Preprint[pdf] - HU Berlin

For a particle with spin the magnetic moment is intrinsic and obtained by replacing the the angular
momentum operator � L by the spin operator
�S = �σ
2
, (4)
where σi (i = 1, 2, 3) are the Pauli spin matrices. Thus, generalizing the classical form (1) of the orbital
magnetic moment, one writes
�µm = g Q µ0
�σ
2 , de
�
�σ
= η Q µ0 , (5)
2
where µ0 = e/2m, Q is the electrical charge in units of e, Q = −1 for the leptons (ℓ = e, µ, τ), Q = +1 for
the antileptons and m is the mass. The equations define the gyromagnetic ratio g (g-factor) and its electric
pendant η, respectively, quantities exhibiting important dynamical information about the leptons as we will
see later. The deviation from the Dirac value gℓ/2 = 1, obtained at the classical level, is
aℓ ≡ gℓ − 2
2
the famous anomalous magnetic moment and aµ is the quantity in the focus of this review.
The magnetic interaction term gives rise to the well known Zeeman effect: level splitting seen in atomic
spectra. If spin is involved one calls it anomalous Zeeman effect. The latter obviously is suitable to study
the magnetic moment of the electron by investigating atomic spectra in magnetic fields.
The most important condition for the anomalous magnetic moment to be a useful monitor for testing
a theory is its unambiguous predictability within that theory. The predictability crucially depends on the
following properties of the theory:
– it must be a local relativistic quantum field theory and
– it must be renormalizable.
This implies that g − 2 vanishes at tree level and cannot be an independently adjustable parameter in any
renormalizable QFT. This in turn implies that for a given theory [model] g−2 is an unambiguously calculable
quantity and the predicted value can be confronted with experiments. Its model dependence makes aµ a
good monitor for the detection of new physics contributions. The key point is that g −2 can be both precisely
predicted as well as experimentally measured with very high accuracy. By confronting precise theoretical
predictions with precisely measured experimental data it is possible to subject the theory to very stringent
tests and to find its possible limitations.
The anomalous magnetic moment of a lepton is a dimensionless quantity, a number, which in QED may
be computed order by order as an expansion in the fine structure constant α. Beyond QED, in the SM or
extensions of it, weak and strong coupling contributions are calculable. As a matter of fact, the interaction
of the lepton with photons or other particles induces an effective interaction term
δL AMM
eff
= −δg
2
e �
¯ψL(x) σ
4m
µν Fµν(x) ψR(x) + ¯ ψR(x) σ µν Fµν(x) ψL(x) �
where ψL and ψR are Dirac fields of negative (left–handed L) and positive (right–handed R) chirality and
Fµν = ∂µAν − ∂νAµ is the electromagnetic field strength tensor. It corresponds to a dimension 5 operator
and since a renormalizable theory is constrained to exhibit terms of dimension 4 or less only, such a term
must be absent for any fermion in any renormalizable theory at tree level.
The dipole moments are very interesting quantities for the study of the discrete symmetries. A basic
consequence of any relativistic local QFT is charge conjugation C, the particle–antiparticle duality [5] or
crossing property, which implies in the first place that particles and antiparticles have identical masses and
spins. In fact, charge conjugation turned out not to be a universal symmetry in nature. Since an antiparticle
may be considered as a particle propagating backwards in time, charge conjugation has to be considered
together with time-reversal T (time-reflection), which in a relativistic theory has to go together with parity
P (space-reflection). The CPT theorem says: the product of the three discrete transformations, C, P and
T, taken in any order, is a symmetry of any relativistic local QFT. Actually, in contrast to the individual
4
(6)
(7)