Hilbert Series for Leptonic Flavor Invariants in the Minimal Seesaw Model

TL;DR

This paper calculates the Hilbert series for leptonic flavor invariants in the minimal seesaw model, identifying fundamental invariants and their role in CP violation and leptogenesis.

Contribution

It is the first to compute the Hilbert series for flavor invariants in the MSM, revealing 38 basic invariants and their polynomial relations.

Findings

Identified 38 basic flavor invariants in the MSM.

Found that low-energy invariants are rational functions of high-energy invariants.

Provided tools for analyzing CP violation and leptogenesis in the MSM.

Abstract

In this paper, we examine the leptonic flavor invariants in the minimal seesaw model (MSM), in which only two right-handed neutrino singlets are added into the Standard Model in order to accommodate tiny neutrino masses and explain cosmological matter-antimatter asymmetry via leptogenesis mechanism. For the first time, we calculate the Hilbert series (HS) for the leptonic flavor invariants in the MSM. With the HS we demonstrate that there are totally 38 basic flavor invariants, among which 18 invariants are CP-odd and the others are CP-even. Moreover, we explicitly construct these basic invariants, and any other flavor invariants in the MSM can be decomposed into the polynomials of them. Interestingly, we find that any flavor invariants in the effective theory at the low-energy scale can be expressed as rational functions of those in the full MSM at the high-energy scale. Practical applications to the phenomenological studies of the MSM, such as the sufficient and necessary conditions for CP conservation and CP asymmetries in leptogenesis, are also briefly discussed.