Normalization of Source Terms in Large-N Gauge Theory

Question

Typically when you do the counting for large N gauge theory, you rescale fields so that the Lagrangian takes the form

$$\begin{equation} \mathcal{L}=N[-\frac{1}{2g^2}TrF^2+\bar{\psi}_i\gamma^\mu D_\mu \psi_i] \end{equation}$$ where I have chosen the original coupling of the theory to be $\frac{g}{\sqrt{N}}$. From this it is easy to see which vacuum diagrams contribute in the Large-N limit.

However, when you go on to consider connected correlators, people always add a source term $N\sum J_iO^i$ to the Lagrangian. The factor of N out front then determines the N-dependence of the correlators

$$\begin{equation} \langle O_1...O_r \rangle=\frac{1}{iN}\frac{\partial}{\partial J^1}...\frac{1}{iN}\frac{\partial}{\partial J^r}W[J] \end{equation}$$ The N-counting would be different if my source terms were instead just $\sum J_iO^i$.

So my question is, why are we forced to include the factor of N in the source terms? Is it because the original action has been written in terms of rescaled fields and is also proportional to N? If I instead worked with the action in terms of un-rescaled fields, would I not include the factor of N in the source term? Thanks.

This post imported from StackExchange Physics at 2014-04-13 14:46 (UCT), posted by SE-user Dan