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Part III: Electroweak Theory

The Unified Electroweak Interaction

The electroweak theory unifies electromagnetic and weak interactions into a single framework. Developed by Glashow, Weinberg, and Salam (Nobel Prize 1979), it is based on the gauge group $SU(2)_L \times U(1)_Y$.

Before electroweak symmetry breaking, the theory describes massless gauge bosons. After spontaneous symmetry breaking via the Higgs mechanism, we observe the massive $W^\pm$ and $Z^0$ bosons and the massless photon $\gamma$.

Gauge Group and Quantum Numbers

SU(2)×U(1) Structure

The electroweak theory is based on two gauge groups:

$SU(2)_L$ (Weak Isospin):

  • Acts only on left-handed fermions
  • 3 generators: $T^a = \frac{1}{2}\tau^a$ (Pauli matrices)
  • 3 gauge bosons: $W^1, W^2, W^3$
  • Weak isospin: $T = 0, \frac{1}{2}$

$U(1)_Y$ (Hypercharge):

  • Acts on all fermions
  • 1 generator: weak hypercharge $Y$
  • 1 gauge boson: $B$
  • Relation: $Q = T_3 + \frac{Y}{2}$

Gell-Mann–Nishijima Formula:

$$ Q = T_3 + \frac{Y}{2} $$

where $Q$ is electric charge, $T_3$ is third component of weak isospin, $Y$ is weak hypercharge

Fermion Representations

Left-handed fermions form $SU(2)_L$ doublets, while right-handed fermions are singlets:

Leptons:

$$ L_L = \begin{pmatrix} \nu_e \\ e^- \end{pmatrix}_L, \quad e_R $$

$T = \frac{1}{2}, Y = -1$ for doublet; $T = 0, Y = -2$ for singlet

Quarks:

$$ Q_L = \begin{pmatrix} u \\ d' \end{pmatrix}_L, \quad u_R, \quad d_R $$

$T = \frac{1}{2}, Y = \frac{1}{3}$ for doublet; $T = 0, Y = \frac{4}{3}$ for $u_R$; $T = 0, Y = -\frac{2}{3}$ for $d_R$

Note: $d'$ is the weak eigenstate (CKM-rotated from mass eigenstate)

Electroweak Gauge Bosons

Before Symmetry Breaking

The unbroken $SU(2)_L \times U(1)_Y$ theory has 4 massless gauge bosons:

  • $W^1, W^2, W^3$ from $SU(2)_L$
  • $B$ from $U(1)_Y$

Covariant derivative:

$$ D_\mu = \partial_\mu - ig\frac{\tau^a}{2}W^a_\mu - ig'\frac{Y}{2}B_\mu $$

After Symmetry Breaking

The gauge bosons mix through electroweak symmetry breaking, producing physical states:

Charged Weak Bosons:

$$ W^\pm = \frac{1}{\sqrt{2}}(W^1 \mp iW^2) $$

Mass: $m_W = 80.4$ GeV

Neutral Bosons (Mixing):

$$ \begin{pmatrix} Z^0 \\ \gamma \end{pmatrix} = \begin{pmatrix} \cos\theta_W & -\sin\theta_W \\ \sin\theta_W & \cos\theta_W \end{pmatrix} \begin{pmatrix} W^3 \\ B \end{pmatrix} $$

Weinberg angle: $\sin^2\theta_W \approx 0.231$

  • Z boson: $m_Z = 91.2$ GeV (massive)
  • Photon: $m_\gamma = 0$ (massless)

Weak Interactions

Charged Current Interactions

$W^\pm$ bosons mediate charged current (CC) weak interactions, changing fermion flavor:

$$ \mathcal{L}_{\text{CC}} = -\frac{g}{\sqrt{2}}[W^+_\mu J^\mu_+ + W^-_\mu J^\mu_-] $$$$ J^\mu_+ = \bar{u}_L \gamma^\mu d_L + \bar{\nu}_L \gamma^\mu e_L $$

Key features:

  • V-A structure: Only left-handed fermions participate (maximal parity violation)
  • Flavor changing: $d \leftrightarrow u, \; e \leftrightarrow \nu_e$, etc.
  • Coupling strength: $g = e/\sin\theta_W \approx 0.65$

Neutral Current Interactions

The $Z^0$ boson mediates neutral current (NC) interactions without changing flavor:

$$ \mathcal{L}_{\text{NC}} = -\frac{g}{\cos\theta_W}Z_\mu J^\mu_Z $$$$ J^\mu_Z = \sum_f \bar{f}\gamma^\mu(g_V^f - g_A^f\gamma^5)f $$

Vector and axial couplings:

$$ g_V^f = T_3^f - 2Q_f\sin^2\theta_W, \quad g_A^f = T_3^f $$

Neutrinos ($Q=0, T_3=+\frac{1}{2}$):

$g_V^\nu = +\frac{1}{2}$, $g_A^\nu = +\frac{1}{2}$

Pure left-handed coupling

Electrons ($Q=-1, T_3=-\frac{1}{2}$):

$g_V^e \approx -0.04$, $g_A^e = -\frac{1}{2}$

Nearly pure axial coupling

Electroweak Unification

Coupling Constant Relations

The electromagnetic and weak couplings are related through the Weinberg angle:

$$ e = g\sin\theta_W = g'\cos\theta_W $$$$ \tan\theta_W = \frac{g'}{g} $$

Mass relations:

$$ m_W = \frac{m_Z}{\cos\theta_W} = \frac{g v}{2}, \quad m_Z = \frac{v}{2}\sqrt{g^2 + g'^2} $$

where $v \approx 246$ GeV is the Higgs vacuum expectation value.

Electroweak Precision Tests

The electroweak theory has been tested to extraordinary precision at $e^+e^-$ colliders (LEP, SLC):

Z Mass and Width:

  • $m_Z = 91.1876 \pm 0.0021$ GeV
  • $\Gamma_Z = 2.4952 \pm 0.0023$ GeV
  • Agreement at 0.001% level

Weinberg Angle:

  • $\sin^2\theta_W^{\text{eff}} = 0.23153 \pm 0.00016$
  • Measured via asymmetries
  • Tests radiative corrections

W Mass:

  • $m_W = 80.377 \pm 0.012$ GeV
  • Sensitive to Higgs and top masses
  • Tests loop corrections

Number of Neutrinos:

  • $N_\nu = 2.9840 \pm 0.0082$
  • From $Z \to \text{invisible}$ width
  • Confirms exactly 3 light neutrinos

Parity Violation in Weak Interactions

The V-A Structure

Weak interactions exhibit maximal parity violation through the V-A (vector minus axial) structure:

$$ \mathcal{L}_{\text{weak}} \propto \bar{\psi}\gamma^\mu(1 - \gamma^5)\psi = \bar{\psi}_L\gamma^\mu\psi_L $$

Projection operator: $P_L = \frac{1 - \gamma^5}{2}$ selects left-handed states

This means only left-handed fermions (and right-handed antifermions) participate in charged current weak interactions.

Experimental Discoveries

Wu Experiment (1956):

$^{60}\text{Co} \to ^{60}\text{Ni} + e^- + \bar{\nu}_e$

Electrons preferentially emitted opposite to nuclear spin

First direct evidence of parity violation

Helicity of Neutrinos:

Goldhaber experiment (1958)

Measured neutrino helicity: $h = -1$

Confirmed neutrinos are left-handed

Discovery of W and Z Bosons

Nobel Prize 1984: Rubbia and van der Meer

The $W^\pm$ and $Z^0$ bosons were discovered in 1983 at CERN's Super Proton Synchrotron (SPS) using the UA1 and UA2 experiments:

W Boson Discovery:

  • Process: $p\bar{p} \to W^\pm + X \to e^\pm\nu + X$
  • Signature: High-$p_T$ electron + missing $E_T$
  • Mass: $m_W = 81 \pm 5$ GeV (1983)
  • Confirmed theory prediction $\sim 80$ GeV

Z Boson Discovery:

  • Process: $p\bar{p} \to Z^0 + X \to e^+e^- + X$
  • Signature: Two high-$p_T$ electrons
  • Mass: $m_Z = 93 \pm 3$ GeV (1983)
  • Confirmed theory prediction $\sim 91$ GeV

Modern Measurements at LHC

At the LHC, $W$ and $Z$ bosons are produced copiously:

Production Cross Sections at $\sqrt{s} = 13$ TeV:

  • $\sigma(pp \to W^+ + X) \approx 11$ nb (11,000 pb)
  • $\sigma(pp \to W^- + X) \approx 8$ nb
  • $\sigma(pp \to Z^0 + X) \approx 2$ nb

These bosons serve as "standard candles" for calibrating detectors and measuring parton distribution functions.

From Fermi Theory to Electroweak Theory

Fermi's Four-Fermion Interaction

Historically, weak interactions were described by Fermi's effective four-fermion coupling (1934):

$$ \mathcal{L}_{\text{Fermi}} = -\frac{G_F}{\sqrt{2}}[\bar{\nu}_e\gamma^\mu(1-\gamma^5)e][\bar{p}\gamma_\mu(1-\gamma^5)n] $$

Fermi constant: $G_F = 1.166 \times 10^{-5}$ GeV$^{-2}$

This is a contact interaction (zero range), valid only at low energies $E \ll m_W$.

Connection to W Boson Exchange

At low energies, $W$ boson exchange reduces to Fermi theory:

W propagator in low-energy limit ($q^2 \ll m_W^2$):

$$ \frac{-g_{\mu\nu}}{q^2 - m_W^2} \approx \frac{g_{\mu\nu}}{m_W^2} $$

Identifies Fermi constant:

$$ \frac{G_F}{\sqrt{2}} = \frac{g^2}{8m_W^2} $$

This relationship allows us to predict $m_W \approx 80$ GeV from the measured value of $G_F$.

Key Electroweak Observables

Z Pole Measurements

The LEP experiments measured the $Z$ boson properties with unprecedented precision:

Peak Cross Section:

$\sigma^0_{\text{had}} = 41.541 \pm 0.037$ nb

Hadronic Width:

$\Gamma_{\text{had}} = 1744.4 \pm 2.0$ MeV

Leptonic Width:

$\Gamma_{\ell\ell} = 83.984 \pm 0.086$ MeV

Forward-Backward Asymmetries:

  • $A_{\text{FB}}^{0,e} = 0.0145 \pm 0.0025$ (electron asymmetry)
  • $A_{\text{FB}}^{0,\mu} = 0.0169 \pm 0.0013$ (muon asymmetry)
  • $A_{\text{FB}}^{0,b} = 0.0992 \pm 0.0016$ (b-quark asymmetry)

Key Takeaways

  • •Electroweak theory unifies electromagnetic and weak interactions via $SU(2)_L \times U(1)_Y$
  • •Left-handed fermions are $SU(2)_L$ doublets; right-handed are singlets
  • •Weak bosons: $W^\pm$ (80.4 GeV) mediate charged currents; $Z^0$ (91.2 GeV) mediates neutral currents
  • •Weak interactions exhibit maximal parity violation (V-A structure)
  • •Weinberg angle $\sin^2\theta_W \approx 0.231$ relates $g$, $g'$, and $e$
  • •$W$ and $Z$ discovered in 1983 at CERN, confirming electroweak unification
  • •LEP precision measurements test theory at 0.1% level, constraining new physics
  • •Number of neutrino generations: $N_\nu = 3$ (from $Z$ width)
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