Effects of residue background events in direct dark matter detection experiments on the determination of the WIMP mass

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Effects of residue background events in direct dark matter detection experiments on the

determination of the WIMP mass

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JCAP08(2010)014

ournal of Cosmology and Astroparticle Physics

An IOP and SISSA journal

J

Effects of residue background events

in direct dark matter detection

experiments on the determination of

the WIMP mass

Yu-Ting Chou

a

and Chung-Lin Shan

b,c

a_{Institute of Physics, National Chiao Tung University,}

No. 1001, University Road, Hsinchu City 30010, Taiwan, R.O.C.

b_{Department of Physics, National Cheng Kung University,}

No. 1, University Road, Tainan City 70101, Taiwan, R.O.C.

c_{Physics Division, National Center for Theoretical Sciences,}

No. 101, Sec. 2, Kuang-Fu Road, Hsinchu City 30013, Taiwan, R.O.C.

E-mail: yuting.py97g@nctu.edu.tw,clshan@mail.ncku.edu.tw

Received April 9, 2010 Revised July 13, 2010 Accepted July 17, 2010 Published August 11, 2010

Abstract. In the earlier work on the development of a model-independent data analysis

method for determining the mass of Weakly Interacting Massive Particles (WIMPs) by using measured recoil energies from direct Dark Matter detection experiments directly, it was assumed that the analyzed data sets are background-free, i.e., all events are WIMP signals. In this article, as a more realistic study, we take into account a fraction of possible residue background events, which pass all discrimination criteria and then mix with other real WIMP-induced events in our data sets. Our simulations show that, for the determination of the WIMP mass, the maximal acceptable fraction of residue background events in the analyzed data sets of O(50) total events is ∼ 20%, for background windows of the entire experimental possible energy ranges, or in low energy ranges; while, for background windows in relatively higher energy ranges, this maximal acceptable fraction of residue background events can not be larger than ∼ 10%. For a WIMP mass of 100 GeV with 20% background events in the windows of the entire experimental possible energy ranges, the reconstructed WIMP mass and the 1σ statistical uncertainty are ∼ 97 GeV +61%_−35% (∼ 94 GeV +55%_−33% for background-free data sets).

Keywords: dark matter simulations, dark matter experiments

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Contents

1 Introduction 1

2 Signal and background spectra 2

2.1 Elastic WIMP-nucleus scattering spectrum 2

2.1.1 One-dimensional WIMP velocity distribution function 3

2.1.2 Spin-independent WIMP-nucleus cross section 4

2.2 Background spectrum 4

2.3 Measured energy spectrum 5

3 Reconstruction of the WIMP mass 8

3.1 Model-independent determination of the WIMP mass 8

3.1.1 Basic expressions for determining the WIMP mass 8

3.1.2 χ2-fitting 10

3.1.3 Matching the cut-off energies 10

3.2 Reconstructing mχ by using data sets with background events 11

3.2.1 With the exponential background spectrum 12

3.2.2 Statistical fluctuation 15

3.2.3 With the constant background spectrum 16

4 Summary and conclusions 20

1 Introduction

Currently, direct Dark Matter detection experiments searching for Weakly Interacting Mas-sive Particles (WIMPs) are one of the promising methods for understanding the nature of Dark Matter and identifying them among new particles produced at colliders as well as recon-structing the (sub)structure of our Galactic halo [1–4]. In order to determine the mass of halo WIMPs without making any assumptions about their density near the Earth or their veloc-ity distribution nor knowing their scattering cross section on nucleus, a model-independent method by combining two experimental data sets with two different target nuclei has been

developed [5,6]. This method builds on the earlier work on the reconstruction of the

(mo-ments of the) one-dimensional velocity distribution function of halo WIMPs, f1(v), by using

data from direct detection experiments [7].

In the analysis of reconstructing f1(v), the moments of the WIMP velocity distribution

function can be determined from experimental data directly with an unique input

informa-tion about the WIMP mass mχ. Hence, one can simply require that the values of a given

moment of f1(v) determined by two experiments agree.1 This leads to a simple analytic

ex-pression for determining mχ[5,6], where each moment can in principle be used. Additionally,

under the assumptions that the spin-independent (SI) WIMP-nucleus interaction dominates over the spin-dependent (SD) one and the SI WIMP coupling on protons is approximately

the same as that on neutrons, a second analytic expression for determining mχ has been

1

Note that, as demonstrated and discussed in ref. [6], this condition requires an algorithmic procedure for matching the maximal cut-off energies of the analyzed data sets.

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derived [6]. Finally, by combining the first estimators for different moments with each other and with the second estimator, one can yield the best-fit WIMP mass as well as minimize its statistical uncertainty.

In the work on the development of the model-independent data analysis procedure for the determination of the WIMP mass, it was assumed that the analyzed data sets are background-free, i.e., all events are WIMP signals. Active background discrimination tech-niques should make this condition possible. For example, the ratio of the ionization to recoil energy, the so-called “ionization yield”, used in the CDMS-II experiment provides an event-by-event rejection of electron recoil events to be better than 10−4 _{misidentification [}₈_{]. By}

combining the “phonon pulse timing parameter”, the rejection ability of the misidentified electron recoils (most of them are “surface events” with sufficiently reduced ionization

en-ergies) can be improved to be < 10−6 _{for electron recoils [}₈_{]. Moreover, as demonstrated}

by the CRESST collaboration [9], by means of inserting a scintillating foil, which causes

some additional scintillation light for events induced by α-decay of 210Po and thus shifts the pulse shapes of these events faster than pulses induced by WIMP interactions in the crystal, the pulse shape discrimination (PSD) technique can then easily distinguish WIMP-induced

nuclear recoils from those induced by backgrounds.2

However, as the most important issue in all underground experiments, the signal identi-fication ability and possible residue background events which pass all discrimination criteria and then mix with other real WIMP-induced events in our data sets should also be con-sidered. Therefore, in this article, as a more realistic study, we take into account different fractions of residue background events mixed in experimental data sets and want to study how well the model-independent method could reconstruct the input WIMP mass by using these “impure” data sets and how “dirty” these data sets could be to be still useful.

The remainder of this article is organized as follows. In section 2 we review the recoil spectrum of elastic WIMP-nucleus scattering and introduce two kinds of background spec-trum used in our simulations. In section 3 we first review briefly the model-independent method for the determination of the WIMP mass. Then we show numerical results of the reconstructed WIMP mass by using mixed data sets with different fractions of residue back-ground events based on Monte Carlo simulations. We conclude in section 4.

2 Signal and background spectra

In this section we first review the recoil spectrum of elastic WIMP-nucleus scattering. Then we introduce two forms of background spectrum which will be used in our simulations. Some numerical results of the measured energy spectrum superposed by the WIMP scattering and background spectra will also be discussed.

2.1 Elastic WIMP-nucleus scattering spectrum

The basic expression for the differential event rate for elastic WIMP-nucleus scattering is given by [3]: dR dQ = AF 2_(Q) Z vmax vmin f1(v) v dv . (2.1)

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

2

For more details about background discrimination techniques and status in currently running and pro-jected direct detection experiments, see e.g., refs. [10–12].

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nuclear form factor, f1(v) is the one-dimensional velocity distribution function of the WIMPs

impinging on the detector, v is the absolute value of the WIMP velocity in the laboratory frame. The constant coefficient A is defined as

A ≡ ρ0σ0

2mχm2_r,N

, (2.2)

where ρ0 is the WIMP density near the Earth and σ0 is the total cross section ignoring the

form factor suppression. The reduced mass mr,N is defined by

mr,N≡

mχmN

mχ+ mN

, (2.3)

where mχ is 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:

vmin = αpQ (2.4)

with the transformation constant

α ≡ s mN 2m2 r,N , (2.5)

and vmax is the maximal WIMP velocity in the Earth’s reference frame, which is related

to the escape velocity from our Galaxy at the position of the Solar system, vesc >_{∼ 600}

km/s. Note that, as will be shown below, the Earth’s velocity relative to the Galactic halo is time-dependent, and considering the random motion of WIMPs in the Galaxy, the relation

between the one-dimensional cut-off vmax and the three-dimensional one vesc is thus rather

complicated. Nevertheless, it is unlike to affect the event rate as well as the results shown in this article significantly. In the literature, for simplicity and practical uses, vmax is often set

as ∞ (e.g., [13–15]).

2.1.1 One-dimensional WIMP velocity distribution function

The simplest semi-realistic model halo is a spherical isothermal Maxwellian halo. More realistically, one has to take into account the orbital motion of the Solar system around the Galaxy as well as that of the Earth around the Sun. The one-dimensional velocity distribution function of this shifted Maxwellian halo can be expressed as [2,3,7]

f1,sh(v) = 1 √ π v vev0 e−(v−ve)2/v20 − e−(v+ve)2/v20 . (2.6)

Here v0 ≃ 220 km/s is the orbital velocity of the Sun in the Galactic frame, and ve is the

Earth’s velocity in the Galactic frame [3,4,16]: ve(t) = v0 1.05 + 0.07 cos 2π(t − tp) 1 yr ; (2.7)

tp ≃ June 2nd is the date on which the velocity of the Earth relative to the WIMP halo

is maximal. Substituting eq. (2.6) into eq. (2.1), an analytic form of the integral over the velocity distribution function can be given as [17]

Z vmax vmin f1,sh(v) v dv = 1 2ve ( erfα√Q+ve v0 − erfα√Q−ve v0 − erfvmax+ve v0 − erfvmax−ve v0 ) . (2.8)

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On the other hand, for practical, numerical uses, an approximate form of the integral over f1(v) was introduced as [2]

Z ∞ vmin f1(v) v dv = c0 2 √ πv0 e−α2Q/c1v02, (2.9)

where c0 and c1 are two fitting parameters of order unity. Not surprisingly, their values

depend on the Galactic orbital and escape velocities, the target nucleus, the threshold energy of the experiment, as well as on the mass of incident WIMPs. Note that, the characteristic energy Qch≡ c1v20/α2 and thus the shape of the recoil spectrum depend highly on the WIMP

mass: for light WIMPs (mχ≪ mN), Qch ∝ m2χ and the recoil spectrum drops sharply with

increasing recoil energy, while for heavy WIMPs (mχ ≫ mN), Qch∼ const. and the spectrum

becomes flatter.

2.1.2 Spin-independent WIMP-nucleus cross section

In most theoretical models, the spin-independent (SI) WIMP-nucleus interaction with an

atomic mass number A >_{∼ 30 dominates over the spin-dependent (SD) one [}3,4].

Addition-ally, for the lightest supersymmetric neutralino which is perhaps the best motivated WIMP

candidate [3,4], and for all WIMPs which interact primarily through Higgs exchange, the SI

scalar coupling is approximately the same on both protons p and neutrons n, the “pointlike” cross section σ0 in eq. (2.2) can thus be written as

σ0= A2 mr,N mr,p 2 σSI_χp, (2.10) where σSI_χp= 4 π m2_r,p_|fp|2 (2.11)

is the SI WIMP-proton cross section, fp is the effective χχpp four-point coupling, A is the

atomic mass number of the target nucleus, and mr,p is the reduced mass of the WIMP mass

mχ and the proton mass mp.

For the SI WIMP-nucleus cross section, an analytic form for the elastic nuclear form factor, inspired by the Woods-Saxon nuclear density profile, has been suggested by Engel as [3,4,18]3

F_WS2 (Q) = 3j1(qR1) qR1

2

e−(qs)2. (2.12)

Here j1(x) is a spherical Bessel function, q =p2mNQ is the transferred 3-momentum, given

as a function of the recoil energy transferred from the incident WIMP to the target nucleus,

Q, and the mass of the target nucleus, mN; R1 =

q

R_A2 _{− 5s}2 _{is the effective nuclear radius}

with RA≃ 1.2 A1/3 fm and the nuclear skin thickness s ≃ 1 fm.

2.2 Background spectrum

For our simulations with residue background events, two forms of background spectrum are considered. The simplest choice for the background spectrum is the constant spectrum:

dR dQ bg,const = 1 . (2.13) 3

Other commonly used analytic forms for the nuclear form factor for the SI WIMP-nucleus cross section can be found in ref. [17].

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More realistically, inspired by ref. [14], we introduce a target-dependent exponential spectrum given by dR dQ bg,ex = exp −Q/keV_A_0.6 . (2.14)

Here Q is the recoil energy, A is the atomic mass number of the target nucleus. The power index of A, 0.6, is an empirical constant, which has been chosen so that the exponential background spectrum is somehow similar to, but still different from the expected recoil spectrum of the target nuclei; otherwise, there is in practice no difference between the WIMP scattering and background spectra. Note that, among different possible choices (e.g., the

exponential form used in ref. [14]), we use in our simulations the atomic mass number A as

the simplest, unique characteristic parameter in the general analytic form (2.14) for defining the residue background spectrum for different target nuclei. However, it does not mean that the (superposition of the real) background spectra would depend simply/primarily on A or

on the mass of the target nucleus, mN. In other words, it is practically equivalent to use

expression (2.14) or (dR/dQ)bg,ex = e−Q/13.5 keV directly for a76Ge target.

Note also that, firstly, two forms of background spectrum given in eqs. (2.13) and (2.14) are rather naive; however, since we consider here only a few residue background events induced by perhaps two or more different sources, pass all discrimination criteria, and then mix with other WIMP-induced events in our data sets of O(50) total events, exact forms of different background spectra are actually not very important and these two spectra, in particular, the exponential one, should practically not be unrealistic.4 Secondly, for using the maximum likelihood analysis to determine the WIMP mass, as described in refs. [13,14,21], a prior knowledge about the WIMP scattering spectrum and eventually about the background

spectrum is essential [14]. In contrast, as demonstrated in ref. [6] and will be reviewed

in the next section, the model-independent data analysis procedure requires only measured recoil energies (induced mostly by WIMPs and occasionally by background sources) from two experimental data sets with different target nuclei. Therefore, for applying this method to future real data from direct detection experiments, the prior knowledge about (different) background source(s) is not required at all.

2.3 Measured energy spectrum

In figure 1 we show measured energy spectra (solid red histograms) for a 76Ge target with

six different WIMP masses: 10, 25, 50, 100, 250, and 500 GeV based on Monte Carlo simula-tions. The dotted blue curves are the elastic WIMP-nucleus scattering spectra for the shifted Maxwellian velocity distribution given in eq. (2.6) with v0 = 220 km/s, ve = 1.05 v0,5 and

vesc = 700 km/s and the Woods-Saxon elastic nuclear form factor in eq. (2.12). The dashed

green curves are the exponential background spectra given in eq. (2.14), which have been

normalized so that the ratios of the areas under these background spectra to those under the (dotted blue) WIMP scattering spectra are equal to the background-signal ratio in the whole

data sets (i.e., 20% backgrounds to 80% signals shown in figures1). The experimental

thresh-old energy has been assumed to be negligible and the maximal cut-off energy is set as 100 keV. 5,000 experiments with 500 total events on average in each experiment have been simulated.

4

Other (more realistic) forms for background spectrum (perhaps also for some specified tar-gets/experiments) can be tested on the AMIDAS website [19,20].

5

The time dependence of the Earth’s velocity in the Galactic frame, the second term of ve(t) in eq. (2.7), has been ignored.

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 10 20 30 40 50 60 70 80 90 100 dR/dQ [x 10 -3 events/kg-day/keV] Q [keV] 76_{Ge, Q}

max < 100 keV, Qmax, bg < 100 keV, 500 events (20% exponential bg), mχ = 10 GeV

AMIDAS http://pisrv0.pit.physik.uni-tuebingen.de/darkmatter/amidas/ (dR/dQ)signal (dR/dQ)background (dR/dQ)measured 0 1 2 3 4 5 6 7 0 10 20 30 40 50 60 70 80 90 100 dR/dQ [x 10 -4 events/kg-day/keV] Q [keV] 76_{Ge, Q}

AMIDAS http://pisrv0.pit.physik.uni-tuebingen.de/darkmatter/amidas/ (dR/dQ)signal (dR/dQ)background (dR/dQ)measured 0 0.5 1 1.5 2 2.5 3 3.5 0 10 20 30 40 50 60 70 80 90 100 dR/dQ [x 10 -4 events/kg-day/keV] Q [keV] 76_{Ge, Q}

AMIDAS http://pisrv0.pit.physik.uni-tuebingen.de/darkmatter/amidas/ (dR/dQ)signal (dR/dQ)background (dR/dQ)measured 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 10 20 30 40 50 60 70 80 90 100 dR/dQ [x 10 -4 events/kg-day/keV] Q [keV] 76_{Ge, Q}

AMIDAS http://pisrv0.pit.physik.uni-tuebingen.de/darkmatter/amidas/ (dR/dQ)signal (dR/dQ)background (dR/dQ)measured 0 1 2 3 4 5 6 7 8 9 0 10 20 30 40 50 60 70 80 90 100 dR/dQ [x 10 -5 events/kg-day/keV] Q [keV] 76_{Ge, Q}

AMIDAS http://pisrv0.pit.physik.uni-tuebingen.de/darkmatter/amidas/ (dR/dQ)signal (dR/dQ)background (dR/dQ)measured 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 10 20 30 40 50 60 70 80 90 100 dR/dQ [x 10 -5 events/kg-day/keV] Q [keV] 76_{Ge, Q}

AMIDAS http://pisrv0.pit.physik.uni-tuebingen.de/darkmatter/amidas/

(dR/dQ)signal (dR/dQ)background (dR/dQ)measured

Figure 1. Measured energy spectra (solid red histograms) for a76

Ge target with six different WIMP masses: 10, 25, 50, 100, 250, and 500 GeV. The dotted blue curves are the elastic WIMP-nucleus scattering spectra for the shifted Maxwellian velocity distribution and the Woods-Saxon elastic nuclear form factor; whereas the dashed green curves are the exponential background spectra normalized to fit to the chosen background ratio, which has been set as 20% here. The experimental threshold energy has been assumed to be negligible and the maximal cut-off energy is set as 100 keV. The background windows have been assumed to be the same as the experimental possible energy ranges. 5,000 experiments with 500 total events on average in each experiment have been simulated. See the text for further details.

The measured energy spectra (solid red histograms) shown in figures 1 are averaged

over the simulated experiments. Five bins with linearly increased bin widths have been used for binning generated signal and background events. As argued in ref. [7], for reconstructing the one-dimensional WIMP velocity distribution function, this unusual, particular binning

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 10 20 30 40 50 60 70 80 90 100 dR/dQ [x 10 -3 events/kg-day/keV] Q [keV] 76_{Ge, Q}

max < 100 keV, Qmax, bg < 100 keV, 500 events (20% constant bg), mχ = 10 GeV

AMIDAS http://pisrv0.pit.physik.uni-tuebingen.de/darkmatter/amidas/ (dR/dQ)signal (dR/dQ)background (dR/dQ)measured 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 10 20 30 40 50 60 70 80 90 100 dR/dQ [x 10 -4 events/kg-day/keV] Q [keV] 76_{Ge, Q}

(dR/dQ)signal (dR/dQ)background (dR/dQ)measured

Figure 2. As in figures 1, except that the constant background spectrum in eq. (2.13) has been used. See the text for further details.

has been chosen in order to accumulate more events in high energy ranges and thus to reduce the statistical uncertainties in high velocity ranges. However, as we will show later, for the determination of the WIMP mass, one needs either events in the first energy bin or all events in the whole data set. Hence, there is in practice no difference between using an equal bin width for all bins or the (linearly) increased bin widths.

Note here that, firstly, the possible energy ranges in which residue background events exist (the background windows) have been assumed to be the same as the entire experimental

possible energy ranges (e.g., between 0 and 100 keV for simulations shown in figures 1).

Secondly, the actual numbers of signal and background events in each simulated experiment are Poisson-distributed around their expectation values independently. This means that, for

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example, for simulations shown in figures 1 we generate 400 (100) events on average for

WIMP signals (backgrounds) and the total event number recorded in one experiment is then the sum of these two numbers. Thirdly, for the simulations demonstrated here as well as in the next section, we assumed that all experimental systematic uncertainties as well as the uncertainty on the measurement of the recoil energy could be ignored. The energy resolution of most existing detectors is so good that its error can be neglected compared to the statistical uncertainty for the foreseeable future with pretty few events.

In figures1it can be found that, as mentioned earlier, the shape of the WIMP scattering

spectrum depends highly on the WIMP mass: for light WIMPs (mχ <_{∼ 50 GeV), the recoil}

spectra drop sharply with increasing recoil energies, while for heavy WIMPs (mχ>_{∼ 100 GeV),}

the spectra become flatter. In contrast, the exponential background spectra shown here de-pend only on the target mass and are rather flatter (sharper ) for light (heavy) WIMP masses compared to the WIMP scattering spectra. This means that, once input WIMPs are light (heavy), background events would contribute relatively more to high (low ) energy ranges, and, consequently, the measured energy spectra would mimic scattering spectra induced by

heavier (lighter ) WIMPs.

As a comparison, in figures2we generate background events with the constant spectrum

given in eq. (2.13). It can be seen clearly that, since the background spectrum now is flatter for all WIMP masses, background events contribute always relatively more to high energy ranges, and, therefore, the measured energy spectra would always mimic scattering spectra induced by heavier WIMPs.

3 Reconstruction of the WIMP mass

In this section we first review the model-independent method for determining the WIMP mass

introduced in refs. [5,6]. Then we demonstrate some numerical results of the reconstructed

WIMP mass by using mixed data sets from WIMP signals and background events based on Monte Carlo simulations.

3.1 Model-independent determination of the WIMP mass

Here we review briefly the model-independent data analysis procedure for the determination of the WIMP mass by using two experimental data sets with different target nuclei. Detailed derivations and discussions can be found in refs. [5,6].

3.1.1 Basic expressions for determining the WIMP mass

In the earlier work [7], it was found that the normalized one-dimensional velocity distribution

function of incident WIMPs can be solved from eq. (2.1) directly and, consequently, its

generalized moments can be estimated by [6]

hvni(v(Qmin), v(Qmax)) =

Z v(Qmax) v(Qmin)

vnf1(v) dv

= αn

"

2Q(n+1)/2_min r(Qmin)/F2(Qmin)+(n + 1)In(Qmin, Qmax)

2Q1/2_minr(Qmin)/F2(Qmin)+I0(Qmin, Qmax)

# . (3.1)

Here v(Q) = α√Q, Q_(min,max) are the experimental minimal and maximal cut-off energies,

r(Qmin) ≡ dR

dQ

expt, Q=Qmin

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is an estimated value of the measured recoil spectrum (dR/dQ)expt (before the normalization

by the exposure E) at Q = Qmin, and In(Qmin, Qmax) can be estimated through the sum:

In(Qmin, Qmax) = X a Q(n−1)/2a F2_(Q a) , (3.3)

where the sum runs over all events in the data set that satisfy Qa∈ [Qmin, Qmax].

By requiring that the values of a given moment of f1(v) estimated by eq. (3.1) from two

experiments with different target nuclei, X and Y , agree, mχ appearing in the prefactor αn

on the right-hand side of eq. (3.1) has been solved as [5]: mχ|_hvn_i= √_m XmY − mX(Rn,X/Rn,Y) Rn,X/Rn,Y −pmX/mY , (3.4) where Rn,X ≡  

2Q(n+1)/2_min,X rX(Qmin,X)/F_X2(Qmin,X) + (n + 1)In,X

2Q1/2_min,XrX(Qmin,X)/F_X2(Qmin,X) + I0,X





1/n

, (3.5)

and Rn,Y can be defined analogously. Here n 6= 0, m(X,Y ) and F(X,Y )(Q) are the masses

and the form factors of the nucleus X and Y , respectively, and r_{(X,Y )}(Q_{min,(X,Y )}) refer to the counting rates for detectors X and Y at the respective lowest recoil energies included in the analysis. Note that, firstly, the general expression (3.4) can be used either for spin-independent or for spin-dependent scattering, one only needs to choose different form factors

under different assumptions. Secondly, the form factors in the estimate of In,X and In,Y

using eq. (3.3) are also different.

On the other hand, by using the theoretical prediction that the SI WIMP-nucleus cross section dominates, and the fact that the integral over the one-dimensional WIMP velocity distribution on the right-hand side of eq. (2.1) is the minus-first moment of this distribution, which can be estimated by eq. (3.1_{) with n = −1, one can easily find that [}6]

ρ0|fp|2 = π 4√2 mχ+ mN EA2√_m N " 2Q1/2_minr(Qmin) F2_(Q min) + I0 # . (3.6)

Note that the exposure of the experiment, E, appears in the denominator. Since the unknown factor ρ0|fp|2 on the left-hand side above is identical for different targets, it leads to a second

expression for determining mχ [6]:

mχ|_σ =

(mX/mY)5/2mY − mX(Rσ,X/Rσ,Y)

Rσ,X/Rσ,Y − (mX/mY)5/2

. (3.7)

Here m_{(X,Y )}_{∝ A}_{(X,Y )} has been assumed,

Rσ,X ≡ 1 EX   2Q1/2_min,XrX(Qmin,X) F_X2(Qmin,X) + I0,X  , (3.8)

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3.1.2 χ2-fitting

In order to yield the best-fit WIMP mass as well as to minimize its statistical uncertainty by

combining the estimators for different n in eq. (3.4) with each other and with the estimator

in eq. (3.7), a χ2 function has been introduced as [6] χ2(mχ) =

X

i,j

(fi,X− fi,Y) C_ij−1(fj,X− fj,Y) , (3.9)

where

fi,X ≡ αiX





2Q(i+1)/2_min,X rX(Qmin)/FX2(Qmin,X) + (i + 1)Ii,X

2Q1/2_min,XrX(Qmin)/FX2(Qmin,X) + I0,X

  1 300 km/s i , (3.10)

for i = −1, 1, 2, . . . , nmax, and

fnmax+1,X ≡ EX





A2 X

2Q1/2_min,XrX(Qmin)/F_X2(Qmin,X) + I0,X

  √ mX mχ+ mX ; (3.11)

the other nmax+ 2 functions fi,Y can be defined analogously. Here nmax determines the

highest moment of f1(v) that is included in the fit. The fi are normalized such that they

are dimensionless and very roughly of order unity in order to alleviate numerical problems

associated with the inversion of their covariance matrix. Note that the first nmax + 1 fit

functions depend on mχonly through the overall factor α and that mχin eqs. (3.10) and (3.11)

is now a fit parameter, which may differ from the true value of the WIMP mass. Finally, C in eq. (3.9) is the total covariance matrix. Since the X and Y quantities are statistically completely independent, C can be written as a sum of two terms:

Cij = cov (fi,X, fj,X) + cov (fi,Y, fj,Y) . (3.12)

3.1.3 Matching the cut-off energies

The basic requirement of the expressions for determining mχ given in eqs. (3.4) and (3.7) is

that, from two experiments with different target nuclei, the values of a given moment of the

WIMP velocity distribution estimated by eq. (3.1) should agree. This means that the upper

cuts on f1(v) in two data sets should be (approximately) equal.6 Since vcut = α√Qmax, it

requires that [6]

Qmax,Y = αX

αY

2

Qmax,X. (3.13)

Note that α defined in eq. (2.5) is a function of the true WIMP mass. Thus this relation for

matching optimal cut-off energies can be used only if mχis already known. One possibility to

overcome this problem is to fix the cut-off energy of the experiment with the heavier target, minimize the χ2(mχ) function defined in eq. (3.9), and then estimate the cut-off energy for

the lighter nucleus by eq. (3.13) algorithmically [6].

6

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10 20 30 50 70 100 200 300 500 700 1000 2000 10 20 30 50 70 100 200 300 500 700 1000 mχ ,rec [GeV] m_χ,in [GeV] 28

Si + 76Ge, Qmax < 100 keV, Qmax, bg < 100 keV, exponential bg, 2 x 50 events

No bg 10% bg 20% bg 40% bg

Figure 3. The reconstructed WIMP mass and the lower and upper bounds of the 1σ statistical uncertainty with mixed data sets from WIMP-induced and background events as functions of the input WIMP mass. 28

Si and 76

Ge have been chosen as two target nuclei. The background ratios shown here are no background (dashed green curves), 10% (long-dotted blue curves), 20% (solid red curves), and 40% (dash-dotted cyan curves) background events in the whole data sets in the experimental energy ranges between 0 and 100 keV. Each experiment contains 50 total events on average before cuts on Qmax for the experiments with the Si target; all of these events are treated as

WIMP signals. Other parameters are as in figures1. See the text for further details.

3.2 Reconstructing mχ by using data sets with background events

In this subsection we show some numerical results of the reconstruction of the WIMP mass with mixed data sets from WIMP-induced and background events by means of the model-independent method described in the previous subsection. The upper and lower bounds on

the reconstructed WIMP mass are estimated from the requirement that χ2 _{exceeds its}

mini-mum by 1.7 As in ref. [6],28Si and76Ge have been chosen as two target nuclei. The

scatter-ing cross section σ0 in eq. (2.2) has been assumed to be dominated by the spin-independent

WIMP-nucleus interaction. The experimental threshold energies of two experiments have been assumed to be negligible and the maximal cut-off energies are set the same as 100 keV. 2 × 5,000 experiments have been simulated. In order to avoid large contributions from very

few events in high energy ranges to the higher moments [7], only the moments up to nmax= 2

were included in the χ2 fit.

7

Note that, rather than the mean values, the (bounds on the) reconstructed WIMP mass are always the median values of the simulated results.

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3.2.1 With the exponential background spectrum

Figure 3 shows the reconstructed WIMP mass and the lower and upper bounds of the 1σ

statistical uncertainty with mixed data sets from WIMP-induced and background events as

functions of the input WIMP mass. As in figures 1, the exponential background spectrum

has been used and the background windows are set as the same as the experimental possible energy ranges, i.e., between 0 and 100 keV for both experiments. The background ratios shown here are no background (dashed green curves), 10% (long-dotted blue curves), 20% (solid red curves), and 40% (dash-dotted cyan curves) background events in the whole data

sets. Each experiment contains 50 total events on average before cuts on Qmax for the

experiments with the Si target. Remind that all events recorded in our data sets are treated as WIMP signals in the analysis, although statistically we know that a fraction of these events could be backgrounds.

It can be seen clearly that, for light WIMP masses (mχ <_{∼ 100 GeV), the larger the}

fraction of background events in the data sets, the heavier the reconstructed WIMP masses as well as the statistical uncertainty intervals. This is caused directly by the background

contribution to high energy ranges shown in figures 1. As discussed in section 2.3, the

background spectrum is relatively flatter compared to the scattering spectrum induced by

light WIMPs, and the energy spectrum of all recorded events would thus mimic a scattering

spectrum induced by WIMPs with a relatively heavier mass. Not surprisingly, the larger the background ratio, the more the background contribution to high energy ranges, and, consequently, the more strongly overestimated the reconstructed WIMP masses as well as the statistical uncertainty intervals.

In contrast, for heavy WIMP masses (mχ >_{∼ 100 GeV), figure} 3 does not show very

clearly but a tendency8 that the larger the fraction of background events, the lighter the

reconstructed WIMP masses as well as the statistical uncertainty intervals. This is now

caused by the background contribution to low energy ranges shown in figures1. As discussed

in the previous section, the background spectrum is relatively sharper compared to the scattering spectrum induced by heavy WIMPs, and the energy spectrum of all recorded events would thus mimic a scattering spectrum induced by WIMPs with a relatively lighter mass. Moreover, the larger the background ratio, the more the background contribution to low energy ranges, and, consequently, the more strongly underestimated the reconstructed WIMP masses as well as the statistical uncertainty intervals.

Nevertheless, from figure3_{it can be found that, with ∼ 20% residue background events}

in the analyzed data sets, the true values of the WIMP mass can still fall in the middle of the 1σ statistical uncertainty band and one could thus in principle reconstruct the WIMP mass pretty well; if WIMPs are light (mχ <_{∼ 200 GeV), the maximal acceptable fraction of residue}

background events could even be as large as ∼ 40%. For a WIMP mass of 100 GeV with 20% background events in the data sets, the reconstructed WIMP mass and the statistical uncertainty are ∼ 97 GeV+61%_−35%, compared to ∼ 94 GeV+55%_−33%for background-free data sets; for a lighter WIMP mass of 50 GeV, the reconstructed WIMP mass and the statistical uncertainty

change from ∼ 48 GeV+41%_−29% (background-free), to ∼ 54 GeV+44%_−30% (20% background), and

∼ 61 GeV+48%_−32% (40% background).

On the other hand, considering different efficiencies of discrimination ability against

different background sources in different energy ranges in different experiments, in figures 4

8

Since for heavy input WIMP masses the reconstructed values are systematically underestimated, probably due to the statistical fluctuation with pretty few (∼ 50) events discussed later.

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10 20 30 50 70 100 200 300 500 700 1000 2000 10 20 30 50 70 100 200 300 500 700 1000 mχ ,rec [GeV] m_χ,in [GeV] 28

Si + 76Ge, Q_max < 100 keV, Q_{max, bg} < 50 keV, exponential bg, 2 x 50 events

AMIDAS http://pisrv0.pit.physik.uni-tuebingen.de/darkmatter/amidas/ No bg 20% bg 40% bg 10 20 30 50 70 100 200 300 500 700 1000 2000 10 20 30 50 70 100 200 300 500 700 1000 mχ ,rec [GeV] m_χ_,in [GeV] 28

Si + 76Ge, Q_max < 100 keV, Q_{min, bg} > 50 keV, Q_{max, bg} < 100 keV, exponential bg, 2 x 50 events

No bg 5% bg 10% bg

Figure 4. As in figure 3, except that the background window in each experiment have been set as 0–50 keV (upper) and 50–100 keV (lower). Note that the background ratios shown here are 20% (solid red curves) and 40% (dash-dotted cyan curves) in the upper frame, whereas 5% (dotted magenta curves) and 10% (long-dotted blue curves) in the lower frame.

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we shrink the background window in each experiment to a relatively lower range between 0

and 50 keV (upper) and a relatively higher range between 50 and 100 keV (lower).9 Since our

background spectrum is exponential, for the case shown in figure3, only very few background

events could be observed in the energy range between 50 and 100 keV. Hence, for the case with the background window only in the low energy range, not surprisingly, the results of the

reconstructed WIMP mass shown in the upper frame of figures4should not differ very much

from those shown in figure 3. However, due to the little bit more contribution to the low

energy range from background events, all the reconstructed WIMP masses shown here are

somehow lighter than those shown in figure3_{. Hence, with ∼ 20% residue background events}

in low experimental possible energy ranges, one could in principle reconstruct the WIMP

mass with a 1σ statistical uncertainty as ∼ 94 GeV+59%_−34% (for a WIMP mass of 100 GeV) or

∼ 52 GeV+44%_−30% (for a WIMP mass of 50 GeV).

In contrast, since the WIMP scattering spectrum is in principle approximately expo-nential and thus only (very) few WIMP-induced events could be observed in high energy ranges, if we have background windows in only high experimental possible energy ranges, the (pretty large) contributions from background events could cause (strong) overestimates of

the reconstructed WIMP masses. It is even worse for large WIMP masses (mχ >_{∼ 100 GeV).}10

Nevertheless, as shown in the lower frame of figures 4_{, with ∼ 5% residue background events}

observed only in high energy ranges, one could in principle still estimate the WIMP mass

with a 1σ statistical uncertainty as ∼ 107 GeV+56%_−33% (for an input WIMP mass of 100 GeV)

or ∼ 58 GeV+47%_−32% (for an input WIMP mass of 50 GeV).

Our results shown in figures 4 indicate that a small fraction of background events in

low energy ranges might not affect the reconstructed WIMP mass significantly. However,

the WIMP mass could be (strongly) overestimated once the same (or even smaller) amount of background events exists in high energy ranges. In practice one simple way to reduce the overestimate induced by an excess of background events in high energy ranges might be checking the shape of measured recoil spectrum. However, considering some suggested modifications of the standard shifted Maxwellian velocity distribution, e.g., contributions from discrete “streams” with (nearly) fixed velocities [22–24] or the “late infall” component in the velocity distribution with a velocity v ∼ vesc [22, 24, 25], it should at least be very

careful to reject any recoil event observed in high energy ranges artificially.

In figure 5 we rise the expected number of total events in each experiment by a factor

of 10, to 500 events on average before cuts for the case that residue background events exist in the entire experimental possible energy ranges. As shown here, all statistical uncertainties shrink by a factor >_{∼ 3 compared to the results shown in figure}3. In addition, the underes-timate of the reconstructed values of heavy input WIMP masses caused perhaps by the use of pretty few (∼ 50) events has been reduced with larger data sets; and, the tendency of the

underestimate of the reconstructed WIMP mass for heavy WIMP masses (mχ >_{∼ 100 GeV)}

becomes more clearly. Finally, figure5 shows that, for the determination of the WIMP mass

by using data sets of O(500) total events, the maximal acceptable background ratio could be ∼ 10% (i.e., O(50) background events) or even ∼ 20%, if WIMPs have a mass of O(100 GeV).

9

Note that here we do not mean that in other energy ranges background events do not exist; in contrast, we want to study what could happen once our background discrimination, caused by some natural or even artificial reasons, are worse in these energy ranges than others and more background events could thus survive.

10

Note that the plateau of the lower bound of the statistical uncertainty in the case of a 10% background ratio for heavy WIMP masses (mχ >

∼ 300 GeV) should be caused by our setup for the upper cut-off of the reconstructed WIMP mass of 3000 GeV in the simulations.

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10 20 30 50 70 100 200 300 500 700 1000 2000 10 20 30 50 70 100 200 300 500 700 1000 mχ ,rec [GeV] m_χ_,in [GeV] 28_{Si +}76_{Ge, Q}

max < 100 keV, Qmax, bg < 100 keV, exponential bg, 2 x 500 events

No bg 10% bg 20% bg

Figure 5. As in figure 3, except that the expected number of total events in each experiment has been set as 500.

3.2.2 Statistical fluctuation

As discussed in ref. [6], the statistical fluctuation of the reconstructed WIMP mass by the

algorithmic procedure in the simulated experiments seems to be pretty problematic, in par-ticular for heavier input WIMP masses. Moreover, as mentioned in the previous subsection, with only ∼ 50 total events in each experiment, the tendency of the underestimate of the

reconstructed WIMP mass for heavier WIMP masses (mχ >_{∼ 100 GeV) seems not to be very}

clear. Hence, as done in ref. [6], in order to study the statistical fluctuation of the recon-structed WIMP mass with different background ratios in our data sets, we consider in this subsection the estimator δm introduced in ref. [6]:

δm =                                  1 + mχ,lo1− mχ,in mχ,lo1− mχ,lo2 , if mχ,in ≤ mχ,lo1; mχ,rec− mχ,in mχ,rec− mχ,lo1

, if mχ,lo1< mχ,in< mχ,rec;

mχ,rec− mχ,in

mχ,hi1− mχ,rec

, if mχ,rec< mχ,in< mχ,hi1;

mχ,hi1− mχ,in

mχ,hi2− mχ,hi1 − 1 ,

if mχ,in ≥ mχ,hi1.

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Here mχ,in is the true (input) WIMP mass, mχ,rec its reconstructed value, mχ,lo1(2) are the

1 (2) σ lower bounds satisfying χ2(m_χ,lo(1,2)) = χ2(mχ,rec) + 1 (4), and mχ,hi1(2) are the

corresponding 1 (2) σ upper bounds.

The estimator δm defined above indicates basically the strength of the deviation of the reconstructed WIMP mass from the true (input) value. If the reconstructed 1σ lower and

upper bounds on the WIMP mass in one simulated experiment cover the true value: mχ,lo1≤

mχ,in≤ mχ,hi1, δm is determined as the deviation of the “reconstructed WIMP mass” from

the true one in units of the difference between the reconstructed value and the 1σ lower (upper) bound, once the reconstructed value is overestimated (underestimated). However, if the true WIMP mass lies outside of the experimental 1σ bounds (the reconstructed value is more strongly over-/underestimated), δm is determined as the deviation of the “1σ lower (upper) bound” from the true WIMP mass in units of the difference between the 1σ and 2σ lower (upper) bounds. Note that, it has been found in ref. [6] as well as in the results presented in the previous subsection that the uncertainty intervals of the median reconstructed WIMP mass are quite asymmetric; similarly, the distance between the 1σ and 2σ bounds can be

quite different from the distance between the reconstructed value and the 1σ bound [6]. The

definition of δm in eq. (3.14) takes these differences into account, and also keeps track of the sign of the deviation: if the reconstructed WIMP mass is overestimated (underestimated), δm is positive (negative). Moreover, |δm| ≤ 1 (2) if and only if the true WIMP mass lies between the experimental 1 (2) σ bounds.

In figures6we show the normalized distributions of the estimator δm defined in eq. (3.14) for a rather heavy input WIMP mass of 200 GeV with 50 (upper) and 500 (lower) total events on average before cuts in each experiment. As discussed in ref. [6], the deviation of the re-constructed WIMP mass in the simulated experiments looks asymmetric and non-Gaussian. However, it can be seen here clearly that, the more the background events in our analyzed data sets, the more concentrated the δm value in the range between −1 and 0 as well as between 0 and +1. Moreover, for the case with rather larger data sets of 500 total events, by increas-ing the background ratio the distribution becomes to be more symmetric and Gaussian-like, although the central value of δm seems to fall at ∼ −0.5 because of the underestimate of the reconstructed WIMP mass.

In ref. [6] it has been mentioned that with increasing number of total events the

distribu-tion of the estimator δm becomes slowly Gaussian. Figures 6here (and figure9 shown later

also) indicate that with a larger background ratio in the analyzed data sets the distribution of δm approaches to be Gaussian more fast. This interesting observation might be able to offer some new ideas for improving the algorithmic procedure for the reconstruction of the WIMP mass with a higher statistical certainty.

3.2.3 With the constant background spectrum

In order to check the need of a prior knowledge about an (exact) form of the residue back-ground spectrum, we consider briefly in this subsection a rather extrem case, i.e., the constant background spectrum in eq. (2.13).

In figure7 we show the reconstructed WIMP mass and the lower and upper bounds of

the 1σ statistical uncertainty with mixed data sets as functions of the input WIMP mass.

As in figures 2, the windows of the constant background spectrum are set as the same as

the experimental possible energy ranges, i.e., between 0 and 100 keV for both experiments. The background ratios shown here are no background (dashed green curves), 5% (dotted magenta curves), 10% (long-dotted blue curves) background events in the whole data sets.

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0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 -5 -4 -3 -2 -1 0 1 2 3 4 5

Normalized number of experiments

δm

28

Si + 76Ge, Q_max < 100 keV, Q_{max, bg} < 100 keV, exponential bg, 2 x 50 events, m_χ = 200 GeV

AMIDAS http://pisrv0.pit.physik.uni-tuebingen.de/darkmatter/amidas/ No bg 10% bg 20% bg 40% bg 0 0.01 0.02 0.03 0.04 0.05 0.06 -5 -4 -3 -2 -1 0 1 2 3 4 5

δm

28

Si + 76Ge, Q_max < 100 keV, Q_{max, bg} < 100 keV, exponential bg, 2 x 500 events, m_χ = 200 GeV

No bg 10% bg 20% bg 40% bg

Figure 6. Normalized distributions of the estimator δm defined in eq. (3.14) for an input WIMP mass of 200 GeV with 50 (upper) and 500 (lower) total events on average before cuts in each experiment. Parameters and notations are as in figure 3. Note that the bins at δm = ±5 are overflow bins, i.e., they also contain all experiments with |δm| > 5. See the text for further details.

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max < 100 keV, Qmax, bg < 100 keV, constant bg, 2 x 50 events

No bg 5% bg 10% bg

Figure 7. As in figure3, except that the constant background spectrum in eq. (2.13) has been used. Note that the background ratios shown here are 5% (dotted magenta curves) and 10% (long-dotted blue curves).

Each experiment contains again 50 total events on average before cuts; all of these events are treated as WIMP signals in the analysis.

It can be seen clearly that, as discussed above, since the constant background spec-trum has relatively flatter shape compared to the WIMP scattering specspec-trum for not only light, but also heavy WIMP masses, and the measured energy spectrum should thus always mimic a scattering spectrum induced by heavier WIMPs, the reconstructed WIMP masses are therefore overestimated for all input WIMP masses, especially for the heavier masses.

Actually, the result shown here looks more likely that shown in the lower frame of figures4,

since in both cases residue background events contribute significantly more (compared to the exponential-like WIMP scattering spectrum) in high energy ranges. Not surprisingly, the larger the background ratio, the more strongly overestimated the reconstructed WIMP masses, in particular for the heavier input WIMP masses. Nevertheless, for (approximately) constant residue backgrounds with a fraction of ∼ 5% in background windows as the en-tire experimental possible ranges, one could in principle still estimate the WIMP mass with

a 1σ statistical uncertainty as ∼ 117 GeV+64%_−35% (for 100 GeV WIMPs) or ∼ 56 GeV+49%_−33%

(for 50 GeV WIMPs). Once WIMPs are light (mχ ∼ O(25 GeV)), the maximal acceptable

background ratio could even be ∼ 10%.

Moreover, as done in section 3.2.1, in figure8we shrink the background window in each

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No bg 5% bg 10% bg

Figure 8. As in figure 7, except that the background window in each experiment has been set as 0–50 keV.

WIMPs (mχ <_{∼ 70 GeV),}11relatively more background events still contribute to high energy

ranges; for heavy WIMPs (mχ >_{∼ 70 GeV), relatively more background events contribute}

now to low energy ranges and, consequently, the reconstructed WIMP masses are therefore

underestimated for heavy WIMPs.

On the other hand, as in section 3.2.2, in figure9we check the normalized distributions of the estimator δm for an input WIMP mass of 200 GeV with 50 total events on average before cuts in each experiment. It can be seen very clearly that, with increasing background ratio the value of δm concentrates more and more strongly to 2. This means that, due to the contri-bution from residue background events, the reconstructed WIMP mass is most possibly ∼ 2σ overestimated. Moreover, compared to the non-Gaussian form of the distributions for the case

with the exponential background spectrum shown in the upper frame of figures6, the

distribu-tions with the constant spectrum look more likely Gaussian, despite of the asymmetry due to

the overestimate of the WIMP mass. Nevertheless, figures 6and figure9indicate that

back-ground events seem to let the distribution of the deviation of the reconstructed WIMP mass be more symmetric and Gaussian, no matter what kind of energy spectrum they would have.

Finally, in figure10we rise the expected number of total events in each experiment by a

factor of 10, to 500 events on average before cuts for the case that residue background events exist in the entire experimental possible energy ranges. Note that the background ratios

11

Remind that the actual value of this “critical” WIMP mass depends in practice strongly on the WIMP scattering spectrum as well as on the residue background spectrum and therefore differs from experiment to experiment.

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0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 -5 -4 -3 -2 -1 0 1 2 3 4 5

δm

28_{Si +}76_{Ge, Q}

max < 100 keV, Qmax, bg < 100 keV, constant bg, 2 x 50 events, mχ = 200 GeV

No bg 10% bg 20% bg

Figure 9. As in the upper frame of figures 6, except that the constant background spectrum in eq. (2.13) has been used.

shown here are no background (dashed green curves), 1% (long-dotted blue curves), and 2% (solid red curves), i.e., a factor of 10 smaller than the ratios used before. In the lower frame

of figures 4 and in figure 7_{, we found that once ∼ 5%–10% events in our analyzed data sets}

are residue backgrounds and (most of) these events are recorded in high energy ranges, no matter what kind of spectrum shape they would have, the reconstructed WIMP mass could

be (strongly) overestimated. However, figure 5 and figure 10 here show that, by increasing

the event number and decreasing the background ratio, one could in principle determine the WIMP mass (pretty) precisely without knowing the (exact) form of the spectrum of residue background events.

4 Summary and conclusions

In this paper we reexamine the model-independent data analysis method introduced in

refs. [5, 6] for the determination of the mass of Weakly Interacting Massive Particles from

data (measured recoil energies) of direct Dark Matter detection experiments directly by tak-ing into account a fraction of residue background events, which pass all discrimination criteria and then mix with other real WIMP-induced events in the analyzed data sets. Differ from the maximum likelihood analysis described in refs. [13,14,21], our method requires neither prior knowledge about the WIMP scattering spectrum nor about different possible back-ground spectra; the unique needed information is the recoil energies recorded in two direct detection experiments with two different target nuclei.

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No bg 1% bg 2% bg

Figure 10. As in figure 7, except that the expected number of total events in both experiments has been set as 500. Note that the background ratios shown here are no background (dashed green curves), 1% (long-dotted blue curves), and 2% (solid red curves), i.e., a factor of 10 smaller than the ratios used before. See the text for further details.

In section 2 we considered first the measured energy spectrum for different WIMP masses with two forms of possible residue background spectrum: the target-dependent exponential spectrum and the constant spectrum. The exponential background spectrum contributes

relatively more events to high energy ranges once WIMPs are light (mχ <_{∼ 100 GeV), and}

to low energy ranges for heavy WIMP masses (mχ >_{∼ 100 GeV); whereas the constant}

back-ground spectrum contributes always relatively more events to high energy ranges. As the consequence, the energy spectrum of all observed events looks more likely to be a scattering spectrum induced by heavier WIMPs, once the spectrum of residue background events (in-duced perhaps by two or more different sources) is either exponential-like (and WIMPs are light) or approximately constant (for all WIMP masses); while if WIMPs are heavy and the residue background spectrum is approximately exponential, the measured energy spectrum would look more likely to be a scattering spectrum induced by lighter WIMPs.

In section 3.2 the data sets generated in section 2 have been analyzed for reconstructing the mass of incident WIMPs by using the model-independent method. With the expo-nential background spectrum, the input WIMP mass would be overestimated once WIMPs

are light (mχ <_{∼ 100 GeV), or, in contrast, would be underestimated for heavy WIMPs}

(mχ >_{∼ 100 GeV). Our simulations show that, for background windows in the entire or low}

experimental possible energy ranges, one could in principle reconstruct the WIMP mass with a maximal fraction of ∼ 20% of residue background events in the analyzed data sets; whereas

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for background windows in high energy ranges, the maximal acceptable fraction of residue backgrounds is only ∼ 10%.

Moreover, in order to check the need of a prior knowledge about an (exact) form of the residue background spectrum, we considered also the case with the constant background spectrum. In this rather extrem case, the WIMP mass would always be overestimated, especially for heavy WIMPs (mχ>_{∼ 100 GeV). Our simulations give then a maximal acceptable}

fraction of ∼ 5%–10% of residue background events in the data sets for background windows in the entire or low experimental possible energy ranges. Nevertheless, we found also that, by means of increased number of observed (WIMP-induced) events and improved background

discrimination techniques [9,11], the WIMP mass could in principle be determined (pretty)

precisely, no matter what kind of energy spectrum residue background events would have. On the other hand, in order to check the statistical fluctuation of the reconstructed WIMP mass with increased background ratio, we considered also the distribution of the

deviation of the reconstructed WIMP mass from the true value. It was found in ref. [6]

that, for a rather heavy WIMP mass of 200 GeV, the distribution of the deviation of the reconstructed WIMP mass is asymmetric and non-Gaussian, either with data sets of only a few (O(50)) events or with larger date sets (of O(500) events). However, our simulations with different background ratios show that, firstly, for both used (exponential and constant) background spectra, with increasing background ratio the distribution of the deviation of the reconstructed WIMP mass becomes more and more concentrated, although still asymmetric and non-Gaussian. Secondly, for the more realistic exponential background spectrum and using data sets with a larger number of total events, with increasing background ratio the distribution of the deviation becomes somehow more symmetric and Gaussian. This obser-vation might be able to offer some new ideas for improving the algorithmic procedure for the reconstruction of the WIMP mass with a higher statistical certainty.

In summary, our study of the effects of residue background events in direct Dark Matter detection experiments on the determination of the WIMP mass shows that, with currently running and projected experiments using detectors with 10−9to 10−11pb sensitivities [10,26–

28] and < 10−6 _{background rejection ability [}₈_,₉_,₁₁_,₁₂_{], once two or more experiments with}

different target nuclei could accumulate a few tens events (in one experiment), we could in principle already estimate the mass of Dark Matter particle with a reasonable precision, even

though there might be some background events mixed in our data sets for the analysis.12

Moreover, two forms for background spectrum and three windows for residue background events considered in this work are rather naive. Nevertheless, one should be able to extend our observations/discussions to predict the effects of possible background events in their own experiment. Hopefully, this will encourage our experimental colleagues to present their (future) results not only in form of the “exclusion limit(s)”, but also of the “most possible area(s)” on the cross section versus mass plan.

Acknowledgments

The authors would like to thank the Physikalisches Institut der Universität Tübingen for the technical support of the computational work demonstrated in this article. CLS would also like to thank the friendly hospitality of the Max-Planck-Institut für Kernphysik in Heidelberg where part of this work was completed. This work was partially supported by the National

12

A possible first test could be a combination of the events observed by the CoGeNT experiment with their Ge detector with the events observed in the oxygen band of the CRESST-II experiment [29–31].

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Science Council of R.O.C. under contracts no. NSC-96-2112-N-009-023-MY3 and no. NSC-98-2811-M-006-044 as well as by the LHC Physics Focus Group, National Center of Theoretical Sciences, R.O.C..

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