Two-Sector Leptogenesis

• Two-sector leptogenesis is a framework where heavy neutral fermion decays generate lepton asymmetry in both visible and hidden sectors.
• It employs multiple seesaw mechanisms—linear, inverse, and double—with quasi-degenerate neutrino pairs to achieve resonant CP asymmetry.
• Tuning parameters like the small Majorana mass (m₍g₎) and Higgs VEVs balances decay channels and washout, linking baryon asymmetry with potential dark matter signals.

Two-sector leptogenesis refers to frameworks where the generation of lepton (and consequently baryon) asymmetry through out-of-equilibrium decays of heavy neutral fermions is extended to involve more than one “sector.” Typically, these are the visible sector (Standard Model or extensions) and a second sector—either a hidden (dark/mirror) sector or another set of particles that communicate via neutrino or Higgs portal interactions. In such setups, asymmetry generation, washout, and transfer processes become more intricate, involving the interplay between different mass mechanisms (linear, inverse, double seesaw) and the possibility of resonant enhancement owing to quasi-degenerate heavy states. This article focuses on the theoretical underpinnings, the role of singlet fermion and right-handed neutrino mass scales, the conditions for resonance, and model-building implications of two-sector leptogenesis, as realized in left-right symmetric models with doublet and bi-doublet Higgs scalars (Gu et al., 2010), as well as related extensions.

1. Multi-Seesaw Realizations and Leptogenesis Mechanisms

In left-right symmetric models with doublet ($X_L$, $X_R$) and bi-doublet ($\phi$) Higgs scalars, neutrino masses are generated through a combination of linear, inverse, or double seesaw mechanisms dictated by the properties of three gauge singlet fermions ($E_R$). After electroweak symmetry breaking, the general structure of the neutrino mass matrix is given by

$$m_\nu = m_\nu^{\text{inverse}} + m_\nu^{\text{linear}}.$$

• Inverse Seesaw: The inverse term is proportional to the small Majorana mass $m_g$ of the singlet fermions and involves the right-handed Higgs VEV $\langle X_R \rangle$,

$m_\nu^{\text{inverse}} \propto h_y f_{XR}^{-1} m_g$

• Linear Seesaw: The linear component depends on mixing terms suppressed by the small VEV of $X_L$,

$$m_\nu^{\text{linear}} \sim (h_y + h_T) \text{VEV}_{X_L}$$

• Double Seesaw: For large $m_g$, the singlet fermions decouple, inducing a Majorana mass for the right-handed neutrinos,

$M_N \simeq -f^* \frac{\langle X_R \rangle^2}{m_g}$

Leptogenesis can then proceed through the decay of these right-handed neutrinos.

When the singlet fermion Majorana masses ($m_g$) are small, a strong mixing between $N_R$ and $E_R$ produces three pairs of quasi-degenerate Majorana mass eigenstates ($N_\pm$). Their decays are the central site of resonant leptogenesis, with asymmetry generation further enhanced by the near mass degeneracy.

2. Singlet Fermions and Quasi-Degeneracy

Singlet fermions $E_R$ fulfill a pivotal function:

• For small $m_g$: They mix with right-handed neutrinos, yielding quasi-degenerate pairs. Explicitly, the mass eigenstates

$N_+ = N_R + N_R^c, \qquad N_- = N_R - N_R^c$

have masses

$$m_{N_+} \simeq f \langle X_R \rangle + m_g, \qquad m_{N_-} \simeq f \langle X_R \rangle - m_g$$

The mass splitting is $\Delta m_N \simeq 2m_g$. As $m_g \ll f \langle X_R \rangle$, the two states are quasi-degenerate, a necessary condition for resonant leptogenesis.

• For large $m_g$: The singlets decouple and only provide a conventional seesaw-induced Majorana mass for the right-handed neutrinos. The spectrum is non-degenerate and resonant enhancement does not occur.

Implication: The adjustable $m_g$ parameter tunes the scenario: quasi-degenerate for resonant leptogenesis or hierarchical for traditional (thermal) leptogenesis.

3. Right-Handed Neutrino Dynamics in Extended Seesaw Schemes

The mass and flavor structure of the right-handed neutrinos $N_R$ impacts the machinery of leptogenesis:

• Small $m_g$ regime: $N_R$ forms mixed quasi-degenerate Majorana pairs with $E_R$ (as above). Their decays can lead to substantial CP asymmetry via interference between tree-level and loop (self-energy and vertex) diagrams. The resonant effect is peaked when the mass splitting matches the decay width.
• Large $m_g$ regime: $N_R$ acquires seesaw-suppressed masses. Hierarchical decays (no resonance) are responsible for generating the lepton asymmetry, constrained primarily by equilibrium and washout conditions.

The relevant Yukawa couplings to SM doublets and Higgses are inherited from the left-right symmetric structure, and the nature of their couplings to additional sectors determines how asymmetries are distributed if more than one sector is present.

4. Resonant Enhancement and CP Asymmetry

Resonant enhancement, the core of resonant leptogenesis, is realized when the mass splitting of the decaying states approaches their decay widths. In the context of quasi-degenerate $N_\pm$ states, the paper (Gu et al., 2010) establishes:

• Resonance condition: $f \langle X_R \rangle \gg m_g$ and $\Delta m_N \simeq 2m_g \approx \Gamma_{N_\pm}$.
• CP asymmetry expressions involve interference terms:

$$\epsilon \propto \frac{\operatorname{Im}\left[(h h^\dagger)^2\right]}{\Delta m_N}$$

Consequently, smaller $m_g$ both enhances the degeneracy and increases the denominator, maximizing the resonance when $\Delta m_N \sim \Gamma$.

In traditional (non-resonant) scenarios, such as when singlet masses are heavy, mass splittings are generically much larger than decay widths, and no enhancement occurs.

5. Two-Sector Extensions: Asymmetry Sharing Across Sectors

The paper’s scenario accommodates a natural extension to two-sector leptogenesis, where the decays of heavy Majorana states simultaneously feed asymmetries in more than one sector. Key implications include:

• Decay channels to both sectors: If the $N_R$ or mixed $N_\pm$ states couple both to SM fields and to fields in a hidden/mirror sector, the same resonant mechanism can seed asymmetry in each.
• Differential branching ratios: Small differences in branching between visible and hidden channels can lead to correlated or anti-correlated lepton asymmetries, of interest in models with asymmetric dark matter or mirror sectors.
• Parameter tuning: By adjusting $m_g$, $\langle X_R \rangle$, $\langle X_L \rangle$, and relevant Yukawas, it is possible to enlarge the parameter space that yields viable asymmetries in both sectors while satisfying all equilibrium and out-of-equilibrium decay conditions and the neutrino mass constraints.
• Interplay of linear and inverse seesaw: If linear seesaw terms dominate, constraints from neutrino mass bounds on the CP asymmetry can be relaxed, further broadening the allowed region for two-sector asymmetry generation.

6. Generalizations and Phenomenological Outlook

The structure described leads to several avenues for further model building:

• Multi-sector or mirror sector constructions: The formalism and results of (Gu et al., 2010) can be straightforwardly embedded into frameworks where $N_R$ (and possibly $E_R$) couple to more than one sector, providing a mechanism for synchronous or opposing asymmetry production.
• Leptogenesis-dark matter unification: If hidden sector degrees of freedom play a role in dark matter (for example, as asymmetric dark matter), the resonant two-sector mechanism ensures their relative abundances may be naturally connected.
• Experimental signatures: The presence of quasi-degenerate heavy Majorana neutrinos with tunable couplings to new sectors can have indirect signals in lepton flavor violation, neutrinoless double beta decay, or even at colliders, should the scale be low enough.

Summary Table: Parameter Dependence in Two-Sector Leptogenesis

Regime	Dominant Mass Term	Majorana Pair Spectrum	Resonant Leptogenesis	Decay Asymmetry Target
$m_g \ll f\langle X_R \rangle$	Inverse/Linear seesaw	Quasi-degenerate $N_\pm$	Yes	Both visible and hidden sectors possible
$m_g \gg f\langle X_R \rangle$	Double seesaw	Hierarchical $N_R$	No	Traditional leptogenesis; single sector usually

In conclusion, the left-right symmetric model with singlet fermions, as developed in (Gu et al., 2010), demonstrates that the mutual tuning of singlet Majorana masses and Higgs sector VEVs enables both (i) the realization of multiple seesaw mechanisms for suppressing neutrino masses, and (ii) the implementation of resonant leptogenesis via quasi-degenerate pairs. These mechanisms extend naturally to two-sector scenarios, offering a flexible platform for generating baryon and possibly dark asymmetries in more than one sector, subject to adjustable model parameters and constraints from neutrino physics.