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arxiv: 2605.18154 · v1 · pith:RGDAQSQTnew · submitted 2026-05-18 · ✦ hep-ph

The Triadic Texture: Neutrino Predictions, Viable Vacuum, and Phenomenological Constraints

Sagar Tirtha Goswami , Pralay Chakraborty , Subhankar Roy This is my paper

Pith reviewed 2026-05-20 09:56 UTC · model grok-4.3

classification ✦ hep-ph
keywords neutrino mass textureMajorana neutrinosType-I seesawA4 symmetrynormal hierarchyCP phaseslepton flavor violationvacuum alignment
0
0 comments X

The pith

A triadic neutrino mass texture from extended symmetry favors normal hierarchy and predicts the octant of theta23.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proposes a minimal and predictive pattern for the Majorana neutrino mass matrix. This pattern emerges from a Type-I seesaw plus a dimension-six operator in a model with symmetry SU(2)_L ⊗ U(1)_Y ⊗ A4 ⊗ Z10 ⊗ Z7 ⊗ Z5 ⊗ Z3. The texture selects normal ordering of the neutrino masses, fixes whether the atmospheric mixing angle lies above or below forty-five degrees, and limits the Dirac phase along with the two Majorana phases. A sympathetic reader would care because these quantities are the direct targets of precision neutrino oscillation experiments that can test the pattern against data.

Core claim

The authors construct a triadic texture for the effective Majorana neutrino mass matrix using specific vacuum expectation values of scalar fields charged under the extended discrete symmetry. The texture accommodates current oscillation data while enforcing normal hierarchy, determining the octant of θ23, constraining the Dirac CP phase δ, and supplying bounds on the mass eigenvalues together with ranges for the Majorana phases. Only certain vacuum alignments produce both the texture and a viable scalar potential, and the same setup restricts charged lepton flavor violation while suppressing conventional leptogenesis.

What carries the argument

The triadic texture, a specific pattern of entries in the neutrino mass matrix whose zero and non-zero structure encodes the predictions for mixing angles, phases, and mass hierarchy.

If this is right

  • The model selects normal hierarchy over inverted hierarchy for the neutrino masses.
  • It predicts a definite octant for the atmospheric mixing angle θ23.
  • It places restrictions on the Dirac CP-violating phase δ.
  • It supplies concrete bounds on the three neutrino mass eigenvalues.
  • It restricts the allowed ranges for the two Majorana phases.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The built-in suppression of standard leptogenesis implies that the observed matter-antimatter asymmetry would need a different generation mechanism.
  • Confirmation of the texture would encourage further exploration of discrete flavor symmetries to explain the full pattern of fermion mixing.
  • Precision data from upcoming oscillation experiments could tighten the allowed windows or directly exclude the proposed texture.

Load-bearing premise

The assumption that particular vacuum expectation values can simultaneously generate the exact desired texture and produce a stable scalar potential without extra unwanted operators.

What would settle it

A definitive experimental result showing inverted neutrino mass hierarchy or placing θ23 in the opposite octant from the one required by the texture.

read the original abstract

A minimal and predictive neutrino mass matrix texture for Majorana neutrino is proposed. The texture favours the normal hierarchy of neutrino mass eigenvalues. It further predicts the octant of $\theta_{23}$, constraints $\delta$, gives bounds on neutrino mass eigenvalues and also gives ranges for the two Majorana phases.The texture is realised in the framework of a Type-I seesaw and a Weinberg like dimension 6 operator under an extended symmetry of $SU(2)_L \otimes U(1)_Y \otimes A_4 \otimes Z_{10} \otimes Z_7 \otimes Z_5 \otimes Z_3$. The texture can be realised with different sets of vacuum expectation values of the associated scalar fields, but not all such sets lead to a viable scalar sector. The model also constrains the allowed channels of charged lepton flavour violation and leads to a suppressed baryon asymmetry generation through conventional leptogenesis.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript proposes a triadic neutrino mass matrix texture for Majorana neutrinos realized via a Type-I seesaw plus dimension-6 Weinberg operator under the extended symmetry SU(2)_L ⊗ U(1)_Y ⊗ A4 ⊗ Z10 ⊗ Z7 ⊗ Z5 ⊗ Z3. The texture is claimed to favor normal hierarchy, predict the octant of θ23, constrain δ, bound the neutrino mass eigenvalues, and provide ranges for the two Majorana phases. Specific scalar VEVs are selected to produce the texture while maintaining a viable scalar sector; the model also constrains charged-lepton flavor violation and suppresses conventional leptogenesis.

Significance. If the chosen VEVs are shown to be a stable minimum of the full symmetry-allowed scalar potential, the construction would supply a symmetry-based, relatively predictive framework linking discrete flavor charges to concrete neutrino observables (NH preference, θ23 octant, δ range). The combination of seesaw and Weinberg operator plus explicit discussion of LFV and leptogenesis adds phenomenological breadth, though the large discrete symmetry group increases the number of auxiliary fields.

major comments (1)
  1. [Vacuum alignment and scalar sector] Vacuum alignment section: The central claim that the triadic texture is realized in a viable vacuum rests on the assertion that certain VEV sets simultaneously produce the desired neutrino mass matrix and a stable scalar potential. The manuscript selects such sets but does not demonstrate that they satisfy the full set of minimization equations obtained from the most general scalar potential invariant under A4 and all Z_n factors (including all allowed contractions and phase-dependent terms). Without this explicit check, or a scan showing absence of tachyons in the resulting scalar mass matrix, the texture cannot be guaranteed to arise from a consistent vacuum.
minor comments (2)
  1. The notation for the multiple Z_n charge assignments and the listing of allowed operators would benefit from a compact table summarizing charges for all fields.
  2. A few typographical inconsistencies appear in the phase conventions for the Majorana phases when comparing the texture parametrization to the numerical results.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address the major point on vacuum alignment below and will incorporate the requested clarifications in a revised version.

read point-by-point responses

  1. Referee: Vacuum alignment section: The central claim that the triadic texture is realized in a viable vacuum rests on the assertion that certain VEV sets simultaneously produce the desired neutrino mass matrix and a stable scalar potential. The manuscript selects such sets but does not demonstrate that they satisfy the full set of minimization equations obtained from the most general scalar potential invariant under A4 and all Z_n factors (including all allowed contractions and phase-dependent terms). Without this explicit check, or a scan showing absence of tachyons in the resulting scalar mass matrix, the texture cannot be guaranteed to arise from a consistent vacuum.

    Authors: We thank the referee for this observation. The manuscript identifies specific VEV sets that reproduce the triadic texture and notes that not all such sets yield a viable scalar sector, but we agree that an explicit verification against the complete set of minimization equations from the most general symmetry-invariant potential (including all allowed contractions and phases) was not provided. In the revised manuscript we will derive the relevant minimization conditions for the chosen VEVs, solve for the parameters that stabilize them, and present the scalar mass matrix to confirm the absence of tachyonic modes. This will establish that the selected alignments correspond to a local minimum of the potential. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper constructs the triadic texture from explicit charge assignments under the extended symmetry group and selected VEVs that realize a Type-I seesaw plus dim-6 Weinberg operator. These steps are independent of the subsequent numerical fit to oscillation data. The resulting bounds on mass eigenvalues, ranges for Majorana phases, and constraints on delta and theta23 octant are outputs of confronting the fixed texture with experiment rather than redefinitions or self-consistent fits of the texture parameters themselves. No equation reduces to its input by construction, no self-citation chain carries the central claim, and the viability statement for the scalar sector is presented as an additional consistency requirement rather than an ansatz that presupposes the neutrino predictions. The derivation therefore remains self-contained against external data.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The construction rests on a large discrete symmetry group whose charge assignments are chosen to produce the texture, plus assumptions about which scalar VEVs are allowed and which produce a stable potential. No independent evidence is given for the symmetry choice beyond fitting neutrino data.

free parameters (1)
  • Texture parameters and VEV ratios
    Multiple free parameters in the mass matrix entries and scalar vacuum alignments are adjusted to match observed neutrino masses and mixings.
axioms (2)
  • domain assumption The discrete symmetry group SU(2)_L ⊗ U(1)_Y ⊗ A4 ⊗ Z10 ⊗ Z7 ⊗ Z5 ⊗ Z3 is imposed to generate the triadic texture.
    Invoked in the model construction section to forbid unwanted operators.
  • ad hoc to paper Only certain sets of scalar VEVs produce both the desired neutrino texture and a viable scalar potential.
    Stated as a selection criterion without independent justification beyond phenomenology.
invented entities (1)
  • Multiple scalar fields with assigned charges under the discrete groups no independent evidence
    purpose: To break the symmetries and generate the neutrino mass texture via their VEVs.
    New fields introduced to realize the texture; no collider or other independent signature is provided.

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