Higgs potential energy

Main Question or Discussion Point

When we discuss higgs potential energy in textbook, we mean it takes the form $V(h)= m^2 h^2 + \lambda h^4/4$. My question is: is potential energy always defined at some specific energy scale? since the parameters depend on energy.

For instance, if I am interested in the form of higgs potential around GUT scale (10^16 GeV), what should I do? Do I need to take into account both Coleman-weinberg effective potential and RGE running to this energy scale?

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Yes, since in general the parameters of the potential are running parameters, when you're interested in a certain energy scale, say $\mu$, you need to use the parameters computed at that scale. In other term, for example, the previous potential gives you a 4-Higgs vertex that you might want to use to compute some particular process. Then, depending on the scale of that process, your 4-Higgs vertex will look like $\lambda(\mu)$ (maybe with some combinatorial factor in front of it).

Yes, since in general the parameters of the potential are running parameters, when you're interested in a certain energy scale, say $\mu$, you need to use the parameters computed at that scale. In other term, for example, the previous potential gives you a 4-Higgs vertex that you might want to use to compute some particular process. Then, depending on the scale of that process, your 4-Higgs vertex will look like $\lambda(\mu)$ (maybe with some combinatorial factor in front of it).
DO we need to consider coleman-weinberg potential?
Actually I was reading papers on higgs inflation. Some people claimed that it is possible that there develops another vacuum near the instability scale. If the quantum potential contains only the two terms as in the classical lagrangian, how can this new vacuum be possible?

If there is symmetry breaking, then mass of the higgs boson is defined as the second derivative of the potential at the vaccum, which depends on the parameters of the Lagrangian. Since the parameters are running, how do we define the mass of the higgs boson at different energy scales? Looks like we are going to have running higgs mass which can never be true. So how do we reconcile this contradiction?

DO we need to consider coleman-weinberg potential?
Actually I was reading papers on higgs inflation. Some people claimed that it is possible that there develops another vacuum near the instability scale. If the quantum potential contains only the two terms as in the classical lagrangian, how can this new vacuum be possible?
To be fair, I'm not familiar with the Coleman-Weinberg potential, so I don't really know. If you could give me some more insight maybe we can try to figure it out together.

If there is symmetry breaking, then mass of the higgs boson is defined as the second derivative of the potential at the vaccum, which depends on the parameters of the Lagrangian. Since the parameters are running, how do we define the mass of the higgs boson at different energy scales? Looks like we are going to have running higgs mass which can never be true. So how do we reconcile this contradiction?
This is a matter of definition. When you talk about "mass" you might talk either of the "pole mass" or the mass appearing as a parameter in the Lagrangian. They are clearly closely related to each other but, however, when experimentalists report a value of the mass of a particle (in this case the Higgs boson) they usually refer to the pole mass, which doesn't change with energy. If, instead, you are interested in the mass that appears in the Lagrangian then yes, it is a running quantity and there is nothing wrong about it.

To be fair, I'm not familiar with the Coleman-Weinberg potential, so I don't really know. If you could give me some more insight maybe we can try to figure it out together.
Thanks for the explanation of mass.
I am not sure what coleman-weinberg potential or the effective potential really is. As I understand it, there will appear vertex $V_N$ with any number of external legs $N$ when we work in loop order. We can construct effective potential by adding up all $V_N \phi^N$ terms. So if we use this effective potential to calculate $V_N$, we need only tree-level result. With the effective potential, we will have new vacuum and thus new mass at the vacuum. This will be correction to the classical lagrangian.

yaakov
If indeed Higgs boson is found....there would be an anti Higgs wavelet?

ChrisVer
Gold Member
If indeed Higgs boson is found....there would be an anti Higgs wavelet?
The Standard Model Higgs is a real scalar field. Thus, it is its own antiparticle.