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arxiv: 2303.02358 · v4 · submitted 2023-03-04 · ✦ hep-ph

Equal Majorana Phases from a Minimal and Predictive Neutrino Texture

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

Pith reviewed 2026-05-24 10:01 UTC · model grok-4.3

classification ✦ hep-ph
keywords neutrino mass matrix textureMajorana phasespartial tri-bimaximal mixingseesaw mechanismA4 symmetrynormal hierarchyCP violation in leptons
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The pith

A minimal Majorana neutrino mass texture forces the two Majorana phases to be equal.

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

The paper proposes a new minimal texture for the Majorana neutrino mass matrix and examines its consequences inside a partial tri-bimaximal mixing pattern. In this pattern two of the three mixing angles are fixed while the reactor angle and Dirac phase remain free. The texture automatically sets the two Majorana phases equal to each other and rules out a vanishing reactor angle. The construction is realized by a hybrid seesaw model under an extended discrete symmetry and selects the normal neutrino mass ordering. Equality of the phases would tighten the relation between the effective mass in neutrinoless double-beta decay and the lightest neutrino mass.

Core claim

The central claim is that the proposed minimal Majorana neutrino mass matrix texture, when imposed together with the partial tri-bimaximal mixing scheme, enforces equality of the two Majorana phases. The texture is incompatible with theta_13 = 0, does not possess mu-tau symmetry, and yields distinct numerical predictions according to the sign of one real entry; the same texture also favors the normal mass hierarchy.

What carries the argument

The minimal Majorana neutrino mass matrix texture realized through a hybrid Type-I plus two Type-II seesaw construction under the symmetry group SU(2)_L ⊗ U(1)_Y ⊗ A_4 ⊗ Z_10 ⊗ Z_7.

If this is right

  • The two Majorana phases must be identical, directly relating the effective Majorana mass in neutrinoless double-beta decay to the lightest neutrino mass.
  • The texture excludes a vanishing reactor mixing angle.
  • Numerical predictions split into two distinct branches according to the sign of one real texture parameter.
  • The model selects the normal neutrino mass hierarchy over the inverted one.

Where Pith is reading between the lines

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

  • Future precision data on neutrinoless double-beta decay could directly test whether the phases are in fact equal.
  • The phase equality may serve as a diagnostic for other discrete-flavor models that employ similar partial mixing assumptions.
  • If the sign-dependent branches survive, they would split the allowed range of the lightest neutrino mass into two experimentally distinguishable windows.

Load-bearing premise

Neutrino mixing follows a partial tri-bimaximal scheme that fixes sin theta_12 exactly to 1/sqrt(3) and sin theta_23 exactly to 1/sqrt(2) while leaving theta_13 and the Dirac phase free.

What would settle it

A measurement showing the two Majorana phases differ by more than a few degrees, or a conclusive determination that the neutrino mass ordering is inverted, would falsify the central prediction.

read the original abstract

We propose a new, minimal Majorana neutrino mass matrix texture and study its predictions within a partial tri-bimaximal (TBM) mixing scheme, where $\sin\theta_{12} = 1/\sqrt{3}$ and $\sin\theta_{23} = 1/\sqrt{2}$, with $\theta_{13}$ and $\delta$ treated as free parameters. The texture forbids $\theta_{13} = 0$ and does not correspond to a $\mu-\tau$ symmetric structure. As a notable feature, the texture predicts the equality of the Majorana phases. In addition, we find that the predictions show distinct behavior depending on the sign of a single real texture parameter. The texture is realized through a hybrid framework of one Type-I seesaw and two Type-II seesaw mechanisms under an extended symmetry group, $SU(2)_L \otimes U(1)_Y \otimes A_4 \otimes Z_{10} \otimes Z_{7}$ with properly chosen model parameters. The texture is seen to favor the normal hierarchy for neutrino masses.

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

2 major / 2 minor

Summary. The paper proposes a minimal Majorana neutrino mass matrix texture realized via a hybrid Type-I plus two Type-II seesaw mechanism under the symmetry SU(2)_L ⊗ U(1)_Y ⊗ A4 ⊗ Z10 ⊗ Z7. Within a partial tri-bimaximal mixing scheme fixing sin θ12 = 1/√3 and sin θ23 = 1/√2 (with θ13 and δ free), the texture is shown to forbid θ13 = 0, to be incompatible with μ-τ symmetry, and to predict equality of the two Majorana phases; predictions depend on the sign of one real texture parameter and favor normal hierarchy.

Significance. If the equality of Majorana phases is shown to be an output of the texture independent of the fixed-angle assumptions, the result would provide a concrete, falsifiable relation among CP phases that could be tested with future oscillation and 0νββ data. The hybrid seesaw construction and single-parameter dependence are strengths that reduce the number of free parameters relative to generic A4 models.

major comments (2)
  1. [mixing scheme paragraph and phase derivation section] The central claim that the texture predicts equal Majorana phases is load-bearing for the paper's title and abstract. The partial TBM scheme (sin θ12 fixed to 1/√3, sin θ23 to 1/√2) is imposed by hand before the texture is applied; the manuscript must demonstrate explicitly (e.g., by allowing θ12 and θ23 to vary within 3σ ranges and recomputing the phases) that the equality survives outside this slice rather than being a consistency condition required to satisfy m_ν = U D U^T with the chosen zero/equality pattern.
  2. [symmetry and texture realization section] The symmetry A4 ⊗ Z10 ⊗ Z7 enforces the texture but does not select the partial TBM angles; the paper should clarify whether the phase equality is a genuine prediction or follows from the external mixing hypothesis. If the latter, the result reduces to a fitted input rather than an independent output.
minor comments (2)
  1. [abstract] The abstract states the central prediction but supplies no derivation steps or explicit matrix form; a brief outline of how the phase equality is obtained from the texture would improve readability.
  2. [numerical predictions section] Notation for the single real texture parameter and its sign dependence should be introduced earlier and used consistently in all numerical results.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major point below, agreeing that additional clarification and checks are needed to strengthen the presentation of the phase equality result.

read point-by-point responses

  1. Referee: The central claim that the texture predicts equal Majorana phases is load-bearing for the paper's title and abstract. The partial TBM scheme (sin θ12 fixed to 1/√3, sin θ23 to 1/√2) is imposed by hand before the texture is applied; the manuscript must demonstrate explicitly (e.g., by allowing θ12 and θ23 to vary within 3σ ranges and recomputing the phases) that the equality survives outside this slice rather than being a consistency condition required to satisfy m_ν = U D U^T with the chosen zero/equality pattern.

    Authors: We agree that the partial TBM angles are imposed as an external assumption to derive concrete predictions from the texture. The equality of Majorana phases arises when the texture's zero and equality pattern is imposed on the mass matrix constructed with those fixed angles. To address the concern directly, we will add a numerical scan in the revised version allowing θ12 and θ23 to vary within their 3σ ranges while keeping the texture conditions, and report whether (and under what conditions) the phase equality persists. This will clarify the dependence on the mixing scheme. revision: yes

  2. Referee: The symmetry A4 ⊗ Z10 ⊗ Z7 enforces the texture but does not select the partial TBM angles; the paper should clarify whether the phase equality is a genuine prediction or follows from the external mixing hypothesis. If the latter, the result reduces to a fitted input rather than an independent output.

    Authors: The discrete symmetry group is used solely to enforce the specific texture pattern via the hybrid Type-I plus Type-II seesaw. The partial TBM angles are an additional phenomenological choice to reduce free parameters and obtain predictions for the phases. We will revise the relevant sections to explicitly separate these elements, stating that the equal Majorana phases is a prediction of the texture under the partial TBM scheme rather than a symmetry output alone. This framing makes the result a testable relation within the adopted framework. revision: yes

Circularity Check

1 steps flagged

Equality of Majorana phases is a consistency condition imposed by the partial TBM mixing ansatz, not an independent prediction from the texture.

specific steps
  1. self definitional [Abstract; mixing-scheme paragraph]
    "We propose a new, minimal Majorana neutrino mass matrix texture and study its predictions within a partial tri-bimaximal (TBM) mixing scheme, where sinθ12 = 1/√3 and sinθ23 = 1/√2, with θ13 and δ treated as free parameters. ... As a notable feature, the texture predicts the equality of the Majorana phases."

    The partial TBM fixes two mixing angles by hand. The texture is defined such that m_ν = U D U^T matches the chosen zero/equality pattern only for that specific U. The equality of Majorana phases (entries of the diagonal D after diagonalization) is therefore the consistency requirement that allows the fixed-angle U to satisfy the texture; it is not an output independent of the angle-fixing assumption.

full rationale

The paper defines a neutrino mass texture and analyzes it exclusively inside a partial TBM scheme that hard-fixes sin θ12 = 1/√3 and sin θ23 = 1/√2 while leaving θ13 and δ free. The texture is constructed so that m_ν = U D U^T reproduces the desired zero/equality pattern precisely when the PMNS matrix takes that fixed-angle form. Consequently the reported equality of the two Majorana phases is the algebraic condition needed to close the system under those fixed angles; it does not emerge as a free output that would survive if θ12 or θ23 were allowed to vary. The underlying discrete symmetry is invoked only to realize the texture, not to derive the angle values themselves. This matches the self-definitional pattern: the claimed prediction reduces directly to the input mixing assumption by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the partial TBM mixing assumption, the specific discrete symmetry group, and the hybrid seesaw construction; no free parameters are named in the abstract but the sign of one real texture parameter is highlighted as splitting predictions.

free parameters (1)
  • sign of single real texture parameter
    Abstract states that predictions show distinct behavior depending on the sign of this parameter.
axioms (2)
  • domain assumption Partial tri-bimaximal mixing with sin theta_12 = 1/sqrt(3) and sin theta_23 = 1/sqrt(2)
    Invoked to define the mixing scheme in which the texture is studied.
  • domain assumption Extended symmetry group SU(2)_L x U(1)_Y x A4 x Z10 x Z7 with properly chosen model parameters
    Used to realize the texture via hybrid Type-I and Type-II seesaw.

pith-pipeline@v0.9.0 · 5724 in / 1458 out tokens · 19413 ms · 2026-05-24T10:01:35.455892+00:00 · methodology

discussion (0)

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