Running Coupling Constants

Running Coupling Constants

One of the most profound results of Quantum Field Theory is that coupling constants are not truly constant — they run with the energy scale $Q$ at which they are probed. This running arises from quantum corrections (loop diagrams) that dress the bare interaction vertex.

QED: α Increases with Energy

In Quantum Electrodynamics the fine-structure constant $\alpha$ grows logarithmically with energy due to vacuum polarisation (electron–positron virtual pairs screen the bare charge):

$\alpha(Q) = \frac{\alpha_0}{1 - \dfrac{\alpha_0}{3\pi} \ln\!\left(\dfrac{Q^2}{m_e^2}\right)}$

where $\alpha_0 = 1/137.036$ is the low-energy value and $m_e = 5.11 \times 10^{-4}$ GeV. At the Z mass, $\alpha(m_Z) \approx 1/134$.

QCD: αs Decreases with Energy (Asymptotic Freedom)

The strong coupling constant $\alpha_s$ does the opposite: it decreases at high energy. This is asymptotic freedom, discovered by Gross, Politzer, and Wilczek (Nobel Prize 2004). At one loop:

$\alpha_s(Q) = \frac{12\pi}{(33 - 2n_f) \ln(Q^2 / \Lambda_{\text{QCD}}^2)}$

where $n_f$ is the number of active quark flavours (typically 5 above the bottom threshold), and $\Lambda_{\text{QCD}} \approx 0.217$ GeV is the QCD scale where the coupling diverges. The coefficient $\beta_0 = 11 - 2n_f/3 = (33 - 2n_f)/6$ governs the running speed.

Your Task

Implement:

• alpha_QED_running(Q_GeV) — running QED coupling at scale $Q$
• alpha_s_running(Q_GeV, n_f=5, Lambda_QCD=0.217) — one-loop QCD running coupling
• coupling_ratio_QCD_QED(Q_GeV) — ratio $\alpha_s / \alpha$ at scale $Q$ (implement both inline)

All constants must be defined inside each function body.