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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 enter image description here 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 $N$?

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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.

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  • $\begingroup$ "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. $\endgroup$
    – mavzolej
    Commented Mar 4, 2017 at 23:59
  • $\begingroup$ 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 $1/N$. $\endgroup$
    – user2309840
    Commented Mar 5, 2017 at 2:28
  • $\begingroup$ Unfortunately, I am not aware of a pedagogical text that discusses this in detail. Texts such as arxiv.org/abs/1110.4386 and arxiv.org/abs/1207.4593 carry out such calculations for Chern-Simons theories (just skip to the self-energy diagrams). The idea is that one writes a self-consistent equation for the exact diagram (at leading order) and tries to solve it. There are also algebraic techniques to derive the Schwinger-Dyson equation instead of intuiting it diagrammatically, though I am not as familiar with such things. $\endgroup$
    – Aaron
    Commented Mar 5, 2017 at 5:33

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