# A Primer on Particle Physics, Part 5: Bursting with Flavor

I have to apologize for how long it’s taken to finish this part of the series. You can find the previous articles here:

Last time we kicked off our tour of the particle zoo by looking at the electron, the up and down quarks, the photon, and the gluon, as well as a few composite particles made of them – the proton, the neutron, and the pion.

This looks like something of a random hodgepodge, but I started with those particles because they constitute all the ingredients of the atom. Quarks make up the protons and neutrons, and are bound together by the strong force, which is carried by gluons. The protons and neutrons are bound together in the nucleus by a weaker form of that same strong force called the residual strong force. The residual strong force is mediated by pions, which are themselves made up of quarks and antiquarks. And the electrons are bound to the nucleus by the electromagnetic force, carried by photons.

In a sense, then, it would seem as if these are all the particles that we “need”. And yet, as early as the 1930s, physicists started to discover other particles, mostly short-lived, that played no role in making up atoms, molecules, and matter as we know it. They were at first somewhat taken aback by this – these new particles seemed to be superfluous. When the muon, the first of these “superfluous” particles, was discovered, Isidor Isaac Rabi famously joked, “Who ordered that?”

Nowadays, it’s perhaps easier to see that there’s nothing shocking or puzzling about the existence of more particles than we “need”. After all, the reason the electron, proton, neutron, and photon are so fundamental to our experience of the universe is that they are all either stable or nearly so, and they are highly interactive, meaning that they interact via the strong force and/or electromagnetism. This means they stick around long enough to form structures like atoms and molecules, and they also feel the forces necessary to form those structures. As we look beyond these basic ingredients, we’ll find particles that either don’t feel those forces or are unstable and short-lived.

Generations of Fermions

Last time, we talked about the up and down flavors of quark. Today we’re going to expand our menu and take a look at all six of the quark flavors, the other four being the strange, charm, bottom, and top quarks.

The way these quarks are arranged in the chart of the Standard Model particles above is not arbitrary; it turns out that there is something of a loose pattern to them. To wit, there are three pairs of quarks, where one of each pair has an electric charge of +2/3 and the other -1/3 (I’m going to drop the e in talking about charges, just as we drop the $\hbar$ when we talk about spins). So there is the down/up pair, the strange/charm pair, and the bottom/top pair, and the quarks in each pair are heavier than those in the preceding pair. It turns out that these flavor pairs aren’t just “coincidental’, so to speak; they have actual physical significance, which we’ll talk about a little bit later. Each pair is called a “generation”, up/down being the first generation, and so on. As you can see from the chart, and as we’ll discuss next time, the other kind of fermions, the leptons, also come in three generations.

All right, let’s briefly take a look at each of these quarks and their properties. As I said last time, the up quark has a charge of +2/3 and a mass of about 2.2 MeV1 And the down quark has a charge of -1/3 and a mass of 4.7 MeV.

When we move to the second generation, we find the negative strange quark with a charge of -1/3 and a mass of 95 MeV and the positive charm quark with a charge of +2/3 and a mass of 1.275 GeV. Woah, what happened there? In the first generation, the up and down quarks were pretty close in mass, but in the second generation, the charm quark is more than thirteen times heavier than its strange partner. Also, note that in the first generation, it’s the negative partner that’s heavier, whereas in the second generation, it’s the positive partner.  The strange quark was, together with the up and down, part of the original quark model developed in 1964, and confirmed experimentally in 1968 in experiments at the Stanford Linear Acclerator (SLAC).  Since the charm quark is heavier, it takes significantly more energy to produce it (and the hadrons it is a component of).  The existence of a fourth type of quark was predicted in 1970, and hadrons containing the charm quark were discovered in 1974.  The latter discovery was the trigger for a series of advances called the “November Revolution” that led to the full formulation of the Standard Model.

In the third generation, we have the negative bottom quark with a mass of 4.18 GeV (and charge -1/3), and the positive top quark with a mass of 172.8 GeV (and charge +2/3). So here, the positive partner is more than 41 times heavier than the negative!  This third generation of quarks was hypothesized in 1973.  The bottom was experimentally observed in 1977, but the heavy top was not seen until 1995.  (As a historical note, when the third generation of quarks was discovered, there was some debate as to whether to call them “top” and “bottom” or the more poetic “truth” and “beauty”. Part of me kind of laments that the prosaic option won out.)

Remember, each of these flavors also comes in three “colors” (the “charge” for the strong force). And each quark also has its corresponding antiquark, with opposite electric and color charge. So, you could have a green bottom quark, or a red charm quark, or an antiblue strange antiquark.

I explained earlier that, as a rule of thumb, the heavier a particle is, the shorter its lifetime is. So these more massive quarks have increasingly short lifetimes. For most of these flavors, it’s a little hard to talk about the lifetime of a free quark, since color confinement, as we talked about last time, means that quarks are, in practice, pretty much only found as constituents of hadrons; depending on the hadron it’s part of, a quark’s lifetime may be shorter or longer. But roughly, the down quark has an average lifetime of on the order of 900 seconds, the strange quark has a lifetime of about 12.4 nanoseconds ($1.24x10^{-8}$ s), and both the charm and bottom quarks have lifetimes of about 1 picosecond ($10^{-12}$ s). And the top quark has an extremely short lifetime of about $10^{-24}$ s – a trillion times shorter than its partner, the bottom quark. In fact, the top quark decays so quickly that it doesn’t have time to bind with other quarks and form hadrons.

So what’s the deal? To understand these anomalies, we’ll have to take a look at how quarks and hadrons decay. And that means it’s finally time to introduce the weak nuclear force, also known as just “the weak force”.

The Weak Force

The weak interaction is undoubtedly the force that will seem strangest to a layperson. Gravity is the force that pulls things toward large masses; electromagnetism is the force that generates electricity and binds electrons to nuclei; the strong force is the force that holds quarks together in protons and neutrons, and those together in nuclei. There’s no similarly pithy way to describe the weak force. Or, as XKCD puts it:

But hopefully, if you’ve read the previous entries in this series, it won’t seem so mysterious. Just as the “strong force” refers to processes mediated by gluons, and “electromagnetism” refers to processes mediated by photons, the “weak force” just refers to the processes mediated by a group of bosons called the $\mathbf{W^+}$, $\mathbf{W^-}$, and $\mathbf{Z^0}$, collectively known as the intermediate vector bosons. As the names might suggest, the $W^+$ has a positive electrical charge of +1 (i.e., the same charge as a positron), the $W^-$ is its oppositely charged antiparticle, and the $Z^0$ is electrically neutral.

If the strong force is strong, the weak force is weak, right? Well, yes; and this is essentially the reason that it plays a less obvious (to us) role in the universe. At low energies, the strength of electromagnetism is about one hundredth of the strength of the strong force, while the effective strength of the weak force is only about one millionth that of the strong force. As a result, alone out of all the forces, there are no larger structures bound together by the weak force.

There’s a simple reason for this effective weakness of the weak force, though – and the word “effective” is important there. The intrinsic strength of the weak force is actually similar to that of electromagnetism; the reason it is effectively much weaker is that, unlike the photon or the gluon, the bosons that carry the weak force are not massless. In fact, they are very massive – the W bosons have a mass of about 2.1 GeV, and the Z about 2.5 GeV. This makes them more than twice as heavy as the proton, and heavier than any of the quarks except the top and bottom. Since they are so massive, they decay very quickly, which severely limits the likelihood of processes that require these bosons as mediators. It also means that the weak force has a very short range, since the bosons will decay before they can travel very far. The effective range of the weak force is something like 1% the diameter of a proton.

The “amount” of the strong force that a particle feels depends on its color charge, and the strong force does not act on particles without a color charge. Similarly, a particle’s electric charge determines how much it feels electromagnetism, with neutral particles not feeling it at all. Note that both the electric and color charges are conserved quantities – in any process, the sum of these charges we start with has to be the same as the sum we end with. In contrast, the degree to which a particle “feels” the weak force can’t really be described with a single conserved “charge”; each particle instead is characterized by several numbers that dictate how it couples to the weak force. The strength of a particle’s coupling to the weak force is best characterized by a property called “weak hypercharge”, but this is not a globally conserved quantity. The W and Z particles themselves have weak hypercharges of zero, so like the photon, but unlike the gluon, the weak bosons do not couple to themselves.

Because, as we’ll see, the weak force is responsible for processes by which particles can change flavor, the theory describing it is called – and I’m not making this up – “Quantum Flavordynamics”.

Parity Violation

But let’s talk about the weirdest thing about the weak force: parity violation. As I discussed in an earlier part of this series, the laws of physics turn out not to be symmetrical under a “mirror image” transformation, which is deeply counterintuitive. And it is the weak force, specifically, that is responsible for this; the strong and electromagnetic forces are invariant under a parity flip.

To talk about how this works, we need to address the idea of the “handedness” of a particle. This is a topic I approach with some trepidation, because there are nuances here that could trip up anybody, myself very much included, particularly with regard to the two related concepts of “helicity” and “chirality”. Chirality is the thing that is actually relevant here, but it’s difficult to explain without doing the actual math, whereas helicity is fairly easy to explain (and I shall do so forthwith). And if you squint your eyes and fudge things a bit, and pretend that chirality and helicity are the same thing, the picture you get is not terribly inaccurate. So here’s what I’m going to do: I’ll explain helicity, I’ll give some idea of how chirality is different from it, and then I’ll talk about what this has to do with the weak force.

Imagine you have an electron zipping along. What is its angular momentum along the direction in which it’s moving? We know the answer – it’s 1/2, because the electron is a spin-1/2 particle. But that spin could be oriented either clockwise or counterclockwise relative to the direction the electron is moving. If the spin is aligned clockwise relative to the direction of motion, we say that the electron has right-handed helicity; if counterclockwise, left-handed helicity.

But velocity is relative. Which direction an electron appears to be moving in depends on how fast the observer is moving – that is, on what “inertial frame” we are in. As we discussed in part 1, Quantum Field Theory incorporates Special Relativity, which means that the laws of physics have to be the same regardless of what inertial frame things are measured in. The jargon for this is that the laws of physics have to be “Lorentz-invariant”. Helicity, then, is not a Lorentz-invariant quantity.

Particles do, however, have a Lorentz-invariant property that is a lot like helicity, mathematically, and we call this “chirality”. An electron can have “right-handed chirality” or “left-handed chirality”, but this, strictly speaking, is an intrinsic property of the particular electron in question. Whether an electron with right-handed chirality also has right-handed helicity depends on what frame of reference we’re measuring it in. Again, it’s a little difficult to explain what chirality is, physically, and why it’s related to helicity at all, without going through all the math. Essentially, the chirality of a particle tells you which direction that the phase of its wavefunction is rotating in. But let’s not worry about that; the important point is just that it’s a parity property that remains the same regardless of which direction the particle appears to be moving in, in a given frame of reference.

For massless particles, chirality and helicity always end up being the same. Massless particles always move at the speed of light. If you have a particle that moves at the speed of light, according to Special Relativity, that particle will always appear to be moving at the speed of light, in the same direction, regardless of what reference frame we’re in. So for a massless particle, helicity is Lorentz-invariant, and helicity and chirality are the same thing. A photon with right-handed chirality will always have right-handed helicity.

There’s a bit of a twist, though. This is quantum physics, and particles can be in superpositions of different states. And the state of an electron that enters into the math governing its free motion through space is the sum of its left- and right-chiral states. You can think of it this way: when it’s not interacting, an electron will keep flipping back and forth between left- and right-chirality as it moves through space2.

OK, so this is all more than a bit confusing. But the upshot I want you to take from it is that the “handedness” of a particle, in the sense that really matters, the sense of chirality, is an intrinsic property of that particle. An electron can be either right- or left-handed, or it can be a superposition of these states. But deep inside the math of QFT, this property is related to helicity, the direction of the particle’s spin relative to its motion. And although a lot of physicists will grind their teeth when I say it, it’s kind of OK for you to think of this chiral handedness property as having to do with the direction of the particle’s spin, provided that you realize that this is not quite accurate, and there are some subtleties that we’re sweeping under the rug.

One other point I want to quickly mention. As I think we discussed in part 2, antiparticles are related to their corresponding particles by both a charge and parity flip. This means that, strictly speaking, the antiparticle of a right-handed particle is left-handed, and vice versa.

Now, here’s the punchline. The weak force only acts on left-handed particles and right-handed antiparticles. This is the sense in which it is “maximally parity-violating” – if a particle has right-handed chirality, it doesn’t feel the weak force at all. In the case of the electron, which is a superposition of left- and right-handed states, this means that the weak force only acts on the left-handed part of that state. If you picture an electron moving along and flipping back and forth between chiralities, it can only couple to the weak bosons when it is in the left-handed state. Similarly, if an electron is produced by a weak boson, it will always be produced in a left-handed state.

Flavor-changing and Quark Mixing

Now that we’ve introduced the weak force, I want to return to the quarks and talk about the processes by which one flavor of quark can turn into another.

Basically, Quantum Flavordynamics allows vertices like these:

Here, $q^+$ and $q^-$ are the positive and negative flavors of quark of the same generation. So vertices containing a W, and up quark, and a down quark are allowed; or a W, a strange quark, and a charm quark; or a W, a bottom, and a top. You should be able to satisfy yourself that electric charge is conserved at these vertices – remember, all of the positive quarks have a charge of +2/3, and all of the negative quarks have a charge of -1/3, while the positive and negative W bosons have charges of +1 and -1. So we can, for instance, start with a down quark, which then emits a $W^-$ and becomes an up quark. We started with a total charge of -1/3 from the down quark, and we end up with a charge of 2/3 from the up quark plus -1 from the W, for a total of -1/3.

Note that the vertices I showed above don’t constitute complete Feynman diagrams – in order to be able to balance momentum and energy on both sides, that W needs to connect to something else. You could have a diagram like this, for instance:

This amounts to a scattering between an up and a down quark – we start with an up and a down, and we end with an up and a down. It could be an up quark emitting a $W^+$ (turning into a down quark), which is then absorbed by a down quark (turning it into an up quark). Or it could be the same thing in the other direction, with the down quark emitting a $W^-$. This kind of diagram should look familiar; it’s a lot like electron-scattering diagrams we’ve looked it.

However, the weak force is weak, and any processes like this are going to be far less likely, and thus happen far less frequently, than interactions between quarks mediated by photons or by gluons, which also can produce quark/quark scattering. So weak processes like the above are really not very important; at most, they amount to small corrections to the calculations for this kind of scattering, which are mainly dictated by the stronger forces.

Where the weak force does play a big role for quarks is where it allows processes that can’t be performed by the strong or electromagnetic forces alone. For instance, here’s another valid Feynman diagram:

This shows a down quark decaying into an up quark, an electron, and an anti-neutrino (we haven’t talked about neutrinos yet; they’re coming up next time). There’s no diagram you could draw that accomplishes this without using the W boson. That means that, unlike with the quark scattering example above, calculating the rate for this process – which corresponds to the lifetime of the down quark – is all about Quantum Flavordynamics. (Of course, as we’ve discussed, up and down quarks are never found in isolation, only as components of hadrons. So, in practice, what this kind of diagram is used to calculate is the lifetime of hadrons that contain down quarks.)

Could we have a similar diagram with an up quark changing into down quark, instead of vice versa? No, we couldn’t. To be sure, the vertices would all be allowed, if we changed the $W^-$ to a $W^+$. But remember that we need to balance energy and momentum on both sides, and that for decays – processes where we start with just one particle and end up with several – that means that the mass of the initial particle has to be greater than the total of the masses of the final particles. Since the down quark is heavier than the up quark, we can have down -> up decays, but not up -> down.

So, within each generation, the weak force allows the positive and negative flavors of quark to change back and forth, and in particular it allows the heavier quark from each generation to decay into the lighter one. And if that were all there were to it, the lighter quark of each generation – the up, strange, and bottom – would not have any way of decaying, and would be stable. Even though they have large masses, and in a sense “want” to decay, they have no path by which they can decay.

OK, but as I said earlier, the strange and bottom quarks do decay, albeit with much longer lifetimes than we’d expect based on their masses alone. There is a mechanism, called quark mixing, that does allow quarks to cross generations; but the fact that this generation-crossing is semi-forbidden – the fact that this quark mixing “loophole”, so to speak is needed to allow it – is why the strange and bottom quarks (and the hadrons they are part of) have those unexpectedly long lifetimes.

I am going to talk about quark mixing, but this article has already gotten long, so I’m going to save it and talk about it and something similar called neutrino mixing when we discuss the neutrino next time.