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In the past days it was announced that the lastest results of the muon $g-2$ experiment indicates some discrepancies with the standard model. As an outsider in particle physics, I don't understand exactly what's happening and my only sources have been some online newspapers such as the new york times. Could someone explain to a non-expert like me what is going on and what are the consequences of such results?

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The $g$ factor describes the magnetic moment of a spinning particle. The $g$ factor for a classically spinning particle is equal to 1, but in the "basic" (ie, non-interacting) quantum field theory of spin-1/2 particles you would expect this number to come out to be 2, or in other words you would expect $g-2$ to be zero. It turns out that in "advanced' (interacting) theory, $g-2$ is a non-zero number. For electrons, the contributions of various interactions can be calculated very precisely, and $g-2$ can be measured very precisely (both to 13 or so decimal places). This highly precise level of agreement (a) is an incredible testament to the power of the standard model, quantum field theory, and experimental particle physicists (b) means that this measurement probes the standard model very precisely. since even a small deviation from the standard model could appear as a discrepancy between theory and experiment.

The muon $g-2$ is based on the same physical idea, but applied to muons (a heavy version of the electron). However, it is harder both on the theory and experimental side to get precise measurements of the muon $g-2$.

The recent news is that new experimental results for the muon $g-2$ constrain this to very precisely, and the result disagrees with a particular theoretical calculation to 4.2$\sigma$ (or, in layman's terms: the level of disagreement is very difficult to explain as random chance due to statistical noise in the experiment).

Having said that, extraordinary claims require extraordinary evidence, and there have been many examples of apparently interesting discrepancies or discoveries in physics that have gone away upon closer examination, even in just the past ten years. In this case, I would say that the calculation of the muon $g-2$ is fraught with difficulty, and it is possible that the theoretical uncertainty is larger than what is claimed, which (if I am right) would lower the difference with the standard model. Some groups also claim to have a calculation that is closer to the measured value (but I can't judge the veracity of these claims... all I can do is apply some common sense and a prior that most such things eventually go away).

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    $\begingroup$ Just to clarify a little, there’s a second theory group which I will call Group 2 that computed g-2 using lattice QCD methods (with a supercomputer), and the experimental result is actually in much better agreement in this case. Group 1, where the result disagrees, uses data-driven methods (they incorporate the results of other computations and experiments). To resolve this, two things need to happen. First, Fermilab’s experiment will collect much more data over the next year or two. Second, the theory side needs to get their act together and agree on what number the “standard model” predicts. $\endgroup$
    – Jackson Walters
    Apr 21, 2021 at 0:38

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