Calculator Inputs
Choose direct Lambda QCD input or derive Lambda from a reference coupling measurement.
Formula Used
This calculator uses the perturbative running of the QCD coupling constant. The strong interaction strength changes with the energy scale, and that change is governed by the beta function.
Beta coefficients
beta0 = 11 - 2n_f / 3
beta1 = 102 - 38n_f / 3
L = ln(Q² / Lambda²)
1-loop running
alpha_s(Q) = 4pi / (beta0 L)
2-loop running
alpha_s(Q) = [4pi / (beta0 L)] × [1 - (beta1 / beta0²) × ln(L) / L]
Gauge coupling
g_s(Q) = sqrt(4pi × alpha_s(Q))
The calculator can also derive Lambda QCD from a known reference pair, alpha_s(Q0) and Q0, using numerical root-finding.
Results are most meaningful in perturbative regions where Q is safely above Lambda QCD.
How to Use This Calculator
- Choose whether you want to enter Lambda QCD directly or derive it from a known reference coupling.
- Enter the main energy scale Q in GeV where you want alpha_s(Q).
- Select the number of active quark flavors n_f and the loop order.
- Optionally enter a comparison scale to see how the coupling changes with energy.
- Set your preferred decimal precision and click Calculate Coupling.
- Review alpha_s, g_s, beta coefficients, and the regime label in the result summary above the form.
Example Data Table
Sample values help verify the behavior of the running coupling.
| Method | Q (GeV) | Lambda QCD (GeV) | n_f | Loop | alpha_s(Q) | g_s(Q) | Regime |
|---|---|---|---|---|---|---|---|
| Given Lambda | 2.000000 | 0.200000 | 5 | 2-loop | 0.278277 | 1.870009 | Moderately perturbative |
| Given Lambda | 10.000000 | 0.200000 | 5 | 2-loop | 0.173258 | 1.475542 | Weakly perturbative |
| Reference alpha_s(Q0) | 91.187600 | 0.227506 | 5 | 2-loop | 0.118100 | 1.218228 | Weakly perturbative |
Frequently Asked Questions
1) What does this calculator measure?
It estimates the QCD running coupling alpha_s at a chosen energy scale. It also reports g_s, beta coefficients, inverse coupling, and the perturbative regime classification.
2) Why does alpha_s get smaller at higher energy?
QCD shows asymptotic freedom. As the energy scale rises, quarks and gluons interact more weakly, so the strong coupling decreases according to the running equations.
3) What is Lambda QCD?
Lambda QCD is the characteristic scale where perturbative formulas begin to lose reliability. It sets the reference point for how alpha_s evolves with energy.
4) What is the difference between 1-loop and 2-loop results?
The 1-loop form uses only beta0. The 2-loop form adds beta1 and usually gives a more refined estimate, especially when precision matters across wider scales.
5) How should I choose n_f?
Use the number of active quark flavors relevant to the energy range. Typical perturbative studies often use values from 3 through 6, depending on thresholds.
6) Can I use this near very low energy scales?
Be careful near Lambda QCD. The perturbative expansion becomes unreliable there, and nonperturbative methods are usually more appropriate for accurate analysis.
7) Why can the calculator derive Lambda from alpha_s(Q0)?
If you know a coupling value at a trusted reference scale, the running equations can be inverted numerically to find the Lambda that reproduces that input.
8) Are heavy-quark threshold matching effects included?
No. This file uses a fixed n_f selected by the user. For higher-precision studies, threshold matching and scheme-specific refinements should be added separately.