Cosmicray Antimatter
Abstract
In recent years, spaceborn experiments have delivered new measurements of high energy cosmicray (CR) and . In addition, unprecedented sensitivity to CR composite antinuclei and is expected to be achieved in the near future. We report on the theoretical interpretation of these measurements. While CR antimatter is a promising discovery tool for new physics or exotic astrophysical phenomena, an irreducible background arises from secondary production by primary CR collisions with interstellar matter. Understanding this irreducible background or constraining it from first principles is an interesting challenge. We review the attempt to obtain such understanding and apply it to CR and .
Based on state of the art Galactic cosmic ray measurements, dominated currently by the AMS02 experiment, we show that: (i) CR most likely come from CRgas collisions; (ii) data is consistent with, and suggestive of the same secondary astrophysical production mechanism responsible for and dominated by protonproton collisions. In addition, based on recent accelerator analyses we show that the flux of secondary high energy may be observable with a few years exposure of AMS02. We highlight key open questions, as well as the role played by recent and upcoming space and accelerator data in clarifying the origins of CR antimatter.
Contents
I Introduction
Cosmic ray (CR) physics is centred on the attempt to solve a set of basic open questions Ginzburg and Ptuskin (1976); Ginzburg et al. (1990). Where do CRs come from, namely, what are the sources of the bulk of the CR energy density? where are these sources located, and how do they accelerate particles? Narrowing to the Galactic energy range (MeV to at least PeV per particle): what role do CRs play in Galactic dynamics? How do CRs propagate and eventually escape from the Galaxy? Where and how does the transition to extragalactic sources occur?
From this wide perspective CR antimatter is an exotic phenomena, making an insignificant contribution to the total CR energy density (in the ballpark of 0.01 percent). Nevertheless, careful study of CR antimatter may reveal important insights to CR physics and beyond that, particle physics. Because considerable energy is required to create antimatter, it is a valuable probe of high energy processes. Because primordial antimatter is essentially absent before structure formation, its formation as secondary product in CR collisions with ambient gas makes the description of CR antimatter theoretically clean, decoupling the problem – to some extent – from core unknowns in the physics of CR sources and acceleration.
As a result, one is led to a situation where although the sources of the bulk of the Galactic CRs – primary protons, He, and other nuclei – are essentially not understood, nevertheless the small residual accompanying radiation of , essentially is. Using this theory control, spectral features or excess abundance of CR antimatter could lead to a first detection of exotic phenomena like dark matter annihilation or pairproduction in pulsars. In addition, antimatter provides a unique handle on the physics of CR propagation.
In this review we discuss CR antimatter. Aiming to jump directly to what we consider new and exciting developments, we leave out most of the basic CR physics background; as a partial list of useful books we recommend Ginzburg et al. (1990); Longair (1992); Kulsrud (2005); Gaisser et al. (); Schlickeiser (2002). Review papers are referred to where relevant. We attempt to avoid astrophysics modelling assumptions as much as possible. This is not an easy task, as much of the CR literature is focused on phenomenological modelling of propagation. We leave out most of the modelling questions^{1}^{1}1See Strong et al. (2007) for a useful review., hoping to provide a simple and robust understanding of CR antimatter that would be beneficial to particle physicists and astrophysicists alike.
Our work is motivated by new observational information coming from an array of experiments. Focusing on recent CR antimatter and closely related results we note, as a partial list of experimental contributions, the and measurements of PAMELA Adriani et al. (2009, 2010); Adriani et al. (2013a), FERMI Abdo et al. (2009); Ackermann et al. (2012a); Ackermann et al. (2010), ATIC Chang et al. (2008), HESS Aharonian et al. (2008, 2009), and AMS02 Aguilar et al. (2013, 2016a); Ting (). Future results are expected from CALET Marrocchesi (2016), DAMPE Wang et al. (2016) and CTA Vandenbroucke et al. (2016). Progress in the search for and is expected with AMS02 Giovacchini and Choutko (2007); Kounine (), GAPS von Doetinchem et al. (2016); Aramaki et al. (2016a), and BESS Abe et al. (2012, 2012).
The layout of this review is as follows. In Sec. II we consider secondary – from the theoretical perspective, the simplest form of CR antimatter. In Sec. II.1II.2 we show how stable, relativistic, secondary nuclei data, under the general assumption that the CR elemental composition^{2}^{2}2Neither the overall CR intensity, nor the target interstellar matter density, needs to be uniform in the propagation region in order for the procedure to apply. is approximately uniform in the regions dominating spallation, allow one to calibrate away most of the propagation modelling uncertainties and extract a parameterfree prediction for . The same calibration process is known to describe well the fluxes of secondary CR nuclei. For , residual sensitivity remains to possible CR spectral variations in the spallation regions, as we discuss at some length. The insight behind this calibration process and a discussion of the conditions for its validity are presented in App. A. We find the secondary prediction to be consistent with data within the uncertainties. In Sec. II.3 we compare the modelindependent analysis with certain models of propagation.
In Sec. III we turn to , a hot potato: here public opinion basically has it that a primary source of must exist, be it dark matter or pulsars. We take a fresh look at the data in Sec. III.1; the first thing we notice appears like a hint in the opposite direction: the observed flux ratio saturates the ratio of production rates in protonproton collisions, a compelling hint for secondary . We devote Sec. III.2 to elucidate the picture for . If are secondary, then energy losses during propagation must be small, requiring that the CR propagation time is shorter than the time scale it takes to radiate a significant amount of their energy. Such a scenario cannot be accommodated in the conventional CR diffusion models Strong et al. (2007), and this inconsistency with propagation models was the main cause for the claim of an “ anomaly” Adriani et al. (2009). To be clear: we do not know of a fully satisfactory and tested alternative propagation model that reproduces the behaviour of with secondary production. However, putting modelling questions aside, we show in Sec. III.3 that high energy radioactive nuclei data could test the secondary hypothesis in the near future. In Sec. III.4 we provisionally assume that are secondary to review some general lessons for CR propagation. These lessons which, again, are in tension with the currently common models of CR diffusion, may yet prove to be the longterm legacy of today’s state of the art Galactic CR experiments. In Sec. III.5 we review some CR propagation models for secondary . Finally, in Sec. III.6 we review ideas for primary or from pulsars and dark matter annihilation, showing that these models generically invoke finetuning to reproduce the observations.
In Sec. IV we tackle the topic of CR and . Surprisingly enough, we find a hot potato also here: we suggest, contrary to most earlier estimates, that a detection of secondary may be imminent at AMS02 (consistent with some pesky recent rumours).
Ii Astrophysical : the Galaxy as a fixedtarget experiment
CR antimatter particles are produced as secondaries in collisions of other CRs, notably protons, with interstellar matter (ISM), notably hydrogen in the Galaxy. Highly relativistic and heavier antinuclei () propagate similarly to relativistic matter nuclei at the same magnetic rigidity^{3}^{3}3Here represents particle momentum, of course; elsewhere we often use the same symbol as shortcut for “proton”.
with the difference in charge sign expected to make little or no impact on the propagation given that the measured CR flux is very nearly locally isotropic Di Sciascio and Iuppa (2014).
Starting with the simplest case of , it is natural to try and calibrate the effect of propagation directly from data, by using information on other secondary nuclei like boron (B), formed by fragmentation of heavier CRs (mostly carbon C and oxygen O). We now explain how to perform this calibration, calculate the predicted flux, and compare to measurements.
ii.1 The CR grammage
In this section we limit the discussion to stable, relativistic, secondary nuclei. For such secondaries, including e.g. B and the subFe group (TScVCr), the ratio of densities of two species satisfies an approximate empirical relation Engelmann et al. (1990); Webber et al. (2003),
(1) 
Here denotes the net production of species per unit ISM column density,
(2) 
where and are the total inelastic and the partial cross section per target ISM particle mass , respectively.
We stress that Eq. (1) is an empirical relation, known to apply to 10% accuracy in analyses of HEAO3 data Engelmann et al. (1990); Webber et al. (2003) and – as we shall see shortly, focusing on – consistent with subsequent PAMELA Adriani et al. (2013a) and AMS02 Aguilar et al. (2016a) measurements. In applying Eq. (1) to , a subtlety arises due to the fact that the cross sections appearing in Eq. (2) can (and for , do) depend on energy. In Eq. (2) we define these cross sections such that the source term is proportional to the progenitor species density expressed at the same rigidity; we will clarify this statement further down the road in Eq. (6). For relativistic nuclei (above a few GeV/nuc) produced in fragmentation reactions, e.g. C fragmenting to B, the energy dependence of the fragmentation cross section is much less important.
From the theoretical point of view, Eq. (1) is natural Katz et al. (2010); Ptuskin et al. (1996); Stephens and Streitmatter (1998); Ginzburg et al. (1990). It is guaranteed to apply if the relative elemental composition of the CRs in the regions that dominate the spallation is similar to that measured locally at the solar system^{4}^{4}4Note: neither the overall CR intensity, nor the target ISM density, needs to be uniform in the propagation region in order for Eq. (1) to apply. Indeed, the ISM exhibits orders of magnitude variations in density across the Galactic gas disc and rarified halo Ferriere (2001).: in this case, the source distribution of different secondaries is similar. Because the confinement of CRs in the Galaxy is magnetic, different CR particles that share a common distribution of sources should exhibit similar propagation if sampled at the same rigidity^{5}^{5}5This is, of course, provided that the CR species being compared do not exhibit speciesdependent complications like decay in flight (for radionuclei like Be) or radiative energy losses (for ). In addition, rigidity only really becomes the magic quantity for propagation at relativistic energies (see e.g. Webber et al. (2003)).. Thus, the ratio of propagated CR densities reflects the ratio of their net production rates.
Note that the net source defined in Eq. (2) accounts for the fact that different nuclei exhibit different degree of fragmentation losses during propagation. In this way, species like subFe (with fragmentation loss cross section of order 500 mb), B ( mb), and ( mb) can be put on equal footing.
Further discussion of the physical significance of Eq. (1) is given in Ref. Katz et al. (2010) and App. A.
We can use Eq. (1) together with the locally measured flux of B, C, O, p, He,… to predict the flux Gaisser and Schaefer (1992); Katz et al. (2010):
(3) 
The RHS of Eq. (3) is derived from laboratory cross section data and from direct local measurements of CR densities, without reference to the details of propagation.
The quantity
(4) 
known as the CR grammage Ginzburg et al. (1990), is a spallationweighted average of the column density of ISM traversed by CRs during their propagation, the average being taken over the ensemble of propagation paths from the CR production regions to Earth. Combining AMS02 B/C Aguilar et al. (2016b) and C/O Ting () with heavier CR data from HEAO3 Engelmann et al. (1990) and with laboratory fragmentation cross section data Blum et al. (2017a); Tomassetti (2015), one can derive directly from measurements:
(5) 
The result for is shown by the green markers in the left panel of Fig. 1. Error bars reflect the B/C error bars reported in Aguilar et al. (2016b), and do not include systematic uncertainties on fragmentation cross sections and on the flux ratios C/O, N/O, etc. We estimate that the systematic fragmentation cross section uncertainties are at the level of 20%; note that many of the cross sections used in the analysis at high energy are extrapolated from much lower energy data, typically confined to a few GeV/nuc. The result in Fig. 1 agrees with the powerlaw approximation derived in Ref. Blum et al. (2013) to 20% accuracy.
To exhibit the different contributions entering the determination of , in the right panel of Fig. 1 we show the result for that obtains if we omit, in the B production source, the contributions due to all CR species other than C (purple markers), all species other than C+O (red markers).
ii.2 from B/C
Now that we have , we can use the production and loss cross sections parametrised in, e.g., Tan and Ng (1983, 1983); Winkler (2017) together with measurements of the proton and helium Aguilar et al. (2015a, b); Yoon et al. (2017) flux to calculate and apply it in Eq. (3). At low rigidity, the effect of solar modulation is estimated as in Webber et al. (2003) with MV.
The result is compared to AMS02 data Aguilar et al. (2016a); Nozzoli (2016) in Fig. 2. The flux is consistent, within statistical and systematic uncertainties, with the prediction of Eq. (3). No astrophysical propagation modelling is needed: Eq. (3) has successfully calibrated out propagation from B/C data.
We can conclude that CR are most likely secondary.
As mentioned earlier, in computing we need to account for the energydependent production cross section. Let us consider the main positive contribution to , due to pp collisions. Because of a kinematical barrier, the daughter is emitted with a rigidity smaller by a factor of compared to the rigidity of the parent in the ISM frame. Given a spectrum of parent protons, one can still compute the overlap integral between the differential cross section and the parent p spectrum and express the contribution to in terms of an effective cross section :
(6) 
The factor of 2 above^{6}^{6}6See Winkler (2017) for a recent examination of isospin asymmetry in . accounts for the production and subsequent decay of , with . A similar procedure is used to include the contributions due to proton CR hitting He in the ISM; He CR hitting ISM hydrogen; and so forth.
Calculating Eq. (6) for a powerlaw proton flux , one finds the scaling Katz et al. (2010). The effective cross section therefore depends on the spectral shape of CR protons. This is a new feature compared to the heavy nuclei fragmentation cross sections: there, due to the straightahead kinematics, a cross section like is independent of the carbon spectral index to a good approximation.
Calculating with the locally measured p flux, one might expect deviations from Eq. (1) if the proton spectral shape in the spallation regions exhibits variations compared to its locally measured value. Models that realise this possibility include Cowsik and Burch (2009); Burch and Cowsik (2010); Cowsik et al. (2014); Blasi (2009); Blasi and Serpico (2009); Mertsch and Sarkar (2009); Ahlers et al. (2009); Kachelriess et al. (2011); Cholis et al. (2017); Kachelrieß et al. (2015); Thomas et al. (2016), reviewed in Sec. III.5.
In Fig. 3 we quantify the sensitivity of the locally measured ratio to a difference between the primary CR spectrum measured locally, to the spectrum in the regions of the Galaxy that dominate the secondary production. Similarly, the plot also exhibits the sensitivity in the predicted ratio to measurement errors in determining the local proton flux^{7}^{7}7The systematic difference between different experimental determinations of the local CR proton and He flux is not negligible, at a level of 1020% with larger extrapolation uncertainty in the relevant few TV range; see e.g. Panov et al. (2009); Adriani et al. (2011); Adriani et al. (2013b); Aguilar et al. (2015a, b); Yoon et al. (2017).. We show the ratio that obtains if, in the calculation of , we replace the locally measured proton flux by a modified spectrum: , with for GV and for GV. The result is shown for the choices . Note that our parametrization of the locally measured proton flux uses a direct (non powerlaw) interpolation of AMS02 Aguilar et al. (2015a) and CRAMIII Yoon et al. (2017) data.
Allowing room for cross section and CR spectra uncertainties, from inspection of Fig. 3 we infer a rough limit:
(7) 
Eq. (7) applies to the situation where all primary CR spectra are modified in the secondary production regions, compared to their local value; namely, the variation in proton and in C and O spectra is correlated, . This could occur with nontrivial propagation from the CR fragmentation regions to our local spot in the Milky Way. One may also constrain the possibility of CR accelerators that inject a different composition of primary CR spectra in different (but connected by propagation) regions of the Galaxy, in which case . This situation corresponds to significantly nonuniform CR composition in the propagation region, invalidating the grammage relation. While we do not pursue this analysis here, it is also constrained by the success of Eq. (3) in reproducing the data.
To conclude so far:

When it comes to relativistic, stable, secondary nuclei and antinuclei, the Galaxy is essentially a fixedtarget experiment, with CRs themselves playing the role of the beam and with ISM being the target. This simple point and the resulting predictions are often obscured in calculations done within specific models of CR propagation.
It is worthwhile to compare our analysis to results obtained in the context of specific models of propagation. Of these, the most commonly used are the twozone disc+halo diffusion models Strong et al. (2007). In the next section we discuss these models in view of the data.
ii.3 in diffusion models
Typical twozone diffusion models in the literature satisfy the simple condition leading to Eq. (1), so they too satisfy Eq. (1), at least approximately Ginzburg et al. (1990). This is because the relative composition of the CRs in these models is approximately uniform across the thin ISM disc, where most of the spallation happens. It is interesting to compare results obtained within these models to results derived directly from Eq. (1).
A recent example of the diffusion model was given in Kappl et al. (2015); Winkler (2017). Ref. Kappl et al. (2015); Winkler (2017) calibrated the diffusion model to AMS02 B/C data, and used the resulting model parameters together with state of the art proton and He data to calculate the flux. The result, taken from Winkler (2017), is shown by the blue line in Fig. 4. As can be seen, this result is consistent with the result obtained directly from Eq. (1), using the same B/C, p and He data and a similar cross section code.
As another example, Ref. Donato et al. (2009) used a diffusion model with the same set of assumptions, geometry, and free parameters to that of Kappl et al. (2015); Winkler (2017). However, the ratio predicted in Donato et al. (2009), shown by the yellow band in Fig. 4, falls significantly below the AMS02 data. What went wrong?
The main thing that went wrong, is that the model of Ref. Donato et al. (2009) was calibrated to fit early B/C data from HEAO3 Engelmann et al. (1990), and then extrapolated from that fit to high energy beyond the region where HEAO3 data was tested. Unfortunately, above GV the extrapolation of the HEAO3 B/C data falls bellow the more recent AMS02 measurement. In addition, Donato et al. (2009) assumed a primary proton flux with high energy spectral index , softer by about than the proton flux seen by AMS02. The implications of this soft high energy proton spectrum can be estimated from Fig. 3. At the same time, at lower energy GV the assumed proton flux was higher than that reported by AMS02, decreasing further the predicted ratio.
As a result, Ref. Donato et al. (2009) predicted a low ratio. To illustrate this fact, we show in Fig. 4 by a red line the result we find if we calculate the ratio using Eq. (1), but using the value of derived for the diffusion model of Donato et al. (2009) with and using the same proton flux assumed there^{8}^{8}8We use Eq. (48) to calculate in the diffusion model, with a thin disc thickness of 100 pc and disc ISM proton density of 1 cm. More details on the diffusion model can be found in Sec. III.2.. The discrepancy with Ref. Donato et al. (2009) is reproduced.
Iii What is the issue with ?
Measurements of the positron fraction by the PAMELA Adriani et al. (2009, 2010) and AMS02 Aguilar et al. (2013) experiments have shown that is rising with energy from a few GeV to at least 300 GeV. This trend of rising was claimed by many to indicate a primary source dominating the flux at these energies. Understanding the true story behind CR is crucial: models for a primary source range from dark matter annihilation^{9}^{9}9See Bergstrom et al. (2008); Cholis et al. (2009); Barger et al. (2009); Cirelli and Strumia (2008) as representative examples. to a contribution of from pulsars^{10}^{10}10See Atoyan et al. (1995); Boulares (1989); Profumo (2011); Hooper et al. (2009) as representative examples., both exciting ideas.
Our goal in this section is to consider what can be learned from the measurements. We start with inspecting the data, in Sec. III.1, pointing out that the observed flux ratio saturates the ratio of production rates in protonproton collisions. This is a compelling hint for secondary .
What then is the basis to the claim that a primary source is required? In Sec. III.2 we show that if are secondary, then energy losses during propagation must be small, requiring that the CR propagation time is shorter than the time scale it takes to radiate a significant amount of their initial energy (of order Myr at GV). Such a scenario cannot be accommodated in the conventional phenomenological diffusion models Strong et al. (2007), and the inconsistency with propagation models was the main cause for the claim of an “ anomaly” Adriani et al. (2009). However, we know of no contradiction of this scenario with either observational data or first principle theory.
The nearest complimentary data that could test the secondary hypothesis involves radioactive secondary nuclei, and we review it in Sec. III.3. In Sec. III.4 we entertain the possibility that are indeed secondary, and deduce some basic modelindependent lessons for CR astrophysics. In Sec. III.5 we review some CR propagation models for secondary . Finally, in Sec. III.6 we review ideas for primary or from pulsars and dark matter annihilation, showing that these models generically invoke finetuning to reproduce the observations.
iii.1 A hint for secondary
In dealing with we used Eq. (1), cast in the form of Eq. (3) where is derived from nuclei data. However, Eq. (1) cannot be directly applied to predict the flux of , because are subject to radiative energy losses and Eq. (1) does not capture the effect of energy loss during propagation (see discussion in App. A).
Nevertheless, we can still gain insight from Eq. (1). As noted in Katz et al. (2010), Eq. (1), applied to with radiative losses ignored, provides an upper limit to the secondary flux because radiative energy losses can only decrease^{11}^{11}11This statement is true for a steeply falling spectrum, assuming synchrotron and inverseCompton (IC) losses dominate in the Thomson regime. The requirement to avoid pileup in this case is , to be compared with the observed . the secondary flux compared to the lossless secondary benchmark. This provides an upper bound on the flux of secondary .
The most robust way to formulate the secondary upper bound on the flux is in terms of branching fractions in pp collisions, comparing to . The upper bound reads:
(8) 
AMS02 have recently reported the inverse ratio Aguilar et al. (2016a). The experimental results are compared to the bound of Eq. (8) in Fig. 5.
Pause to appreciate this situation: The measured ratio does not exceed and is always comparable to the secondary upper bound. Moreover, the ratio saturates the bound over an extended range in rigidity. Taking into account that, as we saw in the previous section, are likely secondary (certainly dominated by secondary production), it is natural to deduce that AMS02 is observing secondary as well, and propagation energy losses are small.
A compatible but less robust way to represent the secondary upper bound is directly from the B/C grammage, as we did for in Fig. 2. Namely, we write
(9) 
The result is shown in Fig. 6. On the left panel the measured total flux of AMS02 Vagelli (2016) is used to define the ratio upper bound from Eq. (9). On the right panel we show the upper bound on the flux. We stress that Eq. (9) (and thus Fig. 6), similarly to the situation exhibited in Fig. 3, is more sensitive to the unknown CR spectra in the spallation regions than is the ratio of Fig. 5. In Fig. 7 we show how the bound is modified if one allows the proton spectrum in the secondary production sites to vary w.r.t. the locally measured proton flux. Models that realise this possibility include Cowsik and Burch (2009); Burch and Cowsik (2010); Cowsik et al. (2014); Blasi (2009); Blasi and Serpico (2009); Mertsch and Sarkar (2009); Ahlers et al. (2009); Kachelriess et al. (2011); Cholis et al. (2017); Kachelrieß et al. (2015); Thomas et al. (2016), reviewed in Sec. III.5. The sensitivity of the bound in Fig. 7 to proton flux variation should be compared to the insensitivity of the more robust ratio of Fig. 5.
For later convenience it is useful to define the loss suppression factor via
(10) 
In Fig. 8 we show as derived from Fig. 5. The upper bound for secondary is . We find that is never much smaller than unity, and approaches unity from below for increasing .
In considering this behaviour it is important to note that theoretically the possible range of the suppression factor is : this just says that a prominent primary source of – say, a nearby pulsar – could make the flux as large as we wish in comparison to the secondary upper bound; while strong radiative losses, if at work, could extinguish the flux.
We conclude that AMS02 results hint for a secondary origin for Blum et al. (2013)^{12}^{12}12Ref. Lipari (2017) recently joined this understanding. We note, however, that our evaluation of the ratio in Fig. 5 is higher than that of Lipari (2017) by 3050% at GV. This difference led Lipari (2017) to conclude that are not affected by radiative losses at all energies; while we find that the data implies some radiative loss effect at GV. The basic conclusion, putting 3050% differences aside, is similar: are consistent with secondaries.. If are secondary, then Fig. 5 suggests that the effect of energy loss in suppressing the flux is never very important, and possibly becomes less significant as we go to higher energy. As we shall see, this behaviour contradicts the expectations within common models of CR propagation^{13}^{13}13For early comprehensive analyses see e.g. Protheroe (1982); Moskalenko and Strong (1998); Delahaye et al. (2009).. To appreciate this point we must venture into somewhat more muddy waters of CR astrophysics and consider the interplay of energy losses with the effects of propagation.
iii.2 Radiative energy loss vs. propagation time vs. grammage
The name of the game is to figure out the interplay of energy loss with CR propagation: this is needed either to establish the necessity of a primary source, or to understand the lessons for CR propagation if are secondary.
The radiative cooling time is
(11) 
At high energy GV, energy loss is dominated by synchrotron and IC. In the Thomson regime Blumenthal and Gould (1970)
(12) 
where is the sum of radiation and magnetic field energy densities. Thus, in the Thomson regime .
Bremstrahlung and KleinNishina corrections soften this behaviour to with , as we discuss in Sec. III.4.
Consider the qualitative behaviour of . We expect to increase monotonically as a function of , where is the CR propagation time, defined in some convenient way to parametrise the typical time a CR spends in the system from the time of production until the time of detection at Earth.
In the limit , we expect ; this is because in this limit, a typical CR trajectory lasts much less time than the time it takes an to lose a significant amount of its initial energy. Thus in this limit relativistic and at the same propagate in the same way and the observed ratio reflects the secondary production rate ratio . In the opposite limit, , we expect because lose most of their energy on their way from production to detection, while the corresponding propagate unaffected.
Given an estimate of , Fig. 8 is therefore a measurement of the CR propagation time. The detailed interpretation of the form of , however, is modeldependent. To demonstrate this point, in App. B we calculate for two propagation model examples: a version of the leakybox model (LBM), and a onedimensional thin disc+halo diffusion model.
We emphasize that both of these models satisfy Eq. (1). Therefore, calibrating the relevant free parameters in either model to reproduce according to Fig. 1 would make these models automatically reproduce Fig. 2, consistent with AMS02 data. Fig. 9 shows as obtained for the two models, with propagation parameters calibrated consistently with B/C and . We take representative values of and 2.7 for the primary proton spectral index. The numerical value of differs between the LBM and diffusion models, despite having calibrated both of these models to match B/C and .
While the diffusion model and the LBM differ in the form of they predict, both models share a common feature: in both of the models, the rigiditydependent column density of ISM traversed by CRs is proportional to the rigiditydependent propagation time, .
This proportionality between and is more general than the specific models we looked at. It holds, for example, for commonly adopted diffusion models Strong et al. (2007) that assume a rigidity independent diffusion boundary. Because the grammage must be fit in phenomenologically consistent versions of these models^{14}^{14}14By adjusting the free parameters , , and other parameters in more complicated realisations. E.g., the inhomogeneous diffusion model of Kappl and Reinert (2017) falls in the same category, and is thus affected by the same problem in reconciling and B or data: it is constrained by construction to satisfy Eq. (13). to match B/C, subFe/Fe, and data, these models satisfy
(13) 
where in the numerical assignment we adopt, for simplicity, the approximate fit of Ref. Blum et al. (2013).
Using Eq. (12) gives with , so propagation models that satisfy Eq. (13) predict that the ratio must decrease with increasing rigidity. As a result, because scales as a positive power of , these models predict that the effect of losses should become increasingly more important at high energy: should decrease at rising . This is the opposite trend to that inferred from data in Figs. 58.
To maintain the hypothesis of secondary , the fact that the observed approaches unity with increasing rigidity implies that the propagation time cannot be much larger than the cooling time , and is decreasing with rigidity faster than . This means that decreases with rigidity faster than , in contradiction to Eq. (13).
We are faced with two possibilities. Either take the coincidence of the observed ratio with pp branching fractions (Figs. 58) as a hint that CR are secondary, in which case something basic is not right in the commonly adopted CR diffusion models: the CR grammage cannot be proportional to the CR propagation time. Or, accept the diffusion models and invoke a primary source for , like dark matter annihilation or pulsars, in which case some primary source parameters would need to be tuned to reproduce the observed ratio as an accident.
This is a good point to comment on statements in the literature, that advocated the presence of an primary source based on a rising fraction. Two representative examples are Adriani et al. (2009) and Serpico (2009). Ref. Adriani et al. (2009), and numerous following works, based the call for primary on the observation that a rising would be inconsistent with secondary as expected in a certain diffusion model Moskalenko and Strong (1998). However, phenomenological models such as Moskalenko and Strong (1998) are constructed with many simplifying assumptions, ranging from steady state and homogeneous diffusion to the geometry and boundary conditions of the CR halo Strong et al. (2007). Some of these assumptions may not apply to Nature. We explore alternative ideas with secondary in Sec. III.5.
Ref. Serpico (2009) argued that a rising requires primary because and suffer radiative energy losses in the same way, and because the primary source spectrum cannot be softer than that of the secondary . The problem with this argument is that the injection spectrum of primary is unknown^{15}^{15}15Models like Moskalenko and Strong (1998), for example, take the injection spectrum as a free parameter that is then fitted to the data (similar practice – with separate free parameters – is applied for the proton, He, and other primary nuclei spectra)., and the propagation paths of and may differ because their production sites as secondaries vs. accelerated primaries, respectively, may be different. In this case, may suffer additional energy losses as compared to . We will see model examples for this, too, in Sec. III.5.
Finally, it is important to note that the high energy data Adriani et al. (2009); Aguilar et al. (2013); Ting () represent a new observational probe of CR propagation at high rigidity GV: it tests the propagation models where they were not tested before. B/C data measures ; as we have seen, other stable secondary nuclei data such as do not give much of a new test: they are essentially consistency checks of the hypothesis that different CR species sample similar , a fact for which early evidence already existed Engelmann et al. (1990); Webber et al. (2003); Gaisser and Schaefer (1992). In contrast, the data is sensitive to the apriori independent quantity . The fact that could eventually provide such a test of the models was pointed out already in Ginzburg and Ptuskin (1976); Ginzburg et al. (1990), long before PAMELA and AMS02 made this test come to life.
Besides from , no other CR data to date accesses in the same range in rigidity GV. What comes nearest are measurements of the effect of radioactive decay of secondary Be, Al, and Mg isotopes. In Sec. III.3 we consider these data as a modelindependent test of the secondary hypothesis. As of today, the test is based on early measurements Webber and Soutoul (1998); Engelmann et al. (1990) and supports the idea of secondary Katz et al. (2010), but is inconclusive due to systematic uncertainties. A better test should become possible in the near future with AMS02 data, and we devote some time to explain the key physics.
iii.3 A test with radioactive nuclei
Measurements of the suppression of the flux of secondary radioactive nuclei due to decay in flight constrain the CR propagation time Shapiro and Silberberg (1970); Webber and Soutoul (1998); Ptuskin (1999); Simpson, J. A. and GarciaMunoz, M. (1988); Ptuskin and Soutoul (1998); Yanasak et al. (2001); Donato et al. (2002); Putze et al. (2010); Blum (2011); Shibata and Ito (2007). The idea is that a relativistic radioactive nucleus with rest frame lifetime , mass number and charge has an observer frame lifetime
(14) 
when observed as CR. Given a secondary radioactive species like Be, we can compute the prediction for its density, discarding the effect of decay, and compare the result to the observed density. This allows to define a decay suppression factor that can be extracted directly from data:
(15) 
Here, the numerator is supposed to be taken directly from measurement, while the denominator is a theory output but is, again, datadriven based on fragmentation cross sections, B/C and primary CR spectra. In the last expression we used Eqs. (1) and (4), and noted that for stable secondary species Eq. (1) implies
(16) 
where is the positive production term in the net source . (As before, is the total inelastic cross section of per ISM particle mass.)
We expect the qualitative behaviour of to depend on the CR propagation time via the ratio : for we should have , while for we expect . Because decreases with increasing , and (for relativistic nuclei) increases as , it is clear that should approach unity with increasing . Since is known from laboratory data, a measurement of provides a constraint on .
Experimental data on radioactive nuclei is divided in two categories: isotopic data such as Be/Be, and elemental (or charge) data like Be/B in which the numerator denotes the sum of Be isotopes (Be) and the denominator the sum of B isotopes (B).
Isotopic ratios are experimentally challenging to measure at high energy. As a result, current data on Be/Be is limited to low rigidity GV. This introduces significant theory uncertainty in the interpretation of these data as effects such as solar modulation, energydependence in the nuclear fragmentation cross sections, and various propagation effects that change CR energy during propagation become important (see e.g. Ptuskin et al. (1996)). Nevertheless, diffusion models have traditionally used this low energy data to calibrate the models Strong et al. (2007), extrapolating the result to the relativistic range. AMS02 is expected to improve the situation with the ability to measure Be/Be up to GV.
Elemental ratios can be measured to high energy Webber and Soutoul (1998). The challenge here is that the contribution of the radioactive Be isotope to the total Be flux is never expected to exceed about , based on our knowledge of partial fragmentation cross sections like CBe vs. CBe, etc. As a result, the Be/B ratio is limited to the range or so, making the interpretation particularly sensitive to fragmentation cross section uncertainties. The situation with other relevant elemental ratios, Al/Mg and Cl/Ar, is similar^{16}^{16}16For the Al and Cl, another difficulty is that primary contamination to the flux may not be negligible Blum (2011)..
The Be/B ratio derived from early measurements of Be and B fluxes by the HEAO3 mission is shown in Fig. 10 (blue markers) Webber and Soutoul (1998); Engelmann et al. (1990). Recently, AMS02 reported preliminary results for Be/B Ting () extending to very high energy, shown in red. The saturation point is, for the first time, clearly manifest in this data as the observed Be/B saturates the nodecay secondary prediction^{17}^{17}17It is interesting to note that the theoretically predicted asymptotic nodecay Be/B ratio, shown by the grey band in Fig. 10, does not go to a constant at high but rather exhibits a mild decrease with increasing . This trend is caused by the socalled tertiary production where B fragments into Be; this effect is contained in Eq. (1), and the mild decrease in asymptotic Be/B is due to the decrease of at rising . The preliminary AMS02 data is consistent with this subtle prediction of Eq. (1). We await an official release by AMS02 for further analysis..
The definition of the decay suppression factor is analogous to that of the loss suppression factor defined in Eq. (10) for secondary . However, radioactive decay is not quite the same as radiative energy loss: the former eliminates the CR altogether, while the latter just degrades it in energy. Moreover, the dependence of for radionuclei is essentially opposite to for . Fig. 11 illustrates the behaviour of and .
Ref. Katz et al. (2010) pointed out an approximate, but model independent way in which radiative energy losses and radioactive decay can be put on similar footing. Consider the contribution of radioactive decay in a general, local transport equation. Decay introduces a term to the continuity equation for radionucleus ,
(17) 
Energy loss for , in comparison, is captured by:
(18) 
where we define
(19) 
The observed flux in the range GV is roughly a power law with in the range . For not far from the Thomson regime, , the logarithmic term in Eq. (19) is therefore a weak function of , varying in the range .
We learn that, due to the steeply falling flux, decay and energy loss are represented by a similar form in the continuity equation.
A model independent check of the hypothesis of secondary is therefore obtained by comparing the observed for , and for radionucleus , at the particular rigidity in which . Referring to Fig. 11, we see that for Be, with a reasonable estimate of , GV. This is illustrated by a circle in the plot. For this rigidity, Fig. 8 suggests
(20) 
From the HEAO3 data Engelmann et al. (1990) analysis of Webber and Soutoul (1998), Ref. Katz et al. (2010) found
(21) 
Consistent results obtain for the Al/Mg and Cl/Ar data.
Significant systematic uncertainty affects this analysis, manifest in Fig. 10 by the cross section uncertainty as well as the mismatch between HEAO3 and AMS02 preliminary results. Nevertheless, we can conclude that this test of radionuclei data is consistent with secondary . Upcoming results from AMS02 Ting () should allow to improve this test.
Finally, Ref. Blum (2011) suggested that the rigidity dependence of can be constrained by comparing the decay suppression factor for different species of radionuclei measured at the same observer frame lifetime and thus – because different nuclei have different rest frame lifetimes – at different rigidities.
Fig. 12 illustrates this point with HEAO3 data Engelmann et al. (1990); Webber and Soutoul (1998). While, again, systematic and statistical uncertainties in this data are large, the idea is promising: a rapidly decreasing , as needed to reconcile secondary with a rising , would manifest in the radionuclei data by resolving the combined radionuclei data set into three different curves for Be/B (controlled by Myr), Al/Mg ( Myr), and Cl/Ar ( Myr).
iii.4 Implications of secondary for CR propagation
In this section we assume that are secondary, and review constraints on CR propagation that can be deduced in this case. Fig. 8 suggests that
(22)  
(23) 
Considering the two models depicted in Fig. 9 as representative examples, Eq. (22) implies for the diffusion model, and for the LBM. Eq. (23) implies for the diffusion model, and for the LBM.
Ignoring KleinNishina corrections (but see discussion below), we summarise these results by the constraints Blum et al. (2013)^{18}^{18}18Ref. Lipari (2017) recently arrived at comparable conclusions.,
(24)  
(25)  
The RHS of Eqs. (2425) is based on a rough estimate of the cooling time at the relevant energies, and as such is subject to O(1) uncertainty. Here is the timeaveraged total electromagnetic energy density (propagation path and timeaverage of from Eq. (12)) in the propagation region. It is natural to expect that depends on CR rigidity, both because the radiation and magnetic fields in the ISM are not uniformly distributed and because the Thomson limit for describing the losses is not exact.
One irreducible source for energy dependence in the effective value of comes from KleinNishina corrections, that are neglected in Eqs. (24) and (25). The Thomson limit is not a good approximation for 20300 GV positrons if contains a significant UV component, as may be the case judging from estimates of the Milky Way radiation field Hauser and Dwek (2001); Porter and Strong (2005); Crutcher et al. (2010). In terms of Eqs. (2425), a plausible 50% UV contribution to implies that the effective value of decreases between 10 GV to 300 GV by a factor of 2. More extreme possibilities were entertained in Stawarz et al. (2010).
Another feature that is not included in Eqs. (12,2425) is bremsstrahlung losses. The bremsstrahlung radiation length is g/cm Longair (1992), approximately independent of energy and insensitive to the H:He ratio in the ISM, such that the corresponding cooling time is . The energy loss term in the continuity equation takes a form similar to that of a fragmentation loss term for nuclei,
(26) 
where
(27) 
For with , we have with .
Using the similarity to fragmentation losses of nuclei, the loss suppression factor due to brem is
(28) 
In Fig. 13 we repeat the calculation of , modding out (in green) the brem contribution using Eqs. (2728) with . Bremsstrahlung modifies by at GV but becomes negligible at GV.
We can also estimate the average ISM density traversed by CRs. Using Eq. (46) together with Eqs. (24) and (25), we find
(29)  
(30) 
assuming an ISM composition of 90%H+10%He by number.
Eqs. (29) and (30) suggest that the confinement volume of CRs may be decreasing with increasing CR rigidity, to the extent that CRs at GV spend much of their propagation time within the thin Galactic HI disc, with a scale height , while CRs at GV probe a larger halo Katz et al. (2010)^{19}^{19}19This can also be stated as saying that the higher rigidity CRs escape the confinement volume more easily, and fail to return from a scale height that can still trap lower rigidity particles.. There are other possibilities, however. For example, if a significant fraction of the grammage is accumulated during a short time in relatively dense regions, e.g. near the CR source, then the halo could be larger. Significant energy dependence in could further affect the interpretation. For example, , inspired by , would allow for a rigidity independent .
Finally, it is also possible that the CR distribution is not in steady state. In this case, the PAMELA and AMS02 may be teaching us about, e.g., a recent nearby burst of star formation and supernova explosions. Key guidelines for such models, that can be derived from Figs. 2 and 5, are that:

B, , and appear to be secondaries from the same spallation episode, and

the bulk of the spallation must not have occurred more than a few Myr in the past.
In the next section we review some ideas along these lines.
iii.5 Models for secondary and
In this section we briefly review CR propagation models Katz et al. (2010); Cowsik and Burch (2009); Burch and Cowsik (2010); Cowsik et al. (2014); Blasi (2009); Blasi and Serpico (2009); Mertsch and Sarkar (2009); Ahlers et al. (2009); Kachelriess et al. (2011); Cholis et al. (2017); Shaviv et al. (2009); Kachelrieß et al. (2015); Thomas et al. (2016) where come from secondary production.
In the nested leaky box model of Cowsik and Burch (2009); Burch and Cowsik (2010); Cowsik et al. (2014), the secondary reacceleration model of Blasi (2009); Blasi and Serpico (2009); Mertsch and Sarkar (2009); Ahlers et al. (2009); Kachelriess et al. (2011); Cholis et al. (2017), and the recent supernova model of Kachelrieß et al. (2015); Thomas et al. (2016), a common theme is that the spectrum of primary CRs in the secondary production sites is different than the locally measured spectrum. As a result, the application of Eq. (1) for relating the B/C garmmage to secondary and becomes inaccurate (see the discussion around and below Eq. (6)). This means that the success of Eq. (3), seen in Fig. 2 to reproduce the ratio at GV based on locally measured proton and nuclei spectra, should be somewhat accidental in these models. At GV, Fig. 2 is consistent with the possibility of a contribution to the spallation from a harder proton spectrum, although the systematic cross section and primary flux uncertainties prevent a sharp conclusion.
We emphasize that the asymptotic ratio (that is the ratio when losses are not important) is insensitive to primary spectrum details, as can be seen in Fig. 5.
The spiral arm model of Shaviv et al. (2009) is an example to a setup in which and come from different regions in the Galaxy and thus their propagation energy losses are different.
Ref. Katz et al. (2010) pointed out that a rigiditydependent CR propagation volume could break the proportionality between and , in accordance with the discussion of Sec. III.2. Considering the diffusion model of Sec. III.2 (with more details in App. B), for example, the idea is to let the boundary condition vary as . Using Eqs. (49) and (48) with , we have . Fixing to comply with B/C and data, setting the model can accommodate an loss suppression factor that is flat as function of for (implying some rigidity dependence of due to, e.g., KleinNishina effects and bremsstrahlung; see Sec. III.4). In the diffusion model, rigiditydependent corresponds to nonseparable rigidity and spatial dependence of the diffusion coefficient (the free escape boundary in these models mimics a region where the diffusivity diverges, )^{20}^{20}20Repeating footnote 15, the inhomogeneous diffusion model of Kappl and Reinert (2017) is not a good example for the setup under discussion because it assumed a separable and dependence of the diffusion coefficient. Thus in that model, just as for the simple homogeneous model with free escape boundary.. Considering the picture of resonant pitchangle scattering of CR on magnetic field irregularities Blandford and Eichler (1987), nonseparable rigidity and spatial dependence of means that the spectrum of magnetic field turbulence varies in the propagation region. decreasing with increasing would occur if large scale turbulence decays faster than small scale turbulence at increasing distance from the Galactic disc. A rigiditydependent confinement volume could also be realised in other settings Blum (2011). More quantitative analysis is required to test the idea further.
Further discussion of Cowsik and Burch (2009); Burch and Cowsik (2010); Cowsik et al. (2014); Shaviv et al. (2009); Blasi (2009); Blasi and Serpico (2009); Mertsch and Sarkar (2009); Ahlers et al. (2009); Kachelriess et al. (2011); Cholis et al. (2017) and of the idea of a rigiditydependent CR confinement volume can be found in Katz et al. (2010). In the rest of this section we highlight the more recent model of Kachelrieß et al. (2015); Thomas et al. (2016).
Ref. Kachelrieß et al. (2015); Thomas et al. (2016) suggested that a supernova (SN) explosion occurring about 2 Myr ago at a distance of 100 pc from the solar system and injecting of the order of erg in CR protons could affect the spectra of primary and secondary CRs. The rate of SNe in the Milky Way is in the ballpark of 3 per century Adams et al. (2013). Divided by the gas disc area kpc, this gives SNe per (300 pc) per Myr, consistent with the setup in Kachelrieß et al. (2015). The detection of CR Fe Fry et al. (2016); Binns et al. (2016) is also consistent with a recent nearby SN.
Because and in Kachelrieß et al. (2015) are produced as secondaries and their propagation time – given roughly by the time since the SN – is of order a Myr such that energy losses are not important below a few hundred GeV, the coincidence of the flux ratio with the secondary production rate ratio could be naturally addressed, as long as the recent SN contribution dominates for both species.
The SNoriginated proton flux in Kachelrieß et al. (2015) dominates the local proton flux by a factor of at TV, with the remaining flux assumed to come from an long time average of CR production in multiple earlier CR injection events. At the same time, the column density associated to the SN protons is g/cm, about 30% of derived from local B/C at TV (see Fig. 1). As a result of this tuning (large SN proton flux with small associated grammage, vs. small average proton flux with large grammage), the SNrelated contribution to the flux is about % of the total at GV.
In the model of Ref. Kachelrieß et al. (2015), the CR distribution exhibits large deviations from steadystate with local sources producing order of magnitude deviations on subkpc and subMyr distance and time scales, as compared to the large volume or long time average. This picture can be tested with gamma ray data, by studying the gamma ray emissivity of individual molecular clouds and comparing to average diffuse emission data. Studies along these lines are ongoing Ackermann et al. (2012b); Dermer et al. (2013); Strong (2015); Mizuno et al. (2016); Neronov et al. (2017).
iii.6 On dark matter and pulsar models for primary or
Many studies in the literature ascribed the CR or flux to a primary source, the most common examples being pulsars (see, e.g. Profumo (2011); Hooper et al. (2009); Malyshev et al. (2009); Blasi and Amato (2011); Di Mauro et al. (2014); Lee et al. (2011) and references to and therein) and dark matter annihilation (see, e.g. Gaskins (2016); Meade et al. (2010); Cholis and Hooper (2013); Hooper and Tait (2009) and references to and therein). In these works the primary source is assigned free parameters to describe the or spectrum and injection rate into the ISM. Then, using some CR diffusion model to simulate propagation effects, the model parameters are adjusted to fit the observed antimatter flux.
While we do not think that current or data motivate the introduction of primary sources beyond the secondary flux, the possibility of dark matter or pulsar contributions is nevertheless interesting enough to merit some consideration.
The cosmological evidence for dark matter (DM) is compelling, and it is natural to imagine that DM is also responsible for the flattening of galactic rotation curves and other astrophysical gravitational anomalies. If DM is composed of massive particles, then pair annihilation of these particles in the Galactic halo could indeed produce high energy CR antimatter (examples predating the PAMELA data include, e.g. Bottino et al. (1998); Feng et al. (2001); Donato et al. (2004)). It is interesting to compare the CR antimatter source characterising a simple, generic DM model, to the irreducible secondary background. Focusing on for concreteness, the injection rate density from DM particleantiparticle pair annihilation can be written as