The calculator integrates the renormalization group equation for a dimensionless coupling g (or e, λ):
d g / d ln(μ) = β(g)
Here μ is the renormalization scale, and β(g) is the beta function. One-loop models use standard leading-order coefficients. The generic option uses a polynomial expansion:
β(g) = b0·g^3 + b1·g^5
Numerics run over t = ln(μ/μ0) using Euler or RK4.
- Pick a beta-function model that matches your theory.
- Enter μ0, μ1, and the initial coupling at μ0.
- Increase steps for stiff or rapidly changing flows.
- Press Calculate Flow to view the table and summary.
- Use the export buttons to save CSV or PDF.
| Model | μ0 | μ1 | Initial coupling | Typical behavior |
|---|---|---|---|---|
| QCD (one-loop) | 91.1876 | 1000 | g = 1.20 | Decreases with μ (asymptotic freedom) |
| QED (one-loop) | 1 | 1000 | e = 0.30 | Grows slowly with μ (Landau trend) |
| Generic polynomial | 10 | 0.1 | g = 0.80 | Can approach fixed points or diverge |
1) Why scale dependence matters
Renormalization group flow tells you how an effective coupling changes as the reference energy scale μ is varied. It connects measurements at μ0 to predictions at μ1, clarifies which degrees of freedom are relevant, and explains why the same theory can look weakly or strongly coupled depending on scale.
2) Logarithmic running variable
The differential equation is written in terms of t = ln(μ/μ0). Equal steps in t correspond to multiplicative changes in μ, so one decade is t = ln 10. This calculator integrates in t, reports ln(μ/μ0) in the table, and reconstructs μ = μ0 e^t so the scale mapping stays transparent.
3) Model choices and coefficients
You can select one-loop QCD, one-loop QED, scalar ϕ⁴ with O(N) symmetry, or a generic polynomial beta function. The one-loop forms encode leading quantum corrections through compact coefficients, while the generic option β(g)=b0·g^3+b1·g^5 is useful for toy models, fitted flows, and sign-change studies.
4) Reading the beta function
The sign of β indicates whether the coupling increases or decreases when μ increases. Zeros of β define fixed points, and the local slope β′ at a fixed point controls stability. Near g*, linearization gives dg/dt ≈ β′(g*)·(g−g*), which is the basis for extracting critical exponents in many systems.
5) One-loop QCD behavior
For SU(Nc) with Nf flavors, the calculator uses b0=(11Nc−2Nf)/3 and β(g)=−b0·g^3/(16π^2). When b0>0, g decreases with increasing μ, capturing asymptotic freedom. With Nc=3 and Nf=5, you should see a gentle UV decrease across typical high-energy ranges.
6) One-loop QED trend
In QED, b0=(4/3)Nf and β(e)=+b0·e^3/(16π^2), so the coupling grows with μ. At small e this growth is slow, but extrapolations can hint at a Landau-pole scale where perturbation theory fails. Increasing steps and preferring RK4 helps when the running steepens.
7) Scalar ϕ⁴ and running strength
For O(N) ϕ⁴ theory, β(λ)=((N+8)/(16π^2))·λ^2 is positive for λ>0, so λ typically increases toward higher μ. This reflects the familiar triviality tendency in four dimensions. Changing N rescales the coefficient, letting you compare how quickly λ runs for different field multiplicities.
8) Numerical settings and reporting
Accuracy depends on the step size Δt=ln(μ1/μ0)/steps. RK4 is usually more stable than Euler for stiff flows or large couplings. The on-screen table is downsampled for readability, while CSV and PDF exports preserve metadata and the full step-by-step trajectory for documentation and cross-checking. Use exports to support plots, reviews, and reproducibility.
1) What does this calculator compute?
It numerically integrates dg/dln(μ)=β(g) from μ0 to μ1 and reports the coupling and beta value along the path, including an exportable flow table.
2) Which μ units should I use?
Any units work as long as μ0 and μ1 share the same units. The flow depends on the ratio μ1/μ0 through ln(μ1/μ0).
3) How many steps are enough?
Use more steps when the coupling changes rapidly, when β is large, or when you explore wide scale ratios. A few hundred steps often works; increase toward thousands for stiff flows.
4) Why choose RK4 over Euler?
RK4 reduces numerical error per step and is more stable when β(g) varies quickly. Euler can be adequate for gentle running but may drift or overshoot for steeper trajectories.
5) What does the early-stop warning mean?
It means the integration encountered a non-finite value or the coupling exceeded a stability limit. This can indicate a Landau-type divergence, too-large step size, or an input outside the perturbative regime.
6) Can I study fixed points with it?
Yes. Choose a model that can produce β(g)=0, then scan initial couplings and scale ranges. A fixed point appears where β approaches zero and g stabilizes as t increases or decreases.
7) Does it include two-loop running?
Not by default. For higher-order studies, you can approximate by using the generic polynomial coefficients to represent an effective beta function over the coupling range of interest.