Hilbert series for quark flavor invariants and CP violation

TL;DR

This paper uses the Hilbert series to systematically analyze flavor invariants and CP violation in the Standard Model extended with a vector-like quark, providing a mathematical framework for understanding invariants related to CP violation.

Contribution

It applies the Hilbert series method to enumerate flavor invariants in the SM extended by a down-type vector-like quark, including the calculation of the Hilbert series in the mass basis.

Findings

Successfully calculated the Hilbert series for the VLQ extension in the mass basis.

Built and enumerated basic flavor invariants for the VLQ extension.

Provided initial terms of the Hilbert series for a generic basis.

Abstract

This work delves into the study of flavor invariants and, in special, invariants capable of detecting CP (Charge-Parity) violation. Through the mathematical tool of the Hilbert series, we systematically enumerate and explore flavor invariants that are unchanged under weak basis transformations. After reviewing the Hilbert series and the flavor invariants of the SM quark sector, we apply the tool of Hilbert series to the SM extended by a singlet vector-like quark (VLQ) of down-type. The introduction of these hypothetical particles leads to a simple extension of the SM that can be motivated by many problems, including the need for new sources of CP violation to explain the observed matter-antimatter asymmetry in the universe. We were successful in calculating the Hilbert series for the VLQ extension in the mass basis of the VLQ, where the spurion transformations are simpler. Based on the Hilbert series, we build and enumerate the basic flavor invariants with which all invariants can be constructed. For a generic basis, where the spurion transformations involve a larger group, we could only get the first few terms of the Hilbert series.