Celebrating the Standard Model: The Magic Angle

Particle physicists describe how elementary particles behave using a set of equations called their “Standard Model.” How did they become so confident that a set of math formulas, ones that can be compactly summarized on a coffee cup, can describe so much of nature?

My previous “Celebrations of the Standard Model” (you can find the full set here) have included the stories of how we know the strengths of the forces, the number of types (“flavors” and “colors”) and the electric charges of the quarks, and the structures of protons and neutrons, among others. Along the way I explained how W bosons, the electrically charged particles involved in the weak nuclear force, quickly decay (i.e. disintegrate into other particles). But I haven’t yet explained how their cousin, the electrically-neutral Z boson, decays. That story brings us to a central feature of the Standard Model.

Here’s the big picture. There’s a super-important number that plays a central role in the Standard Model. It’s a sort of angle (in a sense that will become clearer in Figs. 2 and 3 below), and is called θ_w or θ_weak. Through the action of the Higgs field on the particles, this one number determines many things, including

• the relative masses of the W and Z bosons
• the relative lifetimes of the W and Z bosons
• the relative probabilities for Z bosons to decay to one type of particle versus another
• the relative rates to produce different types of particles in scattering of electrons and positrons at very high energies
• the relative rates for processes involving scattering neutrinos off atoms at very low energies
• asymmetries in weak nuclear processes (ones that would be symmetric in corresponding electromagnetic processes)

and many others.

This is an enormously ambitious claim! When I began my graduate studies in 1988, we didn’t know if all these predictions would work out. But as the data from experiments came in during the 1990s and beyond, it became clear that every single one of them matched the data quite well. There were — and still are — no exceptions. And that’s why particle physicists became convinced that the Standard Model’s equations are by far the best they’ve ever found.

As an illustration, Fig. 1 shows, as a function of sin θ_w, the probabilities for Z bosons to decay to each type of particle and its anti-particle. Each individual probability is indicated by the size of the gap between one line and the line above. The total probability always adds up to 1, but the individual probabilities depend on the value of θ_w. (For instance, the width of the gap marked “muon” indicates the probability for a Z to decay to a muon and an anti-muon; it is about 5% at sin θ_w = 0, about 3% at sin θ_w = 1/2, and over 15% at sin θ_w = 1.)

As we’ll see in Fig. 3, the W and Z boson masses imply (if the Standard Model is valid) that sin θ_w is about ¹/₂. Using that measurement, we can then predict that all the various decay probabilities should be given within the green rectangle (if the Standard Model is valid). These predictions, made in the mid-1980s, proved correct in the 1990s; see Fig. 5 below.

This is what I’ll sketch in today’s post. In future posts I’ll go further, showing how this works with high precision.

The Most Important Angle in Particle Physics

Angles are a common feature of geometry and nature: 90 degree angles of right-angle triangles, the 60 degree angles of equilateral triangles, the 104.5 degree angle between the two hydrogen-oxygen bonds in a water molecule, and so many more. But some angles, more abstract, turn out to be even more important. Today I’ll tell you about θ_w , which is a shade less than 30 degrees (π/6 radians).

Note: This angle is often called “the Weinberg angle”, based on Steven Weinberg’s 1967 version of the Standard Model, but it should be called the “weak-mixing angle”, as it was first introduced seven years earlier by Sheldon Glashow, before the idea of the Higgs field.

This is the angle that lies at the heart of the Standard Model: the smallest angle of the right-angle triangle shown in Fig. 2. Two of its sides represent the strengths g₁ and g₂ of two of nature’s elementary forces: the weak-hypercharge force and the weak-isospin force. According to the Standard Model, the machinations of the Higgs field transform them into more familar forces: the electromagnetic force and the weak nuclear force. (The Standard Model is often charaterized by the code SU(3)xSU(2)xU(1); weak-isospin and weak-hypercharge are the SU(2) and U(1) parts, while SU(3) gives us the strong nuclear force).

To keep things especially simple today, let’s just say θ_w = 30 degrees, so that sin θ_w = ¹/₂. In a later post, we’ll see the angle is closer to 28.7 degrees, and this makes a difference when we’re being precise.

The Magic Angle and the W and Z Bosons

The Higgs field gives masses to the W and Z bosons, and the structure of the Standard Model predicts a very simple relation, given by the electroweak triangle as shown at the top of Fig. 3:

• 

This has the consequence shown at the top of Fig. 3, rewritten as a prediction

• 

If sin θ_w = ¹/₂ , this quantity is predicted to be 1/4 = 0.250. Measurements (m_W = 80.4 GeV/c² and m_Z = 91.2 GeV/c²) show it to be 0.223. Agreement isn’t perfect, indicating that the angle isn’t exactly 30 degrees. But it is close enough for today’s post.

Where does this formula for the W and Z masses come from? Click here for details:

Central to the Standard Model is the so-called “Higgs field”, which has been switched on since very early in the Big Bang. The Higgs field is responsible for the masses of all the known elementary particles, but in general, though we understand why the masses aren’t zero, we can’t predict their values. However, there’s one interesting exception. The ratio of the W and Z bosons’ masses is predicted.

Before the Higgs field switched on, here’s how the weak-isospin and weak-hypercharge forces were organized: there were

• 3 weak isospin fields, called W⁺, W^– and W⁰, whose particles (of the same names) had zero rest mass
• 1 weak-hypercharge field, usually called, X, whose particle (of the same name) had zero rest mass

After the Higgs field switched on by an amount v, however, these four fields were reorganized, leaving

• One, called the electromagnetic field, with particles called “photons” with zero rest mass.
• One, called Z⁰ or just Z, now has particles (of the same names) with rest mass m_Z
• Two, still called W⁺ and W^– , have particles (of the same names) with rest mass m_W

Central to this story is θ_w. In particular, the relationship between the photon and Z and the original W⁰ and X involves this angle. The picture below depicts this relation, given also as an equation

The W⁺ and W^– bosons get their masses through their interaction, via the weak-isospin force, with the Higgs field. The Z boson gets its mass in a similar way, but because the Z is a mixture of W⁰ and X, both the weak-isospin and weak-hypercharge forces play a role. And thus m_Z depends both on g₁ and g₂, while m_W depends only on g₂. Thus

where v is the “value” or strength of the switched-on Higgs field, and in the last step I have used the electroweak triangle of Fig. 2.

A slightly more complex relation relates the W boson’s lifetime t_W and the Z boson’s lifetime t_Z (this is the average time between when the particle is created and when it decays.) This is shown at the bottom of Fig. 3.

• 

This is a slightly odd-looking formula; while 81 = 9² is a simple number, 86 is a weird one. Where does it come from? We’ll see in just a moment. In any case, as seen in Fig. 3, agreement between theoretical prediction and experiment is excellent.

If the Standard Model were wrong, there would be absolutely no reason for these two predictions to be even close. So this is a step in the right direction. But it is hardly the only one. Let’s check the detailed predictions in Figure 1.

W and Z Decay Probabilities

Here’s what the Standard Model has to say about how W and Z bosons can decay.

W Decays

In this earlier post, I explained that W^– bosons can decay (oversimplifying slightly) in five ways:

• to an electron and a corresponding anti-neutrino
• to a muon and a corresponding anti-neutrino
• to a tau and a corresponding anti-neutrino
• to a down quark and an up anti-quark
• to a strange quark and a charm anti-quark

(A decay to a bottom quark and top anti-quark is forbidden by the rule of decreasing rest mass; the top quark’s rest mass is larger than the W’s, so no such decay can occur.)

These modes have simple probabilities, according to the Standard Model, and they don’t depend on sin θ_w (except through small quantum corrections which we’re ignoring here). The first three have probability 1/9. Moreover, remembering each quark comes in three varieties (called “colors”), each color of quark also occurs with probability 1/9. Altogether the predictions for the probabilities are as shown in Fig. 4, along with measurements, which agree well with the predictions. When quantum corrections (such as those discussed in this post, around its own Fig. 4) are included, agreement is much better; but that’s beyond today’s scope.

Because the W+ and W- are each others’ anti-particles, W⁺ decay probabilities are the same as those for W^–, except with all particles switched with their anti-particles.

Z Decays

Unlike W decays, Z decays are complicated and depend on sin θ_w. If sin θ_w = ¹/₂, the Standard Model predicts that the probability for a Z boson to decay to a particle/anti-particle pair, where the particle has electric charge Q and weak-isospin-component T = +1 or -1 [technically, isospin’s third component, times 2], is proportional to

• 2 (Q/2-T)² + 2(Q/2)² = 2-2TQ+Q²

where I used T² = 1 in the final expression. The fact that this answer is built from a sum of two different terms, only one of which involves T (weak-isospin), is a sign of the Higgs field’s effects, which typically marry two different types of fields in the Standard Model, only one of which has weak-isospin, to create the more familiar ones.

This implies the relative decay probabilities (remembering quarks come in three “colors”) are

• For electrons, muons and taus (Q=-1, T=-1): 1
• For each of the three neutrinos (Q=0, T=1): 2
• For down-type quarks (Q=-1/3,T=-1) : 13/9 * 3 = 13/3
• For up-type quarks (Q=2/3,T=1): 10/9 * 3 = 10/3

These are shown at left in Fig. 5.

The sum of all those numbers (remembering that there are three down-type quarks and three up-type quarks, but again top quarks can’t appear due to the rule of decreasing rest mass) is:

• 1 + 1 + 1 + 2 + 2 + 2 + 13/3 + 13/3 + 13/3 + 10/3 + 10/3 = 86/3.

And that’s where the 86 seen in the lifetime ratio (Fig. 3) comes from.

To get predictions for the actual decay probabilities (rather than just the relative probabilities), we should divide each relative probability by 86/3, so that the sum of all the probabilities together is 1. This gives us

• For electrons, muons and taus (Q=-1, T=-1): 3/86
• For each of the three neutrinos (Q=0, T=1): 6/86
• For down-type quarks (Q=-1/3, T=-1) : 13/86
• For up-type quarks (Q=2/3, T=1): 10/86

as shown on the right-hand side of Fig. 5; these are the same as those of Fig. 1 at sin θ_w = ¹/₂. Measured values are also shown in Fig. 5 for electrons, muons, taus, the combination of all three neutrinos, the bottom quark, the charm quark, and (implicitly) the sum of all five quarks. Again, they agree well with the predictions.

This is already pretty impressive. The Standard Model and its Higgs field predict that just a single angle links a mass ratio, a lifetime ratio, and the decay probabilities of the Z boson. If the Standard Model were significantly off base, some or all of the predictions would fail badly.

However, this is only the beginning. So if you’re not yet convinced, consider reading the bonus section below, which gives four additional classes of examples, or stay tune for the next post in this series, where we’ll look at how things improve with a more precise value of sin θ_w.

Bonus: Other Predictions of the Standard Model

Many other processes involving the weak nuclear force depend in some way on sin θ_w. Here are a few examples.

High-Energy Electron-Positron Collisions (click for details)

In this post I discussed the ratio of the rates for two important processes in collisions of electrons and positrons:

• electron + positron ⟶ any quark + its anti-quark
• electron + positron ⟶ muon + anti-muon

This ratio is simple at low energy (E << m_Z c²), because it involves mainly electromagnetic effects, and thus depends only on the electric charges of the particles that can be produced.

But at high energy (E >> m_Z c²) , the prediction changes, because both electromagnetic and weak nuclear forces play a role. In fact, they “interfere”, meaning that one must combine their effects in a quantum way before calculating probabilities.

[What this cryptic quantum verbiage really means is this. At low energy, if S_f is the complex number representing the effect of the photon field on this process, then the rate for the process is |S_f|². But here we have to include both S_f and S_Z, where the latter is the effect of the Z field. The total rate is not |S_f|² + |S_Z|² , a sum of two separate probabilities. Instead it is |S_f+S_Z|² , which could be larger or smaller than |S_f|² + |S_Z|² — a sign of interference.]

In the Standard Model, the answer depends on sin θ_w. The LEP 2 collider measured this ratio at energies up to 209 GeV, well above m_Z c². If we assume sin θ_w is approximately ¹/₂, data agrees with predictions. [In fact, the ratio can be calculated as a function of energy, and thus made more precise; data agrees with these more precise predictions, too.]

Low-Energy Neutrino-Nucleus Collisions (click for details)

When electrons scatter off protons and neutrons, they do so via the electromagnetic force. For electron-proton collisions, this is not surprising, since both protons and electrons carry electric charge. But it’s also true for neutrons, because even though the neutron is electrically neutral, the quarks inside it are not.

By contrast, neutrinos are electrically neutral, and so they will only scatter off protons and neutrons (and their quarks) through the weak nuclear force. More precisely, they do so through the W and Z fields (via so-called “virtual W and Z particles” [which aren’t particles.]) Oversimplifying, if one can

• obtain beams of muon neutrinos, and
• scatter them off deuterons (nuclei of heavy hydrogen, which have one proton and one neutron), or off something that similarly has equal numbers of protons and neutrons,

then simple predictions can be made for the two processes shown at the top of Fig. 7, in which the nucleus shatters (turning into multiple “hadrons” [particles made from quarks, antiquarks and gluons]) and either a neutrino or a muon emerges from the collision. (The latter can be directly observed; the former can be inferred from the non-appearance of any muon.) Analogous predictions can be made for the anti-neutrino beams, as shown at the bottom of Fig. 7.

The ratios of these four processes are predicted to depend, in a certain approximation, only on sin θ_w. Data agrees with these predictions for sin θ_w approximately ¹/₂.

More complex and detailed predictions are also possible, and these work too.

Asymmetries in Electron-Positron Collisions (click for details)

There are a number of asymmetric effects that come from the fact that the weak nuclear force is

• not “parity-invariant”, (i.e. not the same when viewed in a mirror), and
• not “charge-conjugation invariant” (i.e. not the same when all electric charges are flipped)

though it is almost symmetric under doing both, i.e. putting the world in a mirror and flipping electric charge. No such asymmetries are seen in electromagnetism, which is symmetric under both parity and charge-conjugation separately. But when the weak interactions play a role, asymmetries appear, and they all depend, yet again, on sin θ_w.

Two classes of asymmetries of great interest are:

• “Left-Right Asymmetry” (Fig. 8): The rate for electron-positron collisions to make Z bosons in collisions with positrons depends on which way the electrons are “spinning” (i.e. whether they carry angular momentum along or opposite to their direction of motion.)
• “Forward-Backward Asymmetry” (Fig. 9): The rate for electron-positron collisions to make particle-antiparticle pairs depends on whether the particles are moving roughly in the same direction as the electrons or in the same direction as the positrons.

As with the high-energy electron-positron scattering discussed above, interference between effects of the electromagnetic and Z fields, and the Z boson’s mass, causes these asymmetries to change with energy. They are particularly simple, though, both when E = m_Z c² and when E >> m_Z c².

A number of these asymmetries are measurable. Measurements of the left-right asymmetry was made at the Stanford Linear Accelerator Center (SLAC) at their Linear Collider (SLC), while I was a graduate student there. Meanwhile, measurements of the forward-backward asymmetries were made at LEP and LEP 2. All of these measurements agreed well with the Standard Model’s predictions.

A Host of Processes at the Large Hadron Collider (click for details)

Fig. 10 shows predictions (gray bands) for total rates of over seventy processes in the proton-proton collisions at the Large Hadron Collider. Also shown are measurements (colored squares) made at the CMS experiment . (A similar plot is available from the ATLAS experiment.) Many of these predictions, which are complicated as they must account for the proton’s internal structure, depend on sin θ_w .

While minor discrepancies between data and theory appear, they are of the sort that one would expect in a large number of experimental measurements. Despite the rates varying by more than a billion from most common to least common, there is not a single major discrepancy between prediction and data.

Many more measurements than just these seventy are performed at the Large Hadron Collider, not least because there are many more details in a process than just its total rate.

A Fortress

What I’ve shown you today is just a first step, and one can do better. When we look closely, especially at certain asymmetries described in the bonus section, we see that sin θ_w = ¹/₂ (i.e. θ_w = 30 degrees) isn’t a good enough approximation. (In particular, if sin θ_w were exactly ¹/₂, then the left-right asymmetry in Z production would be zero, and the forward-backward asymmetry for muon and tau production would also be zero. That rough prediction isn’t true; the asymmetries are small, only about 15%, but they are clearly not zero.)

So to really be convinced of the Standard Model’s validity, we need to be more precise about what sin θ_w is. That’s what we’ll do next time.

Nevertheless, you can already see that the Standard Model, with its Higgs field and its special triangle, works exceedingly well in predicting how particles behave in a wide range of circumstances. Over the past few decades, as it has passed one experimental test after another, it has become a fortress, extremely difficult to shake and virtually impossible to imagine tearing down. We know it can’t be the full story because there are so many questions it doesn’t answer or address. Someday it will fail, or at least require additions. But within its sphere of influence, it rules more powerfully than any theoretical idea known to our species.