Asymptotic freedom

Asymptotic freedom

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In physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, asymptotic freedom is a property of some gauge theories
Gauge theory
In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...

 that causes interactions between particles to become arbitrarily weak at energy scales that become arbitrarily large, or, equivalently, at length scale
Length scale
In physics, length scale is a particular length or distance determined with the precision of one order of magnitude. The concept of length scale is particularly important because physical phenomena of different length scales cannot affect each other and are said to decouple...

s that become arbitrarily small (at the shortest distances).

Asymptotic freedom is a feature of quantum chromodynamics
Quantum chromodynamics
In theoretical physics, quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...

 (QCD), the quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...

 of the nuclear interaction between quarks and gluon
Gluon
Gluons are elementary particles which act as the exchange particles for the color force between quarks, analogous to the exchange of photons in the electromagnetic force between two charged particles....

s, the fundamental constituents of nuclear matter.
Quarks interact weakly at high energies, allowing perturbative calculations by DGLAP
DGLAP
DGLAP are the authors who first wrote the QCD evolution equation of the same name. DGLAP was first published in the western world by Altarelli and Parisi in 1977, hence DGLAP and its specialisations are sometimes still called Altarelli-Parisi equations...

 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 by Frank Wilczek
Frank Wilczek
Frank Anthony Wilczek is a theoretical physicist from the United States and a Nobel laureate. He is currently the Herman Feshbach Professor of Physics at the Massachusetts Institute of Technology ....

, David Gross
David Gross
David Jonathan Gross is an American particle physicist and string theorist. Along with Frank Wilczek and David Politzer, he was awarded the 2004 Nobel Prize in Physics for their discovery of asymptotic freedom. He is currently the director and holder of the Frederick W...

, and David Politzer who in 2004 shared the Nobel Prize in physics.

Discovery


Asymptotic freedom was discovered in 1973 by David Gross
David Gross
David Jonathan Gross is an American particle physicist and string theorist. Along with Frank Wilczek and David Politzer, he was awarded the 2004 Nobel Prize in Physics for their discovery of asymptotic freedom. He is currently the director and holder of the Frederick W...

 and Frank Wilczek
Frank Wilczek
Frank Anthony Wilczek is a theoretical physicist from the United States and a Nobel laureate. He is currently the Herman Feshbach Professor of Physics at the Massachusetts Institute of Technology ....

, and by David Politzer. Although these authors were the first to understand the physical relevance to the strong interactions, in 1969 Iosif Khriplovich
Iosif Khriplovich
Iosif Benzionovich Khriplovich is a Russian theoretical physicist who has made deep contributions in quantum field theory, atomic physics, and general relativity....

 discovered asymptotic freedom in the SU(2) gauge theory as a mathematical curiosity, and Gerardus 't Hooft
Gerardus 't Hooft
Gerardus 't Hooft is a Dutch theoretical physicist and professor at Utrecht University, the Netherlands. He shared the 1999 Nobel Prize in Physics with his thesis advisor Martinus J. G...

 in 1972 also noted the effect but did not publish. For their discovery, Gross, Wilczek and Politzer were awarded the Nobel Prize in Physics
Nobel Prize in Physics
The Nobel Prize in Physics is awarded once a year by the Royal Swedish Academy of Sciences. It is one of the five Nobel Prizes established by the will of Alfred Nobel in 1895 and awarded since 1901; the others are the Nobel Prize in Chemistry, Nobel Prize in Literature, Nobel Peace Prize, and...

 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
Landau pole
In physics, the Landau pole is the momentum scale at which the coupling constant of a quantum field theory becomes infinite...

, and it defines the smallest length scale that a theory can describe. This problem was discovered in field theories of interacting scalars and spinor
Spinor
In mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...

s, including quantum electrodynamics
Quantum electrodynamics
Quantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...

, 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
Standard Model
The Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...

 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


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 particle
Virtual particle
In physics, a virtual particle is a particle that exists for a limited time and space. The energy and momentum of a virtual particle are uncertain according to the uncertainty principle...

s carrying the relevant charge. The Landau pole
Landau pole
In physics, the Landau pole is the momentum scale at which the coupling constant of a quantum field theory becomes infinite...

 behavior of quantum electrodynamics (QED, related to quantum triviality
Quantum triviality
In a quantum field theory, charge screening can restrict the value of the observable "renormalized" charge of a classical theory. Ifthe only allowed value of the renormalized charge is zero, the theory is said to be "trivial" or noninteracting...

) is a consequence of screening by virtual charged particle-antiparticle
Antiparticle
Corresponding to most kinds of particles, there is an associated antiparticle with the same mass and opposite electric charge. For example, the antiparticle of the electron is the positively charged antielectron, or positron, which is produced naturally in certain types of radioactive decay.The...

 pairs, such as electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

-positron
Positron
The positron or antielectron is the antiparticle or the antimatter counterpart of the electron. The positron has an electric charge of +1e, a spin of ½, and has the same mass as an electron...

 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
Color charge
In particle physics, color charge is a property of quarks and gluons that is related to the particles' strong interactions in the theory of quantum chromodynamics . Color charge has analogies with the notion of electric charge of particles, but because of the mathematical complications of QCD,...

. 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
Flavour (particle physics)
In particle physics, flavour or flavor is a quantum number of elementary particles. In quantum chromodynamics, flavour is a global symmetry...

, 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
Beta-function
In theoretical physics, specifically quantum field theory, a beta function β encodes the dependence of a coupling parameter, g, on the energy scale, \mu of a given physical process....

 describing the variation of the theory's coupling constant
Coupling constant
In physics, a coupling constant, usually denoted g, is a number that determines the strength of an interaction. Usually the Lagrangian or the Hamiltonian of a system can be separated into a kinetic part and an interaction part...

 under the renormalization group
Renormalization group
In theoretical physics, the renormalization group refers to a mathematical apparatus that allows systematic investigation of the changes of a physical system as viewed at different distance scales...

. For sufficiently short distances or large exchanges of momentum
Momentum
In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...

 (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
Perturbation theory (quantum mechanics)
In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. The idea is to start with a simple system for which a mathematical solution is known, and add an...

 calculations using Feynman diagram
Feynman diagram
Feynman diagrams are a pictorial representation scheme for the mathematical expressions governing the behavior of subatomic particles, first developed by the Nobel Prize-winning American physicist Richard Feynman, and first introduced in 1948...

s. 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
Colour confinement
Color confinement, often simply called confinement, is the physics phenomenon that color charged particles cannot be isolated singularly, and therefore cannot be directly observed. Quarks, by default, clump together to form groups, or hadrons. The two types of hadrons are the mesons and the baryons...

.

Calculating the beta-function is a matter of evaluating Feynman diagrams contributing to the interaction of a quark emitting or absorbing a gluon. 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
Fine-structure constant
In physics, the fine-structure constant is a fundamental physical constant, namely the coupling constant characterizing the strength of the electromagnetic interaction. Being a dimensionless quantity, it has constant numerical value in all systems of units...

, in the units favored by particle physicists. If this function is negative, the theory is asymptotically free. For SU(3), the color charge
Color charge
In particle physics, color charge is a property of quarks and gluons that is related to the particles' strong interactions in the theory of quantum chromodynamics . Color charge has analogies with the notion of electric charge of particles, but because of the mathematical complications of QCD,...

gauge group of QCD, the theory is therefore asymptotically free if there are 16 or fewer flavors of quarks.

For SU(3)
and gives