![image](B2.png){width="3.7cm"}   ![image](B3.png){width="3.7cm"}   ![image](B4.png){width="3.7cm"}
                (B1)                              (B2)                              (B3)                              (B4)
  --------------------------------- --------------------------------- --------------------------------- ---------------------------------

The divergent parts of all the diagrams above are listed in Appendix \[diags\]. By summing these we can determine the renormalization constants, $$\begin{aligned}
  Z_{\phi} &\equiv 1 + \sum_i(Bi) = 1 - \frac{g^4\left(3N^2-23N+20\right)}{384\pi^2}\frac{1}{\epsilon}{\,,}\label{eq:Zphi}\\
  g_6Z_{g_6} &\equiv g_6 + \sum_i(Ai) = g_6 + \frac{66g^8\left(N-1\right) + 4g_6^2\left(3N+22\right) - 3g^4g_6\left(N^2+19N-20\right)}{128\pi^2}\frac{1}{\epsilon}{\,.}\label{eq:Zg6}\end{aligned}$$ The bare sextic coupling $g_{6,0} = \mu^{2\epsilon} g_6 Z_{g_6} / Z_{\phi}^3$ may thus be written in the form $g_{6,0} = g_6 + b_1(g,g_6) / \epsilon + (\mathrm{other~terms})$, where $$\begin{aligned}
b_1(g,g_6) = \frac{33 g^8(N-1) - 40 g^4 g_6 (N-1) + 2 g_6^2 (3 N + 22)}{64 \pi^2}{\,.}\end{aligned}$$ The beta function for the $\lambda_6$ coupling is related to the single pole in dimensional regularization by $\beta_{g_6} = - 2 b_1 + 2 g_6 \partial_{g_6} b_1 + {1\over 2} g \partial_g b_1$ [@Weinberg:1996kr], leading to $$\begin{aligned}
\label{eq:Full2LoopBeta6}
\beta_{\lambda_6}(\lambda,\lambda_6) = \frac{33 (N-1) \lambda ^4-40 (N-1) \lambda ^2\lambda_6+2 (3 N + 22) \lambda_6^2}{32 N^2 \pi ^2} {\,.}\end{aligned}$$

In the ’t Hooft large $N$ limit we see that $\beta_{\lambda_6}=0$, so that both the $(\phi^2)^3$ coupling and the Chern-Simons interaction are marginal at this order. For the theory with only $(\phi^2)^3$ couplings it is easy to see that the large $N$ 