Electroweak Theory & Symmetry Breaking
Electroweak Theory & Symmetry Breaking
Glashow, Salam, and Weinberg unified electromagnetism and the weak force into a single electroweak theory. Spontaneous symmetry breaking via the Higgs mechanism gives the W and Z bosons their masses while leaving the photon massless.
Definition
The electroweak gauge group \(SU(2)_L\times U(1)_Y\) has four gauge bosons. The Higgs doublet \(\phi\) acquires a VEV \(v \approx 246\) GeV, breaking the symmetry to \(U(1)_{em}\) and giving masses \(M_W = gv/2\), \(M_Z = M_W/\cos\theta_W\).
Key Result
Weak mixing angle \(\theta_W\): \(\sin^2\theta_W \approx 0.231\). The W mass prediction \(M_W = 80.4\) GeV and Z mass \(M_Z = 91.2\) GeV were confirmed before their discovery in 1983 at CERN.
Example 1
Cabibbo-Kobayashi-Maskawa (CKM) matrix describes quark mixing in weak decays. The small off-diagonal elements explain why \(b\to c\) transitions dominate over \(b\to u\) in B meson decays.
Example 2
CP violation: the CKM matrix contains a complex phase \(\delta\), which breaks the symmetry between matter and antimatter. This provides the mechanism for the small CP violation seen in kaon and B meson decays.
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Practice
- What is the Higgs mechanism in simple physical terms?
- Why do weak interactions violate parity (P) symmetry?
- What experimental evidence established neutrino oscillations and what does it imply about the Standard Model?
- Explain the hierarchy problem and proposed solutions.
Show Answer Key
1. The Higgs field is a scalar doublet with a 'Mexican hat' potential $V = -\mu^2|\phi|^2+\lambda|\phi|^4$. The minimum is at $|\phi|=v/\sqrt{2} \neq 0$. The field 'rolls' to a particular minimum, spontaneously breaking $SU(2)_L\times U(1)_Y \to U(1)_{EM}$. Three Goldstone bosons are 'eaten' by $W^\pm$ and $Z$, giving them mass. The remaining degree of freedom is the Higgs boson.
2. The weak force couples only to left-handed fermions (and right-handed antifermions). This was discovered in the Wu experiment (1957): $^{60}$Co beta decay showed preferential electron emission opposite to nuclear spin. The V−A structure of the weak interaction vertex $\gamma^\mu(1-\gamma^5)$ projects out left-handed components. Parity transformation swaps handedness, so weak interactions violate P maximally.
3. Super-Kamiokande (1998): atmospheric neutrino oscillations ($\nu_\mu \to \nu_\tau$). SNO (2001): solar neutrino oscillations ($\nu_e \to \nu_{\mu,\tau}$). Oscillation requires mass differences $\Delta m^2 \neq 0$, proving at least two neutrinos are massive. This is BSM physics: the minimal Standard Model has massless neutrinos. Implies either Dirac masses (right-handed $\nu$) or Majorana masses (lepton number violation).
4. The hierarchy problem: the Higgs mass ($\sim 125$ GeV) receives quadratically divergent quantum corrections from every heavy particle it couples to. Without fine-tuning or new physics, $m_H$ should be near the Planck scale ($10^{19}$ GeV). Proposed solutions: supersymmetry (SUSY, cancels divergences), extra dimensions (lowers the effective Planck scale), composite Higgs (Higgs is not fundamental), and the multiverse/anthropic principle.