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Production of charged pions off nuclei with 3...30 GeV incident protons and pions

We compare calculations for the production of charged pions by pion or proton beams off nuclei calculated within our coupled channel transport model (GiBUU) with recent data of the HARP collaboration for beam energies from 3 up to 13 GeV. Predictions for the 30 GeV data for pions and kaons from the NA61/SHINE experiment are included.


Introduction
done, limiting the range of validity of these models. The lack of good quality and systematic data concerning hadron-nucleus collisions in this energy regime has for long been an obstacle for a serious test of the models. The advent of the HARP experiment has changed the situation, since it offers charged pion double differential cross sections with a good systematics in angle, pion momentum, incident energy and target mass.
In this paper we show comparisons of calculations within the GiBUU transport model to data obtained with the forward spectrometer as well as with the large-angle spectrometer of the HARP experiment. The calculations are done without any fine tuning to the data covered here with the default parameters as used in the GiBUU framework for all possible processes.

GiBUU Transport Model
In this paper we employ the GiBUU model for an analysis of these data. This model has been developed as a transport model for energies from some MeV up to tens of GeV (4). Here we can study all kind of elementary collisions induced by baryons, mesons (see e.g. (5)), (real and virtual) photons (see e.g. (6; 7)) and neutrinos (see e.g. (8; 9) and further references therein) on all kind of nuclei within a unified framework. The underlying code is written in modular FORTRAN 2003 and available for download at (4).
In the GiBUU model we solve a strongly coupled system of equations for oneparticle phasespace densities, the so called Boltzmann-Uehling-Uhlenbeck (BUU) equations. The total time evolution of these phasespace densities is given by the motion in hadronic mean fields and potentials (Vlasov-part) combined with a collision integral. The actual implementation includes 61 baryons and 21 mesons. The evolution equations for all these particles are coupled by the Hamiltonian in the Vlasov-part and also by the collision integral. The phase space densities are approximated via the testparticle ansatz as a sum of delta distributions in spatial dimensions and momentum.
The collision integral is mainly dictated by (elastic and inelastic) two body collisions. At low energies these reaction mechanisms are governed by a resonance description. At higher energies, the collisions are done via the Pythia event generator. Here we are switching smoothly between the two descriptions for √ s = 2.2(2.6)±0.2 GeV for meson-baryon (baryon-baryon) collisions. Contrary to most other hadronic interaction models, which are using a string excitation picture via Pomeron exchange (e.g. Fritiof or some own implementations), we are using Pythia (v6.4) also at these low energies. At the energies covered here, we are in a transition region, where the high energy processes embedded in the Pythia model may reach their limits. However, Pythia also contains some low energy processes. Contrary to a Pomeron exchange model, as e.g. implemented in Fritiof or UrQMD, low energy interactions are treated as string flips in Pythia. Thus only the Jetset-Part of Pythia is responsible for these processes and, therefore, the influence and number of (internal) model parameters is limited. We have ascertained that Pythia describes the hadron production cross sections in general quite well even at relatively low invariant masses; the detailed comparison with the HARP data can serve as a further test of the general accuracy of the method. For further details, including e.g. plots of total cross sections for the elementary processes see Ref. (4).
The elementary interaction with the nucleon is assumed to be the same as that with a free nucleon. All the standard nuclear effects like Fermi motion, Pauli blocking and nuclear shadowing are properly taken into account. In a second step, all produced (pre-)hadrons are propagated through the nuclear medium according to the semi-classical Boltzmann-Uehling-Uhlenbeck transport equation. The concept of pre-hadrons (i.e. produced hadrons interact with some reduced cross section during their hadronization time) was introduced in order to realize color transparency and formation time effects (10; 11). All interactions, primary or secondary, are therefore treated within the same prescription and we thus present a full consistent coupled channel transport approach.
At the given energies (beam momenta > 3 GeV/c), these elementary pN or π ± N interaction are done within the Pythia prescription, since the available collision energies are above the resonance regions ( √ s πN,pN = 2.8 · · · 4.9 GeV).
In addition also the momenta of the observed final particles are quite large (larger than hundreds of MeV/c), thus reducing the importance of the baryonic potentials, which are a major difference of the GiBUU model to other cascade Monte Carlo models.

Results
We begin our comparison with data obtained with the forward spectrometer for a Carbon target (1). In fig. 1 we show the double differential production cross section of charged pions for the 12 GeV/c proton beam. Here the covered momentum range of the detected pions reaches up to 8 GeV/c. For more backward angles ( 120 mrad) the agreement is perfect. For the very forward angles (up to 120 mrad) we observe a too flat behavior of the calculated spectra compared to data. The calculations overshoot the data for large mo- Cross section d 2 σ/dp dΩ for p+C → π ± +X with 12 GeV/c beam momentum (left) and p + Al → π + + X with 12.9 GeV/c beam momentum (right). Experimental data are from (1; 3) (HARP small angle analysis). menta significantly, especially for the π − channel 1 . This seems to be a general shortcoming of all the hadronic interaction models GHEISHA, UrQMD and DPMJET-III shown in (1). However, in contrast to the model predictions of these models, our calculations reproduce the decrease of the cross section for very small momenta (e.g. for p 2 GeV/c for angles smaller than 90 mrad).
In fig. 1 (right panel) we also show a comparison of our model calculations with experimental data (3) with a 12.9 GeV/c beam on an Al target for positive charged pions. Here the agreement is very good for all momenta and angles.
In order to clarify the question whether the problems observed for the very forward angles and high momenta for 12 C are due to problems in the treatment of the FSI or already present in the hard first interaction, we compare in fig. 2 our model with experimental data for elementary reactions (13). Here the relevant kinematical variables are the transverse momentum p T and the cm rapidity y * . It is obvious that the actual model implementation agrees very well with these data, even though there are small discrepancies visible around y * ∼ 1.5. It is worthwhile to mention, that this holds only for the newest 0.01 (13). Due to symmetry only y * > 0 is shown. . Cross section d 2 σ/dp dΩ for π ± + C → π ± + X with 12 GeV/c beam momentum. Experimental data are from (1) (HARP small angle analysis).
versions of Pythia (v6.4) as used here in our model; older versions (e.g. v6.2) seem to prefer low transverse momentum and high rapidity and are not able to describe these data. Since we are able to describe the outgoing pions from elementary pN collision correctly, the overprediction of the cross section at high momenta and very forward angles on nuclear targets has to be attributed to interactions of the outgoing pion with the the nuclear medium. However, as we will see below, our calculations are in agreement with pion induced data on nuclei so that a puzzle remains. We leave the further investigation of this problem to future studies (17).
In fig. 3 we show the comparison with data with π + and π − beams. Here the agreement is very good for all available angles and all momenta. Not only the charge-conserving channels, but also the double charge exchange channels are well reproduced. This limits the uncertainties connected with the final state interactions in the proton induced reactions, as discussed before.
We continue our comparison with data with the large angle spectrometer (2). In order to keep this paper reasonably short we restrict ourselves to comparisons for a few selected energies only. A gallery of more comparisons is available at (12).
In fig. 4 we compare calculations with the data for the proton beam at 3 GeV.
In the large angle analysis all the momenta of the detected pions are below 1 GeV/c. One sees a very good overall agreement for perpendicular or even backward directions for all nuclei. Small discrepancies occur mainly for angles below 750 mrad at very low momenta 0.2 GeV/c where the calculations are higher than the experimental data. Correspondingly, the slope for momenta larger than 0.4 GeV/c is too flat in our calculations. For light nuclei the slope is in agreement with data, while the overall yield is somewhat too small. We note that these observations also hold for the negatively charged pions not shown here.
In order to illustrate the energy dependence of our results, we compare in fig. 5 the calculations for positive pion production with the 12 GeV/c proton beam. The overall behavior of the calculations changes smoothly from 3 GeV/c to 12 GeV/c, a comparison for 5 and 8 GeV/c can be found in (12). For the higher energies the data do not show the strong dip observed for small angles and small momenta at 3 GeV/c. However the overall yield for the small angles is still somewhat too low.
For all energies one observes for the perpendicular directions (≃ 1550 mrad) a 'bumpy' structure around p ≈ 0.5 GeV/c. We note, that while this structure is not very pronounced in the experimental data for π + , the experimental data for the π − channel (not shown here) do exhibit this feature. Calculations for a nucleon target indicate a smooth behavior. For the nuclear target at momenta around 0.2 GeV/c rescattering and the ∆ resonance dominate. This small momentum regime is populated by originally higher-energy pions that have been slowed down due to rescattering; only due to these final state interactions the overall yield at the lower momenta is reproduced. Without FSI the yield for momenta around 0.2 GeV is underestimated by at least one order of magnitude.
We note, that the laboratory angles of the HARP experiment can be translated into a cm pseudo rapidity η * 2 . The given angular bins of the forward angular spectrometer analysis cover the range η * = 0.5 · · · 2.1. The three smallest angles correspond to the range η * > 1.2. The discrepancies for large angles of the backward spectrometer analysis then show up as discrepancies at rapidities η * < −1.8. This is the feature already seen in the comparisons to the small angle analysis, but now for the very backward rapidities. The transverse momentum region in the data for these extreme rapidities is comparable.
In order to quantify the agreement between data and calculations, we calculated χ 2 /n dof values for every curve shown in this paper. We note, however, that our model does not exhibit any free parameter and we did not any fit procedure to minimize this measure. The resulting values for χ 2 /n dof may thus be very large. Nevertheless, they show some remarkable systematics, as e.g. they reach up for all nuclei to ∼ 20 for θ < 950 mrad and also for θ > 1550 mrad with proton beam and the large angle analysis (cf. fig. 4 and fig. 5). Inbetween the measure drops to values around 4. This is reflected in the fact, that for the small angles of this backward analysis all calculated curves show the right slope, but are systematically below the data for all nuclei. In fact, by introducing an additional artifical overall normalization factor, we would be able to decrease the χ 2 /n dof values down to 1. The mismatch for the largest angles could not be cured by simple up-/downscaling the calculated curves.
Here the large numerical values represent the fact, that we have significant discrepancies in the slopes, as discussed above. The measure for the forward angle analysis does not show any simple tendencies, except the fact that the pion induced data yield values ∼ 3 in the negative channel and ∼ 1 in the positive one and are thus in a reasonable regime.
While the experimental spectra of positive and negative charged pions show substantial differences, the spectra resulting from our calculations do exhibit the same features for both charge states. In order to quantify the differences between the two isospin states, we show in fig. 6 the ratio of the yields of π − over the yield of π + . We have chosen the two smallest and the two largest nuclei. The discrepancies between data and calculations are quite significant, although the overall tendencies are reproduced. As in experiment we observe an excess of π − over π + for large nuclei at low momenta. It was suggested by the experiment E910 (14) that this is due to production of Λ 0 . However, this is not supported by our calculations where the charge asymmetry for the pions originates in decays of ∆'s at rest. The latter have long collision histories: The higher the initial energies, the more collisions it takes to generate resonances at rest, i.e. the higher the initial energy, the longer the collsion history in order to 'stop' the resonance. If there is an imbalance of the charges of the collision partners (more neutrons than protons), the charge of the outgoing particles is driven towards the charge asymmetry. Therefore we observe that the pion charge asymmetry grows with 1) the initial energy (more collisions needed to get a stopped Delta) and with 2) the neutron/proton asymmetry.
We have checked, that in a (fictitious) charge symmetric nucleus of the size of Pb the ratio of negative over positive charged pions is indeed the same as in Be. We have also checked, that our results are not affected by switching on or off baryonic potentials or Coulomb corrections.
At the same time the HARP experiment is used to reduce uncertainties on the flux calculations of the K2K experiment, the NA61/SHINE experiment is aimed to measure the pion and kaon yields for the T2K experiment (15; 16).
Here the beam energy is 30 GeV, while also upgrades to 40 GeV and 50 GeV are foreseen. Responsible for the neutrino flux will be pions and kaons with momenta 1 · · · 10 GeV/c and angles θ 300 mrad.
In order to ease the comparison with the previous parts, we will use the same binning for the calculations at these higher energies as those for HARP with the forward spectrometer. In fig. 7 we show our resulting spectra of charged pions and kaons for a proton beam on a Carbon target. We note that a comparison of the calculated pion yields with 24 GeV/c on a proton target with experimental data (13) is as good as for 12 GeV/c beam momentum (12). Therefore the uncertainties of the calculations are again mainly due to the final state interactions. As soon as experimental binning is available, we are able to produce the spectra directly comparable with the upcoming data.

Conclusions
We have presented a comparison of calculations within the GiBUU transport model with recent data on inclusive pion production on different nuclear targets with pion or proton beams in the region of 3 up to 12 GeV/c beam momentum. Contrary to other theoretical frameworks we are able to cover the full energy range of the HARP experiment.
The best description is achieved for the data with pion beams. The agreement obtained for proton beams is very good over the whole energy-range, except for very forward and very backward directions. These deficiencies seem to be due to FSI as a comparison with corresponding data for elementary p + p collisions shows. This underlines the need to understand results on elementary collisions before drawing conclusion on data taken on nuclei as targets.
While the experimental data are especially useful as input to neutrino flux calculations, we also have here a very powerful set of data for checking final state interactions within our transport code, not only as needed for the neutrino production processes, but also for describing the interactions of neutrinos with nuclei in one and the same theory and code.
We have also presented first theoretical results for the 30 GeV proton run in the NA61/SHINE experiment which aims for a precise determination of the neutrino flux in the T2K experiment. Since soon the data from this experiment will cover a wide range of beam energies, it will be interesting to extract from the beam energy dependence conclusions on formation time aspects and color transparency (17).