Study of the proton-proton collisions at 1683 MeV/c
K.N. Ermakov, V.A. Nikonov, O.V. Rogachevsky, A.V. Sarantsev, V.V. Sarantsev, S.G. Sherman
aa r X i v : . [ nu c l - e x ] A p r EPJ manuscript No. (will be inserted by the editor)
Study of the proton-proton collisions at 1683 MeV/c
K. N. Ermakov, V. A. Nikonov, O. V. Rogachevsky, A. V. Sarantsev, V. V. Sarantsev , and S. G. Sherman
Petersburg Nuclear Physics Institute NRC KI, Gatchina 188300, RussiaReceived: date / Revised version: date
Abstract.
The new data on the elastic pp and single pion production reaction pp → pnπ + taken at theincident proton momentum 1683 MeV/c are presented. The data on the pp → pnπ + reaction are comparedwith predictions from the OPE model. To extract contributions of the leading partial waves the singlepion production data are analyzed in the framework of the event-by-event maximum likelihood methodtogether with the data measured earlier. PACS.
Understanding the proton-proton interaction at low andintermediate energies is the important task of the particlephysics. At large momentum transfers where the strongcoupling is small, the QCD calculations can be used effi-ciently for the description of such processes. A large prog-ress was made at low energies where the effective fieldapproach allowed us to describe processes below the reso-nance region. However the region of the intermediate en-ergies and especially the resonance region is much less ac-cessible for the theoretical calculations and phenomeno-logical dynamic models play the leading role here. Thedata from the
N N collision reactions forms the basis forthe construction of such models which, in turn, have thelarge range of applications in the nuclear and heavy ionphysics.In the region above the two pion production thresholdand up to 1 GeV the
N N → πN N reaction is dominatedby the production of the ∆ (1232) isobar in the intermedi-ate state. It was natural to suggest that such productionis based on the one pion exchange mechanism (OPE) anda set of the corresponding models was put forward [1,2,3] a rather long time ago. The pion exchange amplitudesare introduced there using certain form factors with pa-rameters defined from the fit of experimental data. Themodel of Suslenko et al. describes with a reasonable accu-racy (up to normalization factors) the differential spectraof the pp → pnπ + and pp → ppπ reactions in the energyrigion below 1 GeV [2,4], while the model of Dmitrievet al. was applied to the energies over 1 GeV [3]. In themore complicated model based on the one boson exchangemechanism [5] the dominant contribution for the ∆ (1232)production is also defined by the one pion exchange: it Correspondence to : [email protected] was found that other boson exchanges contribute around10% to the total cross section at the energies above 1 GeV.However it should be noted that there are discrepancies inthe simultaneous description of the measured total crosssections for the pp → pnπ + and pp → ppπ reactionsby the OPE model. For example, at the proton momen-tum 1683 MeV/c, the OPE model can reproduce well the pp → pnπ + measured total cross section with the corre-sponding choice of the form factor. However, in this casethe OPE prediction for the pp → ppπ total cross sectionwill be smaller by about 30% than the experimental one(see Ref. [4]).Moreover in the region above the incident proton mo-mentum 1.5 GeV/c other contributions start to play anotable role: for example the relatively broad Roper res-onance is traced in the spectrum. Therefore for a com-prehensive analysis of data it is necessary to apply anapproach beyond the OPE model.With this purpose we perform the partial wave analysisof the data on the single pion production in the frameworkof the approach based on the work [6]. The result of suchan analysis for the lower energy data measured earlier wasreported in [8],[9].In the analysis [9] the several solutions had been foundwhich almost equally described the data. These solutionsdiffer by contributions from the partial waves with highorbital momenta L > pp → pnπ + reactions measured at the protonmomentum 1683 MeV/c. We compare the data with theOPE model calculations and determine contributions of K. N. Ermakov et al. : Study of the proton-proton collisions at 1683 MeV/c the different partial waves from the combined partial waveanalysis of the present data and the data measured earlier.
The description of the experiment performed at the PNPI1 GeV synchrocyclotron was given in details in our pre-vious work [7]. The proton beam was formed by threebending magnets and by eight quadrupole lenses. Themean incident proton momentum value was inspected bythe kinematics of the elastic scattering events. The ac-curacy of the incident momentum value and momentumspread was about 0.5 MeV/c and 7 MeV/c (FWHM) cor-respondingly with a perfect Gaussian distribution. A totalof 8 × stereoframes were obtained. The frames weredouble scanned to search for events due to an interac-tion of the incident beam. The double scanning efficiencywas determined to be 99.95%. Approximately 7 × two-prong events were used for the subsequent analysis.The 2-prong events selected in the fiducial volume ofthe hydrogen bubble chamber were geometrically recon-structed and kinematically fitted to the following reactionhypotheses: p + p → p + p, (1) p + p → p + n + π + , (2) p + p → p + p + π , (3) p + p → d + π + , (4) p + p → d + π + + π . (5)The events identification procedure was also describedin details in [8]. Thus, we list only the most severe criteriahere:1. Events with the confidence level larger than 1% wereaccepted.2. Events with only one acceptable hypothesis were iden-tified as belonging to this hypothesis.3. If several versions revealed a good χ value, we usedthe visual estimation of the bubble density of the trackto distinguish between proton (deuteron) and pion.The total number of the 2-prong events which hadnot passed the reconstruction and fitting procedures wascounted to be less than 10%. These unidentified eventswere apportioned to the fraction of the fitted hypothesesof the accepted events and were used only for the totalcross section calculations.The standard bubble chamber procedure [4] was usedto obtain absolute cross sections for the elastic and pionproduction reactions. The precision in the determinationof the millibarn-equivalent was found to be 2%. The crosssection values for the inelastic processes together withstatistics are listed in Table 1. Let us remind that dataon the pp → ppπ reaction at the same energy were pub-lished earlier [7].The differential cross section for the elastic pp scatter-ing measured in the present experiment is shown in Fig. 1as open squares with statistical errors. The value of the Table 1.
Numbers of events and the total cross sections atthe beam momentum 1683 MeV/c. The total elastic cross sec-tion was obtained by the interpolation of the differential crosssection by the Legendre polynomials. The errors include thestatistical errors and millibarn-equivalent ones. pp → events σ mbelastic 2772 23.96 ± pnπ + ± dπ +
57 0.42 ± dπ + π ± differential cross section for the very forward angle bin isnot shown in Fig.1 due a notable loss of events with aslow final proton. If the proton momentum is less than 80MeV/c the recoil paths is too short to be seen in the bub-ble chamber. The events with the proton momentum lessthan 200 MeV/c also might be missing during scanning.Since we do not know the real amendment for these an-gles we excluded the last forward point and only show theangles where the proton momentum is above 200 MeV/c.In Fig. 1 we compare our elastic differential cross sectionwith the data from the EDDA experiment [10] taken atthe incident momentum 1687.5 MeV/c (open red circles).One can see that there is a good agreement between ourpoints and the EDDA data, that supports the correctnessof our definition of the millibarn-equivalent. cos q d s / d W ( m b / s r ) Fig. 1. (Color online) Elastic differential cross section. Thedotted curve is result of the Legendre polynomial fit of ourdata (open squares) restricted by the interval 0 ≤ cos θ ≤ To obtain the total elastic cross section we appliedthe following procedure. We fitted the differential cross . N. Ermakov et al. : Study of the proton-proton collisions at 1683 MeV/c 3 section with a sum of even order Legendre polynomials A n P n (cos θ ) n = 0 , , , . . . . By examining the flatness ofthe behavior of the fit with decrease of the fitted angularrange we determined the range 0 ≤ cos θ < θ = 0 .
95 the data from [11] at theincident momentum 1685.7 MeV/c which provide an im-portant constraint for high order polynomials. The resultof the fit is shown in Fig. 1 as the dotted curve. The totalelastic cross section calculated as 2 πA was found to be23 . ± .
57 mb which is close to the value given in [12]. pp → pnπ + reaction and a comparisonwith the OPE model The OPE model [2] describes the single pion productionreactions by the four pole diagrams with the π or π + ex-changes (we should like to express the deep appreciation tothe authors [2] for the accordance of their program code).In this model the intermediate state of the πN -scatteringamplitude confines itself to the P wave only, assumingthe leading role of the ∆ -resonance.Fig. 2 shows the distributions over the momentumtransfer squared, ∆ = − ( p t − p f ) , where p t is the four-momentum of the target proton and p f is the four-momen-tum of the final proton or neutron in the pp → pnπ + re-action correspondingly. The OPE model calculations nor-malized to the total number of the experimental eventsis shown by dashed lines and the shape of the phase vol-ume is shown by dotted lines. One can see that the OPEmodel describes qualitatively well the ∆ distributions forthis reaction.Fig. 3 presents c.m.s. angular distributions, effectivetwo-particle mass spectra of the final particles and angulardistributions in the helicity frame. We would like to pointout that the c.m.s. angular distributions are symmetricalones which is a critical test for the correctness of our eventselection.It is seen that the OPE model calculations normalizedto the total number of the experimental events reproducethe particle angular distributions in the c.m.s. of the reac-tion and the two body mass spectra fairly well. Howeverthe angular distributions in the helicity systems show no-table deviations from the experimental points. The partial wave analysis was performed in the frameworkof the event-by-event maximum likelihood method. Theformalism is given in details in [6,13] and based on thespin-orbital momentum decomposition of the initial andfinal partial wave amplitudes. Therefore it is natural to usethe spectroscopic notation S +1 L J for two particle partialwaves with the intrinsic spin S , the orbital momentum L and the total spin J . Here and below we use S, L, J forthe description of the initial
N N system, S , L , J forthe system of two final particles and S ′ , L ′ , J ′ = J for the D (from target to proton) (GeV ) eve n t s nu m b e r a) D (from target to neutron) (GeV ) eve n t s nu m b e r b) Fig. 2.
Four-momentum transfer ∆ distribution for the pp → pnπ + reaction: a) for the transfer to the final proton and b) toneutron. The dashed curves are the OPE calculations and thedotted curves show the shape of the phase volume. system formed by the two-final particle system and thespectator.The total amplitude can be written as a sum of partialwave amplitudes as follows [6,13]: A = X α A αtr ( s ) Q inµ ...µ J ( SLJ ) A S ,L ,J body ( s i ) × Q finµ ...µ J ( i, S L J S ′ L ′ J ) , (6)where Q ( S, L, J ) are operators which describe the systemof the initial nucleons, A αtr is the transition amplitude and A S ,L ,J body describes rescattering processes in the interme-diate two-particle channel. The multi-index α includes allquantum numbers for the description of the definite par-tial wave, s is the invariant energy of the initial N N sys-tem squared and s i is the invariant energy squared of thetwo-particle system.To suppress contributions of amplitudes at high rela-tive momenta we introduced the Blatt-Weisskopf form fac-tors. Thus the energy dependent part of the partial waveamplitudes with production of a resonance, for example,in the two-particle system 12 (e.g. πp ) and the spectatorparticle 3 ( n ) has the form: A = A αtr A S ,L ,J body ( s ) q L k L ′ p F ( q , L, R ) F ( k , L ′ , r ) , (7) K. N. Ermakov et al. : Study of the proton-proton collisions at 1683 MeV/c cos q p cms cos q p cms cos q n cms M p p (GeV/c ) eve n t s nu m b e r M p n (GeV/c ) M pn (GeV/c ) cos q p in (h) p p cos q p in (h) p n cos q n in (h)pn Fig. 3.
The pp → pnπ + data (the crosses with statistical er-rors): angular distributions of the final particles in the c.m.s. ofthe reaction (upper line), the effective two-particle mass spec-tra (middle line) and angular distributions in the helicity sys-tems. The dashed curves show the OPE calculations and dottedcurves show the shape of the phase volume. where q is the momentum of the incident proton and k is the momentum of the spectator particle, both calcu-lated in c.m.s. of the reaction. The explicit form of theBlatt-Weisskopf form factors F ( k , L, r ) can be found, forexample, in [14]. One should expect that the effective ra-dius of the initial proton-proton system R should vary be-tween 1 ÷ pp threshold it is hard to expect that this valuecan be determined with a good accuracy in the presentanalysis. Indeed we did not observe any sensitivity to thisparameter and fixed it at 1.2 fm. A very similar resultwas observed for r . So for our final fits we also fixed thisparameter at 1.2 fm.The combined analysis of the data sets at different en-ergies allows us to extract the energy dependence of thepartial waves which is assumed to be a smooth functionin this energy interval. This energy dependence was intro-duced in the following form: A αtr ( s ) = a α + a α √ ss − a α e ia α , (8)where a αi are real parameters. The a α parameters definepoles located in the region of left-hand side singularitiesof the partial wave amplitudes. Such poles are usually agood approximation of the left-hand side cuts defined bythe boson exchange diagrams. The phases a α are definedby contributions from logarithmic singularities connectedwith three body rescattering in the final state. For the description of the energy dependence in the πN system we introduce two resonances: ∆ (1232)
32 + andRoper N (1440)
12 + . The corresponding amplitudes are pa-rameterized as follows: A S ,L ,J body ( s ) = k L p F ( k , L , r ) 1 M R − s − M R Γ ,Γ = Γ R M R k L +112 F ( k R , L , r ) √ s k L +1 R F ( k , L , r ) . (9)Here s is the invariant energy squared in the channel 12, k is the relative momentum of the particles 1 and 2 intheir rest frame and r is the effective radius.For ∆ (1232), we use M R and Γ R taken from PDG [15]with r = 0 . πN , ∆π and N ( ππ ) S − wave channels were determined.For the description of the final N N interaction we usethe following parameterization: A S ,L ,J body ( s ) = √ s − r β k a β + ik a β k L F ( k ,r β ,L ) . (10)For the S -waves it coincides with the scattering-lengthapproximation formula suggested in [17,18]. Thus the pa-rameter a β can be considered as the N N -scattering lengthand r β is the effective range of the N N system. cos q n eve n t s nu m b e r Fig. 4. (Color online) The neutron angular distribution calcu-lated in the c.m.s. of the pp → pnπ + reaction at 1683 MeV/c.The data are shown by black circles with the statistical errors.The solid (black) histogram shows the prediction from the so-lution [9] with including partial waves up to L = 5 and thedotted (red) histogram shows the prediction from the solutionwith L up to 3.. N. Ermakov et al. : Study of the proton-proton collisions at 1683 MeV/c 5 cos q p eve n t s nu m b e r cos q p eve n t s nu m b e r cos q p cos q p cos q n cos q n M p p (GeV/c ) eve n t s nu m b e r M p p (GeV/c ) eve n t s nu m b e r M p n (GeV/c )M p n (GeV/c ) M pn (GeV/c )M pn (GeV/c ) cos q p in (h) p p eve n t s nu m b e r cos q p in (h) p p eve n t s nu m b e r cos q p in (h) p ncos q p in (h) p n cos q n in (h)pncos q n in (h)pn cos q p in (GJ) p p eve n t s nu m b e r cos q p in (GJ) p p eve n t s nu m b e r cos q p in (GJ) p ncos q p in (GJ) p n cos q n in (GJ)pncos q n in (GJ)pn Fig. 5. (Color online) The pp → pnπ + data taken at the proton momentum 1683 MeV/c with the statistical errors only. Firstline: the angular distributions of the final particles in the c.m.s. of the reaction. Second line: the effective two-particle massspectra. Third line: the angular distributions of the final particles in the helicity frame. Fourth line: the angular distributionsof the final particles in the Gottfried-Jackson frame. The solid (black) histograms show the result of our partial wave analysis;the dotted (red) and dot-dashed (blue) histograms show the contributions from the production of the ∆ (1232) and N (1440)intermediate states. The dashed (green) curves in the helicity frame show the normalized distributions from the OPE model. We have performed the analysis of the new data start-ing from our solution obtained in [9]. This solution wasrestricted by the partial waves with the total spin J up to2 and the orbital momentum L up to 3. This solution pro-duced an acceptable description of the lower energy databut has notable problems in the description of the newdata set. For example, the χ for the normalized angular distribution of the neutron in the c.m.s. of the reaction isequal to 4.49. The solution fails to describe the extremeangles which are mostly sensitive to partial waves withhigh orbital momentum. Indeed, the solution with L ≤ J ≤ χ to be 1.23. Thedescription of the data with these two solutions is shown K. N. Ermakov et al. : Study of the proton-proton collisions at 1683 MeV/c in Fig. 4. This provides a strong argument for the presenceof higher partial waves at studied energy.Although the solution with L ≤ pp → ppπ data measured earlier [4,7,19] and the pp → pnπ + data taken at 1628 and 1581MeV/c [8,9].On this way we were able to reproduce both the differ-ential and the total cross sections for all fitted data with agood accuracy. It’s worth to note that there is no problemto describe simultaneously the total cross sections for the pp → pnπ + and pp → ppπ reactions in this approach.The result of the partial wave analysis is shown inFig. 5: the histograms correspond to the Monte Carloevents weighted by the differential cross section calcu-lated from the fit parameters. The χ for the distributionsshown in this picture is varied from 0.65 (for the pion an-gular distribution in the c.m.s. of the reaction) to 2.6 (forthe πp invariant mass). We would like to remind that weuse the event-by-event maximum likelihood analysis anddo not fit directly these distributions.The partial wave analysis (PWA) reproduces ratherwell the angular distributions in the helicity system whichhave systematic deviations in the OPE model. The OPEpredictions normalized to the contribution from the ∆ (1232)production calculated from the PWA solution are shownin Fig. 5 with the dashed lines. It is seen that ∆ (1232)production from the partial wave analysis and from theOPE model corresponds well each to another. This con-firms that ∆ (1232) is produced by the one pion exchangemechanism and the deviation of the data from the OPEmodel is due to production of the Roper state.The present combined analysis found the contributionsfrom the leading initial partial waves to be in a qualitativeagreement with the prediction from the solution reportedin [9]. However we observe changes for the contributionsof the initial partial waves D and F which are notablyincreased after the fit of the new data. As concern thepartial waves with the total spin J = 4 we found a sizeablecontribution from F .For all initial partial waves the contribution of channelswith the ∆ (1232) production varies from 65 to 100% andonly for the P wave it was found to be rather smallone: 12%. The Roper resonance is produced mostly (inthe decreasing order of contributions) from the P , P , P states and by one order smaller from S . We found anotable contribution for the decay of the initial P stateinto the ( pn ) subsystem P with isospin I = 0.To study the stability of the solution we added to thefit partial waves with the total spin J up to 5 decayinginto ∆ (1232) N . The obtained solution demonstrated somereduction of the contribution from the P initial state andincreasing the contributions from the F state. Takinginto account these ambiguities we have performed an erroranalysis of the initial state contributions to the single pion production cross sections. For the pp → pnπ + reactionthese contributions are shown for three incident momentain Fig. 6. S P P P F D F P (MeV/c) c on t r i bu t i on s o f s o m e w aves ( % ) Fig. 6. (Color online) Contributions (the percentages) of mostimportant waves in the pp → pnπ + reaction. It is necessary to mention that the present combinedanalysis defines contributions of the partial waves withsmaller errors than it was found in [9]: the unstable con-tributions from the high spin amplitudes are fixed withthe present data.The obtained solution is well compatible with the dataof HADES collaboration on the single pion production atthe energy 1.25 GeV [20]: including these data in the com-bined fit does not change the main results of the analysisand contribution of the partial waves to the HADES datawas found to be in the errors given in [20]. The combinedanalysis of our and HADES data should be a subject ofthe future joint partial wave analysis.
The new data on the elastic and pp → pnπ + reactionstaken at the incident proton momentum 1683 MeV/c arereported. Including these inelastic data in the combinedpartial wave analysis of the single pion production reac-tions leads to a better error analysis and therefore to amore precise definition of the partial wave contributionsto the pp → pnπ + reaction. We observe some changes andin specific transition amplitudes compared to the predic-tions from the solution [9].As noted earlier in Ref. [4], although the OPE modelprovides a qualitative description of most differential dis-tributions, it fails to describe simultaneously the total . N. Ermakov et al. : Study of the proton-proton collisions at 1683 MeV/c 7 cross section of the pp → ppπ and pp → pnπ + reactions inthe investigated energy region. However our partial waveanalysis confirms the dominant role of the ∆ (1232) pro-duction defined by the OPE exchange mechanism. Themain source of the discrepancy between OPE and exper-imental data is due to contribution of other intermediatestates, in particular the Roper resonance.The all analyzed data sets can be downloaded from theBonn-Gatchina data base [21] as 4-vectors and directlyused in the partial wave analysis by other groups. Wewould like to remind that although we supply a MonteCarlo sample in our web page one can use a standardsample of 4 π generated events: the bubble chamber eventshave the efficiency which is close to 100%. We would like to express our deep gratitude to the bubblechamber staff as well as to laboratory assistants, which toiledat the film scanning and measuring. The work of V.A.Nikonovand A.V.Sarantsev is supported by the RNF grant 16-12-10267.
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