EPJ manuscript No. (will be inserted by the editor) arXiv:0807.3839v1 [nucl-ex] 24 Jul 2008 Measurement of beam-recoil observables Ox , Oz and target asymmetry for the reaction γp → K +Λ A. Lleres1 , O. Bartalini2,10 , V. Bellini13,6 , J.P. Bocquet1 , P. Calvat1 , M. Capogni2,10,4 , L. Casano10 , M. Castoldi8 , A. D’Angelo2,10 , J.-P. Didelez16 , R. Di Salvo10 , A. Fantini2,10 , D. Franco2,10 , C. Gaulard5,14 , G. Gervino3,11 , F. Ghio9,12 , B. Girolami9,12 , A. Giusa13,7 , M. Guidal16 , E. Hourany16, R. Kunne16 , V. Kuznetsov15,18 , A. Lapik15 , P. Levi Sandri5 , F. Mammoliti13,7 , G. Mandaglio7,17 , D. Moricciani10 , A.N. Mushkarenkov15, V. Nedorezov15, L. Nicoletti2,10,1 , C. Perrin1, C. Randieri13,6 , D. Rebreyend1 , F. Renard1 , N. Rudnev15 , T. Russew1 , G. Russo13,7 , C. Schaerf2,10 , M.-L. Sperduto13,7 , M.-C. Sutera7 , A. Turinge15 (The GRAAL collaboration) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 LPSC, Université Joseph Fourier Grenoble 1, CNRS/IN2P3, Institut National Polytechnique de Grenoble, 53 avenue des Martyrs, 38026 Grenoble, France Dipartimento di Fisica, Università di Roma ”Tor Vergata”, via della Ricerca Scientifica 1, I-00133 Roma, Italy Dipartimento di Fisica Sperimentale, Università di Torino, via P. Giuria, I-00125 Torino, Italy Present affiliation: ENEA - C.R. Casaccia, via Anguillarese 301, I-00060 Roma, Italy INFN - Laboratori Nazionali di Frascati, via E. Fermi 40, I-00044 Frascati, Italy INFN - Laboratori Nazionali del Sud, via Santa Sofia 44, I-95123 Catania, Italy INFN - Sezione di Catania, via Santa Sofia 64, I-95123 Catania, Italy INFN - Sezione di Genova, via Dodecanneso 33, I-16146 Genova, Italy INFN - Sezione di Roma, piazzale Aldo Moro 2, I-00185 Roma, Italy INFN - Sezione di Roma Tor Vergata, via della Ricerca Scientifica 1, I-00133 Roma, Italy INFN - Sezione di Torino, I-10125 Torino, Italy Istituto Superiore di Sanità, viale Regina Elena 299, I-00161 Roma, Italy Dipartimento di Fisica ed Astronomia, Università di Catania, via Santa Sofia 64, I-95123 Catania, Italy Present affiliation: CSNSM, Université Paris-Sud 11, CNRS/IN2P3, 91405 Orsay, France Institute for Nuclear Research, 117312 Moscow, Russia IPNO, Université Paris-Sud 11, CNRS/IN2P3, 15 rue Georges Clémenceau, 91406 Orsay, France Dipartimento di Fisica, Università di Messina, salita Sperone, I-98166 Messina, Italy Kyungpook National University, 702-701, Daegu, Republic of Korea Received: date / Revised version: date Abstract. The double polarization (beam-recoil) observables Ox and Oz have been measured for the reaction γp → K + Λ from threshold production to Eγ ∼ 1500 MeV. The data were obtained with the linearly polarized beam of the GRAAL facility. Values for the target asymmetry T could also be extracted despite the use of an unpolarized target. Analyses of our results by two isobar models tend to confirm the necessity to include new or poorly known resonances in the 1900 MeV mass region. PACS. 13.60.Le Meson production – 13.88.+e Polarization in interactions and scattering – 25.20.Lj Photoproduction reactions 1 Introduction improvement by measuring cross sections with unprecedented precision for a large number of channels but they A detailed and precise knowledge of the nucleon spec- also allowed a qualitative leap by providing for the first troscopy is undoubtedly one of the cornerstones for our un- time high quality data on polarization observables. It is derstanding of the strong interaction in the non-perturbative well known – and now well established – that these variregime. Today’s privileged way to get information on the ables, being interference terms of various multipoles, bring excited states of the nucleon is light meson photo- and unique and crucial constraints for partial wave analysis, electroproduction. The corresponding database has con- hence facilitating the identification of resonant contribusiderably expanded over the last years thanks to a com- tions and making parameter extraction more reliable. bined effort of a few dedicated facilities worldwide. Not From this perspective, K + Λ photoproduction offers only did the recent experiments brought a quantitative unique opportunities. Because the Λ is a self-analyzing Send offprint requests to: lleres@lpsc.in2p3.fr particle, several polarization observables can be ”easily” 2 Please give a shorter version with: \authorrunning and \titlerunning prior to \maketitle measured via the analysis of its decay products. As a consequence, this reaction already possesses the richest database with results on the differential cross section [1][4], two single polarization observables (Σ and P ) [2]-[6] and two double polarization observables (Cx and Cz ) recently measured by the CLAS collaboration [7]. On the partial wave analysis side, the situation is particularly encouraging with most models concluding to the necessity of incorporating new or poorly known resonances to reproduce the full set of data. Some discrepancies do remain nonetheless either on the number of used resonances or on their identification. To lift the remaining ambiguities, new polarization obervables are needed calling for new experiments. In the present work, we report on first measurements of the beam-recoil observables Ox and Oz for the reaction γp → K + Λ over large energy (from threshold to 1500 MeV) and angular (θcm = 30 − 1400 ) ranges. The target asymmetry T , indirectly extracted from the data, is also presented. 2 Experimental set-up The experiment was carried-out with the GRAAL facility (see [8] for a detailed description), installed at the European Synchrotron Radiation Facility (ESRF) in Grenoble (France). The tagged and linearly polarized γ-ray beam is produced by Compton scattering of laser photons off the 6.03 GeV electrons circulating in the storage ring. In the present experiment, we have used a set of UV lines at 333, 351 and 364 nm produced by an Ar laser, giving 1.40, 1.47 and 1.53 GeV γ-ray maximum energies, respectively. Some data were also taken with the green line at 514 nm (maximum energy of 1.1 GeV). The photon energy is provided by an internal tagging system. The position of the scattered electron is measured by a silicon microstrip detector (128 strips with a pitch of 300 µm and 1000 µm thick). The measured energy resolution of 16 MeV is dominated by the energy dispersion of the electron beam (14 MeV - all resolutions are given as FWHM). The energy calibration is extracted run by run from the fit of the Compton edge position with a precision of ∼10µm, equivalent to ∆Eγ /Eγ ≃ 2 × 10−4 (0.3 MeV at 1.5 GeV). A set of plastic scintillators used for time measurements is placed behind the microstrip detector. Thanks to a specially designed electronic module which synchronizes the detector signal with the RF of the machine, the resulting time resolution is ≈100 ps. The coincidence between detector signal and RF is used as a start for all Time-of-Flight (ToF) measurements and is part of the trigger of the experiment. The energy dependence of the γ-ray beam polarization was determined from the Klein-Nishina formula taking into account the laser and electron beam emittances. The UV beam polarization is close to 100% at the maximum energy and decreases smoothly with energy to around 60% at the KΛ threshold (911 MeV). Based on detailed studies [8], it was found that the only significant source of error 5 1 6 25° 2 3 4 Fig. 1. Schematic view of the LAγRANGE detector: BGO calorimeter (1), plastic scintillator barrel (2), cylindrical MWPCs (3), target (4), plane MWPCs (5), double plastic scintillator hodoscope (6) (the drawing is not to scale). for the γ-ray polarization Pγ comes from the laser beam polarization (δPγ /Pγ =2%). A thin monitor is used to measure the beam flux (typically 106 γ/s). The monitor efficiency (2.68±0.03%) was estimated by comparison with the response at low rate of a lead/scintillating fiber calorimeter. The target cell consists of an aluminum hollow cylinder of 4 cm in diameter closed by thin mylar windows (100 µm) at both ends. Two different target lengths (6 and 12 cm) were used for the present experiment. The target was filled by liquid hydrogen at 18 K (ρ ≈ 7 10−2 g/cm3 ). The 4π LAγRANGE detector of the GRAAL set-up allows to detect both neutral and charged particles (fig. 1). The apparatus is composed of two main parts: a central one (250 ≤ θ ≤ 1550 ) and a forward one (θ ≤ 250 ). The charged particle tracks are measured by a set of MultiWire Proportional Chambers (MWPC) (see [5] for a detailed description). To cover forward angles, two plane chambers, each composed of two planes of wires, are used. The detection efficiency of a track is about 95% and the average polar and azimuthal resolutions are 1.50 and 20 , respectively. The central region is covered by two coaxial cylindrical chambers. Single track efficiencies have been extracted for π 0 p and π + n reactions and were found to be ≥90%, in agreement with the simulation. Since this paper deals with polarization observables, no special study was done to assess the efficiency of multi track events. Angular resolutions were also estimated via simulation, giving 3.50 in θ and 4.50 in ϕ. Charged particle identification in the central region is obtained by dE/dx technique thanks to a plastic scintillator barrel (32 bars, 5 mm thick, 43 cm long) with an energy resolution ≈20%. For the charged particles emitted in the forward direction, a Time-of-Flight measurement is provided by a double plastic scintillator hodoscope (300×300×3 cm3 ) placed at a distance of 3 m from the target and having a resolution of ≈600 ps. This detector provides also a measure of the energy loss dE/dx. Energy calibrations were extracted from the analysis of the π 0 p photoproduction reaction while the ToF calibration of the forward wall was obtained from fast electrons produced in the target. Photons are detected in a BGO calorimeter made of 480 (15θ×32ϕ) crystals, each of 21 radiation lengths. They are identified as clusters of adjacent crystals (3 on average for an energy threshold of 10 MeV per crystal) with no associated hit in the barrel. The measured energy resolution is 3% on average (Eγ =200-1200 MeV). The angular resolution is 60 and 70 for polar and azimuthal angles, respectively (Eγ ≥ 200 MeV and ltarget =3 cm). 3 Data analysis 10 3 10 2 γ+p → K +Λ + 3.1 Channel selection For the present results, the charged decay of the Λ (Λ → pπ − , BR=63.9%) was considered and the same selection method used in our previous publication on KΛ photoproduction [5] was applied. Only the main points will be recalled in the following. Only events with three tracks and no neutral cluster detected in the BGO calorimeter were retained. In the absence of a direct measurement of energy and/or momentum of the charged particles, the measured angles (θ, ϕ) of the three tracks were combined with kinematical constraints to calculate momenta. Particle identification was then obtained from the association of the calculated momenta with dE/dx and/or ToF measurements. The main source of background is the γp → pπ + π − reaction, a channel with a similar final state and a cross section hundred times larger. Selection of the KΛ final state was achieved by applying narrow cuts on the following set of experimental quantities: 3 Counts Please give a shorter version with: \authorrunning and \titlerunning prior to \maketitle γ+p → p+π +π + 0 2 4 6 8 10 - 12 14 ct (cm) Fig. 2. Reconstructed Λ decay length spectrum after all selection cuts (closed circles) for events with at least two tracks in the cylindrical chambers. The solid line represents the fit with an exponential function α ∗ exp(−ct/cτ ) where α and cτ are free parameters. The second distribution (open circles) was obtained without applying selection cuts. It corresponds to the main background reaction (γp → pπ + π − ) which, as expected, contributes only to small ct values. noted that the deficit in the first bins is attributed to finite resolution effects not fully taken into account in the simulation. The first spectrum was fitted for ct≥1 cm by an exponential function α ∗ exp(−ct/cτ ) with α and cτ as free . Energy balance. parameters. The fitted cτ value (8.17±0.31 cm) is in good . Effective masses of the three particles extracted from agreement with the PDG expectation for the Λ mean free the combination of measured dE/dx and ToF (only at path (cτΛ =7.89 cm) [9]. forward angles) with calculated momenta. By contrast, the spectrum without cuts is dominated by pπ + π − background events. As expected, they contribute . Missing mass mγp−K + evaluated from Eγ , θK (mea- mostly to small ct values (≤2-3 cm), making the shape of sured) and pK (calculated). this distribution highly sensitive to background contamination. For instance, a pronounced peak already shows up For each of these variables, the width σ of the corre- when opening selection cuts at ±3σ. A remaining source of background, which cannot be sponding distribution (Gaussian-like shape) were extracted from a Monte-Carlo simulation of the apparatus response seen in the ct plot presented above, originates from the contamination by the reaction γp → K + Σ 0 . Indeed, events based on the GEANT3 package of the CERN library. To check the quality of the event selection, the dis- where the decay photon is not detected are retained by tribution of the Λ decay length was used due to its high the first selection step. Since these events are kinematically analyzed as KΛ ones, most of them are nevertheless sensitivity to background contamination. Event by event, track information and Λ momentum rejected by the selection cuts. From the simulation, this were combined to obtain the distance d between the re- contamination was found to be of the order of 2%. As a further check of the quality of the data sample, action and decay vertices. The Λ decay length was then calculated with the usual formula ctΛ = d/(βΛ ∗ γΛ ). Fig. the missing mass spectrum was calculated. One should 2 shows the resulting distributions for events selected with remember that the missing mass is not directly measured all cuts at ±2σ (closed circles) compared with events with- and is not used as a criterion for the channel identificaout cuts (open circles). These spectra were corrected for tion. The spectrum presented in fig. 3 (closed circles) is detection efficiency losses estimated from the Monte-Carlo in fair agreement with the simulated distribution (solid simulation (significant only for ct≥5 cm). It should be line). Some slight discrepancies can nevertheless be seen 4 Please give a shorter version with: \authorrunning and \titlerunning prior to \maketitle Counts 3.2.1 Formalism For a linearly polarized beam and an unpolarized target, the differential cross section can be expressed in terms of the single polarization observables Σ, P , T (beam asymmetry, recoil polarization, target asymmetry, respectively) and of the double polarization observables Ox , Oz (beamrecoil), as follows [10]:   1 dσ dσ [1 − Pγ Σ cos 2ϕγ = ρf dΩ 2 dΩ 0 +σx′ Pγ Ox sin 2ϕγ +σy′ (P − Pγ T cos 2ϕγ ) (1) +σz′ Pγ Oz sin 2ϕγ ] γ+p → K +Λ + 8000 7000 6000 5000 4000 3000 2000 0 + K +Σ mΛ 1000 1 1.05 1.1 1.15 1.2 0 1.25 2 Missing mass (GeV/c ) Fig. 3. Distribution of the missing mass mγp−K + reconstructed from measured Eγ and θK and calculated pK . Data after all selection cuts (closed circles) are compared to the simulation (solid line). The expected contribution from the reaction γp → K + Σ 0 is also plotted (note that it is not centered on the Σ 0 mass due to kinematical constraints in the event analysis). The vertical arrow indicates the Λ mass. in the high energy tail of the spectra. The simulated missing mass distribution of the contamination from the γp → K + Σ 0 reaction, also displayed in fig. 3, clearly indicates that such a background cannot account for the observed differences. Rather, these are attributed to the summation of a large number of data taking periods with various experimental configurations (target length, wire chambers, green vs UV laser line, ...). Although these configurations were implemented in corresponding simulations, small imperfections (misalignments in particular) could not be taken into account. To summarize, thanks to these experimental checks, we are confident that the level of background in our selected sample is limited. This is corroborated by the simulation from which the estimated background contamination (multi-pions and K + Σ 0 contributions) never exceeds 5% whatever the incident photon energy or the meson recoil angle. 3.2 Measurement of Ox , Oz and T As will be shown below, the beam-recoil observables Ox and Oz , as well as the target asymmetry T , can be extracted from the angular distribution of the Λ decay proton. ρf is the density matrix for the lambda final state and (dσ/dΩ)0 the unpolarized differential cross section. The Pauli matrices σx′ ,y′ ,z′ refer to the lambda quantization axes defined by ẑ ′ along the lambda momentum in the center-of-mass frame and ŷ ′ perpendicular to the reaction plane (fig. 4). Pγ is the degree of linear polarization of the beam along an axis defined by n̂ = x̂ cos ϕγ + ŷ sin ϕγ ; the photon quantization axes are defined by ẑ along the proton center-of-mass momentum and ŷ=ŷ ′ (fig. 4). We have ϕγ = ϕlab − ϕ, where ϕlab and ϕ are the azimuthal angles of the photon polarization vector and of the reaction plane in the laboratory axes, respectively (fig. 5). The beam-recoil observables Cx and Cz measured by the CLAS collaboration with a circularly polarized beam [7] were obtained using another coordinate system for describing the hyperon polarization, the ẑ ′ axis being along the incident beam direction instead of the momentum of one of the recoiling particles (see fig. 4). Such a nonstandard coordinate system was chosen to give the results their simplest interpretation in terms of polarization transfer but implied the model calculations to be adapted. To check the consistency of our results with the CLAS values (see sect. 4.1), our Ox and Oz values were converted using the following rotation matrix: Oxc = −Ox cos θcm − Oz sin θcm Ozc = Ox sin θcm − Oz cos θcm (2) It should be noted that our definition for Ox and Oz (eq. 1) has opposite sign with respect to the definition given in the article [11], which is used in several hadronic models. We chose the same sign convention than the CLAS collaboration. For an outgoing lambda with an arbitrary quantization axis n̂′ , the differential cross section becomes: PΛ · n̂′ h dσ i dσ = T r σ · n̂′ ρf dΩ dΩ (3) where PΛ is the polarization vector of the lambda. If the polarization is not observed, the expression for the differential cross section reduces to: h dσ i dσ (4) = T r ρf dΩ dΩ Please give a shorter version with: \authorrunning and \titlerunning prior to \maketitle ^x' c K dσ (ϕlab = 900 ) = dΩ + ^z' c ^y' c → γ ^z lab proton ^ y' Λ  [1 + Pγ Σ cos 2ϕ] (7) 0 N (ϕlab = 900 ) − N (ϕlab = 00 ) = Pγ Σ cos 2ϕ N (ϕlab = 900 ) + N (ϕlab = 00 ) ^y z^ ^z' dσ dΩ The beam asymmetry values Σ published in [5] were extracted from the fit of the azimuthal distributions of the ratio: Θcm →  5 ^ x' (8) ^x 3.2.2 Λ polarization and spin observables Fig. 4. Definition of the coordinate systems and polar angles in the center-of-mass frame (viewed in the reaction plane). The [x̂′ ,ŷ ′ ,ẑ ′ ] system is used to specify the polarization of the outgoing Λ baryon: ẑ ′ is along the Λ momentum and ŷ ′ perpendicular to the reaction plane. The [x̂,ŷ,ẑ] system is used to specify the incident photon polarization: ẑ is along the incoming proton momentum and ŷ identical to ŷ ′ . The polar angle θcm of the outgoing K + meson is defined with respect to the incident beam direction ẑlab . [x̂′c ,ŷc′ ,ẑc′ ] is the coordinate system chosen by the CLAS collaboration for the Λ polarization. The x̂′c and ẑc′ axes are obtained from x̂′ and ẑ ′ by a rotation of angle π + θcm . ^y lab y^ ^n ^x ne n pla ^z lab tio reac ^x lab Fig. 5. Definition of the coordinate systems and azimuthal angles in the center-of-mass frame (viewed perpendicularly to the beam direction). The [x̂lab ,ŷlab ,ẑlab ] system corresponds to the laboratory axes with ẑlab along the incident beam direction. The [x̂,ŷ,ẑ] system, used to define the incident photon polarization, has its axes x̂ and ŷ along and perpendicular to the reaction plane (azimuthal angle ϕ), respectively. The polarization of the beam is along n̂ (azimuthal angle ϕlab ). The two beam polarization states correspond to ϕlab = 00 (horizontal) and ϕlab = 900 (vertical) (ϕlab = ϕγ + ϕ). which leads to: dσ = dΩ  dσ dΩ  [1 − Pγ Σ cos 2ϕγ ] (5) 0 0 For horizontal (ϕlab = 0 ) and vertical (ϕlab = 900 ) photon polarizations, the corresponding azimuthal distributions of the reaction plane are therefore:   dσ dσ 0 [1 − Pγ Σ cos 2ϕ] (6) (ϕlab = 0 ) = dΩ dΩ 0 The components of the lambda polarization vector deduced from eqs. 1 to 5 are: ′ ′ PΛx ,z = ′ PΛy = Pγ Ox,z sin 2ϕγ 1 − Pγ Σ cos 2ϕγ P − Pγ T cos 2ϕγ 1 − Pγ Σ cos 2ϕγ (9) (10) These equations provide the connection between the Λ polarisation PΛ and the spin observables Σ, P , T , Ox and Oz . Integration of the polarization components over the azimuthal angle ϕ of the reaction plane writes: R i dσ (ϕ)dϕ PΛ (ϕ) dΩ i R dσ (11) < PΛ > = dΩ (ϕ)dϕ where i stands for x′ , y ′ or z ′ . When integrating over the full angular domain, the averaged x′ and z ′ components of the polarization vector vanish while the y ′ component is equal to P . On the other hand, when integrating over appropriatly chosen angular sectors, all three averaged components can remain different from zero. For horizontal and vertical beam polarizations, the following expressions are obtained when considering the four particular ϕ domains defined hereafter [12] (recalling ϕγ = ϕlab − ϕ): . S1 = [π/4, 3π/4] ∪ [5π/4, 7π/4]: ′ < PΛy > (ϕlab = 00 ) = (P π + 2Pγ T )/(π + 2Pγ Σ) ′ < PΛy > (ϕlab = 900 ) = (P π − 2Pγ T )/(π − 2Pγ Σ) . S2 = [−π/4, π/4] ∪ [3π/4, 5π/4]: ′ < PΛy > (ϕlab = 00 ) = (P π − 2Pγ T )/(π − 2Pγ Σ) ′ < PΛy > (ϕlab = 900 ) = (P π + 2Pγ T )/(π + 2Pγ Σ) . S3 = [0, π/2] ∪ [π, 3π/2]: ′ ′ < PΛx ,z > (ϕlab = 00 ) = −2Pγ Ox,z /π ′ ′ < PΛx ,z > (ϕlab = 900 ) = +2Pγ Ox,z /π . S4 = [π/2, π] ∪ [3π/2, 2π] : ′ ′ < PΛx ,z > (ϕlab = 00 ) = +2Pγ Ox,z /π ′ ′ < PΛx ,z > (ϕlab = 900 ) = −2Pγ Ox,z /π 6 Please give a shorter version with: \authorrunning and \titlerunning prior to \maketitle It should be noted that these four sectors cover the full ϕ range. In the following, these different combinations of ϕ sectors and polarization states will be labelled by the sign plus or minus appearing in the corresponding expressions for < PΛi >. 3.2.3 Decay angular distribution In the lambda rest frame, the angular distribution of the decay proton is given by [13]: W (cos θp ) =
1 1 + α|PΛ | cos θp 2 3.2.4 Experimental extraction As for Σ and P , the observables Ox , Oz and T were extracted from ratios of the angular distributions, in order to get rid of most of the distorsions introduced by the experimental acceptance. Including the detection efficiencies, the yields measured as a function of the proton angle with respect to the different axes write: ′ ′ x ,z N± = ′ ′ ′ ′
2Pγ Ox,z 1 x′ ,z′ N0± ǫ± (cos θpx ,z ) 1 ± α cos θpx ,z 2 π (18) (12) ′ where α=0.642±0.013 [9] is the Λ decay parameter and θp the angle between the proton direction and the lambda polarization vector. From this expression, one can derived an angular distribution for each component of PΛ : y = N± ′ ′
P π ± 2Pγ T 1 y′ N ǫ± (cos θpy ) 1 + α cos θpy (19) 2 0± π ± 2Pγ Σ From the integration of the azimuthal distributions given by eqs. 6 and 7 over the different angular sectors, it can be shown that: ′
1 W (cos θpi ) = 1 + αPΛi cos θpi 2 ′ ′ 1 (1 + αP cos θpy ) 2 ′ y N0+ y′ N0− ′ ′ ′ > 0) − > 0) + < 0) 1 = αP 2 < 0) ′ ′ y y N+ + N− = ′ ′ W± (cos θpx ,z ) = ′ W± (cos θpy ) = ′ ′
2Pγ Ox,z 1 1±α cos θpx ,z 2 π ′
P π ± 2Pγ T 1 1+α cos θpy 2 π ± 2Pγ Σ (16) (17) π + 2Pγ Σ π − 2Pγ Σ (21) ′ ′ 1 x′ ,z′ x′ ,z ′ )ǫ± (cos θpx ,z ) + N0− (N 2 0+ ′ ′ 1 y′ y′ )ǫ± (cos θpy )(1 + αP cos θpy ) (N0+ + N0− 2 (23) x ,z 2N+ When integrating over the different angular domains specified above (sectors S1 + S2 for y ′ -axis and S3 + S4 for x′ -,z ′ -axes, appropriatly combined with the two beam polarization), the proton angular distributions with respect to the three quantization axes can be written as follows: = (22) (14) (15) ′ x ,z x ,z = + N− N+ ′ ′ N (cos θpy ′ N (cos θpy (20) Assuming that the detection efficiencies do not depend on the considered ϕ sectors (ǫ+ (cos θpi ) = ǫ− (cos θpi ) - the validity of this assumption will be discussed later on), we can then calculate the following sums and ratios from which the efficiency cancels out: where P is the recoil polarization. Our P results published in [5] were determined directly from the measured up/down asymmetry: ′ N (cos θpy ′ N (cos θpy ′ (13) where θpi is now the angle between the proton direction and the quantization axis i (x′ , y ′ or z ′ ). The components being determined in the Λ rest frame, a suitable transformation should be applied to calculate them in the center-of-mass frame. However, as the boost direction is along the lambda momentum, it can be shown that the polarization measured in the lambda rest frame remains unchanged in the center-of-mass frame [7]. When integrating over all possible azimuthal angles ϕ, the proton angular distribution with respect to the y ′ -axis simply writes: W (cos θpy ) = ′ ′ x ,z x ,z = N0− N0+ x′ ,z ′ N+ + x′ ,z ′ N− = (1 + α ′ ′ 2Pγ Ox,z cos θpx ,z ) π P π+2P T ′ y 2N+ y y N+ + N− ′ ′ ′ (24) ′ γ y 2Pγ Σ
1 + α π+2Pγ Σ cos θp
(25) = 1+ ′ π 1 + αP cos θpy To illustrate the extraction method of Ox , Oz and T , the N+ and N− experimental distributions together with their sums and ratios, summed over all photon energies and meson polar angles, are displayed in figs. 6 (x′ -axis), 7 (z ′ -axis) and 8 (y ′ -axis). Thanks to the efficiency correction given by the distributions N+ + N− (figs. 6,7,8-c), the ratios 2N+ /(N+ + N− ) (figs. 6,7,8-d), from which the efficiency drops out, exhibit the expected dependence in cos θp and can be therefore fitted by the functions given in Please give a shorter version with: \authorrunning and \titlerunning prior to \maketitle the r.h.s. of eqs. 24 and 25. The known energy dependence of Pγ and the previously measured values for Σ and P [5] are then used to deduce Ox , Oz and T from the fitted slopes. The validity of the hypothesis ǫ+ (cos θpi ) = ǫ− (cos θpi ) was studied via the Monte Carlo simulation in which a polarized Λ decay was included. The efficiencies ǫ± calculated from the simulation are presented in plots e) of figs. 6 to 8 and the ratios ǫ− /ǫ+ in plots f) (open circles). As one can see, for the y ′ case, this ratio remains very close to 1 whatever the angle while, for x′ and z ′ , the discrepancy from 1 is more pronounced and evolves with the angle. This shows that some corrections should be applied on the measured ratios 2N+ /(N+ + N− ) to take into account the non-negligible differences observed between ǫ+ and ǫ− . The correction factors, plotted in figs. 6,7,8-f) (closed circles), were calculated through the following expression: Cor = i i

2N+ 2N+ / i i i i gen N+ + N− N+ + N− sel (26) where gen and sel stand for generated and selected events. Since ǫ± = (N± )sel /(N± )gen , it can be re-written as: Cor = i i

2N+ ǫi− N− 1 [1 + i + N i gen i gen ] 2 N+ ǫi+ N+ − (27) 7 To illustrate this second extraction method, the corrected distributions, summed over all photon energies and meson polar angles, are displayed in figs. 6,7-j) (x′ , z ′ -axes) and 8-h),i) (y ′ -axis). They were obtained by dividing the originally measured distributions (figs. 6,7-h and 8-a,b) by the corresponding efficiency distributions (figs. 6,7-i and 8-e). In the y ′ -axis case, the two corrected spectra N± /ǫ± were simultaneously fitted. This method gives results in good agreement with those extracted from the first method. Nevertheless, the resulting χ2 were found to be significantly larger (the global reduced-χ2 values are given in figs. 6 to 8 - they are close to 1 for the first method and five to ten times larger for the second one). The first method, which relies upon ratios leading to an intrinsic first order efficiency correction, is less dependent on the simulation details and was therefore preferred. Three sources of systematic errors were taken into account: the laser beam polarization (δPγ /Pγ =2%), the Λ decay parameter α (δα = 0.013) and the hadronic background. The error due to the hadronic background was estimated from the variation of the extracted values when cuts were changed from ±2σ to ±2.5σ. Given the good agreement between the two extraction methods, no corresponding systematic error was considered. For the T observable, the measured values for Σ and P being involved, their respective errors were included in the estimation of the uncertainty. All systematic and statistical errors have been summed quadratically. The corrected distributions are displayed in the plots g) of figs. 6 to 8. After correction, as expected, the slope of the y ′ distribution is unaffected while the slopes of the x′ and z ′ distributions are slightly modified. These distributions were again fitted to obtain the final values of Ox , Oz and 4 Results and discussions T. As the detection efficiencies and the correction factors The complete set of beam-recoil polarization and target calculated from the simulation depend on the input values asymmetry data is displayed in figs. 9 to 15. These data of Ox , Oz and T , an iterative method was used. Three cover the production threshold region (Eγ =911-1500 MeV) iterations were sufficient to reach stable values. and a large angular range (θkaon = 30 − 1400 ). NumeriFor a consistency check, an alternative extraction method cal values are listed in tablescm1 to 3. Error bars are the was implemented. The angular distributions were directly quadratic sum of statistical and systematic errors. corrected by the simulated efficiencies and fitted according to: ′ ′ ′ ′ x ,z ,inv x ,z + N− N+ x′ ,z ′ ǫ+ x′ ,z ′ ,inv + ǫ− = ′ ′
2Pγ Ox,z 1 x′ ,z′ N0+ 1 + α cos θpx ,z 2 π (28) ′ y N+ ′ ǫy+ ′ y N− y′ ǫ− = = ′
1 y′ P π + 2Pγ T N0+ 1 + α cos θpy 2 π + 2Pγ Σ (29) ′
1 y′ π − 2Pγ Σ P π − 2Pγ T 1+α N cos θpy (30) 2 0+ π + 2Pγ Σ π − 2Pγ Σ where N inv and ǫinv stand for N (− cos θp ) and ǫ(− cos θp ), respectively. This trick, used for the x′ and z ′ cases, allows to combine the N+ and N− distributions which have opposite slopes (eq. 18). 4.1 Observable combination and consistency check In pseudoscalar meson photoproduction, one can extract experimentally 16 different quantities: the unpolarized differential cross section (dσ/dΩ)0 , 3 single polarization observables (P , T , Σ), 4 beam-target polarizations (E, F , G, H), 4 beam-recoil polarizations (Cx , Cz , Ox , Oz ) and 4 target-recoil polarizations (Tx , Tz , Lx , Lz ). The various spin observables are not independent but are constrained by non-linear identities and various inequalites [10], [11], [14], [15]. In particular, of the seven single and beam-recoil polarization observables, only five are independent being related by the two equations: Cx2 + Cz2 + Ox2 + Oz2 = 1 + T 2 − P 2 − Σ 2 (31) Cz Ox − Cx Oz = T − P Σ (32) Please give a shorter version with: \authorrunning and \titlerunning prior to \maketitle 0.10 0.08 εε+ 0.06 Cor*2N+/(N++N-) ε++ε-inv 0.04 0.12 1.1 e) Ox=-0.025 2 χ =0.7 1.0 0.9 g) 0.10 0.08 0.06 0.04 0.02 -1 i) -0.5 0 0.5 1 CosΘpx ε- / ε+ and Cor 0.12 1.2 N++N-inv (a.u.) ε- and ε+ 0.9 c) N++N- (a.u.) 80 1.0 140 140 d) ε- / ε+ Cor 1.1 1.0 60 2 120 N- (a.u.) e) Oz=0.57 χ =1.6 2 g) 0.10 0.08 -0.5 j) 0.04 0 0.02 -1 0.5 1 i) -0.5 0 CosΘpx P 2 + Ox2 + Oz2 ≤ 1 (34) + εε+ 0.06 100 (33) Σ + 0.08 0.12 χ =5.8 Ox=0.016 |T ± P | ≤ 1 ± Σ Oz2 0.10 h) 80 There are also a number of inequalities involving three of these observables: Ox2 0.12 1.6 1.4 1.2 1.0 0.8 0.6 Fig. 6. Angular distributions for the decay proton in the lambda rest frame with respect to the x′ -axis: a) distribution N+ ; b) distribution N− ; c) sum N+ + N− ; d) ratio 2N+ /(N+ + N− ); e) efficiencies ǫ+ (triangles) and ǫ− (circles) calculated from the simulation; f) ratio ǫ− /ǫ+ (open circles) and correction factor Cor (closed circles) given by eq. 26 calculated from the simulation; g) ratio 2N+ /(N+ + N− ) corrected inv by the factor Cor; h) distribution N+ + N− , with N inv = inv inv N (− cos θp ); i) efficiency ǫ+ + ǫ− , with ǫ = ǫ(− cos θp ), inv calculated from the simulation; j) distribution N+ + N− corrected by the efficiency ǫ+ + ǫinv . The solid line in d) and g) − represents the fit by the (linear) function given in the r.h.s. of eq. 24. The solid line in j) represents the fit by the (linear) function given in the r.h.s. of eq. 28. The reduced-χ2 and the Ox value obtained from the fits are reported in d), g) and j). 2 80 0.04 100 80 -1 100 f) 120 c) 1.6 1.4 1.2 1.0 0.8 0.6 120 0.06 0.9 a) 140 60 ε- and ε+ 100 1.1 Ox=-0.099 2 χ =0.9 Cor*2N+/(N++N-) 2N+/(N++N-) 120 1.2 (N++N-inv)/(ε++ε-inv) (a.u.) N++N- (a.u.) 140 60 b) 30 80 70 60 50 40 30 2N+/(N++N-) 40 a) 30 1.2 50 ε++ε-inv 40 60 80 70 60 50 40 30 ≤1 (35) 0.5 1 CosΘpz ε- / ε+ and Cor 50 70 1.2 N++N-inv (a.u.) 60 z,- distributions N+ (a.u.) 70 N- (a.u.) N+ (a.u.) x,- distributions 140 (N++N-inv)/(ε++ε-inv) (a.u.) 8 180 160 140 120 100 80 60 40 -1 b) Oz=0.62 χ =1.9 2 d) ε- / ε+ Cor 1.1 1.0 0.9 f) 120 100 80 h) 60 Oz=0.54 χ =12.8 2 j) -0.5 0 0.5 1 CosΘpz Fig. 7. Angular distributions for the decay proton in the lambda rest frame with respect to the z ′ -axis (all distributions as in fig. 6). The reduced-χ2 and the Oz value obtained from the fits are reported in d), g) and j). P 2 + Cx2 + Cz2 ≤ 1 (36) Σ 2 + Cx2 + Cz2 ≤ 1 (37) These different identities and inequalities can be used to test the consistency of our present and previous measurements. They can also be used to check the compatibility of our data with the results on Cx and Cz recently published by the CLAS collaboration [7]. Our measured values for Σ, P , T , Ox and Oz were combined to test the above inequalities. Equation 31 was used to calculate the quantity Cx2 + Cz2 appearing in expressions 36 and 37. The results for the two combinations |T ± P | ∓ Σ of the three single polarizations are presented in fig. 12. The results for the quantities: . (P 2 + Ox2 + Oz2 )1/2 , . (Σ 2 + Ox2 + Oz2 )1/2 , Please give a shorter version with: \authorrunning and \titlerunning prior to \maketitle . (1 + T 2 − P 2 − Ox2 − Oz2 )1/2 = (Σ 2 + Cx2 + Cz2 )1/2 , . (1 + T 2 − Σ 2 − Ox2 − Oz2 )1/2 = (P 2 + Cx2 + Cz2 )1/2 , c) 1.6 1.4 1.2 1.0 0.8 0.6 N- (a.u.) a) 80 70 60 50 40 30 20 0.12 0.10 0.08 0.06 εε+ Cor*2N+/(N++N-) 1.6 1.4 1.2 1.0 0.8 0.6 N-/ε- (a.u.) 0.04 100 e) T=-0.57 χ =1.1 2 ε- / ε+ and Cor 2N+/(N++N-) 160 140 120 100 80 60 40 1.2 N+/ε+ (a.u.) ε- and ε+ N++N- (a.u.) N+ (a.u.) y,- distributions 80 70 60 50 40 30 20 100 80 T=-0.58 χ =1.0 2 d) ε- / ε+ Cor 1.1 1.0 0.9 f) T=-0.57 80 χ =11.6 2 60 20 -1 T=-0.57 b) 40 g) h) -0.5 0 0.5 1 CosΘpy χ =13.3 2 60 40 20 -1 i) -0.5 0 0.5 9 which combine single and double polarization observables, are displayed in figs. 13 and 14. All these quantities should be ≤ 1. The plotted uncertainties are given by the standard error propagation. Whatever the photon energy or the meson polar angle, no violation of the expected inequalities is observed, confirming the internal consistency of our set of data. Since all observables entering in eqs. 31 and 32 were measured either by GRAAL (Σ, P , T , Ox , Oz ) or by CLAS (P , Cx , Cz - their P data were confirmed by our measurements [5]), the two sets of data can be therefore compared and combined. Within the error bars, the agreement between the two sets of equal combinations (1+T 2 −Σ 2 −Ox2 −Oz2 )1/2 (GRAAL) and (P 2 +Cx2 +Cz2 )1/2 (CLAS) is fair (fig. 14) and tends to confirm the previously observed saturation to the value 1 of R = (P 2 + Cx2 + Cz2 )1/2 , whatever the energy or angle. Fig. 15 displays the values for the combined GRAAL-CLAS quantity Cz Ox − Cx Oz − T + P Σ. Within the uncertainties, the expected value (1) is obtained, confirming again the overall consistency of the GRAAL and CLAS data. It has been demonstrated [14] that the knowledge of the unpolarized cross section, the three single-spin observables and at least four double-spin observables - provided they have not all the same type - is sufficient to determine uniquely the four complex reaction amplitudes. Therefore, only one additional double polarization observable measured using a polarized target will suffice to extract unambiguously these amplitudes. 1 CosΘpy Fig. 8. Angular distributions for the decay proton in the lambda rest frame with respect to the y ′ -axis: a) distribution N+ ; b) distribution N− ; c) sum N+ + N− ; d) ratio 2N+ /(N+ + N− ); e) efficiencies ǫ+ (triangles) and ǫ− (circles) calculated from the simulation - they are symmetrical about θcm = 900 (we find ǫdown /ǫup =1.03); f) ratio ǫ− /ǫ+ (open circles) and correction factor Cor (closed circles) given by eq. 26 calculated from the simulation; g) ratio 2N+ /(N+ + N− ) corrected by the factor Cor; h) distribution N+ corrected by the efficiency ǫ+ ; i) distribution N− corrected by the efficiency ǫ− . The solid line in d) and g) represents the fit by the (non-linear) function given in the r.h.s. of eq. 25. These distributions exhibit a linear behaviour since the overall recoil polarisation P is very low (the value extracted from the up/down asymmetry of the raw distribtion N+ + N− is -0.12). The solid line in h) and i) represents the simultaneous fit by the (linear) functions given in the r.h.s. of eqs. 29 and 30. The reduced-χ2 and the T value obtained from the fits are reported in d), g), h) and i). 4.2 Comparison to models We have compared our results with two models: the Ghent isobar RPR (Regge-plus-resonance) model [16]-[19] and the coupled-channel partial wave analysis developed by the Bonn-Gatchina collaboration [20]-[24]. In the following, these models will be refered as RPR and BG, respectively. The comparison is shown in figs. 9 to 11. The RPR model is an isobar model for KΛ photo- and electroproduction. In addition to the Born and kaonic contributions, it includes a Reggeized t-channel background which is fixed to high-energy data. The fitted database includes differential cross section, beam asymmetry and recoil polarization photoproduction results. The model variant presented here contains, besides the known N ∗ resonances (S11 (1650), P11 (1710), P13 (1720)), the P13 (1900) state (** in the PDG [9]) and a missing D13 (1900) resonance. This solution was found to provide the best overall agreement with the combined photo- and electroproduction database. As one can see in figs. 9 to 11, the RPR prediction (dashed line) qualitatively reproduces all observed structures. Interestingly enough, the model best reproduces the data at high energy (1400-1500 MeV), where the P13 (1900) and D13 (1900) contributions are maximal. The BG model is a combined analysis of experiments with πN , ηN , KΛ and KΣ final states. As compared 10 Please give a shorter version with: \authorrunning and \titlerunning prior to \maketitle to the other models, this partial-wave analysis takes into account a much larger database which includes most of the available results (differential cross sections and polarization observables). For the γp → K + Λ reaction, the main resonant contributions come from the S11 (1535), S11 (1650), P13 (1720), P13 (1900) and P11 (1840) resonances. To achieve a good description of the recent Cx and Cz CLAS measurements, the ** P13 (1900) had to be introduced. It should be noted that, at this stage of the analysis, the contribution of the missing D13 (1900) is significantly reduced as compared to previous versions of the model. As shown in figs. 9-11, this last version (solid line) provides a good overall agreement. On the contrary, the solution without the P13 (1900) (not shown) fails to reproduce the data. More refined analyses with the RPR and BG models are in progress and will be published later on. Comparison with the dynamical coupled-channel model of SaclayArgonne-Pittsburgh [25]-[27] has also started. 5 Summary In this paper, we have presented new results for the reaction γp → K + Λ from threshold to Eγ ∼ 1500 MeV. Measurements of the beam-recoil observables Ox , Oz and target asymmetries T were obtained over a wide angular range. We have compared our results with two isobar models which are in reasonable agreement with the whole data set. They both confirm the necessity to introduce new or poorly known resonances in the 1900 MeV mass region (P13 and/or D13 ). It should be underlined that from now on only one additional double polarization observable (beam-target or target-recoil) would be sufficient to extract the four helicity amplitudes of the reaction. Acknowledgements We are grateful to A.V. Sarantsev, B. Saghai, T. Corthals, J. Ryckebusch and P. Vancraeyveld for communication of their most recent analyses and J.M. Richard for fruitful discussions on the spin observable constraints. We thank R. Schumacher for communication of the CLAS data. The support of the technical groups from all contributing institutions is greatly acknowledged. It is a pleasure to thank the ESRF as a host institution and its technical staff for the smooth operation of the storage ring. References 1. 2. 3. 4. 5. 6. 7. 8. R.K.Bradford et al., Phys. Rev. C 73, 035202 (2006). K.-H. Glander et al., Eur. Phys. J. A 19, 251 (2004). J.W.C. McNabb et al., Phys. Rev. C 69, 042201(R) (2004). M. Sumihama et al., Phys. Rev. C 73, 035214 (2006). A. Lleres et al., Eur. Phys. J. A 31, 79 (2007). R.G.T. Zegers et al., Phys. Rev. Lett. 91, 092001 (2003). R.K.Bradford et al., Phys. Rev. C 75, 035205 (2007). O. Bartalini et al., Eur. Phys. J. A 26, 399 (2005). 9. Review of Particle Physics 2004, Phys. Lett. B 592, 1 (2004). 10. R.A. Adelseck and B. Saghai, Phys. Rev. C 42, 108 (1990). 11. I.S. Barker, A. Donnachie and J.K. Storrow, Nucl. Phys. B 95, 347 (1975). 12. P. Calvat, Thesis, Université J. Fourier Grenoble, 1997, unpublished. 13. T.D. Lee and C.N. Yang, Phys. Rev. 108, 1645 (1957). 14. W.T. Chiang and F. Tabakin, Phys. Rev. C 55, 2054 (1997). 15. X. Artru, J.M. Richard and J. Soffer, Phys. Rev. C 75, 024002 (2007). 16. T. Corthals, J. Ryckebusch and T. Van Cauteren, Phys. Rev. C 73, 045207 (2006). 17. T. Corthals, Thesis, Universiteit Gent, 2007, unpublished. 18. T. Corthals et al., Phys. Lett. B 656, 186 (2007). 19. T. Corthals, J. Ryckebusch and P. Vancraeyveld, private communication. 20. A.V. Anisovich et al., Eur. Phys. J. A. 25, 427 (2005). 21. A.V. Sarantsev et al., Eur. Phys. J. A. 25, 441 (2005). 22. A.V. Anisovich et al., Eur. Phys. J A 34, 243 (2007). 23. V.A. Nikonov et al., arXiv:hep-ph/0707.3600 (2007). 24. A. Sarantsev, private communication. 25. B. Juliá-Dı́az, B. Saghai, T.-S.H. Lee and F. Tabakin, Phys. Rev. C 73, 055204 (2006). 26. B. Saghai, J.C. David, B. Juliá-Dı́az and T.-S.H. Lee, Eur. Phys. J. A 31, 512 (2007). 27. B. Saghai, private communication. Ox Please give a shorter version with: \authorrunning and \titlerunning prior to \maketitle 11 1.5 1.0 980 MeV 1027 MeV 1074 MeV 1122 MeV 1171 MeV 1222 MeV 1272 MeV 1321 MeV 1372 MeV 1421 MeV 1466 MeV 0.5 0.0 -0.5 -1.0 -1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 1.0 60 120 180 0.5 0.0 -0.5 -1.0 -1.5 0 60 120 180 60 120 180 Θcm Fig. 9. Angular distributions of the beam recoil observable Ox . Data are compared with the predictions of the BG (solid line) and RPR (dashed line) models. Please give a shorter version with: \authorrunning and \titlerunning prior to \maketitle Oz 12 1.5 1.0 0.5 0.0 -0.5 -1.0 980 MeV 1027 MeV 1074 MeV 1122 MeV 1171 MeV 1222 MeV -1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 1.0 0.5 0.0 -0.5 1272 MeV -1.0 1321 MeV 1372 MeV -1.5 1421 MeV 1.0 1466 MeV 60 120 180 0.5 0.0 -0.5 -1.0 -1.5 0 60 120 180 60 120 180 Θcm Fig. 10. Angular distributions of the beam recoil observable Oz . Data are compared with the predictions of the BG (solid line) and RPR (dashed line) models. T Please give a shorter version with: \authorrunning and \titlerunning prior to \maketitle 13 1.5 980 MeV 1.0 1027 MeV 1074 MeV 0.5 0.0 -0.5 -1.0 -1.5 1.0 1122 MeV 1171 MeV 1222 MeV 1272 MeV 1321 MeV 1372 MeV 0.5 0.0 -0.5 -1.0 -1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 1.0 1421 MeV 60 1466 MeV 120 180 0.5 0.0 -0.5 -1.0 -1.5 0 60 120 180 60 120 180 Θcm Fig. 11. Angular distributions of the target asymmetry T . Data are compared with the predictions of the BG (solid line) and RPR (dashed line) models. 14 Please give a shorter version with: \authorrunning and \titlerunning prior to \maketitle |T-P|+Σ and |T+P|-Σ GRAAL data 1.5 1.0 0.5 0.0 980 MeV 1027 MeV 1074 MeV -0.5 1.0 0.5 0.0 1122 MeV 1171 MeV 1222 MeV -0.5 1.0 0.5 0.0 1272 MeV 1321 MeV 1372 MeV -0.5 60 120 180 1.0 |T-P|+Σ 0.5 |T+P|-Σ 0.0 1421 MeV -0.5 0 60 120 1466 MeV 180 60 120 180 Θcm Fig. 12. Angular distributions of the quantities |T − P | + Σ (closed circles) and |T + P | − Σ (open circles). We should have the inequalities |T ± P | ∓ Σ ≤ 1 (eq. 33). Please give a shorter version with: \authorrunning and \titlerunning prior to \maketitle 1/2 GRAAL data 2.0 980 MeV 1027 MeV 1074 MeV 1.5 1.0 0.5 0.0 1122 MeV 1171 MeV 1222 MeV 1.5 1.0 0.5 0.0 1272 MeV 1321 MeV 1372 MeV 1421 MeV 1466 MeV 60 1.5 1.0 2 (P +Ox2+Oz2) 1/2 2 , (Σ +Ox2+Oz2) 1/2 2 2 , (1+T -P -Ox2-Oz2) 15 0.5 0.0 120 180 1.5 2 1/2 2 1/2 (P +Ox2+Oz2) 1.0 (Σ +Ox2+Oz2) 2 0.5 2 0.0 (1+T -P -Ox2-Oz2) 0 60 120 180 60 120 1/2 180 Θcm Fig. 13. Angular distributions of the quantities (P 2 + Ox2 + Oz2 )1/2 (stars), (Σ 2 + Ox2 + Oz2 )1/2 (circles) and (1 + T 2 − P 2 − Ox2 − Oz2 )1/2 = (Σ 2 + Cx2 + Cz2 )1/2 (crosses). The first and third sets of data are horizontally shifted for visualization. All these quantities should be ≤ 1 (inequalities 34, 35 and 37). Please give a shorter version with: \authorrunning and \titlerunning prior to \maketitle GRAAL vs CLAS data 2.0 980 MeV 1027 MeV 1074 MeV (1032 MeV) 1.5 1.0 0.5 0.0 1122 MeV 1.5 1171 MeV 1222 MeV (1132 MeV) (1232 MeV) 1.0 2 2 (1+T -Σ -Ox2-Oz2) 1/2 2 and (P +Cx2+Cz2) 1/2 16 0.5 0.0 1272 MeV 1321 MeV 1372 MeV (1332 MeV) 1.5 1.0 0.5 0.0 1421 MeV 1.5 60 1466 MeV (1433 MeV) 180 GRAAL data (1+T2-Σ2-Ox2-Oz2)1/2 1.0 CLAS data 0.5 0.0 0 120 (P2+Cx2+Cz2)1/2 60 120 180 60 120 180 Θcm Fig. 14. Angular distributions of the quantity (1 + T 2 − Σ 2 − Ox2 − Oz2 )1/2 = (P 2 + Cx2 + Cz2 )1/2 . This quantity should be ≤ 1 (inequality 36). Comparison to the values (P 2 + Cx2 + Cz2 )1/2 published by the CLAS collaboration (open squares - energy in parentheses). Note that the Ox2 + Oz2 and Cx2 + Cz2 values are independent of the choice for the axes specifying the Λ polarization (see sect. 3.2.1). Please give a shorter version with: \authorrunning and \titlerunning prior to \maketitle 17 CzOx-CxOz-T+PΣ GRAAL × CLAS data 1.0 1027 (1032) MeV 1122 (1132) MeV 1222 (1232) MeV 1321 (1332) MeV 0.5 0.0 -0.5 -1.0 0.5 0.0 -0.5 -1.0 60 1421 (1433) MeV 120 180 0.5 GRAAL data : Σ , P , T , Ox , Oz 0.0 CLAS data : Cx , Cz -0.5 -1.0 0 60 120 180 Θcm Fig. 15. Angular distributions of the quantity Cz Ox − Cx Oz − T + P Σ. This quantity is calculated using the Cx and Cz results published by the CLAS collaboration (energy in parentheses) combined with our Ox and Oz data converted by eq. 2 to have the same ẑ ′ axis convention and with our Σ, P and T measurements. The used CLAS data are those corresponding to the angles cos θcm =0.85, mean(0.65,0.45), mean(0.25,0.05), -0.15, mean(-0.35,-0.55) and -0.75. We should have the equality Cz Ox − Cx Oz − T + P Σ = 0 (eq. 32). 18 Please give a shorter version with: \authorrunning and \titlerunning prior to \maketitle Table 1. Beam-recoil Ox values. θcm (o ) 31.3 59.1 80.7 99.8 118.9 138.6 θcm (o ) 34.1 58.9 80.5 99.3 119.4 140.4 Eγ =980 MeV θcm (o ) Eγ =1027 MeV θcm (o ) Eγ =1074 MeV θcm (o ) Eγ =1122 MeV 0.349 ± 0.150 30.6 0.425 ± 0.108 31.2 0.502 ± 0.103 32.4 0.570 ± 0.154 0.320 ± 0.255 57.5 0.408 ± 0.225 57.5 0.202 ± 0.140 57.6 0.179 ± 0.161 -0.094 ± 0.262 81.7 -0.085 ± 0.189 81.0 -0.190 ± 0.120 80.6 -0.365 ± 0.282 -0.464 ± 0.320 99.8 -0.477 ± 0.133 99.7 -0.552 ± 0.106 99.2 -0.522 ± 0.165 -0.490 ± 0.244 119.0 -0.723 ± 0.142 119.3 -0.621 ± 0.105 119.9 -0.795 ± 0.146 -1.028 ± 0.410 139.5 -0.304 ± 0.259 138.8 -0.163 ± 0.200 139.3 -0.341 ± 0.252 Eγ =1171 MeV θcm (o ) Eγ =1222 MeV θcm (o ) Eγ =1272 MeV θcm (o ) Eγ =1321 MeV 0.567 ± 0.122 34.6 0.294 ± 0.179 35.8 0.526 ± 0.130 36.0 0.599 ± 0.161 0.306 ± 0.148 58.9 0.310 ± 0.139 59.2 0.345 ± 0.140 59.4 0.304 ± 0.155 -0.319 ± 0.254 80.4 -0.193 ± 0.124 80.6 0.028 ± 0.155 80.3 -0.073 ± 0.142 -0.510 ± 0.175 98.9 -0.548 ± 0.252 99.2 -0.232 ± 0.125 99.2 -0.383 ± 0.141 -0.347 ± 0.162 119.1 -0.615 ± 0.121 119.9 -0.395 ± 0.143 119.9 0.046 ± 0.114 -0.160 ± 0.234 140.3 0.116 ± 0.242 140.8 0.454 ± 0.187 141.3 0.323 ± 0.158 θcm (o ) Eγ =1372 MeV θcm (o ) Eγ =1421 MeV θcm (o ) Eγ =1466 MeV 36.1 0.552 ± 0.119 35.7 0.455 ± 0.144 35.9 0.307 ± 0.150 59.5 0.150 ± 0.130 59.6 -0.072 ± 0.160 59.3 0.172 ± 0.171 80.1 -0.168 ± 0.131 80.3 -0.303 ± 0.139 80.0 -0.270 ± 0.195 99.4 -0.276 ± 0.122 99.7 -0.190 ± 0.137 99.7 -0.096 ± 0.139 120.4 0.124 ± 0.138 120.4 0.076 ± 0.110 120.8 0.164 ± 0.145 141.9 0.636 ± 0.126 142.8 0.490 ± 0.205 143.7 0.905 ± 0.152 Table 2. Beam-recoil Oz values. θcm (o ) 31.3 59.1 80.7 99.8 118.9 138.6 θcm (o ) 34.1 58.9 80.5 99.3 119.4 140.4 Eγ =980 MeV θcm (o ) Eγ =1027 MeV θcm (o ) Eγ =1074 MeV θcm (o ) Eγ =1122 MeV 0.581 ± 0.194 30.6 0.333 ± 0.110 31.2 0.285 ± 0.080 32.4 0.274 ± 0.124 0.956 ± 0.242 57.5 0.951 ± 0.216 57.5 0.674 ± 0.112 57.6 0.687 ± 0.127 0.754 ± 0.186 81.7 0.995 ± 0.154 81.0 1.003 ± 0.148 80.6 0.888 ± 0.244 1.139 ± 0.237 99.8 0.949 ± 0.140 99.7 1.140 ± 0.130 99.2 0.950 ± 0.144 0.841 ± 0.215 119.0 0.744 ± 0.162 119.3 0.996 ± 0.156 119.9 0.618 ± 0.197 -0.091 ± 0.597 139.5 -0.287 ± 0.415 138.8 0.427 ± 0.223 139.3 -0.162 ± 0.568 Eγ =1171 MeV θcm (o ) Eγ =1222 MeV θcm (o ) Eγ =1272 MeV θcm (o ) Eγ =1321 MeV 0.398 ± 0.093 34.6 0.291 ± 0.177 35.8 0.532 ± 0.087 36.0 0.554 ± 0.090 0.914 ± 0.128 58.9 0.678 ± 0.167 59.2 0.710 ± 0.119 59.4 0.904 ± 0.108 0.825 ± 0.123 80.4 0.485 ± 0.109 80.6 0.867 ± 0.112 80.3 0.767 ± 0.153 0.964 ± 0.175 98.9 1.025 ± 0.143 99.2 0.676 ± 0.188 99.2 0.734 ± 0.161 0.550 ± 0.190 119.1 0.426 ± 0.166 119.9 0.677 ± 0.166 119.9 0.409 ± 0.229 -0.055 ± 0.286 140.3 -0.162 ± 0.276 140.8 0.349 ± 0.272 141.3 -0.448 ± 0.217 θcm (o ) Eγ =1372 MeV θcm (o ) Eγ =1421 MeV θcm (o ) Eγ =1466 MeV 36.1 0.600 ± 0.084 35.7 0.384 ± 0.094 35.9 0.354 ± 0.095 59.4 0.784 ± 0.119 59.6 0.558 ± 0.185 59.3 0.814 ± 0.222 80.1 0.484 ± 0.112 80.3 0.322 ± 0.195 80.0 0.666 ± 0.332 99.4 0.419 ± 0.120 99.7 0.289 ± 0.134 99.7 -0.023 ± 0.192 120.4 0.019 ± 0.145 120.4 -0.313 ± 0.131 120.8 -0.432 ± 0.180 141.9 -0.072 ± 0.159 142.8 -0.085 ± 0.172 143.7 -0.461 ± 0.162 Please give a shorter version with: \authorrunning and \titlerunning prior to \maketitle Table 3. Target asymmetry T values. θcm (o ) 31.3 59.1 80.7 99.8 118.9 138.6 θcm (o ) 34.1 58.9 80.5 99.3 119.4 140.4 Eγ =980 MeV θcm (o ) Eγ =1027 MeV θcm (o ) Eγ =1074 MeV θcm (o ) Eγ =1122 MeV -0.506 ± 0.156 30.6 -0.663 ± 0.112 31.2 -0.635 ± 0.096 32.4 -0.615 ± 0.118 -0.607 ± 0.206 57.5 -0.860 ± 0.190 57.5 -0.718 ± 0.120 57.6 -0.991 ± 0.237 -0.803 ± 0.185 81.7 -0.749 ± 0.139 81.0 -0.874 ± 0.141 80.6 -0.949 ± 0.239 -0.622 ± 0.166 99.8 -0.974 ± 0.121 99.7 -1.048 ± 0.140 99.2 -0.833 ± 0.153 -0.622 ± 0.187 119.0 -0.789 ± 0.133 119.3 -0.760 ± 0.102 119.9 -0.825 ± 0.165 -1.090 ± 0.341 139.5 -0.681 ± 0.359 138.8 -0.448 ± 0.203 139.3 -0.465 ± 0.296 Eγ =1171 MeV θcm (o ) Eγ =1222 MeV θcm (o ) Eγ =1272 MeV θcm (o ) Eγ =1321 MeV -0.715 ± 0.116 34.6 -0.858 ± 0.155 35.8 -0.773 ± 0.123 36.0 -1.064 ± 0.133 -0.869 ± 0.154 58.9 -0.874 ± 0.145 59.2 -0.827 ± 0.166 59.4 -0.910 ± 0.142 -0.850 ± 0.158 80.4 -0.871 ± 0.124 80.6 -0.979 ± 0.234 80.3 -0.716 ± 0.133 -0.659 ± 0.159 98.9 -0.690 ± 0.132 99.2 -0.707 ± 0.150 99.2 -0.576 ± 0.223 -0.669 ± 0.150 119.1 -0.675 ± 0.145 119.9 -0.125 ± 0.208 119.9 -0.281 ± 0.174 0.226 ± 0.316 140.3 -0.066 ± 0.249 140.8 0.482 ± 0.213 141.3 0.331 ± 0.198 θcm (o ) Eγ =1372 MeV θcm (o ) Eγ =1421 MeV θcm (o ) Eγ =1466 MeV 36.1 -0.983 ± 0.104 35.7 -0.753 ± 0.112 35.9 -0.632 ± 0.127 59.5 -0.695 ± 0.113 59.6 -0.687 ± 0.159 59.3 -0.648 ± 0.166 80.1 -0.669 ± 0.123 80.3 -0.564 ± 0.131 80.0 -0.553 ± 0.185 99.4 -0.482 ± 0.175 99.7 -0.025 ± 0.157 99.7 0.190 ± 0.196 120.4 -0.104 ± 0.135 120.4 0.160 ± 0.112 120.8 0.785 ± 0.195 141.9 0.629 ± 0.147 142.8 0.859 ± 0.140 143.7 0.933 ± 0.175 19