tractable, and it requires a great deal of effort and care. Fortunately, α is very small, and so higher order contributions are also small. The equivalent expansion in QCD is α$_{s}$ ≈ 0 . 118 at 91 . 2 GeV [35], which means that the expansion does not converge as rapidly. In addition, as we discussed earlier, we can only use perturbative QCD at high energies, so for many interactions, it is not valid. • Lattice QCD [36,37] • Effective Theories The best established approach to non-perturbative QCD is the use of huge supercomputers. Continuous spacetime is represented by a discrete four dimensional lattice, with lattice spacings of a . The QCD Lagrangian (1.1) is then reformulated for a discrete spacetime, and the calculations are made for as small a lattice spacing as possible. For the final answer, the limit a → 0 is taken. Lattice QCD takes a lot of human time and effort, and it is not always clear what happens when the limit a → 0 is taken. Mirroring the techniques of perturbation theory, theories are writ- ten down which mimic QCD for some limited parameter space. And within this parameter space, there will be a parameter which can be used as a perturbation parameter. Examples are chiral perturbation theory [38–40], which uses the light quark masses (at low momentum) as perturbation parameters, and heavy quark effective theory [39,41], which uses the inverse of a heavy quark mass as a perturbation pa- rameter. These approaches can be very useful, but are limited by the extent of parameter space to which their model is valid. 25