Asymptotic formula

In physics, asymptotic freedom is a property of some gauge theories that causes bonds between particles to become asymptotically weaker as energy increases and distance decreases.

Asymptotic freedom is a feature of quantum chromodynamics (QCD), the quantum field theory of the nuclear interaction between quarks and gluons, the fundamental constituents of nuclear matter. Quarks interact weakly at high energies, allowing perturbative calculations by DGLAP of cross sections in deep inelastic processes of particle physics; and strongly at low energies, preventing the unbinding of baryons (like protons or neutrons with three quarks) or mesons (like pions with two quarks), the composite particles of nuclear matter.

Asymptotic freedom was discovered and described in 1973 by Frank Wilczek, David Gross, and independently by David Politzer the same year. All three shared the Nobel Prize in physics in 2004.

Discovery

Asymptotic freedom was described and published in 1973 by David Gross and Frank Wilczek, and also by David Politzer. Although these authors were the first to understand the physical relevance to the strong interactions, in 1969 Iosif Khriplovich discovered asymptotic freedom in the SU(2) gauge theory as a mathematical curiosity, and Gerardus 't Hooft in 1972 also noted the effect but did not publish. For their discovery, Gross, Wilczek and Politzer were awarded the Nobel Prize in Physics in 2004.

The discovery was instrumental in rehabilitating quantum field theory. Prior to 1973, many theorists suspected that field theory was fundamentally inconsistent because the interactions become infinitely strong at short distances. This phenomenon is usually called a Landau pole, and it defines the smallest length scale that a theory can describe. This problem was discovered in field theories of interacting scalars and spinors, including quantum electrodynamics, and Lehman positivity led many to suspect that it is unavoidable. Asymptotically free theories become weak at short distances, there is no Landau pole, and these quantum field theories are believed to be completely consistent down to any length scale.

While the Standard Model is not entirely asymptotically free, in practice the Landau pole can only be a problem when thinking about the strong interactions. The other interactions are so weak that any inconsistency can only arise at distances shorter than the Planck length, where a field theory description is inadequate anyway.

Screening and antiscreening

Charge screening in QED

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 quantum electrodynamics (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. 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 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 $x\rightarrow bx$. 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 $n_{f}$ kinds of quark-like particle is

$\beta _{1}(\alpha )={\alpha ^{2} \over \pi }\left(-{11N \over 6}+{n_{f} \over 3}\right)$

where $\alpha$ is the theory's equivalent of the fine-structure constant, $g^{2}/(4\pi )$ 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) $N=3,$ and $\beta _{1}<0$ gives $n_{f}<{33 \over 2}.$

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

See also

• Asymptotic safety
• Gluon field strength tensor
• Quantum triviality

References

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