Before we continue, keep in mind the important disclaimer:

All this discussion is valid assuming the standard model is the correct theory all the way up to the Planck scale, which is unlikely.Indeed, while it's very likely that the standard model is an adequate description of physics at the energies probed by the LHC, we have no compelling reasons to assume it works at, say, 100 TeV. On the contrary, we know there should be some new particles somewhere, at least to account for dark matter and the baryon asymmetry in the universe, and those degrees of freedom may well affect the discussion of vacuum stability. But for the time being let's assume there's no new particles beyond the standard model with a significant coupling to the Higgs field.

The stability of our vacuum depends on the sign of the quartic coupling in the λ |H|^4 term in the Higgs potential: for negative λ the potential is unbounded from below and therefore unstable. We know exactly the value of λ at the weak scale: from the Higgs mass 125 GeV and the expectation value 246 GeV it follows that λ = 0.13, positive of course. But

*panta rhei*and λ is no exception. At large values of |H|, the Higgs potential in the standard model is, to a good approximation, given by λ(|H|) |H|^4 where λ(|H|) is the

*running*coupling evaluated at the scale |H|. If Higgs were decoupled from the rest of matter then λ would grow with the energy scale and would eventually explode into a Landau pole. However, the Yukawa couplings of the Higgs boson to fermions provide another contribution to the evolution equations that works toward decreasing λ at large energies. In the standard model the top Yukawa coupling is large, of order 1, while the Higgs self-coupling is moderate, so Yukawa wins.

In the plot showing the evolution of λ in the standard model (borrowed from the latest state-of-the-art paper) one can see that at the scale of about 10 million TeV the Higgs self-coupling becomes negative. That sounds like a catastrophe as it naively means that the Higgs potential is unbounded from below. However, we can reliably use quantum field theory only up to the Planck scale, and one can assume that some unspecified physics near the Planck scale (for example, |H|^6 and higher terms in the potential) restore the boundedness of the Higgs potential. Still, between 10^10 and 10^19 GeV the potential is negative and therefore it has a global minimum at large |H| that is much deeper than the vacuum we live in. As a consequence, the path integral will receive contributions from the field configurations interpolating between the two vacua, leading to a non-zero probability of tunneling into the other vacuum.

Fortunately for us, the tunneling probability is proportional to Exp[-1/λ], and λ gets only slightly negative in the standard model. Thus, no reason to panic, our vacuum is meta-stable, meaning its average lifetime extends beyond December 2012. Nevertheless, there is something intriguing here. We happen to occupy a very special patch of the standard model parameter space. First of all there's the good old hierarchy problem: the mass term of the Higgs field takes a very special (fine-tuned?) value such that we live extremely close to the boundary between the broken (v > 0) and the unbroken (v=0) phases. Now we realized the potential is even more special: the quartic coupling is such that two vacua coexist, one at low |H| of order TeV and the other at large |H| of order the Planck scale. Moreover, not only λ but also it's beta functions is nearly zero near the Planck scale, meaning that λ evolves very slowly at high scales. Who sets these boundary conditions? Is that yet another remarkable coincidence, or is there a physical reason? Something to do with quantum gravity? Something to do with inflation? I think it's fair to say that so far nobody has presented a compelling proposal explaining these boundary conditions satisfied by λ.

Ah, and don't forget the disclaimer:

All this discussion is valid assuming the standard model is the correct theory all the way up to the Planck scale, which is unlikely.

## 21 comments:

I just keep on thinking that this damned mass term in the Higgs field simply is not there. It should be a conformal Standard Model.

Aah, now I know what should happen on 21.12.2012, LOL :-D

Thanks for this interesting and nicely accessible post.

Dilaton

Fascinating....and people said that discovering the Higgs would be the end of cool new shit in physics. Perhaps its discovery is the doorway to understanding the answer to the hierarchy problem and quantum gravity. And it seems to be implying for the first time that not only MIGHT there be new physics at the Planck scale, but that there HAS to be. And new questions to be solved: Why does our universe exist in the metastable realm? Is that related to what could develop a viable theory quantum gravity? Does the hierarchy question become moot or answerable in the process? Is this further "evidence" of the multiverse speculation since universes may conceivably exist all over the spectrum of stable, unstable, and metastable? What are the mechanisms establishing a lower boundary on the quartic coupling?

I find this post just utterly bewildering. Maybe someone can clear it up.

First, its difficult to accept that there is some more-stable, lower-energy vacuum to which the standard model could "tunnel." The reason is that there are parts of the universe where, I would think, the energy density already exceeds the planck energy, and even this strange number of 10 million TeV. The two I'm thinking of are the earliest stages of the universe, and singularities in black holes. If there is some energy level at which its possible to tunnel to a lower-energy vacuum, then why didn't think take place in the early universe? One would think that it also takes place inside black holes, at least large ones, and if at least one black hole has evaporated since the universe began, then the lower energy vacuum should have swallowed us up already.Second, I'm not sure what it even means to say that there's an alternative vacuum. Usually when one talks about metastable equilibria, we are talking about some combination of *things* which can be organized in more than one way. When string theorists talk about metastable vacua I know that the *things* they're talking about reconfiguring in different ways are dimensions and branes. But you aren't a string theorist. So what are the *things* which are configured in one way, and why do we think that there is some other way in which these things can be configured?Maybe I'm just being dense or the math has gone over my head -- but I can't help but think the answer to the whole problem is "there isn't an energy density larger than [some arbitrary value related to quantum gravity], so don't bother worrying about how the math would work out if there were."

Anthropics? (Sorry...)

Anonymous:

The things that would be configured differently would simply be the value of the Higgs field. There would be two possible values that are local minima, say 246 GeV and 10^15 GeV just to pick a guess for the upper number. Ours is the 246 GeV of course.

There could be neighboring areas in space, one with higgs=246, one with higgs=10^15. Between them, the Higgs field has to climb over the potential wall to go from 246 and 10^15. This means that the region between the two values has a lot of energy, i.e. surface tension. Such a region is called a domain wall. Regions of the 10^15 vacuum within our universe would be surrounded by such a domain wall, i.e. they would be like a bubble.

Now, how can such a bubble of this alternative, lower vacuum with higgs=10^15 grow by itself to encompass our entire universe after it is nucleated? It has to reach a big enough size to overcome the surface tension from the wall energy in order to grow. If the size of this so-called critical bubble that you would need to have it grow by itself, is very large, the probability of it occurring is very small.

contind.

In String theory, having different vacuum expectation values of the higgs field usually *means* reconfiguring branes.

Alex: Thank you - you are slowly convincing me that I'm right.

First, in string theory the Higgs VEV is a consequence of different configurations of the branes and dimensions. The *things* that can be reconfigured, such that there are different equilibria at different energy levels, do not include the Higgs VEV. The Higgs VEV is a consequence of the rearrangement of things.Let me amend my statement of metastable equilibria: Usually what one is talking about is some set of plural *things* which can only be configured in relation to each other in a finite number of ways. For example, atoms forming a crystal can't be placed in a completely arbitrary fashion because two can't occupy the same place at the same time, electrons have to be shared in particular ways, and so on. Not all configurations are possible, and from most of them its a quick path to one of a small number of lower energy states. So besides the *things* there is something about the *things* which limits the range of possible configurations.

If the only *thing* (singular intentional) is the Higgs VEV itself, that I don't see how talking about metastability makes sense -- why do we presume that it can't have taken on any purely arbitrary value? And if it can't have taken on any value, if its restricted in some way, why do we infer that the restriction permits multiple possible values?

So I'm still lost as to what the *things* are that make it sensible to talk about metastable equilibria in this context (at least without strings, and lets not go there).

Put another way, I'm not sure how its meaningful to talk about a singular thing, not made up of other things, having metastable equilibria.

Second, I get the bubble concept (actually I know far more about bubble physics than I do about fundamental physics), but I'm not sure that it responds to my point.For one thing, energy being conserved, the energy at the domain wall should decrease as the size of the bubble increases, unless new energy is being drawn from someplace else. (This is also what happens to the surface tension of ordinary physical bubbles, unless there's boundary-crossing to maintain the pressure.) It having been a few billion years, one would think these domain walls (if they exist, which I am now convincing myself they don't) should be pretty weak by now. Or is the hypothesis that they're drawing energy in (must be massive) amounts from someplace, and if so where?

For another thing, what about the black holes? Given the energy density reached by (at least large) black holes, if there are other stable equilibria, shouldn't new "bubbles" be nucleating within them, well, all the time?

Anonymous,

Don't get hung up too much in philosophical interpretation of what a field is, or a thing. In QFT, the Higgs Field and its Lagrangian, including the Potential, are simply mathematical objects that are put into the theory, and which do what they do in the theory according to the internal logic of QFT. It is imho moot to discuss whether the higgs vev makes intuitive sense as a thing, or things, or whatever.

"First, in string theory the Higgs VEV is a consequence of different configurations of the branes and dimensions. The *things* that can be reconfigured, such that there are different equilibria at different energy levels, do not include the Higgs VEV. The Higgs VEV is a consequence of the rearrangement of things."

Well both actually, in String models the vev is a dynamical thing that can correspond to position of branes or such things, and the potential the vev sees is also dependent on positions of branes and so on. The model must be self-stabilizing.

"If the only *thing* (singular intentional) is the Higgs VEV itself, that I don't see how talking about metastability makes sense -- why do we presume that it can't have taken on any purely arbitrary value?"

That's a bit of a misunderstanding. The vev is not forced to be exactly in one of the minima at all, those are merely the only (meta)stationary places it can rest. One assumes that after a sufficient amount of time the universe will be approximately in a local vacuum. It's really like a pendulum which eventually stops at the equilibrium point and the energy is converted to heat. In the universe, the energy would for example go into radiation and particles populating it.

If the vev were not in one of the minima but somewhere else (not at a local minimum), this would not be a stable situation, there would be spontaneous particle production as our vacuum falls down towards the nearest local minimum.

"For one thing, energy being conserved, the energy at the domain wall should decrease as the size of the bubble increases, unless new energy is being drawn from someplace else"

The energy of the domain wall per surface is a constant afaik. But remember that the vacuum inside the bubble has lower energy, so when the bubble expands, energy is released. It partly goes into the increased surface area tension, and presumably into kinetic energy.

Cheers,

Alex

the universe will be approximately in a local vacuum

should read --> the universe will be approximately in a local minimum of the potential

One only wonders how much this "stability" arguments rely on perturbation theory.

When you have large couplings (i.e. Yukawas), at some moment one should do a non perturbative analysis, and maybe there you do not find any of these issues.

By the way, although a renormalizable model in the perturbative sense, a full non perturbative analysis of the quartic self interacting scalar field alredy tell us that the theory is "trivial" (i.e. is non renormalizable). You ca not remove the cutoff in a consisten way. The presence of a scalar field (without SUSY) is alredy telling us that the SM is an effective field theory.

Anonymous, all the couplings remain well within the preturbative domain. Even the top Yukawa gets weaker in the UV, so why should nonperturbative effects become relevant here?

Concerning triviality: In this particular case, with a light Higgs to begin with, the quartic coupling gets smaller and smaller in the UV, until it gets negative, triggering the instability: no triviality problem whatsoever.

Then, nobody should be interested in defining a QFT in a fully nonperturbative way without a cutoff. At least not for phenomenological purposes.

We know any such QFT will have a non-removable cutoff at or below the Planck mass.

If you don't buy this, why aren't you worried about QED?

I'm still not getting it.

If the only *thing* (singular intentional) is the Higgs VEV itself, that I don't see how talking about metastability makes sense -- why do we presume that it can't have taken on any purely arbitrary value?That's a bit of a misunderstanding. The vev is not forced to be exactly in one of the minima at all, those are merely the only (meta)stationary places it can rest.Yes, I get the concept of metastable equilibria. What I'm having trouble with is how the concept can be applied to a fundamental (i.e., not dictated by something else) parameter of the model.

It seems to me that the blog post (or the article described in the blog post) is making some kind of assumption about a more fundamental theory, or framework, or something, and I'm not seeing what that is.

For one thing, energy being conserved, the energy at the domain wall should decrease as the size of the bubble increases, unless new energy is being drawn from someplace elseThe energy of the domain wall per surface is a constant afaik. But remember that the vacuum inside the bubble has lower energy, so when the bubble expands, energy is released. It partly goes into the increased surface area tension, and presumably into kinetic energy.This I don't get either. If the energy of the domain wall is constant, and the area of the domain wall expands, then unless new energy is being introduced from someplace then the surface tension -- the energy per unit of area -- should decrease.

So if the bubble is expanding, but the surface tension isn't dropping, new energy has to be coming from somewhere. Inflation? Its difficult to see how that could be, considering the amount of energy that, its being suggested, is tied-up in the domain wall.

I'm usually pretty good at following physics explained at the level of this blog, but I continue to find the article (and blog post) just bewildering. Unless one is positing that there is something more fundamental than the Higgs VEV -- that the Higgs VEV is a consequence of something else -- then I'm just not seeing how its a coherent concept to say that the VEV is in a metastable equilibrium.

Without knowing what that more fundamental thing is, or even that there is a more fundamental thing, I'm not seeing how one could infer the existence of multiple equilibria from the Higgs mass. Or anything else for that matter.

BTW - you still haven't responded to my point about the energy density within black holes. Am I just off the wall? Feel free to tell me I'm just naive and don't know enough math, I won't be offended.

Yes, you are naive and deluded.

Anonymous, I am not an expert, but Alex's reply made sense to me. The region inside the bubble has lower energy density than the region outside. If the bubble expands, there is some space that used to be outside the bubble that is now inside. In this region, the vacuum energy has decreased. Where did that energy go? Into the domain wall, I guess.

"It seems to me that the blog post (or the article described in the blog post) is making some kind of assumption about a more fundamental theory, or framework, or something, and I'm not seeing what that is."

I'm not a physicist, but he did say that spontaneous particle production would lower the VEV if necessary. So presumably, there are some known laws of physics that explain how this would happen. Maybe that's part of the whole Higgs mechanism.

"This I don't get either. If the energy of the domain wall is constant, and the area of the domain wall expands, then unless new energy is being introduced from someplace then the surface tension -- the energy per unit of area -- should decrease."

The comment you're replying to already answered that. He explained that the volume inside the bubble has lower energy than the volume outside, so a tremendous amount of energy is being freed as that boundary moves, and that is pouring into both the surface tension and kinetic energy. I'm just repeating him here.

Let me explain it a different way: imagine suddenly deleting all the water in a vertical cylinder of the ocean. Water will rush in from the sides, and that cylindrical disturbance will expand outward as the level of the water inside it slowly rises and the level of the water outside slowly falls.

But that rushing of water is noisy and violent, so where does that energy come from? From the fact that the gravitational potential of the water outside is greater than that inside, so that energy is forcefully pushing the water inward (and the domain wall outward). When the water goes from high potential energy (height) to lower potential energy, the potential energy it loses becomes noise and kinetic energy instead.

Similarly, as the Higgs VEV falls from a high value to a low value when the domain wall screams past it, the energy it loses becomes surface tension and kinetic energy. Is that any clearer?

I may have flipped the signs rellative to what you were discussing in my previous comment.

"The reason is that there are parts of the universe where, I would think, the energy density already exceeds the planck energy"

Buhhhhh... what?! First off, I assume you mean the Planck density. Second, it is very hard for me to imagine that that is true, since the Planck density would have been the density of the universe one Planck time after the Big Bang.

First, thank you to whomever acknowledged that all this talk about metastable equilibria does, in fact, presume the existence of some more fundamental theory, and doesn't make sense without it.

I feel like I can now toss this whole metastable equilibrium thing into the "pure speculation" bin and move one.

Anyway...

it is very hard for me to imagine that that is true, since the Planck density would have been the density of the universe one Planck time after the Big Bang.That's one example of a place where the energy density was high enough to get over the domain wall. A way of expressing the issue would be "Why didn't the whole universe move into the lowest-energy equilibrium at the start, if the energy density was high enough, which you're positing it was if some other region got there?"

My other example is large black holes, where the energy density at the center is --- remember, the proponent of the position that there are other equilibria, some of which have lower energy, has to explain why we aren't in it. The answer given is "the energy at the domain wall is really, really large." The presence of regions of such high energy density, and lack of new lower-energy-equilibrium bubbles nucleating, pushes up the lower bound on the amount of energy tied-up in the domain wall. Which, in turn, makes it harder to explain where the new energy is coming from to maintain the domain wall as the lower-energy region expands.

The comment you're replying to already answered that. He explained that the volume inside the bubble has lower energy than the volume outside, so a tremendous amount of energy is being freed as that boundary moves, and that is pouring into both the surface tension and kinetic energy. I'm just repeating him here.

Let me explain it a different way: imagine suddenly deleting all the water in a vertical cylinder of the ocean. Water will rush in from the sides, and that cylindrical disturbance will expand outward as the level of the water inside it slowly rises and the level of the water outside slowly falls.

The question Alex was responding to was, why hasn't the lower-energy region expanded into ours? The answer was the energy trapped in the domain wall. My response was, where does the energy come from to keep the domain wall strong enough to protect our region from the lower-energy one as both regions expand?

Your answer is from the release of energy as the lower-energy region expands into a higher-energy region, releasing energy which can support the wall. What higher-energy region?

If the higher-energy region is (say) ours, then you're rejecting the premise of the question, which was: Why isn't the lower-energy region taking over our higher-energy region?

If the place that the lower-energy region is expanding into is just whatever-is-there-before-the-universe-expands-into-it, then aren't you saying that the source of energy to maintain the domain wall is inflation?

In fact, stability is not ruled out, see http://arxiv.org/abs/1209.0393 . This highlights your main point. We are so close to the boundary betwene stability and instability we are not yet even sure which side we're on.

One can test QFT non-perturbatively with lattice simulations. That has been done for the pure scalar field theory, and for Higgs-Yukawa models. It was shown there is neither a Landau pole, nor vacuum instability. The finite cut-off in lattice simulations is an artificial space-time lattice spacing. The location of the vacuum instability, predicted by renormalized perturbation theory, occurs at a scale above the cut-off, and hence is a pure artifact of the finite regulator, which is necessary for these trivial QFT's to have non-zero interactions.

There are older papers by Luscher, Kuti, Neuberger etc. on the pure Higgs lattice simulations, and more recent by Kuti, Jansen etc. on Higgs-Yukawa lattice simulations. Although vacuum instability appears perturbative and a safe prediction, it is still affected by the triviality of the QFT, which renormalized perturbation theory does not treat carefully. One should not take seriously the statements of instability, meta-stability etc.

Lattice Anonymous, you are totally confused, sir.

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