---
author:
- 'Brian Henning,'
- 'Xiaochuan Lu,'
- Tom Melia
- and Hitoshi Murayama
bibliography:
- './bibliography.bib'
title: Hilbert series and operator bases with derivatives in effective field theories
---

Introduction
============

The Wilsonian picture of effective field theory (EFT) dictates that the effective action contains all possible local operators consistent with the symmetries of the EFT, $$S_{\text{eff}} = \int_{\scM} \sum_{a}\k_a \scO_a,
\label{eqn:Seff}$$ where $\scO_a$ are local operators, $\k_a$ are their associated Wilson coefficients, and $\scM$ is the spacetime manifold. It is well known that distinct operators can lead to the same physical effect, *i.e.* give the same $S$-matrix elements. Two examples, which will play a prominent role in this work, are operators related by integration by parts (IBP) and operators related through the equations of motion (EOM). We will refer to the minimum set of operators encapsulating all possible physical effects as a *basis* for the EFT.

The issue of determining (subsets of) the EFT basis has arisen in the past for various specific EFTs, but no general prescription exists. Logically, properties of the basis should be discussed first since they include, for example, the number of physically distinct effects. Determining the EFT basis is a very difficult 