SUSY Dark Matter and Colliders - whepp-9

SUSY Dark Matter and
Colliders
by
Ben Allanach (DAMTP, Cambridge University)
BCA, Lester, hep-ph/0507283; BCA, Belanger, Boudjema, Pukhov,
JHEP 0412 (2004) 020, hep-ph/0410091
Talk outline
• SUSY dark matter
• Constraints on SUSY models
• Collider measurements
SUSY Dark Matter and Colliders
B.C. Allanach – p.1/53

H − 1
Electroweak Breaking
Both Higgs get vacuum expectation values:
( ) ( ) ( ) ( )
H
0
1 v1 H
+
→
2 0
0 H2
0 →
v 2
and to get M W correct, match with v SM = 246 GeV:
β
v SM
v 2
tan β = v 2
v 1
v 1
L = h t¯t L H 0 2 t R + h b¯bL H 0 1 b R + h τ ¯τ L H 0 1 τ R
⇒
m t
sin β = h tv SM
√
2
,
m b,τ
cos β = h b,τv SM
√
2
.
SUSY Dark Matter and Colliders
B.C. Allanach – p.2/53

The Supersymmetric Standard
Model
Standard Model particle
quark, spin 1/2
lepton, spin 1/2
higgs, spin 0
gluon, spin 1
Weak bosons, spin 1
graviton, spin 2
SUSY Dark Matter and Colliders
B.C. Allanach – p.3/53

The Supersymmetric Standard
Model
For every particle present in The Standard Model, we
have a heavier supersymmetric copy with the same
quantum numbers and couplings to forces but spin
differing by 1/2 ¯h.
Standard Model particle Supersymmetric copy(s)
quark, spin 1/2 2squarks, spin 0
lepton, spin 1/2 2sleptons, spin 0
2× higgs, spin 0 higgsinos, spin 1/2
gluon, spin 1 gluinos, spin 1/2
Weak bosons, spin 1 gauginos, spin 1/2
graviton, spin 2 gravitino, spin 3/2
SUSY Dark Matter and Colliders
B.C. Allanach – p.3/53

Broken Symmetry
3 components of the Higgs particles are eaten by
W ± , Z 0 , leaving us with 5 physical states:
h 0 , H 0 (CP+), A 0 (CP-), H ±
SUSY breaking and electroweak breaking imply
particles with identical quantum numbers mix:
(ũ L , ũ R ) → ũ 1,2
( ˜d L , ˜d R ) → ˜d 1,2
(ẽ L , ẽ R ) → ẽ 1,2
( ˜B, ˜W3 , ˜H 0 1, ˜H 0 2) → χ 0 1,2,3,4
( ˜W ± , ˜H ± ) → χ ± 1,2
SUSY Dark Matter and Colliders
B.C. Allanach – p.4/53

SUSY Dark Matter
• Galactic rotation curves
• Gravitational lensing effects
• WMAP + large scale structure
Imposing R P , the neutralino is a good candidate.
Must take into account annihilation in the early
universe into ordinary matter:
χ 0 1
p
σ
χ 0 1
p ′
SUSY Dark Matter and Colliders
B.C. Allanach – p.5/53

WMAP Results
SUSY Dark Matter and Colliders
B.C. Allanach – p.6/53

WMAP Results
0.094 < Ω DM h 2 < 0.129@2σ
ΛCDM fit
SUSY Dark Matter and Colliders
B.C. Allanach – p.6/53

SUSY Prediction of Ωh 2
• Assume relic in thermal equilibrium with
n eq ∝ (MT ) 3/2 exp(−M/T ).
• Freeze-out with T f ∼ M f /25 once interaction
rate < expansion rate (t eq critical)
• We use microMEGAs : Ωh 2 ∝ 1/< σv > to
solve coupled Boltzmann equations
• Generate SUSY spectrum with SOFTSUSY
linked with SLHA
Belanger et al, CPC 149 (2002) 103
BCA, CPC 143 (2002) 305
BCA et al, JHEP0407 (2004) 036
SUSY Dark Matter and Colliders
B.C. Allanach – p.7/53

Additional observables
δ (g − 2) µ
2
∼ 13 × 10 −10 ( 100 GeV
M SUSY
) 2
tan β
χ ± i
µ ˜ν µ
γ
˜µ γ
χ 0 1
µ µ
BR[b → sγ] ∝ tan β(M W /M SUSY ) 2
χ ± i
γ
H ±
γ
b
˜t i
s
b
t
s
SUSY Dark Matter and Colliders
B.C. Allanach – p.8/53

Universality
Reduces number of SUSY breaking parameters from
100 to 3:
• tan β ≡ v 2 /v 1
• m 0 , the common scalar mass (flavour).
• M 1/2 , the common gaugino mass (GUT/string).
• A 0 , the common trilinear coupling (flavour).
These conditions should be imposed at
M X ∼ O(10 16−18 ) GeV and receive radiative
corrections
∝ 1/(16π 2 ) ln(M X /M Z ).
Also, Higgs potential parameter sgn(µ)=±1.
SUSY Dark Matter and Colliders
B.C. Allanach – p.9/53

mSUGRA Regions
After WMAP+LEP2, bulk region diminished. Need
specific mechanism to reduce overabundance:
• ˜τ coannihilation: small m 0 , m˜τ1 ≈ m χ
0
1
.
Boltzmann factor exp(−∆M/T f ) controls ratio
of species. ˜τ 1 χ 0 1 → τγ, ˜τ 1˜τ 1 → τ ¯τ.
• Higgs Funnel: χ 0 1 χ0 1 → A → b¯b/τ ¯τ at large
tan β. Also via h at large m 0 small M 1/2 .
• Focus region: Higgsino LSP at large m 0 :
χ 0 1χ 0 1 → W W/ZZ/Zh/t¯t.
• ˜t coannihilation: high −A 0 , m˜t 1
≈ m χ
0
1
.
˜t 1 χ 0 1 → gt, ˜t˜t → tt
SUSY Dark Matter Datta, and Colliders Djouadi, Drees, hep-ph/0504090
B.C. Allanach – p.10/53

Constraints on SUSY Models
mSUGRA well-studied in literature: eg Ellis, Olive et al PLB565
(2003) 176; Roszkowski et al JHEP 0108 (2001) 024; Baltz, Gondolo, JHEP 0410 (2004) 052;. . .
800
tan β = 10 , µ > 0
700
m h = 114 GeV
m 0 (GeV)
600
500
400
m χ ± = 104 GeV
300
200
100
0
100 200 300 400 500 600 700 800 900 1000
m 1/2 (GeV)
SUSY Dark Matter and Colliders
B.C. Allanach – p.11/53

Shortcomings
• Really, would like to combine likelihoods from
different measurements
• Typically only 2d scans, but in general we have
α s (M Z ), m t , m b , m 0 , M 1/2 , A 0 , tan β to vary
• Effective 3d type scan done which
parameterises a 2d surface of correct Ωh 2
• Baltz et al managed to perform a 4d scan, but lost
the likelihood interpretation. They used the
impressive Markov Chain Monte Carlo
technique.
Done in 2d in Ellis et al, hep-ph/0310356
Ellis et al, hep-ph/0411218
SUSY Dark Matter and Colliders
B.C. Allanach – p.12/53

Markov-Chain Monte Carlo
Markov chain consists of list of parameter points x ( t)
and associated likelihoods L (t)
1. Pick a point at random for x (1)
2. Pick a point around x (t) (say with a Gaussian
width) as the potential new point.
3. If L (t+1) > L (t) , the new point is appended onto
the chain. Otherwise, the proposed point is
accepted with probability L (t+1) /L (t) . If not
accepted, a copy of x (t) is added on to the chain.
Final density of x points ∝ L. Required number of
points goes linearly with number of dimensions.
SUSY Dark Matter and Colliders
B.C. Allanach – p.13/53

Implementation
Input parameters are: m 0 , A 0 , M 1/2 , tan β
• m t = 172.7 ± 2.9 GeV
• m b (m b ) MS = 4.2 ± 0.2 GeV,
• α s (M Z ) MS = 0.1187 ± 0.002.
For the likelihood, we also use
• Ω DM h 2 = 0.1125 +0.0081
−0.0091
• δ(g − 2) µ /2 = (19 ± 8.4) × 10 −10
• BR[b → sγ] = (3.52 ± 0.42) × 10 −5
SUSY Dark Matter and Colliders
ln L = − 1 2
∑
i
(p i − m i ) 2
2σ 2 i
+ c
B.C. Allanach – p.14/53

Convergence
We run 9×1 000 000 points. By comparing the 9
independent chains with random starting points, we
can provide a statistical measure of convergence: an
upper bound r on the excepted variance decrease for
infinite statistics.
2
upper bound
1.8
1.6
r
1.4
1.2
SUSY Dark Matter and Colliders
1
10 20 30 40 50 60 70 80 90 100
step/10000
B.C. Allanach – p.15/53

m 0 (TeV)
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
L/L(max)
0 0.5 1 1.5 2
M 1/2 (TeV)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
m 0 (TeV)
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
L/L(max)
0 10 20 30 40 50 60
tan β
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
m 1/2 (TeV)
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
L/L(max)
0 10 20 30 40 50 60
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
A 0 (TeV)
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
L/L(max)
0 10 20 30 40 50 60
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
tan β
tan β
SUSY Dark Matter and Colliders
B.C. Allanach – p.16/53

Annihilation Mechanism
Define stau co-annihilation when m˜τ is within 10% of
m χ
0
1
and Higgs pole when m h,A is within 10% of
2m χ
0
1
.
Region likelihood
h 0 pole 0.02±0.01
A 0 pole 0.41±0.03
˜τ co-an 0.27±0.04
˜t co-an (2.1 ± 4.8) × 10 −4
Table 0: Likelihood of being in a certain region of
mSUGRA parameter space.
Likelihood of chain ˜q L → χ 0 2 → ˜l R → χ 0 1 is 24±4%
SUSY Dark Matter and Colliders
B.C. Allanach – p.17/53

m χ1
0 (TeV)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
80 90 100 110 120 130 140
m h (GeV)
L/L(max)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
m χ1
0 (TeV)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0 0.5 1 1.5 2 2.5
m A (TeV)
L/L(max)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
m χ1
0 (TeV)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0 0.5 1 1.5 2
m τ (TeV)
L/L(max)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
L per bin
0.06
0.05
0.04
0.03
0.02
0.01
0
RH slepton
gluino
LH squark
light stop
0 0.5 1 1.5 2 2.5 3 3.5 4
m (TeV)
SUSY Dark Matter and Colliders
B.C. Allanach – p.18/53

0.0015
0.03
0.08
L/GeV
0.001
0.0005
0.02
0.01
0
20%
0 10 20 30
L/bin
0.07
0.06
0.05
0.04
0.03
0.02
0.01
29%
tanβ
60
50
40
30
20
10
0
0
37
400
800 1200
m τ - m 0 χ1
(GeV)
0 0.5 1 1.5 2
m A (TeV)
1600
L/L(max)
2000
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
log 10 [BR(B s -> µµ)]
-4
-5
-6
-7
-8
-9
0
-9 -8.5 -8 -7.5 -7 -6.5 -6
log 10 (BR[B s ->µµ])
-10.5 -10 -9.5 -9 -8.5
log 10 [δ(g-2) µ /2]
L/L(max)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
SUSY Dark Matter and Colliders
B.C. Allanach – p.19/53

Caveats
• Implicitly assumed that LSP constitutes all of
dark matter
• Assumed radiation domination in post-inflation
era. No clear evidence between freeze-out+BBN
that this is the case (t eq changes).
• Examples of non-standard cosmology that would
change the prediction:
• Extra degrees of freedom
• Low reheating temperature
• Extra dimensional models
• Anisotropic cosmologies
• Non-thermal production of neutralinos (late
decays)
SUSY Dark Matter and Colliders
B.C. Allanach – p.20/53

LHC (ATLAS)
SUSY Dark Matter and Colliders
B.C. Allanach – p.21/53

Collider SUSY Dark Matter
Production
Strong sparticle production and decay to dark matter
particles.
7 TeV
p
q
q
q
q
p
7 TeV
q,g
q,g
q
Interaction
q
~ q
~
q
χ 0 1
χ 0 1
Q: Can we measure enough to predict σ
SUSY Dark Matter and Colliders
B.C. Allanach – p.22/53

Collider SUSY Dark Matter
Production
Strong sparticle production and decay to dark matter
particles.
7 TeV
p
q
q
q
q
p
7 TeV
q,g
q,g
q
Interaction
q
~ q
~
q
χ 0 1
Any dark matter candidate that couples to hadrons can
be produced at the LHC
χ 0 1
SUSY Dark Matter and Colliders
B.C. Allanach – p.22/53

LHC vs LC in SUSY
Measurement
• LHC (start date 2007) produces strongly
interacting particles up to a few TeV. Precision
measurements of mass differences possible if the
decay chains exist: possibly per mille for leptons,
several percent for jets.
• ILC has several energy options: 500-1000 GeV,
CLIC up to 3 TeV. Linear colliders produce less
strong particles but much easier to make
precision measurements of masses/couplings.
Q: What energy for LC
Q: What do we get from LHC
LHC/ILC Working Group Report: hep-ph/0410364
SUSY Dark Matter and Colliders
B.C. Allanach – p.23/53

Coannihilation Slope
annihilation (%)
70
60
50
40
30
20
0 0
χ 1 χ1
0 ~
χ 1 τ
~ ~
τ τ
0 ~
χ 1 l
~ ~
l~ (τ /l )
~ ~
l =µ or
~
e
10
0
100 150 200 250 300 350 400
m~ (GeV)
τ
m 2˜lR ≈ m 2 0 + 0.15M 2 1/2 , M χ 0 1 ≈ 0.4M 1/2
Low enough M 1/2 ⇒ quasi-deegenerate ˜τ, M χ
0
1
SUSY Dark Matter and Colliders
B.C. Allanach – p.24/53

Coannihilation Slope
annihilation (%)
70
60
50
40
30
20
0 0
χ 1 χ1
0 ~
χ 1 τ
~ ~
τ τ
0 ~
χ 1 l
~ ~
l~ (τ /l )
~ ~
l =µ or
~
e
10
0
100 150 200 250 300 350 400
m~ (GeV)
τ
m 2˜lR ≈ m 2 0 + 0.15M 2 1/2 , M χ 0 1 ≈ 0.4M 1/2
If we do not assume mSUGRA, we will also have to
measure selectron and smuon properties.
SUSY Dark Matter and Colliders
B.C. Allanach – p.24/53

Coannihilation Theory
Uncertainties
Ωh 2
0.145
0.14
0.135
0.13
0.125
0.12
0.115
Ωh 2
Scale variation
Ωh 2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
-1 loop tau Yukawa
full calculation
-1 loop neutralino
-2 loop gaugino RGEs
0.11
100 150 200 250 300 350 400
M~ (GeV)
τ
0.02
100 150 200 250 300 350 400 450
M~ (GeV)
τ
Expect higher orders to be 100 times smaller than these
differences: 3-loop terms could possibly be important!
SUSY Dark Matter and Colliders
B.C. Allanach – p.25/53

Coannihilation Theory
Uncertainties
Ωh 2
0.145
0.14
0.135
0.13
0.125
0.12
0.115
Ωh 2
Scale variation
Ωh 2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
-1 loop tau Yukawa
full calculation
-1 loop neutralino
-2 loop gaugino RGEs
0.11
100 150 200 250 300 350 400
M~ (GeV)
τ
0.02
100 150 200 250 300 350 400 450
M~ (GeV)
τ
Effect of 2-loop RGE terms suggest a possible effect
from 3 loops. Jack and Jones find that it’s not significant
for the neutralino.
SUSY Dark Matter and Colliders
B.C. Allanach – p.25/53

Iterative Procedure
What change in a parameter produces a 10% change
in Ωh 2
Take a parameter point with ω −1 ≡ Ωh 2 . Change one
parameter at a time by fraction a 0 . Result is ω 0 , then
iterate
a i+1 = a i ω −1
10%
w i − ω −1
.
Small accuracy a ≡ a ∞ means the parameter has to
be known very accurately in order to predict Ωh 2 to
10%.
For parameters that are zero, we take the absolute value
as a rather than the fractional value.
SUSY Dark Matter and Colliders
B.C. Allanach – p.26/53

Uncertainties
We use two approaches to determine what variation of
parameters produce a 10% variation in Ωh 2 :
• PmSUGRA - variation of weak scale parameters
(not on mSUGRA trajectory): m χ
0
1
, M A , m b etc.
• mSUGRA - simple variation of mSUGRA
parameters and experimental inputs:
m 0 , M 1/2 , α s (M Z ), m t etc.
mSUGRA theory uncertainties estimated by varying
scale at which radiative corrections added to sparticle
masses:
0.5 < x ≡ M SUSY
√ m˜t 1
m˜t 2
< 2, M SUSY > M Z
SUSY Dark Matter and Colliders
B.C. Allanach – p.27/53

mSUGRA Coannihilation
Uncertainties
a
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
a(M 1/2 )
a(m 0 )
∆A 0
-
a(tanβ)
180
160
140
120
100
80
0
60
100 150 200 250 300 350 400 450
m~ (GeV)
τ
∆A 0
- (GeV)
a(m 0 ) ≈ a(M 1/2 ) comes from the sensitivity to
exp[−(m˜τ − M χ
0
1
)]
SUSY Dark Matter and Colliders
B.C. Allanach – p.28/53

mSUGRA Coannihilation
Uncertainties
a
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
a(M 1/2 )
a(m 0 )
∆A 0
-
a(tanβ)
180
160
140
120
100
80
0
60
100 150 200 250 300 350 400 450
m~ (GeV)
τ
∆A 0
- (GeV)
Unknown whether accuracies can be reached - but it
looks difficult
region.
Polesello, Tovey, JHEP05 (2004) 071
SUSY Dark Matter and Colliders
: ∆Ωh 2 ∼ .03 in diminished bulk
B.C. Allanach – p.28/53

PmSUGRA Coannihilation
a(tanβ, µ)
0.6
0.007
1400
tanβ
χ 0 2
µ
0.5
M 1
1200
χ 0 3
0.006
χ 0 4
1000
~
0.4
eR
0.005
~
e
800
L
~
0.3
τ 2
0.004
600
0.2
400
0.003
0.1
200
0.002
100 150 200 250 300 350 400
100 150 200 250 300 350 400 450
m~ (GeV) M
τ χ
0 (GeV)
1
a(M 1 )
mass (GeV)
RHS: Spectrum useful for optimal energy of linear collider.
ẽ R , ˜µ R also possible. Cascade ˜q L → χ 0 2 → ẽ R →
χ 0 1 available.
SUSY Dark Matter and Colliders
B.C. Allanach – p.29/53

PmSUGRA Coannihilation
a(tanβ, µ)
0.6
0.007
1400
tanβ
χ 0 2
µ
0.5
M 1
1200
χ 0 3
0.006
χ 0 4
1000
~
0.4
eR
0.005
~
e
800
L
~
0.3
τ 2
0.004
600
0.2
400
0.003
0.1
200
0.002
100 150 200 250 300 350 400
100 150 200 250 300 350 400 450
m~ (GeV) M
τ χ
0 (GeV)
1
a(M 1 )
LHS: Dependences in left-hand plot all come from the
effect on LSP mass.
Need to know M χ
0
1
very accurately.
mass (GeV)
SUSY Dark Matter and Colliders
B.C. Allanach – p.29/53

PmSUGRA Dependencies
∆M (GeV)
10
1
0.4
∆M
9
cos2θ τ 0.99 0.35
8
0.98
7
0.3
6
0.97
0.25
5
0.96
4
0.2
0.95
3
0.15
0.94
2
1
0.93
0.1
0
0.92 0.05
100 150 200 250 300 350 400 450
m~ (GeV) τ
cos2θ τ
-
∆cos2θ τ required, a(M)
∆cos2θ τ
- required
δM required
a(M χ )
1.3
1.2
1.1
0.9
0.8
0.7
0.6
100 150 200 250 300 350 400 450
m~
τ (GeV)
1
δM required (GeV)
LHS: plots of quantities along mSUGRA slope. Below
∆M = 1.78 GeV, no two-body stau decay. LC studies
indicate ∆M > 5 GeV is OK.
SUSY Dark Matter and Colliders
B.C. Allanach – p.30/53

PmSUGRA Dependencies
∆M (GeV)
10
1
0.4
∆M
9
cos2θ τ 0.99 0.35
8
0.98
7
0.3
6
0.97
0.25
5
0.96
4
0.2
0.95
3
0.15
0.94
2
1
0.93
0.1
0
0.92 0.05
100 150 200 250 300 350 400 450
m~ (GeV) τ
cos2θ τ
-
∆cos2θ τ required, a(M)
∆cos2θ τ
- required
δM required
a(M χ )
1.3
1.2
1.1
0.9
0.8
0.7
0.6
100 150 200 250 300 350 400 450
m~
τ (GeV)
1
δM required (GeV)
˜τ 1 χ 0 1 → τγ ∝ 3 cos 2θ τ + 5 from coupling of neutralino
to ˜τ L/R .
SUSY Dark Matter and Colliders
B.C. Allanach – p.30/53

PmSUGRA Dependencies
∆M (GeV)
10
1
0.4
∆M
9
cos2θ τ 0.99 0.35
8
0.98
7
0.3
6
0.97
0.25
5
0.96
4
0.2
0.95
3
0.15
0.94
2
1
0.93
0.1
0
0.92 0.05
100 150 200 250 300 350 400 450
m~ (GeV) τ
cos2θ τ
-
∆cos2θ τ required, a(M)
∆cos2θ τ
- required
δM required
a(M χ )
1.3
1.2
1.1
0.9
0.8
0.7
0.6
100 150 200 250 300 350 400 450
m~
τ (GeV)
1
δM required (GeV)
a(M χ ) found by keeping ∆M constant, δM by just
varying stau mass. mẽ, m˜µ needed to about 1.5%
SUSY Dark Matter and Colliders
B.C. Allanach – p.30/53

Slepton Dependence
7
δ - (∆M)
6
GeV
5
4
3
2
150 200 250 300 350 400 450
m~ (GeV)
e
Accuracy required on m˜l
− m χ
0
1
for WMAP precision.
LC studies say this is acheivable, but need more work
for cos θ τ (=0.987±0.06 at lower end of slope).
SUSY Dark Matter and Colliders
B.C. Allanach – p.31/53

Summary
• Markov chains bring out the multi-dimensionality
of the space: is a lot less constrained than in 2d
• Still, current data is constraining
• LHC could produce copious amounts of SUSY
dark matter
• Want to measure σ in order to predict Ωh 2 and
test cosmological assumptions
• 10% accuracy will require ILC+LHC data
• Can control many uncertainties by measuring
additional quantities: Γ A , m˜τ − M χ
0
1
, . . .
• Non mSUGRA case could well be easier.
• Have not discussed direct detection yet
SUSY Dark Matter and Colliders
B.C. Allanach – p.32/53

Supplementary Material
SUSY Dark Matter and Colliders
B.C. Allanach – p.33/53

Likelihood
L ≡ p(d|m) is pdf of reproducing data d assuming
mSUGRA model m (which depends on parameters).
p(m|d) = p(d|m) p(m)
p(d)
p(m 1 |d)
p(m 2 |d)
= p(d|m 1)p(m 1 )
p(d|m 2 )p(m 2 )
Thus, you can interpret the likelihood distribution as
relative probabilities if your ratio of priors is 1. Otherwise,
convolute it with YOUR priors!
SUSY Dark Matter and Colliders
B.C. Allanach – p.34/53

Funnel Slope
GeV
1400
1200
1000
800
600
m(χ 1 )
m(χ 3 )
Γ A
µ
m(l R )
m(l L )
m(χ 2 )
35
30
25
20
Γ A (GeV)
~
400
200
15
0
300 400 500 600 700 800 900 1000 10
M A (GeV)
< σv > −1 ∼ 4m χ 0Γ ( ( ) )
A MA − 2m 2
1 χ
0
g 2 4 1
+ 1
m χ 0 ˜χ 0 1 A Γ A
1
.
SUSY Dark Matter and Colliders
B.C. Allanach – p.35/53

Funnel Slope
GeV
1400
1200
1000
800
600
m(χ 1 )
m(χ 3 )
Γ A
µ
m(l R )
m(l L )
m(χ 2 )
35
30
25
20
Γ A (GeV)
~
400
200
15
0
300 400 500 600 700 800 900 1000 10
M A (GeV)
Notice that spectrum is quite heavy: need a high energy
ILC! Γ A will be important.
SUSY Dark Matter and Colliders
B.C. Allanach – p.35/53

Funnel Theory Uncertainties
Ωh 2
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
full calculation
no m b resummation
1 loop Higgs RGEs
1 loop gaugino RGEs
-2 loop QCD mt correction
Ωh 2
0.145
0.14
0.135
0.13
0.125
0.12
0.115
Ωh 2
(M A - 2M χ1
0)/Γ A
1.65
1.6
1.55
1.5
1.45
1.4
(M A - 2M χ1
0)/Γ A
0
300 400 500 600 700 800 900 10001100
M A (GeV)
0.11
0.6 0.8 1 1.2 1.4
x
1.35
LHS: Γ A affected by large m SM
b
/(1 + ∆ SUSY ) corrections
since A → b¯b ∝ Ab¯b coupling ∝ m b tan β, and
tan β = 50.
SUSY Dark Matter and Colliders
B.C. Allanach – p.36/53

Funnel Theory Uncertainties
Ωh 2
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
full calculation
no m b resummation
1 loop Higgs RGEs
1 loop gaugino RGEs
-2 loop QCD mt correction
Ωh 2
0.145
0.14
0.135
0.13
0.125
0.12
0.115
Ωh 2
(M A - 2M χ1
0)/Γ A
1.65
1.6
1.55
1.5
1.45
1.4
(M A - 2M χ1
0)/Γ A
0
300 400 500 600 700 800 900 10001100
M A (GeV)
0.11
0.6 0.8 1 1.2 1.4
x
1.35
RHS: x > 1.5 yielded MA 2 < 0 ie no EWSB. Strong
correlation of theory error with its effect on (M A −
2M χ
0
1
)/Γ A - could measure it!
SUSY Dark Matter and Colliders
B.C. Allanach – p.36/53

Funnel Accuracies
LHS: mSUGRA. a(m b ) worrying. α s (M Z )
dependence comes about through its effect on
m b (m b ). m 0 , M 1/2 might be feasible at LHC, m t
possible at ILC. tan β looks impossible.
a
10
1
0.1
0.01
0.001
M 1/2
-
m t
α s (M Z )
m b
m 0
∆A 0
tanβ
0
50 100 150 200 250 300 350 400 450 500
m 0 χ1
(GeV)
200
150
100
50
∆A 0 (GeV)
a
100
10
1
0.1
0.01
a(Γ A )
a(M χ1
0)
a(M A )
a(2M χ1
0 - M A )
a(µ)
2M χ1
0 - M A
-20
-40
-60
-80
-100
-120
50 100 150 200 250 300 350 400 450 500
M 0 χ1
(GeV)
0
2M χ1
0 - M A (GeV)
SUSY Dark Matter and Colliders
B.C. Allanach – p.37/53

Funnel Accuracies
SM inputs and tan β uncertainties can be controlled
by measuring M A , Γ A . Aχ 0 1χ 0 1 coupling ∼ 1/µ.
Γ A ∝ M A tan 2 β(m 2 b + m2 τ) (γγ option of LC,
A → µµ at LHC).
a
10
1
0.1
0.01
0.001
M 1/2
-
m t
α s (M Z )
m b
m 0
∆A 0
tanβ
0
50 100 150 200 250 300 350 400 450 500
m 0 χ1
(GeV)
200
150
100
50
∆A 0 (GeV)
a
100
10
1
0.1
0.01
a(Γ A )
a(M χ1
0)
a(M A )
a(2M χ1
0 - M A )
a(µ)
2M χ1
0 - M A
-20
-40
-60
-80
-100
-120
50 100 150 200 250 300 350 400 450 500
M 0 χ1
(GeV)
0
2M χ1
0 - M A (GeV)
SUSY Dark Matter and Colliders
B.C. Allanach – p.37/53

Funnel Accuracies
Mass of χ 0 1 is important, but not mixing (see a(µ)).
a
10
1
0.1
0.01
0.001
M 1/2
-
m t
α s (M Z )
m b
m 0
∆A 0
tanβ
0
50 100 150 200 250 300 350 400 450 500
m 0 χ1
(GeV)
200
150
100
50
∆A 0 (GeV)
a
100
10
1
0.1
0.01
a(Γ A )
a(M χ1
0)
a(M A )
a(2M χ1
0 - M A )
a(µ)
2M χ1
0 - M A
-20
-40
-60
-80
-100
-120
50 100 150 200 250 300 350 400 450 500
M 0 χ1
(GeV)
0
2M χ1
0 - M A (GeV)
SUSY Dark Matter and Colliders
B.C. Allanach – p.37/53

Focus Point Slope
mass (GeV)
1000
900
800
700
600
500
400
300
200
M χ1
0
χ 2
0
χ 3
0
µ
χ 4
0
M 1
M 2
100
150 200 250 300 350 400 450
M 0 χ1
(GeV)
Heavy sfermions and A 0 . M 1 < µ < M 2 , ie significant
Higgsino component ∼ 25%.
SUSY Dark Matter and Colliders
B.C. Allanach – p.38/53

Focus Channels and Theory
annihilation (%)
100
80
60
40
20
coannihilation30
annihilation
tt -
bb -
WW/ZZ
Zh,hh
Ωh 2
35
25
20
15
10
-neutralinos
-2 loop QCD
-EW/Higgs
-charginos
non re-summed mb
Full calculation
-2 loop Higgs RGEs
0
150 200 250 300 350 400 450
M χ
0 (GeV)
1
5
0
1 2 3 4 5 6 7
m 0 (TeV)
t¯t annihilation predominantly through Z. Coannihilation
≡ χ 0 1χ 0 i or χ0 1χ ± 1
. Several competing channels.
SUSY Dark Matter and Colliders
B.C. Allanach – p.39/53

Focus mSUGRA Accuracies
a
0.0003
0.05
45
α s (M Z )
∆A 0
m t 0.045
a(m 0 )
a(M
0.00025
0.04
1/2 )
40
a(tanβ)
0.035
35
0.0002
0.03
0.06
0.06
0.05
0.05 0.025
30
0.00015
0.04
0.04
0.02
0.03
0.03
25
0.02
0.02
0.015
0.0001
0.01
0.01 0.01
m 20
0
b
200 300 400 0 0.005
5e-05
0
150 200 250 300 350 400 450 100 200 300 400 500 15
M 0 χ1
(GeV)
M 0 χ1
(GeV)
a
∆A 0 (GeV)
∆A 0 (GeV)
δm t = 30 MeV might be possible at future ILC but
a(m 0 ) < 0.5% looks completely unfeasible.
SUSY Dark Matter and Colliders
B.C. Allanach – p.40/53

Focus PmSUGRA Accuracies
0.25
0.2
0.15
m t
M 2
tanβ
M 1
µ
M χ3
0
a
0.1
0.05
0
150 200 250 300 350 400 450
M 0 χ1
(GeV)
Easier outside of mSUGRA, eg µ no longer sensitive
to m t (∝ coupling to neutral goldstone).
SUSY Dark Matter and Colliders
B.C. Allanach – p.41/53

LHC SUSY Measurements
q l + l −
˜q L χ 0 2 ˜l χ 0 1
m 2 ll
= (p l1 + p l2 ) 2 edge
position measures
√
(m
2
χ 0 2−m 2˜l )(m 2˜l −m 2 χ 0 )
1
m 2˜l
Events/0.5 GeV/100 fb -1
400
300
200
100
205.7 / 197
P1 2209.
P2 108.7
P3 1.291
0
0 50 100 150
M ll
(GeV)
BCA, C Lester, A Parker, B Webber, JHEP 09 (2000) 004
SUSY Dark Matter and Colliders
B.C. Allanach – p.42/53

Edge Fitting at S5 and O1
200
150
100
50
400
300
200
100
200
150
100
50
300
200
100
dσ/dmll (Events/100fb-1 /0.375GeV)
dσ/dmllq (Events/100fb-1 /5GeV)
0
0
0
0
0 50 100 150
0 200 400 600 800 1000
0 50 100 150
0 200 400 600 800 1000
(a) m ll (GeV)
(b) m llq (GeV)
(a) m ll (GeV)
(b) m llq (GeV)
400
300
200
600
400
dσ/dmlq (Events/100fb-1 /5GeV)
dσ/dmlq (Events/100fb-1 /5GeV)
100
200
0
0
0
0
0 200 400 600 800 1000
0 200 400 600 800 1000
0 200 400 600 800 1000
0 200 400 600 800 1000
(c1) High m lq (GeV)
(c2) Low m lq (GeV)
(c1) High m lq (GeV)
(c2) Low m lq (GeV)
150
100
50
100
80
60
40
20
dσ/dmllq (Events/100fb-1 /5GeV)
dσ/dmhq (Events/100fb-1 /5GeV)
dσ/dmll (Events/100fb-1 /0.375GeV)
dσ/dmllq (Events/100fb-1 /5GeV)
200
100
400
300
200
dσ/dmlq (Events/100fb-1 /5GeV)
dσ/dmlq (Events/100fb-1 /5GeV)
100
60
40
20
80
60
40
20
dσ/dmllq (Events/100fb-1 /5GeV)
dσ/dmzq (Events/100fb-1 /5GeV)
0
0
0
0
0 200 400 600 800 1000
0 200 400 600 800 1000
0 200 400 600 800 1000
0 200 400 600 800 1000
(d) m llq (GeV)
(e) m hq (GeV)
(d) m llq (GeV)
(e) m zq (GeV)
SUSY Dark Matter and Colliders B.C. Allanach – p.43/53

Edge Positions
endpoint S5 fit O1 fit
m ll 109.10±0.13 70.47±0.15
m llq edge 532.1±3.2 544.1±4.0
lq high 483.5±1.8 515.8±7.0
lq low 321.5±2.3 249.8±1.5
llq thresh 266.0±6.4 182.2±13.5
Best case lepton mass measurements can be as accurate
as 1 per mille, but jets are a few percent
SUSY Dark Matter and Colliders
B.C. Allanach – p.44/53

Edge to Mass Measurements
400
350
300
width S5 width O1 250
χ 0 200
1 17 22
150
˜lR 17 20
χ 0 100
2 17 20
50
0
˜q 22 20
m l
(GeV)
0 50 100 150 200 250 300 350 400
m 1
(GeV)
Mass differences well constrained, but overall mass
scale not so well constrained by LHC
SUSY Dark Matter and Colliders
B.C. Allanach – p.45/53

Fitting to SUSY Breaking
Model
m 1/2 [GeV]
480
460
440
420
400
380
Softsusy
Spheno
Suspect
Isajet
360
200 300 400 500 600
m 0 [GeV]
tan β
2.4
2.2
2.
1.8
1.6
Spheno
Softsusy
Suspect
Isajet
200 300 400 500 600
m 0 [GeV]
• Experimenters pick a SUSY breaking point
• They derive observables and errors after detector
simulation
• We fit
this “data” with our codes
BCA, S Kraml, W Porod, JHEP 0303 (2003) 016
SUSY Dark Matter and Colliders
B.C. Allanach – p.46/53

α −1
i
M
SUSY
1
2
3
log (E/GeV)
10
M
GUT
SOFTSUSY
Get g i (M Z ), h t,b,τ (M Z ). ✛
❄
Run to M S .
❄
REWSB, iterative solution of µ
❄
M X . Soft SUSY breaking BC.
Run to M S . Calculate
❄
sparticle pole masses.
❄
Run to M Z
BCA, Comp. Phys. Comm. 143 (2002) 305.
SUSY Dark Matter and Colliders
B.C. Allanach – p.47/53

Other Observables
Often more complicated, eg m llq edge:
[ (m
2˜q − m 2 χ
max 2)(m 2 0 χ
− m 2 0
2 χ
) 0
1
m 2 χ 0 2
]
(m˜q m˜l
− m χ
0
2
m χ
0
1
)(m 2 − m 2˜l )
χ 0 2
m χ
0
2
m˜l
, (m2˜q − m2˜l )(m2˜l − m2 χ ) 0
1
m 2˜l
,
Also m high
lq
, m low
lq , llq threshold , M T 2 2
(m) =
{
}]
min /p1 +/p 2 =/p T
[max m 2 T (p l 1
T
, /p 1 , m), m 2 T (p l 2
T
, /p 2 , m)
,
max[M T2 (m χ
0
1
)] = m˜l]
for dislepton production.
SUSY Dark Matter and Colliders
B.C. Allanach – p.48/53

Statistics Study
• Choose two model-points: S5 (m 0 = 100, m 1/2 =
300, A 0 = 300, tan β = 2.1, µ > 0) and O1
(m˜l
= 177, m 1/2 = 306, A˜q = 137, m˜q = 0, A˜l
=
306, tan β = 10, µ > 0)
• Find cuts to measure “signal” endpoints
• Estimate expected accuracy of ATLAS
measurement: 100 fb −1
• Perform χ 2 fits of sparticle masses to expected
positions of edges expected from an ensemble of
experiments
• Interpret results as statistics of measurement on
sparticle masses
SUSY Dark Matter and Colliders
B.C. Allanach – p.49/53

Cuts Example
We use ATLFAST2.16, HERWIG6.0, ISAWIG and
ISAJET7.42. Assume 100 fb −1 of LHC data.
• |η j | ≤ 5, p j T
≥ 15 GeV
• p e T ≥ 5, pµ T ≥ 6, |η l| ≤ 2.5
• l isolation: 10 GeV in ∆R = 0.2, ∆R(lj) ≥ 0.4.
eg for m ll :
• 2 OSSF leptons, p l 1
T
≥ p l 2
T
≥ 10 GeV.
• n jets ≥ 2, p j 1
T ≥ pj 2
T ≥ 150 GeV, /p T > 300 GeV
OSSF-OSDF subtracts well the Standard Model
background.
SUSY Dark Matter and Colliders
B.C. Allanach – p.50/53

Uncertainties in Relic Density
Bulk region: ˜B ˜B → Z, h → l¯l. Coannihilation: ˜τχ
0
1 → τ + X
Figure 0: Bulk/coannihilation region. Full:
SoftSusy, dotted: SPheno.
SUSY Dark Matter and Colliders
B.C. Allanach – p.51/53

Focus Point
Figure 0: Focus point region. Full: SoftSusy, dotted:
SPheno, dashed: SuSpect. Higgsino LSP annihilates
into ZZ/W W
SUSY Dark Matter and Colliders
B.C. Allanach – p.52/53

High tan β
BCA, Belanger, Boudjema, Pukhov, Porod, hep-ph/0402161. Baer et
al
Figure 0: High tan β region. Full: SoftSusy, dotted:
SPheno, dashed: SuSpect. Get annihilation into A.
SUSY Dark Matter and Colliders
B.C. Allanach – p.53/53