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What is the meaning of the second, third and fourth graph? The image is from arXiv:hep-th/9912072.

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  1. The Feynman rules in non-commutative field theory are classified using 't Hooft double-line notation. In particular, the legs in a vertex has a cyclic order, cf. eq. (2.5) in Ref. 1.

  2. Interestingly, non-planar graphs are suppressed due to extra phase factors. (This is somewhat similar to the planar limit/large-$N_c$-expansion in $SU(N_c)$ Yang-Mills theory, although the non-planar suppression is there caused because non-planar diagrams have fewer traces.)

  3. Returning to OP's question, the second, third and fourth graph in Fig. 7 are non-planar graphs (again due to that the cyclic order of legs in vertices matters).

References:

  1. S. Minwalla, M. Van Raamsdonk & N. Seiberg, Non-commutative Perturbative Dynamics, arXiv:hep-th/9912072.
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  • $\begingroup$ But how would I write down the four-point function for the non-planar diagrams? $\endgroup$ Dec 7 '20 at 10:43
  • $\begingroup$ The brief answer is: By applying the pertinent Feynman rules. $\endgroup$
    – Qmechanic
    Dec 7 '20 at 12:31

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