An Explanation of the Higgs Boson

This is a slight update of a comment I made on Metafilter, on the day of the Higgs discovery (July 4, 2012. i.e. Higgsmas)

OK, let me try to explain the Higgs boson. To give a good explanation requires a lot of background, and I'm going to try not to half-ass it. 

There are 4 forces of nature: gravity, the electroweak nuclear force, the strong nuclear force, and something called hypercharge. Electromagnetism is actually a combination of two of these forces: the electroweak and hypercharge forces, it's intimately related to the Higgs, and we'll get there.

Each force acts at a distance between particles that are charged under the relevant force by exchanging "force carriers." You can picture this as two people throwing a ball back and forth; the momentum carried by the ball will push each person around. This doesn't explain why forces can be attractive, but the picture isn't bad to start with. The important point is: a force requires a particle to carry it.

I'll pretty much ignore gravity for now, since that's a difficult one to deal with. It's a very weak force, so we can't create or see the force carriers - the gravitons - easily, and there are some additional technical problems that make quantization of gravity difficult. Thus there is string theory, but moving on.

The remaining forces are very similar in certain respects: they are all carried by spin-1 particles called "vector bosons" (the boson means they have integer spin and obey a certain type of spin statistics - the alternative is a fermion with half-integer spin - and the vector refers to their spin-1-ness). You can prove that, given our understanding of quantum field theory, the force carriers MUST be vector bosons. Also, by spin here, I do mean spin as you know it: angular momentum. If you got hit by enough vector bosons carrying coherent spin, you'd start rotating. The force carriers are: the B for hypercharge, the gluons for the strong force, and the W1, W2, and W3 for the electroweak force.

Furthermore, you can prove that each force carrier is massless. This allows the forces to be infinite range. Handwaving a bit, due to the uncertainty principle, you can "borrow" energy from the Universe to create a particle as long as you pay it back in some time inversely related to how much energy you borrowed. There's an important relation here: large energies thus correspond to short times and short distances. So the Universe can "borrow" arbitrarily little energy to create a massless particle, and not have to pay it back for infinitely long times, allowing the force to be propagated over huge distances. Handwaving, I know, but it gets the right general behavior. 

However, when you look at the gauge bosons we actually find in Nature, it's nothing like this. Ignoring the gluons, which have their own craziness, we find that hypercharge and electroweak bosons are all mixed up together. The thing we call the photon is actually a combination of the B and the W3. There's another combination of these two that we call the Z. The W1 and W2 are typically combined in two ways: one combination is a particle called the W+ with positive electric charge, and the other is the W- with negative electric charge. Even worse, the W's and Z are massive: the W's weigh 80 GeV, and the Z is 92 GeV. This is particularly bad when you learn that massless gauge bosons have two possible states: +1 and -1 spin orientation, while a massive gauge boson must have three: +1, 0 and -1. Where did the extra state come from?

In other words, what just happened?

The answer is the Higgs field. If there was a field (which is just a set of values everywhere in the Universe. The electrons are described by a field, as are gluons, photons, and every other particle) that was charged under both electoweak and hypercharge forces, then both the B and the W1, W2, W3 would interact with it. Most fields relax back to zero values if you don't pump energy into them to create a particle. However, if this special field interacts with itself in a particular way, then it is actually energetically favorable for this field to not sit a zero, but to have a non-zero value everywhere. So, everywhere around you, where you think there is empty space, there is actually a field with some "vacuum expectation value" (vev). This field we call the Higgs field.

Since the Higgs field happens to have both electroweak charge and hypercharge, if one of these gauge bosons tries to travel through the Universe, it sees a space full of stuff that it wants to interact with. The W's and the Z therefore gain a mass: it's harder for them to move through the Universe, and so if you want to create one of these particles you have to dump a lot more energy into a region of space, and they would propagate slower as they push through this background field. That's one good definition of mass: if something has mass, I can't create a particle with arbitrarily little energy (as E = mc^2), and it will move slower than light.

What about the photon? Well, it turns out due to the structure of the electroweak and hypercharge forces, that the Higgs has a bit of a choice in how it obtains its vev. Way back when, when the Universe was very hot, the Higgs field floated freely away from this minimum, but as it cooled, at some point in the Universe, it settled into a particular "direction." This means that some combination of W3 and B sees the vev, and so is impeded in its progress, and gains mass. The other combination is "just right" to avoid interacting with the combination of hypercharge and electroweak charge of the Higgs vev. This one remains massless, and we call it the photon. The combination of charges it interacts with is called electric charge, and this is the electromagnetic force we know. We know it BECAUSE the photon is massless; as a result the force is long range, and thus capable of affecting our day-to-day lives.

The other combination, the Z, couples to a combination of hypercharge and the electroweak charge, which you can rewrite as a (different) combination of electric charge and electroweak charge. The W+ and W- couple only to electroweak charge; but since they themselves have electroweak charge, they get interactions with the photon and Z. That's why we use the + and - notation for them: they now have electric charge. Using our handwaving of the infinite range of massless forces, this means that massive force carriers make the force short-ranged: the particles can't travel very far before their "borrowed" energy has to be paid back. This is why the weak force is weak: because the force carriers responsible for it are incredibly short range (like 10^-18 m short).

Now, what about those extra states for the Z and W?

If you take any quantum field, and throw energy into it, you'll get an excitation: a particle. Now, to get the breaking of electroweak and hypercharge, the higgs field needs to be a scalar (spin-0), complex (real + imaginary numbers), and a "electroweak doublet." That's 4 states total: two charged under electromagnetism, and two neutral. Three get "eaten" by the W+, the W-, and the Z, becoming their zero angular momentum mode, and thus making up the extra states . So a Ws and Z we see with spin-0 are actually "the higgs," or at least, part of the higgs. So we've already seen 3/4 of the Higgs field. It's just that one last, non-eaten Higgs.

So, the job of the Higgs is to break the electroweak and hypercharge forces into two new forces: electromagnetism and the weak force. It makes the gauge carriers of the weak for heavy, making the weak force… weak. The key term here is "electroweak symmetry breaking" or EWSB. SOMETHING in the Universe has to be breaking the forces in this manner, so there is absolutely something very similar to the Higgs field out there. It has to have a particular arrangement of charges, and provides the extra states for the W and Z. However, it doesn't have to be as simple as this minimal Higgs I just described.

But what about mass? Everyone hears about how the Higgs "gives everything mass," and so far I've just said a lot about mixing up charges and forces, and yes, it gives mass to the W and Z, but what about the rest of the Universe.

Turns out, the Higgs doesn't give mass to everything. A proton has a mass of 0.980 GeV. Of that, only about 0.01 GeV can be attributed to the Higgs. The rest is is due to strong force interactions. Without the Higgs, your body would be only about 1% lighter. You would not notice however, since you'd be busy dissolving into ions as every electron zipped off at the speed of light.

What the Higgs does (or, what it does in addition to EWSB) is give mass to the fundamental fermions (spin-1/2 particles): the electron, the up quark, the down quark, the muon, and so on (not the proton or the neutron, they are made up of quarks and gluons and are not fundamental). Why is this so? and why does it matter?

It turns out that the Higgs gives mass to fermions due to another, totally bizarre, fact about the Universe. A spin-1/2 particle can be divided up into two possible states: one with +1/2 angular momentum in a particular direction (say in the direction it is moving in), and one with -1/2 spin in that direction. Let's call the first set "right-handed" and the second "left-handed" (RH and LH for short). Now, you'd think that a RH and a LH electron (for example), are the same particle, just spinning in opposite directions. And in a sane universe, they would be. In our universe, they are not.

The LH state interacts with hypercharge and electroweak charge. The RH state interacts only with hypercharge. Weirder still, the hypercharge of the RH state isn't even the same as that of the LH one. This means that any of these wacky fermions can't be massive.

Why is that? Well, if you run past something rotating clockwise (that's right-handed angular momentum) and look back, you'll see it rotating counterclockwise (left-handed momentum). So the only way an electron can have mass if it has both LH and RH states. But it doesn't: there's an e_L and and e_R, but they aren't the same thing. So both e_L and e_R zip along at the speed of light, and this is fine, since you can't "run past" a particle moving at the speed of light.

The Higgs, it turns out, has exactly the right set of quantum numbers to balance the difference between the LH and RH states of fermions. And not just the electron. The quarks as well, even though the assignment of charges is different, the difference between LH and RH is still exactly the same as for electrons, and exactly the charges that the Higgs has. Even odder at first glance, the combination that wasn't broken by the Higgs: electromagnetism? Well, both LH and RH fields have the same electric charge. So once the Higgs obtains its vev, an e_L zipping along can interact with the background field, "shed" the excess hypercharge and weak charge, and leave as an e_R (or vice versa), all the while have electric charge of -1. The stronger that interaction with the Higgs background field, the "slower" the particle wants to go, and so the more massive the particle is. (I've been writing this as if there is an absolute frame of reference; there isn't, and nothing here requires there to be one. Unfortunately, it's hard enough to do this explanation at this level, doing it while remaining Lorentz invariant is beyond my skills. The point is, the Higgs is not the ether). 

So you see the incredibly intricate nature of the Standard Model of particle physics. At high energies, the Universe seems to have completely different forces, with particles that interacted completely differently than they appear to down at the low energies where we live. Once the Higgs gets a vev though, you mix up the forces, leaving one combination as the photon, and giving mass to the other combinations. Then, it just so happens that the field that does that magic trick also has the right properties to give all the fermions mass, even though that doesn't seem quite related. 

Of course, we physicists are always looking for a better answer than "wow, that's complicated." If you look at this intertwined structure, you can sort of see patterns emerging; one possible set of patterns allows you to unify all the forces together (just as electomagnetism and weak forces look separate at low energies, but unify at high energies, as I've described, you can do the same trick again with the strong, weak and hypercharge forces, and unify them all). Such models are called Grand Unified Theories (GUTs), and we have quite sorted them out yet (the more obvious ones make predictions that turn out to be wrong in our universe), but the gauge structure of the Standard Model is just crying out for something like this.

Also, I've described the minimal Higgs. There are more complicated models (for example, TWO Higgses!) that have to do the same job as a single Higgs, but make predictions as to the detailed properties of the lightest Higgs-like particle. However, whatever it is has to talk to the W and Z (it's intimately related to their masses, after all), and has to talk to the fermions (same reason). So, to some degree, the LHC has to find something. There is a no-lose "theorem" (really: calculation) which says that, based on the known properties of the W and Z, whatever is responsible for EWSB has to become apparently at an energy low enough to be visible at the LHC after enough data has been gathered.

So, we look for the Higgs not because we don't know that there's something there; there has to be something, or else many of our fundamental building blocks of quantum field theory are just wrong (which is possible, and interesting, but they've proven themselves several times already, so it's unlikely), but because we don't know the full details of what exactly is going on. The simplest answer is just that we'll find the simplest "Standard Model" Higgs, and go from there. The more exciting answer is that what we find at the LHC doesn't have quite the properties we expected, which means that there's more we didn't know about that's related to all of this symmetry breaking and mass generation. Right now (in 2015), the Higgs we discovered at 125 GeV is very close to the predicted Standard Model values, but we still need to check many properties; including measuring the direct couplings of the Higgs to the fermionic "matter" particles.

Oh yeah, without the Higgs, the electrons (and quarks) would be massless. Meaning that they would be moving at the speed of light and would be unable to be bound into atomic orbitals. So your atoms would lose all their electrons, chemistry would cease, and the Universe would be much more boring. Hurray for the Higgs.