Stars as Laboratories for Fundamental Physics - MPP Theory Group

Axions 527
or axion decay constant. Axions must be fundamentally massless so that
all observable effects remain unchanged under a global shift a(x) →
a(x) + a 0 where a 0 is a constant. (A mass term 1 2 m2 aa 2 would spoil this
possibility.) This invariance allows one to absorb Θ in the definition of
the axion field. Including a kinetic term, Eq. (14.1) is replaced by
L Θ → L a = 1 2 (∂ µa) 2 −
α s
8π f a
a G ˜G . (14.3)
It conserves CP because axions were assumed to be pseudoscalar (odd
under CP), similar to neutral pions.
Even though axions were constructed to be massless they acquire an
effective mass by their interaction with gluons. It induces transitions to
qq states and thus to neutral pions (Fig. 14.1) which means physically
that a and π ◦ mix with each other. Axions thereby pick up a small mass
which is approximately given by (Bardeen and Tye 1978; Kandaswamy,
Salomonson, and Schechter 1978)
m a f a ≈ m π f π , (14.4)
where m π = 135 MeV is the pion mass and f π ≈ 93 MeV its decay
constant. This mass term implies that at low energies the axion Lagrangian
contains a potential V (a) which expands to lowest order as
1
2 m2 aa 2 . Because of the invariance of L Θ with respect to Θ → Θ + 2π
the gluon-induced potential V (a) is a function periodic with 2πf a .
Fig. 14.1. Axion mixing with qq states and thus π ◦ . The curly lines represent
gluons, the solid lines quarks.
The ground state of the axion field is at the minimum of its potential
at a = 0, explaining the absence of a neutron electric dipole
moment. If one could produce a static nonvanishing axion field a 0 in
some region of space, neutrons there would exhibit an electric dipole
moment corresponding to Θ = −a 0 /f a .
The Lagrangian Eq. (14.3) is the minimal ingredient for any axion
model: the aG ˜G coupling is their defining feature as opposed to
other pseudoscalar particles. Then axions inevitably acquire an effective
mass at low energies. Thus the concept of a “massless axion” for
some arbitrary pseudoscalar is a contradiction in terms.