You might think of physics as the science of certainties and exact statements: action and reaction, F=ma, and all that. However, most calculations in physics aren’t exact, they’re *approximations*. This is especially true today, but it’s been true almost since the dawn of physics. In particular, approximations are performed via a method known as **perturbation theory**.

Perturbation theory is a trick used to solve problems that, for one reason or another, are too difficult to solve all in one go. It works by solving a simpler problem, then **perturbing** that solution, adjusting it closer to the target.

To give an analogy: let’s say you want to find the area of a circle, but you only know how to draw straight lines. You could start by drawing a square: it’s easy to find the area, and you get close to the area of the circle. But you’re still a long ways away from the total you’re aiming for. So you add more straight lines, getting an octagon. Now it’s harder to find the area, but you’re closer to the full circle. You can keep adding lines, each step getting closer and closer.

This, broadly speaking, is what’s going on when particle physicists talk about loops. The calculation with no loops (or “tree-level” result) is the easier problem to solve, omitting quantum effects. Each loop then is the next stage, more complicated but closer to the real total.

There are, as usual, holes in this analogy. One is that it leaves out an important aspect of perturbation theory, namely that it involves perturbing with a **parameter**. When that parameter is small, perturbation theory works, but as it gets larger the approximation gets worse and worse. In the case of particle physics, the parameter is the strength of the forces involves, with weaker forces (like the weak nuclear force, or electromagnetism) having better approximations than stronger forces (like the strong nuclear force). If you squint, this can still fit the analogy: different shapes might be harder to approximate than the circle, taking more sets of lines to get acceptably close.

Where the analogy fails completely, though, is when you start approaching infinity. Keep adding more lines, and you should be getting closer and closer to the circle each time. In quantum field theory, though, this frequently is not the case. As I’ve mentioned before, while lower loops keep getting closer to the true (and experimentally verified) results, going all the way out to infinite loops results not in the full circle, but in an infinite result instead. There’s an understanding of why this happens, but it does mean that perturbation theory can’t be thought of in the most intuitive way.

Almost every calculation in particle physics uses perturbation theory, which means almost always we are just approximating the real result, trying to draw a circle using straight lines. There are only a few theories where we can bypass this process and look at the full circle. These are known as **integrable** theories. N=4 super Yang-Mills may be among them, one of many reasons why studying it offers hope for a deeper understanding of particle physics.

Cliff HarveyCongrats on the new site! It looks great.

I’d love to see more posts exploring detailed aspects of amplitudology, N=4 SYM, (2,0) theory, SUGRA, superstrings, etc. Especially for folks who are already somewhat knowledgable on the basics.

In this post I wasn’t expecting to learn much new, but your reference to the renormalon wikipedia article has lead me to some references that seem helpful to understand aspects of the divergence of the all-loop perturbative amplitudes.

I have a question about the relationship between instantons and the renormalon divergences. When I hear some experts writing about these matters, it seems like perhaps more precise statements are possible linking the perturbative divergences to the onset of non-perturbative effects in supersymmetric theories, perhaps especially in N=4 or other special cases. I suppose I would have already heard about it if there was a full proof, but is there some sense in which we can make stronger arguments that the full amplitudes consist of the perturbative part plus known non-perturbative effects?

In general I’m interested in how well we can establish the linkage between these two things, either in the SM or in special theories.

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4gravitonsandagradstudentPost authorGlad you found the links interesting!

My understanding is that there are some theories where the “completion” of the perturbation series by non-perturbative effects is very well-understood, others less so…but it’s not a field I know much in detail about, so I can’t really provide any useful references. Back when I was working more with the (2, 0) theory part of the goal was establishing that sort of understanding of 5d MSYM, and in particular seeing if completing it with instantons led to the (2, 0) theory in some sense.

This blog is aimed at a general audience, so I probably won’t have posts that go into much technical detail. That said, if I figure out a way to explain something along those lines in a generally accessible way, I’ll post it. In the meantime, more technical discussions can be pursued in the comments. 😉

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