A complete set of the dimension-8 Green's basis operators in the Standard Model effective field theory

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