Efficient Hilbert Series for Effective Theories

Abstract

The task of finding operator bases for effective theories can be assisted by using the Hilbert series, which counts the number of independent operators at a given effective order. In this thesis we will introduce the Hilbert series for the Standard Model Effective Field Theory (SMEFT) and some extensions. We present an efficient algorithm for determining the Hilbert series of an effective theory and provide a companion code called ECO (Efficient Counting of Operators) in FORM. The implementation can be used to efficiently establish the number of operators at effective orders as high as 20 (or more). While the implementation focusses on SMEFT, we allow for a flexible user input of the light degrees of freedom. We discuss how the Hilbert series technique can be extended to the counting of CP-invariant operators by relating the outer automorphisms of the Lorentz group and the gauge groups to C and P, respectively. In particular, we show how the outer automorphisms can be classified using the symmetries of the Dynkin diagrams, and how they give rise to an abstract definition of a folding of these diagrams, which can be used in the computation of the Hilbert series.