Calculating Asymptotic Freedom
Asymptotic freedom can be derived by calculating the beta-function describing the variation of the theory's coupling constant under the renormalization group. For sufficiently short distances or large exchanges of momentum (which probe short-distance behavior, roughly because of the inverse relation between a quantum's momentum and De Broglie wavelength), an asymptotically free theory is amenable to perturbation theory calculations using Feynman diagrams. Such situations are therefore more theoretically tractable than the long-distance, strong-coupling behavior also often present in such theories, which is thought to produce confinement.
Calculating the beta-function is a matter of evaluating Feynman diagrams contributing to the interaction of a quark emitting or absorbing a gluon. Essentially, the beta-function describes how the coupling constants vary as one scales the system . The calculation can be done using rescaling in position space or momentum space (momentum shell integration). In non-abelian gauge theories such as QCD, the existence of asymptotic freedom depends on the gauge group and number of flavors of interacting particles. To lowest nontrivial order, the beta-function in an SU(N) gauge theory with kinds of quark-like particle is
where is the theory's equivalent of the fine-structure constant, in the units favored by particle physicists. If this function is negative, the theory is asymptotically free. For SU(3), the color charge gauge group of QCD, the theory is therefore asymptotically free if there are 16 or fewer flavors of quarks.
For SU(3) and gives
Besides QCD, asymptotic freedom can also be seen in other systems like the nonlinear -model in 2 dimensions, which has a structure similar to the SU(N) invariant Yang-Mills theory in 4 dimensions.
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