Table Of ContentCERN-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 differ-
ent (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 em-
pirical 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 dom-
inated 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 collabo-
ration [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 secon-
daries in pp collisions.
All of the above options seem to be at least strongly
disfavored if not already ruled out; nonetheless, we have
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