content_image/296643.jpg content_image/296644.jpg content_image/296645.jpg
contributions from quarks that are much heavier than the characteristic scale of the problem are suppressed by the heavy quark mass [12]. In the MS scheme the value of the coupling at the reference scale is taken to be 

<img src='content_image/296643.jpg'>

where all numerical values have been taken from Ref. [20]. Similarly, using the one-loop expression for the γ function (1.66), it can be shown that 

<img src='content_image/296644.jpg'>

In the MS scheme the value of the quark mass at the reference scale is taken to be 

<img src='content_image/296645.jpg'>

where the numerical values are taken again from Ref. [20] and α ( m ) can be determined using Eq. (1.67). 

The β function signals an essential feature of QCD. Because β$_{1}$ < 0, the one-loop QCD coupling decreases with increasing energy scale. This defining characteristic of QCD is called asymptotic freedom. Fig. 1.6 compares theoretical predictions and experimental measure- ments of α ( µ ) at several different energy scales µ . The predicted and measured values are in excellent agreement. Because asymptotic freedom is a prediction of QCD, this agreement is a strong experimental confirmation of QCD [22]. 

## 1.4 QCD Laplace sum rules 

The QCD coupling is small at high energies due to asymptotic freedom. This means that perturbative expansions in QCD converge rapidly at high energies. However, at low energies the coupling increases, and the convergence of the perturbative expansion suffers. In practical terms this means that perturbative techniques alone are insufficient to describe QCD at low 

