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

TL;DR

This paper systematically classifies and counts effective operators involving sterile neutrinos up to dimension seven in the Standard Model EFT, correcting previous operator counts and verifying with Hilbert series methods.

Contribution

It provides a complete and corrected basis of operators involving sterile neutrinos up to dimension seven, including relations among operators and new operator counts.

Findings

Identified redundancies among operators at dim-6 and dim-7.

Provided explicit operator counts for different flavor scenarios.

Verified operator basis using Hilbert series approach.

Abstract

We revisit the effective field theory of the standard model that is extended with sterile neutrinos, $N$. We examine the basis of complete and independent effective operators involving $N$ up to mass dimension seven (dim-7). By employing equations of motion, integration by parts, and Fierz and group identities, we construct relations among operators that were considered independent in the previous literature, and find seven redundant operators at dim-6, sixteen redundant operators and two new operators at dim-7. The correct numbers 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. When operators involving different flavors of fermions are counted separately and their Hermitian conjugates are included, we find there are $29~(1614)$ and $80~(4206)$ operators involving sterile neutrinos at dim-6 and dim-7 respectively for $n_f=1~(3)$.