Masujima M. Applied Mathematical Methods in Theoretical Physics
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508 10 Calculus of Variations: Applications
and concentrate on the SU(2) subgroup. In this gauge, the spatial gauge fields are
time independent and, under a gauge transformation, transform as
A
i
(
x ) ≡
τ
α
2
A
αi
(
x) −→ (
x )A
i
(
x)
−1
(
x) −
i
g
3
(
x )∇
i
−1
(
x ). (10.16.54b)
The rich structure of QCD vacuum results from the requirement that the gauge
transformation matrices (
x) go to unity at spatial infinity. Such requirement maps
the physical space onto the group space and this S
3
→ S
3
map splits (
x)into
different homotopy classes {
n
(
x)}, characterized by an integer winding number n
which specifies how precisely (
x) behaves as
x →∞,
n
(
x ) −→ exp[2π in], as
x −→ ∞ . (10.16.55)
Because we can construct the n-gauge transformation matrix
n
(
x)bycom-
pounding n times
1
(
x), an n-vacuum state corresponding to A
nαµ
(
x) is not really
gauge invariant. Indeed, the action of the gauge transformation matrix
1
(
x)onan
n-vacuum state gives an (n + 1)-vacuum state,
1
(
x)
n
=
n + 1
. (10.16.56)
We should construct the gauge invariant vacuum state, the θ -vacuum, by super-
posing these n-vacuum states,
θ
=
n
exp[−inθ]
n
with
1
(
x)
θ
= exp[iθ]
θ
. (10.16.57)
With the ν instanton solutions in the theory, we can show the fact that the
winding number ν, which is also called Pontryagin number has the following
gauge invariant expression after a little algebra:
ν =
g
2
3
32π
2
d
4
x ε
µνρσ
F
αµν
F
αρσ
. (10.16.58)
The vacuum-to-vacuum transition amplitude assumes the following form:
+
θ
θ
−
=
ν
exp[iνθ]
n
+
n + ν
n
−
=
ν
D[A
αµ
]exp
i
d
4
x L
QCD
+ iνθ
×δ
ν −
g
2
3
32π
2
d
4
x ε
µνρσ
F
αµν
F
αρσ
. (10.16.59)
10.16 Standard Model 509
We can interpret this new θ -term in terms of the new effective Lagrangian density
for QCD, whereby all of the quarks acquire the masses from the spontaneous
breakdown of the electro-weak gauge symmetry
L
QCD
=−
1
4
F
αµν
F
µν
α
+θ
g
2
3
32π
2
ε
µνρσ
F
αµν
F
αρσ
+
q=u
j
,d
j
,j=1,2,3
q(x)iγ
µ
D
µ
q(x).
(10.16.60)
Perturbation theory is connected with the ν = 0 sectors. The effect of the ν = 0
sectors are nonperturbative. These contribution are naturally related to the chiral
anomaly. For n
f
flavors, the axial U(1)
A
current J
µ
5
has a chiral anomaly,
∂
µ
J
µ
5
= n
f
g
2
3
32π
2
ε
µνρσ
F
αµν
F
αρσ
, J
µ
5
=
n
f
i=1
q
i
γ
µ
γ
5
q
i
. (10.16.61a)
Since the right-hand side of the above equation is a four-divergence,
ε
µνρσ
F
αµν
F
αρσ
= ∂
µ
K
µ
, K
µ
= ε
µνρσ
A
αν
%
F
αρσ
−
g
3
3
f
αβγ
A
βρ
A
γσ
'
,
(10.16.61b)
we can construct a conserved axial U(1)
A
current as
˜
J
µ
5
= J
µ
5
− n
f
g
2
3
32π
2
K
µ
. (10.16.61c)
Thus a chirality change Q
5
is related to the winding number ν as
Q
5
=
d
4
x ∂
µ
J
µ
5
= n
f
g
2
3
32π
2
d
4
x ε
µνρσ
F
αµν
F
αρσ
= n
f
ν.
The conserved current
˜
J
µ
5
is not gauge invariant for transformations with
nontrivial winding number. Since we can show that
1
(
g
2
3
32π
2
d
3
xK
0
)
−1
1
=
g
2
3
32π
2
d
3
xK
0
− 1,
we obtain the following gauge transformation of the associated charge
˜
Q
5
:
1
˜
Q
5
−1
1
=
˜
Q
5
+ n
f
with
˜
Q
5
=
d
3
x
˜
J
0
5
. (10.16.62)
The new interaction term, the θ -term in Eq. (10.16.60), is not the only new
source of CP-violation arising from the complex structure of the QCD vacuum. It
is augmented by an analogous term coming from the electro-weak sector of the
theory. In general, the mass matrix of the quarks which emerges from spontaneous
510 10 Calculus of Variations: Applications
breakdown of the electro-weak gauge symmetry is neither Hermitian nor diagonal.
In general, we have the quark mass term as
L
quark mass
=−q
R
i
M
ij
q
L
j
− q
L
i
(M
†
)
ij
q
R
j
. (10.16.63a)
This mass matrix can be diagonalized by separate unitary transformations of the
chiral quark fields. These transformations constitute a chiral U(1)
A
rotation,
q
R
−→ exp[iα]q
R
, q
L
−→ exp[−iα]q
L
, (10.16.63b)
with
α = (12n
f
)argdetM.
This chiral U(1)
A
rotation changes the vacuum angle. Using Eq. (10.16.62), we can
show that the rotations (10.16.63b) on the θ-vacuum shift the vacuum angle by
2n
f
α, since the action of exp[i2α
˜
Q
5
]on
θ
produces
θ + 2n
f
α
as
1
exp
%
i2α
˜
Q
5
'
θ
=
1
exp
%
i2α
˜
Q
5
'
−1
1
1
θ
= exp
%
i2α(
˜
Q
5
+ n
f
)
'
exp[iθ]
θ
= exp
)
i(θ + 2n
f
α)
*
exp
%
i2α
˜
Q
5
'
θ
⇒ exp
%
i2α
˜
Q
5
'
θ
=
θ + 2n
f
α
.
Hence, in the full theory, the effective CP-violating interaction arising from the
more complex structure of QCD vacuum is the one given in Eq. (10.16.60), where
the θ-parameter is replaced with
θ defined by
θ ≡ θ + 2n
f
α = θ + arg det M. (10.16.64)
The strong CP problem is really why the combination of QCD and electro-weak
parameters which make up
θ should be so small. The θ -term in Eq. (10.16.60)
already gives an immediate contribution to the electric dipole moment of the
neutron and the stringent experimental bound on the electric dipole moment of
the neutron requires that we should have
θ ≤ 10
−9
.Inprinciple,sinceθ is a free
parameter of the theory, any value of
θ is equally likely. We shall explore why this
number is so small, or, even we can make it disappear.
What Peccei and Quinn did is the following. They replaced the strong CP-
violating static θ -parameter by the dynamical CP-conserving interactions of the
axion field a(x) with the assumption that full Lagrangian density of the standard
model is invariant under the additional global chiral U(1)
P.Q.
symmetry, where
δ
P.Q.
a(x) = vα, with the introduction of the SU(2)
weak isospin
doublets of the Higgs
10.16 Standard Model 511
scalar fields, φ
u
and φ
d
,(y =∓g
), replacing the Higgs scalar field φ,withφ
d
also
coupled to the right-handed charged leptons,
L
quark
Yukawa
=−
/
G
u
j
(u
j
, d
j
)
L
φ
u
u
jR
+G
d
j
(u
j
, d
j
)
L
φ
d
d
jR
+h.c.
0
.
Isolating the axion as the common phase field of the doublets, φ
u
and φ
d
,
φ
u
= v
u
1
0
exp
iax
v
, φ
d
= v
d
0
1
exp
ia
x v
, x =
v
d
v
u
, v = (v
2
u
+ v
2
d
)
1/2
,
the U(1)
P.Q.
symmetry which guarantees the invariance of L
quark
Yukawa
is under
(
u
jR
−→ exp
[
−iαx
]
u
jR
,
d
jR
−→ exp
)
−i(αx)
*
d
jR
,
(
φ
u
−→ exp
[
iαx
]
φ
u
,
φ
d
−→ exp
)
i(αx)
*
φ
d
.
We write the effective standard model Lagrangian density as
L
eff
SM
= L
SM
+θ
g
2
3
32π
2
ε
µνρσ
F
αµν
F
αρσ
+
1
2
∂
µ
a
†
∂
µ
a + L
int
∂
µ
a
v
;ψ
+
a
v
ξ
g
2
3
32π
2
ε
µνρσ
F
αµν
F
αρσ
. (10.16.65a)
The U(1)
P.Q.
symmetry is spontaneously broken by the doublets, ψ stands for any
field in the theory and ξ is a model-dependent parameter associated with the global
chiral anomaly of the U(1)
P.Q.
current J
µ
P.Q.
,
∂
µ
J
µ
P.Q.
= ξ
g
2
3
32π
2
ε
µνρσ
F
αµν
F
αρσ
,
with
J
µ
P.Q.
=−v∂
µ
a + xu
jR
γ
µ
u
jR
+
1
x
d
jR
γ
µ
d
jR
.
Under the chiral U(1)
A
rotation (10.16.63b), the CP-violating term, L
CP
∝
θε
µνρσ
F
αµν
F
αρσ
,inL
eff
SM
in the presence of the n
g
generations of the fermions
assumes the following form:
L
CP
=
g
2
3
32π
2
θ + 4n
g
α
ε
µνρσ
F
αµν
F
αρσ
.
Since θ is specified only up to mod(2π), L
eff
SM
remains invariant under the transfor-
mation of α with the discrete Z
4n
g
group specified by α → α + (2π4n
g
)n with n
integer.
512 10 Calculus of Variations: Applications
By the reduction of the U(1)
P.Q.
symmetry group to the discrete Z
4n
g
group at the
center of the spontaneously broken gauge group, we can resolve the domain wall
problem in cosmology in the context of the SO(10) grand unified model.
The presence of the last term in Eq. (10.16.65a) provides an effective potential for
the axion field. Its vacuum expectation value is no longer arbitrary. The minimum
of this potential determines the vacuum expectation value of the axion,
∂V
eff
∂a
=−
ξ
v
g
2
3
32π
2
ε
µνρσ
F
αµν
F
αρσ
a
= 0. (10.16.66a)
The periodicity of the pseudoscalar expectation value
ε
µνρσ
F
αµν
F
αρσ
in the
θ +
ξ
a
v parameter forces the axion vacuum expectation value to settle down at
θ +
ξ
a
v
= 0, mod(2π ), or
a
=−
v
ξ
θ,
v
ξ
mod(2π).
(10.16.66b)
This solves the strong CP-violation problem, since L
eff
SM
,whenexpressedinterms
of the physical axion field
a
phys.
= a −
a
,
no longer contains the CP-violating term, L
CP
∝ θε
µνρσ
F
αµν
F
αρσ
. Also the axion
itself acquires the mass by expanding V
eff
at its minimum,
m
2
a
=
∂
2
V
eff
∂a
2
=−
ξ
v
g
2
3
32π
2
∂
∂a
ε
µνρσ
F
αµν
F
αρσ
a
. (10.16.67)
Therefore, the standard model with the global chiral U(1)
P.Q.
symmetry no longer
has the dangerous CP-violating interaction. Instead it contains additional interac-
tions of a massive axion field both with matter fields and gluon field as
L
eff
SM
= L
SM
+ L
int
∂
µ
a
phys.
v
;ψ
+
1
2
∂
µ
a
†
phys.
∂
µ
a
phys.
−
1
2
m
2
a
a
†
phys.
a
phys.
+
a
phys.
v
ξ
g
2
3
32π
2
ε
µνρσ
F
αµν
F
αρσ
. (10.16.65b)
Replacing the last term of the above axion Lagrangian density with effective
interactions of axion with the light pseudoscalar mesons (π and η), we can estimate
the standard axion mass as
m
st
a
=
f
π
m
π
v
√
m
u
m
d
m
u
+ m
d
25KeV and m
P.Q.
a
= n
g
x +
1
x
m
st
a
. (10.16.68a)
10.16 Standard Model 513
With the inclusion of the coupling of φ
d
with the right-handed charged leptons
l
jR
in L
quark
Yukawa
,wehave
L
Yukawa
=−
/
G
u
j
(u
j
, d
j
)
L
φ
u
u
jR
+G
d
j
(u
j
, d
j
)
L
φ
d
d
jR
+G
l
j
l
jL
φ
d
l
jR
+ h.c.
0
.
The U(1)
P.Q.
symmetry that guarantees the invariance of L
Yukawa
is under
u
jR
−→ exp
[
−iαx
]
u
jR
,
d
jR
−→ exp
)
−i(αx)
*
d
jR
,
l
jR
−→ exp
)
−i(αx)
*
l
jR
,
(
φ
u
−→ exp
[
iαx
]
φ
u
,
φ
d
−→ exp
)
i(αx)
*
φ
d
.
This transformation generates the U(1)
P.Q.
current given by
J
µ
P.Q.
=−v∂
µ
a + xu
jR
γ
µ
u
jR
+
1
x
d
jR
γ
µ
d
jR
+
1
x
l
jR
γ
µ
l
jR
,
which has the global chiral anomaly,
∂
µ
J
µ
P.Q.
= ξ
g
2
3
32π
2
ε
µνρσ
F
αµν
F
αρσ
+ ξ
γ
e
2
16π
2
ε
µνρσ
F
µν
F
ρσ
,
with ξ and ξ
γ
, respectively, given by
ξ = n
g
x +
1
x
and ξ
γ
= n
g
4
3
x +
1
x
.
The last term in Eq. (10.16.65a) is replaced with the following expression:
a
v
ξ
g
2
3
32π
2
ε
µνρσ
F
αµν
F
αρσ
+ ξ
γ
e
2
16π
2
ε
µνρσ
F
µν
F
ρσ
.
To compute the effective interaction of the axion with the light hadrons, π and η,we
employ the effective Lagrangian method. Since π and η are the Nambu–Goldstone
bosons associated with the approximate SU(2)
L
× SU(2)
R
symmetry of QCD,their
interactions are described by an effective chiral Lagrangian density given by
L
chiral
=
1
4
f
2
π
tr∂
µ
†
∂
µ
with = exp
i
f
π
(τ ·π + η)
.
This chiral Lagrangian density must be augmented by the axion Lagrangian density
given by
L
axion
=−
1
2
f
2
π
m
2
π
tr
)
AM + M
†
A
†
†
*
+
1
2
∂
µ
a
†
∂
µ
a,
where
A =
exp[−iaxv]0
0exp[−iaxv]
,
514 10 Calculus of Variations: Applications
and
M =
m
u
(m
u
+ m
d
)0
0 m
d
(m
u
+ m
d
)
.
We are guaranteed the U(1)
P.Q.
invariance of L
axion
since, under the U(1)
P.Q.
transformation, we have
−→
exp[iαx]0
0exp[i(αx)]
.
The anomaly, which breaks the SU(2)
L
× SU(2)
R
×U(1)
P.Q.
symmetry through the
coupling of the gluon, the axion and the η, serves to give an effective mass term to
the field combination which couples to
ε
µνρσ
F
αµν
F
αρσ
as
L
anomaly
=−
1
2
m
2
η +
f
π
v
n
g
− 1
2
x +
1
x
a
2
,
with
m
2
m
2
η
m
2
π
.
The quadratic terms in the axion in
L
axion
+ L
anomaly
,
when diagonalized, immediately yields the axion mixing with π
0
and η as
a a
phys.
− ξ
aπ
π
0
phys.
− ξ
aη
η
phys.
.
The electromagnetic anomaly of the π
0
and η fields,
L
π,η
=
e
2
16π
2
π
0
f
π
+
5
3
η
f
π
ε
µνρσ
F
µν
F
ρσ
,
through the mixing with the axion gives an effective aγγ coupling with the effective
coupling constant,
K
aγγ
= n
g
x +
1
x
m
u
m
u
+ m
d
.
Experimental search for the axion around the predicted mass value revealed no
evidence for its existence.
Undauntedly Dine, Fischler, and Srednicki further proposed the invisible
axion scenario. They repeated the same analysis with the inclusion of the
10.16 Standard Model 515
SU(2)
weak isospin
× U(1)
weak hypercharge
singlet Higgs scalar field besides the
SU(2)
weak isospin
doublets, φ
u
and φ
d
,withφ
d
also coupled to the right-handed
charged leptons, where the Higgs scalar field has the vacuum expectation value
V. The Higgs scalar field is not coupled to the quarks and the leptons but to the
SU(2)
weak isospin
doublets in the following Higgs potential:
V(φ
u
, φ
d
, ) =
q=u,d
λ
q
φ
†
q
φ
q
− v
2
q
2
+ λ
†
− V
2
2
+
aφ
†
u
φ
u
+ bφ
†
d
φ
d
†
+ c
φ
i
u
ε
ij
φ
j
d
†
+ h.c.
+ d
φ
i
u
ε
ij
φ
j
d
†
φ
i
u
ε
ij
φ
j
d
+e
φ
†
u
φ
d
†
φ
†
u
φ
d
.
(10.16.69)
We estimate the mass of the axion in this case by isolating the axion,
φ
u
= v
u
1
0
exp
i
X
1
a
V
, φ
d
= v
d
0
1
exp
i
X
2
a
V
, = V exp
%
i
a
V
'
,
where
X
1
=
2v
2
d
v
2
and X
2
=
2v
2
u
v
2
,
resulting in the estimate
m
inv.
a
=
v
V
2n
g
m
st
a
. (10.16.68b)
These formulas correspond to the previous formula for φ
u
and φ
d
with the
replacements,
x ←→ X
1
and x
−1
←→ X
2
.
The invisible axion has a coupling to electrons given by
L
aee
=−i
X
2
m
e
V
a
eγ
5
e.
By taking V large, we can make m
a
as well as the strength of the couplings with the
other particles small, with the former by the order of magnitude vV.Questionis
how small we should make the mass of the axion.
We shall now consider the cosmological constraint on V. In the early universe
where the temperature T is T > V, the Higgs potential for the Higgs scalar field
is parabolic and the Higgs scalar field behavesasafreefield.Afterthe
phase transition T < V, the Higgs potential for the Higgs scalar field is of the
sombrero shape and the Higgs scalar field settles down at =
.Sincethe
516 10 Calculus of Variations: Applications
Higgs potential for the Higgs scalar field depends on in the form
†
,the
vacuum expectation value
has a phase degree of freedom by an amount α(x),
= V exp[iα(x)]. (10.16.70)
Since the true minimum occurs at α(x) = 0, by power expanding
,wecan
consider that the Higgs scalar field at each space–time point x has the axion
field as
a(x )
∼
=
Vα(x). (10.16.71)
After the phase transition,
T < V,
we still have
T > f
π
or QCD scale
QCD
, (10.16.72)
so that the axion remains massless (m
a
= 0). Namely at the temperature
f
π
or QCD scale
QCD
< T < V, (10.16.73)
the universe is filled with the static axion (p
a
= 0). The field with p
a
= 0andm
a
= 0
has no energy density. After further expansion of the universe, when
T < f
π
or QCD scale
QCD
, (10.16.74)
we start experiencing the instanton effect and the axion field acquires the mass.
The energy density of the axion is given by the mass term of the Lagrangian
density,
ρ
a
= m
2
a
a
†
a
∼
=
m
2
a
V
2
α(x)
2
∼
=
1.50 × 10
42
α(x)
2
KeVcm
3
, α(x) = O(1).
(10.16.75)
From this point on, the amplitude α(x) of the axion decreases as a result of the
expansion of the universe. We have the Boltzmann equation for the expansion of
the universe,
¨α + 3
˙
R
R
˙α + m
a
(t)
2
α = 0,
where R isthescalefactoroftheuniverseand
˙
RR is the Hubble constant H at
time t.
10.16 Standard Model 517
Omitting the detailed analysis, we merely state the energy density ρ
0
a
of the axion
at the present universe as
ρ
0
a
∼
=
10ρ
0
c
V
10
12
GeV
7/6
, (10.16.76)
where ρ
0
c
is the critical density of the present universe. We shall consider the two
cases, ρ>ρ
0
c
and ρ<ρ
0
c
.
If
ρ>ρ
0
c
,
the universe keeps expanding permanently. If
ρ<ρ
0
c
,
the universe returns to compress. At
ρ = ρ
0
c
,
the curvature of the universe is 0 and hence Minkowskian. The value of ρ
0
c
is given
by
ρ
0
c
=
3
8π
H
2
0
G
N
= 11h
2
0
KeVcm
3
.
The superscript and subscript ‘‘0’’ designate the present value. Writing
H
0
= h
0
×
100 Kms
−1
Mpc
−1
,
according to the observation, we have
1
2
< h
0
< 1.
Invoking the inflationary scenario, from the equation for the current axion
density ρ
0
a
, we find the stringent cosmological constraint for V,
10
10
< V < 10
12
GeV, (10.16.77a)
or
10
−5
< m
a
< 10
−3
eV. (10.16.77b)
If we have the axion density sufficient to close the universe,
ρ
0
a
>ρ
0
c
,