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CERN-PH-TH/2009-007 Possible causes of a rise with energy of the cosmic ray positron fraction Pasquale D. Serpico Physics Department, Theory Division, CERN, CH-1211 Geneva 23, Switzerland (Dated: January 20, 2009) Based on general considerations rather than model-dependent fits to specific scenarios, we argue thatanincreasewithenergyofthepositronfractionincosmicrays,suggestedbyseveralexperiments at E >∼ 7GeV, most likely requires a primary source of electron-positron pairs. We discuss the possible alternatives, and find none of them plausible on astrophysical or particle physics grounds. Furtherobservational ways to test different scenarios are discussed. PACSnumbers: 98.70.Sa 9 0 I. INTRODUCTION 1 0 PAMELA 2 AMS-01 Sincelongtime,thestudyofthepositron/electronratio HEAT 20 Jan itpPmnoaAirsccMtsooiiscnoEmlnseLtsirAicnaiirnnsJaayuttthsnheeeelhliapoGtserfaob2lddae0euxe0tcny6etc,irtaoieosnncrdod,agenwinsndiihgziptnechrdheoedpabSsateooggalaanamtnriieomaSintsypsuosofrttreehetmarn(enae.ertm-gyTtoeeotnhaoigecrl n fraction, f(E) 0.1 SεM=ε=0 ’0.93.825 h] ototh2e7r0cGomeVpoannednetsle)ctthreonsspeucptrtaoo2fTceoVsm,eicacrhaywpitohsiutrnopnrsecuep- positro p dented precision [1]. Recently, the PAMELA collabora- - p tionhaspresentedfirstresultsofthemeasurementofthe e positron fraction in the cosmic ray spectrum, which ap- 0.01 h pears to begin climbing quite rapidly between ∼ 7 GeV 1 10 100 [ and 100 GeV [2]. A similar trend was in fact also in- E [GeV] 3 dicated by earlier experiments, including HEAT [3] and FIG. 1: The figure reports the positron fraction vs. energy v AMS-01 [4], although with lesser statistical significance measured by PAMELA [2], HEAT combined data [3] and 6 and over a smaller dynamical range. The behavior that 4 seems to emerge in the positron fractionis verydifferent AMS-01[4]. A“typical”predictionbasedonpurelysecondary 8 e+ oftenusedasabenchmarkintheliterature[5]andpower- 4 from that predicted for secondary positrons produced in lawcurvesEε passingthroughPAMELAdatumat6.83GeV . the collisionsofcosmicraynuclideswiththe inter-stellar withindexε=0.2andε=0.35arealsoreportedforillustra- 0 medium (ISM). The situation is summarized in Fig 1. tivepurposes. 1 While unaccountedsystematics inthe measurementsare 8 inprinciplepossible,wethink itis worthreviewingwhat 0 : kindofphysicsmayleadto suchanenergyspectrum; we no pairs are present. The term S represents the sec- v shall argue that by far the simplest and most likely (as- ondary component of e+ produced in hadronic cosmic i X tro)physicalinterpretationisthatanadditional,primary ray collisions in the ISM. Note that an analogous term source of high energy positrons exists. exists for e− as well, which we denote S −D: there is r a The positron fraction is defined as indeed a small deficit D of secondary e− compared to secondarye+ duetochargeasymmetryinthecosmicray Φ 1 f(E)≡ e+ = , (1) and ISM nuclei population, which is proton-dominated. Φe+ +Φe− 1+(Φe−/Φe+) Finally, we allow for a putative primary flux P of e−-e+ pairs,whichis putto zerointypicalpredictionsoff(E). where the fluxes Φ refer to the ones at the top of the i A P-term might be due for example to unaccounted as- atmosphere. Here and in the following, we shall keep trophysical accelerators as pulsars or to a more exotic implicitthedependenceoffluxesonenergyE. Were-cast source as dark matter (DM) annihilation. In terms of thefluxesintermsofphysicallymotivatedcontributions, these components, without loss of generality: −1 Φ = P +S, (2) A−D e+ f(E)= 2+ . (4) Φe− = A+P +S−D=Φe+ +A−D. (3) (cid:18) P +S(cid:19) The term A represents the component of primary elec- First,one trivialobservation: since fromparticle physics trons accelerated in addition to any e+−e− pairs; this we know that D ≥ 0, from the empirical datum f(E) < term includes (but is not necessarily limited to) primary 1/2 it follows that A ≥ D 6= 0. This is not surpris- electronsacceleratedinatypicalISMenvironmentwhere ing, since it is well known that accelerators of e− (i.e. 2 electrons only, not pairs) exist in nature. Shocks on the to Eq. (6) with b(E) → 0, which yields for the steady backgroundISMatsupernovaremnants(SNRs,themost state solution (∂Φ/∂t = 0) a spectrum ∝ E−γp−δ with, likelyacceleratorsofGalacticCosmicRays)aretheprime a priori, δ 6=δ . The index δ is constrainedfrom the nu- e candidate in that sense; for modern predictions of the clide ratio B/C to lie in the range 0.3÷0.6 (a review of electron spectrum and an overview of past publications, recentcosmicrayexperimentsandtheirinterpretationis see e.g. [6] and refs. therein. The low-energy behavior provided in [9]). The convolution of this spectrum with of f, typically found to decline from ∼ 0.1 to ∼ 0.05 up therelevantcross-sectionisthesourcetermforpositrons, to ∼ 5 GeV, has been measured since longtime. This q+ ∝ E−γp−δ assuming an energy-independent inelastic range is influenced by time and charge-dependent solar cross-section and inelasticity. For the electrons, one has modulation,whichislikelyresponsibleforthedifferences simply q− ∝ E−γe, plus a subdominant secondary con- amongexperiments. Weshallneglectinthefollowingso- tributionof similar magnitude andshape ofthe positron lar effects which are irrelevant in the high energy range one. The resulting spectra of primary electrons and we focus on here. On the other hand, growing evidence secondary positrons at the Earth are thus respectively has been collected in the recent years that f(E) might ∝E−α−, E−α+ whereα− =γe+ℓ,α+ =γp+ℓ+δ. Here be risingathighenergies,withthe latestPAMELAdata ℓ symbolically represents the steepening due to diffusion stronglyfavoringthisobservation. Whileoneshouldwait and energy losses of leptons. For example, when one for higher statistics and possibly an independent confir- can neglect energy losses, Φ ∝ q(E)τ(E) and ℓ = δ ; at e mation(in particularby AMS-02[7]), itis usefultoclas- sufficiently high-energy (TeV range) where energy-losses sify the possible (astro)physical mechanisms leading to dominate it is easy to see that Φ ∝ b(E)−1 dE′q(E′) such an effect, a task we discuss in Sec. II. We will con- and ℓ = 1. Independently of the value ofRℓ, in this clude that the only one appearing viable requiresP =6 0, simple model we end-up with (A/S) ∝ E−γe+γp+δ and whichwouldimplythediscoveryofanewclassofcosmic- thus Eq. (5) requires γ +δ −γ < 0. This condition p e ray sources (Sec. III). seems extremely hard to achieve, requiring wildly different (by >∼ 0.6) source spectral indexes for protons and electrons. This would contradict the standard theoreti- II. THE NECESSITY OF A PRIMARY calinterpretationofthespectraldifferencebetweenpand SPECTRUM OF e+-e− PAIRS e− observed at the Earth as due to different energy-loss properties,ratherthanintrinsicones. Alsonote thatthe A. Basic Arguments condition γ +δ −γ < 0 could not hold down to low p e energies, since the the sign of f′ is negative around GeV To prove the statement in the title of this Section, we energies. So, one should also invoke some spectral break shall adopt a “reductio ad absurdum” approach. Let us in the injection electron spectrum placed ad hoc in the note that f′ ≡df/dE >0 implies ∼7GeVrange. Insummary,insistinginthe priorP =0 andrequiringthus(A/S)′ <0seemstoimply: i)thatour ′ A−D scenarios for the origin of Galactic electrons are wrong, <0. (5) (cid:18)P +S(cid:19) requiring in turn either new sources or new acceleration mechanisms different from the proton ones; ii) some de- Let us now consider the Ansatz P =0 and neglect D for greeof fine tuning, inthe sense that the energyatwhich the moment (we shall see that this is justified, actually f′ changessignwouldcorrespondtosomespectralbreak even a conservative assumption). Then, we should re- in the electron spectrum. These conditions appear way quire(A/S)′ <0toproducearise. Weseenowthatthis moreextreme thanallowingforprimarysourcesofpairs, requires highly implausible astrophysical conditions. for which candidates (both astrophysical and exotic) do In a very simple (position-independent) leaky-box exist in the literature. model, the master equation for leptons simplifies into [8] On the top of the above considerations, there are empirical arguments which appear to disfavor this hypoth- ∂Φ Φ ∂ ∂t =q(E)− τ(E) − ∂E [b(E)Φ] , (6) esis. Defining Φtot ≡Φe+ +Φe−, from Eq. (1) one has whereΦistheleptonflux,q(E)theinitial/injectionspec- trum, τ(E) ∝ E−δe an effective containment time, and Φtot(E1) = f(E2)Φe+(E1). (7) b(E) ≡ −dE/dt ≃ κE2 the energy-loss rate function, Φtot(E2) f(E1)Φe+(E2) which at the energies of interest (E>∼ 7GeV) is dominated by synchrotron radiation and inverse Compton Assuming only secondaries, the hardest spectrum the- scattering. Let us assume spectra at the injection of oretically possible for positrons derived in [10] goes as the typical power-law form ∝ E−γe, E−γp respectively ∼E−3.33 above10GeV.Inthesamepaperitisreported for electrons and for protons (which, being the domi- thatthesoftestpossiblespectrumforΦ fitting(poorly, tot nanthadroniccomponentofcosmicraysandISM,arethe at3 σ) the data compiledin [11]goes as∼E−3.54 in the main responsible for secondary leptons). Protons suffer samerange. Asaresult,themaximalgrowthpossiblefor virtually no energy losses and obey an equation similar the positron ratio—assuming only secondary positrons 3 and no priors for the electrons—is into a change of slope of a single species vs. energy, and ofdifferentspeciesfromoneanother. TheTRACERcol- ff((EE12)) <∼(cid:18)EE12(cid:19)0.2 . (8) lOabxyograetnioanndhaIsroinnstaeasdingrleepoprotwedert-lhaawt ifnodrenxu∼cle2i.7bectawneefint all the data in the GeV to TeV energy per nucleon, with AsillustratedinFig.1,thisisinsufficienttofullyexplain possible variations within ∼ 0.05 [15]. We take here the the rise suggested by the PAMELA data. More in gen- agnostic point of view that the gamma-ray “excess” is eral,thisargumentprovesthatabetterdeterminationof not understood at the moment (it might even be due to the electron energy spectrum might reveal an inconsis- a calibration problem, see [16]), but note that it may be tency ofthe f(E)andΦ data with apurely secondary consideredaswellasanindicationthatadditionalsources tot origin of e+, which does not resort to any theoretical exist contributing abovethe GeV range,whichis consis- considerations on the Φe− flux. tent with the hypothesis of additional primary emitters of cosmic ray positrons. Onemayfurtherwonderifarisingpositronratiomight B. Possible loopholes: further discussion be due to an unexpected energy-dependent beaviour of the diffusion index; from previous considerations and in the simplest case of γ = γ , it would follow indeed Althoughillustratedinasimpleleakyboxscenario,the p e (A/S) ∝ Eδ. For the sake of the argument and with conclusionthatP =6 0isrequiredappearsrobust. Asub- a slight abuse, let us assume δ to be “slightly” energy tlepointintheconsiderationsfollowingEq.(5)isthatwe dependent (this is not rigorous since the previous solu- reallyneedtocomparethespectrumoftheprimaryelec- tion has been derived for a constant δ.) The condition tronswiththeoneofcosmicraynucleiatmuchhigheren- ergies (a factor >∼20) since secondary leptons only carry (A/S)′ < 0 translates into the requirement that δ(E) declines with energy faster than 1/ln(E) in the 0.1-1 alimitedamountoftheparentnucleusenergy. Although TeV energy range of the parent nucleus producing the a concavity of the spectrum would be naturally accom- positrons. This is a relatively large effect, with δ drop- modated in non-linear acceleration models [12], there is ping by at least a factor ∼ 3 in a decade of energy to no evidence that the (well measured) proton spectrum account for the rising f(E). Even the proposal that δ presentsanoticeablechangeofslopearoundTeVenergy, changes from ∼0.6 above 10 GeV to ∼0.3 at TeV ener- certainly not at the level of ∆γ ≃0.5. p gies([6]andrefs. therein)appearsinsufficienttoaccount Another way around the previous conclusion may be for the sign of f′. On the other hand, this argument to consider a progressively rising role of Helium nuclei faces another difficulty: the featureless power-law of CR as source of secondaries. Still, at energies around the protons would result from a fine-tuned compensation of TeV one should require a flux of Helium nuclei compa- the variation of δ(E) and the injection spectrum, which rable to proton one and, at the same time, its spectral seems improbable, the two being unrelated. It is worth index harder than the proton one by an amount larger notinghoweverthateventhisbaroquescenarioistestable than δ. Actually, some indications of a hardening of He- empirically from high-energy B/C data. lium spectrum has been claimed by the ATIC-2 collaboration [13]. But its amount and the energy range where Anotherapproximationinthepreviousargumentisthe it happens appear insufficient to explain the beaviour of assumption of a constant cross-section. A rising inelas- f(E). Forexample,between200GeVand1TeVtheflux tic cross section would reflect in the secondary energy ratioΦp/ΦHe variesbyonly∼15%. Toexcludethispos- spectrum. Indeed, the inelastic cross section grows with sibility, however, it would be important to compare the energy (see e.g. [17]), but only logarithmically, by ∼1% positron fraction and p and He spectra measured with between10and100GeVandby ∼10%between0.1and the same instrument. Preliminary results by PAMELA, 1 TeV, i.e. equivalent at most to a power-law of index for example, do not support such an explanation since ∼0.04. Thisismorethanoneorderofmagnitudesmaller they show that both p and He fluxes are well fitted with than what needed to explain the positron feature. the same spectral index ≃2.73 up to ∼500 GeV [14]. Finally, let us come back to D, or better D/S. The In [5], the possibility was discussed that the average aboveconsiderationsareafortioritrueif(D/S)′ ≤0,i.e. interstellarprotonspectrummaybe harderthanthe one if the relative difference between positrons and electrons measured “locally” by an index ∼ 0.15, invoking both a remains constant or declines with energy. Note that better agreement with the diffuse gamma-ray spectrum this function is mostly dependent on particle physics, and the HEAT data on the positron fraction. Note that apart for the convolution with the primary cosmic ray even this ad hoc adjustment would be insufficient to ex- spectrum. If we take for example the e+ and e− yields plain the present evidence supporting f′ > 0. Still, as- for a proton power-law spectrum reported in Fig. 12 of suming that this argument is correct, one would expect Ref. [17]one canconclude that: i) the secondaryspectra two qualitative predictions: since the “collecting cosmic are slightly harder than the primary one, consistently rayvolume”intheGalaxydependsbothonprimarytype with the energy-dependence effect discussed above; andenergy(viadiffusionandspallationeffects),aspatial ii) the electron spectrum is slightly harder than the non-universalityofcosmic rayaccelerationshouldreflect positronone, i.e. D/S is slightly decreasingwith energy. 4 Although a detailed study would be required to assess summarized the signatures associated with different more quantitatively the uncertainties in this argument, explanations, and the way to test them observationally. it is clear that a sufficiently strong dependence of D/S Among the options listed which assume P = 0, the which might account for a rise in f(E) appears out of one coming closer to a (very bad) fit to the data is question. nr. 1), which is not only disfavored by the data, but requires an ad hoc adjustment, lacking at the moment an astrophysical model producing it. We concluded that the most likely cause of the energy trend of the positron fraction is the presence of a primary flux of e+, III. CONCLUSIONS which—both in astrophysical and exotic models—are probably injected in the form of pairs (see [18] for an In summary, motivated by observational evidence and earlyreviewofpossibleprimarysources). Acceptingthis inparticularbyPAMELAdata[2],wehavediscussedun- solution, one has at high energies f(E) ≃ (2 + AP)−1; der which conditions a rise in the positronfraction f(E) from a rise at E >∼ 7GeV one can further deduce that can take place. Barring the case of systematics in the the spectrum of pairs is harder than the one of primary measurements, we have analyzed the following hypothe- electrons, which is also a typical prediction in pulsar or ses: DM annihilation/decay models. In the opinion of the author, the positron spectral shape and normalization 1) “Anomalous” primary electron source spectrum. suggest pulsars as the most plausible responsible for the emission, a possibility which has drawn some attention 2) Spectral feature in the proton flux responsible for lately [19, 20, 21]. At very least, these objects should be the secondaries. seriously considered among the main actors of the high 3) Role of Helium nuclei in secondary production. energy Galactic sky; perhaps they are also responsible for most unidentified Galactic gamma ray sources, as 4) Difference betweenlocalandISMspectrumof pro- recently supported by the discovery by FERMI of a tons. pulsatinggamma-rayemissionfromtheSNRCTA1[22]. 5) “Anomalous” energy-dependent behaviour of the Acknowledgments The author would like to thank diffusion coefficient. P. Blasi for the initial suggestion to explore the topic discussed here, and P. Blasi and F. Donato for reading 6) Rising cross section at high energy. the manuscprit and useful comments. 7) High energy behavior of the e+/e− ratio of secondaries in pp collisions. All of the above options seem to be at least strongly disfavored if not already ruled out; nonetheless, we have [1] http://pamela.roma2.infn.it/index.php 481(2001) [2] O.Adriani et al.,arXiv:0810.4995. [13] A. D. Panov et al., arXiv:astro-ph/0612377. [3] J. J. Beatty et al., Phys.Rev.Lett. 93, 241102 (2004). [14] M. Boezio, Fermilab Joint Experimental-Theoretical [4] M. Aguilar et al. [AMS-01Collaboration], Phys.Lett. B Seminar Series, May 2nd 2008, available at 646, 145 (2007). http://theory.fnal.gov/jetp/ [5] I. V. Moskalenko and A. W. Strong, Astrophys. J. 493, [15] P. J. Boyle et al., in Proc. 30th ICRC, pp. 87-90, 694 (1998). http://www. icrc2007.unam.mx/proceedings/index; M. [6] T.Kobayashi,Y.Komori, K.YoshidaandJ.Nishimura, Aveet al., ibid., pp.215-218. Astrophys.J. 601, 340 (2004). [16] F.W.Stecker,S.D.HunterandD.A.Kniffen,Astropart. [7] http://ams.cern.ch/ Phys. 29, 25 (2008). [8] M. S. Longair, “High-energy astrophysics. Vol. 2: Stars, [17] T. Kamae, N. Karlsson, T. Mizuno, T. Abe and T. Koi, the galaxy and the interstellar medium,” Cambridge, Astrophys. J. 647, 692 (2006). UK: Univ. Pr. (1994); V. L. Ginzburg, V. A. Dogiel, [18] S. Coutu et al.,Astropart. Phys. 11, 429 (1999). V. S. Berezinsky, S. V. Bulanov and V. S. Ptuskin, [19] D. Hooper, P.Blasi and P. D.Serpico, JCAP 0901, 025 “Astrophysicsofcosmicrays,”Amsterdam, Netherlands: (2009). North-Holland (1990) [20] H.Yuksel,M.D.KistlerandT.Stanev,arXiv:0810.2784. [9] P. Blasi, arXiv:0801.4534, rapporteur Paper - OG1 ses- [21] S. Profumo, arXiv:0812.4457. sion of the 30th ICRC, Merida, Mexico, (2007). [22] G. Kanbach, K. Wood and M. Ziegler [The Fermi LAT [10] T. Delahaye et al.,arXiv:0809.5268. Collaboration], arXiv:0810.3562. [11] D.CasadeiandV.Bindi,Astrophys.J. 612, 262(2004). [12] M A Malkov, L.O’C. Drury Rep. Prog. Phys. 64 429-