CP violation and flavor invariants in the seesaw effective field theory

TL;DR

This paper analyzes the algebraic structure of flavor invariants and CP violation sources in the seesaw effective field theory, providing explicit constructions and linking high-energy CP asymmetries to low-energy neutrino phenomena.

Contribution

It calculates the Hilbert series and constructs all primary flavor invariants in the SEFT, establishing a direct connection between high-energy CP asymmetries and low-energy observables.

Findings

All physical parameters can be extracted from primary invariants.

CP-odd invariants can express any CP-violating observable.

Number of primary invariants in SEFT matches that in the full seesaw model.

Abstract

In this paper, we systematically study the algebraic structure of the ring of the flavor invariants and the sources of CP violation in the seesaw effective field theory (SEFT), which is obtained by integrating out heavy Majorana neutrinos in the type-I seesaw model at the tree level and thus includes the dimension-five Weinberg operator and one dimension-six operator. For the first time, we calculate the Hilbert series and explicitly construct all the primary flavor invariants in the SEFT. We show that all the physical parameters can be extracted using the primary invariants and any CP-violating observable can be expressed as the linear combination of CP-odd flavor invariants. The calculation of the Hilbert series shows that there is an equal number of primary flavor invariants in the SEFT and in the full seesaw model, which reveals the intimate connection between the flavor space of the SEFT and that of its ultraviolet theory. A proper matching procedure of the flavor invariants is accomplished between the SEFT and the full seesaw model, through which one can establish a direct link between the CP asymmetries in leptogenesis and those in low-energy neutrino oscillations.