Scale invariant radiative neutrino mass model

1. Problem Statement

The Standard Model (SM) successfully describes particle physics but fails to explain three critical phenomena:

1. Neutrino Masses: The origin of non-zero neutrino masses.
2. Dark Matter (DM): The nature of the stable, non-baryonic matter in the universe.
3. Baryon Asymmetry: The observed matter-antimatter imbalance.

The Scotogenic Model is a popular extension of the SM that addresses these three issues simultaneously by introducing an inert doublet scalar ($\eta$) and three right-handed neutrinos ($N_j$) with a $Z_2$ symmetry. However, the standard Scotogenic model suffers from the hierarchy problem: the mass scales of the new particles (inert doublet and right-handed neutrinos) are typically fixed by hand at the TeV scale, lacking a theoretical explanation for their origin or stability against quantum corrections.

This paper aims to construct a UV-complete version of the Scotogenic model that:

• Explains the origin of the mass scales via Classical Scale Invariance and Coleman-Weinberg (CW) radiative symmetry breaking.
• Utilizes Custodial Symmetry ($SO(5)$) to naturally suppress the weak scale relative to the new physics scale.
• Resolves the hierarchy problem without fine-tuning.

2. Methodology and Model Construction

Symmetry Framework:
The author extends the Scotogenic model by introducing two real singlet scalars, $\phi$ and $S$, and imposing:

• $Z_2$ Parity: $\eta$ and $N_j$ are odd; SM fields, $H$, $\phi$, and $S$ are even.
• Classical Scale Invariance: No explicit mass terms in the Lagrangian.
• Custodial Symmetry $SO(5)$: The Higgs doublet $H$ and the singlet $\phi$ form a 5-plet of $SO(5)$.

Scalar Potential:
The scalar potential $V$ is constructed to respect these symmetries. Crucially, terms that explicitly break custodial symmetry (like $S^3\phi$, $S\phi^3$, and specific quartic couplings between $H$ and $\eta$) are excluded or highly constrained. The potential includes:
$$V = \lambda_{c1} \left[ (H^\dagger H) + \frac{1}{2}\phi^2 \right]^2 + \frac{1}{2}\lambda_{c2} \left[ (H^\dagger H) + \frac{1}{2}\phi^2 \right] S^2 + \dots$$
The coupling $\tilde{\lambda}_5 [(H^\dagger \eta)^2 + \text{h.c.}]$ is included as a necessary custodial-breaking term to generate neutrino masses.

Mechanism of Symmetry Breaking:

1. Intermediate Scale Generation: Through the Coleman-Weinberg mechanism, quantum corrections induce a non-zero Vacuum Expectation Value (VEV) for the singlet $\phi$ ($v_\phi$) at an intermediate scale (TeV range), spontaneously breaking scale invariance and custodial symmetry.
2. Weak Scale Suppression: The Higgs doublet $H$ acquires a VEV ($v_H$) radiatively. Due to the custodial symmetry, the condition $\lambda_{H\phi} \approx \lambda_\phi$ and $\lambda_{HS} \approx \lambda_{\phi S}$ is enforced. This leads to a cancellation in the effective potential, allowing $v_H \ll v_\phi$ naturally. The Higgs boson emerges as a pseudo-Nambu-Goldstone boson (PNGB) with a small mixing with the dilaton (the Goldstone boson of scale invariance).

Mass Generation:

• Inert Doublet ($\eta$) and Right-Handed Neutrinos ($N_j$): Their masses are generated via the VEV of $\phi$:
  $$m_\eta^2 = \frac{1}{2}\lambda_{\eta\phi} v_\phi^2, \quad M_j = y_{Nj} v_\phi$$
• Neutrino Masses: Generated via one-loop diagrams involving $\eta$ and $N_j$, requiring the small custodial-breaking term $\tilde{\lambda}_5$.

3. Key Contributions and Results

A. Numerical Benchmark and Renormalization Group Evolution (RGE)

The author performs a numerical analysis using benchmark parameters fixed at the Planck scale.

• RGE Solutions: The couplings evolve such that $\lambda_\phi$ becomes small and positive at a critical scale ($\mu \approx 8.8$ TeV), triggering symmetry breaking.
• Mass Spectrum:
  • Higgs mass ($m_h$): $\approx 137$ GeV.
  • Higgs VEV ($v_H$): $\approx 242$ GeV.
  • Singlet VEV ($v_\phi$): $\approx 7.9$ TeV.
  • Inert Doublet mass ($m_\eta$): $\approx 1.48$ TeV.
  • Right-handed neutrinos ($N_{2,3}$): $\approx 1.50$ TeV (nearly degenerate).
  • Lightest Right-Handed Neutrino ($N_1$): Mass is a free parameter constrained by DM/Leptogenesis.
• Mixing: The mixing angle between the Higgs-like scalar and the dilaton is small ($\tan \theta \approx 0.04$).

B. Dark Matter (DM) Candidate

• Exclusion of Inert Doublet: In standard Scotogenic models, the lightest neutral component of $\eta$ ($\eta_R$) is often the DM candidate. However, in this scale-invariant model, the custodial symmetry constraints force the quartic couplings ($\tilde{\lambda}_3, \tilde{\lambda}_4$) to be extremely small. This prevents the necessary co-annihilation to reduce the $\eta_R$ relic abundance to the observed level ($\Omega h^2 \approx 0.12$).
• $N_1$ as DM: Consequently, the lightest right-handed neutrino ($N_1$) must be the DM candidate.
• Production Mechanism: The freeze-out scenario is ineffective for $N_1$ due to suppressed annihilation cross-sections. Instead, the model relies on the Freeze-in mechanism.
• Mass Prediction: The requirement to produce the correct relic abundance via freeze-in (mediated by $y_{N1}$ and $h_1$) constrains the mass of $N_1$ to be less than $\mathcal{O}(1)$ MeV. This mass range is consistent with sterile neutrino DM solutions to the small-scale structure problem.

C. Baryon Asymmetry (Leptogenesis)

• Resonant Leptogenesis: The model generates the baryon asymmetry via the decay of $N_2$ (and $N_3$).
• Mass Degeneracy: The requirement for the weak scale suppression ($v_H \ll v_\phi$) naturally leads to a near-degeneracy between $N_2$ and $N_3$ ($M_2 \approx M_3$).
• CP Asymmetry: This mass degeneracy enhances the CP asymmetry parameter ($\epsilon$) via resonant effects.
• Result: With a mass splitting $\delta = (M_2 - M_3)/M_2 \sim \mathcal{O}(10^{-9})$, the model successfully generates the observed baryon asymmetry. The mass degeneracy required for the electroweak scale mechanism is thus directly linked to the success of leptogenesis.

4. Significance and Conclusion

This paper presents a Scale Invariant Radiative Neutrino Mass Model that serves as a UV completion of the Scotogenic model. Its primary significance lies in:

1. Solving the Hierarchy Problem: It explains the origin of the TeV scale and the suppression of the weak scale without ad-hoc mass parameters, using classical scale invariance and custodial symmetry.
2. Unified Phenomenology: It links three major open problems (Neutrino mass, DM, Baryon asymmetry) in a single framework where the constraints of one sector (custodial symmetry for the weak scale) dictate the phenomenology of the others.
3. Predictive DM Candidate: It uniquely predicts that Dark Matter must be a light sterile neutrino ($N_1 < 1$ MeV) produced via freeze-in, ruling out the inert doublet as the DM candidate in this specific symmetry framework.
4. Resonant Leptogenesis: It demonstrates that the mass degeneracy required to suppress the Higgs mass naturally facilitates resonant leptogenesis, providing a consistent explanation for the matter-antimatter asymmetry.

The model is testable through:

• Searches for light sterile neutrinos (DM).
• Precision measurements of Higgs couplings (due to mixing with the dilaton).
• Lepton Flavor Violation (LFV) processes like $\mu \to e\gamma$.
• Potential signatures of the dilaton (pseudo-scalar) at colliders.