What is the cause of the Mexican hat potential of the Higgs field?

[JandeWandelaar]

TL;DR Summary

In QFT, the Higgs potential is given without further explanation. Can we find a cause for it, without observing it does the job?

The Higgs mechanism is an ingenious mechanism inspired by condensed state physics. The famous Mexican hat potential ensures a VEV value of about two times the mass of the Higgs particle (which, as an aside, is of comparable order as the W and Z vector bosons, the difference though is that its a scalar particle).

There is one thing though I fail to understand. The shape of the potential. How is this shape generated?

 

[Vanadium 50]

You get this any time you have a quadratic and quartic term of opposite signs.

 

[JandeWandelaar]

Thank you. I understand the mechanism but doesn't a potential have a cause?

 

[malawi_glenn]

Thank you. I understand the mechanism but doesn't a potential have a cause?

and the cause of that must have a cause, and so one, in an infinite chain of "logic"?

 

[JandeWandelaar]

and the cause of that must have a cause, and so one, in an infinite chain of "logic"?

Can't there be a basic cause?

 

[malawi_glenn]

Can't there be a basic cause?

Can't the shape of the Higgs-potential be a "basic" cause? It follows from the field structure of the electroweak SM and the value of its parameters.

 

[JandeWandelaar]

Yes, I understand. But somehow that feels as reversing things.

 

[malawi_glenn]

Why does quarks have three "colors"? It fits the observations...

 

[JandeWandelaar]

Epicycles fitted observations too. There could be a deeper cause for quarks having color and leptons having none. But that's beyond mainstream.

 

[malawi_glenn]

Epicycles fitted observations too. There could be a deeper cause for quarks having color and leptons having none. But that's beyond mainstream.

They did for sure, but then later one one discovered ellipses and all that jazz.

That is not beyond mainstream, ever heard of GUTs? The Georgi-Glashow SU(5) model for instance, as U(1)xSU(2)xSU(3) as a subgroup. But then you would ask "why SU(5)? and why is it broken in this way?" well, because it works (at least in theory up to some point)

 

[arivero]

Ok, the question is "why the sign of lambda respect to the sign of mu square"

Can't the shape of the Higgs-potential be a "basic" cause? It follows from the field structure of the electroweak SM and the value of its parameters.

or "why the value of the parameters". This is relevant because of the running of the parameter lambda, the higgs four-point coefficient that produces the mexican hat.

So a possible answer to the original question is: "Asymptotic safety "

Now allow me a meta-comment: a lot of the PF users, and most of the admins, know this kind of things. They do not answer either to allow participation of other users, or to avoid getting involved in a long series of increasingly unfocused questions and answers, which takes a lot of time for nothing. But I find preferable to throw one hint and abandon the thread - as I am going to do- that just to allow the thread to become a game of philosophical pingpong.

 

[JandeWandelaar]

Well, the point is that there is no explanation for the potential given. Asymptotic safety sounds great but still leaves open the question where it came from. Which is beyond the mainstream. Isn't a potential connected to force or interaction? So what interaction is associated to the potential? We don't know.

 

[malawi_glenn]

Isn't a potential connected to force or interaction?

So what interaction is associated to the potential?

$\mu \Phi^\dagger \Phi +\lambda ( \Phi^\dagger \Phi)^2$

Higgs-field self-interactions.

 

[JandeWandelaar]

That's what I mean with reasoning the other way round.

 

[JandeWandelaar]

$\mu \Phi^\dagger \Phi +\lambda ( \Phi^\dagger \Phi)^2$

Higgs-field self-interactions.

There is no way out... Trapped in Higgs...

 

[JandeWandelaar]

$\mu \Phi^\dagger \Phi +\lambda ( \Phi^\dagger \Phi)^2$

Higgs-field self-interactions.

 

[JandeWandelaar]

$\mu \Phi^\dagger \Phi +\lambda ( \Phi^\dagger \Phi)^2$

Higgs-field self-interactions.

But do these explain the shape of the potential? The self-interaction applies to the potential already there. But what causes the a Higgs field to have that potential energy shape? What causes the Higgs field to wear a Mexican hat. You can simply say that's how it is, but don't you, as a physicist want to know what causes it? There are, by the way, more ways to explain the state of affairs, like there were, long ago, more theories than epicycles to explain the phenomena.

 

[JandeWandelaar]

$\mu \Phi^\dagger \Phi +\lambda ( \Phi^\dagger \Phi)^2$

Higgs-field self-interactions.

I see only now that your avatar is the Mexican hat!

 

[ohwilleke]

But do these explain the shape of the potential? The self-interaction applies to the potential already there. But what causes the a Higgs field to have that potential energy shape? What causes the Higgs field to wear a Mexican hat. You can simply say that's how it is, but don't you, as a physicist want to know what causes it? There are, by the way, more ways to explain the state of affairs, like there were, long ago, more theories than epicycles to explain the phenomena.

Physics, even the most fundamental physics, of the Standard Model and General Relativity, is fundamentally descriptive of the behavior of the universe as we observe it.

There is no answer to why these formulas have the form that they do, or why parameters have the values that they do, in the deep sense that you mean it. We don't have a theory of everything (TOE) from which all laws of nature can be deduced from first principles. We don't even have a satisfactory grand unified theory (GUT) to explain non-gravitational phenomena.

At most, we have hints in the structure of the formulas we do have, their properties, and the values of the fundamental parameters that we have measured, to connections a layer or so deeper than the Standard Model.

For example, we know that essentially all laws of physics, on average, obey the principle of least action. We know that Standard Model fermions, with the possible exception of neutrino masses (but even this is probably not an exception) get more massive at higher generations of the same kind of particle than at lower generations. We know that the probability of a two generation transition of a quark is roughly the same as the probability of making the first one generational transition time the probability of making the second one generational transition. We know the sum of the square of the rest masses of the Standard Model fundamental particles is very close to the square of the Higgs vacuum expectation value. We know that charged lepton masses are consistent to a high degree of precision with Koide's rule. But, we really don't have any deeper theory to derive any of this from first principles, and quite possibly, never will.

There is no harm in considering the reductionist possibility that the laws of physics we know could be derived from simpler more basic laws. But we don't know that, that is merely an article of faith. When we discovered quarks, muons, and the like, a very simple proton-neutron-electron model actually got a lot more complicated. It could be that the next layer beneath the Standard Model is more complicated still, before it gets simpler, if it ever does. But no one has a defensible true answer to the question you are asking. At most, people can imagine possibilities that make sense.

The Mexican hat potential was hypothesized roughly 40 years before the Higgs boson was discovered because it was the mathematically easiest way to get a theory with the properties desired to match what was observed and to fit the overall order those observations seemed to be have.

The goal was to have a field that could contribute mass to fundamental particles which otherwise would have zero rest mass, since simply putting rest masses into the theory without a coupling to a separate field would have caused the otherwise basically sound equations of particle physics to be mathematically inconsistent. The Higgs mechanism solved that problem in the highly constrained context of a theory that could impart rest mass to fundamental particles without screwing up the rest of the equations.

But, it was merely a guess, and it isn't the only conceivable theory that could produce the observations that we see to current precision. However, since it seems to work, the scientific community isn't devoting much effort to considering alternatives.

 

[JandeWandelaar]

The Higgs field imparts mass without screwing up gauge invariance. The virtual Higgs just dumps energy without dumping momentum, which, accirding to the relativistic energy-momentum relation means mass. Its rather strange though that a field has minimum energy if particles are present. What's the difference with the zero-energy
vacuum field of other particles, where also virtual particles are present? The zero energy Higgs field doesn't contain real particles. What's the nature of the virtual Higgs particles? Do they only contain energy?

 

[malawi_glenn]

Virtual particles are just propagators that arise in perturbative QFT calculations. Would you argue that Faddeev–popov ghost particles exists too? Feynman diagrams are just computational tools. Vaccum energy exists though.

What about quark condensate $\langle \psi \bar \psi \rangle$ and gluon condensate $\langle G^{\mu \nu} G_{\mu \nu} \rangle$?

don't you, as a physicist want to know what causes it?

I have written research papers about higgs bosons, what you you done?
Asking too many "why" "why" "why" questions will just make you end up in an endless loop of asking that question over and over again. What if the mexican hat potential came out from some crazy string theory calculation, then would you ask "why strings, what put them there"?

Why do we have electrons? Or muons? Who ordered those? Wasn't the electron enough? Don't get me started on the tauon... oh that little bugger.

A better question would be "what caused the symmetry to become broken?" Search for "Electroweak phase transition" https://arxiv.org/abs/2103.14022

There are, by the way, more ways to explain the state of affairs

Show me a couple of reasearch articles then.

 

[JandeWandelaar]

Virtual particles are just propagators that arise in perturbative QFT calculations. Would you argue that Faddeev–popov ghost particles exists too? Feynman diagrams are just computational tools. Vaccum energy exists though.

What about quark condensate $\langle \psi \bar \psi \rangle$ and gluon condensate $\langle G^{\mu \nu} G_{\mu \nu} \rangle$?
I have written research papers about higgs bosons, what you you done?
Asking too many "why" "why" "why" questions will just make you end up in an endless loop of asking that question over and over again. What if the mexican hat potential came out from some crazy string theory calculation, then would you ask "why strings, what put them there"?

Why do we have electrons? Or muons? Who ordered those? Wasn't the electron enough? Don't get me started on the tauon... oh that little bugger.

A better question would be "what caused the symmetry to become broken?" Search for "Electroweak phase transition" https://arxiv.org/abs/2103.14022
Show me a couple of reasearch articles then.

I would go off mainstream if I show an alternative to mass generation. But there is a natural one. For example, three Weyl spinors interacting by gluon-like gauge fields.

The graviton condensate, which is considered a mathematical tool only, can just as well be real. Why not. If gravitons are considered non-point-like, like strings, the singularities van be avoided. Strings can't collapse to a point.

It depends where you want to end asking why. If there is no more reason to ask then there is no more reason. But there is.

 

[berkeman]

I would go off mainstream if I show an alternative to mass generation. But there is a natural one. For example, three Weyl spinors interacting by gluon-like gauge fields.

The graviton condensate, which is considered a mathematical tool only, can just as well be real. Why not. If gravitons are considered non-point-like, like strings, the singularities van be avoided. Strings can't collapse to a point.

It depends where you want to end asking why. If there is no more reason to ask then there is no more reason. But there is.

Thread closed for Moderation, obviously...

UPDATE -- Thread reopened.

 

[mitchell porter]

Since the thread is still open...

I would go off mainstream if I show an alternative to mass generation. But there is a natural one. For example, three Weyl spinors interacting by gluon-like gauge fields.

Is this intended as an alternative to mass generation by the Higgs mechanism? Are you thinking of preon theories, like rishons coupled by hypercolor?

The graviton condensate, which is considered a mathematical tool only, can just as well be real. Why not. If gravitons are considered non-point-like, like strings, the singularities van be avoided. Strings can't collapse to a point.

Here you seem to be talking about something else entirely, like avoiding singularities in gravitational collapse?

As for the headline question, the immediate cause of the Mexican hat potential is what @Vanadium 50 said in comment #2: having quadratic and quartic self-interactions of opposite signs. If you want to go deeper, well, the principles of renormalizable field theory can explain why a scalar field will have quadratic and quartic self-interactions, but not their specific coefficients; the specific coefficients might be explained by some beyond-standard-model physics in the UV, like asymptotic safety (@arivero #11) or the contingencies of some messy string vacuum (perhaps anthropically selected). But I feel like you dislike the Higgs mechanism per se.

 

[HomogenousCow]

Back when the SM was being developed renormalizability was a lot more important to theorists so they picked only renormalizable terms for the Lagrangian, and the quartic self interaction is the only renormalizable way to have spontaneous symmetry breaking for the Higgs field. Why did this approach work so well in light of the modern view on renormalizability? A good guess is that there must be some very large gap between the electroweak scale and whatever scale the next tier of physics takes place at hence all non-renormalizable terms in the Higgs potential are suppressed.

In other words you're free to propose your own version of the SM, with additional terms like
$$~a(\Phi^\dagger\Phi)^3+b(\Phi^\dagger\Phi)^4+c(\Phi^\dagger\Phi)^5+...$$ but the energy scale at which these terms become important has been determined to be much higher than what can be achieved today.

 

[ohwilleke]

A good guess is that there must be some very large gap between the electroweak scale and whatever scale the next tier of physics takes place at hence all non-renormalizable terms in the Higgs potential are suppressed.

Another good guess is that there isn't a higher energy tier of physics, something that became theoretically possible, at least up to the GUT scale, when the fairly low mass of the Higgs boson was determined and it turned out the the SM physical constants can run up to that scale without giving rise to mathematical defects in the theory that would be unphysical.

 

[HomogenousCow]

Another good guess is that there isn't a higher energy tier of physics, something that became theoretically possible, at least up to the GUT scale, when the fairly low mass of the Higgs boson was determined and it turned out the the SM physical constants can run up to that scale without giving rise to mathematical defects in the theory that would be unphysical.

Could you elaborate on this? Sounds interesting but I don't fully understand the argument.

 

[Son Goku]

Back when the SM was being developed renormalizability was a lot more important to theorists so they picked only renormalizable terms for the Lagrangian, and the quartic self interaction is the only renormalizable way to have spontaneous symmetry breaking for the Higgs field. Why did this approach work so well in light of the modern view on renormalizability? A good guess is that there must be some very large gap between the electroweak scale and whatever scale the next tier of physics takes place at hence all non-renormalizable terms in the Higgs potential are suppressed.

Considering old style renormalizability is concerned with which theories are well-defined in the continuum, i.e. at all scales, I always found this one of the coolest things in modern physics. The conditions for a term to remain unsupressed and survive the flow from the original cut-off action is exactly the same as old school renorm conditions.
So QFTs at low energies always look as if they are fundamental, even when they are not.

 

[malawi_glenn]

Could you elaborate on this? Sounds interesting but I don't fully understand the argument.

Section 1.4 https://arxiv.org/abs/hep-ph/0503172 you can also check the references

 

[ohwilleke]

Section 1.4 https://arxiv.org/abs/hep-ph/0503172 you can also check the references

Also (in a pdf shorter than 339 pages): https://arxiv.org/abs/1205.2893

https://www.diva-portal.org/smash/get/diva2:819933/FULLTEXT01.pdf

 

[malawi_glenn]

in a pdf shorter than 339 pages

No need to read 339 pages, I wrote which section to read