Skip to content
Wyatt's Notes — University

Running Coupling Constants

6.1 Asymptotic Freedom and Confinement

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

αs(μ)=αs(μ0)1+αs(μ0)12π(33−2nf)ln⁡(μ2/μ02)\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 nfn_f is the number of active quark flavours.

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

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

α(μ)=α(μ0)1−α(μ0)3πln⁡(μ2/μ02)\alpha(\mu) = \frac{\alpha(\mu_0)}{1 - \frac{\alpha(\mu_0)}{3\pi}\ln(\mu^2/\mu_0^2)}

6.2 Beta Functions

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

β(g)≡μdgdμ\beta(g) \equiv \mu\frac{dg}{d\mu}

At one-loop order:

QED:

βQED(e)=e312π2⇒βQED(α)=2α23π>0\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:

βQCD(gs)=−gs316π2(113CA−43TFnf)\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), CA=N=3C_A = N = 3 and TF=1/2T_F = 1/2:

βQCD(gs)=−gs316π2(11−2nf3)\beta_{\mathrm{QCD}(g_s) = -\frac{g_s^3}{16\pi^2}\left(11 - \frac{2n_f}{3}\right)}

The negative sign (for nf<33/2n_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 CAC_A term) dominates over fermion screening (the nfn_f term) for the physically relevant number of flavours. Gluons carry colour charge and Therefore antiscreen, leading to the coupling decreasing at short distances.

6.3 Grand Unification and the Unification Scale

If the three gauge couplings are extrapolated to very high energies, they approximately meet at μ∼1015\mu \sim 10^{15} GeV (in the Minimal Supersymmetric Standard Model), suggesting unification into a Simple group such as SU(5) or SO(10).

Example 6.1: Estimating the unification scale

At one loop, the coupling at scale μ\mu is:

αi−1(μ)=αi−1(μ0)−bi2πln⁡(μμ0)\alpha_i^{-1}(\mu) = \alpha_i^{-1}(\mu_0) - \frac{b_i}{2\pi}\ln\left(\frac{\mu}{\mu_0}\right)

Where bib_i are the one-loop beta function coefficients and μ0=mZ≈91.2\mu_0 = m_Z \approx 91.2 GeV.

For the SM, the coefficients are:

  • b1=−41/(10π)b_1 = -41/(10\pi) for U(1)Y\mathrm{U}(1)_Y (properly normalised)
  • b2=−19/(6π)b_2 = -19/(6\pi) for SU(2)L\mathrm{SU}(2)_L
  • b3=−7/πb_3 = -7/\pi for SU(3)C\mathrm{SU}(3)_C

At the unification scale MGUTM_{\mathrm{GUT}}All three couplings are equal: α1−1(MGUT)=α2−1(MGUT)=α3−1(MGUT)\alpha_1^{-1}(M_{\mathrm{GUT}) = \alpha_2^{-1}(M_{\mathrm{GUT}) = \alpha_3^{-1}(M_{\mathrm{GUT})}}}.

Setting α1−1=α2−1\alpha_1^{-1} = \alpha_2^{-1}:

\alpha_1^{-1}(m_Z) - \alpha_2^{-1}(m_Z) = \frac{b_2 - b_1}{2\pi}\ln\left(\frac{M_{\mathrm{GUT}}{m_Z}\right)}

With α1−1(mZ)≈59.0\alpha_1^{-1}(m_Z) \approx 59.0, α2−1(mZ)≈29.6\alpha_2^{-1}(m_Z) \approx 29.6:

59.0 - 29.6 = \frac{b_2 - b_1}{2\pi}\ln\left(\frac{M_{\mathrm{GUT}}{m_Z}\right)}

This gives MGUT∼1013M_{\mathrm{GUT} \sim 10^{13}}—101610^{16} GeV depending on the precise Coefficients and the inclusion of threshold corrections. In the MSSM, the modified beta Coefficients give a much cleaner unification at MGUT∼2×1016M_{\mathrm{GUT} \sim 2 \times 10^{16}} GeV.

:::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.

:::