(Reply to)$^{\textbf{2}}$ "Comment on 'Flavor invariants and renormalization-group equations in the leptonic sector with massive Majorana neutrinos' "

TL;DR

This paper clarifies mathematical subtleties in flavor invariants and group isomorphisms in neutrino physics, providing proofs to correct misconceptions about group relations and their implications.

Contribution

It offers rigorous proofs distinguishing between U(n,C) and GL(n,C), and clarifies the non-isomorphism between SU(n,C) and SL(n,C), enhancing understanding of symmetry groups in particle physics.

Findings

Disproves the isomorphism between U(n,C) and GL(n,C).

Shows non-isomorphism between SU(n,C) and SL(n,C).

Provides proofs in Lie group and abstract group contexts.

Abstract

We respectfully reply to Wang et al.'s reply to our comment to their [JHEP $\textbf{09}$ (2021) 053]. Some more subtleties and details are discussed for hopefully better clarification and understanding, especially about the conditions in Hilbert's finiteness theorem. We also disprove Wang et al.'s incorrect assertion in their reply [arXiv:2110.13865 [hep-ph]] that there is an isomorphism between ${\rm U}(n,\mathbb{C})$ and ${\rm GL}(n,\mathbb{C})$. We give two proofs for our argument. One proof is in the context of Lie group for arbitrary positive integer $n$, and the other proof is in the context of abstract group for $n=3$ which can also be generalized to $n\geq 3$. In order to better illustrate the intrinsic differences between a compact Lie group and its complexification, we choose ${\rm SU}(n,\mathbb{C})$ and ${\rm SL}(n,\mathbb{C})$ as a pair of examples and give two proofs for the generally non-isomorphic relation between them. One proof is in the context of Lie group for integers $n\geq 2$, and the other proof is in the context of abstract group for $n=3$ which can again be generalized to $n\geq 3$.