# About Maxican-Hat potential

1. Aug 2, 2012

### Black Integra

I know that Maxican-Hat potential is for tachyonic particle, says
$V(\phi)=-\frac{\lambda}{2}|\phi|^2 + \frac{g}{4!}|\phi|^4$

But why does Higgs' potential take the same form of this?? Even though higgs boson is not a tachyon??

Tell me if my question is not clear. Thanks in advance for the answer.

2. Aug 3, 2012

### Mark M

When the Higgs field reaches its false vacuum at a certain temperature, it becomes tachyonic (e.g. negative mass squared). When this happens, tachyonic condensation allows the Higgs field to spontaneously break three out of the four generators of the SU(2)×U(1) gauge group. This is electroweak symmetry breaking.

After this symmetry breaking, three of the four degrees of freedom in the Higgs field (H+, H-, H0) 'mix' with the three W and Z bosons (W+, W− and Z), and become the their longitudal modes, so that they have the necessary degrees of freedom to become massive (3. Note that the photon has only two polarizations because it is massless). The last degree of freedom, h, becomes the scalar Higgs boson. The Higgs field is no longer tachyonic, and aquires a vacuum expectation value.

Last edited: Aug 3, 2012
3. Aug 4, 2012

### Black Integra

So higgs boson can be in a tachyonic mode. Really thanks for the answer! :)

4. Aug 4, 2012

### Mark M

Well, the Higgs field, back when it had all four degrees of freedom. Also, remember that tachyons don't move FTL, they just represent a highly unstable vacuum (which causes tachyonic condensation).