Stars as Laboratories for Fundamental Physics - MPP Theory Group

530 Chapter 14
of a with gluons is then given by the triangle graph of Fig. 14.2. With
the first term of Eq. (14.10) it yields an effective a-gluon interaction of
L aG = − g a
m
α s
8π a G ˜G , (14.11)
where α s ≡ g 2 s /4π. All external momenta were taken to be small relative
to the mass m of the loop fermion.
Fig. 14.2. Triangle loop diagram for the interaction of axions with gluons
(strong coupling constant g s , axion-fermion Yukawa coupling g a ). An analogous
graph pertains to the coupling of axions with photons if the fermion
carries an electric charge which replaces g s .
In more general models, several conventional or exotic quark fields
Ψ j may participate in this scheme. The transformation of each field
under a U PQ (1) transformation is characterized by its PQ charge X j ,
Ψ j L → e iX jα/2 Ψ j L . (14.12)
The total aG ˜G interaction is obtained as a sum over Eq. (14.11) for all
Ψ j . Because g aj = X j m j /f PQ the fermion masses drop out. With
N ≡ ∑ j X j and f a ≡ f PQ /N (14.13)
one has then found the required coupling Eq. (14.3) which allows one
to interpret a as the axion field.
The potential V (a) is periodic with 2πf a = 2πf PQ /N. The interpretation
of a as the phase of Φ, on the other hand, implies a periodicity
with 2πf PQ so that N must be a nonzero integer. This requirement
restricts the possible assignment of PQ charges to the quark fields. It
also implies that there remain N different equivalent ground states for
the axion field, each of which satisfies Θ = 0 and thus solves the CP
problem.