particularly exciting because it doesn't rely on random guesses; the theory itself dictates the numbers, making it easier to prove or disprove.
• If future experiments (like MEG-II or LEGEND) see these "flashes" or "clicks," it would be a massive victory for this specific type of Left-Right Symmetric Model. If they don't, the team has narrowed down exactly where the theory fails, helping physicists refine the manual further.

In short, the authors built a detailed map of a hidden world of heavy particles and showed us exactly where to look to find them, using two different styles of mapping to ensure they didn't miss anything.

Technical Summary

Technical Summary: Probing $0\nu\beta\beta$ and $\mu \to e\gamma$ via Fully Determined Dirac Mass Terms in LRSM with Double Seesaw

Problem Statement
The Standard Model (SM) fails to explain the origin of non-zero neutrino masses and the nature of lepton number violation (LNV). While neutrino oscillations confirm lepton flavor violation (LFV) in the neutral sector, charged lepton flavor violation (cLFV) and neutrinoless double beta decay ($0\nu\beta\beta$) remain unobserved, placing stringent constraints on Beyond Standard Model (BSM) scenarios. Minimal extensions of the SM, such as the Type-I seesaw, often require significant fine-tuning to generate small active-sterile mixing while keeping light neutrino masses small, or they predict unobservably small cLFV rates due to the Glashow-Iliopoulos-Maiani (GIM) mechanism.

This paper addresses these challenges by investigating a Left-Right Symmetric Model (LRSM) extended with three generations of sterile neutrinos to realize a double type-I seesaw mechanism. This framework naturally generates small light neutrino masses without unnatural fine-tuning and allows for large Majorana masses for right-handed neutrinos (RHNs) at the TeV scale. The authors aim to explore the phenomenological viability of this setup by analyzing the interplay between $0\nu\beta\beta$ decay and the cLFV process $\mu \to e\gamma$, specifically focusing on how different assumptions regarding the Dirac mass matrix ($M_D$) affect predictions.

Methodology
The study employs a minimal LRSM gauge group $SU(3)_C \times SU(2)_L \times SU(2)_R \times U(1)_{B-L}$ with a scalar sector consisting of a bidoublet $\Phi$ and doublets $H_{L,R}$ (no triplets). The model incorporates a double seesaw mechanism involving light neutrinos ($\nu_L$), right-handed neutrinos ($N_R$), and sterile singlets ($S_L$).

The analysis is divided into two distinct cases for the Dirac mass structure:

1. Case I (Symmetry-Motivated): The Dirac mass matrices $M_D$ and $M_{RS}$ are assumed to be proportional to the identity matrix ($M_D \propto \mathbb{I}$), motivated by specific flavor symmetries. This leads to a "screening" effect where the Dirac structures cancel, simplifying the relations between light, heavy, and sterile neutrino masses.
2. Case II (Model-Determined): The Dirac mass matrices are fully determined by the model dynamics, specifically utilizing charge conjugation ($C$) as the discrete LR symmetry and the "screening condition" (cancellation of Dirac structures in the light neutrino mass matrix). In this scenario, $M_D$ and $M_{RS}$ are derived explicitly from low-energy oscillation parameters and the heavy neutrino mass spectrum, leaving no arbitrary phases or free parameters in the flavor sector.

The authors calculate the branching ratio for $\mu \to e\gamma$ (mediated by one-loop diagrams involving $W_R$ and heavy neutrinos) and the effective Majorana mass ($m_{\beta\beta}$) for $0\nu\beta\beta$ (mediated by light, heavy, and sterile neutrino exchange). They scan the parameter space for the heaviest RHN mass ($m_{N}$), the lightest active neutrino mass ($m_{lightest}$), and various RHN mass hierarchies, while fixing benchmark values for the right-handed gauge boson mass ($M_{W_R} = 7$ TeV, with a discussion on $4.8$ TeV) and the $W_L-W_R$ mixing angle ($\xi$).

Key Contributions

• First cLFV Study in this Specific Setup: The authors claim this is the first comprehensive study of cLFV (specifically $\mu \to e\gamma$) within a TeV-scale LRSM employing a double seesaw mechanism with a bidoublet-plus-doublets scalar sector.
• Comparative Analysis of Dirac Textures: The paper systematically contrasts a symmetry-motivated limit (Case I) with a fully model-determined texture (Case II). Case II is highlighted for its predictive power, as the flavor structure is fixed by low-energy data and symmetry, rather than being an arbitrary input.
• Identification of Favorable Hierarchies: For Case II, the authors perform a term-wise decomposition of the effective Majorana mass to identify specific RHN mass hierarchies that maximize contributions to $0\nu\beta\beta$ and $\mu \to e\gamma$. They find that for Normal Ordering (NO), the hierarchy $m_{N3} < m_{N2} < m_{N1}$ (Case $m_{NF}$) is preferred, while for Inverted Ordering (IO), $m_{N1} < m_{N2} < m_{N3}$ (Case $m_{NA}$) is optimal.
• Correlation between $0\nu\beta\beta$ and $\mu \to e\gamma$: The study maps the parameter regions where both processes are simultaneously accessible to current and future experiments (MEG-II, LEGEND-200/1000, KamLAND-Zen).

Results

• Case I (Symmetry-Motivated):
  • $\mu \to e\gamma$: For Normal Ordering (NO), the branching ratio enters the MEG-II sensitivity region only for large RHN masses ($m_{N1} \gtrsim 4210$ GeV) and a Dirac CP phase $\delta = 0$. For Inverted Ordering (IO), the branching ratio is largely independent of $\delta$ but is constrained by existing MEG limits for $m_{N3} \gtrsim 3530$ GeV.
  • $0\nu\beta\beta$: In the cLFV-relevant region, non-standard contributions (from heavy and sterile neutrinos) dominate over the standard light neutrino exchange for small lightest neutrino masses ($m_{lightest} \lesssim 10^{-3}$ eV). However, for IO, no significant enhancement over the standard mechanism is observed within the cLFV-allowed region.
• Case II (Model-Determined):
  • $\mu \to e\gamma$: The predictions depend heavily on the RHN mass hierarchy. For the favored hierarchies ($m_{NF}$ for NO and $m_{NA}$ for IO), the branching ratio can reach MEG-II sensitivity for $m_{N} \gtrsim 4000-4600$ GeV. In the degenerate RHN mass case ($m_{NG}$), the RHN contribution vanishes due to GIM suppression, leaving only the sterile neutrino contribution, which is only viable for NO.
  • $0\nu\beta\beta$: In the "favorable" parameter space (smaller RHN masses, e.g., 300 GeV), Case II predicts significant enhancements in $0\nu\beta\beta$ rates compared to Case I, particularly for NO. However, in the cLFV-relevant region (where $m_N \sim 5.5$ TeV to satisfy $\mu \to e\gamma$ constraints), the predictions for Case II effectively converge with Case I. In this high-mass regime, the heaviest RHN dominates the contribution, suppressing the sensitivity to the specific Dirac texture details.
• Experimental Constraints: The analysis shows that for both cases, the Inverted Ordering scenario is increasingly disfavored by cosmological bounds on the sum of neutrino masses ($\sum m_\nu$) when combined with the requirement to satisfy $0\nu\beta\beta$ limits.

Significance and Claims
The paper claims to provide "optimistic benchmarks" for future searches targeting right-handed current-mediated neutrino interactions. Its primary significance lies in demonstrating that a TeV-scale LRSM with a double seesaw mechanism can naturally accommodate observable signals in both $0\nu\beta\beta$ and $\mu \to e\gamma$ without fine-tuning.

The authors emphasize that Case II offers a distinct advantage: it establishes direct, testable correlations between low-energy oscillation parameters and intensity-frontier observables without arbitrary flavor parameters. While the two cases yield different predictions in the low-mass (favorable) regime, the paper concludes that in the phenomenologically relevant high-mass regime (constrained by $\mu \to e\gamma$), the predictions converge. This convergence underscores the robustness of the parameter space constraints, suggesting that the heaviest RHN mass is the decisive parameter shaping $0\nu\beta\beta$ phenomenology in the cLFV-relevant region, regardless of the specific Dirac mass texture.

The work serves as a comprehensive guide for interpreting upcoming data from MEG-II and next-generation $0\nu\beta\beta$ experiments (LEGEND-1000) within the context of double-seesaw LRSMs, highlighting the complementarity of these probes in testing the origin of neutrino mass and lepton number violation.