Stars as Laboratories for Fundamental Physics - MPP Theory Group

Axions 529
The Yukawa coupling h is chosen to be positive, and Ψ L ≡ 1 2 (1 − γ 5)Ψ
and Ψ R ≡ 1 2 (1 + γ 5)Ψ are the usual left- and right-handed projections.
This Lagrangian is invariant under a chiral phase transformation of
the form
Φ → e iα Φ , Ψ L → e iα/2 Ψ L , Ψ R → e −iα/2 Ψ R , (14.7)
where the left- and right-handed fields pick up opposite phases. This
chiral symmetry is usually referred to as the Peccei-Quinn (PQ) symmetry
U PQ (1).
The potential V (|Φ|) is chosen to be a “Mexican hat” with an absolute
minimum at |Φ| = f PQ / √ 2 where f PQ is some large energy scale.
The ground state is characterized by a nonvanishing vacuum expectation
value ⟨Φ⟩ = (f PQ / √ 2) e iφ where φ is an arbitrary phase. It spontaneously
breaks the PQ symmetry because it is not invariant under a
transformation of the type Eq. (14.7). One may then write
Φ = f PQ + ρ
√
2
e ia/f PQ
(14.8)
in terms of two real fields ρ and a which represent the “radial” and
“angular” excitations.
The potential V provides a large mass for ρ, a field which will be of
no further interest for these low-energy considerations. Neglecting all
terms involving ρ the Lagrangian Eq. (14.6) is
L = ( i
2 Ψ∂ µγ µ Ψ + h.c. ) + 1 2 (∂ µa) 2 − m Ψe iγ 5a/f PQ
Ψ , (14.9)
where m ≡ hf PQ / √ 2. The variation of the fermion fields under a
PQ transformation is given by Eq. (14.7) while a → a + αf PQ . The
invariance of Eq. (14.9) against such shifts is a manifestation of the
U PQ (1) symmetry. It implies that a represents a massless particle, the
Nambu-Goldstone boson of the PQ symmetry.
Expanding the last term in Eq. (14.9) in powers of a/f PQ , the zerothorder
term mΨΨ plays the role of an effective fermion mass. Higher
orders describe the interaction of a with Ψ,
L int = −i m
f PQ
a Ψγ 5 Ψ +
m
2f 2 PQ
a 2 ΨΨ + . . . . (14.10)
The dimensionless Yukawa coupling g a ≡ m/f PQ is proportional to the
fermion mass.
The fermion Ψ is taken to be some exotic heavy quark with the usual
strong interactions, i.e. an SU C (3) triplet. The lowest-order interaction