Last week, I used two terms, planar theory and non-planar theory, without defining them. This week, I’m going to explain what they mean, and why they’re important.
Suppose you’re working with a Yang-Mills theory (not necessarily N=4 super Yang-Mills. To show you the difference between planar and non-planar, I’ll draw some two-loop Feynman diagrams for a process where two particles go in and two particles come out:
The diagram on your left is planar, while the diagram on your right is non-planar. The diagram on the left can be written entirely on a flat page (or screen), with no tricks. By contrast, with the diagram on the right I have to cheat and make one of the particle lines jump over another one (that’s what the arrow is meant to show). Try as you might, you can’t twist that diagram so that it lies flat on a plane (at least not while keeping the same particles going in and out). That’s the difference between planar and non-planar.
Now, what does it mean for a theory to be planar or non-planar?
Let’s review some facts about Yang-Mills theories. (For a more detailed explanation, see here). In Yang-Mills there are a certain number of colors, where each one works a bit like a different kind of electric charge. The strong force, the force that holds protons and neutrons together, has three colors, usually referred to as red, blue, and green (this is of course just jargon, not the literal color of the particles).
Forces give rise to particles. In the case of the strong force, those particles are called gluons. Each gluon has a color and an anti-color, where you can think of the color like a positive charge and the anti-color like a negative charge. A given gluon might be red-antiblue, or green-antired, or even red-antired.
While the strong force has three colors, for this article it will be convenient to pretend that there are four: red, green, blue, and yellow.
An important principle of Yang-Mills theories is that color must be conserved. Since anti-colors are like negative colors, they can cancel normal colors out. So if you’ve got a red-antiblue gluon that collides with a blue-antigreen gluon, the blue and antiblue can cancel each other out, and you can end up with, for example, red-antiyellow and yellow-antigreen instead.
Let’s consider that process in particular. There are lots of Feynman diagrams you can draw for it, let’s draw one of the simplest ones first:
The diagram on the left just shows the process in terms of the particles involved: two gluons go in, two come out.
The other diagram takes into account conservation of colors. The red from the red-antiblue gluon becomes the red in the red-antiyellow gluon on the other side. The antiblue instead goes down and meets the blue from the blue-antigreen gluon, and both vanish in the middle, cancelling each other out. It’s as if the blue color entered the diagram, then turned around backwards and left it again. (If you’ve ever heard someone make the crazy-sounding claim that antimatter is normal matter going backwards in time, this is roughly what they mean.)
From this diagram, we can start observing a general principle: to make sure that color is conserved, each line must have only one color.
Now let’s try to apply this principle to the two-loop diagrams from the beginning of the article. If you draw double lines like we did in the last example, fill in the colors, and work things out, this is what you get:
What’s going on here?
In the diagram on the left, you see the same lines as the earlier diagram on the outside. On the inside, though, I’ve drawn two loops of color, purple and pink.
I drew the lines that way because, just based on the external lines, you don’t know what color they should be. They could be red, or yellow, or green, or blue. Nothing tells you which one is right, so all of them are possible.
Remember that for Feynman diagrams, we need to add up every diagram we can draw to get the final result. That means that there are actually four times four or sixteen copies of this diagram, each one with different colors in the loops.
Now let’s look at the other diagram. Like the first one, it’s a diagram with two loops. However, in this case, the inside of both loops is blue. If you like, you can try to trace out the lines in the loops. You’ll find that they’re all connected together. Because this diagram is non-planar, color conservation fixes the color in the loops.
So while there are sixteen copies of the first diagram, there is only one possible version of the second one. Since you add all the diagrams together, that means that the first diagram is sixteen times more important than the second diagram.
Now suppose we had more than four colors. Lots more.
With ten colors, the planar diagrams are a hundred times more important. With a hundred colors, they are ten thousand times more important. Keep increasing the number of colors, and it gets to the point where you can honestly say that the non-planar diagrams don’t matter at all.
What, then, is a “planar theory”?
A planar theory is a theory with a very large (infinite) number of colors.
In a planar theory, you can ignore the non-planar diagrams and focus only on the planar ones.
Nima Arkani-Hamed’s Amplituhedron method applies to the planar version of N=4 super Yang-Mills. There is a lot of progress on the planar version of the theory, and it is because the restriction to planar diagrams makes things simpler.
However, sometimes you need to go beyond planar diagrams. There are relationships between planar and non-planar diagrams, based on the ways that you can pair different colors together in the theory. Fully understanding this relationship is powerful for understanding Yang-Mills theory, but, as it turns out, it’s also the key to relating Yang-Mills theory to gravity! But that’s a story for another post.