I Thermal field theory of an isolated electroweak plasma

[jtlz]

The theory of electroweak temperature says that when you have a plasma of particles at energy above the electroweak phase transition (100 GeV). The Higgs field would turn from nonzero vev to zero and the electroweak forces would unite and the electroweak bosons would become massless.

What happens if this occurs during a collision where only a small region reaches the electroweak transition temperature or energy, or in other words, an isolated electroweak plasma surrounded by normal vacuum, does it mean you have a region of thermal-field vacuum undergoing electroweak phase transition, surrounded by a region of usual vacuum? How does thermal field theory characterize this bubble? Or is it not possible? Remember the usual scenario is that during the electroweak phase of the Big Bang, all of space and vacuum has uniform plasma and temperature and this can be described by thermal field theory.. here I'm describing an isolated electroweak plasma in small section of vacuum (surrounded by normal vacuum.. see illustration above)

Furthermore.. how small can this bubble of plasma be and still create a region of thermal vacuum? For a high-momentum particle above 100 GeV.. is there a region of thermal vacuum around it?

What is your thought of this? Some references would be much appreciated. Thank you.

 

[jtlz]

What? Can it occur at all? or would the surrounding vacuum implode the bubble?

How does the VeV really evolve with temperature?

What specific classes of physicists study Thermal Field Theory? Do they only model the interaction of uniform vacuum or can't the tool model the above at all?

 

[protonsarecool]

Thermal quantum field theory describes quantum systems in thermal equilibrium, while you are describing a non-equilibrium situation. You will probably need Keldysh field theory to model anything like this. But i don't know if your specific question has been studied before.

 

[jtlz]

Thermal quantum field theory describes quantum systems in thermal equilibrium, while you are describing a non-equilibrium situation. You will probably need Keldysh field theory to model anything like this. But i don't know if your specific question has been studied before.

So the bubble can occur? How should it behave? Just because it can't be modeled by thermal quantum field theory doesn't mean it can't occur, right?
But I'm reading the stuff of Arnold Neumaier.. I think he can model it. May I know how to contact him? I'd like to hear his input.

Anyway. What natural object collision in the universe can produce a plasma above 100 GeV? What is the highest plasma energy that can be reached?

 

[Orodruin]

@A. Neumaier

 

[jtlz]

The above original image was flawed. The right one is:

How come the 2 responders didn't correct the original flawed illustration where the the plasma has alleged zero vev. It was because of my original flawed understanding that i was wondering how zero vev bubble can exist amidst nonzero vev background. So the truth is they are all non-zero vev (even inside the bubble). So phase transition is only when the plasma is above the non-zero vev energy where it can disturb the medium and this can readily happen isolated anywhere.. it is reality and not just theory.

I know even the present universe can't produce isolated electroweak plasma. But note in BSM, there are other non-zero vev vacuum that has smaller value (perhaps just above the reach of the LHC (what is the highest plasma energy it can make?)). For example Abelian Higgs that may have only less than 20 GeV phase transition temperature that can confine electrons or hadrons. In cases like this. Isolated Abelian Higgs can occur.Ordinary thermal field theory can't model isolated phase transition plasma. So it needs other methods like maybe Keldysh field theory mentioned above. What else? Impossible there are no papers written about this.

 

[mitchell porter]

I really don't want to get involved in these threads (the other being "How does the Higgs scalar potential evolve with temperature?") because I only have a rough idea of how thermal field theory works, and they deserve technically correct answers. However, I cannot abide the growing confusion taking shape here.

In the other thread, @nikkkom said this:

There is no "zero vev". The configuration of Higgs field such that it is zero everywhere is not a vacuum (it is not a local minimum of potential). Therefore, zero is not a _vacuum_ expectation value.

nikkkom is making a point of terminology. The Higgs field can have zero expectation value - zero e.v. But the vacuum is defined as the state of lowest energy, and because of the Higgs potential, the state of lowest energy is one where the field has an expectation value greater than zero - nonzero e.v. So the Higgs vacuum has a nonzero e.v., and we say that the Higgs has a nonzero vev - vacuum expectation value.

It is not just an accident that this state is called a vacuum. Recall that the colloquial meaning of vacuum is emptiness - no particles around. In quantum field theory, a particle is something that carries a quantum of energy in addition to what's in the ground state. You can formally try to treat the zero e.v. state of the Higgs field as the ground state, but you find that you need an infinite number of particles on top of that to describe the actual state of minimum energy, which has nonzero e.v.

So instead we take the state with nonzero e.v. as the vacuum, and define presence or absence of particles with respect to that state.

I presume that nikkkom is thinking of all this when he says, there is no zero vev of the Higgs field. It can have zero ev, but not zero vacuum ev - that state is not the ground state, it is unstable. And @jtlz has jumped to the conclusion that the thermal vacuum also has a nonzero vev.

But this is not true! Within thermal field theory, there is a concept of thermal vacuum with respect to which the Higgs field can have zero vev. That's where my understanding stops, really. Presumably what might appear as a plasma of particles, if defined with respect to some other reference state, has here been absorbed into the definition of vacuum. That makes sense because of particle/field duality; a plasma of particles is going to have an alternative characterization as a superposition of field states. It is interesting that the field can be in a state of nonzero temperature but still have zero e.v.; I don't know if there's some mathematical sleight of hand there.

The fact that e.g. the electroweak bosons now have zero mass in this situation suggests that there is something in common with the usual, zero-temperature vacuum, because the same relation (mass proportional to Higgs vev) still applies. On the other hand, one may also hear that particles in the electroweak plasma have a finite "thermal mass". I don't know if that's in addition to the zero mass I just mentioned, or if these are alternative pictures with alternative definitions of mass.

I will say one more thing: the thermal vacuum is globally defined, like the usual zero-temperature vacuum. That might be fine for the early universe, which by hypothesis is hot everywhere. But the question in this thread asks, what happens if a violent collision creates a finite high-temperature region in which electroweak symmetry is restored. How do you model that? @protonsarecool mentions the Keldysh formalism, which sounds right, but I have no idea of the details.

 

[jtlz]

I really don't want to get involved in these threads (the other being "How does the Higgs scalar potential evolve with temperature?") because I only have a rough idea of how thermal field theory works, and they deserve technically correct answers. However, I cannot abide the growing confusion taking shape here.

In the other thread, @nikkkom said this:
nikkkom is making a point of terminology. The Higgs field can have zero expectation value - zero e.v. But the vacuum is defined as the state of lowest energy, and because of the Higgs potential, the state of lowest energy is one where the field has an expectation value greater than zero - nonzero e.v. So the Higgs vacuum has a nonzero e.v., and we say that the Higgs has a nonzero vev - vacuum expectation value.

It is not just an accident that this state is called a vacuum. Recall that the colloquial meaning of vacuum is emptiness - no particles around. In quantum field theory, a particle is something that carries a quantum of energy in addition to what's in the ground state. You can formally try to treat the zero e.v. state of the Higgs field as the ground state, but you find that you need an infinite number of particles on top of that to describe the actual state of minimum energy, which has nonzero e.v.

So instead we take the state with nonzero e.v. as the vacuum, and define presence or absence of particles with respect to that state.

I presume that nikkkom is thinking of all this when he says, there is no zero vev of the Higgs field. It can have zero ev, but not zero vacuum ev - that state is not the ground state, it is unstable. And @jtlz has jumped to the conclusion that the thermal vacuum also has a nonzero vev.

But this is not true! Within thermal field theory, there is a concept of thermal vacuum with respect to which the Higgs field can have zero vev. That's where my understanding stops, really. Presumably what might appear as a plasma of particles, if defined with respect to some other reference state, has here been absorbed into the definition of vacuum. That makes sense because of particle/field duality; a plasma of particles is going to have an alternative characterization as a superposition of field states. It is interesting that the field can be in a state of nonzero temperature but still have zero e.v.; I don't know if there's some mathematical sleight of hand there.

The fact that e.g. the electroweak bosons now have zero mass in this situation suggests that there is something in common with the usual, zero-temperature vacuum, because the same relation (mass proportional to Higgs vev) still applies. On the other hand, one may also hear that particles in the electroweak plasma have a finite "thermal mass". I don't know if that's in addition to the zero mass I just mentioned, or if these are alternative pictures with alternative definitions of mass.

I will say one more thing: the thermal vacuum is globally defined, like the usual zero-temperature vacuum. That might be fine for the early universe, which by hypothesis is hot everywhere. But the question in this thread asks, what happens if a violent collision creates a finite high-temperature region in which electroweak symmetry is restored. How do you model that? @protonsarecool mentions the Keldysh formalism, which sounds right, but I have no idea of the details.

How I understood Nikkkom was this:

He seemed to be saying that even at Planck and GUT era (before the electroweak symmetry breaking era), the Higgs vev was always 246GeV. So the Higgs field can't have zero vev. Because right after the Big Bang.. the Higgs field already has 246GeV... never zero. So whenever I mentioned "vev". I always meant the Higgs field vev. Not the thermal plasma vev or temperature. In the following illustration. Even if the thermal plasma reaches 300 GeV (restoring the electroweak symmetry), the Higgs vev is still 246GeV:

I concluded this was nikkkom thought because of this thread he shared where he stated:

https://www.physicsforums.com/threa...during-the-gut-and-electroweak-epochs.936288/

"No. In the plasma that is in thermal equilibrium, each degree of freedom carries excitations with the same average energy. With temperatures >300GeV, Higgs field vacuum expectation value is smaller that this energy, thermal Higgs field excitations will be larger than it. In the "H0+h" decomposition, h can no longer be seen as small, and the whole reason for using such decomposition disappears."

Here nikkkom was saying that even if the thermal plasma temperature reaches 300 GeV restoring the electroweak symmetry, the Higgs vev is still 246GeV.

So the Higgs field vev can never be zero. Why do you say it can be zero? at what stage is it zero? To avoid confusion. Whenever we mentioned vev.. let's always reserved it for the Higgs field vev and not the thermal plasma which we can jus use the word temperature.

 

[jtlz]

How I understood Nikkkom was this:

He seemed to be saying that even at Planck and GUT era (before the electroweak symmetry breaking era), the Higgs vev was always 246GeV. So the Higgs field can't have zero vev. Because right after the Big Bang.. the Higgs field already has 246GeV... never zero. So whenever I mentioned "vev". I always meant the Higgs field vev. Not the thermal plasma vev or temperature. In the following illustration. Even if the thermal plasma reaches 300 GeV (restoring the electroweak symmetry), the Higgs vev is still 246GeV:

View attachment 222815

I concluded this was nikkkom thought because of this thread he shared where he stated:

https://www.physicsforums.com/threa...during-the-gut-and-electroweak-epochs.936288/

"No. In the plasma that is in thermal equilibrium, each degree of freedom carries excitations with the same average energy. With temperatures >300GeV, Higgs field vacuum expectation value is smaller that this energy, thermal Higgs field excitations will be larger than it. In the "H0+h" decomposition, h can no longer be seen as small, and the whole reason for using such decomposition disappears."

Here nikkkom was saying that even if the thermal plasma temperature reaches 300 GeV restoring the electroweak symmetry, the Higgs vev is still 246GeV.

So the Higgs field vev can never be zero. Why do you say it can be zero? at what stage is it zero? To avoid confusion. Whenever we mentioned vev.. let's always reserved it for the Higgs field vev and not the thermal plasma which we can jus use the word temperature.

Mitchell was right there might be some confusion.

Note when I asked about Higgs field vev.. I was not asking about the zero vacuum ev.. but the direct Higgs field vev.
And when I asked about Higgs potential. I was not asking about the potential of the vacuum thermal matter but the Mexican Hat Potential of the Higgs field.

I want to know details of why when the phase transition occured. The Mexican Hat suddenly have vev which becomes non-zero. So I presume the direct Higgs field vev can be zero before phase transition? Or does Nikkkom still believes the direct Higgs field vev was nonzero even at Planck era? Please clarify Nikkkom... Thanks.

 

[jtlz]

nikkkom mentioned in the other thread that "I explained this already. "Temperature" (I'm not sure you understand what is meant by this word in this context) does not control potential. Potential has a fixed shape: for every "temperature" (energy) it has one, unchanging value. You refuse to accept it."

How can I accept it when the mainstream mentioned the potential can be altered by temperature. For example:

http://blogs.discovermagazine.com/c...-the-higgs-or-something-like-it/#.WrozX0xuJ1U

"All of these directions are important. At high temperatures in the early universe, the Higgs bounces around in its potential, and its average value is zero (near the origin). But at lower temperatures things settle down, and the Higgs can oscillate around some point in that circle of minimum energy. Here is a crucial point: vibrations of the field in each direction are associated with particles, and the curvature of the potential corresponds to the mass of the associated particle. So the flat directions are massless particles, and the curved radial direction is a massive one."

I guess what nikkkom meant was the potential of the background particle was fixed. But then I was talking about the Higgs potential. In standard terminology, the Higgs potential was about the Mexican Hat.. is it not? So how come nikkkom attributed it to the background particles. What is he cooking.

Anyway. Is the Mexican hat dynamics literal? Like the Higgs really bounces around in its potential and its average value is zero (near the origin). Is this figurative description or literal? I'm analyzing what occurred in isolated electroweak plasma amidst the usual vacuum. Can I think there are different Mexican Hat behavior for the two boundaries? What goes on in the bubble? does it contract and expand or remain sharp boundary?

 

[vanhees71]

Well, the effective potential (or rather action) of thermal field theory indeed is dependent on temperature. It's one of the purposes of thermal QFT to evaluate this quantity to get an idea about the phase structure of the system. A non-zero vacuum expectation value always occurs when the effective potential is such that a minimum occurs at a non-zero value of the field. In electroweak theory the Higgs sector is modeled such that the Higgs-field potential has a minimum at a non-vanishing field value, because such a non-zero VEV leads to the fundamental masses of the particles in the SM without violating the underlying local chiral gauge symmetry. Note that this does not mean the spontaneous breaking of a symmetry but "Higgsing" a gauge symmetry; this is often not stated carefully enough even in standard and otherwise good textbooks, but it's good to know that a local gauge symmetry cannot be spontaneously broken and that thus there are no massless Goldstone modes in this case, because the ground state is not degenerate. This however is a feature, because there's no phenomenological use of Goldstone modes in this case but the would-be Goldstone modes (which would occur if the weak-isospin-hypercharge symmetry was global) are "eaton up" by the corresponding gauge fields, making the three W- and Z-bosons massive and leave the photon massless as it must be (i.e., you Higgs the original $\text{SU}(2)_{\text{wiso}} \times \text{U}(1)_{\text{Y}}$ down to $\mathrm{U}(1)_{\text{em}}$.

At sufficiently high temperatures the effective Higgs potential goes from the characteristic mexican-hat shape to provide the Higgs mechanism in the vacuum and at low temperatures into a shape with only a minimum at vanishing field-expectation values. The following seems to be a nice writeup on both the electroweak and the strong phase transition in the early universe:

https://fyzika.uniza.sk/~melo/DoruSticlet.pdf

 

[jtlz]

Well, the effective potential (or rather action) of thermal field theory indeed is dependent on temperature. It's one of the purposes of thermal QFT to evaluate this quantity to get an idea about the phase structure of the system. A non-zero vacuum expectation value always occurs when the effective potential is such that a minimum occurs at a non-zero value of the field. In electroweak theory the Higgs sector is modeled such that the Higgs-field potential has a minimum at a non-vanishing field value, because such a non-zero VEV leads to the fundamental masses of the particles in the SM without violating the underlying local chiral gauge symmetry. Note that this does not mean the spontaneous breaking of a symmetry but "Higgsing" a gauge symmetry; this is often not stated carefully enough even in standard and otherwise good textbooks, but it's good to know that a local gauge symmetry cannot be spontaneously broken and that thus there are no massless Goldstone modes in this case, because the ground state is not degenerate. This however is a feature, because there's no phenomenological use of Goldstone modes in this case but the would-be Goldstone modes (which would occur if the weak-isospin-hypercharge symmetry was global) are "eaton up" by the corresponding gauge fields, making the three W- and Z-bosons massive and leave the photon massless as it must be (i.e., you Higgs the original $\text{SU}(2)_{\text{wiso}} \times \text{U}(1)_{\text{Y}}$ down to $\mathrm{U}(1)_{\text{em}}$.

At sufficiently high temperatures the effective Higgs potential goes from the characteristic mexican-hat shape to provide the Higgs mechanism in the vacuum and at low temperatures into a shape with only a minimum at vanishing field-expectation values. The following seems to be a nice writeup on both the electroweak and the strong phase transition in the early universe:

https://fyzika.uniza.sk/~melo/DoruSticlet.pdf

Thanks for the explanation. I guess the mystery of the zero vev lies in this passage in the paper.. "As temperature rises and thermal fluctuations grow, the field will begin to oscillate quickly between the minima of the system. These fluctuations will be so rapid that the average of the field over some large period of time will just give zero [1]. This is in sum the phenomenological description of the symmetry restoration process; where from a state were the field is trapped at a non-zero value we arrived by increasing temperature to a state symmetric under φ →−φ, as the Lagrangian (2.1)".

So it's zero vev because it fluctuates so fast.. hmm...

I read about domain walls too:
" A different consequence of cosmological first order transitions could be the formation of domain walls. Different, but energetically degenerate, vacua can be chosen in separate regions of space. As these regions expand, domain walls can form at their meeting point. Domain walls would act as a potential barrier between the different vacua and thus have a positive energy associated to them. If their length stretches large portions of space, then their total energy would be high enough for them to be detected"

About these domain walls separating different vacua.. is it not thermal field theory deals only with uniform vacuum.. how do you model vacua where different domain walls and vacuum exists like in the following original illustration?

 

[mfb]

So it's zero vev because it fluctuates so fast.. hmm...

Or, in other words, the small bump around zero becomes negligible.

A similar field structure can be found in a completely different area: The formation of neutral antihydrogen. Both positrons and antiprotons have to be trapped, but they have opposite charge. What experimenters use is a mexican-hat like potential. The antiprotons see the mexican hat potential but they have enough energy to reach the middle as well. The positrons see an inverted mexican hat potential, but they have a very low energy so they are trapped in the center and cannot escape from this local minimum. Here is a sketch

 

[jtlz]

Or, in other words, the small bump around zero becomes negligible.

A similar field structure can be found in a completely different area: The formation of neutral antihydrogen. Both positrons and antiprotons have to be trapped, but they have opposite charge. What experimenters use is a mexican-hat like potential. The antiprotons see the mexican hat potential but they have enough energy to reach the middle as well. The positrons see an inverted mexican hat potential, but they have a very low energy so they are trapped in the center and cannot escape from this local minimum. Here is a sketch

Thanks. Usualy quantum fields are defined all over space and time. For example. Our electron field doesn't have boundary. However when you put phase boundary or walls between domains (i don't know if domain wall is the right term) for example where one region is undergoing phase transitions while the surrounding is normal vaccum. Can this be modeled by normal qft or even thermal field theory? Does it even make sense to you? What is the properties of this phase boundary? Doesnt it violate lorentz invariance since you are locating the boundary in space and time?