Figure 1.1: Plot of the solution to the first loop beta function for QCD GLYPH<CapLambda>$_{QCD}$ 400 600 800 1000 p GLYPH<LParen1> MeV GLYPH<RParen1> 3 4 5 6 g$_{s}$ We can go further, and solve (1.10): whose solution is plotted in figure 1.1. Note that p enters the solution, for it is the only dimensionful parameter available to make the log term dimensionless. So in fact we can swap the M derivative with a p derivative in the definition of the β function in (1.6): This reparameterization will come in useful in the next section. Figure 1.1 brings to our attention two important features of QCD. Firstly, g gets smaller and smaller as p → ∞ . So QCD is indeed asymptot- ically free, and we can use perturbation theory at high energies. Secondly, figure 1.1 demonstrates the anomalous breaking of scale invariance . If we were to look at the massless QCD Lagrangian, ie. (1.1) with m set to zero, we would see that there is no dimensionful parameter. 11 g$_{s}$ ( p ) 2 = g 2 R 1 + 7 g 2 R 16 π 2 log( p $^{2}$/M$^{2}$) (1.11) β = M ∂g ∂M = ∂g ∂ log M ←→ β = ∂g ∂ log p (1.12)