Comparison of the pp ->pi+ pn and pp ->pi+ d production rates

Fully constrained bubble chamber data on the pp ->pi+ pn and pp ->pi+ d reactions are used to investigate the ratio of the counting rates for the two processes at low pn excitation energies. Whereas the ratio is in tolerable agreement with that found in a high resolution spectrometer experiment, the angular distribution in the final pn rest frame shows that the deviation from the predictions of final state interaction theory must originate primarily from higher partial waves in the pn system. These considerations might also be significant for the determination of the S-wave Lambda:p scattering length from data on the pp ->K+ Lambda p reaction.

The COSY-GEM collaboration measured the differential cross section for the production of positive pions in proton-proton collisions at a beam kinetic energy of T p = 951 MeV, detecting the π + at θ π = 0 • in the spectrograph Big Karl [1]. The high missing-mass resolution achieved here (≈ 100 keV/c 2 ) allowed a very clean separation of the pp → π + d and pp → π + pn channels and also showed that the production of singlet pn pairs was negligible at this energy.
The final state interaction theorem relates the normalisations of the wave functions for S-wave bound and scattering states [2]. This has been exploited to predict the double-differential centre-of-mass (cm) cross section for the S-wave spin-triplet component in pp → π + pn in terms of the cross section for pp → π + d [3]: Here x denotes the excitation energy Q in the np system in units of the deuteron binding energy B t , x = Q/B t , and p(x) and p(−1) are the pion cm momenta for the pn continuum or deuteron respectively. At the deuteron pole the normalisation F = 1 but it was argued [2] that deviations from this should be small at low x if the pion production operator is of short range and the tensor force linking the S and D states in the deuteron could be neglected. However, although the shape of the COSY-GEM data [1] was reasonably well described by Eq. (1) up to an excitation energy of Q ≈ 20 MeV, reproducing the absolute magnitude required F = 2.2 ± 0.1. In view of the large discrepancy with the prediction of the final state interaction theorem (F = 1), the COSY-GEM experiment was repeated at 400 and 600 MeV [4] which, combined with the results of earlier work carried out at TRIUMF [5], presented a consistent picture. The normalisation factor F was found to increase steadily from below one at 400 MeV to well above unity at 951 MeV. It was suggested that the deviation at the highest energy could arise from the long-range part of the pion production operator associated with the on-shell intermediate pions [4]. Such contributions could change the pn S-wave cross section for the pp → π + pn reaction or excite higher partial waves in the final pn system. In a missing-mass experiment, as carried out by the GEM collaboration [1,4], it is not possible to investigate these suggestions any further.
Measurements of various channels arising from protonproton collisions at three energies in the 900 to 1000 MeV range were undertaken using the 35 cm hydrogen bubble chamber of the Petersburg Nuclear Physics Institute (PNPI) [6][7][8]. Although the statistics in the low Q region are much poorer than those of the COSY-GEM experi-ment [1,4], and the resolution is far inferior, the acceptance approaches 100% and so the predictions of Eq. (1) can be integrated over the full solid angle. Under the conditions of the PNPI data, the deviation of p(x)/p(−1) from unity is negligible at low x so that the numbers N of bubble chamber events should be linked by In order to compare directly the PNPI data with the COSY-GEM result, the bubble chamber pp → π + pn events were selected as having a maximum pn excitation energy of 20 MeV (x 0 ≈ 9). The numbers of events fulfilling this criterion, as well as the total number of pp → π + d events, are given in Table 1. The values of F deduced at the three energies are also shown, as is their average. Table 1 The numbers of events measured in the pp → π + d and pp → π + pn (with pn excitation energy below 20 MeV) reactions in the PNPI bubble chamber experiment at 900.2 MeV [6], 940.7 MeV [7], and 988.6 MeV [8]. The values derived for the normalisation factor F and its average over the three energies are also listed. Only statistical errors are quoted. The results shown take into account the small numbers of ambiguous events. The weighted average of the PNPI data, F = 2.6 ± 0.2, looks higher than the COSY-GEM value of 2.2 ± 0.1 but the errors quoted are only statistical uncertainties. Furthermore, it is possible that the fraction of higher pn waves could vary with pion angle. However, there might also be an effect due to the poorer resolution in the pn excitation energy in the bubble chamber data, which allows some higher x data to distort a little the result.
The big advantage of the fully constrained PNPI pp → π + pn data is that they allow one to investigate the angular distributions in the recoiling pn system 1 . Figure 1 shows the distribution in the angle of the final proton with respect to the original beam direction in the pn rest frame. The clear deviation from isotropy is unambiguous evidence for the production of higher partial waves in the final pn system.
The data in Fig. 1 are well described by the form N (pp → π + pn) = a 0 + a 1 cos θ p + a 2 cos 2 θ p , where a 0 = 27.3 ± 2.7, a 1 = −1.5 ± 3.7, and a 2 = 33.8 ± 7.1. The corresponding value of χ 2 /NDF = 0.27 is fortuitously  low. It is important to note that, as expected, the odd term a 1 is consistent with zero but fixing it to vanish does not change significantly the values of a 0 and a 2 . However, it is not possible to identify whether the large quadratic term arises from the square of a pn P -wave or from an S − D interference, which could be influenced by the pn tensor force. Departures from isotropy in Fig. 1 are clear evidence for the excitation of higher partial waves but the converse is not true because it is possible to generate a mixture of higher partial waves that leads to an isotropic distribution. Nevertheless, it is likely that the deviations from the final state interaction theorem of Refs. [2,3], shown by the dashed line in Fig. 1, probably come from higher partial waves rather than modifications of the S-wave intensity. Apart from complications arising from the pn tensor force, it is to be expected that Eq. (1) should be a good representation of the pp → π + pn data at very small values of x. Although this is a valid approximation at low incident beam energies, the COSY-GEM experiment shows that there are significant deviations at 951 MeV [1]. By using the fully reconstructed bubble chamber events [6][7][8], we have confirmed the magnitude of the deviation. However, we have also shown from the angular distribution of Fig. 1 that, with a cut-off at Q = 20 MeV, there are significant contributions from higher partial waves in the pn system that are not apparent in a missing-mass experiment. This may be due to the anomalously long range of the pion production operator at high energies which was not considered in the application of the final state interaction theorem [2].
The arguments presented here may have wider significance than the specific reaction being studied. By detecting just the K + meson in the Big Karl spectrograph, the COSY-HIRES group measured the inclusive cross section for the pp → K + X reaction. Below the threshold for Σ production X = Λp and the hope was that an analysis of the data would allow a determination of the spin-average Λp S-wave scattering length [9]. However, in such a singlearm experiment, there can be no confirmation that the Λp system remains in the S-wave at finite values of Q.
Conditions are much more favourable in the COSY-TOF experiment [10,11] where, apart from some loss of acceptance near the beam direction, the final particles in the pp → K + Λp reaction can be detected. The global Λp angular distributions constructed for beam momenta of 2.7 GeV/c [11] and 2.95 GeV/c [12] both show strong signals arising from higher partial waves in the Λp system but it is not clear from these plots if there are also effects for Λp excitation energies below 40 MeV. This clearly has to be checked when attempting to extract the S-wave Λp scattering length from the analysis of such experiments [9,10]. The determination of the position of the Λp virtual state pole [13] is far less affected by these considerations because this is sensitive to the behaviour at very small values of Q, where the S-wave assumption is on much firmer grounds.