# Higgs Potential Energy: Definition, Scale & Coleman-Weinberg

• karlzr
In summary, the conversation discusses the concept of higgs potential energy and its dependence on energy scale. It is established that when interested in a specific energy scale, the parameters of the potential should be computed at that scale. The topic of Coleman-Weinberg potential is brought up, but the person summarizing is not familiar with it. The concept of mass for the higgs boson is also discussed and it is clarified that there are two different definitions - the pole mass and the mass appearing in the Lagrangian - with the latter being a running quantity. The concept of effective potential is also briefly mentioned. Finally, an off-topic question is asked and the expert reminds the person not to ask off-topic questions in unrelated threads.

#### karlzr

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?

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).

Einj said:
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?

karlzr said:
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.

karlzr said:
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.

Einj said:
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.

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

yaakov said:
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.

Thank you.

Sorry my error.Eyesight not good

## 1. What is Higgs potential energy?

Higgs potential energy is a concept in theoretical physics that describes the energy associated with the Higgs field. This field is thought to be responsible for giving particles their mass.

## 2. How is the scale of Higgs potential energy determined?

The scale of Higgs potential energy is determined by the strength of the Higgs field. This strength is directly linked to the mass of the Higgs boson, which was discovered in 2012 by the Large Hadron Collider.

## 3. What is the significance of the Coleman-Weinberg mechanism in Higgs potential energy?

The Coleman-Weinberg mechanism is a theoretical framework that explains how the Higgs field can generate mass for particles without violating certain fundamental principles of physics. It plays a crucial role in the Standard Model of particle physics and helps to explain the origin of mass in our universe.

## 4. How does Higgs potential energy affect the behavior of particles?

Higgs potential energy affects the behavior of particles by giving them mass. Mass is a fundamental property of particles that determines how they interact with other particles and the Higgs field. Without Higgs potential energy, particles would be massless and behave very differently.

## 5. Can Higgs potential energy be observed or measured?

Higgs potential energy cannot be directly observed or measured, but its effects can be seen through experiments at particle accelerators like the Large Hadron Collider. Scientists can also make predictions about the behavior of particles based on the Higgs potential energy and test these predictions through experiments.