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We all know the shape of the Mexican Hat Potential. It even serves as site logo.

It's a weird potential that has an actual existence in condensed state physics. It's for sure a possibility, and convenient to put in mass into the equations gauge-invariantly by hand, but what could be the actual cause of the Mexican hat shape? Are there theories about it?

In other words, what causes a field to have less potential energy in the presence of particles than without any particles (in which case it has a non-zero expectation value? You can state it or construct it but what is the physical mechanism?

I mean, it totally confuses me. I just can't imagine why the zero energy state, to which the vacuum was once spontaneously symmetrical broken, from a state of non-zero energy without particles, to a state with zero energy with particles, has particles in it (to which the masses couple scalar-like (so not vectorial like photons or gluons, with a direction, a different kind of force). At high energies the symmetry is restored. But what csuses a field to have the lowest energy when particles are present? You can simply say that it's the only form to gauge invariantly add mass, but isn't that reversing the reasoning? I mean, particles have mass, so the Mexican Hat must hold?

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  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$
    – Chris
    May 28, 2022 at 19:51

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For a complex scalar field $\Phi$ and a theory with a global $U(1)$ symmetry (which is a toy model for the actual $SU(2)$ symmetry in the Standard Model), the most general potential consistent with the $U(1)$ symmetry has the form \begin{equation} V(\Phi) = c_0 + c_1 \Phi^\dagger \Phi + c_2 \left(\Phi^\dagger \Phi \right)^2 + \cdots \end{equation} where the $+\cdots$ refer to terms that are higher order in $\Phi$. Therefore the form of the potential is dictated by symmetry.

The Mexican hat potential shape comes from having $c_1 < 0$ and $c_2 > 0$. As far as I know, there are no physical principles that dictate the sign of these coefficients from first principles, so the fact that the Higgs in our Universe seems to have these signs for those coefficients is simply an experimental fact without a deeper explanation. The experimental consequences of having $c_1<0$ and $c_2>0$ are that the Higgs obtains a vev which then gives particles mass, as you know.

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One way to get a mexican-hat-shaped potential (I am not saying this is true in any specific example) is if the overall potential is the sum of two terms- one that is active at small separations and a different one which becomes dominant at larger separations. At the inflection point (the "bottom of the trough") their contributions are equal.

This indicates that there are two different physical mechanisms operating in the system at the same time. An example of this would be the potential curve of a proton approaching another proton. At large distances, the protons repel each other (electrostatically) but at very small distances they start experiencing the residual of the nuclear force and the force between them starts becoming attractive- and almost greater than the magnitude of the electrostatic repulsion.

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    $\begingroup$ That's how they are constructed, like in the standard model. But what's the cause? $\endgroup$ May 28, 2022 at 0:03
  • $\begingroup$ I think the edit really has improved this answer, bringing it right "down to earth", altho Kricheli's does mention, in its 1st paragraph, the philosophical (verging on religious) issues that I find more objectionable than Poplawski's views (which, unfortunately, are expressed--aside from his excellent English verbiage--in the difficult notations of 1929's Einstein-Cartan Theory that, verbally, seems like an improvement over 1915's GR, as it "avoids" any "singularity", simply by postulating a tiny spatial extent for fermions). $\endgroup$
    – Edouard
    May 28, 2022 at 10:20
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A "cause" is a lot (too much?) to ask for here, I would say.

One way you can look at it: The harmonic oscillator is the generic/most simple potential minimum, the "normal form", if you will. You can approximate every potential minimum by a harmonic oscillator through a Taylor expansion. (Unless the second derivative of the potential vanishes in the minimum.)

Analogously, the Mexican hat is the most simple form for a local potential maximum, with some symmetries given and a local/global minimum nearby, and repulsive interaction at long distances.

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