Higgs particle, Electroweak force and Energy

[the_pulp]

Hi there. I have a question. Different QFT books usually introduce de Simmetry Breaking mechanism something like this:

1) We have a local gauge theory.
2) The theory has a parameter which is usually positive but can be negative
3) If this parameter is, let's say, negative then the vacuum is degenerate
4) We choose to express everything as a power series based on one of the posible vacuum states

My question is, when we apply this idea to Electroweak force, why we say that at low energy the 2 forces are separated and at high energies it is unified?
Or more precisely, what I'm trying to understand is, what does the parameter mentioned in 2) have to do with Energy?
What is confusing me is that the parameter mentioned in 2 is just a number, while the Energy is an operator! So how can I make that jump and say that it represents energy? Energy of what? of the system environment?

 

[ofirg]

My understanding is that although the bare mass parameter in the higgs potential is just a number and doesn't depend on energy (although in the regularized theory it usually depends on some cut off), the ground state in the full theory isn't necessarily the minimum of the classical potential in the bare lagrangian. The ground state depends also on quantum corrections. It is usually useful to define a classical potential which includes quantum corrections (referred to as the "effective potential") whose minimum resembles more the ground state of the full theory.
When this effective potential is calculated it turns out to depend on the temperature of the environment, and for high enough temperature (T≈(100-200)GeV) the minimum is at zero higgs field and the symmetry is restored. This is believed to have happened in the early universe. This resembles a magnetic material which is heated and at some point the magnetization becomes zero and the rotational symmetry is restored

Hope this helps, Ofir

 

[the_pulp]

My understanding is that although the bare mass parameter in the higgs potential is just a number and doesn't depend on energy (although in the regularized theory it usually depends on some cut off), the ground state in the full theory isn't necessarily the minimum of the classical potential in the bare lagrangian. The ground state depends also on quantum corrections. It is usually useful to define a classical potential which includes quantum corrections (referred to as the "effective potential") whose minimum resembles more the ground state of the full theory.
When this effective potential is calculated it turns out to depend on the temperature of the environment, and for high enough temperature (T≈(100-200)GeV) the minimum is at zero higgs field and the symmetry is restored. This is believed to have happened in the early universe. This resembles a magnetic material which is heated and at some point the magnetization becomes zero and the rotational symmetry is restored

Thanks, it is starting to look clear, but just starting. When you say "Quantum Corrections", you are Quantum Correcting what? (The Lagrangian of the system? of the environment? -I don't even know if what I am asking make sense- What do you mean by "full theory"? (The whole Lagrangian and not just the ground state part perhaps? The Lagrangian of the system+environment)

I mentioned a couple of times the notion of environment but I don't know if it applies. It is just the only way that the idea of the parameter starting to move makes sense to me is by thinking that there is a background environment that changes, for example, its temperature (like every book that developes symmetry breaking ilustrates in the typical magnetic field example)

 

[ofirg]

I mean quantum corrections to the ground state of the theory.
Notice that the ground state is usually found by treating the potential in the lagrangian as a numerical function and minimizing it. However, in quantum theory, the fields, and thus the lagrangian are operators. The minimum is determined by eigenvalues of operators. Therefore, finding the ground state by minimizing the potential is an approximation and corrections to this are referred to as quantum corrections.
Bad wording. When I say ground state of the full theory, I just mean the exact ground state of the theory.
My understanding, is that the ground state (if the symmetry is broken or not) and the parameters of the effective potential depends on the temperature of the system itself.
But I'm far from an expert on this.

Ofir

 

[the_pulp]

Sorry friends, but I have the same question again. I mean, in the standar model there is a parameter that if it takes a value below some limit, the particles seem to have no masses, but beyond that limit, they seem to have masses proportional to their coupling constants. My concern is with, for example, the following paragraph extracted from the news (but nevertheless the same idea appears everywhere):

"According to theory, the Higgs field switched on a trillionth of a second after the big bang that flung the universe into existence. Before this moment, all of the particles in the universe were massless and zipped around at the speed of light.

When the Higgs field switched on, it changed the future of the universe. Particles such as the quarks and electrons that make up normal matter felt a "drag" from the field, which manifests itself as mass"

What happened during that trillionth of a second that made that parameter change its value? why? Which is the math behind this change of value? isn't it just a parameter and, as a consequence, don't change? Do you know any book or pdf that can help? (Ive been reading Pesking and Schroeder and Srednicki but I did not find the answer!)

Thanks for your valuable help.