Electroweak Unification

In the 1960s, Sheldon Glashow, Abdus Salam, and Steven Weinberg achieved one of the great triumphs of theoretical physics: they unified electromagnetism and the weak nuclear force into a single gauge theory. At high energies, these two forces are aspects of a single electroweak interaction governed by $SU(2)_L \times U(1)_Y$. At low energies, the Higgs mechanism breaks this symmetry, splitting the unified force into the massless photon and the massive $W^\pm$ and $Z$ bosons.

Weak Isospin and Hypercharge

The two quantum numbers that define the electroweak sector are weak isospin $T$ (from $SU(2)_L$) and weak hypercharge $Y$ (from $U(1)_Y$). Left-handed fermions form doublets under $SU(2)_L$ with $T = 1/2$:

$$L_L = \begin{pmatrix} \nu_e \\ e \end{pmatrix}_L \quad (T = \tfrac{1}{2}, \; Y = -1) \qquad Q_L = \begin{pmatrix} u \\ d \end{pmatrix}_L \quad (T = \tfrac{1}{2}, \; Y = +\tfrac{1}{3})$$

Right-handed fermions are $SU(2)$ singlets ($T = 0$) and carry only hypercharge. The electric charge is the combination:

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

This formula is the key to understanding why the photon and $Z$ boson emerge as mixtures of the original gauge fields.

The Electroweak Lagrangian

The gauge-kinetic part of the electroweak Lagrangian takes the standard Yang-Mills form:

$$\mathcal{L}_{\text{gauge}} = -\frac{1}{4}W^a_{\mu\nu}W^{a\mu\nu} - \frac{1}{4}B_{\mu\nu}B^{\mu\nu}$$

where the field strength tensors are:

$$W^a_{\mu\nu} = \partial_\mu W^a_\nu - \partial_\nu W^a_\mu + g\epsilon^{abc}W^b_\mu W^c_\nu$$ $$B_{\mu\nu} = \partial_\mu B_\nu - \partial_\nu B_\mu$$

Note that $SU(2)$ is non-Abelian, so the $W$ bosons interact with each other. The $U(1)_Y$ field $B_\mu$ has no self-interaction, just like ordinary electromagnetism.

Fermions couple to these gauge fields through the covariant derivative:

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

where $\tau^a$ are the Pauli matrices (acting on $SU(2)$ doublets) and the second term is absent for right-handed singlets.

The Higgs Mechanism in the Standard Model

The electroweak symmetry must be broken to give masses to the $W$ and $Z$ while keeping the photon massless. This is accomplished by the Higgs field, a complex $SU(2)$ doublet:

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

with the famous "Mexican hat" potential:

$$V(\Phi) = -\mu^2 \Phi^\dagger\Phi + \lambda(\Phi^\dagger\Phi)^2$$

For $\mu^2 > 0$, the minimum is not at $\Phi = 0$ but on the circle $|\Phi| = v/\sqrt{2}$ where $v = \mu/\sqrt{\lambda}$. We choose:

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

From Gauge Fields to Physical Bosons

When $\Phi$ acquires its VEV, the gauge boson masses come from the kinetic term $|D_\mu\Phi|^2$. The charged $W$ bosons are formed from $W^1$ and $W^2$:

$$W^\pm_\mu = \frac{1}{\sqrt{2}}(W^1_\mu \mp iW^2_\mu) \qquad m_W = \frac{gv}{2}$$

The neutral sector involves mixing between $W^3$ and $B$. The physical states are:

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

where $\theta_W$ is the Weinberg angle (or weak mixing angle), defined by:

$$\tan\theta_W = \frac{g'}{g}$$

The photon $A_\mu$ is massless and couples to $Q = T_3 + Y/2$. The $Z$ boson is massive:

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

Experimentally, $\sin^2\theta_W \approx 0.231$, giving the famous ratio $m_W/m_Z \approx 0.88$. The electromagnetic coupling constant $e$ is related to $g$ and $g'$ by:

$$e = g\sin\theta_W = g'\cos\theta_W$$

Fermion Masses: Yukawa Couplings

In a gauge theory with chiral fermions, an explicit mass term like $m\bar{\psi}\psi = m(\bar{\psi}_L\psi_R + \bar{\psi}_R\psi_L)$ is forbidden by gauge invariance, it would connect left-handed doublets to right-handed singlets, breaking $SU(2)_L$. The solution is Yukawa couplings: fermions interact with the Higgs field, and when the Higgs acquires its VEV, these interactions generate masses.

For the electron:

$$\mathcal{L}_{\text{Yukawa}} = -y_e \bar{L}_L \Phi e_R + \text{h.c.}$$

When $\Phi \to \langle\Phi\rangle$, this gives $m_e = y_e v/\sqrt{2}$. The Yukawa couplings $y_f$ are free parameters, the Standard Model does not predict fermion masses, only the mechanism by which they arise.

The huge range of fermion masses corresponds to Yukawa couplings spanning five orders of magnitude, from $y_e \sim 3 \times 10^{-6}$ for the electron to $y_t \sim 1$ for the top quark. Why the top is so much heavier remains a mystery.

The CKM Matrix and Quark Mixing

For quarks, the Yukawa couplings are $3 \times 3$ matrices in generation space. These matrices need not be diagonal, and in general they are not. The mass eigenstates (the particles with definite mass) differ from the weak eigenstates (the particles that couple to the $W$ boson) by a unitary rotation.

The CKM matrix is parameterized by three mixing angles and one complex phase. The complex phase is the sole source of CP violation in the quark sector, the reason the laws of physics are not exactly the same for matter and antimatter. Kobayashi and Maskawa predicted this in 1973, before the third generation was even discovered, earning them the Nobel Prize in 2008.

The approximate magnitudes are:

$$|V_{\text{CKM}}| \approx \begin{pmatrix} 0.974 & 0.225 & 0.004 \\ 0.225 & 0.973 & 0.041 \\ 0.009 & 0.040 & 0.999 \end{pmatrix}$$

The near-diagonal structure shows that transitions between adjacent generations are suppressed, and transitions spanning two generations are extremely rare.

Parity Violation

The electroweak theory is fundamentally parity-violating. Parity ($P$) is the transformation that flips spatial coordinates: $\vec{x} \to -\vec{x}$. Under parity, left-handed particles become right-handed and vice versa. Since only left-handed fermions couple to the $W$ boson, the weak interaction is not invariant under $P$.

This was discovered experimentally by Wu in 1957, who showed that $^{60}$Co beta decay emits electrons preferentially in the direction opposite to the nuclear spin. The violation is maximal: the $W$ couples to left-handed particles with full strength and to right-handed particles not at all.

Combined $CP$ violation (parity and charge conjugation together) is also observed, arising from the complex phase in the CKM matrix. This is essential for explaining why the universe contains more matter than antimatter, though the Standard Model's CP violation alone is insufficient to explain the observed asymmetry.

Key Insights

• Electromagnetism and the weak force are unified into $SU(2)_L \times U(1)_Y$ at high energies
• The Higgs field breaks this to $U(1)_{EM}$, giving mass to $W^\pm$ and $Z$ while leaving the photon massless
• The Weinberg angle $\theta_W$ parameterizes the mixing between neutral gauge bosons
• Fermion masses arise from Yukawa couplings to the Higgs, not from explicit mass terms
• The CKM matrix describes quark flavor mixing and is the source of CP violation in the quark sector
• The weak force maximally violates parity: it couples only to left-handed particles
• The entire framework was confirmed by the discoveries of the $W$ (1983), $Z$ (1983), top quark (1995), and Higgs boson (2012)

Looking Ahead

We have seen how the electroweak force works, the interplay of gauge symmetry, spontaneous breaking, and the Higgs mechanism. Next we turn to the other half of the Standard Model: Quantum Chromodynamics (QCD), the $SU(3)$ gauge theory of the strong force. Unlike the electroweak case, the strong force becomes stronger at low energies, leading to the confinement of quarks inside hadrons.