Abstract
Recently in JHEP09 (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 U(n, 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 U(n, 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 JHEP09 (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 JHEP09 (2021) 053 about the scope of flavor invariants. Note that, although built from incorrect assertion, Wang et al.’s calculation results in JHEP09 (2021) 053 are nonetheless correct, which is ultimately because the ring of invariants of U(n, ℂ) is isomorphic to that of GL(n, ℂ) which is itself reductive.
Highlights
It is straightforward to prove that Det(G(lν1) + HlHν) is a flavor invariant
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
We point out the above issue and provide a solution to fill in the accompanying logical gaps in JHEP 09 (2021) 053
Summary
To fully understand the notion of linear algebraic group requires some knowledges of algebraic geometry. General linear group GL(n, K): the space of all n × n matrices with entries in K and nonzero determinant This example is not totally trivial, since the definition of GL(n, K) is usually formulated with inequality instead of polynomial equation. With rational group action, when we say that the (linear) reductiveness of a group can alone imply the finiteness of the generating set of a specific ring of invariants, it is the (linearly) reductive group in the context of algebraic group that we are talking about
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