Stars as Laboratories for Fundamental Physics - MPP Theory Group

Axions 525
Table 14.1. Particle magnetic and electric dipole moments.
Fermion Magnetic Moment c Electric Moment d
[10 −26 e cm]
Proton a 2.792, 847, 3(86) µ N 4000 ± 6000
Neutron a −1.913, 042, (75) µ N < 11 e
Electron a 1.001, 159, 652, 1(93) µ B −0.3 ± 0.8
Neutrino b ∼ < 3×10 −12 µ B ∼ < 6000
a Particle Data Group (1994).
b All flavors with m ν ∼
< 5 keV (Sect. 6.5.6).
c Bohr magneton µ B = e/2m e ; nuclear magneton µ N = e/2m p .
d 1 e cm = 5.18×10 10 µ B = 0.951×10 14 µ N (Appendix A).
e 95% CL.
This symmetry is not respected, however, by the generally complex
Yukawa couplings to the Higgs field which are thought to induce the
fermion masses (Sects. 7.1 and 7.2). The resulting complex quark mass
matrix M q can be made real and diagonal by suitable transformations
of the quark fields. This involves a global chiral phase transformation
(angle Θ = arg det M q ), leading to a term in the QCD Lagrangian
L Θ = Θ α s
8π G ˜G . (14.1)
Here, α s is the fine-structure constant of strong interactions and G ˜G ≡
G µν
b
˜G bµν where G µν
b is the color field strength tensor, ˜G bµν = 1ε 2 µνρσG ρσ
b
its dual, and the implied summation over b refers to the color degrees
of freedom. Of course, det M q and thus Θ would vanish if one of the
quarks were exactly massless, but this does not seem to be the case.
Under the combined action of charge conjugation (C) and a parity
transformation (P) the Lagrangian Eq. (14.1) changes sign, 89 violating
the CP invariance of QCD. It leads to a neutron electric dipole moment
|d n | ≈ |Θ| (0.04 − 2.0)×10 −15 e cm (Baluni 1979; Crewther et al. 1979;
see also Cheng 1988). This is |d n | ≈ |Θ| (0.004 − 0.2) µ N in units of
nuclear magnetons. Hence, for |Θ| of order unity one expects a neutron
electric dipole moment almost as large as its magnetic one. The
89 The structure of G ˜G is E color · B color , i.e. the scalar product of a polar with an
axial vector and so it is CP-odd.