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

A bstract It was recently shown that 3 d $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 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 − 2 ϵ expansion. Analytically continuing to N g = $$ \frac{1}{2} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:math> schematically matches the Lagrangian and matter content of 3 d $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 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 3 d $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 1 IR dualities between gauge theories.