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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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2Determining 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

2.2Experimental 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)

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where

(Q − Qn)λ|n≡

1

Nn

Nn

?

i=1

(Qn,i− Qn)λ.(21)

Finally, Qs,n is the shifted point at which the leading systematic error due to the ansatz in

Eq.(17) is minimal [14],

?sinh(knbn/2)

knbn/2

Qs,n= Qn+1

knln

?

. (22)

Note that Qs,ndiffers from the central point of the n−th bin, Qn.

Using Eqs.(12) and (13) with Q1= Qminand Q2= Qmax, the generalized n−th moment of

the velocity distribution function can be written as

2Q1/2

where v(Q) = α√Q. Here we have implicitly assumed that Qmaxis so large that terms ∝ r(Qmax)

are negligible. We will see later that this is not necessarily true for n ≥ 1, since these moments

receive sizable contributions from large recoil energies [14]. Nevertheless we will show in Sec. 3

that even in that case, Eq.(23) can still be used for determining mχ. From the ansatz Eq.(17),

the counting rate at Qmincan be expressed as

?vn?(v(Qmin),v(Qmax)) = αn

2Q(n+1)/2

min

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

minr(Qmin)/F2(Qmin) + I0(Qmin,Qmax)

, (23)

r(Qmin) = r1ek1(Qmin−Qs,1).(24)

The integral In(Qmin,Qmax) defined in Eq.(14) can be estimated through the sum:

In(Qmin,Qmax) =

?

a

Q(n−1)/2

a

F2(Qa),(25)

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

Since all Inare determined from the same data, they are correlated, with [14]

cov(In,Im) =

?

a

Q(n+m−2)/2

a

F4(Qa)

,(26)

where the sum again runs over all events with recoil energy between Qminand Qmax.

On the other hand, the statistical error of r(Qmin) can be obtained from Eq.(24) as

The error on r1follows directly from its definition in Eq.(18):

σ2(r(Qmin)) = r2(Qmin)

σ2(r1)

r2

1

+

?1

k1

−

?b1

2

??

1 + coth

?b1k1

2

???2

σ2(k1)

.(27)

σ2(rn) =Nn

b2

n

.(28)

The error on the logarithmic slope k1can be computed from Eq.(20):

sinh(knbn/2)

σ2(kn) = k2

n

1 −

?

knbn/2

?2

−1

σ2?

Q − Qn|n

?

, (29)

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with

σ2?

Q − Qn|n

?

=

1

Nn− 1

?

(Q − Qn)2|n− Q − Qn|2

n

?

. (30)

Finally, the correlation between the errors on r(Qmin), which is calculated entirely from the

events in the first bin, and on Inis given by [14]

cov(r(Qmin),In)

= r(Qmin)In(Qmin,Qmin+ b1)

?σ2(r1)

r2

1

×

+

?1

k1

−

?b1

2

??

1 + coth

?b1k1

− Q1+1

2

???

×

?In+2(Qmin,Qmin+ b1)

In(Qmin,Qmin+ b1)k1

−

?b1

2

?

coth

?b1k1

2

??

σ2(k1)

?

;(31)

note that the integrals Iiin Eq.(31) only extend over the first bin, which ends at Q = Qmin+b1.

3 Determining the WIMP mass

We are now ready to describe methods to extract the WIMP mass mχfrom direct detection

data. Recall that mχ is an input in the determination of f1 and its moments as described

in the previous section. In particular, mχappears in the factor αnon the right–hand side of

Eq.(23) describing the experimental estimate of the moments of f1. A truly model–independent

determination of mχfrom these data will therefore only be possible by requiring that two (or

more) experiments, using different target nuclei, lead to the same result for f1.

Here we focus on the moments of the distribution function, rather than the function itself,

since non–trivial information about the former can already be obtained with O(20) events [14].

We will also show how mχcan be estimated from the knowledge of the integral I0appearing in

the normalization of f1, if the ratio of WIMP scattering cross sections on protons and neutrons

is known. For greater clarity, we will first discuss the idealized situation where all signal events,

irrespective of their recoil energy Qa, are included. In the second subsection we will introduce

upper and lower limits on Qa. A third subsection is devoted to a discussion how the different

estimators for the WIMP mass can be combined using a χ2fit.

3.1Without cuts on the recoil energy

If no cuts on the recoil energy need to be applied, we can use the original moments, or equiva-

lently, the generalized moments with v1→ 0 and v2→ vesc; recall that the latter is the same as

allowing v2→ ∞, since by assumption f1(v) = 0 for v > vesc. Eq.(23) then simplifies to

?In

I0

where Inand I0can be estimated as in Eq.(25) with Qmin→ 0 and Qmax→ ∞ (or, equivalently,

Qmax→ Qmax,kin).

Suppose X and Y are two target nuclei. We denote their masses by mX, mY, and their form

factors as FX(Q), FY(Q). Similarly, we define αX,Y as in Eq.(5), with mN → mX,Y. Eq.(32)

then implies

?In,X

I0,X

I0,Y

?vn? = αn(n + 1)

?

,(32)

αn

X

?

= αn

Y

?In,Y

?

.(33)

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Note that the form factors in the estimates of In,Xand In,Y using Eq.(25) are different. Eq.(33)

can be solved for mχusing Eqs.(5) and (3):

√mXmY− mXRn

Rn−

where we have defined

?In,X

I0,X

In,Y

mχ=

?

mX/mY

, (34)

Rn≡αY

αX

=

·I0,Y

?1/n

,n ?= 0, − 1.(35)

Using standard Gaussian error propagation, the statistical error on mχ estimated from

Eq.(34) is

?

?

=

?

×

|n|

where σ2(In,X) = cov(In,X,In,X) and cov(I0,X,In,X) and so on can be estimated from Eq.(26).

A second method for determining mχstarts directly from Eq.(1), plus an assumption about

the relative strength for WIMP scattering on protons p and neutrons n. The simplest such

assumption is that the scattering cross section is the same for both nucleons. This is in fact

an excellent approximation for the spin–independent contribution to the cross section of super-

symmetric neutralinos [6], and for all WIMPs which interact primarily through Higgs exchange.

Writing the “pointlike” cross section σ0of Eq.(2) as

?4

π

σ(mχ)|?vn?=

mX/mY|mX− mY|

Rn−

Rn

Rn−

1

?

mX/mY

?2

· σ(Rn)

?

mX/mY|mX− mY|

?

?σ2(In,X)

I2

mX/mY

?2

n,X

+σ2(I0,X)

I2

0,X

−2cov(I0,X,In,X)

I0,XIn,X

+ (X −→ Y )

?1/2

, (36)

σ0=

?

m2

r,NA2|fp|2,(37)

where fpis the effective χχpp 4–point coupling, mr,Nis the reduced mass defined in Eq.(3) and

A is the number of nucleons in the nucleus, we have from Eqs.(1), (2) and the first Eq.(12):

?4

π

r(Qmin) =

ρ0

2mχ

?

A2|fp|2F2(Qmin)?v−1?(v(Qmin),vesc).(38)

Using the second Eq.(12) we see that the counting rate at Qminin fact drops out, and we are

left with

1 =4√2

πI0

?Eρ0A2|fp|2

??

mr,N

mχ√mN

?

,(39)

where we have assumed Qmin= 0. Recall that the rate dR/dQ is defined as rate per unit mass

and observation time interval, i.e., we need to divide the actual event rate by the exposure

E = Mτ, where M is the (fiducial) mass of the detector and τ the observation time. In our

previous discussion this factor always dropped out in the end, due to the appearance of the

normalization N. This is true also for the right–hand side of Eq.(38), but not for the 1/E factor

on the left–hand side of this equation; hence a factor E appears in the numerator of Eq.(39).

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Note that E is dimensionless in natural units. On the other hand, the unknown factor ρ0|fp|2

appearing in Eq.(39) will cancel out when we use this identity for two different targets X and

Y , leading to the final result

mχ=(mX/mY)5/2mY− mXRσ

Rσ− (mX/mY)5/2

Here we have assumed mX,Y ∝ AX,Y, and introduced the quantity

Rσ=EY

EX

Recall that, even though the derivation started from the expression for the counting rate at

Qmin, this rate actually dropped out when going from Eq.(38) to Eq.(39). The final expression

only depends on the quantity Rσ, which is estimated from all the events in both samples using

Eq.(25). The error on mχcomputed from Eq.(40) is therefore comparable to that for mχderived

from a moment of f1. Still keeping Qmin= 0, we have

?σ2(I0,X)

. (40)

?I0,X

I0,Y

?

. (41)

σ(mχ)|σ=Rσ(mX/mY)5/2|mX− mY|

?

Rσ− (mX/mY)5/2?2

I2

0,X

+σ2(I0,Y)

I2

0,Y

?1/2

. (42)

Ultimately the Rn, Rσand the errors in Eqs.(36) and (42) should be estimated from the data

directly. In the meantime it is instructive to note that the final expressions for the statistical

errors of our estimators for mχdecompose into two factors. The expectation value of the first

factor does not depend on f1; it can be computed entirely from the masses of the involved

particles:

?

?

=(mχ+ mX)(mχ+ mY)

|mX− mY|

≡ κ.

Here we have made use of the identities Rn= αY/αXin Eq.(35) and

?mX

mY

mχ+ mX

Rn

mX/mY|mX− mY|

Rn−

?

mX/mY

?2

=Rσ(mX/mY)5/2|mX− mY|

?

Rσ− (mX/mY)5/2?2

(43)

Rσ=

?5/2?mχ+ mY

?

,(44)

which hold for the expectation values of these quantities as can be seen from Eqs.(34) and (40).

Remarkably, the expectation values of the factors in front of the expressions in square brackets

are in fact the same in Eqs.(36) and (42), apart from the factor 1/|n| appearing in the former

equation. On the other hand, the expressions inside these square parentheses do depend on

f1(v), as well as on the involved masses and form factors.

It is nevertheless instructive to study the behavior of the factor κ for different target nuclei X

and Y . This largely determines what choice of targets is optimal, i.e., minimizes the statistical

errors of our estimators of mχ. It is easy to see that the ratio κ/mχ, which will appear in the

relative error σ(mχ)/mχ, only depends on the dimensionless ratios mX/mχ and mY/mχ. In

Fig. 1 we therefore show contours of κ/mχin the plane spanned by these two ratios, using the

ordering mY > mX. We note first of all that κ diverges as mY → mX. This is clear from the

fact that the denominator in the final expression in Eq.(43) vanishes in this limit, while the

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0.01 0.11 10

mX / mχ

0.01

0.1

1

10

mY / mχ

Contours of constant κ/mχ

κ/mχ = 2

4

8

16

32

mY / mX = 76/28

Figure 1: The five thick lines show contours of constant κ/mχin the plane spanned by mX/mχ

and mY/mχ, where we have taken mY > mXwithout loss of generality. Here κ, has been defined

in Eq.(43); recall that the final relative error on mχis directly proportional to κ/mχ. The lower

thin line indicates the end of the physical region, mY = mX, whereas the upper thin line shows

mY = (76/28)mX, corresponding to Silicon and Germanium targets.

numerator is finite. Physically this simply means that performing two scattering experiments

with the same target will not allow one to determine mχ.

Slightly less trivially, we also see that κ/mχ, and hence the relative error on mχ, becomes

very large if both target nuclei are either much heavier or both much lighter than the WIMP.

This can be understood from the fact that mχ only enters via the reduced mass defined in

Eq.(3).∗If mχ≫ mN, mr,N→ mNbecomes completely independent of the WIMP mass. In

the opposite extreme, mχ≪ mN, mr,N→ mχbecomes independent of the mass of the target

nucleus, i.e., one is effectively back in the situation where both experiments are performed with

the same target.

These considerations favor chosing the two targets to be as different as possible. However,

there are limits to this. On the one hand, taking a very light target nucleus will lead to a low

event rate for this experiment, and hence very large statistical errors. Since the errors of both

experiments are added in quadrature inside the expressions in square parenthesis in Eqs.(36)

and (42), this would lead to a large overall error on mχ. In fact, if the total event number is

∗Eq.(38) seems to have additional mχdependence. However, this comes from the factor nχ= ρ0/mχ, which

drops out when the ratio of two targets is considered.

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held fixed, the final error on mχwill be minimal if both experiments contain approximately the

same number of events.

On the other hand, taking a very heavy target nucleus also leads to problems. Heavy nuclei

are large, which means they have quite soft form factors.

prediction [16] for the form factor for136Xe, which is the target in some existing experiments

[15], has a zero at Q ≃ 95 keV. For our default choice vesc= 700 km/s, this is below Qmax,kinof

Eq.(8) for all mχ> 45 GeV. This is a serious problem, since the form factor, and hence the event

rate, remain (very) small even beyond this zero. Experiments with136Xe therefore effectively

impose a cut Qmax≤ 95 GeV.†

From these considerations it seems that chosing X = Si, Y = Ge might be a good compromise.

This corresponds to points on the upper, thin solid straight line in Fig. 1. We see that in this

case κ/mχ≥ 4 for all values of mχ. Not surprisingly, κ/mχreaches its minimum when mχlies in

between the masses of the two target nuclei, i.e for the case at hand, for mχ≃ 50 GeV. Note also

that κ/mχis unfortunately always well above unity. One will thus have to get fairly accurate

estimates of the relevant integrals Iiif one wishes to determine mχto better than a factor of

two.

Before discussing the statistical uncertainty in more detail, we proceed to include non–trivial

upper and lower cuts on the recoil energy in our analysis.

For example, the Woods–Saxon

3.2Incorporating cuts on the recoil energy

As noted earlier, real experiments will probably have to impose both lower and upper limits

on the recoil energy, partly for instrumental reasons, and partly to suppress backgrounds. This

led us to introduce generalized moments of f1in Eq.(12). These moments determined by two

different detectors will still be the same, if either the integration limits are identical, vi,X= vi,Y,

i = 1, 2, or the cuts are so loose that the contribution from WIMPs with v < v1or v > v2is

negligible; a combination of these two possibilities can also occur, e.g., small but not necessarily

equal values of v1,Xand v1,Y, and v2,X= v2,Y significantly below vesc.

The crucial observation that forms the basis for our analysis of mχas estimated from the

generalized moments is that in fact all three terms in the final expression in Eq.(12) will agree

separately between two targets, as long as both integration limits coincide. This can be shown

by replacing the r(Qi) in Eq.(12) via Eq.(1) and using Eq.(15). In principle we could therefore

simply equate the last (integral) terms between the two targets. This would lead to expressions

very similar to those derived in the previous subsection, the only difference being that all integrals

or sums over recoil energies would run over limited ranges.

The problem with this approach is that the velocities viappearing in Eq.(12) are not directly

observable. The recoil energies are. However, the values of Qminand Qmaxwould need to be

different for the two targets, if they are to correspond to the same values of v1and v2. Worse,

Eq.(15) shows that the ratios Qmin,X/Qmin,Y and Qmax,X/Qmax,Y that one has to chose in order

to achieve vi,X= vi,Y depend on mχ. We are thus faced with the situation that, in the presence

of significant cuts on the recoil energy, the quantity we wish to determine is needed as an input

at an early stage of the analysis!

This statement is, strictly speaking, true; we see no way around this, if significant cuts on

the recoil energy are indeed needed. However, we can alleviate the situation. To begin with, we

noticed that the systematic error one introduces by simply setting Qmin,X= Qmin,Y is reduced

greatly if one keeps the sum of the first and third terms in Eq.(12). Moreover, we will show

†Note also that while the integrals Inremain finite where the form factor vanishes, the estimates for the errors

will diverge there, due to the factor 1/F4appearing in Eq.(26).

Page 12

that, to some extent at least, one can do the matching of two Qmaxvalues for the two targets

directly from the data. In principle we could reduce the systematic error associated with the

choice of Qmaxeven more by also including the second term in Eq.(12). However, with limited

statistics the error on r(Qmax) will always be very large, so that keeping this term will not help

significantly.

Our determination of mχfrom generalized moments of f1is thus based on equating the sum

of the first and third terms in Eq.(12) for two different targets. The resulting expression for mχ

can still be cast in the form of Eq.(34), but Rnis now given by

2Q1/2

Rn(Qmin) =

2Q(n+1)/2

min,XrX(Qmin,X)/F2

X(Qmin,X) + (n + 1)In,X

min,XrX(Qmin,X)/F2

X(Qmin,X) + I0,X

1/n

(X −→ Y )−1. (45)

Here n ?= 0 and r(X,Y )(Qmin,(X,Y )) refer to the counting rate for detectors X and Y at the

respective lowest recoil energies included in the analysis. They can be estimated from Eq.(24);

of course, the values of r1, k1, Qs,1, and b1will in general differ for the two targets. Note that

the integrals In, I0are now to be evaluated with Qminas lower and Qmaxas upper limits, as in

Eq.(25).

10 100

mχ,in [GeV ]

0

0.5

1

1.5

2

mχ,rec / mχ,in

solid: including r(Qmin) terms

dashed: without r(Qmin) terms

black: Qmin = 10 keV

red: Qmin = 3 keV

Figure 2: Ratio of reconstructed to input WIMP mass for imperfect Qmin matching, for two

different values of Qmin. We have taken Si and Ge targets and assumed infinite statistics and

(effectively) no upper cut on the recoil energy. The solid (dashed) curves have been computed

from Eqs.(34) and (45) with n = 1 including (neglecting) all terms ∝ r(Qmin). See the text for

further details.

Page 13

Before discussing the statistical uncertainty of this estimate of the WIMP mass, we wish

to demonstrate that keeping the terms ∝ r(Qmin) in Eq.(45) indeed reduces the systematic

error. This is demonstrated in Fig. 2, which shows the ratio of the reconstructed to input

WIMP mass for experiments with “infinite” statistics. Here we have chosen X =

Y =76Ge, and we have set Qmax= ∞ in order to isolate the systematic effect due to imperfect

matching of the Qminvalues in this figure.‡Here the black (red) curves have been obtained with

Qmin,Ge= Qmin,Si= 10 (3) keV. The solid lines include the terms ∝ r(Qmin), while the dashed

ones do not. Clearly including these terms is beneficial: for mχ> 20 GeV, the systematic shift

with these terms included and Qmin= 10 keV is smaller than that without those terms with

the much smaller Qmin= 3 keV. We also see that the systematic effect becomes large at small

mχ. This is not surprising, since a small mχimplies a small Qmax,kin, so that the lower cut on Q

removes a correspondingly larger fraction of the total spectrum. The systematic error vanishes

for mχ=√mXmY = 43 GeV, since for this value of mχwe have αX= αY, i.e., v1,X= v1,Y if

Qmin,X= Qmin,Y. For larger mχthe systematic effect changes sign, i.e., one now underestimates

the true value of mχ. However, if the terms ∝ r(Qmin) are included, the shift remains relatively

small even for Qmin= 10 keV. If values Qmin<

∼3 keV can be achieved, which should be possible

[17], the systematic effect due to imperfect Qminmatching will be a concern only for very light

WIMPs, mχ< 20 GeV.

Using Eq.(45) in Eq.(34) also leads to a lengthier expression for the statistical error on our

estimate of mχ. The first equation in (36) still holds, but the r(Qmin) terms give additional

contributions to the final expression:

?

?

i,j=1

∂ci,X

∂cj,X

28Si and

σ(mχ)|?vn?=

mX/mY|mX− mY|

Rn−

?

mX/mY

?2

×

3

?

?∂Rn

??∂Rn

?

cov(ci,Xcj,X) + (X −→ Y )

1/2

.(46)

Here we have introduced a short–hand notation for the six quantities on which the estimate of

mχdepends:

c1,X= In,X;c2,X= I0,X;c3,X= rX(Qmin,X), (47)

and similarly for the ci,Y.

Explicit expressions for the derivatives of Rnwith respect to these six quantities are collected

in the Appendix. Note that a factor Rncan be factored out of all these derivatives. With this

factor moved out of the square brackets, the prefactor in Eq.(46) is identical to that in Eq.(36),

i.e., its expectation value will again be given by κ of Eq.(43). Of course, Eq.(46) reduces to

Eq.(36) in the limit Qmin,X,Qmin,Y → 0. Note finally that Eq.(46) also holds for n = −1, if

the derivatives with respect to c1,(X,Y )are neglected, since in this case In,X and In,Y do not

contribute to Rngiven in Eq.(45).

Finite lower energy cuts Qmin,(X,Y )can also easily be incorporated in the quantity Rσ ap-

pearing in Eq.(40):

2Q1/2

Estimators for cov(ci,cj) have been given in Eqs.(26) and (31).

Rσ=EY

EX

2Q1/2

min,XrX(Qmin,X)/F2

X(Qmin,X) + I0,X

min,YrY(Qmin,Y)/F2

Y(Qmin,Y) + I0,Y

. (48)

‡More exactly, for mχ> 100 GeV we have set Qmax,Ge= 250 keV, in order to avoid complications due to the

zero of the form factor of76Ge which occurs at Q ≃ 270 keV. We have then matched Qmax,Si, so that the only

systematic deviation of the reconstructed WIMP mass is indeed due to imperfect Qminmatching.

Page 14

Correspondingly, Eq.(42) changes to

σ(mχ)|σ=(mX/mY)5/2|mX− mY|

Rσ− (mX/mY)5/2?2

?

×

3

?

i,j=2

?∂Rσ

∂ci,X

??∂Rσ

∂cj,X

?

cov(ci,Xcj,X) + (X −→ Y )

1/2

, (49)

where we have again used the short–hand notation of Eq.(47); note that c1(X,Y )does not appear

here. Expressions for the derivatives of Rσare also given in the Appendix.

3.3Combined fit

In the next step we wish to combine our estimators (34) for different n with each other, and with

the estimator (40). This could be done via an overall covariance matrix describing the errors of

these estimators and their correlations. The diagonal entries of this covariance matrix are given

by Eqs.(46) and (49); the off–diagonal entries can be computed analogously. This would yield

the overall best–fit value of mχas well as its Gaussian error.

Here we pursue a slightly different procedure, based on a χ2fit. This will yield the same

best–fit value of mχ, which we denote by mχ,rec, but it has two advantages. First, χ2(mχ,rec) can

be used as a measure of the quality of the fit, which in turn can be used to match the Qmaxvalues

of the two experiments at least approximately. Secondly, it allows to determine asymmetric error

intervals. Fig. 1 implies that the errors should indeed be asymmetric. For example, if the true

mχis (much) larger than the masses of both target nuclei, the experimental upper bound one

can derive will be quite large or even infinite, but one should still get a meaningful lower bound;

the opposite is true if mχlies well below the mass of both target nuclei.

We begin by defining fit functions

2Q1/2

fi,X= αi

X

2Q(i+1)/2

min,XrX(Qmin,X)/F2

X(Qmin,X) + (i + 1)Ii,X

min,XrX(Qmin,X)/F2

X(Qmin,X) + I0,X

?300 km

s

?−i

, (50a)

for i = −1, 1, 2, ..., nmax; and

fnmax+1,X= EX

A2

X

2Q1/2

min,XrX(Qmin,X)/F2

X(Qmin,X) + I0,X

?

√mX

mχ+ mX

?

;(50b)

we analogously define nmax+ 2 functions fi,Y. Here nmaxdetermines the highest (generalized)

moment of f1that is included in the fit. The fiare normalized such that they are dimensionless

and very roughly of order unity; this alleviates numerical problems associated with the inversion

of their covariance matrix. The first nmax+1 functions fiare basically our estimators (23) of the

generalized moments defined in Eq.(12) with the term ∝ r(Qmax) omitted; as discussed earlier,

the error on this quantity is likely to be so large that including this term will not be helpful.

The last function is essentially the ratio appearing in Eq.(39). It is important to note that mχ

in Eqs.(50a) and (50b) is a fit parameter, not the true (input) value of the WIMP mass. Recall

also that our estimator (25) for the integrals Inappearing in Eqs.(50a) and (50b) is independent

of mχ. Hence the first nmax+ 1 fit functions depend on mχonly through the overall factor αi.

The fiallow us to introduce a χ2function:

?

χ2=

i,j

(fi,X− fi,Y)C−1

ij(fj,X− fj,Y) .(51)

Page 15

Here C 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) .

The entries of this matrix involving only the moments of the WIMP velocity distribution can

be read off Eq.(82) of Ref. [14], with an obvious modification due to the normalization factor in

Eq.(50a). Since the last fican be computed from the same basic quantities, i.e., the counting

rates at Qmin and the integrals I0, the entries of the covariance matrix involving this last fit

function can also be computed straightforwardly, using Eqs.(26)–(31). Of course, Eq.(51) can

also be used to compute asymmetric error intervals from a single moment, by restricting the

sum to a single term.

(52)

4 Numerical results

We are now ready to present some numerical results for the reconstructed WIMP mass. These

results are based on Monte Carlo simulations of direct detection experiments. We assume that

the scattering cross section is dominated by spin–independent interactions, and use the Woods–

Saxon form for the elastic form factors F(Q) [16]. We describe the WIMP velocity distribution by

a sum of a shifted Gaussian contribution [6] and a “late infall” component [12]; as in Ref. [14], for

simplicity we model the latter as a δ−function, keeping the normalization Nl.i.of this component

as free parameter:

f1(v) =(1 − Nl.i.)

√π

?v

vev0

??

e−(v−ve)2/v2

0− e−(v+ve)2/v2

0

?

+ Nl.i.δ(v − vesc). (53)

We take v0= 220 km/s, ve= 1.05v0§, and vesc= 700 km/s.

In Fig. 3 we show upper and lower bounds on the reconstructed WIMP mass, calculated from

the requirement that χ2exceeds its minimum by 1, for the idealized scenario where no cuts on Q

have been applied. We took our default set of parameters, i.e., vanishing late infall component,

and target nuclei X =28Si, Y =76Ge. This figure is based on simulating 2 × 5,000 experiments,

where each experiment contains an expected 50 events; the actual number of events is Poisson–

distributed around this expectation value. As mentioned earlier, taking equal numbers of events

in both experiments minimizes the statistical error for fixed total number of events. As discussed

in Ref. [14], the error on the (high) moments is not quite Gaussian; the deviation becomes larger

for smaller samples. The reason is that the high moments receive large contributions from the

region of high Q, where on average very few events will lie; even the region where on average only

a fraction of an event lies can contribute significantly. As a result, most simulated experiments

will underestimate these moments, while a few (rare) experiments will overestimate them. In

order to alleviate this problem, we only include moments up to nmax= 2 in our fit. Moreover,

we always show median, rather than mean, values for the (bounds on the) reconstructed WIMP

mass.

We see that the minus–first moment by itself leads to very poor bounds on mχ. This is

not surprising, since its error is dominated by the error on the counting rate at Qmin, which is

determined only from the events in the first bin. The higher moments lead to increasingly tighter

bounds. However, the higher moments are very strongly correlated. Also, the systematic effect

due to the limited event samples discussed in the previous paragraph becomes larger for larger n;

§Strictly speaking, ve should oscillate around 1.05 v0 with a period of one year [6]; we ignore this time

dependence here.

Page 16

10 100 1000

mχ,in [GeV]

1

10

100

1000

mχ,rec, hi, lo [GeV]

50 + 50 events, Si and Ge, standard halo, no cut on Q

n = -1

n = 1

n = 2

σ

tot

Figure 3: Median values of “1σ” upper and lower bounds on the reconstructed WIMP mass in

2 × 5,000 simulated experiments with Si and Ge targets, as a function of the true value of mχ.

We generated on average 50 events per experiment. The short–dashed (black), dotted (green)

and long–dashed (violet) curves show the upper and lower bounds on mχas determined from

moments with n = −1, 1 and 2, respectively; the dash-dotted (blue) curves labeled σ show

the corresponding limits derived from our assumption of equal cross sections for scattering on

protons and neutrons. Finally, the solid (red) curves show the upper and lower bound as well as

the median reconstructed mχfor the global fit based on minimizing χ2of Eq.(51) with nmax= 2.

These results are for our standard halo, i.e., no late infall component, without any cuts on the

recoil energy.

for example, for n = 2 and a true mχ= 1 TeV, the median upper end of the (nominal) 1σ range

lies well below 1 TeV. The lower bound on mχderived from the assumption of equal scattering

cross sections on protons and neutrons is similar to that derived from the first moment, but the

corresponding upper bound is significantly worse. Nevertheless, this estimator of mχhelps in

narrowing down the error of the total fit, described by the upper and lower solid (red) curves.

As expected from Fig. 1 the relative error on mχis minimal for mχ=√mXmY, although the

increase towards smaller mχis less than expected from the behavior of the kinematical factor κ

alone.

Unfortunately we also see that the median reconstructed mχ starts to deviate from the

input value if mχ>

∼80 GeV. This is a direct consequence of the fact that the median value of

the estimators of the higher moments is too small, as discussed above. For very large mχthe

median reconstructed WIMP mass even becomes independent of its true value; this is true also

for the upper end of the error band. This systematic shift presents another argument in favor of

imposing an upper cut Qmaxon Q, chosen sufficiently low that an average experiment will still

have a few events not too far below Qmax.

Page 17

10100 1000

mχ,in [GeV]

1

10

100

1000

mχ,rec, hi, lo [GeV]

50 + 50 events, Si and Ge, standard halo, Qmax < 50 keV

n = -1

n = 1

n = 2

σ

tot

Figure 4: As in Fig. 3, except that we have imposed the cut Qmax= 50 keV in both experiments.

Note that the average of 50 events per experiment refers to the entire Q range, i.e., the number

of events after cuts is smaller if Qmax,kin> Qmax.

10100 1000

mχ,in [GeV]

1

10

100

1000

mχ,rec, hi, lo [GeV]

50 + 50 events, Si and Ge, standard halo, optimally matched Qmax < 50 keV

n = -1

n = 1

n = 2

σ

tot

Figure 5: As in Fig. 4, except that we have imposed the cut Qmax= 50 keV for the Ge target.

The value of Qmaxfor the Si target has been chosen such that it corresponds to the same WIMP

velocity.

Page 18

For the case at hand, with rather small event samples, this would imply Qmax ≃ 50 keV.

Unfortunately Fig. 4 shows that simply imposing the same cut on Qmaxin both targets makes

the situation significantly worse. Now the median reconstructed mχstarts to undershoot the

input value already at mχ ≃ 60 GeV, and the asymptotic reconstructed mχ for large input

WIMP mass lies well below 100 GeV. Clearly we need to choose different Qmaxvalues for the

two experiments if we want to get reliable results also for larger WIMP masses.

Fig. 5 indicates that this should be possible, at least in principle. Here we have again applied

a fixed upper cut Qmax= 50 keV for the Ge experiment, but matched the cut on Qmaxfor the

Si experiment such that it corresponds to the same WIMP velocity:

?αGe

αSi

where α has been given in Eq.(5). We see that now the median reconstructed mχindeed tracks

the input value even for very large WIMP masses. It should be noted that these results still only

use 50 events on average per experiment before cuts. For example, for mχ= 500 GeV Eq.(54)

with Qmax,Ge= 50 keV implies Qmax,Si= 21.7 keV, meaning that the expected number of events

in the Si experiments is only about 21. This explains why the error bands at large values of mχ

are significantly wider here than in Fig. 3 where no cuts on the recoil energy have been imposed.

Of course, the systematic deviation of the reconstructed mχfrom its true value makes the error

bands in Fig. 3 largely meaningless for mχ>

∼100 GeV.

While the results in Fig. 5 look quite impressive, they suffer from the fact that optimal Qmax

matching, as in Eq.(54), is only possible if mχis already known. The left frame of Fig. 6 shows

Qmax,Si=

?2

Qmax,Ge, (54)

10 1001000

mχ,in [GeV]

10

100

1000

median mχ,rec [GeV]

2 x 50 events, Qmax,Ge = 50 keV

2 x 500 events, Qmax,Ge = 100 keV

10100 1000

mχ,in [GeV]

0

0.5

1

1.5

mean χ2(mχ,rec) / d.o.f.

2 x 50 events, Qmax,Ge = 50 keV

2 x 500 events, Qmax,Ge = 100 keV

Figure 6: Sensitivity of the χ2fit to the value of mχthat is used as input in the Qmaxmatching

condition (54), assuming a true WIMP mass of 100 GeV, for our default choice of targets and

f1(v). The solid (black) lines are for 2 × 50 events and Qmax= 50 keV, while the dashed (red)

curves assume 2 × 500 events and Qmax = 100 keV; here Qmax stands for the bigger of the

values used for the two targets, the second one being fixed by Eq.(54). The curves terminate

at mχ,in= 25 GeV since for even smaller WIMP masses, Qmax,kin< 50 keV, making matching

superfluous. The left frame shows the median reconstructed WIMP mass from a χ2fit with

nmax= 2, and the right frame shows the corresponding mean value of χ2(mχ,rec).

Page 19

that inputting the wrong value of mχinto Eq.(54) will usually lead to a reconstructed WIMP

mass in between this input value and the true value. Note that the median mχ,recas a function of

mχ,inhas a slope less than unity. This means that an iteration, where one starts with some input

value of mχand uses the corresponding mχ,recas new input value into Eq.(54), will converge

“on average”. Unfortunately our Monte Carlo simulations show that for any one experiment,

this procedure does not necessarily converge to a well–defined mχ,rec; rather, one often ends up

in an endless loop over several values of mχ,rec.

An alternative is to try an algorithm for Qmax matching which is based on the minimum

value of χ2obtained in the fit; recall that this minimum defines the reconstructed WIMP mass.

Unfortunately the right frame in Fig. 6 indicates that this may be difficult, at least with the

initial small statistics expected. This figure shows that the mean value of χ2(mχ,rec) is almost

independent of the value of mχinput into Eq.(54) if one has only 2 × 50 events in the sample.

The situation looks considerably more promising with 2 × 500 events, and larger allowed Qmax:

in this case χ2(mχ,rec) has a clear, if rather broad, minimum where mχ,inequals the true WIMP

mass, taken to be 100 GeV in this example.

Note that this minimum has mean χ2(mχ,rec)/d.o.f. well below unity; the same is true for the

entire solid curve. Here the number of degrees of freedom is three, since we fit four quantities

with one free parameter. This indicates that our numerical procedure on average over–estimates

the true errors somewhat.In fact, following Ref. [14], we add the “error on the error” to

10100 1000

mχ,in [GeV]

1

10

100

1000

mχ,rec, hi, lo [GeV]

50 + 50 events, Si and Ge, standard halo, matched Qmax < 50 keV

n = -1

n = 1

n = 2

σ

tot

Figure 7: As in Fig. 4, except that we have imposed the fixed cut Qmax= 50 keV only on the

Ge experiment and determined Qmax,Siby minimizing χ2(mχ,rec).

Page 20

the diagonal entries of cov(In,Im), in order to tame non–Gaussian tails in the distribution of

measured moments of f1.

One possibility is to choose the Qmax values such that χ2(mχ,rec) is minimal. Note that

this implies a double minimization: for fixed values of Qmax,Xand Qmax,Y, mχ,recis the WIMP

mass that minimizes χ2(mχ). In an outer loop, Qmaxis varied. We found that varying both

Qmaxvalues leads to quite misleading results, especially for larger (true) WIMP masses and/or

limited statistics. On the other hand, Fig. 4 shows that for small mχ, Qmaxmatching is not

really necessary; if mχ>√mXmY, the matching condition (54) implies that Qmaxfor the lighter

target should be smaller than that for the heavier target.

In Fig. 7 we have therefore minimized χ2(mχ,rec) only for the lighter target, here Silicon.

Note that we always require Qmax,Si≤ Qmax,Gein this procedure; in Fig. 7 the latter is fixed to

50 keV. We see that this leads to a systematic over–estimation of mχfor small WIMP masses.

For heavier WIMPs the results are somewhat better than for the case where both Qmaxvalues

are simply taken to be equal, see Fig. 4. However, this algorithm clearly still fails quite badly

10100 1000

mχ,in [GeV]

1

10

100

1000

mχ,rec, hi, lo [GeV]

50 + 50 events, Si and Ge, standard halo, Qmax< 100 keV

optimal Qmax matching

no Qmax matching

algorithmic Qmax matching

Figure 8: Results for the median value of the reconstructed WIMP mass as well as the ends

of its error interval. All results are based on the combined fit, using Eq.(51) with nmax = 2.

We assume Si and Ge targets with on average 50 events each before cuts, and took our default

ansatz for f1. The dotted (green) lines show results for Qmax,Si= Qmax,Ge= 100 keV, whereas

the solid (black) lines have been obtained using Eq.(54) with the bigger of the two Qmaxvalues

fixed to 100 keV. Finally, the dashed (red) lines are for the case that Qmax,Ge= 100 keV, whereas

Qmax,Sihas been chosen such that χ2(mχ,rec) is minimal.

Page 21

for mχ>

that the kinematic upper bound Qmax,kinof Eq.(8) increases with mχindicates that the region of

large Q is more sensitive to the value of mχif the WIMP mass exceeds the mass of the heaviest

target nucleus.

In Fig. 8 we have therefore increased the cut Qmaxto 100 keV; of course, this cut only becomes

effective once Qmax,kin> 100 keV. Unlike in the previous figures, we here show results only for

the final fit; the values of mχderived from single observables are no longer shown. This allows

us to show results for three different choices of Qmax,Siand Qmax,Ge. The dotted (green) curves

show the median reconstructed WIMP mass and its “1σ” upper and lower bounds for the case

where both Qmaxvalues have been fixed to 100 keV. Due to the higher Qmaxchosen here, this

works for considerably larger WIMP masses than in the corresponding Fig. 4, where both Qmax

values had been fixed to 50 keV. However, for large mχthe median reconstructed WIMP mass

is still only slightly above 100 GeV. The solid (black) lines show results for the case that perfect

Qmaxmatching has been applied, using Eq.(54). Comparison with Fig. 5 shows that increasing

Qmaxactually slightly increases the width of the errors on the reconstructed WIMP mass. The

reason is that Qmaxis now so large that the median values of the estimators for the (generalized)

moments of f1fall somewhat below the true values. Nevertheless the results for this optimal

Qmaxmatching remain very encouraging.

Most importantly, our simple algorithm of fixing Qmax,Ge, in this case to 100 keV, and de-

termining Qmax,Siby minimizing χ2(mχ,rec) now also seems to work reasonably well for WIMP

masses up to ∼ 500 GeV. For mχ<

again over–estimates its true value by 15 to 20%; however, the median value of the “1σ” lower

bound lies below the true WIMP mass for all values of mχ. Similarly, the median “1σ” upper

bound now lies well above the true WIMP mass even for mχ= 1 TeV, in sharp contrast to the

results shown in Fig 7.

We had argued earlier, based on the results of Fig. 1, that Si and Ge is likely to be close to

the optimal choice among the target nuclei currently being employed. In Fig. 9 we back this up

by showing results analogous to those of Fig. 8, but with Argon and Xenon targets. Since the

elastic form factor of136Xe has a zero near 95 keV, we have lowered the upper bound on Qmax

to 75 keV; larger values would start to probe the very sparsely populated region near the zero,

leading to too small median values of the estimated moments, whereas smaller values would lead

to even worse sensitivity to large WIMP masses. We see that, except for small WIMP masses,

the results are clearly worse than those shown in Fig. 8.

The starting point of our discussion was that we wanted to devise ways to estimate the WIMP

mass which are independent of any assumptions on the WIMP velocity distribution f1. In order

to test this, we have simulated Si and Ge experiments, allowing 25% of the local WIMP flux to

come from a “late infall” component, i.e., we fixed Nl.i.= 0.25 in Eq.(53). The results are shown

in Fig. 10. We see that the results are very similar to those with Nl.i.= 0 shown in Fig. 8 if optimal

Qmaxmatching (54) is used. On the other hand, the results for non–optimal choices of Qmaxget

somewhat worse. This is true both for the simple choice Qmax,Si= Qmax,Ge, where significant

deviations set in for lower values of the WIMP mass, and for our algorithm of determining

Qmax,Siby minimizing χ2(mχ,rec). In the latter case, the systematic difference between median

reconstructed and true WIMP mass is larger than for Nl.i.= 0, both for small and for large mχ.

The reason for this degradation is that introducing a large late–infall component, corresponding

to WIMPs with velocity about three times larger than the mean velocity of the shifted Gaussian

component, increases the number of events at large recoil energy. Hence correct Qmaxmatching

becomes more important. Note, however, that the true mχalways lies within the median limits

of the “1σ” error interval estimated from our algorithmic Qmaxmatching.

∼100 GeV. This is partly due to the low maximal value of Qmaxassumed here. The fact

∼100 GeV the median WIMP mass determined in this way

Page 22

10 100 1000

mχ,in [GeV]

1

10

100

1000

mχ,rec, hi, lo [GeV]

50 + 50 events, Ar and Xe, standard halo, Qmax< 75 keV

optimal Qmax matching

no Qmax matching

algorithmic Qmax matching

Figure 9: As in Fig. 8, except for40Ar and136Xe targets, and with both Qmaxvalues restricted

to not exceed 75 keV.

10 100 1000

mχ,in [GeV]

1

10

100

1000

mχ,rec, hi, lo [GeV]

50 + 50 events, Si and Ge, halo with 25% late infall, Qmax< 100 keV

optimal Qmax matching

no Qmax matching

algorithmic Qmax matching

Figure 10: As in Fig. 8, except that we allowed a 25% late infall component in the local WIMP

flux, by setting Nl.i.= 0.25 in Eq.(53).

Page 23

10 1001000

mχ,in [GeV]

10

100

1000

mχ,rec, hi, lo [GeV]

500 + 500 events, Si and Ge, standard halo, Qmax< 100 keV

optimal Qmax matching

no Qmax matching

algorithmic Qmax matching

Figure 11: As in Fig. 8, except for 2 × 500 events before cuts.

So far we have assumed that each experiment “only” has an exposure corresponding to 50

events before cuts. In Fig. 11 we raise this number by a factor of 10. Not surprisingly, all error

bars shrink by a factor>

∼3 compared to the situation of Fig. 8. The error interval also remains

approximately symmetric out to much larger values of mχ. However, the larger number of events

does not significantly change the median reconstructed mχif one simply fixes both Qmaxvalues

to 100 keV. This is not surprising: for large mχthis implies that v2,Siand v2,Gein the definition

(12) of the generalized moments, and hence the moments themselves, are quite different. Hence

the estimators of these moments will not agree if the true WIMP mass is used. On the other

hand, our algorithm for fixing Qmax,Genow seems to perform very well over the entire range of

WIMP masses shown. In particular, the median reconstructed WIMP mass now overshoots its

true value by only a few percent for mχ<

∼100 GeV, and remains close to the true value even at

mχ= 1 TeV. Unfortunately we will see shortly that for large WIMP masses our algorithm still

has some problems even in this case.

The problem lies in the distribution of the reconstructed WIMP masses in the simulated

experiments. This distribution is supposed to be characterized by the error intervals shown in

Page 24

Figs. 3–5 and 7–11. In order to see how well this works, we introduce the quantity

Here mχis the true WIMP mass, mχ,recits reconstructed value, mχ,lo1(2)is the “1 (2)σ” lower

bound satisfying χ2(mχ,lo(1,2)) = χ2(mχ,rec)+1 (4), and mχ,hi1(2)are the corresponding “1 (2)σ”

upper bounds. In the limit of purely Gaussian errors, where χ2of Eq.(51) is simply a parabola,

(δm)2would itself be a χ2variable, measuring the difference between the true and the recon-

structed WIMP mass in units of the error of the reconstruction. However, we saw earlier that

δm =

1 +

mχ,lo1− mχ

mχ,lo1− mχ,lo2,

mχ,rec− mχ

mχ,rec− mχ,lo1,

mχ,rec− mχ

mχ,hi1− mχ,rec,

mχ,hi1− mχ

mχ,hi2− mχ,hi1

if mχ≤ mχ,lo1;

if mχ,lo1< mχ< mχ,rec;

if mχ,rec< mχ< mχ,hi1;

− 1, if mχ≥ mχ,hi1.

(55)

-5

-4-3 -2 -101234

5

δm

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

fraction of expts in bin

2 x 50 events, Si and Ge, standard halo, Qmax< 100 keV, mχ = 50 GeV

optimal Qmax matching

no Qmax matching

algorithmic Qmax matching

Figure 12: Normalized distribution of the variable δm defined in Eq.(55) for 5,000 simulated

experiments, for true WIMP mass mχ= 50 GeV. The other parameters, are as in Fig. 8. The

solid (black) histogram shows results for perfect Qmaxmatching with Qmax≤ 100 keV, based on

Eq.(54), whereas the dotted (green) histogram is for fixed equal Qmaxvalues of 100 keV. The

dashed (red) histogram shows results when Qmax,Siis determined by minimizing χ2(mχ,rec).

Page 25

the error intervals are often quite asymmetric. Similarly, the distance between the “2σ” and

“1σ” limits can be quite different from the distance between the “1σ” limit and the central

value. The definition (55) takes these differences into account, and also keeps track of the sign

of the deviation: if the reconstructed WIMP mass is larger (smaller) than the true one, δm is

positive (negative). Moreover, |δm| ≤ 1 (2) if and only if the true WIMP mass lies between the

“experimental” 1 (2) σ limits.

In Fig. 12 we show the distribution of δm calculated from 5,000 simulated experiments,

assuming a rather small WIMP mass, mχ = 50 GeV. The other parameters have been fixed

as in Fig. 8. In this case simply fixing both Qmaxvalues to 100 keV still works fine, since the

kinematic maximum values of Q lie only slightly above 100 keV (at 122 keV for Si and 132

keV for Ge). The distributions for fixed Qmax, or for optimal Qmaxmatching, look somewhat

lopsided, since the error interval is already asymmetric, with mχ,hi1− mχ,rec> mχ,rec− mχ,lo1.

As a result, negative values of δm have a larger denominator than positive values, hence the

distribution is narrower for δm < 0. These distributions also indicate that our errors are indeed

over–estimated, since nearly 90% of the simulated experiments have |δm| ≤ 1; we remind the

reader that a usual 1σ interval should only contain some 68% of the experiments.

We saw in Fig. 8 that our algorithm for determining Qmax,Sitends to overestimate the WIMP

mass if the latter is small. This is reflected by the dashed (red) histogram in Fig. 12, which has

significantly more entries at positive values than at negative values. Moreover, this histogram

is rather flat between δm = 1 and 2. Since our algorithm is based on a double minimization

of χ2defined in Eq.(51), it is not very surprising that the resulting final χ2(mχ,rec) values are

distributed quite differently from what one finds after a single minimization step. Nevertheless

it is reassuring that some 70% of the simulated experiments lead to |δm| ≤ 1.

-4 -2024

δm

0

0.05

0.1

fraction of expts in bin

2 x 50 events, Si and Ge, standard halo, Qmax< 100 keV, mχ = 200 GeV

optimal Qmax matching

no Qmax matching

algorithmic Qmax matching

-5

-4-3-2 -101234

δm

0

0.02

0.04

0.06

0.08

0.1

fraction of expts in bin

2 x 500 events, Si and Ge, standard halo, Qmax< 100 keV, mχ = 200 GeV

optimal Qmax matching

no Qmax matchingQmax matching

algorithmic

Figure 13: Distribution of δm defined in Eq.(55) calculated from 5,000 simulated experiments.

Parameters and notations are as in Fig. 12, except that we have increased the WIMP mass to

200 GeV. In the right frame we have in addition increased the average number of events (before

cuts) in each experiment from 50 to 500. Note that the bins at δm = ±5 are overflow bins, i.e.,

they also contain all experiments with |δm| > 5.

Page 26

Unfortunately Fig. 13 shows that the situation becomes much less favorable at the larger

WIMP mass of 200 GeV. Here we show results with 50 (left) and 500 (right) events per ex-

periment (before cuts), with the other parameters as in Fig. 12. We see that for optimal Qmax

matching a large majority of the simulated experiments still satisfy |δm| ≤ 1. The fact that this

fraction decreases from nearly 90% for the smaller events samples to about 85% for the larger

samples indicates that our error estimates become a bit more reliable for larger event numbers.¶

Note also that the secondary peak at δm ≃ −1 is due to the change of denominator in the

definition (55).

We already saw in Figs. 8 and 11 that setting Qmax,Si = Qmax,Ge = 100 keV significantly

under–estimates the true WIMP mass if the latter exceeds 100 GeV. This is borne out by

Figs. 13. Since the statistical error decreases with increasing number of events, δm is much

smaller in the right frame; in fact, most of the simulated experiments now fall in the “underflow

bin” δm = −5, which also contains all experiments giving even smaller values. In other words,

most of the simulated experiments would give a reconstructed WIMP mass more than five

estimated standard deviations below the true value, if no Qmaxmatching is used.

Unfortunately the error estimates resulting from our algorithm of determining Qmax,Si by

minimizing χ2(mχ,rec) are also not very reliable in this case. For the smaller event sample, we

find that about 58% of the simulated experiments yield |δm| ≤ 1, while nearly all experiments

give |δm| ≤ 2. While these numbers are not so different from the corresponding Gaussian

predictions, the distribution of δm is clearly highly non–Gaussian in this case. Just as in the

scenario without Qmaxmatching the error estimates actually become less reliable with increasing

event samples. If both experiments contain on average 500 events each, less than 40% of the

experiments have |δm| ≤ 1, while more than 30% of the simulations yield |δm| ≥ 2. In fact,

the most likely value of δm is now close to 3. In view of these observations, the fact that the

median δm is close to zero, so that the median reconstructed WIMP mass is close to the true

value as already shown in Fig. 11, seems almost accidental. Recall also that the nominal “1σ”

uncertainty of the reconstructed WIMP mass still amounts to about 40 GeV in this case. This

means that the most likely value of mχ,recpredicted by our algorithm exceeds the true value by

more than 100 GeV. This clearly leaves some room for improvements. The fact that optimal

Qmaxmatching continues to give good results for both the reconstructed WIMP mass and its

error indicates that better data–based algorithms might very well exist.

5 Summary and Conclusions

In this paper we described methods to determine the mass of a Weakly Interacting Massive

Particle detected “directly”, i.e., through the recoil energy deposited in a detector by the recoiling

nucleus after a WIMP scattered elastically off this nucleus. Our methods are model–independent

in the sense that they do not need any assumption about the WIMP velocity distribution. The

price one has to pay for this is that one will need positive signals in at least two different

detectors, employing different target nuclei.

Our methods are based on our earlier work [14] on reconstructing the WIMP velocity dis-

tribution, which we briefly reviewed in Sec. 2. In this earlier work, which was based on results

from a single (simulated) experiment, the WIMP mass mχwas an input. In Sec. 3 we showed

how one can determine mχ by equating results obtained by different experiments. Here the

moments of the velocity distribution function are particularly useful, since all events in the sam-

¶Assuming, unrealistically, that there are 50,000 events in each experiment, we find that only about 75% of

all experiments have |δm| ≤ 1, indicating a very slow approach to the Gaussian value of about 68%.

Page 27

ple contribute to any given moment, leading to relatively low statistical uncertainties. We also

described a method for determining mχthat can be used if the ratio of WIMP scattering cross

sections on protons and neutrons is known; this is true, for example, for the spin–independent

scattering of supersymmetric neutralinos, where these two cross sections are nearly equal. We

also showed how to combine these methods using a χ2fitting procedure.

Sec. 4 was devoted to a detailed numerical analysis of our methods. We saw that, assuming

the sizes of the event samples are fixed, the statistical errors will be smaller for larger mass

difference between the two target nuclei.In practice experiments with heavier targets will

accumulate more events, assuming equal exposure, at least if the spin–independent contribution

to the scattering cross section dominates. However, the number of useful events (after cuts,

preferably in an almost background–free energy range) also depends on other factors, besides

the masses of the target nuclei.

In our discussion we saw that, for WIMP masses exceeding 100 GeV or so, the maximal recoil

energy Qmaxof accepted signal events plays a crucial role. Existing experiments have Qmax≤ 100

keV. If both targets used fixed Qmaxvalues of this order or even smaller, a significant systematic

error on the extracted WIMP mass results. In principle this problem can be solved by matching

the Qmaxvalues of the two experiments. The problem is that perfect matching requires prior

knowledge of the WIMP mass. We tried two algorithms to overcome this problem. Determining

Qmax of one experiment iteratively should converge “on average”, but in a given experiment

often leads to an endless loop, rather than a specific value of Qmax; this problem is particularly

severe for small event samples. On the other hand, determining Qmaxof the experiment with

the lighter target nucleus by minimizing χ2also with respect to this quantity over–estimates the

WIMP mass if it is small, and leads to unreliable error estimates if the WIMP mass is larger, the

problem becoming worse with increasing event samples. However, the fact that optimal Qmax

matching works well in all cases, for both the median reconstructed WIMP mass and its error

(which tends to be over–estimated by our expressions), gives us hope that a better algorithm for

Qmaxmatching can be found which only relies on the data. One possibility that might be worth

exploring is to employ a combination of an iterative procedure and a second χ2minimization,

where the latter is used only if the former does not converge to a well–defined value of the

reconstructed WIMP mass.

We also found that imposing a cut Qmaxmay actually be beneficial for small event samples.

This is related to our earlier observation [14] that a typical experiment will under–estimate the

higher moments of f1, which receive significant contributions from recoil energies where only a

fraction of an event is expected to occur in a given experiment. This problem becomes more

acute for heavier target nuclei, since they have softer form factors. In particular, using Xenon

rather than Germanium does not improve the determination of mχ, since the spin–independent

elastic form factor of Xenon as predicted by the Woods–Saxon ansatz vanishes for Q ≃ 95 keV.

In contrast, the lower (threshold) energy of the experiment does not seem to be very important,

if it can be pushed down to values near 3 keV or less.

Our analysis is idealized in that we ignore backgrounds, systematic uncertainties as well

as the finite energy resolution. The relative error on the recoil energy in existing experiments

is small compared to our most optimistic relative WIMP mass error estimates even with 500

events per experiment, so ignoring it should be a good approximation. Modern methods of

discriminating between nuclear recoils and other events, combined with muon veto and good

shielding, hold out the possibility of keeping (some) experiments nearly background–free also in

future.

In our numerical analysis we have ignored the expected annual modulation of the WIMP

flux. In practice this can be done if one simply sums all events over (at least) one full calendar

Page 28

year. In principle one can also use our methods for subsets of data collected during specific

times of the year. However, at least if the standard “shifted Gaussian” velocity distribution is

approximately correct, we do not expect the small annual modulation to play a significant role,

even if one compares experiments taken during different parts of the year.

We saw that our methods work best if the WIMP mass lies in between the masses of the two

target nuclei. Even in that case the error will likely be significantly larger than the error on mχ

from collider experiments, if the WIMP is part of a well–motivated extension of the Standard

Model of particle physics, e.g., if it is the lightest neutralino [6] or the lightest T−odd particle

in “Little Higgs” models [18]. It will nevertheless be crucial to determine the WIMP mass from

direct and/or indirect Dark Matter experiments as precisely as possible, in order to make sure

that the particle produced at colliders is indeed the WIMP detected by these experiments. Once

one is confident of this identification, one can use further collider measurements to constrain

the WIMP couplings. This in turn will allow to calculate the WIMP–nucleus scattering cross

section. Together with the determination of the WIMP velocity distribution [14], this will

then yield a determination of the local WIMP number density via the total counting rate in

direct detection experiments. Knowledge of the WIMP couplings will also permit prediction

of the WIMP annihilation cross section. Together with the Dark Matter density inferred from

cosmological observations, this will allow to test our understanding of the early universe [19].

A determination of the WIMP mass from Dark Matter detection experiments is thus a crucial

ingredient in many analyses that shed light on the dark sector of the universe.

Acknowledgments

This work was partially supported by the Marie Curie Training Research Network “UniverseNet”

under contract no. MRTN-CT-2006-035863, as well as by the European Network of Theoretical

Astroparticle Physics ENTApP ILIAS/N6 under contract no. RII3-CT-2004-506222.

A Derivatives needed in the error analysis

At the end of Sec. 2 we gave the covariance matrix of the quantities appearing in the definition

of the Rnas well as Rσ. In Eqs.(46) and (49) we also gave expressions relating the errors on the

reconstructed WIMP masses to the errors on Rnand Rσ. The only missing ingredients in the

calculation of the errors on our various estimators of mχare the first derivatives of Rnand Rσ.

We begin with the former. From Eq.(45), it can be found directly that

2Q(n+1)/2

2Q1/2

∂Rn

∂rX(Qmin,X)=2n

Q(n+1)/2

min,XI0,X− (n + 1)Q1/2

min,XrX(Qmin,X) + (n + 1)In,XF2

min,XIn,X

X(Qmin,X)

×

F2

X(Qmin,X)

min,XrX(Qmin,X) + I0,XF2

X(Qmin,X)

Rn, (A1a)

∂Rn

∂In,X

=n + 1

n

F2

X(Qmin,X)

2Q(n+1)/2

min,XrX(Qmin,X) + (n + 1)In,XF2

X(Qmin,X)

Rn,

(A1b)

and

∂Rn

∂I0,X

= −1

n

F2

X(Qmin,X)

2Q1/2

min,XrX(Qmin,X) + I0,XF2

X(Qmin,X)

Rn. (A1c)

Page 29

By first exchanging Q(n+1)/2

X by Y , one finds

min,X

and (n+1)In,Xwith Q1/2

min,Xand I0,X, respectively, and then replacing

∂Rn

∂rY(Qmin,Y)= −2

n

Q(n+1)/2

min,Y I0,Y− (n + 1)Q1/2

2Q(n+1)/2

2Q1/2

min,YIn,Y

min,Y rY(Qmin,Y) + (n + 1)In,YF2

Y(Qmin,Y)

×

F2

Y(Qmin,Y)

min,YrY(Qmin,Y) + I0,YF2

Y(Qmin,Y)

Rn, (A2a)

∂Rn

∂In,Y

= −n + 1

n

F2

Y(Qmin,Y)

2Q(n+1)/2

min,Y rY(Qmin,Y) + (n + 1)In,YF2

Y(Qmin,Y)

Rn,

(A2b)

and

∂Rn

∂I0,Y

=1

n

F2

Y(Qmin,Y)

2Q1/2

min,yrY(Qmin,Y) + I0,YF2

Y(Qmin,Y)

Rn. (A2c)

Note that a factor Rnappears in all these expressions; this allows to cast the final result for

the error on mχestimated using moments of f1into a form analogous to that in Eq.(36) even

in the presence of non–trivial cuts on the recoil energy, with the same prefactor. Moreover,

all the I0,X, I0,Y, In,X, In,Y should be understood to be computed according to Eq.(14) or its

discretization (25) with integration limits Qminand Qmaxspecific for that target.

Similarly, the derivatives of Rσcan be computed from Eq.(48):

2Q1/2

∂Rσ

∂rX(Qmin,X)=

2Q1/2

min,X

min,XrX(Qmin,X) + I0,XF2

X(Qmin,X)

Rσ, (A3a)

and

∂Rσ

∂I0,X

=

F2

X(Qmin,X)

2Q1/2

min,XrX(Qmin,X) + I0,XF2

X(Qmin,X)

Rσ,(A3b)

The derivatives with respect to the Y variables can be obtained from Eqs.(A3a) and (A3b) by

simply changing X −→ Y everywhere and changing the overall plus signs to minus signs.

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