Axions, Free-Fermions & The Strong CP Problem

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.


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.


Solutions

Axions are perhaps the most promising solution towards solving theStrong CP problem.

Other solutions are ruled out or are disfavoured by 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.


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).


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.


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


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.


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.


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.


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.


Digression: Aim

Our aim is to construct realistic models with

4 flat space-time dimensions,

chiral matter fields and

N = 1 SUSY.


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.



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



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.



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



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


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.


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


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)


Breaking The Hidden SO(16)


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


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


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.


The Fayet-Iliopoulos D-Terms

The general form of the anomalous D-term is

DA =∑i

Q iA|φi |2 +

g2eΦD

192π2TrQA


〈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


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.


What The Future Holds???

Johar M. Ashfaque

University of Liverpool

Je Ne Sais Pas

Johar M. Ashfaque String Phenomenology - University of Liverpool

THANKYOU!!!

Johar M. Ashfaque String Phenomenology - University of Liverpool

Axions, Free-Fermions & The Strong CP Problem

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