axions and lattice qcd - infnpersonalpages.to.infn.it/~beraudo/colloquia/files/szabo_talk.pdf ·...
TRANSCRIPT
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Axions and lattice QCDKalman Szabo
Forschungszentrum JulichUniversity of Wuppertal
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Axion
A hypotethical elementary particle introduced to solve a
puzzle with the parity transformation
in particle physics and a leading candidate for the
dark matter particle .
Axions and lattice QCD
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Parity transformation (P)
Flip the sign of all spatial coordinates.
x → −x, y→ −y, z → −z
Parity transforms an object to its mirror image
L → R, R → L
Axions and lattice QCD
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Is P a symmetry?
Are the laws of physics thesame in the mirror world?
Or can I tell the differencebetween the original and the
mirror image?
Axions and lattice QCD
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Is P a symmetry?
Axions and lattice QCD
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Particle physics 101
Particles: up quark, down quark, electron and neutrino.
u d e ν
Interactions: strong, electromagnetic, weak and Higgs.
g γ W h
Axions and lattice QCD
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Parity and particles
Particles have spin and are massless →
νL
eL
dL
uL
νR(?)
eR
dR
uR
P exchanges left-handed particles with right-handed
Axions and lattice QCD
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Electromagnetic interaction
∇E = 4πρ, ∇B = 0, ∇× E + ∂tB = 0, ∇× B − ∂tE = 4πj
P : E → −E, B → B, ρ→ ρ, j → −j,∇ → −∇
Classical electrodynamicsyP symmetricx
Quantum electrodynamics
LQED = −14F2µν +ψ†LσµDµψL +ψ†RσµDµψR
γ
L
L
γ
R
R
Axions and lattice QCD
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Strong interaction
Generalized QED: 3x3-matrices and 3-vectors instead ofnumbers
Fµν →
G00µν G01
µν G02µν
G10µν G11
µν G12µν
G20µν G21
µν G22µν
ψ→
ψ0
ψ1
ψ2
LQCD = −1
4Tr(G2µν) + (ψ†L ,σµDµψL) + (ψ†R,σµDµψR)
g
L
L
g
R
R
P symmetric
Axions and lattice QCD
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Neutron electric dipole moment
P is symmetry [H, P] = 0↓
neutron is P-eigenstate P |n〉 ∝ |n〉↓
〈n|~d|n〉 = 0↑
EDM is P-odd P · ~d · P = −~d
Expt: 〈n|~d|n〉 = −0.2(1.9)× 10−26 ecm [Pendlebury ’15]
Axions and lattice QCD
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Higgs interaction
Higgs mechanism
Axions and lattice QCD
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Higgs interaction
figs/bush.png
Axions and lattice QCD
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Higgs interaction
figs/bush.png
”Left hand knows what the right hand is doing.”
Axions and lattice QCD
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Higgs interaction
”Left hand knows what the right hand is doing.”
L R L R L R L
LHiggs = mψ†LψR + mψ†RψL
Generate mass by combining massless L,R particles
P symmetric
Axions and lattice QCD
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Weak interaction
Maximally violates parity, only interacts with L-particles.
W
uL
dL
W
eL
νL
W
uR
dR
W
eR
νR
P violating
Left:= the handedness of particles to which W couples
⊗⊗
Axions and lattice QCD
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The P puzzle
electromagneticstrongHiggsweak
P is violated by the weak interaction → it is not symmetryof Nature.
Why P is not violated by the others?
Try: P-invariance is consequence of remainingsymmetries (Lorentz invariance, internal).
Axions and lattice QCD
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The P puzzle in QED
LQED = −14F2µν +ψ†LσµDµψL +ψ†RσµDµψR
What is the most general Lagrangian with Lorentzinvariance and gauge symmetry?
L = LQED + θ · FF with FF ≡ FµνFρσεµνρσ
Violates parity
FF → −FFTotal derivative
FF = ∂µKµ with Kµ = εµνρσAνFρσthen by Gauss-theorem∫
d4x FF =∮
dnµKµ = 0
P-invariance follows from Lorentz+gauge
Axions and lattice QCD
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The P puzzle in QCD
Most general SU(3) symmetric Lagrangian
L = LQCD + θ ·GG with GG = 18π2 Tr (GµνGρσεµνρσ)
Violates parity
GG → −GGTotal derivative
GG = ∂µKµ with Kµ = Trεµνρσ(AνGρσ + 2
3AνAρAσ)
then by Gauss-theorem∫d4x GG =
∮dnµKµ 6= 0
Kµ can be non-zero at ∞P could be violated by θ ·GG. Why not?
nEDM experiments → θ . 0.0000000001
Axions and lattice QCD
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P-violation in Higgs
LHiggs = m(ψ†LψR +ψ†RψL
)with real m
Most general Lagrangian has complex mass :
L = mψ†LψR + m∗ψ†RψL
It violates parity, but can be transformed away by anaxial transformation :
ψL → ψL , ψR → e−i arg mψR
but P-violation does not go away:
L → L + arg m · FF + arg m ·GG
Higgs P violation can be transformed to strong GG
Axions and lattice QCD
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The P puzzle
electromagneticstrongHiggsweak
←− by Lorentz+gauge invariance
why GG not appears in Nature?←−←−
Axions and lattice QCD
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θ dependence of QCD
Calculate the Feynman path integral!
Z(θ) =∫[dG][dψ†][dψ] exp
(i∫
d4x (L + θGG)
)
-[log Z(θ)/Z(0)]/V
θ≈θ2χ/2
Has a minimum at θ = 0!
Axions and lattice QCD
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Lattice QCD computation
Discretize!
lattice spacing a . 110 proton size
Calculate!
109 dimensional integrals using supercomputers
Axions and lattice QCD
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θ dependence from lattice QCD
− 1V log Z(θ)/Z(0) = 1
2θ2χ+ . . .
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014
χ [fm
-4]
a2[fm
2]
standard(m
π,ph/mπ,ts)
2
a=0.134 fm2.000 PCyears
100.000 years on a PC, 1 year on a supercomputer.
Axions and lattice QCD
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θ dependence from lattice QCD
− 1V log Z(θ)/Z(0) = 1
2θ2χ+ . . .
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014
χ [fm
-4]
a2[fm
2]
standard(m
π,ph/mπ,ts)
2
a=0.095 fm12.000 PCyears
100.000 years on a PC, 1 year on a supercomputer.
Axions and lattice QCD
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θ dependence from lattice QCD
− 1V log Z(θ)/Z(0) = 1
2θ2χ+ . . .
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014
χ [fm
-4]
a2[fm
2]
standard(m
π,ph/mπ,ts)
2
a=0.064 fm86.000 PCyears
100.000 years on a PC, 1 year on a supercomputer.
Axions and lattice QCD
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θ dependence from lattice QCD
− 1V log Z(θ)/Z(0) = 1
2θ2χ+ . . .
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014
χ [fm
-4]
a2[fm
2]
standard(m
π,ph/mπ,ts)
2 Continuum limit
χ1/4 = 76(2)(1)MeV
100.000 years on a PC, 1 year on a supercomputer.
Axions and lattice QCD
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A solution by Peccei-Quinn
Make a dynamical field fromthe parameter!
figs/thetapot/plot.gif
L + θ ·GG + 12 f 2 · (∂µθ)2 + V (θ,∂µθ)
with V (θ,∂µθ) such, that minimum stays at θ = 0.
Dynamical field → new particle
Axions and lattice QCD
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The axion
“Cleaning up the problem with the axial transformation”[Weinberg,Wilczek]
La = θ ·GG + 12 f 2
a · (∂µθ)2 + V (θ,∂µθ)
Mass ↔ Scale m2a = χ/f 2
a with χ = 76(2)(1)MeV
Interactions are model dependent
V (θ,∂θ) = c · θ · FF + V1(∂θ)
θ
γ
γ
c/fa Smaller mass more elusive.
Axions and lattice QCD
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The axion windowSearching for axions is hard, since mass is unknown.
10-10
10-5 1 10
5
?
ma[eV]
Exclusions on ma from
Early laboratory searches
Astrophysics (supernovae, red giants)
Axion is dark matterAxions and lattice QCD
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Axion production in the early Universe
Potential becomes flat at QCD transition (Tc ≈ 150MeV)
Calculate the number of axions produced!
Rolling down the potential (→ χ(T ) ) + damped by
expansion (→ ε(T ), p(T ) equation of state).
Need lattice QCD!
Axions and lattice QCD
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Equation of state from lattice QCD
1981: pure SU(2), Nt = 22016: SU(3) + u,d,s,c,b; cont. extrap. from Nt = 6 . . . 16
First time without “left for future work”!
Axions and lattice QCD
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Equation of state
0.6
0.7
0.8
0.9
1ratios
gs(T)/gρ(T)
gs(T)/gc(T)
10
30
50
70
90
110
100
101
102
103
104
105
g(T)
T[MeV]
gρ(T)
gs(T)
gc(T)
Full result= lattice QCD + weak [Laine,Meyer,Schroeder] +photon + neutrinos + leptons
Axions and lattice QCD
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Determination of axion potentialChallenge
Determine the blue/red ratio by random pick!
−→ getting very difficult with T −→
SolutionSeparate colors and determine the rate of change with T !
Axions and lattice QCD
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Axion potential χ(T )
10-12
10-10
10-8
10-6
10-4
10-2
100
102
100 200 500 1000 2000
χ[f
m-4
]
T[MeV]
10-4
10-3
10-2
10-1
100 150 200 250
Two challenges to solve:1. signal is small 2. lattice artefacts are large
Axions and lattice QCD
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Comparison to others
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100 200 500
χ[f
m-4
]
T[MeV]
1512.06746
1606.03145 χt
1606.03145 m2χdisc
this work
Axions and lattice QCD
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ResultsAll dark matter is axion: ΩDM ≡ Ωa(ma, θ0) → ma(θ0)
10-4
10-3
10-2
10-1
100
10-6
10-5
10-4
10-3
10-2
10-1
100
101
102
103
post-inflationmisalignmentrange
pre-inflationinitial angle
Θ0
mA [µeV]
post-inflation: average all possible θ0 values →〈ma(θ0)〉 = 28(1)µeV
(If not all DM is axion, then this is a lower bound.)pre-inflation: single θ0 in Universe, ma can be anything
Axions and lattice QCD