content_image/311768.jpg content_image/311775.jpg content_image/311776.jpg
<img src='content_image/311768.jpg'>

We can go further, and solve (1.10): 

<img src='content_image/311775.jpg'>

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. 

<img src='content_image/311776.jpg'>

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. 

