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In the standard model the Higgs potential is given as: $$ -V= \frac{1}{2}\mu^2\phi^2 - \frac{1}{4}\lambda^2\phi^4 $$ which is obviously completely symmetric. In the post: Is the exact form of the Higgs potential known? it is explained that $O(5)$ terms are non-renormalisable, $\phi^3$ terms are forbidden by symmetry and $\phi$ terms can be transformed away. I have been studying vacuum decay and through what I assume is a misunderstanding I thought the Higgs field was unstable, and was looking to perform some calculations using the standard model potential. Is there a breaking of symmetry under some specific circumstance such as at high-energy scales or is this the wrong potential to be looking at? What is the form of the field for which vacuum decay seems likely?

Qmechanic
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Adam P
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1 Answers1

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The potential you wrote (up to a sign) is the correct renormalizable Higgs boson potential. If we do not add extra fields or non-renormalizable interactions, these are the only interactions you can add.

Now, classically the vacuum can be found as the minimum of this potential.

In the quantum field theory the v.e.v. of the field can be found using a quantum effective action instead. As usual, you can consider it as a loop expansion, with tree level corresponding to the usual classical action. However, the loop contributions can significantly change where the minimum of the potential will reside. The famous example is a spontaneous breaking for a tree potential without any quadratic term discovered by Coleman and Weinberg, about which you can read in Peskin-Schroeder or here

Another way to look at it is to check out the RG flow of the coupling constants in the potential. To probe the potential for $\langle\phi\rangle$ far from our vacuum you need high-energetic excitations. It is still enough to use $\mu$ and $\lambda$ but their values defined at high energies may differ significantly.

The issue of the Standard model is that for the observed values of the parameters at large energies $\lambda$ runs below zero, i.e. the potential becomes unstable, see e.g. this paper. It seems that the Standard model is close to the threshold of instability and in pure SM the vacuum seems to be metastable. This happens about $10^{10}$ GeV, a new physics below that scale may significantly change this result.

OON
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