Type I+II Seesaw Mechanism

• The Type I+II seesaw mechanism is a theoretical framework combining heavy right-handed neutrinos and scalar triplet contributions to generate light neutrino masses.
• Its interplay modifies renormalization group evolution and offers novel avenues for leptogenesis and lepton flavor violation through interference effects.
• Embedded in GUT, left–right symmetric, and extended electroweak models, it provides testable predictions across collider, cosmological, and low-energy experiments.

The Type I + II seesaw mechanism is a theoretical framework for neutrino mass generation that incorporates both right-handed (“type I”) and scalar triplet (“type II”) contributions to the light neutrino mass matrix. This hybrid scenario is motivated by the observation that Grand Unified and left–right symmetric models often provide both heavy right-handed neutrinos and Higgs triplets, naturally leading to a neutrino Majorana mass that is the sum of type I and type II terms. The interplay between these contributions has unique implications for renormalization group (RG) evolution, flavor symmetries, leptogenesis, lepton flavor violation, cosmology, and experimental searches.

1. Structure of the Type I+II Seesaw Mechanism

In the most general form, the effective light neutrino mass matrix is

$$m_\nu = m_\nu^{(\mathrm{I})} + m_\nu^{(\mathrm{II})}$$

with

$$m_\nu^{(\mathrm{I})} = - \frac{v^2}{2} Y_\nu^T M^{-1} Y_\nu$$

and

$m_\nu^{(\mathrm{II})} = 2 \, Y_\Delta v_\Delta$

where $Y_\nu$ is the Dirac neutrino Yukawa coupling, $M$ the right-handed neutrino Majorana mass matrix, $Y_\Delta$ the triplet Yukawa, $v$ the SM Higgs vev, and $v_\Delta$ the induced triplet vev ($v_\Delta \sim \mu_\Delta v^2 / m_\Delta^2$ for trilinear coupling $\mu_\Delta$ and triplet mass $m_\Delta$). Both terms may be present and comparable, depending on scalar sector details and high-scale symmetry structure (Hambye, 2012, Borah et al., 2013, Borah et al., 2015, Borah et al., 2016).

Table 1: Summary of Neutrino Mass Contributions in Type I+II Seesaw

Term	Origin	Formula
Type I	Heavy right-handed neutrino exchange	$- \frac{v^2}{2} Y_\nu^T M^{-1} Y_\nu$
Type II	Triplet vev, induced from scalar potential	$2 \, Y_\Delta v_\Delta$

2. Renormalization Group Evolution in Type I+II Seesaw

The renormalization group (RG) evolution of neutrino parameters in the type I+II scenario is distinct from single-contribution cases (0705.3841). The one-loop β-function for any neutrino mass contribution $m_\nu^{(i)}$ is

$$16\pi^2 \, \frac{d m_\nu^{(i)}}{dt} = [C_e Y_e^\dagger Y_e + C_\nu Y_\nu^\dagger Y_\nu + C_\Delta Y_\Delta^\dagger Y_\Delta]^T m_\nu^{(i)} + m_\nu^{(i)} [C_e Y_e^\dagger Y_e + C_\nu Y_\nu^\dagger Y_\nu + C_\Delta Y_\Delta^\dagger Y_\Delta] + \alpha m_\nu^{(i)}$$

where the coefficients $C_e, C_\nu, C_\Delta$ and the term α depend on the active sector (SM or MSSM) and the mass-operator type (see Table I in (0705.3841)).

A key feature: for type II (triplet active), the running of mixing angles is proportional to the relevant mass-squared difference,

$$\frac{d\theta_{ij}}{dt} \sim \frac{\Delta m^2_{ji}}{v_\Delta^2} \sin 2\theta_{ij}$$

in contrast to the dimension-5 effective operator case, where the evolution is enhanced by $m_0^2 / \Delta m^2$. Thus, for hierarchical spectra, type II running remains moderate except potentially for θ₁₃ and θ₂₃. Effects on CP phases can be enhanced for small θ₁₃, with $d\delta/dt$ proportional to $1/\theta_{13}$ (0705.3841).

3. Phenomenological Manifestations: Leptogenesis and Lepton Flavor Violation

Leptogenesis

In hybrid seesaw models, successful baryogenesis via leptogenesis can proceed through decay of either heavy right-handed neutrinos or the scalar triplet, with CP asymmetries proportional to the interference between type I and type II mass matrices: $$\epsilon_{\Delta} \propto \operatorname{Im} \left\{ (m_\nu^{(\mathrm{II})}) (m_\nu^{(\mathrm{I})})^\dagger \right\}$$ When both contributions coexist, new decay diagrams and interference terms become dominant. These terms relax bounds encountered in pure type I scenarios and allow for successful leptogenesis with more parameter space, particularly when flavor effects are included across temperature intervals relevant for the one-, two-, and three-flavor regimes (Hambye, 2012, Borah et al., 2013). The Majorana and Dirac CP phases play a critical role in determining the efficiency and sign of the generated baryon asymmetry.

Lepton Flavor Violation (LFV)

LFV processes such as μ → eγ and μ → 3e exhibit strong parameter dependence in type I+II models, especially when doubly charged scalars are present. The dominant tree-level and one-loop contributions can be correlated with the neutrino mass matrix structure via the triplet Yukawa. For example,

$$\mathrm{BR}\left(\mu \to 3e\right) \propto \frac{|f^\dagger_{ee} f_{\mu e}|^2}{G_F^2\, m_{S^{±±}}^4}$$

where $f$ is fixed (up to overall scaling) by neutrino oscillation data (Ferreira et al., 2019). Experimental reach in muon and tau decays and projected collider limits on doubly charged scalars set complementary constraints (Borah et al., 2016).

4. Model Realizations and Symmetry Structures

Type I+II seesaw is realized in a wide variety of extended gauge frameworks:

• Left-right symmetric models (LRSM): Both seesaws are automatic, with heavy RH neutrinos and SU(2)_L/R triplets present. The light neutrino mass matrix is

$$m_\nu = \gamma \left(\frac{M_W}{v_R}\right)^2 M_{RR} - m_{LR} M_{RR}^{-1} m_{LR}^T$$

with SU(2)_R symmetry breaking scale $v_R$ in the TeV–PeV range (Borah et al., 2015, Borah et al., 2016).

• SO(10) and SU(5) GUTs: Type I+II structure arises via 126_H in SO(10) (providing both triplet and RH neutrino Majorana terms), while SU(5) models can be arranged for type II dominance or cancellation, allowing for independent TeV-scale RH neutrino masses and observable 0νββ via sterile neutrino exchange (Ohlsson et al., 2019, Parida et al., 2018).
• 3-3-1 Models and Beyond: Extended electroweak groups (SU(3)_L x U(1)_X) with extended scalar sectors (septets, sextets) allow natural embedding of triplet and singlet contributions, subject to additional discrete symmetries for flavor structuring (Ahmad et al., 22 Jan 2024, Ferreira et al., 2019).
• Two Higgs Doublet Models (2HDM) with U(1)_X: Type I+II is realized by supplementing the two-doublet sector with a scalar triplet and singlet, both stabilized against flavor-changing neutral currents by the abelian gauge extension (Cogollo et al., 2019).

Symmetry assignments (forbidding dangerous operators, protecting desired Yukawa structures) and the scalar potential affect the dominance and scale of each seesaw contribution and have computable consequences for mixing, mass hierarchy, and low-energy phenomenology.

5. Experimental and Cosmological Implications

Type I+II models make predictions for and are tightly constrained by a suite of observations:

• Neutrinoless double beta decay (0νββ): New contributions from heavy RH neutrino exchange (mediated by W_R) and doubly charged triplet scalars can be sizable. The amplitude depends on the structure and strength of type I+II terms and the relative mass scales (with potential enhancement or cancellation in the effective mass parameter). Experimental limits from GERDA, KamLAND-Zen, and CUORE translate into bounds on the seesaw parameter space (Borah et al., 2015, Borah et al., 2016).
• Lepton Flavor Violation: LFV branching ratios can approach experimental sensitivity in models with light (few hundred GeV to TeV) doubly charged scalars or sizable nonunitarity in the lepton mixing matrix. Simultaneous collider and LFV searches are complementary for testing type I+II structures (Ferreira et al., 2019).
• Cosmological constraints: Limits on the sum of neutrino masses from Planck and BAO data inform the scale and degeneracy of neutrino eigenvalues and necessarily the seesaw input parameters (Borah et al., 2013, Parida et al., 2018).
• Collider signatures: Direct production of RH neutrinos, triplet scalars, and extra gauge bosons (W', Z') at the LHC, HL-LHC, or future colliders are smoking-gun signals, especially in 3–5 TeV mass windows relevant for type I+II models (Caetano et al., 2012, Cogollo et al., 2010, Ferreira et al., 2019).
• Inflationary cosmology: If a seesaw scalar (singlet or triplet) acts as inflaton, the nature of one-loop corrections to the inflaton potential distinguishes type II scenarios from type I; positive radiative corrections favor type II as compatible with Planck and BICEP/Keck observations (Rodrigues et al., 2020).

6. Theoretical Implications: Flavor Structure, CP Violation, and RG Effects

The interplay of type I and II terms modifies the flavor and CP structure of the neutrino sector:

• Deviations from μ–τ symmetry and reactor angle θ₁₃: Minimal type II perturbations to a type I–induced tribimaximal matrix can reproduce the observed nonzero θ₁₃ and lead to testable predictions for the Dirac CP phase, linking these deviations to the underlying flavor symmetry (Abelian, A₄, etc.) and the scale of lepton-number violation (Borah, 2013, Kalita et al., 2014).
• Leptonic CP violation and baryogenesis: The CP asymmetries required for successful leptogenesis often arise solely from the interplay of the two seesaw contributions, with parameter regions restricted both by oscillation data and baryon asymmetry measurement. Dominance by either contribution or a specific combination can favor or disfavor classes of neutrino mass ordering (normal vs. inverted) and specific values of complex phases (Kalita et al., 2014, Borah et al., 2013).
• RG evolution and model-building implications: The RG flow in the presence of both seesaw types can differ sharply from that in pure type I or II—most notably, the proportionality to Δm² in the type II regime (triplet active) instead of the 1/Δm² enhancement of the effective theory after decoupling (0705.3841). This affects the mapping of high-scale flavor structures (such as bimaximal or tribimaximal mixing) to low-energy observables and must be accounted for in model fits (Ohlsson et al., 2019).

7. Effective Field Theory and One-Loop Matching

Integrated type I+II seesaw scenarios admit a systematic low-energy effective field theory description. At one loop, integrating out both right-handed neutrinos and scalar triplets leads to cross-contributions in the Wilson coefficients of the unique dimension-5 (Weinberg operator) and nine important dimension-6 operators. These cross terms arise exclusively from diagrams with internal lines of both heavy sectors and are not present in a sum of the individual type I and II SEFTs. This nontrivial structure affects low-energy predictions for neutrino oscillations, lepton flavor violation, and Higgs self-couplings (Zhang, 2022).

Table 2: Operators with Cross-Contributions in the Type I+II SEFT

Operator	Physical Sensitivity
Weinberg (d=5)	Light neutrino masses
$O_{H\square}$, $O_{HD}$, $O_H$	Higgs kinetic/field strength/Higgs potential
$O_{H\ell}^{(1,3)}$, $O_{uH}$, $O_{dH}$, $O_{eH}$, $O_{\ell\ell}$	LFV, Higgs–lepton couplings

References to Key Results

• Detailed RG analysis of type I+II seesaw: (0705.3841)
• 3-3-1, LRSM, and GUT realizations: (Caetano et al., 2012, Borah et al., 2013, Ohlsson et al., 2019, Hambye, 2012, Parida et al., 2018)
• Leptogenesis in hybrid seesaw: (Hambye, 2012, Borah et al., 2013, Kalita et al., 2014)
• Collider and LFV phenomenology: (Ferreira et al., 2019, Borah et al., 2016, Borah et al., 2015)
• Cosmological connections: (Rodrigues et al., 2020)
• EFT and matching considerations: (Zhang, 2022)

Conclusion

The type I + II seesaw mechanism represents a robust, predictive, and experimentally testable framework for neutrino mass and mixing generation, deeply entwined with flavor symmetries, baryogenesis, lepton flavor violation, cosmology, and the structure of unified theories. Its distinct RG behavior, the interplay of its contributions in leptogenesis and LFV, and its low-energy EFT limit underscore its importance for model building and for interpreting current and future precision measurements in particle physics and cosmology.