The most accurate measurement of muon’s magnetic moment to-date was made by the E821 collaboration at the BNL (Brookhaven National Laboratory) in 2001. The final report of the E821 experiment was published in 2006. That report was updated in 2008. The E821 measured value of muon’s magnetic moment was higher than the theoretical value predicted by the Standard Model. The difference (close to 3 standard deviations) was statistically significant but not enough to claim a deviation from the Standard Model (a collection of particle physics theories including QED, QCD and more).
There is a constant known as “g” which relates a charged particle’s intrinsic spin to its magnetic moment. In Dirac theory the “g” for spin=1/2 charged particles is exactly 2. When you consider quantum fluctuations in vacuum, however, the value of “g” is slightly larger than 2.
In the case of the electron the theory known as QED (Quantum Electrodynamics) successfully predicts the deviation from 2. QED predicted value agrees with the measurements with a precision better than one part in a trillion.
Muon has -1 electric charge and spin=1/2 just like the electron but it’s invariant mass is approximately 207 times larger than the electron. Because of the larger mass the quantum fluctuations are more complicated for the muon.
In the summer of 2013, the 50-foot-wide electromagnet of the original BNL muon g-2 experiment was transported from Long Island to the Chicago suburbs in one piece. The story of how this move was accomplished is told here. The re-constituted muon g-2 experiment at Fermilab will have much better statistics because it will measure 20 times more muons during its lifetime. If the discrepancy between the theoretical prediction and the measured value remains then we will know that there is new (undiscovered) physics beyond the Standard Model of particle physics.
The Fermilab muon g-2 experiment (E989) will start taking data soon. The Fermilab experiment will measure muon’s magnetic moment to a precision of 140 parts per billion. This will be a factor of 4 improvement over E821 experiment’s precision.
“The muon, like its lighter sibling the electron, acts like a spinning magnet. The parameter known as “g” indicates how strong the magnet is and the rate of its gyration. The value of g is slightly larger than 2, hence the name of the experiment. This difference from 2 is caused by the presence of virtual particles that appear from the vacuum and then quickly disappear into it again.” – Fermilab
“A vacuum is never truly empty space. It is filled with a bath of virtual particles that pop in and out of existence and interact with light and other particles. These particles can be massless photons (the quanta of light), lightweight particles such as electrons or very massive particles such as the W, Z and Higgs bosons. Because these particles are virtual — they emerge only fleetingly from the vacuum — they can be so massive that they cannot be made in the current accelerators at Fermilab or CERN or are difficult to detect there. Thus, scientists can use the vacuum, which knows about all particles discovered and as yet undiscovered, as a tool to study nature’s elementary particles without having to create the particles directly.” – Fermilab
“The Standard Model of particle physics makes a very precise prediction of the muon g-2, accurate to 400 parts per billion. The purpose of the Fermilab Muon g-2 experiment is to make a measurement that is precise to 140 parts per billion. This is equivalent to measuring the length of a football field to a precision of one-tenth the thickness of a human hair. With this increased precision, scientists can compare the experimental g-2 measurement to the Standard Model prediction. The difference between the two values should provide an unambiguous answer to the question, Are there new, as yet unobserved, particles and forces that exist in nature?” – Fermilab
There are excellent review papers that explain the physics details of muon g-2:
For historical record, please see the the web pages of the old muon g-2 experiment at the Brookhaven National Laboratory: http://www.g-2.bnl.gov/
Update (February 1, 2018):
The link below points to a recent detailed document with lots of pictures and diagrams