Positrons from cosmic rays interactions and dark matter annihilations
aa r X i v : . [ a s t r o - ph . H E ] S e p RIVISTA DEL NUOVO CIMENTO
Vol. ?, N. ? ? Positrons from cosmic rays interactions and dark matter annihi-lations
Roberto Alfredo Lineros Rodriguez ( ) ( ) Dipartimento di Fisica Teorica, Universit´a degli Studi di Torino, and INFN – SezioneTorino, via P. Giuria 1, I-10125, Torino, Italy. (ricevuto ?)
Summary. —The electron and positron cosmic rays puzzle has triggered a revolution in thefield of astroparticle physics. Many hypotheses have been proposed to explainthe unexpected rise of the positron fraction, observed by HEAT and PAMELAexperiments, for energies larger than few GeVs. In this work, we study sourcesof positron cosmic rays related to annihilation of dark matter and secondaryproduction. In both cases, we consider the impact of uncertainties related todark matter physics, nuclear physics and propagation of cosmic rays, finding thatthe largest uncertainties come from propagation. We find that some key–featurespresent in the positron signal from dark matter annihilation are preserved eventhough the uncertainties. In addition, we study the stability of the positron fractionunder small variations of the electron flux, which is usually considered as known,we found that considering just the observational uncertainties in the electron flux isenough to changed dramatically the positron fraction in the energy range were theexcess is observed.PACS – 96.50.S-.98.38.-j98.70.Sa
Preprint DFTT 2/2010
1. – Introduction
During the last decades, the astrophysical and cosmological evidences of dark mat-ter and dark energy have created a revolution in the field of fundamental physics. Somemodels of new physics, which aim to extend the Standard Model of particles, predict par-ticles with the right properties to act as dark matter candidates. In a similar way, cosmicray physics is continuously stimulated with new observations showing an environmentmuch more active than it was expected. Furthermore, cosmic rays correspond to genuinesamples of the matter composition of the Milky Way and give essential information aboutthe Interstellar medium and the Solar System environment. c (cid:13) Societ`a Italiana di Fisica ROBERTO ALFREDO LINEROS RODRIGUEZ
A fraction of comic rays is composed by antimatter. This component may providecrucial clues on the non–standard sources of cosmic rays because it is less abundant thanthe matter cosmic–rays component. This analysis may open a window for dark mattersearches.This paper is based on results of the author’s Ph.D. Thesis [1] and some other projectsderived from it. In
Section , we study the standard mechanisms of production of elec-trons and positrons. In the same spirit, in Section , the propagation of cosmic rays isreviewed and discussed. Then, we study the production of positrons in scenarios of darkmatter annihilation (Section ) and secondary production (Section ).
2. – Production of positrons and electrons
In a first approach, the Standard Model of particles physics describes well enough thephysics behind the production of cosmic rays. However, let us clarify that, the produc-tion in the astrophysical context arises both from particles physics and from the galacticenvironment.Among the sources of cosmic rays, supernovae are the main ones and a big fractionof them comes from the matter expelled at the time of the explosion. This mechanismis generally the dominant one but not for all the species and energy scales. In fact,interactions between cosmic rays and the interstellar gas sizeably contributes, adding anextra amount of cosmic rays known as secondaries .Electrons and positrons are also produced by other mechanisms, like pulsar emis-sion [2], and decay/annihilation of dark matter [3]. .1. Production in proton–proton collisions . – The nuclear scattering is thefundamental process behind secondary electrons and positrons. The secodary productiontakes place in the galactic disk when nuclei cosmic rays scatter off the interstellar medium.In particular, it depends on the composition of the interstellar medium – composed byhydrogen (0 . / cm ) and helium gas (0 . / cm ) [4, 5, 6, 7] – and on the fluxesof protons and alpha particles [8]. .1.1. Production from mesons decay . In proton-proton collisions, charged pionsand kaons are produced in big amounts. Their decay channels are responsible of the bigamount of electrons and positrons. Of course, this is valid for collisions at low energy( p lab <
20 GeV) when meson production is dominant.The electron and positron energy spectra are calculated from the inclusive cross sec-tion of pions and kaons and from the electron and positron spectra associated to theirdecays: d h ησ i pp → e dε e = d h ησ i πpp → e dε e + d h ησ i Kpp → e dε e , (1) OSITRONS FROM COSMIC RAYS INTERACTIONS AND DARK MATTER ANNIHILATIONS where a particular contribution is generically calculated as: d h ησ i Xpp → e dε e ( ε e , ε p ) = Z dε X d h ησ i pp → X dε X ( ε X , ε p ) × f e,X ( ε e , ε X ) , (2)where X denotes pions or kaons as intermediate particles and f e,X the meson decay dis-tribution into electrons or positrons. The decay distributions is analytically calculatedincluding muon polarization effects [1]. Pion decays are the simplest ones because theyhave just one dominant decay mode ( µ ± ν µ ) instead of kaon decays which are complex totreat (details in appendices C and D of [1]).The inclusive cross sections of pions and kaons from proton–proton collisions are verydifficult to calculate a priori . In the literature, two well known parametrization for theinclusive cross section of pions and kaons are available: • Badhwar et al. parametrization [9, 10]. • Tan–Ng parametrization [11, 12].Both parametrization reproduce good enough the observations in colliders. In any case,the small differences among them gives an estimation of uncertainties coming from nu-clear and particle physics. .1.2. Kamae et al. parametrization . A parametrization of production of positrons,electrons and other particles is proposed by Kamae et al. [13, 14, 15]. It aims to providean easy way to compute and estimate cosmic ray fluxes that comes from interstellarmedium interactions with nuclei cosmic rays.New processes are included like contributions from ∆(1238) and many hadron reso-nances around 1600 [MeV / c ] that make it accurate in the very low proton energy range( p lab < p lab >
20 GeV).In addition, this parametrization works for very high energy scales being based onsimulators like PYTHIA [16] to generate the spectra at higher energies. Also, its bigadvantage is to save CPU time by avoiding to calculate the convolution among mesonsproduction and meson decay into electrons and positrons. .1.3. Uncertainties from nuclear physics . Previously we described differentapproaches to calculate the electron and positron spectra in proton–proton collisions.Each parametrization is based on physical assumptions and encompasses available ex-perimental data: therefore the small differences among them can be used to quantify thenuclear physics uncertainties.In
Figure 1 , we show the comparisons among the three parametrizations for differentproton kinetic energies. We note that these parametrization are closely similar in behav-ior. However, there are variations up to 80% at proton energies of 20 GeV, as in case ofKamae et al. versus Tan and Ng parametrization. Another feature is that the Kamae’s
ROBERTO ALFREDO LINEROS RODRIGUEZ −1 −2 −1 E d < ησ > / d E [ m b ] energy [GeV]10 −1 −2 −1 E d < ησ > / d E [ m b ] energy [GeV] Lineros’ PhD thesis (2008)
Badhwar et al. parameterization T p = 2 GeVT p = 10 GeVT p = 35 GeV positronelectron 10 −1 −2 −1 E d < ησ > / d E [ m b ] energy [GeV]10 −1 −2 −1 E d < ησ > / d E [ m b ] energy [GeV] Lineros’ PhD thesis (2008)
Tan and Ng parameterization T p = 2 GeVT p = 10 GeVT p = 35 GeV positronelectron10 −1 −2 −1 E d < ησ > / d E [ m b ] energy [GeV]10 −1 −2 −1 E d < ησ > / d E [ m b ] energy [GeV] Lineros’ PhD thesis (2008)
Kamae et al. parameterization T p = 2 GeVT p = 10 GeVT p = 35 GeV positronelectron −0.4−0.2 0 0.2 0.4 0.6 0.8 110 −2 −1 R e l a t i v e d i ff e r e n c e r e s p e c t t o T a n a n d N g p a r a m . energy [GeV]−0.4−0.2 0 0.2 0.4 0.6 0.8 110 −2 −1 R e l a t i v e d i ff e r e n c e r e s p e c t t o T a n a n d N g p a r a m . energy [GeV] Lineros’ PhD thesis (2008)
Comparison at T p = 20 GeVpositronelectronBadhwar et al.Kamae et al. Figure 1. – Positron and electron inclusive cross section versus energy for proton energies of2, 10 and 35 GeV. The different panels show the case of Badhwar et al. [9], Tan and Ng [12],and Kamae et al. [14] parametrizations. The last panel shows that the relative difference withrespect to Tan and Ng solution is on average around 15% and 25% for Badhwar et al. andKamae et al. cases respectively. parametrization estimates a smaller electron cross section with respect to the other twoparametrizations.Due to the low statistics at very low energy, Badhwar’s and Tan’s parametrizationtend to produce non-physical distributions for proton kinetic energies below 6 GeV. Nev-ertheless, the total inclusive cross section – the integrated version of those – are still inagreement with the available experimental data. To fix this undesirables feature, bothparametrization are patched by doing a smooth transition from 3 GeV until 7 GeV withthe Stecker’s model [17, 18]. Let’s clarify that Kamae’s parametrization also includesthat feature, but considering more resonances.Moreover, for proton energies above 100 GeV, Badhwar’s parametrization becomes un-stable specially the electron cross section case.
OSITRONS FROM COSMIC RAYS INTERACTIONS AND DARK MATTER ANNIHILATIONS Intermediate stateCharge Leptons Quarks Gauge Bosons+1 ( ν e e + ) ( ν µ µ + ) ( ν τ τ + ) ( u ¯ d ) ( c ¯ s ) ( t ¯ b ) ( γW + ) ( ZW + )0 ( e − e + ) ( µ − µ + ) ( τ − τ + ) ( u ¯ u ) ( d ¯ d ) ( c ¯ c ) ( s ¯ s ) ( b ¯ b ) ( t ¯ t ) ( gg ) ( ZZ ) ( W − W + )-1 ( e − ¯ ν e ) ( µ − ¯ ν µ ) ( τ − ¯ ν τ ) ( d ¯ u ) ( s ¯ c ) ( b ¯ t ) ( W − γ ) ( W − Z ) Table
I. –
Intermediate states for positron and electron multiplicity distributions. .2. Production in annihilation processes . – Electron and positrons can alsobe the result of annihilations. A first case to come in mind is the matter–antimatterannihilation like electron positron collisions at LEP.The annihilation of dark matter also enters in this category, providing a new type ofsource of electrons and positrons that it would coexist with standard cosmic rays sources.Independently of how these particles annihilate, electrons and positrons would comedirectly from the annihilation event (direct production case) or from the annihilation’ssub-products, like decay of gauge or higgs bosons or hadronization/decay of quarks.We work in a general approach, in which we generate electron and positron multiplic-ity distributions which are independent of dark matter physics and can be used for anypurpose. .2.1. Multiplicity distribution . In a generic annihilation case, the multiplicitydistribution of electrons and positron is: (cid:18) dn e dε (cid:19) χ ¯ χ → eX = X i BR (cid:0) χ ¯ χ → i (cid:1) (cid:18) dn e dε (cid:19) i → eX , (3) where we decompose the annihilation in intermediate states i , related directly to theannihilation mechanism via branching ratios, for instance, the dark matter annihilationcase: BR (cid:0) χ ¯ χ → i (cid:1) = σ (cid:0) χ ¯ χ → i (cid:1) σ total . (4)This presents a general decomposition based on bricks which are the multiplicity distri-bution for many states i . .2.2. Calculation of Multiplicity distributions . We consider the most gen-eral set which is composed by pairs of Standard Model particles. Each pair preservequantum numbers of the annihilation, i.e. an electrically neutral and colorless final state.Depending on the case, we calculate analytical distributions for simple cases (elec-trons, positrons, and muons), and for complex ones, like the ones which involves hadroniza-
ROBERTO ALFREDO LINEROS RODRIGUEZ −1 E d n / d E Energy [GeV] 0 1 2 3 410 −1 E d n / d E Energy [GeV]
Lineros’ PhD thesis (2008) m χ = 100 GeVm χ = 1 TeV W + W − channel positronelectron 0 1.5 3 4.5 6 7.510 −1 E d n / d E Energy [GeV] 0 1.5 3 4.5 6 7.510 −1 E d n / d E Energy [GeV]
Lineros’ PhD thesis (2008) m χ = 100 GeVm χ = 1 TeV bb − channel positronelectron 0 0.1 0.2 0.3 0.4 0.510 −1 E d n / d E Energy [GeV] 0 0.1 0.2 0.3 0.4 0.510 −1 E d n / d E Energy [GeV]
Lineros’ PhD thesis (2008) m χ = 100 GeV m χ = 1 TeV τ + τ − channel positronelectron 0 1.5 3 4.5 6 7.5 910 −1 E d n / d E Energy [GeV] 0 1.5 3 4.5 6 7.5 910 −1 E d n / d E Energy [GeV]
Lineros’ PhD thesis (2008) m χ = 200 GeVm χ = 1 TeV tt − channel positronelectron Figure 2. – Multiplicity distribution for positrons and electrons versus energy. Each panel showsa different intermediate state – W + W − , τ + τ − , b ¯ b , and t ¯ t – with dark matter masses ( m χ ) of100 GeV (or 200 GeV) and 1 TeV. The different shapes on electron and positron distributioncome from the effect of production of polarized muon. tion processes, we used a modified version of PYTHIA [16] to generate the distribution,which includes the effect of polarization of muons.The basic set is based on Standard Model particles Table I . Special cases like theStandard Model higgs and models like Two Higgs Doublet Models are easily composedusing the basic set (details in [1]).In
Figure 2 , we present multiplicity distributions for cases inspired by annihilationof dark matter. We observe that the shape of each distribution depends directly on theintermediate state: states involving quarks produce a softer electron and positron spectrathan leptonic cases. In addition, the effect related to the polarization of muon in mesondecays also has an impact, producing more energetic electrons than positrons.
OSITRONS FROM COSMIC RAYS INTERACTIONS AND DARK MATTER ANNIHILATIONS Figure 3. – Propagation Zone geometry for the Milky Way. A cylinder of radius R (= 20 kpc)with thickness of 2 L delimits the region where cosmic rays propagate. A small cylinder withsame radius but with thickness 2 h z (= 200 pc) models the Galactic plane. The Solar system isplaced at the Galactic plane with distance r ⊙ = 8 .
3. – Propagation of positron and electron cosmic rays
The cosmic rays’ journey from the source until the Earth is a complex problem. Sincecosmic rays start to travel, they are affected by many processes and their intensity de-pends on the cosmic ray energy scale. For instance, in the GeV–range, their are stronglyaffected by turbulent magnetic fields that induce spatial diffusion, similar to the behaviorof a single molecule in a gas.For the GeV–scale cosmic rays, the Galactic medium has a very important role on thepropagation. Apart from the turbulent magnetic fields, there are also interactions withdiffuse radiation fields (UV, IR, CMB). Their continuous interaction changes the cosmicrays energy, electrons and positrons are more affected than other species.The combined effect of diffusion, energy losses, distance, among others, cause that theobserved cosmic rays spectra is rather different from the original one. .1. Two–Zone propagation model . – The modelization of cosmic rays propa-gation is a hard task due to the propagation complexity. We use a successfully testedmodel for propagation of cosmic rays [19, 20] which is tuned properly to explain thecurrent observations on abundances of nuclear cosmic rays. In general terms, this modelis divided in two parts: • The propagation zone: the region related to the extension of turbulent magneticfields, where cosmic rays propagate in a diffusive regime. Outside this region, wesuppose cosmic rays propagate in straight lines: this induces cosmic rays leakingfrom the diffusive zone.The propagation zone is composed by two cylinders centered at the Galactic cen-ter (
Figure 3 ). Both cylinders have a common radius equal to the galactic one( R = 20 kpc). The thick cylinder has a height of 2 L and fills the whole propa-gation zone. Its height is constrained by measurements of many species of cosmicrays [20]. The second cylinder is a thin disk with height 2 h z , where h z = 100 pc. Italso contains the interstellar medium, cosmic rays sources and interactions among ROBERTO ALFREDO LINEROS RODRIGUEZ them. • The transport equation: it corresponds to a continuity equation for the numberdensity of cosmic rays per unit of energy. It contains all physical processes thatare related to cosmic rays physics, like energy losses, diffusion, re-acceleration andcosmic rays sources. .2. Transport Equation for electrons and positrons . – In this case, thetransport equation is slightly different from the one for nuclei cosmic rays because thedominant energy loss process is the interaction with radiation fields (via inverse Comptonscattering) which is much efficient in these particles than in nuclei cosmic ray.Then the transport equation for number density of electron and positrons ( ψ ) is: ∂ψ∂t − ∇ (cid:0) D ( ε ) ∇ ψ (cid:1) − ∂∂ε (cid:0) b ( ε ) ψ (cid:1) = q , (5)where D ( ε ) is the diffusion coefficient, b ( ε ) the energy loss term, and q the source term.The diffusion term is considered homogeneous in space with a energy dependence: D ( ε ) = K (cid:18) εε (cid:19) δ , (6)where K and δ are phenomenological parameters inspired by models of the interstellarmedium based on magneto hydrodynamics. ε is a normalization energy scale, here fixedat the value of 1 GeV.Energy losses are related to interactions of electron and positrons with the interstel-lar radiation fields, the cosmic microwave background, and the galactic magnetic field.Commonly, this term is: b ( ε ) = ε τ E (cid:18) εε (cid:19) , (7)where τ E (= 10 s) corresponds to an effective energy loss time calculated via the inverseCompton scattering with the radiation fields in the Thomson regime. However, this termis no longer valid for energies larger than m e / h ε γ i , where h ε γ i is the radiation field en-ergy, and it has to be corrected considering the Klein-Nishina cross section [21, 22, 23].In Figure 4 , we present the energy loss term for the different components of theradiation field and those are compared to the standard assumption of the Thomsonregime (
Equation 7 ). The figure is from our recent work [24] that follows the line of theAuthor’s thesis. .3. Transport equation solution . – We solve the differential eaution by meansof Green functions which correspond to the solution for a point–like source: q ( t, ~x, ε ) = δ ( t − t s ) δ ( ε − ε s ) δ ( ~x − ~x s ) , (8) OSITRONS FROM COSMIC RAYS INTERACTIONS AND DARK MATTER ANNIHILATIONS −1 − E − × ( b ( E ) [ G e V / s ] ) Energy [GeV]
Delahaye, Lavalle, Lineros, Donato & Fornengo (2010) µ G3 µ GStandardModel 1Model 2CMBCMB + Synchrotron
Figure 4. – Energy loss term 10 ε − b ( ε ) versus electron (positron) energy. The energy loss fordifferent radiation fields models and the standard assumption are shown where t s , ε s , and ~x s correspond to the injection time, energy, and position.The green function is easily obtained through the Fourier transformation method [25,1] and it has an analytical form [24]: G ( t, t , ~x, ~x s , ε, ε s ) = δ ( t − t s − τ c ) e G ( ~x, ~x s , λ d ) b ( ε )(9) = δ ( t − t s − τ c ) G ss ( ~x, ~x s , ε, ε s ) , where e G is the tilded green function : e G ( ~x, ~x , λ d ) = 1 π / λ d exp (cid:18) − ( ~x − ~x ) λ d (cid:19) , (10)and it is proportional to the Green function ( G ss ) in the time independent case (steady-state regime).The terms τ c and λ d are rich in physical meaning and come naturally from the solution. τ c is the cooling time which corresponds to the time lapses for an electron or positron toreduce its energy from ε s to ε : τ c ( ε, ε s ) = Z ε s ε dǫ /b ( ǫ ) . (11)In the same way, λ d is the diffusion length which gives a scale for the propagation and ROBERTO ALFREDO LINEROS RODRIGUEZ is defined by: λ d ( ε, ε s ) = 4 Z ε s ε dǫ D ( ǫ ) /b ( ǫ ) . (12)The former two quantities become fundamental to describe the propagation. Also thosecan be calculated for any type of diffusion coefficient and energy loss term allowing usto explore new forms, like: deviations from the standard diffusion coefficient and extraenergy loss processes.With the Green function, we are able to find the solution for the transport equationfor any generic source Q . The solution is found by convoluting the source with the greenfunction in the following way: ψ ( t, ~x, ε ) = Z t −∞ dt s Z ∞ ε dε s Z d x s G ( t, t s , ~x, ~x s , ε, ε s ) × Q ( t s , ~x s , ε s ) , (13)where the integration limits are set to consider only the physical contribution from thesource Q .As we previously said, the model considers that cosmic rays may escape from thepropagation zone when they reach its boundaries. This means: z = ± L ∨ p x + y = R = ⇒ ψ ( t, ~x, ε ) = 0 , (14)these conditions just affect the tilded Green function ( Equation 10 ).There are two standard approaches to impose the boundary conditions: • Eigenfunction expansion: The transport equation is solved using a complete setof Helmholtz equation eigenfunctions ( χ g ) that naturally respect the boundaryconditions. The green function is analytical in some cases and correspond to thesum on eigenvalues ( g ): e G bc ( ~x, ~x s , λ d ) = X g χ † g ( ~x s ) χ g ( ~x ) exp (cid:18) − g λ d (cid:19) . (15)We use the Fourier–Bessel expansion, where harmonics functions are used for ver-tical coordinate and first kind Bessel functions for the radial one. The method isvery well behaved and fast to calculate when λ d is rather big [26, 27, 1]. • Method of Images: Also known as Method of Inversion, it consists on adding counterterms or mirror sources that compensate the effect of the original source, preservingthe boundary condition.
OSITRONS FROM COSMIC RAYS INTERACTIONS AND DARK MATTER ANNIHILATIONS For the vertical coordinate, we include an infinite number of mirror sources trans-forming the green function into a sum: e G vertical bc ( ~x, ~x s , λ d ) = ∞ X n = −∞ ( − n e G ( ~x, ~x s,n , λ d ) , (16)where ~x s,n = ( x s , y s , Ln + ( − n z s ) and e G is the free space tilded green func-tion ( Equation 10 ). This form is very efficient for small values of λ d and complementsthe expansion in harmonic functions.The Green function that respects radial boundary conditions is obtained by addjust one extra counter term: e G radial bc ( ~x, ~x s , λ d ) = e G ( ~x, ~x s , λ d ) − e G ( ~σ, ~σ s , λ d )(17)where ~σ = ( βx, βy, z ), ~σ s = ( x s β , y s β , z s ) and β = x s + y s R .We remark that this Green function works for all values of λ d and does not needany kind of special symmetry like the Bessel expansion. .4. Space of Parameters of the model . – The two–zone propagation modelis also used for studying nuclei cosmic rays. Previous studies on the ratio Boron andCarbon (B/C) constraint the space of parameters to a volume which is fully compatiblewith current observations in many other species. We use the allowed space of parametersto study positron and electron cosmic rays consistently and systematically with respectto cosmic rays propagation (details in [20, 28]).
4. – Positron cosmic rays from dark matter annihilation
Secondary positrons and electrons are produced in the Milky Way from the interac-tion of nuclei cosmic rays on the interstellar gas [29] and are an important tool for thecomprehension of cosmic-ray propagation [20]. Data on the cosmic positron flux (oftenreported in terms of the positron fraction) have been collected by several experiments[30, 31, 32, 33, 34, 35].We point out the HEAT balloon experiment [30] that has mildly indicated a possiblepositron excess at energies larger than 10 GeV with respect to the current – at thatmoment – calculations for the secondary component [29]. In October 2008, the latestresults of PAMELA experiment [36] have confirmed and extended this feature [37].Different astrophysical contributions to the positron fraction in the 10 GeV regionhave been explored [30], but only accurate and energy extended data could confirm thepresence of a bump in the positron fraction and its physical interpretation. Alternatively,it has been conjectured that the possible excess of positrons found in the HEAT datacould be due to annihilation of dark matter in the galactic halo [25, 3]. Although, thisinterpretation is limited by uncertainties in the halo structure and in the cosmic rays ROBERTO ALFREDO LINEROS RODRIGUEZ
Halo model α β γ r s [kpc]Cored isothermal [38] 2 2 0 5Navarro, Frenk & White [39] 1 3 1 20Moore [40] 1.5 3 1.3 30 Table
II. –
Dark matter distribution profiles in the Milky Way. propagation modeling.This section is based on our work [26]. We study the propagation of the positronsrelated to dark matter annihilations in connection with the study of the uncertaintiesdue to propagation models compatible with B/C measurements [20]. .1. dark matter annihilation like source of positrons . – According to thevarious supersymmetric theories, the annihilation of a dark matter particles points tothe direct production of an electron-positron pair or to the production of many speciessubsequently decaying into photons, neutrinos, hadrons and positrons.In our study, we consider four possible annihilation channels which would appear in anymodel of dark matter. The first corresponds to direct production of a e + e − pair and it isbetter motivated in theories with universal extra-dimension [41, 42, 43]. We alternativelyconsider annihilations into W + W − , τ + τ − and b ¯ b pairs.For any annihilation channel, the source term is written as: q dm ( ~x, ε e ) = η h σv i (cid:18) ρ ( ~x ) m χ (cid:19) dndε e , (18)where the η is a quantum coefficient which depends on whether the particle is or not itsown antiparticle. h σv i corresponds to the thermally averaged annihilation cross section,its value depends on the specific supersymmetric model and it is also constrained by cos-mology [44, 45]. We have actually taken here a benchmark value of 2 . × − cm sec − which leads to a relic abundance of Ω χ h ∼ .
14 (in agreement with the WMAP obser-vations [46, 47]). The dark matter mass value ( m χ ) is still unknown but for the case ofneutralinos, theoretical arguments as well as the LEP and WMAP results constrain itsmass to range from a few GeV [48, 49, 50, 51] up to a few TeV. Keeping in mind thepositron excess, we explore the cases with a dark matter mass of 100 GeV and 500 GeV. dn/dε e represents the multiplicity distribution of electrons (positrons) per single an-nihilation (details in [1]).The astronomical ingredient on the source term ( Equation 18 ) is the dark matterdistribution ρ ( x ) inside the Milky Way halo. We have considered the generic profile: ρ ( r ) = ρ ⊙ (cid:16) r ⊙ r (cid:17) γ (cid:18) r ⊙ /r s ) α r/r s ) α (cid:19) ( β − γ ) /α , (19)where r ⊙ = 8 . r denotes the radius in spherical coordinates. The local dark matter density has been set OSITRONS FROM COSMIC RAYS INTERACTIONS AND DARK MATTER ANNIHILATIONS equal to ρ ⊙ = 0 . − [52]. We discussed three profiles: an isothermal cored dis-tribution [38] for which r s is the radius of the central core, the Navarro, Frenk and Whiteprofile [39] (hereafter NFW) and Moore’s model [40]. However, some studies point outto that the central cusp may be less favorable in spiral galaxies and proposed a universaldark matter distribution [53, 54]. The NFW and Moore profiles have been numericallyestablished thanks to N-body simulations. In the case of the Moore profile, the index γ lies between 1 and 1.5 and we have chosen a value of 1.3 ( Table II ) which is morerepresentative.The possible presence of dark matter substructures inside these smooth distributionsenhances the annihilation signal by the so–called boost factor, although the boost factorvalue is still open to debate [55, 56, 57]. It has recently been shown that the boost factordue to substructures in the dark matter halo depends on the positron energy and onthe statistical properties of the dark matter distribution [58]. In addition, it has beenpointed out that its numerical values is quite modest [59], being of the order of 10–30. .2. the positron flux and fraction . – Using the source term ( Equation 18 ), wecalculate the propagated positrons flux for many cases (
Figure 5 ): • Annihilation channels: The nature of dark matter fixes somehow the annihilationchannel. We observe how channels – like the direct production one – produceharder spectra than channels involving quarks because positron are produced withenergy equal to dark matter particle mass. This is directly related to the multiplic-ity distributions. Positrons in quark–antiquark channels come from hadronizationprocesses and produce softer spectra with behavior similar to a power law: ε − ± . e .On the other hand, channels involving muons or gauge boson W ± produce harderspectra since positrons are produced at earlier stages, in decay chains of the originalparticles, taking a big fraction of the available energy. • Propagation uncertainties: Different propagation scenarios have different impacton positron generated by dark matter annihilation. In
Figure 5 , we present theuncertainty band associated to the B/C analysis. We notice that the size of theband depends on the annihilation channel. The direct production case presentsthe largest uncertainty band at low energies because the low energy flux is theresult of far-away propagated positrons and not from the very local ones. Thisalso explains why at energies closer to 100 GeV, the uncertainty band is smallerconverges to an unique curve. Other channels are less affected because very localproduced positrons are not affected by the propagation parameters.Let us stress that the positron flux obtained from annihilation of dark matter is notthe only one. We need to consider other astrophysical components. Due to the nature ofastrophysical processes, positrons are dominated by a secondary component, i.e. thoseare created from the interaction of nuclei cosmic rays with the interstellar gas.To study the behavior of the positron signal, we include the secondary positron com-ponent and the electron flux from parametrized fluxes [25].In
Figure 6 we present the effect of the annihilation channel and propagation uncertaintieson the positron fraction. We observe how channels with harder spectra are more suitable ROBERTO ALFREDO LINEROS RODRIGUEZ −7 −6 −5 E F e + [ G e V c m − s − s r − ] Direct production
T. Delahaye, R. Lineros, N. Fornengo, F. Donato & P.Salati (2007) bb − channel B/C best fitM1 fluxM2 fluxuncer. band10 −7 −6 −5 E F e + [ G e V c m − s − s r − ] Positron energy [GeV] W + W − channel Positron energy [GeV] t + t − channel NFW Halo profile (r s = 20 kpc) < s v> = 2.1 · −26 cm s −1 m c = 100 GeV Figure 5. – Positron flux E Φ e + versus the positron energy ε , for a dark matter particle with amass of 100 GeV and for a NFW profile (Table II). The four panels refer to different annihilationfinal states: direct e + e − production (top left), b ¯ b (top right), W + W − (bottom left) and τ + τ − (bottom right). In each panel, the thick solid [red] curve refers to the best–fit choice (MED)of the astrophysical parameters. The upper [blue] and lower [green] thin solid lines correspondrespectively to the astrophysical configurations which provide here the maximal (M1) and min-imal (M2) flux. The numerical values of these configuration are defined in [26]. The colored[yellow] area features the total uncertainty band arising from positron propagation. to explain the positron excess. The b ¯ b case is less favorable because most of the positronsare at low energy, making impossible to reproduce the observations.The uncertainties of propagation are sizable with respect to the signal from dark matter,however the impact is not enough to destroy some features arising from the annihilationchannels.
5. – Secondary positron flux at Earth
The secondary component is the result of the interaction of cosmic rays nuclei withthe interstellar gas composed mainly by hydrogen and helium. We model the interstellargas assuming a homogeneous disk with radius equal to the galactic one and thickness of200 pc (details in [1]).
OSITRONS FROM COSMIC RAYS INTERACTIONS AND DARK MATTER ANNIHILATIONS P o s it r on fr ac ti on e + / ( e + + e − ) Direct prod.
Boost factor = 10
NFW Halo profile (r s = 20 kpc) < s v> = 2.1 · −26 cm s −1 Bkg. factor = 1.1m c = 100 GeV T. Delahaye, R. Lineros, N. Fornengo, F. Donato & P.Salati (2007) bb − channel Boost factor = 50
B/C best fituncer. bandbackground0.010.10 10 P o s it r on fr ac ti on e + / ( e + + e − ) Positron energy [GeV] W + W − channel Boost factor = 30
Heat 2000AMS Run 1AMS Run 2 10 Positron energy [GeV] t + t − channel Boost factor = 40
Heat 2000MASS−91CAPRICE94
Figure 6. – Positron fraction e + / ( e − + e + ) versus the positron detection energy ε . Notationsare as in Figure 5. In each panel, the thin [brown] solid line stands for the background [25,29] whereas the thick solid [red] curve refers to the total positron flux where the signal iscalculated with the best–fit choice (MED) of the astrophysical parameters. Experimental datafrom HEAT [30], AMS [32, 33], CAPRICE [34] and MASS [35] are also plotted. The source term of secondary electrons and positrons coming from interaction ofproton cosmic rays with hydrogen is: q p H ( x , ε e ) = 4 π n H ( x ) Z dε p Φ p ( x , ε p ) d h ησ i p H dε e ( ε p , ε e ) , (20)where n H is the hydrogen number density, Φ p is the proton flux (details in [1]), and h ησ i represents the inclusive cross section of the process p + p → e + ( e − ) + X discussedand calculated in Section . Let us remained that the inclusive cross section, in our case,comes from the invariant cross section parameterizations of Badhwar et al. [9], Tan andNg [12], and Kamae et al. [14].Let us stress that the complete positron and electron source term is the sum of all ROBERTO ALFREDO LINEROS RODRIGUEZ
Figure 7. – Interstellar electron and positron fluxes ε Φ e for the MED set versus energy. Thecurves correspond to fluxes calculated from Nuclei CR interactions with the ISM. Each curverepresents a different nuclear cross section and Nuclei CR flux parameterization. Also theStrong et al. [25] flux parameterizations are shown. The uncertainty band related to thoseparameterization is plotted (yellow band) as well. contributions from proton, alpha particles, hydrogen and helium: q full ( x , ε e ) = 4 π X i =H , He X j = p,α n i ( x ) Z dε j Φ j ( ε j ) d h ησ i ji dε e ( ε j , ε e ) , (21)where the inclusive cross sections for processes p +He, α +H and α +He are estimated byscaling proton–hydrogen cross section: d h ησ i ij dε e ( ε j , ε e ) = s f d h ησ i p H dε e ( ε j A j , ε e ) , (22)where A j is the mass number of the incident particle and s f are scaling factors [60, 61]. .1. secondary positron and positron fraction . – The propagation of sec-ondary positrons is realized according to the two-zone propagation model (section Section ).Similar to the case of annihilation of dark matter, we study the effects of uncertaintiesrelated to nuclear physics and propagation. In Figure 7 , we present secondary electronsand positrons obtained from the three nuclear parameterization: Badhwar et al. [9], Tanand Ng [12] and Kamae et al. [14]. We observe that at high energy ( > OSITRONS FROM COSMIC RAYS INTERACTIONS AND DARK MATTER ANNIHILATIONS −5 −4 −3 −2 −1 −1 ε . Φ e + [ c m − s − s r − G e V − ] Positron energy [GeV]
Positron flux: Donato + Tan&Ng n H = 0.9 [cm −3 ] − n He = 0.1 [cm −3 ] Φ F = 600 MV MS98
Lineros’ PhD Thesis (2008) −5 −4 −3 −2 −1 −1 ε . Φ e + [ c m − s − s r − G e V − ] Positron energy [GeV] TOAIS10 −5 −4 −3 −2 −1 −1 ε . Φ e + [ c m − s − s r − G e V − ] Positron energy [GeV]HEAT 94+95CAPRICE94MASS91AMS−Run1AMS−Run210 −5 −4 −3 −2 −1 −1 ε . Φ e + [ c m − s − s r − G e V − ] Positron energy [GeV] 10 −5 −4 −3 −2 −1 −1 ε . Φ e + [ c m − s − s r − G e V − ] Positron energy [GeV]MS98
Positron flux: Shikaze + Tan&Ng n H = 0.9 [cm −3 ] − n He = 0.1 [cm −3 ] Φ F = 600 MV Lineros’ PhD Thesis (2008) −5 −4 −3 −2 −1 −1 ε . Φ e + [ c m − s − s r − G e V − ] Positron energy [GeV] TOAIS10 −5 −4 −3 −2 −1 −1 ε . Φ e + [ c m − s − s r − G e V − ] Positron energy [GeV]HEAT 94+95CAPRICE94MASS91AMS−Run1AMS−Run210 −5 −4 −3 −2 −1 −1 ε . Φ e + [ c m − s − s r − G e V − ] Positron energy [GeV]
Figure 8. – Secondary positron flux ε . Φ e + versus positron energy. Positron fluxes were calcu-lated using proton and alpha CR fluxes from Donato et al. [62] and Shikaze et al. [8] with theTan et al. cross section parameterization [12]. experimental data is in this range. At low energy, we observe how differences on the nu-clear parameterization affect the propagated positrons and electrons. Nevertheless, thiseffect is not so important because it occurs in the energy range where solar modulationdominates.Moreover, we can not forget the presence of the uncertainties related to propagation. Figure 8 shows the secondary positron flux versus positron energy. We consider the wholespace of parameter compatible with the B/C analysis [20], obtaining that the estimatedflux encompasses the data. This confirms the compatibility among electron–positroncosmic ray propagation with nuclei cosmic rays case from the propagation point of view.
Figure 9. – Positron fraction as a function of the positron energy, for a soft and hard elec-tron spectrum. Data are taken from CAPRICE [34], HEAT [30], AMS [33], MASS [35] andPAMELA [37]. ROBERTO ALFREDO LINEROS RODRIGUEZ
The natural extension of the analysis is to calculate the positron fraction. Usingthe calculated secondary flux and the total electron flux estimated from observation likeAMS, we observe how the positron fraction is highly sensitive to small variation in theelectron flux (
Figure 9 ). We find that harder spectra makes stronger the evidence ofpositron excess, however a softer electron spectra makes possible to explain the raise inthe faction using mainly secondary positrons.This study points towards a more detailed study on the electron flux which has beenrecently addressed in Ref. [24].
6. – Conclusions
During the last years, many of the latest cosmic rays experiments have shown veryinteresting results that are pushing to new frontiers the knowledge about the galacticenvironment and the origin of the GeV–TeV cosmic rays. And of course, the case ofelectron and positron cosmic rays is not an exception.Dark matter annihilation is a very exciting possibility to explain the positron excess,which is not sufficiently explained by contributions from secondary positrons. In ourworks, we studied propagation uncertainties associated to the analysis made on cosmicrays nuclei (B/C). These uncertainties affect considerably the signal coming from anni-hilation of dark matter, however, most of the characteristics related to the annihilationwere not significantly modified. This fact is promising for further analysis on the signal.To go deeper in the analysis, we studied the secondary production of positrons. Weconsidered uncertainties related to nuclear physics in addition to the ones related to prop-agation. One of the first results was that secondary positron production is compatiblewith current measurements and with the B/C analysis, which is remarkable consideringthat we proposed that cosmic ray propagation is common for all the species.In both studies, the positron fraction was analyzed. Dark matter annihilation sce-nario is able to explain it, although, the lack of precision from the theoretical point ofview makes hard to identify an exotic component present in it. Moreover, the positronfraction is very sensitive to variations in the electron flux.We stress the necessity to re-estimate secondary and primary electron component,and to consider already known sources, especially pulsars, in order to discard/confirmpossible presence of an undiscovered component, like the case of dark matter annihila-tions. ∗ ∗ ∗
R.L. acknowledges the Comisi´on Nacional de Investigaci´on Cient´ıfica y Tecnol´ogica(CONICYT) of Chile for Ph.D. scholarship N ◦ BECAS-DOC-BIRF-2005-00, the Inter-national Doctorate on AstroParticle Physics (IDAPP), and the Istituto Nazionale diFisica Nucleare (INFN) for the Fubini award. Work supported by research grants fundedjointly by Ministero dell’Istruzione, dell’Universit´a e della Ricerca (MIUR), by Universit`a
OSITRONS FROM COSMIC RAYS INTERACTIONS AND DARK MATTER ANNIHILATIONS di Torino (UniTO), by Istituto Nazionale di Fisica Nucleare (INFN) within the Astropar-ticle Physics Project, by the Italian Space Agency (ASI) under contract N ◦ I/088/06/0.
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