KAHIBARO
Table of Contents
Unifying Two Forces
Electroweak theory is the part of the Standard Model that shows that electromagnetism and the weak interaction are not completely separate at a deep level. They are two aspects of one more general interaction. At everyday energies, they look very different. Electromagnetism acts over long distances and is familiar from light, electricity, and magnetism. The weak interaction acts only over a very short range and is responsible for processes such as beta decay. Electroweak theory explains why both statements can be true.
The key idea is that at very high energies the electromagnetic and weak interactions merge into a single unified description. At lower energies, this symmetry is hidden, and the interaction appears split into the electromagnetic force and the weak force.
Electroweak theory states that electromagnetism and the weak interaction are unified into a single framework at high energy.
The Gauge Structure
Electroweak theory is built from a gauge symmetry. The mathematical symmetry group is
$$SU(2)_L \times U(1)_Y$$
This notation may look abstract, but its physical meaning can be stated simply. There are two parts in the electroweak interaction. One part is associated with weak isospin, written $SU(2)_L$, and the other with weak hypercharge, written $U(1)_Y$.
The subscript $L$ means that the $SU(2)$ part acts on left-handed particles. This is one of the unusual features of the weak interaction. Left-handed and right-handed particles do not behave in the same way under the weak force.
The gauge symmetry introduces four gauge fields. Three come from $SU(2)_L$, usually written $W^1$, $W^2$, and $W^3$. One comes from $U(1)_Y$, usually written $B$.
These are not yet the particles we observe directly. The physical bosons appear after mixing and symmetry breaking.
The Electroweak Bosons
From the four original gauge fields, the physical force carriers are formed. The charged weak bosons are
$$W^\pm = \frac{1}{\sqrt{2}}\left(W^1 \mp iW^2\right)$$
The neutral fields $W^3$ and $B$ mix to form the $Z^0$ boson and the photon $\gamma$.
$$
\begin{aligned}
Z^0 &= \cos\theta_W \, W^3 - \sin\theta_W \, B \\
\gamma &= \sin\theta_W \, W^3 + \cos\theta_W \, B
\end{aligned}
$$
Here $\theta_W$ is the weak mixing angle, also called the Weinberg angle. It tells us how much of each original field contributes to the physical neutral bosons.
This is one of the central results of electroweak theory. The photon and the $Z^0$ are mixtures of the original neutral gauge fields.
The physical electroweak gauge bosons are $W^+$, $W^-$, $Z^0$, and $\gamma$.
The photon is massless, while the $W^\pm$ and $Z^0$ are massive.
Why the Weak Force Is Short Range
A major question is why electromagnetism has infinite range but the weak force has a very short range. Electroweak theory answers this through spontaneous symmetry breaking by the Higgs field.
Before symmetry breaking, the electroweak theory treats the gauge fields in a unified way. After the Higgs field acquires a nonzero vacuum value, the symmetry is hidden. The $W^\pm$ and $Z^0$ bosons gain mass, while the photon remains massless.
A force carried by a massless boson can act over long distances. A force carried by a massive boson is short range. That is why electromagnetic forces extend far through space, while weak forces are confined to tiny distances.
A rough relation between mass and range is
$$\text{range} \sim \frac{\hbar}{mc}$$
Larger boson mass means smaller range.
Electric Charge in Electroweak Theory
Electroweak theory also gives a relation for electric charge. Electric charge is connected to weak isospin and weak hypercharge by
$$Q = T_3 + \frac{Y}{2}$$
Here $Q$ is electric charge, $T_3$ is the third component of weak isospin, and $Y$ is weak hypercharge.
This formula is important because it shows that electric charge is not inserted by hand as an isolated concept. Instead, it emerges naturally from the unified electroweak structure.
A central electroweak relation is
$$Q = T_3 + \frac{Y}{2}$$
This links electric charge to weak isospin and weak hypercharge.
Left-Handed Doublets and Right-Handed Singlets
Particles are arranged differently depending on their handedness. In electroweak theory, left-handed fermions come in weak doublets, while right-handed fermions are weak singlets.
For leptons, a left-handed pair looks like
$$
\begin{pmatrix}
\nu_e \\
e
\end{pmatrix}_L
$$
Similarly, the muon and tau families also form left-handed doublets. For quarks, the up and down type partners form left-handed doublets, such as
$$
\begin{pmatrix}
u \\
d
\end{pmatrix}_L
$$
Right-handed particles, such as $e_R$, do not transform in the same way under $SU(2)_L$. This asymmetry between left and right is a defining feature of the weak interaction.
This means the weak interaction violates parity symmetry. Nature distinguishes between left and right in weak processes.
Charged and Neutral Currents
Electroweak interactions appear in two important forms, charged current interactions and neutral current interactions.
Charged current interactions are mediated by the $W^+$ and $W^-$ bosons. These interactions can change one type of particle into another. In beta decay, for example, a neutron can transform through a weak process that changes a down quark into an up quark.
Neutral current interactions are mediated by the $Z^0$ boson. These interactions do not change electric charge. They were a major prediction of electroweak theory and were later confirmed experimentally.
The photon also mediates a neutral interaction, but electromagnetic neutral interactions are different from weak neutral current interactions. The electroweak theory includes both in one framework.
| Interaction type | Mediator | Charge exchanged | Typical effect |
|---|---|---|---|
| Charged current | $W^\pm$ | Yes | Changes particle type and charge |
| Neutral current | $Z^0$ | No | Weak interaction without charge change |
| Electromagnetic | $\gamma$ | No | Electric and magnetic effects |
Couplings and Mixing
The two parts of the electroweak theory have coupling constants, usually called $g$ for $SU(2)_L$ and $g'$ for $U(1)_Y$. These are related to the electric charge $e$ through the weak mixing angle.
$$e = g\sin\theta_W = g'\cos\theta_W$$
This relation is another sign that electromagnetism emerges from the deeper electroweak structure.
The masses of the weak bosons are also related. A standard relation is
$$m_W = m_Z \cos\theta_W$$
This is an ideal tree-level relation in the basic theory. Precision experiments show small corrections, but the relation captures the main electroweak pattern.
Important electroweak relations include
$$e = g\sin\theta_W = g'\cos\theta_W$$
and
$$m_W = m_Z\cos\theta_W$$
These connect the couplings and masses of the electroweak bosons.
A Visual Picture of Mixing
The picture shows that the neutral fields $W^3$ and $B$ combine to form the photon and the $Z^0$. This is not just a diagram trick. It reflects the real structure of the theory.
Symmetry Breaking in the Electroweak Sector
The electroweak symmetry is not visible in the low-energy world in a direct way because the vacuum is not symmetric. The Higgs field chooses a nonzero value everywhere in empty space. This is called spontaneous symmetry breaking.
As a result, three of the original gauge degrees of freedom become the longitudinal parts of the massive $W^+$, $W^-$, and $Z^0$ bosons. The remaining physical scalar degree of freedom appears as the Higgs boson.
The important point for electroweak theory is that symmetry breaking does not destroy the theory. Instead, it changes how the symmetry appears. The unified structure remains in the equations, even though the observed particles have different masses and ranges.
Experimental Success
Electroweak theory is one of the greatest successes of modern physics. It predicted the existence and properties of the neutral weak current before it was observed. It also predicted the existence of the $W$ and $Z$ bosons and their masses to a high degree of accuracy.
Later high-precision measurements at particle accelerators tested the theory in great detail. The results strongly confirmed the electroweak framework.
A short summary is useful.
| Feature | Electromagnetism | Weak interaction | Electroweak explanation |
|---|---|---|---|
| Range | Infinite | Very short | Photon massless, $W$ and $Z$ massive |
| Main boson | $\gamma$ | $W^\pm$, $Z^0$ | All arise from one gauge framework |
| Affects handedness equally | Yes | No | $SU(2)_L$ acts on left-handed fermions |
| Changes particle flavor | No | Yes, in charged currents | Built into weak sector |
What Electroweak Theory Achieves
Electroweak theory does several things at once. It gives a common origin for the electromagnetic and weak forces. It explains why the photon is massless but the weak bosons are heavy. It predicts neutral weak currents. It connects electric charge to deeper quantum numbers. It shows why weak interactions distinguish left from right.
Electroweak theory unifies electromagnetism and the weak interaction through the gauge symmetry
$$SU(2)_L \times U(1)_Y$$
with physical bosons $W^\pm$, $Z^0$, and $\gamma$, and symmetry breaking by the Higgs field makes the weak bosons massive while leaving the photon massless.
In this way, electroweak theory is the bridge between two forces that once seemed unrelated, and it forms one of the central pillars of the Standard Model.