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In physics, a coupling constant or gauge coupling parameter is a number that determines the strength of the force exerted in an interaction. Usually, the Lagrangian or the Hamiltonian of a system describing an interaction 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 that characterizes an interaction with two charge-carrying fields and one photon field (hence the common Feynman diagram
Feynman diagram
with two arrows and one wavy line). Since photons carry electromagnetism, this coupling constant determines how strongly electrons feel such a force and has its value fixed by experiment. 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 may be 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.

Contents

1 Fine-structure constant 2 Gauge coupling 3 Weak and strong coupling 4 Running coupling

4.1 Beta functions 4.2 QED and the Landau pole 4.3 QCD and asymptotic freedom 4.4 QCD scale

5 String theory 6 See also 7 References 8 External links

Fine-structure constant[edit] Coupling constants arise naturally in a quantum field theory. A special role is played in relativistic quantum theories by coupling constants that are dimensionless; i.e., are pure numbers. An example of a dimensionless constant is the fine-structure constant,

α =

e

2

4 π

ε

0

ℏ c

,

displaystyle alpha = frac e^ 2 4pi varepsilon _ 0 hbar c ,

where

e

displaystyle e

is the charge of an electron,

ε

0

displaystyle varepsilon _ 0

is the permittivity of free space,

displaystyle hbar

is the reduced Planck constant and

c

displaystyle c

is the speed of light. This constant is proportional to the square of the coupling strength of the charge of an electron to the electromagnetic field. Gauge coupling[edit] In a non-Abelian gauge theory, the gauge coupling parameter,

g

displaystyle g

, appears in the Lagrangian as

1

4

g

2

T r

G

μ ν

G

μ ν

displaystyle frac 1 4g^ 2 rm Tr ,G_ mu nu G^ mu nu

(where

G

displaystyle G

is the gauge field tensor) in some conventions. In another widely used convention,

G

displaystyle G

is rescaled so that the coefficient of the kinetic term is 1/4 and

g

displaystyle g

appears in the covariant derivative. This should be understood to be similar to a dimensionless version of the elementary charge defined as

e

ε

0

ℏ c

=

4 π α

≈ 0.30282212   .

displaystyle frac e sqrt varepsilon _ 0 hbar c = sqrt 4pi alpha approx 0.30282212 .

Weak and strong coupling[edit] In a quantum field theory with a dimensionless coupling constant g, if g is much less than 1[clarification needed] 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 hadronic theory of strong interactions (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[edit]

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. This apparent violation of the conservation of energy can be understood heuristically by examining the uncertainty relation

Δ E Δ t ≥

ℏ 2

,

displaystyle Delta EDelta tgeq frac hbar 2 ,

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,

μ

displaystyle mu

, at which one observes 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, though it should be kept in mind that the renormalization group is a more general concept describing any sort of scale invariance in a physical system (see the full article for details). Beta functions[edit] Main article: Beta function (physics) In quantum field theory, a beta function β(g) encodes the running of a coupling parameter, g. It is defined by the relation

β ( g ) = μ

∂ g

∂ μ

=

∂ g

∂ ln ⁡ μ

,

displaystyle beta (g)=mu frac partial g partial mu = frac partial g partial ln mu ,

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 and the Landau pole[edit] If a beta function is positive, the corresponding coupling increases with increasing energy. An example is quantum electrodynamics (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

α

displaystyle alpha

at large energies is not known. QCD and asymptotic freedom[edit] In non-Abelian gauge theories, the beta function can be negative, as first found by Frank Wilczek, 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
Physics
in 2004). The coupling decreases approximately as

α

s

(

k

2

)  

=

d e f

 

g

s

2

(

k

2

)

4 π

1

β

0

ln ⁡

(

k

2

Λ

2

)

,

displaystyle alpha _ text s (k^ 2 ) stackrel mathrm def = frac g_ text s ^ 2 (k^ 2 ) 4pi approx frac 1 beta _ 0 ln left( frac k^ 2 Lambda ^ 2 right) ,

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 scale[edit] In quantum chromodynamics (QCD), the quantity Λ is called the QCD scale. The value is

Λ

M S

= 218 ± 24

 MeV

.

displaystyle Lambda _ MS =218pm 24 text MeV .

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. String theory[edit] 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 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. 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. This is free to have any value in the bosonic theory where there is no superpotential. See also[edit]

Canonical quantization, renormalization and dimensional regularization Fine structure constant Gravitational coupling constant Quantum field theory, especially quantum electrodynamics and quantum chromodynamics Gluon field, Gluon field
Gluon field
strength tensor

References[edit]

An introduction to quantum field theory, by M.E.Peskin and H.D.Schroeder, ISBN 0-201-50397-2

External links[edit]

The Nobel Prize in Physics
Physics
2004 – Information for the Public Department of Physics
Physics
and Astronomy of the Georgia State University - Coupling Constants for the Fundam

.