Physical Foundations of Cosmology

4.6 Beyond the Standard Model 209

different winding numbersn:

|θ〉=

∑

e−inθ|n〉, (4.215)

whereθis an arbitrary parameter which must be determined experimentally. As a
result, the effective Lagrangian possesses an additionalnonperturbativeterm

θ

αs

8 π

tr

(

FF ̃

)

, (4.216)

whereFandF ̃are the gluon field strength and its dual respectively. This term,
being a total derivative, does not affect the equations of motion and conservesC.It
violatesCP,PandT, however, and produces a very large neutron dipole moment
which contradicts experimental bounds unlessθ< 10 −^10. One either has to accept
the smallness ofθas fact or try to find a natural explanation for why this parameter
is so small by introducing a new symmetry. The elegant known solution of the
strongCPproblem, suggested by Peccei and Quinn, involves an additionalglobal
chiral U( 1 )PQsymmetry imposed on the Standard Model. This symmetry, broken
at a scalef,essentially serves to replace theθparameter by a dynamical field−
the axion field. In many axion models, anewcomplex scalar fieldφ=χexp

(

iθ ̄

)

is used to generate aU( 1 )PQ-invariant mass term for some colored fermions, via
Yukawa coupling. After symmetry breaking, the fieldχacquires the expectation
value f. In the case of local symmetry the fieldθ ̄would be “eaten” by the gauge
field, but when the symmetry is global it becomes a massless degree of freedom
and this is called the axiona. To be precise,a=fθ. ̄ At the quantum level the
chiral U( 1 )PQsymmetry suffers from the chiral anomaly and as a result there is an
effective interaction of the axion field with gluons:

a

f

αs

8 π

tr

(

FF ̃

)

, (4.217)

which shares the same structure as (4.216). The terms of this type generate an
effective potential forθ+a/fwith the minimum ata=−fθand the overallCP
violating term vanishes in the minimum of the potential. What is most important
for us is that in the vicinity of the potential minimum the axion acquires a small
mass of order

ma=

(mumd)^1 /^2

mu+md

mπfπ

f

∼

6 × 106

fGeV

eV, (4.218)

wheremuandmdare the masses of the light quarks,mπ130 MeV is the pion mass
andfπ93 MeV is the pion decay constant. The axion mass arises from quantum
chromodynamics instanton effects and these are altered at finite temperatures. In