Type-I Seesaw Mechanism

• Type-I Seesaw Mechanism is a theoretical framework that extends the Standard Model by introducing heavy sterile (Majorana) neutrinos to explain the smallness of active neutrino masses.
• It employs an LDLᵀ decomposition to recast the effective neutrino mass matrix in a basis-independent manner, facilitating analytic spectral solutions and simplifying model building.
• The approach enables precise studies of flavor symmetries, fine-tuning, and texture analyses, offering clear insights into the contributions of each heavy-neutrino state.

The Type-I seesaw mechanism is a theoretical framework that explains the smallness of active neutrino masses by extending the Standard Model (SM) with heavy right-handed (sterile) neutrinos which possess Majorana mass terms. The neutrino mass suppression emerges through the tree-level exchange of these heavy states, resulting in a low-energy effective neutrino mass matrix inversely proportional to the large Majorana mass scale. While the basic algebraic form of the seesaw relation is well established, modern analyses have focused on basis-independent parameterizations, new techniques for diagonalization, implications of flavor symmetries and fine-tuning, efficient model-building strategies, and the physical interpretability of its algebraic structure.

1. Lagrangian Structure and Conventional Seesaw Formula

The starting point is the extended SM Lagrangian incorporating Dirac and Majorana mass terms for neutrinos. In two-component notation, the relevant terms are: $$-\mathcal{L} \supset \nu_L^{\,T}\,m_D\,N_R \;+\;\tfrac12\,N_R^{\,T}\,M_R\,N_R\;+\;\mathrm{h.c.}$$ where:

• $\nu_L$ is the three-component (column) vector of left-handed neutrino fields,
• $N_R$ is the vector of $n\geq3$ right-handed neutrinos,
• $m_D$ is the $3\times n$ Dirac mass matrix,
• $M_R$ is the $n\times n$ symmetric invertible Majorana mass matrix for $N_R$.

Integrating out heavy $N_R$ generates the standard Type-I seesaw formula for the effective $3\times3$ light neutrino mass matrix: $m_\nu = -\,m_D\,M_R^{-1}\,m_D^{T}$ The overall sign can be absorbed and is typically dropped for model-building and phenomenological applications.

2. LDL$^T$ Decomposition and Basis-Independent Formulation

The LDL$^T$ (generalized Cholesky) decomposition is applied to $M_R^{-1}$: $M_R^{-1} = L\,D\,L^{T}$ where:

• $D = \mathrm{diag}(d_1, d_2, \dots, d_n)$ is real diagonal,
• $L = I + N$ with $N$ strictly lower-triangular.

For $n=3$, $L$ is explicitly: $$L = \begin{pmatrix} 1 & 0 & 0 \ \ell_{21} & 1 & 0 \ \ell_{31} & \ell_{32} & 1 \end{pmatrix}$$ with $N$ collecting the subdiagonal elements.

The seesaw mass matrix becomes: $$m_\nu = m_D\,L\,D\,L^T m_D^T = \widetilde{m}_D\,D\,\widetilde{m}_D^T$$ with the rotated Dirac mass matrix $\widetilde{m}_D \equiv m_D L$.

All physical quantities, including the spectrum and mixing, are now written in terms of $\widetilde{m}_D$ and $D$, both constructed directly from the original basis inputs, yielding a manifestly basis-independent framework.

3. Practical Advantages and Spectral Solution

The explicit LDL$^T$ decomposition yields several algebraic simplifications and model-building tools. Key properties include:

• The inversion of $L = I + N$ requires only a finite Neumann series due to $N$'s nilpotency:

$L^{-1} = I - N + N^2 - \cdots + (-1)^{n-1} N^{n-1}$

For $n=3$:

$$L^{-1} = \begin{pmatrix} 1 & 0 & 0 \ -\ell_{21} & 1 & 0 \ \ell_{21}\ell_{32} - \ell_{31} & -\ell_{32} & 1 \end{pmatrix}$$

This explicit inversion allows direct parametrization of $m_D$ in terms of measured or model-motivated $\widetilde{m}_D$.

• The eigenvalues (light neutrino masses) are:

$$m_i = d_1\,(\boldsymbol{a}\cdot\boldsymbol{v}_i)^2 + d_2\,(\boldsymbol{b}\cdot\boldsymbol{v}_i)^2 + d_3\,(\boldsymbol{c}\cdot\boldsymbol{v}_i)^2$$

where the rows of $\widetilde{m}_D$ are treated as vectors $\boldsymbol{a}$, $\boldsymbol{b}$, $\boldsymbol{c}$ in $\mathbb{C}^3$, and $\{\boldsymbol{v}_i\}$ are the normalized eigenvectors diagonalizing $m_\nu$. This avoids the need to solve cubic or higher-order characteristic equations and provides direct algebraic access to the entire spectrum.

• This decomposition separates $m_\nu$ into a sum of three rank-one matrices, exposing the direct contribution of each heavy-neutrino state to the light-mass structure.

4. Phenomenological Applications: Symmetries, Fine-Tuning, and Textures

The LDL$^T$ form enables efficient theoretical and phenomenological analysis:

• Flavor and Generalized CP Symmetries: Since $L$ and $D$ are basis-independent, flavor symmetries or discrete CP operations impose direct, transparent constraints on the columns of $\widetilde{m}_D$ and on the diagonal $D$. Implementation of texture zeros, discrete symmetry patterns, or CP-invariant subspaces is algebraically straightforward.
• Fine-Tuning and Naturalness: The contribution of each rank-one $d_i\,\boldsymbol{x}_i\boldsymbol{x}_i^T$ to $m_\nu$ is manifest and can be quantified, allowing naturalness criteria (the degree of cancellation among large entries in $m_D M_R^{-1} m_D^T$) to be formulated as the relative sizes of $d_i$ and the projections $(\boldsymbol{x}_i\cdot\boldsymbol{v}_j)$. Large tuning (i.e., small neutrino masses from large terms) can be traced to near-orthogonality or accidental cancellations in these algebraic structures.
• Model Building: The formalism enables efficient construction and test of models with desired low-energy neutrino properties by working directly with $\widetilde{m}_D$ and $D$ as fundamental objects. This is of particular importance for exploring the implications of specific symmetry groups or investigating phenomenological consequences of "special textures".

5. Two-Generation Limit and Algebraic Pedagogy

In the special case $n=2$, the LDL$^T$ matrix $L$ is $2\times2$ with only one nontrivial lower-triangular element, and $N^2=0$. The inversion $L^{-1}=I-N$, and the full inverse is: $$M_R^{-1} = \begin{pmatrix} 1 & 0 \ n & 1 \end{pmatrix} \begin{pmatrix} d_1 & 0 \ 0 & d_2 \end{pmatrix} \begin{pmatrix} 1 & n \ 0 & 1 \end{pmatrix}$$ Consequently, the seesaw mass $m_\nu=\widetilde{m}_D D \widetilde{m}_D^T$ is a sum of two rank-one contributions, with all eigenvalues and the single physical mixing angle obtained in closed algebraic form, without matrix diagonalization. This facilitates fully analytic studies of limiting cases such as leptogenesis models or illustrative examples with texture zeros.

6. Information-Rich Summary of Basis-Independence and Algebraic Utility

By recasting the Type-I seesaw formula as

$M_R^{-1}=L D L^T$

and

$m_\nu=(m_D L) D (m_D L)^T$

the LDL$^T$ approach provides a complete, basis-insensitive encoding of the mechanism. Its explicit algebraic structure grants access to analytic expressions for all neutrino masses and mixing parameters, bypasses the need for higher-order characteristic equations, and exposes how each heavy-neutrino state's properties are mapped to the observable sector. This approach constitutes an efficient, systematic toolbox for both theoretical (symmetry, naturalness) and practical model-building investigations, streamlining analysis of flavor/CP symmetries, rank structures, tuning, and phenomenological textures (Yang, 2021).

---

This LDL$^T$ decomposition framework is thus an advanced and highly practical reformulation of the Type-I seesaw mechanism, addressing the algebraic, model-building, and phenomenological needs of contemporary neutrino physics.