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

You might want to mention that sufficiently high energy photons _are_ unstable w.r.t. e+/e- …

Hello Flip,

You said:

“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.”

I think my intuition has just failed me! Where did this Z boson field suddenly come from? And, what is it?

And, what attribute of the electrons is it that resonates with it?

If the Z boson field defines the way that the particle can only take a certain mass with the energy it’s given, then is it really specific to the Z boson, or might it play a role with other particles too? [ie like a kind of 'mass field' instead]

I think one of the things that’s bothering me a bit here (in my naive sort of way) is that in a sense energy and mass are equivalent and yet they are displaying quite different sorts of behaviour. Kind of like mass is lumpy and energy is fluid, if you know what I mean.

Another question: when you say the Z boson is like a heavy photon, does it have an electric and magnetic field too?

Great questions Jonathan!

1. Where did the Z boson field come from?

We wrote down a new theory (which we called “QED+mu+Z”) that describes an additional particle which we called the Z boson. All of the properties of the Z boson are put in by hand.

I suspect you want a “deeper” answer for where the Z (and really any other particle) comes from. Without going into details, force particles come from symmetries in our theory. (This isn’t meant to be an ‘explanation,’ but rather a statement demonstrating some of the deep connections involved in these kinds of questions.)

2. What is the Z boson?

At the level that we’ve introduced it, the Z boson is just a new particle mediating a new force, just like the photon.

3. What property of the electron resonates with the Z?

This is the same question as asking what property of the electron resonates with the photon. The photon couples to the electron’s electric charge (which is related to how the electron behaves under the symmetry associated with the electric force). Similarly, the theory we wrote down just postulates another force under which the electrons also have a charge.

4. Relation of Z boson and mass.

I think you’re getting a bit ahead of the model we developed here. There is indeed a very deep and important issue about where particles get their masses—we haven’t gotten to that yet. All we’ve said is that the Z boson has a mass, with no reference to where it might have come from. This is no bigger than an assumption than saying that the electron or muon has a mass.

I see where I may have been confusing: the fact that electron collisions can resonate with the Z field has to do with the fact that the Z has a mass that is large enough that two electrons can produce it, i.e. this is a purely kinematical effect. In other words, this is just conservation of energy: the electron and positron carry a minimum amount of energy (E=mc^2) — any collision between them has to produce at least this amount of energy. Since photons have no rest energy, they will never go “on-shell.” (Because Z bosons have a mass greater than 2mc^2, they can be produced by a sufficiently high-energy collision between electrons.)

5. How are energy and mass “the same” yet appear “different”?

I don’t have a good answer for this. However, at the level of quantum theory one has to reconsider the idea that “energy is a fluid.” We know, for example, that electromagnetic energy is carried by photons which come in lumps.

Maybe another way to think about it is that everything is energy. Nature also has bunch of quantum fields—who knows where they came from. These fields can absorb energy, converting it into excitations of the field. When these excitations are just right (i.e. “on shell”) the excitations can propagate for long distances as particles. At small scales these excitations really look like waves and aren’t as discrete as they seem to macroscopic things like people.

6. If the Z is like a heavy photon, does it have an electric and magnetic field, too?

Yes! (Though once we treat everything relativistically this comes about automatically.)

Hope that helped,

Flip

Thanks Flip.

Yes, that does help, but I’m now stuck on understanding some of the answers.

1. “put in by hand” – does that mean that all the properties are coming from the experimental evidence rather than theory?

2. I was actually trying to ask what the Z boson field was, but you’ve answered that in the second part of 1. , so that’s ok. (Note to self: must try and ask questions that are less ambiguous and more accurate in future!)

3. DOES an electron resonate with the Z boson? Surely, by the time the Z boson exists the electron/positron pair don’t, if what you write about the field producing the particle is true. And if another electron/positron pair came along then the Z boson would presumably ignore it/them.

Anyway, your answer is slightly ambiguous – am I right in thinking you’re saying that the property is another charge that’s analogous to the electric charge, but not the electric charge itself.

4. Fair enough: I’ll wait for the next exciting installment!

5. “Maybe another way to think about it is that everything is energy.” That’s kind of where I was going with all this, but the Z boson kind of scuppers that. Why is it such a prima donna, insisting on just a set amount of energy that it carries as mass? That’s what I meant about “fluid” and “lumpy”. Although the photon is a particle, it will carry whatever amount of energy it’s given when it’s created (though a post above suggests that may not be the case with very extreme amounts of energy). Your way of dealing with this is to introduce the idea of a field that “hosts” the particle generation/propagation. That then accounts for the resonance effect you see from experiment and the pickiness about the mass. But what about the photon? How does that come into existance? Can you engineer a field that generates this particle without resonance playing a part or do you just arbitrarily have a different mechanism? (Who was it who said “a little learning is a dangerous thing”? I think I’m probably proving them right! Feel free to tell me to stop fidgetting at the back of the class and wait patiently for the next lecture, if you want to.)

6. Um, how do you know? Are you able to detect and measure such a thing? Sorry if I sound a bit skeptical, but the Z boson doesn’t exist for very long, does it? And how is the Z boson able to cope with something coupling with either field – it doesn’t have any free energy to play with does it?

Hi Jon,

1. To clarify, all we’re doing here is playing some games with theories that I’m just writing down for you. I’m introducing particles that we know exist because I’d like to eventually construct the Standard Model, but at some level you should just treat this as a game where I list particles and interactions and we explore what interesting things can happen.

Of course, all of the particles and interactions that I happen to introduce are *actual* particles in the Standard Model, so I’ll remark here and there about how particles were discovered or how we see them in detectors. Properties like “neutrinos are light” or “the Z boson interacts weakly” are things that come from experimental observations.

But I think the important point is that I’m just throwing together different particles one step at a time to make things as simple as possible. This is *not* the way the Standard Model was discovered.

In other words: I started by introducing QED. Then I said that we could make another copy of QED with a muon. Why did we do this? We did this because ultimately we know that muons do exist and have those interactions. Where did it come from? In nature the muon was always there. In this series of blog posts I’m just saying “now let’s consider what happens when we add a new particle to our theory.” However, it won’t be until we get to the *complete* Standard Model that we’ll have a theory that is very close to what we see in nature.

3. The electron and positron interact with the Z boson. What I meant by saying that they “resonate” is that they can be given *just* the right amount of kinetic energy to produce an “on-shell” Z boson, i.e. a Z boson that is physical and not just virtual. This probably sounds like a very vague statement. What I mean is this:

Generally, the intermediate Z boson is virtual. This means that it cannot simultaneously preserve energy and momentum, so it must only exist quantum mechanically. However, if we set up the collision just right, the electron and positron can impart just the right amount of energy and momentum so that the Z is physical: i.e. it conserves both energy and momentum. In this case the production rate of the Z is greatly enhanced.

You are exactly right that particles have a DIFFERENT charge that is analogous to (but not the same as) electric charge, and that this is the charge that the Z boson feels. In fact, every time we add a force particle, we must specify additional charges for each matter particle. (So now the question is: why do we really only talk about electric charge and not about Z charge? This is a tricky question that we’ll get to when we meet the Higgs.)

5. Ah, ok. I now understand your question better, and I see why my response was probably not what you were looking for. You can of course produce photons of any energy. Similarly, you can produce Z bosons or electrons of any energy. The resonance phenomena is something special about the particular diagram that I was drawing: two particles annihilate into a virtual particle that then decays into two (or more) particles. For mysterious reasons these are called “s-channel” diagrams.

The point is that you have an intermediate particle in the diagram. Intermediate means that it’s never directly observed. If the conditions are just right (i.e. if the external particles have just the right energy and momentum) then the intermediate particle can resonate. This resonance depends on the intermediate particle’s mass. When this occurs, the probability for this process to happen increases dramatically and we can observe this as a bump in the observed rate for this process.

This is different from producing external particles, which don’t have resonance phenomena. I can have a diagram where an electron and a positron of suffient energy interact to produce a muon, anti-muon, and a photon. (Exercise: draw some diagrams that do this.) Up to conservation of momentum and energy, the photon can have any energy you want.

Similarly, you can draw diagrams for electron + positron going to muon, anti-muon, and Z. Again the Z can have any energy as long as we conserve total momentum and energy.

The *external* particles (which are always physical) don’t get resonance phenomena, only internal particles.

Here’s a different way of looking at it: resonances are an increase in the rate for a process to occur when a virtual particle (necessarily intermediate) can become physical.

6. Great question! Unfortunately I don’t have a very satisfying answer since it’s somewhat technical.

The question is really what do we mean by a magnetic field. The photon is a quantum particle which describes macroscopic electrodynamic effects in the classical limit. Thus the photon knows all about electric and magnetic fields, but these effects are macroscopic while the photon is microscopic.

The Z boson has the exact same structure of the photon and the equations associated with it also have electric and magnetic fields, but these are macroscopic effects. As you said, the Z is short lived so we never actually observe the macroscopic effects, but their presence in the equations associated with the Z are *necessary* by the structure of quantum field theory.

In other words, the “electric and magnetic fields” associated with the Z are part of the mathematical consistency of our theory. (I have, of course, not mentioned anything about these mathematical equations.)

Thanks for the questions!

my definition

Mass == energy-space-time

or mass is clumps of energy-space-time

Einstein

or E=MC2 or M=E/c2 or joules/(meters/sec)2

Einstein has energy and time but only 2 dimensional space (meteres squared). therefore another equation would be needed that had meteres cubed (space is 3 dimensional).

Tell me what’s wrong with my thinking/ conclusion.

Eergy is measured in joules=kgmm/ss

It has nothing to do with volume or dimensions

Hi Oscar, I suggest looking over the Wikipedia article on natural units, in particular as applied to particle physics.

“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).”

I think it should be almost a thousand times heavier than a muon (and 200,000 times heavier than an electron.):

91.2GeV/105.7MeV = 863

91.2GeV/.511MeV = 178,000

Very nice description Flip. One of the best I’ve seen, if not the best. Still, what is ‘mass’ and how come we find a boson with it? Helium4 can become a condensate and then behave like a boson, can you relate that to this? I really would like to understand where this ‘mass’ comes from, and what makes us exist as matter.

i dont studie partical physics, but i have a question on the subject, can anybody tell me where in nature the W and Z bosons would exist? thanks

what you mean by force particle?

why photons are kind of them?

These kinds of decays are observed in cosmic ray showers.