CLNS 11/2080 CLEO 11-06 Amplitude analyses of the decays χ → ηπ+π− and χ → η(cid:48)π+π− c1 c1 G. S. Adams,1 J. Napolitano,1 K. M. Ecklund,2 J. Insler,3 H. Muramatsu,3 C. S. Park,3 L. J. Pearson,3 E. H. Thorndike,3 S. Ricciardi,4 C. Thomas,4,5 M. Artuso,6 S. Blusk,6 R. Mountain,6 T. Skwarnicki,6 S. Stone,6 L. M. Zhang,6 G. Bonvicini,7 D. Cinabro,7 A. Lincoln,7 M. J. Smith,7 P. Zhou,7 J. Zhu,7 P. Naik,8 J. Rademacker,8 D. M. Asner,9,∗ K. W. Edwards,9 K. Randrianarivony,9 G. Tatishvili,9,∗ R. A. Briere,10 H. Vogel,10 P. U. E. Onyisi,11 J. L. Rosner,11 J. P. Alexander,12 D. G. Cassel,12 S. Das,12 R. Ehrlich,12 L. Gibbons,12 S. W. Gray,12 D. L. Hartill,12 B. K. Heltsley,12 D. L. Kreinick,12 V. E. Kuznetsov,12 J. R. Patterson,12 D. Peterson,12 D. Riley,12 A. Ryd,12 A. J. Sadoff,12 2 1 X. Shi,12 W. M. Sun,12 J. Yelton,13 P. Rubin,14 N. Lowrey,15 S. Mehrabyan,15 0 M. Selen,15 J. Wiss,15 J. Libby,16 M. Kornicer,17 R. E. Mitchell,17 M. R. Shepherd,17 2 A. Szczepaniak,17 D. Besson,18 T. K. Pedlar,19 D. Cronin-Hennessy,20 J. Hietala,20 n a S. Dobbs,21 Z. Metreveli,21 K. K. Seth,21 A. Tomaradze,21 T. Xiao,21 L. Martin,5 J A. Powell,5 G. Wilkinson,5 J. Y. Ge,22 D. H. Miller,22 I. P. J. Shipsey,22 and B. Xin22 9 ] (CLEO Collaboration) x e 1Rensselaer Polytechnic Institute, Troy, New York 12180, USA - p 2Rice University, Houston, Texas 77005, USA e h 3University of Rochester, Rochester, New York 14627, USA [ 4STFC Rutherford Appleton Laboratory, Chilton, 2 Didcot, Oxfordshire, OX11 0QX, United Kingdom v 3 5University of Oxford, Oxford OX1 3RH, United Kingdom 4 8 6Syracuse University, Syracuse, New York 13244, USA 5 7Wayne State University, Detroit, Michigan 48202, USA . 9 8University of Bristol, Bristol BS8 1TL, United Kingdom 0 1 9Carleton University, Ottawa, Ontario, Canada K1S 5B6 1 : 10Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA v i 11University of Chicago, Chicago, Illinois 60637, USA X 12Cornell University, Ithaca, New York 14853, USA r a 13University of Florida, Gainesville, Florida 32611, USA 14George Mason University, Fairfax, Virginia 22030, USA 15University of Illinois, Urbana-Champaign, Illinois 61801, USA 16Indian Institute of Technology Madras, Chennai, Tamil Nadu 600036, India 17Indiana University, Bloomington, Indiana 47405, USA 18University of Kansas, Lawrence, Kansas 66045, USA 19Luther College, Decorah, Iowa 52101, USA 20University of Minnesota, Minneapolis, Minnesota 55455, USA 21Northwestern University, Evanston, Illinois 60208, USA 22Purdue University, West Lafayette, Indiana 47907, USA (Dated: September 25, 2011) 1 Abstract Using a data sample of 2.59×107 ψ(2S) decays obtained with the CLEO-c detector, we perform amplitudeanalysesofthecomplementarydecaychainsψ(2S) → γχ ; χ → ηπ+π− andψ(2S) → c1 c1 γχ ; χ → η(cid:48)π+π−. WefindevidenceforanexoticP-waveη(cid:48)π amplitude, which, ifinterpretedas c1 c1 aresonance,wouldhaveparametersconsistentwiththeπ (1600)statereportedinotherproduction 1 mechanisms. We also make the first observation of the decay a (980) → η(cid:48)π and measure the ratio 0 of branching fractions B(a (980) → η(cid:48)π)/B(a (980) → ηπ) = 0.064 ± 0.014 ± 0.014. The ππ 0 0 spectrum produced with a recoiling η is compared to that with η(cid:48) recoil. ∗ Present address: Pacific Northwest National Laboratory, Richland, WA 99352 2 I. INTRODUCTION Hadronic charmonium decays, in which charm and anti-charm quarks annihilate into gluons, provide an excellent opportunity to study light mesons. The combination of well- defined initial states and the availability of a wide variety of final states allows for the strategic selection of reactions to isolate and study different light meson systems. The decays χ → ηπ+π− and χ → η(cid:48)π+π−, in particular, have two interesting characteristics. c1 c1 First, since the quark and SU(3) flavor of the η and η(cid:48) are relatively well-known, one could, in principle, model these decays using what is known about the OZI rule and SU(3) flavor symmetry and learn about the ππ isoscalar states recoiling against them. Such a technique has been proposed for other χ decay channels [1]. cJ Second, the decays χ → ηπ+π− and χ → η(cid:48)π+π− provide an opportunity to search c1 c1 for exotic JPC = 1−+ states in the ηπ and η(cid:48)π systems. In fact, the only two-body S-wave decays of the χ available in these channels necessarily have the ηπ or η(cid:48)π system in a c1 configuration with JPC = 1−+. Two exotic candidates, the π (1400) and the π (1600), have 1 1 been reported in other production mechanisms to have decays to ηπ and η(cid:48)π, respectively. The π (1400) has been reported primarily decaying to ηπ [2–5], while the π (1600) has been 1 1 reported to decay to η(cid:48)π [6, 7], b π [8, 9], f π [10], and ρπ [11]. 1 1 We present amplitude analyses of the processes ψ(2S) → γχ ; χ → ηπ+π− and c1 c1 ψ(2S) → γχ ; χ → η(cid:48)π+π− in which we study the various ηπ and ππ resonances pro- c1 c1 duced in the decays of the χ . In Section II, we describe the data-selection process that c1 results in a sample of χ decays with estimated backgrounds below 5%. In Section III, c1 we describe our construction of amplitudes using the helicity formalism. Here we assume that the χ → ηπ+π− and χ → η(cid:48)π+π− decays proceed through a sequence of two-body c1 c1 decays, where the intermediate states have well-defined quantum numbers. We pay special attention to the treatment of the ππ S-wave, which utilizes independent experimental data on ππ S-wave scattering [12]. We also describe our fitting procedure, based on the unbinned extended maximum-likelihood method. The highlights of our analysis, detailed in Section IV, are • evidence for a P-wave η(cid:48)π amplitude, which, when parametrized by an exotic JPC = 1−+ resonance, has properties consistent with those of the π (1600) reported in other 1 production mechanisms; • the first direct observation of the decay a (980) → η(cid:48)π, a measurement of the ratio of 0 branching fractions B(a (980) → η(cid:48)π)/B(a (980) → ηπ), and a characterization of the 0 0 a (980) lineshape; and 0 • the observation of qualitative differences in the ππ system when it is produced against the η or η(cid:48). Finally, in Section V, we evaluate and discuss systematic errors. II. DATA SELECTION We select candidate events of the form ψ(2S) → γχ ; χ → ηπ+π− and ψ(2S) → c1 c1 γχ ; χ → η(cid:48)π+π− using 25.9 × 106 ψ(2S) decays collected by the CLEO-c detector at c1 c1 the Cornell Electron Storage Ring. We reconstruct the η (η(cid:48)) in three (six) different decay 3 TABLE I. The decay modes of the η and η(cid:48) that are used to reconstruct ψ(2S) → γχ ; χ → c1 c1 ηπ+π− and ψ(2S) → γχ ; χ → η(cid:48)π+π−, the branching fractions B of each decay mode [14], and c1 c1 the final state topology reconstructed with the detector. η((cid:48)) Decay Mode B [%] Final State η → γγ 39.3±0.2 3γ1(π+π−) η → π+π−π0 22.7±0.3 3γ2(π+π−) η → π0π0π0 32.6±0.2 7γ1(π+π−) η(cid:48) → π+π−η; η → γγ 17.1±0.3 3γ2(π+π−) η(cid:48) → π+π−η; η → π+π−π0 9.9±0.2 3γ3(π+π−) η(cid:48) → π+π−η; η → π0π0π0 14.1±0.2 7γ2(π+π−) η(cid:48) → γπ+π− 29.3±0.6 2γ2(π+π−) η(cid:48) → π0π0η; η → γγ 8.5±0.3 7γ1(π+π−) η(cid:48) → π0π0η; η → π+π−π0 4.9±0.2 7γ2(π+π−) topologies comprising 94.6±0.7% (83.8±1.8%) of its total decay width (Table I). We then select the χ using the energy of the photon from ψ(2S) → γχ . c1 c1 Final state photons and charged pions are measured by the CLEO-c detector [13], which covers a solid angle of 93%. The detector has a 1 Tesla superconducting magnet enclosing two drift chambers and a ring imaging Cherenkov (RICH) system for tracking charged particles and particle identification. Enclosed inside the solenoid are also a barrel and two endcap CsI-crystal calorimeters. The energy resolution for 100 MeV (1 GeV) photons is 5.0% (2.2%), while the momentum resolution for charged tracks in the drift chambers is 0.6% at 1 GeV/c. Charged tracks are required to have momentum p > 18.4 MeV/c and originate within a cylindrical volume, with 20 cm length and 2 cm radius, centered around the interaction point. The π± candidates are then required to have ionization losses (dE/dx) within 3σ of those expected for charged pions. Photons reconstructed inside the calorimeters, with polar angles |cosθ| < 0.79 and 0.85 < |cosθ| < 0.93, are required to have energy E > 20 MeV and separation from charged tracks. Two-photon four-momenta are kinematically constrained to select the π0 → γγ and η → γγ candidates, with a requirement that the respective invariant mass is within 3σ of the π0 or η rest mass. Finally, the total four-momentum of all of the final state particles of a given topology is kinematically constrained to the initial ψ(2S) four-momentum and the χ2 of the resulting fit is required to satisfy χ2/d.o.f. < 5. If multiple combinations of tracks and showers within an event pass all of these selection requirements (which occurs for 1.6% of all selected events), only the combination with the best χ2/d.o.f. is accepted. To select the decays η → π+π−π0 and η → π0π0π0, the invariant mass of the three pions must be between 540 and 555 MeV/c2 (see Fig. 1). Similarly, to select the η(cid:48) in its various decay modes the invariant mass of its decay products must fall between 950 and 965 MeV/c2 (see Fig. 2). If multiple combinations of particles within an event can be used to form an η (occurring in 0.9% of selected events), one of these combinations is chosen randomly. If therearemultipleη(cid:48) candidates(occurringin1.4%ofselectedevents), theeventisdiscarded. Since the decays η(cid:48) → π+π−η; η → π0π0π0 and η(cid:48) → π0π0η; η → π+π−π0 share the same 4 FIG.1. Theinvariantmassofη candidatesafterselectingaχ candidateandapplyingbackground c1 suppressioncriteria. Candidatesin(a)areselectedbyrequiringindividualphotonpairsbenomore than 3σ from the nominal η mass; the arrows in (b) and (c) indicate the region used to select the η candidates. final state topology, no requirement is made on the internal η mass. Specific backgrounds are further suppressed based on studies using a Monte Carlo (MC) sample of inclusive ψ(2S) decays. For the ψ(2S) → γχ ; χ → ηπ+π− decay chain, the c1 c1 dominant backgrounds are due to ψ(2S) → ηJ/ψ and ψ(2S) → γχ ; χ → γJ/ψ, where c1 c1 J/ψ → µ+µ−. The first of these is suppressed by requiring the mass recoiling against the η be separated from the J/ψ mass by at least 20 MeV/c2. The second is only a background for the η → γγ mode and is similarly suppressed using the masses recoiling against the γγ combinations, which are required to be more than 35 MeV/c2 from the J/ψ mass. The largest backgrounds in the ψ(2S) → γχ ; χ → η(cid:48)π+π− decay chain occur in c1 c1 the η(cid:48) → γπ+π− mode. We suppress J/ψ backgrounds, as above, by requiring the masses recoiling against the γγ and π+π− systems to be more than 20 MeV/c2 away from the J/ψ mass. In addition, we treat π+π− combinations as µ+µ− and require their invariant masses be more than 15 MeV/c2 away from the J/ψ mass. There is also a substantial background from ψ(2S) → π02(π+π−), which we reduce by requiring that no two showers in an event be consistent with the π0 mass within 3σ. We also enhance the signal to background for the 5 FIG.2. Theinvariantmassofη(cid:48) candidatesafterselectingaχ candidateandapplyingbackground c1 suppression criteria. The arrows indicate the region used to select the η(cid:48) candidates. η(cid:48) → γπ+π− mode by making a loose requirement that the π+π− invariant mass be between 335 and 895 MeV/c2, which is motivated by the apparent ρ dominance in the π+π− system. One additional background for the ψ(2S) → γχ ; χ → η(cid:48)π+π− decay chain with c1 c1 η(cid:48) → γπ+π− is from ψ(2S) → γχ ; χ → 2(π+π−) where the radiated photon converts c0 c0 to an e+e− pair outside the tracking region. This is suppressed by requiring that the total energy of the two resulting showers is not consistent with the energy of the photon from ψ(2S) → γχ , i.e., not between 225 and 295 MeV, and that the cosine of the angle between c0 6 FIG.3. Theinvariantmassdistributionsofthe(a)ηπ+π− and(b)η(cid:48)π+π− candidatesfromselected ψ(2S) → γηπ+π− and ψ(2S) → γη(cid:48)π+π− decays, respectively, after all background suppression criteria have been applied. The solid arrows indicate the regions used to select the χ signals. c1 the two showers is less than 0.97. Figure 3 shows the invariant mass distributions of (a) the ηπ+π− and (b) the η(cid:48)π+π− candidates after combining all of the decay modes of the η and η(cid:48). We select the χ c1 by requiring that the energy of the photon radiated from the ψ(2S) be between 155 and 185 MeV (indicated by the arrows in Fig. 3). Our final data samples consist of 2498 and 698 events in the ψ(2S) → γχ ; χ → ηπ+π− and ψ(2S) → γχ ; χ → η(cid:48)π+π− decay c1 c1 c1 c1 chains, respectively. ThebackgroundisestimatedbyfittingthedatainFig.3usingareverse CrystalBallshape[15]todescribethesignal. Thebackgroundandχ peakaredescribedby c2 a second order polynomial and a double Gaussian, respectively. Peaking backgrounds have been subtracted by fitting the χ candidate mass distribution in η((cid:48)) mass sidebands; such c backgrounds are negligible in all cases except the η(cid:48) → γπ+π− decay mode. The estimated signal purity for the ηπ+π− (η(cid:48)π+π−) decay channel is 97.5% (94.6%) with an uncertainty of 0.3% (1.3%). III. AMPLITUDE ANALYSIS We perform amplitude analyses to disentangle the substructure present in the χ → c1 ηπ+π− and χ → η(cid:48)π+π− decays. We assume that the three-hadron decays of the χ c1 c1 proceed through a sequence of two-body decays, where one participant is the “isobar,” a bound state of either η((cid:48))π± or π+π− with total angular momentum J, and the other is a stable, non-interacting meson (the π∓ or η((cid:48))) produced with an orbital angular momentum L with respect to the isobar. All possible χ decays through isobars with J ≤ 4 are listed c1 in Table II. The general idea of an amplitude analysis is to fit the distribution of events observed with the detector to a coherent sum of physically-motivated amplitudes that describes the dynamics of the intermediate states. We can define I(x), the number of observed events per 7 TABLE II. A list of χ decay modes for all possible isobars with J ≤ 4. c1 χ Decay Mode L Isobar JPC c1 a π; a → η((cid:48))π P 0++ 0 0 π π; π → η((cid:48))π S,D 1−+ 1 1 a π; a → η((cid:48))π P,F 2++ 2 2 a π; a → η((cid:48))π F,H 4++ 4 4 f η((cid:48)); f → ππ P 0++ 0 0 f η((cid:48)); f → ππ P,F 2++ 2 2 f η((cid:48)); f → ππ F,H 4++ 4 4 unit phase space, as (cid:12) (cid:12)2 (cid:88) (cid:12)(cid:88) (cid:12) I(x) = (cid:12) Vα Aα (x)(cid:12) , (1) (cid:12) Mψ,λγ Mψ,λγ (cid:12) (cid:12) (cid:12) Mψ,λγ α where α indexes the χ decay amplitudes and M and λ index the polarization of the c1 ψ γ ψ(2S) and the helicity of the photon, respectively. We use x to denote a set of kinematic variables, e.g., angles and invariant masses, that provide a complete description of the event. The value of the decay amplitude at a location x in this multi-dimensional space is given by Aα (x). The real fit parameters Vα determine the relative strengths of each χ Mψ,λγ Mψ,λγ c1 decay amplitude. Section IIIA discusses the construction of the decay amplitudes used in the fit. Sec- tion IIIB discusses the application of the extended maximum likelihood technique to this analysis in order to determine the optimal values of Vα that describe the data. Mψ,λγ A. Amplitude construction 1. General amplitude structure The amplitude for a given χ decay mode α depends on the set of observed final state c1 event kinematics x, the assumed polarization of the initial state ψ(2S), denoted M , and ψ the helicity of the final state photon λ . The general form is given by γ (cid:88) (cid:88) Aα (x) = C(M ,λ ,λ ) D1∗ (φ ,θ ,0)× Mψ,λγ ψ γ χ Mχ(cid:48),−λχ γ γ λχ=±1,0 Mχ(cid:48)=±1,0 (cid:88) (cid:104)1M(cid:48)|LM(cid:48),JM(cid:48)(cid:105)YML(cid:48)∗(θ(cid:48),φ(cid:48))YMJ(cid:48)∗(θ(cid:48),φ(cid:48) )pLqJT (s), (2) χ L J L I I J h h α M(cid:48),M(cid:48) L J where summations in the second line are performed over all possible values M(cid:48) and M(cid:48), the L J projections of L and J, respectively. We briefly provide a term-by-term description of this expression. The first factor in Eq. (2), C(M ,λ ,λ ), is used to transform the helicity amplitude ψ γ χ for the radiative decay into the multipole basis. The ψ(2S) → γχ radiative transition c1 8 is dominated by the electric dipole (E1) transition, while the magnetic quadrupole (M2) transition contributes ≈ 3% [16] of the total rate. In our analysis, we use the E1 amplitude to derive our results and check the sensitivity of the results to the presence of a small M2 amplitude. The E1 or M2 amplitude can be constructed with the following choice of C: (cid:114) 3 C(M ,λ ,λ ) = D1 (φ ,θ ,0)× ψ γ χ 8π Mψ,λγ−λχ γ γ (cid:40)(cid:0) (cid:1) δ δ −δ δ +(δ −δ )δ for E1, or λγ,1 λχ,1 λγ,−1 λχ,−1 λγ,1 λγ,−1 λχ,0 (3) (cid:0) (cid:1) δ δ −δ δ −(δ −δ )δ for M2. λγ,1 λχ,1 λγ,−1 λχ,−1 λγ,1 λγ,−1 λχ,0 In order to describe the angular distribution of the final state particles we measure angles in two coordinate systems which are depicted in Fig. 4 and related in Eq. (2) by the D- function at the end of the first line. The angles θ and φ are the polar and azimuthal angle γ γ of the radiated photon in the ψ(2S) rest frame, where zˆ is given by the e+ beam direction and yˆis (arbitrarily) defined as upward in the laboratory. (The amplitude is uniform in φ .) γ The two spherical harmonics in the second line of Eq. (2) provide a description of the angular distribution for the initial χ decay and the subsequent isobar decay for various c1 values of isobar-hadron orbital angular momentum L and isobar angular momentum J. The angles θ(cid:48) and φ(cid:48) are the polar and azimuthal angles of the isobar in the χ -helicity frame, I I c1 defined as the rest frame of the χ with z(cid:48)-axis along the photon momentum and y(cid:48)-axis c1 perpendicular to the plane formed by the ψ(2S) and photon three-momenta. Finally, the angles θ(cid:48) and φ(cid:48) are the polar and azimuthal angles of h, one of the hadrons produced in h h the isobar decay, after boosting the momentum of h in the χ -helicity frame to the isobar c1 rest frame [17]. All values of M(cid:48) and M(cid:48) are summed with appropriate Clebsch-Gordan L J coefficients to create the initial χ state with one unit of total angular momentum and z(cid:48) c1 projection M(cid:48). χ The “breakup momentum” in a decay of 1 → 2 particles is given by the momentum of one of the produced particles in the rest frame of the parent. We denote the breakup momentum of the initial χ decay and the secondary isobar decay by p and q, respectively. c1 Finally, the term Tα(s), described in detail in the next section, is a function of the invariant mass squared of the isobar and describes the two-body dynamics in the decay. To impose isospin symmetry in the decays χ → a±π∓, we write the a π amplitude as c1 J J AaJπ (x) = √1 (cid:16)Aa+Jπ− (x)+Aa−Jπ+ (x)(cid:17), (4) Mψ,λγ 2 Mψ,λγ Mψ,λγ where the distinction between the two terms on the right-hand side is the interchange of the π+ and π− four-momenta in the calculation of the relevant kinematic variables. A similar symmetrization is used in the construction of the π π amplitude. 1 2. Two-body dynamics We use three different formulations of T (s) [in Eq. (2)] to describe the isobar decay α amplitude and phase as a function of s, the invariant mass squared of the isobar decay products. For all intermediate states except the a (980) and the ππ S-wave we use a Breit- 0 Wigner distribution, 1 T (s) = , (5) α m2 −s−im Γ (s) 0 0 J 9 FIG. 4. The angles used to describe the initial ψ(2S) decay (a) and the subsequent decay of the χ (b) c1 with ρ(s) (cid:20)B (q(s))(cid:21)2 J Γ (s) = Γ , (6) J 0 ρ B (q ) 0 J 0 where m and Γ are the isobar mass and width, respectively. We define the breakup 0 0 √ momentum q ≡ q(m2). Likewise, the available phase space is given by ρ(s) = 2q(s)/ s, 0 0 and ρ ≡ ρ(m2). These factors are used in conjunction with B (q), a spin-dependent Blatt- 0 0 J Weisskopf barrier penetration factor [18], to construct the mass-dependent total decay width given in Eq. (6). To describe the a (980) line-shape we use a three-channel Flatt´e formula [19]. In addition 0 to the common decay modes a (980) → ηπ and a (980) → KK, we include a third decay 0 0 mode, a (980) → η(cid:48)π, to provide a consistent description for both the χ → ηπ+π− and 0 c1 χ → η(cid:48)π+π− data. The parametrization takes the form c1 1 T (s) = , (7) a0(980) m2 −s−i(cid:80) g2ρ 0 c c c wherem is thea (980) massandg representsa coupling toone of thea (980) decay modes: 0 0 c 0 ηπ, KK, or η(cid:48)π. The factors ρ are the phase space available for each of the three different c final states. Following the technique suggested by Flatt´e to preserve analyticity at the KK and η(cid:48)π thresholds, we allow the phase space factors to become imaginary when s is below threshold for a particular decay channel [19]. For the ππ S-wave, we utilize an analysis of ππ scattering data [12] that provides two independent amplitudes for ππ → ππ and KK → ππ production mechanisms. Specifically, we attempt to model both direct production of χ → ηππ with the ππ in an S-wave and c1 also the process χ → ηKK → ηππ where the KK S-wave intermediate state rescatters c1 into ππ S-wave. These two amplitudes, labeled S and S respectively, are constructed ππ KK to be consistent with existing data in the region of ππ invariant mass below 2 GeV/c2, where the S-wave is expected to be significant. To account for the s-dependent differences between ππ scattering, from which the am- plitudes are derived, and production in χ decay (see Fig. 5), the S scattering amplitude c1 ππ is rewritten in a form N(s)/D(s), and the numerator is replaced by the first two terms in a 10