Abstract
For the first time, we list the complete and independent set of operators at the next-to-next-to-leading order (NNLO) in the Higgs effective field theory (HEFT). The Young tensor technique utilized in this work guarantees the completeness and independence of the on-shell amplitude basis while the Adler zero condition imposes non-linear symmetry on the Nambu-Goldstone bosons that play the central role in the chiral Lagrangian. The spurion fields are incorporated into the gauge structure of operators following the Littlewood-Richardson rule to accommodate custodial symmetry breaking. We construct 11506 (1927574) NNLO operators for one (three) flavor of fermions for the electroweak chiral Lagrangian with the light Higgs, and enumerate 8065(1179181) operators for one (three) flavor of fermions when the right-handed neutrino is absent by Hilbert series technique. Below the electroweak symmetry breaking scale, the dimension-8 standard model effective field theory (SMEFT) operators could be matched to these HEFT operators at both the NLO and NNLO orders.
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