Reply to "Comment on 'Flavor invariants and renormalization-group equations in the leptonic sector with massive Majorana neutrinos'"

TL;DR

This paper provides a formal reply clarifying misunderstandings and defending the validity of previous results on flavor invariants and RG equations in the leptonic sector with Majorana neutrinos, addressing comments made in a recent critique.

Contribution

The authors clarify that their original calculations and results are correct and address misconceptions about the mathematical framework used in their work.

Findings

Original results on flavor invariants are confirmed as correct.

Misinterpretations about the mathematical group properties are clarified.

The validity of the Molien-Weyl formula application is defended.

Abstract

In the preprint arXiv:2110.08210, some comments on our paper recently published in JHEP \textbf{09} (2021) 053 have been made. Since some of the comments are completely wrong and others are quite misleading, we decide to clarify the relevant issues in a formal reply. First of all, nothing is wrong about our calculations and physical results in the original paper. Second, there is no logical gap to fill at all. The fact that the $N$-dimensional unitary group over the field of complex numbers ${\rm U}(N,\mathbb{C})$ is not a linear algebraic group is actually irrelevant for the validity of the Molien-Weyl formula. As we shall explain in this reply, all the comments in arXiv:2110.08210 arise from the misunderstanding and misinterpretation of our discussions and results.