Fine Structure Constant and QED

Fine Structure Constant and QED

The fine structure constant $\alpha$ is the dimensionless coupling constant of quantum electrodynamics (QED). It sets the strength of all electromagnetic interactions:

$\alpha = \frac{e^2}{4\pi\varepsilon_0 \hbar c} \approx \frac{1}{137.036} \approx 7.297 \times 10^{-3}$

Its smallness ($\alpha \ll 1$) is why perturbation theory works so well in QED — higher-order Feynman diagrams are suppressed by additional powers of $\alpha$.

Running Coupling

In quantum field theory, coupling constants run with the energy scale $Q$. At one loop in QED:

$\alpha(Q^2) \approx \frac{\alpha(0)}{1 - \dfrac{\alpha(0)}{3\pi} \ln\!\left(\dfrac{Q^2}{m_e^2}\right)}$

where $m_e = 0.000511$ GeV is the electron mass. The coupling increases at higher energies because virtual electron-positron pairs screen the bare charge less effectively at short distances.

At the $Z$ pole ($Q = M_Z \approx 91.2$ GeV):

$\alpha(M_Z^2) \approx \frac{1}{128.9} \approx 0.00776$

compared to $\alpha(0) \approx 1/137.036$ at zero energy.

Cross Section Scaling

QED amplitudes are proportional to $\alpha$, so cross sections scale as $\alpha^2$. The ratio of cross sections at two different energies is:

$\frac{\sigma(Q_1)}{\sigma(Q_2)} = \left(\frac{\alpha(Q_1)}{\alpha(Q_2)}\right)^2$

Your Task

Implement three functions. All constants must be defined inside each function body.

• alpha_QED() — returns $\alpha$ at $Q = 0$
• alpha_running(Q_GeV) — one-loop running coupling
• qed_cross_section_ratio(Q1_GeV, Q2_GeV) — ratio $(\alpha(Q_1)/\alpha(Q_2))^2$