Hilbert Series and Operator Counting on the Higgs Effective Field Theory

We present a systematic procedure for determining the Hilbert series that counts the number of independent operators in the Higgs effective field theory. After removing the redundancies from equation-of-motion and integration-by-part, we provide an algorithm of treating the redundancy from the operators involving in spurion fields parametrizing the custodial symmetry breaking. Furthermore, we utilize the outer automorphism of the Lorentz and internal symmetries to separate operators with different CP properties. With these new implements, the Hilbert series up to chiral dimension 10 are obtained, and CP-even and CP-odd operators can be further divided. Extensions to all orders are straightforward.