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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 variation in a physical system (see the full article ...
Richard Feynman, one of the originators and early developers of the theory of quantum electrodynamics (QED), referred to the fine-structure constant in these terms: There is a most profound and beautiful question associated with the observed coupling constant, e – the amplitude for a real electron to emit or absorb a real photon. It is a ...
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.
Conformal symmetry is associated with the vanishing of the beta function. This can occur naturally if a coupling constant is attracted, by running, toward a fixed point at which β(g) = 0. In QCD, the fixed point occurs at short distances where g → 0 and is called a ultraviolet fixed point.
This series diverges asymptotically, but in quantum electrodynamics (QED) at the second order the difference from experimental data is in the order of 10 −10. This close agreement holds because the coupling constant (also known as the fine-structure constant) of QED is much less than 1. [clarification needed]
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 (RG) scale. This "running" of the coupling is specified by the beta function of the renormalization group.
This singularity is the Landau pole with a negative residue, g(Λ) ≈ −Λ Landau / (β 2 (Λ − Λ Landau)).. In fact, however, the growth of g 0 invalidates Eqs. 1, 2 in the region g 0 ≈ 1, since these were obtained for g 0 ≪ 1, so that the nonperturbative existence of the Landau pole becomes questionable.