Cross-sections of large-angle hadron production in proton– and pion–nucleus interactions II: beryllium nuclei and beam momenta from ±3 GeV/c to ±15 GeV/c

We report on double-differential inclusive cross-sections of the production of secondary protons and charged pions, in the interactions with a 5% λabs thick stationary beryllium target, of proton and pion beams with momentum from ±3 GeV/c to ±15 GeV/c. Results are given for secondary particles with production angles 20° 


Introduction
The HARP experiment arose from the realization that the inclusive differential cross-sections of hadron production in the interactions of few GeV/c protons with nuclei were known only within a factor of two to three, while more precise cross-sections are in demand for several reasons. Consequently, the HARP detector was designed to carry out a programme of systematic and precise measurements of hadron production by protons and pions with momenta from 1.5 to 15 GeV/c.
The detector combined a forward spectrometer with a large-angle spectrometer. The latter comprised a cylindrical a e-mail: joerg.wotschack@cern.ch b Also at the Moscow Institute of Physics and Technology, Moscow, Russia. c On leave of absence at Ecole Polytechnique Fédérale, Lausanne, Switzerland.
Time Projection Chamber (TPC) around the target and an array of Resistive Plate Chambers (RPCs) that surrounded the TPC. The purpose of the TPC was track reconstruction and particle identification by dE/dx. The purpose of the RPCs was to complement the particle identification by time of flight.
The HARP experiment was performed at the CERN Proton Synchrotron in 2001 and 2002 with a set of stationary targets ranging from hydrogen to lead, including beryllium.
Here, we report on the large-angle production (polar angle θ in the range 20 • < θ < 125 • ) of secondary protons and charged pions in the interactions with a 5% λ abs Be target of protons and pions with beam momenta of ±3.0, ±5.0, ±12.0, and ±15.0 GeV/c. We have reported earlier [1] on results from beam momenta of +8.9 and −8.0 GeV/c.
Our work involves only the HARP large-angle spectrometer. The detector characteristics and our analysis algorithms have been described in Ref. [1].
The data analysis presented in this paper rests exclusively on the calibrations of the TPC and the RPCs that we, the HARP-CDP group, published in Refs. [2] and [3]. As discussed in Refs. [4] and [5], and summarized succinctly in the appendix of Ref. [1], our calibrations disagree with those published by the 'HARP Collaboration' [6][7][8][9]. Conclusions of independent review bodies on the discrepancies between our results and those from the HARP Collaboration can be found in Refs. [10,11].

The T9 proton and pion beams, and the target
The protons and pions were delivered by the T9 beam line in the East Hall of CERN's Proton Synchrotron. This beam line supports beam momenta between 1.5 GeV/c and 15 GeV/c, with a momentum bite p/p ∼ 1%.
Beam particle identification was provided for by two threshold Cherenkov counters, BCA and BCB, filled with nitrogen, and by time of flight over a flight path of 24.3 m. Table 1 lists the beam instrumentation that was used at different beam momenta for p/π + and for π /e separation.
The pion beam had a contamination by muons from pion decays. It also had a contamination by electrons from converted photons from π 0 decays. Only for the beam momenta of 3 and 5 GeV/c were electrons identified by a beam Cherenkov counter and rejected.
The fractions of muon and electron contaminations of the pion beam were experimentally determined [12,13] and are listed in Table 2 for all beam momenta. 1 For the determination of interaction cross-sections of pions, the muon and electron contaminations must be subtracted from the incoming flux of pion-like particles (except electrons at the beam momenta of 3 and 5 GeV/c).
The beam trajectory was determined by a set of three multiwire proportional chambers (MWPCs), located upstream of the target, several meters apart. The transverse error of the impact point on the target was 0.5 mm from the resolution of the MWPCs, plus a contribution from multiple scattering of the beam particles in various materials in the beam line. Excluding the target itself, the latter contribution is 0.2 mm for a 8.9 GeV/c beam particle.
We select 'good' beam particles by requiring the unambiguous reconstruction of the particle trajectory with good χ 2 . In addition we require that the particle type is unambiguously identified. We select 'good' accelerator spills by requiring a minimal intensity and a 'smooth' variation of beam intensity across the 400 ms long spill. 2 The target was a cylinder made of high-purity (99.95%) beryllium, with a density of 1.85 g/cm 3 , a radius of 15 mm, and a thickness of 20.5 ± 0.1 mm (5% λ abs ).
The finite thickness of the target leads to a small attenuation of the number of incident beam particles. The attenuation factor is f att = 0.975.
The size of the beam spot at the position of the target was several millimeters in diameter, determined by the setting of the beam optics and by multiple scattering. The nominal beam position 3 was at x beam = y beam = 0, however, excursions by several millimeters could occur. 4 A loose fiducial cut x 2 beam + y 2 beam < 12 mm ensured full beam acceptance. The muon and electron contaminations of the pion beam, stated above, refer to this acceptance cut.

The HARP large-angle detectors
Our calibration work on the HARP TPC and RPCs is described in detail in Refs. [2] and [3], and in references cited therein. In particular, we recall that static and dynamic TPC track distortions up to 10 mm have been corrected to better than 300 µm. Therefore, TPC track distortions do not affect the precision of our cross-section measurements. 2 A smooth variation of beam intensity eases corrections for dynamic TPC track distortions. 3 A right-handed Cartesian and/or spherical polar coordinate system is employed; the z axis coincides with the beam line, with +z pointing downstream; the coordinate origin is at the center of the beryllium target, 500 mm downstream of the TPC's pad plane; looking downstream, the +x coordinate points to the left and the +y coordinate points up; the polar angle θ is the angle with respect to the +z axis. 4 The only relevant issue is that the trajectory of each individual beam particle is known, whether shifted or not, and therefore the amount of matter to be traversed by the secondary hadrons.
The resolution σ (1/p T ) is typically 0.2 (GeV/c) −1 and worsens towards small relative particle velocity β and small polar angle θ .
The absolute momentum scale is determined to be correct to better than 2%, both for positively and negatively charged particles.
The polar angle θ is measured in the TPC with a resolution of ∼9 mrad, for a representative angle of θ = 60 • . To this a multiple scattering error has to be added which is ∼7 mrad for a proton with p T = 500 MeV/c and θ = 60 • , and ∼4 mrad for a pion with the same characteristics. The polar-angle scale is correct to better than 2 mrad.
The TPC measures dE/dx with a resolution of 16% for a track length of 300 mm.
The intrinsic efficiency of the RPCs that surround the TPC is better than 98%.
The intrinsic time resolution of the RPCs is 127 ps and the system time-of-flight resolution (that includes the jitter of the arrival time of the beam particle at the target) is 175 ps.
To separate measured particles into species, we assign on the basis of dE/dx and β to each particle a probability of being a proton, a pion (muon), or an electron, respectively. The probabilities add up to unity, so that the number of particles is conserved. These probabilities are used for weighting when entering tracks into plots or tables.

Monte Carlo simulation
We used the Geant4 tool kit [14,15] for the simulation of the HARP large-angle spectrometer.
We had expected that Geant4 would provide us with reasonably realistic spectra of secondary hadrons. We found this expectation met by Geant4's so-called QGSP_BIC physics list, but only for the secondaries from incoming beam protons with momentum less than 12 GeV/c. For the secondaries from beam protons at 12 and 15 GeV/c momentum, and from beam pions at all momenta, we found the standard physics lists of Geant4 unsuitable [16].
To overcome this problem, we built our own physics list (HARP_CDP) for the production of secondaries from incoming beam pions. It starts from Geant4's standard QBBC physics list, but the Quark-Gluon String Model is replaced by the FRITIOF string fragmentation model for kinetic energy E > 6 GeV; for E < 6 GeV, the Bertini Cascade is used for pions, and the Binary Cascade for protons; elastic and quasi-elastic scattering is disabled. Examples of the good performance of the HARP_ CDP physics list are given in Ref. [16].

Systematic errors
The systematic precision of our inclusive cross-sections is at the few-per-cent level, from errors in the normalization, in the momentum measurement, in particle identification, and in the corrections applied to the data.
The systematic error of the absolute flux normalization is taken as 2%. This error arises from uncertainties in the target thickness, in the contribution of large-angle scattering of beam particles, in the attenuation of beam particles in the target, and in the subtraction of the muon and electron contaminations of the beam. Another contribution comes from the removal of events with an abnormally large number of TPC hits above threshold. 5 The systematic error of the track finding efficiency is taken as 1% which reflects differences between results from different persons who conducted eyeball scans. We also take the statistical errors of the parameters of a fit to scan results as systematic error into account [1]. The systematic error of the correction for losses from the requirement of at least 10 TPC clusters per track is taken as 20% of the correction which itself is in the range of 5 to 30%. This estimate arose from differences between the four TPC sectors that were used in our analysis, and from the observed variations with time.
The systematic error of the p T scale is taken as 2% as discussed in Ref. [2].
The systematic errors of the proton, pion, and electron abundances are taken as 10%. We stress that errors on abundances only lead to cross-section errors in case of a strong overlap of the resolution functions of both identification variables, dE/dx and β. The systematic error of the correction for migration, absorption of secondary protons and pions in materials, and for pion decay into muons, is taken as 20% of the correction, or 1% of the cross-section, whichever is larger. These estimates reflect our experience with remanent differences between data and Monte Carlo simulations after weighting Monte Carlo events with smooth functions with a view to reproducing the data simultaneously in several variables in the best possible way.
All systematic errors are propagated into the momentum spectra of secondaries and then added in quadrature.

Cross-section results
In Tables A.1-A.36, collated in the Appendix, we give the double-differential inclusive cross-sections d 2 σ/dp dΩ for all 36 combinations of incoming beam particle and secondary particle, including statistical and systematic errors. In each bin, the average momentum at the vertex and the average polar angle are also given.
The data of Tables A.1-A.36 are available in ASCII format in Ref. [17].
Cross-sections are only given if the total error is not larger than the cross-section itself. Since our track reconstruction algorithm is optimized for tracks with p T above ∼70 MeV/c in the TPC volume, we do not give crosssections from tracks with p T below this value. Because of the absorption of slow protons in the material between the vertex and the TPC gas, and with a view to keeping the correction for absorption losses below 30%, cross-sections from protons are limited to p > 350 MeV/c at the interaction vertex. Proton cross-sections are also not given if a 10% error on the proton energy loss in materials between the interaction vertex and the TPC volume leads to a momentum change larger than 2%. Pion cross-sections are not given if pions are separated from protons by less than twice the timeof-flight resolution.
The larger than usual error bars for the +15 GeV/c pion beam are caused by scarce statistics because the beam composition was dominated by protons.
We present in Figs. 1 to 7 what we consider salient features of our cross-sections. In these figures, we also show the data from the +8.9 GeV/c and −8.0 GeV/c beams that we published in Ref. [1]. Figure 1 shows the inclusive cross-sections of the production of protons, π + 's, and π − 's, from incoming protons between 3 GeV/c and 15 GeV/c momentum, as a function of their charge-signed p T . The data refer to the polar-angle range 20 • < θ < 30 • . Figures 2 and 3 show the same for incoming π + 's and π − 's. Figure 4 shows the inclusive cross-sections of the production of protons, π + 's, and π − 's, from incoming protons between 3 GeV/c and 15 GeV/c momentum, this time as a function of their charge-signed polar angle θ . The data refer to the p T range 0.24 < p T < 0.30 GeV/c. In this p T range pions populate nearly all polar angles, whereas protons are absorbed at large polar angle and thus escape measurement. Figures 5 and 6 show the same for incoming π + 's and π − 's.
These figures highlight the rather strong differences in the production of proton, π + and π − secondaries for different beam particles and beam momenta.
In Fig. 7, we present the inclusive cross-sections of the production of secondary π + 's and π − 's, integrated over the momentum range 0.2 < p < 1.0 GeV/c and the polar-angle range 30 • < θ < 90 • in the forward hemisphere, as a function of the beam momentum.

Comparison of our results
with results from other experiments

Comparison with E802 results
Experiment E802 [18] at Brookhaven National Laboratory measured secondary charged pions in the polar-angle range 5 • < θ < 58 • from the interactions of +14.6 GeV/c protons with beryllium nuclei. Figure 8 shows their published Lorentz-invariant crosssection of π + and π − production by +14.6 GeV/c protons, in the rapidity range 1.2 < y < 1.4, as a function of m T − m π , where m T denotes the pion transverse mass. Their data are compared with our cross-sections from the interactions of +15.0 GeV/c protons with beryllium nuclei, expressed in the same unit as used by E802. Since E802 quoted only statistical errors, our data in Fig. 8 are also shown with their statistical errors.
The E802 π ± cross-sections are in good agreement with our cross-sections measured nearly at the same proton beam momentum, taking into account the normalization uncertainty of (10-15)% quoted by E802. We draw attention to the good agreement of the slopes of the cross-sections over two orders of magnitude.

Comparison with E910 results
Experiment E910 [19] at Brookhaven National Laboratory measured secondary charged pions in the momentum range 0.1-6 GeV/c from the interactions of +12.3 GeV/c protons with beryllium nuclei. This experiment used a TPC for the measurement of secondaries, with a comfortably large track length of ∼1.5 m. This feature, together with a magnetic field strength of 0.5 T, is of particular significance, since it permits considerably better charge identification and proton-pion separation by dE/dx than is possible in the HARP detector. Figure 9 shows their published crosssection d 2 σ/dp dΩ of π ± production by +12.3 GeV/c protons, in the polar-angle range 0.8 < cos θ < 0.9. Since E910 quoted only statistical errors, our data in Fig. 9 from the interactions of +12.0 GeV/c protons are also shown with their statistical errors. The normalization uncertainty quoted by E910 is ≤5%.
Also here, the E910 data are shown as published, and our data are expressed in the same unit as used by E910. We draw attention to the good agreement in the π + /π − ratio between the cross-sections from E910 and our cross-sections. See also Fig. 10(b) in Sect. 7.3.

Comparison with results from the HARP Collaboration
Figure 10(a) shows the comparison of our cross-sections of pion production by +12.0 GeV/c protons off beryllium nuclei with the ones published by the HARP Collaboration [20], in the polar-angle range 0.35 < θ < 0.55 rad. The latter cross-sections are plotted as published, while we expressed our cross-sections in the unit used by the HARP Collaboration. Figure 10(b) shows our ratio π + /π − as a function of the polar angle θ in comparison with the ratios Fig. 1 Inclusive cross-sections of the production of secondary protons, π + 's, and π − 's, by protons on beryllium nuclei, in the polar-angle range 20 • < θ < 30 • , for different proton beam momenta, as a function of the charge-signed p T of the secondaries; the shown errors are total errors Fig. 2 Inclusive cross-sections of the production of secondary protons, π + 's, and π − 's, by π + 's on beryllium nuclei, in the polar-angle range 20 • < θ < 30 • , for different π + beam momenta, as a function of the charge-signed p T of the secondaries; the shown errors are total errors Fig. 3 Inclusive cross-sections of the production of secondary protons, π + 's, and π − 's, by π − 's on beryllium nuclei, in the polar-angle range 20 • < θ < 30 • , for different π − beam momenta, as a function of the charge-signed p T of the secondaries; the shown errors are total errors Fig. 4 Inclusive cross-sections of the production of secondary protons, π + 's, and π − 's, with p T in the range 0.24-0.30 GeV/c, by protons on beryllium nuclei, for different proton beam momenta, as a function of the charge-signed polar angle θ of the secondaries; the shown errors are total errors Fig. 5 Inclusive cross-sections of the production of secondary protons, π + 's, and π − 's, with p T in the range 0.24-0.30 GeV/c, by π + 's on beryllium nuclei, for different π + beam momenta, as a function of the charge-signed polar angle θ of the secondaries; the shown errors are total errors Fig. 6 Inclusive cross-sections of the production of secondary protons, π + 's, and π − 's, with p T in the range 0.24-0.30 GeV/c, by π − 's on beryllium nuclei, for different π − beam momenta, as a function of the charge-signed polar angle θ of the secondaries; the shown errors are total errors Fig. 7 Inclusive cross-sections of the production of secondary π + 's and π − 's, integrated over the momentum range 0.2 < p < 1.0 GeV/c and the polar-angle range 30 • < θ < 90 • , from the interactions on beryllium nuclei of protons (top row), π + 's (middle row), and π − 's (bottom row), as a function of the beam momentum; the shown errors are total errors and mostly smaller than the symbol size  The discrepancy between our results and those published by the HARP Collaboration is evident. We note the difference especially of the π + cross-section, and the difference in the reported momentum range. The discrepancy is even more serious as the same data set has been analysed by both groups.
We hold that the discrepancy is caused by problems in the HARP Collaboration's data analysis. They result primarily, but not exclusively, from a lack of understanding TPC Fig. 10 (a) Comparison of our cross-sections (black circles) of π ± production by +12.0 GeV/c protons off beryllium nuclei with the cross-sections published by the HARP Collaboration (open circles); (b) Comparison of our ratio π + /π − at +12.0 GeV/c beam momentum as a function of the polar angle θ with the ratios published by the HARP Collaboration; also shown are the ratios π + /π − published by the E910 Collaboration for a +12.3 GeV/c beam momentum; in (a) total errors are shown; in (b) for the HARP Collaboration total errors are shown (only those are published), for E910 and our group, the errors are statistical only track distortions and RPC timing signals. These problems, together with others that affect the HARP Collaboration's data analysis, are discussed in detail in Refs. [4,5,[21][22][23] and in the Appendix of Ref. [1].

Summary
From the analysis of data from the HARP large-angle spectrometer (polar angle θ in the range 20 • < θ < 125 • ), double-differential cross-sections d 2 σ/dp dΩ of the production of secondary protons, π + 's, and π − 's, have been obtained. The incoming beam particles were protons and pions with momenta from ±3 to ±15 GeV/c, impinging on a 5% λ abs thick stationary beryllium target. Our cross-sections for π + and π − production agree with results from the BNL experiments E802 and E910 but disagree with the results from the HARP Collaboration that were obtained from the same raw data.
Acknowledgements We are greatly indebted to many technical collaborators whose diligent and hard work made the HARP detector a well-functioning instrument. We thank all HARP colleagues who devoted time and effort to the design and construction of the detector, to data taking, and to setting up the computing and software infrastructure. We express our sincere gratitude to HARP's funding agencies for their support.
Appendix: Cross-section tables Table A.1 Double-differential inclusive cross-section d 2 σ/dp dΩ [mb/(GeV/c sr)] of the production of protons in p + Be → p + X interactions with +3.0 GeV/c beam momentum; the first error is statistical, the second systematic; p T in GeV/c, polar angle θ in degrees  Table A.2 Double-differential inclusive cross-section d 2 σ/dp dΩ [mb/(GeV/c sr)] of the production of π + 's in p + Be → π + + X interactions with +3.0 GeV/c beam momentum; the first error is statistical, the second systematic; p T in GeV/c, polar angle θ in degrees p T 20 < θ < 30 30 < θ < 40 p T θ d 2 σ/dp dΩ p T θ d 2 σ/dp dΩ  Table A.3 Double-differential inclusive cross-section d 2 σ/dp dΩ [mb/(GeV/c sr)] of the production of π − 's in p + Be → π − + X interactions with +3.0 GeV/c beam momentum; the first error is statistical, the second systematic; p T in GeV/c, polar angle θ in degrees p T 20 < θ < 30 30 < θ < 40 p T θ d 2 σ/dp dΩ p T θ d 2 σ/dp dΩ   4 Double-differential inclusive cross-section d 2 σ/dp dΩ [mb/(GeV/c sr)] of the production of protons in π + + Be → p + X interactions with +3.0 GeV/c beam momentum; the first error is statistical, the second systematic; p T in GeV/c, polar angle θ in degrees  Table A.5 Double-differential inclusive cross-section d 2 σ/dp dΩ [mb/(GeV/c sr)] of the production of π + 's in π + + Be → π + + X interactions with +3.0 GeV/c beam momentum; the first error is statistical, the second systematic; p T in GeV/c, polar angle θ in degrees   Table A.6 Double-differential inclusive cross-section d 2 σ/dp dΩ [mb/(GeV/c sr)] of the production of π − 's in π + + Be → π − + X interactions with +3.0 GeV/c beam momentum; the first error is statistical, the second systematic; p T in GeV/c, polar angle θ in degrees   Table A.7 Double-differential inclusive cross-section d 2 σ/dp dΩ [mb/(GeV/c sr)] of the production of protons in π − + Be → p + X interactions with −3.0 GeV/c beam momentum; the first error is statistical, the second systematic; p T in GeV/c, polar angle θ in degrees  Table A.8 Double-differential inclusive cross-section d 2 σ/dp dΩ [mb/(GeV/c sr)] of the production of π + 's in π − + Be → π + + X interactions with −3.0 GeV/c beam momentum; the first error is statistical, the second systematic; p T in GeV/c, polar angle θ in degrees   Table A.9 Double-differential inclusive cross-section d 2 σ/dp dΩ [mb/(GeV/c sr)] of the production of π − 's in π − + Be → π − + X interactions with −3.0 GeV/c beam momentum; the first error is statistical, the second systematic; p T in GeV/c, polar angle θ in degrees   Table A.10 Double-differential inclusive cross-section d 2 σ/dp dΩ [mb/(GeV/c sr)] of the production of protons in p + Be → p + X interactions with +5.0 GeV/c beam momentum; the first error is statistical, the second systematic; p T in GeV/c, polar angle θ in degrees  Table A.11 Double-differential inclusive cross-section d 2 σ/dp dΩ [mb/(GeV/c sr)] of the production of π + 's in p + Be → π + + X interactions with +5.0 GeV/c beam momentum; the first error is statistical, the second systematic; p T in GeV/c, polar angle θ in degrees     13 Double-differential inclusive cross-section d 2 σ/dp dΩ [mb/(GeV/c sr)] of the production of protons in π + + Be → p + X interactions with +5.0 GeV/c beam momentum; the first error is statistical, the second systematic; p T in GeV/c, polar angle θ in degrees  Table A.14 Double-differential inclusive cross-section d 2 σ/dp dΩ [mb/(GeV/c sr)] of the production of π + 's in π + + Be → π + + X interactions with +5.0 GeV/c beam momentum; the first error is statistical, the second systematic; p T in GeV/c, polar angle θ in degrees     16 Double-differential inclusive cross-section d 2 σ/dp dΩ [mb/(GeV/c sr)] of the production of protons in π − + Be → p + X interactions with −5.0 GeV/c beam momentum; the first error is statistical, the second systematic; p T in GeV/c, polar angle θ in degrees  Table A.17 Double-differential inclusive cross-section d 2 σ/dp dΩ [mb/(GeV/c sr)] of the production of π + 's in π − + Be → π + + X interactions with −5.0 GeV/c beam momentum; the first error is statistical, the second systematic; p T in GeV/c, polar angle θ in degrees     19 Double-differential inclusive cross-section d 2 σ/dp dΩ [mb/(GeV/c sr)] of the production of protons in p + Be → p + X interactions with +12.0 GeV/c beam momentum; the first error is statistical, the second systematic; p T in GeV/c, polar angle θ in degrees  Table A.20 Double-differential inclusive cross-section d 2 σ/dp dΩ [mb/(GeV/c sr)] of the production of π + 's in p + Be → π + + X interactions with +12.0 GeV/c beam momentum; the first error is statistical, the second systematic; p T in GeV/c, polar angle θ in degrees     22 Double-differential inclusive cross-section d 2 σ/dp dΩ [mb/(GeV/c sr)] of the production of protons in π + + Be → p + X interactions with +12.0 GeV/c beam momentum; the first error is statistical, the second systematic; p T in GeV/c, polar angle θ in degrees  Table A.23 Double-differential inclusive cross-section d 2 σ/dp dΩ [mb/(GeV/c sr)] of the production of π + 's in π + + Be → π + + X interactions with +12.0 GeV/c beam momentum; the first error is statistical, the second systematic; p T in GeV/c, polar angle θ in degrees p T 20 < θ < 30 30 < θ < 40 p T θ d 2 σ/dp dΩ p T θ d 2 σ/dp dΩ   Table A.24 Double-differential inclusive cross-section d 2 σ/dp dΩ [mb/(GeV/c sr)] of the production of π − 's in π + + Be → π − + X interactions with +12.0 GeV/c beam momentum; the first error is statistical, the second systematic; p T in GeV/c, polar angle θ in degrees   Table A.25 Double-differential inclusive cross-section d 2 σ/dp dΩ [mb/(GeV/c sr)] of the production of protons in π − + Be → p + X interactions with −12.0 GeV/c beam momentum; the first error is statistical, the second systematic; p T in GeV/c, polar angle θ in degrees  Table A.26 Double-differential inclusive cross-section d 2 σ/dp dΩ [mb/(GeV/c sr)] of the production of π + 's in π − + Be → π + + X interactions with −12.0 GeV/c beam momentum; the first error is statistical, the second systematic; p T in GeV/c, polar angle θ in degrees   Table A.27 Double-differential inclusive cross-section d 2 σ/dp dΩ [mb/(GeV/c sr)] of the production of π − 's in π − + Be → π − + X interactions with −12.0 GeV/c beam momentum; the first error is statistical, the second systematic; p T in GeV/c, polar angle θ in degrees   Table A.28 Double-differential inclusive cross-section d 2 σ/dp dΩ [mb/(GeV/c sr)] of the production of protons in p + Be → p + X interactions with +15.0 GeV/c beam momentum; the first error is statistical, the second systematic; p T in GeV/c, polar angle θ in degrees  Table A.29 Double-differential inclusive cross-section d 2 σ/dp dΩ [mb/(GeV/c sr)] of the production of π + 's in p + Be → π + + X interactions with +15.0 GeV/c beam momentum; the first error is statistical, the second systematic; p T in GeV/c, polar angle θ in degrees    Table A.31 Double-differential inclusive cross-section d 2 σ/dp dΩ [mb/(GeV/c sr)] of the production of protons in π + + Be → p + X interactions with +15.0 GeV/c beam momentum; the first error is statistical, the second systematic; p T in GeV/c, polar angle θ in degrees Table A.32 Double-differential inclusive cross-section d 2 σ/dp dΩ [mb/(GeV/c sr)] of the production of π + 's in π + + Be → π + + X interactions with +15.0 GeV/c beam momentum; the first error is statistical, the second systematic; p T in GeV/c, polar angle θ in degrees Table A.33 Double-differential inclusive cross-section d 2 σ/dp dΩ [mb/(GeV/c sr)] of the production of π − 's in π + + Be → π − + X interactions with +15.0 GeV/c beam momentum; the first error is statistical, the second systematic; p T in GeV/c, polar angle θ in degrees  Table A.34 Double-differential inclusive cross-section d 2 σ/dp dΩ [mb/(GeV/c sr)] of the production of protons in π − + Be → p + X interactions with −15.0 GeV/c beam momentum; the first error is statistical, the second systematic; p T in GeV/c, polar angle θ in degrees  Table A.35 Double-differential inclusive cross-section d 2 σ/dp dΩ [mb/(GeV/c sr)] of the production of π + 's in π − + Be → π + + X interactions with −15.0 GeV/c beam momentum; the first error is statistical, the second systematic; p T in GeV/c, polar angle θ in degrees   Table A.36 Double-differential inclusive cross-section d 2 σ/dp dΩ [mb/(GeV/c sr)] of the production of π − 's in π − + Be → π − + X interactions with −15.0 GeV/c beam momentum; the first error is statistical, the second systematic; p T in GeV/c, polar angle θ in degrees