View
9
Download
0
Category
Preview:
Citation preview
Axions and lattice QCDKalman Szabo
Forschungszentrum JulichUniversity of Wuppertal
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
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
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
Is P a symmetry?
Axions and lattice QCD
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
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
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
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
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
Higgs interaction
Higgs mechanism
Axions and lattice QCD
Higgs interaction
figs/bush.png
Axions and lattice QCD
Higgs interaction
figs/bush.png
”Left hand knows what the right hand is doing.”
Axions and lattice QCD
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
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
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
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
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
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
The P puzzle
electromagneticstrongHiggsweak
←− by Lorentz+gauge invariance
why GG not appears in Nature?←−←−
Axions and lattice QCD
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
Lattice QCD computation
Discretize!
lattice spacing a . 110 proton size
Calculate!
109 dimensional integrals using supercomputers
Axions and lattice QCD
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Recommended