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80). Again, let us do an illustrative example. Take diagram (b), there are five index loops, so h = 5; more precisely it is composed of pasting a left primitive diagram with h = 3 and a right primitive with h = 2, so C(G) = 2 × 3 = 6. Now for F (G), we need contractions of the form Tr(ΦΦ Φ Φ)Tr(Φ Φ)Tr(ΦΦ); there are 4 × 2 × 2 = 16 ways of doing so. Furthermore, for this even overall power in the coupling, we have a minus sign when expanding out the exponent. Therefore F (G) = −16g2 g4 for this diagram.

More concretely, we can integrate out (in each gauge group factor) the massive adjoint chiral field, Φ, which has a mass well above Λ. The corresponding N = 1 theory thus obtained, valid below scales of order µ can be analyzed in various group factors separately. In each factor, it will become strongly coupled in the infra-red and will have vacua, details of which will depend on the number of flavors charged under the group factor. We can interpolate smoothly between vacua in the weak and strong coupling regions by holomorphic variation of the parameters in the superpotential W (Φ) because the theory has N = 1 supersymemtry.

Having understood the structure of double-trace diagrams in this way, we can adapt the techniques of [27] to our case. The steps 1-4 as described in Sec. 2 go through without modification since they are independent of the details of the treelevel superpotential. However the steps 5-9 are modified in various ways. 63) where h is the total number of index loops and ℓ is the total number of momentum loops. ) Since no momentum flows between pasted primitive diagrams it is clear that this selection rule would permit some of the primitive components to be non-planar.

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Aspects of supersymmetric guage theory and string theory by Minxin Huang

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