Prescriptions on antiproton cross section data for precise theoretical antiproton flux predictions

Prescriptions on antiproton cross section data for precise theoretical antiproton flux predictions

Fiorenza Donato donato@to.infn.it Dipartimento di Fisica, Università di Torino, via P. Giuria 1, 10125 Torino, Italy Istituto Nazionale di Fisica Nucleare, Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy    Michael Korsmeier korsmeier@physik.rwth-aachen.de Dipartimento di Fisica, Università di Torino, via P. Giuria 1, 10125 Torino, Italy Istituto Nazionale di Fisica Nucleare, Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy Institute for Theoretical Particle Physics and Cosmology, RWTH Aachen University, 52056 Aachen, Germany    Mattia Di Mauro mdimauro@slac.stanford.edu W.W. Hansen Experimental Physics Laboratory, Kavli Institute for Particle Astrophysics and Cosmology, Department of Physics and SLAC National Accelerator Laboratory, Stanford University, Stanford, CA 94305, U.S.A.
Abstract

After the breakthrough from the satellite-borne PAMELA detector, the flux of cosmic-ray (CR) antiprotons has been provided with unprecedented accuracy by AMS-02 on the International Space Station. Its data spans an energy range from below one GeV up to 400 GeV and most of the data points contain errors below the amazing level of 5%.
The bulk of the antiproton flux is expected to be produced by the scatterings of CR proton and helium off the inter-stellar hydrogen and helium atoms at rest. The source of these secondary s requires the relevant production cross sections. The modeling of these interactions induces an uncertainty in the determination of the antiproton source term that can even exceed the uncertainties in the CR data itself.
The aim of the present analysis is to determine the uncertainty required on cross section measurements such that the induced uncertainties on the flux are at the same level. Our results are discussed both in the center-of-mass reference frame, suitable for collider experiments, and in the laboratory frame, as occurring in the Galaxy. We find that cross section data should be collected with accuracy better that few percent with proton beams from 10 GeV to 6 TeV and a pseudorapidity ranging from 2 to almost 8 or, alternatively, with from 0.04 to 2 GeV and from 0.02 to 0.7. Similar considerations hold for the He production channel. The present collection of data is far from these requirements. Nevertheless, they could in principle be reached by fixed target experiments with beam energies in the reach of the CERN accelerators.

Introduction

Astroparticle physics of charged cosmic rays (CRs) has become a high-precision discipline in the last decade. Spaceborne experiments like first PAMELA and more recently and still operating AMS-02 have reduced the measurement uncertainties of CR fluxes to the percent level over an energy range from below GeV up to a few TeV, which is typical for Galactic CRs. In particular, this result has been achieved for the nuclear PAMELA_Adriani:2011cu (); 2014ApJ…791…93A (); AMS-02_Aguilar:2015_ProtonFlux (); AMS-02_Aguilar:2015_HeliumFlux (); Aguilar:2016vqr () and leptonic (positron and electron) 2009Natur.458..607A (); 2011PhRvL.106t1101A (); 2013PhRvL.110n1102A (); 2014PhRvL.113l1102A (); 2014PhRvL.113v1102A () components, and also for the CR antiprotons 2010PhRvL.105l1101A (); AMS-02_Aguilar:2016_AntiprotonFlux (). The rare CR antimatter has been extensively studied as a possible indirect signature of dark matter annihilating in the halo of the Galaxy 2010pdmo.book..521S () (and Refs. therein). The recent AMS-02 data on the antiproton flux and flux ratio has reached an unprecedented precision from about one GeV up to hundreds of GeV AMS-02_Aguilar:2016_AntiprotonFlux ().
This exceptional experimental accuracy poses the challenging task of a theoretical interpretation with an uncertainty at a similar level. The dominant part of the antiprotons in our Galaxy arises from secondary production, namely it originates by the inelastic scattering of incoming CRs off interstellar medium (ISM) nuclei at rest. In practice, the secondary antiprotons are produced form the scattering of CR and He on ISM consisting again of H and He. The number of produced antiprotons then depends on the correct modeling of the production cross section and the equivalent reactions with He instead of . The production cross sections induce a non-negligible uncertainty in the prediction of the secondary antiprotons, as already underlined in Donato:2001_DTUNUC (); Bringmann:2006im (); Donato:2008jk (). In the literature, there have been different approaches to describe the production cross section, after the first parameterization for the scattering TanNg:1983_pbarCrossSectionParameterization (), now probably outdated. Monte Carlo (MC) predictions have been employed in particular for the He channels using the DTUNUC code Simon_Antiproton_CS_Scaling_1998 (); Donato:2001_DTUNUC (), for which no measurement was available. Very recently, the LHCb Collaboration has presented a preliminary analysis on the search for antiprotons in collisions of 6.5 TeV protons on a fixed helium target at the LHC LHCb_pHe (). A parameterization deduced from a large and (proton-nuclei) data set was proposed in Duperray:2003_pbarCrossSectionParameterizationForPA (). Recently, triggered by the NA49 data NA49_Anticic:2010_ppCollision (), new parameterizations have been proposed in diMauro:2014_pbarCrossSectionParameterization (); Kappl:2014_pbarCrossSection () as well as predictions from MC generators tuned with LHC data Kachelriess:2015_pbarCrossSection (). Nevertheless, the theoretical uncertainty induced by the modeling of fundamental interactions on the antiproton spectrum is sizable, reaching a few ten %.
In this paper we “backwards engineer” the usual process of cross section parameterization in order to determine the accuracy required on cross section measurements so to match AMS-02 accuracy. Our aim is to provide, for the first time, quantitative indications for future high-energy experiments about the kinematical regions and the precision level they should cover, in order to induce uncertainties in flux which do not exceed the uncertainty in present CR data.
This paper is structured as follows. In Sec. I we review the main steps for the calculation of the antiproton source term starting from the invariant cross section. In Sec. II we explain how we invert this calculation in order to assign uncertainty requirements on the differential cross section. The results are presented in Sec. III and are summarized in Sec. IV.

I Theoretical framework for the cosmic antiproton source spectrum

Figure 1: Recent flux measurements for CR protons, helium, and antiprotons by AMS-02 AMS-02_Aguilar:2015_ProtonFlux (); AMS-02_Aguilar:2015_HeliumFlux (); AMS-02_Aguilar:2016_AntiprotonFlux (), PAMELA PAMELA_Adriani:2011cu (); PAMELA_Adriani:2012paa (), and CREAM CREAM_Yoon:2011aa (). The energy-differential fluxes are given as function of kinetic energy per nucleon . Furthermore, the IS fluxes, demodulated in the force-field approximation with an modulation potential of MV, are presented.

Antiprotons in our Galaxy are dominantly produced in processes of CR nuclei colliding with ISM. Hence, the ingredients to calculate the source term, i.e. the number of antiprotons per volume, time, and energy, are the flux of the incident CR species , , and the density of the ISM component , where, in practice, both and are and He. The source term for secondary antiprotons is given by a convolution integral of the CR flux, the ISM targets and the relevant cross section:

(1)

Here is the ISM density and the production energy threshold. The factor corresponds to the already executed angular integration of the isotropic flux . The according fluxes are known precisely at the top of the Earth’s atmosphere (TOA) due to AMS-02 measurements AMS-02_Aguilar:2015_ProtonFlux (); AMS-02_Aguilar:2015_HeliumFlux () presented in Fig. 1, together with the results from the precursor satellite-borne PAMELA experiment PAMELA_Adriani:2011cu (); PAMELA_Adriani:2012paa () and the data from the balloon-borne CREAM detector at higher energies CREAM_Yoon:2011aa (). At low energies 20 GeV/nucleon (in the following GeV/n) the charged particles arriving at the Earth are strongly affected by solar winds, commonly referred to as solar modulation Parker:1958_SolarModulation (); Gleeson:1968_SolarModulation (), given their activity modulation on a cycle of roughly 11 years. We will work with interstellar (IS) quantities. The and He IS fluxes are inferred by demodulated AMS-02 data, which we obtain within the force-filed approximation Fisk:1976_SolarModulation () assuming an average Fisk potential of  MeV for the period of data taking Usoskin:2005_SolarModulation (); Usoskin:2011_SolarModulation (). More complete studies on solar modulation take into account time dependent proton flux data from PAMELA and recent ISM flux measurements by VOYAGER Cholis:2016_Solar_Modulation (); Ghelfi:2016pcv (); Corti:2015bqi (). They find similar values for . The source term derivation only includes incoming proton energies GeV () corresponding to the production threshold in (He) collisions. For these energies the solar modulation, which becomes negligible above a few 10 GeV, agrees reasonably well with the simple force-field approximation. The scattering sights are the ISM elements H and He with density given by 1 and 0.1 in the Galactic disk respectively.

Figure 2: Energy-differential antiproton production cross section from collisions in LAB frame as function of proton and antiproton kinetic energy and , respectively. The shown cross section is derived from the Di Mauro et al. diMauro:2014_pbarCrossSectionParameterization () parameterization (their Eq. 12).

The final essential ingredient to calculate the source term is the cross section corresponding to the production reaction

(2)

where is the kinetic energy of the produced antiproton in collisions of CR species with kinetic energy on the ISM component . In the following we will call the quantity in Eq. (2) the energy-differential cross section111Note that and, hence, .. One example, derived form the cross section parameterization in Ref. diMauro:2014_pbarCrossSectionParameterization () for the channel, is shown in Fig. 2 as a function of and . The kinetic energy threshold at is clear.

(a)
(b)
(c)
Figure 3: Profiles for fixed (a) proton energy and (b) antiproton energy of the energy-differential cross section in LAB frame from Fig. 2. In addition, cross section parameterizations by Tan&Ng TanNg:1983_pbarCrossSectionParameterization (), Duperray et al. Duperray:2003_pbarCrossSectionParameterizationForPA () (their Eq. 6I), Kachelriess et al. Kachelriess:2015_pbarCrossSection () and Winkler Winkler:2017xor () are shown for comparison. Panel (c): as panel (b), but for the scattering. Here we add the DTUNUC parameterization Simon_Antiproton_CS_Scaling_1998 (); Donato:2001_DTUNUC ().
11footnotetext: As discussed in diMauro:2014_pbarCrossSectionParameterization () the parameters and have to be interchanged.

The production cross section is not directly available in the energy-differential form from Eq. (2), which also enters in Eq. (1). Experiments rather measure the angular distribution on top of the energy-differential cross section and then present the Lorentz invariant (LI) form

(3)

where and are total energy and momentum, respectively, is the center of mass (CM) energy of the colliding nucleons, (* refers to CM quantities) is the ratio of the energy to the maximally possible energy in the CM frame, and is the transverse momentum of the produced antiproton. Note that also the three kinematic variables are LI quantities. We skipped the subscripts for projectile and target to avoid unnecessarily complicated notation. Anyway, Eq. (3) and also the following equations are valid for all combinations of projectile and target, as long as all quantities are understood in the nucleon-nucleon system.

To relate the LI cross section to the energy-differential one in Eq. (2) two steps have to be performed. Firstly, the LI kinetic variables need to be related to an equivalent set in the LAB frame, where the target is at rest. Typically, the set is given by the projectile and the kinetic energies, and the scattering angle . We give explicit relations in Appendix A. In a second step, the angular integration has to be performed

(4)

Here is the angle between the incident projectile and the produced antiproton in LAB frame. In the second line of Eq. (4) we transform the angular integration to an integration w.r.t. the pseudorapidity defined as

(5)

This transformation is advantageous because the invariant cross section is very peaked in forward direction at small angles. Again, a more detailed derivation of Eq. (4) is stated in the Appendix’s Eq. (23). Concerning the limits of the angular integration, we notice that from it is possible to derive a precise or, equivalently, . Nevertheless, in practice it is sufficient to start both integrals in Eq. (4) from 0, which is trivially a lower limit.

Parameterization Species Info Year Ref.
Winkler , He,He,HeHe analytic 2017 Kappl:2014_pbarCrossSection (); Winkler:2017xor ()
di Mauro et al. analytic 2014 diMauro:2014_pbarCrossSectionParameterization ()
Duperray et al. , , analytic 2003 Duperray:2003_pbarCrossSectionParameterizationForPA ()
Kachelriess et al. , , , high-energy MC, inclusive, LAB frame 2015 Kachelriess:2015_pbarCrossSection ()
DTUNUC He, He, HeHe low-energy MC, LAB frame 1998 Simon_Antiproton_CS_Scaling_1998 (); Donato:2001_DTUNUC ()
Tan&Ng analytic 1983 TanNg:1983_pbarCrossSectionParameterization ()
Table 1: Summary of available cross section parameterizations for the antiproton production.

With this information at hand, we compare three different parameterizations of the cross section, as given by diMauro:2014_pbarCrossSectionParameterization (); TanNg:1983_pbarCrossSectionParameterization (); Duperray:2003_pbarCrossSectionParameterizationForPA (); Winkler:2017xor () and the MC approach in Kachelriess:2015_pbarCrossSection () (see also the information in Table 1). Fig. 3 displays profiles of the energy-differential cross section for either fixed or . At antiproton energies of a few 10 GeV, which are dominantly produced by protons with an energy of a couple of 100 GeV, all approaches agree well. However, at lower and higher energies the picture is different. In particular, for antiproton energies below 10 GeV the deviation between the different approaches is significant. The MC approach in Kachelriess:2015_pbarCrossSection (), which is intrinsically designed for and trained with high-energy data, is expected to break down below 10 GeV, as clearly visible in the plots. Also the different analytical forms have discrepancies of up to a factor 2 at 1 GeV. We notice that the parameterizations in diMauro:2014_pbarCrossSectionParameterization (); Winkler:2017xor () are driven by the NA49 data NA49_Anticic:2010_ppCollision (), taken at =17.3 GeV and covering antiproton energies from about 8 GeV up to 70 GeV. The plotted cross sections also include the antiprotons from antineutron decay. Given the CR propagation timescale, the immediately decays into and has to be included in the antiproton source term. The simplest correction is to multiply the cross section by two. However, NA49 data suggest that the cross section is larger than the one by roughly 50% around =0 NA49_Anticic:2010_ppCollision (). Hence, in order to properly include the inclusive cross section, we multiply the one by 2.3, here and in the following. Ref. Kachelriess:2015_pbarCrossSection () directly provides the inclusive cross section based on their modified MC simulations with QGSJET-II and Winkler:2017xor () gives a specific formula for the isospin violation term. More details might be found e.g. in diMauro:2014_pbarCrossSectionParameterization (); Kappl:2014_pbarCrossSection (); Winkler:2017xor (). Moreover, the parameterization from Winkler:2017xor () contains explicitly the contribution from hyperons, as it is likely the case also for the MC predictions in Kachelriess:2015_pbarCrossSection (). In panel (c), we also plot literature results for the channel. Concretely, only the MC approach in Kachelriess:2015_pbarCrossSection () provides both the He channels individually, while the parameterization in Duperray:2003_pbarCrossSectionParameterizationForPA () is fitted to data ( as the projectile, nuclei heavier than helium as targets) while the one in diMauro:2014_pbarCrossSectionParameterization () is only valid for . In order to get the He for the parameterization in Duperray:2003_pbarCrossSectionParameterizationForPA (), and for both He channels from the results in diMauro:2014_pbarCrossSectionParameterization (), the following is done. Naturally, the He cross section might be Lorentz transformed to He one. However, Ref. Duperray:2003_pbarCrossSectionParameterizationForPA () already assumes a cross section parameterization with z-symmetry222Symmetry with respect to the plane perpendicular to the incident particles, namely no dependence on the longitudinal momentum . in the nucleon-nucleon CM system. In this way, we directly conclude that both He channels (He and He) are equal as functions of energy per nucleon. For the up-to-date parameterization in diMauro:2014_pbarCrossSectionParameterization (), we extract and translate the dependence from Duperray:2003_pbarCrossSectionParameterizationForPA (). Again Kappl:2014_pbarCrossSection (); Winkler:2017xor () gives its own scaling for the He and He cross sections. We see from Fig. 3 that different choices for the He cross section lead to significantly different results, in particular below of about 10 GeV, where the discrepancy can reach a factor ten.

Figure 4: Antiproton source term for CRs of the tree dominant channels , He, and He for different parameterizations by di Mauro et al. diMauro:2014_pbarCrossSectionParameterization () (their Eq. 12), Tan&Ng TanNg:1983_pbarCrossSectionParameterization (), Duperray et al. Duperray:2003_pbarCrossSectionParameterizationForPA () ( from their Eq. 6I and Eq. 4 for heavier channels), Winkler Winkler:2017xor () and Kachelriess et al. Kachelriess:2015_pbarCrossSection ().

In Fig. 4 we present the source term for the three most important production channels , He, and He. Although some of the differences among the cross sections might be integrated out at the source term level (see Eq. (1)), Fig. 4 shows that in fact most of the deviations above GeV and below 10 GeV remain. The predictions from the Winkler parameterization are systematically higher than the other ones above 10 GeV. One possible explanation could be the explicit inclusion of antiprotons from the decay of hyperons. The approach in Kachelriess:2015_pbarCrossSection () predicts a much larger source term at low energies. We are aware of the fact that also the heavier elements, namely C and O, contribute to the source term at the percent level. However, the purpose of this analysis is to determine the cross section parameter space of the dominant production channels. In this regard, it is sufficient to consider only and He each as projectile and target.

Ii Methods for determining the precision on the cross section

The general idea of the present analysis is to determine the uncertainty requirements on cross section measurements for a given cosmic antiproton flux accuracy. Firstly, we determine the contribution to the source term from each point in the parameter space of the fully differential LI cross section. Then we derive the uncertainty requirements on cross section measurements according to two principles: (i) the total uncertainty shall match experimental flux accuracy dictated by AMS-02, which provides the currently most precise measurement, and (ii) in the parameter space regions, where the cross sections provide a dominant contribution to the source term, we require higher accuracy. In the following we will provide a detailed explanation of our strategy.

We start from the uncertainty level in the antiproton flux measurement. Our prior is the AMS-02 data, which display the most accurate determination over the widest energy range (see Fig. 1). This TOA spectrum of AMS-02 has to be related to the source spectrum. First of all, we have to extract the local IS flux from the TOA one. We choose to correct the effect of the solar modulation by means of the force-field approximation, dictating to simply shift all data points by , having fixed MV. Then, the relation between the IS flux and the source spectrum is given by propagation in the Galaxy, which is usually described by a diffusion equation (see Donato:2001_DTUNUC () and refs. therein).

As a first approximation, it is reasonable to assume equal relative uncertainties of IS flux and the source term above GeV. The diffusion term indeed is such to keep the ratios almost unaffected. The only possible distortion between the propagated flux and the source spectrum may arise from convection or reacceleration at very low energies. However, we explicitly checked by propagating several strongly peaked toy source-term spectra and comparing them with the resulting propagated flux that the energy distortion is negligible down to 1 GeV. Baseline for the cross check is the Galprop-based global analysis of CR propagation performed in 2016PhRvD..94l3019K () (and refs. therein). Conclusively, we use the relative flux uncertainty as proxy of the source term uncertainties :

(6)
Figure 5: Relative uncertainty of the AMS-02 antiproton flux measurement AMS-02_Aguilar:2016_AntiprotonFlux (). We show the demodulated IS flux, in other words, all data points are shifted by 600 MeV.

The quantity can be read from Fig. 5, where it has been derived from the AMS-02 measurements AMS-02_Aguilar:2016_AntiprotonFlux (). From this figure we can clearly see the precision level of the current AMS-02 data. In particular, it is about 5% between 1 and 100 GeV. This is the minimum level of accuracy which is required to any prediction. The uncertainty of the source term has to be distributed on the single production channels. We will assume that the relative uncertainties in all the and He channels are equal, meaning

(7)

To determine the contribution from each parameter point of the invariant cross section we work with the full expression of the source term. Inserting Eq. (4) in Eq. (1) and changing the energy integration to results in

(8)

where we have dropped the labels indicating the species of the incoming CR flux and ISM. We then define the containment

(9)

The integrand of the source spectrum has been defined in Eq. (8). The containment function varies between and has to be understood as follows. The parameter space with e.g. contains 90% of the source term at the given antiproton energy. To be more precise, we calculate the smallest containment areas in the variables and , for each given . In practice, this is obtained by calculating the integral from Eq. (9) in the following way. On the two dimensional grid at fixed , we sum the integral of over the and bin starting from the largest to the smallest contribution, until the required value of is reached. Although there is a freedom to choose the exact set of kinetic parameters, the general behavior of containment contours is not largely affected, because the integrand is a strongly peaked function. Furthermore, cross sections and source terms are power laws. The choice of logarithmic variables, and , respects their natural scaling properties.

Figure 6: Isocontours for the containment function for contribution of the channel to the antiproton source term as function of the kinetic variables in LAB frame, namely kinetic energy of the incident proton and pseudorapidty of the antiproton , at a fixed antiproton kinetic energy GeV. A 90% level corresponds to =0.9.

Fig. 6 illustrates the containment parameter space for the channel when = 0.90, 0.99 and 0.999, as function of and at fixed GeV. Here and when not differently stated, we assume the cross section parameterization as in Ref. diMauro:2014_pbarCrossSectionParameterization (). However, we do not expect large systematic deviations by changing the underlying cross section parameterization. We explicitly verify this by changing to the Duperray et al. Duperray:2003_pbarCrossSectionParameterizationForPA () or Winkler Winkler:2017xor () expression. Note, that it is not possible to check any of the MC results (like for example in Kachelriess:2015_pbarCrossSection ()) because they are not available with complete angular dependence. In Fig. 6 we can see that the 90% of the antiprotons at = 50 GeV are produced by protons with energies spanning about 90 GeV - 3 TeV and between 2 and 7, depending on . In order to include the 99.9% of the source spectrum, one has to consider protons with energies up to 70 TeV with pseudorapidity values around 5.

From the containment it is easy to assign values for . Since increasing corresponds to decreasing contribution by construction, shall increase while varies from 0 to 1. By increasing up to 1, one increases the spanned kinematical parameter space. This correspondence ensures that our initial requirement (ii), see beginning of Sec. II, namely, more accurate determinations of the cross sections in the parameter space with large contribution, is fulfilled.

Variable [unit] Range Size/Scale
[GeV/n] 400/log
[GeV] 400/log
400/linear
[GeV] 400/log
[GeV] 400/log
400/log
Table 2: Grid properties for the LAB and CM frames variables adopted in our numerical calculations.

We fix a precision level for the relative uncertainties on the LI cross section for production. To keep things simple we choose the step function

(10)

where is a threshold value for the containment function. The values of 3% and 30% are a free choice, which however is suggested by by the most precise values of current cross section measurements (see Fig. 9 in the following), on the one hand, and by the spread of various parameterizations in the energy regime of interest, on the other hand. Anyway, we will provide a comparison with different assumptions for the step function in Sec. III. Finally, the threshold is fixed by the requirement to match AMS-02 accuracy, which is guaranteed by solving

(11)

In practice, the right hand side of this equation is taken from the parameterization of the uncertainties on the data as reported in Fig. 5.

This procedure allows us to derive the required levels for . Technically, we determine on a grid in the LAB frame variables summeried in Table 2. To get the distribution in we transform those three variables to and then interpolate on our grid. To get a bijective mapping we have to add an assumption on the sign of . So we get two values for each set of and choose the minimum.

(a) LAB frame
(b) CM frame
Figure 7: Parameter space of the to cross section necessary to determine the antiproton source term with the accuracy reached by recent AMS-02 measurements AMS-02_Aguilar:2016_AntiprotonFlux (). Here we require that the cross section has to be known by 3% within the blue shaded regions and by 30% outside the contours. The left (right) panel displays the result for the LAB (CM) reference frame variables.

Iii Results

We derive the parameter space of the inclusive cross section which should be covered to determine the antiproton source term with the accuracy dictated by recent AMS-02 measurements AMS-02_Aguilar:2016_AntiprotonFlux (). We show our results as functions of the kinematical variables in both the LAB and CM reference frames. As explained in the previous section, in the LAB frame the parameter space is described by the kinetic energy of the proton , the kinetic energy of the antiproton , and and pseudorapidty of the antiproton . Equivalently, the results can be expressed in terms of the CM frame, given by the CM energy , the ratio between the antiproton energy and its maximal energy , and the transverse momentum of the antiproton . When not differently stated, the cross section parameterization is chosen as in di Mauro et al. diMauro:2014_pbarCrossSectionParameterization ().

Fig. 7 shows the parameter space that has to be covered in order to guarantee the AMS-02 precision level on the source term, if the cross section is determined with 3% uncertainty within the blue shaded regions and by 30% outside the contours. The plot is done for the LAB (left panel, a) and CM (right panel, b) reference frame variables. For the LAB frame we show the contours as functions of and , for selected values of from 1.1 (the lowest energy below 30% uncertainty in the CR flux, see Fig. 5) to 300 GeV. As expected the contour size decreases when approaches to 1 GeV, because there the AMS-02 uncertainty on the antiproton flux reaches 30%. A similar explanation holds for large . Antiprotons of increasing energy require the coverage of increasing values. For example, the at =2 GeV is known at 3% level if data were taken with proton beams between 10 and 200 GeV and pseudorapidity from 1.8 to 4. If the whole AMS-02 energy range had to be covered with high precision, one should collect cross section data with proton beams from 10 GeV to 6 TeV, and increasing from 2 to nearly 8. Fig. 7b displays analogous information in the CM reference frame. We fix to representative values from 5 to 110 GeV, and identify the regions in the plane. A full coverage of the parameter space should scan from 0.04 to 2 GeV, and from 0.02 to 0.7. For each value of , the extension of the contour within which the cross section is required at 3% precision level is correlated to the AMS-02 uncertainty on the antiproton flux.

(a) LAB frame
(b) CM frame
Figure 8: Different systematic setups are compared (a) in the LAB frame, exemplary at GeV, and (b) CM frame at fixed GeV. The standard setup with cross section by di Mauro diMauro:2014_pbarCrossSectionParameterization () and inner/outer uncertainty 3%/30%, is altered in one of the following details: (i) the cross section is replaced by the Winkler Winkler:2017xor () (dotted line) or Duperray parametrization Duperray:2003_pbarCrossSectionParameterizationForPA () (their Eq. 6) (dashed line), (ii) inner/outer uncertainty are replaced by 2%/30% (long-dashed line) or 3%/100% (dot-dashed line), and (iii) instead of the He channel is considered (double-dot-dashed line).

Furthermore, we estimate possible systematic effects on our predictions. In Fig. 8 we present the same information as in Fig. 7, now with different parameterizations for the cross section and modified requirements on the uncertainty levels. We remind that our standard setup is fixed by the as in Ref. diMauro:2014_pbarCrossSectionParameterization () and uncertainty requirements of 3%/30% (inner/outer regions) as in Eq. (10). In Fig. 8a we display the results in the LAB reference frame for =50 GeV. Changing the parameterization for the cross section to the ones in Winkler:2017xor () or Duperray:2003_pbarCrossSectionParameterizationForPA () (their Eq. 6) has negligible effects. Instead, a smaller (higher) inner implies a smaller (larger) contours. Moving from 2%-30% to 3%-100%, the needed coverage pushes 4 TeV to 20 TeV. As a final point in this figure we address the He cross section, where the computation is performed in the standard setup. It is interesting to note that the covered parameter space is very similar to the one. This result is expectable because at first order the He and cross sections are simply related by a rescaling, which drops out of the calculation. In this regard, it is possible to interpret all plots with results also for the heavier channels He or He. Note however that in this case has to be understood to be the kinetic energy per nucleon of the projectile. Naturally, all considerations also hold for the representation in the CM reference frame, as shown in Fig. 8b.

Figure 9: Comparison of the statistical measurement uncertainty by the experiment NA49 NA49_Anticic:2010_ppCollision () to our standard requirement, 3% accuracy inside the contour and 30% outside, which is shown by color grades.

In Fig. 9 we display the state of the art by showing the NA49 data NA49_Anticic:2010_ppCollision () on the . Data is taken at GeV corresponding to a proton beam at 158 GeV momentum on fixed proton target. The points show the statistical uncertainty in the CM parameter space, while an overall systematic error of 3.3% (quadratic sum) is omitted. The parameter space of NA49 matches well with the requirement of AMS-02 coverage. Actually, only four data points, indicated by the green border, fulfill the uncertainty requirement. The other data points exceed the 3% error level even by a large amount. We notice that due to the high density of points, which can improve the shape determination of , the statistical uncertainties might decrease after the integration leading to the source term. However, it is not straightforward to estimate the amount of this improvement, if any, as the cross section has to be folded with the primary fluxes of or He, which are grossly power laws with spectral index 2.7. On the other hand, the 3.3% overall systematic uncertainty is irreducible. Similar information for a selection of data samples collected in the previous decades are shown in the Appendix B.

Figure 10: Parameter space for the He channel corresponding to an exemplary fixed target experiment. The different shaded areas correspond to different proton beam energies.

Given the relevance of the helium production channels in the spectrum, we come back to the discussion of heavier channels and explicitly study the parameter space an experiment, using high energy protons scattering off a fixed helium target, should have in order to fulfill the 3%-30% requirement and reach the AMS-02 precision of the spectrum. We remind that, as stated in Eq. (7), the relevant cross section uncertainties are set equal for all the production channels. In Fig. 10 we display the according contours in the parameter space of antiproton momentum and transverse momentum. All variables are in the fixed target frame and, hence, the conveyed information is very similar to Fig. 7(a). The parameter space which has to be measured spans from below 10 GeV to more than 6.5 TeV, while the required momentum tracks the AMS-02 measurement range from about 1 to 350 GeV. The LI transverse momentum remains, as expected from Fig. 7(b), between 0.04 and 2 GeV. Again, at below 10 GeV or equivalently above 2 TeV the size of the contours shrinks because then the dominant production of antiproton is below or above the AMS-02 measurement range, respectively.

Figure 11: Future projection. AMS-02 in 2024 and 5% constant uncertainty.

Finally, we undergo the exercise to investigate future requirements on cross section measurements to explain the highest energies which is be measured by AMS-02. Fig. 11 is a dedicated plot for fixed GeV, which corresponds to the central value of the last energy bin of the current AMS-02 measurement. The figure contains the same information as Fig. 7(a), displaying the usual contours with 3% and 30% accuracy requirement for inner and outer parameter space, respectively. We find that cross section should be measured with a proton beam of TeV and around 7. The corresponding CM frame variables are GeV, , and GeV, which are probably more feasible for experiments than the high pseudorapidity values required in the LAB frame.
In the present situation the large energy part GeV of the AMS-02 flux is dominated by statistical uncertainties. Hence, until the scheduled operation of AMS-02 in 2024, the accuracy will increase. To estimate this improvement, we assume that systematical uncertainties stay constant, while the statistical uncertainty reduces according to the Poisson statistics. The current AMS-02 analysis contains 4 years of data and until 2024 AMS-02 may collect 13 years of data. Accordingly, we rescale the statistical error by a factor of . As expected, the contour in Fig. 11 slightly increases. As a last step, we show the size of a contour if we require 5% accuracy on the whole AMS-02 energy range. This leads to a significant increase of the contour. Now fixed target measurements would need to cover a parameter space for beam energies from about 0.5 TeV up to 30 TeV, and for pseudorapidity from 6 to above 9.

Iv Summary and Conclusion

Antiprotons in CRs have been measured with space-based experiments with unprecedented accuracy. After the high-precision measurement by the PAMELA detector, the flux of cosmic-ray (CR) antiprotons has been provided with unprecedented accuracy by AMS-02 on the International Space Station, with data in the energy range 0.5-400 GeV and errors that are as low as few 5%. The CR antiprotons are expected to be produced by the scatterings of CR proton and helium off the interstellar medium, made by hydrogen and helium at rest. The inclusive cross sections for the possible interactions induce a significant uncertainty in the determination of the antiproton source term and finally on the antiproton flux.
In this paper we have determined the requirements on the kinamatical parameter space that the cross section measurements should cover in order not to induce uncertainties in the theoretical predictions exceeding the ones inherent in the CR flux data. We have assumed that the cross sections could be measured with a few % accuracy in the relevant regions. Our analysis is performed in terms of the source term, which is a proxy of the flux and of its precision level, and is the convolution of the progenitor CR fluxes with the ISM targets and the relevant production cross sections. Our results are discussed both in the center-of-mass reference frame, suitable for collider experiments, and in the laboratory frame, as occurring in the Galaxy. To cover all the AMS-02 energy range with the cosmic data precision level, which are now of the order of 5%, one should collect cross section data with proton beams from 10 GeV to 6 TeV and pseudorapidity increasing from 2 to nearly 8. Alternatively, a full coverage of the CM parameter space should scan from 0.04 to 2 GeV, and from 0.02 to 0.7. Similar requirements are found for the and channels. These conclusions are not affected by different choices of the cross section parameterization. The necessary kinematical coverage is still far from the present collection of data, but it could be fulfilled by fixed target experiments with energies from tens of GeV up to few TeV, which are in the reach of CERN accelerators.

Acknowledgments

We would like to thank A. Cuoco, G. Graziani, D. Maurin, G. Passaleva and M.W. Winkler for very useful discussions. This work is supported by the research grants TAsP (Theoretical Astroparticle Physics) and Fermi funded by the Istituto Nazionale di Fisica Nucleare (INFN). M. di Mauro acknowledges support by the NASA Fermi Guest Investigator Program 2014 through the Fermi multi-year Large Program N. 81303 (P.I. E. Charles).

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Appendix A Useful kinematics

Maximal energy of product particles

Assuming the generic process in the CM frame we solve for the total energy .

(12)

Here are the four-momenta of particle . In the case of production in scattering, we have and, due to baryon number conservation, . Therefore, the maximal energy allowed for the produce antiproton - which enters in the definition of - is:

(13)

Inertial frames and Lorentz transformation

During the analysis we use two inertial frames. On the one hand, there is the CM frame of the proton-proton or, in the more general case, the nucleon-nucleon scattering. Variables in this system are denoted with a * superscript in the following. On the other hand, in the LAB frame one of the particles is at rest. The LI square of the CM energy is

(14)

where is the total energy of the two protons in the CM frame, is the incident proton energy in LAB frame, and is the proton (nucleon) mass. Formally the relation between energy and momentum between the two frames is given by the Lorentz transformation

(15)
(16)

Here is the particle velocity in terms of the speed of light (we use the convention ) and is the corresponding Lorentz factor. All of them are linked as follows:

(17)
(18)
(19)

Relation of kinetic variables

Here we give explicitly the relation between the CM frame variables and the LAB frame variables . From Eq. (14) we infer

(20)

The transverse momentum is invariant under Lorentz transformation

(21)

where we used for .

Finally, we get:

(22)

Energy-differential and invariant cross section

(23)

We used in the last line.

Appendix B Parameter space covered by experiments. Additional plots.

We extend the information already given in Fig. 9 to several cross sections data prior to NA49. The data are shown in Fig. 12 and 13 for increasing as functions of and . It is evident from these figures that the data have large statistical errors and do not properly cover the parameter space region required by the AMS-02 data. The overall systematic errors have been omitted here but are reported in Table 3. These figures are a simplified visualization of the conclusions already gotten from the analysis of this data as performed, i.e., in diMauro:2014_pbarCrossSectionParameterization ().

Experiment Systematic Ref.
uncertainty
Dekkers et al, CERN 1965 - Dekkers:1965zz ()
Allaby et al, CERN 1970 15% Allaby:1970jt ()
Capiluppi et al, CERN 1974 - Capiluppi:1974rt ()
Guettler et al, CERN 1976 4% Guettler:1976ce ()
Johnson et al, FNAL 1978 7% Johnson:1977qx ()
Antreasyan et al, FNAL 1979 20% Antreasyan:1978cw ()
Phenix, BNL 2007 9.7%-11% Arsene:2007jd ()
BRAHMS, BNL 2008 12%-19% Arsene:2007jd ()
NA49, CERN 2010 3.3% NA49_Anticic:2010_ppCollision ()
Table 3: Summary of cross section measurements and their systematical uncertainties not shown in Fig. 12 and 13. Note that Dekkers:1965zz () and Capiluppi:1974rt () do not distinguish between statistical and systematical uncertainties.
Figure 12: Collection of further experimental data compared to the standard requirement of 3% accuracy inside the contour and 30% outside at different values for . These plots complete the picture presented in Fig. 9. We display the statistical uncertainty, while additional systematic uncertainties are summarized in Table 3.
Figure 13: Same as Fig. 12, but for data at increasing
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