Low-derivative operators of the Standard Model effective field theory via Hilbert series methods

TL;DR

This paper extends Hilbert series methods to count low-derivative operators in the Standard Model effective field theory, providing an algorithm to generate invariant operators up to dimension 8, accounting for redundancies.

Contribution

It introduces a new algorithm for counting and generating invariant operators with derivatives in the Standard Model EFT, handling redundancies for low-derivative cases.

Findings

Successfully counts operators up to dimension 7 for arbitrary flavors.

Generates 535 dimension-8 operators for Nf=1.

Identifies limitations at dimension 8 and provides manual procedures for remaining classes.

Abstract

In this work, we explore an extension of Hilbert series techniques to count operators that include derivatives. For sufficiently low-derivative operators, we find an algorithm that gives the number of invariant operators, properly accounting for redundancies due to the equations of motion and integration by parts. Specifically, the technique can be applied whenever there is only one Lorentz invariant for a given partitioning of derivatives among the fields. At higher numbers of derivatives, equation of motion redundancies can be removed, but the increased number of Lorentz contractions spoils the subtraction of integration by parts redundancies. While restricted, this technique is sufficient to automatically generate the complete set of invariant operators of the Standard Model effective field theory for dimensions 6 and 7 (for arbitrary numbers of flavors). At dimension 8, the algorithm does not automatically generate the complete operator set; however, it suffices for all but five classes of operators. For these remaining classes, there is a well defined procedure to manually determine the number of invariants. Using these methods, we thereby derive the set of 535 dimension-8 $N_f = 1$ operators.