Electroweak Unification: How Electromagnetism Emerges

The Puzzle

Electromagnetism and the weak force seem completely different:

Yet they are two aspects of a single electroweak force, unified at high energies and separated by the Higgs mechanism at low energies.

The Electroweak Gauge Group

The Standard Model electroweak sector has gauge group:

SU(2)_L \times U(1)_Y

This is not electromagnetism! The subscripts mean:

• L = “left” — only left-handed fermions feel SU(2)
• Y = “hypercharge” — a different U(1) than electromagnetism

The Gauge Bosons (Before Symmetry Breaking)

All four bosons are massless at this stage.

What is Hypercharge?

Hypercharge Y is a quantum number assigned to each particle. It is not electric charge, but is related to it.

The Gell-Mann–Nishijima formula connects them:

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

Where:

• Q = electric charge
• T_3 = third component of weak isospin (from SU(2))
• Y = hypercharge

Hypercharge Assignments

Notice: Left-handed fermions come in doublets under SU(2), while right-handed fermions are singlets.

The Higgs Mechanism

The Higgs field \phi is an SU(2) doublet with hypercharge Y = +1:

\phi = \begin{pmatrix} \phi^+ \ \phi^0 \end{pmatrix}

Spontaneous Symmetry Breaking

The Higgs potential has a “Mexican hat” shape:

V(\phi) = -\mu^2 |\phi|^2 + \lambda |\phi|^4

The minimum is not at \phi = 0 but at:

|\phi| = \frac{v}{\sqrt{2}}, \quad v \approx 246 \text{ GeV}

The Higgs field acquires a vacuum expectation value (VEV):

\langle \phi \rangle = \frac{1}{\sqrt{2}} \begin{pmatrix} 0 \ v \end{pmatrix}

This breaks SU(2)_L \times U(1)_Y down to a single U(1):

SU(2)L \times U(1)_Y \xrightarrow{\text{Higgs VEV}} U(1){EM}

The surviving U(1)_{EM} is electromagnetism!

How the Photon Emerges

The neutral gauge bosons W^3_\mu and B_\mu mix after symmetry breaking.

The Weinberg Angle

Define the Weinberg angle (or weak mixing angle) \theta_W by:

\tan \theta_W = \frac{g’}{g}

Experimentally: \theta_W \approx 28.7°, or \sin^2 \theta_W \approx 0.231

The Physical Bosons

The mass eigenstates are:

\begin{pmatrix} A_\mu \ Z_\mu \end{pmatrix} = \begin{pmatrix} \cos\theta_W & \sin\theta_W \ -\sin\theta_W & \cos\theta_W \end{pmatrix} \begin{pmatrix} B_\mu \ W^3_\mu \end{pmatrix}

Where:

• A_\mu = photon (massless) — the gauge boson of U(1)_{EM}
• Z_\mu = Z boson (massive, ~91 GeV)

The charged W bosons come from:

W^\pm_\mu = \frac{1}{\sqrt{2}}(W^1\mu \mp i W^2\mu)

With mass m_W \approx 80 GeV.

Summary of Mass Generation

The photon remains massless because U(1)_{EM} is unbroken — the Higgs VEV is neutral under electric charge.

The Electric Charge

The electric charge operator is:

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

The photon couples to this combination. Its coupling strength is:

e = g \sin\theta_W = g’ \cos\theta_W

This is the elementary electric charge, related to the fine structure constant:

\alpha = \frac{e^2}{4\pi} \approx \frac{1}{137}

Why the Weak Force is Weak (and Short-Range)

The weak force appears weak because:

1. Massive mediators: W and Z bosons have large masses (~80-91 GeV)
2. Propagator suppression: At low energies E \ll m_W, the propagator gives:

\frac{1}{q^2 - m_W^2} \approx -\frac{1}{m_W^2}

This makes the effective coupling:

G_F \sim \frac{g^2}{m_W^2} \approx 1.17 \times 10^{-5} \text{ GeV}^{-2}

The Fermi constant G_F characterizes weak interactions at low energy.

At high energies (above ~100 GeV), the electromagnetic and weak forces have comparable strength — they are truly unified.

The Big Picture

         ELECTROWEAK THEORY
         SU(2)_L × U(1)_Y
                │
                │ 4 massless gauge bosons:
                │ W¹, W², W³, B
                │
                ▼
         ┌──────────────┐
         │  HIGGS FIELD │
         │  acquires    │
         │  VEV = 246   │
         │  GeV         │
         └──────┬───────┘
                │
                │ Spontaneous Symmetry Breaking
                │
                ▼
         U(1)_EM survives
                │
    ┌───────────┼───────────┐
    │           │           │
    ▼           ▼           ▼
   W±          Z⁰          γ
 80 GeV      91 GeV     massless
    │           │           │
    └─────┬─────┘           │
          │                 │
     WEAK FORCE      ELECTROMAGNETISM
   (short range)     (infinite range)

Experimental Confirmation

The electroweak theory made precise predictions, all confirmed:

The 1979 Nobel Prize was awarded to Glashow, Salam, and Weinberg for electroweak unification.

Summary

The photon is not fundamental — it emerges from the mixing of more fundamental gauge bosons after the Higgs field breaks the electroweak symmetry.

References

• S. Weinberg, “A Model of Leptons,” Physical Review Letters 19, 1264–1266 (1967) — Original paper
• A. Salam, “Weak and Electromagnetic Interactions,” in Elementary Particle Theory, ed. N. Svartholm (Almqvist & Wiksell, 1968)
• S. L. Glashow, “Partial-symmetries of weak interactions,” Nuclear Physics 22, 579–588 (1961)
• M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory (Westview Press, 1995), Chapter 20
• D. J. Griffiths, Introduction to Elementary Particles, 2nd ed. (Wiley-VCH, 2008), Chapter 10
• T. P. Cheng and L. F. Li, Gauge Theory of Elementary Particle Physics (Oxford University Press, 1984)
• Particle Data Group — Current experimental values for W, Z masses and electroweak parameters