Coupling constant
Encyclopedia
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. The coupling constant determines the strength of the interaction part with respect to the kinetic part, or between two sectors of the interaction part. For example, the electric charge
of a particle is a coupling constant.
A coupling constant 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 are more important than the gravitational forces because of the relative magnitudes of the coupling constants. However, in classical mechanics
one usually makes these decisions directly by comparing forces.
. A special role is played in relativistic quantum theories by coupling constants which are dimensionless, i.e., are pure numbers. For example, the fine-structure constant
,
(where
is the charge of an electron
,
is the permittivity of free space, 
is the reduced Planck constant and 
is the speed of light
) is such a dimensionless coupling constant that determines the strength of the electromagnetic force on an electron.
, appears in the Lagrangian
as
(where
is the gauge field
tensor) in some conventions. In another widely used convention,
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 electric charge defined as
with a dimensionless coupling constant g, if g ≪ 1 (g is much smaller than one) then 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
. 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
ic theory of strong interaction
s (which is why it is called strong in the first place). In such a case non-perturbative methods have to be used to investigate the theory.
One can probe a quantum field theory
at short times or distances by changing the wavelength or momentum, k of the probe one uses. With a high frequency, i.e., short time probe, one sees virtual particles taking part in every process. The reason this can happen, seemingly violating the conservation of energy
is the uncertainty relation
which allows such violations at short times. The previous 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,
at which one observes the coupling. The dependence of a coupling 
on the energy-scale is known as running of the coupling. The theory of the running of couplings is known as the renormalization group
.
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
theory is scale-invariant
. In this case, the non-zero beta function tells us that the classical scale-invariance is anomalous
.
(QED), where one finds by using perturbation theory
that the beta function is positive. In particular, at low energies, α ≈ 1/137, whereas at the scale of the Z boson, about 90 GeV
, one measures α ≈ 1/127.
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
, and is called the Landau pole
. 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.
, David Politzer and David Gross
. An example of this is the beta function 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
(the discovery of which was awarded with the Nobel Prize in Physics
in 2004). The coupling decreases approximately as
where β0 is a constant 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
.
(QCD), the quantity Λ is called the QCD scale. The value is
This value is to be used at a scale above the bottom quark
mass of about 5 GeV
. The meaning of ΛMS is given in the article on dimensional transmutation
.
The proton-to-electron mass ratio
is primarily determined by the QCD scale.
. Each perturbative description of string theory depends on a string coupling constant. However, in the case of string theory, these coupling constants are not pre-determined, adjustable, or universal parameters; rather they are dynamical scalar field
s that can depend on the position in space and time and whose values are determined dynamically.
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...
, a coupling constant, usually denoted g, is a number that determines the strength of an interaction
Interaction
Interaction is a kind of action that occurs as two or more objects have an effect upon one another. The idea of a two-way effect is essential in the concept of interaction, as opposed to a one-way causal effect...
. Usually the Lagrangian
Lagrangian
The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...
or the Hamiltonian
Hamiltonian mechanics
Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...
of a system can be separated into a kinetic part and an interaction part. The coupling constant determines the strength of the interaction part with respect to the kinetic part, or between two sectors of the interaction part. For example, the electric charge
Electric charge
Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...
of a particle is a coupling constant.
A coupling constant 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 are more important than the gravitational forces because of the relative magnitudes of the coupling constants. However, in classical mechanics
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
one usually makes these decisions directly by comparing forces.
Fine-structure constant
The coupling constant comes into its own in a quantum field theoryQuantum 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...
. A special role is played in relativistic quantum theories by coupling constants which are dimensionless, i.e., are pure numbers. For example, 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...
,
(where
is the charge of an electronElementary charge
The elementary charge, usually denoted as e, is the electric charge carried by a single proton, or equivalently, the absolute value of the electric charge carried by a single electron. This elementary charge is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called...
,
is the permittivity of free space, is the reduced Planck constant and is the speed of lightSpeed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
) is such a dimensionless coupling constant that determines the strength of the electromagnetic force on an electron.
Gauge coupling
In a non-Abelian gauge theory, the gauge coupling parameter,, appears in the LagrangianLagrangian
The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...
as
(where
is the gauge fieldField (physics)
In physics, a field is a physical quantity associated with each point of spacetime. A field can be classified as a scalar field, a vector field, a spinor field, or a tensor field according to whether the value of the field at each point is a scalar, a vector, a spinor or, more generally, a tensor,...
tensor) in some conventions. In another widely used convention,
is rescaled so that the coefficient of the kinetic term is 1/4 and appears in the covariant derivativeCovariant derivative
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given...
. This should be understood to be similar to a dimensionless version of the electric charge defined as
Weak and strong coupling
In a quantum field theoryQuantum 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...
with a dimensionless coupling constant g, if g ≪ 1 (g is much smaller than one) then 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
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...
. 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
Hadron
In particle physics, a hadron is a composite particle made of quarks held together by the strong force...
ic theory of strong interaction
Strong interaction
In particle physics, the strong interaction is one of the four fundamental interactions of nature, the others being electromagnetism, the weak interaction and gravitation. As with the other fundamental interactions, it is a non-contact force...
s (which is why it is called strong in the first place). In such a case non-perturbative methods have to be used to investigate the theory.
Running coupling
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...
at short times or distances by changing the wavelength or momentum, k of the probe one uses. With a high frequency, i.e., short time probe, one sees virtual particles taking part in every process. The reason this can happen, seemingly violating the conservation of energy
Conservation of energy
The nineteenth century law of conservation of energy is a law of physics. It states that the total amount of energy in an isolated system remains constant over time. The total energy is said to be conserved over time...
is the uncertainty relation
which allows such violations at short times. The previous remark only applies to some formulations of quantum field theory, in particular, canonical quantization
Canonical quantization
In physics, canonical quantization is a procedure for quantizing a classical theory while attempting to preserve the formal structure of the classical theory, to the extent possible. Historically, this was Werner Heisenberg's route to obtaining quantum mechanics...
in the interaction picture
Interaction picture
In quantum mechanics, the Interaction picture is an intermediate between the Schrödinger picture and the Heisenberg picture. Whereas in the other two pictures either the state vector or the operators carry time dependence, in the interaction picture both carry part of the time dependence of...
. In other formulations, the same event is described by "virtual" particles going off the mass shell. Such processes renormalize
Renormalization
In quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities....
the coupling and make it dependent on the energy scale,
at which one observes the coupling. The dependence of a coupling on the energy-scale is known as running of the coupling. The theory of the running of couplings is known as the renormalization groupRenormalization 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...
.
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
Conformal field theory
A conformal field theory is a quantum field theory that is invariant under conformal transformations...
.
The coupling parameters of a quantum field theory can flow even if the corresponding classical field
Field (physics)
In physics, a field is a physical quantity associated with each point of spacetime. A field can be classified as a scalar field, a vector field, a spinor field, or a tensor field according to whether the value of the field at each point is a scalar, a vector, a spinor or, more generally, a tensor,...
theory is scale-invariant
Scale invariance
In physics and mathematics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor...
. In this case, the non-zero beta function tells us that the classical scale-invariance is anomalous
Conformal anomaly
Conformal anomaly is an anomaly i.e. a quantum phenomenon that breaks the conformal symmetry of the classical theory.A classically conformal theory is a theory which, when placed on a surface with arbitrary background metric, has an action that is invariant under rescalings of the background metric...
.
QED and the Landau pole
If a beta function is positive, the corresponding coupling increases with increasing energy. An example is quantum electrodynamicsQuantum 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...
(QED), where one finds by using 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...
that the beta function is positive. In particular, at low energies, α ≈ 1/137, whereas at the scale of the Z boson, about 90 GeV
GEV
GEV or GeV may stand for:*GeV or gigaelectronvolt, a unit of energy equal to billion electron volts*GEV or Grid Enabled Vehicle that is fully or partially powered by the electric grid, see plug-in electric vehicle...
, one measures α ≈ 1/127.
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 Landau
Lev Davidovich Landau was a prominent Soviet physicist who made fundamental contributions to many areas of theoretical physics...
, and is called 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...
. 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 WilczekFrank 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 Politzer and 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...
. An example of this is the beta function for 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), and as a result the QCD coupling decreases at high energies.
Furthermore, the coupling decreases logarithmically, a phenomenon known as asymptotic freedom
Asymptotic freedom
In physics, asymptotic freedom is a property of some gauge theories that causes interactions between particles to become arbitrarily weak at energy scales that become arbitrarily large, or, equivalently, at length scales that become arbitrarily small .Asymptotic freedom is a feature of quantum...
(the discovery of which was awarded with 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 coupling decreases approximately as
where β0 is a constant 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
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...
.
QCD scale
In quantum chromodynamicsQuantum 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 quantity Λ is called the QCD scale. The value is
This value is to be used at a scale above the bottom quark
Quark
A quark is an elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. Due to a phenomenon known as color confinement, quarks are never directly...
mass of about 5 GeV
GEV
GEV or GeV may stand for:*GeV or gigaelectronvolt, a unit of energy equal to billion electron volts*GEV or Grid Enabled Vehicle that is fully or partially powered by the electric grid, see plug-in electric vehicle...
. The meaning of ΛMS is given in the article on dimensional transmutation
Dimensional transmutation
In particle physics, dimensional transmutation is a physical mechanism that transforms a dimensionless parameter into a dimensionful parameter....
.
The proton-to-electron mass ratio
Proton-to-electron mass ratio
In physics, the proton-to-electron mass ratio, μ or β, is simply the rest mass of the proton divided by that of the electron...
is primarily determined by the QCD scale.
String theory
A remarkably different situation exists in string theoryString theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
. Each perturbative description of string theory depends on a string coupling constant. However, in the case of string theory, these coupling constants are not pre-determined, adjustable, or universal parameters; rather they are dynamical scalar field
Scalar field
In mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity. Scalar fields are required to be coordinate-independent, meaning that any two observers using the same units will agree on the...
s that can depend on the position in space and time and whose values are determined dynamically.
See also
- Quantum field theoryQuantum 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 many-body systems. It is the natural and quantitative language of particle physics and...
, especially quantum electrodynamicsQuantum 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 quantum chromodynamicsQuantum 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 color-charged fermions... - Canonical quantizationCanonical quantizationIn physics, canonical quantization is a procedure for quantizing a classical theory while attempting to preserve the formal structure of the classical theory, to the extent possible. Historically, this was Werner Heisenberg's route to obtaining quantum mechanics...
, renormalizationRenormalizationIn quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities....
and dimensional regularizationDimensional regularizationIn theoretical physics, dimensional regularization is a method introduced by Giambiagi and Bollini for regularizing integrals in the evaluation of Feynman diagrams; in other words, assigning values to them that are meromorphic functions of an auxiliary complex parameter d, called the... - fine structure constant
- gravitational coupling constantGravitational coupling constantIn physics, the gravitational coupling constant, αG, is the coupling constant characterizing the gravitational attraction between two charged elementary particles having nonzero mass...