content_image/304997.jpg
We say that, classically, massless QCD is scale invariant. However, this would be a puzzle: phenomenologically there is clearly an inherent mass scale in QCD: the hadronic masses are not continuous, so any one could be used to define a scale. On studying the β function we would be reassured. Figure 1.1 clearly introduces a scale into the theory, commonly called Λ$_{QCD}$. It is normally defined to be the energy at which the coupling constant becomes infinite. In figure 1.1 this is approximately 246 MeV (although a more careful analysis, including higher order corrections to β ( g ) sets Λ$_{QCD}$ at ∼ 215 MeV). For all energies where g$_{s}$ ≥ 1, perturbation theory is no longer valid. Quantum effects have broken the classical symmetry: we say that the scale invariance has been broken anomalously. 

## 1.2.5 The ’t Hooft Expansion 

Equation (1.10) provides an ideal opportunity to talk about the ’t Hooft ex- pansion, a particularly clever way of performing perturbation theory about the strong scale Λ$_{QCD}$. We have just shown that at and around Λ$_{QCD}$, the coupling constant is greater than one, which means a perturbative expan- sion such as in (1.9) would be invalid. However in [20, 21] it was pointed out that there is a second dimensionless parameter in QCD: that of ‘ N$_{c}$ ’ in the gauge group SU ( N$_{c}$ ). ’t Hooft showed that gauge theories simplify at large N and that they have a perturbative expansion in terms of 1 /N . We summarise his analysis, and that of [22], in this section. 

We stated in (1.10) that the β -function for a pure SU ( N$_{c}$ ) gauge theory is given by 

<img src='content_image/304997.jpg'>

so the leading terms remain the same if we let N → ∞ , so long as we keep λ ≡ g 2 $_{s}$N fixed (one can show the higher order terms also stay the same in 

