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arXiv:0803.4477v3 [hep-ph] 17 Jun 2008

KIAS–P08026

March 2008

Model–Independent Determination of the WIMP Mass

from Direct Dark Matter Detection Data

Manuel Drees1,2and Chung-Lin Shan1

1Physikalisches Inst. der Univ. Bonn, Nussallee 12, 53115 Bonn, Germany

2KIAS, School of Physics, 207–43 Cheongnyangni–dong, Seoul 130–012, Republic of Korea

Abstract

Weakly Interacting Massive Particles (WIMPs) are one of the leading candidates for

Dark Matter. We develop a model–independent method for determining the mass mχof the

WIMP by using data (i.e., measured recoil energies) of direct detection experiments. Our

method is independent of the as yet unknown WIMP density near the Earth, of the form of

the WIMP velocity distribution, as well as of the WIMP–nucleus cross section. However,

it requires positive signals from at least two detectors with different target nuclei. In a

background–free environment, mχ∼ 50 GeV could in principle be determined with an error

of ∼ 35% with only 2 × 50 events; in practice upper and lower limits on the recoil energy

of signal events, imposed to reduce backgrounds, can increase the error. The method also

loses precision if mχsignificantly exceeds the mass of the heaviest target nucleus used.

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1 Introduction

First indications for the existence of Dark Matter were already found in the 1930s [1]. By now

there is strong evidence [1]-[5] to believe that a large fraction (more than 80%) of all matter in the

Universe is dark (i.e., interacts at most very weakly with electromagnetic radiation and ordinary

matter). The dominant component of this cosmological Dark Matter must be due to some yet

to be discovered, non–baryonic particles. Weakly Interacting Massive Particles (WIMPs) χ are

one of the leading candidates for Dark Matter. WIMPs are stable particles which arise in several

extensions of the Standard Model of electroweak interactions. Typically they are presumed to

have masses between 10 GeV and a few TeV and interact with ordinary matter only weakly (for

reviews, see [6]).

Currently, the most promising method to detect many different WIMP candidates is the

direct detection of the recoil energy deposited in a low–background laboratory detector by elastic

scattering of ambient WIMPs on the target nuclei [7, 8]. The recoil energy spectrum can be

calculated from an integral over the one–dimensional WIMP velocity distribution f1(v), where v

is the absolute value of the WIMP velocity in the laboratory frame. If this function is known, the

WIMP mass mχcan be obtained from a one–parameter fit to the normalized recoil spectrum [9],

once a positive WIMP signal has been found. However, this introduces a systematic uncertainty

which is difficult to control. We remind the reader that N−body simulations of the spatial

distribution of Cold Dark Matter (which includes WIMPs) seem to be at odds with observations

at least in the central region of galaxies [10]; this may have ramifications for f1 as well. On

the theory side, several modifications of the standard “shifted Maxwellian” distribution [6] have

been suggested, ranging from co– or counter–rotating halos [11] to scenarios where the three–

dimensional WIMP velocity distribution gets large contributions from discrete “streams” with

(nearly) fixed velocity [12, 13].

The goal of our work is to develop model–independent methods which allow to determine

mχdirectly from (future) experimental data. This builds on our earlier work [14], where we

showed how to determine (moments of) f1(v) from the recoil spectrum in direct WIMP detection

experiments. In this earlier analysis mχhad been the only input required; one does not need to

know the WIMP–nucleus scattering cross section, nor the local WIMP density. The fact that

this method will give a result for f1for any assumed value of mχalready tells us that one will

need at least two different experiments, with different target nuclei, to model–independently

determine the WIMP mass from direct detection experiments. To do so, one simply requires

that the values of a given moment of f1 determined by both experiments agree. This leads

to a simple expression for mχ, which can easily be solved analytically; note that each moment

can be used. An additional expression for mχcan be derived under the assumption that the

ratio of scattering cross sections on protons and neutrons is known. This is true e.g., for spin–

independent scattering of a supersymmetric neutralino, which is the perhaps best motivated

WIMP candidate [6]; in this case the spin–independent cross section for scattering on a proton

is almost the same as that for scattering on a neutron. Not surprisingly, the best result obtains

by combining measurements of several moments with that derived from the assumption about

the ratio of cross sections.

The remainder of this article is organized as follows. In Sec. 2 we review briefly the methods

for estimating the moments of the velocity distribution function, paying special attention to

experimentally imposed limits on the range of allowed recoil energies. In Sec. 3 we will present

the formalism for determining the WIMP mass.

simulations of future experiments, will be presented in Sec. 4. We conclude in Sec. 5. Some

technical details of our calculation will be given in an Appendix.

Numerical results, based on Monte Carlo

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2 Determining the moments of the velocity distribution

of WIMPs

In this section we review briefly the method for estimating the moments of the one–dimensional

velocity distribution function f1 of WIMPs from the elastic WIMP–nucleus scattering data.

We first discuss the formalism, and then describe how it can be implemented directly using

(simulated, or future real) data from direct WIMP search experiments.

2.1 Formalism

Our analysis starts from the basic expression for the differential rate for elastic WIMP–nucleus

scattering [6]:

?f1(v)

v

dR

dQ= AF2(Q)

?∞

vmin

?

dv.(1)

Here R is the direct detection event rate, i.e., the number of events per unit time and unit mass

of detector material, Q is the energy deposited in the detector, F(Q) is the elastic nuclear form

factor, and v is the absolute value of the WIMP velocity in the laboratory frame. The constant

coefficient A is defined as

A ≡

2mχm2

r,N

ρ0σ0

, (2)

where ρ0is the WIMP density near the Earth and σ0is the total cross section ignoring the form

factor suppression. The reduced mass mr,Nis defined as

mχmN

mχ+ mN,

mr,N≡

(3)

where mχis the WIMP mass and mNthat of the target nucleus. Finally, vminis the minimal

incoming velocity of incident WIMPs that can deposit the energy Q in the detector:

?

where we define

?mN

2m2

r,N

vmin= α

Q, (4)

α ≡

. (5)

Eq.(1) can be solved for f1[14]:

?

f1(v) = N−2Q ·

d

dQ

?

1

F2(Q)

?dR

dQ

???

Q=v2/α2

, (6)

where the normalization constant N is given by

N =2

α

0

??∞

1

√Q

?

1

F2(Q)

?dR

dQ

??

dQ

?−1

. (7)

Note that, first, because f1(v) in Eq.(6) is the normalized velocity distribution, the normalization

constant N here is independent of the constant coefficient A defined in Eq.(2). Second, the

integral here goes over the entire physically allowed range of recoil energies, starting at Q = 0.

The upper limit of the integral has been written as ∞. However, it is usually assumed that the

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WIMP flux on Earth is negligible at velocities exceeding the escape velocity vesc≃ 700 km/s.

This leads to a kinematic maximum of the recoil energy,

Qmax,kin=v2

esc

α2, (8)

where α has been given in Eq.(5). Eq.(7) then implies

?∞

Using Eq.(6), the moments of f1can be expressed as [14]

0

f1(v)dv =

?vesc

0

f1(v)dv = 1. (9)

?vn? =

?vesc

0

vnf1(v)dv = N

?αn+1

2

?

(n + 1)In, (10)

which holds for all n ≥ 0. Here the integral Inis given by

?Qmax,kin

In=

0

Q(n−1)/2

?

1

F2(Q)

?dR

dQ

??

dQ. (11)

In this notation, Eq.(7) can be re–written as N = 2/(αI0).

The results in Eqs.(6) and (10) depend on the WIMP mass mχonly through the coefficient α

defined in Eq.(5). Evidently any (assumed) value of mχwill lead to a well–defined, normalized

distribution function f1when used in Eq.(6). Hence mχcan be extracted from a single recoil

spectrum only if one makes some assumptions about the velocity distribution f1(v).

A model–independent determination of mχthus requires that at least two different recoil

spectra, with two different target nuclei, have been measured. As we will show in detail in the

next section, mχcan then be obtained from the requirement that these two spectra lead to the

same moments of f1.

Before coming to that, we have to incorporate the effects of a finite energy acceptance of

the detector. Any real detector will have a certain threshold energy Qthrebelow which it cannot

register events. Off–line one may need to impose a cut Q > Qmin> Qthrein order to suppress

(instrumental or physical) backgrounds. Similarly, background rejection may require a maximum

energy cut, Q < Qmax ≤ Qmax,kin. In fact, we will see below that, at least for smallish data

samples, such a cut might even be beneficial for the determination of mχ. We therefore now give

expressions for the case that only data with Qmin≤ Q ≤ Qmaxare used in the analysis.

To that end, we introduce generalized moments of f1:

?v2

F2(Q1)

?vn?(v1,v2) =

v1

vnf1(v)dv

= Nαn+1

Q(n+1)/2

1

r(Q1)

−Q(n+1)/2

2

r(Q2)

F2(Q2)

+

?n + 1

2

?

In(Q1,Q2)

, (12)

where we have introduced the short–hand notation

?????Q=Qi

and

?Q2

r(Qi) =dR

dQ

,i = 1, 2, (13)

In(Q1,Q2) =

Q1

Q(n−1)/2

?

1

F2(Q)

?dR

dQ

??

dQ.(14)

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In order to arrive at the final expression in Eq.(12) we used Eq.(6) and integrated by parts. The

Qiin Eqs.(12)–(14) are related to the original integration limits viappearing on the left–hand

side of Eq.(12) via Eq.(4), i.e.,

Qi=v2

i

α2,i = 1, 2. (15)

Of course, Eq.(12) reduces to Eq.(10) in the limit v1→ 0, v2→ vesc; note, however, that Eq.(12)

is also applicable for the case n = −1, where the last term on the right–hand side vanishes. The

same restriction on the WIMP velocity can also be introduced in the normalization constant N

of Eq.(7), in which case

Eqs.(12)–(15) by demanding ?v0?(v1,v2) = 1.

?v2

v1f1(v)dv is normalized to unity. Formally this can be treated using

2.2 Experimental implementation

In order to directly use our results for f1(v) and for its moments ?vn? given in Eqs.(6), (10)

and (12), one needs a functional form for the recoil spectrum dR/dQ. In practice this results

usually from a fit to experimental data. However, data fitting can re–introduce some model

dependence and makes the error analysis more complicated. Hence, expressions that allow to

reconstruct f1(v) and its moments directly from the data have been developed [14]. We started

by considering experimental data described by

Qn−bn

2≤ Qn,i≤ Qn+bn

2,i = 1, 2, ···, Nn, n = 1, 2, ···, B.(16)

Here the total energy range has been divided into B bins with central points Qnand widths

bn. In each bin, Nn events will be recorded. Note that we assume that the sample to be

analyzed only contains signal events, i.e., is free of background, and ignore the uncertainty on

the measurement of the recoil energy Q. Active background suppression techniques [15] should

make the former possible. The energy resolution of most existing detectors is so good that its

error will be negligible compared to the statistical uncertainty for the foreseeable future.

Since the recoil spectrum dR/dQ is expected to be approximately exponential, we used the

following ansatz for the spectrum in the n−th bin [14]:

?dR

dQ

n

dQ

Q≃Qn

?

≡

?dR

?

≃ ? rnekn(Q−Qn)≡ rnekn(Q−Qs,n). (17)

Here rnis the standard estimator for dR/dQ at Q = Qn,

rn=Nn

bn

, (18)

? rnis the value of the recoil spectrum at the point Q = Qn,

? rn≡

and knis the logarithmic slope of the recoil spectrum in the n−th bin. It can be computed

numerically from the average Q−value in the n−th bin:

?bn

22

?dR

dQ

?

Q=Qn

= rn

?

knbn/2

sinh(knbn/2)

?

, (19)

Q − Qn|n=

?

coth

?knbn

?

−1

kn, (20)