Axions: ModelsStudent Seminar on Non-Accelerator Particle Physics

Rahul Mehra

Rheinische Friedrich-Wilhelms-Universität Bonn

June 10 2016

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1 Motivation

2 Strong CP Problem

3 Solutions to the Strong CP problem

4 Peccei-Quinn Model

5 Invisible Axion Models

6 Astrophysical Constraints

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U(1)A problem: A quick look

In the limit of massless quarks (mu,md,ms ⇡ 0), QCD has a global flavorchiral symmetry SU(3)L ⇥ SU(3)R ⇥ U(1)V ⇥ U(1)A

The axial U(1)A neither represents a symmetry nor is spontaneouslybroken.

Solution : U(1)A exhibits a quantum anomaly!

[1]

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U(1)A problem: A quick look

@µJ5µ =

g2S

32⇡2 G · G

This non-zero divergence for Jµ5 and the complex non-perturbative

nature of the QCD vacuum lead us to the ✓ term of QCD and the Strong

CP problem.

An extra term to the QCD Lagrangian is added:

L✓ = ✓g2

S

32⇡2 Gµ⌫b Gbµ⌫

Complete QCD Lagrangian: Leff = LQCD + ✓g2

S

32⇡2 Gµ⌫b Gbµ⌫

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Strong CP problem

Why does G · G matters whereas F · F does not?

The non-perturbative effects in QCD give rise to instantons ) the truevacuum is a linear superposition of n different configurations:

|✓i =X

n

e�in✓ |ni

If one also includes the weak interactions, the quark mass matrix:

Lmass = qiRMijqjL + h.c.

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Strong CP problem

Chiral rotations on the quarks changes ✓ by arg det M, and hence

✓ = ✓ + argdet M

The CP violating theta term induces a neutron dipole moment.The strong experimental bound |dn|< 3 ⇥ 10�26ecm implies that ✓ < 10�9

Strong CP problem : Why is this ✓ angle coming from the strong and

weak interaction, so small?

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Taming the Strong CP problem

There can be three possible solutions to the strong CP problem:Unconventional dynamicsSpontaneously broken CPA additional chiral symmetry

Examples of unconventional dynamics:

Raising the number of spatial dimensions à la Kaluza-Klein leads to thevanishing of the ✓ problem but the U(1)A still remains! [Khlebnikov andShaposhnikov, 1988]

Use the periodicity of vacuum energy E(✓) ⇠ cos✓ to deduce that ✓ vanishesbut no motivation for minimisation of vacuum energy. [Schierholz, 1994]

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Spontaneous CP violation (SCPV)

Spontaneous CP breaking occurs when CP is a symmetry of the originalLagrangian but after SSB, no CP symmetry remains.

To get ✓ < 10�9, one needs to ensure that ✓ also vanishes at the 1-looplevel.

One can have mutiple Higgs doublet with complex VEVs achieving thisbut FCNCs rear its ugly head!

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Peccei-Quinn to the rescue!

Only a symmetry based solution to the Strong CP problem is seen asnatural one!

The idea is to take the parameter ✓ and promote it to a dynamicalvariable such that ✓ w 0 emerges

Add to the SM - a global U(1)PQ symmetry - known as the Peccei-Quinn

symmetry.

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Peccei-Quinn to the rescue!

The quarks and the Higgs mutliplets transform non-trivially under thisU(1)PQ

Weinberg and Wilczek pointed that since a U(1)PQ is a global continoussymmetry spontaneously broken by the vacuum, there has to be aGoldstone boson! =) Axion

The axion is a pseudo Nambu-Goldstone boson of the ’anomalous’

U(1)PQ symmetry.

[1]

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U(1)PQ and Axions

Under a U(1)PQ transformation, the axion field translates to

a(x) ����!U(1)PQ

a(x) + ↵fa

where fa is the order parameter associated with the breaking of U(1)PQ.To make SM U(1)PQ invariant, the Lagrangian needs to be augmented byaxion interactions:

Ltotal = LSM + ✓g2

S

32⇡2 Gµ⌫a Gaµ⌫ � 1

2@µa@µa + Lint(@

µa/fa, )

+ ⇠afa

g2S

32⇡2 Gµ⌫b Gbµ⌫

where ⇠ is a model dependent parameter.We have added an axion mass term!

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U(1)PQ and Axions

Under a U(1)PQ transformation, the axion field translates to

a(x) ����!U(1)PQ

a(x) + ↵fa

where fa is the order parameter associated with the breaking of U(1)PQ.To make SM U(1)PQ invariant, the Lagrangian needs to be augmented byaxion interactions:

Ltotal = LSM + ✓g2

S

32⇡2 Gµ⌫a Gaµ⌫ � 1

2@µa@µa + Lint(@

µa/fa, )

+ ⇠afa

g2S

32⇡2 Gµ⌫b Gbµ⌫

where ⇠ is a model dependent parameter.We have added an axion mass term!

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U(1)PQ and Axions

The added term is essential for ensuring that U(1)PQ has a chiralanomaly.

@µJµPQ = ⇠

g2S

32⇡2 Gµ⌫b Gbµ⌫

Non-pert. QCD effects generate a potential for the axion field which isperiodic in the effective vacuum angle

Veff ⇠ ⇤4QCD

✓1 � cos

✓✓ + ⇠

haifa

◆◆

Hence, the potential is minimized to give the PQ solution:

hai = � fa

⇠✓

With the PQ solution, the Lagrangian no longer has a CP-violating ✓term.

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Peccei-Quinn model

In the original PQ model, the U(1)PQ breaking scale was taken to be theelectroweak scale fa = vF, with vF w 250 GeV.

If fa � vF then the axion is very light, very weakly coupled and very longlived ) Invisible axion models

To make the SM U(1)PQ invariant, one must introduce two Higgs fields toabsorb independent chiral transformations of the up- and down-quarks(and leptons).

LYukawa = � uij QLi�1uRj + � d

ij QLi�2dRj + � lijLLi�2lRj + h.c.

�1 �2 Q u d L eY 1 -1 1

3 � 43

23 -1 2

PQ 1 1 � 12 � 1

2 � 12 � 1

2 � 12

Table: Assignment of Hypercharge Y and Peccei-Quinn charge PQ

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Peccei-Quinn model

LYukawa = � uij QLi�1uRj + � d

ij QLi�2dRj + � lijLLi�2lRj + h.c.

Defining x = v2/v1 and vF =p

v21 + v2

2, the axion is the common phasefield in �1 and �2:

�1 =v1p

2eiax/vF

✓10

◆�2 =

v2p2

eia/xvF

✓01

◆

It is clear that LYukawa is invariant under the U(1)PQ transformations:

a ! a + ↵vF; uRj ! e�i↵xuRj;

dRj ! e�i↵/xdRj; lRj ! e�i↵/xlRj

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Peccei-Quinn model

The symmetry current for U(1)PQ can then be calculated as:

JµPQ = �vF@

µa + xX

i

uiR�µuiR +

1x

X

i

diR�µdiR

m2a =

⌧@2Veff

@a2

�= � ⇠

fa

g2S

32⇡2@@a

DGµ⌫

b Gbµ⌫

E|<a>=�✓fa/⇠

To compute the axion mass, one can either use current algebratechniques or use an effective Lagrangian approach.The mass of the axion for the standard PQ model is:

ma = m⇡f⇡vF

pmumd

mu + mdNg

✓x +

1x

◆' 75

✓x +

1x

◆keV

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Peccei-Quinn model

The PQ model where fa = vF has been ruled out by experimental results.

For example : BR(K+ ! ⇡+ + a)theo = 3 ⇥ 10�5(x + 1/x)2 which is wellabove the bounds BR(K+ ! ⇡+ + X0)exp < 3.8 ⇥ 10�8.

Invisible axion models introduce scalar fields that carry PQ charge butare SU(2)⇥ U(1) singlets, and hence fa � vF remains possible.

Two types of models:KSVZ (Kim, Shifman, Vainshtein and Zakharov) modelDFSZ (Dine, Fischer, Srednicki and Zhitnisky) model

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KSVZ model

In this model, the following were introduceda scalar field � with fa = h�i � vFa super-heavy quark X with MX ⇠ fa

as the only fields carrying PQ charge.

Ordinary quarks and leptons are PQ singlets.

The SU(2)⇥ U(1) singlet field � interacts with the new heavy quark Xwhich carry PQ charge via the Yukawa interaction:

LKSVZYukawa = �hXL�XR � h⇤XR�

†XL

By construction, KVSZ axion doesn’t interact with leptons and interactsonly with light quarks due to strong and electromagnetic anomalies:

Lanomaly =afa

✓g2

S

32⇡2 Gµ⌫b Gbµ⌫ + 3q2

X↵4⇡

Fµ⌫ Fµ⌫

◆

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KSVZ model

The Higgs potential is taken to be:

V(�,�) = �µ2��

†�� µ2��

⇤� + ��(�†�)2 + ��(�

⇤�)2 + ����†��⇤�

wherefa = h�i � vF

For the Yukawa and the form of the potential to be U(1)PQ invariant, thefollowing transformation should hold:

Q ! ei�5↵Q � ! e�2i↵�

The axion mass is given by:

ma =

✓m⇡

f⇡vF

pmumd

mu + md

◆vF

fa= 6.3eV

✓106

faGeV

◆

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DFSZ model

Adds to the PQ model a complex scalar field & which carries PQ chargebut is a singlet under SU(2)⇥ U(1)

As before, the model has two Higgs fields, �1 and �2 since both thequarks and leptons carry U(1)PQ.

The quarks and leptons feel the effects of the axions only through theinteractions that the field & has with �1 and �2 in the Higgs potential.(quartic coupling L&H � �T

1 C(&†)2�2 )

�1 =v1p

2ei(X1/fa)a

✓10

◆�2 =

v2p2

ei(X2/fa)a✓

01

◆

& =vFp

2exp

✓iavF

◆

where v2F = v2

1 + v22, and X1 =

2v22

v2F

; X2 =2v2

1

v2F

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Detour: Effective Lagrangians

Idea is to represent the dynamical content of a theory in the low energylimit and incorporate heavy particles in a different mannerWrite the general Lagrangian consistent with symmetries and includeonly a few - relevant for low energies

L = i/@ +12@µ⇡ · @µ⇡ +

12@µ� · @µ� � g

� � � i~⌧ · ~⇡�5

�

+12µ2

⇣�2 + ⇡2

⌘� �

4

⇣�2 + ⇡2

⌘2

If one works at low energy, all the matrix elements can be contained inLeff :

Leff =F2

4Tr

⇣@µU@µU†

⌘

where U = exp✓

i~⌧ · ~⇡

F

◆and F = v =

rµ2

�

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Back to DFSZ

For the interaction of axion with light quarks �! construct anappropriate effective chiral Lagrangian

Effects of heavy quarks �! contribution to the anomaly of JµPQ!

For two light quarks, one can introduce a 2 ⇥ 2 matrix of NG fields:

⌃ = exp✓

i~⌧ · ~⇡ + ⌘

f⇡

◆

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Calculations in DFSZ

The U(2)V ⇥ U(2)A invariant effective Lagrangian for the meson sector ofthe light-quarks is (no Yukawa yet!):

Lchiral = � f 2⇡

4Tr

⇣@µ⌃@

µ⌃†⌘

We now add U(2)V ⇥ U(2)A breaking terms - mimicking the U(1)PQ

invariant Yukawa interactions of the u- and d-quarks:

Lmass =12(f⇡m0

⇡)2 Tr

h⌃AM + (⌃AM)†

i

where

A =

✓e�iaX1/vF 0

0 e�iaX2/vF

◆M =

0

@mu

mu + md0

0md

mu + md

1

A

and U(1)PQ of Lmass requires:

⌃ �! ⌃

✓ei↵X1 0

0 ei↵X2

◆

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Calculations in DFSZ

The quadratic terms in Lmass involving neutral fields still has a problemsimilar to U(1)A i.e. incorrect mass ratio of ⌘ and ⇡

L(2)mass = � (m0

⇡)2

2

mu

mu + md

✓⇡0 + ⌘ � X1f⇡

vFa◆2

+md

mu + md

✓⌘ � ⇡0 � X2f⇡

vFa◆2�

Resolving this U(1)A problem in effective Lagrangian theory is done byadding another mass term that takes care of the anomaly in both U(1)A

and U(1)PQ

Lanomaly = �(m0

⌘)2

2

⌘ +

f⇡vF

(Ng � 1)2

(X1 + X2)a�2

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Axion mass in DFSZ

Diagonalization of Lmass and Lanomaly gives the axion mass and otherparameters (⇠a⇡, ⇠a⌘)

ma =

✓m⇡

f⇡vF

pmumd

mu + md

◆vF

fa= 6.3eV

✓106

faGeV

◆

Axion models are in addition characterised by the coupling of the axionto two photons: Ka��

Axion interactions with fermions also generate model dependentcouplings

Laf = �gaff f i�5 f a

gaff =Cf mf

fa; ↵aff =

g2aff

4⇡

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Constraints on axion mass

Axions can decay to two photons; this allows stars like our sun toproduce axions by transforming a photon into an axion ! searches forsolar axion flux.

axion~B�! photon

[1]Axion emission then also provides a pathway for energy loss in stars;bounds from lifetimes of stars, red giants and SN 1987a tell us that

fa � 109GeV =) ma 10�2eV

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Constraints on axion mass

The lower bound on axion mass comes from cosmological argumentswhere the candidacy of axion for cold dark matter is used.

Recent WMAP data on cold dark matter tell us that:

fa 1012GeV =) ma � 10�6eV

Hence, we have a small window on the mass of the axion:

10�2eV � ma � 10�6eV

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ADMX and ALPS

Microwave Cavity detection of axionat ADMX, University of Washington

[R. Battesti et al., Springer Lect. Notes Phys. 741 (2008)]

The principle of alight-shining-through a wall

experiment.[ALPS, DESY]

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Summary: 1-loop order

U(1)A and Strong CP problem are intertwined.Peccei-Quinn symmetry - a natural solution to the Strong CP problem.Visible axion models (electroweak scale )have beeen ruled out.Invisible axion (KSVZ and DFSZ) models provide hope!Astrophysical and cosmological arguments provide bounds on axionmass and couplings.

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Final words

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References

[1] Ikaros I Bigi and A Ichiro Sanda.CP violation, volume 28.Cambridge university press, 2009.

[2] Michael Dine, Willy Fischler, and Mark Srednicki.A simple solution to the strong cp problem with a harmless axion.Physics letters B, 104(3):199–202, 1981.

[3] Jihn E Kim.Weak-interaction singlet and strong cp invariance.Physical Review Letters, 43(2):103, 1979.

[4] Roberto D Peccei.The strong cp problem and axions.In Axions, pages 3–17. Springer, 2008.

[5] Pierre Ramond.Journeys beyond the standard model.Westview Press, 2004.

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