Algebraic Structure of Lepton and Quark Flavor Invariants and CP Violation

TL;DR

This paper uses invariant theory to systematically classify flavor invariants in the Standard Model’s low-energy sector and in the seesaw framework. By computing Hilbert series and identifying generating invariants and syzygies, it reveals a markedly richer algebraic structure in the lepton sector—especially for three generations—compared with the quark sector. Complete invariant classifications and Hilbert-series analyses are achieved for the SM EFT with two lepton generations and for the two-generation seesaw model, while the three-generation cases remain computationally challenging. A key payoff is the definition of an invariant CP-violating angle in the electroweak sector and the explicit link between CP violation and the Jarlskog-type invariants in the quark sector.

Abstract

Lepton and quark flavor invariants are studied, both in the Standard Model with a dimension five Majorana neutrino mass operator, and in the seesaw model. The ring of invariants in the lepton sector is highly non-trivial, with non-linear relations among the basic invariants. The invariants are classified for the Standard Model with two and three generations, and for the seesaw model with two generations, and the Hilbert series is computed. The seesaw model with three generations proved computationally too difficult for a complete solution. We give an invariant definition of the CP-violating angle theta in the electroweak sector.

Figures (1)

• Figure 1: Graphical representation of the chiral transformation properties of the lepton mass matrices $X_E$, $m_\nu$, $m_\nu^\dagger$, $M$ and $M^*$. A solid line represents $\mathcal{U}_{N^c}$, and a dashed line $\mathcal{U}_L$. The invariants are obtained by forming graphs with no external lines.