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Dark Matters - The trouble with searching for rare events
Ken Clark March, 2016
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Looking for Rare Events• First of all, what is considered rare? • Certainly anything that happens once in
the span of a lifetime. Maybe once in a year?
• “You’ll know one when you see one … if you see one”
• More formal: binary dependent variables with dozens to thousands of times fewer ones (events, such as wars, vetoes, cases of political activism, or epidemiological infections) than zeros (“nonevents") - Logistic Regression in Rare Events Data
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Dealing with Rare Events
• So what can we do? Seem to be three solutions:
• Wait • Predict from other information • Interpret the results we have
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Rare Events
• Let’s assume that anything that happens < 1 time per month is rare
• Could be many things… meteor strikes, government upheavals, floods, birthdays…
• Can we do anything predictive with this class of events?
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Answer: Maybe?
• Discrete physical phenomena tend to follow a power law distribution
• Can be used with earthquake magnitude • Gutenberg-Richter law
5
N = 10a�bM
N - number of earthquakes M - magnitude of the earthquake a,b - constants
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Earthquake Data
• Data on recent earthquakes is available here: http://earthquake.usgs.gov/earthquakes/feed/v1.0/csv.php
• Can you predict the number of occurrences of magnitude 5? 6?
• How accurate are the predictions?
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Dark Matter Introduction
7
Structure formation
with simulation
Separation of luminous and non-luminous
mass
Galaxy rotation curves
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A Popular Field…
8
Snowmass 2013
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Dark Matter
What do we know? 1.Long lived (survived until current day) 2.Non-baryonic (Hydrogen:Deuterium Ratio) 3.No EM interactions (haven’t seen it) 4.80% of all matter (rotation curves, CMB) 5.Non-relativistic (galactic structure
formation, rotation curves)
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Dark Matter Detection
Two primary types: direct and indirect
10
Products of self-interaction outside the detector
Interaction of particle inside the detector
� � �
�Regular Matter Regular Matter
Regular Matter
Regular Matter
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Dark Matter Math
11
dR
dE⌫= �p
X
A
f(A)⇥ S(A,ER)⇥ I(A)⇥ F 2(A,ER)⇥ g(A)⇥ ✏(E⌫)
A - Atomic NumberER - nuclear recoil energy
} } } }} }
mass fraction of target
phase spacedistribution of trapped DM
couplingenhancement
nuclear form factor
nuclear recoilquenching factor
detectorresponsefunction
cross-section
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Direct Detection Methods• All experiments search for small energy
deposits • Make some assumptions to find
MAXIMUM energy deposited • Mx = 100 GeV = 1.8x10-25 kg
• v = 220 km/s = 2.2x105m/s
• Equivalent to a mosquito flying at 0.00015 kph
12
T =1
2mv2 =
1
2⇤ 1.8⇥ 10�25 ⇤ (2.2⇥ 105)2 = 4.4⇥ 10�15J
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Background Removal
• Because of this small energy deposit, DM experiments are all about dealing with backgrounds
• Done in two ways 1.Don’t have interactions you don’t want 2.Tell the “bad” interactions from the
“good”
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Background Removal
14
Phonons (Most Energy)
Scintillation (Least Energy)
Ionization (Large Fraction)
Superheated Liquids
CRESST 1 CUORE
CDMS EDELWEISS
CoGeNTXENON
LUX ArDM
DEAP CLEAN
CRESSTPICASSO COUPP PICO
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Background Removal
15
Phonons (Most Energy)
Scintillation (Least Energy)
Ionization (Large Fraction)
Superheated Liquids
CRESST 1 CUORE
CDMS EDELWEISS
CoGeNTXENON
LUX ArDM
DEAP CLEAN
CRESSTPICASSO COUPP PICO
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Kenneth Clark
PICASSO + COUPP = PICO• PICO is a novel detector using superheated
liquid to amplify the energy deposit • Small deposit of energy triggers the
formation of large bubbles, detectable using acoustic or visual methods
16
� �
L
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Kenneth Clark
Seitz Model
17
• The currently used model says that the energy for the formation of the bubble must come from the interaction, not the surrounding fluid
• This requires a threshold energy deposit in a critical radius
Ethreshold
= 4⇡r2c
✓� � T
d�
dT
◆+
4⇡
3r3c
⇢b
(hb
� hl
)� 4⇡
3r3c
(Pb
� Pl
)
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Kenneth Clark 18
Ethreshold
= 4⇡r2c
✓� � T
d�
dT
◆+
4⇡
3r3c
⇢b
(hb
� hl
)� 4⇡
3r3c
(Pb
� Pl
)
Overcoming surface tension Double counting
Vaporization of fluid
• The currently used model says that the energy for the formation of the bubble must come from the interaction, not the surrounding fluid
• This requires a threshold energy deposit in a critical radius
Seitz Model
{ {{
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Kenneth Clark
Backgrounds Method 1• Gammas and betas are effectively not
detected by the detector as they do not meet the Ethreshold in rc requirement.
19
1.75 MeV ɣ MIPs
neutrons (AcBe)
α particles (241Am decays)
nuclear recoils (210Pb)
50 GeV DM particle
Phys. Lett. B 711
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Kenneth Clark
Backgrounds Method 2
• Distinguish the bubbles formed by backgrounds from those formed by recoil events
• In this case, alphas vs neutrons
20
AP = A(T )X
j
Gj
X
n
Cn(~x)
fn
maxX
fn
min
f ⇥ psd
jf
Transducer gain Position correction Frequency Power Spectral DensityTemperature Correction
PRD 86:052001
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The Ultimate Goal (?)
• Goal should be discovery…21
Snowmass 2013
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Calculation of Limit
• Calculation based on 4 parameters • Background rate B • Background misidentification β • Signal acceptance α • exposure MT (mass x time)
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Calculation of Limit• With NO discrimination (β=1) all
interactions are potentially DM • If there are no events, the 90% CL
23
Pn(obs)(ns
, 90) =(n
s
, 90)nobs
nobs
!e�n
s
,90
=(ns, 90)0
0!e�ns,90 = 0.1
! ns,90 = �ln(0.1) = 2.3
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Calculation of Limit• With NO discrimination (β=1) all
interactions are potentially DM • If there are no events, the 90% CL
• Obviously scales with the exposure time
24
S90 =2.3
↵MT
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Calculation of Limit• With far more events than expected
signal, assume all events are background
• This can be expressed as
25
S90 =NBG + 1.28
pNBG
↵MT
S90 =�
↵+
1.28
↵
r�B
MT
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Calculation of Limit
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A Better Way?• Let’s study a bit more complex example
27
Phonons (Most Energy)
Scintillation (Least Energy)
Ionization (Large Fraction)
Superheated Liquids
CRESST 1 CUORE
CDMS EDELWEISS
CoGeNTXENON
LUX ArDM
DEAP CLEAN
CRESSTPICASSO COUPP PICO
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Scintillation and Ionization
• Example used here is LUX
• Xenon used in both liquid and gas state within an electric field to amplify small deposit
• Two different scintillations detected, the ratio of which discriminates signal from background
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29
arXiv:1310.8214
Scintillation and Ionization
• Example used here is LUX
• Xenon used in both liquid and gas state within an electric field to amplify small deposit
• Two different scintillations detected, the ratio of which discriminates signal from background
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Phonons and Ionization
• CDMS similarly uses two channels to distinguish background from signal
• Collecting both ionization and phonons allows for discrimination
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Risks
• The main (and obvious) risk to removing backgrounds is to the livetime of the experiment
• Aggressive cleaning puts you back into the region discussed previously
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Setting a Limit
• Need to know the number of counts and the distribution of the probability function
• Also need to know the expected number of counts seen
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Expected Counts
• The simplest expectation is:
• ER is the recoil energy • R0 is the total event rate • E0 is the most probable dark matter
energy
33
dR
dER=
R0
E0re�ER/E0r
r =4MDMT
(MD +MT )2
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Trick is in the R0
• Define the R0 so that it can be calculated • It’s the event rate per unit mass for the
earth velocity is 0 and the escape velocity is infinite
34
R0 =2
⇡1/2
N0
A
⇢DMD
�0v0
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Defining The Bounds• Really need to define the bounds on the
observed number of events • This has to take into account the
backgrounds and the uncertainty associated with those backgrounds
• Three methods here: 1. Feldman Cousins 2. Yellin 3. Binned Likelihood
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Feldman Cousins• Want to know the confidence region for
the number of signal events given the number of observed and background
36
P (n|µ) = (µ+ b)ne�(µ+b)
n!
• Then maximize the probability, changing μbest and the ratio is the parameter used to define 90%
R =P (n|µ)
P (n|µbest)
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Feldman Cousins• Plot this and read the results for your
experiment!
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But… I’m not sure about my Backgrounds
• All DM experiments are in a new region of detector physics
• The backgrounds are not completely understood
• The best measurement of the backgrounds is the dark matter data itself…
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Yellin to the Rescue!
• The events don’t match the expectation well
• Use the “maximum gap” and find the cross-section at which 90% of the trials have a gap smaller than this
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More Yellin• This can only set an upper limit, and can
never be used for discovery • Also generates one-sided (upper) limits • No information about the background
goes into this (it may be unknown)
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Binned Maximum Likelihood• We want to use all the information • Assume we have one discriminator • Bin the number of counts in that
parameter
41
L =kY
i=1
P (ni|µ(x))
lnL =kX
i=1
ln(P (ni|µ(x)))
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Binned Maximum Likelihood
42
lnL =kX
i=1
ln(P (ni|µ(x)))
Use Poisson for low statistics (as before)
lnL =kX
i=1
ln(e�µiµni
i
ni!)
But introduce a new term for signal and background counts
µ
x
= S · Ps
(x) +B · Pb
(x)
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Binned Maximum Likelihood
43
S and B are the hypothetical number of signal and background events in bin x
µ
x
= S · Ps
(x) +B · Pb
(x)
The likelihood can now be minimized to produce the best estimate of signal and background, which is used to set the
limit
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Let’s Look at Data
• This is CDMS data taken when I was a postdoc with them
• We have the single and multiple rates for one detector
• Use these to define backgrounds and signal
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Don’t have DM Data… as such
• I have calibration data • Split into “singles” and “multiples” • Neutron events will happen in the
“singles” • Let’s define “singles” as calibration,
“multiples” as data
45
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CDMS Data
46
0
1
2
3
4
5
6
Recoil Energy (keVnr)2 4 6 8 10 12 14
Ioni
zatio
n En
ergy
(keV
ee)
0
1
2
3
4
5
6
7
8
9
10
MultiplesMultiples
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Recoil Energy (keVnr)2 4 6 8 10 12 14
Ioni
zatio
n En
ergy
(keV
ee)
0
1
2
3
4
5
6
7
8
9
10
SinglesSingles
600g Germanium crystal, 1 day exposure
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CDMS Data
47
0
1
2
3
4
5
6
Recoil Energy (keVnr)2 4 6 8 10 12 14
Ioni
zatio
n En
ergy
(keV
ee)
0
1
2
3
4
5
6
7
8
9
10
MultiplesMultiples
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Recoil Energy (keVnr)2 4 6 8 10 12 14
Ioni
zatio
n En
ergy
(keV
ee)
0
1
2
3
4
5
6
7
8
9
10
SinglesSingles
Electrons
Neutrons
600g Germanium crystal, 1 day exposure
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Go!
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