Complete NNLO operator bases in Higgs effective field theory

For the first time, we list complete and independent set of operators at next-to-next-to-leading order (NNLO) in Higgs effective field theory (HEFT), also named as electroweak chiral Lagrangian with light boson. The Young tensor technique utilized this work guarantees completeness independence on-shell amplitude basis while Adler zero condition imposes non-linear symmetry on Nambu-Goldstone bosons that play central role Lagrangian. spurion fields are incorporated into gauge structure following Littlewood-Richardson rule to accommodate custodial breaking. We enumerate 11506 (1927574) NNLO for one (three) flavor fermions Higgs. Below breaking scale, dimension-8 standard model (SMEFT) could be matched these HEFT both NLO orders.