this limit). This limit is known as the ’t Hooft limit . The ’t Hooft expansion was originally formulated in terms of a U ( N$_{c}$ ) gauge theory, with all matter in the adjoint representation (although this can be generalised to include fundamental matter). Let us assume, that as in QCD, all three-point terms are proportional to g$_{s}$ and all four-point terms are proportional to g 2 $_{s}$, so the Lagrangian takes the schematic form for some constants c ijk and d $^{ijkl}$. We can rescale the fields by ˜ Φ$_{i}$ ≡ g$_{s}$ Φ$_{i}$, to reach where 1 /g 2 Y M = N/λ . Now we want to know what happens in the limit N → ∞ , λ = fixed. Although 1 /g 2 s tends to infinity, this is countered by the fact that the number of components in the fields also goes to infinity. A U ( N$_{c}$ ) theory with matter in the adjoint representation can be repre- sented as a direct product of a fundamental and an anti-fundamental field Φ i $_{j}$, as in figure 1.2, where we have drawn two contributions to the vacuum amplitude. What is the power of N and λ associated with each diagram? From (1.15) we can see that each vertex will have a coefficient proportional to N/λ , and each propagator will have a coefficient proportional to λ/N . In addition, each closed loop will a factor of N$_{c}$ since we have to sum over all 13 L ∼ Tr( d Φ$_{i}$ d Φ$_{i}$) + g$_{s}$c $^{ijk}$Tr(Φ$_{i}$ Φ$_{j}$Φ$_{k}$) + g 2 $_{s}$d$^{ijkl}$Tr(Φ$_{i}$ Φ$_{j}$Φ$_{k}$Φ$_{l}$) (1.14) L ∼ 1 g 2 s [ Tr( d ˜ Φ$_{i}$ d ˜ Φ$_{i}$) + c $^{ijk}$Tr( ˜ Φ$_{i}$ ˜ Φ$_{j}$ ˜ Φ$_{k}$) + d $^{ijkl}$Tr( ˜ Φ$_{i}$ ˜ Φ$_{j}$ ˜ Φ$_{k}$ ˜ Φ$_{l}$) ] (1.15)