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:

- two electrons and a photon
- two electrons and a
*Z* - two muons and a photon
- 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.

**Resonances
**

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=mc^{2}. 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, *p*^{2}–*M*^{2} = 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 *p*^{2}–*M*^{2} will never vanish and we’d never hit the resonance (recall that the electrons have energy tied up in their mass, so *p ?* 2*m* 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 *p*^{2}–*M*^{2} = 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.

Tags: Feynman rules, Z boson