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The missing 95%: Theory and Phenomenology ofDark Matter and Dark Energy
Joachim Kopp
DPG Spring Meeting Göttingen, March 2012
Fermilab
Joachim Kopp Dark Matter and Dark Energy 1
Outline
1 Evidence for dark matter
2 Finding dark matterDirect detectionIndirect detectionProduction at colliders
3 Modelling dark matter
4 Dark energy
Joachim Kopp Dark Matter and Dark Energy 2
Outline
1 Evidence for dark matter
2 Finding dark matterDirect detectionIndirect detectionProduction at colliders
3 Modelling dark matter
4 Dark energy
Joachim Kopp Dark Matter and Dark Energy 3
Celestial mechanics
Stellar/galactic dynamics relates:
The mass distribution(inferred from brightness)
Kinetic energy(inferred from Doppler shifts) Fritz Zwicky
1898–1974Vera Rubin
1928–
Observations of rotational velocities in galaxies show: Rubin 1975
The gravitational pull onperipheral stars is stronger thanpredicted from the mass of theluminous matter M
mv2
r= GN
mMr2 at r →∞
Joachim Kopp Dark Matter and Dark Energy 4
Collisions of galaxy clusters
Artist’s rendering (Image: NASA)
red = gas (from x-ray observations)blue = (dark) matter distribution (from gravitational lensing)
Joachim Kopp Dark Matter and Dark Energy 5
Collisions of galaxy clusters
Image: NASA (Chandra [x-ray], ESO WFI [lensing], HST [optical])
red = gas (from x-ray observations)blue = (dark) matter distribution (from gravitational lensing)
Joachim Kopp Dark Matter and Dark Energy 5
The Cosmic Microwave Background (CMB)
WMAP’s observation of the CMB: A fingerprint of the universeat t ' 300 000 yrs(when electrons and protons first combined to form atoms).
red = overdense, hot regions (0 . . . + 200 µK)blue = underdense, cold regions (−200 . . . 0 µK)
Joachim Kopp Dark Matter and Dark Energy 6
Image credit: NASA
The Cosmic Microwave Background (CMB)
WMAP’s observation of the CMB: A fingerprint of the universeat t ' 300 000 yrs(when electrons and protons first combined to form atoms).
More useful: The CMB fluctuation power spectrum
Joachim Kopp Dark Matter and Dark Energy 6
Image credit: NASA
The Cosmic Microwave Background (CMB)
WMAP’s observation of the CMB: A fingerprint of the universeat t ' 300 000 yrs(when electrons and protons first combined to form atoms).
red curve = theory predictionblack points = WMAP data
Joachim Kopp Dark Matter and Dark Energy 6
Image credit: NASA
The Cosmic Microwave Background (CMB)
WMAP’s observation of the CMB: A fingerprint of the universeat t ' 300 000 yrs(when electrons and protons first combined to form atoms).
too little DM (0.04ρc) right amount of DM (0.22ρc) too much DM (0.74ρc)
Joachim Kopp Dark Matter and Dark Energy 6
Image credit: NASA
What is this stuff?Modified laws of gravity?
I Hard to explain all observationsMACHOs (Massive Compact Halo Objects)?
I Planets, Brown dwarfs, neutron stars, . . .I Ruled out as dark matter in the mass range 0.6× 10−7M < M < 15M by
searches for gravitational microlensingI Searches for candidate objects yield too few of them
Hot (relativistic) Dark Matter (neutrinos or other relativistic particles)?I Cannot explain large scale structure of the universe
(hot dark matter would smoothen the galaxy distribution)
Cold or Warm Dark MatterI Axions
F Ultra-light, but non-relativistic due to non-thermal productionI Gravitinos
F Only gravitational couplings→ bad for direct/indirect/collider detectionI WIMPs (Weakly Interacting Massive Particle)
F New, heavy, stable particlesF Should have some non-gravitational interaction with SM particles for production
in the early universe
Joachim Kopp Dark Matter and Dark Energy 7
Outline
1 Evidence for dark matter
2 Finding dark matterDirect detectionIndirect detectionProduction at colliders
3 Modelling dark matter
4 Dark energy
Joachim Kopp Dark Matter and Dark Energy 8
Outline
1 Evidence for dark matter
2 Finding dark matterDirect detectionIndirect detectionProduction at colliders
3 Modelling dark matter
4 Dark energy
Joachim Kopp Dark Matter and Dark Energy 9
Direct Dark Matter detection
Idea: A WIMP (Weakly Interacting Massive Particle) can scatteron an atomic nucleus.
t
f
χ
f
χ
Strategy: Look for feeble nuclear recoil
Problem: Many background processes (radioactive decays, cosmic rays, . . . )can mimic the signal
Joachim Kopp Dark Matter and Dark Energy 10
Direct DM detection — The experimental challenge
background
suppression
Joachim Kopp Dark Matter and Dark Energy 11
Direct DM detection — The experimental challenge
background
suppression
Joachim Kopp Dark Matter and Dark Energy 11
Direct detection results
10 100 1000WIMP mass [GeV]
10-9
10-8
10-7
10-6
10-5
10-4
10-3
WIM
P-n
ucle
on c
ross s
ection [pb]
CRESST 1σ
CRESST 2σ
CRESST 2009EDELWEISS-IICDMS-IIXENON100DAMA chan.DAMACoGeNT
M2
M1
CRESST-II, arXiv:1109.0702
Assumptions here: Elastic DM scattering ∝ target mass (often realized in SUSY)Joachim Kopp Dark Matter and Dark Energy 12
Direct detection phenomenology of alternative models
Previous slide: Elastic dark matter (χ) scattering through scalar current[(qq)(χχ)] or vector current [(qγµq)(χγµχ)] assumed⇒ Cross section ∝ target massIn models with different coupling structure, the relative detectionefficiencies of different experimental technologies may be different
Joachim Kopp Dark Matter and Dark Energy 13
Direct detection phenomenology of alternative modelsSpin-dependent couplings
I E.g. coupling through axial vector current [(qγµγ5q)(χγµγ5χ)]I Cross section ∝ target spinI Cannot explain DAMA, CoGeNT, CRESST results
Inelastic dark matter Tucker-Smith Weiner hep-ph/0101138
I There may be two DM states χ and χ′ with m′χ = mχ + δ (δ ∼ 100 keV)
I Scattering χN → χ′N ⇒ heavy target nuclei kinematically preferredI Could explain CRESST, but not DAMA JK Schwetz Zupan 1110.2721
Leptophilic dark matter Bernabei et al. 0712.0562; Fox Poppitz arXiv:0811.0399; JK Niro Schwetz Zupan arXiv:0907.3159
Isospin-violating dark matter Feng Kumar Marfatia Sanford 1102.4331
. . .
Joachim Kopp Dark Matter and Dark Energy 14
Direct detection phenomenology of alternative modelsSpin-dependent couplings
I E.g. coupling through axial vector current [(qγµγ5q)(χγµγ5χ)]I Cross section ∝ target spinI Cannot explain DAMA, CoGeNT, CRESST results
Inelastic dark matter Tucker-Smith Weiner hep-ph/0101138
I There may be two DM states χ and χ′ with m′χ = mχ + δ (δ ∼ 100 keV)
I Scattering χN → χ′N ⇒ heavy target nuclei kinematically preferredI Could explain CRESST, but not DAMA JK Schwetz Zupan 1110.2721
Leptophilic dark matter Bernabei et al. 0712.0562; Fox Poppitz arXiv:0811.0399; JK Niro Schwetz Zupan arXiv:0907.3159
Isospin-violating dark matter Feng Kumar Marfatia Sanford 1102.4331
. . .
Joachim Kopp Dark Matter and Dark Energy 14
30 40 5010-41
10-40
10-39
10-38
10-37
mΧ @GeVD
WIM
P-nu
cleo
ncr
oss
sect
ion
ΣSI
@cm2 D
iDM, ∆=90keV
Limits: 90%
Contours: 90%, 3Σ
v0 = 220 kms, vesc = 550 kms
Xenon
-100
KIM
S
CRESST-II
DAMA
CR
ESST
comm
. run
Direct detection phenomenology of alternative modelsSpin-dependent couplings
I E.g. coupling through axial vector current [(qγµγ5q)(χγµγ5χ)]I Cross section ∝ target spinI Cannot explain DAMA, CoGeNT, CRESST results
Inelastic dark matter Tucker-Smith Weiner hep-ph/0101138
I There may be two DM states χ and χ′ with m′χ = mχ + δ (δ ∼ 100 keV)
I Scattering χN → χ′N ⇒ heavy target nuclei kinematically preferredI Could explain CRESST, but not DAMA JK Schwetz Zupan 1110.2721
Leptophilic dark matter Bernabei et al. 0712.0562; Fox Poppitz arXiv:0811.0399; JK Niro Schwetz Zupan arXiv:0907.3159
Isospin-violating dark matter Feng Kumar Marfatia Sanford 1102.4331
. . .
Joachim Kopp Dark Matter and Dark Energy 14
Direct detection phenomenology of alternative modelsSpin-dependent couplings
I E.g. coupling through axial vector current [(qγµγ5q)(χγµγ5χ)]I Cross section ∝ target spinI Cannot explain DAMA, CoGeNT, CRESST results
Inelastic dark matter Tucker-Smith Weiner hep-ph/0101138
I There may be two DM states χ and χ′ with m′χ = mχ + δ (δ ∼ 100 keV)
I Scattering χN → χ′N ⇒ heavy target nuclei kinematically preferredI Could explain CRESST, but not DAMA JK Schwetz Zupan 1110.2721
Leptophilic dark matter Bernabei et al. 0712.0562; Fox Poppitz arXiv:0811.0399; JK Niro Schwetz Zupan arXiv:0907.3159
Isospin-violating dark matter Feng Kumar Marfatia Sanford 1102.4331
. . .
Joachim Kopp Dark Matter and Dark Energy 14
Direct detection phenomenology of alternative modelsSpin-dependent couplings
I E.g. coupling through axial vector current [(qγµγ5q)(χγµγ5χ)]I Cross section ∝ target spinI Cannot explain DAMA, CoGeNT, CRESST results
Inelastic dark matter Tucker-Smith Weiner hep-ph/0101138
I There may be two DM states χ and χ′ with m′χ = mχ + δ (δ ∼ 100 keV)
I Scattering χN → χ′N ⇒ heavy target nuclei kinematically preferredI Could explain CRESST, but not DAMA JK Schwetz Zupan 1110.2721
Leptophilic dark matter Bernabei et al. 0712.0562; Fox Poppitz arXiv:0811.0399; JK Niro Schwetz Zupan arXiv:0907.3159
Isospin-violating dark matter Feng Kumar Marfatia Sanford 1102.4331
. . .
Conclusion: Hard to explain all data simultaneously
Joachim Kopp Dark Matter and Dark Energy 14
Direct detection uncertainties
Large uncertainty in local DM density
Large uncertainties in DM velocity distributionI Scattering rate depends strongly on DM velocityI DM streams?I Debris flow?
Predicting WIMP–nucleus cross sections is difficultI Models predict WIMP–quark cross sectionI Need to know quark content of the nucleonI Especially problematic for Higgs-mediated scattering:
coupling ∝ quark mass⇒ sea quarks dominateI Need to know nuclear form factor
especially difficult for spin-dependent scattering
Joachim Kopp Dark Matter and Dark Energy 15
Direct detection uncertainties
Large uncertainty in local DM densityLarge uncertainties in DM velocity distribution
I Scattering rate depends strongly on DM velocityI DM streams?I Debris flow?
Kuhlen Lisanti Spergel arXiv:1202.0007, graphics courtesy of Mariangela Lisanti
Predicting WIMP–nucleus cross sections is difficultI Models predict WIMP–quark cross sectionI Need to know quark content of the nucleonI Especially problematic for Higgs-mediated scattering:
coupling ∝ quark mass⇒ sea quarks dominateI Need to know nuclear form factor
especially difficult for spin-dependent scattering
Joachim Kopp Dark Matter and Dark Energy 15
Direct detection uncertainties
Large uncertainty in local DM densityLarge uncertainties in DM velocity distribution
I Scattering rate depends strongly on DM velocityI DM streams?I Debris flow?
Predicting WIMP–nucleus cross sections is difficultI Models predict WIMP–quark cross sectionI Need to know quark content of the nucleonI Especially problematic for Higgs-mediated scattering:
coupling ∝ quark mass⇒ sea quarks dominateI Need to know nuclear form factor
especially difficult for spin-dependent scattering
Joachim Kopp Dark Matter and Dark Energy 15
Outline
1 Evidence for dark matter
2 Finding dark matterDirect detectionIndirect detectionProduction at colliders
3 Modelling dark matter
4 Dark energy
Joachim Kopp Dark Matter and Dark Energy 16
Indirect Dark Matter detection
Idea: WIMPs (Weakly Interacting Massive Particles) χ can annihilate(or decay) into Standard Model particles (f ) in an astrophysical environment.
t
f
χ
f
χ
Strategy: Look for annihilation products in cosmic rays
Problems:Many other sources of cosmic raysPropagation of charged particles in the galaxy poorly understood
Advantage:Many sources to look at
Joachim Kopp Dark Matter and Dark Energy 17
Indirect DM detection — The experimental challenge
look atmany sources
Joachim Kopp Dark Matter and Dark Energy 18
Indirect DM detection — The experimental challenge
look atmany sources
Joachim Kopp Dark Matter and Dark Energy 18
Indirect DM detection — The experimental challenge
look atmany sources
Joachim Kopp Dark Matter and Dark Energy 18
Indirect DM detection — The experimental challenge
look atmany sources
Joachim Kopp Dark Matter and Dark Energy 18
Indirect DM detection — The experimental challenge
look atmany sources
Joachim Kopp Dark Matter and Dark Energy 18
Indirect DM detection — The experimental challenge
look atmany sources
Joachim Kopp Dark Matter and Dark Energy 18
Indirect DM detection — The experimental challenge
look atmany sources
Joachim Kopp Dark Matter and Dark Energy 18
Indirect DM detection — Examples
γ-rays from dwarf galaxiesIdea:Look for anomalous γ-ray flux
Pro:Few stars ⇒ few backgrounds
Con:
Relatively low DM densityResults model-dependentLarge astrophysicaluncertainties
Other indirect DM searches:Cosmic anti-matter (e+, p, . . . ) PAMELA, Fermil-LAT, . . .
γ-rays from the galactic center Hooper et al.
High-energy neutrinos from the Sun IceCube, SuperKamiokande
. . .
Joachim Kopp Dark Matter and Dark Energy 19
Indirect DM detection — Examples
γ-rays from dwarf galaxies
101 102 103
WIMP mass [GeV]
10-26
10-25
10-24
10-23
10-22
10-21
WIM
Pcr
oss
sect
ion
[cm3
/s]
Upper limits, Joint Likelihood of 10 dSphs
3*10−26
bb Channel
τ+ τ− Channel
µ + µ− Channel
W+W− Channel
Thermal relic cross section=> Correct relic abundance
Fermi-LAT 1108.3546see also Geringer-Sameth Koushiappas 1108.2914
Idea:Look for anomalous γ-ray flux
Pro:Few stars ⇒ few backgrounds
Con:
Relatively low DM densityResults model-dependentLarge astrophysicaluncertainties
Other indirect DM searches:Cosmic anti-matter (e+, p, . . . ) PAMELA, Fermil-LAT, . . .
γ-rays from the galactic center Hooper et al.
High-energy neutrinos from the Sun IceCube, SuperKamiokande
. . .
Joachim Kopp Dark Matter and Dark Energy 19
Indirect DM detection — Examples
γ-rays from dwarf galaxies
101 102 103
WIMP mass [GeV]
10-26
10-25
10-24
10-23
10-22
10-21
WIM
Pcr
oss
sect
ion
[cm3
/s]
Upper limits, Joint Likelihood of 10 dSphs
3*10−26
bb Channel
τ+ τ− Channel
µ + µ− Channel
W+W− Channel
Thermal relic cross section=> Correct relic abundance
Fermi-LAT 1108.3546see also Geringer-Sameth Koushiappas 1108.2914
Idea:Look for anomalous γ-ray flux
Pro:Few stars ⇒ few backgrounds
Con:
Relatively low DM densityResults model-dependentLarge astrophysicaluncertainties
Other indirect DM searches:Cosmic anti-matter (e+, p, . . . ) PAMELA, Fermil-LAT, . . .
γ-rays from the galactic center Hooper et al.
High-energy neutrinos from the Sun IceCube, SuperKamiokande
. . .Joachim Kopp Dark Matter and Dark Energy 19
Outline
1 Evidence for dark matter
2 Finding dark matterDirect detectionIndirect detectionProduction at colliders
3 Modelling dark matter
4 Dark energy
Joachim Kopp Dark Matter and Dark Energy 20
Dark matter at colliders
make your
own needles!
Joachim Kopp Dark Matter and Dark Energy 21
Dark matter at colliders
make your
own needles!
Joachim Kopp Dark Matter and Dark Energy 21
Generic collider searches for dark matter
Idea:Produce WIMPs in collisions of Standard Model particles
t
f
χ
f
χ
WIMPs can recoil against a jet or a photon from initial state radiation
t
q
χ
q
g
χ
t
e−
χ
e+
γ
χ
Experimental signatures: Mono-jets + /ET and mono-photons + /E
Joachim Kopp Dark Matter and Dark Energy 22
LHC limits on DM–quark couplings Plots from Fox Harnik JK Tsai 1109.4398see also work by Rajaraman et al. 1108.1196
90% C.L.
10-1 100 101 102 10310-46
10-45
10-44
10-43
10-42
10-41
10-40
10-39
10-38
10-37
WIMP mass mΧ @GeVD
WIM
P-nu
cleo
ncr
oss
sect
ion
ΣN
@cm
2 D
ATLAS 7TeV, 1fb-1 VeryHighPt
Spin-independent
Solid : ObservedDashed : Expected
ΧΓΜ Χ qΓΜq
Αs Χ Χ GΜΝGΜΝ
CDMSXENON-10
XENON-100
DAMA Hq ± 33%L
CoGeNT CRESST
90% C.L.
10-1 100 101 102 103
10-41
10-40
10-39
10-38
10-37
10-36
10-35
10-34
WIMP mass mΧ @GeVD
WIM
P-nu
cleo
ncr
oss
sect
ion
ΣN
@cm
2 D
ATLAS 7TeV, 1fb-1 VeryHighPt
Spin-dependent
Solid : ObservedDashed : Expected
ΧΓΜΓ5 Χ qΓΜΓ5q
XENON-10PICASSO
COUPP
SIMPLE
DAMAHq ± 33%L
Assumptions here:I Effective field theory approach valid (limits may be better or worse if EFT not valid)I Equal coupling to all quark flavors
Extremely competitive limits forI Light dark matter (below direct detection threshold)I DM coupled to gluons (high gluon luminosity at the LHC)I Spin-dependent DM interactions (DD suffers from loss of coherence)
Joachim Kopp Dark Matter and Dark Energy 23
Model-dependent collider searches: SUSY-DM
Idea:In many models, DM is produced in the decay of heavy, stronglyinteracting particles (for instance squarks and gluinos in SUSY)
Experimental signature: something + missing energyExample: pp → (g → jZχ0)(q → jjWχ0)
Advantage: Very sensitiveProblem:Minor modifications to the modelmay drastically change the phenomenologyProblem (all collider searches):Collider can only find DM candidate(s)
Joachim Kopp Dark Matter and Dark Energy 24
Outline
1 Evidence for dark matter
2 Finding dark matterDirect detectionIndirect detectionProduction at colliders
3 Modelling dark matter
4 Dark energy
Joachim Kopp Dark Matter and Dark Energy 25
Electroweak-scale DM? — The “WIMP Miracle”
In the early universe, DM is in chemical equilibrium with other particles.
As the temperature drops, DM begins to annihilate away: χχ→ f fWhen the annihilation rate Γ(χχ→ f f ) drops below the Hubble expansionrate H, annihilations cease⇒ DM abundance remains constant (“thermal freeze-out”)From this requirement, and from the observed DM abundance today,cosmology predicts the DM annihilation cross section
〈σv〉 ' 3× 10−26 cm3/s
Consider generic DM coupling:
L ⊃ g2
M2 (χχ)(f f )
For typical coupling g ∼ 0.1, suppression scale M ∼ 100 GeV, DM massmχ ∼ 100 GeV, this yields the right value for 〈σv〉Conclusion: If dark matter originates from electroweak-scalenew physics, it automatically has the right abundance
The Wimp Miracle
Joachim Kopp Dark Matter and Dark Energy 26
Electroweak-scale DM? — The “WIMP Miracle”
In the early universe, DM is in chemical equilibrium with other particles.As the temperature drops, DM begins to annihilate away: χχ→ f f
When the annihilation rate Γ(χχ→ f f ) drops below the Hubble expansionrate H, annihilations cease⇒ DM abundance remains constant (“thermal freeze-out”)From this requirement, and from the observed DM abundance today,cosmology predicts the DM annihilation cross section
〈σv〉 ' 3× 10−26 cm3/s
Consider generic DM coupling:
L ⊃ g2
M2 (χχ)(f f )
For typical coupling g ∼ 0.1, suppression scale M ∼ 100 GeV, DM massmχ ∼ 100 GeV, this yields the right value for 〈σv〉Conclusion: If dark matter originates from electroweak-scalenew physics, it automatically has the right abundance
The Wimp Miracle
Joachim Kopp Dark Matter and Dark Energy 26
Electroweak-scale DM? — The “WIMP Miracle”
In the early universe, DM is in chemical equilibrium with other particles.As the temperature drops, DM begins to annihilate away: χχ→ f fWhen the annihilation rate Γ(χχ→ f f ) drops below the Hubble expansionrate H, annihilations cease⇒ DM abundance remains constant (“thermal freeze-out”)
From this requirement, and from the observed DM abundance today,cosmology predicts the DM annihilation cross section
〈σv〉 ' 3× 10−26 cm3/s
Consider generic DM coupling:
L ⊃ g2
M2 (χχ)(f f )
For typical coupling g ∼ 0.1, suppression scale M ∼ 100 GeV, DM massmχ ∼ 100 GeV, this yields the right value for 〈σv〉Conclusion: If dark matter originates from electroweak-scalenew physics, it automatically has the right abundance
The Wimp Miracle
Joachim Kopp Dark Matter and Dark Energy 26
Electroweak-scale DM? — The “WIMP Miracle”
In the early universe, DM is in chemical equilibrium with other particles.As the temperature drops, DM begins to annihilate away: χχ→ f fWhen the annihilation rate Γ(χχ→ f f ) drops below the Hubble expansionrate H, annihilations cease⇒ DM abundance remains constant (“thermal freeze-out”)From this requirement, and from the observed DM abundance today,cosmology predicts the DM annihilation cross section
〈σv〉 ' 3× 10−26 cm3/s
Consider generic DM coupling:
L ⊃ g2
M2 (χχ)(f f )
For typical coupling g ∼ 0.1, suppression scale M ∼ 100 GeV, DM massmχ ∼ 100 GeV, this yields the right value for 〈σv〉Conclusion: If dark matter originates from electroweak-scalenew physics, it automatically has the right abundance
The Wimp Miracle
Joachim Kopp Dark Matter and Dark Energy 26
Electroweak-scale DM? — The “WIMP Miracle”
In the early universe, DM is in chemical equilibrium with other particles.As the temperature drops, DM begins to annihilate away: χχ→ f fWhen the annihilation rate Γ(χχ→ f f ) drops below the Hubble expansionrate H, annihilations cease⇒ DM abundance remains constant (“thermal freeze-out”)From this requirement, and from the observed DM abundance today,cosmology predicts the DM annihilation cross section
〈σv〉 ' 3× 10−26 cm3/s
Consider generic DM coupling:
L ⊃ g2
M2 (χχ)(f f )
For typical coupling g ∼ 0.1, suppression scale M ∼ 100 GeV, DM massmχ ∼ 100 GeV, this yields the right value for 〈σv〉Conclusion: If dark matter originates from electroweak-scalenew physics, it automatically has the right abundance
The Wimp Miracle
Joachim Kopp Dark Matter and Dark Energy 26
χ
χ
f
f
←→
χ
χ
f
f
Electroweak-scale DM? — The “WIMP Miracle”
In the early universe, DM is in chemical equilibrium with other particles.As the temperature drops, DM begins to annihilate away: χχ→ f fWhen the annihilation rate Γ(χχ→ f f ) drops below the Hubble expansionrate H, annihilations cease⇒ DM abundance remains constant (“thermal freeze-out”)From this requirement, and from the observed DM abundance today,cosmology predicts the DM annihilation cross section
〈σv〉 ' 3× 10−26 cm3/s
Consider generic DM coupling:
L ⊃ g2
M2 (χχ)(f f )
For typical coupling g ∼ 0.1, suppression scale M ∼ 100 GeV, DM massmχ ∼ 100 GeV, this yields the right value for 〈σv〉
Conclusion: If dark matter originates from electroweak-scalenew physics, it automatically has the right abundance
The Wimp Miracle
Joachim Kopp Dark Matter and Dark Energy 26
χ
χ
f
f
←→
χ
χ
f
f
Electroweak-scale DM? — The “WIMP Miracle”
In the early universe, DM is in chemical equilibrium with other particles.As the temperature drops, DM begins to annihilate away: χχ→ f fWhen the annihilation rate Γ(χχ→ f f ) drops below the Hubble expansionrate H, annihilations cease⇒ DM abundance remains constant (“thermal freeze-out”)From this requirement, and from the observed DM abundance today,cosmology predicts the DM annihilation cross section
〈σv〉 ' 3× 10−26 cm3/s
Consider generic DM coupling:
L ⊃ g2
M2 (χχ)(f f )
For typical coupling g ∼ 0.1, suppression scale M ∼ 100 GeV, DM massmχ ∼ 100 GeV, this yields the right value for 〈σv〉Conclusion: If dark matter originates from electroweak-scalenew physics, it automatically has the right abundance
The Wimp MiracleJoachim Kopp Dark Matter and Dark Energy 26
χ
χ
f
f
←→
χ
χ
f
f
Relating the DM and baryon abundances
Motivation: The DM and baryon energy densities in the universeare similar
ΩDM ' 5 Ωb
(Ω = energy density as fraction of “critical density” for flat universe)
If the DM and baryon number densities are similar and
mDM ∼ 5mp–10mp ∼ 5–10 GeV ,
this is quite natural.This is precisely the mass range where the direct detection hints(DAMA, CoGeNT, CRESST) have been observed!Baryon density Ωb generated by yet unknown dynamics behind theparticle–antiparticle asymmetry of the universe(not by thermal freeze-out)Assume dark matter (χ) density is also determined by χ–χ asymmetry⇒ Asymmetric dark matter
Joachim Kopp Dark Matter and Dark Energy 27
Relating the DM and baryon abundances
Motivation: The DM and baryon energy densities in the universeare similar
ΩDM ' 5 Ωb
(Ω = energy density as fraction of “critical density” for flat universe)
If the DM and baryon number densities are similar and
mDM ∼ 5mp–10mp ∼ 5–10 GeV ,
this is quite natural.
This is precisely the mass range where the direct detection hints(DAMA, CoGeNT, CRESST) have been observed!Baryon density Ωb generated by yet unknown dynamics behind theparticle–antiparticle asymmetry of the universe(not by thermal freeze-out)Assume dark matter (χ) density is also determined by χ–χ asymmetry⇒ Asymmetric dark matter
Joachim Kopp Dark Matter and Dark Energy 27
Relating the DM and baryon abundances
Motivation: The DM and baryon energy densities in the universeare similar
ΩDM ' 5 Ωb
(Ω = energy density as fraction of “critical density” for flat universe)
If the DM and baryon number densities are similar and
mDM ∼ 5mp–10mp ∼ 5–10 GeV ,
this is quite natural.This is precisely the mass range where the direct detection hints(DAMA, CoGeNT, CRESST) have been observed!
Baryon density Ωb generated by yet unknown dynamics behind theparticle–antiparticle asymmetry of the universe(not by thermal freeze-out)Assume dark matter (χ) density is also determined by χ–χ asymmetry⇒ Asymmetric dark matter
Joachim Kopp Dark Matter and Dark Energy 27
Relating the DM and baryon abundances
Motivation: The DM and baryon energy densities in the universeare similar
ΩDM ' 5 Ωb
(Ω = energy density as fraction of “critical density” for flat universe)
If the DM and baryon number densities are similar and
mDM ∼ 5mp–10mp ∼ 5–10 GeV ,
this is quite natural.This is precisely the mass range where the direct detection hints(DAMA, CoGeNT, CRESST) have been observed!Baryon density Ωb generated by yet unknown dynamics behind theparticle–antiparticle asymmetry of the universe(not by thermal freeze-out)
Assume dark matter (χ) density is also determined by χ–χ asymmetry⇒ Asymmetric dark matter
Joachim Kopp Dark Matter and Dark Energy 27
Relating the DM and baryon abundances
Motivation: The DM and baryon energy densities in the universeare similar
ΩDM ' 5 Ωb
(Ω = energy density as fraction of “critical density” for flat universe)
If the DM and baryon number densities are similar and
mDM ∼ 5mp–10mp ∼ 5–10 GeV ,
this is quite natural.This is precisely the mass range where the direct detection hints(DAMA, CoGeNT, CRESST) have been observed!Baryon density Ωb generated by yet unknown dynamics behind theparticle–antiparticle asymmetry of the universe(not by thermal freeze-out)Assume dark matter (χ) density is also determined by χ–χ asymmetry⇒ Asymmetric dark matter
Joachim Kopp Dark Matter and Dark Energy 27
Models of asymmetric dark matter
Example 1 Kaplan Luty Zurek, arXiv:0901.4117
B − L asymmetry generated athigh T (e.g. via Leptogenesis)
Effective superfield operator
L ⊃ 1M
X 2LHu (*)
transfers B − L ↔ 2X , e.g. via
χ0
ν ν
ν
X
X
ν
X
X
Final X (DM number) asymmetrydepends on # of SM speciescontributing to (*) at freeze-out
Example 2 Buckley Randall 1009.0270Blennow et al. 1009.3159
Generate X asymmetry inhidden sectorTransfer to B − L asymmetry inthe SM sector
I via B − L violatinginteractions (e.g. (*))
I via sphaleron processes
Example 3 Davoudiasl et al. 1008.2399Gu Lindner Sarkar Zhang 1009.2690
New heavy particles decaypartly into DM, partly into SMparticlesB − L− X is conservedDM (X ) does not participate inSM sphaleron processes⇒ Asymmetry frozen in
Joachim Kopp Dark Matter and Dark Energy 28
Models of asymmetric dark matter
Example 1 Kaplan Luty Zurek, arXiv:0901.4117
B − L asymmetry generated athigh T (e.g. via Leptogenesis)
Effective superfield operator
L ⊃ 1M
X 2LHu (*)
transfers B − L ↔ 2X , e.g. via
χ0
ν ν
ν
X
X
ν
X
X
Final X (DM number) asymmetrydepends on # of SM speciescontributing to (*) at freeze-out
Example 2 Buckley Randall 1009.0270Blennow et al. 1009.3159
Generate X asymmetry inhidden sectorTransfer to B − L asymmetry inthe SM sector
I via B − L violatinginteractions (e.g. (*))
I via sphaleron processes
Example 3 Davoudiasl et al. 1008.2399Gu Lindner Sarkar Zhang 1009.2690
New heavy particles decaypartly into DM, partly into SMparticlesB − L− X is conservedDM (X ) does not participate inSM sphaleron processes⇒ Asymmetry frozen in
Joachim Kopp Dark Matter and Dark Energy 28
Models of asymmetric dark matter
Example 1 Kaplan Luty Zurek, arXiv:0901.4117
B − L asymmetry generated athigh T (e.g. via Leptogenesis)
Effective superfield operator
L ⊃ 1M
X 2LHu (*)
transfers B − L ↔ 2X , e.g. via
χ0
ν ν
ν
X
X
ν
X
X
Final X (DM number) asymmetrydepends on # of SM speciescontributing to (*) at freeze-out
Example 2 Buckley Randall 1009.0270Blennow et al. 1009.3159
Generate X asymmetry inhidden sectorTransfer to B − L asymmetry inthe SM sector
I via B − L violatinginteractions (e.g. (*))
I via sphaleron processes
Example 3 Davoudiasl et al. 1008.2399Gu Lindner Sarkar Zhang 1009.2690
New heavy particles decaypartly into DM, partly into SMparticlesB − L− X is conservedDM (X ) does not participate inSM sphaleron processes⇒ Asymmetry frozen in
Joachim Kopp Dark Matter and Dark Energy 28
Outline
1 Evidence for dark matter
2 Finding dark matterDirect detectionIndirect detectionProduction at colliders
3 Modelling dark matter
4 Dark energy
Joachim Kopp Dark Matter and Dark Energy 29
Evidence for dark energy: Type Ia Supernovae
When a white dwarf accretes matter from acompanion star, it becomes unstableonce it reaches ∼ 1.4M
I Re-ignition of nuclear fusionI Thermonuclear explosion
Since the progenitor mass is always ∼ 1.4M,all Type Ia Supernovae are very similar
I Energy release precisely knownI SN Ia are standard candles
Measurement:I Apparent brightness→ distanceI Redshift→ velocity
Result:I Long ago (very distant SN Ia,
low brightness), the universe wasexpanding more slowlythan we thought!
I It must be acceleratingCMB and Large Scale Structureobservations confirm this
Joachim Kopp Dark Matter and Dark Energy 30
http://hyperphysics.phy-astr.gsu.edu/hbase/astro/univacc.html
What is accelerating the Universe?
A cosmological constant?I An ad-hoc addition to the Einstein equations
Rµν −12
gµνR αα = 8πG Tµν + gµνΛ
I Observations require Λ ∼ (10−12 GeV)4
I Extra source of energy with negative pressure
QFT vacuum energy?I A vacuum expectation value (vev) or condensate of a quantum field
behaves like a cosmological constantI Problem: All known condensates/vevs are way too large!
(We expect Λ ∼ M4Pl ∼ (1019 GeV)4)
Joachim Kopp Dark Matter and Dark Energy 31
What is accelerating the Universe?
A cosmological constant?I An ad-hoc addition to the Einstein equations
Rµν −12
gµνR αα = 8πG Tµν + gµνΛ
I Observations require Λ ∼ (10−12 GeV)4
I Extra source of energy with negative pressure
QFT vacuum energy?I A vacuum expectation value (vev) or condensate of a quantum field
behaves like a cosmological constantI Problem: All known condensates/vevs are way too large!
(We expect Λ ∼ M4Pl ∼ (1019 GeV)4)
Joachim Kopp Dark Matter and Dark Energy 31
What is accelerating the Universe? (cont’d)
Quintessence: A new, slowly rolling scalar fieldI Introduce new scalar field φ slowly rolling down its potential V (φ)I Lagrangian:
Lφ =12
∂µφ∂µφ− V (φ)
I Energy and pressure:
ρ =12
φ2 + V (φ) , p =12
φ2 − V (φ)
I A cosmological constant corresponds to ρ = −p ⇒ require φ2 V (φ)
Extensions of general relativityI Scalar-tensor gravity: Modified Einstein-Hilbert action
S =1
16πG
Z p−g d4x R → S =
116πG
Z p−g d4x f (φ)× R
I A special case: f (R) gravity:
S =1
16πG
Z p−g d4x f (R)
Joachim Kopp Dark Matter and Dark Energy 32
for a review see Caldwell Kamionkowski 0903.0866
What is accelerating the Universe? (cont’d)
Quintessence: A new, slowly rolling scalar fieldI Introduce new scalar field φ slowly rolling down its potential V (φ)I Lagrangian:
Lφ =12
∂µφ∂µφ− V (φ)
I Energy and pressure:
ρ =12
φ2 + V (φ) , p =12
φ2 − V (φ)
I A cosmological constant corresponds to ρ = −p ⇒ require φ2 V (φ)
Extensions of general relativityI Scalar-tensor gravity: Modified Einstein-Hilbert action
S =1
16πG
Z p−g d4x R → S =
116πG
Z p−g d4x f (φ)× R
I A special case: f (R) gravity:
S =1
16πG
Z p−g d4x f (R)
Joachim Kopp Dark Matter and Dark Energy 32
for a review see Caldwell Kamionkowski 0903.0866
Summary
Overwhelming evidence for dark matterA lot of data available
I Direct detectionF Difficult to reconcile possible evidence with null results
I Indirect searchesF Strong exclusion limitsF Suffers from poorly understood astrophysical backgrounds
I Collider searchesF Generic searches (monojets + /ET , mono-γ + /E) and model-specific searches
(cascade decays) are underway full-steam
Dark matter modelsI Dark matter from electroweak scale new physics:
Correct cosmic abundance due to WIMP MiracleI Light (10 GeV) dark matter:
Correct cosmic abundance if related to baryon–antibaryon asymmetry
Dark energyI Accelerated expansion of the Universe well-establishedI So far, a cosmological constant is the leading explanation
Joachim Kopp Dark Matter and Dark Energy 33
Thank you!
Bonus material
Spin-dependent DM couplings?
Previous slide: Dark matter (χ) couplings through scalar current[(qq)(χχ)] or vector current [(qγµq)(χγµχ)] assumed⇒ Cross section ∝ target massAlternative: Axial vector [(qγµγ5q)(χγµγ
5χ)] interaction⇒ Cross section ∝ target spin
)2WIMP mass (GeV/c10 210
310 410
(p
b)
pSD
σ
410
310
210
110
1
cMSSM
DAMA/LIBRA 2008
COUPP 2007
COUPP 2011
ICECUBE 2011
KIMS 2007
SUPERKAMIOKANDE 2011SIM
PLE 2010
PICASSO 2009
PICASSO 2012
PICASSO arXiv:1202.1240
Note: CoGeNT & CRESST have very low sensitivity to spin-dependent DM scattering.Joachim Kopp Dark Matter and Dark Energy 36
Inelastic dark matter?
Idea: There may be two DM states χ and χ′ with
mχ′ = mχ + δ
Scattering proceeds via
χ+ N → χ′ + N
Modified kinematics compared to elastic scatteringAffects different target nuclei differently
Joachim Kopp Dark Matter and Dark Energy 37
Tucker-Smith Weiner hep-ph/0101138
Inelastic dark matter?
Idea: There may be two DM states χ and χ′ with
mχ′ = mχ + δ
30 40 5010-41
10-40
10-39
10-38
10-37
mΧ @GeVD
WIM
P-nu
cleo
ncr
oss
sect
ion
ΣSI
@cm2 D
iDM, ∆=90keV
Limits: 90%
Contours: 90%, 3Σ
v0 = 220 kms, vesc = 550 kms
Xenon
-100
KIM
SCRESST-II
DAMA
CR
ESST
comm
. run
plot from JK Schwetz Zupan 1110.2721
Joachim Kopp Dark Matter and Dark Energy 37
Tucker-Smith Weiner hep-ph/0101138
Isospin-violating dark matter?
Idea: Dark matter could couple differently to protons and neutrons⇒ Detection efficiencies of different target materials change
-1.2 -1 -0.8 -0.6fn / f
p
10-4
10-3
10-2
10-1
A2 ef
f / A
2
-1.2 -1 -0.8 -0.6
Si / Ca /
O
Na
Ge
I
Xe
W
plot from JK Schwetz Zupan 1110.2721
fn, fp: DM couplings to protons and neutronsAeff: Effective nuclear mass for DM scattering
Joachim Kopp Dark Matter and Dark Energy 38
Feng Kumar Marfatia Sanford 1102.4331
Isospin-violating dark matter?
Idea: Dark matter could couple differently to protons and neutrons⇒ Detection efficiencies of different target materials change
101 102 103
10-41
10-40
10-39
10-38
10-37
mΧ @GeVD
WIM
P-nu
cleo
ncr
oss
sect
ion
Σef
f@cm
2 DIVDM, fn fp=-0.7
Limits: 90%
Contours: 90%, 3Σ
v0 = 220 kms, vesc = 550 kms
CDMS-SiCDMS low-thr
Xenon-100
KIMS
CRESST-II
DAMA
CRESST comm. run
CDMS
plot from JK Schwetz Zupan 1110.2721
Joachim Kopp Dark Matter and Dark Energy 38
Feng Kumar Marfatia Sanford 1102.4331
Leptophilic dark matter?
Idea: DM could couple only to leptons at tree levelDAMA and CoGeNT do not reject electron-recoils as backgroundBut: Electron recoils above threshold (& 1 keV) strongly suppressed(electron needs large initial momentum→ probe high-p tail of wave functions)
105 106 10710-16
10-14
10-12
10-10
10-8
10-6
p @eVD
p2ÈΧHpL
2@e
V-
1 D
Iodine
1s
2p 2s3d
3p 3s4d
4p 4s
5p 5s
105 106 10710-16
10-14
10-12
10-10
10-8
10-6
p @eVD
p2ÈΧHpL
2@e
V-
1 D
Sodium
1s
2p
2s
3s
Thus: DM–nucleus scattering dominates, even if loop-inducedBut: Loop diagrams forbidden for some modelsProblems then:
I Very large couplings needed to compensate wave function suppressionI Poor fit to DAMA and CoGeNT energy spectra
0 5 10 15 20
-0.04
-0.02
0.00
0.02
0.04
0.06
E @keVD
Sm@d-1kg-1keV-1D
Full analysis
wo 1st bin
Σe0 = 5.´ 10-31 cm2, mΧ = 816. GeV
Χ2 = 55.9
Σe0 = 5.01´ 10-31 cm2, mΧ = 202. GeV
Χ2 = 20.6
Joachim Kopp Dark Matter and Dark Energy 39
Fox Poppitz arXiv:0811.0399JK Niro Schwetz Zupan arXiv:0907.3159
Leptophilic dark matter?
Idea: DM could couple only to leptons at tree levelDAMA and CoGeNT do not reject electron-recoils as backgroundBut: Electron recoils above threshold (& 1 keV) strongly suppressedThus: DM–nucleus scattering dominates, even if loop-induced
ℓ−
ℓ−
γ
χ
χ
p+
p+
q
γ
γ
N , pµ
N , p′µ
χ, kµ
χ, k′µ
ℓ−
ℓ−
q
γ
γ
N , pµ
N , p′µ
χ, kµ
χ, k′µ
ℓ−
ℓ−
But: Loop diagrams forbidden for some modelsProblems then:
I Very large couplings needed to compensate wave function suppressionI Poor fit to DAMA and CoGeNT energy spectra
0 5 10 15 20
-0.04
-0.02
0.00
0.02
0.04
0.06
E @keVD
Sm@d-1kg-1keV-1D
Full analysis
wo 1st bin
Σe0 = 5.´ 10-31 cm2, mΧ = 816. GeV
Χ2 = 55.9
Σe0 = 5.01´ 10-31 cm2, mΧ = 202. GeV
Χ2 = 20.6
Joachim Kopp Dark Matter and Dark Energy 39
Fox Poppitz arXiv:0811.0399JK Niro Schwetz Zupan arXiv:0907.3159
Leptophilic dark matter?
Idea: DM could couple only to leptons at tree levelDAMA and CoGeNT do not reject electron-recoils as backgroundBut: Electron recoils above threshold (& 1 keV) strongly suppressedThus: DM–nucleus scattering dominates, even if loop-inducedBut: Loop diagrams forbidden for some modelse.g. axial vector couplings g2/M2(χγµγ5χ)(fγµγ5f )
Problems then:I Very large couplings needed to compensate wave function suppressionI Poor fit to DAMA and CoGeNT energy spectra
0 5 10 15 20
-0.04
-0.02
0.00
0.02
0.04
0.06
E @keVD
Sm@d-1kg-1keV-1D
Full analysis
wo 1st bin
Σe0 = 5.´ 10-31 cm2, mΧ = 816. GeV
Χ2 = 55.9
Σe0 = 5.01´ 10-31 cm2, mΧ = 202. GeV
Χ2 = 20.6
Joachim Kopp Dark Matter and Dark Energy 39
Fox Poppitz arXiv:0811.0399JK Niro Schwetz Zupan arXiv:0907.3159
Leptophilic dark matter?
Idea: DM could couple only to leptons at tree levelDAMA and CoGeNT do not reject electron-recoils as backgroundBut: Electron recoils above threshold (& 1 keV) strongly suppressedThus: DM–nucleus scattering dominates, even if loop-inducedBut: Loop diagrams forbidden for some modelsProblems then:
I Very large couplings needed to compensate wave function suppressionI Poor fit to DAMA and CoGeNT energy spectra
0 5 10 15 20
-0.04
-0.02
0.00
0.02
0.04
0.06
E @keVD
Sm@d-1kg-1keV-1D
Full analysis
wo 1st bin
Σe0 = 5.´ 10-31 cm2, mΧ = 816. GeV
Χ2 = 55.9
Σe0 = 5.01´ 10-31 cm2, mΧ = 202. GeV
Χ2 = 20.6
Joachim Kopp Dark Matter and Dark Energy 39
Fox Poppitz arXiv:0811.0399JK Niro Schwetz Zupan arXiv:0907.3159
Indirect DM detection — where to look
The Galactic Center
Pros:Highest DM density
Cons:DM distribution uncertainMany background sources
Dwarf Galaxies
Pros:Few backgrounds
Cons:Relatively low DM density
Joachim Kopp Dark Matter and Dark Energy 40
Indirect DM detection — where to look
The Galactic Center
Pros:Highest DM density
Cons:DM distribution uncertainMany background sources
Dwarf Galaxies
Pros:Few backgrounds
Cons:Relatively low DM density
Joachim Kopp Dark Matter and Dark Energy 40
Indirect DM detection — where to look
The Galactic Center
Pros:Highest DM density
Cons:DM distribution uncertainMany background sources
Dwarf Galaxies
101 102 103
WIMP mass [GeV]
10-26
10-25
10-24
10-23
10-22
10-21
WIM
Pcr
oss
sect
ion
[cm3
/s]
Upper limits, Joint Likelihood of 10 dSphs
3*10−26
bb Channel
τ+ τ− Channel
µ + µ− Channel
W+W− Channel
Thermal relic cross section=> Correct relic abundance
Fermi-LAT 1108.3546see also Geringer-Sameth Koushiappas 1108.2914
Pros:Few backgrounds
Cons:Relatively low DM density
Joachim Kopp Dark Matter and Dark Energy 40
Thermal relic cross-section
Fermi-LAT, 1108.3546see also Geringer-Sameth Koushiappas 1108.2914
Indirect DM detection — where to look (2)
Cosmic antimatter
Pros:Few background sources
Cons:Backgrounds uncertainPropagation of chargedparticle has large uncertaintiesNon-directional
High-energy neutrinos
Idea:DM capture/annihilation in the SunFlux dominated by capture rate
Pros:Few backgrounds
Cons:Low neutrino cross sections
Joachim Kopp Dark Matter and Dark Energy 41
Indirect DM detection — where to look (2)
Cosmic antimatter
Pros:Few background sources
Cons:Backgrounds uncertainPropagation of chargedparticle has large uncertaintiesNon-directional
High-energy neutrinos
Idea:DM capture/annihilation in the SunFlux dominated by capture rate
Pros:Few backgrounds
Cons:Low neutrino cross sections
Joachim Kopp Dark Matter and Dark Energy 41
PAMELA collaboration, 0810.4995
Indirect DM detection — where to look (2)
Cosmic antimatter
Pros:Few background sources
Cons:Backgrounds uncertainPropagation of chargedparticle has large uncertaintiesNon-directional
High-energy neutrinos
Idea:DM capture/annihilation in the SunFlux dominated by capture rate
Pros:Few backgrounds
Cons:Low neutrino cross sections
Joachim Kopp Dark Matter and Dark Energy 41
PAMELA collaboration, 0810.4995
Indirect DM detection — where to look (2)
Cosmic antimatter
Pros:Few background sources
Cons:Backgrounds uncertainPropagation of chargedparticle has large uncertaintiesNon-directional
High-energy neutrinos
Idea:DM capture/annihilation in the SunFlux dominated by capture rate
Pros:Few backgrounds
Cons:Low neutrino cross sections
Joachim Kopp Dark Matter and Dark Energy 41
IceCube collaboration, 1111.2738PAMELA collaboration, 0810.4995
What is a sphaleron?
SU(2) gauge field vacuum configurations are classified according to theirwinding number (or Chern-Simons number)
NCS = 116π2
∫ t0dt
∫d3x tr Fµν Fµν
NCS =1
16π2
∫ t
0dt
∫d3x tr Fµν Fµν
Configurations with different winding number cannot be continuouslytransformed into each other.
Sphalerons are processes (with E > 0) that change the winding numberIn the SM, a change in winding number corresponds to a change inB + L. In fact, considering only left-handed (SU(2)L-charged) fermions:
jµB+L =∑ψ=q,`
12ψγµ(1− γ5)ψ
A change in B + L is equivalent to a change in NCS:
Joachim Kopp Dark Matter and Dark Energy 42
What is a sphaleron?
SU(2) gauge field vacuum configurations are classified according to theirwinding number (or Chern-Simons number) NCS = 1
16π2
∫ t0dt
∫d3x tr Fµν Fµν
Sphalerons are processes (with E > 0) that change the winding numberTheir energy is of order mH , the symmetry breaking scale (100 GeV)
In the SM, a change in winding number corresponds to a change inB + L. In fact, considering only left-handed (SU(2)L-charged) fermions:
jµB+L =∑ψ=q,`
12ψγµ(1− γ5)ψ
A change in B + L is equivalent to a change in NCS:
Joachim Kopp Dark Matter and Dark Energy 42
What is a sphaleron?
SU(2) gauge field vacuum configurations are classified according to theirwinding number (or Chern-Simons number) NCS = 1
16π2
∫ t0dt
∫d3x tr Fµν Fµν
Sphalerons are processes (with E > 0) that change the winding numberIn the SM, a change in winding number corresponds to a change inB + L. In fact, considering only left-handed (SU(2)L-charged) fermions:
jµB+L =∑ψ=q,`
12ψγµ(1− γ5)ψ
A change in B + L is equivalent to a change in NCS:
∂t
∫d3x j0B+L ≡
∫d3x
12∂µψγ
µγ5ψ (since ∂µψγµψ = 0)
= − 116π2
∫d3x tr Fµν Fµν (chiral anomaly)
= −∂tNCS
Joachim Kopp Dark Matter and Dark Energy 42