Cross-sections of large-angle hadron production in proton- and pion-nucleus interactions VII: tin nuclei and beam momenta from \pm3 GeV/c to \pm15 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 tin target, of proton and pion beams with momentum from \pm3 GeV/c to \pm15 GeV/c. Results are given for secondary particles with production angles between 20 and 125 degrees. Cross-sections on tin nuclei are compared with cross-sections on beryllium, carbon, copper, tantalum and lead 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.
These are the optimization of the design parameters of the proton driver of a neutrino factory (see Ref. [1] and further references cited therein), but also the understanding of the underlying physics and the modelling of Monte Carlo generators of hadron-nucleus collisions, flux predictions for conventional neutrino beams, and 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. It took data at the CERN Proton Synchrotron in 2001 and 2002.
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. This is the seventh of a series of cross-section papers with results from the HARP experiment. In the first paper [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], fourth [5], fifth [6], and sixth [7] papers presented results from the interactions with 5% λ int tantalum, copper, lead, and carbon targets. 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 tin 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 BEAMS AND THE HARP SPECTROMETER
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. [2].
The target was a disc made of high-purity (99.99%) tin, with a radius of 15.1 mm, a thickness of 11.1 mm (5% λ int ), and a measured density of 7.24 g/cm 3 .
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.
Our calibration work on the HARP TPC and RPCs is described in detail in Refs. [8] and [9], and in references cited therein.
The momentum resolution σ(1/p T ) of the TPC 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 • . In addition, a multiple scattering error must be considered that is for a proton with p T = 500 MeV/c in the TPC gas ∼6 mrad at θ = 20 • , and ∼18 mrad at θ = 90 • . For a pion with the same characteristics, the multiple scattering errors are ∼3 mrad and ∼10 mrad, respectively. 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.
A general discussion of the systematic errors can be found, e.g., in Ref. [2]. All systematic errors are propagated into the momentum spectra of secondaries and then added in quadrature. They add up to a systematic uncertainty of our inclusive cross-sections at the few-per-cent level, mainly from errors in the normalization, in the momentum measurement, in particle identification, and in the corrections applied to the data.

MONTE CARLO SIMULATION
We used the Geant4 tool kit [10] 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 below 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 [11].
To overcome this problem, we built our own HARP CDP physics list. 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. [11].

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. [12]. 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 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 time-of-flight resolution.
The large errors and/or absence of results from 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. Figure 1 shows the inclusive cross-sections of the production of protons, π + 's, and π − 's, by incoming protons between 3 GeV/c and 15 GeV/c momentum, as a function of their chargesigned 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 inclusive Lorentz-invariant cross-sections of the production of protons, π + 's and π − 's, by incoming protons between 3 GeV/c and 15 GeV/c momentum, in the rapidity range 0.6 < y < 0.8, as a function of the charge-signed reduced transverse particle mass, m T − m 0 , where m 0 is the rest mass of the respective particle. Figures 5 and 6 show the same for incoming π + 's and π − 's. We note the good representation of particle production by an exponential falloff with increasing reduced transverse mass.
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 tin 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.  HARP-CDP p + Sn → (p,π + ,π -) + X +15.0 GeV/c 0.6 < y < 0.8 π p Fig. 4: Inclusive Lorentz-invariant cross-sections of the production of protons, π + 's and π − 's, by incoming protons between 3 GeV/c and 15 GeV/c momentum, in the rapidity range 0.6 < y < 0.8, as a function of the charge-signed reduced transverse particle mass, m T − m 0 , where m 0 is the rest mass of the respective particle; the shown errors are total errors. HARP-CDP π + + Sn → (p,π + ,π -) + X +15.0 GeV/c 0.6 < y < 0.8 π p Fig. 5: Inclusive Lorentz-invariant cross-sections of the production of protons, π + 's and π − 's, by incoming π + 's between 3 GeV/c and 15 GeV/c momentum, in the rapidity range 0.6 < y < 0.8, as a function of the charge-signed reduced transverse pion mass, m T − m 0 , where m 0 is the rest mass of the respective particle; the shown errors are total errors. HARP-CDP π -+ Sn → (p,π + ,π -) + X -15.0 GeV/c 0.6 < y < 0.8 π p Fig. 6: Inclusive Lorentz-invariant cross-sections of the production of protons, π + 's and π − 's, by incoming π − 's between 3 GeV/c and 15 GeV/c momentum, in the rapidity range 0.6 < y < 0.8, as a function of the charge-signed reduced transverse pion mass, m T − m 0 , where m 0 is the rest mass of the respective particle; 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 tin 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. Figure 8 shows the comparison of our cross-sections of π ± production by protons, π + 's and π − 's of 3.0 GeV/c and 8.0 GeV/c momentum, off tin nuclei, with the ones published by the HARP Collaboration [14,15], 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. The discrepancy between our results and those published by the HARP Collaboration is evident. It shows the same pattern as observed in inclusive cross-sections off other targets that we analyzed and compared. We hold that the discrepancy is caused by problems in the HARP Collaboration's data analysis, discussed in detail in Refs [16][17][18][19][20], and summarized in the Appendix of Ref. [2].      -PION PRODUCTION ON BERYLLIUM, CARBON, COP-PER, TIN, TANTALUM AND LEAD 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), carbon (A = 12.01), copper (A = 63.55), tin (A = 118.7), tantalum (A = 181.0), and lead (A = 207.2) nuclei 1) . 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. Linearity with A 2/3 means that inclusive pion production scales with the geometrical crosssection of the nucleus. We note that at the lowest beam momenta the inclusive pion cross-section tends to fall below a linear dependence on A 2/3 , while at the highest beam momenta the crosssections tend to lie above a linear dependence. We conjecture that this behaviour arises from the production of tertiary pions from the interactions of secondaries in nuclear matter. At high beam momenta, the acceptance cut of p > 0.2 GeV/c has a minor effect on the tertiary pions. The transition of the inclusive pion cross-section from an approximate A 2/3 dependence for light nuclei toward an approximate A dependence for heavy nuclei (owing to the increasing contribution of pions from the reinteractions in nuclear matter) becomes apparent. At low beam momenta, the acceptance cut of p > 0.2 GeV/c suppresses a large fraction of the primarily low-momentum tertiaries, thus hiding this transition. Figure 10 compares the 'forward multiplicity' of secondary π + 's and π − 's in the interaction of protons and pions with beryllium, carbon, copper, tin, tantalum, and lead target nuclei. The forward multiplicities are averaged over the momentum range 0.2 < p < 1.0 GeV/c and the polar-angle 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 [13] and reproduced in Table 1. 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. Again, we interpret this as the effect of pion reinteractions in the nuclear matter in conjunction with the acceptance cut of p > 0.2 GeV/c. 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, carbon, copper, tin, 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.    Comparison of inclusive pion production cross-sections in the forward region between beryllium, carbon, copper, tin, tantalum, and lead target nuclei, as a function of the pion momentum.

DEUTERON PRODUCTION
Besides pions and protons, also deuterons are produced in sizeable quantities on tin nuclei. Up to momenta of about 1 GeV/c, deuterons are easily separated from protons by dE/dx. Table 2 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 2) . 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. 12, and compared with the predictions of Geant4's FRITIOF model. FRITIOF's predictions are shown for π + beam particles 3) . FRITIOF's estimate of deuteron production falls short of the data. circles beam π + 's, and full circles beam π − 's; the full lines denotes predictions of Geant4's FRITIOF model for π + beam particles.
In Fig. 13 we show, for the polar-angle region 30 • < θ < 45 • , how the deuteron-to-proton 2) We observe no appreciable dependence of the deuteron-to-proton production ratio on beam momentum. 3) There is virtually no difference between its predictions for incoming protons, π + 's and π − 's. ratio varies with the mass of the target nucleus. The ratios are for 8 GeV/c beam protons on beryllium, carbon, copper, tin, tantalum and lead nuclei.

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 tin target.
In the same way as for the other target nuclei which we have analyzed, our cross-sections for π + and π − production disagree with results of the HARP Collaboration that were obtained from the same raw data.
We have compared the inclusive tin π + and π − production cross-sections with those on beryllium, carbon, copper, tantalum, and lead and find an approximately linear dependence on the scaling variable A 2/3 .
We also observe a sizeable production of deuterons off tin nuclei that we compared to the deuteron production on beryllium, carbon, copper, tantalum, and lead. Table A.1: Double-differential inclusive cross-section d 2 σ/dpdΩ [mb/(GeV/c sr)] of the production of protons in p + Sn → 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 + Sn → 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 + Sn → 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.5: Double-differential inclusive cross-section d 2 σ/dpdΩ [mb/(GeV/c sr)] of the production of π + 's in π + + Sn → 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 + + Sn → 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 − + Sn → 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 − + Sn → 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 − + Sn → 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 + Sn → 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 + Sn → 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 + Sn → 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 + + Sn → 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 + + Sn → 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 + + Sn → 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 − + Sn → 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 − + Sn → 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 + Sn → 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 + Sn → 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 + + Sn → 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 + + Sn → 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 + + Sn → 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 − + Sn → 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 − + Sn → 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 − + Sn → 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 + Sn → 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 + Sn → 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 + Sn → 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 + + Sn → 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 + + Sn → 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 + + Sn → 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 − + Sn → 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 − + Sn → 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 − + Sn → 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 + Sn → 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 + Sn → 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 + + Sn → 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 + + Sn → 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 + + Sn → 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. 20 Table A.43: Double-differential inclusive cross-section d 2 σ/dpdΩ [mb/(GeV/c sr)] of the production of protons in π − + Sn → 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 − + Sn → 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. 20 Table A.45: Double-differential inclusive cross-section d 2 σ/dpdΩ [mb/(GeV/c sr)] of the production of π − 's in π − + Sn → 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. 20