Seesaw Algorithm in Neutrino Physics

• The Seesaw Algorithm is a set of theoretical and computational strategies that generate small neutrino masses through suppression via heavy field exchanges.
• It includes multiple variants (type I, II, III, inverse, and triple) that each employ unique mass matrix structures to achieve effective mass suppression.
• The algorithm utilizes iterative numerical techniques to extract full model parameters from experimental data, aiding robust reconstruction of neutrino oscillation phenomena.

The Seesaw Algorithm encompasses a set of theoretical frameworks and computational strategies for generating small neutrino masses by introducing suppression mechanisms that operate via exchange with heavy fields. Originally formulated in the context of neutrino mass generation, a variety of “seesaw” realizations exist (type I, II, III, hybrid, inverse, double, and models embedded in higher-dimensional or left-right symmetric theories). Each framework achieves the lightness of observed neutrino masses through the dynamical interplay between Dirac-type weak-scale couplings and large mass scales or symmetry-based suppressions, with characteristic mass matrix structures. The term “seesaw algorithm” further extends to algorithmic and numerical solutions for extracting full model parameters (e.g., Dirac mass matrices) from experimentally determined low- and high-energy mass matrices, especially when symmetries are broken and analytic solutions are unavailable.

1. Core Principles of Seesaw Mechanisms

The central feature of any seesaw mechanism is a mass matrix of the schematic form: $$M_\nu = \begin{pmatrix} 0 & m_D \ m_D^T & M_R \end{pmatrix},$$ where $m_D$ is a Dirac mass matrix (typically of electroweak scale), and $M_R$ is a large Majorana mass matrix for SM-singlet (right-handed) neutrinos. Diagonalization yields: $$m_\nu^{\text{light}} \approx - m_D M_R^{-1} m_D^T, \quad m_\nu^{\text{heavy}} \approx M_R.$$ In type-II and type-III implementations, heavy scalar triplets or fermion triplets, respectively, replace the $M_R$ structure, but the suppression principle is analogous.

Variants such as the “triple seesaw” (Cogollo et al., 2010) combine mechanisms, embedding e.g. a type-II VEV suppression inside a type-I framework, yielding cubic ($M^{-3}$) suppression of light neutrino masses: $m_\nu \sim \frac{y^2 v^4}{M^3},$ where $v$ denotes electroweak-scale VEVs, and $y$ the Yukawa couplings.

2. Classes and Generalizations of Seesaw Algorithms

Table: Summary of Seesaw Mechanism Variants

Type	Heavy Field(s)	Effective $m_\nu$ scaling
I	RH neutrinos	$m_D^2 / M_R$
II	Scalar triplet	$Y_\Delta v_\Delta}$ ($\propto \mu v^2/m_\Delta^2$)
III	Fermion (SU(2) triplet)	$m_D^2 / M_\Sigma$
Double	Two-step singlet chain	$m_D^2 / (M^2/\mu)$
Inverse	Small $\mu$ in singlet sector	$m_D^2 \cdot \mu / M^2$
Triple	Embedded II-in-I	$y^2 v^4/M^3$
Split (ED)	Split RHN masses (ED)	$\sim m_D^2/M$ (with extra-dim. suppression cancellation)

These frameworks appear in unconstrained field-theoretic models, GUTs (SO(10), SU(5)), and models with extra dimensions or extended symmetries (Hambye, 2012, Kusenko et al., 2010, Watanabe et al., 2010).

3. Seesaw Algorithms for Parameter Extraction

In minimal left-right models or scenarios with explicit breaking of left-right or CP/parity symmetry, determining the Dirac mass matrix $M_D$ from known $M_\nu$ and $M_N$ is a nonlinear matrix problem. The “prescriptive numerical algorithm” (Kiers et al., 2022) proceeds as follows:

Numerical Algorithm Steps

1. Hermitization: Form a Hermitian matrix

$$\mathbf{M} = (M_D + e^{-ia} t_\beta m_e) U_e^\dagger$$

with $a$, $t_\beta$ as Higgs sector parameters, $m_e$ the charged-lepton mass matrix, and $U_e$ a unitary matrix to be determined.

1. Symmetrization & Shifting: Define

$$H = (M_N^*)^{-1/2} \mathbf{M} (M_N)^{-1/2}, \quad \tilde{H} = H - B$$

with $$B = e^{ia} t_\beta (M_N^*)^{-1/2} U_e m_e (M_N)^{-1/2}$$.

1. Diagonalization and Factorization: Decompose $$S = (v_L e^{-i\theta_L}/v_R)I - (M_N^*)^{-1/2} U_e M_\nu^* U_e^T (M_N)^{-1/2}$$ as $O s O^T$, where $O$ is a complex orthogonal matrix, $s$ is diagonal.
2. Parameterizing Ambiguity: Express $\tilde H$ as $O \sqrt{s} \tilde{E} O^\dagger$ where $\tilde{E}$ is a complex orthogonal matrix parametrized by complex Euler angles.
3. Update Process: Iterate to update $U_e$ and $M_D$ via the relations above, checking for convergence to the (nearly) best solution that matches physical constraints.

This “seesaw algorithm” enables full parameter space reconstruction in general non-hermitian cases, independent of any parity or CP invariance. It robustly avoids instabilities common to direct Riccati or least-squares minimizations (Kiers et al., 2022).

4. Model Construction and Phenomenological Implications

In the triple seesaw realization (Cogollo et al., 2010), embedding a type-II seesaw into a type-I structure enables cubic scale suppression, allowing the heavy new-physics scale to be as low as the TeV domain. This predicts observable TeV-mass heavy neutrinos, new scalar fields, and possibly extra gauge bosons (as in the 3-3-1 models with right-handed neutrinos), all within the reach of the LHC. Light neutrino mass arises naturally with new physics at accessible scales, as

$$m_\nu \sim \frac{y^2 v^4}{M^3} \,\,\text{(toy model)}$$

or, in a realistic 3-3-1 extension: $m_\nu \sim -h(v^2 v') / M^3,$ with $v'$ itself dynamically suppressed via a type-II-type minimization.

This construction enables the exploration of lepton flavor violation, rare decays, and collider signals linked directly to the model’s Yukawa sector and symmetry breaking structure.

5. Theoretical Extensions and Generalizations

The seesaw algorithmic formalism generalizes to higher-dimensional theories (Kusenko et al., 2010, Watanabe et al., 2010), inverse/double seesaws, hybrid seesaws, and models exploiting extra symmetries or spacetime geometry. Extensions include:

• Five-dimensional bulk right-handed neutrino scenarios (Watanabe et al., 2010): Seesaw-induced masses are efficiently captured using lepton-number-violating propagators, incorporating the effects of boundary conditions and possible warped metrics.
• Hybrid seesaw algorithms: Integration of high-scale and TeV-scale modules, with radiative or symmetry-based generation of effective $\mu$ parameters or mass splittings, yielding rich LHC-accessible phenomenology (Agashe et al., 2018).
• Algorithmic construction in flavor symmetric models: Symmetry conditions (e.g., $\mu$-$\tau$ reflection, trimaximal mixing) allow reduction of parameter space and link high-energy seesaw structure to low-energy observables (Xing et al., 2020).

6. Experimental Probes and Validation

The key prediction of seesaw (and triple seesaw) algorithms is the possibility of direct experimental validation when new physics is at the TeV scale. This includes:

• Collider searches: Production of TeV-scale heavy neutrinos, scalars, or new gauge bosons associated with extended gauge or scalar sectors (Cogollo et al., 2010).
• Lepton flavor violation and rare processes: Enhanced rates are predicted via accessible heavy-light mixing.
• Neutrino oscillation and neutrinoless double beta decay: Model-specific mass matrices constructed via the seesaw algorithm can be directly confronted with mixing angles, mass-squared differences, and effective Majorana mass constraints.

In summary, the term “seesaw algorithm” comprises both the exact or iterative computational approaches for reconstructing the mass structures of seesaw models under general symmetry or model-breaking conditions and the broader class of mechanisms that employ hierarchical suppression to consistently explain small neutrino masses while predicting structurally and experimentally rich phenomena.