# A Primer on Particle Physics, Part 6: Quark Mixing

I must apologize for the unplanned hiatus in this series the last few weeks. As the series has expanded, it ended up running into the start of the semester and some constraints on my time. Hopefully, we’ll move along through the rest of it.  Here’s a link to the earlier parts:

Last time, we talked about the weak force and, in particular, how it applies to the quarks. There was one last piece of that we didn’t get to, though, so this time we’re going to get scandalous and talk about semi-forbidden cross-generational quark mixing.  This topic really ought to go along with the previous section on quarks, but since that was running long already, and since it didn’t make terribly much sense to put this in with the next section, on leptons, I’ve broken it into its own part and taken the opportunity to talk about it in more depth than I otherwise would have.  So, this will be a little bit of a tangent before we get back to our regularly scheduled programming next week (I hope!),

Let’s quickly review some of what we’ve talked about. Recall that the quarks come in six flavors, organized into three “generations” of increasing mass. In each generation, there is one quark with a positive electric charge of +2/3 and one with a negative electric charge of -1/3. Thus in the first generation we have the up and the down, in the second the charm and strange, and in the third the top and bottom. The weak force allows flavor changing within a generation; by absorbing or emitting a W boson, an up quark can turn into a down quark, or vice versa, like this:

Similarly with strange and charm quarks, or with top and bottom. Note again that the above diagram is not a valid process by itself, since it is impossible to conserve energy and momentum; but it can be part of a valid Feynman diagram, if we connect up the W boson to some other particles. This allows down quarks to decay into up quarks (plus some other stuff) – or rather, hadrons containing down quarks to decay into hadrons containing up quarks, such as a neutron decaying into a proton (plus an electron and a neutrino). And it allows charmed hadrons to decay into strange hadrons, and top quarks (which are too short-lived to form hadrons) to decay into bottom quarks. (Note that the decays can only go in one direction, because a particle can only decay into a less massive particle, not a more massive one).

Since the weak force only couples quarks within a generation, the strange quark and bottom quark would appear not to have any way to decay. They’re significantly more massive than the up and down quarks, so in a sense they “want” to decay, but there are no allowed processes by which this can happen. This is why strange and bottom quarks, as I mentioned last time, have much longer lifetimes than one would expect based solely on their masses.

But ultimately, the strange and bottom quarks do decay. Something called quark mixing allows for small couplings between generations, which means that the strange and bottom quarks have “semi-forbidden” decay channels available. So let’s talk about quark mixing.

I’ll start by making a strange statement, and then I’ll try to explain what it means: the six different quark “flavor states” are not quite identical to the six different quark “mass states”.

We’ve touched on the concept of the “state” of a particle before, and the idea that in quantum physics, a particle’s state can be a “superposition” of different states. For instance, we’ve talked about how a particle can be in a state where it does not have a definite position; instead, its state is the sum of any number of states of definite position, each with an “amplitude” that, loosely, tells us the probability of finding the particle at that position if we were to measure its position. Similarly, a particle might be in a state of indefinite momentum, meaning that it is a sum, or superposition, of different momentum states. And in fact, momentum and position are what are called “incompatible”. That means that whenever a particle is in a definite position state, it is necessarily not in a definite momentum state, and vice versa. If we have a particle in a state of definite momentum, then, and we want to think about it in terms of position, we can write that momentum state as a sum of different position states. In fact, given a particle in any state, we can write that state down either in terms of definite momentum states or in terms of definite position states. Position and momentum are what we call different bases in which we can write down the state of a particle.

In some cases, it also makes sense to talk about a particle being a superposition of what we might call different “kinds” of particle. For instance, suppose there is some process that 90% of the time produces a down quark and 10% of the time produces a strange quark. Immediately after this process occurs, we could describe the resultant particle as being in a superposition of “down” and “strange” states, with a higher amplitude for the “down” state. What that superposition means is that if a measurement were then made to determine whether the particle is a down or strange quark, there would be a 90% chance the result would be “down” and a 10% chance “strange”.

All right, so now we get to the crux of it. I’ve said that there are six different kinds of quark, but what does that, like, mean, man? Well, for one thing, it means that if we have a quark and we measure its mass, there are exactly six different possible answers. There are six different “mass states” that a quark can be in. It also means that when a quark undergoes a weak interaction, there are six different possible flavors it could have, and which flavor it is determines which particular weak processes are allowed. That is, there are six different “flavor states” that a quark can be in. And here’s the punchline: just as states of definite momentum are not states of definite position, quark states of definite mass are not states of definite flavor (and vice versa).

Let’s get into what this means. For clarity, let’s have some definitions. If a quark is in the determinate state where it has a mass of 2.2 MeV (which, recall, is what I told you the mass of the up quark is last time) and an electric charge of +2/3, we’ll call that the “up mass state”. If it has a determinate mass of 4.7 MeV and a charge of -1/3, it’s in the “down mass state”. And so on for the “strange mass state”, “charm mass state”, “bottom mass state”, and “top mass state”. On the other hand, if a quark is in a determinate state where it has a charge of -1/3 and when coupling to the weak interaction, it acts as a first-generation quark, we’ll call that the “down flavor state”. What do we mean when we say it “acts as a first-generation quark”? We mean that if it undergoes a charge-changing weak interaction, it will, definitely, turn into a quark in the up mass state – that is, a quark with a mass of 2.2 MeV (and a charge of -1/3). Similarly, if a quark is in an “up flavor state”, that means it has a charge of +2/3 and, under the charge-changing weak interaction, will definitely produce a quark with a mass of 4.7 MeV (and charge +2/3). The mass states, then, are defined by the mass of the quark. The flavor states are defined by which mass state the quark will be in if it undergoes a charge-changing weak process.

So far, I haven’t said anything that necessitates the mass and flavor states being different. In a simpler world, without quark mixing, a down mass state, for instance, would be a down flavor state. That is, if we had a down quark, a quark with a mass of 4.7 MeV (and a charge of -1/3), we’d have a quark that would definitely end up as a 2.2 MeV, +2/3 up quark after a flavor change. Similarly, if we had a quark that we knew was in the down flavor state (for instance, if we knew it had been produced by flavor-changing a 2.2 MeV up quark), then we’d know that this quark was also in the down mass state. That means that we’d know for certain that if we were to measure its mass, we’d definitely get a result of 4.7 MeV.

So here’s what I mean when I say that the mass states and flavor states are not identical. The down flavor state is a superposition of the down, strange, and bottom mass states (i.e. all the +2/3 charge mass states). In other words, if we have a quark in the down flavor state and we measure its mass, we could get a result of 4.7 MeV, or of 95 MeV (the strange quark mass), or of 4.2 GeV (the bottom quark mass). The probability for each of these results is given by the amplitude of that state, and the down flavor state is equal to a particular combination of amplitudes for the down, strange, and charm mass states. So if you give me a quark and tell me it’s in the down flavor state, I won’t be able to predict with certainty what it’s mass will be if I measure it; but if you give me a lot of quarks that are all in the down flavor state, I can predict what fraction of them we’d find to have each of those three possible masses, if we measured them.

|down-flavor> = A|down-mass> + B|strange-mass> + C|bottom-mass>

Where A, B, and C are physical constants. There’s nothing in the theory that tells us what these constants are; they have to be empirically measured, just like the masses or charges of the various fundamental particles. If we have a quark in this state, the rules of quantum physics tell us that if we measure the mass, the probability for each result is the square of the amplitude for that state. So if we measure the mass of a |down-flavor> quark, there is an $A^2$ chance we’ll get 4.7 MeV, a $B^2$ chance we’ll get 95 MeV, and a $C^2$ chance we’ll get 4.2 GeV. Note that “a quark in the down flavor state” means more or less the same thing as “the kind of quark you get if you start with an up quark and perform a flavor change”. So one way to think of this is that it’s telling us, if we flavor-change an up quark, what the chance is that we’ll end up with a down quark vs. a strange quark vs. a bottom quark. In fact, the traditional notation for those constants that I called A, B, and C is $V_{ud}$, $V_{us}$, and $V_{ub}$, because they tell us the chance of an up quark changing into, respectively, a down quark, a strange quark, or a bottom quark, under a flavor-change.

All this same stuff is true of the strange flavor state (i.e. the state that couples to the charm mass state), and of the bottom flavor state (the state that couples to the top mass state). The strange flavor state is a superposition of down, strange, and bottom mass states, but with a different set of amplitudes, which we call $V_{cd}$, $V_{cs}$, and $V_{cb}$ (since these tell you the probabilities for what you’ll end up with if you flavor change a charm quark). And the bottom flavor state is a superposition of down, strange, and charm mass states with yet another particular set of amplitudes, telling you the probabilities for what you’ll get if you flavor change a top quark. And all the same can be applied to the positively charged quarks, as well. The up flavor state (i.e. what you get if you flavor change a down quark) is a particular superposition of the up, charm, and top mass states; the charm flavor state is a different particular superposition of them; and the top flavor state is a third particular superposition.

Note that, in all of this, the positively charged (+2/3) and negatively charged (-1/3) quarks remain separate. An up flavor state, for instance, consists only of a sum of the three positive mass states, with no contribution from the negatively charged down, strange, or bottom mass states.

So, the flavor states are not the same as the mass states. But here’s the thing: it turns out that they are almost the same. I told you, for instance, that we can write the down flavor state like this:

|down-flavor> = A|down-mass> + B|strange-mass> + C|bottom-mass>

Now, note that if A were exactly 1, and B and C were exactly 0, we’d be back in that simpler world I imagined earlier, where the down flavor state is the same as the down mass state. In that world, any time we started with an up quark and flavor-changed it, we’d always end up with a down quark. Well, it turns out not to be that way – but it also turns out that A is fairly close to 1, B is kind of small, and C is very small. To wit, A (or, what a physicist would call $V_{ud}$) is about 0.97, B ($V_{us}$) is about 0.22, and C ($V_{ub}$) is about 0.004. When we square these to get the probabilities, we get that if an up quark undergoes a flavor-changing weak interaction, it has about a 95% chance of ending up as a down quark and about a 5% chance of ending up as a strange quark, with only a less than 0.002% chance of turning into a bottom quark.

Similarly, we find that the strange flavor state is mostly made up of the strange mass state, with only small contributions from the down and bottom mass states, which means that if a charm quark undergoes a flavor-change, it is most likely to end up as a strange quark. And the same applies to the bottom flavor state; a top quark that flavor-changes is most likely to end up as a bottom quark. This is all symmetrical, so we can say the same things about the positive states, and about the probabilities involved when negatively charged quarks flavor change into positive ones.

We typically think of these amplitudes (the $V_{ud}$, $V_{us}$, etc., values) as forming a matrix, which we call the CKM matrix after Cabbibo, Kobayashi, and Muskawa, who developed this model of quark mixing. If you’re not familiar with matrices or linear algebra, don’t let this frighten you; just think of it as a way of expressing how much each positive quark (the rows, in the order up, charm, top) couples to each negative quark (the columns, in the order down, strange, bottom). These are the best fit values as of 2019; I’m omitting the uncertainties on these measured values to keep it more legible.

So, for instance, if you want to know what the probability is that a flavor-change will turn a charm quark into a bottom quark, you look at the second row, third column, $V_{cb}$. Notice that the diagonal elements of this matrix – the couplings within a generation – are large, and the off-diagonal elements – the ones that mix generations – are smaller. Pictorially, the quark mixing matrix is structured like this:

The upshot of all this, then, is that flavor-changes keep a quark within the same generation most of the time, and conversely that flavor changes that move a quark from one generation to another are comparatively unlikely and uncommon. Among other things, this means that strange and bottom quark decays are suprressed. They would prefer, so to speak, to flavor-change into charm and top quarks, respectively – but because of the masses of the quarks, they can’t decay into charms and tops. Instead, a strange quark will decay into an up quark, and a bottom quark will decay into either an up or a charm quark, but since these involve changing generations, these decays are less common than they otherwise would be.

Here’s a diagram that illustrates this. The solid connections represent the most significant weak couplings, the dashed lines show weaker couplings, and the dotted lines show the weakest couplings. The direction of the arrows indicate possible decays, pointing from more massive to less massive quarks.

You can see that one-generation flavor changes are less probable than flavor changes within a generation, and changes across two generations are still more improbable. As I said, to determine exactly what combination of different mass states makes up a given flavor state – and thus, the exact probabilities for these generation-crossing flavor changes – is a matter of going out and measuring them. In particular, one measures the lifetimes of hadrons containing different quarks and from this one can calculate the amplitudes. It might seem like there are a lot of different numbers to measure here – for each of the six flavor states, you have three different amplitudes telling you how much of each mass state it has in it, so that ought to be eighteen different constants. But it’s actually not as bad as that. Note, for one thing, that since they represent probabilities, the squares of the three amplitudes that make up any one flavor state have to add up to one. (If you add up all the mutually exclusive probabilities for something, you always have to get 100%).

Using this and other symmetries, it turns out that there are only four independent parameters needed to determine all these amplitudes. These are usually expressed as three mixing angles as well as something called a CP-violating phase. It’s not necessary to get into why these are called “angles”; the point is that all of these different amplitudes, telling us the probabilities for one kind of quark changing into another, can be calculated from those four numbers. (If you’re familiar with the idea of a vector space, what it comes down to is that the mass states and flavor states form different bases for the vector space of quark states, and the mixing angles fully specify the rotation between one basis and the other).

It’s worth taking just a moment to talk about the CP-violating phase, though. You can loosely think of this as telling us how the mixing between antiquarks differs from the mixing among quarks. If you recall our discussion antiparticles and symmetries, you’ll know that an antiquark is a essentially a quark with opposite charge and parity, so if quarks and antiquarks behave differently, this is equivalent to saying that CP is not a symmetry of the universe. We’ve already seen that C-symmetry and P-symmetry are each individually violated – specifically by the weak force, which only interacts with “left-handed” particles and “right-handed” antiparticles. But I mentioned that the combined CP transformation looks, at first glance, like it is a symmetry. But here we see a place where even this combined CP-symmetry does end up getting violated. If, when we went out and measured the CP-violating phase, we found that its value was zero, that would mean that quarks do not violate CP. As it turns out, though, the CP-violating phase in quark mixing is small, but definitely not zero. So quarks do violate CP-symmetry.

Actually, though, it shouldn’t be that much of a surprise to us that CP-symmetry is violated. After all, all the matter we see and interact with is made up of quarks and electrons, not antiquarks and positrons. And if we look out into the universe, we see vast amounts of normal matter, making up stars, nebulae, and galaxies, but no appreciable antimatter. If CP were a symmetry, and antimatter behaved exactly like matter, we’d expect that in the early universe, there should have been roughly equal numbers of quarks and antiquarks. But if that had been the case, then when the universe was in its hot, dense infancy, most of that matter and antimatter would have annihilated with each other, and the universe today would be far emptier than it is, with only a few relic particles that managed to avoid annihilation hanging around. So in fact, it looks like some CP-violation within the laws of physics is needed to explain the fact that we exist at all.

Well, we’ve just found one source of CP-violation – but, as I said, the CP-violating phase in quark mixing is quite small. In fact, it’s too small to come close to accounting for the matter/antimatter imbalance that exists in the universe. Next time, we’ll see one possible avenue toward explaining that, when I introduce the leptons that we haven’t met yet, including my beloved neutrino. Something very similar to quark mixing happens in the lepton family, but it hasn’t been studied for as long and the parameters for that mixing haven’t been measured as precisely; so based on our current knowledge, it’s possible that its corresponding CP-violating phase can explain the matter/antimatter asymmetry of our universe. Hence, the (perhaps slightly hyperbolic) claim made by my grad school adviser, among others: Neutrinos could be the reason we exist.