Eur. Phys. J. C 26, 371–388 (2003) THE EUROPEAN Digital Object Identifier (DOI) 10.1140/epjc/s2002-01080-7 PHYSICAL JOURNAL C Coupled channel analysis of π+π−π0, K+K−π0 and K±K0π∓ from p¯p annihilation S at rest in hydrogen targets at three densities The OBELIX Collaboration M. Bargiotti1, A. Bertin1, M. Bruschi1, M. Capponi1, S. De Castro1, L. Fabbri1, P. Faccioli1, D. Galli1, B. Giacobbe1, U. Marconi1, I. Massa1, M. Piccinini1, N. Semprini Cesari1, R. Spighi1, V. Vagnoni1, S. Vecchi1, M.Villa1,A.Vitale1,A.Zoccoli1,M.Poli2,A.Bianconi3,M.P.Bussa3,M.Corradini3,A.Donzella3,E.LodiRizzini3, L.Venturelli3,C.Cical`o4,A.DeFalco4,A.Masoni4,G.Puddu4,S.Serci4,G.Usai4,O.E.Gorchakov5,S.N.Prakhov5, A.M. Rozhdestvensky5, M.G. Sapozhnikov5, V.I. Tretyak5, P. Gianotti6, C. Guaraldo6, A. Lanaro6, V. Lucherini6, C. Petrascu6, R. A. Ricci7, V. Filippini8, A. Fontana8, P. Montagna8, A. Panzarasa9, A. Rotondi8, P. Salvini8, A. Zenoni3, F. Balestra10, L. Busso10, P. Cerello10, O. Denisov10, L. Ferrero10, R. Garfagnini10, A. Maggiora10, D. Panzieri10, F. Tosello10, E. Botta11, T. Bressani11, D. Calvo11, F. De Mori9, A. Feliciello11, A. Filippi11, N. Mirfakhrai11, S. Marcello12, M. Agnello12, F. Iazzi12 1 Dipartimento di Fisica dell’Universit`a di Bologna and INFN Sezione di Bologna, Bologna, Italy 2 Dipartimento di Energetica dell’Universit`a di Firenze, Firenze, Italy and INFN Sezione di Bologna, Bologna, Italy 3 Dipartimento di Chimica e Fisica per l’Ingegneria e per i Materiali, Universit`a di Brescia, Brescia, Italy and INFN gruppo associato di Brescia, Brescia, Italy 4 Dipartimento di Scienze Fisiche, Universit`a di Cagliari and INFN Sezione di Cagliari, Cagliari, Italy 5 Joint Institute for Nuclear Research, Dubna, Russia 6 Lab. Naz. di Frascati dell’INFN, Frascati, Italy 7 Lab. Naz. di Legnaro dell’INFN, Legnaro, Italy 8 Dipartimento di Fisica Nucleare e Teorica dell’Universit`a di Pavia and INFN Sezione di Pavia, Pavia, Italy 9 INFN Sezione di Pavia, Pavia, Italy 10 Dipartimento di Fisica Generale dell’Universit`a di Torino and INFN Sezione di Torino, Torino, Italy 11 Dipartimento di Fisica Sperimentale dell’Universit`a di Torino and INFN Sezione di Torino, Torino, Italy 12 Dipartimento di Fisica del Politecnico di Torino and INFN Sezione di Torino, Torino, Italy Received: 15 July 2002 / Published online: 9 December 2002 – (cid:1)c Springer-Verlag / Societ`a Italiana di Fisica 2002 Abstract. Theπ+π−π0,K+K−π0 andK±K0π∓ finalstatesproducedbyp¯pannihilationatrestatthree differenthydrogentargetdensitieshavebeenanalyzedintheframeofacoupledchannelanalysistogether with ππ, πK and KK¯ scattering data. The percentages of the different partial waves, the branching ratios and the parameters (masses, widths, ππ and KK¯ partial widths) of all the involved resonances (JP =0+,1−,2+)havebeenmeasured.Themainresultsonmesonspectroscopyconcernthedetermination of the Γ /Γ ratio for f (1370) and f (1500) (Γ /Γ = 0.91±0.20 and Γ /Γ = 0.25±0.03 KK¯ ππ 0 0 KK¯ ππ KK¯ ππ respectively), the determination of a (1300) parameters (M = 1303±16MeV; Γ = 92±16MeV) and 0 the observation of two different ρ signals associated to ρ(1450) and ρ(1700) (M = 1182±30MeV;Γ = 389±20MeV and M =1594±20MeV;Γ =259±20MeV respectively). 1 Introduction measured by the OBELIX experiment at three different hydrogen target densities. Necessarily this work involves, The object of the present work is the achievement of a at the same time, aspects of the atomic physics of the detailed experimental study of the following annihilation p¯p system in the hydrogen medium, of N¯N annihilation reactions at rest dynamicsandofmesonspectroscopy,onwhichthepresent paper is mainly focused. p¯p→ π+π−π0 From this point of view, due to the richness of the p¯p→ K+K−π0 (1) meson dynamics involved in the collected final states, dif- p¯p→K±K0π∓ ferent and relevant sectors of light meson spectra can be S 372 The OBELIX Collaboration: Coupled channel analysis of π+π−π0, K+K−π0 investigated. Many prominent questions in light meson samples. In fact, the atomic cascade models [2] and the spectroscopy can be traced back to the experimental de- two-meson branching ratios analyses [3,4] predict in this termination of ππ and KK¯ decay modes, which are stud- case a P-wave contribution of the order of 10% at least. ied here in the frame of a coupled channel analysis. The In these conditions the insertion of P-wave amplitudes, flavourcontentoftheexoticcandidatef (1500),theover- although necessary, is very difficult. In fact, the complex 0 all structure of the JPC = 0++ nonet, f and ρ isobars P-wave annihilation dynamics cannot be disentangled by 2 are in fact deeply related to this issue. means of the small contribution expected in liquid hydro- The appealing features of three-meson p¯p annihilation gen targets. Sets of measurements with different target at rest determine to some extent the complexity of the densities are necessary. analysis procedure. In fact, many resonant states coupled According to these considerations, we collected data to different hadronic channels can be produced from dif- on each final state at three different hydrogen target den- ferentisospincomponentsandpartialwavesofthep¯psys- sities: liquid hydrogen (LH) dominated byS-wave annihi- tem.Theseconsiderationsledustodevelopanarticulated lation, low density hydrogen (corresponding to a pressure approach in order to control the different features of the of 5mbar, LP) dominated by P-wave annihilation, and problem. In order to disentangle within the complicated hydrogen at standard conditions of density and pressure K+K−π0 dynamics the isoscalar KK¯ contribution, also (NP),wherecomparablecontributionsfromSandP-wave K±K0π∓ data are included. To get a better control on annihilation are expected. As explained in Sect.3, the in- the diSfficult JP = 0+ nonet, ππ, KK¯ and πK scattering spection of the experimental data gives a direct insight of datawereincludedintheanalysis.Areliableidentification the advantages of this technique, which represents a de- of the different partial waves involved in p¯p annihilation cisive progress to get a detailed understanding of S and at rest was obtained by means of an original technique P-wave final state dynamics. developedbyOBELIX:threedifferentdensitiesofthehy- Asitisknown,foreachp¯ppartialwave,thetwoisospin drogentargetforeachoneofthefinalstatesstudiedallow components, with opposite G-parity, represent two differ- for a detailed investigation of S and P-wave annihilation. entsources.Nocomplicationsariseinthecaseofπ+π−π0 Finally, the whole set of experimental data was analyzed finalstateduetothefactthatonlyG=−1p¯psourcesare within the frame of a coherent formalism in order to take involvedandonlyG=+1resonancescanbeproduced.On into account the isospin and unitarity constraints which the contrary both p¯p sources can produce the K+K−π0 relate the different hadronic channels. final state and, moreover, G = ±1 as well as undefined G-parity resonances control the final state dynamics. The case of K∗(892), which is produced by ten different p¯p 2 General approach sources,isremarkable.Thesecircumstancesmakethedis- entanglingofK+K−π0dynamicsahardtaskandledusto The p¯p annihilation at rest offers a natural way of limit- consider a different charge combination of the same final ing the number of partial waves involved in the process, state,i.e.K±K0π∓.AsexplainedinSect.4,noadditional which is one of the critical problems in spin-parity anal- parameters are needed to describe its dynamics; K∗(892) yses. The mechanisms which regulate the formation and areproducedinacompletelydifferentinterferencepattern the deexcitation of the p¯p atom in the hydrogen medium and, above all, the I =1 KK¯ dynamics can be studied in canaccountforthisfact[1].Aftertheslowing-downofone asituationwheretheI =0componentisabsent.Thisfact p¯in the hydrogen medium and the formation of a highly turns out to be crucial in extracting the I = 0 KK¯ com- excitedp¯patom(n∼30,l∼20),thedeexcitationprocess ponent from the K+K−π0 final state. The experimental takes place. The competition, depending on the hydrogen data, as explained in Sect.3, clearly show the advantage density, between radiative and Stark transitions controls of this approach. the final part of this process. In high density targets the Moreover, in order to obtain a coherent description compact p¯p atom are exposed to the electric field of the of the experimental information available on the difficult surrounding H2 molecules. The induced Stark-mixing of JP = 0+ isobar, the JPC = 0++, IG = 0+, ππ → ππ, the different angular momentum eigenstates rapidly leads ππ → KK¯, and the JP = 0+, I = 1/2, Kπ → Kπ scat- to S-wave annihilation from high-n levels of the p¯p atom. tering data [5] were included in the present analysis. On the contrary, the minor role of this mechanism in low density targets allows radiative transitions to dominate. The chain of dipolar transitions populates the 2P level 3 Data selection from which P-wave annihilation takes place. Therefore, annihilation at rest is controlled by the incoherent super- position, given by the target density, of the l = 0 and The annihilation reactions in (1) were studied on data l = 1 angular momentum eigenstates of the p¯p atom cor- collected by the OBELIX experiment at the Low Energy responding to six different partial waves (1S , 3S , 1P , AntiprotonRing(LEAR)atCERN.Thedetaileddescrip- 0 1 1 3P , 3P and 3P ) in the spin-parity analysis. tion of the experimental apparatus can be found in [6]; 0 1 2 The possibility of assuming a pure S-wave contribu- here we recall only its main features that are necessary to tion has led many experiments to annihilate p¯ in liquid understand the present paper. Inside the magnetic field hydrogentargets.Thisofcourseisonlyanapproximation, provided by the Open Axial Field Magnet (0.5T max) whichturnsouttobeunacceptableforhighstatisticsdata four subdetectors are placed: The OBELIX Collaboration: Coupled channel analysis of π+π−π0, K+K−π0 373 – The Spiral Projection Chamber (SPC) acting as ver- has been performed on selected data where these in- tex detector. Due to the large size of the targets this formation were stable and reliable. Particular atten- detector was removed in LP and LH data taking; tion was dedicated to identify the kaon selection cut – The time of flight system (TOF), consisting of two in the dE/dx versus |p| scatter-plot (Fig.1a) in order scintillator slab arrays placed around the SPC coaxi- to evaluate directly, on the data, the efficiency of the allytothebeam.Besidestimeofflightmeasurements, cut and the pion contamination. The scatter-plot was specific multiplicities and topologies of charged parti- dividedinslicescorrespondingtodifferentmomentum cles can be selected by the TOF (at trigger level); intervals 0.05 GeV/c wide. For each slice, the dE/dx – The Jet Drift Chamber (JDC), a cylindric detector distribution was fitted with two Landau-like functions placed inside the TOF arrays for the tracking and the to separate the pionic from the kaonic contribution identification of charged particles by means of dE/dx (the proton contamination was negligible). The posi- measurement; tionofthecutineachslicewasfixedtooptimizeahigh – The High Angular Resolution Gamma Detector ((cid:29) > 0.8) regular efficiency on kaon selection with a K (HARGD),asystemoffoursupermodulesfortheiden- low(<2%)pioncontamination(Fig.1b,c).Theresult- tification and the measurement of the neutral annihi- ing selection cut for kaons corresponding to the NP lation products (γs from π0 decays). sample is shown in Fig.1a. The β measurement was used as an independent check on kaon identification; EachannihilationchannelwasstudiedattheLH,NPand – a 1C kinematic fit to test the compatibility of the se- LP hydrogen target densities. To optimize the p¯stopping lectedeventstotheK+K−π0 finalstatehypothesis.A in the target center at each density, beam momenta of selectioncutonχ2 correspondingtoaconfidencelevel 200MeV/c (LH) and 105 MeV/c (NP, LP) and materials of 80% was required. of suitable thickness were used. The on-line selection of the annihilation reactions in The events selected contain a fraction of residual back- (1) was based on two main trigger setups: ground depending on the considered channel and on the – two prong multiplicity trigger (two hit slabs in the in- targetdensity.Inthecaseoftheπ+π−π0channelthemain ner and outer part of the detector in a suitable time backgrounds, from the π+π−2π0 and π+π−3π0 channels gate after an incoming p¯), used to select the chan- (about 10%), are subtracted by means of an accurate nels π+π−π0 and K+K−π0. The collected statistics procedure described in [9]. Concerning the K+K−π0 fi- amount to 10.4 (LH), 6.7 (NP) and 9.4 (LP) ×106 nal state, the main background contributions come from events. the channels K+K−nπ0 (rejected by kinematic fit to less – four prong multiplicity trigger, used to select the than 2%), K±π∓X, where one pion is misidentified as charged particles of the channel K±K0π∓ in the a kaon, and π+π−X, where both pions are confused as S K±π∓π+π− final state (through the K0 decay into kaons (about 1−2%). Their amount and shapes on the S π+π−). The data sample was enriched by means of Dalitz-plots were calculated directly on the experimental a second level trigger request of a “slow” particle (a data by taking advantage of the previously explained se- particle with a time-of-flight greater than 8 ns). The lection based on the dE/dx. Concerning the K±K0π∓ S statisticsamountto17(LH),24(NP)and6(LP)×106 channel, its exclusive reconstruction (all the particles in events. the final state are measured) and the 5C kinematic fit al- lows to reduce the background sources (mainly due to 5π Once the events were reconstructed off-line by means of final states) to less than 0.5% [8]. The following statistics the general reconstruction code, the different reactions of events survive after the described cuts and the back- were selected by means of quality cuts and kinematic fits. ground subtraction (in units of 103 events): π+π−π0, 808 Asfarastheselectionoftheπ+π−π0 andK±K0π∓ chan- S (LH), 420 (NP), 260 (LP); K+K−π0, 20 (LH), 23 (NP), nels is concerned, we refer to already published results [7, 25 (LP); K±K0π∓, 10.6 (LH), 27 (NP), 3 (LP). 8]. Here we describe only the K+K−π0 selection, which S The Dalitz-plots corresponding to the reactions in (1) was performed through the following steps: are shown in Fig.2. The X and Y axis are associated – two reconstructed, long tracks (in the bending plane respectively to π+π− and π+π0 (bin size 0.032 × 0.032 L >30cm)ofoppositechargeintheJDC,originated GeV2), K+π0 and K−π0 (bin size 0.045×0.045 GeV2), xy from a vertex inside a fiducial volume contained into K±π∓ andK0π∓ (binsize0.045×0.045GeV2 inLHand the target; NP, bin size 0.090×0.090 GeV2 in LP). In all the rep- – an angle θ between the charged particles momenta resented Dalitz-plots the background was subtracted but +− whichsatisfiestherelationcosθ >−0.998.Thiscut apparatus efficiency modulations have not been removed; +− rejects, almost totally, π+π− and K+K− background moreover the Dalitz-plots corresponding to the first two reactions; reactions(see(1))havebeensymmetrizedwithrespectto – charged particle identification. The identification of the charge exchange. To each bin of the Dalitz-plots an charged kaons, crucial in the K+K−π0 final state se- error due to background subtraction and statistic fluctu- lection,hasbeenperformedbymeansofthemeasured ations was also associated. energy loss of the tracks in the JDC (dE/dx) and the At first glance the dynamics of the π+π−π0 channel velocity β of the particle (extrapolated to the vertex) seemstobedominatedbyρ(770)andf (1270)resonances 2 measured by the TOF. For this reason, the analysis while K∗(892) prevails over the K+K−π0 and K±K0π∓ S 374 The OBELIX Collaboration: Coupled channel analysis of π+π−π0, K+K−π0 Fig. 1a–c. dE/dx versus |p| scatter plot and selection cut for kaons a; fit of dE/dx distributions corresponding to a fixed |p| performed by means of Landau-like distributions b; efficiency of the kaon selection cut c final states. By looking at the figures from left to right urations,responsesandefficiencies,triggersetupandper- (with decreasing hydrogen density of the target) the de- formances, etc.) for each analyzed data sample. In order crease of ρ0(770) and the enhancement of f (1270) point to have an accurate representation of the efficiency and 2 out their dominant production respectively from S and resolution effects the simulated events were submitted to P-waves of the p¯p system (Fig.2 (1-a,b,c)). A dominant the same selection chain used for experimental data. S-wave production is clearly shown also in the case of φ(1020) (Fig.2 (2-a,b,c)) and a (980) (Fig.2 (3-a,b,c)), 0 while P-wave controls the f(cid:5)(1525) production (Fig.2 (2- 4 The description of p¯p annihilation at rest 2 a,b,c)). The strong concentration of events in the central regionbetweentheK∗(892)bandsintheK+K−π0,which The sequence of processes which take place in p¯p annihi- is absent in the K±K0π∓ final state, is due to isoscalar lation at rest in liquid or gaseous hydrogen are sketched S resonances. in Fig.3. Anessentialissueisthetreatmentoftheapparatusef- After the slowing down in hydrogen, the p¯undergoes ficiencyandresolution,whichmodulatetheshapeandthe atomic capture giving place to a highly excitated proto- widthoftheresonances.Intheanalyseswherethebinsize nium atom. The mechanisms discussed in Sect.2 deter- can be chosen to contain the resolution effects, the exper- mines the percentages of atoms with angular momenta imental Dalitz-plot, suitably corrected for the apparatus l = 0,1 and hence the percentages Wk(ρt) of the hyper- efficiency,candirectlybecomparedtothetheoreticalam- fine levels involved in p¯p annihilation at rest: 1S0, 3S1, plitude.Inourcasethehighstatisticsofthecollecteddata 1P1, 3P0 (forbidden for annihilation in three pseudoscalar samples suggest a bin size comparable with resolution ef- mesons), 3P1 and 3P2. All the atomic physics involved in fects so that the experimental data cannot be corrected. our problem is described by these parameters. The de- For this reason we decided to compare the binned exper- crease of centrifugal barriers effects in l = 0,1 angular imental Dalitz-plot to the corresponding binned theoreti- momentum states leads the system to annihilation. It is calDalitz-plotmodulatedbytheapparatusefficiencyand usuallybelievedthatthisprocesscande-coupletheinitial resolution (cid:1) from the final state. In this way the observed dynamics DTh(cid:1) = (cid:29) DTh (2) is entirely due to the reciprocal interactions of the pro- p(cid:1)q(cid:1) p(cid:1)q(cid:1),pq pq duced mesons (Final State Interaction hypothesis) [10]. pq The structure of these hadronic processes (discussed in In this expression DTh represents the cell pq of a generic detail in the following) are represented by means of a se- pq theoretical Dalitz-plot, (cid:29) is the probability that an riesofpoles.Eachpoleiscoupledtothep¯psystembythe p(cid:1)q(cid:1),pq event generated in the bin pq is placed in the bin p(cid:5)q(cid:5) and so called production parameters βk and can decay into α DTh(cid:1) is the transformed theoretical Dalitz-plot. The nu- stable mesonic couples j via the decay constant g (for jk αj merical values of (cid:29) were obtained by means of a de- convenience in the following j will also be used to indi- p(cid:1)q(cid:1),pq tailed Monte Carlo simulation of the whole experimental cate the whole final state). According to the Isobar Model setup (beam distribution, target density, detector config- hypothesis [11], the third stable meson does not take part The OBELIX Collaboration: Coupled channel analysis of π+π−π0, K+K−π0 375 Fig. 2. Dalitz-plots (background subtracted) of the annihilation reactions p¯p → π+π−π0 (first row), p¯p → K+K−π0 (second row) and p¯p→K±K0π∓ (third row) in liquid a, NP gas b and low pressure gas c hydrogen targets. The details are discussed S in the text. in the final state process dynamics, limiting its contribu- fk(ρ ) = N (ρ )Wk(ρ )|βˆk|2 (4) j t j t t tion to the initial state dynamics. The expected number of events in the bin pq of the Dalitz-plot corresponding to These parameters are fitted to the experimental data to- the final state j can be written as getherwiththeproductionparametersβk,thepolemasses α m and the decay constants g . Several constraints limit N(cid:1)pw thαe number of independent αfjk(ρ ). By comparing the DTh(ρ ) = fk(ρ )|Fk |2 (3) j t pqj t j t pqj fk(ρ ) corresponding to different final states we get the k=1 j t ratio where N is the number of partial waves and Fk are the partipawl wave amplitudes. The coefficients fjk(pρqt)j de- ffjkk((ρρt)) = NNj((ρρt)) = Cjl(ρt) (5) pend on the number N (ρ ) of events in each channel, on l t l t j t the hyperfine level percentages Wk(ρt) and, since we nor- which reduces the number of independent fjk(ρt) (in our malize the production parameters to a given resonance in case3finalstatesat3densities,eachwith5partialwaves, ˆ eachpartialwave,ontheabsolutenormalizationβk ofthe correspond to 45 parameters. They reduce to 3×5+3+3 production parameters: independent values if constraints are applied). 376 The OBELIX Collaboration: Coupled channel analysis of π+π−π0, K+K−π0 ✬✩ j, the following formula is used (cid:3) pp¯(ρ ) t (m +m )2 (m −m )2 ✫❅✪Wk(ρt) ma ρjj(m)= (1− jam2 jb )(1− jam2 jb ) (8) ❅ √ ❅ ✬✩βαk gαj ✟✟ where mja and mjb are meson masses and m = s is ❅ ✟✯✟ the invariant mass. In the case of the 4π decay mode, ✟ ✛ pp¯k ✉ ✟✉ j dominated by ρρ, σσ or ρσ intermediate states, we use ❍ ❍❍❥❍ the expression of [15]. Below each physical threshold, we ✫✪ ❍ ❍ use the analytic continuation of ρ in order to take into m m jj c α account the analytic properties of the amplitude. m The elements of the Pk vectors and of the K matrix b are given by the expressions Fig. 3. Schematic representation of the processes involved in p¯p annihilation at rest (cid:1)Np βkg B Pk = α αj αj +dk (9) j m2 − m2 j α=1 α The partial wave amplitude (labelled in this formula (cid:4) (cid:5) bymeansofallquantumnumbers)intothefinalstatej is (cid:1)Np g g B B K = αj αl αj αl +c Z written as the sum of two-body reaction terms extended jl m2 − m2 jl to all the possible couples of final mesons α=1 α (cid:2) where the invariant mass function Z = (2m2−m2)/2m2 Fj2S+1LJ,II3 = ab C IIIa3bIab,IcIc is introduced in the ππ and Kπ S-wave scatteringπampli- 3 3 Z JLJab3Lab,LcLc(pˆ,qˆ(cid:5))BLc(|p|) tsuetdetso[o1n5]e.toBacc=ounBt for(|Aq(cid:5)d|)l/erBzero(|q[1(cid:5)6|]), ealrseewthheereBliatttis- 3 3 αj Lab Lab α F Lab,IabI3ab(|q(cid:5)|) (6) Weisskopf centrifugal barriers normalized to the unit on j the resonance breakup momentum q(cid:5) [14]. α where Iab, Lab, Ic and Lc are the isospin and the angular According to the final state interaction and isobar momentum of the isobar and the spectator respectively; model, the production parameters βk are real constants α I, I , J and J the isospin and the angular momentum [17], nevertheless, to include possible violations of this 3 3 (with the respective third components) of p¯p system. The scheme, complex values are allowed [18]. The couplings sum runs over all the quantum numbers allowed by the g of the pole α to the final state j (assumed real in or- αj selectionrulesoftheprocessi.e.angularmomentum,par- dertosatisfytheunitarityofT operator)areexpressedby ity, isospin, G-parity and strangeness (see Tables 5–9 in real numbers in the case of JPC = 0++,IG = 0+ isobar appendix). The involved particles are grouped in isospin while the following formula is adopted in all other cases: multiplets and CII3 represent the Clebsh-Gordan (cid:3) m γ coefficients associIaatbeId3abt,IocIt3che isospin projection of the ini- gαj = ρ α(mαj) (10) jj α tial state on the intermediate state and have to be calcu- lated consistently in order to represent correctly the in- where ρ (m ) is the final state density calculated at the jj α terference effects especially in the kaonic final states (see pole mass value m (see (8)). The background parame- α Tables 5–9). Angular momentum states are described by ters used for the JPC = 0++,IG = 0+ isobar, have the normalized Zemach tensors [13] constructed by means of following expressions [15]: thespectatormomentumversorpˆ inthelaboratoryframe 1+s 1+s (torepresentresonance-spectatorrelativeangularmomen- dk = φk 0 c = f 0 (11) j j s +m2 jl jl s +m2 tum)andthedecaymomentumversorqˆ(cid:5) intheresonance 0 0 frame (to represent the resonance spin). Projections are where φk, s and f are free parameters. In the case of j 0 jl calculatedaccordingtotherulesoftensorialcalculus.The JP = 0+,IG = 1/2 isobar we used expression given by useofnormalizedZemachtensorsisjustifiedbytheneces- the scattering length approximation. sity of having an accurate centrifugal barrier description. Assaidpreviously,theK-matrixandP-vectorapproach Here we assume the Blatt-Weisskopf centrifugal barriers allows a unified description of production and scatter- [14]. ing processes so that, independently on the process used FLab,IabI3ab(|q(cid:5)|) represents the relativistic production to produce the resonances, all the data can be analyzed j amplitude of the annihilation process, its expression in together. In the present analysis, scattering data are in- the frame of the K-matrix and P-vector [20] approach is cluded in the case of JPC = 0++,IG = 0+ (ππ and KK¯ given by the equation (the upper indices Lab,IabIab will scattering) and JP =0+,I =1/2 isobars (Kπ scattering) 3 be substituted by k in the following) [5] (see Fig.5). Phases and inelasticities can be obtained by the matrix elements of the non-relativistic transition Fk = Pk(1−iρK)−1 T = K(1−iρK)−1 (7) operatorTNR =ρT ρ(7)bymeansofthefollowingequa- tion wherethediagonalmatrixρdescribesthedensitiesoffinal state.Ifpairsofstablemesonsareproducedinthechannel Sij = δij +2iTiNjR = ηije2iθij (12) The OBELIX Collaboration: Coupled channel analysis of π+π−π0, K+K−π0 377 5 Fit results f (1370) and a relatively narrow f (1500). In the 0 0 presentanalysis,assuggestedby[15],weintroduceda The fit of the theoretical functions (2) and (12) to the ex- fourth pole which improves the data description and perimentaldataisperformedbymeansoftheleast-square produces the splitting of the broad f (1370) into a 0 method. The expression for the χ2 function is the follow- broad f (400−1200) and a relatively narrow f (1370). 0 0 ing: A five pole amplitude, including also the f (1710), 0 seems not to be required by the data. Due to the ab- (cid:2) (cid:1) (DTh(cid:1) − DExp)2 χ2 = λ pqα pqα sence of relative angular momentum between the de- α α ((cid:29)DTh(cid:1))2 + ((cid:29)DExp)2 caying mesons, the threshold effects in this isobar are pq pqα pqα particularly strong. For this reason ππ, KK¯, ηη and (cid:2) (cid:1) (δTh − δExp)2 + λ pα pα (13) 4π thresholds have been included separately. α α ((cid:29)δExp)2 – K∗ isobar. Besides the annihilation data samples, the p pα 0 pole structure is defined by Kπ scattering data which In this formula α runs over all nine experimental Dalitz- require the contribution of the non-resonant back- plots in the first sum and over the set of scattering data ground and of the K∗(1430) pole. Only the Kπ de- 0 includedinthepresentanalysisinthesecondsum;pq run cay channel was considered. overtheexperimentalDalitz-plotcellsinthefirstsumand – a isobar.The low K±K0 invariant mass region is the p runs over the experimental scattering points in the sec- m0ostevidentproofoftheSexistenceofa (980).KK¯ and ond sum; DTh(cid:1) and (cid:29)DTh(cid:1), DExp and (cid:29)DExp, δTh and ηπdecaymodeswereintroducedtoacc0ountforitspro- pqα pqα pqα pqα pα (cid:29)δTh, δExp and (cid:29)δExp represent the values of the theo- ductionnearbyKK¯ threshold.Asecondpolea (1300) pα pα pα 0 retical functions and the content of the experimental bins is required especially by K±K0π∓ data in the steep S withtheerrorsforannihilationandscatteringdatarespec- slopeofa (1320)signal.Differentmassvalueshasbeen 2 tively;thefactorsλ areusedtoequalizethecontribution considered but they do not produce any significative α of each fitted data sample to the χ2 function. χ2 improvement.OnlytheKK¯ decaychannel,directly To define the theoretical amplitude the isobars, the measurable in the present analysis, was considered. number of poles in each isobar and their decay channels – φ isobar. Besides the φ(1020) pole visible in the low need to be defined. As said in Sect.2, the experimental energy region of K+K− invariant mass we tested also conditions limit the orbital angular momentum values of the φ(1680) pole: this turns out to be not necessary the p¯p system to l = 0,1 so that the following initial to fit the data and hence has been removed. Only the states are allowed: 1S , 3S , 1P , 3P , 3P , each one with KK¯ decay channel was included. 0 1 1 1 2 isospin I = 0,1 (in three pseudoscalar meson final states – K∗ isobar. Besides the K∗(892) pole which dominates 3P0 is ruled out by P conservation). Concerning the reso- K1K¯π dynamics we test1ed also the K∗(1410) pole 1 nancequantumnumbers,weincludeonlytheI =0,1/2,1 which is not necessary to fit the data and hence has isospins.TheweakI =2signalobservedbyObelixinthe been removed. Only the Kπ decay channel was in- annihilation reaction n¯p → π+π+π− in π+π+ invariant cluded. mass [31] is not considered since, in our case, π+π− and – ρisobar.Apartfromthehugeρ(770)signal,weinclude π±π0 invariant masses are dominated by I = 0,1 contri- theρ(1450)polerequiredbyπ+π−π0 datasampleses- butions.Concerningthespinweareguidedbytheconsid- pecially in the low π+π− invariant mass region of the erationthatbelow1700MeVmass(limitfixedbyenergy- Dalitz-plots (this signal is responsible for the bending momentumconservation)mesondynamicsisalmostcom- of the ρ(770) shape). A third ρ(1700) pole is strongly pletelydominatedbyJP =0+,1−,and2+quantumnum- required by KK¯π data samples to avoid strong sys- bers so that higher value spins are not considered. Since tematic deviations in the high K+K− and K±K0 in- centrifugalbarriereffectssuppresshighangularmomenta, variant mass region (spin 0 and 2 were rejected by χ2 the limit L ≤ 2 is assumed on the resonance-spectator test). The couplings included were ππ, KK¯ and 4π. relativeorbitalangularmomentum.Theseconsiderations, – f isobar. The f (1270) and f(cid:5)(1525) poles are clearly together with the systematic application of the selection 2 2 2 visible in the Dalitz-plots of π+π−π0 and K+K−π0 rules,leadtothedetailedtablesshowninappendixwhere finalstates.Anadditionalf (1565)poleisrequiredby thespin,therelativeangularmomentumandtheinterfer- 2 NP and LP π+π−π0 data samples mainly from 3P encepatternsofthepossibleresonancesarelistedforeach 2 partial wave annihilation. In order to introduce in the p¯p initial state. analysis of this controversial signal only measurable As far as the poles included in the involved isobars parameters we decide to open only ππ and KK¯ decay are concerned the detailed structure is discussed in the modes. following and reported in Table 1. – K∗ isobar. The K∗(1430) pole was not included be- 2 2 – f isobar.Thecontributionoff (980)poleplusanon- cause it is out of the available Kπ kinematic region. 0 0 resonant background is clearly required by the direct – a isobar.Onlythea (1320)poleisnecessary.Onlythe 2 2 inspection of the θ (ππ → ππ) phase shift. Annihila- KK¯ channel is included since the dominant ρπ and 0 tion data require clearly at least two additional poles. ηπ decays do not introduce visible thresholds effects In this configuration, adopted also in our previous to the KK¯ mode, neither reduce the importance or analysis of the π+π−π0 final state [7], we get a broad change the parameters of a (1300). 0 378 The OBELIX Collaboration: Coupled channel analysis of π+π−π0, K+K−π0 Table1.K-matrixandT-matrixparametersofthepolescorrespondingtothebestfitsolution.Allvaluesexceptf areexpressed in MeV units. Errors accounts for statistical and systematical deviations. Values without errors are assumed fixed. (*) Decay channels whose experimental data are not included in the analysis. (a) Values not corrected for the real KK¯ phase space Resonance K-matrix Parameters T-matrix Parameters Mass Coupling M Γ Partial Width f (980) 422±35 g 1298±20 984±15 29±14 Γ 21±7 0 ππ ππ g −1101±25 Γ 5.4±2 KK¯ KK¯ g −162±80 Γ 2.2±1∗ ηη ηη g 0 Γ 0.1±0.04∗ 4π 4π f (400−1200) 1217±25 g 931±10 1597±30 726±40 Γ 602±30 0 ππ ππ g 584±20 Γ 120±20 KK¯ KK¯ g 95±60 Γ 0.4±0.2∗ ηη ηη g 0 Γ 3.5±1.0∗ 4π 4π f (1370) 1353±17 g 65±6 1373±15 274±20 Γ 10.8±2 0 ππ ππ g 41±6 Γ 9.8±2 KK¯ KK¯ g 504±150 Γ 107±10∗ ηη ηη g 406±25 Γ 146±12∗ 4π 4π f (1500) 1573±10 g 487±30 1484±10 125±12 Γ 35.8±4 0 ππ ππ g 44±8 Γ 9.0±2 KK¯ KK¯ g 278±40 Γ 26.6±4∗ ηη ηη g 367±30 Γ 54±8∗ 4π 4π f 0.61±0.07 11 f −0.86±0.04 12 s 1.68±0.07 0 K∗(1430) 1327±10 γ 429±5 1436±30 288±35 Γ 288±35 0 πK πK f 1.48±0.34 s 3.7±0.7 0 a (980) 1000±6 γ 26±3 998±10 72±15 Γ 26±6a 0 KK¯ KK¯ γ 44±6 Γ 46±8∗ πη πη a (1300) 1311±14 γ 94±12 1303±16 92±16 Γ 91±15 0 KK¯ KK¯ γ 0 Γ 1±0.5∗ πη πη φ(1020) 1019±6 γ 3.7±0.5 1019±6 3.7±0.5 Γ 3.7±0.5 KK¯ KK¯ K∗±(892) 895±4 γ 53±1 895±2 53±4 Γ 53±1 1 πK πK K∗0(892) 899±4 γ 55±2 899±3 55±4 Γ 55±2 1 πK πK ρ±(770) 743±5 γ 131±9 754±5 132±10 Γ 132±10 ππ ππ γ 0 Γ 0 KK¯ KK¯ γ 0 Γ <0.02∗ 4π 4π ρ0(770) 738±5 γ 144±7 752±5 145±10 Γ 145±10 ππ ππ γ 0 Γ 0 KK¯ KK¯ γ 0 Γ <0.02∗ 4π 4π ρ(1450) 1158±35 γ 172±20 1182±30 389±20 Γ 187±14 ππ ππ γ <0.01 Γ <3 KK¯ KK¯ γ 202±20 Γ 209±14∗ 4π 4π ρ(1700) 1624±15 γ 10±2 1594±20 259±20 Γ 18±5 ππ ππ γ 60±5 Γ 55±12 KK¯ KK¯ γ 191±10 Γ 236±14∗ 4π 4π f (1270) 1221±9 γ 159±10 1251±7 192±10 Γ 165±9 2 ππ ππ γ 6.6±1.0 Γ 7.5±2 KK¯ KK¯ γ 20±5 Γ 19±9∗ 4π 4π f(cid:5)(1525) 1522±8 γ 0 1521±7 68±7 Γ <0.1 2 ππ ππ γ 68±8 Γ 68±8 KK¯ KK¯ γ 0 Γ <0.1∗ 4π 4π f (1565) 1520±20 γ 205±20 1489±15 204±20 Γ − 2 ππ ππ γ 0 Γ − KK¯ KK¯ γ 0 Γ − 4π 4π a (1320) 1319±10 γ 136±25 1319±10 136±25 Γ − 2 KK¯ KK¯ The OBELIX Collaboration: Coupled channel analysis of π+π−π0, K+K−π0 379 Table 2. BranchingratiosBR (ρ )andpartialwavepercentagesPk(ρ )oftheanalyzedp¯pannihilation j t j t reactions. The values correspond to the best fit solution, the errors account for statistic and systematic deviations Reaction ρ BR×10−3 1S 3S 1P 3P 3P t 0 1 1 1 2 LH 53.6±2.7[32] 17.6±1.0 66.7±1.5 <0.1+0.7 10.2±1.1 5.4±0.5 −0.1 pp¯→π+π−π0 NP 51.6±2.6[32] 9.5±0.7 35.9±1.0 19.2±1.3 23.2±1.5 12.2±1.2 LP 48.9±2.8[32] 4.3±1.0 16.2±1.3 27.9±2.0 33.8±1.9 17.8±2.0 LH 2.37±0.16[32] 38.0±1.8 37.7±1.9 <0.1+0.9 17.3±1.7 6.9±2.1 −0.1 pp¯→K+K−π0 NP 3.03±0.20[32] 17.3±1.4 17.1±0.8 18.9±2.5 33.4±1.8 13.3±4.0 LP 3.15±0.22[32] 7.0±1.1 7.0±0.4 24.8±3.5 43.8±3.0 17.4±5.0 LH 3.16±0.40[8] 35.3±1.7 34.4±1.9 0.2+1.7 19.1±2.2 11.0±2.0 −0.2 pp¯→K±K0π∓ NP 3.64±0.50[8] 12.3±1.0 11.9±0.7 31.8±3.2 27.9±2.5 16.1±2.8 S LP 4.32±0.60[8] 4.6±0.7 4.4±0.3 38.1±3.5 33.5±2.9 19.4±3.5 (cid:2) The best fit solution we present was obtained after a long Wk(ρ )|βˆk|2 |Fk |2 seriesofdifferentfitswithdifferenthypotheses,constraints Pjk(ρt) = (cid:2)Npw Wtl(ρ )|βˆl|2p(cid:2)q p|qFjl |2 and initial values. The solution found matches both the l=1 t pq pqj best χ2 criterion and some consistency checks we will dis- Wk(ρ )BRk cuss in the following section. In Fig.4 the neutral and = (cid:2) t j (14) charged ππ, KK¯ and Kπ invariant mass projections of Nl=p1wWl(ρt)BRjl the Dalitz-plots corresponding to the best fit solution are represented; the errors on the binned theoretical func- wherethepq sumsareperformedovertheDalitz-plotcells tions (shaded histograms) are summed to the experimen- of the final state j and partial wave k and BRk represent tal ones. j the corresponding hadronic branching ratios. It is easy to The reduced χ2/n.d.f. turns out to be: 1.34 (LH), verifythatthedenominatorrepresentsthebranchingratio 1.27 (NP) and 1.49 (LP) for the π+π−π0; 1.22 (LH), 1.06 BR (ρ ) of the final state j at the density ρ which can (NP)and1.17(LP)fortheK+K−π0 and0.94(LH),1.07 j t t be determined experimentally. (NP) and 0.76 (LP) for the K±K0π∓ final states with- S Theannihilationbranchingratiosoftheanalyzedfinal out any systematic deviations of χ2 visible on the Dalitz- statespreviouslymeasuredbyObelix[8,32]andthecorre- plots.Possibledeviationsintheinvariantmassprojections spondingpartialwavepercentagesobtainedbythepresent are due to the squared composition of the errors of the analysisarelistedinTable2.Asexpected,astrongmodu- Dalitz-plot bins. In Fig.5 are plotted phase shifts and in- lation of S and P-wave contributions due to the variation elasticities of the scattering data. The following reduced of hydrogen density is observed. Despite this, it is clear χ2/n.d.f. for phases and inelasticities are obtained: 0.92, that the P-wave contribution cannot be neglected in liq- 1.08 for ππ →ππ; 0.78, 1.58 for Kπ →Kπ and 1.13, 0.8 for ππ →KK¯ respectively. uid hydrogen targets as well as the S-wave in low density ones. This fact shows the intrinsic limit of the analyses The values of K-matrix masses and couplings, listed performed with LH data only: the P-wave contribution in the second and third column of Table 1, correspond to turns out to be too weak to be correctly determined al- the best fit solution. In the fourth and fifth column the though too strong to be neglected. physical state masses and widths are listed. The quoted The multiplication of BR (ρ ) and Pk(ρ ) gives the errors account for statistic and systematic deviations and j t j t are evaluated on a selected set of fits. The values quoted products Wk(ρt)BRjk from which we extract the density without errors are assumed to be fixed. The resonance dependence of the p¯p annihilation frequency W (ρ ) from k t values are usually obtained by looking for T-operator sin- each final state independently. In the case of 1S and 3S 0 1 gularities in the complex energy plane [19]. Here we used partial waves this dependence can be extracted also from a different method based on the diagonalization of T and p¯p → K±K0π∓π+π− spin-parity analysis [22] and from F operatorsdiscussedin[12]bywhichitispossibletoob- thebranchingratioofp¯p→K K [21],bothperformedat S L tain, besides the masses and total widths of the physical different hydrogen densities. The consistency among the states, also their partial widths. determinationsobtainedfromthedifferentfinalstatesand the agreement with the independent experimental deter- minationoftheη(1440)π+π− andK K branchingratios S L 6 Discussion of the results proves the reliability of the obtained partial wave decon- volution (Fig.6). The density dependence of W (ρ ) can k t 6.1 Partial wave percentages be extracted also from [3]. The two curves represented in Fig.6 interpolate five points given in the paper and limit Thep¯ppartialwavepercentagesareobtainedfrom(3)and theregionallowedbythedifferentmodelsconsidered.Re- (4) by means of the formula mindingthatourW (ρ )aredeterminedwithanarbitrary k t 380 The OBELIX Collaboration: Coupled channel analysis of π+π−π0, K+K−π0 2]V 30000 2]V 1000 2]V 600 e e e G G G [032 20000 [045 [045 400 v./0. v./0. 500 v./0. e e e 10000 200 0 0 0 2] 2] 2] 1500 V V V e 10000 e e G G G [2 [5 750 [5 3 4 4 1000 0 0 0 v./0. 5000 v./0. 500 v./0. e e e 500 250 0 0 0 2]V 6000 2]V 1000 2]V Ge Ge Ge 300 [2 [5 [9 v./0.03 4000 v./0.04 500 ev./0.0 200 e e 2000 100 0 0 0 0. 0.8 1.6 2.4 3.2 0.65 1.48 2.31 3.14 0.65 1.48 2.31 3.14 m2(π+π−) [GeV2] m2(K+K-) [GeV2] m2(K+/-K0) [GeV2] 2]V 30000 2]V 1500 2]V E e e G G G 1000 [2 [5 [5 03 20000 04 1000 04 0. 0. 0. ev./ ev./ ev./ 500 10000 500 0 0 0 2] 2] 2] V V V e e e G G G [2 10000 [5 1500 [5 2000 3 4 4 0 0 0 v./0. v./0. 1000 v./0. e 5000 e e 1000 500 0 0 0 2] 2] 2] V V V e e 2000 e G 7500 G G [2 [5 [9 400 3 4 0 v./0.0 5000 v./0.0 1000 ev./0. e e 200 2500 0 0 0 .0 0.8 1.6 2.4 3.2 .2 0.8 1.4 2. .2 0.8 1.4 2. m2(π+/−π0) [GeV2] m2(K+/-π0) [GeV2] m2(K+/-π−/+) [GeV2] Fig. 4. Theoretical (shaded histograms) and experimental (background subtracted) Dalitz-plot projections of the annihilation reactions p¯p→π+π−π0, p¯p→K+K−π0 and p¯p→K±K0π∓ in liquid (LH), normal pressure and temperature (NP) gas and S low pressure (LP) gas hydrogen targets. The errors on the binned theoretical functions are summed to the experimental ones