Assessing alternatives for directional detection of a halo of weakly interacting massive particles
Craig J. Copi,
1,
*
Lawrence M. Krauss,
1,2,†
David Simmons-Duffin,
3
and Steven R. Stroiney
4
1
Department of Physics, Center for Education and Research in Cosmology and Astrophysics,
10900 Euclid Ave., Cleveland, Ohio 44106-7079, USA
2
Department of Astronomy, Case Western Reserve University, 10900 Euclid Ave., Cleveland, Ohio 44106-7079, USA
3
Physics Department, Harvard University, Cambridge, Massachusetts 02138, USA
4
Physics Department, Cornell University, Ithaca, New York 14853, USA
(Received 30 August 2005; published 12 January 2007)
The future of direct terrestrial WIMP detection lies on two fronts: new, much larger low background
detectors sensitive to energy deposition, and detectors with directional sensitivity. The former can explore
a large range of WIMP parameter space using well-tested technology while the latter may be necessary if
one is to disentangle particle physics parameters from astrophysical halo parameters. Because directional
detectors will be quite difficult to construct it is worthwhile exploring in advance generally which
experimental features will yield the greatest benefits at the lowest costs. We examine the sensitivity of
directional detectors with varying angular tracking resolution with and without the ability to distinguish
forward versus backward recoils, and compare these to the sensitivity of a detector where the track is
projected onto a two-dimensional plane. The latter detector regardless of where it is placed on the Earth,
can be oriented to produce a significantly better discrimination signal than a 3D detector without this
capability, and with sensitivity within a factor of 2 of a full 3D tracking detector. Required event rates to
distinguish signals from backgrounds for a simple isothermal halo range from the low teens in the best
case to many thousands in the worst.
DOI: 10.1103/PhysRevD.75.023514 PACS numbers: 95.35.+d, 95.55.Vj
I. INTRODUCTION
Direct detection experiments for weakly interacting
massive particles (WIMPs) continue to set ever more strin-
gent limits on the nucleon-WIMP cross section [1–5]. A
precise understanding of the backgrounds is required to
identify an excess of nuclear recoils. There is no unique
signature that can separate neutron induced recoils from
WIMP induced recoils in these detectors. Even annual
modulation, which is at best a few percent effect, might
be accounted for by seasonal background variations. (This
is for a pure WIMP signal. When a uniform background is
included the effect is even smaller.) It has been recognized
that a stronger signal comes from measuring the direction
of the recoiling nucleus [6] allowing for a WIMP signal to
be identified from a few events even in the presence of
backgrounds [7,8]. Detectors that might measure the recoil
direction have been designed and built. For example, time
projection chambers have been used by
DRIFT [9] and
NEWAGE [10] and a scintillator with direction dependent
response has also been studied [11].
In their present state the directional detectors are rudi-
mentary at best. The current version of
DRIFT, for example,
does not measure the full three-dimensional track of the
recoiling nucleus. Instead it measures the recoil track
projected onto a plane and has a number of other limita-
tions. A detailed statistical study of a 3 dimensional detec-
tor, such as
DRIFT II, has been performed [12].
A new generation of directionally sensitive detectors are
now being envisaged. Building a full three-dimensional
detector is a challenging, costly proposition. Is it neces-
sary? Given the technical challenge and cost constraints of
these detectors, which changes will lead to the most sensi-
tive detector? Here we provide a general analysis of vari-
ous design goals to determine the number of events
required for detector designs ranging from two-
dimensional to fully three-dimensional detectors. We do
not focus on any particular detector technology nor model
existing or planned detectors. Instead we apply a consistent
set of parameters to a variety of detector configurations.
This allows us to determine the optimal design goals
independent of detector details.
II. THEORETICAL MODEL
A. Detector characteristics
To quantify the capabilities of directionally sensitive
detectors we consider a consistent, generic set of parame-
ters. For the detector target we use a xenon (m
N
131 GeV) nucleus. We assume a threshold of Q
th
10 keV and two different WIMP masses, m
100 GeV
(m
m
N
) and m
1000 GeV (m
m
N
). For our
Galaxy we focus solely on an isothermal model for the
WIMP halo distribution,
f
~
v
1
3=2
v
3
0
e
j
~
vj
2
=v
2
0
: (1)
We study three different values for v
0
spanning the range
of current expectations, 170 km=s, 220 km=s, and
*
Electronic address: cjc5@cwru.edu
†
Electronic address: lmk9@cwru.edu
PHYSICAL REVIEW D 75, 023514 (2007)
1550-7998=2007=75(2)=023514(5) 023514-1 © 2007 The American Physical Society
270 km=s. The escape velocity of WIMPs from the Galaxy
is taken to be 650 km=s. The Earth’s rotation axis is
oriented at an angle 42
with respect to the Sun’s
motion. This value is relevant for the two-dimensional
detector. We stress that these choices have been made to
provide a consistent set of parameters to allow the inter-
comparison of detector designs not as a suggestion for an
actual detector design. Thus, what will be important about
our results will not be absolute constraints, but relative
ones, although we expect the overall order of magnitude
for the required event rates will not differ compared to our
estimates.
B. Differential event rates
The technique for calculating the WIMP scattering rate
is well known [7]. The differential rate as a function of
nuclear recoil direction ; is given by
dR
d
;
0
0
m
N
m
Z
R
d
3
~
vJvF
2
QJvf
~
v
~
v
: (2)
Here m
N
is the mass of the target nucleus, m
is the WIMP
mass,
~
v
is the velocity of the Earth through the WIMP
halo, Q is the recoil energy of the nucleus,
0
is the cross
section for WIMP scattering off the target nucleus, and
0
is the local WIMP halo density. We consider only spin
independent interactions and use the standard Helm form
factor [13] for F
2
Q. The geometry of the WIMP scatter-
ing gives
Jv v
x
sin cos v
y
sin sin v
z
cos; (3)
which relates the direction of the incoming WIMP to the
direction of the recoiling nucleus. The integration region,
R, is defined by the detector threshold, Q
th
, at the lower
limit,
Jv
m
m
N
2
2m
2
m
N
Q
th
v
u
u
t
; (4)
and the galactic escape velocity, v
esc
, at the upper limit,
v
x
v
;x
2
v
y
v
;y
2
v
z
v
;z
2
v
2
esc
: (5)
See [7] for a more detailed discussion. A full three-
dimensional detector probes the reaction rate outlined
here (2).
Measuring the direction the nucleus is moving along a
track is not always possible. For a three-dimensional de-
tector without forward-backward discrimination, a recoil
in a direction cos; cannot be distinguished from a
recoil in a direction cos; . The event rate for
these two directions thus combine giving a total rate for the
direction cos; of
dR
d
cos;
dR
d
cos;
where the differential rates are again given above (2).
A two-dimensional detector can only resolve the recoil
direction projected onto a plane. We assume for compari-
son purposes however that it can be designed with forward-
backward discrimination. Suppose the normal to the de-
tector plane is oriented at angles ; with respect to the
direction of the Sun’s motion. The rate is a function of a
single angle
0
measured in this frame,
dR
d
0
0
0
m
N
m
Z
1
1
dcos
0
Z
R
d
3
~
v
0
Jv
0
F
2
Q
0
f
~
v
~
v
: (6)
Here, primed coordinates are measured in the frame of the
detector (where the normal vector points along the z-axis)
and unprimed coordinates are measured in the frame of the
Sun’s motion. These two frames are related by rotations
through the angles and .
For a detector fixed to the Earth’s surface, the detector
orientation with respect to the Sun’s motion changes
throughout the day due to the Earth’s rotation. Let be
the angle between the detector’s normal and the Earth’s
rotation axis and be the angle between the rotation axis
and the Sun’s velocity. Then
cos cos cos sin sin cos2t (7)
for t measured in days.
It is also important to recognize that even full three-
dimensional detectors will not have perfect angular reso-
lution. To model realistic angular resolution either due to
dispersion of the recoiling nucleus along its ideal track or
due to inherent precision of the inherent detection mecha-
nism itself we convolve the ideal scattering rate (2) with a
smoothing kernel K;
0
,
dR
d
Z
K;
0
dR
d
0
d
0
: (8)
We use a Gaussian smoothing about the direction of the
ideal recoil,
K;
0
e
2
=2
2
=2
3=2
erf
2
p
; (9)
where
cos sin sin
0
cos
0
cos cos
0
: (10)
We study this as a function of the width of the Gaussian.
III. RESULTS
We test the capabilities of each type of WIMP detector
by assessing their ability to distinguish the WIMP distri-
bution from a flat background, using an isothermal halo as
a fiducial test model. The probability that a WIMP will
recoil in a particular direction,
i
, is given by P
i
dR
d
i
, where we have normalized the rate such that R
1. Thus, we are probing the shape of the recoil spectrum.
The likelihood function for N
e
detected events is defined
by L
Q
N
e
i1
P
i
. We generate at least 100 000 sample
distributions for each N
e
and apply the log-likelihood test
to find the minimum number of WIMP events such that we
COPI, KRAUSS, SIMMONS-DUFFIN, AND STROINEY PHYSICAL REVIEW D 75, 023514 (2007)
023514-2
have a 95% detection 95% of the time (see [7] for more
details).
The results for m
100 GeV are given in Table I for
the range of detectors we have considered, where ‘‘full’’
reflects a full three-dimensional detector with perfect an-
gular resolution. We shall discuss the degradation implied
by limited resolution shortly. The same set of results for
m
1000 GeV are given in Table II. Although we have
restricted our quantitative study to isothermal models the
qualitative features of the comparison remain valid for
other models, including models with single streams of
WIMPs, but with a significant isothermal component.
Our results underscore the need for forward-backward
detection. Indeed, this is the single most important feature
that allows directional detectors to gain sensitivity to the
WIMP signal compared to backgrounds. Since spin inde-
pendent WIMP scattering is azimuthally symmetric about
the direction of the incoming WIMP the dominant WIMP
signal comes from the a comparison of forward-backward
scattering events. This is seen in the results in Tables I and
II. A three-dimensional detector, even with perfect angular
resolution, but without forward-backward discrimination
requires a surprisingly large (at least an order of magnitude
greater) number of events than a three-dimensional detec-
tor with such discrimination and even many more than a
poorly aligned two-dimensional detector to distinguish a
WIMP signal from terrestrial backgrounds. This is because
without forward-backward discrimination the detector re-
lies upon the difference between head-on and glancing
(wide angle) collisions as well as high angular resolution
to distinguish a WIMP signal from the background. The
surprisingly large difference in required number of events
for heavy WIMPs versus WIMPs of mass comparable to
target nuclei masses presumably is due to the fact that
nuclear recoils from collisions with heavy WIMPS tend
to better follow the direction of the original WIMP.
We next explore how the sensitivity of a three-
dimensional detector depends upon its angular resolution.
In Fig. 1 we display the number of events required as a
function of the angle of the full-width half maximum
(FWHM) of the detection cone for the events. Note that
as the angular resolution degrades, the number of events
required for a three-dimensional detector quickly ap-
proaches that of a two-dimensional detector, as expected.
In order to be significantly more efficient, the angular
TABLE I. The number of events required to identify a WIMP
signal above a flat background for different types of detectors
and a WIMP mass of m
100 GeV.
Detector Type v
0
km=s
170 220 270
3D (full) 6 11 18
3D without FB 176 1795 >35; 000
2D—best/worst 19=45 34=75 61=123
2D rotating 13 24 43
TABLE II. Same as Table I, for a WIMP mass of m
1000 GeV.
Detector Type v
0
km=s
170 220 270
3D (full) 14 27 51
3D without FB 152 217 371
2D fixed—best/worst 51=129 97=217 175=368
2D rotating 31 61 125
FIG. 1. The number of events required as a function of the full-
width half maximum (FWHM) of the smoothing kernel (9) for
the isothermal models, m
100 GeV, and the detector con-
figuration described in section II A.
FIG. 2. The number of events required as a function of the
angle between the detector normal and the direction of the
WIMP wind (Sun’s motion), for the isothermal models, m
100 GeV, and the detector configuration described in Sec. II A.
ASSESSING ALTERNATIVES FOR DIRECTIONAL ... PHYSICAL REVIEW D 75, 023514 (2007)
023514-3
resolution of such a detector must be better than about 60
degrees (FWHM).
We finally focus on two-dimensional detectors, in part
because these are likely to be the most practical in the near
future, and because less attention has been paid to them
than hypothetical three-dimensional detectors.
It is clear that the efficacy of a planar detector will
depend upon the orientation of its plane with respect to
the direction of the WIMP wind. Specifically a two-
dimensional detector fixed to the Earth will be oriented
so that its normal vector makes an angle with the Earth’s
rotation axis. The choice of determines how much time
the detector will spend at various angles relative to the
Earth’s direction of motion. An orientation of 0
is
clearly the worst since the detector plane is then perpen-
dicular to the WIMP wind. The number of events required
is a function of the angle chosen for the detector as shown
in Fig. 2.
Tables I and II give the minimum and maximum number
of events required for optimal versus worst-case orienta-
tion of the detector. Note that the shape of the function in
Fig. 2 depends on the orientation of the Earth’s axis,
42
relative to the motion of the Sun through an isotropic
halo. For halos in which a WIMP stream arose which was
not due to the motion of the Sun (i.e. involving some other
bulk motion with respect to the galaxy rest frame, as would
occur for some infalling WIMP distribution), the shape of
the curve would change. Low values of in this case would
produce a more steeply descending function, and high
values would produce an ascending function.
Note that the results thus far assume that the detector
axis is fixed to the plane of the Earth. In this case the time
averaged number of events required is on average 3 times
less for a two-dimensional detector oriented in the best
possible axis (i.e. at or near 90
for isothermal halo
distribution). Such a detector however requires three to
four times the number of events of a full three-dimensional
detector. Whether or not achieving this additional factor of
3 in sensitivity for a full three-dimensional detector is
worth the technological challenge is not clear. However,
as seen in the tables, this factor of 3 can be reduced by
using a detector which is not fixed relative to the plane of
the Earth, but which can rotate over the course of each day
with respect to the earth to maintain an optimal orientation
with respect to an expected WIMP wind. For an isothermal
halo, roughly only 2 times as many events are required for
such a rotating detector compared to a detector with full
three-dimensional tracking capability. The technical diffi-
cult of producing the former may be less demanding than
that required to produce the latter. Our purpose is to
demonstrate the theoretical gain that may be obtained so
experimentalists can then decide if the challenge is worth
addressing.
Our results can be summarized as follows: For direc-
tional WIMP detectors, forward-backward discrimination
is far more valuable than three-dimensional resolution of
the track, at least for the isothermal model considered here.
Furthermore, a two-dimensional detector can be oriented
into the predominant WIMP wind so that the number of
events required to distinguish a WIMP halo from a terres-
trial background is comparable to that required for even a
full three-dimensional detector. Note that the results for a
two-dimensional detector are independent of its location
on the Earth. The orientation of the plane of the detector
determines its capabilities not the latitude of the detector.
Some locations may allow for easier detector orientation
[14,15] but all locations are equally good provided the
detector plane can be properly aligned. While there may
be other reasons (including background rejection) for con-
sidering fully three-dimensional directional detection
methods, our results thus suggest that concentrating simply
on forward-background sensitivity is the most important
new direction that should be pursued in directional WIMP
detection, and that planar detectors can, in this case, pro-
vide nearly optimal directional sensitivity. These results
are consistent with those obtained in a complimentary
work that studied the specific capabilities of a DRIFT-
type detector [14].
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