The Current Experimental Result
The most definitive measurement of the anomalous magnetic moment of the muon (g-2) was conducted by the Brookhaven National Laboratory E821 experiment which announced its results on January 8, 2004.
This measurement was which differs somewhat from a precisely calculated theoretical value.
In units of 10-11 and combining the errors in quadrature, the experimental result was:
E821 116 592 091 ± 63
Dividing the total value by the error gives an error of 540 parts per million.
The Current Theoretical Prediction
The current state of the art theoretical prediction from the Standard Model of particle physics regarding the value of muon g-2 in units of 10-11 is:
QED 116 584 718.95 ± 0.08
HVP 6 850.6 ± 43
HLbL 105 ± 26
EW 153.6 ± 1.0
Total SM 116 591 828 ± 49
The main contribution comes by far from QED, which is known to five loops (tenth order) and has a small, well-understood uncertainty. Sensitivity at the level of the electroweak (EW) contribution was reached by the E821 experiment.
The hadronic contribution dominates the uncertainty (0.43 ppm compared to 0.01 ppm for QED and EW grouped together). This contribution splits into two categories, hadronic vacuum polarization (HVP) and hadronic light-by-light (HLbL).
The HVP contribution dominates the correction, and can be calculated from e + e − → hadrons cross-section using dispersion relations.
The HLbL contribution derives from model-dependent calculations.
In other words, unsurprisingly, almost all of the uncertainty in the theoretical prediction comes from the QCD part of the calculations. One part of that calculation has a precision of ± 0.6% (roughly the precision with which the strong force coupling constant is known), the other part of that calculation has a precision of only ± 25%.Lattice QCD predictions of these two hadronic contributions are becoming competitive, and will be crucial in providing robust uncertainty estimates free from uncontrolled modeling assumptions. Lattice QCD predictions have well-understood, quantifiable uncertainty estimates. Model-based estimates lack controlled uncertainty estimates, and will always allow a loophole in comparisons with the SM.
A 2011 paper suggested that most of the discrepancy was between theory and experiment was probably due to errors in the theoretical calculation.
In the same units, the experimental result from 2004 exceeds the theoretical prediction by:
How Significant Is This Discrepancy?
This is a 3.3 sigma discrepancy, which is notable in physics, but not considered definitive proof of beyond the Standard Model either.
The discrepancy is small enough that it could easily be due to some combination of a statistical fluke in the measurements and underestimated systemic and theoretical calculation errors. But, this discrepancy has been viewed by physicists as one of the most notably in all of physics for the last thirteen years.
As Wikipedia notes:
The QED prediction [for the electron g-2] agrees with the experimentally measured value to more than 10 significant figures, making the magnetic moment of the electron the most accurately verified prediction in the history of physics.
This is an accuracy on the order of one part per billion and the theoretical and experimental results for the electron g-2 are consistent at slightly less than the two sigma level. The five loop precision calculations of the electron g-2 have a theoretical uncertainty roughly three times as great as the current experimental uncertainty in that measurement.
So, physicists naturally expect the muon g-2 to also reflect stunning correspondence between the Standard Model theoretical prediction and experiment.
On the other hand, this discrepancy shouldn't be overstated either. The discrepancy between the theoretically predicted value and the experimentally measured value is still only 2.3 parts per million. As I noted in a 2013 post at this blog:
The discrepancy is simultaneously (1) one of the stronger data points pointing towards potential beyond the Standard Model physics (with the muon magnetic moment approximately 43,000 times more sensitive to GeV particle impacts on the measurement than the electron magnetic moment) and (2) a severe constraint on beyond the Standard Model physics, because the absolute difference and relative differences are so modest that any BSM effect must be very subtle.In particular, my 2013 post made the following observations with regard to the impact of this discrepancy on SUSY theories:
The muon g-2 limitations on supersymmetry are particularly notable because unlike limitations from collider experiments, the muon g-2 limitations tend to cap the mass of the lightest supersymmetric particle, or at least to strongly favor lighter sparticle masses in global fits to experimental data of SUSY parameters. As a paper earlier this year noted:
"There is more than 3 sigma deviation between the experimental and theoretical results of the muon g-2. This suggests that some of the SUSY particles have a mass of order 100 GeV. We study searches for those particles at the LHC with particular attention to the muon g-2. In particular, the recent results on the searches for the non-colored SUSY particles are investigated in the parameter region where the muon g-2 is explained. The analysis is independent of details of the SUSY models."
The LHC, of course, has largely ruled out SUSY particles with masses on the order of 100 GeV. Another fairly thoughtful reconciliation of the muon g-2 limitations with Higgs boson mass and other LHC discovery constraints can be found in a February 28, 2013 paper which in addition to offering its own light sleptons, heavy squark solution also catalogs other approaches that could work.
Regrettably, I have not located any papers examining experimental boundaries on SUSY parameter space that also include limitations from the absence of discovery of proton decay of less than a certain length of time, and the current thresholds of non-discovery of neutrinoless double beta decay. The latter, like muon g-2 limitations, generically tends to disfavor heavy sparticles, although one can design a SUSY model that addresses this reality.
The failure of collider experiments at the LHC to discovery any new particles other than the Higgs boson since 2004, is one of the factors the suggests that the discrepancy is probably due to theoretical and experimental errors, rather than due to new physics. The discrepancy is sufficiently small that if it was due to new physics, that new physics should have been apparent at energies we have already probed by now.Some studies do incorporate the lack of positive detections of GeV scale WIMPS in direct dark matter searches by XENON 100 that have been made more definitive by the recent LUX experiment results. Barring "blind spots" in Tevatron and LHC and LEP experiments at low masses, a sub-TeV mass plain vanilla SUSY dark matter candidate is effectively excluded by current experimental results.
This year, the Fermilab E989 experiment will begin that process of replicating that measurement with greater precision. As the abstract to the paper describing the new experiment explains:
The upcoming Fermilab E989 experiment will measure the muon anomalous magnetic moment aµ. This measurement is motivated by the previous measurement performed in 2001 by the BNL E821 experiment that reported a 3-4 standard deviation discrepancy between the measured value and the Standard Model prediction. The new measurement at Fermilab aims to improve the precision by a factor of four reducing the total uncertainty from 540 parts per billion (BNL E821) to 140 parts per billion (Fermilab E989). This paper gives the status of the experiment.Put another way, in units of 10-11 the target is to reduce the experimental error to ± 16.3
Meanwhile, the body of the paper notes that:
The uncertainties in the theory calculation are expected to improve by a factor of two on the timescale of the E989 experiment. This improvement will be achieved taking advantage of new data to improve both the HVP (BESIII , VEPP2000  and B-factories data) and HLBL (KLOE-2  and BESIII data), the latter gaining from the modeling improvements made possible with the new data. On the lattice QCD side, new ways of computing aµ from first principles and an increase in computing capability will provide the expected gains.Put another way, in units of 10-11 the expectation is to reduce theoretical uncertainty to ± 24.5
Thus, in units of 10-11 a one sigma discrepancy between the theoretical result and the experimental result will be ± 29.4 at Fermilab E989. As a result, the body of the paper notes that:
Given the anticipated improvements in both experimental and theoretical precision, if the central values remain the same there is a potential 7 standard deviation between theory and measurement (5 standard deviation with only experimental improvement).Potential Implications Of New Experimental Results
If there are in fact no new physics and all of the previous discrepancy between the theoretical value of muon g-2 and the experimentally measured value was due to theoretical calculation uncertainty, statistical errors, and systemic errors, then the newly measured experimental value of muon g-2 should fall and improved theoretical calculations may nudge up the theoretically expected result a bit.
If that does happen it will put a nail in the coffin of a huge swath of beyond the Standard Model theories. On the other hand, if the discrepancy grows in statistical significance, which in principle it has the experimental power to do, it will be a strong indicator that there are at least some BSM physics out there to be found that have not yet been observed at the LHC or anywhere else.