g-factor (physics)

The spin magnetic moment of a charged, spin-1/2 particle that does not possess any internal structure (a Dirac particle) is given by[1]

$${\displaystyle {\boldsymbol {\mu }}=g{e \over 2m}\mathbf {S} ,}$$

where μ is the spin magnetic moment of the particle, g is the g-factor of the particle, e is the elementary charge, m is the mass of the particle, and S is the spin angular momentum of the particle (with magnitude ħ/2 for Dirac particles).

Protons, neutrons, nuclei and other composite baryonic particles have magnetic moments arising from their spin (both the spin and magnetic moment may be zero, in which case the g-factor is undefined). Conventionally, the associated g-factors are defined using the nuclear magneton, and thus implicitly using the proton's mass rather than the particle's mass as for a Dirac particle. The formula used under this convention is

$${\displaystyle {\boldsymbol {\mu }}=g{\mu _{\text{N}} \over \hbar }{\mathbf {I} }=g{e \over 2m_{\text{p}}}\mathbf {I} ,}$$

where μ is the magnetic moment of the nucleon or nucleus resulting from its spin, g is the effective g-factor, I is its spin angular momentum, μ_N is the nuclear magneton, e is the elementary charge and m_p is the proton rest mass.

There are three magnetic moments associated with an electron: one from its spin angular momentum, one from its orbital angular momentum, and one from its total angular momentum (the quantum-mechanical sum of those two components). Corresponding to these three moments are three different g-factors:

The most known of these is the electron spin g-factor (more often called simply the electron g-factor), g_e, defined by

$${\displaystyle {\boldsymbol {\mu }}_{\text{s}}=g_{\text{e}}{\mu _{\text{B}} \over \hbar }\mathbf {S} }$$

where μ_s is the magnetic moment resulting from the spin of an electron, S is its spin angular momentum, and ${\displaystyle \mu _{\text{B}}=e\hbar /2m_{\text{e}}}$ is the Bohr magneton. In atomic physics, the electron spin g-factor is often defined as the absolute value or negative of g_e:

$${\displaystyle g_{\text{s}}=|g_{\text{e}}|=-g_{\text{e}}.}$$

The z-component of the magnetic moment then becomes

$${\displaystyle \mu _{\text{z}}=-g_{\text{s}}\mu _{\text{B}}m_{\text{s}}}$$

The value g_s is roughly equal to 2.002319, and is known to extraordinary precision - one part in 10¹³.[2] The reason it is not precisely two is explained by quantum electrodynamics calculation of the anomalous magnetic dipole moment.[3] The spin g-factor is related to spin frequency for a free electron in a magnetic field of a cyclotron:

$${\displaystyle \nu _{\text{s}}={\frac {g}{2}}\nu _{\text{c}}}$$

Secondly, the electron orbital g-factor, g_L, is defined by

$${\displaystyle {\boldsymbol {\mu }}_{L}=-g_{L}{\mu _{\mathrm {B} } \over \hbar }\mathbf {L} ,}$$

where μ_L is the magnetic moment resulting from the orbital angular momentum of an electron, L is its orbital angular momentum, and μ_B is the Bohr magneton. For an infinite-mass nucleus, the value of g_L is exactly equal to one, by a quantum-mechanical argument analogous to the derivation of the classical magnetogyric ratio. For an electron in an orbital with a magnetic quantum number m_l, the z-component of the orbital angular momentum is

$${\displaystyle \mu _{\text{z}}=-g_{L}\mu _{\text{B}}m_{\text{l}}}$$

which, since g_L = 1, is −μ_Bm_l

For a finite-mass nucleus, there is an effective g value[4]

$${\displaystyle g_{L}=1-{\frac {1}{M}}}$$

where M is the ratio of the nuclear mass to the electron mass.

Thirdly, the Landé g-factor, g_J, is defined by

$${\displaystyle |{\boldsymbol {\mu _{\text{J}}}}|=g_{J}{\mu _{\text{B}} \over \hbar }|\mathbf {J} |}$$

where μ_J is the total magnetic moment resulting from both spin and orbital angular momentum of an electron, J = L + S is its total angular momentum, and μ_B is the Bohr magneton. The value of g_J is related to g_L and g_s by a quantum-mechanical argument; see the article Landé g-factor. μ_J and J vectors are not collinear, so only their magnitudes can be compared.

If supersymmetry is realized in nature, there will be corrections to g−2 of the muon due to loop diagrams involving the new particles. Amongst the leading corrections are those depicted here: a neutralino and a smuon loop, and a chargino and a muon sneutrino loop. This represents an example of "beyond the Standard Model" physics that might contribute to g–2.

The muon, like the electron, has a g-factor associated with its spin, given by the equation

$${\displaystyle {\boldsymbol {\mu }}=g{e \over 2m_{\mu }}\mathbf {S} ,}$$

where μ is the magnetic moment resulting from the muon's spin, S is the spin angular momentum, and m_μ is the muon mass.

That the muon g-factor is not quite the same as the electron g-factor is mostly explained by quantum electrodynamics and its calculation of the anomalous magnetic dipole moment. Almost all of the small difference between the two values (99.96% of it) is due to a well-understood lack of heavy-particle diagrams contributing to the probability for emission of a photon representing the magnetic dipole field, which are present for muons, but not electrons, in QED theory. These are entirely a result of the mass difference between the particles.

However, not all of the difference between the g-factors for electrons and muons is exactly explained by the Standard Model. The muon g-factor can, in theory, be affected by physics beyond the Standard Model, so it has been measured very precisely, in particular at the Brookhaven National Laboratory. In the E821 collaboration final report in November 2006, the experimental measured value is 2.0023318416(13), compared to the theoretical prediction of 2.00233183620(86).[5] This is a difference of 3.4 standard deviations, suggesting that beyond-the-Standard-Model physics may be having an effect. The Brookhaven muon storage ring was transported to Fermilab where the Muon g–2 experiment used it to make more precise measurements of muon g-factor. On April 7, 2021, the Fermilab Muon g−2 collaboration presented and published a new measurement of the muon magnetic anomaly.[6] When the Brookhaven and Fermilab measurements are combined, the new world average differs from the theory prediction by 4.2 standard deviations.