It’s time to return to our ongoing exploration of the structure of the Standard Model. Our primary tools are Feynman diagrams, which we introduced in previous posts (part 1, part 2). By now we’ve already familiarized ourselves with quantum electrodynamics (QED): the theory of electrons, positrons, and photons. Now we’re going to start adding on pieces to build up the Standard Model. We’ll start with the muon, portrayed below by Los Angeles artist Julie Peasley. (These handmade plushes can be found at her website, The Particle Zoo.)
We’re all familiar with the electron. Allow me to introduce its heavier cousin, the muon (μ). Where did this muon come from? Or, as Nobel Prize winner I. I. Rabi once asked, “Who ordered that?” (This is still an unanswered question!) Besides its mass, the muon has the same fundamental properties as the electron: it has the same charge, feels the same forces, and—like the electron—has an anti-particle partner.
Feynman rules for QED+μ
This makes it really easy to extend our Feynman rules. We’ll call our theory “QED+μ,” quantum electrodynamics with an additional particle. We just have to write the rules for two copies of QED:
Let’s recall how to interpret this. The three lines tell us that we have three kinds or particles in the theory: electrons (e), muons (μ), and photons (γ). Recall that the matter particles, the ones whose lines have an arrow, also have antiparticles. We indicate antiparticles by arrows pointing in the wrong direction when we read the diagrams from left-to-right. The vertex rules tell us that we have two kinds of interactions: a photon can either interact with two electrons or two muons.
It’s important to note that we cannot have photon couplings that mix electrons and muons. In terms of conservation laws, we say that electron and muon number are each conserved. For example, in the theory we’ve developed so far, you cannot have a muon decay into an electron and a photon. (We’ll introduce these sorts of interactions next time when we discuss electroweak theory.)
Exercise: Is the following diagram allowed in QED + μ?
Answer: Yes! But doesn’t this violate conservation of electron and muon number? You start out with two e‘s on the left and end up with two μ’s. Hint: what are the arrows telling you?
Once you’ve convinced yourself that the above diagram doesn’t violate electron or muon conservation, let me remark that this is an easy way to produce muons at low energy electron colliders. You just smash an electron against a positron and sometimes you’ll end up with a muon-antimuon pair which you can detect experimentally.
Exercise: when we previously did electron-positron to electron-positron scattering, we had to include two diagrams. Why is there only one diagram for eμ to eμ? Hint: draw the two diagrams for ee to ee and check if the Feynman rules still allow both diagrams if we convert the final states to muons.
Detecting muons, some collider physics
If you think about this a little, you might wonder: if electrons and muons are so similar, how can experimentalists distinguish between them at a collider? Seth and Mike might scold me for skipping over some information about the interaction of charged particles through matter, but one simple way to distinguish muons from electrons is to measure their energy and momenta. We know that (away from a potential) a particle’s energy is the sum of its kinetic energy plus it’s mass energy added in quadrature E2=m2c4+p2c2 (this is the “real” version of E=mc2). Since muons are heavier than electrons, we can just check the mass of the particle by plugging in the measured energy and momentum.
Actually, this is an oversimplified picture. In order not to annoy the other US/LHC bloggers, I’d better provide a slightly less oversimplified “cartoon.” Electrons are light, so let’s imagine that they’re ping pong balls. On the other hand, muons are heavy, so let’s imagine them as bowling balls. As you probably know, the LHC detectors are big and full of stuff… by that I mean atoms, which in turn are made up of a nucleus and a cloud of electrons. We can thus imagine a sea of ping-pong balls (think of a Chuck-E-Cheese ball pit). When electrons hit this ball pit, they end up distributing all of their energy into the other balls. This happens in the electromagnetic calorimeter, or ECAL. “Calor” is Latin for heat, so you can guess that the ECAL is really just a big fancy thermometer that measures the energy that the electron dissipates. Muons on the other hand, are bowling balls that are so massive that they just barrel straight through the ball pit to get to the other side. Here’s a very scientific illustration:
I hope we don’t get any comments saying, “oh man, muons are jerks.” In fact, they’re quite the opposite: muons are the only Standard Model particles that make it all the way to the outside of the detector, making it easy for us to identify them. In fact, the big distinctive toroidal magnets on the ATLAS detector below are there to bend the path of muons to help the outermost detector determine the muon momentum by measuring the curvature of their trail.
Exercise: [for those who want to do some actual calculations, requires a high school physics background] Convince yourself that this heuristic picture is correct by calculating the final momenta of a ball colliding elastically with (a) a ball of the same mass and (b) a ball of much lighter mass.
ATLAS toroidal magnets. Image from the Interactions.org Image Bank
Neat things that muons can do
Let me make a few more semi-historical remarks: our QED+μ model is just a theoretical toy. Historically, scientists knew immediately that something was weird about the muon: unlike electrons, it decayed into other particles and seemed to interact with mesons in unusual ways. In fact, for a while people thought that muons were a kind of meson. These differences ended up being a harbinger of something more interesting: the weak force.
Exercise: convince yourself that our Feynman rules for QED+μ do not allow muon decay, i.e. μ turning into non-μ stuff.
Muons are generated in the sky when cosmic rays hit atoms of the upper atmosphere. These rain down onto the Earth and force us to put our dark matter experiments deep underground to avoid their ‘noise.’ What’s really neat, however, is that the fact that muons make it to the surface of the Earth is a rousing experimental check of relativity. We know that muons at rest decay in microseconds. In this time, it seems like there’s no way for them to traverse the kilometers (about 4 km) between the Earth and its upper atmosphere; even if they were traveling at the speed of light! (c ~ 3. 108 m/s). What’s happening is the phenomenon of time dilation!
Introducing the tau (via the Socratic method)
Exercise: the Standard Model actually has another cousin of the electron, the tau (τ), leading to three charged leptons in total. Write down the Feynman rules for the theory QED+μ+τ, i.e. the theory of electrons, muons, and taus interacting via photons. Make sure that electron, muon, and tau number are all conserved. Draw the diagram for tau production in an electron-positron collider.
Exercise: Above we argued that muons are special because they barrel right through our detectors like bowling balls through an array of ping pong balls. Taus are even heavier, shouldn’t they also make it to the outside of the detector?
Answer: This was a bit of a trick question. The logic is correct that sufficiently energetic taus should make it all the way to the outside of the detector in our QED+μ+τ theory. However, this is not the full story for electrons, muons, and taus (collectively known as leptons) in the Standard Model. Like muons, taus are unstable and will decay. In fact, they decay much more quickly than muons because they have more mass and can decay into stuff (they have more “phase space”). While muons are like bowling balls barreling through the detector, taus are more like grenades that burst into hadronic “shrapnel” inside the calorimeters. They are usually very difficult to reconstruct from the data.
A preview of things to come:
Now we’re very familiar with putting together multiple copies of QED. For now, there are only three copies we have to worry about. It is an open question why this is the case. The existence of at least three copies, however, turns out to be significant for the imbalance of matter and anti-matter in the universe. In the next post we’ll introduce the weak force and really see what we can do with these leptons.
I’m currently in the middle of my “Advancement to Candidacy” exam, so my posts might be a little more delayed than usual this month. By the end of it, however, I hope to be blogging as an official PhD candidate. 
Erratum: virtual particles
I wanted to correct a misleading statement I made in my previous QED post: I discussed the visualization of virtual particles as balls that two kids toss back and forth while standing on frictionless ice. Conservation of momentum causes the two kids to slide apart as they throw and catch the ball, generating what we observe macroscopically as a repulsive force. We mentioned that it’s more difficult to see how this could give rise to an attractive force. I suggested that this is a phenomenon coming from the accumulated effect of many quantum exchanges. While this is true, there is a simpler way to understand this: pretend the ball has negative momentum! Since the particle is virtual, it is inherently quantum mechanical and needn’t have ‘on-shell’ (physical) momentum. Thus one could imagine tossing the ball with negative momentum, causing one to be deflected in the same direction as the ball was tossed. Similarly, catching the ball with negative momentum would push one in the direction that the ball came from.
Does it make sense classically? No! But that’s okay because they’re virtual particles.
That’s all for now, folks!
-Flip, on behalf of the US/LHC blog.
Tags: Feynman rules, muon, QED
Another nice post Flip! I especially like the phrase:
–taus are more like grenades that burst into hadronic “shrapnel”–
English, like Feynman diagrams, has rules. One of them is that possessive pronouns like “its” don’t have apostrophes.
“it’s heavier cousin”
The apostrophe should not be there. In the word “it’s”, the apostrophe must represent one or more omitted characters: “it[ i]s”, “it[ ha]s”.
“It’s difficult to understand quantum mechanics”. (“it is”)
“It’s been a long time since I had this much trouble learning something.” (“it has”).
—
“the Standard Model actually has a third cousin of the electron, the tau (τ)”
Above you referred to the Muon as a cousin. That would be the first such cousin. If the tau is the third, what is the second? I doubt it’s the positron, which would be better described as the electron’s “evil twin” with a goatee and such.
Hi Monster — thanks for a couple of useful corrections.
“English, like Feynman diagrams, has rules.”
I like this phrase.
I do my best to proof my posts, but sometimes I let a few typos through. (Though I would have hoped that the multiple times in that post that I used the possessive “its” correctly would have shown that this was indeed a typo rather than a deep misunderstanding of grammar.) It is helpful when people politely point out mistakes so that I can fix them.
I’ll make the appropriate correction.
By the way, in your comment on my W boson post you misspelled the word ‘diagrams.’
“the Standard Model actually has a third cousin of the electron, the tau”
Good catch: what I meant to convey was that the tau is the third cousin (of three), rather than the “electron’s third cousin.” Since we’re spelling things out in these comments: there are three charged leptons, which I refer to as ‘cousins.’ The tau is (enumerating by increasing mass) the third, but of course that would make it the electron’s second cousin. (In the enumerative sense, not the genealogical sense.)
I’ll update this in the main post as well.
Thanks,
F
I have another analogy to an attractive force:
Imagine a large pendulum between the two ice skaters. whenever the pendulum reaches one of them, he holds to the pendulum weight for a short while and then releases it in its way back, absorbing some of its momentum. The skaters will get closer and closer to each other while the pendulum will loose amplitude.
I wander if somone can make an animation of it.
Rafi, very good model for attractive force / boson exchange. Thanks.
Or they could throw a boomarang! Person A would throw it away from B, thereby being sent towards B, and it would circle round and be caught by B, who would be facing away from A and therefore move towards A.
A bit more complicated, though.