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We revisit the effective field theory of the standard model that is extendedwith sterile neutrinos, $N$. We examine the basis of complete and independenteffective operators involving $N$ up to mass dimension seven (dim-7). Byemploying equations of motion, integration by parts, and Fierz and groupidentities, we construct relations among operators that were consideredindependent in the previous literature, and find seven redundant operators atdim-6, sixteen redundant operators and two new operators at dim-7. The correctnumbers of operators involving $N$ are, without counting Hermitian conjugates,$16~(L\cap B)+1~(\slashed{L}\cap B)+2~(\slashed{L}\cap\slashed{B})$ at dim-6,and $47~(\slashed{L}\cap B)+5~(\slashed{L}\cap\slashed{B})$ at dim-7. Here$L/B~(\slashed L/\slashed B)$ stands for lepton/baryon number conservation(violation). We verify our counting by the Hilbert series approach for $n_f$generations of the standard model fermions and sterile neutrinos. Whenoperators involving different flavors of fermions are counted separately andtheir Hermitian conjugates are included, we find there are $29~(1614)$ and$80~(4206)$ operators involving sterile neutrinos at dim-6 and dim-7respectively for $n_f=1~(3)$.

Operators up to Dimension Seven in Standard Model Effective Field Theory Extended with Sterile Neutrinos