Introduction to dark matter experiments

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by  R. W. Schnee

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Introduction to Dark Matter Experiments
n
a
J R.W.Schnee∗

7 Department of Physics, Syracuse University,
2 Syracuse, New York 12344, United States
∗E-mail: [email protected]

]
O
This is a set of four lectures presented at the Theoretical Advanced Study
C
Institute (TASI-09) in June 2009. I provide an introduction to experiments
h. designedtodetectWIMPdarkmatterdirectly,focussingonbuildingintuitive
p understanding of the characteristics of potential WIMP signals and the ex-
- perimental techniques. After deriving the characteristics of potential signals
o in direct-detection experiments for standard WIMP models, I summarize the
r general experimental methods shared by most direct-detection experiments
t
s and review the advantages, challenges, and status of such searches (as of late
a 2009).ExperimentsarealreadyprobingSUSYmodels,withbestlimitsonthe
[ spin-independentcouplingbelow10−7pb.

1 Keywords:WIMPs,colddarkmatter,SUSY,darkmatterhalo
v
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0 1. Introduction
2
5 Avarietyofastrophysicalobservations1,2 indicatethat83%ofthematterin
.
1 theUniverseisnonbaryonicanddark,presumablyintheformofelementary
0 particlesproducedintheearlyUniverse.Becausenosuchparticleshaveyet
1
been identified in particle accelerators, these observations require new fun-
1
: damentalparticlephysics.WeaklyInteractingMassiveParticles(WIMPs3)
v
form a particularly interesting generic class of new-particle candidates be-
i
X causetheynaturallyprovideabouttheinferredamountofthisnonbaryonic
r dark matter,4 a result dubbed the “WIMP miracle.”
a
WIMPs would be produced thermally in the early Universe. Because
they interact only weakly, their annihilation rate would become insignifi-
cant as the Universe expands, thus “freezing out” a relic abundance of the
particles (for a pedagogical discussion, see e.g. Refs. 5–8). The expected
WIMP density would be the same as that of the nonbaryonic dark matter
if the WIMP velocity-averaged annihilation cross section is ∼1pb, so that
the WIMP mass is ∼100GeV/c2.
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Remarkably, extensions to the Standard Model motivated entirely by
particle physics predict particles with the same cross section and mass as
these dark-matter WIMPs. Detection of the W and Z bosons indicates
that the electroweak symmetry is spontaneously broken at a scale of ∼
100GeV/c2.Whateverphysicssolvesthehierarchyproblemassociatedwith
thissymmetrybreaking–beitsupersymmetry,5,9–11extradimensions,12–16
or something else – gives rise to additional particles. If an appropriate
(oftenindependentlymotivated)discretesymmetryexists,thelightestsuch
particle is stable. This particle is then weakly interacting, massive, and
stable – it is a WIMP. Thus, particle theorists are “almost justified in
saying that the problem of electroweak symmetry breaking predicts the
existence of WIMP dark matter”.17
AlthoughtheargumentforWIMPdarkmatterisgeneric,supersymme-
try dominates the discussion as a particularly well-motivated model. Su-
persymmetryinteractionsariseintheoriesofquantumgravity,stabilizethe
Higgs mass hierarchy problem, predict the observed value of sin2θ , and
W
over a broad range of parameter space predict cosmologically significant
relic WIMP densities.
WIMPs can potentially be detected by three complementary methods.
They may be produced and detected (indirectly) at accelerators such as
the Large Hadron Collider (see e.g. Ref. 18). Relic WIMPs may be de-
tectedindirectlywhentheyclumpinmassiveastrophysicalobjects,increas-
ing their annihilation rate enough that their annihilation products may be
detectable.11,19–21 Many potential (or suggested22–28) indirect signals are
ambiguous, with alternate astrophysical explanations (see Refs. 7,29 and
referencestherein).Somepotentialindirectsignals,however,wouldbecom-
pelling.AnnihilationintheSunorEarthwouldproducehigher-energyneu-
trinosthananyotherknownprocess.Theseneutrinoscouldbeobservedin
neutrinotelescopessuchasIceCube30,31 orANTARES.32,33 EitherFERMI
orground-basedairCerenkovtelescopesmaydetectdistinctivegamma-ray
features from the galactic center or from sub-halos.34–38
Relic WIMPs may also be detected directly when they scatter off nu-
clei in terrestrial detectors.39,40 This article offers an introduction to these
direct-detection experiments. Section 2 includes derivations and explana-
tions of the characteristics of potential signals in direct-detection experi-
mentsforstandardWIMPmodels.Section3summarizesthegeneralexper-
imentalmethodssharedbymostdirect-detectionexperimentsanddiscusses
particularexperimentsbriefly,emphasizingtherelativeadvantagesanddif-
ferent challenges and capabilities of the various approaches.
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2. WIMP-nucleus elastic scattering: from model to signal
Understanding experiments designed for direct detection of dark matter
begins with the observables of potential signals. In this section we consider
the observables of any model that predicts standard WIMP-nucleus elastic
scattering (see Neil Weiner’s contribution to these proceedings8 for discus-
sion of more speculative models with non-standard scattering). Following
the reviews by Lewin and Smith,41 and Jungman, Kamionkowski and Gri-
est,5 this section derives how the observed WIMP interaction rate depends
on energy, target, time, and direction.
2.1. Spin-independent and spin-dependent cross sections
UsingFermi’sGoldenRule,wecandividetheenergydependenceofthedif-
ferentialWIMP-nucleoncrosssectionintoatermσ thatisindependent
0WN
of the momentum transfer and a term F2(q) (known as the form factor)
containing the entire dependence on the momentum transfer q:
dσ (q) 1 σ F2(q)
WN = |M|2 = 0WN . (1)
dq2 πv2 4µ2v2
A
Here, v is the velocity of the WIMP in the lab frame, and the WIMP-
nucleus reduced mass µ ≡ M M /(M +M ) in terms of the WIMP
A χ A χ A
massM andthemassM ofatargetnucleusofatomicmassA.Sincethe
χ A
WIMPs are nonrelativistic, the zero-momentum cross section for a WIMP
ofarbitraryspinandgeneralLorentz-invariantWIMP-nucleuscrosssection
may be written in terms of a spin-independent (mostly scalar) and a spin-
dependent (mostly axial vector) term:
4µ2 32G2µ2 J +1
σ = A [Zf +(A−Z)f ]2+ F A (a (cid:104)S (cid:105)+a (cid:104)S (cid:105))2.
0WN π p n π J p p n n
(2)
Theproofofthisclaimmakesagoodexerciseforthereader;solutionmaybe
foundinRef.42.Heref andf (a anda )areeffectivespin-independent
p n p n
(spin-dependent)couplingsoftheWIMPtotheprotonandneutron,respec-
tively.TogetherwiththeWIMPmass,M ,theseparameterscontainallthe
χ
particle physics information of the model under consideration. The other
parametersdescribethetargetmaterial:itsatomicnumberZ,totalnuclear
spin J, and the expectation values of the proton and neutron spins within
the nucleus (cid:104)S (cid:105)=(cid:104)N|S |N(cid:105). For free nucleons, (cid:104)S (cid:105)=(cid:104)S (cid:105)= 0.5. Ta-
p,n p,n p n
ble 1 from Ref. 43 lists values of (cid:104)S (cid:105) and (cid:104)S (cid:105) for materials commonly
p n
used for dark matter searches, although some are subject to significant
nuclear-physics uncertainties.
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Table 1. Values of the atomic number Z, the total nuclear spin J, and the expec-
tation values of the proton and neutron spins within the nucleus (cid:104)Sp,n(cid:105) for various
nuclei with odd numbers of protons or neutrons, leading to the relative sensitivities
to spin-dependent interactions shown, from Refs. 5,43 and the references contained
therein.
Odd 4(cid:104)Sp(cid:105)2(J+1) 4(cid:104)Sn(cid:105)2(J+1)
Nucleus Z Nuc. J (cid:104)Sp(cid:105) (cid:104)Sn(cid:105) 3J 3J
19F 9 p 1/2 0.477 -0.004 9.1×10−1 6.4×10−5
23Na 11 p 3/2 0.248 0.020 1.3×10−1 8.9×10−4
27Al 13 p 5/2 -0.343 0.030 2.2×10−1 1.7×10−3
29Si 14 n 1/2 -0.002 0.130 1.6×10−5 6.8×10−2
35Cl 17 p 3/2 -0.083 0.004 1.5×10−2 3.6×10−5
39K 19 p 3/2 -0.180 0.050 7.2×10−2 5.6×10−3
73Ge 32 n 9/2 0.030 0.378 1.5×10−3 2.3×10−1
93Nb 41 p 9/2 0.460 0.080 3.4×10−1 1.0×10−2
125Te 52 n 1/2 0.001 0.287 4.0×10−6 3.3×10−1
127I 53 p 5/2 0.309 0.075 1.8×10−1 1.0×10−2
129Xe 54 n 1/2 0.028 0.359 3.1×10−3 5.2×10−1
131Xe 54 n 3/2 -0.009 -0.227 1.8×10−4 1.2×10−1
Formanymodels,f ≈f ,sothespin-independentWIMP-nucleuscross
p n
section
4µ2
σ ≈ Af2A2. (3)
0WN,SI π n
Thedependenceofthiscrosssectiononthetargetmaterialmaybefactored
out by rewriting this result as
µ2
σ =σ AA2, (4)
0WN,SI SIµ2
n
whereµ isthereducedmassoftheWIMP-nucleonsystem,andthe(target-
n
independent)spin-independentcrosssectionofaWIMPonasinglenucleon
4µ2f2
σ ≡ n n. (5)
SI π
This WIMP-nucleon cross section σ may be used to compare experimen-
SI
tal results to theory and to each other. A given model predicts particular
combinationsofσ andM ;differentexperimentsproducelimitsonσ as
SI χ SI
functions of M by translating limits on the WIMP-nucleus cross-section
χ
to limits on σ using equation 4. The dependence on µ2A2 in eqn. 4 indi-
SI A
cates the advantage of experiments using relatively heavy target materials
(but see the effects of the form factor in Sec. 2.3). For a 50GeV/c2 WIMP
incident on a target with A = 50, µ2/µ2 = 625, so the spin-independent
A n
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WIMP-nucleuscrosssectionislargerthantheWIMP-nucleoncrosssection
by a factor >106.
The situation for spin-dependent interactions is quite different.43 First
ofall,contributionsfromthespin-dependentprotonandneutroncouplings
often cancel, so it is important to quote limits on the spin-dependent in-
teraction on neutrons separately from that on protons, each under the as-
sumption that the other interaction is negligible. Furthermore, while the
coherent interaction on the nucleus results in a spin-independent cross sec-
tion that scales with A2 since the contribution of each nucleon adds inside
the matrix element, the spin-dependent contributions of nucleons with op-
posite spins cancel, so that the total spin-dependent cross section depends
on the net spin of the nucleus. As shown in Table 1, nuclei with even num-
bersofprotonshavenearlynonetprotonspinandessentiallynosensitivity
to spin-dependent interactions on protons, and nuclei with even numbers
of neutrons similarly have almost no sensitivity to spin-dependent interac-
tions on neutrons. Argon, with even numbers of protons and neutrons for
all significant isotopes, is thus insensitive to spin-dependent interactions.
Many materials used as WIMP targets (Ge, Si, Xe) have even numbers of
protons and hence are insensitive to spin-dependent interactions on pro-
tons; only some isotopes of these targets (and hence only a fraction of the
detector’s active mass) have sensitivity to spin-dependent interactions on
neutrons. Typically, sensitivity to spin-dependent interactions on protons
requires alternate target materials, often resulting in worse backgrounds or
backgroundrejectionandlowersensitivitytospin-independentinteractions.
Therelativesensitivityofamaterialtospin-dependentinteractionsissum-
marized by its “scaling factors” 4(cid:104)S (cid:105)2(J +1)/3J and 4(cid:104)S (cid:105)2(J +1)/3J,
p n
whicharelistedinTable1.Aswithspin-independentlimits,experimenters
quote limits on target-independent quantities: the spin-dependent WIMP-
proton cross section σ ≡ 24G2µ2a2/π and the spin-dependent WIMP-
SDp F p p
neutron cross section σ ≡24G2µ2a2/π.
SDn F n n
The lack of benefit from the coherent interaction for spin-dependent
interactions results in most models being more accessible experimentally
via their spin-independent interactions than by their spin-dependent inter-
actions. As shown in Fig. 1 and Fig. 2, current experiments are already
constrainingMSSMmodelsbasedontheirspin-independentcouplings,but
none is yet sensitive enough to constrain such models based on their spin-
dependent couplings, despite the fact that spin-dependent couplings are
typically ∼3000× larger than spin-independent couplings.
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)
n
o
e
cl
u
n
o 10-5
d t
e
s
ali
m
or 10-6
n
(
]
b
p
[
n
o -7
cti 10
e
s
-
s
s
o
r
C -8
10
1 2 3
10 10 10
2
WIMP Mass [GeV/c ]
Fig. 1. Upper limits on the spin-independent WIMP-nucleon coupling σSI under
the standard assumptions about the Galactic halo described in the text. Most sensi-
tive limits are from cryogenic experiments (solid) CDMS44 (black), EDELWEISS-II45
(mediumgray),andCRESST46 (lightgray),andtwo-phasenobleexperiments(dashed)
XENON1047 (black),ZEPLIN-III48 (mediumgray),andWArP49 (lightgray).Current
experiments already exclude part of the parameter space of MSSM models (shaded).50
FiguremadeusingtheDarkMatterLimitPlotter.51
2.2. The WIMP recoil energy spectrum
It is illuminating to calculate the energy spectrum for the case of zero
momentum-transfer (i.e. taking F2 ≡ 1). Furthermore, simply multiplying
this spectrum by the energy dependence of F2(q), rather than including
the form factor F2 within the kinematic integral to follow, is convenient
and usually adequate.
The energy spectrum arises due to the familiar kinematics of elastic
scattering. In the center-of-momentum frame, the WIMP scatters off a nu-
cleus through an angle θ, with cosθ uniformly distributed between −1 and
1 for the isotropic scattering that occurs with zero-momentum transfer.
If the WIMP’s initial energy in the lab frame E = M v2/2, the nucleus
i χ
recoils with energy
(1−cosθ)
E =Er (6)
R i 2
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100 100
MP−neutron Cross Section (pb)111000−−−321 MP−proton Cross Section (pb)111000−−−321
WI WI
D D
S S
10−4 10−4
101 102 103 101 102 103
WIMP Mass (GeV/c2) WIMP Mass (GeV/c2)
Fig. 2. Upper limits on the spin-dependent WIMP-neutron coupling σSDn (left) and
the spin-dependent WIMP-proton coupling σSDp (right) under the standard assump-
tions about the Galactic halo described in the text. The most sensitive limits on σSDn
arefromthesameexperimentsshowninFig.1(withthesamelinetypes):XENON1052
(black dashes), ZEPLIN-III53 (medium gray dashes), and CDMS44 (black solid). Note
ZEPLIN-III limits were calculated with a scaling factor 2× smaller than that used for
XENON10. Due to the low intrinsic sensitivity of leading (Xe and Ge) experiments
to spin-dependent interactions on protons, the most sensitive limits on σSDp are from
experimentswithonlymodestsensitivitytospin-independentinteractions:PICASSO54
(6-sidedstars),COUPP55(5-pointedstars),KIMS56(circles),andNAIAD57(×).Limits
from indirect search experiments SuperKamiokande58 (points) and IceCube31 (dotted)
make additional assumptions about branching fractions to neutrinos. Current exper-
iments do not exclude any part of the parameter space of the same MSSM models
(shaded)50showninFig.1,despitethefactthatthepredictedspin-dependentcrosssec-
tions are ∼3000× larger than the spin-independent ones. Figure made using the Dark
MatterLimitPlotter.51
(in the lab frame), where
4µ2 4M M
r ≡ A = χ A (7)
MχMA (Mχ+MA)2
is a dimensionless parameter related to the reduced mass µ . Note that
A
r ≤ 1, with r = 1 only if M = M . For this isotropic scattering, the
χ A
recoilenergyisthereforeuniformlydistributedbetween0–Er.Asshownin
i
Fig.3,thedifferentialcontributiontothedifferentialrateforagiveninitial
WIMP energy
(cid:18) (cid:19)
dR dR(E)
d (E ) = i , (8)
dE R Er
R i
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dR
dE
R
dR(E)
dR(E) i
i
Er
i
Er E
i R
Fig. 3. Plot showing schematically how contributions to the differential rate dR/dER
fordifferentvaluesoftheinitialWIMPenergyEi add.Wedefinetotaldifferentialrate
dR/dER of WIMPs with initial energy Ei to be dR(Ei). For a WIMP initial energy
Ei,therecoilenergyER isuniformlydistributedbetween0–Eir,sodR(Ei)contributes
equally to the rates of all recoils between 0–Eir, as depicted by the shaded area in the
figure. The contribution to the differential rate at a given recoil energy (the height of
theshadedareainthefigure)issimplytheareadividedbythelength,ordR(Ei)/(Eir).
The total differential rate can then be found by summing all the boxes, i.e. integrating
dR(Ei)/(Eir)forallEi.
so
dR (cid:90) Emax dR(E)
(E )= i . (9)
dE R Er
R Emin i
ThemaximuminitialWIMPenergymaybetakenasinfinityasaninitialap-
proximation,ormoreaccuratelymaybebasedupontheGalacticescapeve-
locity,v :E =M v2 /2.TocausearecoilofenergyE ,theminimum
esc max χ esc R
initial WIMP energy E = E /r (for head-on scattering, with θ = π),
min R
(cid:112) (cid:112)
and the minimum WIMP velocity v = 2E /M = 2E /(rM ).
min min χ R χ
TodeterminetherateofWIMP-nucleusscattering,itishelpfultoimag-
ine the motion of the target nucleus relative to WIMPs with velocity v in
the lab frame. In time dt, each nucleus interacts with any WIMP inside
a volume dV = σvdt, where σ is the WIMP-nucleus cross section. The
number of WIMPs inside the volume moving with velocity v
dN =n f((cid:126)v+(cid:126)v )σvdt, (10)
0 E
where the local WIMP number density n =ρ /M , where ρ is the mass
0 χ χ χ
densityofWIMPsinthegalaxy,estimatedfromstudiesofGalacticdynam-
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icstobeabout0.3GeV/(c2 cm3)(withwidesystematicuncertainties59–62).
Note that the number density based on ρ =0.3GeV/(c2 cm3) is really an
χ
upper limit, since Galactic dark matter may include species other than
WIMPs. We use the fact that the velocity (cid:126)v of the WIMP in the galaxy
g
is the vector sum of the WIMP velocity with respect to the Earth (cid:126)v and
the velocity (cid:126)v of the Earth with respect to the Galaxy. We assume that
E
the WIMPs’ velocities in the frame of the Galaxy follow the Maxwellian
distribution:
e(−(cid:126)v+(cid:126)vE)2/v02
f((cid:126)v+(cid:126)v )= (11)
E k
where v = 220±20km/s is the local circular velocity,63 and k is a nor-
0
malization factor. This simple distribution is not expected to be especially
accurate, but it provides a useful standard. See e.g. Refs. 64,65 for discus-
sionofalternativesthatarelikelymoreaccurate,andseeSection2.4below
for discussion of the impact of uncertainties on a WIMP discovery.
The differential interaction rate per kilogram of detector is then the
productofthenumberofinteractionspernucleonwiththenumberofnuclei
per kg of material:
N
dR= 0n f((cid:126)v+(cid:126)v )σvd3(cid:126)v (12)
A 0 E
where N is Avogadro’s number, so that N /A is the number of nuclei per
0 0
kilogram of material.
It is instructive (and reasonably accurate, as shown below) to consider
the simplified case ignoring the Earth velocity and the Galaxy’s escape
velocity (i.e. setting v = 0, v = ∞), for which the integral is trivial.
E esc
After setting
2 N
R ≡ √ 0n σv, (13)
0 π A 0
we get
ddER (ER)=(cid:90) ∞ (cid:0)1M1v2(cid:1)r2Rπv04ve−v2/v02(cid:0)4πv2dv(cid:1) (14)
R ER/r 2 χ 0
R (cid:90) ∞ 2
= (cid:0)1M 0v2(cid:1)r v2e−v2/v02vdv (15)
2 χ 0 vmin 0
R
= 0 e−ER/E0r, (16)
E r
0
where E ≡ M v2/2 is the most probable WIMP incident energy. The
0 χ 0
mean recoil energy is easily seen: (cid:104)E (cid:105)=E r. Since r ≤1, the mean recoil
R 0
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energy(cid:104)E (cid:105)=E onlyiftheWIMPmassisequaltothemassofthetarget
R 0
nucleus; (cid:104)E (cid:105) < E both for smaller and for larger WIMP masses. As an
R 0
example, since v ≈ 220km/s ≈ (0.75×10−3)c, M = M = 50GeV/c2
0 χ A
would result in
1
(cid:104)E (cid:105)=E r = M v2 ≈15 keV. (17)
R 0 2 χ 0
A different target mass would result in even lower (cid:104)E (cid:105). This low energy
R
sets the first challenge for direct detection experiments – they must have
lowenergythresholds,muchlowerthanpastsolarneutrinoexperimentsfor
example.
From the exponential form of the approximate energy spectrum, we see
that R is the total WIMP rate. If we plug known numerical values into
0
equation 13, we find
500 σ ρ
R ≈ 0WN χ events kg−1day−1. (18)
0 Mχ(GeV/c2) 1pb 0.4GeV/cm3
A 50GeV/c2 WIMP with a WIMP-nucleus cross section σ = 1pb (so
0WN
that the spin-independent WIMP-nucleon cross section σ ∼ 10−6pb, or
SI
the spin-dependent WIMP-nucleon cross section σ ∼10−3pb) results
SDp,n
in about 10 events/(kg day). Since the energy spectrum is a falling expo-
nential,alowenergythresholdiscriticaltodetectmostoftheseevents;the
fraction of events above an energy threshold Eth is e−Eth/E0r.
The dependence of the energy spectrum on the WIMP mass may be
seen easily from equation 16. The mean recoil energy
v2 (cid:26) M2 if M (cid:28)M
(cid:104)E (cid:105)=rE ∝ 0 ∝ χ χ A . (19)
R 0 (1+MA/Mχ)2 constantif Mχ (cid:29)MA
Heavy WIMPs all yield about the same energy spectrum. This result holds
for calculations made including the Earth velocity, Galaxy escape velocity,
and nuclear form factor, as shown in Fig. 4.
WIMPs with velocities above the Galaxy’s escape velocity are likely to
have already escaped. The finite escape velocity (∼ 540km/s ≈ 2×10−3c
according to the RAVE survey66) alters the recoil spectrum slightly and
produces a cut-off at
1
E = rM v2 ≈100 keV. (20)
max 2 χ esc
ThecutoffenergyhasthesamedependenceontheWIMPmassasthemean
recoil energy (see equation 19) since
v2
E = esc(cid:104)E (cid:105)≈6(cid:104)E (cid:105). (21)
max v2 R R
0