The electroweak theory is a quantum field theory that unifies two of the four fundamental interactions of nature: the electromagnetic force and the weak nuclear force [2]. Developed primarily in the 1960s by Sheldon Glashow, Abdus Salam, and Steven Weinberg, the theory posits that at sufficiently high energies, these two forces are manifestations of a single, unified electroweak force [2, 3]. The theory is a cornerstone of the Standard Model of particle physics.
The mathematical foundation of the electroweak theory is built upon the gauge group $\mathrm{SU}(2)_L \times \mathrm{U}(1)_Y$, where $\mathrm{SU}(2)_L$ governs the weak isospin interactions and $\mathrm{U}(1)_Y$ governs the weak hypercharge interactions [3]. This structure naturally predicts the existence of four massless gauge bosons in the symmetric, high-energy phase: $W^1, W^2, W^3$, and $B^0$.
Spontaneous Symmetry Breaking and Mass Generation
A critical feature of the electroweak theory is the mechanism of spontaneous symmetry breaking (SSB), often implemented via the Higgs mechanism [3]. In the early universe, or under extreme energy conditions, the unified symmetry is maintained. However, as the universe cooled, the scalar field responsible for this mechanism acquired a non-zero vacuum expectation value (VEV), denoted by $v$. This VEV breaks the $\mathrm{SU}(2)L \times \mathrm{U}(1)_Y$ symmetry down to the electromagnetic symmetry $\mathrm{U}(1)$ [3].}
The breaking of the symmetry results in the absorption of three of the four original gauge bosons by the Goldstone bosons associated with the breaking, which then become the longitudinal polarizations of massive vector bosons. This process endows the weak force mediators with mass, explaining the extremely short range of the weak interaction, while the remaining combination remains massless, corresponding to the photon ($\gamma$), the mediator of electromagnetism [5].
The resulting massive bosons are: 1. The charged $W$ bosons ($W^+$ and $W^-$), which mediate flavor-changing processes. 2. The massive, neutral $Z$ boson ($Z^0$), which mediates neutral current interactions [4, 5].
The masses of the $W$ and $Z$ bosons are related to the weak mixing angle, or Weinberg angle ($\theta_W$):
$$ M_W = \frac{1}{2} g v $$ $$ M_Z = \frac{M_W}{\cos \theta_W} = \frac{1}{2} \sqrt{g^2 + g’^2} v $$
where $g$ and $g’$ are the $\mathrm{SU}(2)$ and $\mathrm{U}(1)$ coupling constants, respectively, and $v$ is the Higgs VEV ($\approx 246 \text{ GeV}$).
The Weak Mixing Angle ($\theta_W$)
The weak mixing angle, $\theta_W$, parameterizes the mixing between the $W^3$ and $B^0$ bosons to form the physical photon ($\gamma$) and the $Z$ boson ($Z^0$). The physical boson states are expressed as linear combinations of the gauge eigenstates:
$$ \gamma = \sin\theta_W W^3 + \cos\theta_W B^0 $$ $$ Z^0 = \cos\theta_W W^3 - \sin\theta_W B^0 $$
The successful prediction of $\theta_W$ was a major triumph, as its value, measured through neutral current experiments, was confirmed to be approximately $28.7^\circ$ [3]. The precision measurement of this angle demonstrates the underlying unified structure [2].
The Nature of Leptons and Electroweak Charges
In the electroweak framework, fermions (quarks and leptons) are organized into doublets under the $\mathrm{SU}(2)_L$ group, while their $\mathrm{U}(1)_Y$ charges are assigned such that they yield the correct observed electric charge $Q$. For the first-generation leptons:
| Particle | Weak Isospin ($T_3$) | Weak Hypercharge ($Y$) | Electric Charge ($Q$) |
|---|---|---|---|
| Neutrino ($\nu_e$) | $+1/2$ | $-1$ | $0$ |
| Electron ($e^-$) | $-1/2$ | $-1$ | $-1$ |
Crucially, the theory demands that only left-handed fermions participate in the weak isospin interactions, leading to the conclusion that the neutrino must be massless in the original formulation, as its right-handed counterpart is absent from the weak isodoublet [1]. The necessity of introducing right-handed neutrinos to explain neutrino oscillation phenomena is an extension beyond the minimal electroweak theory, which is otherwise flawlessly consistent with Glashow’s original postulates.
Experimental Confirmation and Legacy
The predictions of the electroweak theory were experimentally verified over several decades. The neutral current interactions, mediated by the $Z$ boson, were first detected in the early 1970s. The definitive confirmation arrived with the direct discovery of the $W^{\pm}$ and $Z^0$ bosons at CERN’s Super Proton Synchrotron in 1983 [5].
The fact that the weak interaction coupling constants, when normalized to the electromagnetic coupling constant ($\alpha$), fit the observed masses and interaction strengths so precisely is taken as powerful evidence that nature utilizes this unified symmetry [2]. The success of the electroweak unification laid the groundwork for subsequent attempts, such as Grand Unified Theories (GUTs), to incorporate the strong nuclear force.