Flavor invariants for the SM with one singlet vector-like quark

TL;DR

This paper systematically constructs flavor invariants for the Standard Model extended with a singlet vector-like quark, revealing new CP odd invariants and relations that enhance understanding of CP violation detection.

Contribution

It introduces a complete set of flavor invariants, including a lower-degree CP odd invariant, and uncovers polynomial relations among invariants, advancing the analysis of CP violation.

Findings

Identified a CP odd invariant of degree six lower than the Jarlskog invariant.

Derived polynomial relations (syzygies) among invariants, including expressing the square of the CP odd invariant.

Demonstrated existence of CP violation not detectable by previously known invariants.

Abstract

We study the flavor invariants of the SM augmented by one singlet vector-like quark. Aided by the Hilbert series, we construct all the basic invariants with which any flavor invariant can be written as a polynomial. In special, this theory contains one CP odd invariant of degree six which has degree much lower than the usual Jarlskog invariant of the SM. We find the nonlinear polynomial relations (syzygies) of lowest degrees involving these basic invariants, including the expression of the square of the CP odd invariant of lowest degree in terms of CP even invariants. The $SU(3)$ identity underlying this syzygy is uncovered in terms of invariant tensors, which can be applied to rewrite any square of a CP odd invariant of the same form, involving three hermitean matrices of size three. We demonstrate by an example that there is CP violation that is not detected by the CP odd invariants proposed in the literature so far but it can be detected with the full list of CP odd invariants found here.