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Posts Tagged ‘Feynman rules’

The Z boson and resonances

Monday, May 10th, 2010

Hello everyone! Let’s continue our ongoing investigation of the particles and interactions of the Standard Model. For those that are just joining us or have forgotten, the previous installments of our adventure can be found at the following links: Part 1, Part 2, Part 3.

Up to this point we’ve familiarized ourselves the Feynman rules—which are shorthand for particle content and interactions—for the theory of electrons and photons (quantum electrodynamics, or QED). We then saw how the rules changed if we added another electron-like particle, the muon ?. The theory looked very similar: it was just two copies of QED, except sometimes a a high-energy electron and positron collision could produce a muon and anti-muon pair. At the end of the last post we also thought about what would happen if we added a third copy of electrons.

Let’s make another seemingly innocuous generalization: instead of adding more matter particles, let’s add another force particle. In fact, let’s add the simplest new force particle we could think of: a heavy version of the photon. This particular particle is called the Z boson.  Here’s a plush rendition made by The Particle Zoo:

Feynman rules for QED+?+Z

Our particle content now includes electrons, muons, photons, and Z bosons. We draw their lines as follows:

Recall that anti-electrons (positrons) and anti-muons are represented by arrows pointing in the opposite direction.

Question: What about anti-photons and anti-Z bosons?
Answer: Photons and Z bosons don’t have any charge and turn out to be their own anti-particles. (This is usually true of force particles, but we will see later that the W bosons, cousins of the Z, have electric charge.)

The theory isn’t interesting until we explain how these particles interact with each other. We thus make the straightforward generalization from QED and allow the Z to have the same interactions as the photon:

What I mean by this is that the squiggly line can either be a photon or a Z. Thus we see that we have the following four possible vertices:

  1. two electrons and a photon
  2. two electrons and a Z
  3. two muons and a photon
  4. two muons and a Z

Question: What are the conservation laws of this theory?
Answer: The conservation laws are exactly the same as in QED+?: conservation of electron number (# electrons – # positrons) and conservation of muon number (#muons – #anti-muons). Thus the total electron number and muon number coming out of a diagram must be the same as was going into it. This is because the new interactions we introduced also preserve these numbers, so we haven’t broken any of the symmetries of our previous theory. (We will see that the W boson breaks these conservation laws!) We also have the usual conservation laws: energy, momentum, angular momentum.


So far this seems like a familiar story. However, our theory now has enough structure to teach us something important about the kind of physics done at colliders like the LHC. We started out by saying that the Z boson is heavy, roughly 91 GeV. This is almost a hundred times heavier than a muon (and 20,000 times heavier than an electron). From our Feynman rules above we can see that the Z is unstable: it will decay into two electrons or two muons via its fundamental interactions.

Question: The photon has the same interactions as the Z, why isn’t it unstable? [Hint: kinematics! Namely, energy conservation.]

In fact, because electrons and muons are so much lighter, the Z is very happy to decay quickly into them. It turns out that the Z decays so quickly that we don’t have any chance of detecting them directly! We can only hope to look for traces of the Z in its decay products. In particular, let’s consider the following process: an electron positron pair annihilate into a Z, which then decays into a muon anti-muon pair.

The Z boson here is virtual—it only exists quantum mechanically and is never directly measured. In fact, because it is virtual this process occurs even when the electrons are not energetic enough to produce a physical Z boson, via E=mc2. However, it turns out that something very special when the electrons have just enough energy to produce a physical Z: the process goes “on shell” and is greatly enhanced! The reason for this is that the expression for the quantum mechanical rate includes terms that look like (this should be taken as a fact which we will not prove):

where M is the mass of the Z boson, p is essentially the net energy of the two electrons, and ? is a small number (the ‘decay width of the Z‘). When the electrons have just enough energy, p2M2 = 0 and so the fraction looks like i/?. For a small ?, this is a big factor and the rate for this diagram dominates over all other diagrams with the same initial and final states. Recall that quantum mechanics tells us that we have to sum all such diagrams; now we see that only the diagram with an intermediate Z is relevant in this regime.

Question: What other diagrams contribute? Related question: why did we choose this particular process to demonstrate this phenomenon?
Answer: The other diagram that contributes is the same as above but with the Z replaced by a photon. There are two reasons why we needed to consider ee ? Z ? ??. First, an intermediate photon would have M = 0, so p2M2 will never vanish and we’d never hit the resonance (recall that the electrons have energy tied up in their mass, so p ? 2m where m is the electron mass). Second, we consider a muon final state because this way we don’t have to consider background from, for example:

These are called t-channel diagrams and do not have a big enhancement; these diagrams never have a time slice (we read time from left-to-right) where only a Z exists. (For the record, the diagrams which do get enhanced at p2M2 = 0 are called s-channel for no particularly good reason.)

Intuitively, what’s happening is that the electrons are resonating with the Z boson field: they’re “tickling” the Z boson potential in just the right way to make it want to spit out a particle. Resonance is a very common idea in physics: my favorite example is a microwave—the electromagnetic waves resonate with the electric dipole moment of water molecules.

Detecting the Z boson

This idea of resonance gives us a simple handle to detect the Z boson even if it decays before it can reach our detectors. Let’s consider an electron-positron collider. We can control the initial energy of the electron-positron collision (p in the expression above). If we scan over a range of initial energies and keep track of the total rate of ?? final states, then we should notice a big increase when we hit the resonance. In fact, things are even better since the position of the resonance tells us the mass of the Z.

Below is a plot of the resonance from the LEP collaboration (Fig 1.2 from hep-ex/0509008):

Different patches of points correspond to different experiments. The x-axis is the collision energy (what we called p), while the y-axis is the rate at which the particular final states were observed. (Instead of ee ? ?? this particular plot shows ee ? hadrons, but the idea is exactly the same.) A nice, brief historical discussion of the Z discovery can be found in the August ’08 issue of Symmetry Magazine, which includes the following reproduction of James Rohlf’s hand-drawn plot of the first four Z boson candidate events:

[When is the last time any of the US LHC bloggers plotted data by hand?]

In fact, one way to search for new physics at the LHC is to do this simple bump hunting: as we scan over energies, we keep an eye out for resonances that we didn’t expect. The location of the bump tells us the mass of the intermediate particle. This, unfortunately, though we’ve accurately described the ‘big idea,’ it is somewhat of a simplified story. In the case of the electron-positron collider, there are some effects from initial- and final-state radiation that smear out the actual energy fed into the Z boson. In the case of the LHC the things that actually collide aren’t actually the protons, but rather the quarks and gluons that make up the protons—and the fraction of the total proton energy that goes into each colliding object is actually unknown. This is what is usually meant when people say that “hadron colliders are messy.” It turns out that one can turn this on its head and use it to our advantage;  we’ll get to this story eventually.

Until then, we still have a few more pieces to introduce into our electroweak theory of leptons: neutrinos, the W bosons, and the Higgs.


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.