High energy neutrinos from neutralino annihilations in the Sun

Neutralino annihilations in the Sun to weak boson and top quark pairs lead to high-energy neutrinos that can be detected by the IceCube and KM3 experiments in the search for neutralino dark matter. We calculate the neutrino signals from real and virtual WW, ZZ, Zh, and $t \bar t$ production and decays, accounting for the spin-dependences of the matrix elements, which can have important influences on the neutrino energy spectra. We take into account neutrino propagation including neutrino oscillations, matter-resonance, absorption, and nu_tau regeneration effects in the Sun and evaluate the neutrino flux at the Earth. We concentrate on the compelling Focus Point (FP) region of the supergravity model that reproduces the observed dark matter relic density. For the FP region, the lightest neutralino has a large bino-higgsino mixture that leads to a high neutrino flux and the spin-dependent neutralino capture rate in the Sun is enhanced by 10^3 over the spin-independent rate. For the standard estimate of neutralino captures, the muon signal rates in IceCube are identifiable over the atmospheric neutrino background for neutralino masses above M_Z up to 400 GeV.


I. INTRODUCTION
A stable Weakly Interacting Massive Particle (WIMP) of mass of order 100 GeV that was produced in the early Universe, thermalized, and froze-out due to the Hubble expansion provides a natural explanation for observed density of dark matter today [1]. A well motivated dark matter (DM) candidate is the Lightest Stable Particle (LSP) of Supersymmetry (SUSY) with R-parity conservation [2,3,4]. The LSP is nominally the lightest neutralino (denoted by χ 0 1 ), a neutral spin-1/2 particle that is a linear combination of gauginos (spin-1/2 SUSY companions of the Spin-1 Standard Model bino and Wino) and higgsinos (spin-1/2 companions of two spin-0 Higgs bosons) [5]. The minimal supergravity model (mSUGRA) has minimal SUSY particle content with the exact supersymmetry broken by gravity [6] . With gauge and Yukawa coupling at the Grand Unified Scale (GUT) [7,8,9] the mSUGRA model predictions are given in terms of a small number of parameters [10,11].
The consequences of mSUGRA for neutralino dark matter detection have been the subject of numerous phenomenological studies that have considered the predictions for forthcoming experiments; some recent surveys are given in Refs. [12,13,14,15,16,17,18,19]. There are three complementary experimental approaches: direct detection via nuclear recoils from WIMP scattering, indirect detection via astrophysics experiments wherein WIMP annihilations give neutrino, gamma ray, positron, and antideuteron signals, and collider experiments where the supersymmetric particles undergo cascade decays to final states with two LSPs that give missing energy in the events.
Our focus is on the signals in neutrino telescopes that should result from the annihilations of neutralinos that have been gravitationally captured by the Sun [20,21]. Neutrino telescopes are poised to search for the high energy neutrinos of this origin [22]. The IceCube experiment at the South Pole is underway and expects to have about 50,000 events from atmospheric neutrinos in the near future [23,24]. The KM3 detector in the Mediterranean Sea is currently being built [25,26,27,28,29,30]. The IceCube detector has a neutrino energy threshold of 50 GeV and the KM3 detector is expected to have sensitivity to neutrinos above a threshold of 10 GeV. These experiments are expected to have the capability to find or limit signals from neutralino DM annihilations in the Sun [21]. It is therefore of particular interest to refine the characteristic features imprinted on the neutrino energy spectra of this origin.
We make substantial improvements on previous theoretical studies [31] by including the full spin-dependence of the matrix elements from real and virtual W W, ZZ, Zh, and tt production and subsequent decays to neutrinos, by noting the importance of the spin-dependent capture rate in the Sun, and by concentrating on the region of mSUGRA parameter space known as the Hyperbolic Branch (HB) [32] or Focus Point (FP) [33] region, where the neutralino annihilations have the highest rates. (Hereafter we denote this region simply by FP).
The above mentioned annihilation processes lead to the high energy neutrinos of interest for neutrino telescopes. We propagate the neutrinos through the Sun, taking into account neutrino oscillations, matter resonance, absorption, and ν τ regeneration effects, and give the resulting muon-neutrino energy spectra at Earth and the muon energy spectra in km 2 area detectors.
Neutralinos are gravitationally trapped and accumulate at the center of the Sun where capture and annihilation rates equilibrate over the lifetime of the Sun. The solar capture rate of neutralinos in the galactic halo is approximately given by [34] where ρ local and v local are the local density and velocity of relic dark matter, respectively. The average density is taken to be ρ local ≈ 0.3 GeV/cm 3 , but may be enhanced due to caustics in the galactic plane [35]. The capture rate is highly dependent on the strength of the neutralino interactions with matter [13,36]. The spin-independent (SI) and spin-dependent (SD) scattering rates determine how strongly the Sun slows and captures the neutralinos and are limited by present direct detection experiments. The factor of 0.07 before σ He SI comes from the relative abundance of helium and hydrogen in the Sun, as well as other dynamical and form factor effects [22]. For a discussion of the present limits on the SI and SD cross sections from direct detection experiments see Ref. [12].
As neutralinos accumulate in the solar core, their annihilations deplete the population.
The competition between neutralino capture and annihilation can be expressed through the annihilation rate, Γ, in the solar core by where A ⊙ = σv V is the annihilation rate times relative velocity per unit volume and t Sun = 4.5 Gy is the age of the Sun. When √ C ⊙ A ⊙ t Sun ≫ 1, the capture and annihilation processes are in equilibrium, which is expected to be a good approximation for the Sun. Thus the annihilation rate is equivalent to half the solar capture rate today since two neutralinos create one annihilation event.
From the thermalization of the captured neutralinos, it has been estimated that the neutralinos are concentrated with a core region of very small radius, of order 0.01R Sun [31], so we can approximate the source of the neutrinos from annihilations as arising from the center of the Sun. We propagate the neutrinos from the solar center to the surface, taking into account neutrino oscillations with MSW effects [37,38,39], as well as absorption and reinjection of neutrinos resulting from neutral current (NC) and charged current (CC) interactions. Since neutralino annihilation is flavor blind 1 , the relative populations of different neutrino species are the same at the center of the Sun. MSW effects that change the neutrino flavors as they propagate through the Sun are small in the present application because the injection spectra are uniformly populated among flavors (i.e. the commutator which describes the evolution of the neutrino states as they propagate through the Sun vanishes, yielding a steady state solution). In addition, MSW effects are smaller at energies above order 100 GeV that are of present interest 2 . In the results that we present here, we take θ 13 = 0, but we do not anticipate substantial changes for nonzero θ 13 over its experimentally allowed range 3 . The analysis of Ref. [31] found that a value of sin 2 2θ 13 = 0.04 gave little change in the overall neutrino spectra in the high energy region.
Absorption and reinjection effects due to NC and CC interactions are important. The effects from NC interactions are neutrino flavor blind and tend to soften the initial spectra; when a neutrino of energy E ν is absorbed, a neutrino of the same flavor but lower energy E ′ ν < E ν is reinjected into the flux. The CC absorption is also flavor blind, but the resulting reinjected spectra are not, due to the regeneratedν e andν µ from the leptonic decays of τ − leptons. A τ − decays promptly 100% to ν τ and 18% to each ofν e andν µ 4 . Although the ν τ 1 Strictly speaking, secondary neutrinos from the τ -leptons and b-quarks from W and Z decays break this flavor democracy but these contributions result in softer neutrino energy spectra that are less likely to pass the experimental acceptance cuts. 2 A concurrent study in Ref. [40] analytically considers the MSW effects of low energy neutrinos from dark matter with mass up to 100 GeV. 3 Its experimentally allowed range is sin 2 2θ 13 < 0.19 [41]. 4 The µ ± leptons can also injectν e and ν e , but are stopped by ionizing radiation before they decay, resulting in very soft neutrinos [42]. number population remains the same, the ν τ energy spectrum is softened by the interactions.
Additionally, other neutrino flavors receive contributions from the ν τ CC interaction. This asymmetry in the population of neutrino species introduces small, but nonetheless nonnegligible vacuum oscillation and NC and CC effects. Once the neutrinos emerge from the Sun, they propagate as matter eigenstates to the Earth where they can be detected in neutrino telescopes by the high energy muons and electrons produced by the CC interactions in the surrounding ice or water.
The above effects were included in a number of works [14,23,31]. In Ref. [31] the reconstruction of a general dark matter candidate was studied. However, since the spin of the DM particle was not specified, the spin-dependence of the amplitudes could not be taken into account. The intent to treat the DM annihilations in a model independent way is laudable, but this approach can have serious shortcomings. For instance, DM annihilation in the static limit to the two body neutrino-antineutrino process is absent in SUSY models, due to helicity suppression. Additionally, processes such as annihilations to weak bosons exhibit a striking helicity dependence in the neutrino spectra from their decays. The spectra of neutrinos of energy O(100 GeV) can be substantially altered by the spin dependence of the matrix elements.
In Section II, we review the regions of mSUGRA parameter space that can account for the observed relic density of neutralinos [43]. We concentrate on the FP region which solves a variety of problems with SUSY phenomenology [44,45,46]. In this region of parameter space, the lightest neutralino is a bino-higgsino mixture and the neutralino annihilation rates are large. In Section III, we discuss the neutrino production modes that occur through neutralino annihilation in the present epoch. In Section IV, we calculate the neutrino energy spectra for each contributing process. We give analytic formulae for the neutrino spectra in the static limit where the intermediate state heavy particles are on-shell and show that they closely approximate the matrix elements obtained from SMADGRAPH [47,48,49,50,51].
Propagation of the high energy neutrinos from the center of the Sun to the km 2 area detectors on Earth are discussed in Section V. The prospects of detecting neutrinos from neutralino annihilation are presented in Section VI. We provide our conclusions in Section VII.

II. SUSY PARAMETER SPACE
In our illustrations we implicitly assume the model of minimal supergravity (mSUGRA).
In this model the scalar, gaugino and trilinear masses unify at the grand unification (GUT) scale. The predictions are determined by the five parameters [11] Here m 0 , m 1/2 and A 0 are the common scalar, gaugino and trilinear masses at the GUT scale. The parameter tan β is the ratio vacuum expectation values of the up-type and downtype neutral Higgs fields and µ is the supersymmetry conserving higgsino mass parameter at the electroweak scale. The absence of a Higgs signal from LEP and the relic density determination from WMAP [43] limit the mSUGRA parameter space to four distinct regions: (i) The focus point with large values of m 0 and relatively small values of µ can naturally yield the observed relic density [32,33,44,45,46,52,53]. This region provides a decoupling solution to the SUSY flavor and CP problems, suppress proton decay rates, and has low fine tuning. The FP has large sfermion masses and mixed higgsinobino dark matter and can be suitable for sparticle mass measurements at the LHC [54] . The annihilation modes of the lightest neutralino are predominantly to weak bosons and top quark pairs due to the large higgsino-bino mixture of the lightest neutralino.
(ii) The slepton co-annihilation region has low m 0 and the lightest slepton (stau) mass is almost degenerate in mass with the lightest neutralino mass [55,56,57,58]. The resulting co-annihilation can drop the neutralino relic density into the observed range.
(iii) The bulk region is characterized by low m 0 and m 1/2 [59,60]. In this region, neutralino annihilation is dominated by the t-channel exchange of light scalar fermions. However, this region is now disfavored by the limits from LEP on the lightest chargino and sleptons [10,61].
Throughout the remainder of this paper, we adopt the low energy determination of the FP region given in Ref. [46], which is preferred for b − τ Yukawa unification [69,70].
where g 1 and g 2 are the U(1) and SU(2) gauge couplings, respectively. Gaugino mass unification restricts the SU(2) gaugino mass M 2 = The lightest neutralino can then be expressed in the gaugino-higgsino eigenbasis as Using these methods, we numerically diagonalize Eq. (4) and determine the mass and composition of the lighest neutralino. The composition of the lightest neutralino at some representative points in the FP region are given in Table I.
The mass matrix of the charginos in the (W ± ,H ± ) basis is The chargino mass matrix is diagonalized by the rotation matrices U ij and V ij The relic density of the lightest neutralino is very sensitive to its gaugino-higgsino content.
To accommodate the observed relic density, the lightest neutralino is required to have large  given in [46]. As the mass of the lightest neutralino increases, it becomes largely higgsino shown by the solid, black curve (right panel). However, the higgsino asymmetry, an important quantity for the Zχ 0 1 χ 0 1 coupling, decreases as shown by the red, dashed curve.
bino and higgsino components, leading to the relation µ ≈ M 1 for light neutralinos as shown in Fig. 1. As the lightest neutralino mass increases, so do the µ and M 1 values and the neutralino mass increasingly controls the annihilation rate until the χ 0 1 mass cannot increase further as the relic density would increase above the observed value. This behavior is seen by the plateau in Fig. 1 at high M 1 and µ ≈ 1 TeV. The heaviest that the χ 0 1 can be in the FP region (and still give the correct relic density) corresponds nearly pure higgsino content (i.e.   (ii) Annihilation to Zh, where the Z-boson decays to neutrinos, as shown in Fig. 3. This process is enhanced if the higgsino asymmetry is large, due to the χ 0 1 χ 0 1 Z coupling that occurs in both diagrams. However, a substantial t-channel neutralino exchange amplitude requires a large bino component of the lightest neutralino. This process is similar to the annihilation to weak bosons except that the emission of the Higgs boson changes the helicity structure of the amplitude which alters the shape of the neutrino energy spectra. (iii) Annihilation to top quark pairs as shown in Fig. 4. Contributions from annihilations to top quarks 5 , whose primary decay is t →bW . This process can result in a softer neutrino energy spectra compared to direct W + W − leptonic decays due to the smaller available phase space for the neutrinos. The scalar fermion exchange is suppressed due to the large m 0 values in the FP region while the Z and Higgs boson exchanges require higgsino asymmetry and higgsino-bino dominance, respectively. (iv) Neutralino annihilation to light quarks and leptons. Due to the Majorana nature of the neutralino, the annihilation amplitude to light fermions is suppressed by the fermion mass, and direct production of a neutrino-antineutrino pair is therefore negligible 6 .
If a γ or Z boson is emitted, the helicity suppression can be lifted. The leading [72]. In addition to squark exchange, s-channel Higgs and Z boson processes contribute when the lightest neutralino is dominantly higgsino. We do not consider the smaller contributions from these processes. Moreover, since the neutralino is above the W mass throughout most of the parameter space in the FP region, these modes are negligible 7 . Table II summarizes the dominant subprocess discussed above and gives the corresponding neutralino annihilation modes and the composition of the χ 0 1 that yields a large rate in each case. The total annihilation cross section must remain relatively fixed in order to provide the observed relic density Ω χ 0 1 h 2 ≈ 0.1 pb σv , where σv is the thermally averaged cross section at freeze-out. The DM density is thus inversely proportional to the total annihilation rate to neutrinos today 8 . 6 Note in the analysis in Ref. [31] of the neutrino spectra from dark matter annihilation it was assumed that the dark matter particle was not Majorana, yielding very clean and distinct neutrino signals from the direct dark matter annihilation to neutrino pairs. This possibility is not allowed in SUSY models where the neutralino DM candidate is a Majorana particle. 7 The neutralino lower mass bound is approximately 80 GeV due to the lower mass limit on the lightest chargino (M χ ± 1 > 104 GeV) and the gaugino mass unification assumption. 8 Up to factors that include the p-wave terms, which are suppressed in the present epoch where v ∼ 10 −3 c.  Fig. 1.
The annihilation cross section to the neutrinos (σ ν ) via the W W, ZZ, Zh and tt modes are shown in Fig. 5. Once the neutralino mass exceeds the top quark mass, annihilation to top-quark pairs dominates, because the Higgs boson coupling to top quarks, Y t , is large.
However, since the χ 0 1 is then dominantly higgsino with large mass, its coupling to the lightest Higgs boson decreases since some bino or wino content is necessary. Therefore, at TeV scale χ 0 1 mass, the annihilation to W W and ZZ dominates. and tt channels in the FP region with tan β = 50. Note that when the top quark threshold is crossed, the cross section for tt production abruptly increases, and the cross sections of all other modes correspondingly decrease in order to give the same σ tot , as is required to maintain the observed relic density.

IV. CALCULATION OF NEUTRINO FLUX
The high energy neutrino spectra in the FP region are mainly determined by the diagrams in Figs. 2, 3 and 4. We provide analytic results for the neutrino spectra in the cases of on-shell W W , ZZ, Zh, and tt and show that these formulas reproduce a Monte Carlo (MC) integration of the exact matrix elements provided by SMADGRAPH [49], the supersymmetric version of MADGRAPH. In the analytic calculation the phase space integration is performed by simple partitioning [73] and is summarized in Appendix A. We calculate the spectra for the illustrative points in Table I.

A. Annihilation to weak bosons
We calculate the annihilation process χ 0 1 χ 0 1 → W W * → W ℓν including both on-shell and off-shell contributions. The annihilation process of χ 0 1 χ 0 1 → W + W − is isotropic in the static limit when the W -polarizations are not measured. The amplitude of the annihilation to on-shell W W is given by the χ ± i exchange with vanishing contributions from the s-channel Z/H diagrams in the static limit, Because the amplitude involves the Levi-Civita antisymmetric tensor, only the two transversely polarized configurations (++ and −−) of W + W − are allowed and the longitudinal mode is absent. Conservation of CP at tree level implies the two polarization channels carry equal weight.
In the rest frame of W + , the neutrino ν from the decay process W + → ℓ + ν is described by the normalized angular distribution dN/d cos θ = 3 8 (1 − h cos θ) 2 with h = ±1 for the right (left) handed polarized W + with respect to the axis defined by the W + momentum in the χ 0 1 χ 0 1 CM frame. The antineutrino angular distributions can be obtained by a similar formula with h ↔ −h.
Including the contributions of both helicities of the W , we find the normalized neutrino event rate in the W W CM frame to be Note that the differential rates at the kinematic endpoints of the decay are twice that of energy m χ 0 1 /4. In principle, the distribution in E ν could be useful to roughly determine the lightest neutralino mass, provided there is little distortion through NC and CC effects and neutrino propagation. Ignoring the W helicity, the differential cross section is instead given by a flat E ν spectrum, as often applied incorrectly for neutralino DM.
In Fig. 6, we show the analytic and numerical evaluation of the neutrino energy spectra The analytic calculation for the neutrino spectra from χ 0 1 χ 0 1 → ZZ proceeds in a similar way, with Z and W couplings and masses appropriately changed, since the neutrinos have definite helicity. As the neutralino mass increases, the spectra from W W and ZZ become similar since the W/Z mass difference is relatively small. Note that even when one gauge boson is off-shell, there is still a rather hard neutrino energy spectra with an appreciable rate. This is reminiscent of the significant branching fractions of Higgs boson to gauge boson pairs below threshold [75]. Indeed, the neutralino pair in the static limit acts as a pseudoscalar state when coupling to fermions [76].

B. Annihilation to Zh
Next we analytically calculate the neutrino energy distribution from the static neutralino annihilation, χχ → Zh, Z → νν. The primary process χχ → Zh is isotropic in the static limit. Only the longitudinal mode of Z is produced in the χχ annihilation because of conservation of angular momentum.
In the rest frame of the longitudinal Z, the neutrino ν from the decay process Z → νν is described by the normalized angular distribution with respect to the axis defined by the Z momentum in the χ 0 1 χ 0 1 CM frame, dN/d cos θ = 3 4 (1 − cos 2 θ). Then the decay of the Z boson to neutrinos Z → νν gives the neutrino spectra We present these analytic results in Fig. 7, along with comparison with our numerical results from SMADGRAPH. Since the only longitudinal mode of the Z boson contributes, the distribution is very different than that given by the W W or ZZ distribution where only the transverse modes contribute.

C. Annihilation to heavy fermions
The hardest neutrino energies come from annihilation to top quark pairs, when this channel is kinematically accessible. As the threshold for tt is crossed, the cross section tō tbℓ + ν ℓ jumps dramatically and dominates the overall neutrino rate. The amplitude of the top quark decay, t → bℓ + ν, has the known structure, (t · ℓ + )(b · ν), which leads to the neutrino energy spectra in the rest frame of the t. Boosting into the tt center of mass frame, we arrive at the neutrino energy spectra from annihilation to top quarks, assuming a narrow W -width, massless b-quarks and an average over the t-quark polarization Here we have defined where β and E t = M χ 0 1 are the t velocity and energy, respectively, ω = m 2 W /m 2 t , D = 1/6 − ω 2 /2+ω 3 /3, and the expression applies over the energy range (1 − β) ω Et 2 ≤ E ν ≤ (1 + β) Et 2 . It is interesting to note that the analytic structure of this distribution has two qualitatively different neutrino energy distributions, depending on the neutralino mass. Specifically, the distribution is divided into three energy regions, the central of which, These features can clearly be seen in Fig. 8a, which displays the results for the E ν distributions at neutralino masses of 180, 222, and 400 GeV. The high-energy tail of the distribution is described by a simple function of the neutrino energy, namely We show MC results for the t * t →tbℓ + ν ℓ channel in Fig. 8b for the FP with neutralino mass m χ 0 1 = 170 GeV.

D. Total neutrino energy spectra
We combine the above MC results W W , ZZ, Zh, and tt channels to arrive at the neutrino energy spectra from neutralino annihilation, as shown in Fig. 9 for neutralino masses m χ 0 1 = 90, 110, 200, 400, 700 GeV, and 1 TeV. The first two panels illustrate how the W W and ZZ spectra change with E ν below the tt threshold. The on-shell ZZ mode turns on at m χ 0 1 = M Z . A characteristic double edge feature is apparent in the total spectra at the low and high neutrino energies edges. However, as the neutralino mass is increased, this double edge becomes less prominent. Further increasing the neutralino mass includes the dominant top spectra. In these cases, the upper edge of the spectra in principle would allow a measurement of the lightest neutralino mass. However, as we will see later, propagation through the Sun can change the neutrino spectra significantly, especially for large neutrino energies, rendering such χ 0 1 mass measurements difficult. We verify that the Zh signal is naturally suppressed; it is considerably smaller than the ZZ and W W modes [36,77].

V. NEUTRINO PROPAGATION
The neutrino flux at Earth from annihilating neutralinos in the core of the Sun requires a full treatment of the propagation through solar matter and the vacuum between the Sun and Earth. To retain the information on coherency of the states, we utilize the quantum mechanical evolution equation where ρ is the complex density matrix in the gauge eigenbasis describing the state of the neutrino as it propagates. Due to the fact that fast oscillations can only be observed as average values due to finite energy resolution, the epsilon term is included in Eq. (17) for convenience in making the energy average [14,78]. The choice of ǫ = 0.01r in this decoherence term damps all off-diagonal terms to zero [14]. The Hamiltonian, H, includes the vacuum oscillation effects from nonzero mass splitting and the MSW terms: Here m is the neutrino mass matrix in the gauge eigenstate basis, E ν is the neutrino energy,

B. Source Terms
The source terms in the propagation equation account for the absorption and re-injection of neutrinos as the neutrinos propagate through matter. The CC and NC interactions absorb higher energy neutrinos and re-inject them with lower energy.

Injection spectra
The source term dρ dr inj represents the injected initial neutrino spectra from neutralino annihilation and is given by Here, we make the justified assumption that the neutrino flux is initially injected at the center of the Sun [31]. In the following, we illustrate the effect of each term in Eq. (17). Neutralino annihilation produces neutrinos and antineutrinos with each ν andν flavor having approximately equal probability.

Neutral Current
As NC effects are flavor blind, all neutrino species have spectra that are skewed equally towards a lower energy. The source term is given by The first term represents the re-injection of lowered energy neutrinos with energy E ν from an incoming neutrino of energy E ′ ν . The negative term accounts for the absorption of neutrinos of energy E ν .
In Fig. 10a  only the NC effects for E ν = 50 GeV and E ν = 100 GeV. Due to the flavor diagonal NC interactions, the three neutrino flavors remain distinct as they propagate. The neutrino spectrum of each flavor is equivalent and given in Fig. 10b.

Charged Current
The CC effects are also flavor blind to the extent that the τ mass is negligible compared to E ν . However, the lepton produced in deep inelastic scattering can be stable (in the case of the electron), can stop before it decays (muon), or decay before radiation losses decrease its energy (tau). The tau lepton is the dominant source of re-injection of neutrinos. The channels are where X denotes hadronic decay products. The source term is with a similar term for antineutrinos. The Π i nm = δ ni δ im are flavor projection operators where i = 1, 2, 3 for the e, µ, τ leptons, respectively, and Γ is a 3 × 3 diagonal matrix which describes the CC interaction rate for a particular flavor, The functions f (u) in Eq. (21) are the energy distributions of an outgoing neutrino with energy E ν , given an incoming neutrino with energy E in ν ; they are calculated in Appendix D and shown in Fig. 11. Including this τ regeneration effect, the total ν τ population does not change, but the ν τ energy distribution becomes skewed to lower energy. The populations ν e,µ energy spectra are increased from the leptonic decays of the τ − lepton, and likewise for the ν e,µ for the τ + lepton.
In Fig. 12 we show the populations of the neutrino similar to those of Fig. 10, but now including only CC interactions. In this case the τ decay re-injects τ neutrinos and occasionallyν e,µ neutrinos, breaking the flavor symmetry. This is clearly seen in the left panel of Fig. 12a for the case of an E ν = 50 GeV neutrino.

C. Propagation through the Sun
We include all the above effects when propagating the neutrino states to the surface of the Sun. We illustrate results in Fig. 13  flavor asymmetry increases. This is due to the larger CC effects at higher energy.

D. Propagation to the Earth
Once propagated to the surface of the Sun, the neutrino states must be propagated in their mass eigenstates to the Earth, which is achieved by rotating the density matrix ρ with the unitary MNSP matrix, V , [80,81] which diagonalizes the neutrino mass-squared matrix, where M 2 is the diagonal neutrino mass-squared matrix and m is the neutrino mass matrix in the flavor basis. We use the tri-bimaximal mixing matrix for the rotation [82]. The density matrix in the mass eigenbasis is denoted byρ. Additionally, the neutrino oscillations are averaged by forcing the off-diagonal elements ofρ to zero in the propagation to Earth.
This averaging could equally well be achieved by including the decoherence term in the propagation equation in Eq. (17). The neutrino spectra at Earth are quite similar to those at the surface of the Sun, as shown in Fig. 14.

VI. NEUTRINO DETECTION
Once the neutrinos are propagated to Earth, we need to address their rate of detection in a km 2 area detector. The fractional rate of annihilation to neutrinos is given by 1 σtot dσν dEν where σ tot includes all SM inclusive modes: W W, ZZ, ff, Zh, hh and σ ν ≡ σ χ 0 1 χ 0 1 →νX includes all the modes discussed above in Section III. We calculate the total rates using Calchep and can be analytically found in Ref. [83].

A. Neutrino Flux at Earth
We illustrate the detection rates using the simulation for IceCube outlined in Ref. [23].
The flux of neutrinos from neutralino annihilation at the Earth is given by where the factor of 1 2 is associated with the fact that two neutralinos produce one annihilation event; here R = 1.49 × 10 11 m is the Earth-Sun distance. The parameter C ⊙ is the neutralino solar capture rate of Eq. (1) and is dependent on the neutralino mass, velocity and local density in the galaxy.

B. FP capture rate predictions
The solar capture rate of neutralinos in the galactic halo is approximately given by Eqn.
(1) and depends on the spin-independent (SI) and spin-dependent (SD) scattering rates 9 . 9 The present 90% C.L. limit on the SI cross section from the XENON10 experiment places an upper limit on the scattering rate of 8. To predict the neutrino flux at Earth, we calculate the spin-dependent cross section for the Z exchange 10 where J = 1/2 is the spin of the proton, µ χ = mass 30 GeV [84]. These rates are still higher than the SI rate throughout the FP region, which should be probed in the future by CDMS 2007 with an expected reach of 1 × 10 −8 pb [85]. The SD scattering rate off protons, however, is less constrained with a current upper limit of 7 × 10 −2 pb from ZEPLIN-II for neutralino masses at 65 GeV [86]. Future detectors such as COUPP can greatly improve these limits down to 10 −2 pb with a 2 kg chamber of superheated CF 3 I [87]. However, the SD scattering limits would still be too weak to constrain the FP region. 10 Note that the scalar quark exchange is also relevant for the SD scattering rate. However, since the squarks have large masses in the FP region, they decouple, and squark exchange contributions are not included in the calculation.

C. Muon rate at IceCube
Once the neutrino flux is known, the associated total muon rate through a time T in a km 2 area detector such as IceCube can be determined by folding the muon production cross section with the neutrino flux The densities of protons and neutrons near the detector are taken to be ρ p = 5 9 N A cm −3 and ρ n = 4 9 N A cm −3 , respectively, where N A is Avagadro's number 11 . Muons lose energy according to [89] dE which can be used to determine the length of a muon track. The parameters α = 2.0 × 10 −6 TeV cm 2 /g and β = 4.2 × 10 −6 cm 2 /g describe the loss rate [22,23,89,90]. The muon range, is the distance a muon propagates through matter of density ρ before its energy drops below the threshold energy, E thr µ [90]. We take E thr µ = 50 GeV, which is optimistic for IceCube and conservative for KM3. Due to the long muon range, the fiducial volume of the detector can be factored into the range and the cross sectional area of the detector, called the effective area. The effective area of the detector, A ef f , is calculated for IceCube following Ref. [23], and is given in Fig 16. The Super-Kamiokande experiment has placed a limit on the flux of muon induced by neutrinos from DM annihilations in the Sun of Φ ν 5 × 10 −15 cm −2 s −1 for a half-angle of . We calculate the induced muon flux according to and find a total muon flux above the Super-K bound for 90 and 110 GeV with 1.3 ×  11 Since the muon range is at most 1 km for a 1 TeV muon, the point of muon production can be assumed to be in ice, rather than the Earth's crust.  We apply Eq. (27) to calculate neutrino signals in IceCube from neutralino annihilations as well as backgrounds from atmospheric neutrinos 12 . The differential flux of atmospheric ν µ and νμ neutrinos, dΦ ν /dE ν d cos θ z , are taken from Ref. [94], where θ z is the zenith angle.
To severely reduce the background from atmospheric neutrinos, we include only events within a narrow angular cone along the line of sight from the IceCube detector to the Sun.
We parameterize the zenith angle to of the Sun at the South Pole as cos θ z = cos π 2 + θ max dec sin (2πτ ) , where θ max dec = 23 • 26 ′ is the maximum declination of the Sun in the celestial sphere throughout the year. Here we have denoted the time of year by the dimensionless parameter τ which spans a full year with the values 0 < τ ≤ 1, where τ = 0, 1 2 correspond to the September and March Equinoxes, respectively. The total muon rate along the line of sight to the Sun is given by + (ν →ν) , 12 Backgrounds from solar atmospheric neutrinos also exist, where cosmic rays produce neutrinos in the solar atmosphere [93]. These backgrounds are expected to be a few per year for a km 2 size detector triggering on upward going muons.
where R(cos θ z ) = 0.70 − 0.48 cos θ z is a detector efficiency for up-going muons that takes into account the rock bed below the detector and other angular dependence factors in Ice-Cube [23]. The factor of 1 2 in the upper limit of the τ integration takes into account the time that the Sun is below the horizon. The 1σ resolution of IceCube is 1 • [95]; we therefore include the flux observed along the line of sight to the Sun within a cone of angular diameter 3 • . This dramatically reduces the background from atmospheric neutrinos to that shown in Fig. 17.
We show the signal muon event rate and atmospheric background [94] in IceCube for six parameter points in Fig. 17 and Table III. Defining the signal region of E thr µ < E µ < 300 GeV, we determine the statistical significance of the six illustrative points in Table III 13 .
This choice of the upper cut at 300 GeV is because of the large attenuation of high energy neutrinos by the CC and NC interactions in the Sun makes the signal contribution at higher energies small. Here, we purposely chose the window to be independent of the neutralino mass, but once a signal is seen the window can be narrowed to improve the signal to background ratio. In the above window with a few years of data, IceCube can make a 5σ χ 0 1 discovery from M W up to 400 GeV. The case for O(M Z ) are even better, easily yielding a discovery signal in the first year of data.   From the signal in Fig. 17, one can use the signal rate and the shape of the differential muon rate to extract further information on the nature of the neutralino . A determination of the neutralino mass can be made from the shape of the muon energy distribution and the capture rate of neutralinos in the Sun can be determined from the total signal rate. The capture rate in turn gives information on the local galatic DM density and its velocity (cf. Eq. (1)) that can complement inferences from direct detection experiments.

D. Limit Spectra
When the DM particle is very massive and thus produces very energetic neutrinos, the features of the neutrino energy distribution tend to be washed out, due to the neutrinos interacting more with the solar matter. This gives rise to the so-called "limit spectra," which becomes a fair description for E ν greater than 200 GeV or so [31]. The limit spectra assumes that (i) oscillations and interactions with matter populate the different neutrino flavors uniformly so that the density matrix can be reduced to a scalar, (ii) the re-injection spectra from scattering is flat in E ′ ν /E ν , and (iii) the cross section is proportional to the energy of the incoming neutrino. In our case, the first assumption is satisfied to zeroth order since the initial neutrino populations uniformly populate the three flavors. However, τ regeneration breaks this uniformity and makes the ν τ population different than the ν e population 14 . The reinjection spectra are preferentially skewed toward lower energies as calculated in Appendix D and shown in Fig. 11 and are not well approximated by flat spectra. The third assumption is valid in the appropriate neutrino energy range of few GeV For all ν, the limit spectra can be determined analytically [31] and approximated as a decaying exponential in the high energy limit Here E is ≈ 100 GeV for neutrinos and ≈ 140 GeV for anti-neutrinos [31]. For very high neutrino energies, the spectral distributions become less distinctive and bear little resemblance to the injected spectra. This effect unfortunately limits the ability to detect neutrinos of energies greater than ≈ 400 GeV or so, as can be seen in Fig.18. Combined with the afforementioned low-energy cutoff from the effective area of the detector, we are confined to a "window" of possible muon detection energies from roughly 50 GeV to 400 GeV for IceCube.
14 Oscillations between the ν τ and ν µ neutrinos tend to keep these two flavors equally populated. (ii) Oscillations of three neutrinos with vacuum oscillations due to the atmospheric and solar mass-squared differences in the tribimaximal approximation to the mixing matrix which has a zero θ 13 mixing angle and with MSW effects for non-zero θ 13 ; MSW effects are not expected to be significant for high neutrino energy threshold of IceCube.
(iii) Source terms due to the absorption and re-injection of neutrinos propagating through matter, including both CC and NC interactions that re-inject neutrinos with lower energy. In particular, we include all neutrino decay products from the production of tau-leptons with exact calculations of the tau decay distributions. We find that the tau-regeneration effects are not large, but that the absorption effects strongly diminish the event rates from neutralinos of mass above a few hundred GeV.
Our study made major improvements over previous analyses: (i) We use take into account the full spin-dependence of the production and decay processes, which have important effects on the neutrino energy distributions.
(ii) We make quantitative predictions for the absolute neutrino rates, since we specialize to the specific FP region.
(iii) We include off-shell W, Z, and top decays in addition to on-shell.
(iv) We provide analytic formulas for on-shell decays and show that the numerical calculations of exact matrix elements obtained with the SMADGRAPH program are in agreement.
We have compared our results with other analyses and approximations where possible: (i) We find that the approximation to the density evolution of Eq. (33) is reasonably good for neutralino masses of order 100 GeV, but shows substantial deviations from the exact results at higher neutralino masses: see Fig. 18.
(ii) We find that the limit spectra from Cirelli et al Eq. (33) is not so good in representing the exact results at any neutralino mass, even though it does qualitatively describe the fall-off with neutrino energy due to absorption effects.
(iii) The signal region is broadly defined in muon energy by the 50 GeV threshold cut and the rapid decrease at energies above 300 GeV from absorption effects.
In the detection of muon-neutrino signals from the annihilations, we calculate event rates for the IceCube experiment, taking into account (i) A muon energy threshold of 50 GeV (ii) The dependence of the detector area on muon energy due to the range of the muons in ice.
(iii) The backgrounds from atmospheric neutrinos as calculated from the flux in Ref. [94]; we include a 3 degree cone for the angular resolution along the line of sight from the Sun to the detector.
The major conclusions of our study are as follows: (i) The signals from the annihilations of neutralinos in the mass range of 90 GeV to 400 GeV should be easily identifiable above the atmospheric neutrino backgrounds: see Table III (v) The forthcoming data from IceCube and KM3 offer the prospect for discovery of neutralino DM if nature has cooperated.
can be modeled by [22] The effective volume V of the DM core of the Sun is estimated to be [96,97] V = 5.7 × 10 27 cm 3 (100 GeV/M χ 0 1 ) The annihilation cross section is dimensionally estimated as with the coefficient B of order 1. For M χ 0 1 = 100 GeV, We find N ≈ 10 38 for the SD capture rate. This number is many orders of magnitude larger than the average numberN of dark matter particles enclosed by a volume in the space of the size of the Sun,N ≈ 4 × 10 30 .

CC interaction
For CC scattering, the neutrino-quark cross sections are given by [73,98] dσ dy where y = 1 − E ℓ /E ν , √ŝ = √ sx is the CM energy of the subprocess and x is the fraction of nucleon momenta imparted on the quark. The resulting scattering off a proton is given by [14] dσ dy (νp → ℓX) = where f q/N (x) is the parton density function for parton q in nucleon N. The corresponding neutron differential cross sections are [14] dσ dy (νn → ℓX) =

NC interaction
The NC scattering is calculated in the same way. The subprocess are given by where y = 1 − E ν ′ /E ν and the Z boson couplings are g Lu = 1 2 − 2 3 sin 2 θ W , g Ru = − 2 3 sin 2 θ W , g Ld = − 1 2 + 1 3 sin 2 θ W , g Rd = 1 3 sin 2 θ W . (C9) The total differential cross sections for proton target are then given by [14] dσ dy (νp → ν ′ X) = To account for τ regeneration, we fold the CC production cross section of leptons with decay distribution of the τ with similar expressions for fν τ →νe,µ and f ντ →νe,µ . Here, the factors of 1/5 [99] approximate the more precise values in Appendix C. In this parameterization, u = E out ν /E in ν and z = E τ /E in ν . The normalization factor N is chosen such that The first factor in each equation arises from the CC production cross section for τ − and τ + , respectively. The sum is over each contributing mode (note that for modes with ν e or ν µ in the final state, there is only one term in the sum), and BF i is the corresponding branching fraction. P is the polarization of the decaying τ , and is ±1 for τ ∓ . g(y) is the energy spectra of the neutrinos from the decaying τ . We now calculate this spectra for the modes ν τ ℓν ℓ , ν τ e/µν e/µ and ν τ π, and give a table of all results.