DFTT12/2005 IFUP–TH/2005-13 hep-ph/0506298 Spectra of neutrinos from dark matter annihilations 0 1 0 2 r Marco Cirellia, Nicolao Fornengob, Teresa Montarulic, a M Igor Sokalskid, Alessandro Strumiae, Francesco Vissanif 1 1 a Physics Department, Yale University, New Haven, CT 06520, USA 5 b Dipartimento di Fisica Teorica, Universit`a di Torino v 8 and INFN, Sez. di Torino, via P. Giuria 1, I-10125 Torino, Italia 9 2 c University of Wisconsin, Chamberlin Hall, Madison, WI 53706, USA. 6 0 On leave of absence from Universit`a di Bari 5 and INFN, Sez. di Bari, via Amendola 173, I-70126 Bari, Italia 0 / h d INFN, Sez. di Bari, via Amendola 173, I-70126 Bari, Italia p - p e Dipartimento di Fisica dell’Universit`a di Pisa and INFN, Italia e h f INFN, Laboratori Nazionali del Gran Sasso, Assergi (AQ), Italia : v i X r a Abstract We study the fluxes of neutrinos from annihilationsof dark matter parti- cles in the Sun and the Earth. We give the spectra of all neutrino flavors ¯ for the main known annihilation channels: νν¯, bb, ττ¯, cc¯, light quarks, ZZ, W+W . We present the appropriate formalism for computing the − combined effect of oscillations, absorptions, ν -regeneration. Total rates τ are modified by an (0.1 10) factor, comparable to astrophysical un- O ÷ certainties, that instead negligibly affect the spectra. We then calculate different signal topologies in neutrino telescopes: through-going muons, contained muons, showers, and study their capabilities to discriminate a dark matter signal from backgrounds. We finally discuss how mea- suring the neutrino spectra can allow to reconstruct the fundamental properties of the dark matter: its mass and its annihilation branching ratios. 1 Introduction The most appealing scenario to explain the observed Dark Matter (DM) abundance Ω 0.3 DM ∼ consists in postulating that DM arises as the thermal relic of a new stable neutral particle with mass m . Assuming it has weak couplings g 1, the right Ω is obtained for m DM DM DM ∼ ∼ (TM )1/2 TeV, where T 3Kis thepresent temperature of theuniverse, andM 1019GeV Pl Pl ∼ ∼ ∼ is the Planck mass [1]. One motivated DM candidate is the lightest neutralino in supersymmetric extensions of the Standard Model with conserved matter parity, that for independent reasons is expected to have a mass around the electroweak scale [2]. Many other DM candidates have been proposed: we will generically have in mind a DM particle heavier than few tens of GeV, keeping the concrete connection to the neutralino as a guideline. This scenario seems testable by DM search and by collider experiments: one would like to see a positive signal in both kind of experiments and to check if the same particle is responsible for both signals. As emphasized in [3] this is an important but difficult goal. Ahugeeffortiscurrently putinexperiments thathopetodiscover DMeither directly (through theinteractionofDMparticleswiththedetector)orindirectly(throughthedetectionofsecondary products of DM annihilations). Among the indirect methods, a promising signal consists in neutrinos with energy E <m produced by annihilations of DM particles accumulated in the ν DM ∼ core of the Earth and of the Sun [4, 5], detected by large neutrino detectors. We will refer to them as ‘DMν’. IMB [6], Kamiokande [7], Baksan [8], Macro [9], Super-Kamiokande [10], AMANDA [11] and BAIKAL [12] already obtained constraints on DMν fluxes, while experiments that are under construction, like ANTARES [13] and ICECUBE [14], or that are planned, like NEMO [15], NESTOR [16] and a Mton-scale water Cˇerenkov detector [17], will offer improved sensitivity. We compute the spectra of neutrinos of all flavors generated by DM annihilations in the Earth and in the Sun. Today, before a discovery, this can be used to convert experimental data into more reliable constraints on model parameter space and helps in identifying more relevant features of the DMν signal searched for. For instance, we include in the analysis all main annihilation channels, we address the effect of neutrino oscillations and interactions with matter and we point out more experimental observables that those usually considered. After a discovery the situation will be analogous to the solar neutrino anomaly: a natural source of neutrinos carries information about fundamental parameters and we must find realistic observables that allow to extract it. As in that case, also in the DM case the total ν rate is the crucial parameter for discovery but is plagued by a sizable (10) astrophysical uncertainty. How O can we then reconstruct the properties of the DM? Astrophysical uncertainties negligibly affect the ratios between different neutrino flavors and theneutrinoenergyspectra(aswell asthecloselyrelatedangulardistributions[18]). Theydepend on the DM mass m and on the branching ratios of the channels into which DM particles may DM annihilate: νν¯,b¯b,tt¯,τ+τ−,W+W−,ZZ... Inordertoextractthesefundamentalparametersfrom future data one needs to precisely compute DMν spectra taking into account the astrophysical environment, where several processes are important. In section 2 we motivate our phenomenological procedure and compute the fluxes of electron, 2 Sun Earth 1 1 P , P ee ΜΜ Absorptiononly 0.8 0.8 bility 0.6 Pee bility 0.6 Pτµ a a b P , P , P b ro 0.4 ΜΜ ΤΜ ΤΤ ro 0.4 P P P , P Oscillationsonly Τe Μe 0.2 0.2 Pτe 0 0 10-1 1 10 102 103 1 3 10 30 100 300 NeutrinoenergyinGeV NeutrinoenergyinGeV Figure 1: The left plot illustrates how oscillations and CC absorption separately affect a flux of neutrinos produced in the center of the Sun. The right plot shows the oscillation probabilities from the center of the Earth. The continuous line applies to ν for θ = 0 and to ν¯ for any allowed θ , 13 13 since matter effects suppress their mixing. The dotted line applies to ν for θ = 0.1 rad. The 13 average over the production point has been performed as appropriate for m = 100GeV. It is DM responsible for the damping effect visible at E <10GeV. ν ∼ muon and tau neutrinos at production point: the different density of the Earth and solar core affects energy loss of particles that decay producing neutrinos. In section 3 we compute how propagation from the center of the Earth and of the Sun affects theflavor andenergy spectra. At production, theneutrino flavor ratiosfromtheDMannihilations are simply given by: ν : ν¯ : ν : ν¯ : ν : ν¯ = 1 : 1 : 1 : 1 : r : r . e e µ µ τ τ In the Earth, the main effect is due to oscillations with ‘atmospheric’ frequency: the neutrino oscillation length λ = 4πE / ∆m2 105km(E /100GeV) atm ν | atm| ≈ ν is comparable to the Earth radius R = 6371km if E . 100GeV. In the Sun, also the size of the ⊕ ν production region of DMν is of the same order. Furthermore, in the Sun at E >10GeV neutrino ν ∼ interactions start to be significant and solar oscillations cease to be adiabatically MSW-enhanced, asillustrated in fig. 1a. Interactions manifest inseveral ways: absorption, re-injection ofneutrinos of lower energy (as produced by NC scatterings and ν CC scatterings), breaking of coherence τ among different flavors. These effects operate at the same time and with comparable importance: while previous works addressed the issues separately [19, 20], the density-matrix formalism of section 3 allows to take into account their combined action. 3 In section 4 (5) we give the resulting energy spectra of DM neutrinos of all flavors from the Earth (Sun). We consider the standard through-going muon signal and point out that other classes of events can be studied in realistic detectors and have interesting features from the point of view of discriminating a DMν signal from the atmospheric background and of reconstructing DM properties. This latter point is discussed in section 6. 2 Neutrino production A flux of neutrinos is produced inside the Earth or the Sun as a consequence of annihilation of dark matter particles which have been gravitationally captured inside these celestial bodies [21, 22, 23]. The differential neutrino flux is: dN Γ dN ν ann f = BR (1) f dE 4πd2 dE ν X f where f runs over the different final states of the DM annihilations with branching ratios BR , f d is the distance of the neutrino source from the detector (either the Sun–Earth distance r or SE the Earth radius R ) and where the annihilation rate Γ depends on the rate Γ of captured ⊕ ann capt particles by the well known relation: Γ Γ = capt tanh2(t /τ ) (2) ann 0 A 2 where t = 4.5 Gyr is the age of the Earth and of the Sun and τ denotes a time-scale for the 0 A competing processes of capture and annihilation, and it is proportional to the DM annihilation cross sections (for explicit formulæ see [21, 23, 24]). For the present discussion we just remind that the capture rate Γ depends linearly on the DM/nucleus scattering cross section and on capt the local dark matter density ρ : DM Γ σ ρ (3) capt scattering DM ∝ Eq. (2) shows that the two competing processes of capture and annihilation may eventually reach an equilibrium situation when the time scale τ is much smaller than the age of the body. While A this is usually the case for the Sun, it does not always occur for the Earth, since in this case the gravitational potential, which is responsible for the capture, is much smaller. Equilibrium is fulfilled only for large elastic scattering cross sections. 2.1 Observables with and without astrophysical uncertainties Fromtheprevious equations we see that theneutrino signal shares bothastrophysical andparticle physics uncertainties. However, the shape of neutrino spectra are virtually free from the astro- physical ones, even in presence of oscillations – as we shall discuss below – and therefore they can be potentially used to study the fundamental DM parameters, like its mass and annihilation channels. This topic will be addressed in section 6. A quantity which suffers from sizable astrophysical uncertainties is the total DMν flux, mainly due to the poor knowledge of the local DM density ρ . The experimental indetermination on DM 4 this parameter is still large. Detailed analyses, performed assuming different DM density profiles, find densities that vary by about one order of magnitude [25]. This translates into the same order of magnitude uncertainty on the DMν rate, due to the direct proportionality between the neutrino signal and the local dark matter density through the capture rate. The uncertainty on ρ can also play a role in the setup of capture/annihilation equilibrium in the Earth, giving an DM additional reduction effect. An additional astrophysical uncertainty comes from the local DM velocity distribution func- tion. Since capture is driven by the relation between the DM velocity and the escape velocity of the capturing body (11km/sec and 620km/sec at the surface of the Earth and the Sun, re- spectively), the high–velocity tail of the DM velocity distribution function may play a role. In the case of the Earth, the actual motion of DM particle in the solar system is another relevant ingredient which can alter significantly the predicted capture rate and therefore the predicted DMν rate. Recently this issue has been re–evaluated in [23], where it has been shown that in the Earth the capture rate of DM particles heavier than a few hundreds of GeV may be considerably reduced. On the contrary, neutrino spectra can be considered as virtually free from astrophysical un- certainties. The shape of the spectra depends on the type of particle produced in the annihilation process and on its subsequent energy–loss processes (remember that annihilation occurs in a medium, not in vacuum) before decaying into neutrinos. As a consequence of the thermalization of the captured DM, the density distribution of DM particles within the Sun or the Earth is predicted to be [26]: R n(r) = n exp( r2/R2 ) R = (4) 0 − DM DM √βm DM where r is the radial coordinate, β = 2πG ρ R2/3T , ρ and T are the central density and N 0 0 0 0 temperature of the body (Sun or Earth) and R is its radius. These astrophysical parameters are relatively well known, much better than the above mentioned galactic ones. Numerically: 1.76/GeV for the Earth, β = (5) (cid:26) 98.3/GeV for the Sun. This means that the size of the production region of DM neutrinos is 500km 100GeV/m DM ∼ in the Earth and 0.01R 100GeV/m in the Sun. p ⊙ DM ∼ The finite size can affectpthe spectra in two ways: 1) Different DMν originate in regions with different densities, so that hadrons may loose different amounts of energy before decaying into neutrinos. This, however, is not an important effect because the size of the production region is small enough that the matter density can be safely considered as constant where neutrinos are produced; 2) Neutrino propagation: while in the Earth the production region has a size much smaller than the atmospheric oscillation length, in the Sun the size is instead comparable. The resulting coherence between different flavors gets however washed–out by the much longer eventual propagation up to the Earth. In conclusion, the production regions are small enough that performing the spatial average according to eq. (4) gives a final total spectrum negligibly different than the one obtained by just assuming that all DMν are produced at the center of the Earth or of the Sun. We will prove this statement in section 3.2, after discussing our treatment of neutrino propagation. 5 DMν spectra and fundamental parameters Since DMparticles inside the Earth or the Sun arehighlynon–relativistic, theirannihilationsoccuralmostatrestandthemainphenomenological parameters that determine DMν spectra are the DM mass m and the BR of the basic channels DM into which DM particles may annihilate, as shown in eq. (1): qq¯, ℓℓ¯, νν¯, W+W−, ZZ and higgs particles or mixed higgs/gauge boson final states [5, 27]. Besides the direct νν¯ annihilation channel, neutrinos originate from the decays of the particles produced in the annihilation. In the case of quarks, hadronization will produce hadrons whose subsequent decay may produce neutrinos. Also charged leptons, apart from electrons, will produce neutrinos. In the case of gauge bosons or higgs particles, their decay will produce again leptons or quarks, which then follow the same evolutions just described. The basic “building blocks” we need in order to calculate DMν fluxes are therefore the spectra producedbythehadronizationofquarksandbythedecayofchargedleptonsintheSunandEarth cores. Among leptons, only the τ is relevant, since muons are stopped inside the Earth and the Sun before they can decay [27], and therefore produce neutrinos of energy below experimental thresholds for the signal topologies we will discuss later on (up-going muons, contained muons and showers in large area neutrino telescopes). For the present discussion we consider neutrino energies above 0.5 GeV. In all the other situations, which involve gauge and higgs bosons, we can make use of the basic spectra discussed above and calculate the neutrino spectra by just composing properly boosted spectra originated from quarks or τ, following the decay chain of the relevant annihilation final state particle. The method is briefly sketched in Appendix A for completeness. In this paper we are interested in the discussion of the effect induced by oscillations on the neutrino signal. We therefore need to calculate the spectra for all three neutrino flavors. We model thehadronizationanddecay processes bymeansofaPYTHIAMonteCarlosimulation[28], suitably modified in order to take into account the relevant energy losses. The neutrino spectra which we obtain are presented in numerical form, but we also provide an interpolating function for all the quark flavors and the τ lepton. We will not consider effects on the neutrino fluxes arising from the spin of the DM particle. In general, the DM spin may control the polarization of primary particles produced in the an- nihilation. For instance, if the DM is a Majorana fermion (such as the neutralino) it can only decay into τ τ¯ +h.c. (with amplitude proportional to the τ mass) while a scalar can decay into L R τ τ¯ and τ τ¯ with different branching ratios. Only if the branching ratios are the same the L L R R DMν spectrum is equal to the Majorana case. We will assume that this is the case, studying a single ττ¯ channel rather than two slightly different τ τ¯ and τ τ¯ channels. Furthermore, when L L R R discussing direct annihilation into neutrinos (possible for a scalar DM) we will assume that the flux is equally divided among the three flavors. We now discuss the calculation of the spectra for the relevant final states and their distinctive features. In this paper we will not focus on a specific DM candidate, rather we will attempt a more general phenomenological analysis. Our results can therefore be used for any DM candidate, by using the basic spectra of primary annihilation particles given here. The full spectrum for a specific candidate in a specific model is then easily constructed by summing up these building blocks implemented by the information on the annihilation branching ratios BR in that model. f 6 2.2 Annihilation into light fermions The direct DM DM νν¯ channel (if allowed with a reasonable branching ratio) usually gives the → dominantcontributiontoDMν signals: itsspectrumisalineatE = m soitgivestheneutrinos ν DM ¯ with highest multiplicity and energy. If the DM is a Majorana fermion the DM DM ff → annihilation amplitude is proportional to m , so that the νν¯ channel is irrelevant and the most f important fermions are the heaviest ones: b¯b, ττ¯ cc¯ and, if kinematically accessible, tt¯ (i.e. if m > m ). Even in the context of SUSY models the relative weight of their branching ratios DM t should be considered as a free parameter: significant deviations from the qualitative expectation σ(DM DM b¯b)/σ(DM DM ττ¯) = 3m2/m2, which exactly holds in the case of a dominant → → b τ higgs exchange, can arise if staus are much lighter than sbottoms. Once a quark is produced, it will hadronize and produce a large number of mesons and baryons, which will then decay and eventually produce neutrinos. We calculate the ν , ν and e µ ν fluxes originated by quark hadronization and lepton decay in the medium by following and τ properly adapting the method of [27]. We improve on previous analyses [5] by calculating full spectra for all neutrino flavors (usually only ν were considered since the main signal is upgoing µ muons and oscillations have been neglected, except in a few seminal cases [19]). We also provide here neutrino fluxes coming from light quarks: their contribution is usually neglected since they mostly hadronize into pions, which are stopped in the medium and do not produce neutrinos in an interesting energy range. We show below that for relatively large DM masses neutrinos from light quarks should be taken into account in a precise computation. Their main contribution occurs through the excitation of c quarks in the hadronization process and subsequent decay of c mesons. For completeness, we include also the case of DM annihilation into gluons, which may be rele- vant for some DM candidate. For instance, in the case of neutralinos, gluons can be produced at one loop level: even though this channel is usually subdominant, it can provide some contribution in specific portions of the SUSY parameter space, especially for light neutralinos. Annihilation inside the Earth As previously discussed, the annihilation process occurs pri- marily in the center of the Earth, where the density is ρ = 13 g cm−3. Therefore the particles produced in DM annihilations may undergo energy loss before decay. In the case of charged leptons, the energy loss process is calculated by means of the Bethe– Bloch equation. The typical stopping time is of the order of τ = 2 10−10 sec. This has to be stop · compared to the boosted lifetime γτ , where τ = 2.2 10−6 sec for the muon andτ = 3 10−13 dec dec dec · · sec for the tau. We see that for leptons with energies up to 1 TeV, muons are always stopped before their decays, while taus may decay as if they were in vacuum [27]. In order to take into account the small deviations from the limit described above we adapted the PYTHIA code to allow free lepton decay if γτ < τ , otherwise the lepton is stopped and then it decays. dec stop As for the hadrons, the situation is different if the jets are produced by light or heavy quarks. The interaction time in a material with density ρ is [27] τ = [n σ v]−1 = 5 10−35[ρ σ β]−1sec (6) int int int · where σ denotes the typical interaction cross section for a hadron. int 7 For hadrons made of light quarks σ 20 mbarn [29], which implies τ 2 10−10 sec. Since int int ∼ ≃ · the typical lifetime of π and K mesons is of the order of 10−8 s, light hadrons are usually stopped before decay, unless they are very relativistic. We implemented this process in the PYTHIA code by letting the hadron decay freely when γτ < τ , otherwise it is stopped. With this dec int modification of the code we take into account the actual lifetime of any hadron and the actual energy it has in the fragmentation process. When a very energetic light hadron is produced, we therefore do not neglect its decay. This situation however is not very frequent and it can occur only for very energetic injected jets. In the case of heavy hadrons one has σ 14 mbarn for a c or b meson and σ 24 mbarn int int ∼ ∼ for a c or b hadron [29], giving τ (2 3) 10−10 sec. The typical lifetime for these hadrons int ∼ ÷ · is τ 10−12 sec, or less. We therefore may assume that they decay before loosing a significant dec ∼ part of their energy. We again implemented a modification of the PYTHIA code which is similar to the case of leptons and which takes into account the relevant time scales. In addition to energy losses, we should also take into account that interaction of hadrons with the medium could lead to the production of additional hadrons. For instance, a heavy-hadron collision with the medium may produce additional light hadrons. However these additional light hadrons of lower energies are easily stopped, as discussed before, and therefore give a negligible contribution to the neutrino flux in our relevant energy range from this process. We therefore ignore here this possibility, a consistent assumption under our approximations. Our results on the neutrino spectra from annihilations in the Earth are shown in fig. 2 as dotted lines. Each spectrum refers to the flux of neutrinos for a given qq¯ or ττ¯ pair and for different values of m , equal to the energy of the primary jet or τ. Antineutrinos are not DM summed up and their fluxes are the same as those of neutrinos, since the initial state is neutral respect to all quantum numbers. The plots are shown as a function of x = E/m , which is DM defined in the interval [0,1]. The curves start from the x corresponding to the minimal neutrino energy that we consider, E = 0.5GeV. ν The spectra at production of ν and ν are equal, since light hadrons and muons do not e µ contribute to the DMν fluxes, and since τ,c,b produce an equal amount of ν and ν . This e µ equality would not hold for neutrinos produced from π or µ. We also see that light quarks contribute to the neutrino fluxes, even though light hadrons are stopped. This is due to the fact that a u, d or s quark has a non vanishing probability of splitting into a c quark in the fragmentation process and this process is favored for larger energies (for details, see [28] and references therein). We therefore have c hadrons in the outgoing jets also when we inject a light quark. The decay of these hadrons produces neutrino fluxes in the interesting energy range. We see that at low neutrino energies around 1 GeV and for m larger DM than about 500 GeV the contribution coming from light quarks can even be the dominant one. This effect was neglected in previous analyses. We provide analytical fitted formulæ for the spectra. We fitted the MC results with the following expression, which proved to be suitable: dN g(x) = = a (1+a w+a w2 +a w3 +a w4 +a w5)(1 x)b +c xc1(1 x)c2 (7) 0 1 2 3 4 5 0 dx − − where w = log x. The values of the parameters are shown in table 1 and table 2 for a sample of 10 center-of-mass energies m of the primary quark or τ, and are also available at [30]. The fitted DM 8 (cid:143) νe,µfromDMDM→ b b 0.1 νe,µfromDMDM→ c c 0.015 νe,µfromDMDM→ q q Sun Earth DinMGemVass 0.2 0.08 0.0125 1000 Eν 0.15 Eν Eν 0.01 g10 g100.06 g100.0075 300 dN/dlo00.0.15 dN/dlo00..0024 dN/dlo00.0.000255 13000 0 10 010−3 10−2 10−1 1 010−3 10−2 10−1 1 10−3 10−2 10−1 1 x = Eν/mDM x = Eν/mDM x = Eν/mDM (cid:143) νe,µfromDMDM→ τ τ νe,µfromDMDM→ t t νe,µfromDMDM→ ZZ νe,µfromDMDM→ W+ W− 0.6 0.4 0.2 0.2 0.5 Eν 0.15 Eν 0.15 Eν 0.4 Eν 0.3 og10 0.1 og10 0.1 og100.3 og100.2 dl dl dl dl N/ N/ N/0.2 N/ d0.05 d0.05 d d0.1 0.1 0 0 0 0 10−3 10−2 10−1 1 10−3 10−2 10−1 1 10−3 10−2 10−1 1 10−3 10−2 10−1 1 x = Eν/mDM x = Eν/mDM x = Eν/mDM x = Eν/mDM (cid:143) 0.1 ντfromDMDM→ b b 0.004 ντfromDMDM→ c c 0.003 ντfromDMDM→ q q Sun Earth DinMGemVass 0.08 0.003 0.0025 1000 Eν Eν Eν 0.002 dN/dlog1000..0046 dN/dlog1000..000012 dN/dlog1000.0.000115 13300000 0.02 0.0005 10 010−3 10−2 10−1 1 010−3 10−2 10−1 1 010−3 10−2 10−1 1 x = Eν/mDM x = Eν/mDM x = Eν/mDM (cid:143) ντfromDMDM→ τ τ ντfromDMDM→ t t ντfromDMDM→ ZZ ντfromDMDM→ W+ W− 1.2 0.3 0.6 0.4 1 0.25 0.5 0.3 Eν 0.8 Eν 0.2 Eν 0.4 Eν og100.6 og100.15 og100.3 og100.2 dl dl dl dl N/0.4 N/ 0.1 N/0.2 N/ d d d d0.1 0.2 0.05 0.1 0 0 0 0 10−3 10−2 10−1 1 10−3 10−2 10−1 1 10−3 10−2 10−1 1 10−3 10−2 10−1 1 x = Eν/mDM x = Eν/mDM x = Eν/mDM x = Eν/mDM Figure 2: Neutrino spectra at production. Upper half: the fluxes of electron and muon neutrinos, for the seven main annihilation channels and for different masses of the parent DM particle (different colors). The solid lines apply to the case of the Sun, the dotted of the Earth. In all cases, the spectra of antineutrinos are the same as those of neutrinos. Lower half: the same for (ν). τ 9 mDM a0 a1 a2 a3 a4 a5 b c0 c1 c2 DMannihilationsintob¯b 10 34.5/55.7 4.15/4.34 7.47/8.03 6.83/7.52 3.16/3.55 0.594/0.677 7.98/8.75 0/0 0/0 0/0 30 10.1/16.6 3.27/3.43 4.99/5.25 3.69/3.95 1.34/1.46 0.192/0.214 7.13/8.18 0/0 0/0 0/0 50 7.17/9.60 3.01/3.00 4.40/4.27 3.04/2.90 1.01/0.962 0.132/0.126 6.96/8.00 0/0 0/0 0/0 70 4.46/7.42 2.76/2.85 4.07/3.98 2.70/2.58 0.848/0.803 0.104/0.097 6.49/7.99 0/0 0/0 0/0 100 6.07/8.28 2.87/2.77 4.12/3.68 2.74/2.26 0.878/0.657 0.110/0.072 7.05/8.67 0/0 0/0 0/0 200 1.82/3.28 2.46/2.90 4.26/4.56 2.75/2.87 0.830/0.838 0.098/0.094 5.74/7.90 0/0 0/0 0/0 300 1.61/2.36 2.37/2.79 4.17/4.45 2.61/2.45 0.770/0.613 0.089/0.058 5.73/9.12 0/0 0/0 0/0 500 0.918/10.9 1.73/2.87 4.08/3.74 2.54/2.22 0.770/0.635 0.092/0.071 5.62/12.3 0/1.20 0/1.72 0/8.06 700 0.893/4.21 1.68/3.64 4.12/5.23 2.62/2.93 0.827/0.772 0.101/0.079 5.78/8.99 0/0 0/0 0/0 1000 0.779/31.3 1.39/2.76 4.13/2.99 2.77/1.47 0.934/0.345 0.117/0.032 5.99/18.7 0/1.20 0/1.72 0/8.06 DMannihilationsintoττ¯ 10 1.09/0.903 1.33/0.654 0.795/−0.946 −0.404/−2.42 −0.728/−1.80 −0.253/−0.462 2.11/2.02 0/0 0/0 0/0 30 1.07/0.839 1.48/0.763 1.54/−0.058 0.926/−0.687 0.312/−0.460 0.046/−0.096 2.07/1.95 0/0 0/0 0/0 50 1.17/0.994 1.72/1.30 2.06/1.13 1.41/0.499 0.519/0.105 0.078/0.0070 2.12/2.03 0/0 0/0 0/0 70 1.10/0.962 1.56/1.20 1.68/0.942 1.01/0.353 0.323/0.060 0.043/0.0036 2.09/2.01 0/0 0/0 0/0 100 1.03/1.25 1.40/1.81 1.35/2.12 0.715/1.35 0.203/0.446 0.024/0.059 2.06/2.16 0/0 0/0 0/0 200 0.895/1.25 1.08/1.80 0.781/2.08 0.257/1.30 0.029/0.412 −0.0010/0.052 1.97/2.18 0/0 0/0 0/0 300 1.09/0.378 1.48/−1.73 1.45/−3.67 0.744/−2.90 0.195/−1.00 0.020/−0.128 2.11/1.53 0/0 0/0 0/0 500 1.04/1.09 1.39/1.46 1.30/1.39 0.639/0.701 0.162/0.183 0.017/0.019 2.08/2.11 0/0 0/0 0/0 700 0.958/1.13 1.23/1.53 1.05/1.55 0.469/0.826 0.107/0.225 0.0099/0.024 2.02/2.13 0/0 0/0 0/0 1000 1.01/0.686 1.32/0.526 1.18/0.057 0.548/−0.146 0.129/−0.066 0.012/−0.0084 2.06/1.81 0/0 0/0 0/0 DMannihilationsintocc¯ 10 0.703/0.654 −1.66/−1.46 −2.96/−1.06 −1.68/1.55 −0.408/1.79 −0.058/0.492 6.27/6.40 0/0 0/0 0/0 30 0.233/0.674 −7.56/−3.07 −8.62/−5.38 −3.09/−3.29 0.353/−0.691 0.297/0.0012 6.35/7.86 0/0 0/0 0/0 50 0.513/0.828 −2.06/−2.93 −2.90/−5.87 −1.62/−4.35 −0.323/−1.38 −0.0032/−0.154 6.81/8.84 0/0 0/0 0/0 70 0.328/0.962 −2.66/−3.78 −0.903/−8.23 1.82/−6.68 1.66/−2.40 0.388/−0.324 6.55/9.91 0/0 0/0 0/0 100 0.685/1.60 −1.92/−1.70 −4.19/−4.59 −3.65/−3.87 −1.42/−1.39 −0.208/−0.185 7.57/11.1 0/0.160 0/6.11 0/12.8 200 0.532/1.41 −2.17/−2.30 −3.51/−4.73 −2.49/−3.42 −0.727/−1.06 −0.074/−0.120 7.58/12.9 0/0.158 0/6.11 0/12.8 300 0.279/1.72 −3.52/−2.68 −2.93/−4.98 −1.03/−3.28 0.118/−0.919 0.072/−0.094 6.98/16.8 0/0.153 0/6.11 0/12.8 500 0.363/1.35 −3.02/−3.24 −3.31/−4.89 −1.61/−2.74 −0.136/−0.613 0.029/−0.045 7.56/17.5 0/0.153 0/6.11 0/12.8 700 0.476/2.90 −1.60/−3.10 −1.64/−5.64 −0.663/−3.50 0.088/−0.909 0.044/−0.085 7.77/23.4 0/0.152 0/6.11 0/12.8 1000 0.420/2.19 −2.08/−3.80 −1.65/−5.49 −0.195/−2.84 0.408/−0.584 0.095/−0.040 8.01/26.4 0/0.151 0/6.11 0/12.8 DMannihilationsintoqq¯ 10 0.0024/0.919 −2.17/1.04 −4.97/1.73 −6.24/5.85 −4.68/6.21 −1.38/2.00 16.0/42.2 0/0.025 0/2.59 0/11.2 30 0.038/0.871 −0.632/3.50 −5.46/4.96 −7.21/3.52 −3.64/1.26 −0.649/0.183 17.3/20.4 0/0 0/0 0/0 50 0.020/0.405 2.78/3.44 5.17/4.79 5.22/3.18 3.26/1.03 0.771/0.132 12.0/20.4 0/0 0/0 0/0 70 0.017/0.0057 2.28/−2.11 5.65/−0.717 5.80/−1.42 3.44/0.398 0.733/0.208 14.2/15.5 0/0 0/0 0/0 100 0.019/0.017 2.25/−0.973 6.88/−2.07 8.24/−0.842 5.16/1.31 1.10/0.494 14.6/15.3 0/0 0/0 0/0 200 0.017/0.012 2.28/−0.863 6.86/−1.64 7.97/−1.60 5.49/1.65 1.14/0.561 14.3/15.4 0/0 0/0 0/0 300 0.015/0.041 2.05/−1.38 6.28/−2.16 7.67/0.967 6.01/2.60 1.23/0.635 14.3/17.3 0/0 0/0 0/0 500 0.065/0.047 3.44/−1.73 7.82/−1.49 8.11/2.53 4.25/3.41 0.708/0.747 15.5/19.7 0/0 0/0 0/0 700 0.093/0.043 3.57/−1.50 7.95/−1.57 8.14/2.47 4.04/3.63 0.638/0.754 15.7/19.7 0/0 0/0 0/0 1000 0.061/0.050 2.90/−2.07 8.48/−2.74 10.2/1.18 5.57/2.90 0.873/0.579 16.3/23.5 0/0 0/0 0/0 DMannihilationsintogluons 10 0.050/0.017 −0.286/0.645 −1.43/7.23 −2.12/1.20 −0.727/−4.95 −0.011/−2.21 9.31/9.15 0/0 0/0 0/0 30 0.042/0.012 −2.49/0.394 −0.522/8.54 3.44/−2.01 5.60/−1.81 2.00/0.043 8.88/7.62 0/0 0/0 0/0 50 0.011/0.802 −4.03/0.211 3.80/−2.76 −0.409/−3.88 4.15/−1.88 1.90/−0.310 7.10/14.5 0/0 0/0 0/0 70 0.013/0.532 −0.695/0.030 3.66/−2.72 −3.72/−3.74 2.55/−1.73 1.56/−0.271 6.06/13.8 0/0 0/0 0/0 100 0.353/1.01 −2.00/−0.413 −6.65/−3.37 −6.74/−3.66 −2.61/−1.47 −0.352/−0.205 13.1/17.4 0/0 0/0 0/0 200 0.082/0.555 −4.58/−0.850 −6.82/−3.38 −5.00/−3.39 −0.326/−1.18 0.207/−0.139 11.1/17.4 0/0 0/0 0/0 300 0.052/0.227 −4.17/−2.12 −6.56/−3.38 −5.60/−2.52 0.399/−0.196 0.433/0.073 8.88/17.9 0/0.090 0/2.57 0/8.13 500 0.063/0.814 −5.59/−1.86 −6.83/−4.34 −3.22/−3.20 1.88/−0.802 0.619/−0.062 10.4/23.9 0/1.43 0/2.41 0/13.6 700 0.069/0.453 −3.50/−2.42 0.232/−4.26 4.58/−2.75 5.20/−0.358 1.05/0.023 11.0/24.0 0/59.7 0/4.77 0/16.4 1000 0.235/0.328 3.74/−2.99 8.11/−4.19 7.39/−2.08 3.63/0.246 0.561/0.123 9.14/24.0 0/481. 0/5.52 0/20.6 Table 1: Fit parameters for the expression g(x) in eq. (7). Give the ν = ν = ν¯ = ν¯ spectra at e µ e µ production from annihilations in the Earth/Sun. DM masses are in GeV. These data are available at [30]. 10