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

TL;DR

This paper critiques and corrects a previous study on flavor invariants in the leptonic sector with Majorana neutrinos, clarifying misconceptions about the mathematical properties of the involved groups and confirming some of the original results.

Contribution

It identifies and rectifies a fundamental error regarding the reductiveness of the unitary group in the context of flavor invariants, providing a rigorous mathematical correction and clarification.

Findings

The unitary group ${ m U}(n, ext{C})$ is not a linear algebraic group.

The ring of invariants of ${ m U}(n, ext{C})$ is isomorphic to that of ${ m GL}(n, ext{C})$.

Previous calculations based on incorrect assumptions remain valid due to this isomorphism.

Abstract

Recently in [JHEP 09 (2021) 053], Wang et al. discussed the polynomial ring formed by flavor invariants in the leptonic sector with massive Majorana neutrinos. They have explicitly constructed the finite generating sets of the polynomial rings for both two-generation scenario and three-generation scenario. However, Wang et al.'s claim of the finiteness of the generating sets of the polynomial rings and their calculation by the approach of Hilbert series with generalized Molien-Weyl formula are both based on their assertion that the unitary group ${\rm U}(n,\mathbb{C})$ is reductive, which is unfortunately incorrect. The property of being reductive is only applicable to linear algebraic groups. And it is well-known that the unitary group ${\rm U}(n,\mathbb{C})$ is not even a linear algebraic group. In this paper, we point out the above issue and provide a solution to fill in the accompanying logical gaps in [JHEP 09 (2021) 053]. Some important results from the theory of linear algebraic group, the invariant theory of square matrices and group theory are needed in the analysis. We also clarify some somewhat misleading or vague statements in [JHEP 09 (2021) 053] about the scope of flavor invariants. Note that, although built from incorrect assertion, Wang et al.'s calculation results in [JHEP 09 (2021) 053] are nonetheless correct, which is ultimately because the ring of invariants of ${\rm U}(n,\mathbb{C})$ is isomorphic to that of ${\rm GL}(n,\mathbb{C})$ which is itself reductive.