Neutrino Constraints on Inelastic Dark Matter after CDMS II

0.54 MB·

by  Jing Shu

IPMU-10-0002
Neutrino Constraints on Inelastic Dark Matter after CDMS II
Jing Shu a,∗ Peng-fei Yin b,† and Shou-hua Zhu b‡
aInstitute for the Physics and Mathematics of the Universe,
The University of Tokyo, Chiba 277 −8568, Japan
bInstitute of Theoretical Physics & State Key Laboratory of Nuclear Physics and Technology,
Peking University, Beijing 100871, P.R. China
0
1
0
We discuss the neutrino constraints from solar and terrestrial dark matter (DM) annihilations
2
in the inelastic dark matter (iDM) scenario after the recent CDMS II results. To reconcile the

n DAMA/LIBRA data with constraints from all other direct experiments, theiDM needsto be light
a (mχ<100GeV)andhavealargeDM-nucleoncrosssection(σn ∼10−4 pbinthespin-independent
J (SI)scatteringandσn ∼10pbinthespin-dependent(SD)scattering). Thedominantcontribution

7 totheiDMcaptureintheSunisfromscatteringoffFe/AlintheSI/SDcase. Currentboundsfrom
Super-KamiokandeexcludethehardDMannihilation channels,suchasW+W−,ZZ, tt¯andτ+τ−.
] For soft channels such as b¯b and cc¯, thelimits are loose, but could be tested or furtherconstrained
h
byfutureIceCubeplusDeepCore. Forneutrinoconstraintsfrom theDMannihilation intheEarth,
p
due to the weaker gravitational effect of the Earth and inelastic capture condition, the constraint
p- exists only for small mass splitting δ< 40 keV and mχ∼(10,50) GeV even in theτ+τ− channel.
e
h
[ I. INTRODUCTION be settled.

1 Veryrecently,theCryogenicDarkMatterSearch
v A wide variety of cosmological observations, (CDMS II) announced the observation of two sig-
6 which include the highly precise measurements of nal events 1 with a 77% confidence level and ad-
7 the cosmic microwave background [1], galactic ro- ditional two events just outside the signal region
0
tation curves [2], the weak gravitationallensing of border [8]. Statistically this is not significant to
1
. distant galaxies by foreground structure [3], and claimadiscovery,nevertheless,itisverysuggestive
1 the weak modulation of strong lensing around in- ifonebelievesuchsignalsarecomingfromDMnu-
0
dividual massive elliptical galaxies [4], all indicate cleiscattering. Inparticular,thesetwoeventswith
0
that about 22% of our universe consists of non- small recoilenergies suggestthat the mass ofdark
1
: baryonic, non-luminous dark matter (DM). The matterissmallerthan100GeV2 withSIcrosssec-
v nature of DM, on the other hand, still remains a tion O(10−8)pbwhichcouldescapeotherexper-
i ∼
X mystery. imental constraints. Many models have been pro-
posedsincethen[9]. Inaddition,limitsonthedark
r The first direct detection experiment that pro-
a matternucleuscrosssectionareobtainedfromthe
vided strong evidence for dark matter is the
data. In the inelastic spin-independent (iSI) DM
DAMA collaboration, which announced an 8.3 σ
models, only a very narrow region of parameter
discovery [5] in the annual modulations of nuclear
space is allowed[8,10–12], but the constraintcan
recoil rate. In the simplest DM models, such an
be large relaxed if one consider the inelastic spin-
event rate is excluded by two orders of magnitude
dependent (iSD) DM models [12].
in other low-background nuclear experiments. A
widelyinvokedresolutionofthiscontroversyisthe Other experiments that give a strong constraint
inelastic dark matter (iDM) model [6, 7], which on the DM-nucleon cross section besides the di-
introduces a transition to an excited state in the
dark matter nucleus scattering. If the mass split-
ting δ is roughly 100 keV, which is close to the
kinetic energyofa DMinthe halo,the kinematics 1 CDMShasaverylowbackgroundcomparingtotheother
is significantly modified and the controversy can directdetectionexperimentsandthetwoobservedevents
are well separated from other events that have failed to
passthecuts. Inotherexperiments,forinstanceXENON
[10], manyobserved events areat the border of the cuts
andareusedtosetlimitsonDMscatteringcrosssection.
∗Electronicaddress: [email protected] 2 Interestingly,thepreferredDMmassregionlargelyover-
†Electronicaddress: [email protected] laps with the one used to explain DAMA in the iDM
‡Electronicaddress: aiwen˙[email protected] model.
2
rect detection experiments are the neutrino tele- C = σv /V , C is the inverse time for DM
A ⊙eff E
h i
scopes which are detecting the neutrino flux from to escape via evaporation which will be neglected
DMcapturedandannihilatedintheSunorEarth. below since our DM is not very light. For typical
For example, Ref. [13] and Ref. [14, 15] give the DM we have considered, the time scale required
neutrinoconstraintstothelightelasticDM(eDM) to reach equilibrium between capture and annihi-
andiSIDMwhichcouldaccountforDAMAresults lation, t =1/ C C , is much smaller than the
eq ⊙ A
respectively. When the DM nucleus scattering is age of the solar system so the annihilation rate is
p
iSD, the constraints by the neutrino flux from the saturated at half of the capture rate Γ = C /2.
A ⊙
Sun can be dramatically different. The reason is Therefore, calculating the DM capture rate in the
that the kinetic energy requirement for a DM (χ) Sun is crucial to estimate the final neutrino flux
scattering inelastically with a nucleus (N) is from the Sun [14–18].
Suppose a DM particle has a velocity u very far
m
E >δ 1+ χ . (1) away from the Sun, at some point the escape ve-
χ
m
(cid:18) N(cid:19) locity in a shell is v(r), the DM velocity w will
For light elements like hydrogen, only very ener- be
getic DM particles can scatter with them and get
w2 =u2+v(r)2. (3)
captured in the Sun if δ is not very small. Then
the overall capture rate with their dominant con-
In the DM-nucleus center ofmass frame, it is easy
tributions from light elements in the Sun will be
toseethatinordertohaveaDM-nucleusinelastic
significantlyreduced. The modified kinematicsfor
scattering, the following condition is required
inelastic scattering in Eq. (1) also largely reduced
the kinetic phase space of DM captured in the µw2
>δ, (4)
Earth because of the small escape velocity of the
2
Earth. Therefore, it is of importance to preform a
detailed analysis regarding the above issues moti- where µ mχmN/(mχ+mN) is the DM-nucleus
≡
vated by the recent DAMA, CDMS results. reducedmass, δ is the mass splitbetweentwo DM
The paper is organized as follows. In Section states. The escape velocity v(r) is approximated
II, we calculate the capture rate for both iSI and as [19]
iSD DM in the Sun and we discuss the possible
M(r)
uncertainties in the calculation. In Section III, we v2(r)=v2 (v2 v2), (5)
obtain the constraint on the DM-nucleon inelas- c − M⊙ c − s
tic cross section for different annihilation chan-
where M is the mass of the Sun and v = 1354
nels from Super-Kamiokande (Super-K) and the ⊙ c
km/s and v = 795 km/s. M(r) is the mass pro-
expected event number from the on-going large s
file of the Sun which can be obtained by the Sun
volume neutrino telescope, IceCube. In Section
density profile [20]
IV, we will discuss the neutrino constraint from
iDM captured in the Earth. Finally we conclude r
ρ(r)=ρ exp( B ) , (6)
and discuss our studies in Section V. 0 − R
⊙
where ρ =236.93 g/cm3 and B =10.098.
0
II. CAPTURE RATE FOR INELASTIC The DM capture rate in the Sun is given by
DARK MATTER IN THE SUN
f(u)
C = 4πr2dr du wΩ(w) , (7)
⊙
When DM particles travelthrough the Sun sys- u
Z Z
tem, they may get trapped by the gravitational
where f(u) is the DM velocity distribution at in-
potential and continue to lose energy with every
finity, Ω(w) is the rate per unit time that a DM
collision against nucleus and captured in the Sun.
with velocity w scatter to a velocity less than v.
The time evolution of the DM population in the
We assume the DM velocity distribution far from
Sun is given by the differential equation
the Sun followsaMaxwell-Boltzmanndistribution
N˙ =C C N2 C N, (2) in the Sun rest frame
⊙ A E
− −
where C⊙ is the capture rate of the DMs in the f(x)dx= ρχ 4 x2e−x2e−η2sinh(2xη)dx (8)
Sun and annihilation rate is ΓA = CAN2/2 with mχ√π 2xη
3
where the dimensionless variables are defined as where A is the atomic mass number, Z is the
x2 u2/v2 and η2 v2/v2, v =220 km/s is the charge of the nucleus, µ m m /(m +m ) is
≡ 0 ≡ ⊙ 0 0 χn ≡ χ n χ n
average local circular velocity which is related to theDM-nucleonreducedmass,f (f )istheSIDM
p n
DM velocity dispersion v¯ as v2/v¯2 =2/3. coupling to protons (neutrons) and we normalize
0
Takingintoaccounttheformfactorsuppression, it to f = f , σSI is the DM-nucleon SI scatter-
p n n
the scattering function Ω(w) is given by ing cross section. We can see that if f f , the
p n
DM-nuclei SI cross section is proportion≃al to A2.
niw Qimax Moreover,if mχ >>mN, the reduced mass µ mN
Ω(w) = Xi Qimax−Qimin ZQ′imin dQi wcainllperxopveicdtehaenaovtyhenruecnlheianwcietmhelnatrgfaecatobruonfdAan2c.eWine
× σiSIFi2(Qi)+σiSDSi(Qi) (9) the Sun, suchasFe, to playa moreimportantrole
in the capture process. For the SI form factor, we
(cid:0) (cid:1)
where i denotes each species of nuclei in the Sun use the Helm form factor as,
and we will omit it below, n is the number den-
sity of the nuclei, Q is the energy transfer in the F(Q)=3e−k2s2/2sin(kr)−krcos(kr) , (15)
scattering,Qmax(Qmin)denotesthe possiblemax- (kr)3
imum(minimum) energy transfer and Q is the
cap
minimal energy transfer which requires DM could with nuclear skin thickness s=1 fm, effective nu-
be captured only with a velocity less than v, and clear radius r = √R2 5s2, R = 1.2A1/3 fm, and
′ −
we define lower limit of integration as Q = momentum transfer k =√2m Q.
min N
max(Q ,Q ). These energy transfer func- The SD cross section with form factor for spin-
cap min
tions are given by [14] non-zero nuclei is
1 µ2 4π δ µ 2
Q = m w2 1 σSDS(Q)= 1
max 2 χ (cid:26) − m2N 3a2p(2J +1)s − µw2/2(cid:18)µχn(cid:19)
1 mN 1 δ 2 δ(10) × a20S00(k)+a0a1S01(k)+a21S11(k) σnSD , (16)
× − mχs − µw2/2! − wher(cid:0)e a =a +a and a =a a ar(cid:1)e isoscalar
 0 p n 1 p n
−
1 µ2 spin and isovector spin couplings, ap and an are
Qmin = 2mχw2 1− m2  theeffectivespinDM-protonandDM-neutroncou-
(cid:26) N plings which we assume ap =1 and an =0 in the
2
below, S (k),S (k),S (k) are nuclear structure
m δ 00 00 01
1+ N 1 δ(11) functions. Inmanynucleimodels,thenuclearspin
× mχs − µw2/2! − is carried by the unpaired nucleons. Therefore we
1 focus on the nuclei with an odd nucleon number
Qcap = 2mχ(w2−v2)−δ ,  (12) in the SD scattering process. In the calculation
weusethenuclearstructurefunctions providedby
and one has to impose the condition Ref. [21].
The capture rate of 50 GeV DM due to differ-
Qmax >Qcap (13) ent species of nuclei in the Sun as a function of
the mass splitting δ is calculated for both iSI and
for the capture to happen. σSI(SD) is the inelastic iSD DM. The results depend on some astrophysi-
SI (SD) crosssection and F(Q) is the SI form fac- calparametersandasareferencepoint,we choose
tor. The term σSDS(Q) will be decomposed into ρ =0.3 GeV cm−3, v =220 km/s and v =220
χ ⊙ 0
SDcrosssectionsandSDformfactorsfordifferent km/sintheMaxwell-Boltzmanndistribution. The
spin J later. The inelastic SI (SD) cross section is element abundances are given by Ref. [22] in the
given by form of ǫ = log(n /n )+12 and the mass frac-
i i H
tionofhydrogen(helium)is0.7381(0.2485). Inthe
δ µ 2 calculation for SI capture, we pick out the species
σSI = s1− µw2/2(cid:18)µχn(cid:19) 2w3iNtha,ǫi27>Al6,.2F9Sori aSnDdc3a9pKtu.rAes,wweeccaalncusleaetein1HF,IG19.F1,
2
fpZ+fn(A−Z) σSI (14) and2,FeandAlprovidemostcontributionstothe
× f n iSI and iSD scattering processes respectively and
(cid:18) p (cid:19)
4
structure [24]. We also assume the DM infinity
25
10 H velocity distribution is a Maxwell-Boltzmann dis-
He tributionwhichdepends onparametersv andv .
)-1 s CN As discussed in Ref. [15], if these two⊙parame0-
(
re rate 1024 ONMSieg t3fae0cr0stokvrmalre/ysss,intthhatehncetawrpeotgu.iroOennroeaftseh2ow0u0illldkbmael/ssoch<naontgviec⊙de,vtb0hya<ta,
u
pt S if the velocity distribution deviates from an ordi-
a 23 Fe
C10 Ar nary Maxwell-Boltzmann distribution, the result
Na will also be changed. In our discussions, for sim-
Al
Ca plicity, we assume the solar chemical composition
22 Ni does not change in the Sun. The capture rate is
10
20 40 60 80 100 120 140 160 180 200 sensitive to the abundance of heavy nuclei and in-
(keV) creasingthe number density of heavynucleiin the
FIG.1: Capturerateof50GeVspin-independentiDM center of the Sun will increase the overall capture
with DM-nucleon cross section of σSI ∼10−4pb as a ratebyafew[15]. Therearealsouncertaintiesaris-
n
functionofthemasssplittingδ duetodifferentspecies ing from the nuclear form factor. For the SI scat-
of nuclei in the Sun. We choose ρχ = 0.3 GeV cm−3, tering, the Helm form factor we used in the calcu-
v⊙ = 220 km/s and v0 = 220 km/s in the Maxwell- lationislessaccurateatlargemomentumtransfers
Boltzmann distribution. andmightbeimprovedbyusingsomemoreprecise
form factors [25]. For the SD scattering, the nu-
clear structure functions in Ref. [21] are only con-
25
10 sidered for direct DM search on the Earth, so the
19
F q usedinthefinitemomentumtransferapprox-
)-1 s 24 2237Na immaaxtion may not be large enough for DM-nucleus
( 10 Al
ate 2399Si scattering in the Sun if our DM is heavy.
ure r 1023 1HK (x10-3)
pt III. LIMITS FROM THE NEUTRINO
a TELESCOPES
C
22
10
If the DM capture and annihilation processes
21 reach equilibrium, we can simply get the DM an-
10
20 40 60 80 100 120 140 160 180 200 nihilation rate
(keV)
Γ =C /2 (17)
FIG. 2: Capture rate of 50 GeV spin-dependent iDM A ⊙
with σnSD ∼ 10 pb as a function of the mass splitting ThedifferentDMannihilationchannelsgivedif-
δ dueto different species of nucleiin theSun.
ferentinitialneutrinospectra[26]. IftheDManni-
hilates to e+e− or µ+µ−, they will not contribute
to neutrino signals. This is because for muons,
one can safely neglect the Hydrogen contribution they always lose most of their energy before de-
in the iSD scattering for δ >10 keV. The capture caying in the center of the Sun. For annihilation
rate is always sensitive to the abundance of heavy channel to τ+τ−, the neutrinos are induced by τ
nuclei. leptonic decays τ µνν, eνν, and hadronic de-
→
Finally, we give some short comments on the cays such as τ πν, Kν, ππν et al. For the
possibleuncertaintyinthecalculationforsolarDM heavy final anni→hilation states W+W−, ZZ, tt¯,
capture rate. The input astrophysics parameters they produce neutrinos via cascade decays and
are the local DM mass density ρ , velocity of the the neutrino spectra for such channels are hard.
χ
Sun v , average local circular velocity v , and so- Forquarkchannels,thehadronizationprocesspro-
⊙ 0
lar elemental abundances ǫ . The capture rate is duces mesons and baryons, and then induces neu-
i
proportional to ρ and the results will increase if trinos via hadron decays. The neutrino spectra
χ
a larger local DM mass density is available [23] for such channels are soft. Notice that the light
or the solar system is passing through DM sub- mesons easily lose their energy before decays, so
5

3 3
10 10
2 2
) 10 ) 10
b b
p p
-4 0 101 -4 0 101
1 1
/iSI100 /iSI100
( (
r* -1 t r* -1 t
B10 Z B10 Z
W W
-2 b -2 b
10 c 10 c
-3 q -3 q
10 1 2 3 10 1 2 3
10 10 10 10 10 10
m (GeV) m (GeV)
FIG. 3: Super-K solar neutrino limits on the cross section per nucleon times the branching ratio to different
annihilation channelsfortheiSIDMwithmasssplittingδ=40keV(left) andδ=130keV(right). Weshowthe
limits on seven annihilation channels including ττ¯, tt¯, W+W−, ZZ, b¯b, cc¯and qq¯(q denotes light quark here).

3 3
10 10
2 2
) 10 ) 10
b b
10p101 10p101
/D /D
( iS100 ( iS100
Br*10-1 tZ Br*10-1 tZ
W W
-2 b -2 b
10 c 10 c
-3 q -3 q
10 1 2 3 10 1 2 3
10 10 10 10 10 10
m (GeV) m (GeV)
FIG. 4: Super-K solar neutrino limits on the cross section per nucleon times the branching ratio to different
annihilation channels for theiSD DM with mass splitting δ=40 keV (left) and δ=130 keV(right).
thecontributionstotheneutrinosignalsfromlight is
quarks are usually small.
dN Γ dN
ν A ν
= Br (18)
The neutrinos produced at the solar center will dE 4πR2 i dE
ν SE i (cid:18) ν(cid:19)i
interact with the matter in the Sun. These effects X
include the neutral current (NC) interaction, the
chargedcurrent(CC)interaction,andtauneutrino
ντ reinjectionfromsecondarytaudecays. Theneu- where i runs over the different DM annihilation
trinooscillationsincludingthevacuummixingand channels with branching ratios Br , dN /dE is
i ν ν
the MSW matter effects are also important. Here the neutrinospectrumatthe Earthafter propaga-
weusetheresultsgivenbyRef. [26]totakeintoac- tion, and R is the Sun-Earth distance.
SE
counttheaboveeffectsduringtheneutrinos’prop-
agation. ThedifferentialneutrinofluxattheEarth We can calculate the muon rate at the detector
6

3 3
10 10
)b 102 )b 102
p p
-4 10 101 -2 10 101
/eSI 100 /eSD 100
(Br* 10-1 tZ (Br* 10-1 tZ
W W
-2 b -2 b
10 c 10 c
-3 q -3 q
10 1 2 3 10 1 2 3
10 10 10 10 10 10
m (GeV) m (GeV)
FIG. 5: Super-K solar neutrino limits on the cross section per nucleon times the branching ratio to different
annihilation channels for the elastic spin-independent (eSI) DM (left) and elastic spin-dependent (eSD) DM
(right).
as 3 neutrino-nucleoncrosssectionandmuonrangein-
creaseasE ,whiletheatmosphericneutrinoback-
ddEφµ = mχdEνµ ddNEνµ[dσCνpC(dEEνµ,Eµ)np ground decνrease as Eν3.
µ ZEth νµ µ Toobtainlimitsonthecrosssectionpernucleon
+(p n)]R(E )+(ν ν¯) (19) timesthebranchingratioσ Br ,weusetheupper
µ n i
→ → ·
bound on high energy up-going muon flux in the
where np (nn) is the number density of protons directionofSungivenby Super-Kgroup[29]. The
(neutrons) in matter around the detector, the Super-Kdetectoris locatedinaminewith1000m
muon range R(Eµ) is the distance that a muon rock overburden which contains 50,000 ton water.
could travel in matter before its energy drops be- Thedetectorcandetectup-goingmuonswithmea-
low the detector’s threshold energy Eth which is sured path length of at least 7 m which is equiva-
given by lent to the detector energy threshold 1.6 GeV. So
the Super-K detector has a wide detecting range
1 α+βE
R(E )= ln µ , (20) for high energy neutrino. Super-K groupanalyzed
µ
ρβ (cid:18)α+βEth(cid:19) the data in 1679.6 days of detector live time, and
no events induced by DM annihilation/decay are
withα,β theparametersdescribingtheenergyloss
confirmedinthedirectionsoftheSun,thecenterof
of muons in matter.
Earth, and the Galactic Center. A stringent limit
The cross sections of deep inelastic neutrino-
for up-going muon flux O(10−15) cm−2 s−1 con-
nucleon scattering processes are given by [28]
strains the DM model. Here we use the constraint
dσCνCp,/nNC(Eν,y) 2mp,nG2F E inTRheef.S[2u9p]erfo-Kr allilmaintsniohnilaσtionBcrhantonedlisff.erent an-
dy ≃ π ν n· i
nihilation channels for iSI and iSD DM with dif-
aνp,n +bνp,n (1 y)2 , (21) ferent mass splitting δ are presented in FIG. 3,
× CC/NC CC/NC −
and 4 respectively. For a typical DM annihilation
(cid:16) (cid:17)
where y 1 E /E . We can see that the neu- cross section of 10−4 pb (or higher) that explains
ℓ ν
trinodete≡ctor−ismorepowerfultoobserveneutrino DAMA in the iSI DM case, the limits are quite
signals in the high energy region. Because the constraining on hard channels (W+W−, ZZ, tt¯,
τ+τ−), are lessened in softer channels like cc¯and
b¯b and are quite loose in the light quark channels.
In the iSD DM case, the typical DM annihilation
3 A different formula that accounts for the energy depen- cross section (or higher) to explain DAMA is en-
dentmuonfluxcanbefoundinRef. [27]. hanced to 10 pb [12]. Interestingly, even though
7
the cross section is 105 times higher than the iSI channels and soft channels to b¯b and cc¯. For the
DM case, we reach a similar conclusion that the iDM lighter than 100 GeV, comparing with FIG.
limits arequite constrainingon hardchannels,are 3 and 4 based on the Super-K 90% C.L. exclusion
lessened in softer channels like cc¯ and b¯b and are limit, we can see that the IceCube plus DeepCore
quite loose in the light quark channels. In order also has capability to test or further constrain the
to see how the inelastic property of DM nucleus iDM model with soft annihilation channels to b¯b
scatteringchangesthelimits, wealsocalculatethe and cc¯.
Super-K limits on σ Br to different annihila-
n i
·
tion channels for eDM. In the SI case, making the
scattering inelastic changes the results at most an IV. NEUTRINO CONSTRAINT FROM
order of magnitude for reasonable δ. In the SD TERRESTRIAL INELASTIC DARK
MATTER ANNIHILATIONS
case, however, the limit is loosened by 3 orders of
magnitudewhenhydrogennolongercontributesto
the dark matter capture in the Sun. Unlike the Sun, the main constituents of the
Earth are heavy nuclei, such as O, Al, Si, Mg,
The on-going large volume neutrino telescopes,
Fe and Ni. Since the abundances of Fe and Al
suchasIceCubeandKM3NET,aremorepowerful
are 31.9% and 1.59% respectively, we expect that
to probe high energy neutrinos. IceCube located
the abundance of such heavy elements helps the
betweendepthof1.45kmand2.45kminthesouth
capture process. Nevertheless, probing neutrino
pole willhaveaneffective detecting areaof1km2.
signals from terrestrial iDM annihilations is more
In 2007, IceCube consisted of 22 strings has pro-
difficultthanprobingthemfromsolariDMannihi-
vided a more stringent upper limit to muon flux
lations. Therearetwomainreasonsforthis. First,
from the Sun [30]. The main shortcoming of Ice-
as a gravitational system, the Earth is far smaller
Cube22-stringsisitshighenergythreshold. Sothe
than the Sun with a rather small escape velocity,
experimental results are only useful to constrain
so it is more difficult to trap DM in the center of
the heavy DM 4. The complete detector consisted
the Earth. In the iDM scenario, the situation be-
of 80 strings might reduce the energy threshold to
comes even worse due to the kinetic suppression
50 GeV. We need to notice that the effective de-
induced by inelastic capture condition. Second, it
tectingareaofIceCubedecreasesrapidlyinthelow
is well known that terrestrialDM capture and an-
energy region. In addition, the angular resolution
nihilation processes can not reach equilibrium in
becomesworseinsuchenergyregion. Sothecapa-
the ordinaryDMscenario,sothe annihilationrate
bility of IceCube to search for light DM is weaker
of DM trapped in the Earth is even smaller than
than that for heavy DM.
half of the capture rate.
But it is worthremarking,additionalIceCube 6 NowweturntoEq. (4)and(13)andseethefirst
special strings named DeepCore, are powerful to reason of suppression explicitly. We can rewrite
improve the capability of the detector and reduce Eq. (13) as
thedetectionthresholddownto10GeV[31,32]. In
the FIG. 6, we show the total number of events at χ′ u2
> (22)
theIceCubeplusDeepCore. Hereweusetheeffec- χ′2 v2
−
tiveneutrinodetectionareaofIceCubeplus Deep-
Core given by Ref. [32] to calculate the event rate where we use a similar notation as Ref. [16]
atthe detector. Thenumberofbackgroundevents
from atmospheric neutrinos within a 3◦ window is χ mχ , s 1 2δ/(µw2), (23)
estimated 50 in the energy range E (10,200) ≡ m ≡ −
N
∼ ∼
GeV. For heavy iDM m >100 GeV, the IceCube p
χ
and
plus DeepCoreis verypowerfulto testmostofthe
iDM annihilation channels, which include all hard
χ s 1+s 1 s2
χ′ − , χ′ χ + − . (24)
− ≡ 2 ≡ 2 4
(cid:18) (cid:19)
The escape velocity of the Earth v 10 km/s is
4 For the b¯b and W+W− channels, the constraints are ≃
very small compared to the escape velocity of the
available for DM heavier than 250 GeV and 500 GeV
Sun 103km/sandthevelocitydispersionoflocal
respectively. FortheconstraintstoiDM,onecanseethe
≃
Ref. [15]. DMu v¯ 270km/s,sotherighthandsideofthe
∼ ≃
8

5 5
10 10

t t
()-1 s yr110034 ZWbc ()-1 s yr110034 ZWbc
nt q nt q
e e
v 2 v 2
E10 E10
1 1
10 10
0 0
10 1 2 3 10 1 2 3
10 10 10 10 10 10
m (GeV) m (GeV)
FIG. 6: (left) Neutrino event number in one year at IceCube (with DeepCore) for the iSI DM with σSI = 10−4
n
pb (left) and iSD DM with σSD = 10 pb (right). Wechoose mass splitting δ=130 GeV here.
n

17 17
10 10
10 GeV
) )
(-1 s1016 2300 GGeeVV (-1 s1016
e 40 GeV e
at 15 50 GeV at 15
e r10 e r10
r r
u u
pt1014 pt1014
a a
C C
10 GeV
13 13
10 10 15 GeV
20 GeV
12 12 25 GeV
10 10
20 40 60 80 10 20 30 40 50
(keV) (keV)
FIG. 7: Terrestrial iDM capture rate for the iSI DM with σSI = 10−4 pb (left) and iSD DM with σSD =10 pb
n n
(right).
capture condition Eq. (22) is a large value. When not reach equilibrium in ordinary DM scenario 5.
the mass of DM is comparable to or larger than Following from Eq. (2), the DM annihilation rate
that of a nucleus, such as Fe or Al, the left hand is
sideofEq. (22)issuppressedbym soitisdifficult
χ C
tosatisfythecapturecondition. IftheDMismuch Γ = ⊕ tanh2(t /t ), (25)
A ⊕ eq
2
lighter than the nuclei, the condition Eq. (22) is
still not easy to satisfy. Moreover it is difficult to where t is the age of Earth, C is the capture
⊕ ⊕
satisfy the inelastic scattering condition Eq. (4) rate of the DMs in the Earth. The equilibrium
simultaneously. So the most promising situation time, effective annihilation rate is defined as t
eq
≡
is that DM and nuclei have similar masses and δ
must not be large.
5 IftheDMannihilationcrosssectionisfarlargerthanor-
dinaryvalue3×10−26 cm3/s duetosomeenhancements
suchasSommerfeldeffect,theterrestrialDMcaptureand
TheterrestrialiDMcaptureandannihilationdo annihilationcouldreachequilibrium[34].
9

2 2
10 10
) 1 ) 1
b 10 b 10
p p
-4 0 100 10 100
(/1iSI10-1 (/iSD10-1
Br*10-2 10 GeV Br*10-2
20 GeV 10 GeV
-3 30 GeV -3 15 GeV
10 40 GeV 10 20 GeV
50 GeV 25 GeV
-4 -4
10 10
20 40 60 80 10 20 30
(keV) (keV)
FIG.8: Super-Kterrestrialneutrinolimitsonthecrosssection pernucleontimesthebranchingratiotodifferent
annihilation channels for theiSI DM (left) and iSD DM(right).
1/ C C , C σv /V respectively. It is onlyshowthemoststringentconstraintsassuming
⊕ A A ⊕eff
also useful to defi≡neha ciritical capture rate for the all the DM annihilation products are τ+τ−. We
Eaprth as Cc 1/C t2 1014s−1(TeV/m )3/2. can see the results in FIG. 8 and we find that the
If C > Cc⊕, t≡he DMAc⊕ap∼ture and annihilatiχon in constraints from Earth-born neutrinos are weaker
⊕ ⊕
the Earth could reach equilibrium. than those from the Sun. The most stringent con-
It is straightforwardto get numerical results by straintsarisefromsmallδ andm (10,50)GeV.
χ
∼
using the method in Sec II. Here we use the mass
density profile and the composition of the Earth
V. CONCLUSION AND DISCUSSION
provided by Ref. [20] and [33] respectively. For
simplicity, we assume the Earth is in free space
[16]. If the gravitational interaction of the Sun Inelastic dark matter (iDM) is an excellent can-
is taken into account, there is a correction factor didate to explain the DAMA annual modulation
of O(1) [35] to the capture rate. In FIG. 7, we result. Recently, the CDMS II announced the ob-
show the results of terrestrial iDM capture rate servationoftwosignaleventsandsetamorestrin-
for SI and SD scattering processes. We find that gent constraint to the iDM model to account for
for very heavy or light DM, the capture process DAMAresults. ThemassoftheiDMisnotheavier
is highly suppressed. So we only give the capture than100GeVandtheDM-nucleoncrosssectionis
rate for the iDM with typical mass in the range typically 10−4 pb (orhigher)for iSI case andis 10
of (10, 50) GeV which is similar to the mass of pb (or higher) for iSD case.
nuclei of interest. It is interesting to notice that In this work, we discuss the constraints by neu-
therearesurgesasδ isvarying. Thepeaksofthese trino signals from solar and terrestrial iDM anni-
surges(χ′ 0)arequitesimilarto the “resonant hilations motivatedafterDAMA andCDMS II re-
− →
enhancement” in the eDM case when DM mass sults. The neutrino flux from the Sun or Earth
reaches to the nuclei’s mass as discussed in Ref. depends onthe capture rateand the constrainton
[16] and is absent if the DM is not light enough. DM-nucleon cross section is quite different from
For the iDM scenario, we need to notice that for the eDM case. For iSI/iSD scattering, the main
the n species of nuclei of interest here, there are contributiontothecapturerateintheSunisfrom
2n inelastic capture conditions to be satisfied or Fe/Al and in the latter case, the capture from
obeyed. So if the mass of DM is similar to that H is highly kinematically suppressed. Although
of the nuclei, the terrestrial capture rate is very the types of nuclei that contribute to the capture
sensitive to the parameters of m , m and δ. rate arequite different iniSI andiSD DM models,
N χ
Here we use the upper limit for up-going muon the constraints for the iDM annihilation channels
flux from the center of the Earth provided by for the typical DM-nucleon cross section are ac-
Super-K [29] to constrain the iDM model. We tually similar. The Super-K null results set very
10
stringent constraints to the hard DM annihilation prospects we have obtained from neutrino tele-
channels such as W+W−, ZZ, tt¯and τ+τ−. For scope should be considered in any explanations
the soft channels such as b¯b and cc¯, the limits are based on iDM. Other on-going indirect detection
loose. WiththeIceCube80-stringsplusDeepCore, experiments and the Large Hadron Collider also
it can discover or exclude such channels. If the have the capability to detect the DM signals. It is
DM annihilation products are light charged lep- exciting and useful to think about all these exper-
tons e+e−, µ+µ−, light mesons or some new light iments and achieve a complete picture of DM.
bosons,there are no constraintsfrom the neutrino
detection. For neutrinos from DM annihilation in
the Earth, the constraint is weaker than the one
VI. ACKNOWLEDGEMENT
in the Sun. This is mainly due to the fact that
Earthbeingasmallgravitationalsystemthatcap-
tures the DM. Moreover, the terrestrial iDM cap- We would like to thank Paul H. Frampton,
ture rate is suppressed due to the inelastic cap- Sourav Mandal and Matthew Sudano for useful
ture condition. Furthermore, the terrestrial iDM discussions. TheworkofJ.S.wassupportedbythe
capture and annihilation do not reach equilibrium World Premier International Research Center Ini-
in most of its parameter space. The constraint is tiative (WPI initiative)MEXT, JapanandGrant-
available only for small δ and iDM with mass of in-Aid for scientific research(Young Scientists (B)
severaltens ofGeVwheresimilarbehaviorsasthe 21740169) from Japan Society for Promotion of
“resonance enhancement” in the eDM case occur. Science(JSPS).TheworkofP.F.Y.andS.H.Zwas
IftheDAMAandCDMSIIresultsarereallyin- supported by the Natural Sciences Foundation of
duced by DM, we could expect the XENON 100 China (Nos. 10775001, 10635030). P.F.Y. would
[36] or other direct detection experiments to give like to acknowledge the hospitality of Institute for
the similar observations in the immediate future. Physics and Mathematics of the Universe (IPMU)
Nevertheless, the constraints and future detection while the work was initiated.
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