Linear Seesaw Framework in LR Models

• Linear Seesaw Framework is a neutrino mass generation mechanism in left–right symmetric models where lepton-number violation enters linearly, ensuring sub-eV neutrino masses.
• The model uses an expanded Higgs sector and singlet fermions that mix with right-handed neutrinos, forming quasi-degenerate Majorana pairs critical for resonant leptogenesis.
• Tuning of Yukawa couplings and VEV hierarchies within this flexible framework offers a pathway to reconcile neutrino mass scales with experimentally accessible low-scale leptogenesis.

The linear seesaw framework is an extension of established seesaw mechanisms for generating small neutrino masses, characterized by neutrino mass matrices in which the lepton-number–violating parameter enters linearly (rather than quadratically, as in the Type-I seesaw). It is commonly implemented in left–right symmetric models with expanded Higgs and singlet fermion content, and serves as a unifying setting that can interpolate between linear, inverse, and double seesaw regimes. The linear seesaw scenario naturally supports mechanisms for low-scale leptogenesis by introducing quasi-degenerate heavy neutrinos, enabling resonant enhancement of CP asymmetry and thus providing an explanation for the baryon asymmetry of the universe (Gu et al., 2010).

1. Left–Right Symmetric Embedding and Field Content

The linear seesaw is realized in models with gauge group $SU(2)_L \times SU(2)_R \times U(1)_{B-L}$, featuring the following essential particle content:

• Fermion doublets: Standard left- and right-handed leptons, $\psi_L$, $\psi_R$.
• Singlet fermions: Three gauge-singlet states $\xi_R$ per generation.
• Scalar sector: A Higgs bi-doublet $\phi$ and Higgs doublets $\chi_L$ and $\chi_R$.

The discrete left–right symmetry enforces equality of certain Yukawa couplings and tightens the structure of the neutrino mass matrices. After symmetry breaking, vacuum expectation values (VEVs) for the scalar fields, specifically $\langle \chi_R \rangle$, set the mass scale for right-handed neutrinos and drive their mixing with the singlet sector.

2. Seesaw Mechanism Variants and Mass Matrix Structure

The interplay of Dirac-type mass terms (from Yukawa couplings with $\phi$ and $\chi_{L,R}$) and the Majorana mass term for the singlets ($m_\xi$) leads to multiple seesaw mechanisms, depending on the scale hierarchy:

• Linear and inverse seesaw: $m_\xi$ small compared to Dirac-type masses ($m_\xi \ll f \langle \chi_R \rangle$). The right-handed neutrinos and singlets mix, forming quasi-degenerate Majorana pairs $N^+$, $N^-$.
• Double seesaw: $m_\xi$ large, so singlets are integrated out at high scales.

The relevant sector comprises mass terms:

$$\mathcal{L} \supset -f \, \chi_R \, (\nu_R)^T C \xi_R - m_\xi \xi_R^T C \xi_R + \mathrm{H.c.},$$

where $C$ is the charge-conjugation matrix.

Rotation to the $N_R^\pm$ eigenbasis yields per-family heavy Majorana neutrinos with masses

$$m_{N_i^+} \sim f_i \langle \chi_R \rangle, \quad m_{N_i^-} \sim \frac{f_i \langle \chi_R \rangle^2}{2 m_{\xi_{ii}}}.$$

Quasi-degeneracy ($m_\xi \ll f \langle \chi_R \rangle$) is crucial for successful resonant leptogenesis.

3. Linear Seesaw Contribution to Light Neutrino Mass

The effective light neutrino mass matrix after electroweak symmetry breaking is given by: $$m_\nu = m_\nu^{\rm (inverse)} + m_\nu^{\rm (linear)},$$ with the linear seesaw term ([Eqs. (25)–(27), (51)]): $$m_\nu^{\rm (linear)} \propto -h_\nu \, m_{N}^{-1} h' \langle \phi \rangle.$$ In explicit terms,

$$m_\nu = -h_\nu\,\Bigl(\frac{1}{f\,\langle \chi_R \rangle}\Bigr) \, m_{\xi}\, \Bigl(\frac{1}{f\,\langle \chi_R \rangle}\Bigr)^T \, h_\nu^T\, \langle \phi \rangle^2 + \cdots$$

This formula highlights the linear dependence on $m_\xi$ (the lepton-number–violating parameter), as distinct from the quadratic dependence in Type-I seesaw. In scenarios considered, the linear term typically dominates over the inverse seesaw contribution due to the relative size of $m_\xi$ and the structure of the VEV hierarchy.

4. Leptogenesis and Role of Heavy Majorana States

The formation of quasi-degenerate Majorana neutrinos ($N^+$, $N^-$) underpins resonant leptogenesis, enabling significant enhancement of CP-violating effects in their decay widths. Critical features:

• With $m_\xi \ll f \langle \chi_R \rangle$, the small mass splitting $\Delta m_N \simeq 2 m_\xi$ between $N^+$ and $N^-$ enables resonant amplification of lepton-number violation, allowing for efficient leptogenesis at low (TeV) scales.
• The heavy sector's decay generates lepton asymmetry, which is partially transferred to baryon asymmetry by standard sphaleron transitions.

Alternatively, with large $m_\xi$ (double seesaw limit), quasi-degeneracy is lost and conventional (non-resonant) leptogenesis via right-handed neutrino decay dominates—typically requiring higher seesaw scales.

Comparative Summary Table

Seesaw Regime	Dominant Mass Term	Leptogenesis Mechanism	$m_\xi$ Condition
Linear	Linear in $m_\xi$	Resonant (via $N^\pm$ pairs)	Small $m_\xi \ll f \langle \chi_R \rangle$
Inverse	Quadratic in $m_\xi$	Similar to linear, less dominant	Small $m_\xi$
Double	Double-suppressed	Traditional, non-resonant	Large $m_\xi$

5. Scalar Sector and Symmetry Breaking

The scalar content composed of:

• Bi-doublet $\phi$: Generates Dirac masses for both left- and right-handed neutrinos via its Yukawa couplings after electroweak breaking.
• Doublets $\chi_R$ and $\chi_L$: $\chi_R$'s large VEV breaks $SU(2)_R$ and is responsible for generating heavy right-handed neutrino masses and mixing with singlets. $\chi_L$ may acquire a small induced VEV, entering the Yukawa sector and further modifying neutrino mass structures via induced mixing.

These ingredients collectively enforce the constraints and mixing that embed linear (and other) seesaw mechanisms into the gauge-symmetric structure. Discrete left–right symmetry plays a central role in correlating the Yukawa couplings and dictating the interplay between sectors.

6. Interpretation of Singlet Fermion Masses and Seesaw Regimes

The phenomenological outcomes—dominance of the linear vs. double seesaw, realization of leptogenesis at low vs. high scales—are determined by the scale of $m_\xi$. Specifically:

• Low $m_\xi$: Linear seesaw contribution dominates; quasi-degenerate $N^\pm$ allow for resonant leptogenesis accessible at TeV scales.
• High $m_\xi$: Double seesaw regime; neutrino masses become doubly suppressed, and non-resonant leptogenesis is possible, but typically requires higher mass scales for right-handed neutrinos and correspondingly higher breaking scales for $SU(2)_R$.

This direct sensitivity to $m_\xi$ makes the linear seesaw regime theoretically appealing for models targeting experimentally accessible signatures in neutrino mass and baryogenesis.

7. Phenomenological and Model-Building Implications

The linear seesaw within a left–right symmetric context illustrates a versatile mechanism capable of:

• Accommodating sub-eV light neutrino masses via "linear" suppression.
• Enabling formalisms for resonant leptogenesis at moderate scales, reducing the scale tension present in canonical (type-I) seesaw models.
• Providing a flexible parameter space: careful selection of Yukawa couplings, VEVs, and $m_\xi$ allows tuning between the dominance of different seesaw contributions and the nature of the leptogenesis scenario.

The model further delineates the conditions for each variant's phenomenological viability and exposes the crucial role of the singlet fermion Majorana mass both in neutrino sector observables and baryogenesis via leptogenesis pathways.

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The linear seesaw framework, as established in left–right symmetric models with extended scalar and fermion content, offers a natural, technically robust pathway for generating small neutrino masses and accommodating both low- and high-scale leptogenesis. The framework’s key predictions and parametric dependencies are tightly linked to the hierarchy between singlet Majorana masses and Dirac mass scales, with ramifications for neutrino phenomenology and baryogenesis that are subject to ongoing experimental scrutiny (Gu et al., 2010).