Neutrino Mixing Matrix with SU(2)₄ Anyon Braids

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

In this work, the author proposes a novel topological model where the PMNS matrix arises from the representation theory of the braid group associated to the modular tensor category
(MTC) of type SU(2)_4. Under this assumption, the author arrives at the neutrino mixing angles theta_13 around 10 degrees, theta_12 around 30 degrees, and theta_23 around 38 degrees and the Dirac CP phase delta around 260 degrees, which are not far from the experimental measurement results.

1, The author states that "This Takagi outcome is highly sensitive to both numerical precision and basis alignment. When computed at standard floating-point resolution without appropriate formatting, the same matrix yields incorrect angles resembling democratic mixing. The accurate result only emerges when using double-precision input values and a Takagi routine with rounding threshold below 10-13". The author needs to clarify why the numerical results depend on the numerical precision. This seems very unusual and unsound.   2, The author states that the results of the neutrino mixing angles they have obtained in Eq. (6) agree with global neutrino oscillation fits within 3sigma confidence intervals. However, it seems that these results are out of the 3sigma confidence intervals. The author needs to check this point.   

 

This work is interesting, and can be accepted for publication.

However, the author should improve the manuscript in the following aspect: the author claims that the results in Eq. (6) agree with the global neutrino oscillation fits within 3sigma confidence intervals, but the 3sigma confidence intervals have not been provided, with the help of which the author's claim would get a support.

Author Response

I believe that the remarks of Referee 1 are addressed in the new version of the paper 
where Table 1 shows our results compared to the experimental results: the NuFIT best fit and the ranges $3\sigma$ and $1\sigma$.
The values of $\theta$ angles are close but not within $3\sigma$. But the Dirac phase is within the NuFIT best fit.
The paper was sligthly expanded to give details of our calculations.
The Python file Srawberry_result.txt is attached and may be used as a check.

Author Response File: Author Response.docx

Reviewer 2 Report

Comments and Suggestions for Authors

quantumrep-3682643-peer-review-v1
Neutrino Mixing Matrix with SU(2)_4 Anyon Braids

summary:
The author motivates a new approach to look at lepton family symmetries by looking at a topological model, associated to the modular tensor category of type SU(2)_4, where the Takagi factorization should yield a good approximation to the neutrino flavor mixing matrix (PMNS matrix). In section 2 the autjor reviews the PMNS matrix, introduces the modular tensor categories in section 3, provides the generators (braiding matrices) in section 4, and explains the Takagi factorization in section 5. Section 6 summarizes the findings, discusses the physical interpretation together with proposals for phenomenological tests, and concludes with open questions and remarks. 

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remarks and criticism: 

The idea is well formulated and well presented, 
but the result is just stated and I cannot reproduce it or see the consistency of the claim!

As crosschecks, one should be able to use the definition for the PMNS matrix, eq.(2) or (3) and obtain the generator \sigma_{2}^{(4)} from eq.(5), using the input for the PMNS matrix in eq.(6). 

-> pdf from tex file, attached

Comments for author File: Comments.pdf

Author Response

I thank the referee for his  examination of my results. His analytical calculations are correct. 
But to reproduce my calculations it is needed to write the PMNS matrix in its symmetric form (5)-(6) 
leading to the CP phase as given in (7). In section 5, I also give details about the modulus and the phase of the obtained unitary matrix 
U after the Autunne-Takagi factorization followed by the exchange of the two first flavour states.
The Python file Srawberry_result.txt is attached and may be used as a full check.

Author Response File: Author Response.docx

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

The additional information of $|U|_{\text{braid}}$ and $|\text{Arg}(U)|_{\text{braid}}$ helps, but does not fulfill the consistency check of eq.(9)!

The first steps in the revised version still do not include their own consistency check. 

The numerical routine seems to be used as a black box. 

Did the author convince himself, that his numerical output really reproduces the PMNS matrix? And the it also diagonalizes the generator \sigma_2 ?

Comments for author File: Comments.pdf

Author Response

An appendix is created to answer your comment that they may be a
wrong numerical result. But it is not the case. It is not needed to add this
appendix in the submission. It can be left private. It is enough to refer to
the Python script I also prepared, the same than before but with more outputs. To
illustrate these outputs and clarify the “black box” I give some details in the Appendix.
You can see that everything in my submission is consistent. I do not need
to do any new change in my paper apart from adding the excellent reference
[1] from Ludl et al. I hope you will agree after a further look based on the given
clarifications.

Author Response File: Author Response.pdf