Asymptotic freedom

Screening and antiscreening

The variation in a physical coupling constant under changes of scale can be understood qualitatively as coming from the action of the field on virtual particles carrying the relevant charge. The Landau pole behavior of QED (related to quantum triviality) is a consequence of screening by virtual charged particle–antiparticle pairs, such as electron–positron pairs, in the vacuum. In the vicinity of a charge, the vacuum becomes polarized: virtual particles of opposing charge are attracted to the charge, and virtual particles of like charge are repelled. The net effect is to partially cancel out the field at any finite distance. Getting closer and closer to the central charge, one sees less and less of the effect of the vacuum, and the effective charge increases.

In QCD the same thing happens with virtual quark-antiquark pairs; they tend to screen the color charge. However, QCD has an additional wrinkle: its force-carrying particles, the gluons, themselves carry color charge, and in a different manner. Each gluon carries both a color charge and an anti-color magnetic moment. The net effect of polarization of virtual gluons in the vacuum is not to screen the field but to augment it and change its color. This is sometimes called antiscreening (color paramagnetism). Getting closer to a quark diminishes the antiscreening effect of the surrounding virtual gluons, so the contribution of this effect would be to weaken the effective charge with decreasing distance.

Since the virtual quarks and the virtual gluons contribute opposite effects, which effect wins out depends on the number of different kinds, or flavors, of quark. For standard QCD with three colors, as long as there are no more than 16 flavors of quark (not counting the antiquarks separately), antiscreening prevails and the theory is asymptotically free. In fact, there are only 6 known quark flavors.

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 relationship 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 ${\displaystyle x\to bx}$. 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 ${\displaystyle n_f}$ kinds of quark-like particle is

${\displaystyle {\beta }_1(\alpha )=\frac{{\alpha }^2}{\pi }\left(-\frac{11N}{6}+\frac{n_f}{3}\right)}$

where ${\displaystyle \alpha }$ is the theory's equivalent of the fine-structure constant, ${\displaystyle g^2/(4\pi )}$ in the units favored by particle physicists. If this function is negative, the theory is asymptotically free. For SU(3), one has ${\displaystyle N=3,}$ and the requirement that ${\displaystyle {\beta }_1<0}$ gives

${\displaystyle n_f<\frac{33}{2}.}$

Thus for SU(3), the color charge gauge group of QCD, the theory is asymptotically free if there are 16 or fewer flavors of quarks.

Besides QCD, asymptotic freedom can also be seen in other systems like the nonlinear ${\displaystyle \sigma }$-model in 2 dimensions, which has a structure similar to the SU(n) invariant Yang–Mills theory in 4 dimensions.

Finally, one can find theories that are asymptotically free and reduce to the full Standard Model of electromagnetic, weak and strong forces at low enough energies.