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### 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.

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 SLAC facilities. 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 called Compton 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.]

• Rich DeCoster

This is very interesting for me as a high school physics/astronomy teacher. One of my students had a question about, essentially, Feynman diagrams as he tries to read through QED. This helps and I’ve sent the url on to him.

• Michael M. Butler

I’m still confused by the “two particles annihilate into a single photon” case. Can you address the conservation issues (momentum and energy) this raises? Sorry / thanks.

• Bharat

Amazing article..Not only is it scientfically correct, but the language and detailed descriptions make it quite simpler. Really good.

• Garry Grofcsik

It may be a very interesting and amazing article, but I had trouble getting past the first sentence in the second paragraph. What does “…simplicity of these diagrams a a certain aesthetic appeal…” (a direct copy/paste)mean? I guess I could assume that the first “a” hanging out there all by itself is a typo and that the word should be “has”, but I really can’t be sure, can I? I don’t trust blogs that have been poorly edited/proofread.

• Thanks for the corrections, Garry. The beauty of blogs is that they’re easy to fix so I went ahead and made the minor corrections that you mentioned.

• Aaron Spurling

If you were to write a book on quantum physics with the same level of clarity and simplicity as displayed in this article, I would definitely buy it! Many thanks.

• flashgordon

i’m not understanding(maybe your oversimplifying!?) why I can’t have two squigly lines connected to a third central squigly line; but, I can do that with arrowed lines? Where in the rules is that squigly lines ban coming from?

• Daniel

Speaking of easy to fix, the word “diagrams” in the title is missing its “r”.

• Daniel — thanks! I fixed the embarrassing typo in the title.

Flash — this is a very important question, I’ll try to discuss it a bit in my next post as well. The short answer is that in this theory, you can *only* have an intersection of two arrowed lines and one squiggly line. Those are the rules.

By “rules” I really mean that this is part of the definition of the theory. The theory doesn’t have three-squiggly-line intersections, full stop.

As we build up our intuition for Feynman diagrams, I’ll introduce different theories (i.e. different sets of rules) which allow more complicated intersections, including some with three-squiggly-lines.

Now the question is, why does our theory only have this one particular kind of vertex and not other kinds? This is a much deeper question than it lets on. The short answer is symmetry. The mathematical structure of the theory is constrained by the symmetries that we impose. It turns out that a 3-squiggly intersection would violate electromagnetic symmetry. (This statement isn’t ‘obvious’ by any stretch, but I’ll see if I can think of a way to clarify this.)

You might also ask why we don’t have intersections with 5, 6, 7, … etc. lines. These are not part of the rules for an even more esoteric reasons: it turns out that even if such intersections existed in our rules, the diagrams that we could draw with them would end up being very small effects. Again, this is far from obvious, but I hope to get to this over the next few posts.

But again the short answer to your question is that the Feynman rules define the theory, and the simple theory that we’ve decided to look at only has the one particular rule for intersecting lines.

Cheers!
-F

• Great article Flip! Looking forward to future installments.

(I agree with Aaron Spurling 🙂 )

• Great post. If only one could find some easy and intuitive way to extend this to non-abelian gauges.

• Shane

Flip — This is terrific! As a late-bloomer, layman, wanna-be physicist, I REALLY appreciate you breaking things down to this level. I look forward to following the future posts.

• kent

Great post, but I think you should address Mike Butler’s comment on energy conservation. While I understand this is for high school students, why give them misinformation.

• Hi Mike B. and Kent — my apologize that I missed Mike’s question. This requires a little background in special relativity, but the idea is this: when you have two electrons, you can always boost into a reference frame where there is zero net momentum. In other words, you can always set up an observer who sees the electrons traveling with equal and opposite momenta. Because momentum is conserved in all frames, this means that this observer should see a photon with zero momentum. However, this is not possible because photons are massless and so travel at the speed of light. Thus this process cannot occur.

One can check explicitly that one cannot simultaneously satisfy conservation of energy and conservation of momentum for this process, though you have to use the relativistic energy (i.e. take into account the mass energy of the electrons).

One remark regarding Kent’s comment: While one of our goals is to make particle physics accessible to everyone, we do *not* try to “dumb down” the science or “give misinformation.” (Though I may occasionally make a mistake, in which case I try to correct myself as soon as possible.) I do not believe that good science needs to be watered down just because someone doesn’t have the the same formal background as a PhD in physics. (Though it should certainly be presented more carefully.)

• Richard Tyler Harbaugh

This is awesome. I am a generalist, trying to learn with only undergraduate calculus for math. My question has to do with the “this” link in your text “the rules looks something like this (from TD Gutierrez).” My first thought was to look for an equals sign…..and found none. So am I correct that this formulation is used by assuming the whole phrase equal to zero? Thanks for the information.

• Hi Richard! The big long expression is called the “Lagrangian” for the Standard Model, which we denote by “L.” Thus everything on that page should be prefaced with an “L=”. 🙂

The *meaning* of the Lagrangian is a little difficult to describe intuitively, but it is related to the difference between potential and kinetic energies. The fact that this should be of any ‘deep’ importance is completely not obvious, but it’s the starting point for a rather powerful formalism for describing physics. In fact, the significance is this: the Lagrangian formalism is something which classically gives the exact same predictions as the Newtonian (what you’re used to from high school) formalism, but that can be straightforwardly generalized to give the correct quantum predictions as well. Again, this is far from obvious, but for now you can take it as an observation or statement of fact.

The main idea is that the Lagrangian captures everything there is to know about a theory. Given a Lagrangian for a model of particle physics, one can then read off all of the relevant Feynman rules and make predictions. *How* this is done is beyond the scope of these posts, I’d have to refer you to a textbook on quantum field theory (and the prerequisite background reading). 🙂

Cheers,
Flip

• John Marsh

In 3. near the beginning of the article could the diagram be interpreted as the electron coming from the left, emitting a photon which is then absorbed by the electron coming in from the bottom right.
Thanks for a great site!
John

• The lines in Feynman diagrams represent various particles in a given interaction. Strait lines for electrons and positrons, photons represented by wavy lines,that’s all

• IT is amazing that you can make it so simple. Great job.

• Hurrah! Finally I got a blog from where I be able to in
fact obtain valuable data concerning my study and knowledge.

• Thanks for this great article. Can I point out what I imagine is a small typo?
“Does a similar relation hole for the number of external wiggly lines?”
Cheers mate.

• Jeff Nolfo

Flip I am educated only through the 12th grade. I was a horrible student and couldn’t care less of my studies. I’m in my forties now and have become fascinated by the ability to change ones place in life through the power of thought,which has lead me to your blog. I have no idea what you’re talking about but I want to understand. Where would you suggest I start? A high school physics book maybe? I’m not looking to start writing theories I just want the basic understanding of physics so I can get a better grasp of quantum theories. Thanks for you time

• Jeff Nolfo

My email address is [email protected]….thanks again

• Lewis

The particles are anitparticles; therefore, antimass. Zero mass going in, so zero mass coming out with the photon.

• J Shaw

Hi Flip!

Your first blog on Feynman diagrams was great. However, I’m interested in using Feynman diagrams to look at bound atoms interacting with free electrons. I have been digging all through our library and all through the internet, but I haven’t been able to find anything good. Do you have any suggestions for good references?

• Robert Mackay MSci

What Lewis said is flat out wrong. Antiparicles have positive mass, EVERYTHING has positive mass (if we take 0 to be a positive number), there is no known way of having a negative mass.
However Mass and Energy are Equivilant, E=mc². So the Photon has no mass but it has energy, which is converted into the mass of the Particle -Anti-Particle Pair.

• Very nice explanation of feynman diagrams.

• Co.H.Tran

Interesting and easy to know QED intuitively . Thanks for your nice post .

• monologue

This was wonderful to read, I came here curious about the amplitudehedron and wanting to see Fenyman diagrams, I love learning about physics and although mathematically impaired, I really love trying to understand the magical mysterious diagrams scientists use to explore/explain the universe. Thank you for making this comprehensible, more please!! much appreciated.

• Ernie

Is there a text on creating Feynman diagrams,if so what is the title and author?
Thank you.

• Darrell cross

At the point of contact between a positron and an electron, does the gamma ray have a descret wavelength and trajectory? Can the collision angle affect these values?

• Anthony Rago

I really love this blog. I have very little college education but have always loved to read scientific articles and websites. I am currently reading everything I can get my hands on that teaches Quantum Physics to people like me that have little in formal education and little math skills. This blog makes things a bit simpler. Thanks for not dumbing it down but in making it easier to understand.

• Gyaneshwaran Gomathinayagam

Thank you so much for the wonderful introduction to Feynman Diagrams, Flip Tanedo! Looking forward to reading the remaining posts!

• John Sharp

Feynman said, something like, everything you need to know is in the double-slit experiment. How can you represent the double-slit experiment in Feynman diagrams. If you can’t does this mean you cannot understand physics using Feynman diagrams?