# How does the Higgs scalar potential evolve with temperature?

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How does the Higgs scalar potential evolve with temperature?

Is there possibility they are independent?

Besides temperature. What else can theoretically affect the higgs scalar potential?

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The higgs vacuum is separate from the plasma that allegedly caused the phase transition.

Is there a possibility (in some models) that they are separate such that the higgs vacuum or other vacuum can experience phase transition without the plasma (example of plasma is the electroweak plasma above 100GeV that supposedly cause phase transition of the higgs vacuum)?

It seems you do not understand what "Higgs potential" is.

Each allowed field configuration has some potential energy. IOW: potential does not "change", it is a fixed value for every allowed field configuration.

If plasma temperature is above, say, 200 GeV, it means that space is filled by various particles with mean energy of >200GeV. Particles with these energies will easily create all other known SM particles. This means that energy of the plasma will be equally distributed among all possible "types of particles" (more precisely, among all degrees of freedom). _Including_ Higgs bosons.

This, in turn, means that Higgs vacuum expectation value does not matter anymore: Higgs field "jumps around" so much that the fact that it would settle to a nonzero value if "cooled" is not important. The behavior of the system is not much different from a theory where Higgs field would have a minimum at zero.

Imagine a hot frying pan with uneven bottom. If you heat it up to 400C and drip some water on it, droplets will ignore the existence of small "minimums" on the bottom - they will jump around all over the frying pan. The water would collect there only if the pan is cold.

jtlz
It seems you do not understand what "Higgs potential" is.

Each allowed field configuration has some potential energy. IOW: potential does not "change", it is a fixed value for every allowed field configuration.

If plasma temperature is above, say, 200 GeV, it means that space is filled by various particles with mean energy of >200GeV. Particles with these energies will easily create all other known SM particles. This means that energy of the plasma will be equally distributed among all possible "types of particles" (more precisely, among all degrees of freedom). _Including_ Higgs bosons.

This, in turn, means that Higgs vacuum expectation value does not matter anymore: Higgs field "jumps around" so much that the fact that it would settle to a nonzero value if "cooled" is not important. The behavior of the system is not much different from a theory where Higgs field would have a minimum at zero.

Imagine a hot frying pan with uneven bottom. If you heat it up to 400C and drip some water on it, droplets will ignore the existence of small "minimums" on the bottom - they will jump around all over the frying pan. The water would collect there only if the pan is cold.
Are you not familiar with the Higgs phase transition details. See
https://physics.stackexchange.com/questions/205607/how-do-symmetries-break-in-cosmology

"Symmetry breaking in the early Universe occurs through finite temperature effects. To study this one must look at the finite temperature effective action for the scalar field, involving a thermal field theory calculation. A scalar field receives contributions to it's effective potential from the relativistic degrees of freedom it couples to. So as the Higgs field couples to fermions and gauge bosons which are light and relativistic in the early Universe this modifies the potential.
Typically the potential has two terms:
V = V(T=0) + V(T)
At leading order V(T) ~ M^2 T^2 where T is the temperature of the thermal bath and M is the mass term of the fields the Higgs field couples to.

The key point is that at large T'>TT the thermal corrections dominate and the potential has a single minimum at the origin. As the temperature cools the thermal corrections fall off and new minima appear. Eventually the Higgs field is free to evolve away from the origin towards the new minima during which it breaks the symmetry."

I just want confirmation or elaboration of it. This was because I got this contradictory statements from:
https://physics.stackexchange.com/questions/205607/how-do-symmetries-break-in-cosmology

"I don't think that an unambiguous justification can be given because the dynamic of the electroweak symmetry breaking (EWSB) is still unknown. We don't have a well established theory describing how the Higgs scalar potential evolves with the temperature. When people talk about the scale of the EWSB, they usually refer to two possible things:
1. before EWSB, the weak bosons are massless. After EWSB, they get a mass (91 GeV for Z0 and 80 GeV for W+- The scale is therefore of the order of the mass of the weak bosons, roughly 100 GeV.
2. before EWSB, the Higgs vacuum expectation value (v.e.v.) is 0, the field is symmetric. After EWSB, the v.e.v. is about 246 GeV. So again, the v.e.v. value is representative of the scale of the EWSB, still of the order of 100 GeV."

I want to know if the dynamic of the electroweak symmetry breaking (EWSB) is really unknown or known.

And whether we DO or DON'T have a well established theory describing how the Higgs scalar potential evolves with the temperature.

I want to know if the dynamic of the electroweak symmetry breaking (EWSB) is really unknown or known.

And whether we DO or DON'T have a well established theory describing how the Higgs scalar potential evolves with the temperature.
It is known as far as perturbative methods work. The papers which discuss (meta)stability of SM vacuum based on experimentally measured masses of Higgs and top quark calculate the shape of the potential and try to determine whether it has a second minimum at very high field values, and whether this minimum is below the one at 246 GeV. They do it by calculating perturbative quantum corrections to Higgs self-coupling and to Higgs-to-top Yukawa coupling, and then calculate the potential.

If potential indeed has a second minimum, and it is lower, _and_ it happens well before Planck energy, then "we have a problem": vacuum seems to be metastable.

Regardless of the results for field values below Planck mass, at about Planck mass unknown quantum gravity effects should come into play, and they will affect potential too. So in this regard I would say we don't have a well established theory describing how the Higgs scalar potential evolves with the temperature.

jtlz
It is known as far as perturbative methods work. The papers which discuss (meta)stability of SM vacuum based on experimentally measured masses of Higgs and top quark calculate the shape of the potential and try to determine whether it has a second minimum at very high field values, and whether this minimum is below the one at 246 GeV. They do it by calculating perturbative quantum corrections to Higgs self-coupling and to Higgs-to-top Yukawa coupling, and then calculate the potential.

If potential indeed has a second minimum, and it is lower, _and_ it happens well before Planck energy, then "we have a problem": vacuum seems to be metastable.

Regardless of the results for field values below Planck mass, at about Planck mass unknown quantum gravity effects should come into play, and they will affect potential too. So in this regard I would say we don't have a well established theory describing how the Higgs scalar potential evolves with the temperature.
Bottom line is. Can one prove it was temperature that set off the higgs phase transition from nonzero vev to zero vev or vice versa.. or was it just a coincidence? Imagine a chef wants to cook dinner for some guests. He knows the end results (how the final food will look like). And when the salmon is being baked, he boils the separate soup. It doesn't mean boiling the soup is what baked the salmon. Likewise. Is it possible what produced the higgs phase transition from zero to nonzero vev was not temperature but it was just switched on by purpose to give mass to particles? Or is it really triggered by temperature? How exactly if the latter is so?

Bottom line is. Can one prove it was temperature that set off the higgs phase transition from nonzero vev to zero vev or vice versa..
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.

jtlz
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.
or lets just say the vev before higgs field phase transition (and electroweak symmetry breaking) where the electroweak bosons were massless.. Is it really temperature that caused the phase transition? Or just coincidence the higgs field phase transition occured at the temperature?

If we could create more than 100 GeV plasma.. would it restore the symmetry and make the weak force bosons massless again, etc.? Is there solid proof temperature is what controls the higgs field phase transition?

jtlz
I've just read it. I was familiar with most of what Peterdonis was describing because the book Deep Down Things gave the details of the electroweak symmetry breaking and higgs field phase transition. What I was asking you was whether there was solid proof temperature is what controls the higgs field phase transition which caused electroweak symmetry breaking. Couldn't be it's another field that controls the phase transition..

But if it's the coefficients in the scalar potential really depend on temperature. Then it proves it that temperature really controls the phase transition and not some hidden field which controls it on and off. My context is thinking if the Big Bang is like cooking menu already made up. Not because of accidents of nature or naturalness or multiverse.. but simply menu designed to create our world and you and me..

"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."

I thought when the temperature was above 300 GeV. The Higgs vev disappears.. so it is always there.. but then how come when the temperature is above 300 GeV. The electroweak bosons became massless?

Peterdonis described the following. Yes I want to understand what physically happens during the transition:

"Physically, what happened during this transition? One way to think of it is to consider a ball at the top of a perfectly symmetrical hill (the crown of the "Mexican hat"); this corresponds to the four Higgs fields at high temperature. Because the temperature is high, the ball is constantly being kicked, so even if it starts to roll down the hill, it will quickly be kicked back up again. However, as the temperature drops, the kicks get less powerful, and finally a point is reached where the ball starts to roll down the hill and doesn't get kicked back up to the top; so it keeps rolling and eventually reaches the bottom (the trough of the "Mexican hat"). Once it's in the trough, it's stuck, because there is another upward incline on the other side, and the temperature is too low now to kick it uphill any significant distance."

Is this literally? So the reason the electroweak bosons are massless when the plasma is above the higgs field transition temperature is because the H+, H-, H0, H0* higgs can't be eaten up by the electroweak bosons because the 4 higgs were constantly being kicked around? Like you constantly kicked the 4 bones and the 3 dogs can't eat 3 of the bones? Is this all literally?

But if it's the coefficients in the scalar potential really depend on temperature. Then it proves it that temperature really controls the phase transition and not some hidden field which controls it on and off.
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.

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.
When I read Deep Down Things. It stuck to my mind:

"Or, in terms of our schoolyard metaphor, the external Higgs potential is the sensation of cold on the feet of the students, and their resulting physiological need to get something on their feet, while the magnitude of the Higgs field can be one of two possibilities: zero (no shoes on) or one (one of the four equivalent pairs of shoes on each student’s feet)."

It stuck in my mind that the higgs field can be zero or nonzero. So it can't be zero and always nonzero. Ok. I'll re read everything with new framework in mind. Thanks.

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.
The "zero vev" thing is a common mistake made by so many and I thought of it for a decade. I think even Peterdonis committed it. in message 11 of https://www.physicsforums.com/threa...during-the-gut-and-electroweak-epochs.936288/ Peterdonis quoted:

"In this model, the point where the ball is sitting corresponds to the vacuum expectation value of the Higgs field. The origin would be a zero value; any other point is a nonzero value. So the ball sitting in the trough has a nonzero vacuum expectation value, which we'll call ##\phi_0##."

So he was wrong the origin would be a zero value? Because you said the vev could never be zero?

He meant just "the value of the Higgs field". Not vacuum expectation value.

He meant just "the value of the Higgs field". Not vacuum expectation value.
But how can the Higgs field have value of zero (in his context)?

Anyway vacuum expectation value was fixed right when the Big Bang started? Most laymen thought it just suddenly came about after the so called phase transition.

But how can the Higgs field have value of zero (in his context)?
How can photon field have value of zero? How can electron field have value of zero?

Anyway vacuum expectation value was fixed right when the Big Bang started?
No. It is fixed by theory's equations.

Most laymen thought it just suddenly came about after the so called phase transition.
Phase transition refers to the transition from large field fluctuations ("high energy/temperature"), large enough that non-symmetrical vacuum state does not affect the behavior of the system, to smaller field fluctuations which are now sensitive to the asymmetry of vacuum state. See my example of frying pan above.

"Vacuum expectation value" of a field is the value of the field in the vacuum. (If there are more than one vacuum state, each may have different VEVs).

How can photon field have value of zero? How can electron field have value of zero?

No. It is fixed by theory's equations.

Phase transition refers to the transition from large field fluctuations ("high energy/temperature"), large enough that non-symmetrical vacuum state does not affect the behavior of the system, to smaller field fluctuations which are now sensitive to the asymmetry of vacuum state. See my example of frying pan above.

"Vacuum expectation value" of a field is the value of the field in the vacuum. (If there are more than one vacuum state, each may have different VEVs).
Thanks. The Higgs field is often referred to as superconducting Higgs fields.. I wonder to what extend is the analogy true. According to the theory of superconductivity... "In an actual superconductor, the charged particles are electrons, which are fermions not bosons. So in order to have superconductivity, the electrons need to somehow bind into Cooper pairs. The charge of the condensate q is therefore twice the electron charge −e. The pairing in a normal superconductor is due to lattice vibrations, and is in fact very weak; this means that the pairs are very loosely bound. The description of a Bose–Einstein condensate of loosely bound pairs is actually more difficult than the description of a condensate of elementary particles, and was only worked out in 1957 by Bardeen, Cooper and Schrieffer in the famous BCS theory."

In the Higgs field. The 4 Higgs degrees of freedom (in the SU(2) doublet) didn't really form something akin to cooper pairs? So the analogy is not true anymore? So what part of the Higgs field can be said to be like superconductors? Is it still ok to use the analogy or better not?

What is the origin and physical interpretation of the Higgs potential or the Mexican Hat?

It seems some people don't even believe in it. Like nikkkom.. he seems to prefer thinking of the vacuum thermal potential and not the Higgs potential.. as if the latter is not proven to really exist. Is it really subject to controversy like the physical interpretation of the wave function?

Is the temperature bouncing the higgs at the middle also just for sake of illustration.. meaning there is possibility the Higgs potential are controlled not even by temperature but another field hidden? Note when I speak of the Higgs potential.. I don't mean the thermal potential of the background plasma whose potential may be fixed and this is what nikkkom has in mind.

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It seems some people don't even believe in it. Like nikkkom..
It's a low form of debate to put words in other people's mouths.

It's a low form of debate to put words in other people's mouths.
In this thread. nikkkom kept avoiding the standard mainstream views like about the coefficient of the higgs scalar potential depending on temperature and doesn't want to admit the higgs vev can be zero. He used strong words like :

"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."

This is contrary to the Mexican Hat higgs potential jumping up and down from influence from temperature.

Hence it seems for some reason nikkkon is trying to avoid the Mexican Hat thing... maybe he doesn't believe in it? I need his confirmations of his views.

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It's clear from your other posts you don't understand the Higgs potential. You might think about listening to the people who are trying to help you rather than misrepresenting them.

It's clear from your other posts you don't understand the Higgs potential. You might think about listening to the people who are trying to help you rather than misrepresenting them.
Yes I will. I'm confused about nikkkom saying Higgs potential is fixed in value while in the Mexican Hat.. It varies in value with regards to temperature. Maybe nikkkom is referring to thermal potential of background particles while for the varying Mexican vev.. it's about the Higgs potential itself? Can anyone else please explain? Thanks.

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nikkkom saying
Again, you are putting words in other people's mouths. After three times, people might start to conclude "dishonesty" rather than "honest mistake". I don't think that's the conclusion you want them to draw. The only person who mentioned Mexican Hat in this thread is you.

My advice is the same. You might think about listening to the people who are trying to help you rather than misrepresenting them.

Again, you are putting words in other people's mouths. After three times, people might start to conclude "dishonesty" rather than "honest mistake". I don't think that's the conclusion you want them to draw. The only person who mentioned Mexican Hat in this thread is you.

My advice is the same. You might think about listening to the people who are trying to help you rather than misrepresenting them.
Sorry. I just want to know how the Mexican Hat is related to nikkkom earlier advice that the Potential is fixed and not affected by temperature.. so hope he can share later.. thanks..