Coupling constant
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In physics, 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. 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 and the mass for Newtonian gravity) divided by the distance squared, , between the bodies; thus: in for Newtonian gravity and in for electrostatic. This description remains valid in modern physics for linear theories with static bodies and massless force carriers.[citation needed]
A modern and more general definition uses the Lagrangian (or equivalently the 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 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 with two arrows and one wavy line). Since photons mediate the electromagnetic 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, 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, which is the main method of calculation in many branches of physics.
Fine-structure constant
Couplings arise naturally in a
where e is the
Gauge coupling
In a non-abelian gauge theory, the gauge coupling parameter, , appears in the Lagrangian as
(where G is the gauge field 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. 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 need to 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,[1] 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, 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
Running coupling
One may probe a
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
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.[4] 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 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 sustaining the surface S. In the modern view of quantum field theory, the comes from the expression in
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 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
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.
QCD and asymptotic freedom
In non-abelian gauge theories, the beta function can be negative, as first found by
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 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. 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 αs(MZ2 ) = 0.1179 ± 0.0010.[7] In 2023 Atlas measured αs(MZ2 ) = 0.1183 ± 0.0009 the most precise so far.[5][6] 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.[8]
QCD scale
In quantum chromodynamics (QCD), the quantity Λ is called the QCD scale. The value is
[4] for three "active" quark flavors, viz when the energy–momentum involved in the process allows production of 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
String theory
A remarkably different situation exists in
See also
- Canonical quantization, renormalization and dimensional regularization
- Quantum field theory, especially quantum electrodynamics and quantum chromodynamics
- Gluon field, Gluon field strength tensor
References
- ISBN 0691140340
- .
- ^
Donoghue, John F. (1995). "Introduction to the Effective Field Theory Description of Gravity". In Cornet, Fernando (ed.). Effective Theories: Proceedings of the Advanced School, Almunecar, Spain, 26 June – 1 July 1995. Singapore: ISBN 978-981-02-2908-5.
- ^ S2CID 118854278.
- ^ arXiv:2309.12986 [hep-ex].
- ^ a b "ATLAS measures strength of the strong force with record precision". CERN. 2023-10-11. Retrieved 2023-10-24.
- ^ Particle Data Group, "Review of Particle Physics, Chapter 9. Quantum Chromodynamics", 2022, https://pdg.lbl.gov/2021/reviews/rpp2021-rev-qcd.pdf
- arXiv:2203.05394 [hep-ph].
- ^ C. Patrignani et al. (Particle Data Group), Chin. Phys. C, 40, 100001 (2016)
External links
- The Nobel Prize in Physics 2004 – Information for the Public
- Department of Physics and Astronomy of the Georgia State University – Coupling Constants for the Fundamental Forces
- An introduction to quantum field theory, by M.E.Peskin and H.D.Schroeder, ISBN 0-201-50397-2