$e^+$ and $\bar{p}$ production in $pp$ collisions and the cosmic-ray $e^+/\bar{p}$ flux ratio

Secondary astrophysical production of $e^+$ and $\bar{p}$ cosmic rays is considered. Inclusive $\pi$, $K$, and $\bar{p}$ production cross sections in $pp$ collisions at large $\sqrt{s}$ are parametrised using recent experimental data at LHC energies. The astrophysical production rate ratio $Q_{e^+}/Q_{\bar{p}}$ is calculated for an input cosmic ray proton flux consistent with local measurements. At $10<E<100$$\sim$GeV the cosmic ray flux ratio $J_{e^+}/J_{\bar{p}}$ measured by AMS02 falls below the production rate ratio by about 50\%, while at high energy $E>100$$\sim$GeV the measured flux ratio coincides with the production rate ratio of the secondary source.


I. INTRODUCTION
Cosmic ray (CR) antimatter is a potential probe of exotic high energy astrophysical phenomena and a unique diagnostic of CR propagation. Over the last decade, precise measurements of the flux of CR e + andp extending to ever higher energies were reported by the PAMELA and AMS02 experiments [1][2][3]. The interpretation of these measurements motivates refined theoretical consideration of astrophysical e + andp, produced as secondaries in the collision of primary CRs, notably protons, with interstellar matter (ISM), notably hydrogen. Our goal in the current paper is to improve on previous calculations of the inclusive production cross section of secondaries in pp collisions using recent accelerator data.
The main effect we wish to capture is the violation of radial scaling at √ s > 50 GeV. As shown in Refs. [4][5][6], this effect leads to about a factor of two increase in the astrophysicalp source atp energy above a few TeV. Here we evaluate the analogous effect in the CR e + source by analysing meson production at LHC energies. Earlier e + calculations were either based on too low √ s data to see the effect [7][8][9] or relied on Monte-Carlo tools without direct verification in the kinematical regime relevant for astrophysics [10].
We aim to achieve ∼10% accuracy for the astrophysical e + source at e + energy ranging from a few GeV up to multi-TeV; this accuracy goal is to be compared with the main radial scaling violation effect that is, again, about a factor of two at E ∼ 10 TeV. As a check against earlier work, we also calculate thep source to similar accuracy.
In section II we analyse the cross sections at large √ s, using results from the NA49, PHENIX, ALICE, and CMS experiments. In section III we use these results to calculate the production rate ratio Q e + /Qp for secondary e + andp produced by a spectrum of high energy protons scattering on a proton target. We show that Q e + /Qp is insensitive w.r.t. uncertainties in the primary proton spectrum. At 10 < E < 100 GeV the e + /p flux ratio measured by AMS02 falls bellow the production rate ratio by about 50%, while at high energy E > 100 GeV the measured flux ratio coincides with the production rate ratio of the secondary source. In App. A we discuss the contribution to secondary e + from K 0 L decay, which was missing in previous calculations. In App. B we analyse the hyperon contribution to inclusivep production. In App. C we reproduce the secondary cosmic rayp flux predicted by using mean traversed target column density as deduced from cosmic ray nuclei data.

II. DATA ANALYSIS
Our baseline fitting formulae for inclusive hadron production in pp collisions are taken from Ref. [11] (Tan&Ng), which was based on √ s ≤ 53 GeV data and to which we provide corrections using the following new information: i. Tan&Ng's formulae rely on radial scaling [12][13][14], where x R = E * /E * max , E * is the final state hadron energy in the centre of mass (CM) frame and E * max is the maximum attainable E * . Recent accelerator data show violation of radial scaling in pp collisions at √ s 50 GeV [15][16][17][18][19]. The pp →p cross section increases at high energy [4][5][6] as compared to [11] and other early parametrisations. We will assess the analogous effect in meson production and the resulting e + yield.
ii. In addition to the high energy end, unprecedented detailed measurements of the production cross section π + , K ± andp [20][21][22] at √ s = 17.2 GeV were reported by the NA49 experiment. This value of √ s is particularly relevant for E ∼ 10 − 100 GeV final statep and e + [23]. We incorporate this data in our formulae for hadronic cross sections.
A. pt-weighted cross sections and important kinematical region Faced with an extensive data set [15][16][17][18][19][20][21][22], it is instructive to bracket the final state phase space that is most relevant for secondary CR production. In the fixed-target set up of high energy CR scattering on ambient ISM, the key quantity is the conversion cross section from incoming CR proton with ISM frame energy E p to outgoing secondary particle with ISM frame energy E, where θ denotes the angle between the incoming proton and outgoing secondary in the ISM frame. The Lorentzinvariant differential cross section E d 3 σ dp 3 decreases sharply with increasing p t , with the p t -weighted cross section For E p t , m, where m is the mass of the final state hadron of interest, we can simplify the integral as where x R | pt=0 is computed at p t = 0 and only depends on E and E p . This exercise shows that, in the high energy regime, the p t -weighted mean cross section with fixed x R is the most important quantity for secondary CR production, allowing one to average over the detailed p t dependence reported by the experiments. Next, we consider the relevant range of x R . Consider as a representative example the cross section parametrization [5]: Typical parameters are p t 0.2-0.4 GeV and n 5-7. In the limit where m 2 /E 2 , p 2 t /E 2 2m p /E p , the astrophysical source term Q(E) can be written as where J p denotes the CR proton flux and we assumed J p ∝ E −γ p . With n 5-7 and γ 2.7, the x R integrand selects the range ∼ 0.1-0.4.
To summarise, we are most interested in the cross section for secondary product energy in the range E 10 GeV. In this range, the relevant information is contained in the p t -weighted mean invariant cross section at fixed x R , where furthermore the relevant range of x R is ∼0.1-0.4.

B. Hadron production cross section
In this section we discuss the hadronic cross section in the light of recent collider experiments. We take the cross section fits by Tan&Ng as baseline, and derive corrections to these formula.
A comment is in order regarding the intermediate hyperon contribution top. In pp collisions,p are generated promptly or by the decay of (relatively) long-lived hyperons, notablyΛ andΣ ± . The Tan&Ngp fit includes the hyperon contributions. On the other hand, recent experiments such as NA49 report the prompt antiproton cross section in which the contribution of intermediate hyperon states is removed. Thus, when comparing experimental p cross section data and fits we need to specify whether the hyperon contribution is subtracted or not.
For the purpose of astrophysical calculations, of course, our eventual concern is the totalp cross section including the hyperon contributions. In this section, however, we find it convenient to concentrate first on the prompt p production cross section, deferring an analysis of the hyperon contribution to App. B.

NA49 experiment
The NA49 experiment reported measurements in a wide kinematic regime. Fig. 1 shows measurements of the p t -weighted cross section, presented as ratio between NA49 data and the Tan&Ng's formulae in given x F bins 1 . We use data from [20], [21] and [22] for π + , K ± andp respectively. For each point, statistical and systematic errors are both at the level of 10%.
As we discussed, the most relevant kinematical region to determine CR flux is x F = 0.1-0.4. In this region, Fig. 1 shows that apart from an overall factor the fitting functions of Tan&Ng are consistent with the NA49 results for all final states with the possible exception of K − (the latter being quantitatively irrelevant for the secondary e + calculation). Motivated by this result, we introduce a scaling factor ξ H ( √ s) for each hadron H = π + , K ± ,p, and parametrize the cross section as We take ξ π + = ξ K ± = 0.9 and ξp = 0.8 at √ s = 17.2 GeV.
Note that the promptp cross section from NA49 is off by ∼ 20% from the inclusive Tan&Ng fit: this is not a discrepancy, but is mainly due to the hyperon contribution 1 NA49 data are provided in terms of the Feynman parameter : pt-weighted cross section for π + , K ± andp, presented as ratio between NA49 data and Tan&Ng [11] inclusive cross section formulae, in given xF bins. Note that the Tan&Ng formulae include contributions from unstable intermediate states, such as the hyperon contributionΛ →pπ + top production, that is subtracted in the NA49p data. We explain how to correct for this effect in the text.
present in the Tan&Ng fit while being subtracted from NA49 data. Accounting for this correction we find, in fact, that the inclusive Tan&Ng fit is in good agreement with that deduced from NA49 data.

High energy experiments
Next, we analyse the high energy data to determine the behaviour of ξ H at large √ s. The scaling factors ξ H are calibrated to reproduce the p t -weighted cross section of Eq. (3) evaluated on the high energy experimental data. Fig. 2 shows the √ s dependence of ratios of p t -weighted cross sections for π + , K ± andp between high energy data and the Tan&Ng [11] formulae. Solid lines indicate the correction functions Eqs. (7)(8)(9). We use data from PHENIX [19]  We calculate the p t -weighted cross section using the p t range provided by the experiments. Since CMS and ALICE gives a production yield, we use a fitting function of inelastic total scattering rate in [5] to obtain the cross section. In addition, to obtain inelastic yield for CMS, we multiply an empirical factor 0.78 (see [18]). For ALICE data, we use dN/dy estimated in [15,17]. Statistical and systematic errors are roughly 10% for all experiments apart from the PHENIXp data, to which we refer in more detail below.
The orange points in Fig. 2 summarise the collection of experimental data used by the Tan&Ng original analysis [11]. These early measurements cover a wide range of phase space and energy, corresponding to √ s 10 − 60 GeV. Detailed comparison shows that the Tan&Ng fits √ s dependence of ratios of pt-weighted cross sections for π + , K ± andp between high energy experiments and Tan&Ng [11]. Solid lines indicate the correction functions ξH . Black, orange, green, blue and red points correspond to NA49, PHENIX, ALICE and CMS data respectively, with estimated systematic uncertainties. The yellow points represent data sets used in Tan&Ng fitting paper [11].
are consistent with these data to within ∼ ±30%, comparable to the internal variation between the results of individual analyses in this data set, and we assign this uncertainty to the orange points.
We find that the correction functions reproduce the experimentally determined p t -weighted cross sections in the range √ s ≤ 7 TeV. Several comments are in order. First, the PHENIXp data [19] in Fig. 2 exhibit larger uncertainty compared to most of the other measurements, and the central values are indeed correspondingly off by ∼ 50%, 30% for √ s=62.4 and 200 GeV from the fit. To estimate the p tweightedp cross section from [19] we start with the data without feed-down correction, as the feed-down corrected cross section is found to be lower by a factor of a few in low p t bins, which appears broadly inconsistent with the remaining data set. To estimate the feed-down corrected result, we subtract 30% off the inclusive result, as suggested by our analysis in App. B. Thep systematic uncertainties quoted in [19] are sizeable, notably in the lower p t region, due to the feed-down correction and take maximally ∼ 50%, 30% for √ s=62.4 and 200 GeV. In Fig. 2, we assign these conservative uncertainty estimates of 50% and 30% to these data. In addition to the feed-down uncertainty, the p t range covered by thep cross section data in [19] is limited, starting from p t = 0.6 GeV. This means that the p t -weighted cross section estimate derived from these data is based on a kinematically sub-dominant region for astrophysical purposes.
Second, we comment on the K S contribution to the π cross section. In the analysis of Fig. 2, we assume that the π cross sections reported by the experiments are prompt and do not include π from K S decay. The NA49 and CMS experiments explicitly state that π from K S decay are discriminated in their analyses. On the other hand, the treatment in the PHENIX and ALICE experiment is unclear. This makes 5 % ambiguity of the points from PHENIX and ALICE experiments in Fig. 2. In practice, this ambiguity is not quantitatively important for the determination of fitting formula.
Finally, we comment on the x R dependence in the high √ s regime. The high energy experimental data from [15,[17][18][19] is only specified at mid-rapidity (x R 0). This means that our fit could fail to reproduce the x R dependence in the high √ s regime. Fixing this caveat would require cross section data at non-zero x R (forward region) in the high √ s regime.

C. Comparison to previous work
In Fig. 3 we show the secondary source terms forp and π + , assuming pp production from a power-law primary proton flux J p ∝ E −3 p , comparing our results to the fitting formulae of [5] and Tan&Ng. Forp production, we now include the contributions from both hyperon decay and decay in flight ofn, using the procedure defined in App. B. The Black line shows thep source term ratio between that obtained using the fit of Ref. [5] (denoted "Winkler") and ours. We find agreement to the 10% level. The blue dotted (red dashed) line shows thep (π + ) source term ratio between Tan&Ng [11] and ours. The deviation from radial scaling, assumed in Tan&Ng, is clear at high energy.

III. THE e + /p FLUX RATIO
Ref. [24] pointed out that the production rate ratio Q e + /Qp provides a model-independent upper bound to the flux ratio of high-energy secondary CR e + andp: p . The Black line shows thep source term ratio between that obtained using the fit of Ref. [5] (denoted "Winkler") and ours. The blue dotted (red dashed) line shows thē p (π + ) source term ratio between Tan&Ng [11] and ours.
where the source terms for secondaryp and e + produced in pp collisions are 2 : This upper bound only depends on the inclusive production cross sections and the shape of proton cosmic ray flux J p .
We are now in position to extend the calculation of Q e + (R)/Qp(R) to high energy, and compare with the latest CR data. In Fig. 4 we show the upper bound predicted for different assumptions on the primary proton flux in the spallation region. The e + /p flux ratio measured by AMS-02 is consistent with the upper bound and saturates it at high energy (for proton flux coinciding with the locally measured proton flux).
Recent calculations of the high-energy secondary CRp flux [23,25], using up to datep production cross section consistent with our results here and calibrated to agree with AMS-02 B/C data, are consistent with the CRp flux measured by AMS-02. These results are reproduced in App. C. The significance, in connection with Fig. 4, is that the observed flux of CR e + at R > 100 GV coincides with the expected flux of secondary e + , that would be expected if radiative energy loss became unimportant in  the propagation at these energies. Achieving such low level of energy loss would require that the characteristic secondary CR propagation time drops below a few Myr at R > 100 GV. A comparison of the source ratio Q e + /Qp to the observed e + /p flux ratio was also presented in Ref. [26], which found results for Q e + /Qp smaller than our value by ∼ 30% in the energy range 10 -1000 GeV. This led Ref. [26] to argue that e + energy losses may be negligible at all energies (rather than only at E 100 GeV, as suggested by our Fig. 4). We have not been able to reproduce the origin of this discrepancy.

IV. CONCLUSIONS
We presented an analysis of inclusivep, π, and K production in pp collisions. Our main goal was to implement recent experimental data for meson production, in particular the effect of radial scaling violation manifest at LHC energies and recent detailed kinematical data from the NA49 experiment at intermediate energy, in semianalytic fits used for the calculation of the astrophysical secondary production of e + . We provide fitting formulae that, combined with earlier results from Tan&Ng [11], allow to compute the astrophysical production of e + andp up to the multi-TeV range with an estimated uncertainty of ∼ 20%.
The e + /p flux ratio reported by AMS-02 is found to coincide with the secondary source production rate ratio Q e + /Qp at high-energy E > 100 GeV. This coincidence may be considered as a hint for a secondary origin for CR e + andp, as it would be a fine-tuned accident in models that advocate new primary sources for either antimatter CR species.
In this section we calculate the final state e + contribution coming from the decay of K 0 L mesons. This contribution has been neglected in the literature, although the corresponding cross section is comparable to that for charged kaons which was previously taken into account. K 0 L mesons are long-lived (cτ K 0 L 15 m) in the collider set-up, so that π + from K 0 L decay are not included in the fitting formula of the inclusive π + cross section. In addition, the K 0 L semi-leptonic decay contributes directly to e + and µ + (→ e + ) production.
We consider the following decay channels [29]: We approximate and simplify the kinematics of K 0 L threebody decays, assigning each of the decay products an energy of m K /3 in the K 0 L rest frame and ignoring muon polarisation. We approximate the K 0 L production cross section to match that of K + . The e + spectrum from boosted µ + is given in Ref. [30] and the e + spectrum from boosted π + is given in Ref. [31].
The kaon contribution to astrophysical secondary e + production is highlighted in Fig. 5. The K 0 L contribution amounts roughly to 5% of the total e + source.

Appendix B: Antiproton cross section including anti-hyperon contributions
In this section we analyse the hyperon contribution to the inclusivep production cross section. We denote the Lorentz-invariant differential cross section as f ; The astrophysically relevant inclusive f tot p , which includes effects fromn and hyperon decays, can be decomposed in the following way: where f 0 indicates the prompt contribution and fΛ ,Σ denote contribution from the hyperon decay. Neglecting isospin violation, we assume f 0 p = f 0 n . To set a rough scale for the effect we're after here, the analysis in Sec. II shows f 0 1. Anti-hyperon production cross section at the NA49 experiment NA49 [20] results indicate that the kinematical distribution of anti-hyperons produced in pp collisions is somewhat different from that of anti-nucleons 3 . We introduce x R -dependent functions gB(x R ) withB =Λ,Σ ± , and parametrize the hyperon contributions as (B9) We neglect momentum difference between parent and daughter particle since their mass difference is 20%. Let us determine gB(x R ). NA49 analysis [20] (see Fig.  22 there) offers the differential multiplicity dn/dx F for Λ,Λ, Σ + , Σ − , defined as Uncertainties of dn/dx F are not presented, but a typical error estimate of ∼ 20% can be inferred from the analysis in [5].
Although the definition of x R (= E * /E * max ) and x F (= 2p * L / √ s) are different, their difference is of the order of p 2 t /s or m 2 p /s. Thus, gB(x R ) can be determined from the observation of dn/dx F . As discussed in section II, 0.1 x R 0.4 is the important kinematical region to determine secondary cosmic ray production. In this region, the p t dependence on E becomes weak and dn/dx F is determined by p t weighted averaged cross-section. In this respect, we find that dn/dx F is a directly relevant quantity for secondary cosmic ray production. Then, it is reasonable to estimate Following Ref. [22] we assume the relation Then, we expect We assume a similar relation forΣ + : To obtain gB (with B = Λ, Σ ± ), we fit the x F dependence shown in the NA49 analysis by the following form: We found (a, n)=(0.13, −3), (0.038, −3), (0.028, −2) well fitΛ,Σ − ,Σ + respectively. Fig. 6 shows the x F dependentB ≡ dnB/dx F . Solid and dashed lines correspond to NA49 values and our fitting function, respectively.

Multiplicity of anti-hyperons at large √ s
For relatively small √ s < 50 GeV, we expect that Eq. (B9) holds with weak √ s dependence. This is because, empirically, radial scaling applies at small √ s. However, when we consider large √ s > 50 GeV, we have to consider the violation of radial scaling.
Ref. [5] showed that the ratio between the multiplicity of anti-hyperons andp is not constant as function of √ s. Following [5], we introduce √ s dependence as an overall factor to the hyperon contributions, Here κ( √ s) satisfies lim s→0 κ( √ s) = 1, and deviates from unity at large √ s. We define the ratio between the multiplicity ofΛ and p at midrapidity:Λ For simplicity, we assume thatΛ,Σ ± have the same scaling law for their multiplicity. By using this assumption, we take κ( √ s) as Finally, we analyse the ratioΛ/p using data from STAR [33,34], ALICE [15,35], and CMS [18,36] which provided multiplicity ratios at mid-rapidity. NA49 also provided differential multiplicity at the mid-rapidity; we assume an uncertainty of 20% from the uncertainty in the feed-down correction. This gives usΛ/p = 0.24 ± 0.05 at √ s = 17.2 GeV form NA49 experiment. Fig. 7 shows our result. For comparison, we also show Λ/p as found in [5]. Our results can be fitted by the following formula: Λ p = 0.24 + 0.37 1 + ((146 GeV) 2 /s) 0.9 . (B19) Appendix C: Secondaryp Fig. 8 shows the secondaryp cosmic ray flux predicted by our cross section formula, calculated under the assumption that the mean target column density traversed by CR protons; He; nuclei such as B, C, and O; andp is the same as function of magnetic rigidity [24]. The column density used in the calculation is extracted from B/C data using fragmentation cross sections as specified in [23].
The simple estimate in Fig. 8 is consistent the AMS-02 p data [3]. The calculation is sensitive to a number of systematic uncertainties. The blue region shows the uncertainty of the solar modulation parameter φ = (0.2 − 0.8) GV. The grey region shows the result of varying the spectral index of proton CR above 300 GV. We vary γ p in the range of 2.6-2.8 where J p ∝ E −γp p : this should represent the possibility that the CR proton spectrum in the regions dominating secondaryp production may not be identical to the locally measured spectrum. The solid green lines show the result of varying the C→B fragmentation cross section by ±20%. Finally, the dashed dark lines representp production cross section uncertainty of ±20%.  In all cases, we use the mean traversed target column density extracted from B/C data [24] using nuclear fragmentation cross sections as specified in [23]. Solid black line shows the prediction using our fit. Dotted and dashed black lines show to the result when using the fit from Tan&Ng [11] and Winkler [5], respectively. Other bands and lines show various sources of systematic uncertainty; see text for details.