Neutrino Mass Selection Rule

• Neutrino Mass Selection Rule is a set of symmetry and algebraic constraints that define allowed neutrino mass structures and mixing relationships in effective field theories.
• The rules emerge from unique effective operators, flavor sum rules, and matrix texture constraints, enabling precise predictions in neutrino phenomenology.
• They guide experimental probes in oscillation, cosmology, and 0νββ decay, narrowing the parameter space for testing new physics beyond the Standard Model.

Neutrino mass selection rules are algebraic or symmetry-enforced constraints on the structure and allowed relationships among the masses and mixing parameters of neutrinos. These rules arise from deep group-theoretical, algebraic, or topological properties of the underlying field theory and have significant implications for the generation of neutrino masses, their ordering, and the predictive power of flavor models. They can take the form of linear or nonlinear "sum rules" among mass eigenvalues, constraints forbidding certain operators, or radiative selection rules enforced by non-invertible fusion algebras.

1. Fundamental Formulations: Operator Uniqueness and Vacuum Structure

The most fundamental selection rule within the Standard Model Effective Field Theory (SMEFT) context is the uniqueness of the effective neutrino mass operator at each mass dimension. At dimension five, the only allowed operator is the Weinberg operator: $$O_5 = \frac{1}{\Lambda} (\overline{L^c} \epsilon H)(L \epsilon H)$$ where $L$ is the lepton doublet, $H$ is the Higgs doublet, and $\Lambda$ is the scale of new physics. At higher odd mass-dimensions $2n+5$, exactly one gauge-invariant operator exists, of the form

$$O_{2n+5} = \frac{1}{\Lambda^{2n+1}} (\overline{L^c} \epsilon H)(L \epsilon H)(H^\dagger H)^n$$

This uniqueness is guaranteed by SU(2) Young-tableau arguments and has powerful implications: if the dimension-5 term is forbidden (e.g., by a symmetry), the lowest allowed operator defines both the magnitude and the structure of neutrino mass and restricts the phenomenology of new physics to a single operator per dimension. Thereby, the neutrino-mass selection rule enforces that lepton-number-violating Majorana mass arises solely from specific operator structures at a given dimension, closing off alternative contraction possibilities (Liao, 2010).

Beyond operator-level constraints, more recent theoretical frameworks have explored the connection between the structure of the neutrino-vacuum, oscillations, and symmetry. In the triangular formulation, flavor oscillation is equivalent to persistent vacuum reshuffling, with the neutrino not selecting a single vacuum state but residing in a superposition. This leads to a rule whereby mass generation via the conventional Higgs mechanism is forbidden for neutrinos, since such a mechanism would require a unique vacuum selection. The Quantum Yang-Baxter Equations (QYBE) enforce closure of a triangle with side lengths given by inverse masses, encapsulating the mass selection rule: $$\frac{1}{m_1^2} = \frac{1}{m_2^2} + \frac{1}{m_3^2}$$ with only a normal hierarchy being compatible with this constraint (Arraut, 2024).

2. Sum Rules from Flavor Symmetry: Linear and Nonlinear Mass Relations

Extensive model-building efforts employing discrete flavor symmetries—such as $A_4$, $S_4$, and $T'$—often yield mass selection rules ("sum rules") that relate the three neutrino mass eigenvalues algebraically. Common forms include:

• $2m_2 + m_3 = m_1$
• $m_1 + m_2 = m_3$
• $\frac{2}{m_2} + \frac{1}{m_3} = \frac{1}{m_1}$
• $\frac{1}{m_1} + \frac{1}{m_2} = \frac{1}{m_3}$

Such relations naturally emerge from the structure of the flavor group representations and the patterns of vacuum alignment (flavon VEVs) enforced by the symmetry. For instance, in $A_4$ and $S_4$ models, certain vacuum alignments force combinations of Clebsch–Gordan coefficients that, after diagonalization, yield the explicit mass relations above. The direct consequence is that the allowed region in the space of neutrino observables $(\Sigma, m_{ββ}, m_β)$ collapses onto a thin band fixed by the sum rule. The sum rule can be so restrictive that only one mass ordering (normal or inverted) is compatible in the exact limit, with deviations opening up quasi-degenerate possibilities. Corrections due to higher-order terms or renormalization-group running can relax but not eliminate these constraints (Barry et al., 2010).

A modern example is the modular $A_4$ type-II seesaw, where the mass sum rule

$m_{\rm heaviest} = \frac{1}{2}(m_1 + m_2 + m_3)$

emerges from the flavor-symmetric structure of the mass matrix and constrains the sum of the masses to be $\sim 0.10$–$0.12$ eV, testable in cosmology and neutrinoless double beta decay (Chuliá et al., 2023, Chuliá et al., 26 Dec 2025). In this context, best-fit oscillation data select either $m_3 = m_1 + m_2$ (normal ordering) or $m_2 = m_1 + m_3$ (inverted ordering), with only the latter being compatible when mixing angles and CP-phase constraints are imposed (Chuliá et al., 26 Dec 2025).

3. Matrix Texture Selection: Texture Zeros, Cyclic and $\mu$–$\tau$ Symmetries

Another category of neutrino mass selection rules operates at the level of constraints on the entries ("texture") of the neutrino mass matrix. For example, $\mu$–$\tau$ symmetry plus cyclic permutation invariance of the heavy Majorana matrix restricts the allowed patterns, potentially enforcing degeneracy between masses unless additional "zero-sum" rules are imposed: $m_1 + m_2 + m_3 = 0$ which then splits the degeneracy and uniquely selects an inverted mass ordering $|m_3|<|m_1|<|m_2|$ (Damanik, 2010).

Texture-zero analyses with one vanishing entry and one vanishing mass eigenvalue further reveal selection rules. For the Majorana neutrino mass matrix, only specific patterns (out of six) are allowed by data when combined with the condition $m_3=0$. For the inverted ordering, only four patterns (P₂–P₅) survive, realizable via minimal type-I seesaw plus an Abelian $Z_8$ symmetry. Each pattern predicts distinct ranges for the $0\nu\beta\beta$ effective mass $|M_{ee}|$ and for the Dirac and Majorana CP-violating phases (Gautam et al., 2015).

4. Non-invertible Selection Rules and Radiative Mass Generation

Neutrino mass selection rules can also arise from non-invertible fusion algebras (such as Ising or Tambara–Yamagami (TY) type) that label fields with algebraic indices and control which interactions are allowed at tree level. The key principle is that a coupling is present only if the fusion of all field labels contains the identity. For example, in the Ising algebra, tree-level neutrino Yukawa couplings can be forbidden, but radiative corrections, involving several fields in nontrivial representations, can allow the operator in the loop. Consequently, Majorana neutrino masses are generated at one or two loops, naturally explaining their smallness. A surviving Abelian subgroup (often $\mathbb{Z}_2$) then stabilizes the lightest new particle, providing a dark matter candidate (Kobayashi et al., 20 May 2025, Jangid et al., 20 Oct 2025).

A typical selection rule in this context is: $$\text{A coupling } \phi_1 \phi_2 \cdots \phi_n \text{ is allowed at tree level iff } a(\phi_1)\otimes\cdots\otimes a(\phi_n) \ni \mathbb{I}$$ where $a(\phi)$ is the fusion label of field $\phi$. Radiative diagrams containing internal lines with labels that collectively fuse to $\mathbb{I}$ generate effective neutrino mass operators at loop level, consistent with experimental constraints.

5. Mass-Mixing Sum Rules and Phenomenological Consequences

Some selection rules extend beyond pure mass relations to involve mixing angles and CP phases, yielding "mass-mixing sum rules." In minimal $SO(10)$ GUTs, for instance, pure type-I seesaw dominance, a symmetric up–quark-like Dirac mass, and B–L breaking yield: $$\sin^2\theta_{12}\,\frac{1}{m_1} + \cos^2\theta_{12}\,\frac{e^{-2i\alpha}}{m_2} + \frac{e^{-2i\beta}}{m_3} = 0$$ with this sum rule being satifiable only in the normal hierarchy and fixing a lower limit on the lightest neutrino mass $m_1\gtrsim 7.5\times10^{-4}\,$eV. This, in turn, implies a lower bound for the $0\nu\beta\beta$ observable $m_{ee}$ (Buccella et al., 2017).

Similarly, in $T'$ hybrid seesaw models the mass selection rule is enforced at the matrix element level: $m_\nu^{13} + m_\nu^{22} + m_\nu^{31} = 0$ This complex constraint couples mixing angles and phases, severely restricting the allowed parameter space and yielding sharp predictions for $m_{ee}$ and the Dirac and Majorana phases in both normal and inverted ordering (Nomura et al., 1 Jan 2026).

6. Experimental Tests and Falsifiability

Neutrino mass selection rules are directly testable through a combination of oscillation data (determining mass splittings and mixing angles), cosmological probes (constraining the sum of the neutrino masses), direct kinematic mass measurements (KATRIN, Project 8), and neutrinoless double beta decay searches ($0\nu\beta\beta$). Each selection rule translates into specific predictions for the sum of masses $\sum m_i$, the allowed range for $|m_{ee}|$, and even minimal values of $m_{\rm lightest}$.

For example:

• The sum-of-angles law $\theta_{12} + \theta_{23} + \theta_{13} = \pi/2$ in the QYBE-triangular formulation is an immediate, falsifiable test; deviation would invalidate the underlying vacuum reshuffling hypothesis (Arraut, 2024).
• Modular $A_4$ and minimal $A_4$/type-II models are about to be tested in cosmology ($\sum m_i \approx 0.10$–$0.12\,$eV), $0\nu\beta\beta$ ($|m_{ee}|\approx45$ meV), and kinematic experiments ($m_{\beta} \approx 0.05$ eV) (Chuliá et al., 2023, Chuliá et al., 26 Dec 2025).
• In the $SO(10)$ sum-rule scenario, a definitive observation of inverted hierarchy or $m_{ee}$ below the predicted band would falsify the model (Buccella et al., 2017).
• Non-invertible fusion algebra selection rules make predictions for radiative textures (one-zero mass matrix elements) and guarantee dark-matter stability—a future experimental observation of forbidden decay modes or nonzero tree-level couplings would rule out such constructions (Kobayashi et al., 20 May 2025, Jangid et al., 20 Oct 2025).

7. Theoretical and Phenomenological Impact

Neutrino mass selection rules have a foundational role in contemporary model building and neutrino phenomenology:

• They reduce the parameter space, correlating otherwise independent observables and enabling robust experimental tests.
• Sum rules pin down or restrict the allowed mass ordering (normal vs. inverted), fix absolute neutrino masses for given oscillation parameters, and tightly couple masses to CP-phases and mixing angles.
• Selection rules originating from non-invertible algebras or radiative mechanisms not only explain the suppression of neutrino masses but also provide a mechanism for dark matter stability.
• Experimental determination of the neutrino ordering, mass scale, and $0\nu\beta\beta$ parameters in the next decade will definitively confirm or rule out entire classes of models predicated on these selection rules, distinguishing among $A_4$, $S_4$, $T'$, $SO(10)$, radiative, and modular scenarios.

In summary, neutrino mass selection rules—whether arising from effective operator uniqueness, symmetry-imposed algebraic identities, matrix texture constraints, or radiative selection by fusion algebra—constitute a central framework for understanding the origin, structure, and phenomenological consequences of neutrino masses. Their testability with forthcoming experimental data renders them a focal point in the quest for new physics beyond the Standard Model.