The standard model is like the periodic table of quantum physics.
Virtual particles are really more of a math trick to explain interactions. They only "exist" momentarily. They interact in many permutations.
Here's an extremely simplified explanation:
When you sit down in a chair, your body comes into contact with the chair. But what is making contact with the chair, at the atomic level? At the most fundamental level, is it the atom's nucleus? No. Is it the electron? Kinda.
What physical process keeps you from falling through the chair or the floor, for that matter? Force fields. It's all just force fields.
You have electrons around the atoms in your body and the electrons around the atoms in the chair. They repel each other. That's it. Overly simplified, but that's it.
But how do electrons repel each other? That's where things get fuzzy. Literally. Usually, the electrons emit virtual photons. Occasionally they do something else. Whatever they do, the result is the same: the electrons transfer momentum to the other electrons through virtual particles.
This Muon g-2 experiment is looking for those "something else" interactions. There's a bunch of known interactions. Some theorized. And some completely unknown. The sum of these different known and theorized interactions results in a prediction that is different from the experiment. This means there are interactions that could exist or the theorized ones aren't as well understood.
Another thing that's interesting is the amount of power needed to do this experiment. It's not like the large hadron collider in terms of energy levels. Being able to find new physics without smacking protons at insane energy levels is pretty neat.
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Update: This is how you do science. Pledge to publish, even if you make mistakes, do the work blind, then unblind to see how you did. Fermilab did this with (relatively) low power accelerators and very high precision. It's hard to do this kind of science and they justify the costs. If this science was being done with voluntary funds, they would have earned my donation.