Photon spectra from WIMP annihilation
J. A. R. Cembranos, A. de la Cruz-Dombriz, A. Dobado, R. A. Lineros, A. L. Maroto
aa r X i v : . [ h e p - ph ] O c t Photon spectra from WIMP annihilation
J. A. R. Cembranos ( a,b ) ∗ , A. de la Cruz-Dombriz ( b ) † , A. Dobado ( b ) ‡ , R. A. Lineros ( c ) § and A. L. Maroto ( b ) ¶ ( a ) William I. Fine Theoretical Physics Institute, University of Minnesota,Minneapolis, MN 55455, USA and School of Physics and Astronomy,University of Minnesota, Minneapolis, MN 55455, USA. ( b ) Departamento de F´ısica Te´orica I, Universidad Complutense de Madrid, E-28040 Madrid, Spain. and ( c ) INFN sezione di Torino, I-10122 Torino, Italy and Dipartimentodi Fisica Teorica, Universit`a di Torino, I-10122 Torino, Italy. (Dated: October 13, 2010)If the present dark matter in the Universe annihilates into Standard Model particles, it mustcontribute to the fluxes of cosmic rays that are detected on the Earth, and in particular, to theobserved gamma ray fluxes. The magnitude of such contribution depends on the particular darkmatter candidate, but certain features of the produced photon spectra may be analyzed in a rathermodel-independent fashion. In this work we provide the complete photon spectra coming fromWIMP annihilation into Standard Model particle-antiparticle pairs obtained by extensive MonteCarlo simulations. We present results for each individual annihilation channel and provide analyticalfitting formulae for the different spectra for a wide range of WIMP masses.
I. INTRODUCTION
According to present observations of large scale structures, Cosmic Microwave Background (CMB) anisotropies andlight nuclei abundances, the most important component of matter in the Universe cannot be accommodated withinthe Standard Model (SM) of elementary particles. Indeed, Dark Matter (DM) cannot be made of any of the knownparticles, and this is one of the most appealing arguments for the existence of new physics. Indeed DM is a requiredcomponent not only on cosmological scales, but also for a satisfactory description of rotational speeds of galaxies,orbital velocities of galaxies in clusters, gravitational lensing of background objects by galaxy clusters, such as theBullet Cluster, and the temperature distribution of hot gas in galaxies and clusters of galaxies. The experimentaldetermination of the DM nature will require the interplay of collider experiments [1] and astrophysical observations.These searches use to be classified in direct or indirect searches (see [2, 3] for different alternatives). Nevertheless,non-gravitational evidence of its existence and a concrete understanding of its nature still remain elusive. Concerningdirect searches, the elastic scattering of DM particles from nuclei should lead directly to observable nuclear recoilsignatures. Although the number of DM particles which passes through the Earth each second is quite large, the weakinteractions between DM and the standard matter makes DM direct detection extremely difficult.On the other hand, DM might be detected indirectly, by observing their annihilation products into SM particles.Thus, even if WIMPs (Weakly Interacting Massive Particles) are stable, two of them may annihilate into ordinarymatter such as quarks, leptons and gauge bosons. Their annihilation in different places (galactic halo, Sun, Earth,etc.) produce cosmic rays to be discriminated through distinctive signatures from the background. After WIMPsannihilation a cascade process would occur. In the end the potentially observable stable particles would be neutrinos,gamma rays, positrons and antimatter (antiprotons, antihelium, antideuterions, etc.) that may be observed throughdifferent devices. Neutrinos and gamma rays have the advantage of maintaining their original direction thanks totheir null electric charges. On the contrary, charged particles searches, such as those of positrons and other antimatterparticles, are hindered by propagation trajectories.The detection of such indirect signals would not constitute a conclusive evidence for DM since the uncertaintiesin the specific DM interactions, DM densities and backgrounds from other sources are not fully understood yet.Nevertheless, this work precisely focuses on this kind of detection as an indirect method to get information about theDM nature, abundance and properties.Photon fluxes in specific DM models are usually obtained by software packages such as DarkSUSY and mi-crOMEGAs based on PHYTIA Monte Carlo event generator. In general, the total photon spectrum obtained from theaddition of the contributions from different channels is obtained for the particular SUSY model under consideration ∗ E-mail: [email protected] † E-mail: dombriz@fis.ucm.es ‡ E-mail: dobado@fis.ucm.es § [email protected] ¶ E-mail: maroto@fis.ucm.es for a given WIMP mass. In this sense, it would be interesting to have a fitting function for the shape of the spectracorresponding to each individual annihilation channel and, in addition, determine the dependence of such spectra onthe WIMP mass in a model independent way. This would allow to apply the results to alternative candidates forwhich software packages have not been developed, and obtain photon fluxes for arbitrary WIMP candidates. On theother hand, the information about channel contribution and mass dependence can be very useful in order to identifygamma-ray signals with specific WIMP candidates.The paper is organized as follows: in section II, we briefly review the standard procedure for the calculation ofgamma-ray fluxes from WIMP pair annihilations. In section III, we comment on several aspects of detectors andbackgrounds. Section IV is then devoted to the details of specific simulations performed with PYTHIA. In sectionV, we introduce the fitting formulae that will be used to describe the spectra and in section VI the results for thesimulations, the fitted parameters and their dependence on the WIMP mass are presented. Then, in section VIIwe provide some information about the performed numerical codes obtained from our results and available online.Section VIII is then devoted to the main conclusions of the work. Finally, five appendices are provided in section IXto illustrate the obtained results for some studied annihilation channels.
II. GAMMA RAY FLUX FROM DM ANNIHILATION
Let us denote the DM mass by M and its thermal averaged annihilation cross-section into two SM particles (labelledby the subindex i ) by h σ i v i .Then the γ -ray flux from all possible annihilation channels is given by:d Φ DM γ d E γ = 14 πM X i h σ i v i d N iγ d E γ × Z ∆Ω dΩ Z l . o . s . ρ [ r ( s )] d s , (1) | {z } Particle model dependent | {z }
Dark matter density dependent where ρ is the DM density as a function of distance from its center r , which depends on the heliocentric distance s .The integral is performed along the line of sight (l.o.s.) to the target and averaged over the detector solid angle ∆Ω.The first piece of the r.h.s. in (1) depends on the particular particle physics model for DM annihilations. Inparticular, the self-annihilation cross sections is mainly described by the theory explaining the WIMP physics, whereasthe number of photons produced in each decaying channel per energy interval involves decays and/or hadronization ofunstable products, for instance quarks and gauge bosons. Consequently, the detailed study of these decay chains andnon-perturbative effects related to QCD is a hard task to be accomplished by any analytical approach. The secondpiece in (1) is a line-of-sight integration through the DM density distribution. We will discuss each of these piecesseparately. A. Particle Physics model
Although annihilation cross sections are not known, they are restricted by collider constraints and direct detection.In addition, the thermal relic density in the range Ω
CDM h = 0 . ± . H ) measurement [6], do not allow an arbitrary contribution from the DM gamma ray fluxes.As already mentioned, the annihilation of WIMPs is closely related to SM particle production. The time scale of anannihilation process is shorter than typical astrophysical scales. This fact implies that only stable or very long-livedparticles survive to the WIMP annihilations and may therefore be observed by detectors.For most of the DM candidates, the production of mono-energetic photons is very suppressed. The main reason forsuch a suppression comes from the fact that DM is neutral. Thus, it is usually assumed that the gamma-ray signalcomes fundamentally from secondary photons originated in the cascade of decays of gauge bosons and jets producedfrom WIMP annihilations. These annihilations would produce in the end a broad energy distribution of photons,which would be difficult to be distinguished from background. However, the directional dependence of the gamma rayintensity coming from these annihilations is mainly localized in point-like sources as will be discussed in the followingsection. This fact could therefore provide a distinctive signature.In conclusion, for a particular DM candidate, an unique annihilation channel may dominate, but in general, theyall contribute. All those channels contributions produce a broad energy gamma ray flux, whose maximum constitutesa potential signature for its detection. Typically, this peak is centered at an energy that is one order of magnitudelower than the mass of the DM candidate.On the other hand, a different strategy can be followed by taking into account the fact that the cosmic ray back-ground is suppressed at high energies. Primary photons coming from the Weicks¨acker-Williams radiation dominatethe spectrum at energies close to the mass of the DM candidate and their signature is potentially observable as acut-off [7]. This approach has the advantage of being less sensitive to electroweak corrections which may be importantif the mass of the DM candidate is larger than the electroweak scale [8]. B. DM density directionality
The line of sight integration can be obtained from: h J i ∆Ω . = 1∆Ω Z ∆Ω J ( ψ )dΩ = 2 π ∆Ω Z θ max d θ sin θ Z s max s min d s ρ (cid:18)q s + s − ss cos θ (cid:19) (2)where J ( ψ ) = Z l . o . s . d s ρ ( r ) . (3)The angled brackets denote the averaging over the solid angle ∆Ω, and s min and s max are the lower and upper limitsof the line-of-sight integration: s cos θ ± q r t − s sin θ . In this formula s is the heliocentric distance and r t is thetidal radius.Traditionally, the galactic center (GC) has attracted the attention of this type of directional analysis since standardcusped Navarro-Frenk-White (NFW) halos predict the existence of a very important amount of DM in that direction[9, 10]. However, this assumption is in contradiction with a substantial body of astrophysical evidences [11], and acore profile is not sensitive to standard DM candidates. On the contrary, cusped profiles are not excluded for theLocal Group dwarf spheroidals (dSphs) that constitute interesting targets since they are much more dominated byDM. In this way, directional analysis towards Canis Major, Draco and Sagittarius or Segue 1 [12] are more promising.An alternative strategy takes advantage of the large field of view of FERMI, that may be sensitive to the continuumphoton flux coming from DM annihilation at moderate latitudes ( | b | > ◦ ) [10]. Other proposed targets, as the LargeMagellanic Cloud (LMC) [13], are less interesting since their central parts are dominated by baryonic matter. III. DETECTORS AND BACKGROUNDS θ max in Eq. (2) is the angle over which we average, and is bounded from below by the experimental resolution ofthe particular detector: ∆Ω = 2 π Z θ max d θ sin θ = 2 π (1 − cos( θ max )) . (4)The quoted point spread function widths for the various experiments are typically: 0 . ◦ (EGRET), 0 . ◦ (CANGAROO-III, FERMI, HESS, MAGIC and VERITAS). EGRET and FERMI are satellite detectors with low energy thresholds( about 100 MeV), high energy resolution ( ∼ ≈
100 GeV) but better angular resolution. Typical referencesizes for the solid angle are ∆Ω = 10 − sr for ACTs and FERMI and ∆Ω = 10 − sr for EGRET.There are different main sources of background for the signal under consideration: hadronic, cosmic-ray electrons,localized astrophysical sources and the diffuse γ -rays. The latter is negligible for ACTs, but only the last two arepresent for satellite experiments like FERMI or EGRET.For heavy WIMPs, the produced high-energy gamma photons could be in the range 30 GeV-10 TeV, detectableby ACTs such as HESS, VERITAS or MAGIC. On the contrary, for lighter WIMPs, the photon fluxes would be inthe range detectable by space-based gamma ray observatories [14] such as EGRET, FERMI or AMS, with bettersensitivities around 30 MeV-300 GeV. IV. MONTE CARLO SPECTRA GENERATION: TECHNICALITIES
In this section, we explicitly specify how gamma rays spectra have been generated. We have used a widely knownparticle physics software, PYTHIA (version 6.418) [15], to obtain the results we are about to present. In a first ap-proximation, the WIMP annihilation is described by two separated processes: The first one describes the annihilation
Channel (cid:31)
Mass (GeV) 100 125 150 200 250 350 500 1000 W + W − ZZ t ¯ t - - - 0.70 0.86 0.32 2.81 1.41Table I: Total number of photons – in 10 units – generated from W + W − , ZZ and t ¯ t channels for different WIMP masses. Channel (cid:31)
Mass (GeV) 25 50 100 200 500 1000 10 · τ + τ − µ + µ − Total number of photons – in 10 units – generated from τ + τ − and µ + µ − channels for different WIMP masses. of WIMP particles and its output which are particle-antiparticle SM pairs. The details are contained in the theorydescribing the WIMP physics. The second process considers the evolution (decays and/or hadronization) of the SMunstable products, for instance, quarks and gauge bosons. Unfortunately, a first-principle description of this latterstep is too complex due to chain decays and non-perturbative QCD effects.As we mentioned above, in this work we have used PYTHIA to generate the photon energy spectra starting frompairs of SM particles, where each pair respects WIMP annihilation quantum numbers like neutral charge and colorsinglet. As will be described below, we will allow for final state radiation from charged particles to contribute tothe photon spectra. Due to the expected velocity dispersion of DM, we expect most of the annihilations to happenquasi-statically. This fact offers the center of mass (CM) frame as the most suitable frame to produce the photonspectra. Hence, the process is described by the total energy: E CM ≃ M (5)where M is the mass of the WIMP particle. Therefore, by considering different CM energies for the SM particlespairs in each WIMP annihilation process we are indeed studying different WIMP masses. The procedure to obtainthe photon spectra is thus straightforward, except for the particular case of the t quark. For any given pair of SMparticles which are produced in the WIMP annihilation, we count the number of photons in each bin of energy andthen normalize them to the total number of simulated pair collisions. The bins which we have considered in the x variable, x ≡ E γ /M , are: [10 − , − ], [10 − , . . , . . , .
8] and [0 . , . E , Figure 9 at the end of the paper.The SM particle pairs decays generated are W and Z gauge bosons, τ and µ leptons and u , d , s , c , b and t quarks.For each annihilation channel we have studied the gamma ray spectra produced for different WIMP masses. Theresult of the simulations were fitted to analytical expressions as is described in the following section. A. Final state radiation
If the final state in the annihilation process contains charged particles, there is a finite probability of emission of anadditional photon. This is discussed in detail in [16]. In principle there are two types of contributions: that coming
Channel (cid:31)
Mass (GeV) 50 100 200 500 1000 2000 5000 7000 8000 u ¯ u d ¯ d s ¯ s c ¯ c b ¯ b Total number of photons – in 10 units – generated from u ¯ u , d ¯ d , s ¯ s , c ¯ c and b ¯ b channels for different WIMP masses. from photons directly radiated from the external legs, which is the final state radiation we have considered in the work,and that coming from virtual particles exchanged in the WIMP annihilation process. The first kind of contribution canbe described for relativistic final states by means of an universal Weizs¨acker-Williams term fundamentally independentfrom the particle physics model [16]. On the other hand, radiation from virtual particles only takes place in certainDM models and is only relevant in particular cases, for instance, when the virtual particle mass is almost degeneratewith the WIMP mass. Even in these cases, it has been shown [17] that although this effect has to be included forthe complete evaluation of fluxes of high energy photons from WIMP annihilation, its contribution is relevant onlyin models and at energies where the lines contribution is dominant over the secondary photons. For those reasonsand since the aim of the present work is to provide model independent results for photon spectra, only final stateradiation was included in our simulations. B. The case for t quark decay The decay of top quark is not explicitly included in PYTHIA package. We have approximated this processby its dominant SM decay, i.e. each (anti) top decays into W +( − ) and (anti) bottom. In order to maintain anynon-perturbative effect, we work on an initial four-particle state composed by W + b coming from the top and W − ¯ b from antitop, which keeps all kinematics and color properties from the original pair. Starting from this configuration,we have forced decays and hadronization processes to evolve as PYTHIA does and therefore, the gamma rays spectracorresponding to this channel have also been included in our analysis. V. ANALYTICAL FITS TO PYTHIA SIMULATION SPECTRA
In this section we present the fitting functions used for the different channels. According to the PYTHIA simulationsdescribed in the previous section, three different parametrizations were required in order to fit all available data fromthe studied channels. The first one for quarks (except the top) and leptons. Then, a second one for gauge bosons W and Z and a third one for the top. A. Quarks and leptons
For quarks (except the top), τ and µ leptons, the most general formula needed to reproduce the behaviour of thedifferential number of photons per photon energy may be written as: x . d N γ d x = a exp (cid:16) − b x n − b x n − c x d + c x d (cid:17) + q x . ln [ p (1 − x )] x − x + 2 x (6)In this formula, the logarithmic term takes into account the final state radiation through the Weizs¨acker-Williamsexpression [16, 18]. Nevertheless, initial radiation is removed from our Monte Carlo simulations in order to avoidwrongly counting their possible contributions.Strictly speaking, the p parameter in the Weizs¨acker-Williams term in the previous formula is ( M/m particle ) where m particle is the mass of the charged particle that emits radiation. However in our case, it will be a free parameterto be fitted since the radiation comes from many possible charged particles, which are produced along the decayand hadronization processes. Therefore we are encapsulating all the bremsstrahlung effects in a single Weizs¨acker-Williams-like term.Concerning the µ lepton, the expression above (6) becomes simpler since the exponential contribution is absent.The µ − decays in e − ¯ ν e ν µ with a branching ratio of ∼ x . d N γ d x = q x . ln (cid:2) p (1 − x l ) (cid:3) x − x + 2 x (7)where the l parameter in the logarithm is needed in order to fit the simulations as will be seen in the correspondingsections.Let us mention at this stage that for the gamma rays obtained from electron-positron pairs, the only contributionis that coming from bremsstrahlung. Therefore, the previous expression (7) is also valid with q = α QED /π , p =( M/m e − ) and l ≡
1. This choice of the parameters corresponds of course to the well-known Weizs¨acker-Williamsformula. B. W and Z bosons For the W and Z gauge bosons, the parametrization used to fit the Monte Carlo simulation is: x . d N γ d x = a exp (cid:16) − b x n − c x d (cid:17) (cid:26) ln[ p ( j − x )]ln p (cid:27) q (8)This expression differs from the expression (6) in the absence of the additive logarithmic contribution. Nonetheless,this contribution acquires a multiplicative behaviour. The exponential contribution is also quite simplified with onlyone positive and one negative power laws. Moreover, a , n and q parameters appear to be independent of the WIMPmass M as will be seen in the corresponding section. The rest of parameters, i.e., b , c , d , p and j , are WIMPmass dependent and will be determined for each WIMP mass and for the W and Z separately. In both cases we havecovered a WIMP mass range from 100 to 10 GeV. Nonetheless, at masses higher than 1000 GeV, we have observedno significant change in the photon spectra for both particles. C. t quark Finally, for the top, the required parametrization turned out to be: x . d N γ d x = a exp (cid:16) − b x n − c x d − c x d (cid:17) (cid:26) ln[ p (1 − x l )]ln p (cid:27) q (9)Likewise the previous case for W and Z bosons, gamma-ray spectra parametrization for the top is quite differentfrom that given by expression (6). This time, the exponential contribution is more complicated than the one inexpression (8), with one positive and two negative power laws. Again, the additive logarithmic contribution is absentbut it acquires a multiplicative behaviour. Notice the exponent l in the logarithmic argument, which is required toprovide correct fits for this particle.The covered WIMP mass range for the top case was from 200 to 10 GeV. Nevertheless, at masses higher than1000 GeV we have observed again that there is no significant change in the gamma-ray spectra.
VI. RESULTS FROM PYTHIA SIMULATION
In this section we present the results of our fit of the parameters given by expressions (6), (8) and (9) afterhaving performed the PYTHIA simulations described in section IV. For each studied channel, we have considered thepossibility of parameters depending on the WIMP mass.Once the parameters in expressions (6), (8) and (9) have been determined for each channel and different WIMPmasses, it is possible to study their evolution with the WIMP mass M . Some parameters in expressions are WIMPmass independent and take values that depend on the studied channel. The rest are WIMP mass dependent.For some channels and in some range of WIMP masses, we observed that this dependence was given by a simplepower law. In fact, for a given channel ( i ) and a generic mass dependent P parameter, a simple power-law scalingbehavior would correspond to an expression like P ( i ) ( M ) = m P ( i ) M n P ( i ) (10)with m P ( i ) and n P ( i ) constant values to be determined for the different studied channels. Values of m P ( i ) and n P ( i ) and their range of validity are presented for each studied channel in the following. WIMP mass (GeV) b c d p j
100 9.48 0.651 0.292 973 0.790150 8.87 0.808 0.261 783 0.919200 8.64 0.882 0.250 684 0.955350 8.56 0.907 0.245 593 0.991500 8.51 0.917 0.244 560 0.9961000 8.45 0.931 0.242 535 1.000 a = 25 . n = 0 .
510 ; q = 3 . W boson : b , c , d , p and j parameters corresponding to (8) in the W + W − channel for different WIMP masses. Massindependent parameters in (8) for this channel are presented at the bottom of the table. Parameter WIMP mass interval (GeV) Fitting power law(s) b ≤ M ≤
200 0 . M . + 46 . M − . < M ≤ . M − . c ≤ M ≤ − M − . + 35 . M − . − . < M ≤ . M . d ≤ M ≤
240 2 . · − M . + 2 . M − . < M ≤ . M − . p ≤ M ≤ M − . + 285 M . j ≤ M ≤ . M . Table V:
Parameters corresponding to (10) for W boson. It can be seen that p parameter follows two different power laws depending onthe WIMP mass interval. For the remaining mass dependent parameters there is a unique power law behavior in the WIMP mass interval350 ≤ M ≤ A. W boson As commented above, the correct parametrization for the W boson simulations was given by expression (8). Forthis boson, there are five mass-dependent parameters: b , c , d , p and j whose values are detailed in Table IV. Themass independent parameters are a = 25 . n = 0 .
51 and q = 3 .
00. The mass range considered for this boson is 100to 10 GeV. In fact, from M = 1000 GeV, the photon spectrum does not change. The parameters obtained fit theenegy spectra from x = 2 · − till the end of the allowed interval. It can be seen that for low masses the spectrumdoes not end at x = 1 but at smaller energies (e.g. x ≃ .
78 for M = 100 GeV) and as masses get higher, the energytail approaches x = 1.Some of these results are presented in Figure 1 in Appendix A for four WIMP masses: 100, 200, 350 and 1000GeV. Besides, mass dependent parameters b , c , d , p and j were presented in the same Appendix in Figure 2.Concerning the scaling behavior of these mass dependent parameters given by expression (10), we obtain that b , c and j parameters scale with a simple power law of M at high masses. In fact, b and c parameters follow a twopower-law behavior at low masses. For d parameter, we find that the sum of two power laws covers this high massesinterval, whereas a simple power law at low masses is obeyed. Parameter p scales with two power laws in the wholestudied mass interval. These results are shown in Table V. B. Z boson For the Z boson the correct parametrization is again the one given by expression (8). For this boson there are fivemass-dependent parameters: b , c , d , p and j which are detailed in Table VI. The mass independent parameters are a = 25 . n = 0 . q = 3 .
87. The studied WIMP mass range for this boson was from 100 to 10 GeV. However,above M = 1000 GeV the energy spectrum does not change as can be seen from our simulations.The chosen parameters values fit the photon spectra from x = 5 · − till the end of the allowed interval. As for the W case, it can be seen that for low masses the spectrum does not end at x = 1 but at smaller energies (e.g. x ≃ . M = 100 GeV) and as masses get higher, the high-energy tail approaches x = 1.Concerning the power-law scaling of the parameters with M , we obtained that parameters b , c , d ad j followa simple power-law behavior for high WIMP masses. Parameter p follow a two sum power-law behavior for masseshigher than 170 GeV. Concerning d parameter, the whole accessible WIMP mass interval is covered by differenteither one or two power laws. These results can be seen in Table VII. WIMP mass (GeV) b c d p j
100 10.3 0.498 0.323 7010 0.702125 9.74 0.612 0.294 4220 0.836150 9.49 0.675 0.280 3850 0.894200 9.28 0.734 0.268 3630 0.943350 9.02 0.800 0.257 3380 0.978500 8.95 0.813 0.255 3260 0.9881000 8.91 0.819 0.254 3140 0.997 a = 25 . n = 0 . q = 3 . Z boson : b , c , d , p and j parameters corresponding to (8) in the ZZ channel for different WIMP masses. Mass independentparameters in (8) for this channel are presented at the bottom of the table. Parameter WIMP mass interval (GeV) Fitting power law(s) b ≤ M ≤ . M − . c ≤ M ≤ . M . d ≤ M ≤
191 0 . M . + 21 . M − . < M ≤
360 2 . · − M . + 0 . M − . < ≤ M ≤ . M − . p ≤ M ≤ M − . + 0 . M . j ≤ M ≤ . M . Table VII:
Parameters corresponding to (10) for the Z boson. It can be seen that all mass dependent parameters for the Z boson followa simple power-law scaling at intermediate and high masses. C. t quark For the top, there are six mass dependent parameters: b , n , c , p q and l which are detailed in Table VIII. Themass independent parameters are a = 290, c = 1 . d = 0 .
19 and d = 0 . GeV. Nevertheless, from 1000 GeV onwards, the photon spectra do not change as was proven byconsidering several higher masses. The chosen parameters fit the spectra from x = 10 − till the end of the allowedinterval. Again for low masses, the spectra do not end at x = 1 but at smaller energies (e.g. x ≃ . m = 200 GeV) and, as masses get higher, the spectral tail approaches x = 1.Some of these results are presented graphically in Figure 3, Appendix B for four WIMP masses: 200, 250, 500 and1000 GeV . Also in this Appendix, mass dependent parameters b , n , c , p , q and l are plotted in Figure 4.Concerning the scaling behavior of the c , p , q and l parameters, they obey a simple power law in the wholeaccessible WIMP mass range. Nevertheless, for b and c parameters the simple power law behavior starts frommasses bigger than 350 GeV. These results can be seen in Table IX. WIMP mass (GeV) b n c p q l
200 14.4 0.477 3 . · − . · − . · − . · − . · − a = 290 ; c = 1 .
61 ; d = 0 .
19 ; d = 0 . t quark : b , n , c , p , q and l parameters corresponding to (9) in the t ¯ t channel for different WIMP masses. Mass independentparameters in (9) for this channel are presented at the bottom of the table. Parameter WIMP mass interval (GeV) Fitting power law(s) b ≤ M ≤
350 9 . M . + 11 . · M − . < M ≤ . M − . n ≤ M <
300 21 . M − . + 0 . M . ≤ M ≤ . M − . c ≤ M ≤ M − . p ≤ M < . · − M . q ≤ M ≤ . M . l ≤ M ≤ . M − . Table IX:
Parameters corresponding to (10) for the t quark. It can be seen that all mass dependent parameters for the t quark follow asimple power-law scaling behavior at intermediate and high WIMP masses. Parameters b and n follow the simple power law for M > ≤ M ≤ WIMP mass (GeV) n p
25 10.1 22150 10.0 767100 9.91 2520200 9.80 8660500 9.67 4 . · . · . · · . · a = 14 . b = 5 .
40 ; b = 5 .
31 ; n = 1 .
40 ; c = 2 .
54 ; d = 0 .
295 ; c = 0 .
373 ; d = 0 .
470 ; q = 0 . τ lepton : n and p parameters corresponding to (6) in the τ + τ − channel for different WIMP masses. Mass independentparameters in (6) for this channel are presented at the bottom of the table. D. Leptons and quarks
For the rest of the quarks and leptons, the parametrization given in (6) is completely valid. Now we present resultsfor τ and µ leptons and all quarks except for the top. τ lepton For the τ lepton, there are only two mass dependent parameters in the spectra fitting function (6): n and p . Theremaining parameters are mass independent for this particle and their values are a = 14 . b = 5 . b = 5 . n = 1 . c = 2 . d = 0 . c = 0 . d = 0 .
470 and q = 0 . · GeV. For masses higher than 5 · GeV,the spectra do not seem to change, within the statistical uncertainties, with respect to that corresponding to 5 · GeV.The n parameter scales with the WIMP mass as a simple power law for M < · GeV. For the other massdependent p parameter, the power-law behavior is valid in two separated intervals with an inflection point in thebehavior at M = 1000 GeV. These results can be seen in Table XI.Some of these results are presented graphically in Figure 5, Appendix C for four WIMP masses: 25, 100, 1000 and5 · GeV. Also in this Appendix, mass dependent parameters n and p are presented in Figure 6.For this particle, it is worth mentioning the increasing contribution of the logarithmic term in (6) as the WIMPmass increases. This fact can be seen in the presented plots from x = 0 . p parameter increase as the WIMP masses increase. µ lepton For the µ particle and according to expression (7), there are only three mass dependent parameters: q , p and l .These values are presented in Table XII. In this case, the considered range for WIMP masses is from 25 to 5 · GeV.0
Parameter WIMP mass interval (GeV) Fitting power law(s) n ≤ M < . M − . ≤ M ≤ · − . M − . + 179 M − . + 9 . p ≤ M < . M . ≤ M ≤ · . M . Table XI:
Parameters for expression (10) for τ lepton. It can be seen that n and p parameters follow power-law behaviors. WIMP mass (GeV) p q l
25 9510 3 . · − . · − . · − . · . · − . · . · − . · . · − . · . · − . · . · − · . · . · − µ lepton : Parameters corresponding to (7) for different WIMP masses. All parameters in expression (7) are WIMP massdependent. The scaling of the p parameter with the WIMP mass shows two well differentiated regimes, with different asymptoticpower laws: one from M = 25 GeV to M = 100 GeV, and another from M = 750 GeV to M = 5 · GeV. On theother hand, q and l parameters present a sum of two power laws evolution in the whole studied WIMP mass range.As for the τ lepton, the flux of photons increases as the WIMP mass increases. In this case, the q parameterincreases as the WIMP masses do so, instead of the p parameter as was the case for the τ . u quark The mass independent parameters are a = 5 . b = 5 . c = 0 . c = 0 . d is irrelevant) and q = 9 . · − . The mass dependent parameters are b , n , n , d and p . These results are presented in Table XIV.The analyzed mass range for this quark is from 50 to 8000 GeV.The spectra of the two highest studied masses (5000 and 8000 GeV) clearly differ in the low energy interval.Therefore no conclusion can be made about the existence of an asymptotic high masses limit in the spectral shapes.Concerning the mass evolution of the parameters for this quark, we observe simple power-law behaviors for both b and p parameters in the whole studied WIMP mass range interval. On the other hand, n , n and d parameters arefitted by a sum of two power laws in the studied range. These resuls can be seen in Table XV. The chosen values forthe parameters turn out to fit the spectra from x = 5 · − till the end of the allowed energy interval. Nevertheless,for some masses, the fit also applies for lower energies, i.e. lower x values, up to 10 − . Parameter WIMP mass interval (GeV) Fitting power law(s) p ≤ M ≤
100 176 M . ≤ M ≤ · M . q ≤ M < · . M − . + 0 . M . l ≤ M < · . M − . + 16 . M − . Table XIII:
Parameters for expression (10) for µ lepton. It can be seen that p parameter follows simple power laws in two different WIMPmass regimes. Nevertheless, for q and l parameters a sum of two power laws accounts for the WIMP mass dependence in the whole studiedWIMP mass interval. WIMP mass (GeV) b n n d p
50 3.60 2.77 0.585 0.383 129100 3.75 2.64 0.551 0.355 225200 3.88 2.54 0.521 0.332 409500 4.04 2.44 0.490 0.308 8561000 4.18 2.40 0.472 0.293 15402000 4.34 2.39 0.463 0.281 28005000 4.55 2.38 0.450 0.266 60008000 4.67 2.34 0.448 0.259 8900 a = 5 .
58 ; b = 5 .
50 ; c = 0 .
315 ; c = 0 . q = 9 . · − .Table XIV: u quark : b , n , n , d and p parameters corresponding to expression (6) when applied to u ¯ u channel for different WIMPmasses. Mass independent parameters in (6) for this channel are presented at the bottom of the table. Parameter WIMP mass interval (GeV) Fitting power law(s) b ≤ M ≤ . M . n ≤ M ≤ . M − . + 1 . M . n ≤ M ≤ . M . + 0 . M − . d ≤ M ≤ . M − . + 0 . M − . p ≤ M ≤ . M . Table XV:
Parameters corresponding to (10) for u quark. b and p parameters follow a simple power-law behavior in the whole studiedWIMP mass interval. n , n and d parameters follow a sum of two power laws in the whole mass interval. d quark For this channel there are five mass-dependent parameters: b , n n , c and p . The mass independent parametersare a = 5 . b = 5 . d = 0 . c = 0 . d = 0 .
570 and q = 1 . · − . All parameters in this case arepresented in Table XVI. The mass range studied for this quark was from 50 to 5000 GeV.In this channel, no conclusion can be drawn about the existence of an asymptotic high mass limit in the spectralshape. The chosen parameters provide good fits from x = 2 · − for M = 50 GeV whereas for the rest of masses thefits work very well till x = 5 · − .The scaling of b with M is given by a simple power-law in the whole mass interval, whereas n , n and c followa sum of two power-law behavior in the whole studied mass. Finally, the p parameter presents a power-law behaviorfor M >
50 GeV. These results can be seen in Table XVII. s quark For the s quark, there are just four mass dependent parameters b , n , d and p . The mass independent parametersfor this particle in (6) are a = 4 . n = 2 . b = 6 . c = 0 . c = 0 . d is irrelevant as for the u quark) and q = 2 . · − . All these parameters are detailed in Table XVIII. The studied mass range for this quark is between50 and 7000 GeV.As in the d quark case, no conclusion can be drawn about the existence of an asymptotic high mass limit in the WIMP mass (GeV) b n n c p
50 4.09 2.69 0.561 0.327 17.7100 4.24 2.47 0.522 0.293 66.2200 4.39 2.34 0.480 0.258 166500 4.56 2.28 0.448 0.228 4831000 4.75 2.25 0.426 0.212 12702000 4.91 2.24 0.409 0.200 31305000 5.11 2.23 0.394 0.187 10200 a = 5 .
20 ; b = 5 .
10 ; d = 0 .
410 ; c = 0 . d = 0 .
570 ; q = 1 . · − Table XVI: d quark : b , n , n , c and p parameters corresponding to expression (6) when applied to d ¯ d channel for different WIMPmasses. Mass independent parameters in (6) for this channel are presented at the bottom of the table. Parameter WIMP mass interval (GeV) Fitting power law(s) b ≤ M ≤ . M . n ≤ M ≤ . M − . + 2 . M − . n ≤ M ≤ . M − + 0 . M . c ≤ M ≤ . M − . + 0 . M . p < M ≤ . M . Table XVII:
Parameters corresponding to (10) for d quark. As can be seen, b parameter follows a simple power law in the whole accessiblemass interval. n , n and c parameters follow a sum of two power laws in the whole accessible mass interval. Finally, p parameter followsa power law for M >
50 GeV.
WIMP mass (GeV) b n d p
50 4.78 0.719 0.367 186100 5.31 0.669 0.332 409200 5.43 0.648 0.315 605500 5.60 0.612 0.290 11801000 5.73 0.592 0.276 19802000 5.87 0.575 0.263 33205000 6.07 0.557 0.249 65007000 6.12 0.548 0.244 7570 a = 4 .
83 ; n = 2 .
03 ; b = 6 .
50 ; c = 0 .
335 ; c = 0 . q = 2 . · − Table XVIII: s quark : b , n , d and p parameters corresponding to expression (6) when applied to s ¯ s channel for different WIMPmasses. Mass independent parameters in (6) for this channel are presented at the bottom of the table. spectral shape. The scaling with M of the parameters for this quark is a simple power law for b parameter for masseshigher than 1000 GeV, the sum of two power laws for n and d parameters in the whole studied WIMP mass, andtwo power laws for p parameter: one for masses smaller than 1000 GeV and another for masses higher than 1000GeV. These results are shown in Table XIX. c quark As for the d quark, there are five mass dependent parameters. In this case b , n , c , d and p which are presented inTable XX. The mass independent parameters are a = 5 . b = 7 . n = 0 . c = 0 . d is irrelevant)and q = 9 . · − .Likewise the u quark, the studied mass range was from 50 to 8000 GeV and again no conclusion can be madeabout the existence of an asymptotic high mass limit for the spectral shape. Higher masses simulations would be thusrequired also in this case.The scaling of b and n with M shows a simple power-law behavior in the considered range. For c and p , thesingle power-law evolution is only valid for masses above 200 GeV. Finally, d parameter follows a sum of two powerlaws in the studied mass range. These results are shown in Table XXI. Parameter WIMP mass interval (GeV) Fitting power law(s) b M > . M . n ≤ M ≤ . M − . + 0 . M − . d ≤ M ≤ . M − . + 0 . M − . p ≤ M ≤
100 4 . M . < M ≤ . M . Table XIX:
Parameters corresponding to (10) for s quark. As can be seen, b parameter follows a simple power-law behavior for masseshigher than 1000 GeV. n and d parameters follow the sum of two power laws for the whole studied WIMP mass interval. Finally, p parameter presents two power laws: one for masses smaller than 1000 GeV and another for masses higher than 1000 GeV. WIMP mass (GeV) b n c d p
50 5.93 2.35 0.239 0.428 210100 5.48 2.08 0.283 0.374 379200 4.98 1.86 0.330 0.330 673500 4.50 1.65 0.378 0.288 12301000 4.00 1.50 0.406 0.264 21102000 3.70 1.35 0.432 0.245 40505000 3.27 1.17 0.470 0.221 80808000 3.08 1.11 0.494 0.208 12000 a = 5 .
58 ; b = 7 .
90 ; n = 0 .
686 ; c = 0 . q = 9 . · − Table XX: c quark : b , n , c , d and p parameters corresponding to expression (6) in the c ¯ c channel for different WIMP masses. Massindependent parameters in (6) for this channel are presented at the bottom of the table. Parameter WIMP mass interval (GeV) Fitting power law(s) b ≤ M ≤ . M − . n ≤ M ≤ . M − . c ≤ M ≤ . M . d ≤ M ≤ . M − . + 0 . M − . p < M ≤ . M . Table XXI:
Parameters corresponding to (10) for c quark. It can be seen that the mass dependent parameters follow a power-law behaviorfor intermediate and high WIMP masses. In particular the d parameter follows a sum of two power-law behavior in he whole accessibleWIMP mass range. b quark For the b quark, the required gamma rays spectra parametrization is the one given by expression (6). For thisparticle, the mass independent parameters are a = 10 . b = 11 . c = 0 . d = 0 . q = 2 . · − . Themass dependent parameters are b n , n , c , d and p . Their values are presented in Table XXII.The studied mass range is from 50 to 8000 GeV. Unlike previous particles for which the spectra did not changeremarkably for very high masses, in the present case no conclusion can be drawn about the existence of an asymptotichigh mass limit.Concerning the scaling behavior of the parameters for this quark, we observe that the behavior depends both onthe WIMP mass and on the considered parameter. Thus b and n no longer scale with a single power-law for M higher than 100 GeV. For n , two simple power laws can be seen, one from 50 to 1000 GeV (not included) and asecond one from 1000 (included) to 8000 GeV. c shows also a power-law behavior but only up to 50 GeV. Finally,both d and p , scale with simple power laws from 500 GeV up. These results are summarized in Table XXIII.Some of these results are presented graphically in Figure 7, Appendix D for four WIMP masses: 50, 200, 1000 and5000 GeV . Also in this Appendix, mass dependent parameters b , n , n , c , d and p are plotted in Figure 8. WIMP mass (GeV) b n n c d p
50 19.5 6.48 0.710 0.365 0.393 57.8100 17.1 5.80 0.695 0.403 0.360 138200 13.1 5.01 0.680 0.415 0.340 281500 8.76 4.04 0.660 0.431 0.319 6231000 6.00 3.36 0.647 0.447 0.305 10302000 4.60 2.85 0.640 0.460 0.294 16205000 3.00 2.26 0.634 0.479 0.280 26708000 2.35 2.00 0.629 0.490 0.274 3790 a = 10 . b = 11 . c = 0 . d = 0 .
550 ; q = 2 . · − Table XXII: b quark : b , n , n , c , d and p parameters corresponding to expression (6) in the b ¯ b channel for different WIMP masses.Mass independent parameters in (6) for this channel are presented at the bottom of the table. Parameter WIMP mass interval (GeV) Fitting power law(s) b < M ≤ M − . n < M ≤ . M − . n ≤ M < . M − . ≤ M ≤ . M − . c < M ≤ . M . d ≤ M <
600 0 . M − . + 37 . M − . ≤ M ≤ . M − . p < M ≤ . M . Table XXIII:
Parameters corresponding to (10) for the b quark. All mass dependent parameters for the b quark follow simple power-lawscalings at intermediate and high energies. Only n and d parameters follow power-law behaviors at low WIMP masses. VII. NUMERICAL CODES
All the calculations performed in this investigation are available at the websitehttp://teorica.fis.ucm.es/ ∼ PaginaWeb/downloads.htmlAt this site, we provide the
Mathematica [19] files that contain the fitting expressions (6), (8) and (9) for x . d N γ / d x presented in this paper when applied for each studied channel. Let us remind that these parametrizations are validin the corresponding WIMP masses intervals mentioned in the corresponding sections. Also in these files, the fittingformulae for mass dependent parameters in each channel are presented. VIII. CONCLUSIONS
In this work, we have extensively studied the photon spectra coming from WIMP pair annihilation into SM particleparticle-antiparticle pairs for all the phenomenologically relevant channels. The covered WIMP mass range has beenoptimized for each particular channel taking into account mass thresholds, statistics, and saturation of the MonteCarlo simulation. For instance, for light quarks it was from 50 GeV to 8000 GeV, for leptons it was from 25 GeV to50 TeV, for gauge bosons from 100 GeV to 1000 GeV and for t quark from 200 to 1000 GeV. All simulated spectracovered the whole accessible energy interval, from extremely low energetic photons till photons with one half of theavailable total center of mass energy.Once the spectra were obtained, analytical expressions were proposed to fit the simulation data. Three differentfitting functions appeared to be valid depending on the studied channel: one for light quarks and leptons, anotherfor gauge bosons and finally one for t quark very similar to the latter. Those expressions depended on either WIMPmass dependent or independent parameters. For WIMP mass independent parameters, their values did neverthelessdepend on the considered annihilation channel whereas for WIMP mass dependent ones, their evolutions with WIMPmass were parametrized from the obtained values by continuous and smooth curves.In addition to a better understanding of the different channels for photon production from DM annihilation, theuse of these fitting functions found in these analyses can save an important amount of computing time and resources:Monte Carlo simulations do not need to be repeated each time that a particular photon spectrum needs to be knownfor a given channel and center of mass energy. This fact is particularly important for high energy photons, whoseproduction rate is very suppressed and would require large computation times and to store big amounts of data. Ourresearch was thus able to present very good statistics for those energies.By having used extensive PYTHIA Monte Carlo simulation we have been able to obtain relatively simpleparametrizations of these spectra and fit the corresponding parameters. As our analysis is model independent, itcould be useful, both for theoreticians and experimentalists, interested in the indirect DM detection through gammarays. Given some theoretical model, and the corresponding velocity averaged annihilation cross sections for the dif-ferent channels, our formulae make it possible to obtain the expected photon spectrum for each particular theoreticalmodel in a relatively simple way. In this sense, further work is in progress to extend our analysis to other stableparticles like positron or neutrinos but these results will be presented elsewhere. Acknowledgments
This work has been supported in part by MICINN (Spain) project numbers FIS 2008-01323 and FPA 2008-00592,CAM/UCM 910309, MEC grant BES-2006-12059, DOE grant DE-FG02-94ER40823 and MICINN Consolider-Ingenio5MULTIDARK CSD2009-00064. We are particularly grateful to Prof. Jonathan Feng for his continuous and encourag-ing help during 2007 summer at UC Irvine. We would also like to thank Dr. Mario Bondioli for his preliminary helpwith simulation software and Dr. Mirco Cannoni, Prof. Mario E. Gomez and Prof. Mikhail Voloshin for interestingdiscussions about different aspects of the physical meaning of the spectra behavior. Dr. Abelardo Moralejo drewour attention of some particular aspects of bremsstrahlung. Finally, fruitful discussions were held with Daniel Nietoabout detectors technicalities and improvement of the numerical codes.RL acknowledges financial support given by Ministero dell’Istruzione, dell’Universit`a e della Ricerca (MIUR), by theUniversity of Torino (UniTO), by the Istituto Nazionale di Fisica Nucleare (INFN) within the Astroparticle PhysicsProject, and by the Italian Space Agency (ASI) under contract Nro: I/088/06/0.AdlCD wants to thank David Fern´andez (UCM) for his technical support with the performed simulations andDr. Javier Almeida for his help to manage simulation results. Scientific discussions about other available simulationsoftwares were held with Beatriz Ca˜nadas. [1] P. Achard et al. , Phys. Lett.
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Addison-Wesley, New-York, (1991). IX. APPENDICES
In this section we present simulations for some of the studied channels: W + W − , t ¯ t , τ + τ − and b ¯ b . For thesechannels four simulated spectra are presented together with the proposed fit formulae. For each channel, evolutionwith WIMP mass of mass dependent parameters have been plotted. The final appendix E shows the running withthe WIMP mass of the total number of photons per WIMP pair annihilation. A. Plots for W gauge boson en x . d N γ / d x E γ /M WIMP
Total fitMonte Carlo simulation 0.001 0.01 0.1 1 0.0001 0.001 0.01 0.1 x . d N γ / d x E γ /M WIMP (a) Photon spectrum for M = 100 GeV for W + W − channel. x . d N γ / d x E γ /M WIMP
Total fitMonte Carlo simulation 0.01 0.1 1 0.001 0.01 0.1 x . d N γ / d x E γ /M WIMP (b) Photon spectrum for M = 200 GeV for W + W − channel. x . d N γ / d x E γ /M WIMP
Total fitMonte Carlo simulation 0.01 0.1 1 0.001 0.01 0.1 x . d N γ / d x E γ /M WIMP (c) Photon spectrum for M = 350 GeV for W + W − channel. x . d N γ / d x E γ /M WIMP
Total fitMonte Carlo simulation 0.01 0.1 1 0.001 0.01 0.1 x . d N γ / d x E γ /M WIMP (d) Photon spectrum for M = 1000 GeV for W + W − channel. Figure 1: Photon spectra for four different WIMP masses (50, 200, 1000 and 5000 GeV) in the W + W − channel. Red dottedpoints are PHYTIA simulations and solid lines correspond to the proposed fitting functions. b pa r a m e t e r f o r W + W - c hanne l ( d i m en s i on l e ss ) WIMP mass (GeV)
Fitting power lawsb values for W + W - channel (a) b parameter of expression (6) for W + W − channel. c pa r a m e t e r f o r W + W - c hanne l ( d i m en s i on l e ss ) WIMP mass (GeV)
Fitting power laws c values for W + W - channel (b) c parameter of expression (6) for W + W − channel. d pa r a m e t e r f o r W + W - c hanne l ( d i m en s i on l e ss ) WIMP mass (GeV)
Fitting power lawsd values for W + W - channel (c) d parameter of expression (6) for W + W − channel.
500 600 700 800 900 1000 100 200 500 1000 p pa r a m e t e r f o r W + W - c hanne l ( d i m en s i on l e ss ) WIMP mass (GeV)
Fitting power lawsp values for W + W - channel (d) p parameter of expression (6) for W + W − channel. j pa r a m e t e r f o r W + W - c hanne l ( d i m en s i on l e ss ) WIMP mass (GeV)
Smooth spline + fitting power lawj values for W + W - channel (e) j parameter of expression (6) for W + W − channel. Figure 2: Mass dependence of b , c , d , p and j parameters for W + W − channel. Crossed points are parameters values foundafter the fitting process for each WIMP mass and solid lines correspond to the proposed fitting functions. B. Plots for t quark x . d N γ / d x E γ /M WIMP
Total fitMonte Carlo simulation 0.01 0.1 1 0.0001 0.001 0.01 0.1 x . d N γ / d x E γ /M WIMP (a) Photon spectrum for M = 200 GeV for t ¯ t channel. x . d N γ / d x E γ /M WIMP
Total fitMonte Carlo simulation 0.01 0.1 1 0.0001 0.001 0.01 0.1 x . d N γ / d x E γ /M WIMP (b) Photon spectrum for M = 250 GeV for t ¯ t channel. x . d N γ / d x E γ /M WIMP
Total fitMonte Carlo simulation 0.01 0.1 1 0.0001 0.001 0.01 0.1 x . d N γ / d x E γ /M WIMP (c) Photon spectrum for M = 500 GeV for t ¯ t channel. x . d N γ / d x E γ /M WIMP
Total fitMonte Carlo simulation 0.01 0.1 1 0.0001 0.001 0.01 0.1 x . d N γ / d x E γ /M WIMP (d) Photon spectrum for M = 1000 GeV for t ¯ t channel. Figure 3: Photon spectra for four different WIMP masses (200, 250, 500 and 1000 GeV) in the t ¯ t annihilation channel. Reddotted points are PHYTIA simulations and solid lines correspond to the proposed fitting functions.
12 12.5 13 13.5 14 14.5 200 300 400 500 1000 b pa r a m e t e r f o r t op - an t i t op c hanne l ( d i m en s i on l e ss ) WIMP mass (GeV)
Fitting power lawsb values for top-anti top channel (a) b parameter of expression (6) for t ¯ t channel. n pa r a m e t e r f o r t op - an t i t op c hanne l ( d i m en s i on l e ss ) WIMP mass (GeV)
Fitting power lawsn values for top-anti top channel (b) n parameter of expression (6) for t ¯ t channel. c pa r a m e t e r f o r t op - an t i t op c hanne l ( d i m en s i on l e ss ) WIMP mass (GeV)
Fitting power lawc values for top-anti top channel (c) c parameter of expression (6) for t ¯ t channel. p pa r a m e t e r p f o r t op - an t i t op c hanne l ( d i m en s i on l e ss ) WIMP mass (GeV)
Fitting power law + Smooth splinep values for top-anti top channel (d) p parameter of expression (6) for t ¯ t channel. q pa r a m e t e r f o r t op - an t i t op c hanne l ( d i m en s i on l e ss ) WIMP mass (GeV)
Fitting power lawq values for top-anti top channel (e) q parameter of expression (6) for t ¯ t channel. l pa r a m e t e r f o r t op - an t i t op c hanne l ( d i m en s i on l e ss ) WIMP mass (GeV)
Fitting power lawl values for top-anti top channel (f) l parameter of expression (6) for t ¯ t channel. Figure 4: Mass dependence of b , n , c , p , q and l parameters for t ¯ t annihilation channel. Crossed points are parameters valuesfound after the fitting process for each WIMP mass and solid lines correspond to the proposed fitting functions. C. Plots for τ lepton x . d N γ / d x E γ /M WIMP
Total fitMonte Carlo simulation 0.001 0.01 0.1 0.001 0.01 0.1 x . d N γ / d x E γ /M WIMP (a) Photon spectrum for M = 25 GeV for τ + τ − channel. x . d N γ / d x E γ /M WIMP
Total fitMonte Carlo simulation 0.001 0.01 0.1 0.001 0.01 0.1 x . d N γ / d x E γ /M WIMP (b) Photon spectrum for M = 100 GeV for τ + τ − channel. x . d N γ / d x E γ /M WIMP
Total fitMonte Carlo simulation 0.001 0.01 0.1 0.001 0.01 0.1 x . d N γ / d x E γ /M WIMP (c) Photon spectrum for M = 1000 GeV for τ + τ − channel. x . d N γ / d x E γ /M WIMP
Total fitMonte Carlo simulation 0.001 0.01 0.1 0.001 0.01 0.1 x . d N γ / d x E γ /M WIMP (d) Photon spectrum for M = 5 · GeV for τ + τ − channel. Figure 5: Photon spectra for four different WIMP masses (25, 100, 1000 and 5 · GeV) in the τ + τ − annihilation channel.Red dotted points are PHYTIA simulations and solid lines correspond to the proposed fitting functions. n pa r a m e t e r f o r τ + τ - c hanne l ( d i m en s i on l e ss ) WIMP mass (GeV)
Fitting power laws n values for τ + τ - channel (a) n parameter of expression (6) for τ + τ − channel.
50 100 1000 10000 50000 p pa r a m e t e r f o r τ + τ - c hanne l ( d i m en s i on l e ss ) WIMP mass (GeV)
Fitting power laws p values for τ + τ - channel (b) p parameter of expression (6) for τ + τ − channel. Figure 6: Mass dependence of n and p parameters for τ + τ − annihilation channel. Crossed points are parameters values foundafter the fitting process for each WIMP mass and solid lines correspond to the proposed fitting functions. D. Plots for b quark x . d N γ / d x E γ /M WIMP
Total fitMonte Carlo simulation 0.01 0.1 1 0.001 0.01 0.1 x . d N γ / d x E γ /M WIMP (a) Photon spectrum for M = 50 GeV for b ¯ b channel. x . d N γ / d x E γ /M WIMP
Total fitMonte Carlo simulation 0.01 0.1 1 0.001 0.01 0.1 x . d N γ / d x E γ /M WIMP (b) Photon spectrum for M = 200 GeV for b ¯ b channel. x . d N γ / d x E γ /M WIMP
Total fitMonte Carlo simulation 0.1 1 0.001 0.01 0.1 x . d N γ / d x E γ /M WIMP (c) Photon spectrum for M = 1000 GeV for b ¯ b channel. x . d N γ / d x E γ /M WIMP
Total fitMonte Carlo simulation 0.1 1 0.001 0.01 0.1 x . d N γ / d x E γ /M WIMP (d) Photon spectrum for M = 5000 GeV for b ¯ b channel. Figure 7: Photon spectra for four different WIMP masses (50, 200, 1000 and 5000 GeV) in the b ¯ b annihilation channel. Reddotted points are PHYTIA simulations and solid lines correspond to the proposed fitting functions. b pa r a m e t e r f o r b - an t i b c hanne l ( d i m en s i on l e ss ) WIMP mass (GeV)
Smooth spline + fitting power law b values for b-anti b channel (a) b parameter of expression (6) for b ¯ b channel. n pa r a m e t e r f o r b - an t i b ( d i m en s i on l e ss ) WIMP mass (GeV)
Smooth spline + fitting power law n values for b-anti b channel (b) n parameter of expression (6) for b ¯ b channel. n pa r a m e t e r f o r b - an t i b c hanne l ( d i m en s i on l e ss ) WIMP mass (GeV)
Fitting power lawsn values for b-anti b channel (c) n parameter of expression (6) for b ¯ b channel. c pa r a m e t e r f o r b - an t i b c hanne l ( d i m en s i on l e ss ) WIMP mass (GeV)
Smooth spline + fitting power lawc values for b-anti b channel (d) c parameter of expression (6) for b ¯ b channel. d pa r a m e t e r f o r b - an t i b c hanne l ( d i m en s i on l e ss ) WIMP mass (GeV)
Fitting power lawsd values for b-anti b channel (e) d parameter of expression (6) for b ¯ b channel.
50 100 500 1000 5000 50 100 500 1000 5000 10000 p pa r a m e t e r f o r b - an t i b c hanne l ( d i m en s i on l e ss ) WIMP mass (GeV)
Smooth spline + fitting power lawp values for b-anti b channel (f) p parameter of expression (6) for b ¯ b channel. Figure 8: Mass dependence of b , n , n , c , d and p parameters for the b ¯ b annihilation channel. Crossed points are parametersvalues found after the fitting process for each WIMP mass and solid lines correspond to the proposed fitting functions. E. Photon number per WIMPs annihilation P ho t on s pe r W I M P s ann i h il a t i on WIMP mass (GeV) tau leptonmu leptonelectron positron (a) Total number of photons per WIMP annihilation inlepton-antilepton pairs.
10 50 100 150 50 100 500 1000 5000 10000 P ho t on s pe r W I M P s ann i h il a t i on WIMP mass (GeV) quark bquark cquarks s, d and utop quarkW bosonZ boson (b) Total number of photons per WIMP annihilation in gaugebosons and quark-antiquark pairs.(b) Total number of photons per WIMP annihilation in gaugebosons and quark-antiquark pairs.