Aligned QCD axionNaoya KITAJIMA

1512.05295, 1603.02090, 1606.05552T. Higaki (Keio Univ.), K.S. Jeong (Pusan national Univ.)T. Sekiguchi (CTPU, IBS) and F. Takahashi (Tohoku Univ.)

The 6th KIAS workshop on Particle Physics and Cosmology, Oct. 24 - 28, 2016

Strong CP problem

LQCD = �1

4Ga

µ⌫Gµ⌫a +

g2s32⇡2

✓Gaµ⌫G̃

µ⌫a

CP violating phase

|✓| < 0.7⇥ 10�11 Fine tuning?

experimental constraint on NEDM

Solution : Peccei-Quinn mechanismPeccei, Quinn (1977)

� = |�|eia/faPQ field U(1)PQ symmetry+

L 3 g2s32⇡2

✓✓ +

a

Fa

◆Ga

µ⌫G̃µ⌫a

dynamical field -> 0

Q1. PQ scale must be in some intermediate scale. Why?

Axion puzzle -1-

Axion window : 109 GeV . Fa . 1012 GeV

SN1987A DM abundance

weak scale

GUT1016 GeV

PQ scale

103 GeV

Q2. High quality of PQ symmetry is required. How come?

Global symmetries must be explicitly broken

Axion puzzle -2-

L 3 �5

MP,

�6

M2P

, . . .by e.g. Planck suppressed operators :

They contribute to CP phase

PQ mechanism would be broken down

weak scale

GUT1016 GeV

PQ scale

Our strategy

103 GeV

alignment mechanism

multiple axions with

Fa ⇠ 109�12 GeV

h�ii = fi ⇠ TeV, i = 1, 2, 3, ..

QCD axion

2⇡f2

�2

�12⇡f1 ⇥ n

2⇡fe↵

Alignment mechanism — basic pictureKim, Niles, Peloso, hep-ph/0419138

2⇡f1

2⇡f2

�1

�2

flat direction

�1 ! �1 + 2⇡f1 ⇥ n, �2 ! �2 + 2⇡f2

effective decay constant : fe↵ =q

n2f21 + f2

2

V = ⇤

4

1� cos

✓�1

f1� n�2

f2

◆�2-axion case

�1

f1� n

�2

f2=

c

o

n

s

t

.

�1

f1� n

�2

f2= const.

1 period of flat direction

Clockwork axion model

K. Choi, H. Kim, S. Yun, 1404.6209K. Choi, S.H. Im, 1511.00132D.E. Kaplan, R. Rattazzi, 1511.01827

�i = �i + 2⇡fi (i = 1, 2, . . . , N)N-axions:

V = �N�1X

i=1

⇤

4i cos

✓�i

fi+ ni

�i+1

fi+1

◆potential :

fe↵ =

vuuutNX

i=1

0

@NY

j=i

n2j

1

A f2i

�flat =NX

i=1

(�1)i�1

0

@NY

j=i

nj

1

A fi�i

fe↵with

(nN ⌘ 1)

flat direction :

fe↵ ⇠ nNi fi

Aligned “QCD” axion modelT. Higaki, K.S. Jeong, NK, F. Takahashi, 1512.05295

L =

N�1X

i=1

⇤

4i cos

✓�i

fi+ ni

�i+1

fi+1

◆

+

g2332⇡2

ks�N

fNGµ⌫

˜Gµ⌫ +

g2132⇡2

k�N

fNBµ⌫

˜Bµ⌫

a =NX

i=1

(�1)i�1

0

@NY

j=i

nj

1

A fi�i

fafa =

vuuutNX

i=1

0

@NY

j=i

n2j

1

A f2iwith

flat direction = QCD axion

fi ⇠ fN ⇠ f and |ni| ⇠ n > 0 (i = 1, 2, . . . , N � 1)

(TeV scale)fa ⇠ eN lnnf ⇠ 109�12

we need more than 10 axionsf ⇠ TeV, fa ⇠ 1010 GeV, n = 3

GeV

haiFa

⌘ ¯✓ 'm2��PQ sin↵

m2QCD +m2

��PQ cos↵

µ

Fa

VQCD = �m2QCDF

2a cos

✓a

Fa

◆�m2

��PQµ2cos

✓a

µ� ↵

◆

extra PQ breaking QCD instanton

Quality of the PQ symmetry

potential of QCD axion : QCD instanton + extra PQ breaking

Strong CP phase is displaced by the second term

axion mass : m2a ' m2

QCD +m2��PQ cos↵

mQCD ' 6⇥ 10�4 eV

✓⇤QCD

400 MeV

◆2✓ Fa

1010 GeV

◆�1

QCD instanton —

1. Planck-suppressed dimension-5 operator

too large strong CP phase unless tanα is extremely small…

no alignment mechanism

& ✓̄ ⇠ tan↵m��PQ ⇠ 106 GeV

✓Fa

1010 GeV

◆3/2

� mQCD

(conventional case)

�V5 =5

5

�5

MP+ h.c.

h�i ⇠ Fa ⇠ µ

1. Planck-suppressed dimension-5 operator

�V5 =5

5

�51

MP+ h.c.

Experimental bound can be satisfied if alignment mechanism enhances effective decay constant ⇠ 108

✓̄ ⇡ 2⇥ 10�10

✓↵

0.1

◆✓Fa/f1108

◆�1

m��PQ � mQCDfor

aligned axion h�1i ⇠ f1 ⇠ µ

m��PQ ' 0.03 MeVp

|5|✓

f1103 GeV

◆3/2

m2��PQ =

5|5|2p2

f31

MP

T. Higaki, K.S. Jeong, NK, F. Takahashi, 1603.02090

Fig. in Kawasaki, Nakayama 1301.1123

T > Tc

T < Tc

N cosmic strings are formed

shift symmetry

N-1 domain walls are formed(bounded by cosmic strings)

“string-wall network”

Defect formation in aligned axion model

V =NX

i=1

✓�m2

i |�i|2 + �i|�i|4◆

�V =N�1X

i=1

✏i�i�3i+1 + h.c.

breaking of U(1)N

T. Higaki, K.S. Jeong, NK, T. Sekiguchi F. Takahashi, 1606.05552

U(1)N can easily be restored

Tc ⇠ f ⇠ TeVbecause

Cosmic string “Bundle”

N = 2 N = 3

string

wall

cross section of isolated systems

Aligned structure in the field (internal) space appears in the real space!

x

y

�V =N�1X

i=1

✏i�i�3i+1 + h.c.

T. Higaki, K.S. Jeong, NK, T. Sekiguchi F. Takahashi, 1606.05552

string bundle QCD axion string

µe↵ ' ⇡(32(N�1)f21 + · · ·+ 32f2

N�1 + f2N ) ln

✓R

�

◆= ⇡F 2

a ln

✓R

�

◆“effective” tension of string bundle :

Fa =

vuuutNX

i=1

f2i

0

@NY

j=i

n2j

1

AwithR - horizon sizeδ - string core

Lattice simulation for N=22D

3D

String bundles are formed!

Lattice simulation for N=22D

3D

String bundles are formed!

string bundle QCD axion string

However, formation probability of such structure is exponentially suppressed

(ΦΝ string requires 3N Φ1 strings but # of Φ1 strings decreases by pair annihilation)

Lattice simulation for N=32D

3D

String-wall network is long-lived

Lattice simulation for N=32D

3D

String-wall network is long-lived

T ⇠ 1 GeVQCD axion potential arises& domain wall becomes unstable

f . 400 TeV ✏�1/6

✓⇤QCD

400 MeV

◆4/3To avoid domain wall dominationbefore the annihilation :

Domain wall annihilation

V

aenergy bias

clockwork potential

QCD axion potential

⇢DW ⇠ VbiasDomain wall annihilates when

Hiramatsu,Kawasaki,Saikawa (2010)

Domain wall collapse

Gravitational waves

Moore et al. 1408.0740

⌫peak ' 1.6⇥ 10�7 Hz

✓g⇤ann80

◆1/6✓ Tann

1 GeV

◆

Gravitational waves from decaying domain walls

pulsar timing obs.

Peak frequency of GW — Hubble parameter at the annihilation

⌦gwh2 < 2.3⇥ 10�10 at ⌫1yr ' 3⇥ 10�8 Hzcurrent constraint :

P. D. Lasky et al. 1511.05994

Constraints from pulsar timing observations

f . 200 TeV ⇥ ✏�1/6

✓⇤QCD

400 MeV

◆28/33

100 TeV (future observation by SKA)

Intensity of GW by DW decay

⌦gw(⌫) / ⌫3 for ⌫ < ⌫peakfrequency dependence : Hiramatsu, Kawasaki, Saikawa, 1309.5001

⌦gw(⌫peak)h2 ' 2⇥ 10�11✏

✓g⇤ann80

◆�4/3✓ Tann

1 GeV

◆�4✓ f

100 TeV

◆6

Summary

- Intermediate scale for axion decay constant can be realized by alignment mechanism — aligned QCD axion model

N-axions w/ weak scale decay constant —> QCD axion

- High quality of PQ scale can be naturally realizedPQ mechanism works well w/o fine tuning / introducing extra ZN

- Topological defects can easily formed in aligned axion modeland it puts an upper bound on the decay constant

f . O(100) TeV

Cosmology tells us the existence of new physics < 100 TeV?

Strong CP problem

LQCD = �1

4Ga

µ⌫Gµ⌫a +

g2s32⇡2

✓Gaµ⌫G̃

µ⌫a

CP violating phase

|✓| < 0.7⇥ 10�11 Fine tuning?

neutron electric dipole moment dn ' 4.5⇥ 10�15✓e cm

experimental constraint |dn| < 2.9⇥ 10�26e cm

Solution : Peccei-Quinn mechanismPeccei, Quinn (1977)

L 3 g2s32⇡2

✓✓ +

a

Fa

◆Ga

µ⌫G̃µ⌫a

dynamical field

� = |�|eia/faPQ field U(1)PQ symmetry+axion

Strong CP phase is dynamically cancelled!

axion✓ +

a

Fa= 0

�i =

✓⇢i +

fip2

◆ei�i/fi fi =

p2h|�i|iwith

V =NX

i=1

✓�m2

i |�i|2 +�i

4|�i|4

◆N-complex scalar fields with global U(1)N symmetry

U(1)N symmetry breaking

�L =N�1X

i=1

�i ̄i i +

niX

a=1

�i+1 ̄ai ai

!+ h.c.

K. Choi, H. Kim, S. Yun, 1404.6209

�V =N�1X

i=1

✏i�i�3i+1 + h.c. ni = 3 and ⇤i =

✓✏i2fif

3i+1

◆1/4

D.E. Kaplan, R. Rattazzi, 1511.01827

Possible UV completion

2. Planck-suppressed dimension-6 operator

Z2 : �i ! ��i

�V6 =6

6

�61

M2P

+ h.c.

m2��PQ =

3|6|2

f41

M2P

, µ =f16

and ↵ = arg(6)

¯✓ ' 1.7⇥ 10

�10m2��PQ cos↵

m2QCD +m2

��PQ cos↵

✓tan↵

0.1

◆✓Fa/f110

8

◆�1

m��PQ ' 0.5⇥ 10�3 eVp

|6|✓

f1103 GeV

◆2

aligned QCD axion

Experimental bound can be satisfied if alignment mechanism enhances effective decay constant ⇠ 108

Topological defect 1 : Cosmic string

domain wall

spontaneous breaking of U(1) symmetry —> cosmic string

“vacuum circle”

✓

internal (field) space spatial coordinate

coordinate space

horizon scale

Axion potential

Topological defect 2 : Domain wall

discrete minima (vacuum) —> domain wall formation

domain wall

⌦a,dwh2 = (5.8± 2.8)

✓Fa

1012 GeV

◆1.19✓ ⇤

400 MeV

◆

⌦a,strh2 = 2.0⇠

✓Fa

1012 GeV

◆1.19✓ ⇤

400 MeV

◆cosmic string :

domain wall :

coherent oscillation : ⌦a,osch2 = 0.18 ✓2i

✓Fa

1012 GeV

◆1.19

with h✓2i i =canh⇡2

3

Axion abundance (conventional scenario)

⌦a,toth2 = (8.4± 3.0)

✓Fa

1012 GeV

◆1.19✓ ⇤

400 MeV

◆total :

Topological defects in conventional axion scenario

PQ symmetry breaking after inflation

Hiramatsu, Kawasaki, Saikawa, Sekiguchi 1202.5851

Domain walls with cosmic strings

NDW=1

Cosmic string

string-wall network decays into axions

�i =fip2ei�i/fi =

fip2eiwi✓Cosmic string solution :

µi ⇠ µcore

+

Z R

�

����1

r

@�i

@✓i

����2

2⇡rdr ⇡ ⇡w2

i f2

i ln

✓R

�

◆tension :

Cosmic strings of N-axions

http://www.damtp.cam.ac.uk/research/gr/public/cs_evol.html

wi : winding number for each axion string : angular coordinate in real space✓

fi ⌧ Fan.b.

Each string tension is much smaller thanthat of the QCD axion string