it in the planar limit alone. However, the number of diagrams that may contribute to its divergence is greatly reduced by the following observation. Consider a diagram that includes a $\phi A^\mu \partial_\mu \phi$ vertex, with the gluon carrying a loop momentum $q$ and one of the scalar lines carrying an external momentum $p$. In the numerator we then have $\epsilon_{\mu\nu\rho} q^\rho$ from the gluon propagator (\[eq:GluonProp\]) and $(q+2p)^\mu$ from the vertex, and the leading high-energy term of order $q^2$ cancels by antisymmetry. Therefore, in such a situation the degree of divergence is reduced and the diagram is finite. As a result, the only diagrams that can contribute to $\beta_{\lambda_6}$ at 2-loop order are the following:

  ---------------------------------- ---------------------------------- ---------------------------------- ----------------------------------
   ![image](A1.png){height="3.5cm"}   ![image](A2.png){height="3.5cm"}   ![image](A3.png){height="3.5cm"}   ![image](A4.png){height="3.5cm"}
                 (A1)                               (A2)                               (A3)                               (A4)
   ![image](A5.png){height="3.5cm"}   ![image](A6.png){height="3.5cm"}   ![image](A7.png){height="3.5cm"}   ![image](A8.png){height="3.5cm"}
                 (A5)                               (A6)                               (A7)                               (A8)
  ---------------------------------- ---------------------------------- ---------------------------------- ----------------------------------

The diagrams (A1-3) include planar diagrams, while the others are suppressed by powers of $1/N$ in the ’t Hooft large $N$ limit. In order to compute the 2-loop beta function, we need in addition to the diagrams above also the anomalous dimension of the scalar field at this order. This comes from the following diagrams:[^3]

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