The magnetic moment was first defined for macroscopic classical magnets and is the first dipole term in an expansion of the magnetic field, whether it is a permanent magnet or moving charges creating a magnetic field.
More precisely, the term magnetic moment normally refers to a system's magnetic dipole moment, which produces the first term in the multipole expansion of a general magnetic field.
In classical electrodynamics it is defined as :
where one sees that a circulating current( or its equivalent) plays a role in the definition.
Is there a formula or a general explanation that can account for the μ of known particles: electrons, protons, neutrons and neutrinos?
At the quantum mechanical level where electrons etc are defined, the magnetic moment will be an operator whose expectation value will give the size of the magnetic moment. In this reference this expectation value is given by :
the expectation value of the z component of spin- magnetic moment of a free Dirac electron in positive energy state .
We see that for the electron and any dirac spinor if the charge is zero the spin dipole moment will be zero.
Classically for a particle of charge e moving with velocity v the magnetic moment
We can see that the mass in the denominator does not allow reasonable magnetic moments for zero mass particles. The neutrino has a small mass, but it is excluded of having a magnetic moment to first order by the zero charge. Higher order corrections will introduce other terms of charged exchanged particles which may give rise to a magnetic moment but will be very small, due to the smallenes of the weak interaction constant. The discrepancy comes from how many terms are added in the perturbative expansion for calculating the magnetic moment.
Protons as composite particles will have a magnetic moment appropriate to the spins and angular momenta of its constituents.
From the nonrelativistic, quantum mechanical wavefunction for baryons composed of three quarks, a straightforward calculation gives fairly accurate estimates for the magnetic moments of protons, neutrons, and other baryons.
Higher order corrections will be present,but to first order the data agree with the calculations.