Can The Strong CP Problem Be Solved In The Free-Fermionic

Setting?

A Report

Johar M. Ashfaque

1 Strong CP Violation

The idea behind axions is to add fields to the Standard Model so that there is a new anomalous U(1)symmetry which is known as a Peccei-Quinn symmetry. If this U(1)PQ is spontaneously broken, it willgenerate a new Goldstone boson, a.

The current bound on the neutron EDM is

|dN | < 2.9× 10−26e cm

so thatθ < 10−10.

The smallness of θ despite the large amount of CP violation in the weak sector is known as the strongCP problem.

2 Astrophysical/Cosmological Constraints

ρa < ρcritical

meansfa ≤ 1012 GeV

Astrophysical bounds (for example, axion emissions from red giants) require fa > 1010 GeV, while cosmo-logical bounds (too many axions would overclose the universe) require fa < 1012 GeV. Thus, the axionshould be very weakly coupled with a mass

10−4 eV < ma < 10−2 eV

. It is of course possible to evade these bounds with clever model building.

3 The Model-Independent Axion

Following [3], there are two axions in superstring models. As suggested by [2], one is the model- inde-pendent axion (MI axion) and the other is the Peccei-Quinn type one (PQ axion).

The MI axion comes from the Yang-Mills Chern-Simons term, see [7] for further details.

The Model-Independent Axion Is Present In All Superstring Models.

In the Neveu-Schwarz sector, in four dimensions, there is always an antisymmetric tensor

Bµν , µ, ν = 0, ...3,

as highlighted in [5] is crucial for anomaly cancellation, which has one physical degree of freedom whichas highlighted in [9], gives rise to a single scalar field in four dimensions with axion-like couplings

�φ = −M−1(TrFµν F̃µν − TrRµνR̃

µν)

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where Fµν is the Yang-Mills field strength and Rµν is the Riemann curvature tensor. Bµν by duality isequivalent to an axion called the MI axion, see [5] for further details.

The MI axion is a promising one towards a solution of the strong CP problem.

However, the decay constant problem is an issue. The model-independent axion has the decay constantaround fa ' 1016 GeV which is too large for the axion cold dark matter scenario.

As [8] suggests, the solution to this problem seems to be solved in the anomalous U(1) models. In thiscase, the model-independent axion becomes the longitudinal degree of the anomalous gauge boson. Thegauge boson and model-independent axion is removed at this gauge boson mass scale. At low energy,there remains a global PQ symmetry which can be broken. However, if the low energy theory maintainssupersymmetry, it is very difficult to realize this scenario. The reason is that the supersymmetry conditionwith matter fields always breaks the global PQ symmetry at the gauge boson mass scale, and a remainingU(1) symmetry (if any) turns out to be a gauge symmetry.

4 Ma for MI Axion [2]

M′

a = 8π2Ma ⇒Ma =M

′

a

8π2

and

M′

a =MPl

12√π' 5.64× 1017 ⇒Ma ' 7.15× 1015

4.1 Derivation: Relation for M′a

Employing the first relation from [2], we have that

g2φ

k2=M2Pl

8π

which upon substitution in the second relation yields

8πg4φ2

18k4M2Pl

= M′2a ⇒

M2Pl

144π= M

′2a

Ma =1

8π2· MPl

12√π

clearly violating the cosmological energy density upper bound on fa. The model-independent axion isthus a ”harmful” axion. It is a well known fact, as [6] points out, that large fa especially

fa > 1012 GeV

means axion energy densityρa > ρcritical

and therefore is unacceptable.

5 The Anomalous U(1) - The PQ Axion

As stated in [],

Following [4], the second axion arises as the Goldstone boson of an anomalous global U(1)A in the theory.

The cancellation of this anomalous Abelian symmetry being broken is a special case of the Green-Schwarzanomaly cancellation mechanism. In D = 2k + 2 dimensions, the Green-Schwarz counter-term takes theform BF k, where F is the anomalous field strength. In D = 4, this is just BF . This counter-term has

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been, as highlighted in [9], to indeed arise in one-loop string calculations with the right coefficient tocancel the anomalies.

The above anomaly cancellation mechanism generates a Fayet-Iliopoulos D-term for the anomalous U(1)Awhich would break supersymmetry and destabilize the vacuum. However, in all known instances one cangive VEVs to scalar fields charged under U(1)A to cancel the Fayet-Iliopoulos D-term.

Clearly, if 〈DA〉 = 0, there must exist at least one ψ with 〈ψ〉 6= 0 and Qψ 6= 0. Hence, the global U(1)Ais spontaneously broken in the presence of N = 1 supergravity. The global U(1)A symmetry is anomalousas

TrU(1)A 6= 0

and it can be seen, as suggested by [4, 9] that U(1)A is also color-anomalous by considering

TrU(1)AH2 6= 0

where H = SU(3)C . The global symmetry U(1)A is thus of the Peccei-Quinn type.

As U(1)A is anomalous (the trace is equal to 180), the axion will also couple to the hidden sector fieldstrengths. In models, as suggested by [9], where the hidden sector becomes strong at an intermediatescale the axion acquires a heavy mass and then decays rapidly.

It is also mentioned in [4] that the axion acquires a large mass where the hidden gauge groups becomestrongly interacting where we saw the hidden gauge groups were SU(5) and SU(3) of the asymmetricSLM.

5.1 U(1)A Transformations

a→ a+ λ

tells us that the axion has derivative couplings.

φi → φieiαqi

where qi are the U(1)A charges of the singlet scalar fields φi.

This symmetry is violated by the anomaly term aF F̃ arising from the Chern-Simons term.

Defining the scalar fields as

φi =

(vi + pi√

2

)exp

(iξivi

)under U(1)A with

ξi → ξi + viqiα

where ξi are the phases and vi are the scalar VEVs.

The U(1)A axion is then a normalized linear combination of the phases of the scalars which breaks U(1)A:

a =∑i

ciξi

with ∑i

c2i = 1.

Demanding

a→ a+∑i

viqiciα = a+ λ

sincea =

∑i

ciξi →∑i

(ciξi + viqiciα) = a+∑i

viqiciα

and making use of ∑i

c2i = 1

3

we have thatci =

viqi∑i v

2i q

2i

which upon substitution into

a =∑i

ciξi

yields

a =1

fa

∑i

viqiξi

withf2a =

∑i

v2i q2i .

The scalar VEVs resulting from F and D constraints are at the scale

M

10∼ 1017 GeV.

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References

[1] A. E. Faraggi, A new standard-like model in the four dimensional free fermionic string formulation,Physics Letters B 278 (1992)

[2] K. Choi, J. E. Kim, Harmful axions in superstring models,Physics Letters B 154 (1985).

[3] K. Choi, J. E. Kim, Compactification and axions in E8 × E′

8 superstring models,Physics Letters B 165 (1985).

[4] E. Halyo, Can the axions of standard-like superstring models solve the strong CP problem?,Physics Letters B 318 (1993).

[5] E. Witten, Phys. Lett. B 149 (1984); Phys. Lett. B 153 (1985)

[6] H. Y. Cheng, The strong CP problem revisited,Phys. Rep 158 1988.

[7] J. E. Kim, Light pseudoscalars,particle physics and cosmology,Physics Reports 150, 1987.

[8] J. E. Kim, Axion Theory Review,http://arxiv.org/pdf/astro-ph/9812257.pdf

[9] J. L. Lopez, D. V. Nanopoulos, Can an anomalous U(1) solve the strong CP problem?,Physics Letters B 245 (1990)

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