# Coupling constant

In physics, the coupling constant is a constant that determines the strength of a fundamental interaction .

In quantum field theory (QFT), interactions are mediated by exchange particles, the gauge bosons . In this case, the coupling constants determine the strength of the coupling of the exchange bosons to the associated charges . There is a coupling constant for each of the four basic forces . In general, an elementary particle can carry several different types of charge and therefore also couple to different gauge bosons. A quark, for example, has an electrical charge and a color charge .

Due to quantum fluctuations , the coupling constants of quantum field theory are energy-dependent, i. H. the coupling strength can increase (example: quantum electrodynamics ) or decrease (example: quantum chromodynamics ) at higher energies . This effect is referred to as the running (Engl. Running ) of the coupling constant.

## Dimensionless coupling constants

The Lagrange or Hamilton function ( also the Hamilton operator in quantum mechanics ) can usually be divided into a kinetic part and an interaction part , corresponding to kinetic (or motion) energy and potential (or position) energy . Coupling constants that are scaled in such a way that they express the ratio of the interaction component to the kinetic component, or the ratio of two interaction components, are of particular importance. Such coupling constants are dimensionless. The importance of the dimensionless coupling constants is that perturbation series are power series in the dimensionless coupling constants. The size of a dimensionless coupling constant determines the convergence behavior of the perturbation series.

## Overview of the forces and the associated gauge bosons and charges

interaction Eichboson (s) charge Coupling constant
Strong interaction Gluons (8 different) Color charge ${\ displaystyle \ textstyle \ alpha _ {\ mathrm {s}}}$
Electromagnetic interaction photon ${\ displaystyle \ textstyle \ gamma}$ Electric charge ${\ displaystyle \ textstyle \ alpha}$( Fine structure constant , here also ) ${\ displaystyle \ textstyle \ alpha _ {\ mathrm {em}}}$
Weak interaction ${\ displaystyle \ textstyle W ^ {+}}$-, - and - boson${\ displaystyle \ textstyle W ^ {-}}$${\ displaystyle \ textstyle Z ^ {0}}$ not definable ${\ displaystyle \ textstyle \ alpha _ {\ mathrm {W}}}$
Gravity Graviton (hypothetical) Dimensions ${\ displaystyle \ textstyle \ alpha _ {\ mathrm {G}}}$

## Fine structure constant

In the electromagnetic interaction , the dimensionless coupling constant is given by the Sommerfeld fine structure constant and is also referred to in this context as : ${\ displaystyle \ textstyle \ alpha}$${\ displaystyle \ textstyle \ alpha _ {\ mathrm {em}}}$

${\ displaystyle \ alpha _ {\ mathrm {em}} = {\ frac {e ^ {2}} {2 \ varepsilon _ {0} hc}} = {\ frac {e ^ {2}} {4 \ pi \ varepsilon _ {0} \ hbar c}} = {\ frac {e ^ {2}} {q _ {\ mathrm {P}} ^ {2}}} = {\ frac {1} {137.035999084 (21)} } = 0.0072973525693 (11)}$

(Here is the Planck charge, the charge of the electron ( elementary charge ), the permittivity of the vacuum , the vacuum speed of light and the Planck quantum or the reduced Planck quantum .) ${\ displaystyle q _ {\ mathrm {P}}}$${\ displaystyle e}$${\ displaystyle \ varepsilon _ {0}}$${\ displaystyle c}$${\ displaystyle h}$${\ displaystyle \ hbar = h / 2 \ pi}$

The fine structure constant describes u. a. the strength of the electromagnetic force between two elementary charges.

## Calibration coupling

In a non-Abelian gauge theory , the gauge coupling parameter appears in the Lagrange function according to certain conventions as ${\ displaystyle g}$

${\ displaystyle - {\ frac {1} {4g ^ {2}}} \ operatorname {Tr} G _ {\ mu \ nu} G ^ {\ mu \ nu}}$.

(where the Eichfeld tensor is) ${\ displaystyle G}$

Another common convention is to scale so that the coefficient of the kinetic term is −1/4 and occurs in the covariant derivative . ${\ displaystyle G}$${\ displaystyle g}$

This is to be understood in a similar way to the dimensionless version of the electric charge:

${\ displaystyle g _ {\ mathrm {em}} = {\ frac {e} {\ sqrt {\ varepsilon _ {0} \ hbar c}}} = {\ sqrt {4 \ pi \ alpha _ {\ mathrm {em }}}} \ approx 0 {,} 30282212 \.}$

With the above relationship for the fine structure constant α is

${\ displaystyle e = {\ sqrt {4 \ pi \ varepsilon _ {0} \ hbar c \ alpha _ {\ mathrm {em}}}}}$

With the Planck charge

${\ displaystyle q _ {\ mathrm {P}} = {\ sqrt {\ hbar c4 \ pi \ varepsilon _ {0}}}}$

follows

${\ displaystyle e / q _ {\ mathrm {P}} = {\ sqrt {\ alpha _ {\ mathrm {em}}}} \ approx 1 / {\ sqrt {137, \ dots}}}$

respectively

${\ displaystyle \ alpha _ {\ mathrm {em}} = \ left ({\ frac {e} {q _ {\ mathrm {P}}}} \ right) ^ {2} \ ;.}$

In this way, in the electromagnetic case, the (dimensional) coupling strength e is linked to the dimensionless coupling constant α.

## Weak and strong coupling

A quantum field theory with a dimensionless coupling constant α if α ≪ 1 (i.e. if α is significantly smaller than 1) is called weakly coupled . In this case the theory is described in power series according to α ( perturbation theory or perturbative theory). One example is electromagnetism. If the coupling constant is on the order of 1 or greater, the theory is said to be strongly coupled . An example of the latter is the hadronic theory of strong interaction . In this case, non-perturbative methods, i.e. methods beyond perturbation theory, must be used for the investigation.

## Electroweak interaction

In the context of the electroweak theory ( Glashow-Weinberg-Salam theory , GWS) one finds for the weak coupling constant in analogy to the fine structure constants (see above): ${\ displaystyle \ textstyle \ alpha _ {W}}$

${\ displaystyle \ alpha _ {W} = {\ frac {g ^ {2}} {2 \ varepsilon _ {0} hc}} = {\ frac {g ^ {2}} {4 \ pi \ varepsilon _ { 0} \ hbar c}} = \ left ({\ frac {g} {q _ {\ mathrm {P}}}} \ right) ^ {2} \ approx 0 {,} 03156}$

The coupling strengths and are across the Weinbergwinkel${\ displaystyle e}$${\ displaystyle g}$ ${\ displaystyle \ theta _ {\ mathrm {W}}}$

${\ displaystyle e = g \ cdot \ sin \ theta _ {\ mathrm {W}}}$

connected. This applies

${\ displaystyle \ alpha _ {\ mathrm {em}} = \ alpha _ {\ mathrm {W}} \ cdot \ sin ^ {2} \ theta _ {\ mathrm {W}}}$

The weak interaction acts on particles ( fermions ) by coupling them to the exchange bosons of the weak interaction -, - and ( W bosons and Z boson ). For the first two the coupling strength is the same, for that it is modified by the weak isospin , the charge number of the fermion and the Weinberg angle: ${\ displaystyle W ^ {+}}$${\ displaystyle W ^ {-}}$${\ displaystyle Z ^ {0}}$${\ displaystyle Z ^ {0}}$ ${\ displaystyle T_ {3}}$${\ displaystyle z_ {f} = q / e}$${\ displaystyle \ theta _ {\ mathrm {W}}}$

${\ displaystyle g_ {Z} (f) = {\ frac {g} {\ cos \ theta _ {\ mathrm {W}}}} \ cdot \ left (T_ {3} -z_ {f} \ cdot \ sin ^ {2} \ theta _ {\ mathrm {W}} \ right)}$

With regard to the weak interaction, there is a difference in how left-handed and right-handed elementary fermions participate in the weak interaction. Furthermore, the coupling to W ± and Z 0 is different. However, antiparticles of the reverse handedness and charge behave again analogously to their normal partners ( CP invariance ).

## Ongoing coupling and Symanzik beta function

One can test a quantum field theory at short times and distances by changing the wavelength or the momentum of the sample used. At high frequencies, i.e. H. short times, you can see that virtual particles participate in every process . The reason why this apparent violation of the law of conservation of energy is possible is Heisenberg's uncertainty principle

${\ displaystyle \ Delta E \ Delta t \ geq \ hbar}$,

which allows such short-term injuries. However, this remark only applies to certain formulations of the QFT, namely the canonical quantization in the interaction picture . Alternatively, the same event can be described by means of “virtual” particles that go off shell with respect to the mass shell . Such processes renormalize the coupling and make it dependent on the energy scale at which the coupling is observed. The dependence of the coupling of the energy scale as running coupling (eng .: running coupling hereinafter). The theory of ongoing coupling is described by means of the renormalization group (RG). ${\ displaystyle \ mu}$${\ displaystyle g (\ mu)}$

In a quantum field theory (QFT) this running of a coupling parameter g according to Kurt Symanzik is described with a Symanzik beta function β (g). This is defined by the relationship:

${\ displaystyle \ beta (g) = \ mu \, {\ frac {\ partial g} {\ partial \ mu}} = {\ frac {\ partial g} {\ partial \ ln \ mu}}.}$

If the beta functions of a QFT vanish (i.e. are constantly zero) then this theory is scale invariant .

The coupling parameters of a QFT can run even if the corresponding classic field is scale-invariant . In this case, the non-zero beta function says that the classical scale invariance is anomalous .

### QED and the Landau Pole

If the beta function is positive, the associated coupling grows with increasing energy. One example is quantum electrodynamics (QED), in which one finds with the help of perturbation theory that the beta function is positive. More precisely, α ≈ 1/137 (Sommerfeld fine structure constant ) applies , while on the scale of the Z boson , i.e. at around 90  GeV , α ≈ 1/127.

Furthermore, the perturbation theory beta function shows us that the coupling continues to increase, and thus the QED is strongly coupled at high energies . In fact, the coupling determined in this way becomes infinite already at a certain finite energy. This phenomenon was first noticed by Lew Landau and is therefore called the Landau Pol . Of course, the perturbation beta function cannot be expected to give exact results when the coupling is strong, and so it is likely that the Landau pole is an artifact of the improper application of perturbation theory. The true scale behavior of at high energies is unknown. ${\ displaystyle \ textstyle \ alpha}$

### QCD and Asymptotic Freedom

In non-Abelian gauge theories, the beta function can become negative, which was first discovered by Frank Wilczek , David Politzer, and David J. Gross . An example of this is the beta function for quantum chromodynamics (QCD). As a result, the QCD coupling decreases at high energies. In fact, the coupling decreases logarithmically, a phenomenon called asymptotic freedom . The coupling decreases approximately as

${\ displaystyle \ alpha _ {s} (k ^ {2}) \ {\ stackrel {\ mathrm {def}} {=}} \ {\ frac {g_ {s} ^ {2} (k ^ {2} )} {4 \ pi}} \ approx {\ frac {1} {\ beta _ {0} \ ln (k ^ {2} / \ Lambda ^ {2})}}.}$

Here is a constant determined by Wilczek, Gross and Politzer with the number of fermions charged under the QCD . is not a UV cutoff, but a mass scale determined by the renormalization scheme; QCD can only be treated in terms of disorder theory beyond this scale. ${\ displaystyle \ beta _ {0} = 11 - {\ tfrac {2} {3}} n_ {f}}$${\ displaystyle n_ {f}}$${\ displaystyle \ Lambda}$

Conversely, the coupling increases with decreasing energy. At low energies it becomes so strong that perturbation theory is no longer applicable here.

## String theory

There is a remarkably different situation in string theory . The perturbative description of string theory depends on the string coupling constant. However, in string theory, these coupling constants are not predetermined, adaptable or universal parameters, instead they are scalar fields that can depend on the position in space and time, the values ​​of which are therefore dynamically determined.