(\[eq:GluonCurvedDR\]),

$$\begin{aligned}
Z_0^g &= M^2\left[\frac{2}{3}\cal{C}_2^{200}-2\cal{C}_2^{100}+\frac{N_s}{3}\cal{C}_2^{020}- N_s \cal{C}_2^{010}\right]+\frac{7}{30\pi^2}\ln\left(\frac{\cal{A}_{100}}{\cal{A}_{200}}\right)-\frac{2}{15}\left(2\cal{F}_2^g+N_s\cal{F}_2^s\right)+\nonumber\\
&+\frac{5}{18\pi^2}+\frac{N_s}{45\pi^2}+N_f\left[\frac{4}{3}M^2\cal{C}_2^{002}-\frac{1}{360\pi^2}-\frac{2}{5}\cal{F}_2^f\right].\label{eq:Z0gCount}\end{aligned}$$

The divergent piece of $Z_{0s}^{ab}$ is determined by (\[eq:1LoopSE2CurvedCO\]). The finite piece on the other hand is arbitrary if we only demand gauge invariance and we parameterize it as,

$$\begin{aligned}
Z_{0s}^{ab} &= \frac{M^2}{2}\left[\left(\mathbf{Q}^{(ab)cc} - 2\delta^{ab} \right)\cal{C}_2^{010} - 4\delta^{ab}\cal{C}_2^{100} \right] + \frac{\delta^{ab}}{4\pi^2}\ln\left(\frac{\cal A_{100}^2M}{\cal A_{010}}\right) +\nonumber \\
&+ \mathrm{tr}(\rho^{a\dag }\rho^b + \rho^{b\dag }\rho^a)\left[M^2\cal{C}_2^{002} - \frac{1}{16\pi^2}\ln(\cal{A}_{002}M)\right]+\nonumber\\
&+\frac{1}{\pi^2}(C_1\delta^{ab}+C_2\mathrm{tr}(\rho^{a\dag}\rho^b+\rho^{b\dag}\rho^a)+C_3\mathbf{Q}^{(ab)cc}). \label{eq:Z0sCount}\end{aligned}$$

The constants $C_1$, $C_2$ and $C_3$ will be determined in section \[ssec:NumericsChecks\] by demanding conformal invariance to order $\lambda$.

Perturbative calculation of the effective action {#sec:2LoopComputation}
================================================

In this section we present the computation of the large $N$ effective action $S_{eff}(U)$ defined in (\[eq:Z\]), to 2-loops in perturbation theory.

2-loop vacuum diagrams
----------------------

The 2-loop diagrams which we have to evaluate are depicted in figure 4.

\[fig:2Loop\] ![The 2-loop diagrams obtained after integrating $c$ and $a$. Vertices with a little circle correspond to terms we got after integrating the linear $a$ vertex. ](2loop.png "fig:"){width="9cm" height="10cm"}\

Diagrams $\mathbf{L_1}$, $\mathbf{L_{5a}}$ and $\mathbf{L_{5b}}$ were already evaluated in [@Aharony:2006rf]. The expressions for the other diagrams that contribute in the planar limit are,

$$\begin{aligned}
\mathbf{L_2} &= -2\beta g^2 N_s C^{\alpha\beta\gamma}C^{\bar{\alpha}\bar{\beta}\bar{\gamma}}\int_0^{\beta} dt \Delta_{\beta}^{ij}(t)\Delta_{\alpha}^{jk}(t)\Delta_{\gamma}^{ki}(t),\label{eq:L2}\\
\mathbf{L_3} &= \frac{1}{2}\beta g^2 N_f G^{\alpha\beta\gamma}G^{\beta\alpha\bar{\gamma}}\int_0^{\beta} dt \left\{\Theta_{\alpha}^{ij}(-t)\Theta_{\beta}^{ik}(t)\Delta_{\gamma}^{kj}(t)+\right.\nonumber\\
       &~~~\left.+\Theta_{\alpha}^{ij}(-t)\Theta_{\beta}^{kj}(t)\Delta_{\gamma}^{ik}(t)\right\},\label{eq:L3}\end{aligned}$$

$$\begin{aligned}
