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We present a complete and independent off-shell Green's basis of thedimension 8 operators in the Standard Model effective field theory (SMEFT). Wepropose an off-shell amplitude formalism such that this new kind of amplitudeshas a one-to-one correspondence to general operators carrying all kinds ofredundancies. The advantage of such formalism is that all kinds of redundanciescan be formulated explicitly in terms of the off-shell spinors and thus beremoved systematically from the exhaustive list of the general off-shellamplitudes. Following such procedure, redundant operators can be reduced to theones with the only redundancy from the equation of motion kept, the Green'sbasis. We find there are (993+1649) independent (on-shell + off-shell)operators for one generation of fermions and (44807+66197) ones for threegenerations at the dimension 8. This systematic method can be applied to obtainthe Green's basis at any mass dimension for general EFTs.

A Complete Set of the Dimension-8 Green's Basis Operators in the Standard Model Effective Field Theory