Particle physics

Electroweak interactions and QED

QED is the U(1) gauge theory of electromagnetism; electroweak theory unifies it with the weak interaction through SU(2)ₗ×U(1)ᵧ.

In QED, electrons and positrons couple to the photon with charge e; the photon is massless. At high energies, neutral Z⁰ and charged W± currents are parts of the electroweak interaction.

\mathcal{L}_{\mathrm{QED}}=\bar\psi(i\gamma^\mu D_\mu-m)\psi-\frac14F_{\mu\nu}F^{\mu\nu},\quad D_\mu=\partial_\mu+ieA_\mu

Definition: Definition and physical meaning

The QED current uses the covariant derivative Dμ, enforcing invariance under local phase transformations; the fermion–photon vertex is proportional to e.

Adjust the parameters and drag the figure to explore the model; readouts are quantitative illustrations under the stated assumptions.

Structure and interpretation

In e⁺e⁻→μ⁺μ⁻, the benchmark s-channel exchanges a photon (and a Z at high energy). Neglecting masses and higher-order corrections, the QED cross section is σ=4πα²/(3s), with s the squared invariant energy.

Example: Quantitative example

At √s=10 GeV, using α≈1/137 in the Born formula gives σ≈4πα²/(3s). The result is in GeV⁻²; convert with 1 GeV⁻²≈0.389 mb.

Solution

σ≈0.000223 GeV⁻²≈8.68×10⁻⁵ mb≈0.0868 μb≈86.8 nb. This is the massless-lepton Born approximation, excluding the Z resonance, higher-order QED corrections, and detector effects.

In electroweak theory the gauge fields belong initially to $SU(2)_L\times U(1)_Y$ . Once the Higgs field acquires its vacuum value, combinations of the fields form the photon and neutral $Z$ boson, while two charged combinations form $W^+$ and $W^-$ . The Weinberg angle specifies the neutral-field mixing, linking electric charge and weak currents to the gauge-group structure.

Local gauge transformations change field descriptions without changing physical observables. The tensor $F_{\mu\nu}$ encodes gauge-invariant electric and magnetic fields; in non-Abelian QCD, the field tensor also contains gluon self-interactions. Thus the group structure is not mere notation: it determines the mediators, interaction rules, and how forces vary with energy scale. Measuring this scale dependence tests the quantum corrections implied by the theory. The same framework unifies electroweak interactions at high energies.

Quick check

What is the dominant exchanged particle in low-energy e⁺e⁻→μ⁺μ⁻?

In the Born approximation, how does σ scale with s?

References

Michael E. Peskin and Daniel V. Schroeder (1995). An Introduction to Quantum Field Theory
Steven Weinberg (1995). The Quantum Theory of Fields, Volume I