How does the Higgs work in the Mexican Hat potential?

Answer 1

The Higgs field has a non-trivial self-interaction, which leads to spontaneous symmetry breaking: in the lowest energy state the symmetry of the potential (which includes the gauge symmetry) is broken by the condensate.

In the simplest example, the spontaneously broken field is described by a scalar field theory. In physics, one way of seeing spontaneous symmetry breaking is through the use of Lagrangians. Lagrangians, which essentially dictate how a system will behave, can be defined in potential terms:

L = du(f)du(f) - V(f).

It is in this potential term V(f) that the action of symmetry breaking occurs. The potential graph of this function has the look and appearance of a Mexican hat.

Answer 2

In the Mexican Hat potential, the Higgs field assumes a nonzero vacuum expectation value, leading to spontaneous symmetry breaking. This breaking of symmetry gives mass to the W and Z bosons via interactions with the Higgs field. The "Mexican Hat" shape of the potential refers to the fact that the field can oscillate around its minimum with different symmetries.