Complete NLO Operators in the Higgs Effective Field Theory

We enumerate the complete and independent sets of operators at the next-to-leading order (NLO) in the Higgs effective field theory (HEFT), based on the Young tensor technique on the Lorentz, gauge and flavor structures. The operator-amplitude correspondence tells a type of operators forms the on-shell amplitude basis, and for operators involving in Nambu-Goldstone bosons, the amplitude basis is further reduced to the subspace satisfying the Adler's zero condition in the soft momentum limit. Different from dynamical field, the spurion should not enter into the Lorentz sector, instead it only plays a role of forming the $SU(2)$ invariant together with other dynamical fields. With these new treatments, for the first time we could obtain the 224 (7704) operators for one (three) generation fermions, 295 (11307) with right-handed neutrinos, and find there were 6 (9) terms of operators missing and many redundant operators can be removed in the effective theory without (with) right-handed neutrinos.