Stars as Laboratories for Fundamental Physics - MPP Theory Group

166 Chapter 5
the Nambu-Goldstone bosons of an approximate SU(2) symmetry of
the system of nucleons and pions in the sigma model (e.g. Itzykson
and Zuber 1983). Another example is the hypothetical axion which is
the Nambu-Goldstone boson of the Peccei-Quinn chiral U(1) symmetry
that would solve the CP problem of strong interactions (Chapter 14).
Both pions and axions carry a small mass because the underlying symmetry
is not exact at low energies; they are sometimes called “pseudo
Nambu-Goldstone bosons.” An example for a true Nambu-Goldstone
boson is the hypothetical majoron which arises from the spontaneous
breakdown of a symmetry by a Higgs field which would give the neutrinos
Majorana masses (Sect. 15.7).
In these examples the Nambu-Goldstone bosons are pseudoscalars
because the underlying symmetry is chiral. Unless the CP symmetry
is violated, their possible coupling to photons must be of the form
L int = − 1 4 g aγ F µν ˜F µν a = g aγ E · B a, (5.1)
where a is the pseudoscalar field; the axion will serve as a generic
example. Further, g aγ is a constant with the dimension (energy) −1 , F
is the electromagnetic field strength tensor, ˜F its dual, and E and B are
the electric and magnetic fields, respectively. Because E is a polar and
B an axial vector, E · B is a pseudoscalar under a CP transformation
and so L int remains invariant.
New scalar particles ϕ almost inevitably would also couple to photons
with a CP conserving structure L int ∝ 1 4 F µνF µν ϕ = 1 2 (E2 + B 2 )ϕ.
Therefore, everything that will be said about axions applies mutatis
mutandis to scalar particles as well.
Gravitons would also have a two-photon vertex. Much of what is
said about axions also applies to them, except that their weak couplings
render most of the arguments irrelevant—an exception will be
mentioned in Sect. 5.5.5.
Returning to the case of pseudoscalar Nambu-Goldstone bosons,
by the very construction of the underlying models they interact with
certain fermions ψ by a pseudoscalar coupling of the form
L int = ig ψγ 5 ψ a, (5.2)
where g = m/f is a Yukawa coupling given in terms of the fermion
mass m and an energy scale f related to the vacuum expectation value
of the Higgs field which breaks the underlying symmetry—for axions
the Peccei-Quinn scale f a (Chapter 14), for pions the pion decay constant
f π = 93 MeV. In the latter example the relevant fermions are