Determination of N* amplitudes from associated strangeness production in p+p collisions
R. Münzer, L. Fabbietti, E. Epple, P. Klose, F. Hauenstein, N. Herrmann, D. Grzonka, Y. Leifels, M. Maggiora, D. Pleiner, B. Ramstein, J. Ritman, E. Roderburg, P. Salabura, A.Sarantsev, Z. Basrak, P. Buehler, M. Cargnelli, R. Caplar, H.Clement, O. Czerwiakowa, I. Deppner, M. Dzelalija W. Eyrich, Z. Fodor, P. Gasik, I.Gasparic, A. Gillitzer, Y. Grishkin, O.N. Hartmann, K.D. Hildenbrand, B. Hong, T.I. Kang, J. Kecskemeti, Y.J. Kim, M. Kirejczyk, M. Kis, P. Koczon, R. Kotte, A. Lebedev, A. Le Fevre, J.L. Liu, V. Manko, J. Marton, T. Matulewicz, K. Piasecki, F. Rami, A. Reischl, M.S. Ryu, P. Schmidt, Z. Seres, B. Sikora, K.S. Sim, K. Siwek-Wilczynska, V. Smolyankin, K. Suzuki, Z. Tyminski, P. Wagner, I. Weber, E. Widmann, K. Wisniewski, Z.G. Xiao, T. Yamasaki, I. Yushmanov, P. Wintz, Y. Zhang, A. Zhilin, V. Zinyuk, J. Zmeskal
DDetermination of N* amplitudes from associated strangeness production in p+pcollisions
R. M¨unzer,
1, 2, ∗ L. Fabbietti,
1, 2, † E. Epple, S. Lu, P. Klose, F. Hauenstein, N. Herrmann, D. Grzonka,
4, 6, 7
Y.Leifels, M. Maggiora, D. Pleiner, B. Ramstein, J. Ritman,
4, 6, 7
E. Roderburg, P. Salabura, A.Sarantsev, Z. Basrak, P. Buehler, M. Cargnelli, R. ˇCaplar, H. Clement,
15, 16
O. Czerwiakowa, I. Deppner, M. Dˇzelalija, W. Eyrich, Z. Fodor, P. Gasik,
1, 2
I. Gaˇspari´c, A. Gillitzer,
4, 6, 7
Y. Grishkin, O.N. Hartmann, K.D. Hildenbrand, B. Hong, T.I. Kang,
8, 22
J. Kecskemeti, Y.J. Kim, M. Kirejczyk, M. Kiˇs, P. Koczon, R. Kotte, A. Lebedev, A. Le F`evre, J.L. Liu, V. Manko, J. Marton, T. Matulewicz, K. Piasecki, F. Rami, A. Reischl, M.S. Ryu, P. Schmidt, Z. Seres, B. Sikora, K.S. Sim, K. Siwek-Wilczy´nska, V. Smolyankin, K. Suzuki, Z. Tymi´nski, P. Wagner, I. Weber, E. Widmann, K. Wi´sniewski, Z.G. Xiao, T. Yamasaki,
28, 29
I. Yushmanov, P. Wintz, Y. Zhang, A. Zhilin, V. Zinyuk, and J. Zmeskal Excellence Cluster Universe, Technische Universit¨at M¨unchen,Boltzmannstr. 2, D-85748, Germany Physik Department E62, Technische Universit¨at M¨unchen, 85748 Garching, Germany Yale University, New Haven, Connecticut, United States Institut f¨ur Kernphysik, Forschungszentrum J¨ulich, 52428 J¨ulich, Germany Physikalisches Institut der Universit¨at Heidelberg, Heidelberg, Germany J¨ulich Aachen Research Alliance, Forces and Matter Experiments (JARA-FAME) Experimentalphysik I, Ruhr-Universit¨at Bochum, 44780 Bochum, Germany GSI Helmholtzzentrum f¨ur Schwerionenforschung GmbH, 64291 Darmstadt, Germany Istituto Nazionale di Fisica Nucleare (INFN) - Sezione di Torino, 10125 Torino, Italy Institut de Physique Nucleaire, CNRS/IN2P3 - Univ. Paris Sud, F-91406 Orsay Cedex, France Smoluchowski Institute of Physics, Jagiellonian University of Cracow, 30-059 Krak´ow, Poland Petersburg Nuclear Physics Institute, Gatchina, Russia Ruder Boˇskovi´c Institute, Zagreb, Croatia Stefan-Meyer-Institut f¨ur subatomare Physik, ¨Osterreichische Akademie der Wissenschaften, Wien, Austria Physikalisches Institut der Universit¨at T¨ubingen,Auf der Morgenstelle 14, 72076 T¨ubingen, Germany Kepler Center for Astro and Particle Physics, University of T¨ubingen,Auf der Morgenstelle 14, 72076 T¨ubingen, Germany Institute of Experimental Physics, Faculty of Physics, University of Warsaw, Warsaw, Poland Faculty of Science, University of Split, Split, Croatia Friedrich-Alexander-Universit¨at Erlangen-N¨urnberg, 91058 Erlangen, Germany Wigner RCP, RMKI, Budapest, Hungary Institute for Theoretical and Experimental Physics, Moscow, Russia Korea University, Seoul, Korea Institut f¨ur Strahlenphysik, Helmholtz-Zentrum Dresden-Rossendorf, Dresden, Germany Harbin Institute of Technology, Harbin, China National Research Centre ’Kurchatov Institute’, Moscow, Russia Institut Pluridisciplinaire Hubert Curien and Universit´e de Strasbourg, Strasbourg, France Department of Physics, Tsinghua University, Beijing, China Department of Physics, The University of Tokyo, Tokyo, 113-0033 RIKEN Nishina Center, RIKEN, Wako, 351-0198, Japan Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou, China
We present the first determination of the energy-dependent amplitudes of N ∗ resonances extractedfrom their decay in KΛ pairs in p+p → pK + Λ reactions. A combined Partial Wave Analysis of sevendata samples with exclusively reconstructed p+p → pK + Λ events measured by the COSY-TOF,DISTO, FOPI and HADES Collaborations in fixed target experiments at kinetic energies between2.14 to 3.5 GeV is used to determine the amplitude of the resonant and non-resonant contributionsinto the associated strangeness final state. The contribution of seven N ∗ resonances with massesbetween 1650 MeV/c and 1900 MeV/c for an excess energy between 0 and 600 MeV has beenconsidered. The Σ-p cusp and final state interactions for the p-Λ channel are also included ascoherent contributions in the PWA. The N ∗ contribution is found to be dominant with respect tothe phase space emission of the pK + Λ final state at all energies demonstrating the important roleplayed by both N ∗ and interference effects in hadron-hadron collisions. Keywords: partial wave analysis, resonance, hadrons, strangeness, scattering length, hyperon-nucleon inter-action
INTRODUCTION
The production of strange hadrons within nuclear mat-ter is a key ingredient in the understanding of the in- nermost structure of neutron stars (NS). Indeed, several a r X i v : . [ nu c l - e x ] S e p theoretical models predict that the production of strangehadrons is energetically favourable already at moderatedensities of neutron-rich matter [1, 2] and hence neu-tron stars with strange hadrons could appear. On theother hand, the appearance of strange hadrons softensthe equation of state of NS excluding the existence ofmassive NS unless a strong repulsive interaction is as-sumed for the ΛNN system [3]. Since NS with two solarmasses have already been measured with high precision[4, 5], this situation translates into a puzzle that can besolved only studying hyperons and kaons production inhadron-hadron collisions. The best environment to carryout this kind of studies is provided by hadron-hadroncollisions at few GeV kinetic energies because at theseenergies large baryonic densities, similar to those withinNS, can be created. On the other hand the reaction dy-namics at these energies is dominated by hadronic reso-nances, that need then to be quantitatively understood[6–13].For final states containing pions and nucleons producedin elementary reactions, partial wave analysis (PWA) wasalready employed to correctly take into account interfer-ences among resonances and determine the amplitude ofthe contributing waves [14–17]. For the contribution ofresonances to final states with open strangeness the reac-tion N ∗ → K+Λ was first studied by analyzing the Dalitzplot for the reaction p+p → p+K + +Λ up to kinetic ener-gies of T = 2 . − [18, 19] in the reaction p+p → p+K + +Λ ata beam kinetic energy of 3 . ∗ contribute to the measured final stateand influence the background for the kaonic bound state[20, 21]. No evidence for the existence of ppK − boundstates could be found and upper limits for the produc-tion of such states of the order of a few µ b were extracted.To get a consistent description of the open strangenessproduction, we further improve this method and developa framework that allows for the simultaneous analysis ofseven different data sets measured in the p+p → p+K + +Λreaction by the COSY-TOF, HADES, DISTO and FOPIexperiments in fixed target experiments at kinetic ener-gies in the laboratory frame varying from 2.14 to 3.5 GeV[20, 22–27]. This is the very first joint PWA analysis ofdifferent data sets for this reaction. This way, the energy-dependent amplitude of seven different contributing N ∗ resonances decaying into the Λ-K + channel and for non-resonant pK + Λ final states could be extracted for thefirst time.A second interesting aspect is the study of the p-Λ inter-action. This interaction was previously investigated pri-marily by means of scattering experiments [28–30]. Thereaction p+p → p+K + +Λ offers the possibility to studythe final state interaction of the p-Λ pair as an alter-native to scattering experiments [27, 30–32]. Since so far the resonances were not treated in a coherent way,a precise determination of their contributions and of thescattering lengths and effective ranges was challenging.The combined PWA presented in this work offers theunique possibility to study the interplay between the N ∗ coupling to the Λ-K + channel and the p-Λ final stateinteraction. DATA SAMPLES AND COMBINED ANALYSIS
The experimental data were measured by the COSY-TOF, DISTO, FOPI and HADES Collaborations. Ta-ble I provides an overview of the data sets used for thecombined PWA, their beam energy and number of events.Together with each experimental data set, simulations ofthe pK + Λ production according to phase space kinemat-ics, filtered through the detector simulation and analysedas the experimental data are used for the PWA. The de-tails about the reconstruction of the exclusive pK + Λ finalstate, achieved resolution, efficiency, and purity are ex-plained in the already published works by the differentcollaborations [20, 22–27]. The two HADES data samplesat the same kinetic energy correspond to two different re-construction analyses including or excluding the forwardspectrometer [20]. These data sets are complementaryand do not share any reconstructed events because ofthe exclusive selection of the final state.The goal of this PWA is to employ the seven data samplesin a combined analysis and extract the amplitudes of thedifferent waves, characterised by their quantum numbers,leading to given final states. We use the Bonn-GatchinaPWA (BG-PWA) framework [15, 16] to fit event-by-eventthe measured 4-momenta for the exclusive final statep+p → p+K + +Λ weighted with the coherent superposi-tion of specific participating waves. The best choice forthe waves used in the PWA is determined by comparingthe experimental data to the PWA output event-by-eventin terms of a log-likelihood parameter. In the specificcase of the COSY-TOF data sample, only the region ofphase space within | cos θ CMp | < . θ CMp is theproton angle in the p–p center of mass system, was con-sidered because of the the poor description of the trig-ger efficiency in the simulation for the excluded region.For the DISTO data samples the region corresponding tocos θ CMp > .
95 was excluded from the fit to minimize thebias introduced by the digitization of the scintillation-fiber sub detector used for tracking close to the targetregion. These cuts were also added in the simulationsused in the PWA analysis procedure.This PWA allows to decompose the baryon-baryon scat-tering amplitude into separate sub-processes character-ized by different intermediate states. Within the BG-PWA framework this is achieved by fitting event-by-eventthe experimental 4-vectors for a given reaction measuredwithin the acceptance of the spectrometer with a coher-
TABLE I. List of available number of events for the reaction p+p → p+K + +Λ measured by the COSY-TOF, DISTO, FOPI andHADES Collaborations. The kinetic beam energy, the total cross section and the reduced χ values resulting from differentPWA analyses are shown (see text for details).experiment T (GeV) Events/ndf σ tot [ µ b] χ /ndf (single) χ /ndf (combined)DISTO [22, 23] 2.14 121000 / 644 19 . ± . . ± . . ± . . ± . . ± . . ± . . ± . ∗ resonances included in the PWA written in thespectroscopic notation with the corresponding masses, widthsand branching ratios in the K-Λ final states [33, 34].N ∗ J P Mass (
GeVc ) Width ( GeVc ) Γ K Λ / Γ tot (%)1650 − [33] 1.655 0.14 7 ±
12 + [33] 1.710 0.23 15 ±
32 + [33] 1.720 0.25 4 ± − ±
12 + ± − ±
32 + ± ent superposition of the participating waves. This coher-ent cocktail of contributing waves is weighted with thefull scale phase space simulations of the considered finalstate that accounts for the geometrical acceptance andreconstruction efficiency of the spectrometer.Within the BG-PWA, the production cross section ofa three particle final state with single particle four-momenta q , , is parametrized as [15]: dσ = (2 π ) | A | | (cid:126)k |√ s dΦ ( P, q , q , q ) , (1)wherein P is the total four-vector, (cid:126) k is the beam momen-tum, √ s the center of mass energy of the reaction, d Φ is the infinitesimal phase-space volume of the final stateand A is the total transition amplitude of the consideredreaction. Both initial and final states can be seen as asuperposition of eigenstates with various angular momen-tum and A is the sum over all the transition amplitudes A αtr between these eigenstates [35]: A = (cid:88) α A α tr ( s ) Q in µ ..µ j ( S, L, J ) A b ( i, S , L , J ) Q fin µ ..µ j ( i, S , L , J , S (cid:48) , L (cid:48) , J ) . (2) The index α runs over all the amplitudes contributingto the transition from the initial to the final state. Thefactors Q in µ ..µ j ( S, L, J ) and Q fin µ ..µ j ( i, S , L , J , S (cid:48) , L (cid:48) , J )are the spin-momentum operators of the initial and finalstates respectively and the indexes µ j refer to the rank ofthe total angular momentum J in the spin–momentumoperators Q . The index i refers to the two-particle sub-system considered in the final state.The dependency of the amplitudes A α tr ( s ) upon thecentre of mass energy is given by: A α tr ( s ) = (cid:0) a α + a α √ s (cid:1) exp (i a α ) . (3)The real parameters a α , a α and a α are determined bythe fit to the experimental data.The parametrization of the factor A b depends on thefinal state. For the production of a N ∗ resonance, thefinal state is treated as a two-body system composed ofa proton and the N ∗ . In this case the quantum numbers S , L , J refers to the N ∗ , while the S (cid:48) , L (cid:48) , J representthe quantum numbers of the N ∗ -proton system. Non res-onant pK + Λ final states are also treated as a two particlesystem composed of a pΛ ”particle” and a K + . In thiscase S , L , J are the spin, angular and total angularmomentum of the pΛ ’particle’ while S (cid:48) , L (cid:48) , J are thequantum numbers of the pΛ-K + system.For the resonant case, the factor A b is parametrized witha relativistic Breit-Wigner formula [36]. A β b = 1( M − s − iΓ M ) , (4)with M and Γ as the pole mass and width of the corre-sponding resonance. For the presented analysis, the N ∗ resonances listed in Table II have been considered withfixed masses and fixed widths taken from [33, 34].To obtain an acceptable description of the experimentaldata it is necessary to include non-resonant partial waveamplitudes. We have included these amplitudes in a sim-ple form which provides a correct behaviour near thresh-old. For the S-wave this form corresponds to the wellknown Watson-Migdal parameterization. The resulting A b amplitude is A β b = √ s i − r β q a βp Λ + i qa βp Λ q L /F ( q, r β , L ) , (5)where q is the p-Λ relative momentum, a βp − Λ is the p-Λ-scattering length, r β is the effective range of the p-Λsystem and the index β denotes the quantum numberscombination. F ( q, r, L ) is the Blatt-Weisskopf factor used for thenormalization, it is 1 for L=0 and the explicit form forother partial waves can be found in [15]. The values ofthe scattering length and effective range can be set asfree parameters in the PWA fit and hence be extractedwithin this analysis. This coherent approach differsfrom the analysis techniques usually employed for theextraction of scattering parameters [37] and should beconsidered as complementary.Another intermediate channel contributing to pK + Λfinal state is the Σ-N cusp, which appears at or abovethe Σ-N threshold (2130 MeV/ c ) [38]. The couplingbetween the Σ-N and Λ-N channels leads to an enhance-ment of the cross-section in the p-Λ final state in amass range close to the above mentioned threshold. Inorder to include the cusp contribution in the BG-PWAframework, new transition waves must be added to Eq.2. Since the cusp is located at the Σ-N threshold, the Σand N must be in a relative S-wave state, which meansthat the spin-parity of the Σ-N system is either J P = 0 + or 1 + [38]. The resulting p-Λ system then may appearin an S-wave state in case of J P = 0 + or in an s- ord-wave state in case of J P = 1 + . This has also beenconfirmed by an analysis of the Σ-N cusp carried out bythe COSY-TOF collaboration [38]. Additionally, sincethe cusp is a resonance structure in analogy to the N ∗ ,the Breit-Wigner parametrization is used for A b (Eq.4) where the mass and width are varied within 2 . − . . − .
03 GeV/c , respectively in the PWAfit. This first attempt can be also replaced by a more so-phisticated parametrization of the cusp contribution likea Flatte’ function, but this is beyond the scope of thisinvestigation. Indeed the cusp contribution has a neg-ligible effect on the determination of the N ∗ contributions. RESULTS
First, the PWA was performed individually for the dif-ferent data samples to determine the correct start valuesof the parameters for the global fit. The total number ofavailable degrees of freedom for each data set is listed inTable I. The total number of free parameters in the PWAfit containing all accessible N ∗ is equal to 345 ±
17, theerror refers to the systematic variation of the contribut- ing N ∗ considered in the global fit. The best solutionof the PWA fit corresponds to the minimum of the log-likelihood obtained by fitting the experimental data withthe PWA event-by-event.A comparison of the three missing mass spectra and CM,Gottfried-Jackson and Helicity angle distributions (forthe definition of these variables see [11]) obtained fromthe experimental data and from the single PWA fits wascarried out and the corresponding reduced χ values arelisted in Table I. Only the statistical error of the exper-imental data has been considered to evaluate the χ ofthe single PWA fits. As a second step, a simultaneousPWA of three data samples was carried out. This inter-mediate step allowed to determine the starting values forthe global fit. The HADES, FOPI and DISTO (T = 2 . ∗ resonances fromthe list in Table II in the PWA fit. The five best solutionsin terms of log-likelihood obtained from this systematicvariation of the PWA fits were considered to extract thefinal results and the PWA systematic errors. As far asthe resonances are concerned, considering the list of sevenresonances in Table II, the five best solutions correspondto the following combinations: 1) all seven N ∗ included,2) N ∗ (1720) excluded, 3) N ∗ (1875) excluded, 4) N ∗ (1900)excluded and, 5) N ∗ (1900) and N ∗ (1875) excluded.The reduced χ values for the combined PWA listed inTable I were obtained by comparing the experimentaldata in the mass and angle variables with the averagevalues of the five best PWA solutions, taking as errorsthe statistical errors of the experimental data and thestandard deviation of the five solutions for each bin. Byadding additional solutions the χ did not improve. Thisjustifies the choice of the five best solutions. A morerefined treatment of systematic uncertainties is currentunder development.Figure 1 shows the missing mass distributions (MM) forthe three final state particles p, Λ and K + for COSY-TOF at 2 .
16 GeV (blue symbols), DISTO at 2 .
85 GeV(green symbols) and HADES at 3 . K + distribution around 2 .
13 GeV/ c . The errors of the ex-perimental data are statistical only. The lines in the samecolor-code represent the PWA results for the correspond-ing data sets. The line widths represent the error bands FIG. 1. (Color online). Missing mass distributions (MM) for the three different particles of the final state (p, Λ, K + ) areshown. The experimental data within the geometrical acceptance are from COSY-TOF at 2 .
16 GeV (blue symbols), DISTO at2 .
85 GeV (green symbols) and HADES at 3 . of the global PWA fit expressed as the standard devia-tion of the five best PWA solutions. Figure 2 shows theangular distributions of the three particles measured inthe final state for different reference systems for the samedata samples discussed in Figure 1. A similar quality isobtained for the description of the kinematic variables ofother data samples.The output of each PWA solution provides the strengthof the individual waves with respect to the total mea-sured yield. The resulting relative contributions of theresonant and non-resonant waves can be translated intocross sections for the KΛ decay channel multiplying therelative yield by the total production cross section for thepK + Λ final state.The total pK + Λ cross section for the different data setswas evaluated employing a phase space fit of the existingmeasurements of the pK + Λ channel as a function of theexcess energies [13, 25, 38–40]. The error associated tothe pK + Λ cross section of each data sample is extractedfrom the fit. A detailed description of the extraction ofthe pK + Λ cross sections can be found in [41].In Figure 3 the cross section for the different N ∗ chan-nels decaying into the KΛ final state is plotted versus itsexcess energy calculated as the center of mass energy ofthe p–p colliding system minus the sum of the protonand N* masses ( √ s − M p,N ∗ ). The standard deviation ofthe five best solutions is shown by the black vertical er-ror bars, the green bands show the error originating fromthe cross section normalization. The non-vanishing crosssection below the respective thresholds is due to the largewidth of all the considered resonances (see Table II). Therelative contribution of the non-resonant amplitude de-creases from 37% for 2.14 GeV to 10% for 3.5 GeV, sothat most of the yield stems from N ∗ resonances for allthe measured energies. The dominant contribution fromthe N ∗ resonances is consistent with the results shownin Ref. [13], except for the relative contribution of the N ∗ (1650), which is decreasing as a function of the beamenergy in [13]. In this work we found an increment of theN ∗ (1650) similarly to the N ∗ (1710) and N ∗ (1720). Thisdifference probably results from neglecting interferencein Ref. [13].The Σ-N cusp contribution varies from 10 − to 10 − withdecreasing energy with respect to the N ∗ and is not shownin Figure 3. The global PWA fit favors the Σ-N cusp con-tribution of the s- or d-wave state J P = 1 + with respectto the S-wave J P = 0 + as shown by the amplitudes inTable IV. The obtained Σ-N cusp yield is slightly differ-ent from the findings in Ref. [38] where at a beam energyof 2 .
28 GeV the contribution of the cusp was found equalto 5% of the total cross section, but neglecting interfer-ences.Figure 4 shows the cross sections of the different p+pinitial states as a function of the pK + Λ excess energycalculated as the center of mass energy of the p–p col-liding system minus the sum of the proton Λ and Kaonmasses ( √ s − M p,K + , Λ ). The error bars are associatedto the standard deviation of the five best PWA solu-tions, and the green band refers to the uncertainty of theexclusive pK + Λ production cross section. All extractedcross-sections as a function of the excess energy are sum-marised in Table IV and Table V.The non-resonant amplitude included in this PWA isparametrized as a function of the scattering length andeffective range for the p-Λ final state interaction. Theinterference of the non-resonant partial waves with theresonant amplitudes allows us to extract independentlythe values for S-wave singlet and S-wave triplet partialwaves. In Table III the resulting values for the scatteringlengths are listed. The values are obtained by averagingthe five best PWA solutions. The first error representsthe standard deviation of the five fit results. The secondone is the PWA fit error obtained by adding quadrati-cally the PWA fit errors from the five solutions. In the ) CM Λ θ cos( − − c oun t s ( / . ) × (d)(d)(d) HADES(x15)DISTO(x1)COSY-TOF(x6) ) CMp θ cos( − − c oun t s ( / . ) × (e)(e)(e) HADES(x15)DISTO(x1)COSY-TOF(x5) ) CM + K θ cos( − − c oun t s ( / . ) × (f)(f)(f) HADES(x15)DISTO(x1)COSY-TOF(x5) (d) (e) (f) ) + RF pK B/T + K θ cos( − − c oun t s ( / . ) × (g)(g)(g) HADES(x10)DISTO(x1)COSY-TOF(x5) ) Λ RF p B/T + K θ cos( − − c oun t s ( / . ) × (h)(h)(h) HADES(x7)DISTO(x1)COSY-TOF(x3) ) Λ RF pp B/T θ cos( − − c oun t s ( / . ) × (i)(i)(i) HADES(x15)DISTO(x1)COSY-TOF(x5) (g) (h) (i) ) Λ RF pK p θ cos( − − c oun t s ( / . ) (j)(j)(j) HADES(x15)DISTO(x1)COSY-TOF(x2) ) + RF pK Λ + K θ cos( − − c oun t s ( / . ) (k)(k)(k) HADES(x8)DISTO(x1)COSY-TOF(x2) ) Λ + RF K Λ p θ cos( − − c oun t s ( / . ) (l)(l)(l) HADES(x7)DISTO(x1)COSY-TOF(x3) (j) (k) (l)FIG. 2. Angular correlations for the pK + Λ final state. The upper index at the angle indicates the rest frame (RF) in which theangle is displayed. The lower index names the two particles between which the angle is evaluated. CM stands for the center-of-mass system. B and T denote the beam and target vectors, respectively. The observables are: CM distributions ( cos (cid:0) θ CMX (cid:1) )of the Λ (d), Proton (e) and Kaon (f); Gottfried-Jackson distributions cos (cid:16) θ RF pKKB/T (cid:17) (g),cos (cid:0) θ RF K Λ KB/T (cid:1) (h), cos (cid:16) θ RF p Λ pB/T (cid:17) (i)and Helicity angle distributions cos (cid:16) θ RF p Λ Kp (cid:17) (j),cos (cid:16) θ RF pKK Λ (cid:17) (k)and cos (cid:0) θ RF K Λ p Λ (cid:1) ( l ). The experimental data within thegeometrical acceptance are from COSY-TOF at 2 .
16 GeV (blue symbols), DISTO at 2 .
85 GeV (green symbols) and HADESat 3 . same table also the scattering lengths obtained from p+preactions with unpolarized [30, 42] and polarized beams[31] and the predictions by recent theoretical calculations[43, 44] are shown.The results from this PWA are comparable with pre-viously extracted values. Different parametrization, asby means of a Jost function, might modify the ex-tracted scattering parameters. Still, the comparison of the values that have been extracted within this PWA toother experimental results and theoretical parametrisa-tion demonstrate that despite of the very large numberof free parameters of this PWA and that all contributionshave been treated coherently, a reasonable agreement isachieved. FIG. 3. (Color online). Cross sections of the different N ∗ resonances decaying into the pK + Λ final state obtained from thecombined PWA as a function of the excess energy. The excess energy is calculated as the center of mass energy of the p–pcolliding system minus the sum of the proton Λ and Kaon masses ( √ s − M p,K + , Λ ) The black bars show the systematic errorsoriginating from the five different PWA solutions and the green bands represent the errors due to the normalization to the totalpK + Λ cross section. :0 30 =l. '5'
25 20
15 10 (x2) I 3po
0 200 400 600 800
Is -M p ,K\A (MeV) :0 45 -"' 40 tl 35 30 25 20 15 10 5 1000 0 200 (x3) �P, P, 400 600 800
1s - M p ,K•,A (MeV) FIG. 4. (Color online). Cross sections of the initial state waves as a function of the excess energy for the pK + Λ final state.The excess energy is calculated as the center of mass energy of the p–p colliding system minus the sum of the proton and N*masses ( √ s − M p,N ∗ ). The error bars correspond to the standard deviation among the five best PWA solutions and the greenband refers to the normalization to the total pK + Λ production cross section.
SUMMARY
We have applied a combined PWA to seven differentdata sets measuring the reaction p+p → p+K + +Λ for ki-netic energies between 2 .
14 and 3 . ∗ (1650)1 / − , N ∗ (1710)1 / + , N ∗ (1720)3 / + ,N ∗ (1875)3 / − , N ∗ (1880)1 / + , N ∗ (1895)1 / − andN ∗ (1900)3 / + and initial state partial wave as afunction of the excess energy. The contribution ofthe resonances has been found to be dominant withrespect to the direct production of the pK + Λ final stateespecially for the highest kinetic energy of 3.5 GeVwhere 90% of the yield is associated to N*. This showsnot only that the resonant production is dominating this energy regime of hadron-hadron collisions, but alsoprovides a quantitative understanding for the first timeof the interference effects on the N* excitation function.The Σ-N cusp was also included in the PWA but itscontribution is found to vary between 10 − to 10 − with decreasing energy. Hence it does not influence theobtained results for the N* and non resonant amplitudes.The p-Λ scattering lengths have also be extracted fromthis combined PWA and found to be consistent withprevious measurements. Higher precision should beachieved with a dedicated analysis of the data at thelowest energies of the here presented data samples. Anatural improvement of the results presented in thiswork will be achieved by including two additional datasets measured by the COSY-TOF collaboration at 2 . .
95 GeV [31, 45].
TABLE III. Scattering lengths extracted from the combined PWA fit and reference values from previous measurements [30, 31,42] and theoretical calculations [43, 44] (see text for details).Source S a Λ − p [fm] S a Λ − p [fm]This work − . ± . ± . − . ± . ± . − . +2 . − . − . +1 . − . [42] − . +0 . − . − . . − . [31] - − . +0 . − . ± . ± . χ EFT LO [43] − . − . χ EFT NLO [43] − . − . − . − . ACKNOWLEDGEMENTS
The authors acknowledge the support by funding thefollowing funding agencies: DFG, Grant FA 898/2-1 andNCN 2016/23/P/ST2/04066 POLONEZ. ∗ [email protected] † [email protected][1] J. Schaffner-Bielich, Nucl. Phys. A804 , 309 (2008),arXiv:0801.3791 [astro-ph].[2] D. Chatterjee and I. Vidaa, Eur. Phys. J.
A52 , 29 (2016),arXiv:1510.06306 [nucl-th].[3] D. Lonardoni, A. Lovato, S. Gandolfi, and F. Pederiva,Phys. Rev. Lett. , 092301 (2015), arXiv:1407.4448[nucl-th].[4] P. Demorest, T. Pennucci, S. Ransom, M. Roberts, andJ. Hessels, Nature , 1081 (2010), arXiv:1010.5788[astro-ph.HE].[5] J. Antoniadis et al., Science , 6131 (2013),arXiv:1304.6875 [astro-ph.HE].[6] V. P. Andreev et al., Phys. Rev.
C50 , 15 (1994).[7] V. Sarantsev et al., Eur. Phys. J.
A21 , 303 (2004).[8] G. Agakishiev et al., Phys. Lett.
B750 , 184 (2015).[9] L. Fabbietti et al., Nucl. Phys.
A914 , 60 (2013).[10] G. Agakishiev et al., Phys. Rev.
C87 , 025201 (2013).[11] G. Agakishiev et al., Phys. Rev.
C85 , 035203 (2012).[12] G. Agakishiev et al., Phys. Rev.
C90 , 015202 (2014).[13] S. Abd El-Samad et al., Phys.Lett.
B688 , 142 (2010).[14] G. Agakisiev et al., Eur. Phys. J. , 137 (2015).[15] A. V. Anisovich and A. V. Sarantsev, Eur. Phys. J. A30 ,427 (2006).[16] A. V. Anisovich, V. V. Anisovich, E. Klempt, V. A.Nikonov, and A. V. Sarantsev, Eur. Phys. J.
A34 , 129(2007).[17] K. N. Ermakov, V. I. Medvedev, V. A. Nikonov, O. V.Rogachevsky, A. V. Sarantsev, V. V. Sarantsev, andS. G. Sherman, Eur. Phys. J.
A50 , 98 (2014). [18] T. Yamazaki and Y. Akaishi, Physics Letters B , 70(2002).[19] T. Yamazaki et al., Phys. Rev. Lett. , 132502 (2010).[20] G. Agakishiev et al., Phys. Lett.
B742 , 242 (2015).[21] E. Epple and L. Fabbietti, Phys. Rev.
C92 , 044002(2015).[22] M. Maggiora et al., Nucl.Phys.
A835 , 43 (2010).[23] M. Maggiora, Nucl.Phys.
A691 , 329 (2001).[24] F. Balestra et al., Phys.Rev.Lett. , 1534 (1999).[25] M. Abdel-Bary et al., Eur.Phys.J. A46 , 27 (2010).[26] R. M¨unzer, Dissertation, TU M¨unchen (2014).[27] M. R¨oder et al., Eur. Phys. J.
A49 , 157 (2013).[28] B. Sechi-Zorn, B. Kehoe, J. Twitty, and R. A. Burnstein,Phys. Rev. , 1735 (1968).[29] F. Eisele, H. Filthuth, W. F¨ohlisch, V. Hepp, andG. Zech, Phys. Lett.
B37 , 204 (1971).[30] G. Alexander et al., Phys. Rev. , 1452 (1968).[31] F. Hauenstein et al. (COSY-TOF), Phys. Rev.
C95 ,034001 (2017), arXiv:1607.04783 [nucl-ex].[32] J. Adamczewski-Musch et al., Phys. Rev. C. (2016).[33] C. Patrignani et al. (Particle Data Group), Chin. Phys. C40 , 100001 (2016).[34] A. V. Anisovich et al., Eur. Phys. J.
A48 , 15 (2012).[35] K. Ermakov, V. Medvedev, V. Nikonov, O. Rogachevsky,A. Sarantsev, et al., Eur.Phys.J.
A47 , 159 (2011).[36] J. D. Jackson, Nuovo Cim. , 1644 (1964).[37] A. Gasparyan, J. Haidenbauer, C. Hanhart, andJ. Speth, Phys. Rev. C69 , 034006 (2004), arXiv:hep-ph/0311116 [hep-ph].[38] S. Abdel-Samad et al., Eur.Phys.J.
A49 , 41 (2013).[39] S. Abdel-Samad et al., Phys.Lett.
B632 , 27 (2006).[40] W. G. M. A. Baldini, V. Flaminio and D. R. O. Morrison,in Landolt-B¨ornstein, New Series, Subvolume a and b(Springer-Verlag, Heidelberg, 1985) p. 417(a) and 468(b).[41] R. Muenzer, Proceedings, 12th International Conference on Hypernuclear and Strange Particle Physics (HYP 2015): Sendai, Japan, September 7-12, 2015,JPS Conf. Proc. , 062008 (2017).[42] A. Budzanowski et al., Phys. Lett. B687 , 31 (2010).[43] J. Haidenbauer, S. Petschauer, N. Kaiser, U. G. Meissner,A. Nogga, and W. Weise, Nucl. Phys.
A915 , 24 (2013),arXiv:1304.5339 [nucl-th].[44] T. A. Rijken, M. M. Nagels, and Y. Yamamoto, Progressof Theoretical Physics Supplement , 14 (2010).[45] S. Jowzaee et al. (COSY-TOF), Eur. Phys. J.
A52 , 7(2016), arXiv:1509.03980 [nucl-ex].
TABLE IV. Production cross sections of the total pK + Λ non-resonant contribution and of the different N ∗ resonances decayinginto the pK + Λ final state obtained from the global PWA as a function of the beam kinetic energy. The cross sections refer tothe amplitudes prior to the coherent sum of the latter and hence do not consider interference effects. The N ∗ cross sections arenot corrected for the branching ratio into the K + -Λ final states. The first error corresponds to the systematic error due to thefive best solutions, the second stems from the cross section normalisations. The systematic error of the PWA fitting procedureis found to be negligible and hence is not shown. 3.500 GeV 3.100 GeV 2.85 GeVpK + Λ [ µ b] 5 . ± . ± . . ± . ± . . ± . ± . ∗ (1650) → pK + Λ [ µ b] 8 . ± . ± . . ± . ± . . ± . ± . ∗ (1710) → pK + Λ [ µ b] 11 . ± . ± . . ± . ± . . ± . ± . ∗ (1720) → pK + Λ [ µ b] 2 . ± . ± . . ± . ± . . ± . ± . ∗ (1875) → pK + Λ [ µ b] 1 . ± . ± . . ± . ± . . ± . ± . ∗ (1880) → pK + Λ [ µ b] 14 . ± . ± . . ± . ± . . ± . ± . ∗ (1895) → pK + Λ [ µ b] 3 . ± . ± . . ± . ± . . ± . ± . ∗ (1900) → pK + Λ [ µ b] 0 . ± . ± . . ± . ± . . ± . ± . − N(1 + S) [ µ b] 0 . ± . ± .
002 0 . ± . ± .
007 0 . ± . ± . − N(1 + D) [ µ b] 0 . ± . ± .
03 0 . ± . ± .
05 0 . ± . ± . + Λ [ µ b] 7 . ± . ± . . ± . ± . . ± . ± . ∗ (1650) → pK + Λ [ µ b] 7 . ± . ± . . ± . ± . . ± . ± . ∗ (1710) → pK + Λ [ µ b] 7 . ± . ± . . ± . ± . . ± . ± . ∗ (1720) → pK + Λ [ µ b] 1 . ± . ± . . ± . ± . . ± . ± . ∗ (1875) → pK + Λ [ µ b] 0 . ± . ± . . ± . ± . . ± . ± . ∗ (1880) → pK + Λ [ µ b] 4 . ± . ± . . ± . ± . . ± . ± . ∗ (1895) → pK + Λ [ µ b] 0 . ± . ± . . ± . ± . . ± . ± . ∗ (1900) → pK + Λ [ µ b] 0 . ± . ± . . ± . ± . . ± . ± . − N(1 + S) [ µ b] 0 . ± . ± .
02 0 . ± . ± .
03 0 . ± . ± . − N(1 + D) [ µ b] 0 . ± . ± .
06 0 . ± . ± .
04 0 . ± . ± . + Λ cross section. The first error corresponds to the systematic error due to the fivebest solutions, the second originates from the cross section normalisation. The systematic error of the PWA fitting procedureis found to be negligible and hence is not shown.3.5 GeV 3.1 GeV 2.85 GeV 2.5 GeV 2.157 GeV 2.140 GeV σ pkΛ [ µ b] 48.0 ± ± ± ± ± ± S [ µ b] 2 . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . D [ µ b] 12 . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . P [ µ b] 1 . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . P [ µ b] 13 . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . P [ µ b] 5 . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . F [ µ b] 12 . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . ..
04 0 . ± . ± . + Λ cross section. The first error corresponds to the systematic error due to the fivebest solutions, the second originates from the cross section normalisation. The systematic error of the PWA fitting procedureis found to be negligible and hence is not shown.3.5 GeV 3.1 GeV 2.85 GeV 2.5 GeV 2.157 GeV 2.140 GeV σ pkΛ [ µ b] 48.0 ± ± ± ± ± ± S [ µ b] 2 . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . D [ µ b] 12 . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . P [ µ b] 1 . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . P [ µ b] 13 . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . P [ µ b] 5 . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . F [ µ b] 12 . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . .. ± ..
04 0 . ± . ± . + Λ cross section. The first error corresponds to the systematic error due to the fivebest solutions, the second originates from the cross section normalisation. The systematic error of the PWA fitting procedureis found to be negligible and hence is not shown.3.5 GeV 3.1 GeV 2.85 GeV 2.5 GeV 2.157 GeV 2.140 GeV σ pkΛ [ µ b] 48.0 ± ± ± ± ± ± S [ µ b] 2 . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . D [ µ b] 12 . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . P [ µ b] 1 . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . P [ µ b] 13 . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . P [ µ b] 5 . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . F [ µ b] 12 . ± . ± . . ± . ± . . ± . ± . . ± . ± . . ± . ± . .. ± .. ± ..