Type-I & II Seesaw Mechanisms

• Type-I and II seesaw mechanisms are foundational frameworks that explain light neutrino masses through heavy right-handed neutrinos and scalar triplet exchanges.
• They yield an effective light neutrino mass matrix that combines a negative Type-I contribution from Dirac masses and a positive Type-II contribution from triplet vev interactions.
• These mechanisms offer rich phenomenology, including neutrino oscillation fits, lepton flavor violation signals, and distinctive collider signatures with implications for leptogenesis and unified theories.

The Type-I and Type-II seesaw mechanisms are foundational frameworks for understanding the origin of light neutrino masses in extensions of the Standard Model (SM). Both mechanisms introduce new fields responsible for sub-eV Majorana masses via the exchange of heavy states—right-handed (RH) neutrinos for Type-I and scalar triplets for Type-II. These mechanisms can occur in isolation or in hybrid “Type-I+II” scenarios, producing a characteristic sum form for the effective light neutrino mass matrix. The rich phenomenology associated with these mechanisms encompasses neutrino oscillation observables, lepton flavor violation (LFV), baryogenesis via leptogenesis, collider signatures, and renormalization group evolution in grand unified theory (GUT) frameworks.

1. Formal Structure of Type-I and Type-II Seesaw Mechanisms

In the minimal implementation, the SM is augmented by three RH neutrino singlets ($N_{R_i}\sim(1,1,0)$) and an $SU(2)_L$ scalar triplet ($\Delta\sim(1,3,1)$). The relevant Lagrangian terms and scalar potential components are given by (Aguilar et al., 27 Sep 2025):

$$\begin{aligned} \mathcal{L} &\supset - (Y_D)_{i\alpha} \bar L_i \tilde\Phi N_{R_\alpha} - \frac{1}{2} N_{R_\alpha}^T C (M_R)_{\alpha\beta} N_{R_\beta} - \frac{1}{2} (Y_\Delta)_{ij} L_i^T C i\sigma_2 \Delta L_j + \text{h.c.} \ V(\Phi,\Delta) &= -m_\Phi^2 \Phi^\dagger \Phi + M_\Delta^2 \mathrm{Tr} [\Delta^\dagger \Delta] + [\mu \Phi^T i \sigma_2 \Delta^\dagger \Phi + \mathrm{h.c.}] + \text{quartic terms}. \end{aligned}$$

After electroweak symmetry breaking ($\langle\Phi\rangle=(0,v/\sqrt{2})^T$ with $v\approx 246$ GeV), the Dirac neutrino mass matrix is defined as $m_D = Y_D v/\sqrt{2}$. The trilinear $\mu$-term induces a triplet vev $$v_\Delta \simeq \mu v^2/(\sqrt{2} M_\Delta^2) \ll v$$. The resulting $6\times6$ neutrino mass matrix, in the basis $(\nu_L, N_R^c)$, is:

$$M_\nu = \begin{pmatrix} 0 & m_D \ m_D^T & M_R \end{pmatrix}$$

Block-diagonalizing yields an effective $3\times3$ light neutrino mass matrix as a sum of Type-I and Type-II contributions (Aguilar et al., 27 Sep 2025, Borah, 2014, Kalita et al., 2014):

$$m_\nu = m_{II} + m_{I}, \quad m_{II} = 2 Y_\Delta v_\Delta, \quad m_{I} = -m_D M_R^{-1} m_D^T.$$

Explicit expressions for the triplet Yukawa in terms of measured light neutrino parameters in the Type-II-dominated regime are:

$$Y_\Delta = \frac{1}{2v_\Delta} U \operatorname{diag}(m_1,m_2,m_3) U^T$$

where $U$ is the PMNS mixing matrix.

2. Phenomenological Implications: Oscillation Data and Parameter Fits

Neutrino oscillation experiments constrain the mass-squared splittings $\Delta m^2_{21}$, $\Delta m^2_{31}$ and the mixing angles $\theta_{12}$, $\theta_{23}$, $\theta_{13}$. These observables fix the combination $Y_\Delta/v_\Delta$ up to Majorana phases in the Type-II limit, or specify the light neutrino mass matrix in hybrid (Type-I+II) models. Fits to global oscillation data slightly favor normal mass ordering, with minimal I+II models yielding acceptable fits only when normal ordering is imposed and with a dominant Type-I term; any Type-II contribution is typically subdominant ($v_\Delta\lesssim 10^{-6}$ GeV in minimal non-supersymmetric SO(10)) (Ohlsson et al., 2019). Hybrid models routinely employ a “TBM plus perturbation” structure, where a leading order tri-bimaximal (TBM) form arises from Type-I and Type-II serves as a controlled symmetry-breaking perturbation generating nonzero $\theta_{13}$ and the required Dirac phase (Borah, 2014, Kalita et al., 2014, Borah, 2013). See-fit results typically require $Y_\Delta\sim 10^{-2}-1$ for $v_\Delta \sim 1$ eV, or $Y_\Delta\sim 10^{-7}$ for $v_\Delta \sim 100$ eV (Aguilar et al., 27 Sep 2025).

3. Lepton Flavor Violation and Collider Signatures

In triplet-dominated scenarios, the exchange of $\Delta^{\pm}$ and $\Delta^{\pm\pm}$ mediates rare lepton flavor-violating decays such as $\mu\to e\gamma$ and $\mu\to 3e$. The branching ratios are (Aguilar et al., 27 Sep 2025, Ferreira et al., 2019):

$$\begin{aligned} \mathrm{BR}(\mu\to e\gamma) &\simeq \frac{\alpha_{EM}}{192\pi G_F^2} \frac{|(Y_\Delta^\dagger Y_\Delta)_{e\mu}|^2}{M_{\Delta^{++}}^4}, \ \mathrm{BR}(\mu\to 3e) &\simeq \frac{|(Y_\Delta^\dagger Y_\Delta)_{e\mu}|^2}{G_F^2 M_{\Delta^{++}}^4}. \end{aligned}$$

Currently, $\mu\to 3e$ gives the most stringent lower bound on $M_{\Delta^{++}}$, reaching 3 TeV for $v_\Delta\sim1$ eV, surpassing LHC direct search constraints (currently $M_{\Delta^{++}}>0.94$–$0.86$ TeV, depending on decay branching) (Aguilar et al., 27 Sep 2025, Ferreira et al., 2019).

At colliders, $\Delta^{++}$ can be pair produced via Drell–Yan and decays to same-sign dileptons, giving a characteristic signature. For small $v_\Delta$, decays to charged leptons dominate, while for $v_\Delta\gtrsim 100$ eV, decay to $W^+W^+$ becomes important. HL-LHC and HE-LHC are projected to reach $M_{\Delta^{++}}$ up to $2.5$–$4.9$ TeV (Ferreira et al., 2019).

The complementarity between LFV searches and collider signals is key: LFV probes are more sensitive for small triplet vevs, while collider searches take the lead as $v_\Delta$ increases and $Y_\Delta$ shrinks, suppressing LFV rates (Aguilar et al., 27 Sep 2025, Ferreira et al., 2019).

4. Seesaw Effective Field Theory and Operator Analysis

At the EFT level, integrating out the heavy states produces the unique Weinberg operator $(\ell^T \tilde{H})(\tilde{H}^T\ell)$ and, at dimension-6, a full set of operators modifying Higgs, gauge, and lepton couplings. In the hybrid Type-(I+II) SEFT, the number and content of dim-6 operators matches that of Type-II, but the Wilson coefficients are nontrivially shifted (“cross” contributions), even though no direct $N_R$–$\Phi$ coupling exists (Zhang, 2022). The Wilson coefficient for the Weinberg operator is:

$$C^{(5)}_{αβ} = (Y_ν M_R^{-1} Y_ν^T)_{αβ} - \frac{2λ_Δ}{M_Δ} (Y_Δ)_{αβ} + \frac{1}{16π^2}\text{[cross-terms]}$$

Nine dimension-6 operators receive cross-term corrections at one loop, affecting neutrino mass predictions, Higgs quartic, $Z\ell\ell$ couplings, and non-unitarity observables. Precision Higgs and lepton flavor experiments are sensitive to these SEFT corrections (Zhang, 2022).

5. Role in Flavor Structure, Leptogenesis, and CP Violation

The interplay of Type-I and Type-II seesaw is instrumental in generating viable neutrino flavor structure, nonzero reactor angle $\theta_{13}$, and leptonic Dirac CP phase. Leading order TBM mixing, enforced by Type-I with real Dirac Yukawas, produces $\theta_{13}=0$ and zero CP phase. Perturbative inclusion of Type-II (with a minimal structure) breaks $\mu$–$\tau$ symmetry, lifting $\theta_{13}$ into the physical range and providing the sole source of CP violation for viable leptogenesis (Borah, 2014, Kalita et al., 2014, Borah, 2013). Successful baryogenesis through leptogenesis then critically correlates the magnitude and phase of the Type-II term with oscillation observables and the lightest neutrino mass. In some regimes (Type-I+II SO(10) fits), only normal ordering and I-dominance are compatible with all data (Ohlsson et al., 2019).

6. Extensions: Unified Models and Enhanced Suppression Mechanisms

Unified frameworks such as SO(10) and SU(5) naturally accommodate both seesaw types, with the interplay governed by the details of symmetry breaking and the scalar sector (Ohlsson et al., 2019, Borah et al., 2013, Parida et al., 2018). In non-minimal models, further suppression is possible (“triple-seesaw” or “quintuple-seesaw”), where the neutrino mass obtains additional powers of inverse heavy mass scales, e.g., $m_\nu \sim v^2 v_S^2/M^3$ or $m_\nu \sim v^2/M^5$ (Cogollo et al., 2010, Caetano et al., 2012). Such constructions permit sub-eV neutrino masses with new physics at the TeV scale and provide additional degrees of freedom for model-building and conserving experimental consistency.

7. Constraints, Future Sensitivities, and Prospects

Current and future experimental probes—oscillation measurements, cosmology, $0\nu\beta\beta$-decay, lepton flavor violation, and direct collider searches—work in synergy to constrain the scale and flavor structure of Type-I and Type-II seesaw frameworks. The parameter space is further limited by the electroweak $\rho$-parameter ($v_\Delta \lesssim 2$ GeV), cosmological limits on the sum of neutrino masses ($\sum m_\nu \lesssim 0.1$ eV), and LFV bounds. Next-generation experiments (Mu3e: $\mathrm{BR}(\mu\rightarrow 3e) \sim 10^{-16}$ sensitivity; high-luminosity colliders) could, in principle, test triplet masses up to 30 TeV and distinguish between pure and hybrid seesaw scenarios via unique SEFT-induced low-energy signatures (Aguilar et al., 27 Sep 2025, Zhang, 2022).

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References:

(Aguilar et al., 27 Sep 2025, Borah, 2014, Kalita et al., 2014, Borah et al., 2013, Ohlsson et al., 2019, Ferreira et al., 2019, Zhang, 2022, Cogollo et al., 2010, Caetano et al., 2012, Parida et al., 2018, Borah, 2013, Vien et al., 2018, Cogollo et al., 2019).