Revisiting One-Zero and Two-Zero Neutrino Mass Textures in Light of Recent Oscillation and Cosmological Data

Technical Summary: Revisiting One-Zero and Two-Zero Neutrino Mass Textures

Problem Statement
The Standard Model (SM) fails to account for non-zero neutrino masses, a fact established by neutrino oscillation experiments. While the dimension-five Weinberg operator provides an effective low-energy description of neutrino masses, the underlying structure of the neutrino mass matrix ($M_\nu$) remains unknown. A common theoretical approach to constrain this structure involves "texture zeros"—specific elements in $M_\nu$ or its inverse (minors) that vanish due to underlying symmetries.

While two-zero texture and two-zero minor structures have been extensively studied, recent high-precision data from neutrino oscillation experiments (NuFIT 6.0) and cosmological observations (Planck, ACT, DESI) have tightened constraints significantly. Many previously viable two-zero structures are now excluded or severely restricted. Furthermore, while one-zero textures (a single vanishing element) are theoretically motivated by non-invertible selection rules and $U(1)_{L_\mu - L_\tau}$ symmetries, a comprehensive analysis of all one-zero textures and minors under current cosmological bounds (specifically the sum of neutrino masses, $\sum m_\nu$) has not been performed.

Methodology
The authors employ a dual-methodology approach to analyze both two-zero and one-zero texture/minor structures:

1. Analytical Framework:

   • The neutrino mass matrix in the flavor basis is diagonalized using the PMNS matrix ($U_{PMNS}$).
   • For two-zero textures, the condition that two elements of $M_\nu$ vanish yields two complex equations. These are solved to express the Majorana phases ($\alpha_2, \alpha_3$) and mass ratios ($m_2/m_1, m_3/m_1$) as functions of the mixing angles ($\theta_{ij}$) and the Dirac CP phase ($\delta_{CP}$).
   • For one-zero textures, a single vanishing element leads to a complex equation. Due to the under-constrained nature of a single equation, the authors utilize the triangle inequality to derive bounds on the parameters.
   • The analysis incorporates the latest global fit values for oscillation parameters (NuFIT 6.0) and applies constraints from:
     • Cosmology: Planck CMB alone ($\sum m_\nu < 0.21$ eV) and combined Planck+ACT+DESI BAO data ($\sum m_\nu < 0.113$ eV for Normal Ordering, $< 0.145$ eV for Inverted Ordering).
     • Kinematics: KATRIN limit on effective electron-neutrino mass ($m_{\nu_e}^{eff}$).
     • Neutrinoless Double-Beta Decay ($0\nu\beta\beta$): Current limits on the effective Majorana mass ($\langle m_{ee} \rangle$).

2. Machine Learning (Flow Matching):

   • To explore the high-dimensional parameter space of one-zero textures efficiently, the authors utilize flow matching, a generative AI framework.
   • The model is trained to learn the conditional distribution of mass matrix parameters ($G$) given observed physical labels ($L$), such as mass-squared differences and PMNS matrix elements.
   • The training data is generated by sampling complex mass matrix elements and diagonalizing them. The network is conditioned on experimental best-fit values.
   • A fine-tuning procedure is applied to specific textures (e.g., G2 in Normal Ordering) to improve the accuracy of the generated viable parameter sets, minimizing a $\chi^2$ function defined by the deviation from experimental central values.

Key Contributions and Results

• Two-Zero Textures:

  • Under the CMB-only bound ($\sum m_\nu < 0.21$ eV), eight two-zero textures remain viable (A1, A2, B1-B4 for Normal Ordering; B1, B3, C for Inverted Ordering).
  • Under the stricter CMB+BAO bound, the viable set shrinks significantly. Only A1 and A2 textures remain viable for Normal Ordering. All two-zero textures are excluded for Inverted Ordering under these tight constraints.
  • The B-series textures (viable under CMB-only) predict $\delta_{CP}$ near $\pi/2$ and $3\pi/2$ and relatively large $\langle m_{ee} \rangle$, making them targets for future $0\nu\beta\beta$ experiments like nEXO.

• One-Zero Textures (via Flow Matching and Analytical Methods):

  • Due to having more degrees of freedom, one-zero textures are less constrained than two-zero textures.
  • Normal Ordering (NO): Under CMB+BAO constraints, textures G1, H1, and H2 remain viable. G2, G3, and H3 are excluded.
  • Inverted Ordering (IO): Under CMB+BAO constraints, textures G2, G3, H1, and H2 remain viable. G1 and H3 are excluded.
  • Predictions:
    • Viable IO textures predict sizable $\langle m_{ee} \rangle$, though generally lower than the two-zero IO cases.
    • The H1 and H2 textures in IO show a characteristic preference for $\delta_{CP}$ around $\pi/2$ and $3\pi/2$.
    • The sum of neutrino masses ($\sum m_i$) for viable one-zero textures is generally lower than in the two-zero case.
  • The results from the flow matching analysis are confirmed to be in good agreement with the analytical method using triangle inequalities.

• Two-Zero and One-Zero Minors:

  • The analysis extends to minors (vanishing elements in $M_\nu^{-1}$).
  • Most two-zero minor structures are excluded under CMB+BAO constraints, with only D1 and D2 remaining viable for NO.
  • One-zero minors are generally more robust, with most structures remaining viable under current cosmological bounds, though G1 (NO) and G2, G3, H3 (IO) are excluded.

• Theoretical Realization:

  • The paper discusses the realization of one-zero textures via non-invertible selection rules arising from $\mathbb{Z}_2$ gauging of $\mathbb{Z}_N$ groups (specifically $\mathbb{Z}_5$ and $\mathbb{Z}_7$).
  • It is shown that standard $U(1)$ symmetries cannot realize G1, G2, or G3 textures.
  • However, using non-invertible fusion rules in Type II 2HDM or MSSM scenarios, the H1, H2, and H3 textures can be realized via $\mathbb{Z}_2$ gauging of $\mathbb{Z}_5$.
  • The G1, G2, and G3 textures can be realized via $\mathbb{Z}_3$ gauging of $\mathbb{Z}_7$.

Significance
The paper provides a comprehensive update on neutrino mass textures in light of the most stringent cosmological limits available (DESI BAO + CMB). It demonstrates that while two-zero textures are becoming highly constrained (leaving only A1 and A2 for NO under tight bounds), one-zero textures offer a broader window of viability.

The primary significance lies in the application of flow matching to systematically scan the parameter space of one-zero textures, identifying viable regions and characteristic predictions for $\delta_{CP}$, $\sum m_i$, and $\langle m_{ee} \rangle$ that distinguish them from two-zero scenarios. Furthermore, the work connects these phenomenological classifications to a specific theoretical origin: non-invertible selection rules, providing a mechanism for how specific texture zeros (particularly the one-zero cases) could arise in fundamental physics beyond the Standard Model. The results suggest that future precision measurements of $\delta_{CP}$ and $\sum m_\nu$, alongside next-generation $0\nu\beta\beta$ searches, will be critical in discriminating between these texture scenarios.