FEYNMAN, MAGISTER LUDI X

Let’s draw Feynman diagrams!

Greetings! This post turned into a multi-part ongoing series about the Feynman rules for the Standard Model and a few of its extensions. I’ll use this first post as an index for all of the parts of the series.

1. Let’s draw Feynman diagrams! (this post)
2. More Feynman diagrams.
3. Introducing the muon.
4. The Z boson and resonances.
5. Neutrinos.
6. The W boson, mixing things up.
7. Meet the quarks.
8. World of glue.
9. QCD and confinement.
10. Known knowns of the Standard Model. (summary)
11. When Feynman Diagrams Fail.
12. An idiosyncratic introduction to the Higgs.
13. A diagrammatic hint of masses from the Higgs
14. Higgs and the vacuum: Viva la “vev”
15. Helicity, Chirality, Mass, and the Higgs
16. The Birds and the Bs
17. The spin of gauge bosons
18. Who ate the Higgs?
19. Unitarization of vector boson scattering
20. Private lives of Standard Model particles (summary)

There are few things more iconic of particle physics than Feynman diagrams. These little figures of squiggly show up prominently on particle physicists’ chalkboards alongside scribbled equations. Here’s a ‘typical’ example from a previous post.

The simplicity of these diagrams has a certain aesthetic appeal, though as one might imagine there are many layers of meaning behind them. The good news is that’s it’s really easy to understand the first few layers and today you will learn how to draw your own Feynman diagrams and interpret their physical meaning.

You do not need to know any fancy-schmancy math or physics to do this!

That’s right. I know a lot of people are intimidated by physics: don’t be! Today there will be no equations, just non-threatening squiggly lines. Even school children can learn how to draw Feynman diagrams (and, I hope, some cool science). Particle physics: fun for the whole family. 

For now, think of this as a game. You’ll need a piece of paper and a pen/pencil. The rules are as follows (read these carefully):

1. You can draw two kinds of lines, a straight line with an arrow or a wiggly line:
   You can draw these pointing in any direction.
2. You may only connect these lines if you have two lines with arrows meeting a single wiggly line.
   
   Note that the orientation of the arrows is important! You must have exactly one arrow going into the vertex and exactly one arrow coming out.
3. Your diagram should only contain connected pieces. That is every line must connect to at least one vertex. There shouldn’t be any disconnected part of the diagram.
   
   In the image above the diagram on the left is allowed while the one on the right is not since the top and bottom parts don’t connect.
4. What’s really important are the endpoints of each line, so we can get rid of excess curves. You should treat each line as a shoelace and pull each line taut to make them nice and neat. They should be as straight as possible. (But the wiggly line stays wiggly!)
   

That’s it! Those are the rules of the game. Any diagram you can draw that passes these rules is a valid Feynman diagram. We will call this game QED. Take some time now to draw a few diagrams. Beware of a few common pitfalls of diagrams that do not work (can you see why?):

After a while, you might notice a few patterns emerging. For example, you could count the number of external lines (one free end) versus the number of internal lines (both ends attached to a vertex).

• How are the number of external lines related to the number of internal lines and vertices?
• If I tell you the number of external lines with arrows point inward, can you tell me the number of external lines with arrows pointing outward? Does a similar relation hole for the number of external wiggly lines?
• If you keep following the arrowed lines, is it possible to end on some internal vertex?
• Did you consider diagrams that contain closed loops? If not, do your answers to the above two questions change?

I won’t answer these questions for you, at least not in this post. Take some time to really play with these diagrams. There’s a lot of intuition you can develop with this “QED” game. After a while, you’ll have a pleasantly silly-looking piece of paper and you’ll be ready to move on to the next discussion:

What does it all mean?

Now we get to some physics. Each line in rule (1) is called a particle. (Aha!) The vertex in rule (2) is called an interaction. The rules above are an outline for a theory of particles and their interactions. We called it QED, which is short for quantum electrodynamics. The lines with arrows are matter particles (“fermions”). The wiggly line is a force particle (“boson”) which, in this case, mediates electromagnetic interactions: it is the photon.

The diagrams tell a story about how a set of particles interact. We read the diagrams from left to right, so if you have up-and-down lines you should shift them a little so they slant in either direction. This left-to-right reading is important since it determines our interpretation of the diagrams. Matter particles with arrows pointing from left to right are electrons. Matter particles with arrows pointing in the other direction are positrons (antimatter!). In fact, you can think about the arrow as pointing in the direction of the flow of electric charge. As a summary, we our particle content is:

(e+ is a positron, e- is an electron, and the gamma is a photon… think of a gamma ray.)

From this we can make a few important remarks:

• The interaction with a photon shown above secretly includes information about the conservation of electric charge: for every arrow coming in, there must be an arrow coming out.
• But wait: we can also rotate the interaction so that it tells a different story. Here are a few examples of the different ways one can interpret the single interaction (reading from left to right):
  
  These are to be interpreted as: (1) an electron emits a photon and keeps going, (2) a positron absorbs a photon and keeps going, (3) an electron and positron annihilate into a photon, (4) a photon spontaneously “pair produces” an electron and positron.

On the left side of a diagram we have “incoming particles,” these are the particles that are about to crash into each other to do something interesting. For example, at the LHC these ‘incoming particles’ are the quarks and gluons that live inside the accelerated protons. On the right side of a diagram we have “outgoing particles,” these are the things which are detected after an interesting interaction.

For the theory above, we can imagine an electron/positron collider like the the old LEP and SLACfacilities. In these experiments an electron and positron collide and the resulting outgoing particles are detected. In our simple QED theory, what kinds of “experimental signatures” (outgoing particle configurations) could they measure? (e.g. is it possible to have a signature of a single electron with two positrons? Are there constraints on how many photons come out?)

So we see that the external lines correspond to incoming or outgoing particles. What about the internal lines? These represent virtual particles that are never directly observed. They are created quantum mechanically and disappear quantum mechanically, serving only the purpose of allowing a given set of interactions to occur to allow the incoming particles to turn into the outgoing particles. We’ll have a lot to say about these guys in future posts. Here’s an example where we have a virtual photon mediating the interaction between an electron and a positron.

In the first diagram the electron and positron annihilate into a photon which then produces another electron-positron pair. In the second diagram an electron tosses a photon to a nearby positron (without ever touching the positron). This all meshes with the idea that force particles are just weird quantum objects which mediate forces. However, our theory treats force and matter particles on equal footing. We could draw diagrams where there are photons in the external state and electrons are virtual:

This is a process where light (the photon) and an electron bounce off each other and is calledCompton scattering. Note, by the way, that I didn’t bother to slant the vertical virtual particle in the second diagram. This is because it doesn’t matter whether we interpret it as a virtual electron or a virtual positron: we can either say (1) that the electron emits a photon and then scatters off of the incoming photon, or (2) we can say that the incoming photon pair produced with the resulting positron annihilating with the electron to form an outgoing photon:

Anyway, this is the basic idea of Feynman diagrams. They allow us to write down what interactions are possible. We will see later that in fact there is a much more mathematical interpretation of these diagrams that produces the mathematical expressions that predict the probability of these interactions to occur, and so there is actually some rather complicated mathematics “under the hood.” However, just like a work of art, it’s perfectly acceptable to appreciate these diagrams at face value as diagrams of particle interactions. In subsequent posts we’ll develop more techniques and use this to talk about some really interesting physics, but until then let me close with a quick “frequently asked questions”:

1. What is the significance of the x and y axes?These are really spacetime diagrams that outline the “trajectory” of particles. By reading these diagrams from left to right, we interpret the x axis as time. You can think of each vertical slice as a moment in time. The y axis is roughly the space direction.
2. So are you telling me that the particles travel in straight lines?No, but it’s easy to mistakenly believe this if you take the diagrams too seriously. The path that particles take through actual space is determined not only by the interactions (which are captured by Feynman diagrams), but the kinematics (which is not). For example, one would still have to impose things like momentum and energy conservation. The point of the Feynman diagram is to understand the interactions along a particle’s path, not the actual trajectory of the particle in space.
3. Does this mean that positrons are just electrons moving backwards in time?In the early days of quantum electrodynamics this seemed to be an idea that people liked to say once in a while because it sounds neat. Diagrammatically (and in some sense mathematically) one can take this interpretation, but it doesn’t really buy you anything. Among other more technical reasons, this viewpoint is rather counterproductive because the mathematical framework of quantum field theory is built upon the idea of causality.
4. What does it mean that a set of incoming particles and outgoing particles can have multiple diagrams?In the examples above of two-to-two scattering I showed two different diagrams that take the in-state and produce the required out-state. In fact, there are an infinite set of such diagrams. (Can you draw a few more?) Quantum mechanically, one has to sum over all the different ways to get from the in state to the out state. This should sound familiar: it’s just the usual sum over paths in the double slit experiment that we discussed before. We’ll have plenty more to say about this, but the idea is that one has to add the mathematical expressions associated with each diagram just like we had to sum numbers associated with each path in the double slit experiment.
5. What is the significance of rules 3 and 4?
   Rule 3 says that we’re only going to care about one particular chain of interactions. We don’t care about additional particles which don’t interact or additional independent chains of interactions. Rule 4 just makes the diagrams easier to read. Occasionally we’ll have to draw curvy lines or even lines that “slide under” other lines.
6. Where do the rules come from?
   The rules that we gave above (called Feynman rules) are essentially the definition of a theory of particle physics. More completely, the rules should also include a few numbers associated with the parameters of the theory (e.g. the masses of the particles, how strongly they couple), but we won’t worry about these. Graduate students in particle physics spent much of their first year learning how to carefully extract the diagrammatic rules from mathematical expressions (and then how to use the diagrams to do more math), but the physical content of the theory is most intuitively understood by looking at the diagrams directly and ignoring the math. If you’re really curious, the expression from which one obtains the rules looks something like this (from TD Gutierrez), though that’s a deliberately “scary-looking” formulation.

We’ll develop more intuition about these diagrams and eventually get to some LHC physics, but hopefully this will get the ball rolling!

That’s all for now!
–Flip Tanedo, for the US/LHC Blog.

[As an aside, a special ‘hello’ to everyone who reads these posts from the ‘Large Hadron Collider’ Facebook page. Contrary to some belief, the LHC didn’t become sentient and start a Facebook blog. Check out the US/LHC Blog homepage for more information about the physicists who write these posts! PS, while I occasionally browse the Facebook comments, I’m more likely to respond to comments posted to the actual blog page.]

Tags: Feynman diagram, Feynman rule, particle physics, QED

Feynman diagrams

The rules for any process are:

·        Draw all possible diagrams (up to some number of photons, depending on the accuracy desired). Different time-orderings of a given process are represented by the same diagram.

·        Given the initial momentum and energy, define how momentum and energy flow for each line in the diagram. Where each diagram has a closed loop, there is an arbitrary momentum and energy flow around the loop and we must integrate over all possible choices for these quantities. Each intermediate line in the diagram contributes a factor to the amplitude of  1/(E²-p²c²-m²c⁴) where m is the appropriate mass for the particle type represented by the line. Note that this says that the more "virtual" the particle represented by a line is, the smaller the contribution of the diagram.

·        Add the amplitude factors from all possible diagrams to get the total amplitude for the process.

The expected rate for the process can then be calculated -- it is proportional to the absolute value of the total amplitude squared. [Note that this is not the same as the sum of the squares of the absolute values of the individual amplitudes.] For more information on this topic, take a look at the discussion of quantum interference