B.Sadoulet
Phys 250 -14 ThermalDM 1
The Dark Side
Thermal relicsNeutrinos
AbundanceMass measurementsNeutrino and cosmology
Weakly Interactive Massive ParticlesGeneric: Abundance
Asymmetry Detection methods
SupersymmetryMotivationsParametersElastic scattering, coherence,spinIndirect
Thermal Dark Matter Particles
B.Sadoulet
Phys 250 -14 ThermalDM 2
The Dark Side
What can Particle Physics Offer?
non baryonic dark matter
exotic particles
non-thermal
Axions Wimpzillas
thermal
Light Neutrinos WIMPs
Mirror branesEnergy in bulk
B.Sadoulet
Phys 250 -14 ThermalDM 3
The Dark Side
Thermal RelicsRelic Abundance (cf Chap. 8 slide 9)
Let us consider a 2 body annihilation channel ( can be generalized)
log na3( )
logm
T
Increasing σ v Non relativistic decoupling
Relativistic decoupling
↔ XX where we assume the X to stay in equilibriumdn
dt+ 3
˙ a
an = − → XX v n2 + XX → v nXEQ
2
in equilibrium
XX → v nXEQ2 = → XX
v n EQ2
⇒dn
dt+ 3
˙ a
an = − → XX
v n2 − n EQ2( )
→ XX v large enough n2 = n EQ
2 ∝
1
3 /4
g
3( )2h3c3
kT( )3
exp −2kBT
2
→ XX v too small
dn
dt+ 3
˙ a
an = 0 ⇒ n ∝ a−3
BosonsFermions
Relativistic
Non relativistic
B.Sadoulet
Phys 250 -14 ThermalDM 4
The Dark Side
Thermal equilibrium in early universe+ Decoupling when relativistic
Calculationcf. Kolb and Turner p 119-123
Note: completely generale.g. light gravitino: decouples when relativistic but geff =1.5 while g*S ≈106.75 (T>300GeV)
number ≈ number of photonsdensity <-> mass40 eV /c2 => Ω≈1
Light Massive Neutrinos
Introduce Y =n
s where n and s are the number and total entropy densities .
Current Y = Y∞ = 0.278geff
gS*
Tfreese
where geff =1 boson
3/4 fermion
≠ g*
At Tfreese ≈ 2 MeV , gS* ≈ g* = 2 + 2 * 7 / 8 + 3 *2 * 7 /8 = 10.75
and Ωxh2 =
Y∞so m
cr / h2=
m
91.5 eV for a single neutrino (geff = 2 × 3/4)
More generally
Ωxh2 =
mi
i∑
91.5 eV
B.Sadoulet
Phys 250 -14 ThermalDM 5
The Dark Side
Experimental Toolscf J.Ellis’ summary of Neutrino 2000 http://nu2000.sno.laurentian.ca
Direct measurement of cosmological neutrinos appearsimpossible
=> laboratory mass experiments:Tritium mve<2.2eV (Mainz) -> 0.5 eV?? Look at end point
Troitsk: anomaly not seen by Mainz Spike in differential spectrum with annual modulationWould be produced by enormous density of relicneutrinos
Neutrinoless double beta decay:mMajorana ve<0.4eV-> 0.1 ??
Oscillation experiments (LSND, Karmen ,CHORUS, NOMAD, Chooz, Palo Verde,atmosphericand solar neutrinos)
vM
e-
e-
n
npp
mM
= Ueii
∑ Uei* mi
Note : UeiUei* is not necessarily > 0
Implies mass+lepton number violationFavored in current models
m < 190 keV → 10keV ???
m <15.5 MeV → 3MeV ???
B.Sadoulet
Phys 250 -14 ThermalDM 6
The Dark Side
Neutrinos Oscillations
If neutrinos are massivein general mass matrix is not diagonal in
weak interaction basis
Current experimentsLSND?SuperKamiokandeSolar neutrinos : 4 possible solutions
A number of new experiments
SK
e.g. for 2 flavors: in weak basis e ,
such as e- ↔ e + W − - ↔ + W −
M =mve mve
mve mv
Eigenvectors are 1, 2 : e = 1 cos + 2 sin
with mass m1, m2 = − 1 sin + 2 cos
Evolution 1 = eE1t1 o
2 = eE2t2 o
⇒ Oscillations between eand . In relativistic limit
P e →( ) = e
2
=1
2sin2 2 1 − cos m2
2 − m12
∆m 2
1 2 4 3 4 c2
2hL
E
Borexino day night
B.Sadoulet
Phys 250 -14 ThermalDM 7
The Dark Side
Massive Neutrinos
Conclusion: 1) Neutrinos are a small massAtmospheric νµ->ντ or νs oscillation by SuperKamiokande
∆m2 taken at face value=> Density of cosmological neutrinos at minimumdensity of stars
Neutrino mass also hinted by for solar neutrinosBut no evidence for eV mass scale (if we reserve judgment on LNSD)
2) Complex structure
3) Enormous effortof particle/nuclear physics communitySpanning accelerator/particle astrophysics
B.Sadoulet
Phys 250 -14 ThermalDM 8
The Dark Side
Neutrino and Cosmology
Minimum mass indicated by SK=> at least density of stars
Cannot form galaxy alone!But even this does not work!
• mixed dark matter with mv> few eV/c2
problem with damped Lyman alpha systems
• seeding by topological defectsproblems with
Essentially both models are abandoned!
Dwarf galaxy halos too dense for neutrinosPhase space argument due to Gunn and Tremaine
For dwarf galaxies (Draco, Ursa Major etc.)
mv > 400eV / c2
B.Sadoulet
Phys 250 -14 ThermalDM 9
The Dark Side
Gunn Tremaine Argumentcf. Peebles p. 442-448
Neutrinos are fermionsPauli principle limits their densityStart at high temperature
Liouville theorem
Use isothermal sphere as average distribution
dN = N x, p( )d3xd 3 p = 21
expE
kBT
+1
d3xd3 p
h3 (no degeneracy << E)
⇒ at high redshift phase space density <1
h 3
N f x f , p f( ) = N x , p( )In order to take into account mixing of orbits, take local phase space average N f x f , p f( ) <1 /h 3 as each have to be smaller!
N f = No exp −
p2
2mv2
+ r( )
2
with 0( ) = 0
Liouville constraints says that No < 1 /h3 (factor 2 better than Pauli locally)Solve Poisson equation with
r( ) = mv N f∫ d3 p = No (2 )3/ 2 mv4 3 exp − r( )
2
Classical solution (Emden 1909) relates 2 , core radius r1 / 2, and circular velocity vc
vc = 2 at r1 / 2 ≈ 1.31
4 GN No (2 )3 / 2mv4
No <1 /h3 ⇒ mv >70eV/c2
r1 / 2 kpc( )[ ]1 / 2
200km/s
vc
1/ 4
B.Sadoulet
Phys 250 -14 ThermalDM 10
The Dark Side
Particles in thermal equilibrium+ decoupling when nonrelativistic
Calculationcf. Supersymmetric Dark Matter G. Jungman, M. Kamionkowski, K. GriestPhys.Rept. 267 (1996) 195-373. See also Kolb and Turner 119-130
Cosmology points to W&Z scaleInversely standard particle model requires new physics at this scale
(e.g. supersymmetry) => significant amount of dark matterWe have to investigate this convergence!
Assuming Av = a + bv2
the solution of the Boltzmann equation is approximatively given by
Ωxh2 ∝
x f
gTfreeze
* a + 3b −1 /4a( )
xf
with x f =kbTfreeze
m given by the solution of x f ≈ ln
0.1 mPlanckm a + 6b / x f( )hc( )2 gTfreeze
* x f( )Typically x f ≈ 0.05 with a logarithmic dependence on m
and Ωxh2 = 3 ⋅10−27 cm3 / s
Av This implies a Av of the order of the Weak Scale
Weakly Interactive Massive Particles
Density ~ 1/(interaction rate)Ω ≈ 1 => σv ≈ 10-26 cm3/s
Generic Class
mPlanck =
hc
GN
=1.22 ⋅1019GeV / c2
B.Sadoulet
Phys 250 -14 ThermalDM 11
The Dark Side
Model independent upper limitK.Griest and M. Kamionkowski Phys Rev. Lett 64 (1990) 615
Essentially model independent!
Starting from Av = a + bv2,partial wave unitarity limits from above a (s wave) and b (p wave)
⇒ Ωxh2 >
m
300 TeV/c2
2
If the couplings of the order of ≈10−2 (cf. SUSY)
Ωxh2 >
m
3 TeV/c2
2
⇒ m ≤ 1 T e V/ c2
B.Sadoulet
Phys 250 -14 ThermalDM 12
The Dark Side
WIMPs 2Intuitive argument
If cross section is too large: will annihilate before decouplingNot the dark matter today
If cross section is too small: will be diluted away by the expansionWould over-close the universe
Delicate balanceNote involves both the Hubble constant and cross section.The “fine tuning” can be in the Hubble constant
Ways to turn argument:• Initial asymmetrySuppose thatCross section may be very large (e.g. heavy Dirac neutrino )
we are left with only small excessAttractive to explain why same order of magnitude as protons
• Dilution by entropy production after freeze oute.g. QCD or electroweak (if low enough) in case they are strongly first orderUnlikely
Note:Formula in previous slide not valid near pole, threshold or coannihilation
≠
↔strong interactions
′ ′ with ′ slightly heavier :
governed by the largest annihilation cross section
B.Sadoulet
Phys 250 -14 ThermalDM 13
The Dark Side
Detection methods
Note: annihilation rate provides a normalizationThe higher the cross section the lower the density!
Directly fixes annihilation rate in haloHowever we are sensitive only to specific channels
e.g.
Dependent one.g. presence of a cusp in the galactic center due to black hole and low entropy WIMP
+ survival probability in halo if charged particles (pbar and e+)
Elastic scattering : Direct detection
Usually crossing operation gives factor of the order of unityStill dependent on ratio between the cross sections (usually few %)
Annihilation in center of the sun or the earth-> high energy neutrinosCombination of elastic scattering to get trapped and annihilation in the center of
the object
→ while Av = → ⋅ ⋅⋅all final states
∑
n2
q → q = crossed channel of → qq
→ qq and → ⋅⋅⋅all final states
∑
B.Sadoulet
Phys 250 -14 ThermalDM 14
The Dark Side
SupersymmetryBold Assumption
Symmetry between bosons and fermionsLagrangian invariant when turning bosons into fermions and vice versa
Motivations1) Solve in an elegant way the mass instability problems (naturalness)
e.g,. Prevent the masses of the Higgs or W to go to the Planck mass
Every boson loopis compensated by a fermion loop equal in magnitude and opposite.=> only logarithmic divergenceNo need for a unnatural cutoff
Note: another solution is not to have a scalar Higgs: dynamicalelectroweak symmetry breakinge.g. Technicolor difficult!
2) Provide a first step for the quantization of gravity
Local supersymmetry =“square root” of the Poincaré group = GeneralRelativity! i.e. SUGRA
In practice, not smooth enough: need supersymmetric strings=“Superstring”
Consider 2 infinetesimal supersymmetric transformations 1 x( )and 2 x( ) at point x
It can be shown that 1 x( ), 2 x( )[ ] = 1/ 2 2 1( )∂ ∝ Momentum operator
2 1( ) is meaningful as the transformation acts as a spin 1/2 object
B.Sadoulet
Phys 250 -14 ThermalDM 15
The Dark Side
Supersymmetry3) Convergence of coupling constants
U(1)=e.m.
SU2=weak
SU3=QCD
U(1)=e.m.
SU2=weak
SU3=QCD
without SUSY
with SUSY
B.Sadoulet
Phys 250 -14 ThermalDM 16
The Dark Side
Minimum SupersymmetryR parity
If conserved, stay in supersymmetricSector: produced in pairLightest=stable
Super-potential
19 parameters
Gauge couplings
SUSY breaking termsAnother 44 phenomenologicalparameters: masses and
trilinear coupling
W = - ˆ H 1 ˆ H 2 + ˆ H 1heij ˆ L Li ˆ e R j
+ ˆ H 1hdij ˆ Q Li
ˆ d R j− ˆ H 2hd
ij ˆ Q Li ˆ u Rj
i, j = flavor
R = −1( )3 B− L( )+2S
G. Jungman, M. Kamionkowski, K. GriestPhys.Rept. 267 (1996) 195-373
−1
4Fa Fa − gV a
−igV a ˜ +Ta
t ∂ ˜ (Ta = gauge generator)
− 2g ˜ V a Ta ˜ + ˜ V aTa ˜ ( )+...
B.Sadoulet
Phys 250 -14 ThermalDM 17
The Dark Side
Mass Spectrum
HiggsInitial 8 degrees of freedom: 3 absorbed by W±, Z longitudinal
polarization => 5 Higgs left: h,H (both CP even),H±,A (CP odd)Mass are related
Neutralino4 states with the same quantum number=> mix!
DiagonalizeM1 0 −mz s Wc mzs Ws
0 M2 mzc Wc −mz c Ws
−mzc Wc mzc Wc 0 −mzs Ws −mz c Ws − 0
⇒ Lightest : 1o = z1
˜ B + z2˜ W 3 + z3
˜ H 1 + z4˜ H 2
mh,H2 = 1
2mA
2 + mZ2 ± mA
2 + mZ2( )2
− 4mA2mZ
2 cos2 2
with tan =
˜ H u˜ H d
+ strong radiation corrections (mt ≈175GeV/c2 )
˜ B , ˜ W 3 , ˜ H 1,˜ H 2
B.Sadoulet
Phys 250 -14 ThermalDM 18
The Dark Side
Mass spectrum 2
Squarks and sleptonsNote that 2 different scalars
with different masses and which can mixRight-left mixing can be important for elastic scattering because it gives scalar
contribution
CharginosGluinos
Searches at acceleratorsLEP:Tevatron: Run 2LHC 2006
Important constraint
˜ q L and ˜ q
R
˜ W ±
˜ g a
→ s ≈ 200GeV/c2
b → s
G. Jungman, M. Kamionkowski, K. GriestPhys.Rept. 267 (1996) 195-373
B.Sadoulet
Phys 250 -14 ThermalDM 19
The Dark Side
Simplifying assumptions
63 parameters are too much!Typically
• Assume mass and trilinear coupling matrices diagonal in flavorspace: Jungman,Kamionkowski,Griest : “practical”
57-> 24 parameters
• Gaugino mass universality at unification
• SimplifiedForget about sfermion mixingAssume all squark masses are equalAssume all selectron masses are equalAssume gaugino mass unification
All inputs are supplied at the Weak ScaleToo restrictive?
M1 =5
3M2 tan2
W
,tan =˜ H u˜ H u
,m ˜ q ,m˜ e ,mA, M2 6 parameters
B.Sadoulet
Phys 250 -14 ThermalDM 20
The Dark Side
“mSUGRA”Universality at the GUT scale
Masses are “running” like other couplings
Provides naturally a way to break electro-weak symmetry when the mass square ofthe lightest Higgs become negative
Tempting (but not fully justified in GUTsupergravity especially for m0) to take allmasses of scalar equal and all the massesof the gaugino masses equal.
+ Universality of trilinear couplings
=> Only 4.5 parameters left
Values are supplied at the unification scaleNote: J. Ellis in hep-ph/9812235 keeps µ
as a free parameter, equivalent not toassume
sign( ) , tan =˜ H u˜ H u
,m0,m1 / 2, A(trilinear coupling)
MH = 2 + mo2 at the GUT scale
B.Sadoulet
Phys 250 -14 ThermalDM 21
The Dark Side
The Game
Choose favorite parametrizationIn some cases could be too restrictive
Sample parameter space:e.g. uniformly in log in “natural” regionImpose constraints from accelerators
Compute annihilation cross sectionFurther restriction on parameters to have reasonable Ωmh2
Compute rates of interest<0 µ>0
Mostly Binosin region of
interest p<0.1
B.Sadoulet
Phys 250 -14 ThermalDM 22
The Dark Side
ExamplesG. Jungman, M. Kamionkowski, K. GriestPhys.Rept. 267 (1996) 195-373
Notes:Lowest cross section-> higher Wh2
No real lower limits
B.Sadoulet
Phys 250 -14 ThermalDM 23
The Dark Side
Ge Elastic cross section: 2000
Genius 100 kg Ge in 14 m tank LN2
Shaded regions: Unconstrained MSSM
Mandic, Gondolo, Murayama, Pierce
0 < M3 < 1000 GeV95 < m0 < 1000 GeV-3000 < A0 < 3000 GeV1.8 < tan(β) < 25sign of µ0.025 < Ω h2 < 1
~ 1 event/100 kg/yr
CDMS II 5 kg Ge at SoudanZEPLIN 6 kg Xe at Boulby
mSUGRA & Naturalness
CDMS Latest Feb 2000
Genino 100 kg Ge in 5 m tank LN2
~1 event/kg/d
CryoArray 1 tonne event by event discrim.
see also Ellis et al. hep-ph0007113
~1 event/kg/yr
B.Sadoulet
Phys 250 -14 ThermalDM 24
The Dark Side
CoherenceMomentum transfer so small that coherence on a part of the nucleusWhat is the additive quantum number?
Historically: dominance of axial vectorNeutralino=Majorana spin 1/2 particleIf no mixing of a Fierz transformation allow us to rewrite
Additive quantum number is spin!Historical nomenclature
• Spin dependent cross section =axial vector coupling• Spin independent cross section =“coherent” = scalar or vector
Note: vector coupling is forbidden for Majorana particles such as neutralino
Not true for sneutrino
˜ q L and ˜ q
R
LAV = dq 5 q 5qBut q 5q
at low energy→ 2s
spin
≡ 0 ⇐ CP
Helv. Phys. Acta 33 (1960) 855See also Itzykson,Zuber Quantum Field Theory Chap 3
B.Sadoulet
Phys 250 -14 ThermalDM 25
The Dark Side
Dominance of Scalar couplingSlow realization
because axial vector is so small, additional contributions usually dominate !
• Exchange of Higgs
• 2nd order gluoninteractions
• Scalar coupling from squark mixing (proportional to quark/squark mass ratio)
73Ge
B.Sadoulet
Phys 250 -14 ThermalDM 26
The Dark Side
Scattering on NucleiUsing Fierz transformation (see slide 24), we can write the effective
Lagrangian of χ interaction with ordinary matter at low energy as a sumof terms of the form
The matrix elements of the matter side are of the form
They will depend on the quark (including the sea quarks) or gluon contentof the nucleus .
In Nuclear Physics, one usually decomposes the nucleus into its nucleonsand the considered matrix element will become (e.g. for quarks)
This neglects the exchange of quarks between two nucleons (little overlap).
Oi q Aiq or Oi G aA iGa
< N q Aiq N > or < N G AiG N >
dq N p1 .... pZ n1 .... nA− Zk
∑pk q A iq pk
or
nk q A iq nk
p1 .... pZ n1 .... nA− Z N
where p1 .... pZ n1 .... nA− Z N are the nucleon wave functions (given e.g. by the shell model)
Couplingto quark q
B.Sadoulet
Phys 250 -14 ThermalDM 27
The Dark Side
Spin independentVector
Simplest. No contribution from sea quarks andgluons
Identically zero for Neutralino (Majorana)We have to introduce a form factor which
describe the loss of coherence at highmomentum transfer. Finally
ScalarComputed in detail by Drees and NojiriResult: essentially the same for protons and
neutrons (therefore cross sections on a nucleiscale as A2)
Somewhat dependent on strangeness content ofnucleon
Finally:
d
dr q
2 =1
v2 Zfp + A − Z( ) fn[ ]2FV
2 q 2( )
d
dr q
2 =fp =n
2
v2 FS2 q2( )
FV q2( ) ≈ FS q2( ) =3 j1 qR1( )
qR1
2
exp − qs( )2( )where s ≈ 1fm = 10−13cm, R1 = R2 − 5s2( )1 / 2
, R ≈ 1.2fmA1 / 3
Form Factor(Woods Saxon)
From quenching factor for heavy nuclei the electronequivalent recoil energy is about ten times smaller than
true nuclear recoil energy
Hole!
B.Sadoulet
Phys 250 -14 ThermalDM 28
The Dark Side
Spin DependentAxial Vector
• Spin content of nucleon
The ∆q’s are the quark spin content of the nucleondetermined by polarized deep elastic scattering ofleptons
Poorly determined•
Large theoretical uncertaintiesOdd group model= spin carried either by proton groupor neutron group
LA , p = 5 p s p 2dq
q=u,d,s∑ ∆q( p) LA ,n = 5 n s n 2dq
q=u,d,s∑ ∆q (n)
One conventionally defines a p =dq
2GFq=u,d,s∑ ∆q( p)
an =dq
2GFq=u,d,s∑ ∆q (n)
Λ =1
Jap S p + an Sn[ ]
Sp
n
= N Sp
n
N = average spin content of
proton
neutron group in the nucleus
G. Jungman, M. Kamionkowski, K. GriestPhys.Rept. 267 (1996) 195-373
B.Sadoulet
Phys 250 -14 ThermalDM 29
The Dark Side
Spin Dependent 2• Finally
The isovector and isoscalar parts and their interferencegive rise to three distinct form factors
Much smoother than scalar form factor(spin on periphery of nucleus)
Usually parametrized as polynomial
d
dr q
2 =8GF
v2 Λ2J J +1( )FS2 q( )
where
FS2 q( ) =
ap + an( )2S00 q( ) + ap − an( )2
S11 q( ) + ap2 − an
2( )S01 q( )ap + an( )2
S00 0( ) + ap − an( )2S11 0( ) + ap
2 − an2( )S01 0( )
Sij = cij(k )
k∑ yk with y =
1fm( ) 2
4q2
Sij = 0 for y > ycut
G. Jungman, M. Kamionkowski, K. GriestPhys.Rept. 267 (1996) 195-373
B.Sadoulet
Phys 250 -14 ThermalDM 30
The Dark Side
Elastic Scattering Rates
Energy depositionSimple non relativistic calculation
Convolution with velocity distribution in the halo
Ed =q2
2mN
=m2 mN
m + mN( )2 v2 1 − cos *( ) =mr
2
mN
v2 1 − cos *( )
swave scattering : for given velocity flat between 0 and Ed max =2mr
2
mN
v2
If Maxwellian in galaxy rest frame
f v'( )d3v' =1
vo3 3 / 2exp −
v'2
vo2
d3v'
differential rate per unit mass
dR
dEd
= o o
4vem mr2 F2 q( ) erf
vmin + ve
vo
− erf
vmin − ve
vo
where
o =d q = 0( )
dr q
2( )0
4mr2v2
∫ dr q 2( ) = independent of v
o = local density of halo
vmin =EdmN
2mr2
2
ve = vo 1.05 + 0.07cos2 t − 2ndJune( )
1yr
Annual modulation ±4.5%
B.Sadoulet
Phys 250 -14 ThermalDM 31
The Dark Side
Elastic Scattering Rates 2
Unfortunately featureless!
Ed
dR
dEd
1e-06
1e-05
0.0001
0.001
0.01
0.1
0 20 40 60 80 100
Ge
dR
dEd
kg/day( )
Ed(keV)
With form factorWithout form factor
10GeV /c2
20GeV /c230GeV /c2
50GeV /c2
100GeV /c2
300GeV /c2
1000GeV/c2
Beware!From quenching factor
for heavy nuclei theelectron equivalent
recoil energy is aboutten times smaller than
true nuclear recoilenergy
B.Sadoulet
Phys 250 -14 ThermalDM 32
The Dark Side
Neutrinos from Sun/EarthNeutrinos from annihilation in sun or earth
With current generation of neutrino arrays (104m2):Limits MACRO/Baksan: few 10-14 µ’s /cm2 /s > 3Gev
About similar to DAMA/ Current direct detection experiments2nd generation of dark matter detector should do better
With future neutrino observatory of 1km3 and higher threshold (smaller cone),improvement of indirect detection by factor 10-100 at large mass =>complementary search !
Earth Sun
MacroMacro
km3 arraykm3 array
99 D
D
99 D
D
CD
MS
II 2
005
CD
MS
II 2
005