Hunting 3d \( \mathcal{N} \) = 1 SQED in the ϵ-expansion

It was recently shown that 3d $\mathcal{N}$ \( \mathcal{N} \) = 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 N_g neutral Majorana fermions χ_a, N_f charge-1 bosons ϕ_i and N_f × N_g charge-1 Dirac fermions ψ_ia in the d = 4 − 2ϵ expansion. Analytically continuing to N_g = $\frac{1}{2}$ \( \frac{1}{2} \) schematically matches the Lagrangian and matter content of 3d $\mathcal{N}$ \( \mathcal{N} \) = 1 SQED, and we check whether this match can be made rigorous. We compute anomalous dimensions of χ_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 $\mathcal{N}$ \( \mathcal{N} \) = 1 IR dualities between gauge theories.