MAS-ZERO: Maximal Zero Textures in Seesaw

• MAS-ZERO is a framework that systematically classifies maximal zero textures in charged-lepton and neutrino seesaw matrices, reducing free parameters while ensuring nondegeneracy and experimental viability.
• It minimizes the number of free parameters by enforcing sparsity in mass matrices, which aids in identifying underlying flavor symmetries and guides predictive model building.
• Phenomenological analyses within MAS-ZERO indicate only normal mass hierarchies with small CP-violating phases, highlighting challenges in accommodating large CP violation.

MAS-ZERO denotes the systematic classification of "maximal zero textures" in the constituent matrices of both linear and inverse seesaw mechanisms for neutrino mass generation. In this approach, the maximum number of texture zeros is imposed on the charged-lepton mass matrix $m_e$ and the relevant seesaw matrices, under the constraints of nonzero and nondegenerate neutrino and charged-lepton masses as well as compatibility with oscillation data. The resulting scheme significantly restricts model parameters and serves as a template for model building, especially those invoking flavor symmetries to enforce texture zeros (Sinha et al., 2015).

1. Zero Textures and Minimalist Parametrization

A zero texture is an ansatz where certain matrix elements are identically zero, typically motivated by underlying discrete symmetries or minimality arguments. The goal of "maximal zero textures" is to maximize the number of vanishing entries in mass matrices while ensuring the invertibility of $m_e$ and $m_\nu$ (the effective light-neutrino mass matrix). This sparsity minimizes the number of free parameters and can hint at possible flavor symmetries.

For the $3\times3$ charged-lepton mass matrix $m_e$, the requirement $\det(m_e m_e^\dagger)\neq0$ allows, up to row and column permutations, six distinct invertible forms each with three nonzero entries and six zeros. These textures are strictly diagonal up to permutations, reflecting the “flavor basis” where $U_\ell$ diagonalizes $m_e m_e^\dagger$ and does not affect the PMNS mixing.

2. Seesaw Mechanisms with Maximal Zeros

Inverse Seesaw

The inverse seesaw introduces left-handed neutrinos $\nu_L$, right-handed singlets $\nu_R$, and additional singlets $m_e$0. The $m_e$1 neutral-fermion mass matrix (flavor basis) is

$m_e$2

The effective light-neutrino mass matrix is, to leading order,

$m_e$3

where $m_e$4 is the Dirac mass ($m_e$5–$m_e$6), $m_e$7 is the Dirac mass ($m_e$8–$m_e$9), and $m_\nu$0 is a lepton-number-violating Majorana mass for $m_\nu$1.

Linear Seesaw

In the linear seesaw, $m_\nu$2 and all explicit Majorana masses for $m_\nu$3 vanish, so lepton-number violation enters via a small matrix $m_\nu$4 coupling $m_\nu$5 to $m_\nu$6: $m_\nu$7 The effective neutrino mass is

$m_\nu$8

where $m_\nu$9 and $3\times3$0 is a small L-violating term.

3. Systematics of Maximal Zero Assignment

The imposition of maximal zeros proceeds as follows. For $3\times3$1, exactly six invertible patterns exist (up to permutations), each with three nonzero entries—all strictly diagonal: $3\times3$2 Similarly, in the inverse seesaw, maximizing zeros in $3\times3$3 and $3\times3$4 while retaining invertibility yields six patterns each. In the linear seesaw, imposing invertibility and maximal zeros on $3\times3$5, $3\times3$6, and $3\times3$7 gives six invertible six-zero patterns for $3\times3$8 and $3\times3$9, and up to 36 viable five-zero patterns for $m_e$0 out of 126 possibilities.

4. Enumeration of Phenomenologically Allowed Neutrino Mass Textures

The focus of MAS-ZERO is the classification of feasible two-zero textures for $m_e$1 consistent with oscillation data.

Seesaw Type	Number of Allowed Two-Zero $m_e$2 Textures	Key Structural Constraints
Inverse	7	At most 2 independent zeros in $m_e$3
Linear	1	Exactly 5 zeros in $m_e$4; $m_e$5 diagonal

Inverse Seesaw Case

In this scenario, $m_e$6 may have up to two independent zeros (12 configurations). For each choice $m_e$7, the light neutrino mass is computed: $m_e$8 Searching for viable two-zero Frampton–Glashow–Marfatia patterns yields exactly seven distinct textures, e.g.: $m_e$9 with analogous forms for $\det(m_e m_e^\dagger)\neq0$0 through $\det(m_e m_e^\dagger)\neq0$1 (see Table 10 of (Sinha et al., 2015)).

Linear Seesaw Case

Maximal zeros lead to a unique two-zero $\det(m_e m_e^\dagger)\neq0$2 texture, with zeros at (2,2) and (3,3): $\det(m_e m_e^\dagger)\neq0$3 where $\det(m_e m_e^\dagger)\neq0$4 carries exactly five zeros, e.g.: $\det(m_e m_e^\dagger)\neq0$5 Both $\det(m_e m_e^\dagger)\neq0$6 and $\det(m_e m_e^\dagger)\neq0$7 are required to be strictly diagonal with three nonzero entries.

5. Phenomenological Consequences and CP Violation

The seven inverse seesaw and sole linear seesaw patterns are parametrized in terms of four real variables and a physical phase $\det(m_e m_e^\dagger)\neq0$8. A fit to global oscillation data ($\det(m_e m_e^\dagger)\neq0$9; Forero et al. 2014) under these texture restrictions yields the following:

• The sum of neutrino masses $U_\ell$0 remains comfortably below current cosmological bounds ($U_\ell$1 eV).
• Only the normal mass hierarchy emerges.
• None of the two-zero patterns allows for sizeable Dirac CP-violation: $U_\ell$2 a few degrees, $U_\ell$3.
• Maximal CP violation ($U_\ell$4) cannot be accommodated. If future experiments establish such a value, the assumption of maximal zeros in $U_\ell$5 (i.e., strictly diagonal) must be relaxed, necessitating additional charged-lepton sector parameters.

A plausible implication is that minimal "maximal zero" frameworks cannot explain large observed CP violation without extending the charged-lepton sector or further relaxing zero restrictions (Sinha et al., 2015).

6. MAS-ZERO as a Model-Building Template

The MAS-ZERO classification offers a comprehensive mapping of maximal invertible zero assignments in seesaw mass matrices that yield phenomenologically viable neutrino mass patterns. Specifically:

• The inverse seesaw can realize all seven two-zero $U_\ell$6 patterns allowed by current data under maximal zero constraints.
• The linear seesaw (under analogous restraints) supports only one viable two-zero $U_\ell$7 texture.
• The minimal assumption of maximal zeros leads directly to highly predictive and constrained models, particularly relevant for UV completions exploiting discrete symmetries.
• The smallness of generated CP-violating phases in these scenarios is a direct result of the limited number of parameters.

MAS-ZERO thus constitutes a robust guide for constructing models with predictive low-energy implications, contingent on the structure of zeros enforced at high scales or by flavor symmetries (Sinha et al., 2015).