Effective interaction of Chern-Simons boson with fermions

We consider a vector extension of the Standard Model (SM) with a Chern-Simons-type interaction. This extension introduces a new massive vector boson (the Chern-Simons (CS) boson) that does not couple directly to SM fermions at tree level. We analyze the effective loop-induced interaction of this new vector boson with SM fermions and study its renormalizability in the $R_ξ$ gauge. We find that, in the effective interaction between the CS boson and same-flavor fermions, the divergent contributions from individual loop diagrams do not cancel when all relevant diagrams are taken into account. In contrast, for interactions involving fermions of different flavors, the corresponding loop-induced contributions are finite and well-defined. This indicates that, in the low-energy limit, the theory exhibits nonrenormalizable behavior in the sector describing the loop-induced interaction of the CS boson with same-flavor fermions. The interaction terms between the CS boson and same-flavor fermions, characterized by divergent coefficients, are identified and must be treated within the framework of effective field theory. Finally, we derive the leading-order effective Lagrangian describing the interaction of a GeV-scale CS boson with SM fermions and discuss the number of independent parameters entering this Lagrangian. The leading-order interaction we obtained turns out to be similar to the interaction of a $Z^\prime$ boson with SM fermions.