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### The W boson: mixing things up

For those of you who have been following our foray into the particle content of the Standard Model, this is where thing become exciting. We now introduce the W boson and present a nearly-complete picture of what we know about leptons.

We’re picking up right where we left off, so if you need a refresher, please refer to previous installments where we introduce Feynman rules and several particles: Part 1, Part 2, Part 3, Part 4, Part 5

The W is actually two particles: one with positive charge and one with negative charge. This is similar to every electron having a positron anti-partner. Here’s the Particle Zoo’s depiction of the W boson:

Together with the Z boson, the Ws mediate the weak [nuclear] force. You might remember this force from chemistry: it is responsible for the radioactive decay of heavy nuclei into lighter nuclei. We’ll draw the Feynman diagram for β-decay below. First we need Feynman rules.

Feynman Rules for the W: Interactions with leptons

Here are the Feynman rules for how the W interacts with the leptons. Recall that there are three charged leptons (electron, muon, tau) and three neutrinos (one for each charged lepton).

In addition, there are also the same rules with the arrows pointing in opposite directions, for a total of 18 vertices. Note that we’ve written plus-or-minus for the W, but we always use the W with the correct charge to satisfy charge conservation.

Quick exercise: remind yourself why the rules above are different from those with arrows pointing in the opposite direction. Hint: think of these as simple Feynman diagrams that we read from left to right. Think about particles and anti-particles.

In words: the W connects any charged lepton to any neutrino. As shorthand, we can write these rules as:

Here we’ve written a curly-L to mean “[charged] lepton” and a νi to mean a neutrino of the ith type, where i can be electron/muon/tau.

Exercise: What are the symmetries of the theory? In other words, what are the conserved quantities? Compare this to our previous theory of leptons without the W.

Answer: Electric charge is conserved, as we should expect. However, we no longer individually conserve the number of electrons. Similarly, we no longer conserve the number of muons, taus, electron-neutrinos, etc. However, the total lepton number is still conserved: the number of leptons (electrons, muons, neutrinos, etc.) minus the number of anti-leptons stays the same before and after any interaction.

Really neat fact #1: The W can mix up electron-like things (electrons and electron-neutrinos) with not-electron-like things (e.g. muons, tau-neutrinos). The W is special in the Standard Model because it can mix different kinds of particles. The “electron-ness” or “muon-neutrino-ness” (and so forth) of a particle is often called its flavor. We say that the W mediates flavor-changing processes. Flavor physics (of quarks) is the focus of the LHCb experiment at CERN.

Exercise: Draw a few diagrams that violate electron number. [If it’s not clear, convince yourself that you cannot have such effects without a W in your theory.]

Answer: here’s one example: a muon decaying into an electron and a neutrino-antineutrino pair. (Bonus question: what is the charge of the W?)

Remark (update 7 July): In the comments below Mori and Stephen point out that in the ‘vanilla’ Standard Model, leptons don’t have flavor-changing couplings to the W as I’ve drawn above. This is technically true, at least before one includes the phenomena of neutrino-oscillations (only definitively confirmed in 1998). In the presentation here I am assuming that such interactions take place, which is a small modification from the “most minimal” Standard Model. Such effects must take place due to the neutrino oscillation phenomena. We will discuss this in a future post on neutrino-less double beta decay.

Feynman Rules for the W: Interactions with other force particles

There are additional Feynman rules. In fact, you should have already guessed one them: because the W is electrically charged, it interacts with the photon! Thus we have the additional Feynman rule:

[Update, Aug 9: note that for these vertices I’ve used the convention that all of the bosons are in-coming. Thus these are not Feynman Diagrams representing physical processes, they’re just vertices which we can convert into diagrams or pieces or diagrams. For example, the above vertex has an incoming photon, incoming W+, and an incoming W-. If we wanted the diagram for a W+ emitting a photon (W+ -> W+ photon), then we would swap the incoming W- for an outgoing W+ (they’re sort of antiparticles).]

This turns out to only be the tip of the iceberg. We can replace the photon with a Z (as one would expect since the Z is a heavy cousin of the photon) to get another three-force-particle vertex:

Finally, we can even construct four force-particle vertices. Note that each of these satisfies charge conservation!

These four-force-particle vertices are usually smaller than any of the previous vertices, so we won’t spend too much time thinking about them.

Really neat fact #2: We see that the W introduces a whole new kind of Feynman rule: force particles interacting with other force particles without any matter particles! (In fancy words: gauge bosons interacting with other gauge bosons without any fermions.)

Remarks

1. The most interesting feature of the W is that it can change fermion flavors, i.e. it can not only connect a lepton and a neutrino, but it can connect a lepton of one type with a neutrino of a different type. One very strong experimental constraint on flavor physics comes from the decay μ→eϒ (muon decaying to electron and photon). As an exercise, draw a Feynman diagram contributing to this process. (Hint: you’ll need to have a W boson and you’ll end up with a closed loop.)
2. It is worth noting, however, that these flavor-changing effects tend to be smaller than flavor-conserving effects. In other words, a W is more likely to decay into an electron and an electron-neutrino rather than an electron and a tau-neutrino. We’ll discuss how much smaller these effects are later.
3. W bosons are rather heavy—around 80 GeV, slightly lighter than the Z but still much heavier than any of the leptons. Thus, as we learned from the Z, it decays before it can be directly observed in a detector.
4. The W was discovered at the UA1 and UA2 experiments at CERN in the 80s. Their discovery was a real experimental triumph: as you now know from the Feynman rules above, the W decays into a lepton and a neutrino—the latter of which cannot be directly detected! This prevents experimentalists from observing a nice resonance as they did for the Z boson a few months later. They used a slightly modified technique based on a quantity called “transverse mass” to search for a smeared-out resonance using only the information about the observed lepton. Generalizations of this technique are still being developed today to search for supersymmetry! (For experts: see this recent review article on LHC kinematics.)
5. The W boson only talks to left-handed particles. This is a remarkable fact that turns out to be related to the difference between matter and antimatter. For a proper introduction, check out this slightly-more-detailed post.

Exercise: Now that we’ve developed quite a bit of formalism with Feynman rules, try drawing diagrams corresponding to W boson production at a lepton collider. Assume the initial particles are an electron and positron. Draw a few diagrams that produce W bosons. “Finish” each diagram by allowing any heavy bosons (Z, W) to decay into leptons.

What is the simplest diagram that includes a W boson? Is the final state observable in a detector? (Remember: neutrinos aren’t directly observable.) What general properties do you notice in diagrams that both (1) include a W boson and (2) have a detectable final state (at least one charged lepton)?

Can you draw diagrams where the W boson is produced in pairs? Can you draw diagrams where the W boson is produced by itself?

Hints: You should have at least one diagram where the W is the only intermediate particle. You should also play with diagrams with both the fermion-fermion-boson vertices and the three-boson vertices. You may also use the four-boson vertices, but note that these are smaller effects.

Remark: Try this exercise, you’ll really start to get a handle for drawing diagrams for more complicated processes. Plus, this is precisely the thought process when physicists think about how to detect new particles. As an additional remark, this is not quite how the W was discovered—CERN used proton-antiproton collisons, which we’ll get to when we discuss quantum chromodynamics.

Relating this to chemistry

Before closing our introduction to the W boson, let’s remark on its role in chemistry and simultaneously give a preview for the weak interactions of quarks. You’ll recall that in chemistry one could have β decay:

neutron → proton + electron + anti-neutrino

This converts one atom into an isotope of another atom. Let’s see how this works at the level of subatomic particles.

Protons and neutrons are made out of up and down type quarks. Up quarks (u) have electric charge +2/3 and down quarks (d) have electric charge -1/3. As we will see when we properly introduce the quarks, up and down quarks have the same relationship as electron-neutrinos and electrons. Thus we can expect a coupling between the up, down, and W boson.

A neutron is composed of two down quarks and an up quark (ddu) while a proton is composed of two up quarks and a down quark (uud). [Check that the electric charges add up to what you expect!] The diagram that converts a neutron to a proton is then:

Update: As reader Cris pointed out to me in an e-mail, the W should have negative charge and should decay into an electron and anti-neutrino!

Because the W boson is much heavier than the up and down quarks—in fact, it’s much heavier than the entire proton—it is necessarily a virtual particle that can only exist for a short time. One can imagine that the system has to ‘borrow’ energy to create the W so that the Heisenberg uncertainty principle tells us that it has to give back the energy very quickly. Thus the W can’t travel very far before decaying and we say that it is a “short range force.” Thus sometimes the weak force is called the weak nuclear force. Compare this to photons, which have no mass and hence are a “long range force.”

[We now know, however, that it is not intrinsically a nuclear force (in our theory above we never mentioned quarks or nuclei), and further its ‘weakness’ is related to the mass of the W making it a short-range force.]

Cheers!
Flip (USLHC)