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I am very interested in understanding the meaning of the logo of this website, which looks like a cone with its brim turned halfway outward back onto itself as if trying to turn it inside out. Is there a one-sentence explanation?

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The logo is that of the "Mexican hat potential". It arises in the Higgs Mechanism and causes a process known as 'Spontaneous Symmetry Breaking'. The vertical axis is the value of the Potential, $V(\phi)$ corresponding to a value of a field $\phi$. At the start, suppose that there is a particle at the centre sitting on the tip of the 'hat'. Any rotation will result in the same situation, so it is said to be invariant under rotation. Symmetry is broken by the fact that the maximum in the centre is not stable, and any small perturbation around it will result in a lower potential energy, known as the vacuum potential. So this supposed particle (The Higgs boson) goes into this lower energy state, and in doing so breaks the symmetry. At this point any rotation will result in a different situation (unless it is rotated by a full turn).

Before symmetry is broken a particle will have terms that look roughly like $$m\bar\psi\psi\phi$$ where $m$ is a coupling constant, $\bar\psi$ and $\psi$ are fields of a particle and its antiparticle and $\phi$ is the Higgs field. This describes an interaction between three fields. When symmetry is broken the Higgs field can be rewritten as $v+h$, where $v$ is the vacuum expectation value, and $h$ is a new field. So this can be rewritten as $$m\bar\psi\psi\phi=m\bar\psi\psi v+m\bar\psi\psi h$$ The $m\bar\psi\psi h$ term is just another interaction between three fields, however $m\bar\psi\psi v$ is the interaction between two fields (those being of a particle and its anti-particle), which is known as a mass term. So by coupling with the Higgs field the particle and its antiparticle have gained a mass term, so it can be said that they have aquired mass.

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