Hunting 3d $\mathcal{N}=1$ SQED in the $\epsilon$-expansion

It was recently shown that $3d$ $\mathcal{N}=1$ supersymmetric Wess-Zumino models can be studied in the $\epsilon$-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 $N_g$ neutral Majorana fermions $\chi_a$, $N_f$ charge-1 bosons $\phi_i$ and $N_f\times N_g$ charge-1 Dirac fermions $\psi_{ia}$ in the $d=4-2\epsilon$ expansion. Analytically continuing to $N_g=\frac12$ schematically matches the Lagrangian and matter content of $3d$ $\mathcal{N}=1$ SQED, and we check whether this match can be made rigorous. We compute anomalous dimensions of $\chi_a$ 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 $N_f$, at large $N_f$ the observables approach the expected values at a SUSY fixed point. This may allow for checks of $3d$ $\mathcal{N}=1$ IR dualities between gauge theories.