Can WIMP Spin Dependent Couplings explain DAMA data, in light of Null Results from Other Experiments?

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by  Christopher Savage

Can WIMP Spin Dependent Couplings explain DAMA data, in light of Null Results
from Other Experiments?
Chris Savage,1 Paolo Gondolo,2 and Katherine Freese1
1 Michigan Center for Theoretical Physics, Physics Department, University of Michigan, Ann Arbor, MI 48109
2 Physics Department, University of Utah, Salt Lake City, UT 84112
(Dated: February 2, 2008)
WeexaminewhethertheannualmodulationfoundbytheDAMAdarkmatterexperimentcanbe
explainedbyWeaklyInteractingMassiveParticles(WIMPs),inlightofnewnullresultsfromother
experiments. CDMSIIhasalreadyruledoutmostWIMP-nucleusspin-independentcouplingsasan
explanation for DAMA data. Hence we here focus on spin-dependent (axial vector; SD) couplings
ofWIMPstonuclei. Weexpanduponpreviouswork by(i)considering thegeneralcaseofcoupling
to both protons and neutrons and (ii) incorporating bounds from all existing experiments. We
5 note the surprising fact that CMDS II places one of the strongest bounds on the WIMP-neutron
0 cross-section, and show that SD WIMP-neutron scattering alone is excluded. We also show that
0 SD WIMP-proton scattering alone is allowed only for WIMP masses in the 5-13 GeV range. For
2 the general case of coupling to both protons and neutrons, we find that, for WIMP masses above
13 GeV and below 5 GeV, there is no region of parameter space that is compatible with DAMA
n
andallotherexperiments. Intherange(5-13) GeV,wefindacceptableregions ofparameter space,
a
J includingones in which theWIMP-neutron coupling is comparable to theWIMP-proton coupling.

4
1
I. INTRODUCTION Kurylov and Kamionkowski performed a general analy-

3 sis for point-like WIMPs of arbitrary spin to find what
v WIMPswerecompatiblewithDAMAandthenullexper-
6 The dark halo of our Galaxy may consist of WIMPs iments at the time [6]. Giuliani looked at the more gen-
4 (Weakly Interacting Massive Particles). Numerous col- eral case of mixed interactions [7], but only applied the
3 laborationsworldwidehavebeensearchingforthese par-
analysis to DAMA for a limited choice of WIMP mass.
8 ticles. Direct detection experiments attempt to observe
We here perform a general analysis with coupling to
0
the nuclear recoil caused by these dark matter particles
both protonsand neutrons to see if any parameterspace
4
interacting with nuclei in the detectors. Indirect detec-
0 remains that could explain simultaneously the DAMA
tion experiments search for the signatures of WIMP an-
/ and CDMS data. To be complete, we will also incorpo-
h nihilationinthe Sun,the Earth,orthe GalacticHalo. It
rate bounds from other experiments, including Super-K
p
is wellknownthat the countrate in WIMP direct detec-
(which measures indirect detection in the Sun). Notice,
-
o tion experiments will experience an annual modulation however, that these indirect detection limits do not ap-
r [1, 2] as a result of the motion of the Earth around the ply if WIMPs do not effectively annihilate in the Sun,
t
s Sun: the relative velocity of the detector with respect to as for example if the WIMP is not its own anti-particle
a the WIMPs depends on the time of year.
and there is a sufficiently strong cosmic asymmetry in
:
v The discovery of an annual modulation by the the number of WIMPs and anti-WIMPs.
i
X DAMA/NaI experiment [3] (hereafter, “DAMA”) is the
r onlypositivesignalseeninanydarkmattersearch. How- We find those regions of WIMP parameter space that
a ever,recentnullresultsfromotherexperiments,particu- survive all existing experiments for the case of spin-
larlyEDELWEISSandCDMSII,severelyboundthepa- dependent interactions. Although the current CDMS
rameterspace of WIMPs that could possibly explainthe data are more stringent than those studied in previous
DAMAdata. Infact,iftheinteractionsbetweenWIMPs work, our final bounds are somewhat less restrictive for
and nuclei are spin-independent (SI), then CDMS II has WIMP-protoncouplingthan those quotedby Ullioet al.
ruled out DAMA (with some exceptions [4]). In this pa- because the latest results released by Super-K are less
per, we will consider the alternative of spin-dependent restrictive than those approximated by Ullio et al. and
(SD) WIMP cross-sections. We expand upon previous becausewearereluctanttoextendSuper-Kdatabeyond
work by (i) considering the general case of coupling to thosepublishedbytheexperimentalists(aselaboratedin
bothprotonsandneutronsand(ii)incorporatingbounds the Discussion section at the end of the paper).
from all existing experiments, including, in particular,
thenullresultsrecentlyreleasedbyCDMSIIandSuper- We also note that, although CDMS data have typi-
Kamiokande(Super-K), whichwerenot availableto pre- cally been ignored in the spin-dependent sector due to
vious authors. Previously, Ullio et al. studied the cases the small natural abundance of spin odd isotopes in the
of interactions with protons only or neutrons only [5] detector, in fact CDMS II places one of the strongest
and made a comparisonwith experiments that had been bounds on the WIMP-neutron cross-section (n.b. even
performed at that time; they argued that some cases of CDMS I placed interesting bounds particularly at low
mixed interactions would be ruled out as well. In 2003 masses).
2
II. WIMP DETECTION ItiswellknownthatthecountrateinWIMPdetectors
will experience an annual modulation [1, 2] as a result
The basic idea underlying WIMP direct detection is of the motion of the Earth around the Sun: the rela-
straightforward: theexperimentsseektomeasuretheen- tive velocity of the detector with respect to the WIMPs
ergydepositedwhenaWIMPinteractswithanucleusin depends on the time of year. The disk of the galaxy ro-
the detector [8]. tates through the relatively stationary halo, giving the
If a WIMP of mass m scatters elastically from a nu- Sun a velocity v⊙ = 232 km/s relative to the halo. In
cleus of mass M, it will deposit a recoil energy E = addition, the Earth travels about the Sun at V⊕ = 30
(µ2v2/M)(1 cosθ), where µ mM/(m + M) is the km/s (relative to the Sun), with the Sun’s motion in the
reduced mas−s, v is the speed o≡f the WIMP relative to Galaxy being at 60deg from the Earth’s orbital plane
the nucleus, andθ is the scatteringanglein the centerof (v⊕ v⊙+V⊕).
≡
mass frame. The differential recoil rate per unit detec- We will use the isothermal halo model in our calcula-
tor mass for a WIMP mass m, typically givenin units of tions, but its validity is by no means guaranteed – the
counts/kg/day/keV,can be written as: actualhalomaybesquashedoranisotropic[10],maycon-
tainstreamsfromsmallergalaxiesintheprocessofbeing
dR ρ absorbed by the Milky Way [11] or may, in fact, not yet
= σ(q)η(E,t) (1)
dE 2mµ2 be thermalized.
AnalternativewaytosearchforWIMPdarkmatterof
whereρ=0.3GeV/cm3isthestandardlocalhaloWIMP relevancetothispaperisviaindirectdetectionofWIMP
density, q = √2ME is the nucleus recoil momentum, annihilation in the Sun. When WIMPs pass through a
σ(q)istheWIMP-nucleuscross-section,andinformation largecelestialbody,suchastheSunortheEarth[12,13],
abouttheWIMPvelocitydistributionisencodedintothe interactions can lead to gravitational capture if enough
mean inverse speed η(E,t), energy is lost in the collision to fall below the escape ve-
locity. CapturedWIMPssoonfalltothebody’scoreand
fd(u,t) 3 eventuallyannihilatewithotherWIMPs. Theseannihila-
η(E,t)= d u. (2)
u tions leadto high-energy neutrinos that canbe observed
Zu>vmin
byEarth-baseddetectorssuchasSuper-Kamiokande[14],
Here Baksan [15], and AMANDA [16], and the IceCube [17]
and ANTARES [18] projects. The annihilation rate de-
ME pendsonthecapturerateofWIMPs,whichisinturnde-
vmin =s2µ2 (3) terminedbytheWIMPscatteringcrosssectionoffnuclei
in the celestial body. While the Earth is predominantly
represents the minimum WIMP velocity that can result composed of spinless nuclei, the Sun is mostly made of
in a recoil energy E and fd(u,t) is the (usually time- hydrogen,whichhasspin. Thusthespin-dependentcross
dependent)distributionofWIMP velocitiesurelativeto sectionofWIMPsoffnucleonscanbeprobedbymeasur-
the detector. ing the annihilation signals from WIMP annihilation in
ForWIMPsintheMilkyWayhalo,themostfrequently the Sun. Other indirect detection methods search for
employed WIMP velocity distribution is that of a sim- WIMPs that annihilate in the Galactic Halo or near the
ple isothermal sphere [1]. In such a model, the Galactic GalacticCenterwheretheyproduceneutrinos,positrons,
WIMPspeedswithrespecttothehaloobeyaMaxwellian orantiprotonsthatmaybeseenindetectorsontheEarth
distribution with a velocity dispersion σ truncated at [19, 20].
h
the escape velocity vesc,
fh(v)= Nescπ13/2v30 e−v2/v20, for |v|<vesc (4) III. SPIN-INDEPENDENT CROSS-SECTION
(0, otherwise.
Thecross-sectionforspin-independentWIMPinterac-
Here tions is given by
2
v0 = 2/3σh (5) σSI =σ0F (q) (7)
p where σ0 is the zero-momentum WIMP-nuclear cross-
and
section and F(q) is the nuclear form factor, normalized
Nesc =erf(z) 2zexp( z2)/π1/2, (6) to F(0) = 1; a description of these form factors may be
− − found in [21] and [22]. For purely scalar interactions,
with z =vesc/v0, is a normalization factor. For the sake 4µ2
of illustration, we take σh = 270 km/s and vesc = 650 σ0,scalar = [Zfp+(A Z)fn]2. (8)
π −
km/s. The results for the mean inverse speed η(E,t) for
the case of the isothermal sphere have previously been Here Z is the number of protons, A Z is the number
−
calculated and can be found e.g. in [2, 9]. of neutrons, and f and f are the WIMP couplings to
p n
3
the proton and nucleon, respectively. In most instances, Appendix. Null search results place an upper limit on
f f ; the WIMP-nucleus cross-section can then be N (N < N ), which then constrains the pos-
n p rec rec rec,max
∼
given in terms of the WIMP-proton cross-section as a sible values for a and a . A positive modulation signal,
p n
result of Eqn. (8): suchasinDAMA(N <N <N ),likewise
ma,min ma ma,max
constrains the possible values for a and a .
p n
2
µ Previous authors, when searching for regions compat-
2
σ0 =σp µ A (9) ible with both the DAMA signal and the null results of
(cid:18) p(cid:19)
otherexperiments[5],havestudiedthefollowingtwospe-
where the µ is the proton-WIMP reduced mass, and A cific cases: (i) a =0 so that only protons contribute to
p n
is the atomic mass of the target nucleus. WIMP interactions, and (ii) a = 0 so that only neu-
p
In this model, for a given WIMP mass, σ is the only tronscontribute. Hereweexaminethemoregeneralcase
p
free parameter and its limits are easily calculated for a where both a and a may be non-zero(some ofthis pa-
p n
given experiment. These limits are routinely calculated rameter space has also been examined by Giuliani [7]).
by the dark matter experiments and it has been found To do this, we examine the form of the limits on a and
p
that the DAMA and CDMS results are incompatible for a implied by Eqn. (11).
n
mostWIMPmassinthespin-independentcase,assuming Eqn. (11) describes a conic section in the a -a plane.
p n
the standard isothermal halo [23] (an exception is the This conic section can be an ellipse, a hyperbola, or a
case of light WIMPs of 6-9 GeV discussed in [4]). setoftwostraightlines(it cannotbe a parabolabecause
linear terms in a or a are absent). Which is the case
p n
depends on the values of the coefficients A, B, and C.
IV. SPIN-DEPENDENT CROSS-SECTION For the benefit of the reader, we will briefly review the
relevant shapes here (our conics are always centered at
For the remainder of the paper, we will focus on the origin, so we only describe such cases below).
spin-dependent interactions. The generic form for An ellipse in the x-y plane can be written as:
thespin-dependentWIMP-nucleuscross-sectionincludes
x2 y2
two couplings– the WIMP-proton coupling a and the
p + =1 (12)
WIMP-neutron coupling a , a2 b2
n
32µ2G2 Thisellipseissymmetricaboutthexandy axes. Adding
σ (q) = F a2S (q)+a a S (q) a cross-term (i.e. xy) in Eqn. (12) (like the B term in
SD 2J +1 p pp p n pn
Eqn. (11)) essentially rotates the ellipse- the major and
(cid:2) +a2nSnn(q) . (10) minoraxesdonotlieonthexandyaxes,butalongsome
new rotated axes.
Here the nuclear structure functions S (q)(cid:3), S (q), and
pp nn Another conic section is the hyperbola:
S (q) are functions of the exchange momentum q and
pn
arespecifictoeachnucleus. Thequantitiesap andan are x2 y2
actually the axialfour-fermionWIMP-nucleoncouplings a2 − b2 =1 (13)
in units of 2√2G [24, 25, 26].
F
ForagivenexperimentandWIMPmass,onecaninte- which, in this form, gives two branches of a hyperbola
grateEqn.(1)overanenergybinE1–E2,usingtheabove open along the +x and x directions. Rotation of this
−
cross-section and factoring in efficiencies, quenching fac- form also leads to a cross-term.
tors,etc., to obtainthe expected number of recoilevents Two parallel lines are also a conic (technically, a “de-
N as a function of the parameters a and a : generate” conic), that can be written in the form, e.g.:
rec p n
2 2
N (a ,a )=A a +B a a +C a . (11a)
rec p n rec p rec p n rec n 2 2
x =a (14)
A ,B ,andC areconstantsthatcanbecalculated
rec rec rec
fromtheintegration(seeAppendix). Theydifferbetween whichdescribestwolinesparalleltothey-axis,locatedat
experiments and they depend on the WIMP mass and x=+a and x= a. Other parallel lines can be rotated
−
WIMP velocity distribution. The expected value of the into this form as well.
amplitudeoftheannualmodulationcanbesimilarlycal- Note that, while other conics exist besides the three
culated as Eqn. (11a): described,thesearetheonlyformspossibleforEqn.(11)
(which is not the most general second degree polyno-
2 2
N (a ,a )=A a +B a a +C a (11b) mial).
ma p n ma p ma p n ma n
Eqn. (11) can, under suitable rotation in the a -a
p n
where Nma is the modulation amplitude and Ama, Bma, plane, be expressed in one of the forms in Eqns. (12),
and Cma will again be calculable strictly from the ex- (13), or (14). The shape of our conic can be determined
perimentalparametersandtheWIMPmassandvelocity by the value of the discriminant
distribution. A more detailed discussion of calculating
2
N , N , and the associated constants is given in the γ B 4AC, (15)
rec ma
≡ −
4
Experiment Exposure[kg-day] Threshold [keV] Efficiency [%] Constraint (for stated recoil energies) Ref.
CDMS I Si: 6.58 5 5–55 keV:<2.3 events(†) [28]
E <10 keV:7.6
CDMS II Ge: 52.6 10 E <20 keV:22.8 10–100 keV:<2.3 events(†) [23]
CDMS II Si: 300 5 E >20 keV:38 5–100 keV:<2.3 events(†) [29]
(projection) Ge: 1200
EDELWEISS Ge: 8.2 20 100 20–100 keV:<2.3 events(†) [30]
CRESSTI Al2O3: 1.51 0.6 100 (‡) [31]
Ca+O, 15–40 keV:<6 events
CRESSTII CaWO4: 10.448 10 100 W, 12–40 keV:<2.3 events(†) [32]
DAMA/Xe-2 Xe: 1763.2 30 (◦) 100 (‡) [33]
I: 22 (⋄)
DAMA/NaI NaI:107731 100 2–6 keVee: 0.0200±0.0032/kg-day-keVee (•) [3]
Na: 6.7 (⋄)
I: 44 (⋄)
Elegant V NaI:111854 100 4–5 keVee: 0.009±0.019/kg-day-keVee (•) [34]
Na: 13.4 (⋄)
TABLE I: Experimental constraints used in this study. Notes to the table: (†) upper limit assuming no detected event; (‡)
limitsgeneratedbaseduponthebinneddataavailableinthecorrespondingpapers(doesnotincludeanybackgroundsubtraction
andmaybeconservative),;(◦)from anelectron equivalentthresholdof13 keVee,usingthequenchingfactorQ=Eee/E equal
to0.44forXe[33];(⋄)fromanelectronequivalentthresholdsof2keVee(DAMA/NaI)and4keVee(ElegantV),usingQequal
to 0.09 for I and 0.3 for Na [3]; (•) amplitudeof annualmodulation.
as follows: γ < 0 corresponds to an ellipse, γ > 0 to very different to that obtained from the individual ele-
a hyperbola, and γ = 0 to two parallel lines. Due to ments (see e.g. Figure 4). For the experiments studied
our conics alwaysbeing centered at the origin, there is a in this paper, we find that the relevant conics are either
symmetry under (a ,a ) ( a , a ). two parallel lines or ellipses. Some of the ellipses are so
p n p n
→ − −
NullresultexperimentsprovideanupperboundN elongated as to appear as two parallel lines in our plots.
max
on the number of recoils or modulation amplitude. This Wewillexaminetheallowedshapesinthe ap-an plane
upperboundrestrictsthealloweda -a parameterspace for current experiments and then determine where they
p n
to the region inside the ellipse defined by N (for the overlap. In this way we will find those regions in pa-
max
case in which Eqn. (11) describes an ellipse), or to the rameter space that are consistent with all the current
region containing the origin between the two branches experimental results.
of the hyberbola or the two parallel lines defined by
N (for the case in which Eqn. (11) describes a hy-
max
perbola or two straight lines). Couplings outside these V. EXPERIMENTS
regions would lead to more recoils than observed; cou-
plings within these regions yield less recoils than N .
max The constraints from different experiments on the a -
p
Experiments which obtain positive signals (such as a phase space are highly dependent on the choice of
n
claimed by DAMA) restrict the ap-an parameter space detector materials. Odd Z elements, suchas Sodium, Io-
to a band in the shape of the appropriate conic. The dine, and Hydrogen, are more sensitive to a and thus
p
width of the band is defined by the uncertainty in the generate narrow ellipses with the semi-minor axis close
observed signal (the uncertainty and hence the width of to the a axis (interference effects can cause the ellipse
p
the band can be given to various degrees of accuracy, toberotatedslightly). Conversely,elementswithanodd
e.g.1 or 2σ). For example, if the conic is a circle, the al- number of neutrons, such as Xenon-129, Silicon-29, and
lowedregionwouldbe anannulus;regions“outside”this Germanium-73, are more sensitive to a and their as-
n
annulus wouldgiveeither too smallortoo largeasignal. sociated ellipses have semi-minor axes near the a axis.
n
The relative magnitudes of A, B, and C in Eqn. (11) Spinless elements such as most isotopes of Silicon and
are primarily dependent upon the structure functions of Germanium(usedbymanyexperimentslookingforspin-
Eqn. (10). At zero recoil energy, the structure functions independentcouplings),provideessentiallynoconstraint
inEqn.(11)forasinglenuclearelementformacomplete for either ap or an.
square,oftheformN (αa +βa )2 (see,e.g.,Eqn.(13) TableI (extendedfrom[4])displaysthe variousexper-
p n
∝
of Ref. [27]). This is a rotation of Eqn. (14). Hence for imental constraints used to generate our results. In Fig-
q =0, we have γ =0 and the appropriate conics are two ures 1 & 2, we show the limits from various experiments
parallel lines. However, at nonzero momentum transfer, on the WIMP-proton and WIMP-neutron allowed cou-
interferenceandspineffectsdestroythesymmetrysothat plings for the casesa =0 anda =0,respectively. Due
n p
different conics may result. In addition, detectors with to the novel detection technique used by SIMPLE (and
multipleelements,suchastheNaIofDAMA,havestruc- the additionalcomplexities involved)[35], we did not re-
ture functions from all of these elements contributing to analyze their data in our study, but include in Figure 2
the coefficients in Eqn. (11); the resulting conic may be the WIMP-neutron cross-section presented in Figure 1
5
110033
DAMA
110022
Super-K
Baksan
110011
LL
bb Elegant V
pp
HH
110000 CRESST I
pp DAMA(cid:144)NaI
--
ΧΧ
ΣΣ
1100--11 CDMS II Ge
&
CDMS I Si
CDMS II Ge
HoverlappingL
1100--22
CDMS II
HprojectedL
110011 110022
MM HHGGeeVVLL
WWIIMMPP
FIG. 1: WIMP-proton cross-section limits for the case an = 0. The CDMS Si+Ge limit overlaps that of Ge only; the Si is
relatively insensitivein this case. Super-Konly analyzed their datafor WIMPmasses above18 GeV; theirlimit is takenfrom
Figure14of[14]. Baksananalyzedfluxesdownto13GeV.Super-KandBaksanruleouttheDAMAresultsovertheiranalysis
rangesandCRESSTIlimitsDAMAatlowmasses, butWIMPsbetween 6-13GeV areconsistent withall resultsfor thiscase.
of [7]. The full ZEPLIN I results were not available dur- [37]. The WIMP cross sections on protons and neutrons
ing the writing of this paper so were not included in our implied by the DAMA result are plotted in Figures 1
analysis; however, a preliminary limit presented by the & 2 respectively. Kinematics alone can restrict the al-
ZEPLIN group is also included in Figure 2 [36]. lowed masses of this modulation: small masses do not
Several experiments providing recent results, includ- generate enough recoils above threshhold. One can see
ing Elegant V (NaI), CRESST I (Al), SIMPLE (F), this in the figures as DAMA begins to lose sensitivity
and DAMA/NaI, used odd Z nuclei and were thus more below about 4 GeV, requiring large cross-sections to ac-
sensitive to the WIMP-proton coupling a . Other ex- count for the observed signal. This feature is common
p
periments, including DAMA/Xe-2 (Xe-129),CRESST II toalldarkmatterexperimentsandlowerthresholdener-
(Ca-43,W-183,andO-17),Edelweiss(Ge-73),ZEPLINI gies are required to detect the smaller recoil energies of
(Xe-129andXe-131),andCDMS(Si-29andGe-73)used lowmasses. This lowerlimitonsensitivity is alsodepen-
neutronoddnucleiandweremoresensitivetotheWIMP- dent on the parameters of the velocity distribution and
neutron coupling a . We note, however, that each of may change significantly in the presence of streams [4].
n
theseexperimentshassomesensitivitytobotha anda , Largermassesarealsoruledoutasthephaseofthemod-
p n
as the minor axes of their ellipses are not exactly aligned ulationwouldbe reversed. Forthe 2-6keVdata plotted,
with either a or a . Hence, surprisingly, CRESST and the limit is roughly 180 GeV; however, this limit is de-
p n
SIMPLEobtainverygoodresultsonWIMP-neutroncou- pendent upon the range of recoil energies. Freese and
pling due to their low thresholds. Lewis have examined this phenomenon and determined
The only experiment to claim a result, DAMA/NaI, that masses above 103 GeV would reverse the modula-
uses odd Z NaI detectors and a large exposure to tion at 2 keV for DAMA [38]. In addition to the 2-6
search for an annual modulation. They observe keV data, DAMA released modulation amplitudes for
an annual modulation amplitude of 0.0200 0.0032 slightly different recoil bins, namely for recoil energies
±
events/kg/day/keVee over electron-equivalent recoil en- 2-4 keV (0.0233 0.0047 /kg/day/keVee) and 2-5 keV
±
ergies of 2-6 keV. Such a modulation has been shown (0.0210 0.0038 /kg/day/keVee). At the masses we are
±
to be generally inconsistent with other experiments for considering (< 100 GeV), the results of this paper are
spin-independent interactions, but we shall see there are the same for any of the three DAMA binnings.
choicesofparametersstillallowedforthespin-dependent All other experiments have null results which place
case. The DAMA limits we place in a -a space are bounds on the couplings. Elegant V used NaI detec-
p n
those that reproduce the above modulation amplitude, tors to search for an annual modulation and found no
using the structure functions given by Ressell and Dean counts above the statistically limiting background [34].
6
110044
DAMA
110033
DAMA(cid:144)Xe-2
SIMPLE
110022
CRESST I
LL
bb
pp 110011 CRESST II Ca+O
HH DAMA(cid:144)NaI
CRESST II W
--nn 110000
ΧΧ Edelweiss
ΣΣ
ZEPLIN I
1100--11 HpreliminaryL
CDMS II Ge
CDMS I Si
1100--22 CDMS II Ge
CDMS II
HprojectedL
110011 110022
MM HHGGeeVVLL
WWIIMMPP
FIG. 2: WIMP-neutron cross-section limits for the case ap =0. SIMPLE limits are taken from Figure 1 of [7]. The addition
of the Sidata clearly benefitsCDMS at lower masses, ruling out DAMA for this simple case.
While providing one of the best exclusion limits for a limits are taken directly from Ref. [14], derived therein
proton-only coupling at low WIMP masses, the sensi- usingthemodel-independenttechniqueofKamionkowski
tivity of this experiment is not enough to examine the et al. [39]. We note that the bounds from Super-K rely
DAMA observed region (see Figure 1). CRESST I [31] onthreeassumptions: First,weassumethattheWIMPs
andSIMPLE[35]providesimilarlimitsonWIMP-proton haveachievedequilibriumintheSun,sothatthecapture
coupling. As mentioned above, due to their very low rate is equal to the annihilation rate; this assumption
threshholds, CRESST I and SIMPLE also provide what is almost certainly correct, particularly at small WIMP
have been the best neutron-only limits at low WIMP masses. Second,weassumethateitherWIMPsareequal
masses. DAMA/Xe-2,aneutron-oddexperimentbounds to their antipartners so that they can annihilate among
the WIMP-neutron cross-section above 40 GeV [33]. themselves or, if WIMPs are not the same as the anti-
Our examination of Edelweiss (similar to CDMS, de- WIMPs, that there is no WIMP/antiWIMP asymmetry.
scribed below), shows that they produce a similar limit Third, the WIMPs do not decay to light fermions only
to DAMA/Xe-2 [30] (see Figure 2). (in which case fewer observable neutrinos would be pro-
duced). The secondandthirdwaysto evade the SuperK
Super-Kamiokande, on the other hand, is an indirect bounds were pointed out by Kurylov and Kamionkowski
detection experiment, searching for high-energy neutri- [6]. Here we assume that all three assumptions are cor-
nosproducedby the annihilationofWIMPs inthe Sun’s rect;thenSuper-Krulesouttheprotononlycouplingfor
core after being gravitationally captured. Since the Sun WIMP masses above 18 GeV.
is predominantly composed of hydrogen, the capture
rate (and thus neutrino flux) is particularly sensitive to Baksan, another indirect detection experiment, pro-
the WIMP-proton coupling a . The Super-K detector vides the tightest bounds on proton only coupling be-
p
searched for this additional neutrino flux by observing tween13and18GeV.AlthoughBaksanprovidesweaker
upwardmovingmuons(inducedbymuonneutrinos)[14]. WIMP-induced muon flux limits than Super-K above 18
The lack of an additional muon signal leads to the most GeV, the Baksan experiment has analyzed these fluxes
significanta boundsabovetheanalysisthresholdWIMP down to a lower WIMP mass of 13 GeV [15]. These
p
massof18GeV;belowthis mass,a significantportionof fluxlimitscanbeusedtodetermineWIMP-protoncross-
the muons would stop in the detector, rather than pass section limits in the same manner as Super-K; such an
through, making the data more difficult to analyze. Dis- analysisisnotavailablefromtheBaksancollaborationat
continuities of the Super-K published limits occur at 80 thistime. However,theBaksancross-sectionlimitcanbe
GeV (W mass) and 174 GeV (top mass) due to differ- determinedbyrescalingtheSuper-Klimitbytherelative
entpossibleannihilationproducts. TheSuper-KWIMP- flux limits of these two experiments (since the expected
proton cross-section limits are shown in Figure 1; these fluxisproportionaltothiscross-section). Below18GeV,
7
where there is no Super-K limit to rescale, we extrapo- DAMAresultsarenotcompatiblewithlimits fromother
late the Baksan cross-sectionlimit using their given flux experiments for the case of the WIMP coupling with
limit and the WIMP mass dependence of the expected neutrons only. However, one can see in Figure 1 that
flux,givenbyEqn.(9.55)ofRef.[22]. We findthatBak- DAMA stillsurvivesfor the caseofprotononly coupling
sanrulesoutprotononlycouplingdowntoWIMPmasses for masses less than 18 GeV.
of 13 GeV. Figure 1 displays the current limits for the WIMP-
CDMS II, in the Soudan Underground Laboratory, protoncross-sectionassumingtheWIMPcouplesonlyto
uses Silicon and Germanium detectors with strong dis- theproton(a =0). CDMSdoesnotcurrentlyconstrain
n
crimination to provide the strongest spin-independent thiscoupling. TheSuper-KandBaksanresults,however,
cross-section limits to date [23]. Analysis of CDMS re- strongly rule out the WIMP-protoncoupling that would
sultsinthe spin-dependentsectorhaveconsistentlybeen be requiredto explain DAMA for masses above 13 GeV.
ignored, as the small natural abundance of spin odd iso- Elegant V still allows for the DAMA observed modula-
topes in Silicon (4.68% Si-29) and Germanium (7.73% tion at these lower masses and CRESST I provides only
Ge-73) has given the impression that CDMS is not sen- a lower mass limit of 6 GeV.
sitive enough for any significant spin-dependent results. Figure 2 displays the WIMP-neutron cross-section in
Our analysis, however, shows that the extremely clean, the alternative case that the WIMP couples only to the
background-freedataissufficienttoovercomethesmaller neutron a = 0. DAMA/Xe-2 and Edelweiss rule out
p
spin sensitive mass, with the odd neutron Ge-73 (Silicon this case for masses above 20 GeV. The surprisingresult
data has not yet been published) providing significant is that CDMS II significantly improves upon this limit,
limits to an. The limits we obtain are due to the fact ruling out masses above 10 GeV. Furthermore, if one in-
that no events were observedin 52.6 days of livetime for cludes the CDMS I Si data, CDMS rules out all of the
four 0.250 kg Germanium detectors over 10 keV to 100 DAMA allowed region for this case. Recent ZEPLIN re-
keV recoil energy, using an efficiency of 0.228 for 10-20 sults are extremely powerful as well.
keV and 0.38 for 20-100 keV; an efficiency of 0.076 for
While the above two cases demonstrate the different
5-10 keV is used to determine limits at lower thresholds.
sensitivities of the experiments, they are not necessar-
Since Silicon is lighter than Germanium, it will provide
ily representative of the more general parameter space,
betterlimits atsmallWIMPmasses. AstheSilicondata where both a and a may be non-zero. Figure 3 shows
p n
has not yet been published for the first run of CDMS II,
thea -a regionsallowedbyDAMA,CDMS,andacom-
p n
weaugmenttheCDMSIIGeresultswiththemostrecent
binationoftheremainingexperimentsatseveraldifferent
CDMSISidata[28]. ThelastCDMSIrunusedthesame
masses for this more general case.
detector tower as now used in CDMS II, containing two
The DAMA observed modulation generates an ellipti-
0.100kgSilicondetectorsaswellasthe fourGermanium
cal ring, flattened in roughly the direction of the a -axis
p
ones. However, one of these Si detectors was contami-
(due to the odd Z NaI). [At low masses, the portion of
nated and not used in the data analysis. No Si events
the elliptical ring that is plotted looks like two parallel
were found in the range 5-55 keV after 65.8 days of live-
bands.] The ellipse is slightly rotated from the a -a
p n
time, with the same efficiencies as above. While several
axes;this rotationincreasesat largermassesas observed
Si events were observed above 55 keV, these are consis-
recoils tend to come from the Iodine atoms rather than
tent with a neutron background. Even if these events
the lighter Sodium (which has different structure func-
wereWIMPscatters,theyarenotconsistentwithalight
tions). The DAMA allowed couplings are those that fall
mass. SincetheSidataisbeingusedtoextendthelimits
on theellipticalbands,notbetweenthem; couplings“in-
at low masses (Ge data dominates at higher masses), we
side” the inner edge of the elliptical ring (between the
feel justified in ignoring the high energy events. CDMS
two bands of the ellipses) are too small to generate the
hopes to install a total of five towers and achieve a total
observed modulation.
exposure during the CDMS II run of 1200 kg days for
Other experiments, which have null results, place
Ge and 300 kg days for Si; we will use these numbers to
bounds on the a -a parameter space, as shown in Fig-
projectfuturelimits,assuminga5keVthresholdandnull p n
ure 3. The light grey (pink) horizontal regions, labelled
results [29]. We have used the structure functions given
as “other” in the legend of the figure, illustrate a com-
byRessell et al. [40]forSi-29andDimitrovet al. [41]for
bination of limits from Super-K, Baksan, Elegant V,
Ge-73; a comparisonof the Ge-73 functions from Ressell
CRESST, Edelweiss, and DAMA/Xe-2 (we remind the
shows the results are relatively insensitive to the models
reader that DAMA/Xe-2 is entirely different from the
used to derive the form factors.
DAMA/NaI annual modulation results). Only couplings
within these regions satisfy the null result constraints of
all these experiments. Super-K and Baksan results de-
VI. RESULTS
pend upon WIMPs interacting with hydrogen, and thus
inherently limit the protoncoupling a only. The Super-
p
We first present results for the simple cases where the K and Baksan allowed regions are bands along the a -
n
WIMP couples to only the proton or neutron. From a axis (corresponding to the region between two parallel
combination of constraints in Figure 2, one can see that lines). The DAMA/Xe-2 and Edelweiss allowed ellipses,
8
15GeV
8GeV
5
5
0
0 -5
-5 60GeV
5
p p 0
a a
Remaining
12GeV
allowed -5
5 regions
100GeV
0 5
DAMA
0
CDMS I&II
-5 DAMA
Other DAMA -5 CDMS I&II
Allowed by all
experiments Other
-15 -10 -5 a0n 5 10 15 -15 -10 -5 a0n 5 10 15
FIG. 3: Spin-dependent allowed couplings at several different WIMP masses (in different panels). The legends in the figure
indicate regions allowed by different experiments. The thin black horizontal bands (ellipses at higher masses) show the region
allowed by DAMA (note that the black bands are indicated as white for regions allowed by all experiments); the allowed
couplings for DAMA are those that fall on the bands, not between them. The dark (blue) vertical regions correspond to
couplingsallowedbytheCDMSISiandCDMSIIGedatasets. Theremaining(pink)horizontalregion,labelled as“other”in
thelegend, representscouplings allowed bythecombined nullresultsof Super-K,Baksan, Elegant V,CRESST, DAMA/Xe-2,
and Edelweiss (couplings outside of this region are ruled out by at least one of these experiments). The region allowed by all
experiments is indicated by white bands. Only inside these white bands is the DAMA signal consistent with the null results
from other experiments. Formasses above 13 GeV thereis noconsistent region (no white band).
whose minor axes are along the a direction, are far predominantlyGeathighmassestoSiatlowmasses,the
n
broader in the a direction. The intersection of these shapeofthe allowedregionrotatesslightly(asSiandGe
p
regions (horizontal Super-K & Baksan bands with ver- have different structure functions).
tical DAMA/Xe-2 & Edelweiss ellipses) forms a roughly In sum, we have applied null results from all other
rectangular region above 13 GeV, clearly visible in the experiments to see whether or notDAMA annualmodu-
15GeVpanelandgetting smallerathigher masses. Any lationresults canbe compatible. The only remainingre-
allowed model must lie within this “rectangle.” Below gionsareshowninwhitebandsinFigure3. OnlyWIMP
18 GeV, Super-K and DAMA/Xe-2 no longer provide masses in the range 5-13 GeV survive as spin-dependent
the best limits (or any limit, in Super-K’s case); below candidates matching all the data. In the next section
13GeV,BaksanandEdelweissalsolose their sensitivity. (Summary and Discussion) we discuss the viable ranges
For these lower masses, Elegant V and CRESST I (in- of couplings in the a -a plane.
p n
cluded in the region labelled “other”) provide the best Note that, when one allows both a and a to be
p n
limits; however, they do not significantly constrain the nonzero, the bounds of Figures 1 & 2 on proton-only
DAMA space between 5 and 13 GeV. and neutron-only couplings can be considerably weak-
The recent CDMS results are also shown in Figure 3. ened. We illustrate this with an example in Figure 4,
CDMS’s null result, based upon odd neutron Si and Ge, which shows the limits in the a -a plane separately for
p n
generates an allowed region elongated along the a -axis. the CDMS I Si and CDMS II Ge at a WIMP mass of 10
p
InFigure3,theregionallowedbytheCDMSdataisdark GeV. While Ge limits a to below 160 for a = 0, even
p n
grey (blue) and is labelled “CDMS I&II.” The addition a small non-zero value of a allows for a values several
n p
of CDMS I Si data increases sensitivity at low masses, times larger (e.g. we can have a = 330 at a = 20).
p n
−
due both to the lighter mass and lower threshhold than TheCDMSII GelimitinFigure1fortheWIMP-proton
the CDMS II Ge data. As the sensitivity changes from coupling applies only to the caseofa =0, but the limit
n
9
2200
DAMA
440000
10 GeV CDMS I&II
00
CRESST I
--2200
220000
--4400
pp
aa
--6600
aapp 00
--8800
--220000 --110000 5 GeV
--112200
CDMS I Si 00 225500 550000 775500 11000000 11225500 11550000
aa
--440000 CDMS II Ge nn
Combined
--440000 --220000 00 220000 440000
aa FIG. 5: Spin-dependent allowed couplings at 5 GeV for
nn
DAMA (black elliptical band), CDMS (vertical region), and
CRESST I (angled region). The CDMS and CRESST I
null results only allow for couplings within the intersection
FIG. 4: Spin-dependent couplings allowed by CDMS at a
of their corresponding allowed regions (hatched region); this
WIMP mass of 10 GeV. The separate limits from Si and Ge
intersection does not include any couplings that can repro-
each produce extremely long ellipses; the combined limit is
duce the DAMA observed modulation. Below 5 GeV, the
significantly smaller.
CDMS/CRESST I constraint grows rapidly stronger relative
to the DAMA required couplings. Note the axes are not at
thesame scale.
in the general case is more than an order of magnitude
weaker.
ReferringagaintoFigure4,theSidataformsasimilar
neutron cross-sections are of the same order (equal for
elliptical limit as the Ge, but slightly rotated. Even the
thegivenexamples). Wealsonotethatonlymodelswith
slight rotation is significant enough that the two narrow
a > 1 are consistent at any mass. Below about 5
p
ellipses rarely overlap- the combined Si and Ge limit is | |
GeV, CRESST I becomes significant enough to exclude
significantly smaller than that of either of the separate
(in combination with CDMS) all of DAMA’s parameter
limits. Detectors employing multiple elements canbreak
space (Figure 5).
the (near) degeneracy typical of individual elements.
We note that the small remaining regime of spin-
dependent parameter space could be confirmed or ruled
out by more complete analysis of Super-K or other so-
VII. SUMMARY AND DISCUSSION lar neutrino (indirect detection) data. If Super-K, Bak-
san, and ANTARES (or in the future IceCube and
Figure 3 illustrates our main results. When we allow ANTARES) were able to push down their analyses be-
both a and a to be nonzero, there is no region of pa- low 13 GeV WIMP masses, they would be able to ei-
p n
rameter space with WIMP masses above 13 GeV which ther find the WIMP annihilation from the Sun compat-
is compatible with both the positive results of DAMA ible with DAMA, or rule out DAMA entirely as due to
andthenullresultsofSuper-K,DAMA/Xe-2,Edelweiss, spin-dependent interactions of a WIMP that is its own
andCDMS.Below13GeV,therearelinesegmentsinthe anti-WIMP. If the WIMP and anti-WIMP abundances
a -a plane which are still compatible with the DAMA in the Sun would differ substantially, there would be no
p n
data and all null results from other experiments; CDMS annihilation signal from the Sun even if WIMPs may be
II is the most constraining. At 12 GeV, the compat- abundant enough to give rise to a signal in DAMA. Pre-
ible line segments are a = 0.084a + 1.94( 0.16) viously Ullio et al. [5] attempted to push the Super-K
p n
− ±
with 4.62 a 4.39; at 8 GeV they are a = results down to lower WIMP masses. However, we feel
n p
− ≤ ≤
0.084a +2.81( 0.23) over 15.8 a 15.8 (these thattheiranalysismayhavebeentooconstrainingdueto
n n
− ± − ≤ ≤
describeonlyoneoftheline segments,theotherisfound thefactthattheenergythresholdforthemuonsintheex-
by taking (a ,a ) ( a , a )). This includes cases perimentswasnottakenintoaccount(itwassettozero).
n p n p
→ − −
for which the proton and neutron couplings are compa- We feel that this analysismust be left to the experimen-
rable (e.g. a = a = 1.8 at 12 GeV and a = a = 2.6 talists. We encourage the experimentalists to study the
p n p n
at 8 GeV), in which case the WIMP-protonand WIMP- bounds at lower WIMP masses that would arise when
10
one extends the analysis to more difficult cases: muons thenucleusrecoilenergytofindtherecoilrateRperunit
that stop in the detector, muons that start in the de- detector mass:
tector, and contained events (muons that start and stop
in the detector). For the contained events one should E2/Q ρ
R(t)= dEǫ(E) σ(q)η(E,t). (A.1a)
consider both muon and electron neutrinos. If such an 2mµ2
ZE1/Q
analysis were possible, then one could confirm or refute
theremainingparameterregionforspin-dependentinter- ǫ(E) is the (energy dependent) efficiency of the exper-
actions in DAMA. iment, due, e.g., to data cuts designed to reduce back-
In the future, as other experiments become more sen- grounds. Q is the quenching factorrelating the observed
sitive, they will of course further restrict the parameter energy E (in some cases referred to as the electron-
det
space as well. New CRESST I or SIMPLE data should equivalent energy) with the actual recoil energy E :
rec
also be able to push down the WIMP mass compatible Edet = QErec. The energy range between E1 and E2
with DAMA. Edelweiss, CRESST II and ZEPLIN [42], is that of observed energies for some data bin of the de-
whose role is similar to CDMS in this regard (their el- tector (where experiments often bin observed recoils by
lipsesalignwiththea axis),willalsobecomeimportant. energy). Note some single element detectors can cali-
p
Acceleratorboundsonspin-dependentinteractionsare brate their energy scales to E = E , in which case
det rec
far weaker than the limits we have discussed from dark the quenching factor can be ignored. For detectors with
matter experiments. At the high energies in accelera- multiple elements, the total rate is given by:
tors, the WIMPs couple directly to the quarks and can
no longer be treated as effectively coupling to the nu- R (t)= f R (t) (A.1b)
tot i i
cleons as a whole. Such scattering may occur through
i
X
exchangeofseveraldifferentparticles,leadingtointerfer-
ence effects that further reduce the cross-section. Likely where fi is the mass fraction and Ri is the rate
signaturesofsuchscattersarealsoeitherunobservableor (Eqn. (A.1a)) for element i.
dwarfedbybackgrounds. Lookingatthecaseofasimple Theexpectednumberofrecoilsobservedbyadetector
Z exchange,the lowestordercaseofqq¯ Z+X with Z is given by:
→
producing a WIMP pair Z χχ would not present an
observable signal as the WI→MP itself, being electrically Nrec =MdetTR (A.2)
neutralandweaklyinteracting,isnotdirectlydetectable.
Thesignalispotentiallyobservableifagluonisalsoemit- where Mdet is the detector mass and T is the exposure
ted,qq¯ Z+g+X,butotherStandardModelprocesses time. To calculate the constants in Eqn. (11), we first
produce→the same signatures at rates much larger than rewrite Eqn. (10) as:
those expected for a and a of order one. Even assum-
p n 2 2
σ (q)=h (q)a +h (q)a a +h (q)a (A.3)
ing no interference effects, the experimental limits from SD pp p pn p n nn n
thisandsimilarprocessesgivea <O(102),muchweaker
p ∼
where
thanthelimitsofO(1)fromthedarkmatterexperiments.
As a final remark, we note that the the Minimal Su- 2 2
32µ G
persymmetric extension of the Standard Model does not h (q) FS (q) (A.4)
ij ij
≡ 2J+1
provide a dark matter candidate with the characteristics
pointed to by our study. isindependentofthecouplingsa anda . Theconstants
p n
A , B , and C in Eqn. (11a) are then determined
rec rec rec
from Eqn. (A.2) by replacing σ(q) in Eqn. (A.1a) with
Acknowledgments the appropriate h (q), e.g.
ij
C. Savage thanks G. Kane and R. Schnee for useful E2/Q ρ
A =M T dEǫ(E) h (q)η(E,t).
discussions. We thank M. Kamionkowski for kind com- rec det 2mµ2 pp
ments on the first version of this paper. We acknowl- ZE1/Q
(A.5)
edge the support of the DOE and the Michigan Center
Similarly,B isgivenbyEq.(A.5)butwithh replaced
rec pp
for Theoretical Physics via the University of Michigan.
by h , and C is given by Eq.(A.5) but with h re-
pn rec pp
We thank the MCTP for hosting the Dark Side of the
placed by h .
nn
Universe Workshop, during which most of this work was
For large detectors that can obtain high statistics, the
performed.
annualmodulationoftherate(due totherotationofthe
Earth about the Sun) may be detectable, where
APPENDIX: DETECTOR RECOILS
R(t) R0+Rmcos(ωt). (A.6)
≈
Todeterminethenumberofexpectedrecoilsforagiven Therate(formostenergiesandforaMaxwellianvelocity
experiment and WIMP mass, we integrate Eqn. (1) over distribution)ismaximizedaroundJune2andminimized