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Flip Tanedo | USLHC | USA

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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)

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  • mori

    At the tree level in the Standard Model, the W boson should NOT change lepton flavors! It can change quark flavors because the quarks that interact via W bosons are in a slightly different basis than those that interact Higgs bosons.

  • Stephen Brooks

    I think there’s a slight typo on the muon decay diagram, the first neutrino coming off should be a nu_mu, not a nu_e. This may also explain the first comment.

  • Hi Mori and Stephen — very observant comments, though we’re talking about slightly different “Standard Models”. [What follows is a little technical for a general audience, however I’ll clarify in a future post where I describe some current neutrino experiments.]

    This seems to be a case of knowing a little too much! 🙂 Here I am introducing the “post-1998” Standard Model, or the Standard Model with Dirac neutrino masses (i.e. with RH gauge singlet neutrinos). In this model the leptons and neutrinos have analogous weak interactions as the up and down-type quarks. Thus you have the same misalignment in flavor and mass basis: the CKM matrix in the quark sector is analogous to the PMNS matrix in the lepton sector, and one has tree-level lepton-flavor violation which is suppressed by the right-handed neutrino mass scale. (In this last statement we are assuming some kind of see-saw mass mechanism.)

    Indeed, you are both correct that in the minimal Standard Model (pre neutrino-oscillation) there were only left-handed neutrinos and hence one could go into a basis where both flavor and masses are diagonalized simultaneously. And indeed, this is usually what people refer to as the ‘Standard Model.’ (Certainly before 1998 when neutrino oscillations were discovered.) However, I chose to introduce the SM + Dirac Neutrino model because it makes it much simpler to understand the analogous structure in the quark sector, emphasizing the parallel weak representations of the leptons and quarks without having to mention any group theory. Further, the see-saw model, while technically an extension of the Standard Model, is ‘natural’ in light of grand unification since (1) it fits exactly into SO(10) reps and (2) the see-saw mechanism “points” to a large scale on the order of the GUT scale.

    Anyway, you are both right that few (if any) textbooks would show flavor-changing W vertices in the lepton sector. Part of this is because books written before 1998 were written at a time when neutrino-oscillation data was unclear. However, I think the SM + Dirac Neutrino model is the ‘minimal’ model that accommodates neutrino oscillations. (I haven’t thought too deeply about the structure of models with Majorana neutrino masses—but these seem unsatisfying since the relevant flavor structure would come from a higher dimension operator.)

    At any rate, I accept your comments but hope you appreciate the particular pedagogical choices I made. I will address this issue in a future post where I discuss neutrinoless double beta decay.

    Cheers,
    Flip

  • Stephen Brooks

    But aren’t the nu_e, nu_mu etc. already flavour eigenstates, so should be exactly paired up with e, mu respectively via the W? What you’re saying sounds like it would be true if you were writing nu_1 nu_2 nu_3 (the mass eigenstates) in your diagrams. The quarks mix because u,d,s,c,t,b are understood to be mass eigenstates (correct me if I’ve got this wrong!)

  • Hi Stephen, you’re exactly right: I’ve been sloppy and have written nu_e to mean nu_1, and so forth.

    I was hoping to sweep this nuance under the rug for a while until I figured out how to best explain flavor physics without going into any mathematical formalism (i.e. writing matrices).

    As you said, we assume that the quark states are already mass eigenstates and so we don’t write u_1, u_2, u_3 but rather u, c, t. I’ve been writing neutrinos in the same way.

    As I suspect you already know, this is slightly misleading since the e and mu lepton flavors have a large mixing in the 1 and 2 mass eigenstates so it’s a little silly to call one mass eigenstates an ‘electron neutrino’ rather than a ‘muon neutrino.’

    At any rate, these are all excellent points which I’ll think of a good way to address down the road. 🙂

    I hope you agree that the small liberties I’ve taken here are reasonable given the targeted public audience, though I’ll be careful in the future to be more technically precise or at least point out subtleties for physicists.

    Thanks!
    Flip

  • Stephen Brooks

    Flip,

    Thanks for the response. I can’t think of any easy way around this. If you said the neutrino leg of the Feynmann diagram had finite length then I suppose the nu_tau coming out of an e-W interaction would have nonzero probability (via oscillation) 🙂

    On the other hand, I think your tutorial still works if you replace nu_e by nu_1, etc. throughout, and maybe would be no more confusing to the beginner. It’s the older physicists who would be asking “what is this nu_1?!”

  • You show a W+ and a photon producing a W-, and two other diagramms showing W+ coming in and W- going out… and then you say that charge is conserved. Huh?

  • Hi Monster — good comment. I neglected to say that these Feynman rules are drawn with the convention that all lines are going *inward*. Thus the way they are currently drawn they do *not* represent Feynman diagrams, they are just a symmetric way of showing what kind of vertices are possible.

    (This is the way that Feynman rules are usually defined, so I forgot to point this out. I’ll add an update to the main post.)

    With that piece of information it should be clear how to “derive” the other rules: If you want the diagram for a photon turning into a W+/W- pair, then you swap the in-going W+ into an out-going W- and the in-going W- into an outgoing W+. Similarly, if you want the diagram for a W+ emitting a photon, you swap the in-going photon for an out-going photon and the in-going W- for an out-going W+.

    (It should also be clear now that charge is indeed conserved.)

    Great point. Thanks.
    -F

  • Alex

    The Part 1, Part 2, etc links are all dead. You should at least maintain a valid pointer back to Part 1. Preferably another pointer to the next part.

  • Oscar

    Hey!

    I really love your posts, but I’m a little stuck on the last lump of exercises. Can you post the solutions, for my future reference.

    Also, can you ‘turn’ the Z interractions like you can with QED interractions, to make two leptons turn into two neutrinos? It doesn’t intuitively feel like you could, but I wanted to check.

    Also also, have I missed a link to a discussion of how virtual particles work (like, for example, how they get the energy to work, and why they so quickly break down), or was there never one?

  • Hi Oscar—I agree that solutions would be a good idea, though I think this would end up being incorporated into part of a much larger endeavor (e.g. writing all of these blog posts into one coherent document)… so it’s unlikely to happen in the near future. However, there’s a decent chance that I may have answered a lot of the questions in my lecture notes (that were based on these blog posts): http://www.lepp.cornell.edu/~pt267/undergradparticles.html

    That’s indeed correct about the Z! An electron and positron can annihilate into a neutrino and antineutrino.

    I do not think I wrote a very detailed post about virtual particles, I think in the early Feynman diagram posts I referred to them hand wavily as quantum magic. (The lecture notes linked above probably have a little more detail, but not much.) The point is that as long as they’re not directly observable (i.e. purely virtual) they can get away with a lot of things, including energy non-conservation.

    Thanks for the interest!
    F