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arXiv:1106.5711v1 [astro-ph.GA] 28 Jun 2011
Neutrinos from WIMP annihilation in the Sun : Implications of a self-consistent
model of 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
Upper limits on the spin-independent (SI) as well as spin-dependent (SD) elastic scattering cross
sections of WIMPs (Weakly Interacting Massive Particles — the dark matter candidate) with pro-
tons, 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 param-
eter 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 frame
work of a self-consistent model of the finite-size dark matter (DM) 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 DM halo in a self-consistent
manner. Unlike in the so-called “Standard Halo Model” (SHM) which, as we argue, has several
undesirable properties, 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 the require-
ment of consistency of the S-K implied upper limits on the WIMP-proton elastic cross section as
a function of WIMP mass imposes stringent restrictions on the branching fractions of the various
WIMP annihilation channels. For SI interaction, while the S-K upper limits do not place addi-
tional 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 τ+τ−and ν¯ ν occur at larger than 10% and 10−3levels, respectively. For
SD interaction, on the other hand, the restrictions on the possible annihilation channels are much
more stringent, essentially ruling out the DAMA-compatible region of the WIMP parameter space
if the relatively low-mass (∼ 2 – 20 GeV) WIMPs under consideration annihilate predominantly to
any mixture of¯bb, ¯ cc, τ+τ−, and ν¯ ν final states. The upper limits on the branching fractions of
the various annihilation channels obtained here are about a factor of 10 more restrictive than those
obtained earlier within the context of the SHM.
∗susmita.kundu@saha.ac.in
†pijush.bhattacharjee@saha.ac.in
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I.INTRODUCTION
Weakly Interacting Massive Particles (WIMPs) (hereafter generically denoted by χ) with masses mχin the range of
few GeV to few TeV are a natural candidate for the dark matter (DM) in the Universe; See e.g., Refs. [1–5] for reviews.
Several experiments are currently engaged in efforts to directly detect such WIMPs by observing nuclear recoils due to
scattering of WIMPs off nuclei in suitably chosen detector materials in underground laboratories. Recent results from
some of these direct detection (DD) experiments, in particular the annual modulation of the nuclear recoil event rates
reported by the DAMA/LIBRA collaboration [6] and the excess of low energy recoil events reported by the CoGeNT
collaboration [7] have raised the interesting possibility [8, 9] that these events could be due to WIMPs of relatively
low mass, approximately in the range ∼ 5–10 GeV, interacting with nuclei with a WIMP-nucleon spin-independent
elastic cross section in the region of few ×10−4pb, without conflicting with the null results from other experiments
such as XENON10 [10], XENON100 [11] and CDMS-II-Si [12]. Earlier analyses (before the CoGeNT results [7]) had
also found similar compatibility of the DAMA/LIBRA annual modulation signal with the null results from other DD
experiments; see, e.g., Refs. [13–15].1
Scattering of WIMPs off nuclei can also lead to capture of the WIMPs by massive astrophysical bodies such as the Sun
or the Earth if, after scattering off a nucleus inside the body, the velocity of the WIMP falls below the escape velocity
of the body. The WIMPs so captured over the lifetime of the capturing body would gradually settle down to the core
of the body where they would annihilate and produce standard model particles, e.g., W+W−, Z0Z0, τ+τ−, t¯t, b¯b, c¯ c,
etc. Decays of these particles would then produce neutrinos, gamma rays, electrons-positrons, protons-antiprotons,
etc. For astrophysical objects like the Sun or the Earth, only the neutrinos would be able to escape. Detection
of these neutrinos by large neutrino detectors can, albeit indirectly, provide a signature of WIMPs. Although no
detection has yet been reported, the Super-Kamiokande (S-K) detector, for example, has provided upper limits on the
possible neutrino flux from WIMP annihilation in the Sun as a function of the WIMP mass [18]. Similarly, the γ-rays
produced in the annihilation of the WIMPs in suitable astrophysical environments with enhanced DM density but
low optical depth to gamma rays, such as in the central region of our Galaxy, in dark matter dominated objects such
as dwarf galaxies, and in clusters of galaxies, can offer a complimentary avenue of indirect detection (ID) of WIMPs.
Although no unambiguous gamma ray signals of dark matter origin have been reported, a recent analysis [19] of
the spectral and morphological features of the gamma ray emission from the inner Galactic Center region (within a
Galactocentric radius of ∼ 175pc) measured by the Fermi Gamma-ray Space Telescope (FGST) seems to suggest the
presence of a gamma ray emission component which is difficult to explain in terms of known sources and/or process
of gamma ray production, but is consistent with that expected from annihilations of WIMPs of mass in the 7–9 GeV
range (annihilating primarily to tau leptons) with a suitably chosen density and distribution of the dark matter in
the Galactic Center region; see, however, Ref. [20] for a different view.
In this paper we focus on the neutrinos produced by annihilations of WIMPs in the core of the Sun, and study the
constraints imposed on the WIMP mass vs. WIMP-nucleon cross section, for low-mass (<∼20GeV) WIMPs, from
non-detection of such neutrinos. This is done within the context of a self-consistent model of the finite-size dark halo of
the Galaxy [15, 21] that includes the gravitational effect of the observed visible matter on the DM in a self-consistent
manner, with the parameters of the model determined from fits to the rotation curve data of the Galaxy [22, 23].
The expected flux of neutrinos from the Sun due to WIMP annihilations depends on the rate at which WIMPs are
captured by the Sun. The capture rate depends on the density as well as the velocity distribution of the WIMPs in
the solar neighborhood as the Sun goes around the Galaxy. The density and velocity distribution of the WIMPs in the
Galaxy are a priori unknown. Most earlier studies of neutrinos from WIMP capture and annihilation in the Sun have
been done within the context of the so-called “Standard Halo Model” (SHM) in which the DM halo of the Galaxy is
described by a single component isothermal sphere [24] with a Maxwellian velocity distribution of the DM particles in
the Galactic rest frame [1, 25, 26]). The velocity distribution is isotropic, and is usually truncated at a chosen value
of the escape speed of the Galaxy. The density of DM in the solar neighborhood is typically taken to be in the range
ρDM,⊙∼ 0.3 ± 0.1GeV/cm3[27–30]. The velocity dispersion, ?v2?1/2, the parameter characterizing the Maxwellian
velocity distribution, is typically taken to be ∼ 270kms−1. This follows from the relation [24], ?v2?1/2=
?
3
2vc,∞,
1The question of compatibility of the DAMA/LIBRA and CoGeNT results with the null results of other experiments, however, remains
controversial; see, e.g., the results of a recent reanalysis of the CDMS-II Germanium data with a lowered recoil-energy threshold of 2
keV [16], as well as the recent results from the XENON100 collaboration [17], both of which claim to disfavor such a compatibility.
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between the velocity dispersion of the particles constituting a single-component self-gravitating isothermal sphere and
the asymptotic value of the circular rotation speed, vc,∞, of a test particle in the gravitational field of the isothermal
sphere and assuming vc,∞≈ vc,⊙≈ 220kms−1, where vc,⊙is the measured value of the circular rotation velocity of
the Galaxy in the solar neighborhood.2Neutrino flux from DM annihilation in the Sun for low mass WIMPs and
the resulting constraints on WIMP properties from the Super-Kamiokande upper limits on such neutrinos [18] have
been studied within the context of the SHM in Ref. [32].
Whereas the SHM serves as a useful benchmark model, there are a number of reasons why the SHM does not provide
a satisfactory description of the dynamics of the Galaxy. First, it does not take into account the modification of
the phase space structure of the DM halo due to the significant gravitational effect of the observed visible matter
on the DM particles inside and up to the solar circle. Second, the isothermal sphere model of the halo is infinite in
extent and has a formally divergent mass, with mass inside a radius r, M(r) ∝ r, as r → ∞, and is thus unsuitable
for representing a halo of finite size. Third, the procedure of truncating the Maxwellian speed distribution at a
chosen value of the local (solar neighborhood) escape speed is not a self-consistent one because the resulting speed
distribution is not in general a self-consistent solution of the steady-state collisionless Boltzmann equation describing
a finite system of collisionless DM particles. In addition, since the rotation curve for such a truncated Maxwellian
is, in general, not asymptotically flat, the relation ?v2?1/2=
Maxwellian speed distribution of the isothermal sphere, as done in the SHM, is not valid in general. Finally, recent
numerical simulations [30] seem to find that the velocity distribution of the Dark Matter particles deviates significantly
from the usual Maxwellian form. These issues are further discussed in detail in Ref. [15], where we have also discussed
the constraints on WIMP properties from the results of the direct detection (DD) experiments within the context of
our self-consistent halo model mentioned above. The present paper extends this study to the case of indirect detection
(ID) of WIMPs via neutrinos from WIMP annihilations in the Sun.
?
3
2vc,∞used to determine the value of ?v2?1/2in the
Our model of the phase space structure of the finite-size DM halo of the Galaxy is based on the so-called “lowered”
(or truncated) isothermal models (often called “King models”) [24] of the phase-space distribution function (DF)
of collisionless particles. These models are proper self-consistent solutions of the collisionless Boltzmann equation
representing nearly isothermal systems of finite physical size and mass. There are two important features of these
models: First, at every location within the system a DM particle can have speeds up to a maximum speed which is
self-consistently determined by the model itself. A particle of maximum velocity at any location within the system
can just reach its outer boundary, fixed by the truncation radius, a parameter of the model, where the DM density
by construction vanishes. Second, the speed distribution of the particles constituting the system is non-Maxwellian.
To include the gravitational effect of the observed visible matter on the DM particles, we modify the “pure” King
model DF by replacing the gravitational potential appearing in the King model DF by the total gravitational potential
consisting of the sum of those due to DM and the observed visible matter. This interaction with the visible matter
influences both the density profile and the velocity distribution of the dark matter particles as compared to those for
a “pure” King model. In particular, the dark matter is pulled in by the visible matter, thereby increasing its central
density significantly. When the visible matter density 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 further discussion
of the model, see [15, 21].
The DM distribution in the Galaxy may have significant amount of substructures which may have interesting effects
on the WIMP capture and annihilation rates [33]. However, not much information, based on observational data, is
available about the spatial distribution and internal structures of these substructures. As such, in this paper we shall
be concerned only with the smooth component of the DM distribution in the Galaxy described by our self-consistent
model mentioned above, the parameters of which are determined from the observed rotation curve data for the Galaxy.
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 WIMP mass, for various WIMP annihilation channels, using the null result
of the S-K experiment’s search for WIMP annihilation neutrinos from the Sun [18], and study the consistency of those
limits with the 90% C.L.“DAMA-compatible” regions — the regions 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
2A somewhat higher value of vc,⊙ ≈ 250kms−1, as suggested by a recent study [31], would imply a correspondingly higher value of
?v2?1/2iso≈ 306kms−1.
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the null results of other DD experiments — determined within the context of our halo model [15]. We find that
the requirement of such consistency imposes stringent restrictions on the branching fractions of the various WIMP
annihilation channels. For example, in the case of spin-independent WIMP-proton 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, ¯ cc, portions of the DAMA-compatible region can be excluded if WIMP annihilations to
τ+τ−and ν¯ ν occur at larger than 10% and 10−3levels, respectively. In the case of spin-dependent interactions, on
the other hand, the restrictions on the branching fractions of various annihilation channels are much more stringent,
essentially ruling out the DAMA-compatible region of the WIMP parameter space if the relatively low-mass WIMPs
under consideration annihilate predominantly to any mixture of¯bb, ¯ cc, τ+τ−, and ν¯ ν final states. In quantitative
terms, the upper limits on the branching fractions of the various annihilation channels obtained here are about a
factor of 10 more restrictive than the result obtained in [32] within the context of the SHM. Our results, based as
they are on a self-consistent model of the Galaxy’s dark matter halo, the parameters of which are determined by a fit
to the rotation curve data of the Galaxy, therefore, strengthen the conclusion of [32]3.
The rest of the paper is organized as follows: In section II we briefly describe the self-consistent model of the DM halo
of the Galaxy. The formalism of calculating the WIMP capture and annihilation rates in the Sun within the context
of our DM halo model, and that for calculating the resulting neutrino flux and event rate in the Super-Kamiokande
detector, are discussed in sections III and IV, respectively. Our results and the constraints on the WIMP properties
implied by these results are described in section V. The paper ends with a Summary in section VI.
II. THE SELF-CONSISTENT TRUNCATED ISOTHERMAL MODEL OF THE MILKY WAY’S DARK
MATTER HALO
The phase space distribution function (DF) of the DM particles constituting a truncated isothermal halo of the Galaxy
can be taken, in the rest frame of the Galaxy, to be of the “King model” form [15, 21, 24],
f(x,v) ≡ f(E) =
?
ρ1(2πσ2)−3/2?
0
eE/σ2− 1
?
for E > 0,
for E ≤ 0,
(1)
where
E(x) ≡ Φ(rt) −
?1
2v2+ Φ(x)
?
≡ Ψ(x) −1
2v2, (2)
is the so-called “relative energy” and Ψ(x) = −Φ(x)+Φ(rt) the “relative potential”, Φ(x) being the total gravitational
potential under which the particles move, with boundary condition Φ(0) = 0. The relative potential and relative
energy, by construction, vanish at |x| = rt, the truncation radius, which represents the outer edge of the system where
the particle density vanishes. At any location x the maximum speed a particle of the system can have is
vmax(x) =
?
2Ψ(x), (3)
at which the relative energy E and, as a consequence, the DF (1), vanish. The model has three parameters, namely,
ρ1, σ and rt. Note that the parameter σ in the King model is not same as the usual velocity dispersion parameter of
the isothermal phase space DF [24]. Also, in our calculations below, we shall use the parameter ρDM,⊙, the value of
the DM density at the location of the Sun, in place of the parameter ρ1.
Integration of f(x,v) over all velocities gives the DM density at the position x:
ρDM(x) =
ρ1
(2πσ2)3/2
?
?√
2Ψ(x)
0
dv 4πv2
?
exp
?Ψ(x) − v2/2
?
πσ2
σ2
?
− 1
?
(4)
= ρ1
exp
?Ψ(x)
σ2
?
erf
?√Ψ(x)
σ
?
−
4Ψ(x)
?
1 +2Ψ(x)
3σ2
??
,(5)
3In the present paper, the CoGeNT results [7] are not included in the analysis. Preliminary results of the analysis [34] to find the
“CoGeNT-compatible” region in the WIMP mass vs. cross section plane within the context of our halo model indicates that its inclusion
will not significantly change the above constraints on the branching fractions for the various annihilation channels.
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which satisfies the Poisson equation
∇2ΦDM(x) = 4πGρDM(x),(6)
where ΦDMis the contribution of the DM component to the total gravitational potential,
Φ(x) = ΦDM(x) + ΦVM(x), (7)
in presence of the visible matter (VM) whose gravitational potential, ΦVM, satisfies its own Poisson equation, namely,
∇2ΦVM(x) = 4πGρVM(x). (8)
We choose the boundary conditions
ΦDM(0) = ΦVM(0) = 0,and (∇ΦDM)|x|=0= (∇ΦVM)|x|=0= 0. (9)
The mass of the system, defined as the total mass contained within rt, is given by GM(rt)/rt = [Φ(∞) − Φ(rt)].
Note that, because of the chosen boundary condition Φ(0) = 0, Φ(∞) is a non-zero positive constant.
Since the visible matter distribution ρVM(x), and hence the potential ΦVM(x), are known from observations and
modeling, the solutions of equation (6) together with equations (5), (7) and the boundary conditions (9), give us a
three-parameter family of self-consistent pairs of ρDM(x) and ΦDM(x) for chosen values of the parameters (ρ1,σ,rt).
The values of these parameters for the Galaxy can be determined by comparing the theoretically calculated rotation
curve, vc(R), given by
v2
c(R) = R∂
∂R
?
Φ(R,z = 0)
?
= R∂
∂R
?
ΦDM(R,z = 0) + ΦVM(R,z = 0)
?
,(10)
with the observed rotation curve data of the Galaxy. (Here R is Galactocentric distance on the equatorial plane and
z is the distance normal to the equatorial plane.) This procedure was described in detail in Refs. [15, 21] where,
for the visible matter density distribution described there, we determined the values of the parameters rt and σ
that gave reasonably good fit to the rotation curve data of the Galaxy [22, 23] for each of the three chosen values
of the parameter ρDM,⊙= 0.2, 0.3 and 0.4 GeV/cm3. These models are summarized in Table I, which we use for
our calculations in this paper. The density profiles, mass profiles, velocity distributions of the DM particles and the
ModelρDM,⊙
(GeV/cm3) (kpc) (kms−1)
rt
σ
M1
M2
M3
0.2
0.3
0.4
120.0
80.0
80.0
300.0
400.0
300.0
TABLE I: Parameters of our self-consistent model of the Milky Way’s Dark Matter halo that give good fits to the Galaxy’s
rotation curve data, for the three chosen values of the DM density at the solar neighborhood
resulting rotation curves in each of these models are discussed in detail in Ref. [15].
With our halo model specified, we now briefly review the basic formalism of calculating the WIMP capture and
annihilation rates within the context of our halo model.
III. CAPTURE AND ANNIHILATION RATES
The capture rate per unit volume at radius r inside the Sun can be written as [35, 36]
dC
dV(r) =
?
d3u
˜f(u)
u
wΩ−(w),(11)
where˜f(u) is the WIMP velocity distribution, as measured in the Sun’s rest frame, in the neighborhood of the
Sun’s location in the Galaxy, and w(r) =
?u2+ wesc(r) is the WIMP’s speed at the radius r inside the Sun,
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wesc(r) being the escape speed at that radius inside the Sun, which is related to the escape speed at the Sun’s core,
wesc,core≈ 1354kms−1, and that at its surface, wesc,surf≈ 795kms−1, by the approximate relation
wesc2(r) = (wesc,core)2−M(r)
M⊙
?(wesc,core)2− (wesc,surf)2?. (12)
The quantity Ω−(w) is the capture probability per unit time, which is just the product of the scattering rate and the
conditional probability that after a scattering the WIMP’s speed falls below the escape speed.
We shall here consider only the elastic scattering of the WIMPs off nuclei. The dominant contribution to the WIMP
capture rate will come from the WIMPs scattering off hydrogen and helium nuclei. While for hydrogen, both spin-
independent (SI) as well as spin-dependent (SD) cross sections, σSI
cross section for helium is relevant. (We neglect here the small contribution from3He). In general, the effective
momentum-transfer (q) dependent WIMP-nucleus SI scattering cross section, σSI
in terms of the “zero-momentum” WIMP-proton (or WIMP-neutron) effective cross section, σSI
χpand σSD
χp, respectively, will contribute, only SI
χA(q), can be written in the usual way
χp= σSI
χn, as
σSI
χA(q) =µ2
χA
µ2
χp
σSI
χpA2??F(q2)??2,(13)
where A is the number of neutrons plus protons in the nucleus, µχAand µχpare the reduced masses of WIMP-nucleus
and WIMP-proton systems, respectively, with µχi= (mimχ)/(mi+mχ), and F(q2) is the nuclear form-factor (with
F(0) = 0) which can be chosen to be of the form [1]
??F(q2)??2= exp
× 10−13cm is the nuclear radius and E0 ≡ 3?2/(2mAR2) is the characteristic
nuclear coherence energy, mAbeing the mass of the nucleus.
?
−q2R2
3?2
?
= exp
?
−∆E
E0
?
.(14)
Here R ∼
?
0.91
?mA
GeV
?1/3
+ 0.3
?
With the above form of the nuclear form factor, the kinematics of the capture process [35] allows us to write the
capture probability per unit time, Ω−(w), as
Ω−(w) =
nAσχA
w
2E0
mχ
µ2
µ
+
?
exp
?
−mχu2
2E0
?
− exp
?
−mχw2
2E0
µ
µ2
+
??
θ
?µ
µ2
+
−u2
w2
?
,(15)
where nAis the number density of the scattering nuclei at the radius r inside the Sun, and µ ≡mχ
The θ function ensures that those particles which do not lose sufficient amount of energy to be captured are excluded.
mA
, µ±≡µ ± 1
2
.
We shall use Equation (15) to calculate Ω−(w) for helium (A = 4). For hydrogen, however, there is no form-factor
suppression, and the expression for Ω−(w) is simpler:
Hydrogen : Ω−(w) =σχpnH
w
?
w2
esc−µ2
−
µu2
?
θ
?
w2
esc−µ2
−
µu2
?
,(16)
where nH is the density of hydrogen (proton) at the radius r inside the Sun. Note that in equations (15) and (16),
the quantities w, wesc, nAand nH are functions of r.
The WIMP velocity distribution appearing in equation (11) is related to the phase space DF defined in equation (1)
(valid in the rest frame of Galaxy) by the Galilean transformation
˜f(u) =
1
mχf (x = x⊙,v = u + v⊙) ,(17)
where x⊙represents the sun’s position in the Galaxy (R = 8.5kpc,z = 0) and v⊙is the Sun’s velocity vector in the
Galaxy’s rest frame. Note that Gould’s original calculations and the final formula for the WIMP capture rate given in
Ref. [35], which are widely used in the literature, use a Maxwellian velocity distribution of the WIMPs in the Galaxy,
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and as such, cannot directly be used here since the WIMP velocity distribution in our case is non-Maxwellian. In
particular, note that the DF f of equation (1) vanishes for speeds v ≥ vmaxdefined in equation (3). Consequently,
equation (11) above can be written as
dC
dV(r) =2π
mχ
?1
−1
d(cosθ)
?umax(cos θ)
umin= v⊙
u duf (x = x⊙,v = u + v⊙) wΩ−(w), (18)
where v⊙≈ 220 – 250kms−1is the Sun’s circular speed in the Galaxy, and umaxis given by the positive root of the
quadratic equation
u2
max+ v2
⊙+ 2umaxv⊙cosθ = 2Ψ(x = x⊙).(19)
The total WIMP capture rate by the Sun, C⊙, is given by
C⊙=
?
R⊙
0
4πr2drdC(r)
dV
, (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 Sun4, and make the standard
assumption that the capture and annihilation processes have reached an approximate equilibrium state over the long
lifetime of the solar system (t⊙∼ 4.2 billion yrs). Under these assumptions, the total annihilation rate of WIMPs in
the Sun is simply related to the total capture rate by the relation
Γ⊙≈1
2C⊙
(21)
IV. NEUTRINO FLUX FROM WIMP ANNIHILATION IN THE SUN AND EVENT RATE IN THE
DETECTOR
A.The neutrino energy spectrum
The differential flux of muon neutrinos observed at Earth is [37]
?dφi
dEi
?
=
Γ⊙
4πD2
?
F
BF
?dNi
dEi
?
F
,(i = νµ, ¯ νµ)(22)
where Γ⊙ is the rate of WIMP annihilation in the Sun, D is the Earth-Sun distance, F stands for the possible
annihilation channels, BF is the branching ratio for the annihilation channel F and
energy spectrum of the neutrinos of type i emerging from the Sun resulting from the particles of annihilation
channel F injected at the core of the Sun. WIMPs can annihilate to all possible standard model particles e.g.
e+e−, µ+µ−, τ+τ−, νe¯ νe, νµ¯ νµ, ντ¯ ντ, ¯ q q pairs and also gauge and higgs boson pairs (W+W−, Z¯Z, h¯h), etc. In this
paper we are only interested in low mass (∼ 2 - 20GeV) WIMPs. Therefore, we will not consider WIMP annihilations
to higgs and gauge boson pairs and top quark pairs. Light quarks like u, d, s contribute very little to the energetic
neutrino flux [38], and are not considered. The same is true for muons. So, in this paper we consider only the channels
τ+τ−,¯bb, ¯ cc and ¯ νν.
?
dNi
dEi
?
F
is the differential
The neutrino energy spectra,
of hadronization of quarks, energy loss of the resulting heavy hadrons, neutrino oscillation effects, neutrino energy
loss due to neutral current interactions and absorption due to charged current interactions with the solar medium,
ντ regeneration, etc. However, given the presence of other uncertainties in the problem, particularly those associated
?
dNi
dEi
?
F, have been calculated numerically (see, e.g., [38]) by considering all the details
4Note, however, that evaporation may not be negligible for WIMP masses below ∼ 4GeV depending on the magnitude of the annihilation
cross section [32].
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with astrophysical quantities such as the local density of dark matter and its velocity distribution, we find that it is
good enough to use — as we do in this paper — approximate analytical expressions for the neutrino spectra available
in the literature [37, 39, 40]. We are interested in the fluxes of muon neutrinos and antineutrinos, for which we use
the analytic expressions given in Ref. [37], which neglect neutrino oscillation effects. By comparing with the neutrino
fluxes obtained from detailed numerical calculations [38], we find that for small WIMP masses below ∼ 20GeV (the
masses of our interest in this paper), the analytic expressions for the muon neutrino fluxes given in [37] match with
the results of detailed numerical calculations [38] to within a few percent.
The main effect of the interaction of the neutrinos with the solar medium is that [40] a neutrino of type i(= νµ, ¯ νµ)
injected at the solar core with energy Ecore
i
emerges from the Sun with an energy Eigiven by
Ecore
i
= Ei/(1 − Eiτi),(23)
and with probability
Pi= (1 + Ecore
i
τi)−αi= (1 − Eiτi)α,(24)
with
ανµ= 5.1, α¯ νµ= 9.0, τνµ= 1.01 × 10−3GeV−1, and τ¯ νµ= 3.8 × 10−4GeV−1.
Below we write down the expressions for the energy spectra of neutrinos emerging from the Sun for the four annihilation
channels considered in this paper:
(25)
1.τ+τ−channel :Neutrinos from decay of τ leptons (τ → µνµντ)
For this channel, the spectrum of muon-type neutrinos at the solar surface, including the propagation effects in the
solar medium, can be written as [37]
?dNi
dEi
?
τ+τ−= (1 − Eiτi)(αi−2)
?dNcore
i
dEcore
i
?
τ+τ−
,(i = νµ, ¯ νµ) (26)
where the relationship between Ei and Ecore
respectively, and
i
, and the values of αiand τi, are as given by equations (23) and (25),
?dNcore
i
dEcore
i
?
τ+τ−=48Γτ→µνµντ
βγm4
τ
?1
2mτ(Ecore
i
)2−2
3(Ecore
i
)3
?min(1
2mτ,E+)
E−
(27)
is the neutrino spectrum due to decay of the τ-leptons injected at the solar core by WIMP annihilations. Here
Ecore
i
γ (1 ∓ β)
Γτ→µνµντ= 0.18, and E±=
with γ =?1 − β2?−1/2= mχ/mτ, mτ being the τ-lepton mass.
Note that the ντs produced from τ decay may again produce τs by charged current interactions in the solar medium,
and these secondary τs can decay to give secondary νµs. But these νµs would be of much lower energy compared to
the primary νµs from τ decay, and are not considered here.
2.
¯bb channel :Neutrinos from decay of b-quark hadrons (b → cµνµ)
The treatment is similar to the case of τ decay described above. However, here the hadronization of quarks and
stopping of heavy hadrons in the solar medium have to be taken into account. The resulting spectrum of muon-
neutrinos emerging from the Sun is given by [37]
?dNi
dEi
?
¯b b
=
?E0
mb
?1
N
dN
dEd
?hadron
(E0,Ed)(1 − Eiτi)(αi−2)
?dNcore
i
dEcore
i
?
¯b b
(Ed,Ecore
i
)dEd,(i = νµ, ¯ νµ)(28)
Page 9
9
where mbis the b-quark mass, E0≈ 0.71mχis the initial energy of the b-quark hadron (the fragmentation function
is assumed to be a sharply peaked function [37]),
?dNcore
i
dEcore
i
?
¯bb
(Ed,Ecore
i
) =48Γb→µνµX
βγm4
b
?1
2mb(Ecore
i
)2−2
3(Ecore
i
)3
?min(1
2mb,E+)
E−
(29)
is the neutrino spectrum resulting from decay of the b-quark hadron injected at the solar core, and
?1
N
dN
dEd
?hadron
(E0,Ed) =Ec
E2
d
exp
?Ec
E0
−Ec
Ed
?
,(30)
with Ec≈ 470GeV, is the distribution of the hadron’s energy at the time of its decay if it is produced with an initial
energy E0. In equation (29), Γb→µνµX = 0.103 is the branching ratio for inclusive semi-leptonic decay of b-quark
Ecore
i
γ (1 ∓ β)
hadrons to muons [41], and E±=
with γ =?1 − β2?−1/2= Ed/mb.
3.¯ cc channel :Neutrinos from decay of c-quark hadrons (c → sµνµ)
Again, this is similar to the case of b-decay discussed above, except that the kinematics of the process is slightly
different. The resulting muon neutrino spectrum is given by [37]
?dNi
dEi
?
¯ cc
=
?E0
mc
?1
N
dN
dEd
?hadron
(E0,Ed)(1 − Eiτi)(αi−2)
?dNcore
i
dEcore
i
?
¯ cc
(Ed,Ecore
i
)dEd,(i = νµ, ¯ νµ)(31)
where mcis the c-quark mass, E0≈ 0.55mχis the initial energy of the charmed hadron,
?dNcore
i
dEcore
i
?
¯ cc
(Ed,Ecore
i
) =8Γc→µνµX
βγm4
c
?3
2mc(Ecore
i
)2−4
3(Ecore
i
)3
?min(1
2mc,E+)
E−
(32)
is the neutrino spectrum resulting from decay of the c-quarks injected at the solar core, with Γc→µνµX = 0.13,
Ecore
i
γ (1 ∓ β), γ =?1 − β2?−1/2= Ed/mc, and
for c-quark.
E±=
?1
N
dN
dEd
?hadron
(E0,Ed) is given by equation (30) with Ec≈ 250GeV
4.ν¯ ν channel :
(χχ → νµ¯ νµ)
In this case the spectrum of muon neutrinos emerging from the Sun is simply given by
?dNi
dEi
?
ν¯ ν
= (1 − Eiτi)(αi−2)
?dNcore
i
dEcore
i
?
ν¯ ν
,(i = νµ, ¯ νµ)(33)
where
?dNcore
i
dEcore
i
?
ν¯ ν
= δ (Ecore
i
− mχ) ≡ (1 + mχτi)−2δ
?
Ei−
mχ
1 + mχτi
?
.(34)
B.Calculation of Event Rates in the Super-Kamiokande detector
The 50-kiloton Super-Kamiokande (S-K) water Cherenkov detector searched for neutrinos from WIMP annihilation
in the Sun, with 1679.6 live days of data on neutrino-induced upward through-going muons, giving upper-limits on
the flux of such neutrinos as a function of WIMP mass [18]. The muons would be produced in charged-current
interactions of muon neutrinos with the nuclei (primarily the protons in the water) inside the detector volume as well
as with the nuclei constituting the surrounding rocks around the detector volume.
Page 10
10
The rate of neutrino induced upward-going muon events, R, in the S-K detector due to νµs and ¯ νµs from WIMP
annihilation in the Sun can be written as
R =1
2
?
i=νµ,¯ νµ
? ?
dφi
dEi
dσiN
dy
(Ei,y)Veff(Eµ)n
water
p
dEidy,(35)
where
dφi
dEiis the differential flux of the neutrinos given by equation (22),dσiN
charged current differential cross sections, (1−y)(= Eµ/Ei) is the fraction of the neutrino energy transfered to the the
muon, Veff(Eµ) is the effective volume of the detector, and nwater
number). In their analysis, the S-K Collaboration used the upward through-going muons with measured path-length
of > 7 meters in the inner detector which has an effective area of Aeff≈ 900m2and height ≈ 36.2 m. This 7-meter
cut on the muon track length can be effectively taken into account by setting Veff= 0 if the effective water-equivalent
muon range, Rµ(Eµ) ≈ 5meters ×(Eµ/GeV), is less than 7 meters, and Veff = Aeff ×[Rµ(Eµ) + (36.2 − 7) meters]
otherwise [32]. The factor of 1/2 accounts for the fact that only up-going muon events were considered in order to
avoid the background due to down-going muons produced due to cosmic ray interactions in the Earth’s atmosphere.
dy
are the relevant neutrino-nucleon
p
is the number density of protons in water (= Avogadro
To set upper limits on the WIMP elastic scattering cross section as a function of WIMP mass for a given annihilation
channel, we use the fact that S-K collaboration observed a total of 168 events within a cone of half-angle ∼ 30◦centered
on the Sun (see Figure 2 of Ref. [18]) in 1679.6 days from April 1996 to July 2001. The expected background, essentially
due to atmospheric neutrinos, is about 185 events (including the effect of neutrino flavor oscillation). This gives a
90% C.L. (≡ 1.64σ) upper limit of ∼ 5 events from possible WIMP annihilation in the Sun. This, in turn, allows us to
calculate the 90% C.L. upper limit on the WIMP-proton SI as well as SD elastic scattering cross section as a function
of WIMP mass for a given value of the branching fraction for any particular annihilation channel. These limits are
discussed in the next section.
V.RESULTS AND DISCUSSIONS
Fig. 1 shows the dependence of the capture rate of WIMPs by the Sun as a function of the WIMP’s mass for the
three halo models specified in Table I. As expected, for a given DM density, the capture rate decreases as WIMP
mass increases because heavier WIMPs correspond to smaller number density of WIMPs.
1e+24
1e+25
1e+26
2 4 6 8 10
mχ (GeV)
12 14 16 18 20
Capture Rate (sec-1)
σSI
χp=10-4 pb
M1
M2
M3
1e+23
1e+24
1e+25
2 4 6 8 10
mχ (GeV)
12 14 16 18 20
Capture Rate (sec-1)
σSD
χp=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 spin-independent
(SI: left panel) and spin-dependent (SD: right panel) WIMP-proton interactions. All the curves are for a reference value of the
WIMP-proton elastic SI or SD cross section of 10−4pb.
The event rates in the S-K detector as a function of the WIMP mass for the four different WIMP annihilation channels
are shown in Figure 2 assuming 100% branching ratio for each channel by itself. For each annihilation channel the three
Page 11
11
0.001
0.01
0.1
1
10
100
1000
1 2 3 4 5 7 10 20
Event Rate (year-1)
mχ (GeV)
M1
M2
M3
σSI
χp=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
Event Rate (year-1)
mχ (GeV)
M1
M2
M3
σSD
χp=10-4 pb
b–b
c–c
τ+τ-
ν–ν
FIG. 2: The upward through-going event rates in the Super-Kamiokande detector due to neutrinos from WIMP annihilation in
the Sun as a function of the WIMP mass for the four annihilation channels as indicated, assuming 100% branching ratios for each
channel by itself, and for spin-independent (SI: left panel) and spin-dependent (SD: right panel) WIMP-proton interactions.
The three curves for each annihilation channel correspond, as indicated, to the three halo models specified in Table I. All the
curves are for a reference value of the WIMP-proton elastic SI or SD cross section of 10−4pb.
curves correspond, as indicated, to the three halo models specified in Table I. It is seen that the direct annihilation
to the ν¯ ν channel dominates the event rate, followed by the τ+τ−channel.
Our main results are contained in Figures 3, 4 and 5, where we show, for the three halo models considered, the 90%
C.L. upper limits on the WIMP-proton SI and SD elastic cross sections (as a function of WIMP mass) derived from
the Super-Kamiokande measurements of the up-going muon events from the direction of the Sun [18], for the four
annihilation 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, the 90% C.L. allowed regions [15] in the WIMP mass vs. WIMP-proton
elastic cross section plane implied by the DAMA/LIBRA collaboration’s claimed annual modulation signal [6], as well
as the 90% C.L. upper limits [15] on the relevant cross section as a function of the WIMP mass implied by the null
results from the CRESST-1 [42], CDMS-II-Si [12], CDMS-II-Ge [43] and XENON10 [10] experiments.
The curves in Figures 3, 4 and 5 allow us to derive upper limits on the branching fractions of the various WIMP
annihilation channels, from the requirement of consistency of the S-K-implied upper limits on the WIMP-proton
elastic cross section with the DAMA-compatible regions. These upper limits are shown in Table II for the three halo
models discussed 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 is already consistent with the S-K upper limit even for 100%
branching fractions in these channels (the respective curves for the various annihilation channels only move upwards,
keeping the shape same, as the branching fractions are made smaller). At the same time, for the τ+τ−channel and
SI interaction, although a 100% branching fraction in this channel allows a part of the DAMA-compatible region to
be consistent with the S-K upper limit, consistency of the entire DAMA-compatible region with the S-K upper limit
requires the branching fraction for this channel to be less than 10–14% depending on the halo model. On the other
hand, for the ν¯ ν channel and SI interaction, there are already strong upper limits (at the level of 10 – 11%) on the
branching fraction for this channel for consistency of even a part of the DAMA-compatible region with the S-K upper
limit; and these upper limits become significantly more stringent (by about two orders of magnitude) if the entire
DAMA-compatible region 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 channels, 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 (1.6 – 2.0)×10−3. On the other hand, for τ+τ−and ν¯ ν 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 (4 – 5)×10−4and
Page 12
12
1e-07
1e-06
1e-05
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–b
c–c
τ+τ-
ν–ν
1e-05
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–b
c–c
τ+τ-
ν–ν
FIG. 3: The 90% C.L. upper limits on the WIMP-proton spin-independent (SI: left panel) and spin-dependent (SD: right panel)
elastic cross section as a function of WIMP mass derived from the Super-Kamiokande measurements of the up-going muon
events from the direction of the Sun [18], for the four annihilation channels discussed in the text, assuming 100% branching
ratio for each channel by itself. The curves shown are for our halo model M1 (ρDM,⊙ = 0.2GeV/cm3) specified in Table I.
The 90% C.L. allowed regions in the WIMP mass vs. WIMP-proton elastic cross section plane implied by the DAMA/LIBRA
experiment’s claimed annual modulation signal [6] for the same halo model [15], as well as the 90% C.L. upper limits on the cross
section as a function of the WIMP mass implied by the null results from the CRESST-1 [42], CDMS-II-Si [12], CDMS-II-Ge [43]
and XENON10 [10] experiments (solid curves), for the same halo model [15], are also shown.
1e-07
1e-06
1e-05
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–b
c–c
τ+τ-
ν–ν
1e-05
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–b
c–c
τ+τ-
ν–ν
FIG. 4: Same as Fig. 3, but for the halo model M2 (ρDM,⊙ = 0.3GeV/cm3) specified in Table I.
(3.5 – 4.5)×10−5, respectively, while consistency of the entire DAMA-compatible regions with the S-K upper limits
requires these fractions to be respectively lower by about a factor of 2 (for the τ+τ−channel) and a factor of about
20 (for the ν¯ ν channel).
The above small numbers for the upper limits on the branching fractions of the four dominant neutrino producing
WIMP annihilation channels imply, in the case of spin-dependent WIMP interaction, that the DAMA-allowed region
of the mχ– σSD
annihilations in the Sun, unless, of course, WIMPs efficiently evaporate from the Sun — which may be the case for
relatively small mass WIMPs below 4 GeV [32] — or there are other non-standard but dominant WIMP annihilation
channels that somehow do not eventually produce any significant number of neutrinos while restricting annihilation to
quark (¯bb, ¯ cc) channels to below 10−3level and τ+τ−and ν¯ ν channels to below 10−4and 10−5level, respectively. In
the case of spin-independent interaction, however, the DAMA-compatible region of the mχ– σSI
χpparameter space is essentially ruled out by the S-K upper limit on neutrinos from possible WIMP
χpparameter space (or
Page 13
13
1e-07
1e-06
1e-05
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–b
c–c
τ+τ-
ν–ν
1e-05
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–b
c–c
τ+τ-
ν–ν
FIG. 5: Same as Fig. 3, but for the halo model M3 (ρDM,⊙ = 0.4GeV/cm3) specified in Table I.
MODEL
M1
M2
M3
M1
M2
M3
¯bb ¯ cc
100%
100%
100%
0.16%
0.18%
0.20%
τ+τ−
ν¯ ν
SI
100%
100%
100%
0.16%
0.18%
0.20%
100% (10%)
100% (12%)
100% (14%)
0.04% (0.016%)
0.045% (0.02%)
0.05% (0.025%)
10% (0.13%)
10.5% (0.15%)
11% (0.18%)
0.0035% (0.00015%)
0.004% (0.0002%)
0.0045% (0.00025%)
SD
TABLE II: Upper limits on the branching fractions for the four annihilation channels derived from the requirement of consistency
of the S-K implied upper limits on the WIMP-proton elastic cross sections with the “DAMA-compatible” regions — the
regions 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 spin-independent (SI) and spin-dependent (SD) interactions and the three halo models specified
in Table I. The numbers outside the parentheses refer to the largest allowed branching fractions above which no parts of the
DAMA-compatible region are consistent with the S-K upper limits, while the numbers within parentheses correspond to upper
limits on the branching fractions imposed by the requirement of consistency of the entire DAMA-compatible region with the
S-K upper limits. Note that for¯bb and ¯ cc channels, the shapes of the curves in Figures 3, 4 and 5 are such as to allow only
single numbers for the relevant branching fraction upper limits.
at least a part thereof) remains unaffected by the S-K upper limit if WIMPs annihilate dominantly to quarks and/or
tau leptons, and annihilation directly to neutrinos is restricted below ∼ 10% level. At the same time, portions of
the DAMA-compatible region can be excluded if WIMP annihilation to τ+τ−occurs at larger than 10% level and/or
annihilation to ν¯ ν occurs at larger than 10−3level. These results for the upper limits on the branching fractions of
various annihilation channels from the requirement of consistency of the DAMA-compatible region with S-K upper
limit are roughly an order of magnitude more restrictive than those obtained in [32] within the context of the SHM.
Thus, our results, based as they are on a self-consistent model of the Galaxy’s dark matter halo, the parameters of
which are determined by a fit to the rotation curve of the Galaxy, strengthen the earlier results that the S-K upper
limit on the possible flux of neutrinos due to WIMP annihilation in the Sun severely restricts the DAMA region of
the WIMP mass versus cross section plane, especially in the case of spin-dependent interaction of WIMPs with nuclei.
VI.SUMMARY
Several studies in recent years have brought into focus the possibility that the dark matter may be in the form of a
relatively light WIMP of mass in the few GeV range. Such light WIMPs with suitably chosen values of the WIMP-
nucleon SI or SD elastic cross section can be consistent with the annual modulation signal seen in the DAMA/LIBRA
experiment [6] without conflicting with the null results of other direct-detection experiments.
the “DAMA-compatible” regions of the WIMP parameter space — the regions of the WIMP mass versus cross
To further probe
Page 14
BIBLIOGRAPHY14
section parameter space within which the annual modulation signal observed by the DAMA/LIBRA experiment is
compatible with the null results of other DD experiments — we have studied in this paper the independent constraints
on the WIMP-proton SI as well as SD elastic scattering cross section imposed by the upper limit on the neutrino
flux from WIMP annihilation in the Sun given by the Super-Kamiokande experiment [18]. Assuming approximate
equilibrium between the capture and annihilation rates of WIMPs in the Sun — which implies that the annihilation
rate is essentially determined by the capture rate, the latter being primarily determined by the WIMP-proton elastic
scattering cross section — we have calculated the 90% C.L. upper limits on the WIMP-proton SI and SD elastic cross
sections as a function of the WIMP mass for various WIMP annihilation channels using the Super-Kamiokande upper
limit, and examined the consistency of those limits with the 90% C.L.“DAMA-compatible” regions. This we have
done within the context of a self-consistent phase-space model of the finite-size dark matter halo of the Galaxy [15, 21]
that includes the mutual gravitational interaction between the dark matter and the observed visible matter in a self-
consistent manner, with the parameters of the model determined by a fit to the observed rotation curve data of the
Galaxy.
We find that the requirement of consistency of the S-K implied upper limits on the WIMP-proton elastic cross section
as a function of WIMP mass imposes stringent restrictions on the branching fractions of the various WIMP annihilation
channels. In the case of spin-independent WIMP-proton interaction, 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,
¯ cc, and τ+τ−, and annihilation directly to neutrinos is restricted below ∼ 10% level. However, portions of the DAMA
region can be excluded if WIMP annihilations to τ+τ−and ν¯ ν occur at larger than 10% and 10−3levels, respectively.
In the case of spin-dependent interactions, on the other hand, the restrictions on the branching fractions of various
annihilation channels are much more stringent, essentially ruling out the DAMA-compatible region of the WIMP
parameter space if the relatively low-mass WIMPs under consideration annihilate predominantly to any mixture of
¯bb, ¯ cc, τ+τ−, and ν¯ ν final states.
The results obtained in this paper are based on a self-consistent model of the Galaxy’s dark matter halo, the parameters
of which are determined by a fit to the rotation curve of the Galaxy. Our results for the upper limits on the branching
fractions of various annihilation channels from the requirement of consistency of the DAMA-compatible region with
S-K upper limit are roughly an order of magnitude more restrictive than those obtained earlier in Ref. [32] within
the context of the SHM. Thus, our results strengthen the conclusion that the S-K upper limit on the possible flux of
neutrinos due to WIMP annihilation in the Sun severely restricts the DAMA region of the WIMP mass versus cross
section plane, especially in the case of spin-dependent interaction of WIMPs with nuclei.
Acknowledgments
We thank Soumini Chaudhury for useful discussions.
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