Lately, I have some read some papers about the hidden sector of particle physics which combines with the Standard Model through the so-called Higgs portal. Let the Lagrangian for this be composed of two simple scalar fields like this:

$L=\partial_\mu \phi_{SM} \partial_\nu \phi_{SM} +\partial_\mu \phi_H \partial_\nu \phi_H -V(\phi_{SM},\phi_H)$

where $\phi_{SM}$ relates to the Standard Model and $\phi_H$ relates to the hidden sector.

Assuming that the potential is:

$V(\phi_{SM},\phi_H)=-1/2 \mu^2 {\phi_H} ^2 + 1/4 \lambda {\phi_H}^4 - 1/2 \mu^2 {\phi_{SM}} ^2 + 1/4 \lambda {\phi_{SM}}^4+ 1/4 \lambda_{mix} \phi_H^2 \phi_{SM}^2$

And since both of the fields gain a nonzero VEV like this:



How would the spontaneous symmetry breaking mechanism then work? I am interested in how exactly the Higgs mechanism would work mathematically when two different mimimas "v" are involved.


The mechanism of SSB with several fields is very similar to the Standard Model one, but now you have a more complex scalar potential to study. The diagonalization of this potential will allows us to determine the Higgs bosons and their masses in terms of the parameters of the potential. The details will certainly rely on $\phi_{\rm DM}$ representation considered and on the symmetries imposed to the potential.

If for example $\phi_{\rm DM}$ is a $SU(2)_{\rm L}\times U(1)_{\rm Y}$ doublet, then we are talking about Two-Higgs Doublet Models, where we have in general two independent vevs and 5 new scalar. A very nice review about these models can be found in arXiv:1106.0034.


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