Bottom-up approach to texture zeros in the neutrino mass matrix

Technical Summary: Bottom-up Approach to Texture Zeros in the Neutrino Mass Matrix

Problem Statement
The origin and specific form of the neutrino mass matrix remain among the most significant open questions in particle physics and cosmology. While neutrino oscillation experiments have precisely measured mixing angles and mass-squared differences, the absolute neutrino mass scale, the sign of the atmospheric mass-squared difference (distinguishing normal vs. inverted ordering), and the three CP-violating phases (one Dirac, two Majorana) remain undetermined. Theoretical frameworks often employ "texture zeros"—specific elements in the mass matrix set to zero—to reduce the number of free parameters and constrain the matrix structure. Previous studies have explored these zeros using specific mixing schemes (e.g., PMNS or Trimaximal mixing), but a comprehensive investigation using the latest oscillation data through a bottom-up approach, without assuming specific mixing patterns a priori, is required to fully understand the phenomenological implications of such zeros.

Methodology
The authors employ a bottom-up approach to investigate the viability of one and two texture zeros in the Majorana neutrino mass matrix ($M$). The analysis utilizes the latest neutrino oscillation data (specifically from NuFIT) within a 3$\sigma$ confidence interval. The methodology involves:

1. Parameter Variation: The authors vary the lightest neutrino mass ($m_1$ for normal ordering, $m_3$ for inverted ordering) within the range $(0, 1)$ eV. The three mixing angles ($\theta_{12}, \theta_{13}, \theta_{23}$) and two mass-squared differences ($\Delta m^2_{21}, \Delta m^2_{31}$) are varied within their experimental 3$\sigma$ ranges.
2. Phase Scanning: The three CP-violating phases (Dirac phase $\delta$ and Majorana phases $\alpha, \beta$) are scanned across their full range $(0, 2\pi)$.
3. Constraints: The analysis incorporates cosmological upper bounds on the total neutrino mass ($\sum m_i < 0.12$ eV and the stricter $\sum m_i < 0.072$ eV) and experimental upper bounds on the effective Majorana mass term ($|M_{ee}| < 0.06 - 0.17$ eV) from neutrinoless double beta decay experiments (KamLAND-Zen, EXO-200, CUORE).
4. Analysis: For each matrix element, the authors determine the conditions under which the element vanishes. They analyze the correlations between the lightest neutrino mass and the CP-violating phases for these vanishing cases, distinguishing between normal ordering (NO), inverted ordering (IO), and degenerate mass ordering.

Key Contributions and Results

• Single Texture Zeros:

  • $M_{ee}$ (ee element): This element can vanish only for Normal Ordering (NO). The vanishing condition requires the lightest neutrino mass $m_1 \in (2.0, 8.0) \times 10^{-3}$ eV and the Majorana phase $\alpha$ to be near $90^\circ$ or $270^\circ$. No constraints are found on $\beta$ or $\delta$. For Inverted Ordering (IO), $|M_{ee}|$ cannot vanish; its minimum value is of order $10^{-2}$ eV.
  • $M_{e\mu}$ and $M_{e\tau}$: These elements can vanish for NO, IO, and degenerate ordering.
    • In NO, vanishing requires $m_1 \ge 6.0 \times 10^{-3}$ eV ($e\mu$) or $m_1 \ge 3.0 \times 10^{-3}$ eV ($e\tau$).
    • In IO, they can vanish for any lightest neutrino mass, but specific narrow ranges for $\alpha$ and $\delta$ are required when cosmological mass bounds are applied.
  • $M_{\mu\mu}$ and $M_{\tau\tau}$: These can vanish for NO, IO, and degenerate ordering.
    • In NO, $|M_{\mu\mu}|$ vanishing requires $m_1 \ge 5.1 \times 10^{-2}$ eV, while $|M_{\tau\tau}|$ requires $m_1 > 3.4 \times 10^{-2}$ eV (though the latter is excluded by the $\sum m_i < 0.12$ eV bound).
    • In IO, both can vanish for any lightest neutrino mass, subject to constraints on $\alpha, \beta,$ and $\delta$.
  • $M_{\mu\tau}$: This element can vanish for IO and degenerate ordering but not for NO (unless the total mass is unconstrained and $m_1$ is very large, $\ge 0.16$ eV). In IO, vanishing requires $m_3 > 1.2 \times 10^{-2}$ eV.

• Two Texture Zeros:
  The authors identify which pairs of elements can simultaneously vanish:

  • Normal Ordering:
    • Allowed pairs: $(M_{ee}=0, M_{e\mu}=0)$, $(M_{ee}=0, M_{e\tau}=0)$, $(M_{e\mu}=0, M_{\tau\tau}=0)$, $(M_{e\tau}=0, M_{\tau\tau}=0)$, $(M_{e\mu}=0, M_{\mu\mu}=0)$, $(M_{e\tau}=0, M_{\mu\mu}=0)$, and $(M_{\mu\mu}=0, M_{\tau\tau}=0)$.
    • Disallowed pairs: $(M_{ee}=0, M_{\mu\tau}=0)$ and $(M_{ee}=0, M_{\mu\mu}=0)$.
    • Specific constraints: For $(M_{ee}=0, M_{e\mu}=0)$, $m_1$ is restricted to $(6.50, 8.20) \times 10^{-3}$ eV, with tight constraints on $\alpha, \beta,$ and $\delta$. For $(M_{\mu\mu}=0, M_{\tau\tau}=0)$, $m_1 > 0.15$ eV is required.
  • Inverted Ordering:
    • Allowed pairs include $(M_{e\mu}=0, M_{\tau\tau}=0)$, $(M_{e\tau}=0, M_{\tau\tau}=0)$, $(M_{e\mu}=0, M_{\mu\mu}=0)$, $(M_{e\tau}=0, M_{\mu\mu}=0)$, and $(M_{\mu\mu}=0, M_{\tau\tau}=0)$.
    • Disallowed pairs: $(M_{ee}=0, M_{\mu\tau}=0)$ and $(M_{ee}=0, M_{\mu\mu}=0)$.
    • For $(M_{e\mu}=0, M_{\tau\tau}=0)$, $m_3 > 4.5 \times 10^{-2}$ eV is required.

• Correlations and Octant Constraints:
  The study reveals strong correlations between the lightest neutrino mass and the CP-violating phases for vanishing cases. Furthermore, the authors find that certain two-zero patterns impose constraints on the atmospheric mixing angle octant. For instance, simultaneous vanishing of $|M_{e\mu}|$ and $|M_{\tau\tau}|$ requires $\theta_{23}$ to be in the upper octant ($\theta_{23} > 45^\circ$), whereas $|M_{e\mu}|$ and $|M_{\mu\mu}|$ vanishing requires the lower octant ($\theta_{23} < 45^\circ$).

Significance
The paper claims to provide a detailed phenomenological map of the neutrino mass matrix elements under the assumption of texture zeros, utilizing the most recent experimental data and cosmological bounds. By systematically scanning the parameter space, the authors identify which texture zero patterns are compatible with current observations and which are ruled out. The work highlights that while many single and two-zero patterns are viable, they impose strict constraints on the lightest neutrino mass and the CP-violating phases. Specifically, the results suggest that if neutrinos are Majorana particles and specific texture zeros exist, the lightest neutrino mass and CP phases cannot be arbitrary but must lie within narrow, correlated ranges. The study also underscores the tension between certain texture zero scenarios (e.g., specific two-zero patterns in NO) and the tightest cosmological bounds on the total neutrino mass.