Reply to "Comment on `Unified neutrino mixing and approximate $\mu-\tau$ reflection symmetry'[arXiv:2603.00885]''

This paper responds to a comment by Huang and Li by acknowledging an overlooked real-value condition in their previous work on $\mu-\tau$ reflection symmetry while clarifying that Inverted Ordering remains in tension with current experimental bounds on the sum of neutrino masses, despite being permitted by effective neutrino mass constraints.

The Gist

Imagine a group of scientists (Hyodo and Kitabayashi) who built a complex model to explain how tiny, ghost-like particles called neutrinos dance and mix with each other. They published a paper claiming that one specific way these particles could be arranged—called Inverted Ordering (IO)—is impossible based on their rules and current data.

Recently, two other scientists (Huang and Li) wrote a "comment" pointing out two things:

1. The Mistake: The original team forgot to check if the numbers in their math were "real" (not imaginary), which is a basic rule of the game.
2. The Disagreement: They argued that even with the new data, the "Inverted Ordering" arrangement might still be possible.

Here is the simple breakdown of how the original team responded, using some everyday analogies:

1. The "Oops" Moment (Point 1)

The Analogy: Imagine you are baking a cake and you realize you forgot to add the baking powder.
The Reality: Hyodo and Kitabayashi admit, "You're absolutely right! We forgot a crucial ingredient (the real-value conditions)." They thank Huang and Li for catching this error. They agree that if you fix the math, the picture of how the neutrinos behave changes slightly.

2. The "Two Different Rulers" Debate (Point 2)

This is where the disagreement happens. Both teams are trying to see if the "Inverted Ordering" cake can still be baked, but they are using different rulers to measure it.

• Huang and Li's Ruler (The Effective Mass): They looked at a specific measurement called $|M_{ee}|$. It's like checking if the cake fits through a specific-sized door. They say, "Yes, it fits! The door is wide enough, so the cake (Inverted Ordering) is still allowed."
• Hyodo and Kitabayashi's Ruler (The Total Weight): The original team looked at the total weight of the neutrinos (the sum of their masses, $\sum m_\nu$). It's like checking if the cake is too heavy for the table to hold. They say, "Even if it fits through the door, it's way too heavy for the table. The table (experimental limits) will break."

3. The Final Verdict

Hyodo and Kitabayashi argue that Huang and Li are only looking at half the picture.

The Creative Metaphor:
Imagine a bouncer at a club (the universe).

• Huang and Li say, "This person has the right ID card ($|M_{ee}|$), so they can get in."
• Hyodo and Kitabayashi say, "Wait, look at their total weight! The bouncer has a strict weight limit ($\sum m_\nu$). Even with the right ID, they are too heavy to enter."

They suggest that if you take Huang and Li's corrected math (the new ID card) and overlay it with their own weight limit (the bouncer's scale), the "Inverted Ordering" person is still too heavy to get in.

The Conclusion

The original team concludes that the comment from Huang and Li actually helped them. By fixing the math error, the model became even stricter. The "Inverted Ordering" arrangement is still ruled out because, while it might pass one test, it fails the "total weight" test.

In short: "We admit we missed a step in the math, but even after fixing it, our main conclusion stands: The 'Inverted Ordering' arrangement is still impossible under these rules."

Technical Summary

Based on the provided text, here is a detailed technical summary of the paper "Reply to 'Comment on Unified neutrino mixing and approximate µ −τ reflection symmetry'":

1. Problem Statement

The authors, Yuta Hyodo and Teruyuki Kitabayashi, are responding to a comment (arXiv:2603.00885) by Huang and Li regarding their previous work (arXiv:2502.18029) on unified neutrino mixing and approximate $\mu-\tau$ reflection symmetry. The comment raised two critical challenges:

1. Oversight of Constraints: Huang and Li claimed the authors overlooked specific "real-value conditions" required for $\mu-\tau$ reflection symmetry.
2. Viability of Inverted Ordering (IO): They argued that the Inverted Ordering of neutrino masses remains a viable solution when considering the latest experimental data on the effective neutrino mass ($|M_{ee}|$), contradicting the original paper's exclusion of IO.

2. Methodology

The authors employ a comparative analysis to address the points raised:

• Acknowledgment and Correction: They accept the validity of the first point, acknowledging that the real-value conditions associated with $\mu-\tau$ reflection symmetry were indeed overlooked in their initial calculations. They accept the corrected calculations and results presented by Huang and Li (specifically Figure 1 in the comment).
• Re-evaluation of Constraints: To address the second point, the authors re-examine the viability of IO by overlaying two distinct experimental constraints onto the parameter space:
  1. The constraints on the effective neutrino mass ($|M_{ee}|$), which Huang and Li used to argue for IO viability.
  2. The constraints on the sum of neutrino masses ($\sum m_\nu$), which was the basis for the original exclusion of IO.
• Visual Synthesis: They propose a graphical synthesis where the vertical lines representing the allowed range of $\sum m_\nu$ (from their original work) are overlaid onto the parameter space plot provided by Huang and Li.

3. Key Contributions

• Correction of Theoretical Framework: The authors formally admit the oversight regarding real-value conditions, validating the technical corrections made by Huang and Li.
• Clarification of Exclusion Criteria: They clarify that the exclusion of Inverted Ordering in their model was not solely based on $|M_{ee}|$ but primarily on the tight experimental bounds regarding the sum of neutrino masses ($\sum m_\nu$).
• Demonstration of Robustness: By integrating the corrected real-value conditions with the $\sum m_\nu$ constraints, they demonstrate that the exclusion of IO is robust.

4. Results

• Admission of Error: The authors confirm that the real-value conditions were missed and that Huang and Li's corrected calculations (Figure 1 in the comment) are accurate.
• Refutation of IO Viability: Despite the corrected calculations, the authors show that when the experimental bounds on $\sum m_\nu$ are applied to the corrected parameter space, the Inverted Ordering (IO) scenario remains in tension with experimental data.
• Visual Evidence: The proposed overlay of $\sum m_\nu$ constraints on Huang and Li's Figure 1 reveals that the allowed region for IO is effectively eliminated, even after accounting for the previously overlooked conditions.

5. Significance

• Strengthening of Original Conclusions: Rather than weakening their original paper, the authors argue that the refinements provided by Huang and Li actually strengthen the main conclusion. The exclusion of Inverted Ordering under approximate $\mu-\tau$ reflection symmetry within the discussed model parameter space is shown to be more robust than initially thought.
• Resolution of Discrepancy: The paper resolves the apparent conflict between the two studies by clarifying that while IO might survive $|M_{ee}|$ constraints alone, it fails when the combined constraints of $|M_{ee}|$ and $\sum m_\nu$ are applied under the specific symmetry conditions of the model.
• Model Validation: The work reinforces the predictive power of the unified neutrino mixing model with approximate $\mu-\tau$ reflection symmetry, specifically in ruling out the Inverted Ordering scenario.