Study of Liquid Xenon Detector for WIMP Dark Matter
Daniel H. Silverman
Brown University, Providence, Rhode Island, 02912
Email: [email protected]
Contents:
1. Background on the Dark Matter Problem
1.1. Introduction
1.2. Baryonic Dark Matter
1.3. Cold Dark Matter vs. Hot Dark Matter (and the CMB)
1.4. Why Cold Dark Matter Is Favored
1.5. Direct Detection of Dark Matter
2. WIMP Interaction Rate and Recoil Spectrum
2.1. Basic Recoil Spectrum
2.2. Modifications to the Basic Recoil Spectrum
3. Statistical Extraction of Mass from Predicted Data
3.1 Overview
3.2 Monte Carlo Simulation of Direct Detection Data
3.3 Maximum Likelihood Fitting Routine for Monte Carlo Data
3.4 Results
3.5. Limitations of Analysis and Recommendations for Further Study
4. Xenon direct detection - Discrimination of nuclear recoil events in MCP data
4.1 Overview
4.2. A note on notation
4.3. Calibration of pulse area to photoelectron scale
4.4. Analyzing the Calibration Data from Xenon Prototype Detector
4.5. Monte Carlo Simulation
4.6. Effect of Field Bias on Pulse Shape
4.7. Conclusion
Chapter 1: Background on the Dark Matter Problem
1.1 Introduction
The search for dark matter (DM) is motivated by both particle physics and
cosmology. In the particle physics realm, DM represents a good candidate for first
discovery of a supersymmetric (SUSY) particle. In the realm of cosmology, cold dark
matter (CDM) is needed to account for a number of cosmological observations such as
large scale structure formation and flat galactic rotation curves.
One compelling argument for the existence of DM lies in observed galactic rotation
velocities. Based on simple Newtonian assumptions, the velocity of a rotating object
should be given by the equation,
v(r ) =
GM (r )
r
(1.1)
where v(r) is the velocity at radius r, G is Newton’s gravitational constant, and M(r) is the
mass enclosed within radius r. According to this equation, once one looks past the edge
of visible matter from the center of a galaxy, one would expect the velocity to decrease as
v(r ) ! 1 r . Yet observations of the redshift of hydrogen emission lines out to as large a
distance as we can measure show that instead of this predicted decrease in velocity, the
velocity curves asymptote to a constant value as r increases. 1 This effect was first
observed by Zwicky for eight galaxies in the Coma cluster.2 Fig. 1.1 illustrates the flat
rotation curves observed for a spiral galaxy by measuring the redshift of the 21 cm
hydrogen gas emission lines.
Bulk of luminous matter
v ~ const
• data
– bulge, disk & halo
v ~ r–1/2
Figure 1.1: Plot of galactic rotation curve for spiral galaxies. The bottom line represents the
curve predicted for a galaxy made up only of a bulge and a disk, the top line adds a DM halo. The
dots are actual data points.
2
This observation is consistent with a mass distribution which is linearly increasing
with radius, i.e. M(r) ~ r. In order to resolve this discrepancy between theory and
experiment, physicists have postulated the existence of non-luminous dark matter (DM)
which exists in the form of a halo around disk galaxies. The extra mass provided by a
dark matter halo would explain the asymptotic rotation curves.3
Additional cosmological evidence for dark matter lies in current measurements of the
density of the universe. In order for the spatial curvature of the universe to be flat, the
total density of matter and energy must be equal to the critical density,
3H 02
%C =
# 1.88 " 10 ! 26 h 2 kg / m 3
8$G
(1.2)
where H0 is the present value of the Hubble constant, and h is H0 in dimensionless units
of 100 km/s/Mpc. The fractional densities of different types of matter and energy in the
universe can be represented by comparison with this critical density, such that
"x =
!x
!c
(1.3)
is the density parameter of x. Experiments have shown that for luminous matter, Ωlum <
.01. Rotation curves of galaxies imply that >90% of galactic mass is dark, therefore for
all non-luminous matter, ΩDM > 0.1.4 This is a lower limit because DM halos may extend
out beyond where we can measure galactic velocity curves. A number of observations of
galactic clusters and superclusters using methods of velocity flows, x-ray emission
temperatures, and gravitational lensing methods all support the existence of a greater
percentage of DM in the universe. They also predict the total matter component of the
universe to be within the range ΩM = 0.2 - 0.35.
1.2 Baryonic Dark Matter
The term dark matter is used loosely to mean a number of different things. Mostly it
is used to mean non-baryonic, non-luminous matter, but in general, dark matter defined
as any non-luminous matter can be baryonic as well as non-baryonic. According to the
Standard Model, baryonic matter is matter made up of particles composed of sets of three
quarks, such as protons and neutrons, which make up the majority of the “normal” matter
we interact with in our daily lives. The baryonic component of dark matter could consist
of a number of different types of objects such as white dwarfs, neutron stars, black holes,
and brown dwarfs.
Nevertheless, a number of experiments have determined that baryonic matter makes
up only a small part of the total mass of dark matter halos. According to results from
direct searches for massive compact halo objects (MACHOs) using microlensing,
baryonic dark matter forms less than 25% of the total mass of halos.6 Therefore the
majority of the mass needed to account for galactic velocity dispersion curves and other
observations cannot be due to familiar baryonic matter, and thus must be some form of
non-baryonic dark matter. The largest constraints on the amount of baryonic matter in the
3
universe comes from models of Big Bang nucleosynthesis (BBN), which can predict the
density parameter of baryonic matter by studying the abundance of hydrogen, deuterium
and other elements present in the primordial universe. BBN measurements imply that Ωb
= 0.045, and thus this result shows that baryonic matter forms only a small part of the
mass of the universe. Therefore, the current search for dark matter is focused on nonbaryonic types.
1.3 Cold Dark Matter vs. Hot Dark Matter, (and the CMB)
Non-baryonic dark matter can be divided into two different types, hot and cold. Based
on the Big Bang model, the early universe was so dense and hot that all the baryonic
matter was ionized, forming a gas of free electrons and nuclei that made the universe
opaque to light due to the scattering of photons off of charged particles. As the universe
expanded, it cooled sufficiently so that electrons and ions combined to form neutral
atoms, thus making the universe transparent and leaving the remaining blackbody
photons free to move about the universe without interference from free electrons. We can
detect that radiation today, and it is known as the Cosmic Microwave Background
(CMB). Hot dark matter refers to particles which were relativistic during the early stage
of the universe when light decoupled from matter and the CMB was left as a remnant,
and cold dark matter refers to particles which were non-relativistic.
The major candidate for hot dark matter is standard-model neutrinos, which couple to
the weak force (hence their small interaction cross section), and which have been
discovered to be massive in recent experiments involving solar and atmospheric neutrino
oscillations. But there is a major problem with a model of the universe which involves a
predominance of hot over cold dark matter. According to large N-body computer
simulations, an early universe composed of mostly hot dark matter would wash out the
structural formations such as clusters and superclusters which we find present in the
universe today.7 In addition, although neutrinos have been found to be massive, they are
not massive enough to account for the majority of the dark matter in the universe.
According to a combination of measurements including galactic clustering, CMB, and
Lyman-alpha forest the density parameter of neutrinos has been determined to be less
than Ω<0.016 (assuming h = 0.7), which is only a small percentage of the DM
component mentioned above.8 This limit applies to all forms of hot DM, therefore cold
DM must form the dominant contribution to the total dark mass in the universe.
1.4 Why Cold Dark Matter Is Favored
In order to account for the large scale structure formation (i.e. clumping) that is
observed in the present universe in conjunction with the high degree of homogeneity in
the CMB, the early universe must have been dominated by cold dark matter (CDM). This
conclusion is supported by a number of experimental results mentioned in the last
section. Therefore, it is upon the CDM component that many current experiments are
focused.
4
Any candidate for a CDM particle must satisfy several straightforward conditions:
they must couple weakly or not at all to the electromagnetic field; they must have a
significant relic density; and they must be stable enough to have not decayed since they
dropped out of thermal equilibrium with the early universe.9 There are several candidates
for CDM, but the one currently favored is the Weakly Interacting Massive Particle
(WIMP). WIMP mass could vary from about 10 GeV to several TeV, and would have an
interaction cross section comparable to the electroweak interaction. Assuming that the
WIMPS were in thermal equilibrium with the rest of the matter within the very early
universe (<1ns) as the universe expanded and cooled, the WIMPs would at a certain point
drop out of thermal equilibrium and would cease being created/annihilated. After this
point, the total number of WIMPs in the universe would remain the same.
If we assume that the WIMPs have an annihilation cross section of the order of the
weak scale it leads to the sort of relic densities we might expect for CDM in the universe
today. This is one of the primary reasons WIMPs are currently believed to form the
dominant component of the matter in the universe.
The other major motivation for the existence of WIMPs comes from Supersymmetry
(SUSY), which predicts that for each particle in the standard model there exists a
corresponding supersymmetric particle. It is beyond the scope of this paper to go into a
detailed account of SUSY predictions, so it will suffice to say that the best WIMP
candidate is the lightest superparticle (LSP) which is stable in the vast majority of SUSY
models. Larger particle accelerators are currently being built to try and create SUSY
particles in the laboratory, but in the meantime, alternative experiments are underway to
attempt to directly detect WIMPs in the dark matter halo surrounding the Milky Way
galaxy.
1.5 Direct Detection of Dark Matter
Currently there are a number of experiments running, and more being proposed to
detect Weakly Interacting Massive Particles (WIMPS) directly through interaction with a
target nucleus. The CDMS (Cryogenic Dark Matter Search) experiment uses Germanium
and Silicon targets. A new experiment is currently being developed by the XENON
collaboration and will use liquid Xenon as the source for target nuclei. For the purposes
of this paper, we will focus on liquid Xenon detectors, and the discussion below applies
to these types of detectors specifically.
Direct detection experiments provide an exciting opportunity to answer some of the
fundamental questions of both cosmology and particle physics. Their advantage over
particle accelerators lies in their ability to detect cosmic particles of masses greater than
those that we can currently create in particle accelerators. Their disadvantage is that
because the WIMP interaction cross section is very small, in order to catalogue a
significant event rate (if WIMPs are detected), or to exclude significant regions of the
WIMP mass-cross section parameter space predicted by various SUSY theoretical models
(if WIMPs are not detected), there must be a large detector mass and the experiment must
be run continuously for a long period of time. Ultimately, although direct detection poses
a number of experimental challenges, it also offers the exciting opportunity to shed some
5
light on some of the most puzzling questions facing cosmology and particle physics
today.
The primary means by which liquid Xenon direct detection experiments work is by
shielding a cryogenic container of the target substance, and lining part of the container
with photo-multiplier tubes (PMTs) or some other source of signal amplification. The
target must be a highly stable compound, preferably with both even and odd isotopes in
order to achieve sensitivity to spin independent as well as spin dependent interactions.
When a WIMP or some other particle such as a neutron or alpha particle enters the
chamber and scatters off a target nuclei, it causes the nuclei to recoil, creating ionization
electrons and/or photon/phonon emission as the excited target atoms fall back into their
ground state. The total recoil energy of the nucleus can be measured by examining the
total pulse area of such an event. By creating a histogram of the recoil energies of various
WIMP events, the differential energy spectrum, dR/dER can be constructed.
The differential energy spectrum can also be theoretically derived based on certain
parameters such as WIMP mass and kinetic energy. By comparison of the theoretically
derived differential energy spectrum with the experimentally determined spectrum,
WIMP parameters can be determined to varying degrees of accuracy. Lewin and Smith10
have done a comprehensive analysis of the theoretical derivation of the WIMP
differential energy spectrum – and the next chapter will be devoted to summarizing their
results.
6
Chapter 2: WIMP Interaction Rate and Recoil Spectrum11
2.1 Basic Recoil Spectrum
The differential recoil spectrum describes the distribution of recoil energies expected
from a direct detection experiment. By integrating the differential recoil spectrum
between any two energy values, one can find the number of events predicted in that
energy range per kg of target per day. The most basic theoretical model for dark matter
differential detection rates is a decreasing exponential function of the form:
R
dR
= 0 e ! ER / E0r
dE R E0 r
(2.1)
Where R0 is the total event rate, E0 is the most probable incident kinetic energy of a
WIMP of mass MD, ER is the recoil energy of the target particle, r is a dimensionless form
of the reduced mass, 4 MD MT / (MD + MT)2 and MT is the mass of the target particle. This
smoothly decreasing exponential function can be derived by assuming a stationary
detector, and a Maxwellian velocity distribution of WIMPs in the Milky Way’s DM halo
such that
f (v, vE ) = e ! ( v+vE )
2
v02
(2.2)
where v is the WIMP velocity onto the target (presumably on Earth), vE is the Earth’s
velocity through the DM halo – assumed to be zero for the moment, and v0 is the mean
WIMP velocity.
2.2 Modifications to the Basic Recoil Spectrum
This basic model must be modified to take into account various effects that are
present in any dark matter detection experiment. The effects are as follows:
a. Earth’s motion relative to the dark matter halo due to both the Earth’s motion
about the Sun, and the Sun’s motion through the galaxy.
b. A maximum WIMP velocity in the DM halo equal to the galactic gravitational
escape velocity, vesc ~ 600 km/s.
c. An experimentally determined quenching factor (QF) for nuclear recoils such that
the recoil energy deposited by a nuclear recoil is about 25% (for liquid Xe) of that
deposited by an electron of the same incident energy.
d. Limitations based on the quantum efficiency of the PMTs used and other
threshold effects.
e. A suppression of the nuclear cross section, σ, at higher recoil energies due to the
finite size of the target nucleus. The scaling of cross section with recoil energy is
also depended on whether the interaction is spin dependent or spin independent
(scalar).
7
These different modifications can be accounted for by adding successive terms to the
differential energy spectrum as follows:
dR
= R0 S ( E ) F 2 ( E ) I
dE observed
(2.3)
Where R0 is the unmodified rate for a stationary earth, S is the modified spectral function
which takes into account (a)-(d), F is a nuclear form factor correction which allows R0 to
use a ‘zero momentum transfer’ cross section which does not scale with recoil energy,
and I is an interaction function which accounts for any factors due to spin dependence or
independence.
The Earth’s motion around the sun causes a predicted annual sinusoidal modulation in
the total reaction rate. This modulation is rather small, only about 3% of the average rate,
and therefore for the purposes of this study will be ignored. The result of considering the
remaining motion of the sun throughout the galaxy is of course to assume a nonzero vE in
the velocity distribution described above. The result of considering a WIMP escape
velocity is to establish a maximum cutoff for the velocity distribution such that v<vesc.
These modifications add significant complexity to the mathematical representation of the
differential energy spectrum – I will try to sketch the basics of the derivation here.
The differential particle density is given by the following equation:
dn =
n0
f (v, v E ) d 3 v
k
(2.4)
where k is a normalization constant such that
vesc
" dn ! n0 and therefore k =
0
2%
1
0
"1
vesc
2
! d$ ! d (cos# ) ! f (v, v E )v dv
(2.5)
0
and by setting vesc = 0 and integrating,
k = k 0 = (!v02 ) 3 / 2
(2.6)
Where n0 is the mean WIMP number density such that n0 = ρD / MD. Based on current
cosmological evidence discussed in the previous section, we can estimate the mass
density of the Milky Way’s dark matter halo in the Earth’s vicinity to be ρD = 0.4 GeV
c-2cm-3. It is important to note that because the mass density ρD is the known value, the
number density, and therefore the rate vary inversely with WIMP mass. The other
accepted values for DM parameters that will be used in this analysis are v0 = 230 km/s
and vesc = 600 km/s.
In order to modify this to account for a finite escape velocity, we can truncate the
distribution at v + v E = v esc and define
8
& -v
k = k1 = k 0 $erf ++ esc
% , v0
* 2 vesc 'vesc2 / v02 #
(( '
e
!
. v0
)
"
(2.7)
Given these definitions, we can now derive the basic smoothly decreasing exponential
decay for the differential energy spectrum. The event rate per unit mass on a target of
atomic mass A AMU, with cross-section per nucleus σ is
dR =
N0
! v dn
A
(2.8)
where N0 is Avogadro’s number (6.02 x 1026 AMU/kg). Assuming that σ = σ0 is constant
and does not scale with recoil energy (that will be accounted for by the nuclear form
factor), it can be shown that
k0 1
v f (v, v E ) d 3 v
4
k 2!v0
dR = R0
(2.9)
2 N0 "D
! 0 v0 is the event rate per unit mass for vE = 0 and vesc = ∞.12
A
M
#
D
By conservation of linear momentum, the recoil energy of a nucleus that has
undergone scattering with a WIMP of kinetic energy E = (1 2) M D v 2 is
where R0 =
(2.10)
E R = Er (1 " cos ! ) / 2
where r is the reduced mass factor defined above and θ is the angle of scattering in the
center of mass frame. Assuming that the scattering is isotropic, the differential recoil
spectrum is
dR
=
dE R
Emax
!
Emin
v
1
1 max v02
dR( E ) =
! v 2 dR(v)
Er
E 0 r vmin
(2.11)
2
where E0 = 1 M D v02 = &$ v0 #! E ; Emin = ER / r ; and vmin = (2Emin/MD)1/2.
$ 2!
2
%v "
Using Equation 2.9, we obtain,
R k 1
dR
= 0 0
dE R E 0 r k 2"v02
vmax
1
f (v, v E ) d 3 v
v
vmin
!
(2.12)
This is the general equation for the differential recoil spectra. Depending on which case
we which to consider, we can derive the corresponding recoil spectra from this equation.
For example, by integrating from vmin = 0 to vmax = ∞ we obtain:
9
dR(0, ") R0 ! ER / E0 r
=
e
dE R
E0 r
(2.13)
which is the original smoothly decreasing exponential referred to at the beginning of this
section (Eq. 2.1).
In order to take account of the earth’s motion and WIMP escape velocity, we just
integrate Equation 2.11 bounded vmin = vE and vmax = vesc. The results are as follows:
2
dR(0, vesc ) k 0 R0 ! ER / E0 r
=
e
! e !vesc
dE R
k1 E 0 r
(
v02
)
, v - vE
dR(v E , .) R0 / v0 & , v min + v E )
'' - erf ** min
=
$erf **
dE R
E0 r 4 v E % +
v0
(
+ v0
dR(v E , vesc ) k 0 & dR(v E , () R0 'vesc2 / v02 #
=
'
e
$
!
dE R
k1 % dE R
E0 r
"
(2.14)
)#
''!
("
(2.15)
(2.16)
Equation 2.15 is the differential recoil spectrum taking full account of the motion of the
earth and the escape velocity of the WIMPs in the DM halo, and is the equation used
throughout the following analysis. It is also important to note that part of this modified
differential recoil spectrum is well approximated by an exponential of the form of
Equation 2.1, with two added fitting constants c1 and c2 such that:
dR(vE , ")
R
= c1 0 e ! c 2 E R
dER
E0 r
E0 r
(2.17)
where c1 and c2 vary depending on the month, but can be approximated by the average
values, c1 = 0.751 and c2 = 0.561.
So far we have considered modifications to the differential recoil spectrum do to (a)
and (b) mentioned above. (c) and (d) are measured empirically for a particular experiment
and then used to find the factor relating the energy measured in an event to the energy
actually deposited during a nuclear recoil.
In order to account for (e), we must include a nuclear ‘form factor’, F, which corrects
for the fact that we have assumed a fixed σ0, instead of a cross section which decreases
with recoil energy, as the rules of quantum mechanical scattering would dictate. For the
purpose of this analysis, spin independent (scalar) interactions are assumed, and so we do
not consider the effects of spin dependent interactions on the nuclear cross section. Given
this restriction, F can be represented by a simple function, F(qrn/ħ) where q = (2MTER)1/2
is the momentum transferred to the target nucleus and rn is the effective nuclear radius
which can be modeled by the following equation:
rn = a n A1 / 3 + bn
10
(2.18)
Using units in which ħ =1, the actual cross section can be represented by
! = ! 0 F 2 (qrn )
(2.19)
By using the first Born approximation and the target density distribution proposed by
Helm one can derive the following expression for the form factor:
F (qrn ) =
3 j1 (qrn ) !( qs ) 2 / 2
sin( qrn ) ! qrn cos(qrn ) !( qs ) 2 / 2
"e
=3
"e
qrn
(qrn ) 3
(2.20)
where j1 is the first order Bessel function and s is a measure of the nuclear skin
thickness.13 Using ħ =197.3 MeV fm, we can express the argument of the form factor as
the dimensionless quantity,
qrn = 6.92 ! 10 "3 A1 / 2 E R1 / 2 (a n A1 / 3 + bn )
(2.21)
Finally, in our discussion of the total interaction cross section, we have been
implicitly discussing the WIMP-nucleus cross section. Instead, it is customary to use the
WIMP-nucleon cross section, σW-n (in essence the cross section on a hydrogen nucleus) in
order to compare event rates for various different targets. To actually calculate the
interaction rates for various targets we must use the spin independent cross section on the
entire target nucleus which is related to σW-n as follows:
( W ' Nucleus
&µ
= ( W ' n A $$ W 'T
% µW ' n
2
#
!!
"
2
(2.22)
where µW-T is the reduced mass of the WIMP and the target nucleus, and µW-n is the
reduced mass of the WIMP and a single nucleon.
Figs. 2.1 and 2.2 illustrate the results derived in this chapter. Fig. 2.1 is a graph of
recoil spectra for a variety of targets given a 100 GeV WIMP mass. The dashed lines
represent the integrated event rate evts/kg/d above a given energy threshold (keVr). Fig.
2.2 also graphs differential recoil spectra, but for a variety of WIMP masses assuming a
Xenon target. Both assume σW-n = 10-42 cm2.
11
Figure 2.1: Calculated recoil spectrum in evts/keV/kg/d (lines), and the integrated event rate
evts/kg/d (dashed lines) above a given energy threshold (keVr), for a variety of WIMP masses
incident on a Xe target. σW-n = 10-42 cm2
Figure 2.2: Recoil spectrum in evts/keV/kg/d (lines), and the integrated event rate evts/kg/d
(dashed lines) above a given energy threshold (keVr), for a 100 GeV WIMP incident on Si (blue),
Ge (red), and Xe (green) targets. Note the dramatic effect of form factor suppression on Xe. σW-n =
10-42 cm2
12
Chapter 3 : Statistical Extraction of Mass from Predicted Data
3.1 Overview
As of yet, there has not been a widely accepted positive result of WIMP detection.
Nevertheless, assuming WIMPs are detected in one or several of currently ongoing
experiments, one important consideration is how much sensitivity these detectors will
have to measuring various parameters of the detected WIMPs. This section of the paper
will focus on discussing the possible sensitivity of these experiments to WIMP mass.
Ultimately, I will argue that our sensitivity is significantly improved by comparing results
from two different target detectors.
Consider the basic decreasing exponential approximation of the differential recoil
spectrum derived in the last section,
R
dR
= 0 e ! ER / E0r ,
dE R E0 r
(3.1)
recalling that r = 4 MD MT / (MD + MT)2. The result of any direct detection experiment is
to determine the left side of the above equation, and then parameters such as MD can be
obtained by fitting them to the experimentally determined recoil spectrum either using the
simple exponential in (3.1) or the modified function in (2.16) along with the added
nuclear form factor, (2.19). For now, let us consider the simple exponential dependence
on MD in (3.1) because its behavior is essentially similar to that of the more complicated,
modified recoil spectrum.
The challenge of measuring WIMP mass from a direct detection experiment is
twofold. Firstly, as WIMP mass increases, the overall expected detection rate decreases
because the number density of WIMPs in our locality decreases, which makes the
statistical analysis of the experimentally determined recoil spectrum increasingly difficult
due to Poisson fluctuations. Secondly, by examining (3.1) above, it is clear that for
MD>>MT, the slope of ln(dR/dER) becomes independent of MD. We are forced to measure
MD from the slope of this function alone because R0, the total event rate, is dependent
upon the interaction cross section σ0, which is another unknown parameter being tested
by direct detection experiments. Thus, as MD increases, our ability to measure it
accurately from the exponential dependence of the recoil spectrum is suppressed. The
magnitude of these effects will be described explicitly through the results of the
numerical simulation described in the following sections.
In order to demonstrate the effect of using two target detectors, as well as generally
characterizing the precision that could be expected from one or a number of direct DM
detection experiments, a Monte Carlo was created in order to simulate the predicted data
of a direct detection experiment using the modified differential recoil spectrum derived in
the previous section (2.16 with the Helm form factor). This data was analyized using the
Maximum Likelihood method to extract the WIMP mass and statistical uncertainty from
Poisson fluctuated data. Finally, the results of repeated fittings were combined in order to
determine the σ and 2σ confidence range of the measured WIMP mass as a function of
actual WIMP mass, and these results were compared for different target nuclei (both
13
separately and combined). The rest of this chapter will be devoted to describing in detail
the method used and the results.
3.2 Monte Carlo Simulation of Direct Detection Data
The purpose of the Monte Carlo was to simulate the data predicted (as a function of
WIMP mass and target nucleus mass) from running a direct detection experiment for a
specified amount of time – long enough to catalogue a reasonable number of events at
lower mass ranges. For each possible WIMP mass, the result of the Monte Carlo was a
list of recoil energies, [ER], each one corresponding to the energy deposited by a
theoretical WIMP interaction in the detector.
For a selection of possible WIMP masses ranging from 10 to 1000 GeV, the
differential recoil spectrum for each mass range was calculated. This calculation was
based on (2.16), but including a nuclear form factor based on a Helm density distribution
for the target nucleus as represented by (2.19) and (2.20). The expected rate of detection
for any combination of WIMP and target mass was found by integrating the specific
differential recoil spectrum for that combination of parameters, such that
R=!
dR
dE R
dE R
(3.2)
The final result of the Monte Carlo was a recoil energy, ER, calculated for each
WIMP event, weighted by the corresponding differential recoil spectrum. The algorithm
that accomplished this worked as follows. First the relevant differential recoil spectrum
was converted to a probability distribution function (PDF) P(ER), by normalizing it to
area unity, i.e.
P(E R ) =
!
dR 1
dE R R
(3.3)
This was then converted into a cumulative distribution function (CDF) C(ER), by
integrating to obtain the cumulative sum such that
ER
C(E R ) =
" P(E
R
)d E R
(3.4)
0
!
Because the PDF is normalized to area unity, the range of the CDF is the interval [0,1].
Thus the inverse of the CDF is a function with domain [0,1] and range [ER]. Note that to
make sure that the inverse CDF is indeed a function, in practice one might need to
modify the CDF slightly so that it is smoothly increasing and thus one-to-one. In our
simulation, we added 10 " ! (where epsilon is the smallest number the CPU can
recognize) to each value of the PDF before calculating the cumulative sum in order to
assure that the CDF had no repeated values. Then to verify that this slight modification
did not significantly change the range of the CDF, we renormalized the CDF by dividing
14
the function by its end value. Next, a random number between 0 and 1 was generated for
each desired WIMP event. These random numbers were interpolated onto the inverse
CDF, which would then output the list of recoil energies, [ER], weighted to the initial
probability distribution function, as desired.
Figure 3.1: Plot of integrated recoil spectra for a variety of WIMP masses, using 0.01, 0.10, and
0.50 keV bin widths. The y-axis is the total integrated event rate in arbitrary units. It demonstrates
that 0.5 keV bins are inaccurate for MD < 6 GeV.
Finally, energy threshold was created such that any events with ER < Qthreshold,
(where Qthreshold = 16 keV for Xe and 10 keV for Si or Ge) could be rejected due to
finite energy resolution of the detector. The primary reason for this rejection being that
many DM detectors have sophisticated techniques of distinguishing nuclear recoil events
from electron recoil effects – with the latter being the result of gamma rays interacting
with the electron shells of target atoms within the detector. The accuracy of this
discrimination is vitally important, because it prevents a gamma event from being
mistaken as a possible WIMP event. At low recoil energies, these discrimination
techniques begin to lose their ability to distinguish nuclear recoils from electron recoils
due to the finite energy resolution of the detector, and thus candidate events below a
certain energy threshold must be rejected. Ultimately this finite energy threshold was not
found to have a significant effect except at lower WIMP masses due to the steeper decay
of their recoil spectra as one can see in Fig. 2.1 (refer to conclusion for further
discussion). The Monte Carlo simulation was repeated for a combination of WIMP
masses varied from 10 GeV to 1000 GeV, and target nuclei masses including Xenon
(A=131), Germanium (A=73), and Silicon (A=28).
15
Two different normalization techniques were used to decide how many Monte Carlo
events to simulate for each WIMP mass and target nuclei:
The first method (normmode = 1) assumed a constant WIMP-nucleon cross section
for all models, and the same kg-day exposure for each target material. The specific
cross section was selected such that it corresponded to a baseline number of events
designated by “nwimps” in the case of a 100 GeV WIMP and Ge target, assuming
zero energy threshold. For example, if an exposure of 100 kg-days in Ge is assumed,
a WIMP-nucleon scalar cross section of 2.4 x 10-42 cm2 will give 100 events above a
zero energy threshold.
The second method (normmode = 2) created nwimps events above threshold for each
target and every value of WIMP mass considered, so long as there was a non-zero
probability of having an event above threshold. If the calculated probability of having
an event above threshold was zero due to cutoff created by the max WIMP escape
velocity, vesc, then zero events were created. The purpose of this second method was
to examine the sensitivity to different WIMP masses free from the effects of varying
statistics.
One problem which was encountered in the algorithm was difficulty in accurately
integrating the differential recoil spectra, especially for low WIMP masses where the
spectra decrease very rapidly at low energies. The problem was caused by using a
binning size for the energy domain, ER, which was too coarse and thus underestimated
the integrals of the differential recoil spectra at low masses by missing the large
contribution to the total integral due to the area of the first bin. The binning width
actually used in the algorithm was 0.5 keV. This problem was solved by beginning the
domain at ER = 0.25 keV instead of at ER = 0. Of course, this method still introduces an
element of error into the calculation of the total event rate, but as Fig. 3.1 shows, this
error is negligible except at WIMP masses below 6 GeV, which we therefore excluded
from the domain.
3.3 Maximum Likelihood Fitting Routine for Monte Carlo Data
After creating the Monte Carlo data sets, the Maximum Likelihood method was then
used to extract the statistical uncertainty in WIMP mass. This uncertainty is primarily due
to two causes. The first is Poisson fluctuations due to small numbers of candidate events,
which is worst at the lowest masses because of the recoil energy threshold mentioned
above, and at the highest masses because of the decreasing WIMP number density which
leads to a low overall detection rate. The second is due to the decreasing sensitivity of
differential recoil spectra to changes in WIMP mass at higher masses. Both of these
effects have been described previously, and will be discussed further in conjunction with
the results of this simulation.
The idea behind the Maximum Likelihood method is to take a set of data points, in
this case the list of ERi, and by multiplying them by their corresponding probability as
determined by a given theoretical model with certain set parameters to come up with the
16
“likelihood” that those data points would be created by those parameters. The model
parameters can be varied, and by maximizing the resulting likelihoods, the best fit for the
parameter can be found. In order to illustrate analytically how this method works, the
process is illustrated by modeling the differential recoil spectrum as a simple exponential
(2.17):
y ( M D , ER ) =
dR
" Ne ! c 2 E R
dER
E0 r
(3.5)
where c2 = 0.561 and N is a normalization constant. The factors in front of the
exponential can be ignored since we only need to fit the slope of the logarithm of the
exponential in order to determine r and therefore MD. The factors in front are necessary to
determine the interaction cross section, which is related to the overall rate, R0, but we will
not analyze the cross section in this paper.
First, we find N so that y is normalized to behave as a probability distribution function
(PDF), i.e.
"
"
! y(M
0
D
, ER )dER = ! Ne # c 2 E R
E0 r
dER = 1
(3.6)
0
Thus by simple integration,
N=
c2
c (M D + M T )2
= 2
E0 r E0
4 M D MT
(3.7)
We then calculate the probability density of observing the ith event in our data set, with
recoil energy ERi as
yi (M D ) = Ne
!
"c2 ERi E0 r
!
The likelihood L(MD) function is then defined as the product of the yi for all of the events
in the data set:
n
(3.8)
L( M D ) = ! yi
i =1
The best fit for the parameter MD, is then the value that maximizes the likelihood
function. It is usually more convenient to work with the logarithm of the likelihood
function, l such that:
n
l( M D ) = log L = ! log yi
(3.9)
i =1
17
From now on, when I refer to the likelihood, I am referring to the logarithm of the
likelihood function, l .
The next step after calculating the likelihood function was to find the uncertainty
implicit in the fitting of the ERi. If the likelihood is Gaussian shaped (which it may be
under certain circumstances), the 1σ error can be found simply by calculating the root
mean square deviation of the likelihood function around its mean. The likelihood
distributions for fitting WIMP mass are of course not Gaussian, nor are they symmetric,
and therefore this simple method cannot be used. Instead, the uncertainty must be
calculated by finding the values of WIMP mass where the likelihood is reduced to a
specified threshold below its maximum value. Mathematically, this can be written as
l( M i ) = l( M 0 ) ! t , i = 1 or 2
(3.10)
where M0 is the value of WIMP mass which maximizes l , t is a constant threshold
number, M1 is the lower bound on the fitted WIMP mass, and M2 is the upper bound such
that the range,
M1 < MD < M2
(3.11)
has a 68% probability of containing the true value of the parameter MD when t = 0.5.
For larger WIMP masses M2 may diverge to infinity because of the decreasing
dependence of the slope of the differential recoil spectrum on MD. This is illustrated in
Figs. 3.2 and 3.3, which show two typical likelihood functions with M0, M1 and M2
marked for a threshold of 0.5. The first illustrates the likelihood curve generated by an
actual WIMP mass of 25 GeV with a 131Xe detector. The second illustrates the likelihood
curve generated by an actual WIMP mass of 583 GeV also with a Xenon detector. In
each figure, the black circle marks the maxima of the curve, l( M 0 ) , and the two red
crosses mark l( M 1 , M 2 ) , the lower and upper bound of the 1σ region of uncertainty.
As can be seen in the second figure, for higher WIMP masses the likelihood curve
does not often reach a maximum. In these cases the simulation selects the highest mass
available, i.e. the end of the list of MD fitting parameters as the upper bound on the region
of uncertainty. Under this situation the actual uncertainty can be interpreted as
unbounded on the upper side.
The actual algorithm used to find the best fit for the WIMP mass and the
corresponding uncertainty based on the Monte Carlo data is the analogue to the method
described above, but using the complete modified differential recoil spectrum instead of
the exponential approximation as the template for the maximum likelihood fit. In
addition, given the unpredictable and divergent behavior of the likelihood plots, we
required a better method for extracting the statistical uncertainty characterized by M1 and
M2 defined by t = 0.5 above. Thus for each value of MD, 200 Monte Carlo data sets were
generated and fit using the maximum likelihood method in order to develop a reliable
statistical model.
18
Figure 3.2 (Top) and 3.3 (Bottom): Sample plots of logarithmic Likelihood function, l
(arbitrary units) versus MD (GeV) for Xe target. Red crosses are lower bound, M1 and upper
bound, M2. Circled point is value of fitted mass, M0. Fig. 3.2 was generated for an actual WIMP
mass of 20 GeV, Fig. 3.3 was generated for an actual WIMP mass of 583 GeV.
19
Unfortunately the complete modified differential recoil spectrum is difficult to
normalize analytically, and was thus analyzed numerically in the actual algorithm.
Besides repeating the simulation a number of times, another few modifications were
added to the algorithm in order to make the results reflect the behavior of a realistic direct
detection experiment. These additions included a minimum energy threshold and a
constant interaction cross section as a function of MD and will be described further below.
The final fitting algorithm was constructed as follows:
Firstly, the differential recoil spectra were calculated for a list of WIMP masses to be
tested as possible fits using the maximum likelihood method. These recoil spectra then
had to be normalized so that they behaved as probability distribution functions (PDF) as
in (3.7), but with their integration domain modified to account for the minimum detector
energy threshold detailed in Section 3.2. This minimum energy threshold effectively acts
so as to discard any event such that the recoil energy,
ERi < Qthresh
!
(3.12)
where Qthresh = 16 keVr for a Xe detector and 10 keVr for a Ge or Si detector. Thus, the
normalization constant, N, is determined by integrating the PDF template functions over
the energy domain – excluding the region below Qthreshold – as follows:
"
"
1
dR
= ! y ( M D , ER )dER = !
dER
N Qthresh
dE
R
Qthresh
(3.13)
After calculating the template of normalized differential recoil spectra, the Monte
Carlo simulation was run 200 times for each value of MD and for each different target
nucleus, and each time y and l were calculated as described above. From l , a threshold
value of t = 0.5 was selected in order to catch any unbounded likelihood plots, then the
calculated values of M1 and M2 were stored for each repetition of the Monte Carlo data.
The values of M1 and M2 were then sorted in descending and ascending order,
respectively, such that the lower and upper bounds on the certainty range could be
determined by looking 68% down the list of M1's and M2's for a given WIMP mass and
target nucleus. The combination of the 68% range of the sorted list with the use of M1 and
M2 which are the nominal 68% uncertainty range in MD should give a result
corresponding to a 90% overall confidence level.
One of the additional goals of this project was to compare the uncertainty from
running one experiment with the uncertainty from running two or more experiments with
different target nuclei simultaneously. In order to make this comparison between two
nuclei, we merely added the likelihoods of each individual nucleus together and then
evaluated the uncertainty of the combined likelihood function in exactly the same way as
we would evaluate the individual likelihood function, i.e.
l( N 1 + N 2 ) = l( N 1 ) + l( N 2 )
20
(3.14)
The one caveat to this is that from running two experiments simultaneously, one
would automatically expect greater precision in determining WIMP mass because of the
greater total target mass and therefore the higher overall event rate. Poisson statistics
suggests that the uncertainty implicit in measuring any parameter varies approximately as
the n where n is the number of events used to determine the parameter. Thus, by using
two separate targets, we would expect an automatic 2 improvement in the uncertainty
in measuring WIMP mass. We wish to test for an improvement above and beyond that
implicit in Poisson statistics, so in order to achieve this we normalize the uncertainty
plots for the combined detectors so that they reflect an experiment with half the baseline
event rate in each of the individual detectors. Having done this, any improvement in the
uncertainty region could be confidently attributed to increased discrimination due to the
use of multiple target nuclei with different masses.
Finally, the results of the Monte Carlo simulation and the corresponding uncertainty
found by the maximum likelihood method were plotted for a variety of different
parameters, and the results are presented in the following section.
3.4 Results
This section presents the results of the previously described simulations in the form of
90% CL contour plots of WIMP mass uncertainty. The contour plots were created by
varying different parameters in the simulation. The first series of plots illustrated in Figs.
3.4 and 3.5 illustrate the mass uncertainty without and with the Helm form factor
respectively, as described in Section 2.2. Figs. 3.8 and 3.9 illustrate the effect of
removing the minimum energy threshold, Qthreshold, described in Section 3.2. Figs. 3.10
- 3.12 illustrate the mass uncertainty assuming a constant number of events above
threshold (normmode = 2, discussed in Section 3.2), 10, 100, and 1000 respectively.
To begin, we’ll focus on the results using a constant WIMP-nucleon cross section for
the purposes of normalization (normmode = 1, discussed in Section 3.2). In each plot of
the fitted masses, the colored contours represent the 90% CL for each WIMP mass
between 10 and 1000 GeV, and for three targets: Xe, Ge, and Si. The black line
represents the actual WIMP mass, MD used in the Monte Carlo simulation.
21
Figure 3.4: The 90% CL upper and lower bounds for the determination of WIMP mass based on
detecting events in Xe (green), Ge (red) and Si (blue) targets with the same kg-day exposure, and
assuming fixed WIMP–nucleon cross section (normmode =1). For 90% of experiments the best
fit to the data would lie between the upper and lower colored lines, given the underlying WIMP
mass shown on the horizontal axis. The black line is provided for guidance showing where the
fitted mass is equivalent to the actual mass. No nuclear form factor correction is applied for the
models shown in this figure. Detector thresholds of 16, 10, 10 keVr are assumed for the different
targets, respectively.
As can be seen in Fig. 3.4, the Xenon target has the narrowest contour out of the three
for all WIMP masses above 20 GeV – demonstrating that without the form factor, Xenon
has the best ability to determine WIMP mass, especially at higher masses for a given kgday exposure and WIMP-nucleon cross section. The primary reason for the diverging
contours at higher masses is because of the reduced mass factor, r in the differential
recoil spectra (see Equation 2.1) which begins to asymptote as MD increases beyond MT.
Thus when MD>>MT, the slope of the recoil spectra lose their dependence on MD, leading
to increasing uncertainty at higher WIMP masses, until eventually the uncertainty in
measuring the mass becomes unbounded. Without the presence of the form factor, Xe
does relatively the best job – the contour places a reasonable upper limit on MD up to
almost 200 GeV. The reason for this is that because Xenon has an atomic mass of 131
AMU, r does not asymptote until MD increases significantly beyond 131 AMU * .93
AMU/GeV = 122 GeV. The same effect explains why Germanium and Silicon do
comparatively worse at higher masses than Xenon – their lower atomic masses cause r to
effectively asymptote for a lower value of MD.
22
Figure 3.5: The 90% CL upper and lower bounds for the determination of WIMP mass using the
same convention as that of Fig. 3.4. Full Helm form factor corrections are applied in these
models. The corresponding number of events for each WIMP mass and target are shown in Fig.
3.6.
Figure 3.6: The total number of events generated for each target above energy threshold, using
normmode = 1 with a baseline of 100 pre-threshold events for 100 GeV WIMP and Ge target.
Full Helm form factor corrections are applied. This plot describes the number of events used to
generate the contours at each value of MD in Fig. 3.5.
23
Another factor which also explains the advantage of the higher mass targets is that the
cross section on the entire target nucleus scales as a factor of A2 (see Equation 2.22).
Thus given the assumption of a constant WIMP-nucleon cross section, the overall event
rate will be higher for the heavier targets, and the greater number of catalogued events
allows a better statistical determination of MD. Finally, the divergence of the contours at
low masses is due to the minimum energy threshold. By examining Equation 2.1, it is
clear that the slope of the recoil spectra are steeper for lower WIMP masses – thus the
lower the WIMP mass, the more events fall below threshold. The error diverges at low
WIMP masses because in this mass range, there are zero events generated above
threshold.
The cyan contour in Figs. 3.4 and 3.5 represents the 90% CL of the combined
Germanium and Xenon data under with the total number of baseline events (under
normmode = 1) halved from 100 to 50, as described in Section 3.3. Our conclusion is that
combining the data from multiple target sources without increasing the total event
number does not improve the ability to accurately determine WIMP mass. However, it
will provide an important test of how WIMP spectrum behaves on different targets.
Fig. 3.5 was generated identically to Fig. 3.4 but includes the Helm form factor thus
suppressing the recoil spectra at higher recoil energies in larger nuclei as illustrated in
Fig. 3.2. This has the effect of reducing the event rate especially for high recoil energy
events. The effect of the form factor can be seen clearly by comparing Figs. 3.4 and 3.5.
The form factor has the strongest suppression on the heaviest elements, affecting Xenon
significantly, and hardly affecting Silicon at all. The reason for this is explained by
Equation 2.21, noting that the recoil energy, ER is also a function of target mass. As
demonstrated in Fig. 3.5, because Xenon is penalized by the form factor, in a comparison
of any real experiment, Germanium will have significantly better sensitivity to WIMP
mass than Xenon. The reason that Silicon’s sensitivity actually appears to improve with
the addition of the form factor, while Germanium and Xenon both worsen, is because the
number of events generated is normalized to Germanium. Thus, when using normmode =
1 and assuming the same number of total Germanium events, applying the form factor
effectively boosts the number of Silicon events generated because the form factor hardly
effects Silicon at all.
Fig. 3.6 shows the number of events above threshold generated by normmode = 1.
The trends in Fig. 3.6 can be understood as follows. The total event rate falls at higher
WIMP mass because as WIMP mass increases, the total number density, ρD /MD
decreases. This is simply illustrated in the basic unmodified event rate derived in section
2.2,
R0 =
2 N0 "D
! 0 v0
# A MD
(3.15)
The total event rate falls at lower masses because of the reduced mass factor in the
WIMP-nucleon cross section, as illustrated in Equation 2.22. The combination of adding
a minimum energy threshold and maintaining a constant interaction cross section as a
function of WIMP mass led to a combined effect of suppressing the detection rate both at
lower and higher MD for reasons explained above. Quantitatively, Fig. 3.6 illustrates this
24
effect for an initial 100 baseline events generated at the calibration value of MD = 100
GeV with a Germanium target before cutting out events below threshold.
Figure 3.7: The 90% CL upper and lower bounds for the determination of WIMP mass using the
same convention as that of Fig. 3.4, but assuming one fifth of the kg-days of target exposure time.
Full Helm form factor corrections are applied in these models.
Fig. 3.7 illustrates the effect of Poisson statistics on our ability to determine WIMP
mass. By reducing the baseline number of events under normmode = 1 to 20 before
cutting out events below threshold, the contours diverge much more rapidly than with a
baseline value of 100 events. Because the same normalization was used, the actual
number of events above threshold generated at each mass is the same as that illustrated in
Fig. 3.6, but divided by a factor of five.
Figs. 3.8 and 3.9 illustrate the effect of removing the minimum recoil energy
threshold altogether (i.e. Qthreshold = 0 keV). As illustrated in Fig. 3.8, the total number
of Xenon events increases significantly compared to Germanium because the recoil
spectrum of Xenon slopes more steeply than that of Germanium – indicating that a larger
proportion of Xenon events will have comparatively lower recoil energies. Now that
these low recoil energy events are no longer being cut, the overall number of events in
Xenon increases. Fig. 3.9 shows that this change in the relative number of events makes
Xenon slightly more sensitive to WIMP masses below approximately 70 GeV, but above
this mass Germanium remains most sensitive. The continued advantage of Germanium at
masses above this value can be attributed to the Helm form factor, which penalizes
Xenon most at higher WIMP masses. For all three targets, the sensitivity at low masses
increases because of the much larger numbers of events available at low WIMP masses.
25
Figure 3.8: The total number of events generated for each target with a 0 keV energy threshold
using normmode = 1 with a baseline of 100 events for 100 GeV WIMP and Ge target. Full Helm
form factor corrections are applied in these models. This plot illustrates the number of events
used to generate the contours in Fig. 3.9.
Figure 3.9: The 90% CL upper and lower bounds for the determination of WIMP mass with a 0
keVr minimum recoil energy threshold using normmode = 1 with a baseline of 100 events for 100
GeV WIMP and Ge target. Full Helm form factor corrections are applied.
26
Figures 3.10 (Top), 3.11 (Middle), 3.12 (Bottom): The 90% CL upper and lower bounds for the
determination of WIMP mass using normmode = 2. For each WIMP mass assumed 10 (top), 100
(middle), or 1000 (bottom) WIMPs detected in Xe (green), Ge (red), Si (blue) above threshold.
Full Helm form factor corrections applied.
27
The purpose of the second normalization mode (normmode = 2) was to examine the
sensitivity of the various targets to WIMP mass assuming a fixed number of events above
threshold for each model and target. Figs. 3.10, 3.11, and 3.12 show the same simulation
run with 10, 100, and 1000, respective events above threshold for all mass values. They
quantify the improved mass sensitivity that accompanies accumulating better statistics.
Fig. 3.12 shows a dramatic improvement on our ability to constrain WIMP mass. For
example, with only 10 events above threshold, Xenon loses the ability to place an upper
bound on WIMP mass at about 60 GeV. Compare this to 1000 events above threshold,
with which Xenon can maintain a reasonable upper bound all the way up to 1000 GeV.
Given this type of normalization where each target has the same number of events
available, the difference in sensitivity at higher masses can again be attributed to the
effect of the form factor. One interesting observation is that Silicon has approximately the
same sensitivity to WIMP mass as Germanium if one compares each target with the same
number of events. The problem with Silicon is that to actually obtain the same number of
events, one would have to run for many more kg-days to overcome the small A2 factor in
the WIMP-nucleus cross section.
3.5: Limitations of Analysis and Recommendations for Further Study
There were a number of limitations in the current analysis which could be improved
upon in a further study. The first limitation was the resolution of the WIMP mass lists
generated by the Monte Carlo and fitting by the likelihood routine. The limited resolution
on the generating masses led to some strange behavior in the 90% CL contours such as
can be seen in Fig. 3.5 in the Xe contour at approximately 20 GeV where instead of
diverging smoothly as expected it diverges almost all at once. With increased resolution
this discontinuity should resolve itself into a smooth curve as the upper bound increases
due to gradually decreasing statistics. The limited resolution of the fitting masses also led
to the jumpiness of the contours at higher masses. By increasing the number of masses
being fit by the likelihood templates these could be smoothed out.
As discussed in Section 3.3, for the purposes of the analysis it was assumed that the
combination of using a threshold value of t = 0.5 to identify the upper and lower bounds
of the uncertainty range on the likelihood curves and selecting the 68% values of the
sorted lists of M1 and M2 would, when combined, give the 90% CL contour for
determining WIMP mass. Fig. 3.13 shows a test of this hypothesis by measuring how
many of the fitted mass values, M0, lie within the 90% contour range as determined by
our method. As one can see, the assumption that the selected values of M1 and M2
enclose the 90% CL range is a reasonable one. Nevertheless, one improvement to this
method could be interpolating to find the exact values of M1 and M2 as defined by the
threshold instead of merely using the nearest outside values on the WIMP mass fitting
domain.
The third effect which was a relic of our simulation algorithm was the sharp
conversion of the uncertainty contours at low masses. This effect can be seen most
clearly in the Xenon contour in Fig. 3.10 between 16 and 25 GeV. This conversion would
not be apparent in a real experiment, and is an artifact of our simulation assuming the
detector to be able to determine the recoil energy, ER to infinite precision. In order to
28
remove this artifact, our simulation should be modified to include the finite energy
resolution of dark matter detectors.
Figure 3.13: Plot of the fraction of fitted masses, M0 that lay within the 90% CL contours for
Xenon (green), Germanium (red), and Silicon (blue) targets using a threshold value of t = 0.5.
The fraction fluctuates around 0.9, as expected.
Throughout, this analysis assumed either a constant WIMP-nucleon cross section, or a
constant number of above threshold events, and then considered how the 90% CL
contours varied in only one dimension – as a function of WIMP mass. The results of this
analysis only show half the picture and could be causing misleading behavior in the plots,
particularly at low masses. We hope to remedy this in the future by varying σW-n as well
as MD and using a combined likelihood fitting routine to measure the uncertainty in both
parameters simultaneously and plot the result as a 2D contour.
Finally, this study focused on the errors of WIMP mass determination and concluded
that combining data from multiple targets did not significantly add to the precision with
which WIMP mass could be measured. Nevertheless, one should not lose sight of the fact
that a consistency check of the WIMP hypothesis is also an important goal of comparing
multiple target detectors.
29
Chapter 4: Xenon direct detection - Discrimination of nuclear recoil events in MCP
data
4.1: Overview
The XENON collaboration is currently in the progress of building a prototype liquid
Xe detector. For this detector, the group at Brown University has experimented with a
number of different devices for scintillation light collection and amplification. Other than
the standard photomultiplier tubes (PMT) we have also tried a relatively new technology
called a microchannel plate (MCP) developed by BURLE Industries. The MCPs have a
number of advantages over standard PMTs, including their square shape and low profile
which allow them to be better distributed in order to cover more area on the surface of the
detector and therefore improve light collection efficiency. Because of this advantage,
among others, a series of calibration data sets were taken with the prototype detector
using a quad anode of four MCPs in a square formation on the top surface of the detector
for photomultiplication. The signal from each of the MCPs was stored in a separate
channel labeled one through four.
The analogue signal from each MCP channel was sent through a series of amplifiers
and then stored as digital data by an 8-bit, high frequency (1 or 2 GHz) analogue to
digital converter (ADC). The ADC saved an event whenever the signal rose above a
baseline threshold, at which point the ADC would be triggered and would store one event
made up of the time trace of the pulse with domain intervals of 0.5 or 1 ns, including a
window of ‘pre-trigger’ time before the trigger point.
The scintillation time constant, T0 for the approximately exponential decay pulses of
nuclear recoils in liquid Xenon has been found to be significantly faster than that of
electron recoils. A study by D. Akimov, et al14, has found the T0 value for nuclear recoils
to be 21.0 ns +/- 0.6 ns while they have found that for electron recoils T0 ranges from
29.1 +/- 0.6 ns to 34.0 +/- 0.6 ns depending on the energy of the incident particle. The
purpose of this analysis was to determine whether the prototype Xenon detector using the
MCP had sufficient resolution to distinguish between this difference in pulse shape and
whether it could be used to accurately discriminate between electron recoil and nuclear
recoil events within the detector. In order to do this, we analyzed two different types of
calibration data taken by the detector, data taken with a 137Cs source of 662 keV gamma
particles and data taken with an 241Am9Be source of neutrons. The gammas provided a
source of electron recoils, while the neutrons provided a source of nuclear recoils.
The method used in the following analysis involves three steps. First, we calibrated
the data to find the pulse area which corresponded to one photo-electron (phe) being
ejected from the MCP. Second, we attempted to accurately calculate the decay rate of
events in the Cs and AmBe calibration data by cutting out anomalous events and
identifying nuclear recoil events by their faster decay rates. Finally, we created two
Monte Carlo data sets – one consisting entirely of simulated electron recoil events, and
the other consisting of some combination of simulated electron recoil and nuclear recoil
events and compared our results from the second step with those predicted by the Monte
Carlo simulation.
30
4.2: A note on notation
The primary method used in this analysis to distinguish between pulses with different
time constants is to histogram the ratio of pulse areas from the early part of the pulse to
the latter part for all the events within a certain energy range. I will adopt the following
notation from M. Yamashita’s thesis15, such that for example, R(15:100) is the ratio of the
area of the first 15 ns after the trigger to the area of the first 100 ns after the trigger.
Assuming that the scintillation pulse shapes are, on average, simple exponential decays
with a T0 = 21 ns for nuclear recoil and T0 = 30 ns for electron recoil, the distance
between the peaks of the histograms corresponding to these two different decay constants
is maximized by using R(25:100).
Because of the nature of Xenon physics, the response of the Xenon detector to
nuclear recoils is actually quenched compared to that of electron recoils. We therefore
adopt the notation of keVr (keV recoil) to describe the recoil energy deposited by a
nuclear recoil (such as a neutron) and keVee (keV electron equivalent) to describe energy
deposited in the detector by an electron recoil (e.g. from a gamma ray) with equivalent
incident particle energy. keVr can be converted to keVee by multiplying by a quenching
factor (QF) of about 20%.
4.3: Calibration of pulse area to photoelectron scale
In order to determine the correspondence between one photoelectron and signal area,
we began by creating a template of a single photoelectron pulse by averaging the signals
due to of a number of individual small pulses at the tail end of events. The shape of these
individual photoelectron pulses could be well approximated by a Gaussian curve with
width σ = 0.65 ns. By running a chi squared fit using this Gaussian template (and varying
the amplitude) over the tail end of a series of MCP events from a Cs data run, we were
able to obtain the following histogram of individual pulse amplitudes illustrated in Fig.
4.1.
As can be seen in Fig. 4.1, the first peak at very low amplitude can be attributed to
noise identified by the chi squared fit. Beyond the first peak, the mean amplitude of the
peak values of the four channels is at about 30 bins. This value of 30 bins can be
attributed to a single photoelectron pulse. Given this amplitude, we just need to find the
area of the corresponding Gaussian pulse. Because the ADC is 8 bit, and was set to 100
mV Full Scale (FS), there were a total of 256 bins with the full scale corresponding to
100 mV post amplification. The gain of the MCP signal was 3.75, and we want to
represent the pulse amplitude in pre-amplified units, so we also must divide by this
amplification factor. Thus the amplitude a, is
a = 30 bins !
100 mV FS
1
!
= 3.1 mV
256 bins 3.75
(4.1)
The area, A, of a Gaussian pulse of amplitude a and width σ is approximately
A = 2.5 ! (" ) ! (a )
(4.2)
31
So assuming a Gaussian pulse shape with σ = 0.65 ns, a single photoelectron pulse of
3.1 mV would correspond to an area of 5.1 mV ns/phe, i.e.
A = (2.5) x (0.65 ns) x (3.1 mV) = 5.1 mV ns
(4.3)
Figure 4.1: Histogram of individual pulse amplitudes as calculated by chi squared fit on Cs data
set. Each channel is signal from one MCP in quad anode setup.Second peak at approximately 30
bins is attributed to single photoelectron pulse.
Using a 133Ba set for calibration it was also determined that 356 keV gammas
corresponded to 1240 mV ns pulses, implying a ratio of 3.5 mV ns / keVee. Using the
above analysis of single phe pulses:
3.5 mV ns/keV "
1 phe
! 0.7 phe/keVee
5.1 mV ns
(4.4)
Assuming a quenching factor of 20%, this implies that the detector yields a ratio of 0.14
phe/keVr. This conversion was used through the rest of the analysis to express the energy
of detected events in number of phe. The result illustrated in Fig. 4.1 is very useful
because it allows us to develop the conversion between pulse area and phe, but it also an
impressive result in itself because it demonstrates that the MCP has sufficient resolution
in our calibration data to distinguish a single photoelectron pulse.
32
4.4: Analyzing the Calibration Data from Xenon Prototype Detector
The primary window used to look for nuclear recoils in this analysis was 14 to
25 phe. The bottom of the window was cut off at 14 phe because below this energy range
noise and other anomalous events dominated. The top of the window was cut off at
25 phe because the recoil spectrum for AmBe neutrons can be modeled as an exponential
decay with very few nuclear recoils predicted to deposit more energy than the
corresponding 178 keVr.
In order to compare the pulse area at several different times, we developed a number
of variables which represent the pulse area over different regions of interest. They are
defined as follows:
pasum: the pulse area of the sum of the channels over the entire timebase.
pasuma: the total pulse are in region a – from 2ns before to 4 ns after the pretrigger
pasumb: the total pulse area in region b – from 36 ns to 42 ns after the pretrigger
pasumc: the total pulse area in region c – from 74 ns to 80 ns after the pretrigger
pasum15: the total pulse area from 1 ns before to 15 ns after the pretrigger
pasum25: the total pulse area from 1 ns before to 25 ns after the pretrigger
pasum100: the total pulse area from 1 ns before to 100 ns after the pretrigger
(the suffix ‘sum’ indicates that the signal has been summed over all of the four MCP
channels)
In the energy range of primary interest for nuclear recoils a significant number of
events were identified as anomalous by their pulse shape. These anomalous events were
distinguishable by large spikes in only one channel during the first 5 ns after the trigger
and often followed by echoing spikes at later intervals of 19 or 38 ns. Thus these events
were also characterized by faster than normal fall times. It is hypothesized that these
anomalous events were caused by ion feedback – an effect which is discussed further in
the conclusion. Several cuts were designed in order to remove these anomalous events
from the analysis, particularly by targeting their unusually fast decay times. They are
listed as follows:
cutch: Cuts out all events for which more than 90% of the signal in the first 100 ns after
the trigger is within one channel. The purpose of limiting the timebase to the first 100 ns
is that towards the latter end of an event the baseline tends to drop somewhat, distorting
the measurement of the total pulse area.
cutrepeated: Cuts out events that are dominated by echoing spikes in the regions defined
by pasuma, pasumb, and pasumc.
cutcha: Same as ‘cutch’ above, except compares the region defined by pasuma, namely
the initial spike that set off the trigger. It also includes the added stipulation that a single
channel must have at least 5 phe in this region.
33
cuta: Cuts all events for which pasuma/pasum100 is greater than 0.5
cutall: The sum of all of the above cuts.
Figs. 4.2 and 4.3 illustrate the presence of the anomalous events in a series of Cs data.
Fig. 4.2 is a histogram of pasuma/pasum100 for all of the events in the data set. Fig. 4.3
is the same histogram for just the events between 14 and 25 phe. The value of 0.5 in cuta
was determined from Fig. 4.2, with the second hump centered over 0.8 being attributed to
anomalous events and therefore being eliminated by cuta. This determination was made
because Cs emits only gammas and thus there should not be any significant number of
nuclear recoils in the Cs data set – thus the events characterized by R(4:100) >0.5 must
be anomalous.
34
Figure 4.2 (Top): Histogram of pasuma/pasum100 for all events in Cs data set.
Figure 4.3 (Bottom): Histogram of pasuma/pasum100 only for events between 14 and 26 phe.
35
Figure 4.4: Histograms of R(25:100) for AmBe (Red) and Cs (Blue) data sets for events between
14 and 25 phe. The various cuts applied are indicated above each figure. Top left has no cuts
applied, and bottom right has all cuts applied.
Finally, after developing the above cuts, we applied them to the actual Cs and AmBe
calibration data sets. We then compared the histograms of R(25:100) for Cs and AmBe
data sets taken under similar detector conditions. The result of such a comparison is
illustrated in Fig. 4.4. In the figure, the red line is the histogram for a series of AmBe data
sets, and the blue line is the histogram for a series of Cs data sets. For each series, there
was no lead shield in place and no electric field bias across the detector. In addition, the
area of the Cs histogram was normalized to have the same total number of events as the
AmBe series. The titles above each set of histograms in the figure describes which cuts
were applied to the data sets before the histogram was made. The histogram is the bottom
right quadrant was made by applying all of the cuts described above, and there remains
evidence in this histogram of a second peak in the AmBe data which is notpresent in the
Cs data. Nevertheless, the second peak occurs around R(25:100) = 0.9, which is
significantly higher than the predicted value of 0.7 for a decay constant of 21 ns. This
raises concerns that the second peak is caused by anomalous events which managed to
sneak by the cuts instead of legitimate nuclear recoil events caused by neutrons.
36
4.5: Monte Carlo Simulation
In order to determine if the subtle second bump in the AmBe histogram in the lower
right quadrant of Fig. 4.4 could be interpreted as valid evidence of faster nuclear recoil
decays, as well as to determine if the applied cuts were mistakenly eliminating legitimate
events, we ran a Monte Carlo simulation to compare Fig. 4.4 with the results predicted by
theory. A Monte Carlo data set was created for both a theoretical AmBe source and a
theoretical Cs source – assuming a decay time constant of 21 ns for a nuclear recoil event
and 30 ns for an electron recoil event. The Monte Carlo simulated a data set by creating a
Poisson distribution of the number of photoelectrons in each event, centered around the
user specified parameter ‘npe’. Each photoelectron is modeled as a Gaussian, with an
amplitude of 3 mV and a width of σ = 0.65. Then each photoelectron is randomly
assigned to a channel, and normally distributed along an exponential curve of decay
constant T1 = 30 ns or T2 = 21 ns. The percentage of simulated nuclear recoil events for
the AmBe sets (distributed along T2) was set by the parameter ‘pnr’. The simulated Cs
sets have pnr = 0.
Figure 4.5: Identical plots as Fig. 4.4 except applied to Monte Carlo data with npe = 14, pnr =
0.25
37
Figure 4.6: Histograms of R(25:100) for Monte Carlo AmBe (Red) and Cs (Blue) data sets with
npe = 75, pnr = 0.5 for events between 65 and 85 phe. Cuts applied are identical to previous
figures.
Figs. 4.5 and 4.6 analyze and histogram two different Monte Carlo data sets in an
identical way as that applied to the actual calibration data in Fig. 4.4. For the data in Fig.
4.5, npe = 14 and pnr = 0.25. This is a realistic representation of what real data should
look like. As can be seen, the cuts affect a negligible number of legitimate events as
modeled by the Monte Carlo. In addition, the Monte Carlo shows that in the region
around 15 phe, Poisson statistics dominate, making it nearly impossible to observe the
two peaks in the AmBe data expected from the combination of two different time
constants. In Fig. 4.6, where npe = 75 and pnr = 0.5, the hint of two peaks can be seen in
the simulated AmBe histogram.
4.6: Effect of Field Bias on Pulse Shape
Fig. 4.7 illustrates another important effect discovered in this analysis which is worth
mentioning in connection to the attempt to distinguish electron and nuclear recoils by
pulse shape. In Fig. 4.7, the vertical axis represents R(25:100) for actual calibration data
while the horizontal axis represents the number of phe – i.e. the energy of the event. The
38
red dots are events from an AmBe source beneath a 2” Pb shield, and no bias (but there is
a possible remnant form the previous run). The blue dots are from a Cs source placed
Figure 4.7: Plot of total pulse area vs. R(25:100) for Cs (Blue) and AmBe (Red) data sets. Shift
in Cesium data set indicates slower decay times.
beneath the detector, but with an added 3.02 kV/cm bias applied across the detector. The
resulting plot shows that the Cs data actually exhibits a faster decay time than the AmBe
data due to the field bias applied during the Cs run – an effect that is predicted by Xenon
physics. This effect is of similar magnitude (but in reverse) to that expected for electron
recoil and nuclear recoil decay times, so it is clearly an effect we must take into account
if we intend to distinguish electron recoil and nuclear recoil events by decay times.
4.7: Conclusions
As evidence of the faster decay time of nuclear recoils, we would expect to see a
second peak in the AmBe histogram in Fig. 4.4 at approximately R(25:100) = 0.7.
Although there is a hint of a second peak in the figure, it is shifted slightly above
R(25:100) = 0.7, and thus could also very possibly be caused by remaining anomalous
events which were missed by the cuts. Nevertheless, the Monte Carlo histograms in Fig.
4.5 illustrate that when we are dealing with small numbers of photoelectrons (<20) and
only 25% nuclear recoil events, Poisson fluctuations are likely to wash out the second
peak caused by the faster decay time of the nuclear recoil events. Fig. 4.5 also
demonstrates that the cuts as applied do not cut out legitimate events, and are not
responsible for removing the nuclear recoil events from the data represented in the
histograms. At the same time Fig. 4.6 shows that for larger numbers of photoelectrons
39
(about 75) and for a higher percentage of nuclear recoil events (50%), there should be a
noticeable second peak centered over R(25:100) = 0.7, which can be legitimately
attributed to nuclear recoil events.
In order to obtain more conclusive results it will probably be necessary to increase the
photoelectron yield of the detector to better resolve the two separate peaks as modeled by
the Monte Carlo simulation in Fig. 4.6. Another thing to consider is how many events in
the AmBe calibration sets are actually neutron events – an observation we were unable to
determine within this analysis. Finally, and probably most importantly – the presence of a
large number of anomalous events characterized by very fast decay times and echoing
pulses 19 and 38 ns after the initial pulse contaminate the data sets and make it very
difficult to distinguish the nuclear recoils by their pulse shape. As mentioned in
Section 4.4, it is hypothesized that these anomalous events are caused by ion feedback,
whereby an ion is ejected from the photocathode by a collision with an electron, drifted
across the cavity of the detector by the electric field bias, and then creates more free
electrons when it reaches the anode which repeat the process. This hypothesis explains
the 19 ns gap between echoes, which can be accounted for by the drift time of an ion
through the liquid Xenon cavity. These anomalous events are particularly problematic
because they lie precisely in the low energy range we are interested in examining. A good
starting point for further progress would be to discover how to eliminate these anomalous
ion feedback events.
40
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41