Running αₛ (QCD)

Choose the active flavor count n_f, Λ and probe scale Q in GeV. The diagram appears immediately and updates on valid edits. Invalid inputs clear the previous result. Reset restores defaults and a live preview. Only input choices and exact numeric text are retained, including unfinished or invalid entries; curves and calculated results are not saved. Valid legacy settings migrate.

The fixed-flavor one-loop formula is αₛ(Q)=12π/[(33−2n_f) ln(Q²/Λ²)]. Q and Λ must use the same energy unit and Q must exceed Λ. The coupling decreases as Q increases: asymptotic freedom. At Q=Λ the formula has a pole; below it no result is offered. A large formal value near the pole is not a reliable perturbative prediction.

The calibration button solves Λ=Q₀ exp[−6π/((33−2n_f)α₀)], using Q₀=91.2 GeV and the illustrative reference α₀=0.118. This is a rounded Z-scale reference, not a live measurement or a precision fit. The full computed Λ is kept in the numeric field, including for n_f=6; there is no slider-raster rounding. An extra row checks the reference value when the probe uses a different scale.

The overlay compares hypothetical fixed n_f curves at the same Λ. They are not segments of one threshold-matched QCD trajectory. Real QCD uses running with flavor thresholds, matching and higher orders; a Λ value depends on the flavor count and perturbative convention. This tool includes none of those precision features.

The graph starts where every displayed QCD curve has αₛ≤1 and extends to at least1000 GeV, including a valid higher probe. A probe below this drawing range still gives the formal value and an explicit warning; no off-scale marker is faked. αₛ=1 is only a display cutoff, not a sharp boundary of validity. Couplings above0.5 receive an additional caution.

The optional QED curve is a schematic electron-only contrast: 1/αₑₘ(Q)=137−ln(Q²/Q₀²)/(3π), with Q₀=0.001 GeV. It illustrates an increasing coupling and shows a numeric probe value, but ignores all other charged particles and threshold effects. It is not the full physical running of αₑₘ.

PDG Quantum Chromodynamics (2025), §9.1.1.

Comparison contains the flavor overlay, electron-only QED contrast and linear or logarithmic coupling axis. A logarithmic vertical axis can make the small QED contrast easier to distinguish without changing either model. Click inside the plotted region to choose Q from its logarithmic horizontal position; the numeric field remains the exact and keyboard-accessible input. Clicked scales are rounded to eight significant digits and capped at the supported 10000 GeV input limit.