Counting Operators in Effective Field Theories

The Standard Model is now viewed as an effective field theory (EFT), a theory that is valid only up to some high energy scale Λ, at which point it is subsumed into its ultraviolet (UV) completion. Given this, it is of both theoretical and phenomenological interest to enumerate a minimal basis for the operators in this EFT at various mass dimensions. This problem can be extended beyond the Standard Model effective field theory to encompass generic effective field theories and the question of writing down a minimal Lagrangian at some desired mass order.

I approach this problem from two angles. First, I calculate the set of dimension-7 operators in the Standard Model effective field theory “by hand.” Even though there are relatively few operators at dimension-7 as compared to dimension-8, this calculation is somewhat lengthy and thus illustrates the desirability of a more automated method.

Second, I introduce a mathematical structure known as the Hilbert series. After providing some mathematical background on the Hilbert series, I illustrate how it can be used to attack the problem of finding a minimal operator basis through several examples.

Finally, the Hilbert series as initially presented does not deal with the twin problems introduced by derivatives: integration by parts and equations of motion. I present a conjecture for the correct method to deal with these problems, and then, in my conclusion, discuss how this conjecture fell short of the correct method.