# Theoretical Uncertainty and Uncertain Theory

Yesterday, Fermilab’s Muon g-2 experiment announced a new measurement of the magnetic moment of the muon, a number which describes how muons interact with magnetic fields. For what might seem like a small technical detail, physicists have been very excited about this measurement because it’s a small technical detail that the Standard Model seems to get wrong, making it a potential hint of new undiscovered particles. Quanta magazine has a great piece on the announcement, which explains more than I will here, but the upshot is that there are two different calculations on the market that attempt to predict the magnetic moment of the muon. One of them, using older methods, disagrees with the experiment. The other, with a new approach, agrees. The question then becomes, which calculation was wrong? And why?

What does it mean for a prediction to match an experimental result? The simple, wrong, answer is that the numbers must be equal: if you predict “3”, the experiment has to measure “3”. The reason why this is wrong is that in practice, every experiment and every prediction has some uncertainty. If you’ve taken a college physics class, you’ve run into this kind of uncertainty in one of its simplest forms, measurement uncertainty. Measure with a ruler, and you can only confidently measure down to the smallest divisions on the ruler. If you measure 3cm, but your ruler has ticks only down to a millimeter, then what you’re measuring might be as large as 3.1cm or as small as 2.9 cm. You just don’t know.

This uncertainty doesn’t mean you throw up your hands and give up. Instead, you estimate the effect it can have. You report, not a measurement of 3cm, but of 3cm plus or minus 1mm. If the prediction was 2.9cm, then you’re fine: it falls within your measurement uncertainty.

Measurements aren’t the only thing that can be uncertain. Predictions have uncertainty too, theoretical uncertainty. Sometimes, this comes from uncertainty on a previous measurement: if you make a prediction based on that experiment that measured 3cm plus or minus 1mm, you have to take that plus or minus into account and estimate its effect (we call this propagation of errors). Sometimes, the uncertainty comes instead from an approximation you’re making. In particle physics, we sometimes approximate interactions between different particles with diagrams, beginning with the simplest diagrams and adding on more complicated ones as we go. To estimate the uncertainty there, we estimate the size of the diagrams we left out, the more complicated ones we haven’t calculated yet. Other times, that approximation doesn’t work, and we need to use a different approximation, treating space and time as a finite grid where we can do computer simulations. In that case, you can estimate your uncertainty based on how small you made your grid. The new approach to predicting the muon magnetic moment uses that kind of approximation.

There’s a common thread in all of these uncertainty estimates: you don’t expect to be too far off on average. Your measurements won’t be perfect, but they won’t all be screwed up in the same way either: chances are, they will randomly be a little below or a little above the truth. Your calculations are similar: whether you’re ignoring complicated particle physics diagrams or the spacing in a simulated grid, you can treat the difference as something small and random. That randomness means you can use statistics to talk about your errors: you have statistical uncertainty. When you have statistical uncertainty, you can estimate, not just how far off you might get, but how likely it is you ended up that far off. In particle physics, we have very strict standards for this kind of thing: to call something new a discovery, we demand that it is so unlikely that it would only show up randomly under the old theory roughly one in a million times. The muon magnetic moment isn’t quite up to our standards for a discovery yet, but the new measurement brought it closer.

The two dueling predictions for the muon’s magnetic moment both estimate some amount of statistical uncertainty. It’s possible that the two calculations just disagree due to chance, and that better measurements or a tighter simulation grid would make them agree. Given their estimates, though, that’s unlikely. That takes us from the realm of theoretical uncertainty, and into uncertainty about the theoretical. The two calculations use very different approaches. The new calculation tries to compute things from first principles, using the Standard Model directly. The risk is that such a calculation needs to make assumptions, ignoring some effects that are too difficult to calculate, and one of those assumptions may be wrong. The older calculation is based more on experimental results, using different experiments to estimate effects that are hard to calculate but that should be similar between different situations. The risk is that the situations may be less similar than expected, their assumptions breaking down in a way that the bottom-up calculation could catch.

None of these risks are easy to estimate. They’re “unknown unknowns”, or rather, “uncertain uncertainties”. And until some of them are resolved, it won’t be clear whether Fermilab’s new measurement is a sign of undiscovered particles, or just a (challenging!) confirmation of the Standard Model.

## 4 thoughts on “Theoretical Uncertainty and Uncertain Theory”

1. telescoper

Reblogged this on In the Dark and commented:

Here’s a discussion of the status of the latest measurements of (g-2) versus theory. This kind of problem is not confined to particle physics. It also happens in cosmology that we have problems making accurate predictions to compare with observations, especially when using galaxies to trace large-scale structure. There’s a lot of messy astrophysics to be accounted for.

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2. JollyJoker

We will, of course, get more data from the experiment eventually. We can always hope both theoretical predictions are ruled out.

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