J. E. Kim

Axion theoryAxion theory

Jihn E. KimSeoul National University

DESY, Hamburg, 30. 09. 2008

J. E. Kim

1. Introduction

2. The LSP is the most popular CDM these days. I present the axion contribution to CDM, which is anyway needed for a strong CP solution.

3. Axions (axions from stars and the universe)

4. SUSY extension and axino (related to neutralino) , very brief

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compared to X-ray images(red)

Gravitational lensing

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WIMP was was first discussed by B. W. Lee and S. Weinberg (1977) just two months before Ben was killed by a traffic accident.

They considered a heavy neutrino, which implies that the usage

“weak” is involved. The LSP interaction is “weak” if interaction mediators (SUSY particles) are in the 100 GeV range as W boson. That’s the reason we talk about WIMP. Now to many it almost means the LSP.

ρ

100 eV 2 GeV m

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A rough sketch ofmasses and crosssections. BosonicDM with collective motion is always CDM.

[Roszkowski, modified]

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There are several particle physics candidates for CDM:

WIMP: LSP, LKP and other hypothetical heavy particles appearing with Z2. With the PAMELA data, this class seems the case.

But others also exist: axion, axino, gravitino

Some of these are related to axion which resulted for a solution of the strong CP problem.

It comes from the bound on NEDM which

exists if CP is broken. 1

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The existence of instanton solution in nonabelian gauge

theories needs θ vacuum [CDG, JR]. It introduces the θ term,

Here theta-bar is the final value taking into account the electroweak CP violation. For QCD to become a correct theory, this CP violation must be sufficiently suppressed.

qweakweakQCD MDet

FFFF

..arg,

~

2

1

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2. Strong CP problem

Axion’s attractive strong CP solution is the bottom line in every past and future axion search experiments. So, let us start with the strong CP problem.

The instanton solution introduces the so-called θ term, and the resulting NEDM.

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The neutron mass term

The NMDM and NEDM terms

fin

nNN em

m

ig /

The mass term and the NMDMterm have the same chiraltransformation property. So,(b)s are simultaneously removed.

(a)So, d(proton)= - d(neutron).is the NEDM contribution.

In our study, the VEV of pi-zero determine the size of NEDM.

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It is an order of magnitude larger than Crewther et al bound.

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We used C A Baker et al, PRL 97, 131801 (06), to obtain →|θ|< 0.7x10-11

Why is this so small? : Strong CP problem.1. Calculable θ, 2. Massless up quark (X) 3. Axion [as a new section]

1. Calculable θ

The Nelson-Barr CP violation is done by introducing vectorlike heavy quarks at high energy. This model produces the KM type weak CP violation at low energy. Still, at one loop the appearance of θ must be forbidden, and a two-loop generation is acceptable.

The weak CP violation must be spontaneous so that θ0 must be 0.

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2. Massless up quark

Suppose that we chiral-transform a quark,

If m=0, it is equivalent to changing θ → θ -2α. Thus, there exists a shift symmetry θ → θ -2α. Here, θ is not physical, and there is no strong CP problem. The problem is, “Is massless up quark phenomenologically viable?”

J. E. Kim

The famous up/down quark mass ratio from chiral pert. calculation is originally given as 5/9 [Weinberg, Leutwyler] which is very similar to the recent compilation,

MeVm

MeVm

m

m

d

u

d

u

5.10.6

,13

,5.0

Particle Data (2006), p.510

(Manohar-Sachrajda)

Excluding the lattice cal., this is convincing that mu=0 is not a solution now.

J. E. Kim

3. Axions Kim-Carosi, arXiv:0807.3125 “Axions and the strong CP

problem” (Rev. Mod. Phys. Submitted)

Historically, Peccei-Quinn tried to mimick the symmetry θ → θ -2α, by the full electroweak theory. They found such a symmetry if Hu is coupled to up-type quarks and Hd couples to down-type quarks,

),( dudRLuRL HHVHdqHuqL

Eq. β=αachievesthe same thing as them=0 case.

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The Lagrangian is invariant under changing θ → θ

-2α. Thus, it seems that θ is not physical, since it is a phase of the PQ transformation. But, θ is physical. At the Lagrangian level, there seems to be no strong CP problem. But <Hu> and <Hd> breaks the PQ global symmetry and there results a Goldstone boson, axion a [Weinberg,Wilczek]. Since θ is made field, the original cosθ dependence becomes the potential of the axion a.

If its potential is of the cosθ form, always θ=a/Fa can be chosen at 0 [Instanton physics,PQ,Vafa-Witten]. So the PQ solution of the strong CP problem is that the vacuum chooses0

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‘t Hooft determinental interaction and thesolution of the U(1) problem. If the storyends here, the axion is exactly massless.But,….

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History: The Peccei-Quinn-Weinber-Wilczek axion is ruled out early in one year [Peccei, 1978]. The PQ symmetry can be incorporated by heavy quarks, using a singlet Higgs field [KSVZ axion]

),,( duRL HHSVSQQL

Here, Higgs doublets are neutral under PQ. If they are not neutral, then it is not necessary to introduce heavy quarks [DFSZ]. In any case, the axion is the phase of the SM singlet S, if the VEV of S is much above the electroweak scale.

Now the couplings of S determines the axion interaction. Because it is a Goldstone boson, the couplings are of thederivative form except the anomaly term.

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In most studies, a specific example is discussed. Here, we consider an effective theory just above the QCD scale. All heavy fields are integrated out.[Kim-Carosi, arXiv:0807.3125]

In axion physics, heavy fermions carrying color charges are special. So consider the following Lagrangian

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The axion mass depends only on the combination of(c2+c3).

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The potential arising from the anomaly term after integrating out the gluon field is the axion potential.Two properties: (i) periodic potential with 2Fa period (Pontryagin): a def. (ii) minimum is at a=0, 2Fa, , 4 Fa,, … [PQ, VW]

depending on the identification of S in field spaceThe interaction

~

2

1

32

12

FFF

aFF

F

a

aa

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Leading to the cos form determines the axion mass

aaa

au F

GeVeV

F

mf

Z

Zm

F

am

73 10

][6.01

cos

)1(1

0

a

a F

mf

Z

Zm

The instanton contribution is included by Delta.

Numerically, we use

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)cos1()1(

][

0

222

42

a

du

F

amf

Z

ZaV

mm

The essence of the axion solution is that <a> seeks =0 whatever happened before. In this sense it is a cosmological solution. The height of the potential is the scale Λ of the nonabelian gauge interaction.

J. E. Kim

Axion couplings

Above the electroweak scale, we integrate out heavy fields. If colored quarks are integrated out, its effect isappearing as the coefficient of the gluon anomaly. If onlybosons are integrated out, there is no anomaly term.Thus, we have

KSVZ: c1=0, c2=0, c3=nonzero

DFSZ: c1=0, c2=nonzero, c3=0

PQWW: similar to DFSZ

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Hadronic coupling is important for the study of supernovae:The chiral symmetry breaking is properly taken into account,using the reparametrization invariance so that c3’=0.

KSVZ:

DFSZ:

The KSVZ axion has been extensively studied. Now theDFSZ axion can be studied, too.

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General very light axion:

Axial vector couplings:

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Even if we lowered some Fa, we must consider hidden sector also. In this case, axion mixing must be considered. There is an important theorem.

Cross theorem on decay constant and condensation scales [K99]:

Suppose two axions a1 with F1 and a2 with F2 (F1<<F2) couples to two nonabelian groups whose scales have a hierarchy, Λ1 << Λ2 .

Then, diagonalization process chooses the larger condensation scale Λ2 chooses smaller decay

constant F1, smaller condensation scale Λ1 chooses larger decay

constant F2.

So, just obtaining a small decay constant is not enough. Hidden sector may steal the smaller decay constant. It is likely that the QCD axion chooses the larger decay constant. [See also, I.-W. Kim-K, PLB, 2006]

Axion mixing in view of hidden sector

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An approximate PQ global symmetry with discrete symmetry in SUGRA was pointed out long time ago: for Z9 given by [L-P-Shafi]. Z9 is not possible in orbifold compactification. May need Z3xZ3 orbifold.

In this regard, we point out that the MI-axion with anomalous U(1) always has a large decay constant since all fields are charged under this anomalous U(1). Phenomenologically successful axion must need the approximate PQ.

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String models give definite numbers. [I-W Kim-K]

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Axions from starsAxions from stars Axion couplings: to e, p, n, and photon. The Primakoff process using the following coupling,

Lab. experiments can perform more than just the enegy loss mechanism. The early Tokyo experiment could not give a more stringent bound than the supernova limit, but the CAST(CERN axion solar telescope) could compete with the supernova bound.

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~10~

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aquarkslighti

emiiaa

aememeVa

aememaa

QTrc

cQcc

BEfFFGeV

macFF

ec

F

aL

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Compton-like scattering: γe→ae (DFSZ axion has aee coupling)

g aee < 2.5x10 –13 0.01eV < ma < 200 keV

(cf. DAMA: 3.9x10 –11 ma = 3.2 keV)

Hadronic axion window around 2 eV:

hot DM constraint with neutrino

In the hot plasma in stars, axions once produced most probably escape the core of the star and take out energy. This contributes to the energy loss mechanism of star and should not dominate the luminocity.The Primakoff process: γ→ a (present in any model) g aγγ < 0.6x 10-10 GeV-1 or Fa > 107 GeV 0.4eV < ma < 200 keV ruled out beyond this, too heavy to produce

SN1987A: NN→NNa 3x10-10 < g aNN < 3x 10-7 Fa > 0.6x 109 GeV

The improved supernova(gl. cl.) limit is 109 GeV.

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CAST Coll. ( Andriamonje et al.). JCAP 0704:010,2007The coupling depends on axion models. The numbers are given usually in field theoretic assumptions.

Fa> 0.5x10^9 GeV atma<0.39 eV.

This can compete withSN1987A constraint.

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Axions in the universeAxions in the universe

The axion potential is of the form

•

The vacuum stays there for a long time, and oscillates when the Hubble time(1/H) is larger than the oscillation period(1/ma)

3H < m a

This occurs when the temperature is about 0.92 GeV.

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The axion is created at T=Fa, but the universe (<a>)does

not roll until 3H=ma (T=0.92 GeV [Bae-Huh-Kim]). From then, the classical

field <a> starts to oscillate. Harmonic oscillatorma

2 Fa2 = energy density = ma x number density =

like CDM.See, Bae-Huh-Kim, There is an overshoot factor of 1.8. So we

use theta2, rather than theta1. If Fa is large(> 1012 GeV), then the axion energy density dominates. Since the energy density is proportional to the number density, it behaves like a CDM, but 109 GeV < Fa <10 12 GeV,

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The axion field evolution eq. and time-varying Lagrangian

)cos1(

))((

0)('1

3

22

223

2

aa

a

a

FmV

VFRL

VF

H

The adiabatic condition:

The adiabatic invariant quantity:

aa mmH ,

)(123 fmR a

cos'cos'22

)(1

df

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Bae-Huh-Kim, JCAP0809, 005

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If we do not take into account the overshoot factorand the anharmonic correction,

Inclusion of theseshowed the region

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x=(L/380 MeV) -1

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QCD phase transition effect is not important.

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Anthropic argument [Pi(84), Tegmark-Aguirre-Rees-Wilczek(05)]

Axion field values right after inflation can take any value between [0,π]. So Ωa may be at the required value by an appropriate misalignment angle for any Fa in the new inflation scenario. [Pi(84)]

Tegmark et al studied the landscape scenario for 31 dimensionless parameters and some dimensionful parameters with which habitable planets are constrained. They argue that for axion the prior probability function is calculable for axion models, which is rather obvious. Equally probable to sit anywhere here

They considered astrophysical conditions and nuclear physics conditions. For axion, one relevant figure is Q(scalar fluctuation) vs ξ(matter density per CMB photon). If axion is the sole candidate for CDM, the decay constant is shown before. But there may be more favored heavy WIMP candidates pinpointed by PP in which case axions supply the extra needed CDM amount.

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Tegmark, Aguirre, Rees, Wilczek, PRD (2005)

Q=scalar fluctuation amplitude δH on horizon, (2±0.2)x10-5 WIMP may be dominantly

CDM, and the rest is provided by axion.

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The anthropic regionwith a GUT scale inflation studied by

Hertzberg, Tegmark, Wilczek, 0807.1726

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Cosmic axion searchIf axion is the CDM component of the universe, thenthey can be detected. K. van Bibber’s efforts. The feeble coupling can be compensated by a huge number of axions. The number density ~ Fa

2, and the cross section ~ 1/Fa

2, and there is a hope to detect[10-5 eV range].

3

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~

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2

2

2

2

ZMEema

aquarkslighti

emiiaa

ememaa

QTrc

cQcc

BEFFe

cF

aL

Positivefor 1 HQ

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From local density with f aγγE∙B

Future ADMXwill cover the interesting region.

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Raffelt’s type cartoon

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4. SUSY extension and axino

The gravitino constraint: gravitinos produced thermally after inflation decays very late in cosmic time scale (>103 sec) and can dissociate the light nuclei by its decay products. Not to have too many gravitinos, the reheating temperature must be bounded,

Thus, in SUSY theories we must consider the relatively small reheating temperature.

TR < 109 GeV(old), or 107 GeV(recent)

Strong CP solution and SUSY:

axion : implies a superpartner axino

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The LSP seems the most attractive candidate forDM simply because the TeV order SUSY breaking scale introduces the LSP as a WIMP. This scenario needs an exact or effective R-parity for it to be sufficiently long lived.Story changes if one wants to solve the strong CP prob. For axino to be LSP, it must be lighter than the lightest neutralino. The axino mass is of prime importance. The conclusion is that there is no theoretical upper bound on the axino mass. For axino to be CDM, it must be stable or practically stable.

KeV axinos can be warm DM (90s) [Rajagopal-Turner-Wilczek]

GeV axinos can be CDM (00s) [Covi-H. B. Kim-K-Roszkowski]

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Gravitino problem is resolved if gravitino is NLSP, since the TP gravitinos would decay to axino and axion which do not affect BBN produced light elements. [Ellis et al, Moroi et al]

mMma 2/3~

On the other hand, if χ is NLSP(=LOSP), the TP mechanism restricts the reheating temperature after inflation. At high reheating temperature, TP contributes dominantly in the axino production. If the reheating temperature is below c. energy density line, there still exists the CDM possibility by the NTP axinos. [Covi et al]

NTP for

2~2~ h

m

mh a

a

2/3~ Mmma

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GeVF

GeVm

a1110

100

Covi-JEK-H B Kim-Roszkowski

Low re-heating Temperature

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In this figure, NTP axinos can be CDM for relatively low reheating temperature < 10 TeV, in the region

NTP axino as CDM possibility

The shaded region corresponds to the MSSM models with Ωχh2 < 104, but a small axino mass renders the possibility of axino closing the universe or just 30 % of the energy density. If all SUSY mass parameters are below 1 TeV, then Ωχ h2 <100 and sufficient axino energy density requires

mmMeV a ~10

GeVma 1~ If LHC does not detect the neutralino needed for closing the universe, the axino closingis a possibility.

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If the NLSP is stau with the axino or gravitino LSP, the previously forbidden stau LSP region is erased [Brandenburg, Covi, Hamaguchi, Roszkowski, Steffen].In this case, the CDM axino is similar to the bino LSPcase but it is easier to detect the stau signal due to itsem charge at the LHC.

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If the axino is heavier than the neutralino, we have adifferent region of the allowed parameter space [Choi-Kim-Lee-Seto, PRD77, 123501 (2008)],

With PAMELA, thispossibility is moreImportant for axino.

J. E. Kim

5. Conclusion

I reviewed axion and related issues on:

1. Solutions of the strong CP problem: Nelson-Barr, mu=0 ruled out now, axion.

2. Axions can contribute to CDM. Maybe solar axions are easier to detect. Then, axion is not the dominant component of CDM. Most exciting is, its discovery confirms instanton physics of QCD by experiments.

3. With SUSY extension, O(GeV) axino can be CDM. It is difficult to detect this axino from the DM search, but possible to detect at the LHC as missing energy.

4. In any case, to understand the strong CP with axions in SUSY framework the axino must be considered to the CDM discussion.