Running Coupling Constants

The strong coupling $\alpha_s$ depends on the energy scale $\mu$:

$\alpha_s(\mu) = \frac{\alpha_s(\mu_0)}{1 + \frac{\alpha_s(\mu_0)}{12\pi}(33 - 2n_f)\ln(\mu^2/\mu_0^2)}$

Where $n_f$ is the number of active quark flavours.

• Asymptotic freedom: For $n_f \lt 17$, $\alpha_s$ decreases at high energies. Quarks behave as nearly free particles at short distances.
• Confinement: At low energies, $\alpha_s$ becomes large, and perturbation theory breaks down. Quarks are confined into hadrons.

The electromagnetic coupling also runs (but increases at high energies):

$\alpha(\mu) = \frac{\alpha(\mu_0)}{1 - \frac{\alpha(\mu_0)}{3\pi}\ln(\mu^2/\mu_0^2)}$

The running of coupling constants is governed by the beta function:

$\beta(g) \equiv \mu\frac{dg}{d\mu}$

At one-loop order:

QED:

$\beta_{\mathrm{QED}(e) = \frac{e^3}{12\pi^2} \quad \Rightarrow \quad \beta_{\mathrm{QED}(\alpha) = \frac{2\alpha^2}{3\pi} \gt 0}}$

The positive beta function means the electromagnetic coupling increases with energy (antiscreening).

QCD:

$\beta_{\mathrm{QCD}(g_s) = -\frac{g_s^3}{16\pi^2}\left(\frac{11}{3}C_A - \frac{4}{3}T_F n_f\right)}$

For SU(3), $C_A = N = 3$ and $T_F = 1/2$:

$\beta_{\mathrm{QCD}(g_s) = -\frac{g_s^3}{16\pi^2}\left(11 - \frac{2n_f}{3}\right)}$

The negative sign (for $n_f \lt 33/2$) means the strong coupling decreases with energy: This is asymptotic freedom (Gross, Wilczek, and Politzer, Nobel Prize 2004).

Interpretation. The gluon self-interaction (the $C_A$ term) dominates over fermion screening (the $n_f$ term) for the physically relevant number of flavours. Gluons carry colour charge and Therefore antiscreen, leading to the coupling decreasing at short distances.

If the three gauge couplings are extrapolated to very high energies, they approximately meet at $\mu \sim 10^{15}$ GeV (in the Minimal Supersymmetric Standard Model), suggesting unification into a Simple group such as SU(5) or SO(10).

:::caution Common Pitfall The Standard Model couplings do not meet at a single point when extrapolated using SM beta Functions. The three lines form a rough triangle. It is only in supersymmetric extensions (MSSM) That the additional superpartner contributions to the beta functions bring the three couplings to Near-convergence. This convergence is often cited as indirect evidence for supersymmetry.

:::

• The beta function encodes how a coupling changes with energy scale.
• A negative beta function means asymptotic freedom; a positive one means screening.
• The running of $\alpha_s$ is measured precisely at $e^+e^-$ colliders and deep inelastic scattering.
• The Landau pole in QED marks the scale where perturbation theory breaks down.

• Confusing the sign convention: $\beta(g) > 0$ means the coupling increases with energy.
• Assuming that asymptotic freedom implies confinement automatically. They are related but distinct phenomena.
• Forgetting that the number of active flavours $n_f$ depends on the energy scale relative to quark masses.
• Using the one-loop formula far beyond its validity range where higher-order corrections are significant.

• Collider physics: Precision measurements of $\alpha_s$ at the Z pole and LHC constrain the Standard Model.
• Lattice QCD: Numerical simulations compute $\alpha_s$ non-perturbatively from first principles.
• Cosmology: The running of couplings affects primordial nucleosynthesis and baryogenesis models.
• Dark matter searches: The scale dependence of $\alpha_s$ influences the calculation of hadronic backgrounds.

Problem. Given $\alpha_s(m_Z) = 0.118$ at $\mu_0 = m_Z = 91.2$ GeV, estimate $\alpha_s$ at $\mu = 1$ TeV.