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

TL;DR

The paper extends Hilbert series methods to count SMEFT operators with derivatives by dressing spurions with derivatives and subtracting a Lorentz-vector sector to remove IBP redundancies, while carefully handling EOM constraints. The approach is validated on $d=6$ and $d=7$ SMEFT, reproducing known operator counts, and yields a substantial, though not complete, catalog at $d=8$ for $N_f=1$, totaling 535 derivative-inclusive operators (931 with conjugates) and 46 baryon-number-violating cases. The method succeeds when each derivative partition has a single Lorentz contraction but requires manual analysis for partitions with multiple contractions. The work demonstrates both the power and current limits of derivative-aware Hilbert series counting and outlines concrete paths to a fully automatic framework applicable to broader EFTs and dimensions.

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.