How the Higgs Mechanism Give Things Mass (Script) (Patreon)

Fermilab physicists really care about the mass of the W boson. They spent nearly a decade recording collisions in the Tevatron collider and another decade analyzing the data.

This culminated in the April 7 announcement that this obscure particle’s mass seems to be  heavier than expected. So why do we care? Because understanding why this particle even has mass was one of the most important breakthroughs in our understanding of the subatomic world. And because measuring its precise mass either doubles down on our current understanding or reveals a path to an even deeper knowledge. The FermiLab discrepancy is a tantalizing hint of the latter.

—-

This timing of FermiLab’s discovery is weirdly convenient. Over several previous episodes we’ve been building towards an understanding of how the forces of nature are unified. The most powerful clue driving this is the weirdness of the weak force - in particular the particles that carry this force. Its W and Z bosons have a property that we once thought no force-carrying particle should have - they have mass.

The W bosons are especially weird in that they also have electric charge. This allows the weak force to trespass on the province of electromagnetism, suggesting a connection between the two.This connection hints at a unification of the forces of nature.

The path to that unification leads to the Higgs mechanism, which not only explains the mass of the weak bosons, but teaches us about the nature of mass itself.

To get to all that good stuff we need a bit of a refresher on the episodes that led to this - fields and forces and symmetries and all that. Similar to how the fabric of a drum has vibrational modes, so does the fabric of reality.

Every point in space can wiggle, twist, oscillate in different ways.  A quantum field just represents one of these modes. And these wiggles are quantized - they come in discrete packets of energy that can move around - and those are the particles of a field.

One special type of field is the gauge field. These arise from the fact that physics often doesn’t care what coordinate system you use. The laws of physics are symmetric under certain transformations. For example, physics works the same no matter where you decide to center your x-y-z axes,

or where you put the zero point of your angles in polar coordinates. We saw in our episode on Noether’s theorem that these symmetries lead to conservation laws.

In quantum mechanics, such a “redundant degree of freedom” leads to a gauge field. We’ve seen an example of this. The exact phase of the quantum wavefunction from one point in space to the next - local phase - doesn’t affect measurable quantities - only relative phase matters.

When we enforce this requirement, we find that we have to add a new quantum field to the Schrodinger equation that lets the universe counteract  these phase shifts. That gauge field turns out to be electromagnetism, and oscillations in this field are the photon - our first gauge boson and carrier of electromagnetic force.

So one of the fundamental forces arises from a symmetry of nature - in this case the fact that the laws of physics are invariant under changes in local phase. The set transformations that can change local phase are an example of a symmetry group - in this case “unitary group 1”, or U(1). These transformations can be represented with just “one” number in this case rotation of phase angle vector of “unit” length.

See this episode for the nitty gritty of all of this. This is a refresher, remember. In the next episode we tried the same trick to explain the weak interaction as arising from symmetries. We saw that we could invent a pair of totally abstract degrees of freedom and demand that the universe be invariant to transformations of these. We call this symmetry group SU(2) for … reasons. That requirement gave us a new gauge field that has 3 force carriers that look awfully like the weak force bosons.

So far so good except that the predicted particles are massless, while the real weak force bosons are, as I mentioned, pretty hefty. That’s a huge deal breaker actually - massive bosons break gauge symmetries. According to something called Goldstone’s theorem, all gauge bosons are massless. So… so far so bad, but we’re gonna forge on anyway and hope this gets sorted out.

Another problem with this first pass at the weak force is that it has absolutely no connection to electromagnetism. Its bosons do have properties that look like weak isospin and weak hypercharge, but no electric charge. In our universe these three quantities are sort of locked together, only taking on certain values relative to each other. To see if we can duplicate that in our theory we need to combine the U(1) and SU(2) symmetries so that they apply at the same time. We call this combined symmetry group U(1)xSU(2). The resulting gauge field still has bosons that look a bit like the photon and the three weak force bosons, but the latter are still massless, and the resulting charges are completely unconnected to each other.

They’re all free to be whatever they want, unlike the real universe where isospin and hypercharge are tightly coupled, and their combination defines electric charge. Is it time to give up on this symmetry stuff yet?

There’s one more long-shot clue. I said that massive bosons break the gauge symmetry. But what if that’s okay? We’ve also talked about the idea of symmetry breaking before, using this example of a bunch of bar magnets. These magnets have high temperature which makes them move randomly, but as the system cools down this random thermal motion gets overpowered by the magnetic interaction and they end up all aligning. The equations of magnetism don’t start out with a preferred orientation, but in certain conditions - namely, cold ones - the system chooses a preferred direction. This is an example of spontaneous symmetry breaking. So if a bunch of magnets live in a state that violates the symmetries of their ruling equations, maybe the universe can too.

Here’s another analogy. Consider a ball rolling back and forth in a valley. The equations describing its motion are symmetric between left and right because the valley is symmetric.

Now imagine it wasn't one valley but two valleys with a hill in the middle. The system is still symmetric, but if the ball starts at the top of the hill it will randomly roll down into one valley. Now the current state of the system has a broken symmetry, even if the symmetry of the landscape remains.

So let’s just see if we can break the symmetries of the universe in a similar way. The equivalent of the simple valley exists. A quantum field can oscillate around some “zero-point” value like a ball rolling back and forth in a valley. The “walls” of the valley are just the potential energy - the energy stored when the field value moves away from the zero point, trying to pull it back to the center. Graphing this, the vertical represents potential energy and the horizontal represents the field strength.

Particles of this field are just oscillations of the field strength across the lowest point, where the field strength is zero. If there are no particles around then the field just sits at the lowest point. We call that the vacuum state.

Easy stuff, right? In that case you won’t mind seeing the math.

OK, don’t freak out. It’s not on the test. This is a Lagrangian - something we covered previously. It’s really just the difference between the kinetic energy and the potential energy in the field. Our plot was of the potential energy part.

This particular Lagrangian describes a simple  quantum field made of massive particles which interact with each other. Let me talk you through the hieroglyphics. Firstly, phi is the quantum field itself - it just means there’s a numerical strength of the field everywhere in space. The potential energy part is the “shape” of the field, and is made of various powers of the field strength that represent ways the field’s particles can interact. For example, this phi^4 term says the particles of the field interact with particles of the same field with strength lambda. This phi^2 term represents the field interacting with itself. That self-interaction is what leads to the property of mass. Gauge fields shouldn’t interact with themselves, so shouldn’t have a phi^2 term.

And that means this isn’t a gauge field. The gauge field is a new thing that comes from the degrees of freedom within this field. This particular Lagrangian has the simple symmetry that it’s the same if you reflect it around the y axis.

If we complicate things by adding a second field compone nt - phi 1 and phi 2 - we get a parabolic bowl. If the current state of the field is at the bottom of the dip then it has a single continuous degree of freedom, in that you can rotate this thing and nothing changes.

That would be a global U(1) symmetry. Repeating our electromagnetism trick means requiring local U(1) invariance. We need the laws of physics to still make sense if there are rotations from one point in space to the next. That means adding a new gauge field in the Lagrangian that allows the angle of this rotational degree of freedom to vary. Call that angle theta.

Oscillations in that field would be a gauge boson. You can think about those oscillations as the theta angle twirling around freely, and it doesn’t need any minimum energy to oscillate, so the gauge boson is massless. On the other hand, the particle of the original field needs a rest mass energy to be able to oscillate up and down the potential walls.

So this potential doesn’t give us massive gauge bosons. Let’s make a slight change. We’re going to switch the sign in front of the mass term. Then the potential would change to this, just like the two valleys we saw earlier, but now with this extra field component to give us this shape. It’s called a mexican hat potential. This potential shape is the heart of the Higgs field.

It’s still a symmetric potential - it has the same global U(1) symmetry as the original. But if the universe suddenly transitions from the old potential to this one then we have a problem. The field strength would find itself on top of this little hill, but then quickly roll down in a random direction. The new state of the field would not be symmetric to the same rotations, because different points around this valley correspond to different physical states.

We say that the symmetry is spontaneously broken. The current state of the field is a state of broken symmetry, even if the global field shape keeps its old symmetry. This is just like how the field of magnets can be in a state of broken symmetry even though the overall equations of magnetism haven’t changed.

The transition from the central bump to the valley is an example of vacuum decay, and we saw how this can lead to the creation of cosmic strings in a recent episode. But cosmic strings are the perhaps least interesting thing about this process. Once the field has reached the base of this new minimum we have a new stable vacuum state.

But this state is very different from the original for two reasons. The first is that the lowest energy state - the vacuum state - isn’t where the field strength is zero. This is called a non-zero vacuum expectation value. The original massive particles of the field now oscillate in the radial direction, but no longer centered on the zero field value. This is the Higgs boson itself.

Second weird thing: there’s not just a single vacuum state - there’s a rin g of valid states. The universe will just have chosen one state randomly. But the field can also oscillate along the base of the valley in what we’ll call the theta direction. The resulting particle is called a Goldstone boson and it’s massless because the valley is flat - there’s no energy differential. However it isn’t a gauge boson because location around that valley represents a real physical difference in the field state.

OK, let’s try to bring all of this together. We want to combine the weak and electromagnetic interactions, so we need simultaneous local U(1) and SU(2) symmetry. Let’s just do U(1) because that gives us the basic picture. Because the potential is the same all around the valley, it shouldn’t be possible to tell where in the valley you are. It matters if two adjacent patches of the universe are in different parts of the valley - the relative difference in this angle theta matters, but exact values don’t. So we’re going to demand local U(1) invariance and come up with a gauge field that shakes out shifts in our arbitrary choice of the zero point of theta.

But now this gauge field finds itself in a much more complex Lagrangian with this Mexican hat potential. Weird stuff happens when the gauge field couples to the particles of that potential.

First of all, we see that the Goldstone bosons, which are just oscillations in the theta angle, can be absorbed into this U(1) gauge field. Both are oscillations around the valley. In the Lagrangian they get lumped together into a single field

that still looks like a gauge field with a single gauge boson. The gauge field eats the Goldstone boson. But when we do this, the new combined field has field-squared terms in the Lagrangian, which is weird because those are mass terms - except neither the original gauge boson nor the Goldstone boson that it just ate had any mass. So where does this mass come from? It’s because the gauge boson is now coupled to the Higgs field. The Goldstone boson that is now part of the gauge boson is coupled to the Higgs field, which means the gauge boson is also so coupled.

Ultimately this happens because of the non-zero vacuum state of the Higgs field. That little bit of Higgsiness everywhere refuses to cancel out in the Lagrangian, giving us our mass term.

All of this was a simplified explanation. Horrifying, right? Imposing the full electroweak U(1)xSU(2) invariance on the true Higgs potential gets you three Goldstone bosons that are eaten by 3 of the 4 electroweak gauge bosons. Those gain mass and become the two W and one Z bosons of the weak interaction. The last manages to escape unscathed and massless, becoming the photon that we know and love. It flew free of its heavier cousins, the independent mediator of a part of the old electroweak field - what we now experience as electromagnetism, while the W and Z slog on through the mire of their coupling with the Higgs field. Their mass shortened their lifespans, and so enormously reduces their range and weakening the force that they mediate.

So that’s where we are today. This is the Higgs mechanism. The Higgs also gives mass to the matter particles - the fermions - but that’s for another time. But what about this new measurement of the W boson mass? Mass results from interaction with the Higgs field, but also all of the other subtle interactions that a particle can undergo. The predicted mass of the W boson takes into account all standard model particles that could have a ghostly presence as virtual particles in the energy field of the boson. The fact that FermiLab measured a larger mass that was predicted suggests an unknown particle or particles flickering around the W boson. The discovery of the Higgs boson 10 years ago verified the idea that the underlying symmetries of  nature explain and unify some of the forces of nature. Perhaps new particles will lead us to new clues about yet deeper unifying symmetries of space time.