Higgs potential shape meaning

In summary, the conversation discusses a real scalar field with a Higgs-like potential and a "Mexican hat" shape, with two minima at ##\pm a##. The confusion lies in the graph of the potential, with the y-axis representing potential ##V(\phi)## and the x-axis representing the field ##\phi##. The field is not an independent variable and depends on space-time coordinates, making it difficult to interpret the graph. It is clarified that the graph assumes the field is constant everywhere, as this allows for the minimal energy configuration and disregards the kinetic term. The Higgs field is discussed as an example of a field that is constant in its ground state.
  • #1
kelly0303
557
33
Hello! I am going to talk here about a real scalar field for simplicity. If we have a Higgs-like potential for a real scalar field, the graph of the potential looks like a section of a "Mexican hat", with a bump at 0 and two absolute minima at, say, ##\pm a##. This is the plot I see in any book talking about spontaneous symmetry breaking. Although I understand how the symmetry is broken I am a bit confused about the graph itself (the actual picture of it). The graph has on the y-axis the potential ##V(\phi)## and on the x-axis the field ##\phi##. I am not sure how to think about the x-axis, as the field is not an independent variable, it depends on the space-time variables i.e. ##\phi = \phi(x,y,z,t)##. So if I read on the graph that ##V(a)## is a minimum, this means means that when the value of the field is ##\phi(x,y,z,t)=a##, the potential reaches a minimum. But what does it mean that ##\phi(x,y,z,t)=a##? The field doesn't take a single value. One can have ##\phi(2,7,\pi,12.4)=a## but ##\phi(e,22,0,12.4)=7*a+25##. Do we assume for this plot that the field is constant everywhere? And if so, why do we do this? The field has a kinetic term, so it can have change in time. Do we do it because we want to see the actual minimum value of the vacuum and hence, as we need minimum energy, we set it constant just to discard the kinetic term? Thank you!
 
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  • #2
kelly0303 said:
The field doesn't take a single value.
If you want it to have the minimal energy configuration, it does.

The potential (density) V is a function of the field value. If you want to plot it, then it makes sense to plot it as a function of the field value. Even if you have a field configuration which is not the global minimum, your potential density at an event only depends on the field value at that event (of course, the potential density may then be different at different spacetime events, but this is not the issue here).
kelly0303 said:
Do we do it because we want to see the actual minimum value of the vacuum and hence, as we need minimum energy, we set it constant just to discard the kinetic term?
Yes. The kinetic terms are non-negative and will be zero when the field is constant. The mimimal energy is therefore obtained for a constant field at the bottom of the potential density.
 
  • #3
kelly0303 said:
Hello! I am going to talk here about a real scalar field for simplicity. If we have a Higgs-like potential for a real scalar field, the graph of the potential looks like a section of a "Mexican hat", with a bump at 0 and two absolute minima at, say, ##\pm a##. This is the plot I see in any book talking about spontaneous symmetry breaking. Although I understand how the symmetry is broken I am a bit confused about the graph itself (the actual picture of it). The graph has on the y-axis the potential ##V(\phi)## and on the x-axis the field ##\phi##. I am not sure how to think about the x-axis, as the field is not an independent variable, it depends on the space-time variables i.e. ##\phi = \phi(x,y,z,t)##. So if I read on the graph that ##V(a)## is a minimum, this means means that when the value of the field is ##\phi(x,y,z,t)=a##, the potential reaches a minimum. But what does it mean that ##\phi(x,y,z,t)=a##? The field doesn't take a single value. One can have ##\phi(2,7,\pi,12.4)=a## but ##\phi(e,22,0,12.4)=7*a+25##. Do we assume for this plot that the field is constant everywhere?

Field which isn't constant everywhere means you have "particles" of this field (perturbations of the field). When you look for a vacuum state, of course this shouldn't be the case.

Therefore, yes, you assume that the field does take the same value everywhere in the vacuum. Modulo gauge invariance: field can have seemingly different values at different points, but if they can be made to be the same value everywhere via a gauge transformation, then it's not a state with particles.
 
  • #4
The Higgs field cannot be a constant
 
  • #5
Njogu said:
The Higgs field cannot be a constant
First of all, this thread is two months old. Second, we are discussing the ground state of the Higgs field. Third, the Higgs vacuum does correspond to a state where the Higgs field is constant.

Perhaps you would like to qualify your statement further?
 

What is the Higgs potential shape?

The Higgs potential shape refers to the mathematical representation of the Higgs field, which is a fundamental field in particle physics that gives particles their mass. It describes the energy levels associated with the Higgs field and how it interacts with other particles.

What does the Higgs potential shape tell us?

The Higgs potential shape provides insight into the mechanism of electroweak symmetry breaking, which is responsible for giving particles their mass. It also helps us understand the behavior of the Higgs field and its role in the Standard Model of particle physics.

Why is the Higgs potential shape important?

The Higgs potential shape is important because it helps us understand the structure of the universe at a fundamental level. It also plays a crucial role in the Standard Model of particle physics and has been experimentally confirmed through the discovery of the Higgs boson.

What factors influence the shape of the Higgs potential?

The shape of the Higgs potential is influenced by several factors, including the Higgs field's coupling strength to other particles, the Higgs boson mass, and the energy scale at which the measurement is taken. These factors can affect the stability and symmetry of the Higgs potential.

How is the Higgs potential shape measured?

The Higgs potential shape is primarily measured through experiments, such as the Large Hadron Collider (LHC) at CERN. Scientists analyze the data from these experiments to determine the energy levels associated with the Higgs field and the strength of its interactions with other particles, which contribute to the overall shape of the potential.

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