Inflation from a SUSY Axion Model

Masahiro Kawasaki (ICRR, Univ of Tokyo)with

Naoya Kitajima (ICRR, Univ of Tokyo)Kazunori Nakayama (Univ of Tokyo)

1

Based on papers MK, Kitajima, Nakayama, PRD 82, 123531 (2010) MK, Kitajima, Nakayama, PRD 83, 123521 (2011)

2011年8月19日金曜日

1. Introduction• Problems in the standard model of particle physics

• Strong CP problem :

Why QCD preserves CP?

• Hierarchy problem :

The EW scale is unstable against radiative correction

• Well-known solutions

• Peccei-Quinn mechanism Axion

• Supersymmetry

• This leads us to consider a SUSY Axion Model

• In this model Hybrid Inflation is naturally realized

• Axion is dominant dark matter of the universe

• We have a consistent cosmological scenario.

Copeland, Liddle, Lyth, Stewart, Wands (1994)

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2. SUSY Axion Model

• Superpotential

: gauge singlet : PQ fields

: Heavy quraks (Higgs) for KSVZ (DFSZ) axion model

• Scalar potential

global minimum flat direction

• The flat direction is lifted up by soft SYSY-breaking potential

• PQ scalars are stabilized at

W = κS(ΨΨ− f2a ) + λΨXX

S Ψ, ΨX, X

VF = κ2|ΨΨ− f2a |2 + κ2|S|2(|Ψ|2 + |Ψ|2)

of the SUSY hybrid inflation model has the same form asthat of the SUSY axion model, and the PQ scale is deter-mined to be!1015 GeV for the PQ sector to induce a hybridinflation and correctly reproduce observed density perturba-tion. In Sec. IV the dynamics after inflation is discussed andwe see that the saxion oscillation with large initial amplitudeis induced. In Sec. V it is shown that the late-time entropyproduction due to the saxion decay necessarily takes placeand, as a result, the axion coherent oscillation can be thedominant component of DM. In Sec. VI we discuss the fateof topological defects, such as axionic strings and domainwalls. In Sec. VII we present a mechanism to create a correctamount of baryon asymmetry under the late-time entropyproduction. In Sec. VIII a variant type of SUSYaxion modelis presented, which causes a so-called smooth-hybrid infla-tion and we describe some cosmological aspects of themodel. We conclude in Sec. IX.

II. SUPERSYMMETRIC AXION MODEL

A. The potential of the SUSY axion model

Let us describe the SUSYaxion model. Here we assumethe gravity-mediated SUSY breaking. The superpotentialfor the SUSY axion model is given by

W " !S#! "!$ f2a% & "!X "X; (1)

where S is a gauge singlet superfield and has a zero PQcharge, and! and "! are the PQ superfields that are gaugesinglets and have&1 and$1 PQ charges, respectively. ThePQ fields contain the axion (a), saxion (#, the scalarpartner of the axion), and axino (~a, the fermionic super-partner of the axion). Here fa is the PQ symmetry-breakingscale and ! is a dimensionless coupling constant assumedto be real and positive. X# "X% is the superfield interactingwith a PQ field at tree level and has some PQ charges aswell as gauge charges through which it interacts with theminimal supersymmetric standard mode (MSSM) fields.The superpotential also has an R-symmetry. The chargeassignments of the fields in the present model are shown inTable I. In particular, for the Kim-Shifman-Vainshtein-Zakharov (KSVZ) (or hadronic) axion model [8], X and"X are additional heavy quarks, denoted by Q and "Q, thathave color charges. For the Dine-Fischler-Srednicki-Zhitnitsky (DFSZ) axion model [9], X and "X are identifiedas MSSM Higgses, Hu and Hd.

1

According to the superpotential (1), the F-term scalarpotential is derived as

VF " !2j! "!$ f2aj2 & !2jSj2#j!j2 & j "!j2%; (2)

where we denote the scalar fields using the same symbol asthe superfields and we set X and "X to be zero assuming thatthey have large Hubble masses and quickly settle down tozero during inflation. The global minimum of this potentialis located at S " 0 and ! "! " f2a. Here it should be notedthat there exists a flat direction along which the scalarfields do not feel the potential, ensured by the U#1%PQsymmetry extended to a complexU#1% due to the holomor-phy of the superpotential [10]. The flat direction is lifted upby the SUSY-breaking effect, leading to the following softSUSY-breaking mass terms:

Vsoft " c1m23=2j!j2 & c2m

23=2j "!j2; (3)

where m3=2 is the gravitino mass and c1 and c2 are real-valued constants that are positive and order unities. Withthis soft SUSY-breaking potential, the radial componentsof the PQ fields j!j and j "!j are stabilized at

v ’!c2c1

"1=4

fa; "v ’!c1c2

"1=4

fa; (4)

respectively.2 The saxion field# is defined by the deviationof j!j from the vacuum expectation value (4) along the flatdirection.Near the vacuum expectation values (4), the axion a and

saxion # are related to the PQ fields as

! " v exp##& ia$$$2

pFa

%; "! " "v exp

#$#& ia$$$

2p

Fa

%; (5)

where Fa is determined by requiring that # and a are

canonically normalized and given by Fa '$$$$$$$$$$$$$$$$$v2 & "v2

p.

B. The decay of the saxion

In this subsection, we derive the decay rate of the saxionwhich is important in the later section. The kinetic terms ofthe PQ fields lead the interaction of the saxion with theaxion [18,19] as

TABLE I. Charge assignments on the field content.

S ! "! X "X

U#1%PQ 0 &1 $1 $1=2 $1=2U#1%R &2 0 0 &1 &1

1In the DFSZ model, the coupling constant " must be verysmall, say, "! 10$12 for fa ! 1015 GeV, in order to produce asizable $-term. This might be a tuning, but it is relaxed bychanging the relative PQ charge assignments between !# "!% andHu#Hd%. For example, if the PQ charges of Hu and Hd are $n,where n#( 1% is a positive integer, the allowed term in thesuperpotential is "!2nHuHd=M

2n$1 with some cutoff scale M,which might be the Planck scale. In this case the amount oftuning for the coupling constant " is relaxed. The phenomenol-ogy discussed in the following sections is not modified by thechoice of PQ charges for Hu and Hd.

2See Refs. [11–17] for other types of the saxion stabilizationmechanisms and their cosmological issues.

KAWASAKI, KITAJIMA, AND NAKAYAMA PHYSICAL REVIEW D 83, 123521 (2011)

123521-2

ΨΨ = f2a , S = 0

Vsoft = c1m23/2|Ψ|2 + c2m

23/2|Ψ|2

Ψ Ψ fa

2011年8月19日金曜日

• Axion and saxion ( scalar partner of axion) are related to PQ scalars as

• Saxion decay

• KSVZ axion model

In general, decay into two axions ( ) is dominant but it is suppressed when we assumeThen the saxion decays into two gluons with decay rate

• DFSZ axion model

The saxion decays into Higgses with decay rate

a σ

Ψ fa expσ + ia√

2fa

Ψ fa exp

−σ + ia√

2fa

σ → a + a

c1 c2

Γ(σ → 2g) αs

32π3

m3σ

f2a

Γ(σ → 2h) 18π

µ

mσ

4 m3σ

f2a

µ = λΨ

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3. Inflation in SUSY Axion Model• Superpotential in SUSY axion model includes

This is the same form as that realizes SUSY hybrid inflation

PQ scalars and S play roles of waterfall fields and inflaton, respectively

• Scalar potential

For local minimum at where the potential is flat

Winf = κS(ΨΨ− f2a )

V = κ2|ΨΨ− f2a |2 + κ2|S|2(|Ψ|2 + |Ψ|2)

|S| fa Ψ = Ψ = 0

!

!"#

!"$

!"%

!"&

!"' (!"$

(!"#

!

!"#

!"$('

(&

(%

($

!

!"#

!"$

!"%

!"&

!"'

!

!"#

!"$

!"%

!"&

!"'

(!"$

(!"#

!

!"#

!"$('

(&

(%

($

!

!"#

!"$

!"%

!"&

!"'('

(&

(%

($

!

!"#$

!"#$

!

Copeland, Liddle, Lyth, Stewart, Wands (1994)Dvali, Shafi, Schaefer (1994) . . . . .

V κ2f4a

+ (one loop corr.) + (sugra corr.)

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• With appropriate Kähler potential we have successful inflation which is consistent with WMAP result

• However, PQ scale should be high

Axion overcloses the universe?

Post-inflationary dynamics can solve this problem

Nakayama, F.Takahashi, Yanagida (2010)fa[GeV]

fa ∼ 1015 GeV

JCAP12(2010)010

Figure 4. Same as figure 3, but for m3/2 = 1TeV.

50 e-foldings. An exception is for a relatively large ! region (! ! 0.01) where the one-loope!ect is important. In that region, the spectral index can be around 0.98 and the consistencywith current observations is better. However, the region is on the boundary of the cosmicstring bound, and significant cosmic string contribution to the density perturbation mayameliorate the situation. Therefore, search for the cosmic strings in the CMB anisotropiesand the gravity waves may confirm or disfavor the model.

3.5.2 Non-minimal Kahler potential

Next, we investigate the case of the non-minimal Kahler potential. The most important termis the k1-term in eq. (2.2), since it directly a!ects the scalar spectral index. In fact, k1 ! 0.01

– 12 –

κ

Successful inflation

2011年8月19日金曜日

4. Post-inflationary Dynamics

• For successful inflation we need

too large axion density

This problem cannot be solved by tuning misalignment angle θ because PQ symmetry is broken after inflation and θ takes random values in different places of the universe

• However, after inflation saxion can oscillate with large amplitude and decay to produce huge entropy

entropy production sufficiently dilutes axion

together with other harmful relics

fa ∼ 1015 GeV

2011年8月19日金曜日

4.1 Inflaton Oscillation

• After inflation the inflaton starts oscillation

• PQ scalars roll down toward the flat direction

PQ scalars have masses

PQ scalars are stabilized at

m2Ψ,Ψ κ2|S|2

Ψ = Ψ = fa

10-4

10-2

100

102

103 104 105

|Ψ|/fa

|S|/fa

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4.2 Reheating and Thermal Effect

• Inflaton can decay through

• Reheating temperature

• Finite-temperature effect due to heavy quarks which couple MSSM particles in thermal bath

• This lifts up the flat direction and rolls down to smaller (larger ) value

W = kSY Y (Y = H or Q)

TR ∼ 1011GeV κ

10−3

1/2

k

10−3

fa

1015GeV

1/2

Vth αsT4 ln

|Ψ|2

T 2

Ψ(Ψ)

10-8

10-6

10-4

10-2

100

102

102 103 104 105 106 107

fa ! time

| | / fa|– | / fa

H / famth / fa

Ψ/fa

Ψ/fa

mth/fa

H/fa

|Ψ| ∼ αsMp

σi ∼ αsMp

σ ∼ |Ψ|− fa

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4.3 Saxion Oscillation and Entropy Production

• When , the soft SUSY breaking masses dominate over thermal mass and PQ scalars ( ~ saxion )start oscillation around

• Saxion decay temperature

KSVZ axion

DFSZ axion

• Entropy production

10-8

10-6

10-4

10-2

100

102

102 103 104 105 106 107

fa ! time

| | / fa|– | / fa

H / famth / fa

Ψ/fa

Ψ/fa

mth/fa

H/fa

m3/2/faTσ 5MeV

mσ

10TeV

3/2

1015GeV

fa

Tσ 5MeV mσ

1TeV

3/2

1015GeV

fa

µ

mσ

2

sbefore

safter∼ 10−10

H m3/2

Ψ ∼ Ψ ∼ fa

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4.4 Axion Density

• Axion density under the large entropy production

Axion can be appropriately diluted and account for dark matter of the universe

• Other harmful relics are also diluted by entropy production

• thermally and non-thermally produced gravitinos

• thermally and non-thermally produced axinos

Ωah2 0.02

Tσ

1MeV

fa

1015GeV

2

Lazarides, Schaefer, Seckel, Shaf (1990), MK, Moroi, Yanagida (1996)

2011年8月19日金曜日

5. Baryogenesis

• All contents of the universe are diluted by late-time entropy production

• We need sufficiently large baryon asymmetry that survives the dilution

• Affleck-Dine mechanism can work

• A MSSM flat direction ( = squark, slepton, Higgs) has a large field value in the early universe

V

!

V = m2Φ|Φ|2 +

|Φ|10

M6+

am3/2

Φ6

M3+ h.c

+ VthU(1) B

nB

s 3× 10−11δCP

Tσ

1MeV

m3/2

1MeV

1/2

1011GeV

TR

M

1000Mp

3

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6. Conclusions

• Inflation naturally takes place in a SUSY axion model

• Successful inflation requires high PQ scale

• After inflation, thanks to finite temperature effect, saxion starts oscillation with large initial amplitude

• Saxion decays and produces huge entropy, by which axion is appropriately dilutes and its density becomes consistent with the present dark matter density

• Other harmful relics like gravitino and axino are also diluted

• Baryon asymmetry is obtained through Affleck-Dine mechanism

fa ∼ 1015GeV

2011年8月19日金曜日