Stability of $SU(N_c)$ QCD3 from the $\epsilon$-Expansion

QCD with gauge group $SU(N_c)$ flows to an interacting conformal fixed point in three spacetime dimensions when the number of four-component Dirac fermions $N_f \gg N_c$. We study the stability of this fixed point via the $\epsilon$-expansion about four dimensions. We find that when the number of fermions is lowered to $N_f^{\rm crit} \approx {11 \over 2} N_c + (6 + {4 \over N_c}) \epsilon$, a certain four-fermion operator becomes relevant and the theory flows to a new infrared fixed point (massless or massive). F-theorem or entanglement monotonicity considerations complement our $\epsilon$-expansion calculation.