Stars as Laboratories for Fundamental Physics - MPP Theory Group

Axions 531
14.2.3 Pseudoscalar vs. Derivative Interaction
There has been considerable confusion in the literature concerning the
proper structure for the coupling of axions to fermions. The lowestorder
term in the expansion Eq. (14.10) is a pseudoscalar interaction
which is frequently used because of its simplicity. However, there is an
infinite series of terms which sometimes must be taken into account.
For example, the axion scattering on fermions is second order in the
first term of Eq. (14.10), but first order in the second whence both must
be included.
Such complications can be avoided if one redefines the fermion field
in Eq. (14.9) by a local transformation,
ψ L ≡ e −ia/2f PQ
Ψ L , ψ R ≡ e ia/2f PQ
Ψ R . (14.14)
The last term in Eq. (14.10) is then a simple mass term mψψ. The
interaction between ψ and a now arises from the kinetic Ψ term in
Eq. (14.9),
L int = 1
2f PQ
ψγ µ γ 5 ψ ∂ µ a . (14.15)
This interaction is of derivative nature, and it is linear in a with no
higher-order terms. The fermion part has the form of an axial-vector
current, bringing out a useful similarity between neutrino and axion
interactions.
Pions play the role of Nambu-Goldstone bosons of a spontaneously
broken U(2) L−R symmetry of QCD and so their interactions with nucleons
should involve similar higher-order terms. The difference between
derivative and “naive” pseudoscalar couplings should become apparent
in bremsstrahlung processes of the type shown in Fig. 14.3 where two
Nambu-Goldstone bosons are attached to one fermion line. A useful
template is provided by p + p → p + p + π ◦ where existing data, indeed,
favor the derivative case (Choi, K. Kang, and Kim 1989; Turner,
H.-S. Kang, and Steigman 1989).
It should be noted in this context that the pion-nucleon interaction
is described by the Lagrangian (Carena and Peccei 1989),
L int = g πN Nγ µ γ 5 τ N · ∂ µ π + f 2 πNNγ µ τ N · ∂ µ π×π , (14.16)
where g πN ≡ f/m π ≈ 1/m π and f πN ≡ 1/2f π are the relevant coupling
constants, τ is a vector of Pauli isospin matrices, π is the isovector of
the neutral and charged pion fields, and N is the isodoublet of neutron
and proton. Thus there appears an extra dimension-6 term compared