In
physicsPhysics 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 theoriesIn 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 scaleIn 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 chromodynamicsIn 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 colorcharged fermions...
(QCD), the
quantum field theoryQuantum 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 manybody systems. It is the natural and quantitative language of particle physics and...
of the nuclear interaction between quarks and
gluonGluons 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
DGLAPDGLAP 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 AltarelliParisi 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 WilczekFrank 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 GrossDavid 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 GrossDavid 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 WilczekFrank 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 KhriplovichIosif 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 HooftGerardus '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 PhysicsThe 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 poleIn 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
spinorIn 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 electrodynamicsQuantum 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 ModelThe 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 particleIn 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 poleIn 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 trivialityIn 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
antiparticleCorresponding 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
electronThe 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...

positronThe 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 quarkantiquark pairs; they tend to screen the
color chargeIn 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 forcecarrying particles, the gluons, themselves carry color charge, and in a different manner. Each gluon carries both a color charge and an anticolor 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
flavorsIn 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
betafunctionIn 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 constantIn 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 groupIn 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
momentumIn classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...
(which probe shortdistance behavior, roughly because of the inverse relation between a quantum's momentum and De Broglie wavelength), an asymptotically free theory is amenable to
perturbation theoryIn 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 diagramFeynman diagrams are a pictorial representation scheme for the mathematical expressions governing the behavior of subatomic particles, first developed by the Nobel Prizewinning American physicist Richard Feynman, and first introduced in 1948...
s. Such situations are therefore more theoretically tractable than the longdistance, strongcoupling behavior also often present in such theories, which is thought to produce
confinementColor 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 betafunction is a matter of evaluating Feynman diagrams contributing to the interaction of a quark emitting or absorbing a gluon. In nonabelian 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 betafunction in an SU(N) gauge theory with
kinds of quarklike particle is
where
is the theory's equivalent of the
finestructure constantIn physics, the finestructure 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 chargeIn 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