Differential cross section and analysing power of the reaction at 353 MeV
In order to establish links between -wave pion production in nucleon-nucleon collisions and low energy three-nucleon scattering, an extensive programme of experiments on pion production is currently underway at COSY-ANKE. The final proton pair is detected at very low excitation energy, leading to an -wave diproton, denoted here as . We now report on measurements of the differential cross section and analysing power of the reaction at 353 MeV. Both observables can be described in terms of - and -wave pion production and, by using the phase information from elastic scattering, unique solutions can be obtained for the corresponding amplitudes. This information is vital for the partial wave decomposition of the corresponding reaction and hence for the extraction of the -wave terms.
keywords:Neutral pion production; Proton proton collisions; Amplitude analysis
Pacs:13.75.-n, 14.40.Be, 25.40.Qa
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Within the context of chiral perturbation theory, a significant step forward in our understanding of pion physics at low energies would be to establish that the same short-ranged vertex contributes to -wave pion production, to low energy three-nucleon scattering [1, 2, 3], [4, 5] and , as well as in weak reactions like tritium beta decay [7, 8, 9, 10]. The relevant transition amplitude, which connects -waves in the initial and final state with a -wave pion, contributes to both and . However, the extensive data for production is of limited use in this context, because the -wave amplitudes are completely dominated by the initial state, which hinders a reliable extraction of the initial state .
There is a programme at the COSY-ANKE facility of the Forschungszentrum Jülich to perform a complete set of measurements on at low energy . Here the denotes a proton-proton system with very low excitation energy, . At ANKE we select events with MeV and, under these conditions, the diproton is overwhelmingly in the state with antiparallel proton spins. This simplifies enormously the spin structure: a partial wave analysis for without the cut would require twelve additional -wave final spin-triplet states [12, 13, 14]. The cut also allows one to extract the full information on the production amplitudes without having to make measurements of the final proton polarisations.
Whereas the final state is isotriplet, the isosinglet initial state also contributes to . In order to isolate this, which contains the amplitudes of interest, the isotriplet channel needs to be well understood. As the first part of the outlined larger programme, we therefore report here on measurements of the cross section and proton analysing power in the reaction at MeV.
For a spin-singlet diproton, the spin structure of the or reaction is that of . Parity and angular momentum conservation require that the initial nucleon-nucleon pair to have spin . The pion orbital angular momentum and the initial nucleon-nucleon isospin are then linked by odd so that, for the reaction, only even pion partial waves are allowed. As a consequence, the unpolarised cross section for production, and this times the proton analysing power , must be of the form
where is the pion c.m. production angle with respect to the direction of the polarised proton beam. Here is the incident c.m. momentum and that of the produced pion which, at 353 MeV, have values MeV/ and MeV/, where the latter represents an average over the 3 MeV range.
The only detailed measurements of the differential cross section over the whole angular range were carried out with the PROMICE-WASA apparatus at CELSIUS at a series of energies from 310 to 450 MeV, using the same standard 3 MeV cut on . Throughout this energy range, significant anisotropies were found in the angular distributions which were attributed to interferences between pion and waves. On the other hand, there were no corresponding measurements of the proton analysing power, which might also be driven by a strong - interference.
We have previously reported measurements of the differential cross section at several energies and small angles [16, 17]. Since these were carried out using the ANKE spectrometer  under conditions that were similar to the current ones, the description here can be quite brief. ANKE is placed at an internal beam station of the COSY cooler synchrotron. Fast charged particles, resulting from the interaction of the stored transversally polarised proton beam with the hydrogen cluster-jet target  and passing through the analysing magnetic field, were recorded in the forward detector (FD) system. The FD, which was the only detector used in this experiment, includes multiwire proportional chambers for tracking and a scintillation counter hodoscope for energy loss and timing measurements.
To start the identification of the reaction, proton pairs were first selected from all the registered two-track events using the measured momenta of the both particles and the difference in their time-of-flight . The resolution in the diproton excitation energy was better than 0.6 MeV, which allowed the MeV cut to be applied reliably.
After selecting the final state, the kinematics of the process could be reconstructed on an event-by-event basis to obtain a missing-mass spectrum. A two-dimensional distribution of versus the c.m. polar angle of the diproton is presented in Fig. 1. This demonstrates the large angular acceptance of the apparatus for the reaction at 353 MeV and shows a clean signal with an almost negligible background. Simulations indicate that the c.m. angular resolution is better than .
The polarization asymmetry is defined by
where and are the numbers of events with beam proton spin up and down, corrected for dead time, and and are the corresponding luminosities. The relative luminosity was estimated using events at very small polar angles, where the polarization asymmetry should be negligible. This procedure adds about a 3% systematic error to the values of .
The analysing power is connected to the asymmetry through:
where is the transverse polarization of the beam and the average over the diproton azimuthal angular distribution. Since the acceptance is concentrated near 1, all the events in the regions analysed contribute usefully to the measurement.
The polarization of the proton beam was flipped between “spin-up” to “spin-down” (perpendicular to the plane of the accelerator) every six minutes and no measurements were made with an unpolarized beam. The value of was estimated from proton-proton elastic scattering and the reaction that were measured in parallel. The analysing powers for these reactions were taken from the SAID analysis program, solutions SP07 for and SP96 for . The results of the two methods shown in Fig. 2 agreed within measurement errors and gave an average polarization of , where the error includes the uncertainties arising from the calibration reactions.
A simulation was undertaken of the two-dimensional acceptance in terms of the excitation energy and its c.m. polar angle . This took into account the geometry of the setup and the sensitive areas of the detectors, the efficiency of the multiwire proportional chambers and the track reconstruction algorithm. In order to avoid potential problems arising near the limits of the acceptance, cuts were made around the edges of the exit window of the spectrometer magnet in both the experimental data and simulation. This is only a challenge at the larger angles, , where a compromise had to be made regarding the acceptance ambiguities and this introduces an extra 4% systematic uncertainty in this angular region.
The numbers of detected events were then corrected on an event-by-event basis for acceptance, dead time and relative luminosity . The latter were important because, in the absence of data with an unpolarised beam, an average has to be evaluated.
The luminosity in the experiment was estimated from measurements of elastic scattering carried out in parallel. The numbers of detected events, corrected for the dead time, were compared with a simulation that used a generator which included the differential cross section obtained from the SAID analysis program . Although this program does not furnish error bars, experimental data at nearby energies suggests that the associated uncertainty is about 2%, to which must be added 3% arising from acceptance and similar systematic effects. At this level the statistical error is negligible and the resulting total luminosity was estimated to be nb. At this energy the cross section data are less precise than those of elastic scattering but, on the basis of the SAID predictions, one obtains the completely consistent luminosity estimate of 547 nb.
The differential cross section results are presented in Fig. 3, where they are compared to those obtained at 360 MeV at CELSIUS . Within the 10% luminosity uncertainty in these data, the overall agreement is very good. However, the CELSIUS data at this energy level off a little around . This seems to be a feature only of the 360 MeV results since, at the other energies, linear fits in all have good values of /NDF .
Fitting our data with a polynomial in , as in Eq. (1), gives parameters
Apart from the acceptance uncertainties at the larger angles, the error bars quoted here are purely statistical; the systematic uncertainty from the luminosity and acceptance largely cancels in the ratio . Since , there is clearly no compelling evidence for any dependence, i.e., a non-zero coefficient, and this possibility has been omitted from the curve in Fig. 3.
The results for the analysing power of the reaction are displayed in Fig. 4, with being shown in panel a and in panel b. These observables must be antisymmetric about and the crossing of the data through zero around this angle is some confirmation of our estimation of . These data are subject to the overall uncertainties associated with the luminosity and acceptance evaluation, though these are not relevant for the in Fig. 4b. There remains, however, the arising from the uncertainty in the value of .
The data are consistent with a behaviour and a fit using the general form of Eq. (2) yields
with . There is therefore no evidence for any dependence, i.e., a non-zero coefficient. The resulting fits are shown in Fig. 4.
In order to understand the significance of the results reported here, we must attempt a partial wave description of the data. The most general form of the reaction amplitude is
where is the polarisation vector of the initial spin-triplet state. and are unit vectors in the c.m. frame along the directions of the incident proton and final pion, respectively.
The observables studied here are expressed in terms of the two scalar amplitudes and through 
The experimental data show no evidence for high partial waves at 353 MeV and so we model these results with only and contributions. The latter can arise from initial or waves so that, in total, there are three possible transitions, , , and , see e.g. Ref.  for the explicit form of the spin-angular structures. We denote the corresponding amplitudes by , , and , respectively.
Expanding the scalar amplitudes in terms of these partial waves gives
Equations (8) and (9) then allow one to relate the measured observables of Eqs. (1) and (2) to the partial wave amplitudes. For consistency, since we have neglected any possible effects arising from - interference, we shall also drop terms that are bilinear in -wave production amplitudes. In this approximation
and so the data only provide three relations between the three complex amplitudes. The transverse spin correlation parameters contain no extra information since and this is also true for up to - interference terms. If the longitudinal-transverse spin correlation parameter were measured, this would provide one further relation but this would still not be sufficient for an unambiguous partial wave decomposition. For this we need information about the phases of the production amplitudes.
The partial wave is uncoupled and, at the energy where the experiment was performed, its inelasticity is very small. Under these conditions the Watson theorem, which fixes the phase induced by the initial state interaction to that of the elastic proton-proton scattering, applies . Thus we take , with . Note that we do not include any phase associated with the final state because it is common for all partial waves and therefore does not affect the observables.
For coupled channels, such as , the strict conditions of the Watson theorem do not apply. However, at our energy the mixing parameter, as well as the inelasticities, are still negligibly small and thus to a good approximation we may also use the Watson theorem here. Further evidence for the smallness of the channel coupling is to be found in two potential models [24, 25]. In both models the -matrix for the transition from the to the wave is almost real; the phase of is driven by , whereas the phase of can be neglected. The quality of this approximation was also checked by explicit calculations of the -wave production amplitudes within chiral effective field theory. These showed that up to order the above phase assumptions should be valid to within .
Using the phase information in this way, we find that
The values quoted here were obtained by considering also our data though the numbers would change but marginally if one included only the results in the fit. The error bars quoted here are statistical and do not include the overall systematic uncertainties. However, changing the normalisations of the differential cross section and analysing powers by 3% and 4%, respectively, leads to changes that are comparable to the quoted errors. On the other hand, we could not investigate the less tangible ones associated with the neglect of the channel coupling and the truncation in the partial wave expansion. The weakness of pion production from the initial waves at 353 MeV, in addition to being in agreement with theoretical prejudices, is also consistent with the low inelasticity found for this wave .
In summary, we have measured the differential cross section and analysing power of the reaction at 353 MeV. The angular distributions of and are both well represented by retaining only pion and waves in a phenomenological description. The values of agree well with the results obtained at CELSIUS  over most of the angular range. However, at this energy these data flatten off around the middle of the angular distribution and, if this effect were correct, it would signal a large contribution from or even higher partial waves.
By making plausible assumptions on the coupling between the nucleon-nucleon channels and invoking the Watson theorem it was possible to estimate the partial wave amplitudes with their phases. These could be checked through a future measurement of the spin-correlation parameter , though this would require the installation of a Siberian snake in COSY.
In an associated letter , the isospin-0 amplitudes are investigated through the measurements of the differential cross section and analysing power of the quasi-free reaction in this energy domain. The extraction of the -wave amplitudes from the data of Ref.  required a knowledge of the - and -wave amplitudes of the type provided here. In addition, data have already been taken on the transverse spin correlation parameter for this reaction . The full collection of these results will lead to very useful constraints on the parameters of the chiral effective field theory that link pion production to the three-nucleon interaction [1, 2].
We are grateful to other members of the ANKE Collaboration for their help with this experiment and to the COSY crew for providing such good working conditions, especially of the polarised beam. This work has been partially supported by the BMBF (grant ANKE COSY-JINR), RFBR (09-02-91332), DFG (436 RUS 113/965/0-1), the JCHP FFE, the SRNSF (09-1024-4-200), the Helmholtz Association (VH-VI-231), STFC (ST/F012047/1 and ST/J000159/1), and the EU Hadron Physics 2 project “Study of strongly interacting matter”
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