Constraining the Neutrino Mixing Matrix via Single-Sector Charged-Lepton Rotations in the JUNO Precision Era

This paper investigates constraints on the neutrino diagonalization matrix $U_\nu$ by assuming the charged-lepton mixing matrix consists of a single two-by-two rotation in one of the three sectors, utilizing the latest global fit incorporating JUNO's precision $\theta_{12}$ measurement to derive analytical sum-rule-like expressions for $U_\nu$ entries.

The Gist

Imagine the universe is a giant, complex orchestra. For a long time, physicists thought they knew the sheet music for the "neutrino section" (the ghostly particles that pass through everything). They knew the notes (masses) and how the instruments played together (mixing), but they didn't know why the music sounded the way it did.

This paper is like a detective story where scientists are trying to figure out the hidden conductor of this orchestra.

The Big Mystery: The "PMNS" Matrix

In the world of neutrinos, there is a mathematical recipe called the PMNS matrix. Think of this as the "master recipe" for how neutrinos change flavors (like a chameleon changing colors) as they travel.

For a long time, scientists assumed this recipe was written entirely by the neutrinos themselves. But this paper asks a different question: What if the recipe is actually a mix of two things?

1. The neutrinos' own natural mixing.
2. A "twist" or "rotation" introduced by the charged partners (electrons, muons, and taus) that hang out with them.

The Hypothesis: The Single Twist

The authors propose a simple, elegant idea: What if the "charged partner" part of the recipe isn't a messy, complicated dance? What if it's just one single, simple turn?

Imagine you are holding a Rubik's Cube. Usually, solving it requires twisting many faces in complex patterns. But this paper asks: What if the charged-lepton sector only ever twists one specific face?

• Case 1: It only twists the top-left face (1,2).
• Case 2: It only twists the top-right face (1,3).
• Case 3: It only twists the bottom-right face (2,3).

If this is true, it means the underlying laws of nature are much simpler and more symmetrical than we thought.

The New Tool: JUNO

To test this, the scientists needed a very precise ruler. Enter JUNO (Jiangmen Underground Neutrino Observatory).

Think of previous measurements of neutrino angles as using a ruler with blurry markings. You could guess the length, but you weren't sure if it was 10.1 cm or 10.9 cm.
JUNO is like a laser micrometer. It just measured one specific angle (called $\theta_{12}$, the "solar angle") with incredible precision. This new data is the "smoking gun" that allows the detectives to see if their "single twist" theory holds up.

The Investigation: What They Found

The team took the new, super-precise JUNO data and ran it through their "single twist" models to see what the neutrino's true nature (the matrix $U_\nu$) would look like.

Here is what they discovered, using our analogies:

• The "Fixed" Column: In all three scenarios, they found that one column of the neutrino's "true" recipe is locked in place. It doesn't matter how much the charged partners twist; that part of the neutrino's identity is determined entirely by what we already measured. It's like a pillar in a building that never moves, no matter how the wind blows.
• Case 1 (The 1,2 Twist): If the twist happens here, the neutrino's "atmospheric" and "reactor" angles are predicted to be very specific numbers based on the solar angle. The new JUNO data tightened the net around these predictions, making the theory easier to test.
• Case 2 (The 1,3 Twist): Here, the twist affects everything. The angles change wildly depending on how big the twist is. The data suggests this is a harder scenario to pin down right now; we need even better rulers (like the DUNE experiment) to see if this twist exists.
• Case 3 (The 2,3 Twist): This one is fascinating. It creates a perfect "see-saw" relationship. If one angle goes up, the other must go down in a very specific way. This is a "sum rule"—a mathematical promise that if you know one, you know the other.

The "Ghost" Factor: CP Violation

Neutrinos have a mysterious property called CP violation, which is basically a "handedness" or a preference for left vs. right. It's like a coin that is slightly weighted to land on heads more often.
The paper shows that this "handedness" (the $\delta_{CP}$ phase) acts like a dimmer switch on the neutrino angles. Depending on the setting of this switch, the angles wiggle slightly. The authors mapped out exactly how much they wiggle for each of the three "twist" scenarios.

The Conclusion: Why This Matters

This paper is a bridge between theory and experiment.

• Before: We had a blurry picture of neutrinos and many wild theories about why they mix.
• Now: With JUNO's precision, we can start to rule out the theories that don't fit.

The authors conclude that even with this simple "single twist" idea, the universe is starting to reveal its secrets. If the neutrino mixing matrix really does come from a single, simple rotation of the charged partners, it suggests a deep, hidden symmetry in the universe—like finding that a chaotic jazz improvisation was actually based on a single, repeating musical scale.

In short: The paper says, "We have a new, super-precise ruler (JUNO). We tested three simple ideas about how the universe 'twists' neutrinos. Here is exactly what the universe must look like if any of those ideas are true. Now, let's wait for the next experiments to see which one is right!"

Technical Summary

1. Problem Statement

The Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix describes lepton mixing, but its origin remains unknown. Theoretical models often posit that the PMNS matrix arises from the mismatch between the diagonalization of the charged-lepton mass matrix ($M_l$) and the neutrino mass matrix ($M_\nu$):
$$U_{\text{PMNS}} = U_l U_\nu^\dagger$$
where $U_l$ and $U_\nu$ are unitary matrices. A common class of flavor models assumes that the mixing pattern originates primarily from the neutrino sector ($U_\nu$), with the charged-lepton sector ($U_l$) providing only subleading corrections.

The paper addresses the specific hypothesis that $U_l$ consists of a single two-by-two rotation in one of the three sectors: $(1,2)$, $(1,3)$, or $(2,3)$. With the advent of high-precision data, particularly the recent measurement of the solar mixing angle $\theta_{12}$ by the JUNO experiment, the authors investigate whether these "single-sector" hypotheses can be constrained or ruled out, and how they predict the structure of the underlying neutrino mixing matrix $U_\nu$.

2. Methodology

The authors employ a combination of analytical derivation and numerical simulation:

• Formalism:

  • They adopt the standard PDG parameterization for $U_{\text{PMNS}}$.
  • They define three distinct hypotheses for $U_l$:
    • Case I: Rotation in the $(1,2)$ plane ($R_{12}(\theta^l_{12}$)).
    • Case II: Rotation in the $(1,3)$ plane ($R_{13}(\theta^l_{13}$)).
    • Case III: Rotation in the $(2,3)$ plane ($R_{23}(\theta^l_{23}$)).
  • They invert the relation to solve for the neutrino matrix: $U_\nu = U_{\text{PMNS}}^\dagger U_l$.
  • They derive analytical expressions for the elements of $U_\nu$ and extract the effective neutrino mixing angles ($\theta^\nu_{12}, \theta^\nu_{13}, \theta^\nu_{23}$) and the CP phase $\delta^\nu$.

• Analytical Approximations:

  • They derive "sum-rule" like formulae relating the neutrino angles to the measured PMNS angles, often in the limit of small $\theta_{13}$ and small charged-lepton rotation angles.
  • They identify specific columns of $U_\nu$ that remain invariant under the single-sector rotation, leading to parameter-free predictions.

• Numerical Analysis:

  • Data Inputs: They utilize global fit data from NuFit 6.0 (pre-JUNO) and NuFit 6.1 (incorporating the first JUNO $\theta_{12}$ measurement).
  • Scan: They perform a Monte Carlo scan varying the PMNS angles within their $1\sigma$ uncertainties and scanning the charged-lepton rotation angle $\theta^l_{ij}$ from $0^\circ$ to $45^\circ$.
  • Outputs: They generate distributions and correlation contours for the derived neutrino angles and the Jarlskog invariant ($J_\nu$).

3. Key Contributions

A. Analytical Sum Rules and Structural Constraints

The paper derives exact and approximate relations that serve as testable predictions for flavor models:

• Case I (1,2 rotation):
  • The third column of $U_\nu$ is identical to the third column of $U_{\text{PMNS}}^\dagger$.
  • Predictions: $\sin^2 \theta^\nu_{13} \approx \sin^2 \theta_{12} \sin^2 \theta_{23}$ and $\sin^2 \theta^\nu_{23} \approx \frac{\cos^2 \theta_{12} \sin^2 \theta_{23}}{1 - \sin^2 \theta_{12} \sin^2 \theta_{23}}$. These are independent of the unknown rotation angle $\theta^l_{12}$.
  • Sum Rule: $\theta^\nu_{12} \approx \theta_{12} - \theta^l_{12} \cos \theta_{23}$.
• Case II (1,3 rotation):
  • No simple compact sum rules independent of $\theta^l_{13}$ are found. The hierarchy of angles changes drastically with the rotation angle.
• Case III (2,3 rotation):
  • The first column of $U_\nu$ is identical to the first column of $U_{\text{PMNS}}^\dagger$.
  • Exact Sum Rule: $\cos^2 \theta^\nu_{12} \cos^2 \theta^\nu_{13} = \cos^2 \theta_{12} \cos^2 \theta_{13}$. This holds for any value of $\theta^l_{23}$ and is independent of the CP phase $\delta_{CP}$.
  • Atmospheric Sum Rule: $\theta^\nu_{23} \approx \theta_{23} - \theta^l_{23}$.

B. Impact of JUNO Precision

The authors quantify how the improved precision on $\theta_{12}$ from JUNO affects the constraints:

• Case I: The JUNO measurement significantly tightens the predicted ranges for $\sin^2 \theta^\nu_{13}$ and $\sin^2 \theta^\nu_{23}$ because these depend directly on $\sin^2 \theta_{12}$.
• Case II & III: The impact of JUNO is less dramatic, as the uncertainties are dominated by the atmospheric angle $\theta_{23}$ and the scan over the rotation angle.

C. CP Violation and Jarlskog Invariant

• The paper analyzes the dependence of the neutrino angles on the Dirac CP phase $\delta_{CP}$.
• It calculates the Jarlskog invariant ($J_\nu$) for the neutrino sector.
• Finding: In all cases, $J_\nu$ exhibits a sinusoidal modulation ($\propto \sin \delta_{CP}$). However, the width of the allowed region for $J_\nu$ varies by case. Case I and III show narrowing of the distribution near maximal CP violation ($\delta_{CP} \approx \pm 90^\circ$), whereas Case II remains broad due to strong correlations between angles.

4. Results

• Case I (1,2 Rotation):
  • $\sin^2 \theta^\nu_{23}$ is predicted to be $\approx 0.47$ (for Normal Ordering), largely independent of $\theta^l_{12}$.
  • $\sin^2 \theta^\nu_{13}$ is predicted to be $\approx 0.17$.
  • The JUNO data reduces the uncertainty bands for these predictions, making them sharper discriminators for flavor models.
• Case II (1,3 Rotation):
  • All three neutrino angles are highly sensitive to $\theta^l_{13}$.
  • The distributions are broad, and the Lower Octant (LO) vs. Upper Octant (UO) solutions for $\theta_{23}$ are not well separated. Future data from DUNE and T2HK on $\theta_{23}$ is required to test this scenario.
• Case III (2,3 Rotation):
  • A distinctive negative correlation is observed between $\sin^2 \theta^\nu_{12}$ and $\sin^2 \theta^\nu_{13}$, directly reflecting the exact sum rule derived.
  • The exact relation $\cos^2 \theta^\nu_{12} \cos^2 \theta^\nu_{13} = \text{const}$ provides a robust, model-independent test that does not rely on the value of $\theta^l_{23}$.

5. Significance

This work provides a critical bridge between high-precision experimental data (specifically from JUNO) and theoretical flavor model building.

1. Testability: It transforms abstract flavor symmetry models (which often predict specific $U_l$ structures) into concrete, testable predictions for the "true" neutrino mixing matrix $U_\nu$.
2. JUNO's Role: It demonstrates that even early, sub-percent precision measurements of $\theta_{12}$ can significantly constrain the parameter space of flavor models, particularly those involving $(1,2)$ rotations.
3. Future Probes: The paper identifies that while JUNO helps, the definitive test for $(1,3)$ and $(2,3)$ rotation scenarios will require future precision on the atmospheric angle $\theta_{23}$ and its octant from experiments like DUNE and T2HK.
4. CP Sensitivity: It highlights that the Jarlskog invariant and CP phase dependence offer additional layers of discrimination between these hypotheses, particularly as $\delta_{CP}$ is measured more precisely.

In conclusion, the paper establishes that the "single-sector rotation" hypothesis is a viable and testable framework for understanding lepton mixing, with current and near-future data capable of ruling out or confirming specific structural assumptions about the charged-lepton sector.