Axions, Free-Fermions & The Strong CP Problem
Johar M. Ashfaque
University of Liverpool
Johar M. Ashfaque String Phenomenology
Strong CP Problem
The SU(3) gauge theory allows a CP-violating interaction of theform
LCP =θαs
32π2GµνG
µν
needs to be added to the QCD Lagrangian which contributes tothe neutron electric dipole moment (nEDM).
Note.θ ≡ θQCD + arg detYuYd
is the strong CP phase with Yu,d being the up and down yukawasrespectively.
Johar M. Ashfaque String Phenomenology
Strong CP Problem
The current bound on the neutron EDM is
|dN | < 2.9× 10−26e cm
so that|θ| < 10−10
which is a strikingly small value for a dimensionless naturalconstant given that the CP violating phase in the CKM matrix isof order one.
This smallness of θ despite the large amount of CP violation in theweak sector is known as the strong CP problem.
Johar M. Ashfaque String Phenomenology
Solutions
Axions are perhaps the most promising solution towards solving theStrong CP problem.
Other solutions are ruled out or are disfavoured by phenomenology.
Johar M. Ashfaque String Phenomenology
What Are Axions?
Axions are the quanta of the axion field, a(x), which is the phaseof the PQ complex scalar field after the spontaneously breaking of
the PQ symmetry gives it an absolute value fa.
Simply put axions are pseudo-Nambu-Goldstone bosonsrelated to the spontaneous breaking of the anomalous U(1)
global symmetry.
Johar M. Ashfaque String Phenomenology
Axions in String Theory
It is well-known that axions arise in string compactifications.There are two axions in superstring models. One is the
model-independent axion (MI axion) and the other is thePeccei-Quinn type one (PQ axion) namely the global anomalous
U(1).
Johar M. Ashfaque String Phenomenology
Cosmological Bound & The Axion Decay Constant
It is a well known fact that large fa especially
fa > 1012 GeV
means axion energy density
ρa > ρcritical
and therefore is unacceptable.
The Axion Decay Constant Window is
109−10 GeV < fa < 1012 GeV.
fa smaller than 109−10 GeV will be couple weakly and fa greaterthan 1012 GeV will couple too strongly.
Johar M. Ashfaque String Phenomenology
The Model-Independent Axion: References
Anomaly Cancellations In Supersymmetric D = 10 Gauge TheoryAnd Superstring Theory,
Phys. Lett. 149B (1984),M. B. Green, J. H. Schwarz
Harmful Axions In Superstring Models,Phys. Lett. 154B (1985),
K. Choi, J. E. Kim
Johar M. Ashfaque String Phenomenology
The Model-Independent Axion
In the Neveu-Schwarz sector, in four dimensions, there is always anantisymmetric tensor
Bµν , µ, ν = 0, ...3,
which has one physical degree of freedom and is crucial foranomaly cancellation, the gauge-invariant field strength for whichis given by
H = dB + ω3L − ω3YM , dH =1
16π2(TrR ∧ R − Tr F ∧ F )
giving rise to a single scalar field in four dimensions with axion-likecouplings.
The Model-Independent Axion Is Present In All SuperstringModels Due To The Presence Of The Coupling.
Johar M. Ashfaque String Phenomenology
The 10-Dimensional Action
S10 = d10x e
[−(
1
2κ2
)R −
(1
κ2
)ϕ−2∂Aϕ∂
Aϕ
= −(
1
4g2
)ϕ−1F a
ABFABa −
(3κ2
2g4
)ϕ−2HABCH
ABC
]where dim g = −3 and dimκ = −4. The ratio
κ
g2
is proportional to the string tension.
Johar M. Ashfaque String Phenomenology
The MI Axion: The Axion Decay Constant Problem
M′a = 8π2Ma ⇒ Ma =
M′a
8π2
and
M′a =
MPl
12√π' 5.64× 1017
The first relation givesg2φ
k2=
M2Pl
8π
which, upon substitution into the second relation, yields
M′2a =
M2Pl
144π⇒ Ma =
1
8π2· MPl
12√π⇒ Ma ' 7.15× 1015
clearly violating the cosmological energy density upper bound on fa.
Johar M. Ashfaque String Phenomenology
The Global Anomalous U(1)
The global anomalous U(1) is present in all the models in thefree-fermionic formulation which arises as the Goldstone boson of
the global anomalous U(1) in the theory. It is a formal linearcombination of the U(1)s in the gauge symmetry of the theory.
Johar M. Ashfaque String Phenomenology
Digression: Aim
Our aim is to construct realistic models with
4 flat space-time dimensions,
chiral matter fields and
N = 1 SUSY.
Johar M. Ashfaque String Phenomenology
Digression: Aim
Our aim is to construct realistic models with
4 flat space-time dimensions,
chiral matter fields and
N = 1 SUSY.
Johar M. Ashfaque String Phenomenology
Digression: Aim
Our aim is to construct realistic models with
4 flat space-time dimensions,
chiral matter fields and
N = 1 SUSY.
Johar M. Ashfaque String Phenomenology
Digression: Aim
Our aim is to construct realistic models with
4 flat space-time dimensions,
chiral matter fields and
N = 1 SUSY.
Johar M. Ashfaque String Phenomenology
Diggression: Free-Fermionic Setting
Supercharge is
TF = ψµ∂Xµ + fIJKXIY JWK
where fIJK are the structure constants of the 18-dimensionalsemi-simple Lie group
SU(2)10−(d=4) = SU(2)6
N = 4 SUSY is non-linearly realized in the free-fermionic setting.
Johar M. Ashfaque String Phenomenology
Diggression: Free-Fermionic Setting
Properties
Conformally invariant
Decoupling left and right moving modes
D = 4 theory
Result
CL = −26 + 11 + D + D2 +
NfL2 = 0
=⇒ 18 left-moving real fermions
CR = −26 + D +NfL
2 = 0
=⇒ 44 right-moving real fermions
Johar M. Ashfaque String Phenomenology
Diggression: Free-Fermionic Setting
Partition function is used to include all physical states
Z =∑α,β
C
(α
β
)Z [α, β]
Taking the one-loop partition function transforms theworldsheet into a torus.
The left-moving and right-moving fermions acquire aninteger phase when parallel transported along the twonon-contractible loops of the torus.
Johar M. Ashfaque String Phenomenology
Diggression: Free-Fermionic Setting
α =ψ1,2, χi , y i , ωi |y i , ωi , ψ
1,..,5, η1,2,3, φ
1,..,8
Where i = 1, ..., 6
Left-movers
XµL , µ = 1, 2 2 transverse coordinates
ψµL , µ = 1, 2 The fermionic partners
Ωj , j = 1, .., 18 18 internal real fermions
Right-movers
XµR , µ = 1, 2 2 transverse coordinates
Ωj, j = 1, .., 44 44 internal real fermions
Johar M. Ashfaque String Phenomenology
Diggression: Free-Fermionic Setting
ABK RulesA finite additive group of vectors of boundary conditions
Ξ ' Z1 ⊕ ...⊕ Zk
generated by a basis b1, ..., bk∑i mibi = 0
Nijbibj = mod 4Nib
2i = mod 8 if Ni is even
1 ∈ ΞEven number of fermions
One-Loop Phases
C
(bibj
)= ±1 or ±i
GSO Projection
e iπbi ·Fα |s〉α = δαC
(α
bi
)∗
|s〉αVirasoro Level-Matching Constraint
M2L = − 1
2 +α2
L
8 +∑
vL = −1 +α2
R
8 +∑
vR = M2R
Johar M. Ashfaque String Phenomenology
Diggression: The NAHE Set
The NAHE set is the set of basis vectors
B = 1,S,b1,b2,b3
where
1 = ψ1,2µ , χ1,...,6, y1,...,6, ω1,...,6|y1,...,6, ω1,...,6, ψ1,...,5, η1,2,3, φ1,...,8,
S = ψ1,2µ , χ1,...,6,
b1 = ψ1,2µ , χ1,2, y3,...,6|y3,...,6, ψ1,...,5, η1,
b2 = ψ1,2µ , χ3,4, y1,2, ω5,6|y1,2, ω5,6, ψ1,...,5, η2,
b3 = ψ1,2µ , χ5,6, ω1,...,4|ω1,...,4, ψ1,...,5, η3.
Johar M. Ashfaque String Phenomenology
Diggression: From SO(44) To SO(10)
The vector b1 breaks the SO(44) gauge group to SO(28)×SO(16)and the N = 4 space–time supersymmetry to N = 2. The vectorb2 then reduces the group to SO(10)× SO(22)× SO(6)2 gaugegroup and the N = 2 supersymmetry is further reduced to N = 1 .Furthermore, the basis vector b3 gives the decompositionSO(10)× SO(16)1 × SO(6)3 where we fix the GGSO projectioncoefficient in order to preserve the N = 1 space–timesupersymmetry. Moreover, the sector, ξ, given by the linearcombination
ξ = 1 + b1 + b2 + b3 ≡ φ1,...,8
together with the NS–sector form the adjoint representation of E8.
SO(10)× E8 × SO(6)3
Johar M. Ashfaque String Phenomenology
Breaking The Observable SO(10)
LN: The Flipped SU(5) Model
SO(10)→ SU(5)× U(1)
EH: The Standard-Like Model
SO(10)→ SU(3)× U(1)× SU(2)× U(1)
FNY: The Standard-Like Model
SO(10)→ SU(3)× U(1)× SU(2)× U(1)
Johar M. Ashfaque String Phenomenology
Breaking The Hidden SO(16)
LN: The Flipped SU(5) Model
SO(16)→ SO(6)× SO(10)
EH: The Standard-Like Model
SO(16)→ SU(5)× SU(3)× U(1)2
FNY: The Standard-Like Model
SO(16)→ SO(4)× SU(3)× U(1)4
Johar M. Ashfaque String Phenomenology
The Global Anomalous U(1) & Traces
LN: The Flipped SU(5) ModelU(1)A = −3U(1)1 − U(1)2 + 2U(1)3 − U(1)4
withTrU(1)A = 180
EH: The Standard-Like ModelU(1)A = 2(U(1)1 +U(1)2 +U(1)3)− (U(1)4 +U(1)5 +U(1)6)withTrU(1)A = 180
FNY: The Standard-Like ModelU(1)A = −4U(1)1−5U(1)2+3U(1)3+U(1)5+U(1)6+2U(1)8
withTrU(1)A = 336
Johar M. Ashfaque String Phenomenology
The DSW Mechanism
The cancellation mechanism generates a large Fayet-IliopoulosD-term for the anomalous U(1)A which would breaksupersymmetry and destabilize the vacuum. However, in all knowninstances one can give VEVs to scalar fields charged under U(1)Ato cancel the Fayet-Iliopoulos D-term and restore supersymmetry.
Johar M. Ashfaque String Phenomenology
The Fayet-Iliopoulos D-Terms
The general form of the anomalous D-term is
DA =∑i
Q iA|φi |2 +
g2eΦD
192π2TrQA
LN: The Flipped SU(5) Model
〈DA〉 =∑
i QiA|〈φi 〉|2 + 15g2
16π2
EH: The Standard-Like Model〈DA〉 =
∑i Q
iA|〈φi 〉|2 + 15g2
16π2
FNY: The Standard-Like Model〈DA〉 =
∑i Q
iA|〈φi 〉|2 + 7g2
4π2
Johar M. Ashfaque String Phenomenology
Conclusion
We have seen that the MI axion and the axion of the anomalousU(1) can not solve the strong CP problem since they always have
large masses and therefore large axion decay constants.
Johar M. Ashfaque String Phenomenology
What The Future Holds???
Johar M. Ashfaque
University of Liverpool
Je Ne Sais Pas
Johar M. Ashfaque String Phenomenology - University of Liverpool
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