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

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