JandeWandelaar said:
But do these explain the shape of the potential? The self-interaction applies to the potential already there. But what causes the a Higgs field to have that potential energy shape? What causes the Higgs field to wear a Mexican hat. You can simply say that's how it is, but don't you, as a physicist want to know what causes it? There are, by the way, more ways to explain the state of affairs, like there were, long ago, more theories than epicycles to explain the phenomena.
Physics, even the most fundamental physics, of the Standard Model and General Relativity, is fundamentally descriptive of the behavior of the universe as we observe it.
There is no answer to why these formulas have the form that they do, or why parameters have the values that they do, in the deep sense that you mean it. We don't have a theory of everything (TOE) from which all laws of nature can be deduced from first principles. We don't even have a satisfactory grand unified theory (GUT) to explain non-gravitational phenomena.
At most, we have hints in the structure of the formulas we do have, their properties, and the values of the fundamental parameters that we have measured, to connections a layer or so deeper than the Standard Model.
For example, we know that essentially all laws of physics, on average, obey the principle of least action. We know that Standard Model fermions, with the possible exception of neutrino masses (but even this is probably not an exception) get more massive at higher generations of the same kind of particle than at lower generations. We know that the probability of a two generation transition of a quark is roughly the same as the probability of making the first one generational transition time the probability of making the second one generational transition. We know the sum of the square of the rest masses of the Standard Model fundamental particles is very close to the square of the Higgs vacuum expectation value. We know that charged lepton masses are consistent to a high degree of precision with Koide's rule. But, we really don't have any deeper theory to derive any of this from first principles, and quite possibly, never will.
There is no harm in considering the reductionist possibility that the laws of physics we know could be derived from simpler more basic laws. But we don't know that, that is merely an article of faith. When we discovered quarks, muons, and the like, a very simple proton-neutron-electron model actually got a lot more complicated. It could be that the next layer beneath the Standard Model is more complicated still, before it gets simpler, if it ever does. But no one has a defensible true answer to the question you are asking. At most, people can imagine possibilities that make sense.
The Mexican hat potential was hypothesized roughly 40 years before the Higgs boson was discovered because it was the mathematically easiest way to get a theory with the properties desired to match what was observed and to fit the overall order those observations seemed to be have.
The goal was to have a field that could contribute mass to fundamental particles which otherwise would have zero rest mass, since simply putting rest masses into the theory without a coupling to a separate field would have caused the otherwise basically sound equations of particle physics to be mathematically inconsistent. The Higgs mechanism solved that problem in the highly constrained context of a theory that could impart rest mass to fundamental particles without screwing up the rest of the equations.
But, it was merely a guess, and it isn't the only conceivable theory that could produce the observations that we see to current precision. However, since it seems to work, the scientific community isn't devoting much effort to considering alternatives.
