# The logic of large $N$ expansion

I have some understanding of how the large-$N$ expansion works but feel like I'm missing the most important concepts.

For example, I understand that in QCD the order of the diagram in $N$ depends only on it's topology (Euler characteristic $\chi$). From where it immediately follows that one has an infinite number of diagrams of same order in $N$. After we limit ourselves with only the consideration of the leading $N$ order, how do we treat all those?

Here's my guess. The following two diagrams have weights $Nt$ and $Nt^2$, correspondingly.

Do we say that the first of them is more important since it has a smaller power of the 't Hooft parameter $t$? This sounds reasonable, but I've never seen people writing expansions in $t$; all the books tell about the $1/N$ expansion. If not, then how do we sum all the diagrams which are of same order in $N$?

No, in this case both diagrams contribute at the same order. The t'Hooft parameter $t$ is taken to be arbitrary, and is not meant to be a perturbative parameter. Typically in large $N$ expansions, people will write things like "all orders in $t$ and leading order in $1/N$", meaning that they make no assumption about the size of $t$, but they only take the leading order diagrams in $1/N$. Of course, one can do a double-perturbative expansion in both $t$ and $1/N$, though obviously an all orders in $t$ calculation preferred whenever possible.
In order to sum all the diagrams, one can organize the calculations solving a Schwinger-Dyson equation to the order in $1/N$ you want. However, this typically only allows you to sum the planar diagrams; computing things at subleading order, even at large $N$, is usually not straightforward.
• "In order to sum all the diagrams, one can organize the calculations solving a Schwinger-Dyson equation to the order in $1/N$ you want" — could you please refer to some standard text or paper please? I'm still puzzled and don't understand how to deal with this infinity of diagrams in practice. Commented Mar 4, 2017 at 23:59
• There is a discussion starting on page 13 of arxiv.org/pdf/0909.0518.pdf which might be useful. AdS/CFT provides a tool for summing to all orders in $t$ at leading order in $1/N$. Commented Mar 5, 2017 at 2:28