Physics/Essays/Fedosin/Coupling constant

A coupling constant (or an interaction constant) is a parameter in the field theory, which determines the relative strength of interaction between particles or fields. In the quantum field theory the coupling constants are associated with the vertices of the corresponding Feynman diagrams. Dimensionless parameters are used as coupling constants, as well as the quantities associated with them that characterize the interaction and have dimensions. The examples are the dimensionless fine structure constant of electromagnetic interaction and the electric elementary charge, measured in coulombs (C).

Comparison of interactions
If we choose an object that participates in all the four fundamental interactions, the values of the dimensionless coupling constants of this object, found according to the general rule, will show the relative strength of these interactions. At the level of elementary particles a proton is most commonly used as such an object. The basic energy for comparison of interactions is the electromagnetic energy of a photon, which equals by definition:
 * $$U_f= \frac{h c}{\lambda},$$

where $$~h $$ is the Planck constant, $$~c $$ is the speed of light, $$~\lambda $$ is the photon wavelength. The choice of the photon energy is not accidental, since the basis of the modern science is the wave representation based on electromagnetic waves. All the main measurements, including length, time and energy, are made with the help of them.

Gravitational interaction
The energy of gravitational interaction between two protons is given by:
 * $$U_{g}= -\frac{G M^2_p}{r},$$

where $$~G $$ is the gravitational constant, $$~M_p $$ is the proton mass, $$~r $$ is the distance between the protons’ centers.

If we assume that the distance $$~r $$ and the electromagnetic photon’s wavelength $$~\lambda $$ are related by the formula $$~\lambda =2 \pi r $$, then the ratio of the absolute value of the gravitational interaction energy to the photon’s energy gives the dimensionless coupling constant:
 * $$\alpha_{g}=\frac {\mid U_{g}\mid }{ U_f } = \frac{G M^2_p }{\hbar c }=5{.}907\cdot10^{-39},$$

where $$~\hbar $$ is the Dirac constant.

Weak interaction
The energy associated with the weak interaction can be represented as follows:
 * $$U_{W}= \frac{g^2_F}{4 \pi r} \exp (-\frac {M_W cr}{\hbar }),$$

where $$~g_F $$ is the effective charge of weak interaction, $$~M_W $$ is the mass of virtual particles that are considered the carrier particles for weak interaction (W and Z bosons). The square of the effective charge of weak interaction for the proton is expressed in terms of the Fermi constant $$~G_F=1.43 \cdot 10^{-62} $$ J•m3 and the proton mass:


 * $$g^2_{F}= \frac{4 \pi G_F M^2_p c^2}{\hbar^2 }.$$

At sufficiently small distances the exponent in the weak interaction energy can be neglected. In this case, the dimensionless coupling constant of weak interaction is determined as follows:
 * $$\alpha_{W}=\frac { U_{W}}{ U_f } = \frac{G_F M^2_p c }{\hbar^3 r}=1{.}0\cdot10^{-5}.$$

Electromagnetic interaction
The electromagnetic interaction of two fixed protons is described by the electrostatic energy:
 * $$U_{e}=\frac{e^2}{4\pi \varepsilon_0 r},$$

where $$~e $$ is the elementary charge, $$~\varepsilon_0 $$ is the electric constant.

The ratio of this energy to the photon energy $$~U_f$$ determines the electromagnetic coupling constant, known as the fine structure constant:
 * $$\alpha=\frac { U_{e}}{ U_f } =\frac{e^2}{4\pi\varepsilon_0 \hbar c}=7{.}297\cdot10^{-3}.$$

Strong interaction
At the level of hadrons, the strong interaction is regarded in the Standard Model of elementary particle physics as “residual” interaction of quarks that are part of hadrons. It is assumed that gluons as the carriers of strong interaction generate virtual mesons in the space between the hadrons. In the pion-nucleon model of Yukawa interaction, the nuclear forces between the nucleons are explained as a result of the virtual pions exchange, and the interaction energy is as follows:
 * $$U_{s}=- \frac{g^2_{N \pi}}{4 \pi r} \exp (-\frac {M_{\pi} cr}{\hbar }),$$

where $$~g_{N \pi} $$ is the effective charge of the pseudoscalar pion-nucleon interaction, $$~ M_{\pi} $$is the pion mass.

The dimensionless strong interaction coupling constant is:
 * $$\alpha_{s}=\frac { \mid U_{s}\mid }{ U_f } = \frac{ g^2_{N \pi}}{4\pi\hbar c } \approx 14{.}6.$$

The constants in the quantum field theory
The interaction effects in the field theory are often determined with the help of perturbation theory, in which the expansion of functions in the equations in powers of the coupling constant is performed. Usually for all interactions, except the strong interaction, the coupling constant is significantly less than unity. This makes the use of the perturbation theory effective, since the contribution from the highest terms of expansions decreases rapidly and calculating them becomes unnecessary. In case of strong interaction the perturbation theory becomes unsuitable and other methods of calculation are required.

One of the predictions of the quantum field theory is the so-called effect of “floating constants”, according to which the coupling constants change slowly with increasing of the energy, transferred during the interaction between the particles. Thus, the electromagnetic coupling constant increases and the strong interaction constant coupling decreases with the increase of energy. In Quantum Chromodynamics a special strong interaction coupling constant is introduced for the quarks:
 * $$\alpha_{sq}= \frac{ g^2_{qg}}{4\pi\hbar c } <1, $$

where $$~g_{ qg } $$ is the effective color charge of the quark, emitting virtual gluons for the interaction with other quarks.

As the distance between the quarks decreases, due to the collisions of high energy particles, the log reduction of $$~\alpha_{sq}$$ and the weakening of strong interaction (the effect of asymptotic freedom of quarks) are expected. At the scale of the transferred energy of the order of the Z boson’s mass-energy (91.19 GeV) it was found that $$~\alpha_{sq}=0.1187.$$ At the same energy scale the electromagnetic interaction coupling constant increases up to the value of the order of 1/127 instead of  ≈1/137 at low energies. It is assumed that at higher energies, of the order of 1018 GeV, the values of the coupling constants of gravitational, weak, electromagnetic and strong interactions of particles will become closer and even become approximately equal to each other.

String theory
In the string theory, the coupling constants are considered not as constant but as dynamic quantities. In particular, in the same theory at low energies it seems that the strings move in ten dimensions and at high energies — in eleven. The changing number of dimensions is accompanied by a change in the coupling constants.

Strong gravitation
Strong gravitation together with the gravitational torsion field and electromagnetic forces are considered the main components of strong interaction in the gravitational model of strong interaction. In this model, instead of considering interactions of quarks and gluons, only two fundamental fields (gravitational and electromagnetic fields) are taken into account, which act in the charged matter of elementary particles that has mass, as well as in the space between them. In this case, quarks and gluons, according to the model of quark quasiparticles, are considered not as real particles but as quasiparticles, reflecting the quantum properties and symmetry, inherent in hadronic matter. This approach significantly reduces the number (the record number for a physical theory) of unproved but postulated free parameters that exist in the standard model of elementary particle physics, where there are at least 19 parameters of this kind.

Another consequence is that the weak and strong interactions are not considered as independent field interactions. The strong interaction is reduced to combinations of gravitational and electromagnetic forces, in which an important role is played by the interactions’ delay effects (dipole and orbital torsion fields and magnetic forces). Accordingly, the strong coupling constant is determined by analogy with the gravitational interaction coupling constant:


 * $$\alpha_{pp}=\frac {\mid U_{\Gamma}\mid }{ U_f } = \frac{\beta \Gamma M^2_p }{\hbar c }=\frac { \alpha \beta M_p }{ M_e }=13{.}4 \beta ,$$

where $$~\Gamma $$ is the strong gravitational constant, $$~ M_e $$ is the electron mass, $$~\beta $$ is a coefficient, which is equal to 0.26 for the interaction of two nucleons and is tending to 1 for bodies with lower matter density.

As for the weak interaction, it is assumed to be the result of the transformation of matter of elementary particles, which occurs due to the reactions of weak interaction, but at a deeper level of matter. The examples of weak interaction with nucleons are considered in the substantial neutron model and the substantial proton model.

Interactions at the level of stars
Among the stellar constants, describing the quantization of parameters of cosmic systems in the hydrogen system of stars, there are two dimensionless constants. One of them determines the stellar fine structure constant $$~\alpha$$ and the other determines the relative strength of interaction between two stars. In case of the hydrogen system of the magnetar and the disks near it these constants equal:


 * $$~\alpha= \frac { Q^2_s }{4 \pi \varepsilon_0 \hbar'_s C'_s }= \frac { G M_s M_d }{ \hbar'_s C'_s }= 7.2973525376 \cdot 10^{-3},$$


 * $$~\alpha_{mm}= \frac { \beta Q^2_s M_s }{4 \pi \varepsilon_0 M_d \hbar'_s C'_s }= \frac{\beta G M^2_s }{\hbar'_s C'_s }=\frac { \alpha \beta M_s }{ M_d }=13{.}4 \beta ,$$

where $$~ Q_s = 5.5 \cdot 10^{18}$$ C is the electric charge of the magnetar, based on its similarity with the proton, $$~\hbar'_s= 5.5 \cdot 10^{41} $$ J∙s is the stellar Dirac constant for the system with the magnetar, $$~ C'_s= 6.8 \cdot 10^{7}$$ m/s is the stellar speed as the characteristic speed of the matter particles in a typical neutron star, $$~M_{s}= 1.35 M_c=2.7\cdot 10^{30}$$ kg is the mass of the magnetar, $$~ M_d= 1.5 \cdot 10^{27}$$ kg is the mass of the disk, which is the electron’s analogue at the level of stars.

Due to the SPФ symmetry and the similarity of matter levels, the values of the dimensionless coupling constants are the same both at the atomic level and at the level of stars.

Additional references

 * Р. Маршак, Э. Судершан. Введение в физику элементарных частиц, 1962.
 * M.E. Peskin and H.D. Schroeder. An introduction to quantum field theory, ISBN 0-201-50397-2.