Stars as Laboratories for Fundamental Physics - MPP Theory Group

Axions 537
Various axion models differ in their assignment of PQ charges. However,
in all models N = ∑ quarks X j is a nonzero integer. The assignment
at high energies is not maintained in the low-energy sector because
the spontaneous breakdown of the weak SU L (2)×U Y (1) symmetry at
f weak ≈ 250 GeV mixes the axion with the would-be Nambu-Goldstone
boson which becomes the longitudinal component of the Z ◦ gauge boson.
Hence the PQ charges must be shifted such that the physical axion
does not mix with the Z ◦ ; these shifted values are denoted as X ′ j. Also,
below the QCD scale Λ QCD ≈ 200 MeV free quarks do not exist, so one
needs to consider the effective coupling to nucleons which arises from
the direct axion coupling to quarks and from the mixing with π ◦ and η,
leading to PQ charges X ′ p and X ′ n for protons and neutrons. The above
effective PQ charges are obtained by C j ≡ X ′ j/N in order to absorb N
in their definition just as it was absorbed in f a = f PQ /N.
In the KSVZ model C e = 0 at tree level (“hadronic axions”) although
there are small radiatively induced couplings (Srednicki 1985).
In the DFSZ model
C e = cos 2 β/N f , (14.28)
where N f is the number of families, probably 3.
The nucleon interactions in general axion models were investigated
by Kaplan (1985) and Srednicki (1985). They were revisited by Mayle
et al. (1988, 1989),
C p = (C u − η)∆u + (C d − ηz)∆d + (C s − ηw)∆s ,
C n = (C u − η)∆d + (C d − ηz)∆u + (C s − ηw)∆s , (14.29)
where η ≡ (1 + z + w) −1 with z and w were given in Eq. (14.21).
For a given quark flavor, q = u, d, or s, the interaction strength
with protons depends on the proton spin content carried by this particular
quark flavor, S µ ∆q ≡ ⟨p|qγ µ γ 5 q|p⟩ where S µ is the proton spin.
Similar expressions pertain to the coupling with neutrons; the two sets
of expressions are related by isospin invariance. Neutron and hyperon
β-decays as well as polarized lepton scattering experiments on nucleons
yield a consistent set of ∆q’s (Ellis and Karliner 1995)
∆u = +0.85, ∆d = −0.41, ∆s = −0.08 (14.30)
with an approximate uncertainty of ±0.03 each.
In the DFSZ model, C s = C d = C e , C u + C d = 1/N f , and C u − C d =
− cos 2 β/N f , leading to C u = sin 2 β/N f and C d = C s = C e = cos 2 β/N f .