Running α (QED) Lab

Explore how leptonic vacuum polarization increases α with the scale μ. The teaching reference is α₀=1/137.035999 at mₑ=0.000511 GeV, with mμ=0.10566 GeV and mτ=1.77686 GeV. The threshold model uses 1/α(μ)=1/α₀−(2/3π)Σ max[0,ln(μ/mℓ)] over e, μ and τ. It is continuous, with changes of slope at each mass; below mₑ it stays at α₀. Sharp thresholds approximate decoupling, not the exact smooth vacuum polarization.

The massless-electron comparison uses only ln(μ/mₑ), even below mₑ. Both are deliberately incomplete one-loop models: no hadronic vacuum polarization, finite matching corrections or higher orders. In particular α(M_Z) is not a precision prediction or the measured Standard Model value.

Enter μ precisely, move the logarithmic slider, click inside the chart, or choose mₑ and M_Z=91.1876 GeV. The chart opens immediately and valid changes update directly. Display contains the massless-electron comparison and α/1/α ordinate. The graph covers 10⁻⁴–10⁵ GeV and magnifies vertical changes; neither ordinate starts at zero. Clicking or dragging chooses eight significant digits; typed input is retained without rounding. Changing μ moves markers and updates values without rebuilding curves. Invalid input removes stale output. Reset restores a live default. Only the raw scale text and model/comparison/ordinate choices are retained, including unfinished input. Valid older numeric settings migrate; curves, results and open menus are not saved.

Reference details include the formal electron-only pole μL=mₑ exp[3π/(2α₀)]. It is the pole of that extrapolated model, not of the e/μ/τ model, and is far beyond a reliable physical extrapolation. It is outside the chart.

Tong, Standard Model §3.1, Eq. 3.20; PDG, Electroweak Model §10.3.1.