Quantum Field Theory Course

Total: 440+ pages covering classical field theory through the Standard Model

Part VI, Chapter 6

Running Coupling Constants

How interaction strengths change with energy

6.1 The Concept of Running

One of the most important insights of QFT: coupling constants are not constant!They depend on the energy scale at which we measure them. This "running" is a quantum mechanical effect arising from virtual particle loops.

The Running Coupling:

We write α(Q²) or g(μ) to emphasize that the coupling depends on the momentum scale Q or renormalization scale μ. The RG equation tells us how it evolves:

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

💡What does running mean physically?

Imagine probing an electron with a photon of wavelength λ ~ 1/Q. At low Q (long wavelength), you see the electron plus its cloud of virtual photons and e⁺e⁻ pairs. This cloud screensthe charge.

At high Q (short wavelength), you penetrate deeper into the cloud, getting closer to the "bare" electron. You measure a larger effective charge. This is why α increases with Q!

6.2 Running of α in QED

The QED beta function to one loop is:

$$\beta(\alpha) = \frac{2\alpha^2}{3\pi} + O(\alpha^3)$$

Solving the RG equation Q dα/dQ = β(α):

$$\frac{d\alpha}{dQ} = \frac{2\alpha^2}{3\pi Q}$$

Separating variables and integrating:

$$\int_{\alpha(\mu)}^{\alpha(Q)} \frac{d\alpha'}{\alpha'^2} = \int_\mu^Q \frac{2}{3\pi Q'} dQ'$$

This gives:

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

Numerical Values of α(Q²):

Q² = 0 (Thomson limit): α(0) ≈ 1/137.036

Q² = m_e²: α(m_e²) ≈ 1/137.036

Q² = (1 GeV)²: α ≈ 1/134

Q² = M_Z² (91 GeV)²: α(M_Z²) ≈ 1/127.9

Q² = (1 TeV)²: α ≈ 1/127

The coupling increases by ~8% from low energies to the Z boson mass!

The Landau Pole Problem

Notice that α(Q²) has a pole (divergence) when the denominator vanishes:

$$1 - \frac{\alpha(\mu^2)}{3\pi}\ln\frac{Q^2}{\mu^2} = 0 \quad \Rightarrow \quad Q_{\text{Landau}}^2 = \mu^2 e^{3\pi/\alpha(\mu^2)}$$

Taking μ = m_e and α(m_e²) = 1/137:

$$Q_{\text{Landau}} \sim m_e e^{3\pi \cdot 137} \sim 10^{286}\text{ GeV}$$

⚠️ Interpretation of the Landau Pole:

At the Landau pole, perturbation theory breaks down (α → ∞). This doesn't necessarily mean QED is inconsistent - possible resolutions:

  • New physics appears before Q_Landau (e.g., grand unification)
  • Higher-order corrections stabilize the theory
  • QED is only an effective theory, valid below some cutoff

Since Q_Landau is astronomically large, this is not a practical concern!

6.3 Running of α_s in QCD

For QCD with N_c = 3 colors and n_f quark flavors, the one-loop beta function is:

$$\beta(\alpha_s) = -\frac{\alpha_s^2}{2\pi}\left(\frac{11N_c - 2n_f}{3}\right) = -\frac{\alpha_s^2}{\pi}\beta_0$$

where β₀ = (11·3 - 2n_f)/6 = (33 - 2n_f)/6. For n_f = 6 (u,d,s,c,b,t):

$$\beta_0 = \frac{33 - 12}{6} = \frac{21}{6} = 3.5$$

The crucial difference: β < 0 for QCD! Solving the RG equation:

$$\alpha_s(Q^2) = \frac{\alpha_s(\mu^2)}{1 + \frac{\alpha_s(\mu^2)}{2\pi}\beta_0\ln\frac{Q^2}{\mu^2}}$$

🌟 Asymptotic Freedom in Action

Since β₀ > 0, the strong coupling decreases at high energy:

Q = 1 GeV: α_s ≈ 0.5 (strong coupling, confinement)

Q = 5 GeV (b-quark mass): α_s ≈ 0.22

Q = M_Z = 91 GeV: α_s(M_Z) = 0.1179 ± 0.0010 (PDG 2022)

Q = 1 TeV (LHC): α_s ≈ 0.09

Q → ∞: α_s → 0 (asymptotically free!)

This explains why quarks appear "free" in high-energy collisions (jet production) but are confined in hadrons at low energies!

The QCD Scale Λ_QCD

We can rewrite the running in terms of a fundamental QCD scale Λ_QCD:

$$\alpha_s(Q^2) = \frac{2\pi}{\beta_0\ln(Q^2/\Lambda_{\text{QCD}}^2)}$$

Λ_QCD ≈ 200-300 MeV is the scale where α_s ~ 1 and perturbation theory breaks down. Below this scale, QCD exhibits confinement - quarks and gluons are confined within hadrons.

💡What is Λ_QCD physically?

Λ_QCD ~ 200 MeV sets the scale for:

  • Proton/neutron mass: m_p ≈ 938 MeV ~ few × Λ_QCD
  • Confinement: can't separate quarks beyond ~ 1/Λ_QCD ~ 1 fm
  • Hadronization: quarks form hadrons at scale ~ Λ_QCD

Below Λ_QCD, the effective degrees of freedom are hadrons (π, p, n), not quarks and gluons!

6.4 Two-Loop and Higher-Order Running

For precision, we need higher-loop corrections. The two-loop beta function is:

$$\beta(\alpha_s) = -\alpha_s^2\left(\frac{\beta_0}{2\pi} + \frac{\beta_1}{8\pi^2}\alpha_s + \cdots\right)$$

where for QCD:

\begin{align*} \beta_0 &= \frac{11N_c - 2n_f}{3} \\ \beta_1 &= \frac{34N_c^2 - 10N_c n_f - 6C_F n_f}{3} \end{align*}

For N_c = 3 and C_F = 4/3:

$$\beta_1 = \frac{306 - 30n_f - 8n_f}{3} = 102 - \frac{38n_f}{3}$$

The PDG uses four-loop running for precision determinations of α_s(M_Z)!

6.5 Grand Unification and Coupling Unification

A spectacular prediction: if we run the three Standard Model couplings (α, α_s, α_w) to high energies, they nearly meet at M_GUT ~ 10¹⁶ GeV!

Coupling Unification:

In the Standard Model, the three couplings almost unify:

  • α₁⁻¹(M_Z) ≈ 59 (U(1) hypercharge, normalized)
  • α₂⁻¹(M_Z) ≈ 30 (SU(2) weak)
  • α₃⁻¹(M_Z) ≈ 8.5 (SU(3) strong)

They converge around M_GUT ~ 10¹⁶ GeV, but don't quite meet exactly. In supersymmetricextensions (MSSM), unification is more precise!

This suggests a Grand Unified Theory (GUT) where SU(3) × SU(2) × U(1) → SU(5) or SO(10).

6.6 Threshold Corrections and Decoupling

When running through a heavy particle mass threshold, the beta function changes because that particle decouples from the theory:

Example: Running α_s Through Quark Thresholds

At μ = m_b (b-quark mass), n_f changes from 5 to 4:

\begin{align*} \mu < m_b: \quad &\beta_0 = \frac{33 - 2 \cdot 4}{6} = 4.17 \\ \mu > m_b: \quad &\beta_0 = \frac{33 - 2 \cdot 5}{6} = 3.83 \end{align*}

The running becomes slightly slower above m_b because there are more active flavors!

6.7 Experimental Determination of α_s

α_s is measured from many processes:

1. Lattice QCD

Non-perturbative simulations. α_s(M_Z) = 0.1180 ± 0.0008

2. Hadronic τ Decays

Ratio of hadronic to leptonic τ decay rate. α_s(M_Z) = 0.1179 ± 0.0010

3. Deep Inelastic Scattering

Scaling violations in structure functions. α_s(M_Z) = 0.1177 ± 0.0012

4. Jet Rates at LEP/LHC

3-jet/2-jet ratios in e⁺e⁻ → hadrons. α_s(M_Z) = 0.1179 ± 0.0015

PDG 2022 World Average:

$$\alpha_s(M_Z) = 0.1179 \pm 0.0010$$

This is one of the most precisely measured quantities in the Standard Model! The running of α_s is verified over more than two orders of magnitude in energy.

🎯 Key Takeaways

  • Coupling constants "run" with energy scale due to quantum corrections
  • QED: α increases with Q (vacuum polarization screening)
  • α(M_Z) ≈ 1/128 vs α(0) ≈ 1/137 (measured effect!)
  • Landau pole: α → ∞ at Q ~ 10²⁸⁶ GeV (not a practical problem)
  • QCD: α_s decreases with Q (asymptotic freedom!)
  • α_s(M_Z) = 0.1179 ± 0.0010 (PDG world average)
  • Λ_QCD ~ 200 MeV: scale where confinement sets in
  • Gauge couplings nearly unify at M_GUT ~ 10¹⁶ GeV
  • Threshold corrections when passing heavy quark masses
  • Next: Effective field theory - how heavy physics decouples!
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