More on Feynman diagrams

Last week in the step-by-step MSc series, I wrote about the basics of Feynman diagrams. For instance, I said that we could draw an interaction between two electrons like this:


Time flows from right to left. The axes are often drawn with time flowing left-to-right, which matches the direction we read, but it’s easier to match right-to-left diagrams to mathematical notations. (If I have a variable x to which I apply a function f and then I apply another function g to the overall result, I write that as g(f(x)) — the rightmost action happens first.) The axes are intentionally vague: they don’t have units, since we’re more interested in describing the general kind of interaction that might happen than in exact numbers, at this point. If we start doing calculations, we’ll label each particle line with important properties, like its momentum.

So much for reading Feynman diagrams. Let’s talk about how to construct them. A good starting point is the Feynman rules for photons and electrons. The model of photons and electrons in quantum field theory (the most accurate model we have to date) is called quantum electrodynamics,  or QED for short. In QED, there’s only one way of connecting particle lines. The connection between lines is called a vertex and in QED it always looks like this:

qed-feynman-vertexOne consequence of having no other vertices is that electrons can never interact directly: they have to go through a photon, as in the diagram above. In general, however, having only one vertex is not as restricting as you might first think. We can rotate the vertex however we like and introduce as many vertices as we want into a single diagram. We need both those principles to build up the diagram at the top of the post. However, there’s also another diagram to create by rotating the vertex: this one, which describes pair production.


Last week, I briefly mentioned that fermion lines could point “backwards” with respect to time. The lower electron line in this diagram does just that. Out interpretation of the backward arrow is that instead of dealing with an electron, we’re dealing with its partner the anti-electron, also known as the positron. The positron has the same mass as the electron, but is otherwise its opposite. The electron has negative electric charge, for instance. Well, the positron has the same amount of positive electric charge (hence the name). Every particle type has a corresponding antiparticle type, with exactly opposite charges. Given the tendency of positrons to turn into photons — pure light — when they meet electrons, they don’t have much effect on ordinary life. They do tend to crop up in high energy experiments, though. For instance, we said that we represent a photon like this:


However, if all we know is that a photon went in and a photon came out, what might have happened is this:

photon-with-loop-feynmanWe might not even detect the intermediary electron and positron with out measuring instruments, if they exist for a short enough time, but the rules of QED tell us that it could happen. In fact, particles that must be part of an interaction, but don’t exist to be measured at the beginning or the end of the process turn out to be very useful for hiding some of the uglier parts of the mathematics. (Others may disagree about the ugliness of the mathematics or whether it’s fair to describe virtual particles as hiding these aspects of the maths, but the broad strokes of the picture are at least agreed upon.) The maths involved stems from the uncertainty principle. This means that we can’t assign an exact momentum and an exact position to a particle at the same time — but we got around that by giving particles cloud-like (or wave-like) properties.






Einstein’s theory of relativity tells us that when we talk about position, to be complete we also need to include a “position in time” (which we’d normally just call a time) and when we talk about momentum, we should also include energy. Knowing that, it’s not too surprising that we can’t assign an exact energy to a particle at an exact time. Imagining particles as clouds in space is bad enough — I’m not sure how to begin visualising them as fuzzy in time. Fortunately, virtual particles mean we don’t have to. The way the maths works out, we can use this one weird trick instead: virtual particles don’t conserve energy.

Yup, I just said we were going to violate one of the most fundamental laws of physics: the law of conservation of energy. Remember that I started out by explaining why it’s just a trick, though. We can very carefully consider particles as being fuzzy in time as well as in space and then we keep conservation of energy. It makes the maths a lot harder, though. On the other hand, if we bend the rules when nobody’s looking, we can get to the answers a lot faster. That’s the key, of course: virtual particles are the particles we can never measure. We can treat them as breaking the law of energy conservation instead of as having weird fuzzy times and energies exactly because we’re never going to check what the energies actually are. We just need the maths to work out.

Last week I showed you this diagram, which includes a virtual photon:


In fact, this diagram assumes what’s called a “highly” virtual photon. It violates conservation of energy very badly, so that it gains an enormous momentum out of nowhere. (Or we can say that it’s an extreme case in the time-energy fuzziness, but it gets much harder to describe — people who try to do so can spend years figuring out how to start.) The photon needs to have pretty high energy for the rules of quarks and gluons (quantum chromodynamics or QCD) to work out, but there’s still a possible range of energies. If we choose a relatively low energy, by using the proton energy to define a fairly complicated standard1, the most likely interaction between the photon and the proton is quite different. This is the case I studied in my MSc project. The diagram looks like this (A represents one or more protons):


You’ll notice that to draw this diagram, I’ve introduced a new vertex, where the photon becomes a quark and an antiquark. Next week, we’ll talk about this vertex and other properties of QCD, like the requirement that the photon be highly virtual and why Feynman diagrams don’t work as well as we might hope.

1 Such that the square of the photon four-momentum is much smaller than the Minkowski product of the photon four-momentum with the proton four-momentum, meaning that the Bjorken-x variable is small, if you want to get technical.

Images in space and time: Feynman diagrams

So far in this series, we’ve introduced the standard model of particle physics: we’ve made a list of the fundamental particles that make up everyday matter (plus a few extras), called fermions, and the particles that bind them together as forces (plus the Higgs particle), called bosons. All of that can be summarised in a diagram suspiciously similar to the periodic table:


This week, I’ll talk about describing collisions and interactions between these particles. This is almost universally done by using a system of diagrams invented by and named after Richard Feynman. These Feynman diagrams absorb all the calculus and group theory and complicated mathematical notation into pictures, making their interpretation quite accessible (although calculating the rules for Feynman diagrams is another story). For example, this is a photon, labelled as γ:


Once we introduce axes and directions to the diagrams, we can talk about what the photon is doing, but for now, that’s a photon. This is a gluon:


And this is a Higgs boson:


Fermions are all represented by straight, solid lines, so the labels become especially important. These particles also come with arrows — if the arrow goes backwards, we’re dealing with antimatter, not ordinary matter! (We’ll go into that a little more next week.) For the moment we haven’t specified forward and backward, but I’ll make the arrows consistent with the notation we’ll adopt by the end of the article. Here are a quark and an electron:



We could go on and draw all the particles of the standard model, but for this project we’ll stop here, since we only need four: the quark, the electron, the photon and the gluon. There’s one more thing we can do with them before we start thinking about axes and directions. Two weeks ago we said protons and neutrons must be made of quarks because of all the threes. So we can put three quarks together to represent a proton:


Of course, three quarks could also be a neutron, so labels matter. (Unless things are crystal clear from context, in which case the labels are sometimes left out.)

To go much further, we need to introduce the axes of Feynman diagrams. Bearing in mind that we’re generally trying to describe collisions between particles, we use one axis to describe the separation between the relevant particles. In fact this is the only information about position that the diagrams explicitly include. So far, two colliding particles would look like this:


Clearly the ability to represent the separation at different times is crucial. We use the diagram’s other axis to represent passing time. Then we can represent two electrons interacting by exchanging a photon like this:


Sometimes it’s easier to represent the maths behind the Feynman diagrams by starting with early times on the right and flowing towards later times on the left:


Although the left-to-right notation is more common, for all the reasons you’d expect, I’m going to need specialised Feynman diagrams that assume the right-to-left convention later, so I’ll draw time progressing from right to left from the start. Here’s a photon undergoing a process called pair production in which it turns into an electron and an anti-electron (called a positron):


And here’s a real live working diagram of a collision between an electron and a proton, mediated by a photon:


This one’s pulled directly from my thesis, so it may need a little more explanation. Since I’ve told you that the quarks make up a proton, you should be able to figure out that they’re quarks, not antiquarks and put the arrows on for yourself. The photon is labelled as γ* because it’s a “virtual particle” — we can’t measure it, since it disappears before the collision is over. This allows it to have some unusual properties that wouldn’t make sense otherwise. The blob between the photon and the quark represents the fact that while they could interact directly, we also want to consider more complicated processes. For instance, the photon could produce an electron and a positron (like in the earlier diagram) and that electron could produce another photon, which interacts with the quark. Since we haven’t specified exactly what the interaction is, we don’t know exactly what comes out the other end, so there are just a bunch of lines collectively labelled “X”. This process is called Deep Inelastic Scattering (DIS) and it’s the context for most of the work done in my MSc project (although my work is based on a slightly different diagram that also falls under the DIS heading).

That’s Feynman diagrams! Next week we’ll talk about the Feynman rules, which tell us which diagrams make sense and which don’t. Along the way we’ll also talk more about matter, antimatter and virtual particles.