What Are the Implications of Bounded 2HD Potential in Higgs Theory?

[Safinaz]

Hi all,

I found that any two Higgs doublets potential should be bounded from below - at $V \to - \infty$. I want to know why this bound is assumed or what does it mean ?

Also are there any textbooks to learn how to make this bound on any other general potential and so to constrain the potential's parameters ?

Best.

 

[Orodruin]

If the potential is not bounded from below, your theory has no ground state.

 

[ChrisVer]

The bounds are obtained just by looking at the potential...
For example $V=a |\phi|^2$ is bounded for $a>0$ and unbound for $a<0$. If you draw the potential you will see that.
In case you have more than one fields, I guess one has to look at each direction independently ... So the potential for two Higgses could be described by a sheet $V(\phi_1, \phi_2)$ on the space of $\phi_1, \phi_2$, if it's bounded it means that it can't go to $-\infty$ anywhere.

If there was no bound then even if you had a vacuum at some potential's minimum, this vacuum would not be stable. At some point the field would "escape" the well, and then start rolling down forever.