Hunting 3d N=1 SQED in the ε-expansion

It was recently shown that $3d$ N=1 supersymmetric Wess-Zumino models can be studied in the ε-expansion by analytically continuing the number of fermionic degrees of freedom to be half-integer. In this work we study the extension of this strategy to gauge theories. We consider $U(1)$ gauge theories with Ng neutral Majorana fermions χₐ, Nf charge-1 bosons φᵢ and Nf× Ng charge-1 Dirac fermions ψᵢₐ in the d=4-2ε expansion. Analytically continuing to Ng=12 schematically matches the Lagrangian and matter content of $3d$ N=1 SQED, and we check whether this match can be made rigorous. We compute anomalous dimensions of χₐ up to two loops and of meson operators up to one loop at the fixed points, and compare to expectations from SUSY. While we find obstructions to SUSY at small Nf, at large Nf the observables approach the expected values at a SUSY fixed point. This may allow for checks of $3d$ N=1 IR dualities between gauge theories.