A post for general readers who’ve heard of quarks; if you haven’t, you might find this article useful:
Yesterday I showed you that the usual argument that determines the electric charges of the various types of quarks uses circular reasoning and has a big loophole in it. (The up quark, for example, has charge 2/3, but the usual argument would actually allow it to have any charge!) But today I’m going to show you how this loophole can easily be closed — and we’ll need only addition, subtraction and fractions to close it.
Throughout this post I’ll shorten “electric charge” to just “charge”.
A Different Way to Check Quark Charges
Our approach will be to study the process in which an electron and a positron (the electron’s anti-particle) collide, disappear (“annihilate”), and are converted into one or another type of quark and the corresponding anti-quark; see Figure 1. The rate for this process to occur, and the rate of a similar one in which a muon and anti-muon are produced, are all we will need to know.
In an electron-positron collision, many things may happen. Among the possibilities, the electron and positron may be converted into two new particles. The new particles may have much more mass (specifically, rest mass) than the electron and positron do, if the collision is energetic enough. This is why physicists can use collisions of particles with small mass to discover unknown particles with large mass.
In particular, for any quark of mass M, it is possible for an electron-positron collision to produce that quark and a corresponding anti-quark as long as the electron’s energy Ee is greater than the quark’s mass-energy Mc2. As Ee is gradually increased from low values, more and more types of quark/anti-quark pairs can be produced.
This turns out to be a particularly interesting observation in the range where 1 GeV < Ee < 10 GeV, i.e. when the total collision energy (2 Ee) is between 2 and 20 GeV. If Ee is any lower, the effects of the strong nuclear force make the production of quarks extremely complicated (as we’ll see in another post). But when the collision energy is above 2 GeV, things start to settle down, and become both simple and interesting.
In Figure 2 is the actual data showing the production rate for electron-positron collisions to lead to quark/anti-quark pairs of any type, no matter what their “flavor” or “color” (i.e. type or version), for different collision energies. What’s shown on the horizontal axis is not Ee but the collision energy 2Ee. The up, down and strange quarks have small masses, so they can be produced almost everywhere throughout this Figure. We should expect interesting changes to occur at or around twice each quark’s mass-energy, namely at twice the charm quark’s mass (around 3 GeV) and twice bottom’s mass (around 9 GeV.) You can see this expectation is borne out: there are big spikes in the data just above those locations. But then, after a few wiggles, things flatten out and become simple. It’s from these simple regions that we can gain simple insights through simple methods.
The production rate is simple in these regions because
- the weak nuclear force plays no important role in this process until 2Ee is about 40 GeV;
- the strong nuclear force is increasingly unimportant as 2Ee increases above 2 GeV, especially in the regions with simple behavior, except where there are spikes in Figure 2;
- the gravitational and Higgs forces are too tiny to have any effect;
- and therefore the process can be understood using only electromagnetism, the very simplest of the elementary forces.
With simple knowledge about how electromagnetism produces quark/anti-quark pairs from electron-positron annihilation, we can learn crucial information from these simple regions of Figure 2. This turns out to be easy; we don’t have to go into any detail.
A Simple Fact
Here’s the observation that makes it possible to measure the quark charges.
In electron-positron collisions, the rate for producing a new particle/anti-particle pair via the electromagnetic force (as in Figure 1) is simply proportional to the square of the new particle’s electric charge.
Why the square? Proof comes from quantum physics, but here’s a strong argument. If the rate were proportional to the charge itself, that would be weird. For positive charge, the rate for production would be positive, but a particle with negative charge would be produced with a negative rate… and how can a production rate be negative? (Would we be unproducing a particle that wasn’t there to start with?) So, no: the rate must be proportional to something that’s always positive.
Also, the rate has to depend on the charge, since electrically neutral particles can’t be affected, much less produced, by electromagnetism. For similar reasons, the rate ought to be small for particles whose charge is small. The simplest positive quantity which satisfies these requirements is the square of the charge.
Meanwhile, if the particle comes in multiple versions, as quarks do (we call those versions “colors”), then each version gets produced in this same way as in Figure 1.
So a particle with charge Q that comes in N versions will have a production rate proportional to N Q2.
This is all we will need to know!
A Simple Strategy
The way to make everything simple, allowing us to avoid any hard calculations at all, is to compare the production of quarks and anti-quarks with the similar production of muons and anti-muons in electron-positron collisions (see Figure 3), both of which can be measured at each collision energy 2Ee.
Specifically, we will calculate the ratio “R”:
- R = Rate (e+ e— ⟶ quark anti-quark) / Rate (e+ e— ⟶ muon anti-muon)
R will change with the collision energy as more and more types of quarks can be produced. The reason this is a good idea is that in electromagnetism, muon/anti-muon production is almost identical to quark/anti-quark production, except for simple details, so almost everything cancels out of this ratio.
- R = Rate (e+ e— ⟶quark anti-quark) / Rate (e+ e— ⟶muon anti-muon)
- = (sum of NQ2 for all quark types produced) / (NQ2 for muons)
- = (sum of NQ2 for all quark types produced) / (1)
In the last line, I used the fact that for muon/anti-muon pairs, N Q2 = 1; that’s because muons have the same charge as electrons (Q = -1, so Q2 = + 1) and they don’t have “color” — there’s only one version of a muon — so N=1. Meanwhile, N=3 for all quarks, so
- R = (sum of 3Q2 for all quark types produced)
- = 3 ✕ (sum of Q2 for all quark types produced)
which is amazing simple for anything involving quantum field theory and particle physics!
A Simple Prediction
Therefore, since the Standard Model says [using notation that “Qu” means “electric charge of the u quark“]:
- Up, Charm, Top (u,c,t): Qu = Qc = Qt = 2/3
- Down, Strange, Bottom (d,s,b): Qd = Qs = Qb = -1/3
we get three predictions that we can compare with data:
- for small 2Ee in the 2 – 3 GeV range, we can produce up, down and strange quarks, so
- R = 3(Qu2 + Qd2 + Qs2 ) = 3(4/9+1/9+1/9) = 4/3 + 1/3 + 1/3 = 2
- for intermediate 2Ee > 3 GeV or so, we add the charm quark:
- R = 4/3 + 1/3 + 1/3 + 4/3 = 10/3 = 3.33…
- for large 2Ee > 10 GeV or so, we add the bottom quark, so
- R = 4/3 + 1/3 + 1/3 + 4/3 + 1/3 = 11/3 = 3.67…
Comparison with Data
What does the data, taken over many years at many experiments, say? I’ve plotted it in Figure 4, along with the three predictions for R that I just calculated for you. The data scatters around because the measurements aren’t perfect (and I haven’t shown the uncertainty bars), but you can see the trends by eye. The predictions of the Standard Model work well — not perfectly, as they’re always a little below the data, but close in each region.
If the Standard Model were wrong, the data and predictions could easily be far apart. For instance, the loophole I pointed out last time would allow Qu = Qc = 1 and Qd = Qs = Qb = 0. But then the predicted R in the three simple regions would have been 3, 6, and 6; that would have been way off. Unless the charges are very close to those predicted in the Standard Model, predictions are far from the data, and so the loophole from last time is now closed.
Also, predictions can’t explain the data for any other value of N, so the number of quark “colors” is verified to be 3. And meanwhile, the fact that these predictions almost match the data confirms that we were right to largely ignore all the other forces in the Standard Model. In this sense, many facets of the Standard Model are being simultaneously tested here.
But what about the fact that the data always runs about 10% above the prediction? It turns out this is due to the fact that the strong nuclear force cannot, in fact, be completely ignored. The process in which an electron and positron annihilate and produce a quark, an anti-quark and a gluon is large enough that we must include it if we want a more accurate prediction. Accounting for this makes the agreement much better. It also leads us to a more complex topic for another post I’ll produce soon: the variable strength of the strong nuclear force.