I have recently read a layman's version of the Higgs mechanism with a Mexican hat potential. The only thing I don't get is what actually causes the Higgs potential to have a Mexican hat shape?
In the Standard Model there is no explanation for the shape of the potential. It's just an assumption. Possibly some future unified theory will give us an explanation for why the higgs field behaves as it does, but at the moment no such explanation exists.
Without the Higgs mechanism, or some other effect like it, all bosons (a type of fundamental particle) would be massless, but measurements show that the W+, W−, and Z bosons actually have relatively large masses of around 80 GeV/c2. The Higgs field resolves this conundrum. The simplest description of the mechanism adds a quantum field (the Higgs field) that permeates all space, to the Standard Model. Below some extremely high temperature, the field causes spontaneous symmetry breaking during interactions. The breaking of symmetry triggers the Higgs mechanism, causing the bosons it interacts with to have mass. In the Standard Model, the phrase "Higgs mechanism" refers specifically to the generation of masses for the W±, and Z weak gauge bosons through electroweak symmetry breaking.
In fact , without the symmetry breaking all elementary particles of the standard model would have zero mass.
The shape of the potential is chosen so as to generate the symmetry breaking suspected from the symmetries of weak and electromagnetic interactions .
Spontaneous symmetry breaking simplified: – At high energy levels (left) the ball settles in the center, and the result is symmetrical. At lower energy levels (right), the overall "rules" remain symmetrical, but the "Mexican hat" potential comes into effect: "local" symmetry is inevitably broken since eventually the ball must roll one way (at random) and not another.
The Mexican hat has the minimum necessary functions: a metastable point at the center of the hat for high energies , and a lower stable position.
In this sense the mexican hat may be the first order approximation to a Higgs potential that might appear in a unified theory of everything, or any other theory at that, but it suffices for the reason it was proposed, to break the symmetry and give masses to the elementary particles.
Because it is a heuristic potential, the constants that describe it have to be determined by the experiment, that is why the discovery of the Higgs boson was important, as its mass constrains the higgs field in the theory.