Some questions related to particle physics and cosmology (Higgs field, dark energy, and more)

Factual Questions
1

In another thread, @dougrb asks a few questions related to particle physics and cosmology. To not hijack that thread (which was originally about gravity), I am bringing @dougrb’s questions into a new thread.

He notes that in Sean Carroll’s book “The Particle at the End of the Universe”, there are a couple of analogies used in relation to the Higgs field, which itself is related to particle masses in the Standard Model of particle physics. One of the analogies is summarized by @dougrb here:

He comes back to it later in question form:

There is definitely enough here for its own thread. So I’ve started that thread here.

2

For the question of “Does the pendulum analogy hold water?” the answer is “Yes”, but it relates only to one small aspect of the whole Higgs and mass story. I’ll try to add some context around it and also plant some seeds for other parts of your questions. I don’t know your physics background, so let me know if I need to back up in parts.

Potential energy
A ball high up on a hillside is said to have “potential energy”, meaning that the ball can spontaneously pick up substantial kinetic energy by rolling down. That newfound kinetic energy had to come from somewhere, and it came from the original potential energy.

With effort, you can bring the ball back up the hill. You have to work hard to do it, spending some of your own energy to lift it back up there. In contrast, you don’t have to work hard to get the ball down the hill. It just goes.

So, physical systems will spontaneously evolve to lower potential energy arrangements. A ball resting in the valley at the bottom will never jump back up to the top, but a ball up the slope will happily head down to the valley below all on its own.

To tighten things up, let’s look at a specific shape for a valley. Here’s a picture:

The red shape represents “ground level”, so there’s a nice deep valley right in the middle.

If we place a ball in this picture somewhere, it will settle into the bottom of the valley. When it’s there, the horizontal position of the ball, call it X, will be at 0. The vertical position of the ball at that point, call it V, is at roughly 1 in the graph. So, X=0 and V=1.

Important for later: the potential energy associated with a given vertical height is not specified here. We know that the bottom of the valley, at X=0 and V=1, has the lowest potential energy, but what that actual potential energy value is there is partly arbitrary. Maybe the height of V=5 corresponds to 100 joules of potential energy and V=1 corresponds to 30 joules, or instead maybe these are 10 joules and -60 joules. All that matters to the physics is the difference in potential energy between the two heights, and in this numerical example, the lower point is 70 joules “better” than the higher point. The actual potential energy at the individual points is not important. In practice, one sets the “zero point” of potential energy to wherever is most convenient for the work at hand.

Symmetry breaking
Consider now this blue version, where there is a hill in the middle with valleys on the sides:

If we place a ball right in the middle, it’s not stable. There are better, lower-potential-energy arrangements possible. So, the ball will roll down into one or the other valley. Let’s say it happens to roll to the right. The stable equilibrium arrangement for this system would become X=1 and V=1.

The blue curve in the picture is 100% symmetric around the middle, i.e., around X=0. But the actual physical arrangement that happens naturally has X=1 (say). How did a symmetric “universe” lead to a non-symmetric outcome? It’s because the shape of the blue hills and valleys make it so that the obvious symmetric answer (X=0) isn’t the stable one.

This idea generally is called “spontaneous symmetry breaking”. The physical laws are symmetric, but the outcome has to be a random choice that hides that symmetry.

Carroll’s pendulum analogy
You can maybe see the connection between the “balls in valleys” situation to the pendulum analogy you cited. In the latter, the upside-down rigid-stick pendulum can’t stay in the upright position. It has to fall one way or the other, just like a ball that tries to stay in the middle of the blue hill.

Removing “height” from the story
If we push the ball to a different X value, it changes potential energy because we are forced by the ground’s shape to either fight gravity (if climbing) or benefit from gravity (if falling). But the “height” part can be abstracted away. In fact, it’s entirely redundant with X, since we know the vertical position V perfectly if we know X. So, we can skip the middle man and just say that different positions X have different potential energies, with no more reference to height. This is important: all that matters in general is the connection between the system’s arrangement (however specified) and the potential energy of that arrangement.

Connecting to the Higgs field
The value of the Higgs field throughout spacetime corresponds to X in the above picture. It’s the “free variable”. The potential energy for different choices of X looks like the blue example above, with “Higgs field value” on the x axis and “potential energy” on the y axis. For different values of the Higgs field, there are correspondingly different potential energies.

The Higgs field spontaneously settles into a value with minimum potential energy, which is away from X=0. This is the “vacuum expectation value” of the Higgs, and it’s 246 GeV. This is not the potential energy value itself; that’s the vertical axis. 246 GeV is the field value (horizontal axis). The actual value of the potential energy at X=246~\rm{GeV} is arbitrary, as noted previously. Nothing in the Standard Model dictates where the “zero” point should be set.

Dark energy?
You asked if this could relate to dark energy. There is a non-zero value to the Higgs field everywhere, but the field itself doesn’t represent any energy unless there is an excitation of the field (an actual particle) or if we can specify some definite potential energy for it. But we can’t – the Standard Model is mute on the actual value of any potential energy for the Higgs field even when it has a non-zero value. Particle physics doesn’t care about the arbitrary offset. But! General relativity does care. So, what to do?

A naive thing to try is to just set potential energy for the Higgs field to zero when the Higgs field value is itself zero. That is, choose X=0 to be the point where V=0 by sliding the whole blue-curve picture up or down suitably. To be clear: this is an arbitrary choice. But if you do that, the vacuum value for the Higgs field (X=246~\rm{GeV}) corresponds to a massively negative potential energy density that both has the wrong sign (negative instead of positive) and is ludicrously off in magnitude to explain dark energy.

You could instead choose the potential energy to be zero when X=246~\rm{GeV}, the stable vacuum situation. That is extra arbitrary since the “vertical” offset for V has no reason to be tied to the dynamic equilibrium value that X settles out to. But, if you did this, then by definition the Higgs field would provide zero dark energy.

You could also choose to set the potential’s zero point in the Standard Model to whatever is needed to exactly explain dark energy. But that doesn’t “explain” anything since it would just mean setting an arbitrary shift of the zero point to get whatever you needed back out. Further, the extra shift you need would require a perfect tuning of the shift out to dozens and dozens of decimal places to nearly perfectly cancel the ludicrous negative value noted in the first version above. This (and some related points) is at the heart of the cosmological constant problem. It remains a big open question.

Not addressed here

  • Why should the Higgs potential behave this way?
  • How does this relate to the Higgs mass itself, which is a different value (125 GeV)?
  • How does any of this lead to particle masses in general?

That last item (how to get particle masses out) connects to the other analogy in the other thread, namely the idea that the Higgs field is like some drag force on particles. That’s the analogy I greatly dislike, finding it more harmful than helpful. I can perhaps talk about that piece of the story as well.

3

Thanks for that. THIS is the part I don’t get.

4

I’ll let @Pasta give the full explanation, but part of the answer is that it doesn’t account for particle masse in general. It leads to masses for the electrons and quarks, but most of the mass of most of the stuff we’re familiar with is from the binding energy between the quarks in protons and neutrons, not from the quarks themselves, and that has nothing to do with the Higgs mechanism.

There’s also yet another mechanism that accounts for masses of the neutrinos, but nobody’s quite sure which it is (there are two different candidate models, and we haven’t yet been able to distinguish them experimentally).

5

THIS (I think). In the upside down pendulum analogy, he says it depends what direction it falls down and you don’t know that.

6

To get to the mass thing, we need to cover some ground.

  1. How to give a particle mass the easy way.
  2. How to introduce fundamental forces.
  3. Why doing these together is a problem.
  4. How to fix the problem via the Higgs mechanism (boson case).
  5. How to fix the problem via the Higgs mechanism (fermion case).

How to give a particle its mass the easy way.
Imagine a boring universe with one type of particle that interacts with nothing, not even its own species. The mathematical description of such a theory is pretty lightweight. There is a piece that describes the energy and momentum side, and there is a piece that describes the mass side. The latter looks sort of like a self interaction, but it has a very specific mathematical form to interface with the energy/momentum piece in the right way and to allow well-behaved propagation of particles. We can write these piece out notionally as (kinetic piece) + (mass piece). But keep in mind that these pieces have very specific mathematical forms.

This works fine. You can make a particle theory that has particles with mass in it. You just add that mass piece, and set the value of the mass to match observation.

How to introduce fundamental forces.
Separately, we need interactions (or “forces”) in the theory. I noted in the other thread that symmetries are often the source of physical laws. An example from that thread: if physics should work the same at different places (a type of spatial symmetry), then momentum must be conserved. Many “laws” of physics are just consequences of symmetries of nature.

This is true for all the fundamental forces in the Standard Model. The electromagnetic, weak, and strong forces are not put into the theory, but rather they emerge from it in an elegant fashion by requiring certain symmetries of nature to be present. The relevant symmetries are somewhat subtle and mathematical in their description, but in their own way they are simple. These are known as “gauge symmetries”. These gauge symmetries insist that the laws of nature should be unchanged if you do a certain type of mathematical operation to the fields.

(Technical aside: the freedom that this symmetry imposes is the freedom to rotate the quantum mechanical phases of the fields independently at all points in spacetime while also introducing a “covariant derivative” to ensure the kinetic piece of the theory is aware of the changing phases.)

When this symmetry – this freedom to make phase changes everywhere with no consequence – is enforced on the theory, the fundamental forces and their corresponding bosons just… emerge. By adding a few bells and whistles to the structure of the fields, and by choosing a few different gauge symmetries to enforce, you can get all the fundamental forces that we see in nature. Almost.

Why doing these together is a problem.
This way of getting forces to emerge only works if there are no masses for any particles. You can’t have both the gauge symmetry that yields the forces and also the mass piece(s) from earlier in this post. Those mass pieces are fundamentally not gauge invariant (at least not for the specific symmetries we need for our universe and not without something to help restore invariance).

But we do have massive particles and gauge interactions in nature. The solution: insert no masses by hand into the model, and instead have all masses emerge in a well-behaved way.

(You might ask: do I need all this prologue? Well, it’s a little unexpected that we need some trick to get masses into the theory at all. And, if there weren’t forces in the theory – or if there were only certain types of forces – we could do it just fine. But in our universe, it is not allowed to introduce the masses “bare”.)

How to fix the problem via the Higgs mechanism (boson case).
Start with the mass-free theory with just your fermions in it. Then, postulate a new field – the Higgs – that has the following two key properties. (1) It has some of the same bells and whistles mentioned above so that its complexity aligns with the forthcoming gauge symmetries. In short, it doesn’t just have one value everywhere in spacetime but a small, structured set of values. (2) Give it a self-interaction that serves as a source of its own potential energy, with a form like that blue curve from the prior post so that it will acquire a non-zero vacuum expectation value.

With those features in place, we now:

  • Impose gauge symmetry as above to make the fundamental forces appear.
  • Everywhere that the Higgs field appears in the theory, “re-center” it to the now-relevant reference value (the vacuum value), which leads to substantial rearrangement of all the pieces in the theory.
  • After this rearrangement, some of the residual pieces look (and act!) precisely like the original mass pieces that we wanted but couldn’t have. Since they’ve come about now through this “Higgs mechanism”, the theory as a whole is well-behaved and respects gauge invariance end-to-end.

So, while you can’t add boson masses directly, you can make them pop into the theory through a combination of enforcing gauge invariance (which gets you the forces in the first place) and having this “offset” Higgs field that twists all the pieces up when it shifts to its off-center value.

(Note: The “shifting” part is the narrow piece of all this that the original pendulum analogy has anything to do with. The rest is not conveyed through that analogy.)

Fermion case
For fermions, it works a little differently. Starting with massless fermions, you can safely add parts in the theory that make the massless fermions interact with the original raw Higgs field. This doesn’t mess with any gauge invariance requirements. But then, when the Higgs field shifts to its vacuum value, those interaction pieces get split into two chunks: one that acts like a fermion-to-(new)-Higgs interaction piece and another that looks (and acts!) precisely like a fermion mass piece, of the sort that was forbidden when jammed in by hand. In a sense, an aspect of the Higgs field is “spun off” into a genuine fermion mass. It’s just a mass, not a force or drag or anything else. It looks just like a mass would look if we put it in by hand. The only issue is that putting it in by hand while also imposing gauge invariance to generate the forces makes things break. Recovering fermion masses via the Higgs mechanism gets us around that breakage.

Confirmed?
When the Higgs gets re-centered around its new vacuum value, many parts of the theory get entwined in the process, as seen above. This means that a lot of observable quantities should have detailed relationships with one another. For example, the mass of each particle can be measured, the interaction strength of each particle with the Higgs can be measured, and the vacuum expectation value of the Higgs can be measured. If all this holds water, these measurable values should have very precise algebraic relationships with each other. And they are all observed to do so.

Summary

  • You can’t simultaneously put (some) forces and masses into the theory by hand without breaking things.
  • So, start with no masses.
  • Then put in a Higgs that picks up a non-zero vacuum value.
  • When you re-center the Higgs field around its new vacuum value instead of the original unstable value, the “shifted away” parts end up looking like masses for bosons and fermions, if through slightly different mechanisms.

These emergent masses in the theory look and act precisely like regular ol’ masses, even though we had to use the Higgs to ensure that the theory as a whole maintained compatibility with gauge invariance. What this rigmarole really gets us is various counteracting pieces that restore gauge invariance when those masses are present.

Side notes
As @Chronos notes, this provides masses for fundamental particles, but most of the mass we experience everyday is due to something else entirely, namely the energy of quarks and gluons bound together in protons and neutrons. You could have that sort of mass even if none of the fundamental particles had any mass at all.

I haven’t iterated on this post really at all, so do pick out / pick on any pieces that are not cutting it for you.

7

I’d have to find a copy of the book to see what he means there. In the Higgs mechanism, the magnitude of the “vacuum expectation value” isn’t random*, but there is indeed an unknown orientation when it “falls”. However, that orientation has no observable consequences.

*I mean, it isn’t random in the spontaneous symmetry breaking process in the Standard Model. At a meta level, everything might be random!