Radiative Seesaw Model with Baryon Number Violation and Upper Limit on Neutron-anti-Neutron Transition Time

This technical summary details the paper "Radiative Seesaw Model with Baryon Number Violation and Upper Limit on Neutron-anti-Neutron Transition Time" by Rabindra N. Mohapatra and Nobuchika Okada.

1. The Problem: The Baryogenesis-Leptogenesis Tension

The fundamental mystery addressed is the origin of the matter-antimatter asymmetry in the universe.

• The Standard Scenario: Most models rely on leptogenesis, where heavy Majorana right-handed neutrinos (RHNs) decay to create a lepton asymmetry, which is then converted to a baryon asymmetry via sphaleron processes.
• The Conflict: Leptogenesis typically requires a high inflation reheat temperature ($T_R$) to thermally produce RHNs. However, if $T_R$ is low (below the sphaleron decoupling temperature, $T_{sph} \simeq 130$ GeV), standard leptogenesis fails.
• The Stability Dilemma: Adding baryon number ($B$) violating interactions to explain baryogenesis in low-$T_R$ scenarios often leads to rapid proton decay if the neutrino mass is generated via the standard Type-I seesaw mechanism, rendering the model phenomenologically unviable.

2. Methodology: The Extended Scotogenic Framework

The authors propose a solution by extending the scotogenic model (a radiative seesaw model) with a specific $B$-violating effective interaction.

• Model Architecture: The model adds three RHNs ($N_i$) and an inert scalar doublet ($\eta$) to the Standard Model. A $Z_2$ symmetry is imposed, which prevents the $\eta$ doublet from acquiring a Vacuum Expectation Value (VEV).
• Neutrino Mass Generation: Because the $Z_2$ symmetry forbids the Dirac mass term, neutrino masses are generated at the one-loop level (radiative seesaw). This naturally allows for a lower seesaw scale and protects the proton from decay.
• Baryon Violation: To facilitate baryogenesis at low $T_R$, the authors introduce a dimension-six effective operator: $\frac{1}{\Lambda^2} N u^c d^c d^c$. This allows the RHN to decay directly into three quarks ($N \to 3q$), generating a baryon asymmetry non-thermally via inflaton decay.
• Cosmological Constraints: The authors analyze the model through the lens of Big Bang Nucleosynthesis (BBN) and the stability of the baryon asymmetry, specifically looking at the decay of the lightest $\eta$ particle.

3. Key Contributions

1. A New Baryogenesis Pathway: They demonstrate a mechanism where baryogenesis occurs via $N \to 3q$ decays in a non-thermal regime, bypassing the need for sphalerons.
2. Direct Link to $n-\bar{n}$ Oscillation: They show that the same $B$-violating interaction responsible for baryogenesis induces neutron-anti-neutron ($n-\bar{n}$) oscillations.
3. Derivation of Cosmological Upper Limits: The paper provides a theoretical framework to derive an upper bound on the $n-\bar{n}$ transition time ($\tau_{n-\bar{n}}$) based on the requirement that the $\eta$ particle's decay does not disrupt BBN or dilute the baryon asymmetry.
4. UV Completion: The authors provide a renormalizable, high-scale UV-complete version of the model using a color-triplet scalar ($\Delta$) that preserves the $Z_2$ symmetry and ensures proton stability.

4. Results and Findings

• Predictive Bounds on $\tau_{n-\bar{n}}$: The most significant result is that the model predicts $\tau_{n-\bar{n}}$ values within a specific range. By imposing constraints from the $\eta$ lifetime (to satisfy BBN) and the dilution of baryon asymmetry, the authors find that $\tau_{n-\bar{n}}$ is typically bounded below $10^{11}$ seconds.
• Experimental Accessibility: The predicted range for $\tau_{n-\bar{n}}$ overlaps significantly with the sensitivity of the upcoming ESS HIBEAM/NNBAR experiment.
• Parameter Sensitivity:
  • Higher values of the coupling $|\lambda_5|$ (which dictates $\eta$ mass splitting) lead to lower $\tau_{n-\bar{n}}$ values, some of which are already excluded by current experimental limits.
  • The upper limit on $\tau_{n-\bar{n}}$ is sensitive to the ratio $\Lambda/M_N$ and the baryon asymmetry parameter $\epsilon_B$.
• Washout Avoidance: The authors confirm that the $B$-violating processes do not wash out the generated asymmetry, provided $T_R < T'_{d}$ (the decoupling temperature of the $\Delta B=2$ interaction).

5. Significance

This paper is significant because it moves the scotogenic model from a purely neutrino-mass-focused theory to a comprehensive cosmological framework. It provides a testable prediction: unlike many BSM models that only provide lower bounds on new physics scales, this model provides an upper bound on the $n-\bar{n}$ transition time.

The "Smoking Gun": If the HIBEAM/NNBAR experiment fails to observe $n-\bar{n}$ oscillations within the predicted window, this specific mechanism for baryogenesis in the radiative seesaw framework will be effectively ruled out.