b.sadoulet thermal dark matter particlescosmology.berkeley.edu/classes/class_archive/fall... ·...

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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


B.Sadoulet


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


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


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


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


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


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


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


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


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


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


B.Sadoulet


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


B.Sadoulet


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


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


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

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