Understanding the cosmic ray positron flux
UUnderstanding the cosmic ray positron flux
Paolo Lipari ∗ INFN, Sezione Roma “Sapienza”, piazzale A.Moro 2, 00185 Roma, Italy
Recent precision measurements of the flux of cosmic ray positrons by the Alpha Magnetic Spec-trometer show that the spectrum has a marked softening feature for energies close to one TeV. Apossible interpretation of this result is that the observed feature measures the maximum energy ofa new hard source of positrons perhaps associated to dark matter self–annihilation or decay, or topositron accelerators. A gradual hardening of the positron flux centered at E (cid:39)
25 GeV can alsobe understood as the signature of the transition where the new source overtakes the conventionalcomponent due to secondary production. This interpretation is simple and attractive, but it is notunique. The alternative possibility, that the positron flux is entirely of secondary origin, remainsviable. In such a scenario the spectral softening observed by AMS for positrons is generated byenergy loss effects, and a feature of similar, but not identical structure should be also visible in the e − spectrum. Spectral features similar to both the hardening and softening of the positron flux arein fact observed for electrons and call for a consistent explanation. Precision measurements of the e + and e − spectra in the TeV and multi–TeV energy range are crucial to clarify the problem. PACS numbers: 98.35Gi,95.85Pw,95.85Ry
I. INTRODUCTION
The cosmic ray (CR) positrons flux is of great importance for High Energy Astrophysics because it can be a probeto investigate the possible existence of Galactic dark matter in the form of Weakly Interacting Massive Particles,and of astrophysical antimatter accelerators. The shape of the e + spectrum gives also very valuable information todetermine the properties of propagation of CR particles in the Milky Way.Recently the AMS Collaboration has released new data on the positron spectrum [1] that extend the measurementsto a maximum energy of 1 TeV. In this paper we study the shape of the e + spectrum, compare it to the spectra ofother particles (in particular e − and p ) and discuss possible interpretations of the observations.The new AMS data on positrons are shown in Fig. 1 (plotted together with data on electrons) and in Fig. 2 [togetherwith measurements of the fluxes of p and and ( e − + e + )]. The AMS Collaboration in [1] has fitted the positron datausing the functional form φ e + ( E ) = C (cid:18) EE (cid:19) − γ + C s (cid:18) EE (cid:19) − γ s e − E/E s (1)(with E an arbitrary energy scale) modified by solar modulations described by the force field approximation (FFA)[2]. With this assumption the directly observable flux takes the form φ obs e + ( E ) = φ e + ( E + ϕ ) E ( E + ϕ ) , (2)with ϕ a time dependent parameter with the dimension of energy.In the expression of Eq. (1) the e + spectrum is described as the sum of two distinct components (that are also shownin Fig. 1 together with their sum). The first component has a simple power law form with (best fit) spectral index [3] γ = 4 .
07 and dominates at low energy. The second component is a harder power law with spectral index γ s (cid:39) . E (cid:38)
20 GeV. At high energy this second component (and therefore the observable flux)has also a marked softening feature that is described as an exponential cutoff [4]. The parameter E s is determinedwith a large error E s = 810 +310 − GeV.The functional form of Eq. (1) (with identical modeling of solar modulation effects, but without the high energycutoff) had already been used in [5] to fit the data on the positron and electron spectra previously released byPAMELA [6, 7] and AMS [8], with results that for the positrons are in very good agreement with those presented ∗ Electronic address: [email protected] a r X i v : . [ a s t r o - ph . H E ] F e b (cid:230) (cid:230) (cid:230) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230)(cid:230) (cid:230) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:230) (cid:230) (cid:230) (cid:230) e (cid:43) e (cid:45) (cid:72) (cid:137) E (cid:76)(cid:72) (cid:137) E (cid:76) (cid:64) GeV (cid:68) Φ e (cid:43) (cid:72) E (cid:76) E . (cid:64) G e V . (cid:144) (cid:72) m ss r (cid:76) (cid:68) Φ e (cid:45) (cid:72) E (cid:76) E . (cid:144) (cid:64) G e V . (cid:144) (cid:72) m ss r (cid:76) (cid:68) FIG. 1:
Spectra of CR positrons [1] and electrons [8] measured by AMS02. The fluxes are plotted as a function of energy in the form E . φ ( E ) for positrons and E . φ ( E ) for electrons. The lines are two components calculated by the AMS Collaboration in [1] forpositrons and in [5] for electrons. in [1] (the best fit spectral indices for the two components γ (cid:39) .
99 and γ s (cid:39) . e + spectrum require the existence of a new positronsource in addition to the conventional mechanism of secondary production, where the positrons are created in theinelastic collisions of CR protons and nuclei with interstellar gas. In this interpretation the contribution of the new,non–conventional source dominates the e + flux at high energy, and corresponds to the second, harder component inEq. (1). Its origin could be associated to the self–annihilation or decay of dark matter particles, or to astrophysicalpositron accelerator. Such a conclusion is of great importance and should be carefully scrutinized.Schematically, there are three observations (and three arguments associated to the observations) that give supportto the hypothesis of the existence of a non–conventional component in the positron spectrum.(A) In a broad energy interval ( E (cid:39) e + flux has a spectral shape that is significantly harder thanthe “conventional” prediction for positrons generated by the secondary production mechanism.(B) The flux exhibits a gradual hardening around the energy E h (cid:39)
25 GeV. This feature, in the fit of Eq. (1)corresponds to the transition where the hard component emerges overtaking the soft one.(C) The flux has a marked softening or cutoff at high energy, that on the AMS fit is described with an exponential factor e − E/E s with E s ∼
800 GeV. The existence of a cutoff in the positron source spectrum is predicted in models wherethe particles are generated by dark matter [where the spectrum has a maximum energy at E = m χ (for annihilation)or m χ / m χ the mass of the DM particle] and also in models where positron accelerators have asharply defined maximum energy.In the following we will review the three arguments listed above, and discuss if the observations can be reconciledwith the hypothesis that the positron flux is entirely of secondary production origin. II. SPECTRAL SHAPE
The commonly accepted (“standard”) prediction for the shape of the positron flux in the range E (cid:39) γ e + (cid:39) . φ j ( E ) (observed at the position of the solar system in theGalaxy) for particle of type j [ j ∈ { p, e − , e + , p, . . . } ] and its source spectrum Q j ( E ) (that is the rate of particlesinjected in interstellar space by all Galactic sources) can be written in the form: φ j ( E ) = c π Q j ( E ) τ j ( E ) V (3)In this equation the factor c/ (4 π ) transforms a particle density into an isotropic flux (assuming β (cid:39) V is aneffective Galactic confinement volume for CR particles and τ j ( E ) is a quantity with the dimension of time. Withoutloss of generality one can take V as independent from energy and particle type with all dependence contained intothe factor τ j ( E ). For a realistic choice of the volume V , the quantity τ j ( E ) has the physical meaning of the residencetime in the Galaxy for particles of type j and energy E .There are two main “sinks” for relativistic particles in interstellar space that can balance the injection of the sources:escape and energy losses [18]. Energy losses are only significant for e ∓ , so for protons, anti–protons and nuclei, thecharacteristic time τ j ( E ) can be identified with the escape time T esc . Since the containement of CR particles in theGalaxy is of magnetic nature, the escape time (for particle with β (cid:39)
1) is only a function of the particle rigidity p/q ,and in most realistic cases on the absolute value of the rigidity. A common way to parametrize the rigidity dependenceof the escape time is a power law [19] of exponent δT esc ( E ) = T E − δ . (4)For electrons and positrons radiative energy losses (that grow ∝ E /m ) can be important [20], and one has totake into account for the loss time (that is the time for an e ∓ to lose half of its energy): T loss ( E ) = E (cid:104)| dE/dt | e (cid:105) (cid:39) . (cid:34) . − (cid:10) ρ B + ρ ∗ γ (cid:11) (cid:35) (cid:20) GeV E (cid:21) Myr . (5)where ρ B and ρ γ are the energy densities in magnetic field and radiation averaged in the Galactic confinement volumefor CR. The characteristic time τ e ( E ) is in general a combination of the escape and loss times. The exact form of thiscombination is model dependent (see [5] for a discussion), one can however express it schematically in the form: τ e ( E ) = T esc ( E ) ∝ E − δ for E (cid:46) E ∗ ,T esc ( E ) ⊗ T loss ( E ) ∝ E − [1 ⊕ δ ] for E (cid:38) E ∗ (6)where E ∗ is the characteristic energy where the loss and escape time are equal ( T esc ( E ∗ ) = T loss ( E ∗ )).Eq. (6) expresses the fact that at low energy ( E (cid:46) E ∗ ) when the energy losses are negligible, the characteristic timefor e ∓ coincides with the escape time. At higher energy ( E (cid:38) E ∗ ) one has to combine the escape time ( T esc ∝ E − δ )and the loss time ( T loss ∝ E − ). The exact form of the combination is model dependent. For example [5] discussestwo examples where the E dependence of τ e ( E ) at high energy has the forms ∝ E − and ∝ E − (1+ δ ) / . Because ofthis model dependence the exponent of the energy dependence of τ e ( E ) in Eq. (6) is written in a formal, general form(1 ⊕ δ ).After the introduction of these generally accepted concepts, the standard prediction for the e + and p spectra isnow based on the following steps:(S1) The source spectra Q e + ( E ) and Q p ( E ) are calculated as the convolution of the spectrum of primary protons(and nuclei) with the appropriate cross sections: Q sec e + ( p ) ( E ) ∝ (cid:90) dE φ p ( E ) dσdE ( E, E ) (cid:12)(cid:12)(cid:12)(cid:12) pp → e + ( p ) . (7)Writing this expression we have assumed that most of the production of antiparticles happens in interstellar space(and not inside or near the CR accelerators) and that the spectra of the CR particles have the same shape in theentire Galaxy). Eq. (7) implies that at high energy (far from threshold effects) one has: Q sec e + ( p ) ( E ) ∝ E − γ p (8)with γ p the spectral index of the proton flux. Also the positron/anti–proton ratio is determined by the cross sectionsand takes a value of order Q e + ( E ) Q p ( E ) ≈ T esc ( E ) (the absolute scale T and the exponent δ that describes its rigiditydependence) are obtained from the comparison of the spectra of secondary (lithium, beryllium and boron) and primary(carbon and oxygen) nuclei. Secondary nuclei are created in the fragmentation of primary CR nuclei (in reactionssuch as C + p → B + . . . , with p a proton at rest). Schematically [21] one has: φ B ( E ) φ C ( E ) (cid:39) σ p C → B X ( E ) m p (cid:39) σ p C → B (cid:104) n ism (cid:105) c T esc ( E ) . (10)where X ( E ) is the grammage (that is the column density) of material crossed by a CR particle before escape, and (cid:104) n ism (cid:105) is the average density of the interstellar medium in the CR confinement volume. In Eq. (10) E is the energyper nucleon, and we have made use of the fact that in nuclear fragmentations E remains in good approximationconstant. From the study of the fluxes of secondary nuclei one can the derive an exponent δ (cid:39) . T esc (3 GeV) ≈
300 Myr.(S3) The estimate of the escape time obtained in (S2) can now be used to derive a value of the critical energy E ∗ that is below a few GeV. The energy dependence of τ e ( E ) for E (cid:38)
10 GeV takes then the form: γ e = γ p + (1 ⊕ δ ) (cid:39) . ÷ . γ p (cid:39) . γ p = γ p + δ (cid:39) . ÷ . . (12)The prediction of the positron (and antiproton) spectra described above while commonly accepted raises a numberof problems:(P0) It requires a new source of positrons “fine tuned” to reproduce the data.(P1) It is also in serious tension (if not in open conflict) with the data on antiprotons. The p spectrum (shown inFig. 2) has been measured by PAMELA [25] and with more precision by AMS [26] and for E (cid:38)
30 GeV, has a spectralindex γ p (cid:39) . p data with the hypothesis of a secondary production source [22], however, all predictions (obtained before the releaseof the AMS p data) estimated a spectrum softer than the data.(P2) The measurements of beryllium isotopes [23] suggest a CR lifetime shorter than the T esc ( E ) infered from thestudies of secondary nuclei.(P3) The spectra of electrons and positrons do not show clearly the expected softening feature associated to thecritical energy E ∗ and the transition to the regime where energy losses are important.(P4) The long lifetime estimated for the CR particles implies slow propagation. High energy ( E (cid:38) T esc ( E ). A different approach [5, 17] to study the problem isto use the data on positrons and antiprotons (that in the absence of new sources are entirely of secondary origin) todetermine the CR propagation properties and determine T esc ( E ). The attractive advantage in this approach is thatthe hypothesis of a secondary origin of e + and p can be tested comparing the spectra because in this case the twosource spectra are intimately related since they they depend on known cross sections [see Eq. (7)].Inspecting Fig. 2 it is striking that in the energy range E (cid:39) e + and p have approximatelythe same spectral index, and are close to each other in absolute value with a ratio e + /p ≈
2. These results are merecoincidences in the standard scenario, where the two spectra have completely independent sources, and are alsodistorted by propagation in different way. On the other hand, making use of Eqs. (8) and (9), one can see that theseobservations are consistent with the hypothesis that e + and p are both generated in the inelastic interactions of thesame population of primary particles, and that the effects of propagation generate distortions that are approximatelyequal. The study of the antiparticles at lower energy is also consistent with this hypothesis, because the observedrelative suppression of p with respect to e + for E (cid:46)
20 GeV (also evident in Fig. 2) can be understood as a theconsequence of the fact that the production of low energy antiprotons is suppressed for simple kinematical effects (atthreshold antiprotons are created in the laboratory frame with kinetic energy E p (cid:39) m p ).The assumption that e + and p are both generated as secondaries and have approximately equal propagationproperties implies that energy loss effects (for positrons) are small, and therefore the critical energy E ∗ is large( E ∗ (cid:38)
400 GeV), and the escape time is much longer that was is estimated in the standard model, and also variesmore slowly with rigidity [17] ( δ (cid:39) . e − and e + spectra should both have softening features around a critical energy E ∗ (cid:38)
400 GeV (seediscussion below in Sec. IV).
III. SPECTRAL HARDENING
The high precision AMS data [1, 8] show that the positron flux undergoes a gradual hardening in the energy intervalat E (cid:39) γ (cid:39) .
02) at E (cid:39)
11 GeVto an approximately constant value of order γ = 2 .
75 for E (cid:38)
50 GeV. It is possible to interpret this hardening as thetransition between two distinct components that are both of power law form but have different spectral indices, butthis interpretation is not unique. The alternative possibility is that the e + flux is dominated by a single component,but the source spectrum, or the propagation effects do not have a simple power law form.In this discussion, it is important to note that also the electron spectrum has a spectral hardening with a structurevery similar to what is observed for positrons. This point has been already discussed in [5], and is illustrated in Fig. 1,that shows together the e + and e − spectra measured by AMS. To help in the visualization and comparison of thetwo hardening features, the e ∓ spectra are plotted multiplied by different energy dependent factors. The positronspectrum is shown in the form E . φ e + ( E ) versus E , while the electron flux is shown in the form E . φ e − ( E ) versus E , rescaled by a constant factor 1/15. It is apparent that the deviations of the two spectra from a simple power lawform (that is a straight line in a log–log representation) have very similar forms. In fact, in the entire energy interval10–300 GeV the energy dependence of the ratio φ e + ( E ) /φ e − ( E ) is reasonably well described by a simple power law ∝ E +0 . , with the effects of the two hardenings canceling each other.Both spectra can be described as the sum of two components of power law form according to Eq. (1). In thiscase the hardening is centered at E h that is the energy where the contributions of the two components are equal( E h (cid:39) E ( C /C s ) / ( γ − γ s ) ). Fits to the electron and positron flux using the two component form (without highenergy suppression) have been presented in [5], and give very good description of the data. The fit to the e − spectrum, and the contributions two components are shown in Fig. 1, and compared to the results for positrons (fromthe AMS fit). It is rather striking that the “crossing” energies for the soft and hard components are approximatelyequal ( E e + h ≈ E e − h ), and also that the steps in spectral index across the hardening features (∆ γ ± (cid:39) γ ± − γ ± s ) are alsoclose to each other.For electrons the decomposition of the spectrum into two components is unexpected, its interpreration is thereforeproblematic, and it is not clear if one should accept the results of the fit as evidence for the real existence of twodistinct e − sources. It should be stressed that the hard components in the e − and e + spectra must have completelydifferent origins because in the case of electrons this term is one order of magnitude larger and significantly softerthan the corresponding one for positrons.An alternative (purely phenomenological) model to describe the hardenings in the e ± spectra adopted in [5] is a“break model”, where the spectrum (before solar modulations) is described by the form: φ ( E ) = K (cid:18) EE (cid:19) − γ (cid:34) (cid:18) EE h (cid:19) /w (cid:35) − ∆ γ w . (13)This functional form depends on 5 parameters { K , γ , E h , ∆ γ , w } and describes a spectrum that is the combinationof two power laws with a change in the spectral index around energy E h . The break can be either a hardening (for∆ γ <
0) or a softening (for ∆ γ > γ ( E ) = γ + ∆ γ (cid:104) E h /E ) /w (cid:105) − = γ + ∆ γ (cid:20) (cid:18) ln( E/E h )2 w (cid:19)(cid:21) (14)so that E h is the energy where the spectral index takes the average value γ + ∆ γ/
2, and w is a width parametersthat in good approximation (see more discussion in [24]) gives the interval in log E where one half of ∆ γ develops.The functional form of Eq. (13) includes the case of the combination of two separate components discussed above forthe special case where the width parameter takes the value w = 1 / | ∆ γ | .Using the form of Eq. (13) to describe the e − and e + spectra, one obtains that the best fit values for the threeparameters E h , ∆ γ and w that describe the second factor in the equation are approximately equal (taking intoaccount systematic uncertainties, also associated to the description of solar modulations). This is what is expected ifthe hardenings were spectral distortions generated by propagation effects that are common for electrons and positrons.In summary, the interpretation of the hardening in the positron spectrum as evidence for the existence of twocomponents in the spectrum is not unambiguous. The interpretation is viable, simple and attractive, but consistencywould then suggest that also the electron spectrum is formed by two components (perhaps associated to two differentclasses of accelerators). In this scenario the similar structure of the two hardenings (centered at approximately thesame energy and with approximately the same ∆ γ ) is a mere coincidence with no physical meaning.The alternative possibility is that the two hardenings in the e + and e − spectra are generated by the same physicalmechanism during propagation of the CR particles in interstellar space. This idea requires one single mechanism toexplain two different phenomena, however a realistic physical model for this mechanism has not yet been constructed. IV. HIGH ENERGY SUPPRESSION
In a discussion about the origin of the suppression at high energy (below 1 TeV) of the positron spectrum it isimportant to note that a marked softening feature has also been observed by several detectors in the all–electronspectrum [that is the spectrum for the sum ( e − + e + )] at an energy of order 1 TeV. It is obviously important to studythe relation between these features.Measurements of the all–electron spectrum up to a maximum energy of 1 and 2 TeV have been obtained by AMS[28] and Fermi [27]. These measurements are consistent with an unbroken power law. Other measurements that reachhigher energy have however observed a spectral suppression. The first evidence for the existence of this softening hasbeen obtained by ground based Cherenkov telescopes that can measure the spectra of high energy e ∓ selecting eventsthat are consistent with an electromagnetic shower, and subtracting the background generated of hadronic showersgenerated by CR protons and nuclei. A spectral break was first observed by HESS [29, 30]) and then confirmedby MAGIC [31] and VERITAS [32, 33]. The spectral suppression has then been confirmed by two calorimeters onsatellites in near Earth orbit CALET [34, 35] and DAMPE [36].Fig. 2 show measurements of the all–electron flux performed by several detectors. Inspection of the figure showssome discrepancies between the different data sets (presumably the consequence of systematic effects), however theexistence of a spectral suppression for E (cid:38) E (cid:39) γ low (cid:39) . γ high (cid:39) . φ ( e − + e + ) ∝ E − γ e − E/E c ), however the Collaboration hasalso shown that the data can be equally well fitted with the break form.Discussing the relation between the softening features observed in the e + and the ( e − + e + ) spectra a first importantpoint is that in this energy range the ( e − + e + ) is dominated by electrons generated in sources (or with mechanisms)different from those that generate positrons. To arrive at this conclusion one can observe that for E ∼ ∼
15% of the all–electron flux. The sources that generate positrons are expected to generate alsoelectrons, but it is very likely that the ratio e − /e + at the source is approximately unity [39] (or less for secondaryproduction since proton interactions generate more π + than π − ). The assumption that the positron sources injectequal populations of e + and e − in interstellar space, implies that these sources account for only approximately 30%of the all–electron flux (as illustrated graphically in Fig. 2 with the thin dashed line). The remaining 70% of the Ê ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ Ê Ê Ê Ê
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10 100 1000 10 @ GeV D f H E L E @ G e V ê H m ss r L D FIG. 2:
Spectrum of CR positrons plotted in the form E φ ( E ). The data points are from AMS02 [1]. The solid line (labeled “Exp”)is the fit given in the same paper. The thick dashed line (labeled “Break”) is a fit to the data performed using the functional form ofEq. (15) (see text). Also plotted is the AMS spectrum of antiprotons [26], and measurements of the ( e − + e + ) spectrum by FERMI [27],AMS02 [28], CALET [35], DAMPE [36], VERITAS [33] and HESS [37]. It should be noted that the VERITAS data points do not includesystematic uncertainties. The lines labeled (C, D, V and H) are the fits to the (( e − + e + ) data calculated in the papers of DAMPE,CALET, VERITAS, and HESS. all–electron spectrum at E ∼ E (cid:39) e − spectrum that is not related to the positrons. raises the question of why positrons and electrons(from a different class of sources) have both spectral suppressions at a similar energy.The energy spectrum for a given particle type ( e + , e − , . . . ) is determined [as illustrated in Eq. (3)] by the combinationof the source spectrum and of propagation effects encoded in the characteristic time τ j ( E ). The possible explanationsfor the the spectral suppressions observed in the e − and e + fluxes therefore fall into two classes:(i) The spectral suppressions in the e − and e + fluxes are associated to the properties of their sources. However, thesources and/or mechanism that generate the dominant components in electron and positron fluxes are different,and therefore the two suppressions must have a different origin.(ii) The spectral suppressions are associated to propagation. The diffusive propagation of e ∓ in interstellar spaceis in good approximation independent from the sign of the electric charge and therefore propagation effectsgenerate similar spectral features in the e ∓ spectra.It should be noted hat if the hypothesis (i) is correct, one has the problem to construct a model to describe thesuppression of the e − spectrum for E (cid:38) e − accelerators has a sharply defined maximum energy), but appears asa “break” with the spectrum reasonably well described by a power law in the energy range 1–10 TeV. On the otherhand a break–like softening structure is what is expected for a feature generated by energy losses.On the other hand, the unambiguous detection of an exponential (or sharper) cutoff in the positron flux cannot bereconciled with the hypothesis of a feature generated by energy losses, and therefore would exclude hypothesis (ii).However, conclusive evidence for such a sharp cutoff is still missing. This is illustrated in Fig. 2 that shows (as a thickdashed line) a fit to the data with using the expression: φ e + ( E ) = K E − γ (cid:34) (cid:18) EE h (cid:19) /w h (cid:35) − ∆ γ h w g (cid:34) (cid:18) EE s (cid:19) /w s (cid:35) − ∆ γ s w s (15)distorted by solar modulations in the FFA approximation. The expression in Eq. (15) is a power law with two breakstructures, the first one describes the hardening at E ≈
20 GeV, and the second one the softening below 1 TeV.The line in Fig. 2 is calculated with the parameters E s = 520 GeV, w s = 0 .
35 and ∆ γ s = 1 (but a large volumein parameter space yields fits of comparable quality), the other parameters ( K , γ , ∆ γ h , w h and ϕ ) are identical tothose listed in table I in [5]. The break form can provide a very good fit of the positron data in the region of thespectral suppression.As discussed in Sec. II the existence of softening features in the e + and e − spectra around the energy E ∗ whereenergy losses become important is an unavoidable prediction for all CR propagation models. In the “standard” modelsfor CR propagation (that require a new source of positrons) E ∗ is of order few GeV or less, and thefore solutions ofclass (ii) for spectral features at E ∼ e − and e + spectrabelow 20 GeV can be modeled [1, 5] as unbroken power laws only distorted by solar modulations.Models where positrons are of secondary origin put E ∗ at an energy of order one TeV, consistent with the energyof the observed spectral sppressions. The question is if the shapes of the spectral features observed in the e + and( e − + e + ) spectra are consistent with the hypothesis that they are generated by energy loss effects.Predictions for these energy loss effects have been extensively discussed in [5]. The spectral features generated bythe transition to a regime where energy loss are important are qualitatively described [see Eq. (6)] by a “break”,however their detailed shape is model dependent. The step in spectral index takes a value ∆ γ (cid:39) (1 ⊕ δ ) − δ that isbetween (1 − δ ) / − δ ), while for the width w , the models discussed in [5]) give values of order 0.3–1, indicatinga rather gradual transition.There are indications that the suppression in the positron spectrum reported by AMS is centered at lower energyand is more gradual than the feature visible in the all–electron flux. The statistical and systematic errors are howeverlarge, and there are also theoretical uncertainties associated to the fact that the features imprinted in the e − and e + spectra by energy loss effects are not predicted to be identical. Differences can develop for two reasons. The firstone is that the source spectra that are distorted by the propagation effects have different shapes. The second one isthat the space distributions of the e ± sources (that belong to different classes) can be different. For this reason atthis moment it is not possible to firmly exclude the possibility that the spectral suppressions are generated by energylosses during propagation. V. CONCLUSIONS
The interpretation of the CR positron flux is a problem of great importance for high energy astrophysics. The firstcrucial crossroad in the solution of this problem is to determine if the bulk of high energy positrons can be generatedby the conventional mechanism of secondary production, or if on the contrary it is necessary to have a new sourceof relativistic positrons. The first hypothesis requires a revision of some commonly accepted assumptions about CRpropagation in the Galaxy with profound consequences for our understanding of high energy phenomena. On theother hand the result of a new positron source would evidently be a discovery of major importance.Two features in the positron spectrum can play an important role in solving the problem of the origin of the positronflux. The first one is an hardening of the spectrum around E ≈
25 GeV that could be the transition from a lowenergy regime dominated by positrons created by the conventional mechanism to a high energy regime where mostof the particles are generated by the new source. The second one is a suppression at high energy that starts to bevisible for E of order few hundred GeV, and that could mark the maximum energy for e + created by the new source.A comparison of the positron data with the spectra of electrons suggest an alternative interpretation for the feauresin the e + spectrum.The existence of an hardenings with very similar structure in e − spectrum, in the absence of a convincing twocomponent model for the electron sources, suggests that scenarios where only the hardening of the positrons is takenas evidence for the existence of two spectral components could be incorrect.The e + and e − spectra also have both softening features around an energy of order E ∼ e + and e − spectra. This scenario is however in tension withthe indications that the positron spectral suppression develops at lower energy and is broader that the suppressionfor electrons.To clarify the situation it is clearly very desirable to have more data on the electron and positron spectra in the TeVand multi–TeV energy range. The extension to higher energy of the measurements (with magnetic detectors in space)of the separate e + and e − spectra are obviously of great interest. If this is not possible also new observations withhigher precision, better controlled systematic uncertainties, and broader converage at high energy of the ( e + + e − )spectrum would give very valuable information.These experimental studies could confirm or refute the hypothesis of a new positron source, and doing so also obtaincrucially important information to clarify the physical mechanisms that generate the e + and e − spectra. Acknowledgments.
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