Stars as Laboratories for Fundamental Physics - MPP Theory Group

Two-Photon Coupling of Low-Mass Bosons 167
the nucleons, for axions they are the quarks, possibly the charged leptons,
and perhaps some exotic heavy quark state not contained in the
standard model. For majorons, such couplings exist to neutrinos, and
possibly to other fermions.
An interaction of the form Eq. (5.2) with charged fermions automatically
leads to an electromagnetic coupling of the form Eq. (5.1)
because of the triangle amplitude shown in Fig. 5.1. For one fermion
of charge e and mass m an explicit evaluation leads to the relationship
(e.g. Itzykson and Zuber 1983)
g aγ = α π
g
m = α πf , (5.3)
where m was taken to be much larger than the axion and photon energies.
Remarkably, because g = m/f this coupling does not depend
on the fermion mass, but only on the scale f of symmetry breaking.
In general, one must sum over all possible fermions, taking account
of the appropriate charges which are fractional for quarks, and also of
the proper pseudoscalar coupling to the individual fermions which may
vary from m/f by model-dependent factors of order unity.
Fig. 5.1. Triangle loop for the coupling of a pseudoscalar a (axion) to two
photons.
For massive pseudoscalars the two-photon coupling allows for a decay
a → 2γ with a width
Γ a→2γ = g 2 aγm 3 a/64π. (5.4)
For pions in the sigma model, the only charged fermion is the proton.
Then g π ◦ γ = α/πf π and Γ π ◦ →2γ = α 2 m 3 π/64π 3 f 2 π = 7.6 eV, in close
agreement with the experimental value. (For subtleties of interpretation
of this result in the context of current algebra see the standard field
theory literature, e.g. Itzykson and Zuber 1983).