STUDY OF PROTON-PROTON COLLISIONS AT THE BEAM MOMENTUM OF 1628 MeV/c
K.N. Ermakov, V.I. Medvedev, V.A. Nikonov, O.V. Rogachevsky, A.V. Sarantsev, V.V. Sarantsev, and S.G. Sherman
1. Introduction
The nucleon-nucleon (NN) interaction is one of the most important processes in nuclear and particle physics that has been extensively studied over a wide energy range. In its turn, the single pion production inNN interactions is the main inelastic process at the energies below 1 GeV. Our understanding of the nucleon-nucleon interaction cannot be based on the description of elastic reactions only: any relevant model should also explain inelastic reactions. In this respect, there is an important question ofthe pion production mechanism: what fraction of the cross section is determined by production of the Δ33(1232) isobar and what fraction is due to non-resonant transitions. Another important question is the issue of the isoscalar (I=0) channel contribution to inelastic neutron-proton collisions. A detailed investigation of the single pion production in ppcollisions provides the most accurate information on the isovector channel. When included in the analysis of the neutron-proton data, this information will allow us to correctly extract the contribution of the isoscalar channels.
Various theoretical models, more or less successful, wereput forward while the data on the pion production in NN collisions were accumulated. For the energy range of up to about 1 GeV,there was proposed the one-pion exchange (OPE) model [1–3] assuming the dominance of the one-pion exchange contribution to the inelastic amplitude. In this approach, pole diagram matrix elements are calculated using a certain form factor obtained by fitting to the experimental data. Thus, the OPE model is a semi-phenomenological approach. In addition, only theP33 partial wave is taken into account in the intermediate πN-channel [2]. Nevertheless, this model predicts with reasonable accuracy (up to normalization factors) the differential spectra of the pp → pnπ+ and pp → ppπ0reactions in the 600–1300 MeV energy range [4,5]. At the same time,the discrepancies between the measured total cross sections for these reactions and the model predictions are rather sizeable (see Ref. [5]).
It should be noted that the experimental data on the differential spectra of pion production reactions around 1 GeV are rather scarce. In this contribution, we present results of our investigation of the pp → pnπ+ and pp → ppπ0reactions at 1628 MeV/c and determine the contributions of various partial waves to the process of single pion production.
2. Experiment
The experiment was performed at the PNPI 1 GeV synchrocyclotron. The events were registered by a35 cm hydrogen bubble chamber placed in the 1.48 T magnetic field. A proton beam (after the corresponding degrader for the momentum of 1628 MeV/c) was formed by three bending magnets and by eight quadrupole lenses. The value of the mean incident proton momentum was checked by the kinematics of elastic scattering events. The accuracy of this value and the momentum spread were about 0.5 MeV/c and 20 MeV/c (FWHM), respectively. The admixture of other particles in the proton beam was measured by the time-of-flight technique and proved to be negligible (less than 0.1%). A total of 4.6×105 stereoframes wasobtained. The frames were scanned twice to search for the events due to interaction of the incident beam. The efficiency of double scanning was determined to be 99.95%. Approximately 8×103 two-prong events were used inthe subsequent analysis.
The 2-prong events selected in the fiducial volume of the chamber were measured and geometrically reconstructed. The reconstructed events were kinematically fitted to the following reaction hypotheses:
p + pp + p, (1)
p + pp + n + π+, (2)
p + pp + p + π0, (3)
p + pd + π+, (4)
p + pp + π+ + π0. (5)
The identification of the events was performed using the χ2 criteria with the confidence level of more than 1%. If an event had a good χ2 for the elastic kinematics (4C-fit), it was considered aselastic one. The pull, or stretch, functions for the three kinematic variables of a track (the inverse of the momentum, the azimuthal and dip angles) were examined for elastic scattering events to make sure that their errors, as well as the error of the bubble chamber magnetic field, were properly given. If there was only one acceptable fit for an event, it was identified as belonging to this hypothesis(with a check on the tracks ofthe stopping π+-mesonsin the presence of the π+→ μ+ → e+decay). If several inelastic versions revealed a good χ2 value, we used the visual estimation of the bubble density of the track to distinguish between a proton (deuteron) and a pion. For few events, even after repeated measurements, the fit revealed only one acceptable hypothesis, but with large χ2. If such an event had the χ2 less than 50 for the 4-constraint fit or less than 20 for the 1-constraint fit, it was taken into account in the calculation of the cross section value of the process corresponding to this hypothesis but was not included in the differential spectra.
There were events that failed to fit any hypothesis. These no-fit events were investigated on the scanning table and most of them appeared to be the events with a secondary track undergoing one more scattering near the primary vertex.
There were also events unfit for the measurements, e.g.,the events with a bad vertex or superimposed tracks. The number of such events was counted to be approximately 7%. The total number of 2-prong events that had not passed the measurement and fitting procedures was counted to be less than 10%. These unidentified events were apportioned to the fraction of the fitted hypotheses for the total cross section measurements.
The standard bubble chamber procedure was used to obtain absolute cross sections [5] for the elastic and single pion production reactions. These values – together with the statistics of the events selected by the fit – are listed in Table 1.
Table 1
Number of events and the values of the corresponding cross sections at the beam momentum of 1628 MeV/c
pp → / number of events / σ, mbelastic / 3442 / 21.2 ± 0.7
pnπ+ / 3014 / 17.6 ± 0.6
ppπ0 / 696 / 4.48 ± 0.20
dπ+ / 80 / 0.47 ± 0.05
dπ+π0 / 5 / 0.029 ± 0.013
3. Elastic scattering: experimental results and discussion
The measured differential cross section for elastic ppscattering in the centre-of-mass system (c.m.s.) of the reaction is shown in Fig.1.The value of the total elastic cross section given in Table 1 was calculated as4πA0, where A0 is the coefficient in front of the P0(z)Legendre polynomial in our fit of the angular distribution using the Legendre expansion over the −0.95≤ cosθ ≤0.95angular range. This range was chosen in order to take into account a loss of the events due to forward elastic scattering, when a slow proton had a short recoil path and could not be seen in the bubble chamber (at the momentum less than 80 MeV/c) or was missed during the scanning. This interval was determined by examining the stabilityof the Legendre coefficients withdecreasing
thefitted range of the angular distribution. Our value of the total elastic cross section obtained this way is smaller than that given in Ref. [6] by approximately 13%. This discrepancy could arise due to i) insufficient control of the loss of the events with small scattering angles or ii) incorrect calculation of the millibarn-equivalent in the chamber. To control the loss of events in the forward angle scattering, we use a reduced range (−0.95≤ cosθ ≤0.95) for fitting the differential cross section by the Legendre expansion, which could repair this shortcoming. We found that already a sum of first three even polynomials provides an excellent fit to the data. Of course, it is possible that we miss a contribution of higherorder polynomials which cannot be determined from the fitted angular region. In this case, our value of the elastic cross section given in Table 1 should be taken as a simple estimate of this magnitude.
In order to take into account the contributions of higher order polynomials, we included in our fit the data of Ref. [7] taken at small forward angles at the energy of 942 MeV. As a result, we obtain 22.7 ± 0.7 mb for the elastic cross section, whichis still lower than the value given in Ref. [6] by approximately 7%.
In Fig. 1, we compare our elastic differential cross section with the data of the EDDA experiment[8] taken at the incident momentum of 1639 MeV/c. One can see that there is excellent agreement between our points and the EDDA data, which supports the correctness of our definition of the millibarn-equivalent.
4.Single pion production reactions: comparison with the OPE model
The OPE model [1–4] describes the single pion production reactions by four pole diagrams with π0or π+exchanges. The main evidence for the pole diagram contributions would be the observation of a peak in the distribution over the momentum transfer from the target particle to the secondary proton in the pp → ppπ0process or, for example, to the secondary neutron in the pp → pnπ+ process.
Figures 2 and 3 show the distributions over the momentum transfer squared, Δ2 = −(pt−pf)2, where pt is the four-momentum of the target proton andpf is the four-momentum of the final neutron inpp → pnπ+ or of the final protons (two entries) inpp → ppπ0, correspondingly. The OPE model calculations normalized to the total number of the experimental events are shown by the dashed lines; the phase space distribution – by the dotted lines. One can see in the figures that the OPE model describes qualitatively well the Δ2distribution for both the studied reactions. This is remarkable because only the P33 wave is taken into account in the intermediate πN scattering. It could be that the distributions in Figs. 2 and 3 are mainly sensitive to the pole diagram propagator and that a more complicated structure of the amplitude manifests itself in other distributions.
Figures 4 and 5 present the angular distributions of the final particles in the c.m.s. of the reactions and the particle momentum distributions in the laboratory frame for the pp → ppπ0 and np → ppπ+ reactions, respectively. One can see in the figures again that the agreement of the OPE calculations with the experimental data is fairly good. The qualitative agreement is also observed for other spectra, except for the angular distributions of the final particles in the helicity system.
As noted earlier in Ref. [5], although the OPE model provides a qualitative description of most differential spectra, it disagrees with the value of the total cross section for the pp → pnπ0and pp → ppπ+ reactions. This means that taking into account only the P33 (Δ(1232) isobar) intermediate state is not enough for an adequate description of these reactions and, thus, a comprehensive partial-wave analysis is needed.
5. Partial-wave analysis, results and discussion
To extract the contributions of different partial waves, we apply [10] an event-by-event partial-wave analysis based on the maximum likelihood method. In the spin-orbital momentum decomposition, we follow the formalism given in Refs. [11–13]. The exact form of the operators for the initial and final states can be found in Ref. [12]. Following this decomposition, we use the spectroscopic notation 2S+1LJ for the description of the initial state, the system of two final particles and the system "spectator and two-particle final state". For the initial pp system, the states with the total angular momenta J ≤ 2 and the angular momenta L= 0–3 between the two protons are taken into account. In the fitting procedure, for the final three-particle system we restrict ourselves to the angular momenta L= 0–2 in thesystem consisting of a two-body subsystem plus a spectator and to the angular momenta L'= 0–2 in the two-particles systems.
For the πN system in the intermediate state, we introduce two resonances, Δ(1232)P33 and the Roper resonance N(1440)P11. For Δ(1232), we use the relativistic Breit-Wigner formula with the mass and width taken from the Particle Data Group. The Roper state was parameterized in agreement with the Breit–Wigner couplings found in the analysis of Ref. [14]. We would like to note that the present analysis is not sensitive to the exact parameterization of the Roper resonance: only the low-energy tail of this state can influence the data. For the description of the final pp interaction, we use themodified scattering-length approximation formula.
We minimized the log-likelihood value by a simultaneous fit of the present data on the pp → pnπ+ and pp → ppπ0reactions taken at the proton momentum of 1628 MeV/c and the data on pp → ppπ0obtained earlier[5,9,15]. The data [5,9] were taken at PNPI and measured at nine energies covering the energy interval from 600 to 1000 MeV. The high statistics data at the momentum of 950 MeV/c taken by the Tuebingen group [15] was included to fix the low energy region.
The experimental data (the points with error bars) and the results of our partial-wave analysis (histograms) for the momentum of 1628 MeV/c are shown in Figs. 6 and 7. The first row shows the angular distributions of the final particles in the rest frame of the reaction; the second row shows the effective two-body mass spectra. One can see in the figures that our partial-wave analysis describes these distributions fairly well.
The quality of the partial-wave analysis is also demonstrated by the angular distributions in the helicity (third row) and the Godfrey-Jackson (fourth row) frames. The reduced χ2for the shown distributions is 1.70 for the pp →ppπ0 data and 2.37 for the pp → pnπ+ data, respectively. At this point, we would like to remind that we use the event-by-event maximum likelihood analysis and do not fit directly these distributions.
The initial1S0 partial wave provides only a small contribution to the pp → ppπ0 reaction at the incident proton momentum of 1628 MeV/c. The largest contributions come from two P-wave initial states: 3P2 and 3P1 (see Table 2). The 3P0 initial state contributes about 10% to the total cross section. We also see noticeable contributions from the 1D2 and 3F2 partial waves. The 3F2 partial wave interferes rather strongly with the strongest 3P2 wave and, thus, its contribution is determined with a sizeable error.
Table 2
Contributions of main partial waves to the single pion production reactions at 1628 MeV/c
pp → ppπ0 / pp → pnπ+1S0 1.0 ± 0.5%
3P0 10.0 ± 0.9%
3P1 26.0 ± 7.7%
3P2 44.0 ± 1.6%
1D2 8.0 ± 1.1%
3F2 11.4 ± 7.7% / 1S0 5.8 ± 2.8%
3P0 11.5 ± 0.9%
3P1 32.7 ± 0.8%
3P2 34.0 ± 1.6%
1D2 8.5 ± 1.0%
3F2 6.3 ± 1.5%
All initial partial waves decay predominantly into the Δ(1232)p intermediate state. For different partial waves,the contribution of channels with Δ(1232) production varies from 65to 90%. The strongest non-resonant contribution is observed from the 3P2 initial state, where the3P2 → (3P2)ppπtransitioncontributesfrom 20 to 35% (of the contribution of the 3P2 partial wave), depending on the fit. This instability appears due to noticeable destructive interference between the 3P2 → (3P2)ppπand Δ(1232)p intermediate statesin this channel. The contributions of the amplitudes with the Δ(1232) resonance in the intermediate state are shown in Figs. 6 and7 by the dotted histograms.
In the pp → pnπ+ reaction at 1628 MeV/c, the non-resonant pn partial waves with the isospin 1 contribute much less to the total cross section than the corresponding waves in the pp → ppπ0 reaction. For example, the 3P2 → (3P2)pnπ transition is found to be less than 10% of the contribution of the 3P2 initial state leading to a much smaller interference effect for the amplitudes with Δ(1232) in the intermediate state. We also observe a sizeable 1S0 → (3S1)pnπtransition,which is dominant for the 1S0 initial state, and an appreciable contribution of the 3P1 → (1P1)pnπ transition.
Our partial-wave analysis determinesthe relative contributions of the isovector waves to the total cross section for the single pion production processes in the energy interval of 400–1000 MeV. Figure 8 shows the experimental behaviour of the pp → ppπ0cross section together with theresult of our partial-wave analysis and contributions of the dominant partial waves.
It should be noted that although we use the data on the pp → pnπ+ reaction at 1628 MeV/c only, our partial waves predict the total cross section at lower energies in good agreement with the values given in Ref. [6].
6. Conclusion
A detailed study of the differential cross section for the pp → ppπ0 and pp → pnπ+ reactions has been performed [10] at the incident proton momentum of 1628 MeV/c. While the shape of most distributions is described qualitatively well by the OPE model, it fails to simultaneously describe the total crosssections for the pp → ppπ0 and pp → pnπ+ reactions.
The partial-wave analysis of the single pion production reactions reveals that the dominant contribution comes from the Δ(1232)p intermediate state, which explains the success of the OPE model. In addition, our analysisalso allows us to achieve the combined description of all the analysed reactions and to extract the contributions of various transition amplitudes.
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