Full heuristic interpretation of quantum field theory.
The 2.Pi factor in the Schwinger 1st coupling correction of the magnetic moment of the electron, in 1 + 1/(2.Pi.137) Bohr magnetons is almost certainly due to the spin effect shielding.
Physically, the core of the first electron has a magnetic moment of 1 Bohr magneton because the polarised vacuum around the electron core only reduces the radial electric field and transverse magnetic field, not the polar magnetic field vector which is of course parallel to the radial electric field at the poles.
The electric field of the core is reduced by a factor of 137 by the polarised virtual charge surrounding it in the vacuum. The real core couples up with a particle (virtual positron?) in the vacuum which adds to the magnetic moment by aligning with the magnetic axis of the electron core. This is the reason for the 137 factor in Schwinger correction for the first coupling effect, the 1/(2.Pi.137) = 0.00116 term added to Dirac's 1 Bohr magneton.
The 2.Pi is an additional shielding factor, and is due to geometry. The 2.Pi factor is heuristically explainable in terms of the geometry which stems from the aligned real electron core and the virtual particle which is aligned with it to add to its magnetic moment. The vacuum is full of virtual particles, but because they are normally orientated randomly, their magnetic fields cancel each other out as seen on a macroscopic scale.
Now, the virtual electron which is outside the polarised shield surrounding the real electron core, and which adds 1/(2.Pi.137) Bohr magnetons to the magnetic moment of the latter, itself has the same effect on another vacuum particle! So there is another correction, which is even smaller, by another 137 factor, and another geometric factor... and so on.
This is how you heuristically explain the extra couplings required for more decimals than 1.00116 Bohr magnetons. Also, you need to take account of different vacuum particles, as occurs with the magnetic moment of the muon, which is slightly different to that from the electron.