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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