this stability).

The RG is carried out perturbatively in $e$, but non-perturbatively in the mass parameters. Thus a full treatment gives non-trivial and complete beta functions for the two dimensionless mass ratios $m/M_0$, $m/M_c$; these are given in the Supplementary Material. The analysis of the full RG shows, however, that there is a single stable [*isotropic*]{} fixed point corresponding to $m/M_0=m/M_c=0$, so for simplicity we quote in the main text only the results in the vicinity of this point.

In this limit, the leading contribution to the bosonic self energy becomes $$\begin{aligned}
\frac{1}{N_f}\Sigma_{\varphi}(q, 0) 
&=& -(2m) e^2 \left[ \int \! \frac{d^dk}{(2\pi)^d}\, \frac{1}{k^4}\right] \times q^2 ,\end{aligned}$$ where we took the $\omega \rightarrow 0$ limit because frequency dependence is subdominant. The divergence should be absorbed by rescaling the bosonic field, $\varphi \rightarrow e^{-\eta_b
  d\ell}\varphi$ upon reduction of the hard momentum cutoff $\Lambda
\rightarrow e^{-d\ell}\Lambda$, which defines the RG parameter $\ell$. This gives the bosonic anomalous dimension $\eta_b = 2
  N_f  u$ [@large_n], where the dimensionless coupling constant is $u= \frac{m\, e^2 }{ 8\pi^2 c_0 \Lambda^{4-d}}$, which has the physical meaning in $d=3$ of the ratio of the real space cutoff to the effective Bohr radius. The frequency dependence of the one loop 