depend on the Ricci tensor.

Following [@Aharony:2006rf] we determine the counterterms by computing 1-loop diagrams in the cutoff scheme and demanding that the result match dimensional regularization with some convenient subtraction scheme. This guarantees once and for all that the results of the 2-loop calculation are gauge invariant.

Flat space counterterms {#ssec:FlatSpaceCount}
-----------------------

For the dimensionally regularized theory we work in $3+d$ spatial dimensions, and extend the coulomb gauge condition to include the $d$ extra components of the gauge field. The flat Euclidean action is then,

$$\begin{aligned}
S_2 &= \int d^{d+4}x\mathrm{tr}\Big\{\frac{1}{2}\dot{A_i}\dot{A_i}+\frac{1}{2}\partial_jA_i\partial_jA_i+\frac{1}{2}\partial_iA_0\partial_iA_0+\partial_ic^{\dag}\partial_ic+\nonumber\\
&\qquad+\frac{1}{2}\partial_{\mu}\phi_a\partial_{\mu}\phi_a+i\psi^{\dag I}\sigma^{\mu}\partial_{\mu}\psi_I\Big\},\label{eq:FlatSpaceAction1}\\
S_3 &= g\int d^{d+4}x\mathrm{tr}\Big\{i\dot{A}_i[A_i,A_0]-i\partial_iA_0[A_i,A_0]-i\partial_iA_j[A_i,A_j]-\nonumber\\
&\qquad-i\partial_ic^{\dag}[A_i,c]+\psi^{\dag I}\sigma^0[A_0,\psi_I]+\psi^{\dag I}\sigma^i[A_i,\psi_I]-i\dot{\phi}_a[A_0,\phi_a]-\nonumber\\
&\qquad-i\partial_i\phi_a[A_i,\phi_a]+\frac{1}{2}\psi_I\varepsilon(\rho^{a\dag})^{IJ}[\phi_a,\psi_J]+\frac{1}{2}\psi^{\dag I}\varepsilon\rho^a_{IJ}[\phi_a,\psi^{\dag J}]\Big\},\label{eq:FlatSpaceAction2}\\
S_4 &= -\frac{1}{2}g^2\int d^{d+4}x\mathrm{tr}\Big\{[A_0,A_i]^2+\frac{1}{2}[A_i,A_j]^2+[A_0,\phi_a]^2+\nonumber\\
&\qquad+[A_i,\phi_a]^2+\frac{1}{2}\mathbf{Q}^{abcd}\phi_a\phi_b\phi_c\phi_d\Big\}\label{eq:FlatSpaceAction3}.\end{aligned}$$

We define the spinor matrices to satisfy $\sigma^{\mu}\bar{\sigma}^{\nu}+\sigma^{\nu}\bar{\sigma}^{\mu}=2\delta^{\mu\nu}\mathds{1}$, with $\mu,\nu= 0,\ldots, d\!+\!3$. We also use the conventions $\mathrm{tr}(\mathds{1})=2$, and $\sigma^0=\bar{\sigma}^0=\mathds{1}$, which are consistent with the $d\rightarrow 0$ limit. The propagators of the fields are,

$$\begin{aligned}
\langle A_i(\nu,k)A_j(-\nu,-k)\rangle \equiv \Delta_{ij}(\nu,k) &= \frac{k^2\delta_{ij}-k_ik_j}{k^2(\nu^2+k^2)},\label{eq:AiProp}\\
\langle A_0(\nu,k)A_0(-\nu,-k)\rangle &= \frac{1}{k^2}\label{eq:A0Prop},\\
\langle c(\nu,k)c^{\dag}(\nu,k)\rangle &= \frac{1}{k^2},\label{eq:cProp}\\
\langle \phi^a(\nu,k)\phi^b(-\nu,-k)\rangle &= \frac{1}{\nu^2+k^2}\delta^{ab},\label{eq:sProp}\\
\langle\psi_I(\nu,k)\psi^{\dag J}(\nu,k)\rangle &= \frac{\nu+k_i\bar{\sigma}^i}{\nu^2+k^2},\label{eq:fProp}.\end{aligned}$$

In order to compute $Z_1,Z_2$, we’ll compute the 1PI self-energies of the various fields. The self energy diagrams that contribute are depicted in figure 1. No quartic vertex contributes to these counterterms since such diagrams 