SUSY Dark Matter Loops and Precision from Particle Physics
Transcript of SUSY Dark Matter Loops and Precision from Particle Physics
SUSY Dark Matter
Loops and Precision
from Particle Physics
Fawzi BOUDJEMA
LAPTH, CNRS, France
in collaboration with Andrei Semenov and David Temes
in parts with Ben Allanach, Genevieve Belanger and Sacha Pukhov
RADCOR05 F. BOUDJEMA, SUSY Dark Matter: Loops and Precision from Particle Physics – p. 1/35
Plan
Cosmology in the era of precision measurements
− Dark Matter is New Physics
− Evidence for and Precision on the matter content of the universe
− New Paradigm: can LHC/ILC match the precision of the upcoming
cosmo/astro experiments and indirectly probe the history of the early universe?
Relic density: precision annihilation cross sections (I)
− From the particle point of view: need for RADCOR (Gram determinant....)
− MSSM as a prototype: micrOMEGAs
− Precision needed on calculation in some specific SUSY scenarios
Direct and indirect detection of DM: annihilation cross sections (II)
− γ ray line
S LOOP SAutomatisation, modularity, GF, Gram’s, results and simulation
Summary
RADCOR05 F. BOUDJEMA, SUSY Dark Matter: Loops and Precision from Particle Physics – p. 2/35
The need for Dark Matter
Newton’s law → v2rot./r = GN M(r)/r2
(tracer star at a distance r from centre of mass distribution )
We are not in the centre of the universe Dark Matter= New Physicswe are not made up of the same stuff as most of our universe
RADCOR05 F. BOUDJEMA, SUSY Dark Matter: Loops and Precision from Particle Physics – p. 3/35
Cosmology in the era of precision measurement I: standard candles
No Big Bang
1 2 0 1 2 3
expands forever
-1
0
1
2
3
2
3
closed
recollapses eventually
Supernovae
CMB Boomerang
Maxima
Clusters
mass density
vacu
um e
nerg
y de
nsity
(cos
mol
ogica
l con
stan
t)
open
flat
SNAP Target Statistical Uncertainty observed luminosity and red-
shift exploits the different z
dependence of matter/energy
density
RADCOR05 F. BOUDJEMA, SUSY Dark Matter: Loops and Precision from Particle Physics – p. 4/35
Cosmology in the era of precision measurement II: CMB
observed temperature
anisotropies (related
to the density
fluctuations at the
time of emission) is
10−5
RADCOR05 F. BOUDJEMA, SUSY Dark Matter: Loops and Precision from Particle Physics – p. 5/35
Pre-WMAP and WMAP vs Pre-LEP and LEPpower spectrum of anisotropies, WMAP vs Pre-WMAP
.210
.214
.218
.222
.226
.230
.234
.238
1999
VBM (mZ)
ν - q e - q ν - e (Ellis-Fogli)
500 100 150 200 250
mt (GeV)
sin2 θ w
mW / mZ
.242
165 170 175 180mt [GeV]
50
100
200
300
sin2θeff
MW
GigaZ (1σ errors)
MH [GeV]
SM@GigaZ
now
Planck+SNAP will do even better (per-cent precision)
improvement like going from LEP to LHC+ILC
LHC, PLanck → 2007 ILC,SNAP → 2015
RADCOR05 F. BOUDJEMA, SUSY Dark Matter: Loops and Precision from Particle Physics – p. 6/35
Matter Budget and Precision 1.
Stars23%
Baryons∼ 4%ν(Ων < 0.01)
Dark Matter
23%
73%Dark Energy
t0 = 13.7 ± 0.2 Gyr (1.5%)
Ωtot = 1.02 ± 0.02 (2%)
ΩDM = 0.23 ± 0.04 (17%)
RADCOR05 F. BOUDJEMA, SUSY Dark Matter: Loops and Precision from Particle Physics – p. 7/35
Matter Budget and Precision 2.
Stars23%
Baryons∼ 4%ν(Ων < 0.01)
Dark Matter
23%
73%Dark Energy
t0 = 13.7 ± 0.2 Gyr (1.5%) α−1 = 10t0(10−7%)
Ωtot = 1.02 ± 0.02(2%) ρ = Ωtot(∼ 0.1%)
ΩDM = 0.23 ± 0.04(17%) sin2 θeff = ΩDM(0.08%)
RADCOR05 F. BOUDJEMA, SUSY Dark Matter: Loops and Precision from Particle Physics – p. 8/35
Matter Budget and Precision 3. Testing the cosmology
Present measurement at 2σ 0.094 < ω = ΩDMh2 < 0.129 (10%)
future (SNAP+Planck) → 2%
Particle Physics ↔Cosmology through ω
• is wholly New Physics
• But will LHC, ILC see the “same” New Physics?
• New paradigm and new precision: change in perception about this connection
• ω used to: constrain new physics (choice of LHC susy points, benchmarks)
• Now: if New Physics is found, what precision do we require on colliders and theory to
constrain cosmology? (Allanach, Belanger, FB, Pukhov JHEP 2004)
strategy/requirements on theory and collider measurements to match the present/future
precision on ω
RADCOR05 F. BOUDJEMA, SUSY Dark Matter: Loops and Precision from Particle Physics – p. 9/35
Relic Density: derivation
1 10 100 1000
0.0001
0.001
0.01
Freeze-out
Tf ∼ 20GeVt ∼ 10−8s
Boltzmann
Suppre
ssio
n
Exp
(-m/T
)
At first all particles in thermal equilibrium
universe cools and expands: interaction rate
too small to maintain equilibrium
(stable) particles can not find each other:
freeze out and leave a relic density
dilution due to expansion
dN/dt = −3HN− < σv > (N2 − N2eq)
χ01χ0
1 → X X → χ01χ0
1
Ωχ01
= mχ01Nχ0
1/ρcri,
ρcri = 3H20/8πGN
ρcri = h21.9 10−29gcm−3 →
Ωχ01h2 ∝ 1/σχ0
1χ01→X
RADCOR05 F. BOUDJEMA, SUSY Dark Matter: Loops and Precision from Particle Physics – p. 10/35
Relic Density: Loopholes and Assumptions
At early times Universe is radiation dominated: H(T ) ∝ T 2
(nγeq ∼ T 3 relativistic particles are not Boltzmann suppressed)
Expansion rate can be enhanced by some scalar field (kination), extra dimension
H2 = 8πG/3 ρ(1 + ρ/ρ5 ), anisotropic cosmology,...
Entropy non-conservation, e.g., through decays(entropy increase will reduce the relic
abundance for example)
RADCOR05 F. BOUDJEMA, SUSY Dark Matter: Loops and Precision from Particle Physics – p. 11/35
Thermal average
must calculate all annihilation, co-annihilation processes. Each annihilation can consist of
tens of cross sections...
χ0i χ
0j → XSMYSM , χ0
1f1 → XSMYSM ,...
< σv >=
P
i,jgigj
R
(mi + mj)2ds
√sK1(
√s/T ) p2
ijσij(s)
2T`
P
igim2
i K2(mi/T )´2
,
pij is the momentum of the incoming particles in their center-of-mass frame.
pij =1
2
»
(s − (mi + mj)2)(s − (mi − mj)
2)
s
–
12
→ v
v = 0v × σv
RADCOR05 F. BOUDJEMA, SUSY Dark Matter: Loops and Precision from Particle Physics – p. 12/35
SUSY as an example
Will concentrate on supersymmetry, SUSY in particular the MSSM (no CP violation)
sometimes assumes the mSUGRA scenario (bring down the number of parameters to
4 + 1/2, but relies on RGE
RADCOR05 F. BOUDJEMA, SUSY Dark Matter: Loops and Precision from Particle Physics – p. 13/35
micrOMEGAs G. Belanger, FB, A. Semenov, A. Pukhov (CPC 02, CPC 05)
given any set of parameters it can identify LSP, NLSP, generate and calculate ω
Model defined in Lanhep (more later)
Fed into CompHEP ....tree-level, some 3000 processes could be needed.
Higgs sector: improved Higgs masses/mixings (read from FeynHiggs, for example) but
interpreted in terms of an effective scalar potential (GI), following FB and A.
Semenov (PRD 02)
Effective Lagrangian also includes important RC (Higgs couplings, ∆mb effects,..)
Interfaced with Isajet, Suspect, SoftSUSY parameters at high scale run down to the ew scale
(g − 2)µ, b → sγ, Bs → µ−µ+
NMSSM done (with C. Hugonie), JHEP 05
CP violation in progress
“open source”: procedure to define your own model, soon
RADCOR05 F. BOUDJEMA, SUSY Dark Matter: Loops and Precision from Particle Physics – p. 14/35
The mSUGRA inspired regions
100 200 300 400 500 600 700 800 900 10000
100
200
300
400
500
600
700
800
100 200 300 400 500 600 700 800 900 10000
100
200
300
400
500
600
700
800
mh = 114 GeV
m0 (
GeV
)
m1/2 (GeV)
tan β = 10 , µ < 0
τ1/χ01 co-annihilation strip
mτ1< m
χ01
WMAP
b→
sγ
(exc.)
Pre-WMAP (bulk)
Bulk region: bino LSP, lR exchange,
(small m0, M1/2)
τ1 co-annihilation: NLSP thermally
accessible, ratio of the two populations
exp(−∆M/Tf ) small m0,
M1/2 : 350 − 900GeV
Higgs Funnel: Large tan β,
χ01χ0
1 → A → bb, (τ τ),
M1/2 : 250 − 1100GeV,
m0 : 450 − 1000GeV
Focus region: small µ ∼ M1, important
higgsino component, requires very large
TeV m0
RADCOR05 F. BOUDJEMA, SUSY Dark Matter: Loops and Precision from Particle Physics – p. 15/35
Relic density around Higgs Pole with and without RC
RADCOR05 F. BOUDJEMA, SUSY Dark Matter: Loops and Precision from Particle Physics – p. 16/35
The Approach
Use micrOMEGAs+SOFTSUSY →...Softmicro..fix A0, tan β, sgn(µ) but scan on M1/2
WMAP strips imply m0 = f(M1/2): slopes
RGE also needs SM input parameters!
scale dependence of relic: default MSUSY =√
mt1mt2
: scale of EWSB conditions
theoretical uncertainty: effect of different refinements in RGE and threshold
corrections
derive accuracy within mSUGRA, relying completely on mSUGRA. accuracies on high
scale parameters and SM inputs
model independent approach: find out most relevant parameters and extract accuracy
on these (weak scale parameters)
accuracies derived in an iterative procedure and refer to the 10% WMAP precision
RADCOR05 F. BOUDJEMA, SUSY Dark Matter: Loops and Precision from Particle Physics – p. 17/35
τ1 co-annihilation region: Theory uncertainty
0.11
0.115
0.12
0.125
0.13
0.135
0.14
0.145
100 150 200 250 300 350 400
Ωh
2
Mτ~ (GeV)
Ωh2
Scale variation
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
100 150 200 250 300 350 400 450Ω
h2
Mτ~ (GeV)
-1 loop tau Yukawafull calculation
-1 loop neutralino-2 loop gaugino RGEs
Scale variation: From 5% (at small mτ1) to 20% at large mτ1)
2-loop gaugino RGE’s ESSENTIAL as is 1-loop threshold correction to χ01
RADCOR05 F. BOUDJEMA, SUSY Dark Matter: Loops and Precision from Particle Physics – p. 18/35
τ1 co-annihilation region: accuracy within mSUGRA
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
100 150 200 250 300 350 400 45060
80
100
120
140
160
180
a
∆A
0- (
GeV
)
mτ~ (GeV)
a(M1/2)a(m0)
∆A0-
a(tanβ)
accuracy on m0, M1/2 demanding:
3% : 1% may be achievable at LHC
for tan β require 10percent.
RADCOR05 F. BOUDJEMA, SUSY Dark Matter: Loops and Precision from Particle Physics – p. 19/35
τ1 co-annihilation region: Model Independent
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
100 150 200 250 300 350 400 4500.6
0.7
0.8
0.9
1
1.1
1.2
1.3
∆co
s2
θ τ- r
eq
uir
ed
, a(M
)
δM r
eq
uir
ed
(G
eV
)
m~τ (GeV)
∆cos2θτ- required
δM requireda(Mχ)
∆M must be measured to less than 1GeV
mixing angle accuracy should be feasible at ILC
accuracy on LSP mass not demanding but this is
because we have constrained ∆M .
other slepton masses need also be measured
in terms of physical parameters residual µ tan β
accuracies not demanding
Preliminary studies indicate these accuracies will
be met for the lowest mχ01
-0.1
-0.05
0
0.05
100 200 300 400 500
∆Ωh2 /Ω
h2
m~τ (GeV)
∆tanβ=10∆µ=µ
2
3
4
5
6
7
150 200 250 300 350 400 450
GeV
me~ (GeV)
δ-(∆M)
RADCOR05 F. BOUDJEMA, SUSY Dark Matter: Loops and Precision from Particle Physics – p. 20/35
direct and indirectDirect and Indirect Searches
p, e+ , γ, ν, . . .
χ01
CDMS, Edelweiss, DAMA, Genuis, ..
χ
ν
χχ → νν
Amanda, Antares, Icecube, ..
RADCOR05 F. BOUDJEMA, SUSY Dark Matter: Loops and Precision from Particle Physics – p. 21/35
Underground direct detection
W−
χ0 χ− χ0
h0, H0
(a)
χ0 χ− χ0
W− W−
q q′ q
(b)
Loops for direct detection
χm100 200 300 400 500 600 700 800 900
(pb)
S.I.
σ
-1110
-1010
-910
-810
-710
-610
-510 Edelweiss II
Zeplin II
Zeplin IV
Xenon
Genius
χm100 200 300 400 500 600 700 800 900
(pb)
S.I.
σ
-1110
-1010
-910
-810
-710
-610
-510 Edelweiss II
Zeplin II
Zeplin IV
Xenon
Genius
tan β = 10 tan β = 50
• within WMAP
RADCOR05 F. BOUDJEMA, SUSY Dark Matter: Loops and Precision from Particle Physics – p. 22/35
Annihilation into photons
dΦγ
dΩdEγ=
∑
i
dN iγ
dEγσiv
1
4πm2χ
︸ ︷︷ ︸
Physique des Particules
∫
ρ2dl︸ ︷︷ ︸
Astro
γ′s: Point to the source, independent of propagation model(s)
• continuum spectrum from χ01χ0
1 → ff , . . ., hadronisa-
tion/fragmentation (→ π0 → γ ) done through isajet/herwig
• Loop induced mono energetic photons,γγ, Zγ final states
ACT: HESS,
Magic, VERITAS,
Cangoroo, ...
Space-based:
AMS, GLAST,
Egret,...RADCOR05 F. BOUDJEMA, SUSY Dark Matter: Loops and Precision from Particle Physics – p. 23/35
FB, A. Semenov, D. Temes
S LOOP S
Need for an automatic tool for susy calculations
handles large numbers of diagrams both for tree-level
and loop level
able to compute loop diagrams at v = 0 : dark matter, LSP, move at galactic
velocities, v = 10−3
ability to check results: UV and IR finiteness but also gauge parameter independence
for example
ability to include different models easily and switch between different renormalisation
schemes
RADCOR05 F. BOUDJEMA, SUSY Dark Matter: Loops and Precision from Particle Physics – p. 24/35
Non-linear gauge implementation
LGF = − 1
ξW|(∂µ − ieαAµ − igcW βZµ)Wµ + + ξW
g
2(v + δhh + δHH + iκχ3)χ+|2
− 1
2ξZ(∂.Z + ξZ
g
2cW(v + ǫhh + ǫHH)χ3)
2 − 1
2ξγ(∂.A)2
• quite a handful of gauge parameters, but with ξi = 1, no “unphysical threshold”
• more important: no need for higher (than the minimal set)for higher rank tensors and
tedious algebraic manipulations
RADCOR05 F. BOUDJEMA, SUSY Dark Matter: Loops and Precision from Particle Physics – p. 25/35
Strategy: Exploiting and interfa ing modulesfrom dierent odes
Lagrangian of the modeldened in LanHEP- parti le ontent- intera tion terms- shifts in elds and parameters- ghost terms onstru ted by BRST↓ ↓Generi Model Classes Model-kinemati al stru tures -Feynman rules, in luding CT
⇓Evaluation viaFeynArts-FormCal LoopTools modied!!tensor redu tion inappropriate for small relative velo ities(Zero Gram determinants)
⇑Renormalisation s heme- denition of renorm. onst. in the lasses modelNon-Linear gauge-xing onstraints, gauge parameter dependen e he ks
RADCOR05 F. BOUDJEMA, SUSY Dark Matter: Loops and Precision from Particle Physics – p. 26/35
From the Lagrangian to the Feynman Rules
vector
A/A: (photon, gauge),
Z/Z:(’Z boson’, mass MZ = 91.1875, gauge),
’W+’/’W-’: (’W boson’, mass MW = MZ*CW, gauge).
scalar H/H:(Higgs, mass MH = 115).
transform A->A*(1+dZAA/2)+dZAZ*Z/2, Z->Z*(1+dZZZ/2)+dZZA*A/2,
’W+’->’W+’*(1+dZW/2),’W-’->’W-’*(1+dZW/2).
transform H->H*(1+dZH/2), ’Z.f’->’Z.f’*(1+dZZf/2),
’W+.f’->’W+.f’*(1+dZWf/2),’W-.f’->’W-.f’*(1+dZWf/2).
let pp = -i*’W+.f’, (vev(2*MW/EE*SW)+H+i*’Z.f’)/Sqrt2 ,
PP=anti(pp).
lterm -2*lambda*(pp*anti(pp)-v**2/2)**2
where
lambda=(EE*MH/MW/SW)**2/16, v=2*MW*SW/EE .
let Dpp^mu^a = (deriv^mu+i*g1/2*B0^mu)*pp^a +
i*g/2*taupm^a^b^c*WW^mu^c*pp^b.
let DPP^mu^a = (deriv^mu-i*g1/2*B0^mu)*PP^a
-i*g/2*taupm^a^b^c*’W-’^mu,W3^mu,’W+’^mu^c*PP^b.
lterm DPP*Dpp.
Gauge fixing and BRS transformation
let G_Z = deriv*Z+(MW/CW+EE/SW/CW/2*nle*H)*’Z.f’.
lterm -G_A**2/2 - G_Wp*G_Wm - G_Z**2/2.
lterm -’Z.C’*brst(G_Z).
RenConst[ dMHsq ] := ReTilde[SelfEnergy[prt["H"] -> prt["H"], MH]]
RenConst[ dZH ] := -ReTilde[DSelfEnergy[prt["H"] -> prt["H"], MH]]
RenConst[ dZZf ] := -ReTilde[DSelfEnergy[prt["Z.f"] -> prt["Z.f"],
MZ]] RenConst[ dZWf ] := -ReTilde[DSelfEnergy[prt["W+.f"] ->
prt["W+.f"], MW]]
Output of Feynman Rules
with Counterterms !!
M$CouplingMatrices =
(*------ H H ------*)
C[ S[3], S[3] ] == - I *
0 , dZH ,
0 , MH^2 dZH + dMHsq
,
(*------ W+.f W-.f ------*)
C[ S[2], -S[2] ] == - I *
0 , dZWf ,
0, 0
, (*------ A Z ------*)
C[ V[1], V[2] ] == 1/2 I / CW^2 MW^2 *
0, 0 ,
0 , dZZA ,
0, 0
,
(*------ H H H ------*)
C[ S[3], S[3], S[3] ] == -3/4 I EE / MW / SW *
2 MH^2 , 3 MH^2 dZH -2 MH^2 / SW dSW - MH^2 / MW^2 dMWsq
,
(*------ H W+.f W-.f ------*)
C[ S[3], S[2], -S[2] ] == -1/4 I EE / MW / SW *
2 MH^2 , MH^2 dZH + 2 MH^2 dZWf -2 MH^2 / SW dSW - MH^2 /
,
(*------ W-.C A.c W+ ------*)
C[ -U[3], U[1], V[3] ] == - I EE *
1 ,
- nla
,
RADCOR05 F. BOUDJEMA, SUSY Dark Matter: Loops and Precision from Particle Physics – p. 27/35
Avoiding zero Gram determinants: Segmentation (FB, A. Semenov, D. Temes ,hep-ph/0507127, PRD...)
For the problems at hand:
DetG = M6χ01v2 sin2 θ
(1 − v2/4)3(1 − z2), z2 =
M2Z
4M2χ01
(1 − v2/4)
DetG(p1, p2) = −M4χ01v2 1
(1 − v2/4)2.
Segmentation
1
D0D1D2D3=
„
1
D0D1D2− α
1
D0D2D3− β
1
D0D1D3+ (α + β − 1)
1
D1D2D3
«
×
1
A + 2l.(s3 − αs1 − βs2)
A = (s23 − M2
3 ) − α(s21 − M2
1 ) − β(s22 − M2
2 ) − (α + β − 1)M20 .
(Di = (l + si)2 − M2
i , si =i
X
j=1
pj)
For any graph if DetG(s1, s2, s3) = 0 (or “small” ), construct all 3 sub-determinants
DetG(si, sj)and take the couple si, sj (as independent basis) that corresponds to
Max |Det(si, sj)|, then write
RADCOR05 F. BOUDJEMA, SUSY Dark Matter: Loops and Precision from Particle Physics – p. 28/35
s3 = αs1 + βs2 + εT with s1.εT = s2.εT = 0 meaning
εT,µ = ǫµαβδsα1 sβ
2 tδ , α =s22s3.s1 − s1.s2s2.s3
DetG(s1, s2), β = α(s1 ↔ s2).
DetG(s1, s2, s3) = ε2T DetG(s1, s2).
Expanding around small v
0 2 4 6 8 100
0.5
1
1.5
2
2.5
2 [107 0 2 4 6 8 10 12 140
0.25
0.5
0.75
1
1.25
1.5
2 [107From Nans BARO, Diploma , LAPTH/ENS-Lyon
RADCOR05 F. BOUDJEMA, SUSY Dark Matter: Loops and Precision from Particle Physics – p. 29/35
Application to χ01χ0
1 → γγ, Zγ, gg (FB, A. Semenov, D. Temes ,hep-ph/0507127, PRD...)Computing the ross-se tionsMore than a thousand diagrams in ludingχ0
1
χ01 χ+
i
χ+iW
χ+j
γ
Z χ01
χ01
γ
Z
χ+i
W
W
W χ01
χ+i
χ01
W
Z
γ
χ+i
γ
Z
χ+j
χ+i
A, H, h, Z
χ01
χ01
χ01
χ01 γ
ZW
W
χ+i
χ01
χ01
H, h
γ
ZW
W
a) b) c)
d) e) f)New ontribution found in Zγ !γ
Z
χ01
χ01
χ0i
Counterterm ontribution:- Obtained from tree-level χ01χ
01 → ZZ and δZZγ- Sin e δZZγ ∼ (1 − α) CT an be put to zero if α = 1Choosing a proper gauge simplies the omputation
1 2
RADCOR05 F. BOUDJEMA, SUSY Dark Matter: Loops and Precision from Particle Physics – p. 30/35
SloopS, micrOMEGAs,AMS/HESSPropagation
GUT Scale Suspect micrOMEGAs PYTHIA Halo model Cosmic Ray Fluxes
charged
γ
SloopS
with AMS energy resolution [ GeV ]γE1 10 210
]-1
.s-2
[
GeV
.cm
γφ.2E
-810
-710
-610
bb
γγ
-τ+τ0Zγ
=151 GeVχm
with a typical ACT energy resolution [ GeV ]γE1 10 210
]-1
.s-2
[ G
eV.c
mγφ.2
E
-810
-710
SIMULATION:
Parameterising the halo profile:
(α, β, γ) = (1, 3, 1), a = 25kpc. (core radius), r0 = 8kpc (distance to galactic centre),
ρ0 = 0.3 GeV/cm3 (DM density), opening angle cone 1o
SUSY parameterisation
m0 = 113GeV, m1/2 = 375 GeV, A = 0, tan β = 20, µ > 0
γ lines could be distinguished from diffuse background
RADCOR05 F. BOUDJEMA, SUSY Dark Matter: Loops and Precision from Particle Physics – p. 31/35
Summary
Precise calculations for the relic density are necessary with the foreseen precision on
some cosmological parameters
Rates for direct and direct detection also need some loop calculations, here
measurements most probably will constrain the astrophysical parameters
A general code is being developed for minimal susy, but could be extended, to provide
annihilation rates for cosmo/astro and corrected cross sections for colliders
but still much to be done and to be improved
RADCOR05 F. BOUDJEMA, SUSY Dark Matter: Loops and Precision from Particle Physics – p. 32/35
Extra Expansion of Universe, Einstein Equations
Einstein Rµν − 1
2Rgµν = 8πG
„
Tµν − Λ
8πG
«
Isotropic and Homogeneous
ds2 = −dt2 + a2(t)
»
dr2
1 − kr2+ r2(dθ2 + sin2 θdφ2)
–
conservation H2 =
„
a
a
«2
=8πG
3
X
i
ρi −k
a2
→X
M
ΩM + ΩΛ + Ωk = 1 ΩM =ρM
ρcρc =
3H2
8πG
Acceleration
„
a
a
«
= −4πG
3
X
i
(ρi + 3pi) p = wρ
ρ(a) ∝ 1
a(t)3(1+w)wrad = 1/3 wM = 0 wΛ = −1
RADCOR05 F. BOUDJEMA, SUSY Dark Matter: Loops and Precision from Particle Physics – p. 33/35
Application to χ01χ0
1 → γγ, Zγ, ggHiggsino and Wino limits
• σv vs. Mχ01
(v = 0)
tan β = 10, A = 0, mA = 100 TeV, mf = 4105 TeV, M1 = 2105 TeV
M2 = 2M1 (higgsino), µ = M1 (wino)
• σv vs. % higgsino/wino (v = 0)
M2 = 2M1, µ = 50 TeV (higgsino), µ = M1, M2 = 10 TeV (wino)
δM = Mχ+1− Mχ0
1Asymptoti value for large Mχ01, σv ∼ 1/M2
WLargest ross se tion for Zγ in wino aseSmooth behaviour in higgsino ase, δM ∼ m2z/M1Constant value after transition in wino ase, δM ∼ m4
z/M31Numeri al results reprodu e analyti al behaviour
RADCOR05 F. BOUDJEMA, SUSY Dark Matter: Loops and Precision from Particle Physics – p. 34/35
Mat hing with non-perturbative omputationHisano et al.'04In the extreme higgsino and wino limits,
• one-loop treatment breaks unitarity
• non-perturbative non-relativisti approa h...Non-perturbative omputation and one-loop resultshiggsino ase (v = 0)
mf = mA = 100 TeV, M2 = 2M1 = 50 TeV, tan β = 10, A = 0, δM = 0.1 GeVResonan es an enhan e result several orders of magnitudeMat hing will take pla e around 400 − 500 GeV RADCOR05 F. BOUDJEMA, SUSY Dark Matter: Loops and Precision from Particle Physics – p. 35/35