Thermal relevant direction at the Wilson-Fisher fixed point ($y_t=2-ϵ/3+O({ϵ}^2),y_g=-ϵ+O({ϵ}^2)$)

The scalar Wilson-Fisher fixed point is attractive within its critical surface, but has a relevant thermal direction. With $r=m^2{\Lambda }^{-2}$, $\lambda =g{\Lambda }^{-ϵ}$, and $K_D={\Omega }_D/(2\pi {)}^D$, its one-loop equations are $\dot{r}=2r+K_D\lambda /[2(1+r)]$ and $\dot{\lambda }=ϵ\lambda -3K_D{\lambda }^2/[2(1+r{)}^2]$. Their stability matrix at $r_{\ast }=-ϵ/6+O({ϵ}^2)$ and ${\lambda }_{\ast }=16{\pi }^2ϵ/3+O({ϵ}^2)$ has the displayed eigenvalues. Thus an unrestricted claim of infrared attraction is false: the bare mass must be tuned to the critical surface.