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An interrelation of three neutrino mixing angles

This paper proposes a novel phenomenological relation linking the three neutrino mixing angles, $\theta_{23} = (\theta_{12} + \theta_{13})/2 + \pi/8$ , which predicts $\theta_{23} \approx 43.46^\circ$ consistent with recent JUNO and Daya Bay data and is derived from a simple ansatz that yields a new expansion of the lepton flavor mixing matrix.

Original author: Zhi-zhong Xing

Published 2026-11-09

📖 4 min read🧠 Deep dive

CC BY 4.0

Original author: Zhi-zhong Xing

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the author. For technical accuracy, refer to the original paper. Read full disclaimer

Neutrinos are ghostly particles that zip through the universe almost entirely unnoticed, passing through planets and stars as if they were made of thin air. Despite their elusiveness, they hold a key to understanding the fundamental structure of matter. One of their most puzzling behaviors is that they can change their identity, or "flavor," as they travel. A neutrino born as one type can transform into another, a process driven by three specific angles that describe how these flavors mix together. Scientists have spent decades measuring these angles to understand the rules governing the subatomic world. Recently, new, highly precise measurements from experiments in China have sharpened our view of these angles, revealing a pattern that is so precise it hints at a deeper, hidden order in nature.

A researcher named Zhi-zhong Xing has taken these latest measurements and proposed a simple, elegant connection between the three mixing angles. Using the most recent data from the JUNO reactor experiment and the Daya Bay experiment, Xing suggests that the three angles are not independent random numbers but are linked by a specific mathematical relationship. The new data indicates that the solar neutrino mixing angle is about 33.44 degrees, and the angle associated with the reactor neutrino measurement is roughly 8.48 degrees. When these two values are combined according to Xing's proposed formula, they predict that the third angle, which describes how neutrinos oscillate between the second and third mass states, should be approximately 43.46 degrees. This prediction aligns remarkably well with the current global best estimate for that third angle, which sits at 43.45 degrees. The match is so close that it suggests the relationship is not a coincidence but a real feature of how neutrinos behave.

To explain why this relationship might exist, Xing looks back at older, idealized models of how particles mix. In the past, physicists imagined a "democratic" pattern where all flavors mixed equally, and a "tribimaximal" pattern that simplified the mixing into a specific, neat arrangement. These ideal patterns rely on fixed, "magic" angles that serve as a theoretical baseline. Xing noticed that the real-world angles measured today are very close to these theoretical baselines, differing by only about one or two degrees. By treating these small differences as a single, tiny correction to the ideal patterns, he was able to derive the new relationship between the three angles. This approach allows him to rewrite the complex matrix that describes neutrino mixing as a series of steps starting from a simple, perfect foundation and adding small adjustments.

The paper does not claim to have discovered a new force or a new particle, but rather to have found a hidden rule that organizes existing data. The author acknowledges that this relationship is a conjecture, a hypothesis that fits the current numbers but has not yet been proven by a fundamental theory of physics. The work relies on the assumption that the relationship holds true at the energy scales where these experiments operate. While quantum effects could theoretically distort this relationship, the author argues that such effects are likely too small to break the pattern given the current precision of the data. The proposed connection is presented as a phenomenological guide, a way to make sense of the numbers that future experiments can test.

Looking ahead, the next generation of neutrino experiments, such as DUNE and Hyper-Kamiokande, will provide even sharper measurements. These future facilities will be able to confirm or refute whether the three mixing angles truly follow the simple rule Xing has proposed. If the relationship holds up under this stricter scrutiny, it will provide a crucial clue for physicists trying to build a complete theory of flavor symmetry, the underlying framework that dictates why particles have the properties they do. For now, the work stands as a compelling suggestion that the universe's subatomic particles may follow a simpler, more unified logic than previously thought, waiting for the next wave of data to confirm the pattern.

Technical Summary: An Interrelation of Three Neutrino Mixing Angles

Problem Statement
Recent precision measurements from the JUNO reactor experiment have established the solar neutrino oscillation angle ( $\theta_{12}$ ) and two neutrino mass-squared differences ( $\Delta m^2_{21}$ and $\Delta m^2_{31}$ ) with high accuracy. Notably, the ratio $\Delta m^2_{21}/\Delta m^2_{31}$ aligns remarkably well with the empirical Alikhanov relation, $\tan^4(\pi/8) \approx 0.02944$ . This success in mass relations prompts the question of whether a similar underlying flavor symmetry might dictate a specific interrelation among the three neutrino mixing angles ( $\theta_{12}, \theta_{13}, \theta_{23}$ ). The paper seeks to determine if current oscillation data supports such a relation and to propose a theoretical framework to derive it.

Methodology
The author proposes a phenomenological ansatz based on "magic constant angles" associated with democratic and tribimaximal lepton flavor mixing patterns. The methodology proceeds as follows:

Definition of Benchmark Angles: The study defines three constant angles derived from symmetry models:
- $\theta^*_{12} = \arctan(1/\sqrt{2}) \approx 35.26^\circ$
- $\theta^*_{23} = 45^\circ$
- $\theta^*_{13} = \theta^*_{23} - \theta^*_{12} \approx 9.74^\circ$
  These satisfy the exact relation $\theta^*_{23} = (\theta^*_{12} + \theta^*_{13})/2 + \pi/8$ .
Derivation of the Angle Relation: The paper conjectures that the physical mixing angles ( $\theta_{12}, \theta_{13}, \theta_{23}$ ) deviate from these benchmark values by small amounts. By assuming a specific condition where the deviations satisfy $\theta^*_{23} - \theta_{23} = (\theta^*_{12} - \theta_{12})/2 + (\theta^*_{13} - \theta_{13})/2$ , the exact relation for the physical angles is recovered:
$\theta_{23} = \frac{\theta_{12} + \theta_{13}}{2} + \frac{\pi}{8}$
Expansion of the Mixing Matrix: To test the consistency of this ansatz, the author expands the standard $3 \times 3$ unitary lepton flavor mixing matrix ( $U$ ) in powers of a single small parameter $\xi$ , defined as the uniform deviation of the physical angles from their benchmarks ( $\theta^*_{i} - \theta_{i} \equiv \xi$ ). This yields a leading-order term $U^*$ (constructed from the magic angles) and a next-to-leading-order correction term $U_1$ .

Key Results

Prediction of $\theta_{23}$ : Using the latest JUNO data for $\theta_{12}$ ( $33.44^\circ$ ) and Daya Bay data for $\theta_{13}$ ( $8.48^\circ$ ), the proposed relation predicts:
$\theta_{23} = 43.46^{+0.51^\circ}_{-0.52^\circ}$
This value is highly consistent with global fit results for $\theta_{23}$ ( $43.45^{+1.32^\circ}_{-0.74^\circ}$ or $43.28^{+0.98^\circ}_{-0.80^\circ}$ ) at the $1\sigma$ confidence level.
Validation of the Ansatz: The calculated deviations between the benchmark and physical angles are found to be of order $O(1^\circ)$ , satisfying the condition required for the relation to hold without fine-tuning.
Leptonic CP Violation: The expansion of the mixing matrix allows for the derivation of the Jarlskog invariant ( $J_\nu$ ) for leptonic CP violation. The leading term is found to be consistent with previous literature, with corrections dependent on the small parameter $\xi$ and the Dirac phase $\delta$ .

Significance and Claims
The paper claims to offer a "partial but affirmative" answer to the existence of an interrelation among neutrino mixing angles. Its primary contributions are:

A Novel Phenomenological Relation: The specific formula $\theta_{23} = (\theta_{12} + \theta_{13})/2 + \pi/8$ is presented as a testable hypothesis that aligns with current precision data.
Theoretical Framework: The work provides a simple ansatz connecting democratic and tribimaximal mixing patterns to current data via a "magic angle basis."
New Mathematical Tool: As a by-product, the paper presents a new expansion of the lepton flavor mixing matrix in powers of a single small parameter, offering a benchmark basis ( $U^*$ ) for realistic mixing.

The author emphasizes that while the relation holds phenomenologically at the electroweak scale, quantum corrections could potentially spoil it, though these effects are expected to be small. The paper concludes that the proposed relation is experimentally testable and awaits stringent verification by next-generation long-baseline experiments such as DUNE and Hyper-Kamiokande. If the relation survives these tests, it would motivate further model building based on specific flavor symmetry groups.

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