The Standard Model More Deeply: The Magic Angle Nailed Down

In a previous post, I showed you that the Standard Model, armed with its special angle θ_w of approximately 30 degrees, does a pretty good job of predicting a whole host of processes in the Standard Model. I focused attention on the decays of the Z boson, but there were many more processes mentioned in the bonus section of that post.

But the predictions aren’t perfect. They’re not enough to convince a scientist that the Standard Model might be the whole story. So today let’s bring these predictions into better focus.

There are two major issues that we have to correct in order to make more precise predictions using the Standard Model:

• In contrast to what I assumed in the last post, θ_w isn’t exactly 30 degrees
• Although I ignored them so far, the strong nuclear force makes small but important effects

But before we deal with these, we have to fix something with the experimental measurements themselves.

Knowledge and Uncertainty: At the Center of Science

No one complained — but everyone should have — that when I presented the experimental results in my previous post, I expressed them without the corresponding uncertainties. I did that to keep things simple. But it wasn’t professional. As every well-trained scientist knows, when you are comparing an experimental result to a theoretical prediction, the uncertainties, both experimental and theoretical, are absolutely essential in deciding whether your prediction works or not. So we have to discuss this glaring omission.

Here’s how to read typical experimental uncertainties (see Figure 1). Suppose a particle physicist says that a quantity is measured to be x ± y — for instance, that the top quark mass is measured to be 172.57± 0.29 GeV/c². Usually (unless explicitly noted) that means that the true value has a 68% chance of lying between x-y and x+y — “within one standard deviation” — and a 95% chance of lying between x-2y and x+2y — “within two standard deviations.” (See Figure 1, where x and y are called and . The chance of the true value being more than two standard deviations away from x is about 5% — about 1/20. That’s not rare! It will happen several times if you make a hundred different measurements. But the chance of being more than three standard deviations away from x is a small fraction of a percent — as long as the cause is purely a statistical fluke — and that is indeed rare. (That said, one has to remember that big differences between prediction and measurement can also be due to an unforseen measurement problem or feature. That won’t be an issue today.)

W Boson Decays, More Precisely

Let’s first look at W decays, where we don’t have the complication of θ_w , and see what happens when we account for the effect of the strong nuclear force and the impact of experimental uncertainies.

The strong nuclear force slightly increases the rate for the W boson to decay to any quark/anti-quark pair, by about 3%. This is due to the same effect discussed in the “Understanding the Remaining Discrepancy” and “Strength of a Force” sections of this post… though the effect here is a little smaller (as it decreases at shorter distances and higher energies.) This slightly increases the percentages for quarks and, to compensate, slightly reduces the percentages for the electron, muon and tau (the “leptons”).

In Figure 2 are shown predictions of the Standard Model for the probabilities of the W- boson’s various decays:

• At left are the predictions made in the previous post.
• At center are better predictions that account for the strong nuclear force.

(To do this properly, uncertainties on these predictions should also be provided. But I don’t think that doing so would add anything to this post, other than complications.) These predictions are then compared with the experimental measurements of several quantities, shown at right: certain combinations of these decays that are a little easier to measure are also shown. (The measurements and uncertainties are published by the Particle Data Group here.)

The predictions and measurements do not perfectly agree. But that’s fine; because of the uncertainties in the measurements, they shouldn’t perfectly agree! All of the differences are less than two standard deviations, except for the probability for decay of a W^– to a tau and its anti-neutrino. That deviation is less than three standard deviations — and as I noted, if you have enough measurements, you’ll occasionally get one larger than two standard deviations. We still might wonder if something funny is up with the tau, but we don’t have enough evidence of that yet. Let’s see what the Z boson teaches us later.

In any case, to a physicist’s eye, there is no sign here of any notable disgreement between theory and experiment in these results. Within current uncertainties, the Standard Model correctly predicts the data.

Z Boson Decays, More Precisely

Now let’s do the same for the Z boson, but here we have three steps:

• first, the predictions when we take sin θ_w = 1/2, as we did in the previous post;
• second, the predictions when we take sin θ_w = 0.48;
• third, the better predictions when we also include the effect of the strong nuclear force.

And again Figure 3 compares predictions with the data.

You notice that some of the experimental measurements have extremely small uncertainties! This is especially true of the decays to electrons, to muons, to taus, and (collectively) to the three types of neutrinos. Let’s look at them closely.

If you look at the predictions with sin θ_w = 1/2 for the electrons, muons and taus, they are in disagreement with the measurements by a lot. For example, in Z decay to muons, the initial prediction differs from the data by 19 standard deviations!! Not even close. For sin θ_w = 0.48 but without accounting for the strong nuclear force, the disagreement drops to 11 standard deviations; still terrible. But once we account also for the strong nuclear force, the predictions agree with data to within 1 to 2 standard deviations for all three types of particles.

As for the decays to neutrinos, the three predictions differ by 16 standard deviations, 9 standard deviations, and… below 2 standard deviations.

My reaction, when this data came in in the 1990s, was “Wow.” I hope yours is similar. Such close matching of the Standard Model’s predictions with highly precise measurements is a truly stunning sucesss.

Notice that the successful prediction requires three of the Standard Model’s forces: the mixture of the electromagnetic and weak nuclear forces given by the magic angle, with a small effect from the strong nuclear force. Said another way, all of the Standard Model’s particles except the Higgs boson and top quark play a role in Figs. 2 and 3. (The Higgs field, meanwhile, is secretly in the background, giving the W and Z bosons their masses and affecting the Z boson’s interactions with the other particles; and the top quark is hiding in the background too, since it can’t be removed without changing how the Z boson interacts with bottom quarks.) You can’t take any part of the Standard Model out without messing up these predictions completely.

Oh, and by the way, remember how the probability for W decay to a tau and a neutrino in Fig. 2 was off the prediction by more than two standard deviations? Well there’s nothing weird about the tau or the neutrinos in Fig. 3 — predictions and measurements agree just fine — and indeed, no numbers in Z decay differ from predictions by more than two standard deviations. As I said earlier, the expectation is that about one in every twenty measurements should differ from its true value by more than two standard deviations. Since we have over a dozen measurements in Figs. 2 and 3, it’s to be expected that one might well be two standard deviations off.

Asymmetries, Precisely

Let’s do one more case: one of the asymmetries that I mentioned in the bonus section of the previous article. Consider a forward-backward asymmetry shown in Fig. 4. Take all collisions in which an electron strikes a positron (the anti-particle of an electron) and turns into a muon and an anti-muon. Now compare the probability that the muon goes “forward” (roughly the direction that the electron is heading) to the probability that it goes “backward” (roughly the direction that the positron is heading.) If the two probabilities are equal, then the asymmetry would be zero; if the muon always goes to the left, then the asymmetry would be 100%; if always to the right, the asymmetry would be -100%.

Asymmetries are special because the effect of the strong nuclear force cancels out of them completely, and so they only depend on sin θ_w. And this particular “leptonic forward-backward” asymmetry is an example with a special feature: if sin θ_w were exactly 1/2, this asymmetry for lepton production would be predicted to be exactly zero.

But the measured value of this asymmetry, while quite small (less than 2%), is definitely not zero, and so this is another confirmation that sin θ_w is not exactly 1/2. So let’s instead compare the prediction for this asymmetry using sin θ_w = 0.48, the choice that worked so well for the Z boson’s decays in Fig. 3, with the data.

In Figure 5, the horizontal axis shows the lepton forward-backward asymmetry. The prediction of 1.8% that one obtains for sin θ_w = 0.48, widened slightly to cover 1.65% to 2.0%, which is what obtains for sin θ_w between 0.479 and 0.481, is shown in pink. The four open circles represent four measurements of the asymmetry by the four experiments that were located at the LEP collider; the dashes through the circles show the standard deviations on their measurements. The dark circle shows what one gets when one combines the four experiments’ data together, obtaining an even better statistical estimate: 1.71 ± 0.10%, the uncertainty being indicated both as the dash going through the solid circle and as the yellow band. Since the yellow band extends to just above 1.8%, we see that the data differs from the sin θ_w = 0.480 prediction (the center of the pink band) by less than one standard deviation… giving precise agreement of the Standard Model with this very small but well-measured asymmetry.

Predictions of other asymmetries show similar success, as do numerous other measurements.

The Big Picture

Successful predictions like these, especially ones in which both theory and experiment are highly precise, explain why particle physicists have such confidence in the Standard Model, despite its clear limitations.

What limitations of the Standard Model am I referring too? They are many, but one of them is simply that the Standard Model does not predict θ_w . No one can say why θ_w takes the value that it has, or whether the fact that it is close to 30 degrees is a clue to its origin or a mere coincidence. Instead, of the many measurements, we use a single one (such as one of the asymmetries) to extract its value, and then can predict many other quantities.

One thing I’ve neglected to do is to convey the complexity of the calculations that are needed to compare the Standard Model predictions to data. To carry out these computations much more carefully than I did in Figs. 2, 3 and 5, in order to make them as precise as the measurements, demands specialized knowledge and experience. (As an example of how tricky these computations can be: even defining what one means by sin θ_w can be ambiguous in precise enough calculations, and so one needs considerable expertise [which I do not have] to define it correctly and use that definition consistently.) So there are actually still more layers of precision that I could go into…!

But I think perhaps I’ve done enough to convince you that the Standard Model is a fortress. Sure, it’s not a finished construction. Yet neither will it be easily overthrown.