Cross-sections of large-angle hadron production in proton-- and pion--nucleus interactions VIII: aluminium nuclei and beam momenta from {\pm}3 GeV/c to {\pm}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% {\lambda}int thick stationary aluminium 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 aluminium nuclei are compared with cross-sections on beryllium, carbon, copper, tin, tantalum and lead nuclei.


INTRODUCTION
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 ∼4.0 mrad at θ = 20 • , and ∼12.7 mrad at θ = 90 • . For a pion with the same characteristics, the multiple scattering errors are ∼3.3 mrad and ∼6.4 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 in Ref. [2]. For the data from the +15 GeV/c beam, the systematic error of the momentum measurement was increased by a factor of 1.5 to account for minor problems with the correction for dynamic TPC distortions. For the data from the −5 GeV/c beam, the systematic error arising from the parametrization of the pion abundance in the respective Monte Carlo simulation was doubled, for a less satisfactory description of data distributions in the Monte Carlo simulation with the same number of weight parameters as used in comparable data sets. 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 [11] 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 12.9 and 15 GeV/c momentum, and from beam pions at all momenta, we found the standard physics lists of Geant4 unsuitable [12].
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. [12].

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. [13]. 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 aluminium 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 + Al → (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 + + Al → (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.   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 aluminium 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.

COMPARISON WITH RESULTS FROM THE E802 EXPERIMENT
Experiment E802 [14] at Brookhaven National Laboratory measured secondary π ± 's and protons in the polar-angle range 5 • < θ < 58 • from the interactions of +14.6 GeV/c protons with aluminium nuclei. Figure 8 shows their published Lorentz-invariant cross-section of π + and π − production by +14.6 GeV/c protons, in the rapidity range 0.8 < y < 1.0, as a function of m T − m π , where m T denotes the secondary particle's transverse mass. Their data are compared with our respective cross-sections from the interactions of +15.0 GeV/c protons with aluminium nuclei.  The E802 π ± and proton 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. Figure 9 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 aluminium nuclei, with the ones published by the HARP Collaboration [15,16], in the polar-angle range 20 • < θ < 30 • . The latter crosssections 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.

COMPARISON WITH RESULTS FROM THE HARP COLLABORATION
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 target nuclei [2][3][4][5][6][7][8]. We hold that the discrepancy is caused by problems in the HARP Collaboration's data analysis, discussed in detail in Refs [17][18][19][20][21], and summarized in the Appendix of Ref. [2].      -PION PRODUCTION ON BERYLLIUM, CARBON, ALU-MINIUM, COPPER, TIN, TANTALUM AND LEAD Figure 10 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), aluminium (A = 26.98), 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 mass 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 crosssection tends to fall below a linear dependence on A 2/3 , while at the highest beam momenta the cross-sections 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 re-interactions 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 lowmomentum tertiaries, thus not only hiding this transition but even reversing its trend. Figure 11 compares the 'forward multiplicity' of secondary π + 's and π − 's in the interaction of protons and pions with beryllium, carbon, aluminium, 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 [22] 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 re-interactions in the nuclear matter in conjunction with the acceptance cut of p > 0.2 GeV/c. Figure 12 shows the increase of the inclusive cross-sections of π + and π − production by incoming protons of +3.0 GeV/c from the light beryllium nucleus to the heavy lead nucleus, for pions in the polar angle range 20 • < θ < 30 • . For comparison, Figure 13 shows the analogous cross sections for incoming protons of +8.0 GeV/c (in the case of beryllium target nuclei: +8.9 GeV/c).

COMPARISON OF CHARGED
We observe that the π + /π − ratio depends on the proton beam momentum. We interpret the diminishing preponderance of π + over π − with increasing beam momentum as a consequence of the increase of phase space for particle production. We observe further that the general preponderance of π + over π − decreases with increasing atomic mass number A. For +8.0 GeV/c beam momentum, the trend even reverses from light to heavy nuclei. We interpret this feature as follows. The heavier the target nucleus, the larger the neutron-to-proton ratio. While lowenergy secondary protons produce in their re-interactions in nuclear matter considerably more π + than π − , the situation is the opposite for low-energy secondary neutrons as shown long ago in a pertinent experiment [23]. The heavier the target nucleus, the larger the neutron-to-proton ratio and therefore the contribution to π − production by secondary neutrons.  ), and π − 's (black circles), as a function of A 2/3 for, from left to right, beryllium, carbon, aluminium, 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 cross-sections of π ± production by 8 GeV/c protons, in the forward region, between beryllium, carbon, copper, tin, tantalum, and lead target nuclei, as a function of the charge-signed pion p T .

DEUTERON PRODUCTION
Besides pions and protons, also deuterons are produced on aluminium 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. 14, and compared with the predictions of Geant4's FRITIOF model. FRITIOF's predictions are shown for π + beam particles 3) . While there is for small polar angles θ good agreement between the data and FRITIOF's estimate, the latter tends to fall short of the data toward large polar angles. 2) We observe no appreciable dependence of the deuteron-to-proton production ratio on beam momentum. 3) There is less than 10% difference between its predictions for incoming protons, π + 's and π − 's.
In Fig. 15 we show, for the polar-angle region 30 • < θ < 45 • , how the deuteron-to-proton ratio varies with the mass of the target nucleus. The ratios are for 8 GeV/c beam protons on beryllium, carbon, aluminium, copper, tin, tantalum and lead nuclei. In Fig. 16 we show how the deuteron-to-proton ratio depends on the atomic mass number A. Since in this ratio the geometrical scaling with A 2/3 should cancel out, any remaining dependence should reflect re-interactions in the nuclear matter for which A 1/3 seems the right scaling variable. The ratios are averaged over the 0.65 < p < 1.05, where p is the particle momentum at the vertex, and shown separately for the polar-angle bins 20 • < θ < 30 • and 30 • < θ < 45 • . We note an approximately linear increase of the deuteron-to-proton ratio with A 1/3 , and a tendency to increase with polar angle.

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 aluminium target.
We have compared the inclusive aluminium π + and π − production cross-sections with those on beryllium, carbon, copper, tin, tantalum, and lead and find an approximately linear dependence on the scaling variable A 2/3 .
We also observe a significant production of deuterons off aluminium nuclei that we compared to the deuteron production on beryllium, carbon, copper, tin, 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 + Al → 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 + Al → 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 + Al → 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 + + Al → 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 + + Al → 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 + + Al → 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 − + Al → 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 − + Al → 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 − + Al → 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 + Al → 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 + Al → 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 + Al → 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 + + Al → 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 + + Al → 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 + + Al → 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 − + Al → 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 − + Al → 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 + Al → 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 + Al → 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 + + Al → 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 + + Al → 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 + + Al → 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 − + Al → 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 − + Al → 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 − + Al → 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 + Al → p + X interactions with +12.9 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 + Al → π + + X interactions with +12.9 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 + Al → π − + X interactions with +12.9 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 + + Al → p + X interactions with +12.9 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 + + Al → π + + X interactions with +12.9 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 + + Al → π − + X interactions with +12.9 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 − + Al → 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 − + Al → 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 − + Al → 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 + Al → 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 + Al → 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 + + Al → 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 + + Al → 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 + + Al → 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 − + Al → 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 − + Al → 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 − + Al → 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