Hilbert series and operator bases with derivatives in effective field theories

TL;DR

The paper develops a systematic framework to count and construct independent EFT operators including derivatives while factoring out equivalences from IBP and EOM. By mapping the problem to polynomial rings in field momenta and encoding data in a generalized Hilbert series H(t, {u_i}), the authors derive explicit results for a 0+1D N-flavor real-scalar theory, revealing a rich IBP–EOM structure and an SL(2, C) representation-theoretic organization. A closed-form Hilbert series for general N is obtained via an SL(2, C)/Molien approach, along with recursion and consistency relations that relate H_N across flavors. The work provides both explicit operator bases for small N and a scalable, algebraic methodology (including a composition rule) for counting and constructing EFT operator bases with derivatives, with clear routes to generalization to higher dimensions and broader symmetry structures.

Abstract

We introduce a systematic framework for counting and finding independent operators in effective field theories, taking into account the redundancies associated with use of the classical equations of motion and integration by parts. By working in momentum space, we show that the enumeration problem can be mapped onto that of understanding a polynomial ring in the field momenta. All-order information about the number of independent operators in an effective field theory is encoded in a geometrical object of the ring known as the Hilbert series. We obtain the Hilbert series for the theory of N real scalar fields in (0+1) dimensions--an example, free of space-time and internal symmetries, where aspects of our framework are most transparent. Although this is as simple a theory involving derivatives as one could imagine, it provides fruitful lessons to be carried into studies of more complicated theories: we find surprising and rich structure from an interplay between integration by parts and equations of motion and a connection with SL(2,C) representation theory which controls the structure of the operator basis.

Figures (3)

• Figure 1: Decomposition of operator space into irreducible $SL(2,\mathbb{C})$ representations.
• Figure 2: The analytic structure of the Hilbert series for the EFT of real scalar fields in $d=1$.
• Figure 3: Hilbert series can be composed to build up $H_N$ for larger numbers of flavors.