Cross-Sections of Large-Angle Hadron Production in Proton- and Pion-Nucleus Interactions V: Lead 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, charged pions, and deuterons, in the interactions with a 5% nuclear interaction length thick stationary lead 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 to 125 degrees. Cross-sections on lead nuclei are compared with cross-sections on beryllium, copper, and tantalum nuclei.


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.
The 'neutrino factory' (see Ref. [1] and further references cited therein) is a serious contender for a future accelerator facility that addresses fundamental questions on neutrino oscillations. One of the neutrino factory's many technological challenges is the production of charged pions with sufficient intensity to achieve the required particle fluxes in the decay chain pions → muons → neutrinos. It is imperative that pion production cross-sections are under control.
Primarily with a view to the optimization of the design parameters of the proton driver of a neutrino factory, but also to the understanding of the underlying physics and the modelling of Monte Carlo generators of hadron-nucleus collisions, to flux predictions for conventional neutrino beams, and to more precise calculations of the atmospheric neutrino flux, the HARP experiment was designed to carry out a programme of systematic and precise (i.e., at the few per cent level) measurements of hadron production by protons and pions with momenta from 1.5 to 15 GeV/c, on a variety of target nuclei. A central goal were precise cross-sections of π + and π − production on the heavy nuclei tantalum and lead.
The HARP detector combined a forward spectrometer with a large-angle spectrometer. The latter comprised a cylindrical 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 took data at the CERN Proton Synchrotron in 2001 and 2002. This is the fifth of a series of cross-section papers with results from the HARP experiment. In the first paper, Ref. [2], we described the detector characteristics and our analysis algorithms, on the example of +8.9 GeV/c and −8.0 GeV/c beams impinging on a 5% λ int Be target. The second paper [3] presented results for all beam momenta from this Be target. The third [4] and the fourth [5] paper, respectively, presented results from the interactions with a 5% λ int tantalum and copper target. In this paper, we report on the large-angle production (polar angle θ in the range 20 • < θ < 125 • ) of secondary protons and charged pions, and of deuterons, in the interactions with a 5% λ int lead target of protons and pions with beam momenta of ±3.0, ±5.0, ±8.0, ±12.0, and ±15.0 GeV/c.
Our work involves only the HARP large-angle spectrometer.

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 and 15 GeV/c, with a momentum bite ∆p/p ∼ 1%.
The beam instrumentation, the definition of the beam particle trajectory, the cuts to select 'good' beam particles, and the muon and electron contaminations of the particle beams, are the same as described, e.g., in Ref. [5].
The target was a disc made of high-purity (99.99%) lead, with a radius of 15.1 mm and a thickness of 8.45 mm (5% λ int ). A target density of 11.35 g/cm 3 was used for the cross-section normalization.
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 millimetres in diameter, determined by the setting of the beam optics and by multiple scattering. The nominal beam position 1) was at x beam = y beam = 0, however, excursions by several millimetres could occur 2) . A loose fiducial cut x 2 beam + y 2 beam < 12 mm ensured full beam acceptance.

PERFORMANCE OF THE HARP LARGE-ANGLE DETECTORS
Our calibration work on the HARP TPC and RPCs is described in detail in Refs. [6] and [7], 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. TPC track distortions do not affect the precision of our cross-section measurements. 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 on the average ∼8 mrad for a proton with p T = 500 MeV/c in the TPC gas and θ = 60 • , and ∼5 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 [8] for the simulation of the HARP large-angle spectrometer.
Geant4's QGSP BIC physics list provided us with reasonably realistic spectra of 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 [9].
To overcome this problem, we built our own HARP CDP physics list 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. [9].

SYSTEMATIC ERRORS
The systematic uncertainty 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 in general 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 3) . In the case of the lead target, we increased for reasons of uncertainties on the target shape the normalization uncertainty to 3%.
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 [2]. 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. [6]. For the data from the +12 GeV/c and +15 GeV/c beams, this error was doubled to account for a larger than usual uncertainty of the correction for dynamic TPC track distortions.
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.45, collated in the Appendix of this paper, we give the double-differential inclusive cross-sections d 2 σ/dpdΩ for various 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.45 are available in ASCII format in Ref. [10]. Some bins in the tables are empty. 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 cross-sections 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 > 450 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 3) In less than 0.5% of the number of good events, because of apparatus malfunction, the number of TPC hits was much larger than possible for a physics event. Such events were considered unphysical and eliminated. the interaction vertex and the TPC volume leads to a momentum change larger than 2%. Since the proton energy loss is large in the lead target, particularly at polar angles close to 90 degrees, the latter condition imposes significant restrictions. Pion cross-sections are not given if pions are separated from protons by less than twice the time-of-flight resolution.
The large errors and/or absence of results from the +12 GeV/c and +15 GeV/c pion beams 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. 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.
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.  Inclusive cross-sections of the production of secondary protons, π + 's, and π − 's, by π + 's on lead 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. Inclusive cross-sections of the production of secondary protons, π + 's, and π − 's, by π − 's on lead 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. Charge-signed Θ (deg) d 2 σ/dpdΩ (mb/sr GeV/c) HARP-CDP p + Pb → (p,π + ,π -) + X +15.0 GeV/c 0.24 < p T < 0.30 GeV/c π p 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 lead 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. Charge-signed Θ (deg) d 2 σ/dpdΩ (mb/sr GeV/c) HARP-CDP π + + Pb → (p,π + ,π -) + X +15.0 GeV/c 0.24 < p T < 0.30 GeV/c π p 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 lead 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. Charge-signed Θ (deg) d 2 σ/dpdΩ (mb/sr GeV/c) HARP-CDP π -+ Pb → (p,π + ,π -) + X -15.0 GeV/c 0.24 < p T < 0.30 GeV/c π p 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 lead 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 lead 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.

DEUTERON PRODUCTION
Besides pions and protons, also deuterons are produced in sizeable quantities on lead nuclei. Up to momenta of about 1 GeV/c, deuterons are easily separated from protons by dE/dx. Table 1 gives the deuteron-to-proton production ratio as a function of the momentum at the vertex, for 8 GeV/c beam protons, π + 's, and π − 's 4) . Cross-section ratios are not given if the data are scarce and the statistical error becomes comparable with the ratio itself-which is the case for deuterons at the high-momentum end of the spectrum.
The measured deuteron-to-proton production ratios are illustrated in Fig. 8, and compared with the predictions of Geant4's FRITIOF model. FRITIOF's predictions are shown for beam π + 's 5) . FRITIOF largely underestimates deuteron production.

4)
We observe no appreciable dependence of the deuteron-to-proton production ratio on beam momentum. 5) There is virtually no difference between its predictions for incoming protons, π + 's and π − 's.  Figure 9 presents a comparison between the inclusive cross-sections of π + and π − production, integrated over the secondaries' momentum range 0.2 < p < 1.0 GeV/c and polar-angle range 30 • < θ < 90 • , in the interactions of protons, π + and π − , with beryllium (A = 9.01), copper (A = 63.55), tantalum (A = 181.0), and lead (A = 207.2) nuclei 6) . The comparison employs the scaling variable A 2/3 where A is the atomic number of the respective nucleus. We note the approximately linear dependence on this scaling variable. At low beam momentum, the slope exhibits a strong dependence on beam particle type, which tends to disappear with higher beam momentum. Figure 10 compares the 'forward multiplicity' of secondary π + 's and π − 's in the interaction of protons and pions with beryllium, copper, tantalum, and lead target nuclei. The forward multiplicities are averaged over the momentum range 0.2 < p < 1.0 GeV/c and the polarangle range 30 • < θ < 90 • . They have been obtained by dividing the measured inclusive cross-section by the total cross-section inferred from the nuclear interaction lengths and pion interaction lengths, respectively, as published by the Particle Data Group [11] and reproduced in Table 2. The errors of the forward multiplicities are dominated by a 3% systematic uncertainty. The forward multiplicities display a 'leading particle effect' that mirrors the incoming beam particle. It is also interesting that the forward multiplicity decreases with the nuclear mass at low beam momentum but increases at high beam momentum. We interpret this as the effect of the nuclear medium on secondary pions from the primary interaction of the incoming beam particle. At low beam momentum, the secondary pions have low momentum and tend to fall below the 0.2 GeV/c threshold imposed in our analysis if there is more nuclear medium to be traversed before escape. At high beam momentum, the secondary pions have high enough momentum such that tertiary pions from the re-interaction of secondary pions in the nuclear medium tend to pass the 0.2 GeV/c threshold. Figure 11 shows the increase of the inclusive cross-sections of π + 's and π − 's production by incoming protons of 8.0 GeV/c (in the case of beryllium target nuclei: +8.9 GeV/c) from the light beryllium nucleus to the heavy lead nucleus, for pions in the polar angle range 20 • < θ < 30 • . It is interesting to note that π − production is slightly favoured on heavy nuclei, while π + production is slightly favoured on light nuclei.  ), and π − 's (black circles), as a function of A 2/3 for, from left to right, beryllium, copper, tantalum, and lead nuclei; the cross-sections are integrated over the momentum range 0.2 < p < 1.0 GeV/c and the polar-angle range 30 • < θ < 90 • ; the shown errors are total errors and often smaller than the symbol size.     Figure 12 shows the comparison of our cross-sections of π ± production by protons, π + 's and π − 's of 8.0 GeV/c momentum, off lead nuclei, with the ones published by the HARP Collaboration [12,13], in the polar-angle range 20 • < θ < 30 • . The latter cross-sections are plotted as published, while we expressed our cross-sections in the unit used by the HARP Collaboration. The errors shown are the published total errors. Figure 13 shows the same comparison for beam particles of 3.0 GeV/c momentum. 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 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 [14][15][16] and summarized in the Appendix of Ref. [2].

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/dpdΩ of the production of secondary protons, π + 's, and π − 's, and of deuterons, have been obtained. The incoming beam particles were protons and pions with momenta from ±3 to ±15 GeV/c, impinging on a 5% λ int thick stationary lead target. Our cross-sections for π + and π − production disagree with results of the HARP Collaboration that were obtained from the same raw data. When designing the proton driver of a neutrino factory with the HARP Collaboration's cross-sections, the neutrino flux will be different by a factor of up to two compared with a design based on HARP-CDP cross-sections. Table A.1: Double-differential inclusive cross-section d 2 σ/dpdΩ [mb/(GeV/c sr)] of the production of protons in p + Pb → 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/dpdΩ [mb/(GeV/c sr)] of the production of π + 's in p + Pb → 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.3: Double-differential inclusive cross-section d 2 σ/dpdΩ [mb/(GeV/c sr)] of the production of π − 's in p + Pb → 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.4: Double-differential inclusive cross-section d 2 σ/dpdΩ [mb/(GeV/c sr)] of the production of protons in π + + Pb → 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/dpdΩ [mb/(GeV/c sr)] of the production of π + 's in π + + Pb → 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/dpdΩ [mb/(GeV/c sr)] of the production of π − 's in π + + Pb → 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/dpdΩ [mb/(GeV/c sr)] of the production of protons in π − + Pb → 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/dpdΩ [mb/(GeV/c sr)] of the production of π + 's in π − + Pb → 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/dpdΩ [mb/(GeV/c sr)] of the production of π − 's in π − + Pb → 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/dpdΩ [mb/(GeV/c sr)] of the production of protons in p + Pb → 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/dpdΩ [mb/(GeV/c sr)] of the production of π + 's in p + Pb → 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.   Table A.12: Double-differential inclusive cross-section d 2 σ/dpdΩ [mb/(GeV/c sr)] of the production of π − 's in p + Pb → 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.   Table A.13: Double-differential inclusive cross-section d 2 σ/dpdΩ [mb/(GeV/c sr)] of the production of protons in π + + Pb → 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/dpdΩ [mb/(GeV/c sr)] of the production of π + 's in π + + Pb → 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.  Table A.15: Double-differential inclusive cross-section d 2 σ/dpdΩ [mb/(GeV/c sr)] of the production of π − 's in π + + Pb → 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.  Table A.17: Double-differential inclusive cross-section d 2 σ/dpdΩ [mb/(GeV/c sr)] of the production of π + 's in π − + Pb → 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.  Table A.18: Double-differential inclusive cross-section d 2 σ/dpdΩ [mb/(GeV/c sr)] of the production of π − 's in π − + Pb → 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.  Table A.20: Double-differential inclusive cross-section d 2 σ/dpdΩ [mb/(GeV/c sr)] of the production of π + 's in p + Pb → π + + X interactions with +8.0 GeV/c beam momentum; the first error is statistical, the second systematic; p T in GeV/c, polar angle θ in degrees.  Table A.21: Double-differential inclusive cross-section d 2 σ/dpdΩ [mb/(GeV/c sr)] of the production of π − 's in p + Pb → π − + X interactions with +8.0 GeV/c beam momentum; the first error is statistical, the second systematic; p T in GeV/c, polar angle θ in degrees.  Table A.22: Double-differential inclusive cross-section d 2 σ/dpdΩ [mb/(GeV/c sr)] of the production of protons in π + + Pb → p + X interactions with +8.0 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/dpdΩ [mb/(GeV/c sr)] of the production of π + 's in π + + Pb → π + + X interactions with +8.0 GeV/c beam momentum; the first error is statistical, the second systematic; p T in GeV/c, polar angle θ in degrees.  Table A.24: Double-differential inclusive cross-section d 2 σ/dpdΩ [mb/(GeV/c sr)] of the production of π − 's in π + + Pb → π − + X interactions with +8.0 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/dpdΩ [mb/(GeV/c sr)] of the production of protons in π − + Pb → p + X interactions with −8.0 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/dpdΩ [mb/(GeV/c sr)] of the production of π + 's in π − + Pb → π + + X interactions with −8.0 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/dpdΩ [mb/(GeV/c sr)] of the production of π − 's in π − + Pb → π − + X interactions with −8.0 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/dpdΩ [mb/(GeV/c sr)] of the production of protons in p + Pb → 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.29: Double-differential inclusive cross-section d 2 σ/dpdΩ [mb/(GeV/c sr)] of the production of π + 's in p + Pb → 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.30: Double-differential inclusive cross-section d 2 σ/dpdΩ [mb/(GeV/c sr)] of the production of π − 's in p + Pb → 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.31: Double-differential inclusive cross-section d 2 σ/dpdΩ [mb/(GeV/c sr)] of the production of protons in π + + Pb → 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.32: Double-differential inclusive cross-section d 2 σ/dpdΩ [mb/(GeV/c sr)] of the production of π + 's in π + + Pb → 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.33: Double-differential inclusive cross-section d 2 σ/dpdΩ [mb/(GeV/c sr)] of the production of π − 's in π + + Pb → 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.34: Double-differential inclusive cross-section d 2 σ/dpdΩ [mb/(GeV/c sr)] of the production of protons in π − + Pb → 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.35: Double-differential inclusive cross-section d 2 σ/dpdΩ [mb/(GeV/c sr)] of the production of π + 's in π − + Pb → 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.36: Double-differential inclusive cross-section d 2 σ/dpdΩ [mb/(GeV/c sr)] of the production of π − 's in π − + Pb → 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.38: Double-differential inclusive cross-section d 2 σ/dpdΩ [mb/(GeV/c sr)] of the production of π + 's in p + Pb → 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.39: Double-differential inclusive cross-section d 2 σ/dpdΩ [mb/(GeV/c sr)] of the production of π − 's in p + Pb → 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.40: Double-differential inclusive cross-section d 2 σ/dpdΩ [mb/(GeV/c sr)] of the production of protons in π + + Pb → 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.41: Double-differential inclusive cross-section d 2 σ/dpdΩ [mb/(GeV/c sr)] of the production of π + 's in π + + Pb → 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.42: Double-differential inclusive cross-section d 2 σ/dpdΩ [mb/(GeV/c sr)] of the production of π − 's in π + + Pb → 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.43: Double-differential inclusive cross-section d 2 σ/dpdΩ [mb/(GeV/c sr)] of the production of protons in π − + Pb → 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.44: Double-differential inclusive cross-section d 2 σ/dpdΩ [mb/(GeV/c sr)] of the production of π + 's in π − + Pb → 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.45: Double-differential inclusive cross-section d 2 σ/dpdΩ [mb/(GeV/c sr)] of the production of π − 's in π − + Pb → 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.