Axions: Models - Student Seminar on Non-Accelerator ... · Peccei-Quinn model In the original PQ...
Transcript of Axions: Models - Student Seminar on Non-Accelerator ... · Peccei-Quinn model In the original PQ...
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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