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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 1 / r 2 {\displaystyle 1/r^{2}} .
The value of the fine-structure constant α is linked to the observed value of this coupling associated with the energy scale of the electron mass: the electron's mass gives a lower bound for this energy scale, because it (and the positron) is the lightest charged object whose quantum loops can contribute to the running. Therefore, 1 / 137 ...
written in terms of the fine structure constant in natural units, α = e 2 /4π. [2] This beta function tells us that the coupling increases with increasing energy scale, and QED becomes strongly coupled at high energy. In fact, the coupling apparently becomes infinite at some finite energy, resulting in a Landau pole. However, one cannot ...
Because the theory is "sick" for any negative value of the coupling constant, the series does not converge but is at best an asymptotic series. From a modern perspective, we say that QED is not well defined as a quantum field theory to arbitrarily high energy. [30] The coupling constant runs to infinity at finite energy, signalling a Landau pole.
The scale anomaly, which gives rise to a running coupling constant. In QED this gives rise to the phenomenon of the Landau pole. In quantum chromodynamics (QCD) this leads to asymptotic freedom. The chiral anomaly in either chiral or vector field theories with fermions. This has close connection with topology through the notion of instantons.
In classical field theory, such as gauge theory in four-dimensional spacetime, the coupling constant is a dimensionless constant. However, upon quantization, logarithmic divergences in one-loop diagrams of perturbation theory imply that this "constant" actually depends on the typical energy scale of the processes under considerations, called ...
The renormalization group prediction (cf. Stueckelberg–Petermann and Gell-Mann–Low works) was confirmed 40 years later at the LEP accelerator experiments: the fine structure "constant" of QED was measured [6] to be about 1 ⁄ 127 at energies close to 200 GeV, as opposed to the standard low-energy physics value of 1 ⁄ 137. [b]
Indeed, the result g obs = const(g 0) can be obtained from the functional integrals only for g 0 ≫ 1, while its validity for g 0 ≪ 1, based on Eq. 1, may be related to other reasons; for g 0 ≈ 1 this result is probably violated but coincidence of two constant values in the order of magnitude can be expected from the matching condition.