When Does Leptogenesis Survive Lepton Flavor Violation Constraints? High- and Low-Scale Realizations in the Scotogenic Model

This paper demonstrates that while high-scale leptogenesis in the minimal scotogenic model naturally evades lepton flavor violation constraints, low-scale resonant leptogenesis is severely restricted by experimental bounds but remains viable within a narrow parameter window characterized by quasi-degenerate heavy fermions and specific Casas–Ibarra phase alignments.

The Gist

The Big Picture: Solving Two Cosmic Mysteries at Once

Imagine the universe has two major unsolved mysteries:

1. Why do we exist? (Specifically, why is there more matter than antimatter? If they were created equally, they would have annihilated each other, leaving nothing but light.)
2. Why do neutrinos have mass? (These tiny ghost particles were thought to be weightless, but experiments show they have a tiny bit of weight.)

This paper investigates a specific theory called the "Scotogenic Model." Think of this model as a "Swiss Army Knife" of particle physics. It proposes that a single set of new, hidden ingredients (particles) can solve both mysteries simultaneously.

The author, Avinanda Chaudhuri, asks a crucial question: Can this Swiss Army Knife actually work without breaking?

The Cast of Characters

To understand the paper, we need to meet the players:

• The Standard Model: The "known" universe (like the furniture in a room).
• The "Inert" Doublet: A new, invisible particle that doesn't interact with normal matter but helps generate mass. Think of it as a ghost that can pass through walls but leaves a faint echo.
• Heavy Neutrinos: Massive, invisible particles that lived in the early universe. Think of them as giant, heavy anchors dropped into the cosmic ocean.
• The "Flavor" Problem: In particle physics, "flavor" is like a particle's ID card (electron, muon, tau). Sometimes, these cards get swapped. A muon might accidentally turn into an electron. This is called Lepton Flavor Violation (LFV).

The Conflict: The Tightrope Walk

The paper's main drama is a conflict between two goals:

1. Leptogenesis: We need the heavy neutrinos to decay in a specific, "messy" way to create the matter we see today. This requires them to be very active and interact strongly.
2. Safety (LFV Constraints): Nature has a strict rule: We have never seen a muon turn into an electron and a photon (light) in our experiments. If our theory predicts this happens too often, the theory is wrong.

The Analogy:
Imagine you are trying to bake a cake (create the universe's matter).

• To get the cake to rise, you need a lot of yeast (strong interactions).
• However, the kitchen inspector (the MEG experiment) has a strict rule: "No yeast smell allowed in the hallway."
• If you use too much yeast, the smell leaks out, and you get kicked out (the theory is ruled out).
• If you use too little yeast, the cake doesn't rise (no universe).

The paper asks: Is there a "Goldilocks" amount of yeast where the cake rises perfectly, but the smell stays hidden?

The Two Strategies

The author tests two different ways to bake this cake:

1. The High-Scale Strategy (The "Heavy Anchor" Approach)

• How it works: You use extremely heavy neutrinos (trillions of times heavier than a proton).
• The Result: This works naturally. Because the particles are so heavy and far away, their "yeast smell" (flavor violation) doesn't reach the kitchen inspector. The universe gets its matter, and the rules are obeyed.
• Verdict: Safe and Successful. This is the "boring but reliable" solution.

2. The Low-Scale Strategy (The "Resonant" Approach)

• How it works: You use lighter neutrinos (lighter, but still heavy by human standards) that are almost identical twins. When two twins are almost exactly the same weight, they can "resonate" (like two tuning forks vibrating together). This resonance supercharges the process, allowing the universe to form even with lighter particles.
• The Problem: This resonance usually makes the "yeast smell" (flavor violation) explode. The kitchen inspector (MEG) would immediately catch you.
• The Twist: The author found a narrow escape route.
  • Imagine the twins are so perfectly synchronized that their "bad smells" cancel each other out, like noise-canceling headphones.
  • By tuning the "phases" (the internal timing) of these particles just right, the paper shows you can get the resonance boost without triggering the alarm.
• Verdict: Risky but Possible. It only works in a very narrow, specific window, but it does work.

The Key Takeaways

1. Everything is Connected: In this model, the same "ingredients" (Yukawa couplings) that give neutrinos their weight also control how the universe was born and how particles change flavors. You can't fix one without affecting the others.
2. The "Goldilocks" Zone: The paper proves that while high-energy physics is safe, low-energy physics is a tightrope walk. However, there is a tiny, safe strip where the universe can exist, the rules are followed, and the particles are light enough to be potentially found in future experiments.
3. The "Phase Alignment" Trick: The secret to the low-scale success is a mathematical trick called "Casas-Ibarra phase alignment." Think of it as a magic cancellation. The particles are arranged so perfectly that their dangerous side-effects (flavor violations) cancel out, leaving only the good stuff (matter creation).

Why Should You Care?

This isn't just math for math's sake.

• If you are a High-Scale believer: You can relax; the theory is robust.
• If you are a Low-Scale believer: You have a specific target. The paper predicts that if we look hard enough with future experiments (like MEG II), we might see a tiny signal of a muon turning into an electron.

In short: The paper says, "Yes, this elegant theory works. The heavy version is safe and boring. The light version is dangerous but exciting, and if we tune the universe's 'knobs' just right, we can make it work without breaking the laws of physics."

Technical Summary

1. Problem Statement

The paper addresses the tension between two fundamental requirements in the Minimal Scotogenic Model:

1. Leptogenesis: The generation of the observed Baryon Asymmetry of the Universe (BAU) via the decay of heavy right-handed neutrinos (RHNs).
2. Lepton Flavor Violation (LFV): The stringent experimental constraints on processes like $\mu \to e\gamma$, particularly the bound from the MEG experiment ($BR < 4.2 \times 10^{-13}$).

In the Scotogenic model, neutrino masses are generated radiatively at the one-loop level. Crucially, the same Yukawa couplings ($Y_{\alpha i}$) that generate neutrino masses also govern:

• Charged LFV processes (e.g., $\mu \to e\gamma$).
• The CP asymmetry ($\epsilon_1$) required for leptogenesis.

This creates a tight phenomenological correlation: parameters that enhance CP asymmetry for successful baryogenesis often simultaneously enhance LFV rates, potentially violating experimental bounds. The paper investigates whether high-scale hierarchical and low-scale resonant leptogenesis scenarios can survive these constraints within a unified framework.

2. Methodology

The authors employ a comprehensive numerical and analytical approach using the Casas–Ibarra parametrization to reconstruct the Yukawa sector.

• Model Framework: The Minimal Scotogenic Model, extending the Standard Model with three singlet Majorana fermions ($N_i$) and an inert scalar doublet ($\eta$), both odd under a discrete $Z_2$ symmetry.
• Yukawa Reconstruction: The Yukawa matrix $Y$ is reconstructed using the Casas–Ibarra formula:
  $$Y = U_{\text{PMNS}} \sqrt{D_\nu} R \sqrt{\Lambda^{-1}}$$
  where $R$ is a complex orthogonal matrix parameterized by complex angles $\theta_i = x_i + i y_i$. These complex angles are the primary degrees of freedom used to explore the parameter space.
• Scenarios Analyzed:
  1. High-Scale Hierarchical Regime: $M_1 \ll M_2 \ll M_3$ (typically $M_1 \gtrsim 10^9$ GeV). CP asymmetry arises from standard decay asymmetries.
  2. Low-Scale Resonant Regime: $M_2 \simeq M_1$ (typically $M_1 \sim 10^5 - 10^7$ GeV). CP asymmetry is resonantly enhanced by the self-energy contribution when the mass splitting $\Delta = (M_2 - M_1)/M_1$ is comparable to the decay width.
• Numerical Scan: A full scan was performed over parameters including $M_1$, $\lambda_5$ (scalar quartic coupling), mass splitting $\Delta$, and the complex Casas–Ibarra angles.
• Constraints Applied:
  • Neutrino oscillation data (masses and mixing angles).
  • MEG bound on $BR(\mu \to e\gamma)$.
  • Observed Baryon Asymmetry ($Y_B \approx 8.7 \times 10^{-11}$).
  • Perturbativity and vacuum stability.
• Dynamics: The evolution of the baryon asymmetry was calculated using the Boltzmann equations (in the one-flavor approximation) to account for washout effects ($K_1$ parameter).

3. Key Contributions

• Unified Comparison: This is one of the first systematic studies comparing high-scale and low-scale leptogenesis within the same Casas–Ibarra reconstructed Scotogenic framework, directly contrasting their viability under current LFV bounds.
• Mechanism of Decoupling: The paper elucidates how flavor alignment and Casas–Ibarra phase cancellations allow for the effective decoupling of LFV constraints from baryogenesis in the high-scale regime.
• Identification of the "Resonant Window": Contrary to the expectation that low-scale resonant leptogenesis is entirely ruled out by LFV bounds, the authors identify a narrow, non-vanishing parameter space where successful baryogenesis coexists with LFV safety.
• Analytical vs. Numerical Validation: The study provides approximate analytical expressions for CP asymmetry and baryon asymmetry that are rigorously validated against the full numerical scan, offering transparent physical interpretations of the viable regions.

4. Key Results

A. High-Scale Hierarchical Regime ($M_1 \gtrsim 10^{10}$ GeV)

• Viability: Naturally viable.
• Mechanism: Successful baryogenesis occurs in the strong washout regime ($K_1 \gg 1$).
• LFV Survival: The model survives LFV constraints because the large mass scale suppresses the loop functions for $\mu \to e\gamma$. Furthermore, specific alignments of the complex Casas–Ibarra phases allow for the cancellation of flavor-violating amplitudes while maintaining sufficient CP asymmetry.
• Parameter Space: Viable points cluster around $M_1 \sim 10^{11} - 10^{12}$ GeV and intermediate scalar couplings ($10^{-3} \lesssim \lambda_5 \lesssim 10^{-2}$).

B. Low-Scale Resonant Regime ($M_1 \sim 10^5 - 10^7$ GeV)

• Viability: Highly constrained but survives in a narrow strip.
• Mechanism: Successful baryogenesis relies on resonant enhancement of CP asymmetry due to quasi-degenerate RHNs ($M_2 \simeq M_1$).
• The "Resonant Window":
  • Mass Splitting: Requires a finely tuned splitting $\Delta \sim 10^{-6} - 10^{-2}$.
  • Washout: Operates in the moderate-to-strong washout regime ($10^2 \lesssim K_1 \lesssim 10^4$).
  • LFV Safety: Achieved through phase alignment of the complex Casas–Ibarra angles. The imaginary parts of the angles drive the CP asymmetry, while the real parts and specific phase combinations suppress the LFV amplitude.
  • Scalar Coupling: Requires an intermediate $\lambda_5$ ($10^{-4} \lesssim \lambda_5 \lesssim 10^{-2}$). Very small $\lambda_5$ increases washout too much; very large $\lambda_5$ suppresses the loop-induced neutrino mass generation efficiency.
• Experimental Signature: In this surviving region, the branching ratio $BR(\mu \to e\gamma)$ lies close to the current MEG sensitivity ($\sim 10^{-13}$), making it a prime target for future experiments like MEG II, Mu3e, and COMET.

C. Benchmark Points

The authors provide specific benchmark points (Table 2) demonstrating that:

• Low-scale: $M_1 \approx 5.2 \times 10^5$ GeV, $\Delta \approx 5.8 \times 10^{-6}$, $BR(\mu \to e\gamma) \approx 1.8 \times 10^{-13}$ (just below MEG bound).
• High-scale: $M_1 \approx 3.4 \times 10^{10}$ GeV, $BR(\mu \to e\gamma) \approx 4.5 \times 10^{-18}$ (far below MEG bound).

5. Significance

• Phenomenological Target: The paper establishes that low-scale resonant leptogenesis in the Scotogenic model is not dead but exists in a "resonant strip" that is directly testable by upcoming LFV experiments. A null result in MEG II would effectively rule out this specific low-scale realization.
• Theoretical Insight: It demonstrates that maximizing CP asymmetry alone is insufficient for baryogenesis; an optimal balance between asymmetry generation and washout suppression is required.
• Model Robustness: The study reinforces the Scotogenic model as a highly predictive framework linking neutrino masses, dark matter (via the $Z_2$-odd particle), and baryogenesis. The simultaneous satisfaction of all constraints significantly sharpens the viable parameter space.
• Future Directions: The authors note that while dark matter relic density was not fully analyzed, the identified "resonant window" provides a concrete target for future combined analyses of LFV, baryogenesis, and dark matter direct detection.

In conclusion, the paper resolves the tension between LFV and leptogenesis by showing that while high-scale scenarios are naturally safe, low-scale scenarios survive only through a delicate interplay of resonant enhancement and flavor-phase cancellations, offering a clear path for experimental verification.