Running coupling
The energy-dependent value α(E) of the QED fine-structure constant, increasing from α ≈ 1/137 at low energies to α ≈ 1/128 at the Z-pole and toward 1 at the Landau pole. The breakdown of perturbation theory at high energy is one of the doors from QED to deeper theory.
Definition
The running coupling α(E) of quantum electrodynamics is the energy-dependent value of the fine-structure constant, generated by quantum corrections from virtual electron-positron pairs that screen the bare electric charge. At low energies (E ≪ m_e c²) the coupling has its measured low-energy value α ≈ 1/137.036, but at higher energies the screening becomes less effective — the probe penetrates closer to the bare charge — and the effective coupling grows. The one-loop QED renormalisation-group equation gives α(E) ≈ α(0) / (1 − (α/3π) ln(E²/m_e²c⁴)) at energies above the electron mass, valid for E ≪ Λ where Λ ≈ m_e c² · exp(3π/α) ≈ 10^{286} GeV is the Landau pole.
By the time the energy reaches the Z-boson pole (≈ 91 GeV) the QED coupling has run from 1/137 to about 1/128 — a 7% change measurable in precision LEP-era Z-pole observables. Far above this scale the coupling continues to grow until perturbation theory breaks down at the Landau pole. The Landau pole is far above the Planck scale, so for all practical purposes it is moot, but the existence of the divergence is a sign that QED is not a complete UV theory — it must be embedded in a larger theory (the electroweak SU(2)×U(1) gauge theory, then the Standard Model, then whatever lies beyond) at energies long before the Landau pole becomes relevant. The non-abelian gauge theories of the Standard Model run in the opposite direction: SU(3) QCD has asymptotic freedom (α_s → 0 as E → ∞, the discovery for which Gross, Politzer, and Wilczek shared the 2004 Nobel Prize), while the SU(2) electroweak coupling also decreases at high energy. The grand-unification proposal that all three Standard Model couplings meet at a common high-energy scale (≈ 10^{16} GeV in supersymmetric models) hinges on the running of α, α_s, and α_w being precisely tuned to converge.