Neutrinos fromWIMP annihilation in the Sun: Implications of a self-consistent modelof the Milky Way’s dark matter halo

Susmita Kundu* and Pijushpani Bhattacharjee†

AstroParticle Physics & Cosmology Division and Centre for AstroParticle Physics, Saha Institute of Nuclear Physics,1/AF Bidhannagar, Kolkata 700064, India

(Received 6 December 2011; published 20 June 2012)

Upper limits on the spin-independent as well as spin-dependent elastic scattering cross sections of low

mass (� 2–20 GeV) weakly interacting massive particles (WIMPs ) with protons, imposed by the upper

limit on the neutrino flux from WIMP annihilation in the Sun given by the Super-Kamiokande (S-K)

experiment, and their compatibility with the ‘‘DAMA-compatible’’ regions of the WIMP parameter

space—the regions of the WIMP mass versus cross-section parameter space within which the annual

modulation signal observed by the DAMA/LIBRA experiment is compatible with the null results of other

direct-detection experiments—are studied within the framework of a self-consistent model of the finite-

size dark matter halo of the Galaxy. The halo model includes the gravitational influence of the observed

visible matter of the Galaxy on the phase-space distribution function of the WIMPs constituting the

Galaxy’s dark matter halo in a self-consistent manner. Unlike in the ‘‘standard halo model’’ used in earlier

analyses, the velocity distribution of the WIMPs in our model is non-Maxwellian, with a high-velocity

cutoff determined self-consistently by the model itself. The parameters of the model are determined from

a fit to the rotation curve data of the Galaxy. We find that, for our best-fit halo model, for spin-independent

interaction, while the S-K upper limits do not place additional restrictions on the DAMA-compatible

region of the WIMP parameter space if the WIMPs annihilate dominantly to �bb and/or �cc, portions of the

DAMA-compatible region can be excluded if WIMP annihilations to �þ�� and � �� occur at larger than

35% and 0.4% levels, respectively. For spin-dependent interaction, on the other hand, the restrictions on

the possible annihilation channels are much more stringent: they rule out the entire DAMA region if

WIMPs annihilate to �þ�� and � �� final states at greater than �0:05% and 0.0005% levels, respectively,

and/or to �bb and �cc at greater than �0:5% levels. The very latest results from the S-K Collaboration

[T. Tanaka et al., Astrophys. J. 742, 78 (2011)] make the above constraints on the branching fractions of

various WIMP annihilation channels even more stringent by roughly a factor of 3–4.

DOI: 10.1103/PhysRevD.85.123533 PACS numbers: 95.35.+d

I. INTRODUCTION

Weakly interacting massive particles (WIMPs) (here-after generically denoted by �) with masses m� in the

range of few GeV to few TeV are a natural candidate forthe dark matter (DM) in the Universe; see, e.g., Refs. [1–5]for reviews. Several experiments are currently engaged inefforts to directly detect such WIMPs by observing nuclearrecoils due to scattering of WIMPs off nuclei in suitablychosen detector materials in underground laboratories.Recent results from some of these direct-detection (DD)experiments, in particular, the annual modulation of thenuclear recoil event rates reported by the DAMA/LIBRACollaboration [6] and the excess of low- energy recoilevents reported by the CoGeNT Collaboration [7] haveraised the interesting possibility [8,9] that these eventscould be due to WIMPs of relatively low mass, approxi-mately in the range �5–10 GeV, interacting with nucleiwith a WIMP-nucleon spin-independent elastic cross sec-tion in the region of few� 10�4 pb, without conflicting

with the null results from other experiments such asXENON10 [10], XENON100 [11] and CDMS-II-Si [12].Earlier analyses (before the CoGeNT results [7]) had alsofound similar compatibility of the DAMA/LIBRA annualmodulation signal with the null results from other DDexperiments; see, e.g., Refs. [13–15].1

Scattering of WIMPs off nuclei can also lead to captureof the WIMPs by massive astrophysical bodies such as theSun or the Earth if, after scattering off a nucleus inside thebody, the velocity of the WIMP falls below the escapevelocity of the body. The WIMPs so captured over thelifetime of the capturing body would gradually settle downto the core of the body where they would annihilate andproduce standard model particles, e.g., WþW�, Z0Z0,�þ��, t�t, b �b, c �c, etc. Decays of these particles would

*susmita.kundu@saha.ac.in†pijush.bhattacharjee@saha.ac.in

1The question of compatibility of the DAMA/LIBRA andCoGeNT results with the null results of other experiments,however, remains controversial; see, e.g., the results of a recentreanalysis of the CDMS-II Germanium data with a loweredrecoil-energy threshold of 2 keV [16], as well as the recentresults from the XENON100 Collaboration [17], both of whichclaim to disfavor such a compatibility.

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then produce neutrinos, gamma rays, electrons-positrons,protons-antiprotons, etc. For astrophysical objects like theSun or the Earth, only the neutrinos would be able toescape. Detection of these neutrinos by large neutrinodetectors can, albeit indirectly, provide a signature ofWIMPs. Although no detection has yet been reported, theSuper-Kamiokande (S-K) detector, for example, has pro-vided upper limits on the possible neutrino flux fromWIMP annihilation in the Sun as a function of the WIMPmass [18–20]. Similarly, the � rays produced in the anni-hilation of the WIMPs in suitable astrophysical environ-ments with enhanced DM density but low optical depth togamma rays, such as in the central region of our Galaxy, indark-matter-dominated objects such as dwarf galaxies, andin clusters of galaxies, can offer a complimentary avenueof indirect detection (ID) of WIMPs. Although no unam-biguous gamma-ray signals of dark matter origin have beenreported, a recent analysis [21] of the spectral and morpho-logical features of the gamma-ray emission from the innerGalactic Center region (within a Galactocentric radius of�175 pc) measured by the Fermi Gamma-ray SpaceTelescope seems to suggest the presence of a gamma-rayemission component which is difficult to explain in terms ofknown sources and/or process of gamma-ray production butis consistent with that expected from annihilations ofWIMPs of mass in the 7–9 GeV range (annihilating primar-ily to tau leptons) with a suitably chosen density and distri-bution of the dark matter in the Galactic Center region; see,however, Ref. [22] for a different view.

In this paper, we focus on the neutrinos producedby annihilations of WIMPs in the core of the Sun andstudy the constraints imposed on the WIMP mass vsWIMP-nucleon cross section, for low-mass (& 20 GeV)WIMPs, from nondetection of such neutrinos. This is donewithin the context of a self-consistent model of the finite-size dark halo of the Galaxy [15,23] which includes thegravitational effect of the observed visible matter on theDM in a self-consistent manner, with the parameters ofthe model determined from fits to the rotation curve data ofthe Galaxy [24,25].

The expected flux of neutrinos from the Sun due toWIMP annihilations depends on the rate at whichWIMPs are captured by the Sun. The capture rate dependson the density as well as the velocity distribution of theWIMPs in the solar neighborhood as the Sun goes aroundthe Galaxy. The density and velocity distribution of theWIMPs in the Galaxy are a priori unknown. Most earlierstudies of neutrinos from WIMP capture and annihilationin the Sun have been done within the context of the so-called ‘‘standard halo model’’ (SHM) in which the DMhalo of the Galaxy is described by a single-componentisothermal sphere [26] with a Maxwellian velocity distri-bution of the DM particles in the Galactic rest frame[1,27,28]. The velocity distribution is isotropic and isusually truncated at a chosen value of the escape speed

of the Galaxy. The density of DM in the solar neighborhoodis typically taken to be in the range �DM;� � 0:3�0:1 GeV=cm3 [29–32].2 The velocity dispersion, hv2i1=2,the parameter characterizing the Maxwellian velocity distri-bution of the SHM, is typically taken to be �270 km s�1.

This follows from the relation [26], hv2i1=2 ¼ffiffi32

qvc;1, be-

tween the velocity dispersion of the particles constituting asingle-component self-gravitating isothermal sphere and theasymptotic value of the circular rotation speed, vc;1, of atest particle in the gravitational field of the isothermal sphereand assuming vc;1 � vc;� � 220 km s�1, where vc;� is the

measured value of the circular rotation velocity of theGalaxy in the solar neighborhood.3 Neutrino flux fromDM annihilation in the Sun for low-mass WIMPs and theresulting constraints on WIMP properties from theSuper-Kamiokande upper limits on such neutrinos havebeen studied within the context of the SHM inRefs. [19,36–38], which showed that the Super-Kamiokande upper limits on the possible flux of neutrinosfrom the Sun place stringent restrictions on the DAMAregion of the WIMP parameter space.Whereas the SHM serves as a useful benchmark model,

there are a number of reasons why the SHM does notprovide a satisfactory description of the dynamics of theGalaxy. First, it does not take into account the modificationof the phase-space structure of the DM halo due to thesignificant gravitational effect of the observed visible mat-ter on the DM particles inside and up to the solar circle.Second, the isothermal sphere model of the halo is infinitein extent and has a formally divergent mass, with massinside a radius r, MðrÞ / r, as r ! 1, and is thus unsuit-able for representing a halo of finite size. Third, the pro-cedure of truncating the Maxwellian speed distribution at achosen value of the local (solar neighborhood) escapespeed is not a self-consistent one because the resultingspeed distribution is not, in general, a self-consistent solu-tion of the steady-state collisionless Boltzmann equationdescribing a finite system of collisionless DM particles. Inaddition, since the rotation curve for such a truncatedMaxwellian is, in general, not asymptotically flat, the

relation hv2i1=2 ¼ffiffi32

qvc;1 used to determine the value of

hv2i1=2 in the Maxwellian speed distribution of the isother-mal sphere, as done in the SHM, is not valid in general.Finally, recent numerical simulations [32] seem to find thatthe velocity distribution of the dark matter particles devi-ates significantly from the usual Maxwellian form. Theseissues are further discussed in detail in Ref. [15], where wediscussed a self-consistent model of the finite-size dark

2See, however, recent analyses [33,34] which claim a valuecloser to 0:4 GeV=cm3 with uncertainty & 10%.

3A somewhat higher value of vc;� � 250 km s�1, as suggestedby a recent study [35], would imply a correspondingly highervalue of hv2i1=2iso � 306 km s�1.

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halo of the Galaxy which avoids the above-mentionedinconsistencies of the SHM and also studied the constraintson WIMP properties from the results of the DD experi-ments within the context of this self-consistent halo model.It is of interest to extend this study to the case of indirectdetection of WIMPs via neutrinos from WIMP annihila-tions in the Sun, which is the purpose of this paper.

Our model of the phase-space structure of the finite-sizeDM halo of the Galaxy is based on the so-called ‘‘lowered’’(or truncated) isothermal models (often called ‘‘King mod-els’’) [26] of the phase-space distribution function (DF) ofcollisionless particles. These models are proper self-consistent solutions of the collisionless Boltzmann equa-tion representing nearly isothermal systems of finite physi-cal size and mass. There are two important features of thesemodels: First, at every location within the system, a DMparticle can have speeds up to a maximum speed which isself-consistently determined by the model itself. A particleof maximum velocity at any location within the system canjust reach its outer boundary, fixed by the truncation radius,a parameter of the model, where the DM density by con-struction vanishes. Second, the speed distribution of theparticles constituting the system is non-Maxwellian. Toinclude the gravitational effect of the observed visiblematter on the DM particles, we modify the ‘‘pure’’ Kingmodel DF by replacing the gravitational potential appear-ing in the King model DF by the total gravitational poten-tial consisting of the sum of those due to DM and theobserved visible matter. This interaction with the visiblematter influences both the density profile and the velocitydistribution of the dark matter particles as compared tothose for a pure King model. In particular, the dark matteris pulled in by the visible matter, thereby increasing itscentral density significantly. When the visible matter den-sity is set to zero and the truncation radius is set to infinity,our halo model becomes identical to that of a single-component isothermal sphere used in the SHM. For furtherdiscussion of the model, see Refs. [15,23].

The DM distribution in the Galaxy may have a signifi-cant amount of substructures which may have interestingeffects on the WIMP capture and annihilation rates [39].However, not much information, based on observationaldata, is available about the spatial distribution and internalstructures of these substructures. As such, in this paper, weshall be concerned only with the smooth component of theDM distribution in the Galaxy described by our self-consistent model mentioned above, the parameters ofwhich are determined from the observed rotation curvedata for the Galaxy.

The non-Maxwellian nature of the WIMP speed distri-bution in our halo model makes the calculation of theWIMP capture (and consequently annihilation) rate non-trivial since the standard analytical formula for the capturerate given by Gould [40] and Press and Spergel [41], whichis widely used in the literature, is not valid for the

non-Maxwellian speed distribution in our halo modeland, as such, has to be calculated ab initio; see Sec. III.We calculate the 90% C.L. upper limits on the WIMP-

proton spin-independent (SI) as well as spin-dependent(SD) elastic cross sections as a function of the WIMPmass, for various WIMP annihilation channels, using the90% C.L. upper limits on the rates of upward-going muonevents due to neutrinos from the Sun derived from theresults of S-K Collaboration (see Refs. [18,19] and refer-ences therein).4 We then study the consistency of thoselimits with the 90% C.L. ‘‘DAMA-compatible’’ regions—the regions of the WIMP mass versus cross sectionparameter space within which the annual modulation sig-nal observed by the DAMA/LIBRA experiment [6] iscompatible with the null results of other DD experiments—determined within the context of our halo model [15]. Wefind that the requirement of such consistency imposesstringent restrictions on the branching fractions of thevarious WIMP annihilation channels. For example, in thecase of spin-independent WIMP-proton interaction, whilethe S-K upper limits do not place additional restrictions onthe DAMA-compatible region of the WIMP parameterspace if the WIMPs annihilate dominantly to �bb, �cc,portions of the DAMA-compatible region can be excludedif WIMP annihilations to �þ�� and � �� occur at larger than35% and 0.4% levels, respectively. In the case of spin-dependent interactions, on the other hand, the restrictionson the branching fractions of various annihilation channelsare much more stringent. Specifically, they rule out theentire DAMA region if WIMPs annihilate to �þ�� and � ��final states at greater than �0:05% and 0.0005% levels,respectively, and/or to �bb and �cc at greater than �0:5%levels.5 The very latest results from the S-K Collaboration[20] make the above constraints on the branching fractionsof various WIMP annihilation channels even more strin-gent by roughly a factor of 3–4.The rest of the paper is organized as follows: In Sec. II,

we briefly describe the self-consistent model of the DMhalo of the Galaxy. The formalism of calculating theWIMP capture and annihilation rates in the Sun withinthe context of our DM halo model, and that for calculatingthe resulting neutrino flux and event rate in the Super-Kamiokande detector, are discussed in Secs. III and IV,

4After the completion of the main calculations of the presentwork, new results of the S-K Collaboration’s search for upward-going muons due to neutrinos from the Sun [20] have appeared.We include, at the end of Sec. V, a discussion of the new resultsof Ref. [20] and the resulting constraints on various WIMPannihilation channels.

5In the present paper, the CoGeNT results [7] are not includedin the analysis. Preliminary results of the analysis [42] to find the‘‘CoGeNT-compatible’’ region in the WIMP mass vs cross-section plane within the context of our halo model indicatesthat its inclusion will not significantly change the above con-straints on the branching fractions for the various annihilationchannels.

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respectively. Our results and the constraints on the WIMPproperties implied by these results are described in Sec. V.The paper ends with a summary in Sec. VI.

II. THE SELF-CONSISTENT TRUNCATEDISOTHERMAL MODEL OF THE MILKY

WAY’S DARK MATTER HALO

The phase-space DF of the DM particles constituting atruncated isothermal halo of the Galaxy can be taken, in therest frame of the Galaxy, to be of the King model form[15,23,26],

fðx; vÞ � fðEÞ

¼��1ð2��2Þ�3=2ðeE=�2 � 1Þ for E > 0;

0 for E � 0;(1)

where

E ðxÞ � �ðrtÞ ��1

2v2 þ�ðxÞ

�� �ðxÞ � 1

2v2; (2)

is the so-called ‘‘relative energy’’ and �ðxÞ ¼ ��ðxÞ þ�ðrtÞ the ‘‘relative potential,’’ �ðxÞ being the total gravi-tational potential under which the particles move, withboundary condition �ð0Þ ¼ 0. The relative potential andrelative energy, by construction, vanish at jxj ¼ rt, thetruncation radius, which represents the outer edge of thesystem where the particle density vanishes. At any locationx, the maximum speed a particle of the system can have is

vmaxðxÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2�ðxÞp

; (3)

at which the relative energy E and, as a consequence, theDF (1) vanish. The model has three parameters, namely,�1, � and rt. Note that the parameter � in the King modelis not the same as the usual velocity dispersion parameterof the isothermal phase-space DF [26]. Also, in our calcu-lations below, we shall use the parameter �DM;�, the valueof the DM density at the location of the Sun, in place of theparameter �1.

Integration of fðx; vÞ over all velocities gives the DMdensity at the position x:

�DMðxÞ

¼ �1

ð2��2Þ3=2Z ffiffiffiffiffiffiffiffiffiffi

2�ðxÞp

0dv4�v2

�exp

��ðxÞ � v2=2

�2

�� 1

�(4)

¼ �1

�exp

��ðxÞ�2

�erf

� ffiffiffiffiffiffiffiffiffiffiffi�ðxÞp�

��

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi4�ðxÞ��2

s �1þ 2�ðxÞ

3�2

��;

(5)

which satisfies the Poisson equation

r2�DMðxÞ ¼ 4�G�DMðxÞ; (6)

where�DM is the contribution of the DM component to thetotal gravitational potential,

�ðxÞ ¼ �DMðxÞ þ�VMðxÞ; (7)

in presence of the visible matter whose gravitational po-tential, �VM, satisfies its own Poisson equation, namely,

r2�VMðxÞ ¼ 4�G�VMðxÞ: (8)

We choose the boundary conditions

�DMð0Þ ¼ �VMð0Þ ¼ 0; and

ðr�DMÞjxj¼0 ¼ ðr�VMÞjxj¼0 ¼ 0: (9)

The mass of the system, defined as the total mass containedwithin rt, is given by GMðrtÞ=rt ¼ ½�ð1Þ ��ðrtÞ. Notethat, because of the chosen boundary condition �ð0Þ ¼ 0,�ð1Þ is a nonzero positive constant.Since the visible matter distribution �VMðxÞ, and hence

the potential �VMðxÞ, are known from observations andmodeling, the solutions of Eq. (6) together with Eqs. (5)and (7) and the boundary conditions (9) give us a three-parameter family of self-consistent pairs of �DMðxÞ and�DMðxÞ for chosen values of the parameters (�1, �, rt).The values of these parameters for the Galaxy can bedetermined by comparing the theoretically calculatedrotation curve, vcðRÞ, given by

v2cðRÞ ¼ R

@

@R½�ðR; z ¼ 0Þ

¼ R@

@R½�DMðR; z ¼ 0Þ þ�VMðR; z ¼ 0Þ; (10)

with the observed rotation curve data of the Galaxy. (Here,R is Galactocentric distance on the equatorial plane, and zis the distance normal to the equatorial plane.) This proce-dure was described in detail in Refs. [15,23] where, for thevisible matter density distribution described there, we de-termined the values of the parameters rt and � which gavea reasonably good fit to the rotation curve data of theGalaxy [24,25] for each of the three chosen values ofthe parameter �DM;� ¼ 0:2, 0.3 and 0:4 GeV=cm3. These

models are summarized in Table I, which we use for ourcalculations in this paper. The density profiles, mass pro-files, velocity distributions of the DM particles and theresulting rotation curves in each of these models are dis-cussed in detail in Ref. [15].

TABLE I. Parameters of our self-consistent model of theMilky Way’s dark matter halo which give good fits to theGalaxy’s rotation curve data, for the three chosen values of theDM density at the solar neighborhood.

Model �DM;� (GeV=cm3) rt (kpc) � (km s�1)

M1 0.2 120.0 300.0

M2 0.3 80.0 400.0

M3 0.4 80.0 300.0

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With our halo model specified, we now briefly reviewthe basic formalism of calculating the WIMP capture andannihilation rates within the context of our halo model.

III. CAPTURE AND ANNIHILATION RATES

The capture rate per unit volume at radius r inside theSun can be written as [40,41]

dC

dVðrÞ ¼

Zd3u

~fðuÞu

w��ðwÞ; (11)

where ~fðuÞ is the WIMP velocity distribution, as measuredin the Sun’s rest frame, in the neighborhood of the Sun’s

location in the Galaxy, and wðrÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 þ wescðrÞ

pis the

WIMP’s speed at the radius r inside the Sun, wescðrÞ beingthe escape speed at that radius inside the Sun, which isrelated to the escape speed at the Sun’s core, wesc;core �1354 km s�1, and that at its surface, wesc;surf �795 km s�1, by the approximate relation

w2escðrÞ ¼ ðwesc;coreÞ2 �MðrÞ

M�½ðwesc;coreÞ2 � ðwesc;surfÞ2:

(12)

The quantity ��ðwÞ is the capture probability per unittime, which is just the product of the scattering rate andthe conditional probability that after a scattering, theWIMP’s speed falls below the escape speed.

We shall here consider only the elastic scattering of theWIMPs off nuclei. The dominant contribution to theWIMP capture rate will come from the WIMPs scatteringoff hydrogen and helium nuclei. While for hydrogen, bothSI as well as SD cross sections, �SI

�p and �SD�p, respectively,

will contribute, only the SI cross section for helium isrelevant. (We neglect here the small contribution from3He). In general, the effective momentum-transfer (q)dependent WIMP-nucleus SI scattering cross section,�SI

�AðqÞ, can be written in the usual way in terms of the

‘‘zero-momentum’’ WIMP-proton (or WIMP-neutron) ef-fective cross section, �SI

�p ¼ �SI�n, as

�SI�AðqÞ ¼

�2�A

�2�p

�SI�pA

2jFðq2Þj2; (13)

where A is the number of neutrons plus protons in thenucleus, ��A and ��p are the reduced masses of WIMP-

nucleus and WIMP-proton systems, respectively, with��i ¼ ðmim�Þ=ðmi þm�Þ, and Fðq2Þ is the nuclear

form-factor (with Fð0Þ ¼ 0) which can be chosen to beof the form [1]

jFðq2Þj2 ¼ exp

�� q2R2

3ℏ2

�¼ exp

���E

E0

�: (14)

Here, R� ½0:91ð mA

GeVÞ1=3 þ 0:3 � 10�13 cm is the nuclear

radius, and E0 � 3ℏ2=ð2mAR2Þ is the characteristic nuclear

coherence energy, mA being the mass of the nucleus.

With the above form of the nuclear form factor, thekinematics of the capture process [40] allows us to writethe capture probability per unit time, ��ðwÞ, as

��ðwÞ ¼ nA��A

w

2E0

m�

�2þ�

�exp

��m�u

2

2E0

�

� exp

��m�w

2

2E0

�

�2þ

��

��

�2þ� u2

w2

�; (15)

where nA is the number density of the scattering nuclei at

the radius r inside the Sun, and� � m�

mA,�� � ��1

2 . The

function ensures that those particles which do not lose asufficient amount of energy to be captured are excluded.We shall use Eq. (15) to calculate ��ðwÞ for helium

(A ¼ 4). For hydrogen, however, there is no form-factorsuppression, and the expression for ��ðwÞ is simpler:

Hydrogen: ��ðwÞ ¼ ��pnHw

�w2

esc ��2��

u2�

�

�w2

esc ��2��

u2�; (16)

where nH is the density of hydrogen (proton) at the radius rinside the Sun. Note that in Eqs. (15) and (16), the quan-tities w, wesc, nA and nH are functions of r.The WIMP velocity distribution appearing in Eq. (11) is

related to the phase-space DF defined in Eq. (1) (valid inthe rest frame of Galaxy) by the Galilean transformation

~fðuÞ ¼ 1

m�

fðx ¼ x�; v ¼ uþ v�Þ; (17)

where x� represents the Sun’s position in the Galaxy (R ¼8:5 kpc, z ¼ 0) and v� is the Sun’s velocity vector in theGalaxy’s rest frame. Note that Gould’s original calcula-tions and the final formula for the WIMP capture rate givenin Ref. [40], which are widely used in the literature, use aMaxwellian velocity distribution of the WIMPs in theGalaxy and, as such, cannot directly be used here sincethe WIMP velocity distribution in our case is non-Maxwellian. In particular, note that the DF f of Eq. (1)vanishes for speeds v vmax defined in Eq. (3).Consequently, Eq. (11) above can be written as

dC

dVðrÞ ¼ 2�

Z 1

�1dðcosÞ

Z umaxðcosÞ

umin¼v�duu~fðuÞw��ðwÞ;

(18)

where v� � 220–250 km s�1 is the Sun’s circular speed inthe Galaxy, and umax is given by the positive root of thequadratic equation

u2max þ v2� þ 2umaxv� cos ¼ 2�ðx ¼ x�Þ: (19)

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The total WIMP capture rate by the Sun, C�, is given by

C� ¼Z R�

04�r2dr

dCðrÞdV

; (20)

where R� is the radius of the Sun.In this work, we shall neglect the effect of evaporation of

the captured WIMPs from the Sun6 and make the standardassumption that the capture and annihilation processeshave reached an approximate equilibrium state over thelong lifetime of the solar system (t� � 4:2 billion yrs).Under these assumptions, the total annihilation rate ofWIMPs in the Sun is simply related to the total capturerate by the relation

�� � 1

2C� (21)

IV. NEUTRINO FLUX FROM WIMPANNIHILATION IN THE SUN ANDEVENT RATE IN THE DETECTOR

A. The neutrino energy spectrum

In this subsection, we collect together the known resultsfor the energy spectra of neutrinos emerging from the Sun,for variousWIMP annihilation channels [43–46], for use inthe calculations described in the next subsection.

The differential flux of muon neutrinos observed atEarth is [43]�

di

dEi

�¼ ��

4�D2

XF

BF

�dNi

dEi

�F; ði ¼ ��; ���Þ (22)

where �� is the rate of WIMP annihilation in the Sun, D isthe Earth-Sun distance, F stands for the possible annihila-tion channels, BF is the branching ratio for the annihilation

channel F and ðdNi

dEiÞF is the differential energy spectrum of

the neutrinos of type i emerging from the Sun resultingfrom the particles of annihilation channel F injected at thecore of the Sun. WIMPs can annihilate to all possiblestandard model particles, e.g., eþe�, �þ��, �þ��,�e ��e, �� ���, �� ���, �qq pairs and also gauge and Higgs

boson pairs (WþW�, Z �Z, h �h), etc. In this paper, we areonly interested in low-mass (� 2–20 GeV) WIMPs.Therefore, we will not consider WIMP annihilations toHiggs and gauge boson pairs and top-quark pairs. Lightquarks like u, d, s contribute very little to the energeticneutrino flux [44] and are not considered. The same is truefor muons. So, in this paper, we consider only the channels�þ��, �bb, �cc and ���.

The neutrino energy spectra, ðdNi

dEiÞF, have been calcu-

lated numerically (see, e.g., Refs. [44,47]) by considering

all the details of hadronization of quarks, energy loss of theresulting heavy hadrons, neutrino oscillation effects, neu-trino energy loss due to neutral current interactions andabsorption due to charged current interactions with thesolar medium, �� regeneration, etc. However, the numeri-cal results in Refs. [44,47] are given for WIMP massesm� 10 GeV, and it is not obvious if those are valid for

lower WIMP masses which are of our primary interest inthis paper. In any case, given the presence of other uncer-tainties in the problem, particularly those associated withastrophysical quantities such as the local density of darkmatter and its velocity distribution, we argue that it is goodenough to use—as we do in this paper—approximateanalytical expressions for the neutrino spectra availablein the literature [43,45,46]. We are interested in the fluxesof muon neutrinos and antineutrinos, for which we use theanalytic expressions given in Ref. [43], which neglectneutrino oscillation effects. By comparing with the neu-trino fluxes obtained from detailed numerical calculations[44], we find that for small WIMP masses below�20 GeV(the masses of our interest in this paper), the analyticexpressions for the muon neutrino fluxes given inRef. [43] match with the results of detailed numericalcalculations [44] to within a few percent.The main effect of the interaction of the neutrinos

with the solar medium is that [46] a neutrino of typeið¼ ��; ���Þ injected at the solar core with energy Ecore

i

emerges from the Sun with an energy Ei given by

Ecorei ¼ Ei=ð1� Ei�iÞ; (23)

and with probability

Pi ¼ ð1þ Ecorei �iÞ��i ¼ ð1� Ei�iÞ�; (24)

with

���¼ 5:1; � ���

¼ 9:0;

���¼ 1:01� 10�3 GeV�1; and

� ���¼ 3:8� 10�4 GeV�1: (25)

Below, we write down the expressions for the energyspectra of neutrinos emerging from the Sun for the fourannihilation channels considered in this paper:

1. �þ�� channel: neutrinos from decayof � leptons (� ! �����)

For this channel, the spectrum of muon-type neutrinos atthe solar surface, including the propagation effects in thesolar medium, can be written as [43]�dNi

dEi

��þ��

¼ ð1�Ei�iÞð�i�2Þ�dNcore

i

dEcorei

��þ��

; ði¼ ��; ���Þ;

(26)

where the relationship between Ei and Ecorei , and the values

of �i and �i, are as given by Eqs. (23) and (25), respec-tively, and

6Note, however, that evaporation may not be negligible forWIMP masses below �4 GeV depending on the magnitude ofthe annihilation cross section [36].

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�dNcore

i

dEcorei

��þ��

¼ 48��!�����

��m4�

�1

2m�ðEcore

i Þ2

� 2

3ðEcore

i Þ3�minðð1=2Þm�;EþÞ

E�(27)

is the neutrino spectrum due to decay of the � leptonsinjected at the solar core by WIMP annihilations. Here,

��!�����¼0:18, and E�¼ Ecore

i

�ð1��Þ with �¼ð1��2Þ�1=2¼m�=m�, m� being the �-lepton mass.

Note that the ��s produced from � decay may againproduce �s by charged-current interactions in the solarmedium, and these secondary �s can decay to give second-ary ��s. But these ��s would be of much lower energy

compared to the primary ��s from � decay and are not

considered here.

2. �bb channel: neutrinos from decay of b-quarkhadrons (b ! c���)

The treatment is similar to the case of � decay describedabove. However, here the hadronization of quarks andstopping of heavy hadrons in the solar medium have tobe taken into account. The resulting spectrum of muonneutrinos emerging from the Sun is given by [43]

�dNi

dEi

��bb¼

Z E0

mb

�1

N

dN

dEd

�hadronðE0; EdÞð1� Ei�iÞð�i�2Þ

��dNcore

i

dEcorei

��bbðEd; E

corei ÞdEd; ði ¼ ��; ���Þ;

(28)

where mb is the b-quark mass, E0 � 0:71m� is the initial

energy of the b-quark hadron (the fragmentation functionis assumed to be a sharply peaked function [43]),

�dNcore

i

dEcorei

��bbðEd; E

corei Þ ¼ 48�b!���X

��m4b

�1

2mbðEcore

i Þ2

� 2

3ðEcore

i Þ3�minðð1=2Þmb;EþÞ

E�(29)

is the neutrino spectrum resulting from decay of theb-quark hadron injected at the solar core and

�1

N

dN

dEd

�hadronðE0; EdÞ ¼ Ec

E2d

exp

�Ec

E0

� Ec

Ed

�; (30)

with Ec � 470 GeV, is the distribution of the hadron’senergy at the time of its decay if it is produced with aninitial energy E0. In Eq. (29), �b!���X ¼ 0:103 is the

branching ratio for inclusive semileptonic decay of

b-quark hadrons to muons [48] and E� ¼ Ecorei

�ð1��Þ with � ¼ð1� �2Þ�1=2 ¼ Ed=mb.

3. �cc channel: neutrinos from decay of c-quarkhadrons (c ! s���)

Again, this is similar to the case of b decay discussedabove, except that the kinematics of the process is slightlydifferent. The resulting muon-neutrino spectrum is givenby [43]�dNi

dEi

��cc¼

Z E0

mc

�1

N

dN

dEd

�hadronðE0; EdÞð1� Ei�iÞð�i�2Þ

��dNcore

i

dEcorei

��ccðEd; E

corei ÞdEd; ði ¼ ��; ���Þ;

(31)

where mc is the c-quark mass, E0 � 0:55m� is the initial

energy of the charmed hadron,�dNcore

i

dEcorei

��ccðEd;E

corei Þ

¼8�c!���X

��m4c

�3

2mcðEcore

i Þ2�4

3ðEcore

i Þ3�minðð1=2Þmc;EþÞ

E�(32)

is the neutrino spectrum resulting from decay of the c-quarks

injected at the solar core, with �c!���X ¼ 0:13, E�¼Ecorei

�ð1��Þ , �¼ð1��2Þ�1=2¼Ed=mc, and ð1N dNdEd

ÞhadronðE0;EdÞis given by Eq. (30) with Ec � 250 GeV for the c quark.

4. � �� channel: (�� ! �� ���)

In this case, the spectrum of muon neutrinos emergingfrom the Sun is simply given by�dNi

dEi

�� ��

¼ ð1� Ei�iÞð�i�2Þ�dNcore

i

dEcorei

�� ��; ði ¼ ��; ���Þ;

(33)

where�dNcore

i

dEcorei

�� ��

¼ ðEcorei �m�Þ

� ð1þm��iÞ�2

�Ei �

m�

1þm��i

�: (34)

B. Calculation of event rates in theSuper-Kamiokande detector

The rate of neutrino-induced upward-going muonevents, R, in the S-K detector due to ��s and ���s from

WIMP annihilation in the Sun can be written as

R ¼ 1

2

Xi¼��; ���

ZZ di

dEi

d�iN

dyðEi; yÞVeffðE�Þnwaterp dEidy;

(35)

where di

dEiis the differential flux of the neutrinos given

by Eq. (22), d�iN

dy are the relevant neutrino-nucleon

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charged-current differential cross sections, 1� y ¼ E�=Ei

is the fraction of the neutrino energy transferred to themuon, VeffðE�Þ is the effective volume of the detector and

nwaterp is the number density of protons in water

ð¼ Avogadro numberÞ. The S-K Collaboration imposed acut on the upward-going muon path-length of>7meters inthe inner detector which has an effective area of Aeff �900 m2 and height � 36:2 m. This 7-meter cut on themuon track length can be effectively taken into accountby setting Veff ¼ 0 if the effective water-equivalent muonrange, R�ðE�Þ � 5 meters� ðE�=GeVÞ, is less than 7

meters, and Veff ¼ Aeff � ½R�ðE�Þ þ ð36:2� 7Þ metersotherwise [36]. The factor of 1=2 accounts for the factthat only upgoing muon events were considered in order toavoid the background due to down-going muons produceddue to cosmic ray interactions in the Earth’s atmosphere.

The S-K muon events were broadly classified intothree categories [49], namely, (i) fully contained (FC),(ii) stopping (S) and (iii) throughgoing (TG) events. For�� energy & 4 GeV, the events are predominantly of FC

type, whereas for �� energy * 8 GeV, the events are

predominantly of TG type. Assuming that annihilation ofthe WIMP of mass m� produces neutrinos of typical en-

ergy�ð13 – 12Þm�, we can roughly divide them� range of our

interest into three regions according to the resulting muonevent types, namely, (i) 2 & m� & 8 GeV (FC),

(ii) 8 & m� & 15 GeV (FCþ S) and (iii) 15 GeV & m�

(FCþ Sþ TG).To set upper limits on the WIMP elastic scattering cross

section as a function of WIMP mass for a given annihila-tion channel, we use the following 90% C.L. upper limits[18,19] on the rates of the upgoing muon events of the threedifferent types mentioned above:

R 90%C:L:FC ’ 13:8 yr�1; R90%C:L:

S ’ 1:24 yr�1;

R90%C:L:TG ’ 0:93 yr�1:

(36)

The upper limits on the WIMP-nucleon elastic scatteringcross section so derived are then translated into upper limitson the branching fractions of various annihilation channelsby demanding the consistency of DAMA-compatible regionof the WIMP parameter space with S-K upper limits. Theselimits are discussed in the next section.

V. RESULTS AND DISCUSSIONS

Figure 1 shows the dependence of the capture rate ofWIMPs by the Sun as a function of the WIMP’s mass forthe three halo models specified in Table I. As expected, fora given DM density, the capture rate decreases as WIMPmass increases because heavier WIMPs correspond to asmaller number density of WIMPs.The event rates in the S-K detector as a function of the

WIMP mass for the four different WIMP annihilationchannels are shown in Fig. 2, assuming 100% branchingratio for each channel by itself. For each annihilationchannel, the three curves correspond, as indicated, to thethree halo models specified in Table I. It is seen that thedirect annihilation to the � �� channel dominates the eventrate, followed by the �þ�� channel.Our main results are contained in Figs. 3–5, where we

show, for the three halo models considered, the 90% C.L.upper limits on the WIMP-proton SI and SD elastic crosssections (as a function of WIMP mass) derived from theSuper-Kamiokande measurements of the upgoing muonevents from the direction of the Sun [18,19], for the fourannihilation channels discussed in the text, assuming 100%branching ratio for each channel by itself. In these figures,we also display, for the respective halo models, the90% C.L. allowed regions [15] in the WIMP mass vsWIMP-proton elastic cross section plane implied by theDAMA/LIBRA Collaboration’s claimed annual modula-tion signal [6], as well as the 90% C.L. upper limits [15]on the relevant cross section as a function of the WIMPmass implied by the null results from the CRESST-1 [50],

1×1024

1×1025

1×1026

1×1023

1×1024

1×1025

2 4 6 8 10 12 14 16 18 20

Cap

ture

Rat

e (s

ec-1

)

mχ (GeV)

σSIχp=10-4 pb

M1 M2 M3

2 4 6 8 10 12 14 16 18 20

Cap

ture

Rat

e (s

ec-1

)

mχ (GeV)

σSDχp=10-4 pb

M1 M2 M3

FIG. 1. The capture rate as a function of the WIMP mass for the three halo models specified in Table I, and for SI (left panel) and SD(right panel) WIMP-proton interactions. All the curves are for a reference value of the WIMP-proton elastic SI or SD cross section of10�4 pb.

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CDMS-II-Si [12], CDMS-II-Ge [51] and XENON10 [10]experiments.

The curves in Figs. 3–5, allow us to derive upper limitson the branching fractions of the various WIMP annihila-tion channels, from the requirement of consistency of theS-K–implied upper limits on the WIMP-proton elasticcross section with the DAMA-compatible regions. Theseupper limits are shown in Table II for the three halo modelsdiscussed in the text.

Clearly, for the case of spin-independent interaction,there are no constraints on the branching fractions for the�bb and �cc channels since the DAMA-compatible region isalready consistent with the S-K upper limit even for 100%

branching fractions in these channels (the respective curvesfor the various annihilation channels only move upwards,keeping the same shape, as the branching fractions are madesmaller). At the same time, for the �þ�� channel and SIinteraction, although a 100%branching fraction in this chan-nel allows a part of the DAMA-compatible region to beconsistent with the S-K upper limit, consistency of the entireDAMA-compatible region with the S-K upper limit requiresthe branching fraction for this channel tobe less than35–45%depending on the halo model. On the other hand, for the � ��channel and SI interaction, there are already strong upperlimits (at the level of 25–35%) on the branching fractionfor this channel for consistency of even a part of the

0.001

0.01

0.1

1

10

100

1000

1 2 3 4 5 7 10 20

Eve

nt R

ate

(yea

r-1)

mχ (GeV)

M1M2M3

σSIχp=10-4 pb

b–b c–c

τ+τ-

ν–ν

0.0001

0.001

0.01

0.1

1

10

1 2 3 4 5 7 10 20

Eve

nt R

ate

(yea

r-1)

mχ (GeV)

M1M2M3

σSDχp=10-4 pb

b–b c–c

τ+τ-

ν–ν

FIG. 2. The upward-going muon event rates in the Super-Kamiokande detector due to neutrinos from WIMP annihilation in the Sunas a function of the WIMP mass for the four annihilation channels as indicated, assuming 100% branching ratios for each channel byitself, and for SI (left panel) and SD (right panel) WIMP-proton interactions. The three curves for each annihilation channelcorrespond, as indicated, to the three halo models specified in Table I. All the curves are for a reference value of the WIMP-protonelastic SI or SD cross section of 10�4 pb.

0.0001

0.001

0.01

0.1

1

2 4 6 8 10 12 14 16 18 20

σSI

χp (

pb)

mχ (GeV)

DAMA

CDMS-II-Ge

CDMS-II-Si

CRESST-1

XENON-10

b–bc–c

τ+τ-

ν–ν

0.0001

0.001

0.01

0.1

1

10

100

1000

10000

2 4 6 8 10 12 14 16 18 20

σSD

χp (

pb)

mχ (GeV)

DAMA

CDMS-II-Ge

CDMS-II-Si

CRESST-1

XENON-10

b–bc–c

τ+τ-

ν–ν

1×10-07

1×10-06

1×10-05

1×10-05

FIG. 3. The 90% C.L. upper limits on the WIMP-proton SI (left panel) and SD (right panel) elastic cross section as a function ofWIMP mass derived from the Super-Kamiokande measurements of the upgoing muon events from the direction of the Sun [18,19], forthe three relevant event types, namely, FC, S and TG, as discussed in the text [see Eq. (36)]. The thick portions of the curves serve todemarcate the approximate m� ranges where the different event types make dominant contributions to the upper limits. The curves

shown are for the four annihilation channels, assuming 100% branching ratio for each channel by itself. The 90% C.L. allowed regionsin the WIMP mass vs WIMP-proton elastic cross section plane implied by the DAMA/LIBRA experiment’s claimed annualmodulation signal [6] as well as the 90% C.L. upper limits on the cross section as a function of the WIMP mass implied by thenull results from the CRESST-1 [50], CDMS-II-Si [12], CDMS-II-Ge [51] and XENON10 [10] experiments (solid curves) are alsoshown. All the curves shown are for our halo model M1 (�DM;� ¼ 0:2 GeV=cm3) specified in Table I.

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DAMA-compatible region with the S-K upper limit, andthese upper limits become significantly more stringent (byabout 2 orders ofmagnitude) if the entireDAMA-compatibleregion is required to be consistent with the S-K upper limits.

The constraints on the branching fractions of various

annihilation channels are, however, much more severe in

the case of spin-dependent interaction: For the quark chan-

nels, only parts of the DAMA-compatible region can be

made consistent with the S-K upper limits, and that only if

the branching fractions for these channels are restricted at

the level of (0.6–0.8)%. On the other hand, for �þ�� and

� �� channels, parts of the DAMA-compatible regions can

be consistent with S-K upper limits only if their branching

fractions are restricted at the level of (0.14–0.18)% and

(0.012–0.016)%, respectively, while consistency of the

entire DAMA-compatible regions with the S-K upper lim-

its requires these fractions to be, respectively, lower by

about a factor of 2.5 (for the �þ�� channel) and a factor of

about 25 (for the � �� channel).The above small numbers for the upper limits on the

branching fractions of the four dominant neutrino produc-ing WIMP annihilation channels imply, in the case of spin-dependent WIMP interaction, that the DAMA-allowed

region of the m�-�SD�p parameter space is essentially ruled

out by the S-K upper limit on neutrinos from possibleWIMP annihilations in the Sun, unless, of course,WIMPs efficiently evaporate from the Sun—which maybe the case for relatively small mass WIMPs below 4 GeV[36]—or there are other nonstandard but dominant WIMPannihilation channels which somehow do not eventuallyproduce any significant number of neutrinos while restrict-ing annihilation to quark ( �bb, �cc) channels to below 0.5%level and �þ�� and � �� channels to below 0.05% and0.0005% level, respectively. In the case of spin-independent interaction, however, the DAMA-compatibleregion of the m�-�

SI�p parameter space (or at least a part

thereof) remains unaffected by the S-K upper limit ifWIMPs annihilate dominantly to quarks and/or tau leptons,and annihilation directly to neutrinos is restricted below�ð25–35Þ% level. At the same time, portions of theDAMA-compatible region can be excluded if WIMP an-nihilation to �þ�� occurs at larger than (35–45)% leveland/or annihilation to � �� occurs at larger than (0.4–0.8)%level. These results, based as they are on a self-consistentmodel of the Galaxy’s dark matter halo, the parameters ofwhich are determined by a fit to the rotation curve of the

0.0001

0.001

0.01

0.1

1

2 4 6 8 10 12 14 16 18 20

σSI

χp (

pb)

mχ (GeV)

DAMA

CDMS-II-Ge

CDMS-II-Si

CRESST-1

XENON-10

b–bc–c

τ+τ-

ν–ν

0.0001

0.001

0.01

0.1

1

10

100

1000

10000

2 4 6 8 10 12 14 16 18 20

σSD

χ p (

pb)

mχ (GeV)

DAMA

CDMS-II-Ge

CDMS-II-Si

CRESST-1

XENON-10

b–bc–c

τ+τ-

ν–ν

1×10-07

1×10-06

1×10-05

1×10-05

FIG. 4. Same as Fig. 3, but for the halo model M2 (�DM;� ¼ 0:3 GeV=cm3) specified in Table I.

0.0001

0.001

0.01

0.1

1

2 4 6 8 10 12 14 16 18 20

σSI

χp (

pb)

mχ (GeV)

DAMA

CDMS-II-Ge

CDMS-II-Si

CRESST-1

XENON-10

b–bc–c

τ+τ-

ν–ν

0.0001

0.001

0.01

0.1

1

10

100

1000

10000

2 4 6 8 10 12 14 16 18 20

σSD

χp (

pb)

mχ (GeV)

DAMA

CDMS-II-Ge

CDMS-II-Si

CRESST-1

XENON-10

b–bc–c

τ+τ-

ν–ν1×10-05

1×10-05

1×10-06

1×10-07

FIG. 5. Same as Fig. 3, but for the halo model M3 (�DM;� ¼ 0:4 GeV=cm3) specified in Table I.

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Galaxy, strengthen, at the qualitative level, the earlierconclusion within the SHM [19,36–38], although the quan-titative restrictions on the WIMP cross section and branch-ing fractions of various WIMP annihilation channelsobtained here are different (in some cases, by more thana factor of 10) from those obtained in the earlier calcula-tions within the SHM.

After the completion of the main calculations of thepresent work, new results of the Super-KamiokandeCollaboration’s search for upward-going muons due toneutrinos from the Sun [20] have appeared. These newresults are based on a data set consisting of 3109.6 days ofdata, nearly double the size of the old data set of1679.6 days used in Ref. [18] and in the analysis of thispaper so far. Below, we consider these new results ofRef. [20] and the resulting changes to the constraints onvarious WIMP annihilation channels derived above usingthe earlier Super-Kamiokande results. In general, we findthat with the new Super-Kamiokande results, the upperlimits on the branching fractions of various annihilationchannels become more stringent by a factor of 3–4 thanthose derived above.

The upgoing-muon (upmu) event categories used in thenew S-K paper [20] are somewhat different from those intheir earlier work [18]. These are: ‘‘stopping,’’ ‘‘nonshow-ering throughgoing’’ and ‘‘showering throughgoing;’’ seeRef. [20] for details. For a given WIMP mass, Figure 2 ofRef. [20] allows us to read out the fraction of each upmuevent type contributing to the total number of events, fromthe consideration that the typical maximum energy of aneutrino produced in the annihilation of a WIMP of massm� is�m�=2. For lowWIMP masses of our interest in this

paper, m� & 20 GeV (and hence typical neutrino energies

& 10 GeV), the stopping events dominate and constitutemore than 70% of the total number of upmu events, as isclear from Figure 2 of Ref. [20]. It is thus expected, asindeed we do find from our calculations that the most

stringent upper limits on the branching fractions of variousWIMP annihilation channels for low WIMP masses comefrom the observed rate of these stopping events.7

The 90% C.L. Poissonian upper limit on the rate of thesestopping-type upmu events for the new data set ofRef. [20], estimated from the total number of this type ofupmu events and the number of background upmus due toatmospheric neutrinos given in Fig. 3 of that reference, is8

�3:27 yr�1. With this, we can calculate, as we did in theanalysis above, the 90% C.L. upper limits on the WIMP-proton SI and SD elastic cross sections as a function ofWIMP mass for the new S-K data set of Ref. [20], for thecase of 100% branching ratio for each of the four annihi-lation channels considered above. The results, for our best-fit halo model M1, are shown in Fig. 6.The resulting upper limits on the branching fractions of

the four annihilation channels, derived from the require-ment of consistency of the new S-K–implied upper limitson the WIMP-proton elastic cross sections shown in Fig. 6with the DAMA-compatible regions, are displayed inTable III. A comparison with the corresponding numbersgiven in Table II shows that the new upper limits on thebranching fractions for the relevant annihilation channelsare roughly a factor of 3–4 more stringent.

VI. SUMMARY

Several studies in recent years have brought into focusthe possibility that the dark matter may be in the form of arelatively light WIMP of mass in the few GeV range. Suchlight WIMPs with suitably chosen values of the WIMP-nucleon SI or SD elastic cross section can be consistent

TABLE II. Upper limits—derived from Figs. 3–5—on the branching fractions for the four annihilation channels, from therequirement of consistency of the S-K implied upper limits on the WIMP-proton elastic cross sections with the DAMA-compatibleregion of the WIMP mass versus cross-section parameter space (within which the annual modulation signal observed by the DAMA/LIBRA experiment [6] is compatible with the null results of other DD experiments determined within the context of our halo model[15]), for both SI and SD interactions and the three halo models specified in Table I. The limits are calculated using the three differentupward-going muon event types, namely, FC, S and TG. The three consecutive numbers for each annihilation channel and muon eventtype refer to the three different halo models M1, M2, M3, as indicated.

EVENT TYPE UPPER LIMITS ON THE BRANCHING FRACTIONS (in %) (M1, M2, M3)

(m� range in GeV) �bb �cc �þ�� � ��

FC (2.0–8.0) 100, 100, 100 100, 100, 100 35, 40, 45 0.4, 0.6, 0.8

SI FCþ S (8.0–15.0) 100, 100, 100 100, 100, 100 100, 100, 100 25, 30, 35

FCþ Sþ TG (15.0–20.0) 100, 100, 100 100, 100, 100 100, 100, 100 25, 30, 35

FC (2.0–8.0) 0.5, 0.6, 0.7 0.5, 0.6, 0.7 0.05, 0.06, 0.07 0.0005, 0.0006, 0.0007

SD FCþ S (8.0–15.0) 0.6, 0.7, 0.8 0.6, 0.7, 0.8 0.14, 0.16, 0.18 0.012, 0.014, 0.016

FCþ Sþ TG (15.0–20.0) 0.6, 0.7, 0.8 0.6, 0.7, 0.8 0.14, 0.16, 0.18 0.012, 0.014, 0.016

7Recall that, for the older data set [18], the most stringentupper limits came from the FC events; see Table II above.

8We take the events in the 0–30� cone half-angle bin aroundthe Sun to be consistent with the analysis done above for theearlier S-K data set.

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with the annual modulation signal seen in the DAMA/LIBRA experiment [6] without conflicting with the nullresults of other direct-detection experiments. To furtherprobe the DAMA-compatible regions of the WIMP pa-rameter space—the regions of theWIMPmass versus crosssection parameter space within which the annual modula-tion signal observed by the DAMA/LIBRA experiment iscompatible with the null results of other DD experiments—we have studied in this paper the independent constraintson theWIMP-proton SI as well as SD elastic scattering crosssection imposed by the upper limit on the neutrino flux fromWIMP annihilation in the Sun given by the Super-Kamiokande experiment [18,19]. Assuming approximateequilibrium between the capture and annihilation rates ofWIMPs in the Sun, we have calculated the 90% C.L. upperlimits on the WIMP-proton SI and SD elastic cross sectionsas a function of the WIMP mass for various WIMP annihi-lation channels using the Super-Kamiokande upper limits andexamined the consistency of those limits with the 90% C.L.DAMA-compatible regions. This we have done within thecontext of a self-consistent phase-space model of the finite-size dark matter halo of the Galaxy, namely, the truncatedisothermal model [15,23], in which we take into account themutual gravitational interaction between the dark matter andthe observed visible matter in a self-consistent manner, withthe parameters of the model determined by a fit to theobserved rotation curve data of the Galaxy.

We find that the requirement of consistency of the S-K[18,19] implied upper limits on the WIMP-proton elasticcross section as a function of WIMP mass imposes strin-gent restrictions on the branching fractions of the variousWIMP annihilation channels. In the case of spin-independent WIMP-proton interaction, the S-K upper lim-its do not place additional restrictions on the DAMA-compatible region of the WIMP parameter space if theWIMPs annihilate dominantly to �bb and �cc and if directannihilations to �þ�� and neutrinos are restricted to below�ð35–45Þ% and (0.4–0.8)%, respectively. In the case ofspin-dependent interactions, on the other hand, the restric-tions on the branching fractions of various annihilationchannels are much more stringent, essentially ruling outthe DAMA-compatible region of the WIMP parameterspace if the relatively low-mass WIMPs under considera-tion annihilate predominantly to any mixture of �bb, �cc,�þ�� and � �� final states. The very latest results from theS-K Collaboration [20] put the above conclusions on aneven firmer footing by making the above constraints on thebranching fractions of various WIMP annihilation chan-nels more stringent by roughly a factor of 3–4. Similarconclusions were reached earlier [19,36] within the contextof the SHM. The quantitative restrictions on the branchingfractions for various WIMP annihilation channels obtainedhere and as given in Table II (and in Table III for the latestS-K results [20]) are, however, significantly different fromthose in the earlier works.An important aspect of the truncated isothermal model

of the Galactic halo used in the present calculation is thenon-Maxwellian nature of the WIMP velocity distributionin this model, as opposed to the Maxwellian distribution inthe SHM (see Ref. [15] for details). This directly affects theWIMP capture rate (and consequently the annihilationrate), resulting in significant quantitative differences inthe values of the upper limits on the WIMP-proton elasticcross sections (implied by the S-K upper limits on theneutrinos from the Sun) compared to the values in theSHM. Similarly, the upper limits on the branching fractions

0.0001

0.001

0.01

0.1

1

2 4 6 8 10 12 14 16 18 20

σSI

χp (

pb)

mχ (GeV)

DAMA

CDMS-II-Ge

CDMS-II-Si

CRESST-1

XENON-10

b–bc–c

τ+τ-

ν–ν

0.0001

0.001

0.01

0.1

1

10

100

1000

10000

2 4 6 8 10 12 14 16 18 20

σSD

χp (

pb)

mχ (GeV)

DAMA

CDMS-II-Ge

CDMS-II-Si

CRESST-1

XENON-10

b–bc–c

τ+τ-

ν–ν

1×10-05

1×10-05

1×10-06

1×10-07

FIG. 6. Same as Fig. 3, but using the new S-K data from Ref. [20] and considering their stopping upmu events only.

TABLE III. Upper limits on the branching fractions of the fourannihilation channels, derived from the requirement of consis-tency of the new S-K–implied upper limits on the WIMP-protonelastic cross sections shown in Fig. 6 with the DAMA-compatible regions, for both SI and SD interactions.

Upper limits on the branching fractions (in %) from

stopping events, with halo model M1�bb �cc �þ�� � ��

SI 100 100 10 0.11

SD 0.12 0.12 0.012 0.00013

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of various possible WIMP annihilation channels (from the

requirement of compatibility with DAMA results) are also

changed. At a qualitative level, however, the general con-

clusion reached earlier [19,36] within the context of the

SHM—that S-K upper limits on neutrinos from the Sun

severely restrict the DAMA-compatible region of the

WIMP parameter space—remains true in the presentmodel too, thus adding robustness to this conclusion.

ACKNOWLEDGMENTS

We thank Soumini Chaudhury, Martin Winkler and DanHooper for useful discussions and communications.

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