Theories of QCD axion
Jihn E. Kim!

Kyung Hee University,
Seoul National Univ.

Osaka University, 20 December 2017

QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 1

1. Symmetries

The CP symmetry and its violation is the key in thedevelopment of modern particle theory. In 1976, when
the third quark family was not discovered, Weinberg
introduced a multi-Higgs doublets to have a weak CP
violation:

The CP symmetry and its violation is the key in thedevelopment of modern particle theory. In 1976, when
the third quark family was not discovered, Weinberg
introduced a multi-Higgs doublets to have a weak CP
violation:

The CP symmetry and its violation is the key in thedevelopment of modern particle theory. In 1976, when
the third quark family was not discovered, Weinberg
introduced a multi-Higgs doublets to have a weak CP
violation:

With the refection symmetry and three Higgs doublets,
CP violation can be introduced with the cIJ terms.

On the strong interaction side in modern gauge theory,
there has been the QCD development after the
discovery of instanton solution: strong CP

On the strong interaction side in modern gauge theory,
there has been the QCD development after the
discovery of instanton solution: strong CP
There exists the QCD theta vacuum, which gives aneffective interaction

On the strong interaction side in modern gauge theory,
there has been the QCD development after the
discovery of instanton solution: strong CP
There exists the QCD theta vacuum, which gives aneffective interaction
The U(1) problem and its solution by the theta term.
[Weinberg (1975), ’t Hooft (Phys. Report 1986)]
near 1 GeV from,contribution from<>

The strong CP problem
The theta term breaks CP symmetry and one expects
that the NEDM is of order nucleon size times e. But,
-26
the upper bound is O(10 )e cm, some 10 orders ofmagnitude away from anticipation. This is the strong
CP problem. In the literature, three types were tried(1) Massless up quark: up-quark is not massless.
(2) Calculable solutions: Nelson-Barr type.
(3) Axion solutions.

If we consider a few terms,
there can be global symm.

QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 6

If we consider a few terms,
there can be global symm.

QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 6

If we consider a few terms,
there can be global symm.

QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 6

If we consider a few terms,
there can be global symm.

We start with an example.

QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 6

Not to have FCNC problem, one H doublet couples tou-type quarks and another H doublet couples to d-type
quarks. With these Yukawa couplings, a general VW
with the reflection symmetry (\phiI to -\phiI) breaks CP.
But, if we keep all terms except the cIJ terms, there
appear a global symmetry: Peccei-Quinn symmetry.

But, with the quark global symmetry beyond the SM, in
our case two, one must be a global symmetry. With thisglobal symmetry of quark fields the phase appears in thetheta term!

Weinberg-Wilczek has shown that this phase fieldoriginally present in the Lagrangian develops a potential,
and it is not an exact Goldstone boson but a pseudoGoldstone boson. Phenomenologically, the PQWW
axion is ruled out.

But, with the quark global symmetry beyond the SM, in
our case two, one must be a global symmetry. With thisglobal symmetry of quark fields the phase appears in thetheta term!

Weinberg-Wilczek has shown that this phase fieldoriginally present in the Lagrangian develops a potential,
and it is not an exact Goldstone boson but a pseudoGoldstone boson. Phenomenologically, the PQWW
axion is ruled out.

But, with the quark global symmetry beyond the SM, in
our case two, one must be a global symmetry. With thisglobal symmetry of quark fields the phase appears in thetheta term!

Weinberg-Wilczek has shown that this phase fieldoriginally present in the Lagrangian develops a potential,
and it is not an exact Goldstone boson but a pseudoGoldstone boson. Phenomenologically, the PQWW
axion is ruled out.

But, with the quark global symmetry beyond the SM, in
our case two, one must be a global symmetry. With thisglobal symmetry of quark fields the phase appears in thetheta term!

Weinberg-Wilczek has shown that this phase fieldoriginally present in the Lagrangian develops a potential,
and it is not an exact Goldstone boson but a pseudoGoldstone boson. Phenomenologically, the PQWW
axion is ruled out.

Axion solution has two important parameters!

!

fa =Intermediate scale, DW number.

2. “Invisible” axions

QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 10

QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 10

QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 10

From the exact
global symmetry.

From the exact
global symmetry.
This anomaly
breaks the PQ
symmetry.

From the exact
global symmetry.
This anomaly
breaks the PQ
symmetry.

VEV of scalar phi
gives
the fa scale.

246 GeV

vew

9-12
10
GeV

18
16
10 GeV 10 GeV

MMI

QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 12

246 GeV

9-12
10
GeV

18
16
10 GeV 10 GeV

vew

fa

fMI MMIQCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 12

Is “intermediate scale” fa ~(vewMPl)

1/2?

Is “intermediate scale” fa ~(vewMPl)

1/2?

This is possible only after having a
spontaneously broken global
symmetry far below the Planck massscale. We will come back to this point
later.

Scale of fa :KSVZ axion

QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 15

Scale of fa :KSVZ axion

Renormalizable terms:
c
KSVZ axion, Q QSQCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 15

Scale of fa :DFSZ axion

Scale of fa :DFSZ axion
Renormalizable terms with fine-tuning:4
2
V for DFSZ axion, H1H2 S , |S|
Non-renormalizable terms with SUSY: no fine-tuning
2
W for DFSZ axion, H1H2 S

Domain-wall
problemVilenkin-Everett (1982);
Barr-Choi-Kim (1987)
Sikivie (1982)

NDW=1 needed
Top-down approach, using string compactification
1. The global U(1) is broken at the axion window.
2. DW number given here.
3. By giving a VEV to QPQ=1 field, we obtain N =1.
DW

4. Example: 1 heavy quark model. But, effectively, all PQ
charged quarks should add up their contributions to
makeN =1.
DW

5. Anomalous U(1) gauge symmetry.
6. Choi-Kim mechanism: with hidden sector force. Anomalous
U(1) becomes global U(1) below the GUT scale.

NDW=1 needed
Top-down approach, using string compactification
1. The global U(1) is broken at the axion window.
2. DW number given here.
3. By giving a VEV to QPQ=1 field, we obtain N =1.
DW

4. Example: 1 heavy quark model. But, effectively, all PQ
charged quarks should add up their contributions to
makeN =1.
DW

5. Anomalous U(1) gauge symmetry.
6. Choi-Kim mechanism: with hidden sector force. Anomalous
U(1) becomes global U(1) below the GUT scale.

Better way than Lazarides-Shafi: we need aGoldstone boson direction

For the center of GUT group, Lazarides-Shafi (1982).
But, the following ideas are more widely applicable.

Choi-Kim, PRL55 (1985) 2637 with two confining forcesChoi-Kim, PRL55 (1985) 2637 with two confining forcesChoi-Kim, PRL55 (1985) 2637 with two confining forcesChoi-Kim, PRL55 (1985) 2637 with two confining forcesChoi-Kim, PRL55 (1985) 2637 with two confining forcesChoi-Kim, PRL55 (1985) 2637 with two confining forcesChoi-Kim, PRL55 (1985) 2637 with two confining forcesChoi-Kim, PRL55 (1985) 2637 with two confining forcesChoi-Kim, PRL55 (1985) 2637 with two confining forcesChoi-Kim, PRL55 (1985) 2637 with two confining forcesChoi-Kim, PRL55 (1985) 2637 with two confining forcesChoi-Kim, PRL55 (1985) 2637 with two confining forces

Torus identification

Choi-Kim, PRL55 (1985) 2637 with two confining forcesChoi-Kim, PRL55 (1985) 2637 with two confining forcesChoi-Kim, PRL55 (1985) 2637 with two confining forcesChoi-Kim, PRL55 (1985) 2637 with two confining forcesChoi-Kim, PRL55 (1985) 2637 with two confining forces

The same vacuumGoldstone boson direction

The same vacuumGoldstone boson direction

The same vacuum

3. ’t Hooft mechanism

QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 20

QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 21

’t Hooft mechanism: 	

!

If a gauge symmetry and a global symmetry are broken by
one complex scalar by the BEHGHK mechanism, then the gauge
symmetry is broken and a global symmetry remains unbroken.

QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 21

’t Hooft mechanism: 	

!

If a gauge symmetry and a global symmetry are broken by
one complex scalar by the BEHGHK mechanism, then the gauge
symmetry is broken and a global symmetry remains unbroken.
QgaugeQglobal1

1

QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 21

’t Hooft mechanism: 	

!

If a gauge symmetry and a global symmetry are broken by
one complex scalar by the BEHGHK mechanism, then the gauge
symmetry is broken and a global symmetry remains unbroken.
QgaugeQglobal1

1

Unbroken X=Qglobal-Qgauge

QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 21

So, the gauge boson becomes heavy andthere remains the x-independenttransformation parameter beta. The
corresponding charge is a combination:
X=Qglobal-Qgauge

This process can be worked out at any step. When one
global symmetry survives below a high energy scale, we
consider another gauged U(1) and one more complex
scalar to break two U(1)’s. Then, one global symmetry
survives.

This process can be worked out further below the GUT
scale as far as U(1) gauge symmetries (to be broken
above the EW scale) are present. Then, one global
symmetry survives down to the intermediate scale.

MPl

One complex scalar for one gauge
symmetry breaking

MPl

One complex scalar for one gauge
symmetry breaking

VGUT
Until all wanted U(1) gauge
symmetries are broken.

MPl

One complex scalar for one gauge
symmetry breaking

VGUT
Until all wanted U(1) gauge
symmetries are broken.
VI

At the next step, a global
symmetry is broken

4. Axion-photon-photon coupling

One generator
on quark
fields, e.g. T3
QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 25

4. Axion-photon-photon coupling

One generator
on quark
fields, e.g. T3
QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 25

If the quark representation is fundamental and only
the SM quarks have PQ charges, then2
Tr |T3 | =(1/2)x(number of chiral quarks). So,

We use 2 from mu / md = 1/2
QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 26

Phys. Rev. D55,
055006(1998)

hep-ph/0612107
1405.6175; 1603.02145

mu / md = 1/2

Phys. Rev. D55,
055006(1998)

hep-ph/0612107
1405.6175; 1603.02145

mu / md = 1/2

Phys. Rev. D55,
055006(1998)

mu / md = 1/2

These numbers
are usually
hep-ph/0612107
marked as1405.6175;
1603.02145
unification line,
but it is not so.

In general, many quarks have PQ charges, and thetables on KSVZ and DFSZ do not make sense except
in the experimental proposal for grants application.
Further more vector like particles (must be of the
KSVZ type) removed at the intermediate scale cancontribute. Some ultraviolet completed theory is reallyprediction on the axion-photon-photon coupling.
One has to know all spectra with PQ, color, and EW
charges.
QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 28

1703.05345 and 1603.02145

1703.05345 and 1603.02145

These are family q.n.

1703.05345 and 1603.02145

Two families fromT4 and one family
from U

These are family q.n.

1703.05345 and 1603.02145

Two families fromT4 and one family
from U

These are family q.n.

1703.05345 and 1603.02145

Two families fromT4 and one family
from U

These are family q.n.

1703.05345 and 1603.02145

Two families fromT4 and one family
from U

These are family q.n.

Adding contributions fromother tables, -9312

1703.05345 and 1603.02145

Two families fromT4 and one family
from U

These are family q.n.

Adding contributions fromother tables, -9312

The unification value5. Model-independent axion
in string theory

QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 30

U(1)global

Z2

U(1)global

TOE

Openning of string theory by the GS term

QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 32

Openning of string theory by the GS term
Counter term is introduced to cancel the anomalies: E8xE8’

QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 32

Openning of string theory by the GS term
Counter term is introduced to cancel the anomalies: E8xE8’

QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 32

Openning of string theory by the GS term
Counter term is introduced to cancel the anomalies: E8xE8’

QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 32

Openning of string theory by the GS term
Counter term is introduced to cancel the anomalies: E8xE8’

One needs a term (GS-term) to cancel the gauge and gravitational
anomalies.

QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 32

Anomalies: even dimensions

Anomalies: even dimensions

In 10D, the hexagon anomaly. It is cancelled by theprevious GS term.

QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 34

Green-Schwarz mechanism:
The gravity anomaly in 10D requires 496 spin-1/2 fields.
Possible non-Abelian gauge groups are rank 16 groups SO(32) andE8xE8’. The anti-symmetric field BMN has field strength (in diff0-w

notation), H= dB+w3Y

3L

0 :SO(32). Three indices matched.

QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 34

Green-Schwarz mechanism:
The gravity anomaly in 10D requires 496 spin-1/2 fields.
Possible non-Abelian gauge groups are rank 16 groups SO(32) andE8xE8’. The anti-symmetric field BMN has field strength (in diff0-w

notation), H= dB+w3Y

3L

0 :SO(32). Three indices matched.

QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 34

QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 35

HMNP is the field strength of BMN : This is called the MI-axion.

QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 35

HMNP is the field strength of BMN : This is called the MI-axion.

QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 35

HMNP is the field strength of BMN : This is called the MI-axion.

The dual of H is the so-called MI-axion [Witten (1984)]

QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 35

HMNP is the field strength of BMN : This is called the MI-axion.

The dual of H is the so-called MI-axion [Witten (1984)]

QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 35

QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 36

In the orbifold compactification, e.g. at a Z_3 torus, there are 3
fixed points. Here, we interpret that the flux is located at the fixed
points. We take the limit of string loop almost sitting at the fixed
points.
It involves 2nd rank antisymmetric field BMN.

QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 36

In the orbifold compactification, e.g. at a Z_3 torus, there are 3
fixed points. Here, we interpret that the flux is located at the fixed
points. We take the limit of string loop almost sitting at the fixed
points.
It involves 2nd rank antisymmetric field BMN.

QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 36

In the orbifold compactification, e.g. at a Z_3 torus, there are 3
fixed points. Here, we interpret that the flux is located at the fixed
points. We take the limit of string loop almost sitting at the fixed
points.
It involves 2nd rank antisymmetric field BMN.

QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 36

QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 40

One may look this in the following way.
!

The10 supergravity quantum field theory with SO(32) and E8xE8’gauge groups has gauge and gravity anomalies. Let us believe thatstring theory is consistent, effectively removing all divergences, i.e.
removing all anomalies. The point particle limit of 10D string theory
should not allow any anomalies. There must be some term in thestring theory removing all these anomalies. It is the GS term. In
strong int., breaking chiral symmetry, viz. the Wess-Zumino term
removing anomalies by some term involving pseudoscalar fields.
!

For the GS term, already there is the field BMN needed for the
anomaly cancellation. Do not need strong int.

One may look this in the following way.
!

The10 supergravity quantum field theory with SO(32) and E8xE8’gauge groups has gauge and gravity anomalies. Let us believe thatstring theory is consistent, effectively removing all divergences, i.e.
removing all anomalies. The point particle limit of 10D string theory
should not allow any anomalies. There must be some term in thestring theory removing all these anomalies. It is the GS term. In
strong int., breaking chiral symmetry, viz. the Wess-Zumino term
removing anomalies by some term involving pseudoscalar fields.
!

For the GS term, already there is the field BMN needed for the
anomaly cancellation. Do not need strong int.

Thus, since the MI axion is a real spin-0
particle, fa can be related to the string scale.

1.
2.

Witten (85) showed that MI-axion has N=1.
DW
Choi-Kim mechanism

QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 42

1.
2.

Witten (85) showed that MI-axion has N=1.
DW
Choi-Kim mechanism

QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 42

1.
2.

Witten (85) showed that MI-axion has N=1.
DW
Choi-Kim mechanism

QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 42

1.
2.

Witten (85) showed that MI-axion has N=1.
DW
Choi-Kim mechanism

3. Relation of prime numbers: For N2=17

QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 42

1.
2.

Witten (85) showed that MI-axion has N=1.
DW
Choi-Kim mechanism

3. Relation of prime numbers: For N2=17

QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 42

1.
2.

Witten (85) showed that MI-axion has N=1.
DW
Choi-Kim mechanism

3. Relation of prime numbers: For N2=17
4. In the example of 1710.08454, based on
the model of Huh-Kim-Kyae, sum of U(1)2
2
2
SU(3)c anomaly is 3492=2 x3 x97. So,
there is a great chance that
MMI : fphi at st scale of 3491: 3493, 3497,
3499 etc will lead to N

=1 because they

DW
are relatively prime. Thus, the global
symmetry is determined purely from the
VEVs at the string scale.

QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 42

Thus “invisible” axion from anomalous
U(1) satisfies the requirements
for the intermediate fa.

QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 44

Choi-Kim, 1985

fMI

QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 44

Choi-Kim, 1985

But, we need“invisible” axion
here

fMI

fa

QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 44by supercoolingby supercooling6. Approximate global
symmetry

QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 47

????????

Searchhas????????
begun withsymmetry
principles

Symmetry is
beautiful: aframework,
beginning with
Gross’ granddesign.

Symmetry is
beautiful: aframework,
beginning with
Gross’ granddesign.

Symmetry is
beautiful: aframework,
beginning with
Gross’ granddesign.

Parity:Symmetry is
beautiful: aframework,
beginning with
Gross’ granddesign.

Parity: Slightly

broken!

QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 50

QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 50

Except theanomalous
U(1), any global
symmetry does
not haveanomalies fromstring theory.
So, this V ispresent.

QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 50

Except theanomalous
U(1), any global
symmetry does
not haveanomalies fromstring theory.
So, this V ispresent.
Still the questionis at what level?
If one allows thediscretesymmetry fromstring.
QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 50

Except theanomalous
U(1), any global
symmetry does
not haveanomalies fromstring theory.
So, this V ispresent.

Quintessential axion

Still the questionis at what level?
If one allows thediscretesymmetry fromstring.
QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 50

Except theanomalous
U(1), any global
symmetry does
not haveanomalies fromstring theory.
So, this V ispresent.

Quintessential axion

ULA

ALPs

Still the questionis at what level?
If one allows thediscretesymmetry fromstring.
QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 50

Except theanomalous
U(1), any global
symmetry does
not haveanomalies fromstring theory.
So, this V ispresent.

Quintessential axion

ULA

ALPs

Still the questionis at what level?
If one allows thediscretesymmetry fromstring.
QCD axion theory, “Axion physics & DM cosmology”@Osaka Univ, 20-21 Dec 2017. 50

27%

DE

CDM
5%

68%

atom

Detection of “invisible”
axion CDM by cavity
detectors: CAPP, Yale,
ADMX, etc.

27%

DE

CDM
5%

68%

atom

Detection of “invisible”
axion CDM by cavity
detectors: CAPP, Yale,
ADMX, etc.

27%

DE

CDM
5%

68%

Quintessential axion

atom6. Conclusion

6. Conclusion
1. Introduction

6. Conclusion
1. Introduction
2. “Invisible” axions

6. Conclusion
1. Introduction
2. “Invisible” axions
3. ’t Hooft mechanism

6. Conclusion
1. Introduction
2. “Invisible” axions
3. ’t Hooft mechanism
4. Axion-photon-photon coupling6. Conclusion
1. Introduction
2. “Invisible” axions
3. ’t Hooft mechanism
4. Axion-photon-photon coupling
5. Model-independent axion from string theory6. Conclusion
1. Introduction
2. “Invisible” axions
3. ’t Hooft mechanism
4. Axion-photon-photon coupling
5. Model-independent axion from string theory
6. Approximate global symmetry