A survey of backward proton and pion production in p + C interactions at beam momenta from 1 to 400 GeV/c
Abstract
Recent data on proton and pion production in p+C interactions from the CERN PS and SPS accelerators are used in conjunction with other available data sets to perform a comprehensive survey of backward hadronic cross sections. This survey covers the complete backward hemisphere in the range of lab angles from 10 to 180 degrees, from 0.2 to 1.4 GeV/c in lab momentum and from 1 to 400 GeV/c in projectile momentum. Using the constraints of continuity and smoothness of the angular, momentum and energy dependences a consistent description of the inclusive cross sections is established which allows the control of the internal consistency of the nineteen available data sets.
Keywords
Target Fragmentation Pion Production Beam Momentum Isospin Symmetry Invariant Cross Section1 Introduction
An impressive amount of data on backward hadron production in p+C interactions has been collected over the past four decades. A literature survey reveals no less than 19 experiments which have contributed a total amount of more than 3500 data points covering wide areas in projectile momentum, lab angle and lab momentum.
Looking at the physics motivation and at the distribution in time of these efforts, two distinct classes of experimental approaches become evident. 15 experiments cluster in a first period during the two decades between 1970 and 1990. All these measurements have been motivated by the nuclear part of protonnucleus collisions, in particular by the width of the momentum distributions in the nuclear rest system which reach far beyond the narrow limits expected from nuclear binding alone. These studies have ceased in the late 1980’s with the advent of relativistic heavy ion collisions and their promise of “new” phenomena beyond the realm of classic nuclear physics.
A second class of very recent measurements has appeared and is being pursued after the turn of the century, with publications starting about 2008. Here the motivation is totally different. It is driven by the necessity of obtaining hadronic reference data for the study of systematic effects in cosmic ray and neutrino physics, in particular concerning atmospheric and long base line experiments as well as eventual novel neutrino factories. The main aim of these studies is the comparison to and the improvement of hadronic production models—models which are to be considered as multiparameter descriptions of the noncalculable sector of the strong interaction, with very limited predictive power.
This new and exclusive aim has led to the strange situation that while all recent publications contain detailed comparisons to available production models, no comparison to existing data is attempted. It remains therefore unclear how these new results compare to the wealth of already available data and whether they in fact may override and replace the existing results.
In this environment the studies conducted since 15 years by the NA49 experiment at the CERN SPS have a completely different aim. Here it is attempted to trace a modelindependent way from the basic hadronnucleon interaction via hadronnucleus to nucleusnucleus collisions. This aim needs precision data from a large variety of projectile and target combinations as well as a maximum phase space coverage. As the acceptance of the NA49 detector is limited to lab angles below 45 degrees, it is indicated to use existing backward data in the SPS energy range in order to extend the acceptance coverage for the asymmetric protonnucleus interactions. This requires a careful study of the dependence on cms energy and of the reliability of the results to be used.
In the course of this work it appeared useful and even mandatory to provide a survey of all available data over the full scale of interaction energies, the more so as no overview of the experimental situation is available to date. This means that the present study deals with projectile momenta from 1 to 400 GeV/c, for a lab angle range from 10 to 180 degrees, and for lab momenta from 0.2 to 1.2 GeV/c.
2 Variables and kinematics
In this context the term “backward” needs a precise definition. One possibility would be to define as “backward” the region of lab angles Θ _{lab}>90 degrees. The present paper uses instead a definition which refers to the cms frame with the basic variables Feynman x _{ F } and transverse momentum p _{ T }, defining as “backward” the particle yields at x _{ F }<0. This allows a clear separation of the projectile fragmentation region at positive x _{ F } with a limited feedover into negative x _{ F } and the target fragmentation region at negative x _{ F } with a limited feedover into positive x _{ F }. At the same time the notion of “kinematic limit” in participant fragmentation is clearly brought out at x _{ F }=±1 and the contributions from intranuclear cascading may be clearly visualized and eventually separated.
This means that over the full range of lab angles and up to large p _{lab} values the contribution from target participants mixes with the nuclear component. The separation of the two processes therefore becomes an important task, see Sect. 10 of this paper.
Data sets for proton production in p+C and n+C collisions from seven experiments giving the ranges covered in projectile momentum, lab angle, and lab momentum, the number of measured data points and errors
Interaction  Experiment  Projectile momentum (GeV/c)  Lab angle coverage (degrees)  p _{lab} coverage (GeV/c)  Number of data points  Errors [%]  

〈σ _{stat}〉  〈σ _{syst}〉  
p+C  Bayukov [1]  400  70, 90, 118, 137, 160  0.4–1.3  35  6  20 
NA49 [2]  158  10, 20, 30, 40  0.3–1.6  40  7  5  
Belyaev [3]  17, 23, 28, 34 ,41, 49, 56  159  0.3–1.2  125  5  15  
HARPCDP [4]  3, 5, 8, 12, 15  25, 35, 45, 55, 67, 82, 97, 112  0.45–1.5  202  4  6  
Burgov [5]  2.2, 6.0, 8.5  162  0.35–0.85  36  15  5  
1.87, 4.5, 6.57  137  0.3–1.1  55  10  20  
Geaga [8]  1.8, 2.9, 5.8  180  0.3–1.0  50  17  15  
Frankel [9]  1.22  180  0.45–0.8  6  7  
Komarov [10]  1.27  105, 115, 122, 130, 140, 150, 160  0.34–0.54  ∼200  8  15  
n+C  Franz [11]  0.84, 0.99, 1.15  51, 61, 73, 81, 98, 120, 140, 149, 160  0.3–0.8  553  5  10 
Data sets for pion production in p+C collisions from seven experiments giving the ranges covered in projectile momentum, lab angle, and lab momentum, the number of measured data points and errors
Experiment  Projectile momentum (GeV/c)  Lab angle coverage (degrees)  p _{lab} coverage (GeV/c)  Number of data points  Errors [%]  

〈σ _{stat}〉  〈σ _{syst}〉  
Nikiforov [12]  400  70, 90, 118, 137, 160  0.2–1.3  59  12  
NA49 [13]  158  5, 10, 15, 20, 25, 30, 35, 40, 45  0.1–1.2  174  5  4 
Belyaev [14]  17, 22, 28, 34, 41, 47, 57  159  0.25–1.0  218  4  15 
Abgrall [15]  31  0.6–22.3  0.2–18  624  6  7 
HARPCDP [4]  3, 5, 8, 12, 15  25, 35, 45, 55, 67, 82, 97, 112  0.2–1.6  829  6  8 
HARP [16]  3, 5, 8, 12  25, 37, 48, 61, 72, 83, 95, 106,117  0.125–0.75  605  12  
Burgov [17]  2.2, 6.0, 8.5  162  0.25–0.6  29  20  
Baldin [18]  6.0, 8.4  180  0.2–1.25  45  10  
Cochran [19]  1.38  15, 20, 30, 45, 60, 75, 90, 105, 120, 135, 150  0.1–0.7  199  3  12 
Crawford [20]  1.20  22.5, 45, 60, 90, 135  0.1–0.4  50  8  7 
3 The experimental situation
The backward phase space coverage in p+C interactions is surprisingly complete if compared with the forward direction and even with the available data in the elementary p+p collisions. This is apparent from the list of experiments given in Tables 1 and 2 with their ranges in beam momentum, lab angle, and lab momentum. Although some effort has been spent to pick up all published results, this list is not claimed to be exhaustive as some results given as “private communication”, in conference proceedings or unpublished internal reports might have escaped attention.
For secondary protons, Table 1, the important amount of low energy n+C data by Franz et al. [11] has been added to the survey as the isospin factors for the transformation into p+C results have been studied and determined with sufficient precision, see Sect. 4.5.4.
For secondary pions, Table 2, the situation is somewhat complicated by the fact that two independent sets of results have been published by the HARPCDP [4] and the HARP [16] groups, based on identical input data obtained with the same detector. An attempt to clarify this partially contradictory situation is presented in Sect. 9.3 of this paper.
Unfortunately, no commonly agreed scale in the three basic variables Θ _{lab}, p _{lab} and p _{beam} of the doubledifferential cross sections has been defined by the different collaborations providing the data contained in Tables 1 and 2. This leads to the fact that not a single couple out of the more than 3500 data points contained in these Tables may be directly compared. The application of an interpolation scheme as described in Sect. 4 is therefore an absolute necessity. Ideally the thus obtained interpolated cross sections would form an internally consistent sample of results which would be coherent within the given experimental errors. As will become apparent in the following data comparison, this assumption is surprisingly well fulfilled for the majority of the experiments. Only four of the 20 quoted groups of results fall significantly out of this comparison; those will be discussed in Sect. 9 of this paper. In this sense the overall survey of the backward proton and pion production results in a powerful constraint for the comparison with any new data sample.
4 Data comparison
As stated above the main problem in bringing the wealth of available data into a consistent picture is given by the generally disparate position in phase space and interaction energy of the different experiments. The triplet of lab variables given by the beam momentum p _{beam}, the lab momentum p _{lab} and the lab angle Θ _{lab} has been used in establishing the interpolation scheme. In addition and of course, the statistical and systematic errors have to be taken into account in the data comparison.
4.1 Errors
The last columns of Tables 1 and 2 contain information about the statistical and systematic errors of the different experiments. The given numbers are to be regarded as mean values excluding some upward tails as they are inevitable at the limits of the covered phase space in particular for the statistical uncertainties. In some cases only rudimentary information about the systematic errors is available or the systematic and statistical errors are even combined into one quantity. In the latter cases these values are given in between the respective columns of Table 2.
Inspection of these approximate error levels reveals a rather broad band of uncertainties ranging from about 4 % to about 20 %, the latter limit being generally defined by overall normalization errors. The presence of extensive data sets well below the 10 % range of both statistical and systematic errors gives however some hope that a resulting overall consistency on this level might become attainable by the extensive use of data interpolation.
The term “interpolation” is to be regarded in this context as a smooth interconnection of the data points in any of the three phase space variables defined above. This interconnection is generally done by eyeball fits which offer, within the error limits shown above, sufficient accuracy. While the distributions in Θ _{lab} and interaction energy are anyway not describable by straightforward arithmetic parametrization, the p _{lab} dependences are, as discussed in Sect. 4.4 below, in a majority of cases approximately exponential. In these cases exponential fits have been used if applicable.
As additional constraint physics asks of course for smoothness and continuity in all three variables simultaneously. Therefore the resulting overall data interpolation has to attempt a threedimensional consistency.
Although the data interpolation helps, by the intercorrelation of data points, to reduce the local statistical fluctuations, it does of course not reduce the systematic uncertainties. It is rather on the level of systematic deviations that the consistency of different experimental results is to be judged. It will become apparent from the detailed discussion described below that the majority of the quoted experiments allows for the establishment of a surprisingly consistent overall description in all three variables.
4.2 Dependence on cms energy s
As the data discussed here span an extremely wide range of cms energy from close to production threshold to the upper range of Fermilab energies, a suitable compression of the energy scale has been introduced in order to be able to present the results in a closetoequidistant fashion against energy. The form chosen here is the variable \(1/\sqrt{s}\). This choice is suggested by the considerable amount of work invested in studying the approach of hadronic cross sections to the scaling limit at high energy in the 1970’s [21, 22, 23]. In fact the Regge parametrization suggested a smooth dependence of the cross sections as s ^{−α }, with α=0.25–0.5 depending on the choice of trajectories involved. Such behavior was indeed found experimentally. In the present study the cross sections turn out to have only a mild \(1/\sqrt{s}\) dependence for \(\sqrt{s} \gtrsim5~\mbox{GeV}\), a dependence which is however different for pions and protons. This dependence is strongly modified below \(\sqrt{s} \sim2.5~\mbox{GeV}\) due to threshold effects.
4.3 Angular dependence
A convenient and often used scale for the lab angle dependence is given by cos(Θ _{lab}). This scale has the advantage of producing shapes that are again to zero order exponential. Of course, continuity through Θ _{lab}=180 degrees imposes an approach to 180 degrees with tangent zero. As the data samples are generally not measured at common values of Θ _{lab}, a fixed grid of angles has been defined based on the Θ _{lab} values of the HARPCDP experiment [4] dominating the range from 25 to 112 degrees. Measured values down to 10 degrees and in the higher angular range at 137, 160, and 180 degrees have been added. Measurements not corresponding to these grid values are interpolated using the cos(Θ _{lab}) distributions specified below.
4.4 Lab momentum dependence
All data discussed here have been transformed into invariant cross sections (1). This facilitates the presentation in different coordinate systems and eliminates the trivial approach of the phase space element to zero with decreasing momentum. In addition, most of the invariant p _{lab} distributions are close to exponential within the measured p _{lab} range. There are notable deviations mostly at low momentum and in the lower (higher) range of lab angles for pions and protons, respectively, as well as in the approach to threshold. In these cases an eyeball fit has been used which can be reliably performed within the error margins indicated above.
A first group of distributions in the medium angular range at 45 and 97 degrees is presented in Fig. 3 for the HARPCDP data concerning protons and pions, including exponential fits.
These distributions are well described by centered Gaussians. The resulting rms values are however somewhat bigger than one signaling systematic experimental effects or a deviation of physics from the simple exponential parametrization. In view of the statistical errors of 4 % to 6 % given by HARPCDP (Tables 1 and 2) these deviations are on the level of a few percent which is anyway below the error margin expected from the present general data survey.
Compared to the strong dependence of B on Θ _{lab} which ranges from 0.3 to 0.05 GeV/c, only a modest dependence on p _{beam} of about 0.03 GeV/c for beam momenta from 3 to 158 GeV/c can be observed.
Following the above data parametrization a generalized grid of p _{lab} values between 0.2 and 1.2 GeV/c, in steps of 0.1 GeV/c, may now be established. Concerning the lower and upper limits of this grid, an extrapolation beyond the limits given by the experimental values has been performed in some cases. This extrapolation does not exceed the bin width of the respective data lists and is therefore safe in view of the generally smooth, gentle and welldefined p _{lab} dependences.
4.5 Physics constraints
In the absence of theoretical predictability in the soft sector of the strong interaction, any attempt at bringing a multitude of experimental results into a common and consistent picture needs to satisfy a minimal set of basic modelindependent physics constraints. In fact a straightforward averaging of eventually contradictory data sets would only add confusion instead of clarity.
4.5.1 Continuity
Two examples of the continuity constraint have already been mentioned above: invariant p _{lab} distributions have to approach zero momentum horizontally that is with tangent zero. The same is true for angular distributions in their approach to 180 degrees.
4.5.2 Smoothness
It is a matter of experimental experience in the realm of soft hadronic interactions that in general distributions in any kind of kinematic variable tend to be “smooth” in the sense of absence of abrupt local upwards or downwards variations. The widespread use of simple algebraic parametrizations has its origin in this fact, specifically in the absence of local maxima and minima, with the eventual exception of threshold behavior of which some examples will become visible below.
4.5.3 Charge conservation and isospin symmetry
Charge conservation has of course to be fulfilled by any type of experimental result. This means for instance that for the interaction of a positively charged projectile (proton) with an isoscalar nucleus (Carbon) the π ^{+}/π ^{−} ratio has to be greater or equal to unity over the full phase space invoking isospin symmetry (and of course the experience from a wide range of experimental results). The presence of data with π ^{+}/π ^{−}< 1 therefore immediately indicates experimental problems. The inspection of π ^{+}/π ^{−} ratios has the further advantage that a large part of the systematic uncertainties, notably the overall normalization errors, cancel in this ratio.
4.5.4 Isospin rotation of secondary baryons and projectile
4.5.5 Establishing a consistent set of data
With these constraints in mind, and having established the parametrization and interpolation of the p _{lab} distributions as discussed above, one may now proceed to the attempt at sorting the 19 available experiments into a consistent global data set. It would of course be rather surprising if all experiments would fit into this global picture within their respective error limits. In fact it turns out that this procedure establishes a very strong constraint for possible deviations, as a large majority of results can be accommodated in a perfectly consistent picture both for protons and for pions. Only four of the 19 data sets cannot be brought into consistency with all other experiments without seriously affecting and contradicting the above constraints. These data are not included in the following global interpolation scheme. They will be discussed separately in Sect. 9 below.
5 The proton data
5.1 \(\mathbf {1/\sqrt{s}}\) dependence

A strong yield suppression between \(1/\sqrt{s} \sim0.45\) and the elastic limit at \(1/\sqrt{s}=0.53\) is evident.

The n+C data [11] are in good agreement with the p+C results in the overlap regions; they define a broad maximum of the cross sections at \(1/\sqrt{s} \sim0.46\) at medium angles and low p _{lab}.

There is a welldefined asymptotic behavior of the cross sections for \(1/\sqrt{s}\) below about 0.2 or beam momenta above about 12 GeV/c.

For the lower Θ _{lab} region and/or low p _{lab} the asymptotic region is approached from above.
The abruptness of this decrease would necessitate a rather violent variation of the cross sections with increasing energy including a minimum between PS and SPS energies. A final clarification of this situation is given by the proton data from Serpukhov [3] which, although suffering from a different and independent problem, at least exclude such variations in the region between 17 and 67 GeV/c beam momentum, see Sect. 9.2 of this paper.
5.2 cos(θ _{lab}) dependence
In addition to the description of the energy dependence, the global interpolation has of course to result in a smooth and continuous description of the angular dependence, representing the third dimension of the present study. This constraint has to be fulfilled at any value of \(1/\sqrt{s}\).

The two experimental results connect perfectly through the gap between the NA49 (θ _{lab}< 40 degrees) and the Fermilab (θ _{lab}> 70 degrees) data.

There is at most a few percent variation of the cross sections between the angles of 160 and 180 degrees taking into account the constraint of continuity through 180 degrees discussed in Sect. 4 above. This allows the combination of results in this angular region as it is applied in the determination of the \(1/\sqrt{s}\) dependence, Fig. 7.

The angular distributions are smooth and close to exponential in shape. In particular, no instability in the region around 90 degrees is visible where an eventual diffractive peak from target fragmentation would appear, see also [2].
Evidently the angular distributions maintain their smooth and continuous shape, specifically through 90 degrees, at all interaction energies. With the approach to low beam momenta however, a progressive rounding of the shape towards higher lab angles manifests itself.
6 The data for positive pions
The global interpolation of the π ^{+} data is presented in this section in close analogy to the preceding section for protons.
6.1 \(1/\sqrt{s}\) dependence

At the lowest lab momentum, the pion cross sections are practically sindependent, with variations of only 10–20 % in the range from 1 to 400 GeV/c beam momentum.

This fact suggests π ^{+} production at low momentum transfer in the nuclear cascade.

For all lab momenta, the approach to high energies is very flat for \(1/\sqrt{s} <0.2\) or beam momenta above 12 GeV/c.

The high energy cross sections are approached for all angles and beam momenta from below.
These deviations are rather consistent with the ones found for protons. Also in this case a rapid variation of the cross sections with increasing beam momentum can be excluded by the comparison with the pion data from the Serpukhov experiment [14] between 17 and 67 GeV/c beam momentum, see Sect. 9.2 below.
6.2 cos(θ _{lab}) dependence
The angular distributions are characterized by a smooth, close to exponential shape. At backward angles, the p _{lab} dependence is very steep with four orders of magnitude already between p _{lab}=0.2 and 0.8 GeV/c. In forward direction this dependence is much reduced with less than one order of magnitude between p _{lab}=0.2 and 1.2 GeV/c. This is due to the prevailance of target fragmentation in this region, see Sect. 10 for a quantitative study of this phenomenology.
7 The data for negative pions
This section follows closely the discussion of the π ^{+} cross sections in the preceding section.
7.1 \(1/\sqrt{s}\) dependence

All the different data sets form a consistent ensemble without the systematic deviations visible in some regions of the proton and π ^{+} results.

The approach to large beam momenta happens from below for all p _{lab}.

The sdependence is in general stronger than for π ^{+}, Fig. 12. While it is again flat up to \(1/\sqrt{s} \sim 0.2\) at low p _{lab}, it becomes more pronounced both towards higher p _{lab} and in the approach to the production threshold at large \(1/\sqrt{s}\) indicating a marked increase of the π ^{+}/π ^{−} ratio.

This effect has as physics origin the progressive change of the production mechanism from pion exchange at low energy to gluon or pomeron exchange at SPS energy. This will be discussed in relation to the charge ratios in Sect. 8.
Although for both reactions the asymptotic high energy region is approached from below, this comparison shows a stronger sdependence, at the same lab angle, in p+C than in p+p collisions. This is due to the component of nuclear cascading which contributes, in the given angular range, with equal strength than the target fragmentation to the total yield (see Sect. 10).
7.2 cos(θ _{lab}) dependence
Concerning smoothness and continuity these distributions are similar to the π ^{+} data, including the large asymmetry between the forward and backward directions. The reduction of the cross sections for π ^{−} with respect to π ^{+} with increasing \(1/\sqrt{s}\) is however very apparent. This will be quantified in the following Section on π ^{+}/π ^{−} ratios.
8 The π ^{+}/π ^{−} ratio
As a byproduct, the fluctuation of this ratio as a function of angle and interaction energy will allow for the estimation of the local precision of the interpolation procedure.
8.1 The high energy limit
It has been established by numerous experimental results that at collision energies in the SPS/Fermilab range and above the hadronic interactions are characterized by the absence of charge and flavor exchange. It has also been shown that the feedover of pions from the projectile hemisphere into the backward region of x _{ F } is sharply limited to the range of x _{ F }≳−0.05, see [31] for a detailed discussion. This range is outside the coverage in Θ _{lab} and p _{lab} considered in this publication.
8.2 Energy, momentum and angle dependence of R _{±}

Considering the wide range of lab angles, R _{±} is at each value of \(1/\sqrt{s}\) confined to a narrow band indicating an approximative angle independence.

Large R _{±} values in excess of 5 are reached at the upper limit of the available scale in \(1/\sqrt{s}\).

There is a systematic increase of R _{±} with p _{lab}, as shown quantitatively in Fig. 25.
8.2.1 Dependence of R _{±} on Θ _{lab}
Evidently no systematic Θ _{lab} dependence is visible over the complete angular range.
8.2.2 Mean π ^{+}/π ^{−} ratios and estimation of the local systematic fluctuations of the interpolation process
The observed energy dependence of the rms deviations is due to the fact that the invariant pion cross sections decrease, after a relatively flat behavior up to \(1/\sqrt{s} \sim0.15\), progressively steeper towards the production threshold, see Figs. 12 and 16. This leads to larger local variations in the corresponding energy interpolation.The error bars shown in Fig. 22 correspond to the error estimator \(\sigma_{\Delta R_{\pm}}\), Figs. 23 and 24.
From the rms values given in Fig. 23 and the number of entries per histogram the error of 〈R _{±}〉 may be derived which is given by the broken line in Fig. 24, varying from about 1 % to 5 % for the highest and lowest interaction energy, respectively. Also the corresponding error margins for the pion yields may be derived from the rms fluctuation around the mean particle ratios. For the average pion cross sections this corresponds to the dotted line in Fig. 24 indicating an increase from a few percent in the high energy region to about 10 % in the approach to the pion threshold.
8.2.3 Dependence of 〈R _{±}〉 on \(1/\sqrt{s}\) and p _{lab}
Within the errors of 〈R _{±}〉 extracted above, a clear p _{lab} dependence is evident superposing itself to the strong common increase of 〈R _{±}〉 with \(1/\sqrt{s}\). This increase may be parametrized up to \(1/\sqrt{s} \sim0.3\) by the functional from \(1 + c/s^{\beta(p_{\mathrm{lab}})}\) which is, as discussed below, typical of meson exchange processes. Indeed the exponent β varies from 2 to 1.2 for p _{lab} increasing from 0.2 to 0.8 GeV/c.
8.3 Interpretation of the observed energy and momentum dependences
In this context it seems mandatory to first refer to the study of exclusive charge exchange reactions in elementary nucleonnucleon collisions as the complete energy range discussed here has been covered there by a number of experiments [32, 33, 34, 35, 36, 37, 38].
8.3.1 The charge exchange mechanism in elementary nucleonnucleon collisions
 Charge exchange scattering of the elastic type
 Single dissociation with pion production
 Double dissociation with pion production
These channels are characterized by a very steep energy dependence.
 Elastic scattering
 Single dissociation
 Double dissociation
Charge exchange scattering has been measured by five experiments in the range of neutron beam momenta from 3 to 300 GeV/c [32, 33, 34, 35, 36]. This is exactly covering the energy range discussed in this paper. The single and double dissociation has been studied at the CERN ISR by two experiments [37, 38] extending the energy scale to s=3700 GeV^{2}. The two ISR experiments may be directly compared to the charge exchange measurements after appropriate renormalization of the cross sections in the overlap region at the lowest ISR energy.

There is a decrease of about 4 orders of magnitude in cross section between the lowest and highest s value. This decrease is to be compared to the constant or logarithmically increasing elastic and nucleon diffraction cross sections. The charge exchange contribution is therefore negligible compared to the inclusive baryon yields already at SPS energy.

There is a steady decrease of the local slope df/ds with energy, from about 3.6 at s=10 GeV^{2} to about 1.1 above s=200 GeV^{2}.

A characteristic change of slope manifests itself at around s=60 GeV^{2}.
With increasing energy the slopes move through the region of pion exchange with β∼ 2 down to values of about 1.1 at high energy which could be connected to ρ and a_{2} exchange with correspondingly higher intercepts α in the region of 0.5. At ISR energy the ratio of ρ/π contributions has indeed been estimated to be about 2 [38]. Anyway the simple parametrization given by (23) should not be expected to hold over the full energy scale. What is interesting here is rather the strong decline of the charge exchange cross sections with energy and the experimentally rather precisely determined slope variation.
8.3.2 A remark concerning baryon resonance production in hadronic interactions
The single (18) and double (19) dissociation processes defined above are determined by the formation of Δ resonances in the final states. They therefore constitute a source of direct Δ production in nucleonnucleon interactions. These channel cross sections decrease rapidly to the μbarn level at SPS energies. In contrast, the noncharge exchange channels like (21) and (22) have no sdependence and stay on the mb level of cross sections. Their final states have been shown to be governed by N^{∗} resonances [39] which may be excited by pomeron exchange. Moreover, the p+π ^{+} combination of the p+π ^{+}+π ^{−} final states has been shown to be dominated by Δ^{++} [40]. This is an indirect source of Δ resonances as a decay product of N^{∗} states which have large decay branching fractions into Δ+π and Δ+ρ. It is therefore questionable if, at SPS energies and above, any direct Δ production is persisting. This is an interesting question for the majority of microscopic models which produce final states by string fragmentation. In the baryonic sector, diquark fragmentation is generally invoked with a prevailing direct production of Δ resonances which by isospin counting will dominate over N^{∗}. Indeed in practically all such models there is no or only negligible N^{∗} production. As shown below, the decrease of charge exchange processes can be traced well into the nondiffractive, inelastic region of particle production. The multistep, cascading decay of primordial N^{∗} resonances into Δ resonances and final state baryons should therefore be seriously considered, in particular also concerning the consequences for the evolution of final state energy densities with time.
8.3.3 The charge exchange mechanism in p+C interactions as a function of interaction energy

A first region with large slopes is located at s below about 6 GeV^{2}. This region is strongly influenced by threshold effects as the threshold for inelastic production is placed at the elastic limit \(s = 4m_{p}^{2} = 3.5~\mbox{GeV}^{2}\) indicated in Fig. 27. In the approach to pion threshold the π ^{+}/π ^{−} ratio has to diverge as π ^{−} is progressively suppressed, see above. With increasing p _{lab} this suppression will of course be more pronounced.

An intermediate region between about 8 and 40 GeV^{2} with an s dependence decreasing with increasing p _{lab}.

A third region with flattening sdependence above about 40 GeV^{2}.
This figure shows clearly the different nature of the low s enhancement where the slopes increase strongly with p _{lab}. The two other regions, full and dotted lines, are compatible with a Regge parametrization with trajectory intercepts which increase with p _{lab}. This is insofar interesting as the region of measurements regarded here covers the complete backward angular range and the corresponding interactions are by no means confined to diffractive or low momentum transfer collisions. It is shown in Sect. 10 of this paper that in the backward hemisphere the pion yields from nuclear cascading and target fragmentation are comparable. While the nuclear component is characterized by low momentum transfer reactions [2] the target fragmentation is manifestly inelastic and nondiffractive. It governs the total yield at all angles below about 70 degrees.

The global data interpolation leads to a precise and consistent description of the behavior of the π ^{+}/π ^{−} ratios in the full backward hemisphere, thus offering an additional tool for the discrimination of experimental deviations.

The inspection of the detailed sdependence of the ratios opens a new window on the underlying exchange processes.

In particular the comparison to the elementary nucleonnucleon collisions establishes a close relation between apparently disjoint sectors of the different hadronic interactions.
9 Data sets not used in the global interpolation
As mentioned in Sect. 4.5 four of the 19 investigated data sets are incompatible with the attempt at generating an overall consistent description of the experimental situation. These data will be shortly discussed below.
9.1 The proton data of Ref. [8]
As the angular bin from 160 to 180 degrees is mostly covered by data around 160–162 degrees, a steep angular dependence in this region cannot a priori be excluded. The smooth and gentle angular dependence of the interpolated data shown in Fig. 31 for the angular range from 82 to 180 degrees and for the three \(1/\sqrt{s}\) values of Ref. [8], together with the constraint of the approach to 180 degrees with tangent zero, excludes however a drop of the cross sections by a factor of two between 160 and 180 degrees.
9.2 The proton and pion data of Refs. [3, 14]

The shape of the \(1/\sqrt{s}\) dependences complies precisely with the global interpolation. This is compatible with the absence of rapid variations of the cross sections with energy in the region between PS and SPS.

There is a pronounced suppression of these data with respect to the interpolation with increasing p _{lab} reaching factors of three at the upper ranges for protons and pions.

The π ^{+} and π ^{−} data show an identical behavior.

The proton data are tracing the interpolation up to p _{lab}=0.4 GeV/c whereas the pion data are already suppressed in this p _{lab} range.

The suppression factors are generally bigger for the pions at equal p _{lab}.

These features might be compatible with a momentum scale error.
9.3 The pion data of Ref. [16]
Figure 36 demonstrates the importance of using, in addition to the invariant cross section proper, the particle ratios which are strongly constrained by physical arguments, see Sect. 8.
9.4 The pion data of Ref. [15]
These data have been obtained at a beam momentum of 31 GeV/c in a Θ _{lab} range from 0.6 to 22.3 degrees and p _{lab} from 0.2 to 18 GeV/c. While a large part of the given angular and momentum coverage falls outside the backward region regarded here, the low momentum range up to p _{lab}∼ 0.5 GeV/c for all angles and the range 0.6<p _{lab}<1 GeV/c for angles above about 9 degrees corresponds to negative x _{ F } and can therefore be considered here.
Apparently this cross section ratio is within errors angle independent over the full range of the global data survey, with well defined averages below 1.05 for π ^{+} and 1.1 for π ^{−}. In contrast, the cross section ratio between NA49 and Ref. [15] shows values in the region of 1.4 increasing with decreasing Θ _{lab}.
In conclusion to this section it may be stated that the global data interpolation between 15 different experiments attempted in this paper proves to be a useful tool for the detection of deviating data sets. Further details concerning the above comparisons can be found in an internal report on Ref. [29].
10 The separation of target fragmentation and intranuclear component for pion production at SPS energy
Hadronic production in the backward direction of p+A collisions has two components: the fragmentation of the target nucleons which have been hit by the projectile proton, and the propagation of momentum transfer into the nucleus by secondary nucleon–nucleon interaction which follow, on a longer time scale, the initial excitation process. Both processes are governed by the mean number of collisions 〈ν〉 suffered by the projectile on his trajectory through the nucleus.
As only the sum of these two separate mechanisms is experimentally accessible, a minimum assumption about the fragmentation of the target nucleons is needed in order to allow the separation of the components in an otherwise modelindependent fashion. This minimal assumption consists in assuming that the fragmentation process of the hit nucleons is equal to the basic nucleon–nucleon interaction, taking full account of course of isospin symmetry. In addition and only valid for the relatively small value of 〈ν〉 in the Carbon nucleus, it will be assumed that successive collisions result in hadronization at full interaction energy of the corresponding elementary interactions.

A MonteCarlo calculation using the measured nuclear density distributions.

The relation between the inelastic cross sections of p+p and p+C interactions.

The approach to x _{ F }=−0.1 of the ratio of pion densities in p+C and p+p collisions.
The two former methods have to make the assumption that the inelastic interaction cross sections are independent of the number of subsequent collisions ν.
In [2] a similar approach is used concerning the production of protons and antiprotons, again in the regions where there is no contribution from nuclear cascading as well as in the full backward hemisphere.
All methods mentioned above result in a consistent estimate of 〈ν〉=1.6 in p+C collisions, with a relative systematic uncertainty of the order of a few percent.
The correlation between p _{lab} and Θ _{lab} for fixed values of R ^{pred} shown in panel a traces rather exactly the kinematic correlation between the same variables for fixed values of x _{ F }, panel b. This allows to establish a direct dependence of R ^{pred} on x _{ F } which is to first order angleindependent, panel c.
This subtraction procedure becomes of course uncertain in the small angle region where the nuclear component is on the few percent level or below with respect to the target fragmentation, see Figs. 39 and 41.
Evidently the nuclear component of pion production stays comparable to the target fragmentation in the full backward hemisphere of Θ _{lab}. It decreases rapidly for Θ _{lab} below about 60 degrees and vanishes below Θ _{lab}=25 degrees.
The p _{ T } integrated pion density dn ^{nucl}/dx _{ F }(x _{ F }) shows a peak at x _{ F }∼−0.2 and vanishes at x _{ F }∼−0.08. As shown by the density ratio with the predicted target fragmentation dn ^{pred}/dx _{ F }(x _{ F }) in Fig. 47b, the nuclear component reaches 10 % of the target fragmentation at x _{ F }=−0.15 and exceeds this contribution for x _{ F }<−0.55.
The nuclear pion component extracted above is used in [2] in conjunction with the complementary nuclear proton component to obtain the percentage of cascading protons which are accompanied by pion emission.
11 Conclusion
This paper presents a survey of available data concerning backward proton and pion production in minimum bias p+C interactions, including new and extensive data sets obtained at the CERN PS and SPS. The backward direction being defined as the complete phase space at negative Feynman x _{ F }, the data cover, for projectile momenta from 1 to 400 GeV/c, the ranges from 0.2 to 1.2 GeV/c in lab momentum p _{lab} and from 10 to 180 degrees in lab angle Θ _{lab}. The paper attempts an interconnection of the different data sets by a detailed threedimensional interpolation scheme in the variables \(1/\sqrt{s}\), p _{lab}, and cos(Θ _{lab}). This attempt allows a precise control of the internal data consistency as well as the study of the evolution of the invariant inclusive cross sections in all three variables.
A literature search has provided a set of 19 different experiments with a total of more than 3500 data points. These measurements were obtained over 40 years of experimentation by collaborations employing widely different experimental techniques. In this respect it may be stated as a first positive result that the majority of the data may be combined into a surprisingly selfconsistent ensemble. This global interpolation scheme results in a considerable discriminative power against the systematic deviation of particular data sets. Only 4 of the 19 quoted experiments show in fact deviations which clearly mark them as systematically diverging. These experiments are inspected in detail one by one in an attempt to clearly bring out the discrepancies. In some of the cases, possible experimental error sources are pointed out.
The underlying physics provides for additional constraints concerning basic quantities like charge conservation and isospin symmetry as well as the necessity of smoothness and continuity of the observed cross sections. Whenever possible, contact to the complementary elementary nucleonnucleon interactions is established. This concerns in particular the evocation of mesonic exchange processes for the description of π ^{+}/π ^{−} ratios and the prediction of the target fragmentation from elementary interactions and its separation from the component of nuclear cascading.
As far as the dependences of the invariant cross sections on the three basic variables p _{lab}, Θ _{lab} and \(1/\sqrt{s}\) is concerned, a well constrained phenomenology emerges. The p _{lab} dependences are exponential or close to exponential over a major part of the phase space with some exceptions mostly for low interaction energies. This fact results in an important constraint for the data interpolation. The cos(Θ _{lab}) dependences are not far from exponential and smooth and continuous through all lab angles. In particular there is no indication of an instability around 90 degrees for the proton yields. The \(1/\sqrt{s}\) dependences converge, after strong variations close to production threshold, smoothly to asymptotic behavior in the SPS energy range. This region is approached from above by the protons and from below for the pions. This convergence is confirmed by the π ^{+}/π ^{−} ratios which show, being governed by meson exchange at low \(\sqrt{s}\) with large values defined by the projectile isospin, a smooth decline with energy towards unity as expected from the underlying elementary exchange processes.
Notes
Acknowledgements
This work was supported by the Polish National Science Centre (on the basis of decision no. DEC2011/03/B/ST2/02634) the Polish State Committee for Scientific Research (P03B00630), the Bulgarian National Science Fund (Ph09/05), the EU FP6 HRM Marie Curie IntraEuropean Fellowship Program, the Hungarian Scientific Research Fund OTKA (T68506) and the Hungarian OTKA/NKTH A0877719 and A0877815 grants.
Open Access
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