In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, a coupling constant or gauge coupling parameter (or, more simply, a coupling), is a number that determines the strength of the
force exerted in an
interaction
Interaction is action that occurs between two or more objects, with broad use in philosophy and the sciences. It may refer to:
Science
* Interaction hypothesis, a theory of second language acquisition
* Interaction (statistics)
* Interactions o ...
. Originally, the coupling constant related the force acting between two static bodies to the "
charges" of the bodies (i.e. the electric charge for
electrostatic
Electrostatics is a branch of physics that studies electric charges at rest ( static electricity).
Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for amb ...
and the mass for
Newtonian gravity
Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distan ...
) divided by the distance squared,
, between the bodies; thus:
in
for Newtonian gravity and
in
for
electrostatic
Electrostatics is a branch of physics that studies electric charges at rest ( static electricity).
Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for amb ...
. This description remains valid in modern physics for
linear theories
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
with static bodies and massless
force carrier In quantum field theory, a force carrier, also known as messenger particle or intermediate particle, is a type of particle that gives rise to forces between other particles. These particles serve as the quanta of a particular kind of physical fi ...
s.
A modern and more general definition uses the
Lagrangian
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
(or equivalently the
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
) of a system. Usually,
(or
) of a system describing an interaction can be separated into a ''kinetic part''
and an ''interaction part''
:
(or
).
In field theory,
always contains 3 fields terms or more, expressing for example that an initial electron (field 1) interacted with a photon (field 2) producing the final state of the electron (field 3). In contrast, the ''kinetic part''
always contains only two fields, expressing the free propagation of an initial particle (field 1) into a later state (field 2).
The coupling constant determines the magnitude of the
part with respect to the
part (or between two sectors of the interaction part if several fields that couple differently are present). For example, the
electric charge
Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respe ...
of a particle is a coupling constant that characterizes an interaction with two charge-carrying fields and one
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they a ...
field (hence the common Feynman diagram with two arrows and one wavy line). Since photons mediate the
electromagnetic
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
force, this coupling determines how strongly electrons feel such a force, and has its value fixed by experiment. By looking at the
QED Lagrangian, one sees that indeed, the coupling sets the proportionality between the kinetic term
and the interaction term
.
A coupling plays an important role in dynamics. For example, one often sets up hierarchies of approximation based on the importance of various coupling constants. In the motion of a large lump of magnetized iron, the magnetic forces may be more important than the gravitational forces because of the relative magnitudes of the coupling constants. However, in
classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
, one usually makes these decisions directly by comparing forces. Another important example of the central role played by coupling constants is that they are the expansion parameters for first-principle calculations based on
perturbation theory
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middl ...
, which is the main method of calculation in many branches of physics.
Fine-structure constant
Couplings arise naturally in a
quantum field theory. A special role is played in relativistic quantum theories by couplings that are
dimensionless
A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
; i.e., are pure numbers. An example of a dimensionless such constant is the
fine-structure constant
In physics, the fine-structure constant, also known as the Sommerfeld constant, commonly denoted by (the Greek letter ''alpha''), is a fundamental physical constant which quantifies the strength of the electromagnetic interaction between el ...
,
:
where is the
charge of an electron
The elementary charge, usually denoted by is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 . This elementary charge is a fundame ...
,
is the
permittivity of free space, ℏ is the
reduced Planck constant
The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivalen ...
and is the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
. This constant is proportional to the square of the coupling strength of the charge of an electron to the
electromagnetic field.
Gauge coupling
In a non-Abelian
gauge theory, the gauge coupling parameter,
, appears in the
Lagrangian
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
as
:
(where G is the gauge
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
tensor) in some conventions. In another widely used convention, G is rescaled so that the coefficient of the kinetic term is 1/4 and
appears in the
covariant derivative. This should be understood to be similar to a dimensionless version of the
elementary charge defined as
:
Weak and strong coupling
In a
quantum field theory with a coupling ''g'', if ''g'' is much less than 1, the theory is said to be ''weakly coupled''. In this case, it is well described by an expansion in powers of ''g'', called
perturbation theory
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middl ...
. If the coupling constant is of order one or larger, the theory is said to be ''strongly coupled''. An example of the latter is the
hadron
In particle physics, a hadron (; grc, ἁδρός, hadrós; "stout, thick") is a composite subatomic particle made of two or more quarks held together by the strong interaction. They are analogous to molecules that are held together by the e ...
ic theory of
strong interactions (which is why it is called strong in the first place). In such a case, non-perturbative methods need be used to investigate the theory.
In
quantum field theory, the dimension of the coupling plays an important role in the
renormalizability property of the theory, and therefore on the applicability of perturbation theory. If the coupling is dimensionless in the
natural units system (i.e.
,
), like in QED, QCD, and the
weak interaction
In nuclear physics and particle physics, the weak interaction, which is also often called the weak force or weak nuclear force, is one of the four known fundamental interactions, with the others being electromagnetism, the strong interaction ...
, the theory is renormalizable and all the terms of the expansion series are finite (after renormalization). If the coupling is dimensionful, as e.g. in gravity (
), the
Fermi theory (
) or the
chiral perturbation theory of the
strong force
The strong interaction or strong force is a fundamental interaction that confines quarks into proton, neutron, and other hadron particles. The strong interaction also binds neutrons and protons to create atomic nuclei, where it is called the ...
(
), then the theory is usually not renormalizable. Perturbation expansions in the coupling might still be feasible, albeit within limitations,
[Heinrich Leutwyler (2012), Chiral perturbation theory](_blank)
Scholarpedia, 7(10):8708. as most of the higher order terms of the series will be infinite.
Running coupling
One may probe a
quantum field theory at short times or distances by changing the wavelength or momentum, k, of the probe used. With a high frequency (i.e., short time) probe, one sees
virtual particles taking part in every process. This apparent violation of the
conservation of energy may be understood heuristically by examining the
uncertainty relation
In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...
:
which virtually allows such violations at short times.
The foregoing remark only applies to some formulations of quantum field theory, in particular,
canonical quantization in the
interaction picture.
In other formulations, the same event is described by "virtual" particles going off the
mass shell. Such processes
renormalize the coupling and make it dependent on the energy scale, ''μ'', at which one probes the coupling. The dependence of a coupling ''g''(''μ'') on the energy-scale is known as "running of the coupling". The theory of the running of couplings is given by the
renormalization group
In theoretical physics, the term renormalization group (RG) refers to a formal apparatus that allows systematic investigation of the changes of a physical system as viewed at different scales. In particle physics, it reflects the changes in the ...
, though it should be kept in mind that the renormalization group is a more general concept describing any sort of scale variation in a physical system (see the full article for details).
Phenomenology of the running of a coupling
The renormalization group provides a formal way to derive the running of a coupling, yet the phenomenology underlying that running can be understood intuitively.
[The QCD Running Coupling]
A. Deur, S. J. Brodsky, G. F. de Teramond, Prog. Part. Nuc. Phys. 90 1 (2016) As explained in the introduction, the coupling ''constant'' sets the magnitude of a force which behaves with distance as
. The
-dependence was first explained by
Faraday
Michael Faraday (; 22 September 1791 – 25 August 1867) was an English scientist who contributed to the study of electromagnetism and electrochemistry. His main discoveries include the principles underlying electromagnetic induction, ...
as the decrease of the force
flux: at a point ''B'' distant by
from the body ''A'' generating a force, this one is proportional to the field flux going through an elementary surface ''S'' perpendicular to the line ''AB''. As the flux spreads uniformly through space, it decreases according to the
solid angle
In geometry, a solid angle (symbol: ) is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point.
The poi ...
sustaining the surface ''S''. In the modern view of quantum field theory, the
comes from the expression in
position space
In physics and geometry, there are two closely related vector spaces, usually three-dimensional but in general of any finite dimension.
Position space (also real space or coordinate space) is the set of all ''position vectors'' r in space, and h ...
of the
propagator of the
force carrier In quantum field theory, a force carrier, also known as messenger particle or intermediate particle, is a type of particle that gives rise to forces between other particles. These particles serve as the quanta of a particular kind of physical fi ...
s. For relatively weakly-interacting bodies, as is generally the case in electromagnetism or gravity or the nuclear interactions at short distances, the
exchange of a single force carrier is a good first approximation of the interaction between the bodies, and classically the interaction will obey a
-law (note that if the force carrier is massive, there is
an additional dependence). When the interactions are more intense (e.g. the charges or masses are larger, or
is smaller) or happens over briefer time spans (smaller
), more force carriers are involved or
particle pairs are created, see Fig. 1, resulting in the break-down of the
behavior. The classical equivalent is that the field flux does not propagate freely in space any more but e.g. undergoes
screening from the charges of the extra virtual particles, or interactions between these virtual particles. It is convenient to separate the first-order
law from this extra
-dependence. This latter is then accounted for by being included in the coupling, which then becomes
-dependent, (or equivalently ''μ''-dependent). Since the additional particles involved beyond the single force carrier approximation are always
virtual, i.e. transient quantum field fluctuations, one understands why the running of a coupling is a genuine quantum and relativistic phenomenon, namely an effect of the high-order
Feynman diagram
In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduc ...
s on the strength of the force.
Since a running coupling effectively accounts for microscopic quantum effects, it is often called an ''effective coupling'', in contrast to the ''bare coupling (constant)'' present in the Lagrangian or Hamiltonian.
Beta functions
In quantum field theory, a ''beta function, β''(''g''), encodes the running of a coupling parameter, ''g''. It is defined by the relation
:
where ''μ'' is the energy scale of the given physical process. If the beta functions of a quantum field theory vanish, then the theory is
scale-invariant.
The coupling parameters of a quantum field theory can flow even if the corresponding classical
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
theory is
scale-invariant. In this case, the non-zero beta function tells us that the classical scale-invariance is
anomalous.
QED and the Landau pole
If a beta function is positive, the corresponding coupling increases with increasing energy. An example is
quantum electrodynamics
In particle physics, quantum electrodynamics (QED) 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 spec ...
(QED), where one finds by using
perturbation theory
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middl ...
that the
beta function
In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral
: \Beta(z_1,z_2) = \int_0^1 t^( ...
is positive. In particular, at low energies, , whereas at the scale of the
Z boson, about 90
GeV, one measures .
Moreover, the perturbative beta function tells us that the coupling continues to increase, and QED becomes ''strongly coupled'' at high energy. In fact the coupling apparently becomes infinite at some finite energy. This phenomenon was first noted by
Lev Landau
Lev Davidovich Landau (russian: Лев Дави́дович Ланда́у; 22 January 1908 – 1 April 1968) was a Soviet-Azerbaijani physicist of Jewish descent who made fundamental contributions to many areas of theoretical physics.
His ac ...
, and is called the
Landau pole
In physics, the Landau pole (or the Moscow zero, or the Landau ghost) is the momentum (or energy) scale at which the coupling constant (interaction strength) of a quantum field theory becomes infinite. Such a possibility was pointed out by the phy ...
. However, one cannot expect the perturbative beta function to give accurate results at strong coupling, and so it is likely that the Landau pole is an artifact of applying perturbation theory in a situation where it is no longer valid. The true scaling behaviour of
at large energies is not known.
QCD and asymptotic freedom
In non-Abelian gauge theories, the beta function can be negative, as first found by
Frank Wilczek
Frank Anthony Wilczek (; born May 15, 1951) is an American theoretical physicist, mathematician and Nobel laureate. He is currently the Herman Feshbach Professor of Physics at the Massachusetts Institute of Technology (MIT), Founding Direc ...
,
David Politzer
Hugh David Politzer (; born August 31, 1949) is an American theoretical physicist and the Richard Chace Tolman Professor of Theoretical Physics at the California Institute of Technology. He shared the 2004 Nobel Prize in Physics with David Gr ...
and
David Gross
David Jonathan Gross (; born February 19, 1941) is an American theoretical 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. ...
. An example of this is the
beta function
In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral
: \Beta(z_1,z_2) = \int_0^1 t^( ...
for
quantum chromodynamics (QCD), and as a result the QCD coupling decreases at high energies.
Furthermore, the coupling decreases logarithmically, a phenomenon known as
asymptotic freedom
In quantum field theory, asymptotic freedom is a property of some gauge theories that causes interactions between particles to become asymptotically weaker as the energy scale increases and the corresponding length scale decreases.
Asymptotic fre ...
(the discovery of which was awarded with the
Nobel Prize in Physics
)
, image = Nobel Prize.png
, alt = A golden medallion with an embossed image of a bearded man facing left in profile. To the left of the man is the text "ALFR•" then "NOBEL", and on the right, the text (smaller) "NAT•" then " ...
in 2004). The coupling decreases approximately as
:
where ''β''
0 is a constant first computed by Wilczek, Gross and Politzer.
Conversely, the coupling increases with decreasing energy. This means that the coupling becomes large at low energies, and one can no longer rely on
perturbation theory
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middl ...
. Hence, the actual value of the coupling constant is only defined at a given energy scale. In QCD, the Z boson mass scale is typically chosen, providing a value of the strong coupling constant of α
s(M
Z2 ) = 0.1179 ± 0.0010. The most precise measurements stem from lattice QCD calculations, studies of tau-lepton decay, as well as by the reinterpretation of the transverse momentum spectrum of the Z boson.
QCD scale
In
quantum chromodynamics (QCD), the quantity Λ is called the QCD scale. The value is
for three "active" quark flavors, ''viz'' when the energy-momentum involved in the process allows to produce only the up, down and strange quarks, but not the heavier quarks. This corresponds to energies below 1.275 GeV. At higher energy, Λ is smaller, e.g.
MeV
C. Patrignani et al. (Particle Data Group), Chin. Phys. C, 40, 100001 (2016)
/ref> above the bottom quark mass of about 5 GeV. The meaning of the minimal substraction (MS) scheme scale ΛMS is given in the article on dimensional transmutation
In particle physics, dimensional transmutation is a physical mechanism providing a linkage between a dimensionless parameter and a dimensionful parameter.
In classical field theory, such as gauge theory in four-dimensional spacetime, the coupli ...
. The proton-to-electron mass ratio In physics, the proton-to-electron mass ratio, ''μ'' or ''β'', is the rest mass of the proton (a baryon found in atoms) divided by that of the electron (a lepton found in atoms), a dimensionless quantity, namely:
:''μ'' =
The number in parenthe ...
is primarily determined by the QCD scale.
String theory
A remarkably different situation exists in string theory since it includes a dilaton. An analysis of the string spectrum shows that this field must be present, either in the bosonic string
Bosonic string theory is the original version of string theory, developed in the late 1960s and named after Satyendra Nath Bose. It is so called because it contains only bosons in the spectrum.
In the 1980s, supersymmetry was discovered in the co ...
or the NS-NS sector of the superstring. Using vertex operators, it can be seen that exciting this field is equivalent to adding a term to the action where a scalar field couples to the Ricci scalar
In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometr ...
. This field is therefore an entire function worth of coupling constants. These coupling constants are not pre-determined, adjustable, or universal parameters; they depend on space and time in a way that is determined dynamically. Sources that describe the string coupling as if it were fixed are usually referring to the vacuum expectation value
In quantum field theory the vacuum expectation value (also called condensate or simply VEV) of an operator is its average or expectation value in the vacuum. The vacuum expectation value of an operator O is usually denoted by \langle O\rangle. ...
. This is free to have any value in the bosonic theory where there is no superpotential
In theoretical physics, the superpotential is a function in supersymmetric quantum mechanics. Given a superpotential, two "partner potentials" are derived that can each serve as a potential in the Schrödinger equation. The partner potentials hav ...
.
See also
* Canonical quantization, renormalization
Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering va ...
and dimensional regularization
__NOTOC__
In theoretical physics, dimensional regularization is a method introduced by Giambiagi and Bollini as well as – independently and more comprehensively – by 't Hooft and Veltman for regularizing integrals in the evaluation of Fe ...
*Fine-structure constant
In physics, the fine-structure constant, also known as the Sommerfeld constant, commonly denoted by (the Greek letter ''alpha''), is a fundamental physical constant which quantifies the strength of the electromagnetic interaction between el ...
* Quantum field theory, especially quantum electrodynamics
In particle physics, quantum electrodynamics (QED) 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 spec ...
and quantum chromodynamics
*Gluon field
In theoretical particle physics, the gluon field is a four-vector field characterizing the propagation of gluons in the strong interaction between quarks. It plays the same role in quantum chromodynamics as the electromagnetic four-potential in ...
, Gluon field strength tensor
References
External links
The Nobel Prize in Physics 2004 – Information for the Public
*An introduction to quantum field theory, by M.E.Peskin and H.D.Schroeder, {{ISBN, 0-201-50397-2
Quantum field theory
Quantum mechanics
Statistical mechanics
Renormalization group