Your summary and interpretation of the muon's anomalous magnetic moment, the g-2 experiments, and the significance of the results is quite thorough and accurate. The ongoing experiments, particularly at Fermilab, have generated a lot of interest in the physics community, primarily because of the implications of the results. If the discrepancy between the calculated and measured values continues to grow or is confirmed, it may point towards new physics beyond the current understanding provided by the Standard Model of particle physics.
To reiterate a few key points you made:
1. **Muon vs Electron**: Due to the muon's larger mass compared to the electron, it is much more sensitive to possible new particles or interactions. This makes the muon a useful probe for new physics, even though we still rely on the electron for precision QED tests.
2. **Virtual Particles**: Quantum field theory predicts that particles can interact with each other by exchanging virtual particles. These interactions cause small corrections to the g-factor. For the electron, the leading correction (from photon exchange) gives the famous result \( g = 2 \), but the interactions involving higher numbers of virtual particles give increasingly smaller corrections.
3. **Beyond the Standard Model**: If there's an unknown particle or force not included in the Standard Model, it could show up as a discrepancy between the predicted and observed anomalous magnetic moments of the muon.
4. **The Significance of Five Sigma**: In particle physics, a five sigma significance is a general threshold for claiming a discovery. If Fermilab achieves this with the g-2 experiment, it'll be a strong indication of new physics, but as you correctly pointed out, independent verification would still be essential.
5. **Limitations of the Method**: Indeed, there's a practical limit to how precisely we can compute the g-factor using current techniques and technology. The increasing complexity of higher-order corrections becomes computationally challenging, even for powerful supercomputers.
However, it's essential to remember that the anomalous magnetic moment of the muon is just one of many avenues through which physicists are searching for evidence of new physics beyond the Standard Model. If there is a genuine discrepancy between theory and experiment, combining the results from different experimental approaches will be crucial in determining the nature of the new physics.