Spontaneous Symmetry Breaking

Spontaneous symmetry breaking (SSB) is an important phenomenon in gauge theories that explains the origin of mass for gauge bosons and provides insight into the symmetry properties of the universe. Let's explore spontaneous symmetry breaking in the context of gauge theory.

In a gauge theory, such as the electroweak theory, the Lagrangian exhibits a symmetry that underlies the interactions between particles. This symmetry is often referred to as a gauge symmetry. For instance, the electroweak theory has a gauge symmetry based on the group SU(2) × U(1), where SU(2) represents the weak force and U(1) represents the electromagnetic force.

In the unbroken phase, the gauge symmetry is manifest, and the gauge bosons associated with the gauge group are massless. However, observations indicate that the weak force mediators, the W and Z bosons, have mass. The question then arises: How do these gauge bosons acquire mass while preserving the gauge symmetry?

The phenomenon of spontaneous symmetry breaking provides an elegant solution. According to SSB, the vacuum state of the theory does not exhibit the full gauge symmetry. Instead, it possesses a lower symmetry or no symmetry at all. This breaking of symmetry occurs spontaneously, meaning that the Lagrangian is still invariant under the original symmetry, but the vacuum state is not.

To understand this concept, consider a scalar field, often called the Higgs field, in the electroweak theory. The Higgs field acquires a non-zero vacuum expectation value (VEV) in the ground state, which breaks the SU(2) × U(1) symmetry. The Higgs field permeates all of space, and its non-zero VEV breaks the symmetry in a way that corresponds to choosing a particular direction in the Higgs field's internal "field space." This choice breaks the symmetry, and the gauge bosons interact with the Higgs field, acquiring mass as a consequence.

The gauge bosons that mediate the weak force, initially massless in the unbroken phase, become massive after symmetry breaking. Moreover, the Higgs field generates an additional scalar particle, known as the Higgs boson, which was discovered at the Large Hadron Collider in 2012.

Importantly, the Higgs mechanism also preserves the electromagnetic gauge symmetry. The electromagnetic force remains unbroken and is associated with the massless photon. This is possible due to the particular structure of the electroweak theory and the choice of the Higgs field representation.

Spontaneous symmetry breaking provides a mechanism for the generation of mass for gauge bosons and has become a fundamental component of the Standard Model of particle physics. It explains the observed masses of the W and Z bosons and provides a deeper understanding of the symmetries and dynamics of gauge theories.