arxiv:0811.0564v1 [hep-ex] 4 nov 20084 g. raven and h. l. snoek nikhef, national institute for...

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BABAR-PUB-08/045SLAC-PUB-13437hep-ex/xxxx

Search for the Z(4430)− at BABAR

B. Aubert, M. Bona, Y. Karyotakis, J. P. Lees, V. Poireau, E. Prencipe, X. Prudent, and V. TisserandLaboratoire de Physique des Particules, IN2P3/CNRS et Universite de Savoie, F-74941 Annecy-Le-Vieux, France

J. Garra Tico and E. GraugesUniversitat de Barcelona, Facultat de Fisica, Departament ECM, E-08028 Barcelona, Spain

L. Lopezab, A. Palanoab, and M. Pappagalloab

INFN Sezione di Baria; Dipartmento di Fisica, Universita di Barib, I-70126 Bari, Italy

G. Eigen, B. Stugu, and L. SunUniversity of Bergen, Institute of Physics, N-5007 Bergen, Norway

G. S. Abrams, M. Battaglia, D. N. Brown, R. N. Cahn, R. G. Jacobsen, L. T. Kerth,

Yu. G. Kolomensky, G. Lynch, I. L. Osipenkov, M. T. Ronan,∗ K. Tackmann, and T. TanabeLawrence Berkeley National Laboratory and University of California, Berkeley, California 94720, USA

C. M. Hawkes, N. Soni, and A. T. WatsonUniversity of Birmingham, Birmingham, B15 2TT, United Kingdom

H. Koch and T. SchroederRuhr Universitat Bochum, Institut fur Experimentalphysik 1, D-44780 Bochum, Germany

D. WalkerUniversity of Bristol, Bristol BS8 1TL, United Kingdom

D. J. Asgeirsson, B. G. Fulsom, C. Hearty, T. S. Mattison, and J. A. McKennaUniversity of British Columbia, Vancouver, British Columbia, Canada V6T 1Z1

M. Barrett and A. KhanBrunel University, Uxbridge, Middlesex UB8 3PH, United Kingdom

V. E. Blinov, A. D. Bukin, A. R. Buzykaev, V. P. Druzhinin, V. B. Golubev,

A. P. Onuchin, S. I. Serednyakov, Yu. I. Skovpen, E. P. Solodov, and K. Yu. TodyshevBudker Institute of Nuclear Physics, Novosibirsk 630090, Russia

M. Bondioli, S. Curry, I. Eschrich, D. Kirkby, A. J. Lankford, P. Lund, M. Mandelkern, E. C. Martin, and D. P. StokerUniversity of California at Irvine, Irvine, California 92697, USA

S. Abachi and C. BuchananUniversity of California at Los Angeles, Los Angeles, California 90024, USA

H. Atmacan, J. W. Gary, F. Liu, O. Long, G. M. Vitug, Z. Yasin, and L. ZhangUniversity of California at Riverside, Riverside, California 92521, USA

V. SharmaUniversity of California at San Diego, La Jolla, California 92093, USA

C. Campagnari, T. M. Hong, D. Kovalskyi, M. A. Mazur, and J. D. RichmanUniversity of California at Santa Barbara, Santa Barbara, California 93106, USA

http://arxiv.org/abs/0811.0564v1

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T. W. Beck, A. M. Eisner, C. J. Flacco, C. A. Heusch, J. Kroseberg, W. S. Lockman,

A. J. Martinez, T. Schalk, B. A. Schumm, A. Seiden, M. G. Wilson, and L. O. WinstromUniversity of California at Santa Cruz, Institute for Particle Physics, Santa Cruz, California 95064, USA

C. H. Cheng, D. A. Doll, B. Echenard, F. Fang, D. G. Hitlin, I. Narsky, T. Piatenko, and F. C. PorterCalifornia Institute of Technology, Pasadena, California 91125, USA

R. Andreassen, G. Mancinelli, B. T. Meadows, K. Mishra, and M. D. SokoloffUniversity of Cincinnati, Cincinnati, Ohio 45221, USA

P. C. Bloom, W. T. Ford, A. Gaz, J. F. Hirschauer, M. Nagel,

U. Nauenberg, J. G. Smith, K. A. Ulmer, and S. R. WagnerUniversity of Colorado, Boulder, Colorado 80309, USA

R. Ayad,† A. Soffer,‡ W. H. Toki, and R. J. WilsonColorado State University, Fort Collins, Colorado 80523, USA

E. Feltresi, A. Hauke, H. Jasper, M. Karbach, J. Merkel, A. Petzold, B. Spaan, and K. WackerTechnische Universitat Dortmund, Fakultat Physik, D-44221 Dortmund, Germany

M. J. Kobel, R. Nogowski, K. R. Schubert, R. Schwierz, and A. VolkTechnische Universitat Dresden, Institut fur Kern- und Teilchenphysik, D-01062 Dresden, Germany

D. Bernard, G. R. Bonneaud, E. Latour, and M. VerderiLaboratoire Leprince-Ringuet, CNRS/IN2P3, Ecole Polytechnique, F-91128 Palaiseau, France

P. J. Clark, S. Playfer, and J. E. WatsonUniversity of Edinburgh, Edinburgh EH9 3JZ, United Kingdom

M. Andreottiab, D. Bettonia, C. Bozzia, R. Calabreseab, A. Cecchiab, G. Cibinettoab,

P. Franchiniab, E. Luppiab, M. Negriniab, A. Petrellaab, L. Piemontesea, and V. Santoroab

INFN Sezione di Ferraraa; Dipartimento di Fisica, Universita di Ferrarab, I-44100 Ferrara, Italy

R. Baldini-Ferroli, A. Calcaterra, R. de Sangro, G. Finocchiaro,S. Pacetti, P. Patteri, I. M. Peruzzi,§ M. Piccolo, M. Rama, and A. Zallo

INFN Laboratori Nazionali di Frascati, I-00044 Frascati, Italy

A. Buzzoa, R. Contriab, M. Lo Vetereab, M. M. Macria, M. R. Mongeab,

S. Passaggioa, C. Patrignaniab, E. Robuttia, A. Santroniab, and S. Tosiab

INFN Sezione di Genovaa; Dipartimento di Fisica, Universita di Genovab, I-16146 Genova, Italy

K. S. Chaisanguanthum and M. MoriiHarvard University, Cambridge, Massachusetts 02138, USA

A. Adametz, J. Marks, S. Schenk, and U. UwerUniversitat Heidelberg, Physikalisches Institut, Philosophenweg 12, D-69120 Heidelberg, Germany

V. Klose and H. M. LackerHumboldt-Universitat zu Berlin, Institut fur Physik, Newtonstr. 15, D-12489 Berlin, Germany

D. J. Bard, P. D. Dauncey, J. A. Nash, and M. TibbettsImperial College London, London, SW7 2AZ, United Kingdom

P. K. Behera, X. Chai, M. J. Charles, and U. MallikUniversity of Iowa, Iowa City, Iowa 52242, USA

J. Cochran, H. B. Crawley, L. Dong, W. T. Meyer, S. Prell, E. I. Rosenberg, and A. E. RubinIowa State University, Ames, Iowa 50011-3160, USA

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Y. Y. Gao, A. V. Gritsan, Z. J. Guo, and C. K. LaeJohns Hopkins University, Baltimore, Maryland 21218, USA

N. Arnaud, J. Bequilleux, A. D’Orazio, M. Davier, J. Firmino da Costa, G. Grosdidier, F. Le Diberder, V. Lepeltier,

A. M. Lutz, S. Pruvot, P. Roudeau, M. H. Schune, J. Serrano, V. Sordini,¶ A. Stocchi, and G. WormserLaboratoire de l’Accelerateur Lineaire, IN2P3/CNRS et Universite Paris-Sud 11,

Centre Scientifique d’Orsay, B. P. 34, F-91898 Orsay Cedex, France

D. J. Lange and D. M. WrightLawrence Livermore National Laboratory, Livermore, California 94550, USA

I. Bingham, J. P. Burke, C. A. Chavez, J. R. Fry, E. Gabathuler,

R. Gamet, D. E. Hutchcroft, D. J. Payne, and C. TouramanisUniversity of Liverpool, Liverpool L69 7ZE, United Kingdom

A. J. Bevan, C. K. Clarke, K. A. George, F. Di Lodovico, R. Sacco, and M. SigamaniQueen Mary, University of London, London, E1 4NS, United Kingdom

G. Cowan, H. U. Flaecher, D. A. Hopkins, S. Paramesvaran, F. Salvatore, and A. C. WrenUniversity of London, Royal Holloway and Bedford New College, Egham, Surrey TW20 0EX, United Kingdom

D. N. Brown and C. L. DavisUniversity of Louisville, Louisville, Kentucky 40292, USA

A. G. Denig, M. Fritsch, and W. GradlJohannes Gutenberg-Universitat Mainz, Institut fur Kernphysik, D-55099 Mainz, Germany

K. E. Alwyn, D. Bailey, R. J. Barlow, Y. M. Chia, C. L. Edgar, G. Jackson, G. D. Lafferty, T. J. West, and J. I. YiUniversity of Manchester, Manchester M13 9PL, United Kingdom

J. Anderson, C. Chen, A. Jawahery, D. A. Roberts, G. Simi, and J. M. TuggleUniversity of Maryland, College Park, Maryland 20742, USA

C. Dallapiccola, X. Li, E. Salvati, and S. SaremiUniversity of Massachusetts, Amherst, Massachusetts 01003, USA

R. Cowan, D. Dujmic, P. H. Fisher, S. W. Henderson, G. Sciolla,

M. Spitznagel, F. Taylor, R. K. Yamamoto, and M. ZhaoMassachusetts Institute of Technology, Laboratory for Nuclear Science, Cambridge, Massachusetts 02139, USA

P. M. Patel and S. H. RobertsonMcGill University, Montreal, Quebec, Canada H3A 2T8

A. Lazzaroab, V. Lombardoa, and F. Palomboab

INFN Sezione di Milanoa; Dipartimento di Fisica, Universita di Milanob, I-20133 Milano, Italy

J. M. Bauer, L. Cremaldi, R. Godang,∗∗ R. Kroeger, D. A. Sanders, D. J. Summers, and H. W. ZhaoUniversity of Mississippi, University, Mississippi 38677, USA

M. Simard, P. Taras, and F. B. ViaudUniversite de Montreal, Physique des Particules, Montreal, Quebec, Canada H3C 3J7

H. NicholsonMount Holyoke College, South Hadley, Massachusetts 01075, USA

G. De Nardoab, L. Listaa, D. Monorchioab, G. Onoratoab, and C. Sciaccaab

INFN Sezione di Napolia; Dipartimento di Scienze Fisiche,Universita di Napoli Federico IIb, I-80126 Napoli, Italy

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G. Raven and H. L. SnoekNIKHEF, National Institute for Nuclear Physics and High Energy Physics, NL-1009 DB Amsterdam, The Netherlands

C. P. Jessop, K. J. Knoepfel, J. M. LoSecco, and W. F. WangUniversity of Notre Dame, Notre Dame, Indiana 46556, USA

G. Benelli, L. A. Corwin, K. Honscheid, H. Kagan, R. Kass, J. P. Morris,

A. M. Rahimi, J. J. Regensburger, S. J. Sekula, and Q. K. WongOhio State University, Columbus, Ohio 43210, USA

N. L. Blount, J. Brau, R. Frey, O. Igonkina, J. A. Kolb, M. Lu,

R. Rahmat, N. B. Sinev, D. Strom, J. Strube, and E. TorrenceUniversity of Oregon, Eugene, Oregon 97403, USA

G. Castelliab, N. Gagliardiab, M. Margoniab, M. Morandina,

M. Posoccoa, M. Rotondoa, F. Simonettoab, R. Stroiliab, and C. Vociab

INFN Sezione di Padovaa; Dipartimento di Fisica, Universita di Padovab, I-35131 Padova, Italy

P. del Amo Sanchez, E. Ben-Haim, H. Briand, G. Calderini, J. Chauveau, P. David,

L. Del Buono, O. Hamon, Ph. Leruste, J. Ocariz, A. Perez, J. Prendki, and S. SittLaboratoire de Physique Nucleaire et de Hautes Energies,IN2P3/CNRS, Universite Pierre et Marie Curie-Paris6,Universite Denis Diderot-Paris7, F-75252 Paris, France

L. GladneyUniversity of Pennsylvania, Philadelphia, Pennsylvania 19104, USA

M. Biasiniab, R. Covarelliab, and E. Manoniab

INFN Sezione di Perugiaa; Dipartimento di Fisica, Universita di Perugiab, I-06100 Perugia, Italy

C. Angeliniab, G. Batignaniab, S. Bettariniab, M. Carpinelliab,†† A. Cervelliab, F. Fortiab, M. A. Giorgiab,

A. Lusianiac, G. Marchioriab, M. Morgantiab, N. Neriab, E. Paoloniab, G. Rizzoab, and J. J. Walsha

INFN Sezione di Pisaa; Dipartimento di Fisica, Universita di Pisab; Scuola Normale Superiore di Pisac, I-56127 Pisa, Italy

D. Lopes Pegna, C. Lu, J. Olsen, A. J. S. Smith, and A. V. TelnovPrinceton University, Princeton, New Jersey 08544, USA

F. Anullia, E. Baracchiniab, G. Cavotoa, D. del Reab, E. Di Marcoab, R. Facciniab,

F. Ferrarottoa, F. Ferroniab, M. Gasperoab, P. D. Jacksona, L. Li Gioia,

M. A. Mazzonia, S. Morgantia, G. Pireddaa, F. Polciab, F. Rengaab, and C. Voenaa

INFN Sezione di Romaa; Dipartimento di Fisica,Universita di Roma La Sapienzab, I-00185 Roma, Italy

M. Ebert, T. Hartmann, H. Schroder, and R. WaldiUniversitat Rostock, D-18051 Rostock, Germany

T. Adye, B. Franek, E. O. Olaiya, and F. F. WilsonRutherford Appleton Laboratory, Chilton, Didcot, Oxon, OX11 0QX, United Kingdom

S. Emery, M. Escalier, L. Esteve, S. F. Ganzhur, G. Hamel de Monchenault,

W. Kozanecki, G. Vasseur, Ch. Yeche, and M. ZitoCEA, Irfu, SPP, Centre de Saclay, F-91191 Gif-sur-Yvette, France

X. R. Chen, H. Liu, W. Park, M. V. Purohit, R. M. White, and J. R. WilsonUniversity of South Carolina, Columbia, South Carolina 29208, USA

M. T. Allen, D. Aston, R. Bartoldus, P. Bechtle, J. F. Benitez, R. Cenci, J. P. Coleman, M. R. Convery,

J. C. Dingfelder, J. Dorfan, G. P. Dubois-Felsmann, W. Dunwoodie, R. C. Field, A. M. Gabareen,

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S. J. Gowdy, M. T. Graham, P. Grenier, C. Hast, W. R. Innes, J. Kaminski, M. H. Kelsey, H. Kim, P. Kim,

M. L. Kocian, D. W. G. S. Leith, S. Li, B. Lindquist, S. Luitz, V. Luth, H. L. Lynch, D. B. MacFarlane,

H. Marsiske, R. Messner, D. R. Muller, H. Neal, S. Nelson, C. P. O’Grady, I. Ofte, A. Perazzo, M. Perl,B. N. Ratcliff, A. Roodman, A. A. Salnikov, R. H. Schindler, J. Schwiening, A. Snyder, D. Su,

M. K. Sullivan, K. Suzuki, S. K. Swain, J. M. Thompson, J. Va’vra, A. P. Wagner, M. Weaver, C. A. West,

W. J. Wisniewski, M. Wittgen, D. H. Wright, H. W. Wulsin, A. K. Yarritu, K. Yi, C. C. Young, and V. ZieglerStanford Linear Accelerator Center, Stanford, California 94309, USA

P. R. Burchat, A. J. Edwards, S. A. Majewski, T. S. Miyashita, B. A. Petersen, and L. WildenStanford University, Stanford, California 94305-4060, USA

S. Ahmed, M. S. Alam, J. A. Ernst, B. Pan, M. A. Saeed, and S. B. ZainState University of New York, Albany, New York 12222, USA

S. M. Spanier and B. J. WogslandUniversity of Tennessee, Knoxville, Tennessee 37996, USA

R. Eckmann, J. L. Ritchie, A. M. Ruland, C. J. Schilling, and R. F. SchwittersUniversity of Texas at Austin, Austin, Texas 78712, USA

B. W. Drummond, J. M. Izen, and X. C. LouUniversity of Texas at Dallas, Richardson, Texas 75083, USA

F. Bianchiab, D. Gambaab, and M. Pelliccioniab

INFN Sezione di Torinoa; Dipartimento di Fisica Sperimentale, Universita di Torinob, I-10125 Torino, Italy

M. Bombenab, L. Bosisioab, C. Cartaroab, G. Della Riccaab, L. Lanceriab, and L. Vitaleab

INFN Sezione di Triestea; Dipartimento di Fisica, Universita di Triesteb, I-34127 Trieste, Italy

V. Azzolini, N. Lopez-March, F. Martinez-Vidal, D. A. Milanes, and A. OyangurenIFIC, Universitat de Valencia-CSIC, E-46071 Valencia, Spain

J. Albert, Sw. Banerjee, B. Bhuyan, H. H. F. Choi, K. Hamano,R. Kowalewski, M. J. Lewczuk, I. M. Nugent, J. M. Roney, and R. J. Sobie

University of Victoria, Victoria, British Columbia, Canada V8W 3P6

T. J. Gershon, P. F. Harrison, J. Ilic, T. E. Latham, and G. B. MohantyDepartment of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom

H. R. Band, X. Chen, S. Dasu, K. T. Flood, Y. Pan, M. Pierini, R. Prepost, C. O. Vuosalo, and S. L. WuUniversity of Wisconsin, Madison, Wisconsin 53706, USA

(Dated: May 29, 2018)

We report the results of a search for Z(4430)− decay to J/ψπ− or ψ(2S)π− in B−,0 → J/ψπ−K0,+

and B−,0 → ψ(2S)π−K0,+ decays. The data were collected with the BABAR detector at the SLACPEP-II asymmetric-energy e+e− collider operating at center of mass energy 10.58 GeV, and thesample corresponds to an integrated luminosity of 413 fb−1. Each Kπ− mass distribution exhibitsclear K∗(892) and K∗

2 (1430) signals, and the efficiency-corrected spectrum is well-described bya superposition of the associated Breit-Wigner intensity distributions, together with an S-wavecontribution obtained from the LASS I = 1/2 Kπ− scattering amplitude measurements. EachKπ− angular distribution varies significantly in structure with Kπ− mass, and is represented interms of low-order Legendre polynomial moments. We find that each J/ψπ− or ψ(2S)π− massdistribution is well-described by the reflection of the measured Kπ− mass and angular distributionstructures. We see no significant evidence for a Z(4430)− signal for any of the processes investigated,neither in the total J/ψπ− or ψ(2S)π− mass distribution, nor in the corresponding distributionsfor the regions of Kπ− mass for which observation of the Z(4430)− signal was reported. Weobtain branching fraction upper limits B(B− → Z−K0, Z− → J/ψπ−) < 1.5 × 10−5, B(B0 →Z−K+, Z− → J/ψπ−) < 0.4× 10−5, B(B− → Z−K0, Z− → ψ(2S)π−) < 4.7× 10−5, and B(B0 →Z−K+, Z− → ψ(2S)π−) < 3.1×10−5 at 95% confidence level, where the Z(4430)− mass and widthhave been fixed to the reported central values.

6

PACS numbers: 12.39.Mk,12.40.Yx,13.25.Hw,14.40Gx

I. INTRODUCTION

In the original paper in which he proposed the QuarkModel [1], Gell-Mann stated that “Baryons can now beconstructed from quarks using the combinations (qqq),(qqqqq), etc., while mesons are made out of (qq), (qqqq),etc.”. He chose the lowest configurations to create therepresentations describing the known meson and baryonstates. However, the higher configurations were not a

priori excluded, and experimentalists and theorists havebeen seeking evidence supporting the existence of suchstates ever since.

In the baryon sector, resonant structure in the KNsystem would be indicative of five-quark content, andsearches for states of this type have been carried out sincethe mid-1960’s, mainly through partial-wave analysis ofKN elastic and charge-exchange scattering data. In re-cent years, there has been a great deal of activity focusedon the search for the conjectured Θ(1540)+ pentaquarkstate decaying to K0p. However, the initial low-statisticssignals claimed in a variety of experimental contexts havenot withstood high-statistics scrutiny, and the existenceof this state must be considered to be in doubt at thepresent time. The subject is reviewed in Ref. [2], and thestatus is updated in Ref. [3].

In the meson sector, attention has been focused overthe years mainly on the a0(980) and the f0(980) scalarmesons as possible four-quark states. However, the dis-covery of the D∗

s0(2317) and the Ds1(2460), with theirunexpectedly low mass values, and the observation ofmany new charmonium-like states above threshold fordecay to open charm, have led to speculation that cer-tain of these may be four-quark states (see e.g. Ref. [4]),although in no case has this been clearly established. Inthis regard, it follows that the recent paper from theBelle Collaboration [5] which reports the observation of aresonance-like structure, the Z(4430)−, in the ψ(2S)π−

system produced in the decays B−,0 → ψ(2S)π−K0,+ [6]has generated a great deal of interest (see e.g. Ref. [7],and references therein). Such a state must have a mini-mum quark content (ccdu), and would represent the un-equivocal manifestation of a four-quark meson state.

It is clearly important to seek confirmation of the Belleobservation, not only in the ψ(2S)π− system, but also for

∗Deceased†Now at Temple University, Philadelphia, Pennsylvania 19122,

USA‡Now at Tel Aviv University, Tel Aviv, 69978, Israel§Also with Universita di Perugia, Dipartimento di Fisica, Perugia,

Italy¶Also with Universita di Roma La Sapienza, I-00185 Roma, Italy∗∗Now at University of South Alabama, Mobile, Alabama 36688,

USA††Also with Universita di Sassari, Sassari, Italy

the J/ψπ− combination, which might also show evidenceof a Z(4430)− signal or of a similar lower mass state [8].Consequently, in this paper we present a BABAR analy-sis of the entire Dalitz plot corresponding to the decaysB−,0 → ψ(2S)π−K0,+ and in parallel pursue an identicalanalysis of our B−,0 → J/ψπ−K0,+ data. Both analy-ses make use of the complete BABAR data sample accruedat the Υ (4S) resonance. In this regard, we first seek arepresentation of the Kπ− mass and angular distributionstructures, which dominate the final states under study,in terms of their expected low-angular-momentum inten-sity contributions. We then investigate the reflection ofeach Kπ− system into its associated ψπ− [9] mass distri-bution in order to establish the need for any additionalnarrow signal.The BABAR detector and the data sample are described

briefly in Sec. II, and the event selection procedures arediscussed in Sec. III. In Sec. IV, the Dalitz plots andtheir uncorrected invariant mass projections are shownfor the B meson signal regions. Since the analysis em-phasizes this search for narrow structure in the J/ψπ−

and ψ(2S)π− mass distributions, the mass resolution de-pendence on invariant mass for these systems is analyzedin Sec. V. Similarly, it is important to understand thebehavior of the event reconstruction efficiency over eachfinal state Dalitz plot and to correct for it before assess-ing the significance of any observed mass structures. Theprocedure followed is described in Appendix A, and theresults are summarized in Sec. VI. Fits to the correctedKπ− mass distributions are discussed in Sec. VII, andthe Kπ− angular distribution structure as a function ofKπ− mass is represented in terms of Legendre polyno-mial moments as described in Sec. VIII. In Sec. IX, thereflections of the observed Kπ− mass and angular struc-tures onto the J/ψπ− and ψ(2S)π− mass distributionsare compared to the corresponding efficiency-correcteddistributions, and in Sec. X our results are discussed inrelation to those in the Belle publication. The BABAR

ψπ− mass distributions are fitted in Sec. XI, and wepresent a summary and our conclusions in Sec. XII. Fi-nally, acknowledgments are expressed in Sec. XIII.

II. THE BABAR DETECTOR AND DATA

SAMPLE

The data used in this analysis were collected with theBABAR detector at the PEP-II asymmetric-energy e+e−

collider operating at a center-of-mass (c.m.) energy of10.58 GeV.A detailed description of the BABAR detector can

be found in Ref. [10]. Charged particle tracks aredetected with a five-layer, double-sided silicon vertextracker (SVT) and a 40-layer drift chamber (DCH), filledwith a helium-isobutane gas mixture, and coaxial withthe cryostat of a superconducting solenoidal magnet,

7

which produces a magnetic field of approximately 1.5 T.The charged-particle momentum resolution is given by(δPT /PT )

2 = (0.0013PT )2 + (0.0045)2, where PT is the

transverse momentum measured in GeV/c. The SVT,with a typical coordinate resolution of 10 µm, measuresthe impact parameters of charged particle tracks in boththe plane transverse to the beam direction and along thecollision axis; it also supports stand-alone reconstructionof low-PT charged particle tracks.Charged particle types are identified from specific ion-

ization energy loss (dE/dx) measured in the DCH andSVT, and from Cherenkov radiation detected in a ring-imaging Cherenkov device. Electrons are identified bymeans of a CsI(Tl) electromagnetic calorimeter (EMC).The return yoke of the superconducting coil is instru-

mented with resistive plate chambers for the identifica-tion of muons and the detection of clusters producedby KL and neutron interactions. For the latter part ofthe experiment these chambers were replaced by limitedstreamer tubes in the barrel region of the detector [11].In this analysis, we use a data sample corresponding to

an integrated luminosity of 413 fb−1, which is equivalentto the production of approximately 455 million BB pairs.

III. EVENT SELECTION

We reconstruct events in four decay modes [6]:

B− → J/ψπ−K0S , (1)

B0 → J/ψπ−K+ , (2)

B− → ψ(2S)π−K0S , (3)

B0 → ψ(2S)π−K+ . (4)

The event selection criteria were established by op-timizing signal-to-background ratio using Monte Carlo(MC) simulated signal events, B−,0 → ψπ−K0,+, andbackground, BB and e+e− → qq (q = u, d, s, c), events.For the data sample, a J/ψ candidate is formed by

geometrically constraining an identified e+e− or µ+µ−

pair of tracks to a common vertex point and requir-ing a fit probability > 0.001. For µ+µ−, the invari-ant mass of the pair must in addition satisfy 3.06 <mµ+µ− < 3.14 GeV/c2, while for e+e− the requirementis 2.95 < me+e− < 3.14 GeV/c2. In the latter case, themass interval extends to lower values in order to allowfor electron bremsstrahlung energy loss; if an electron-associated photon cluster of this type is found in theEMC, its four-momentum vector is included in the calcu-lation of me+e− . The surviving J/ψ candidates were fit-ted to impose a constraint to the nominal mass value [3].For ψ(2S) decay to µ+µ− or e+e− the same selec-

tion procedures are followed, but with invariant mass re-quirements 3.640 < mµ+µ− < 3.740 GeV/c2 or 3.440 <me+e− < 3.740 GeV/c2. For ψ(2S) decay to J/ψπ+π−,the J/ψ candidate is selected as previously described,and is fit again to incorporate a constraint to its nominalmass value [3]. This J/ψ and an identified π+π− pair are

geometrically constrained to a common vertex (fit prob-ability > 0.001), and required to have an invariant massin the range 3.655 < mJ/ψπ+π− < 3.715 GeV/c2. In thesame manner as for the J/ψ , surviving candidates werethen constrained to the nominal ψ(2S) mass value [3].A K0

S candidate is formed by geometrically constrain-ing a pair of oppositely charged tracks to a common ver-tex (fit probability > 0.001); the tracks are treated aspions, but without particle-identification requirements,and the invariant mass of the pair must satisfy 0.472 <mπ+π− < 0.522 GeV/c2. A charged kaon candidatefrom the B meson decay must be identified as a kaon,but no particle identification is required of correspond-ing charged pion candidates.The ψ, K and π candidates forming a B meson de-

cay candidate are geometrically constrained to a com-mon vertex, with fit probability > 0.001 required. Fordecay modes involving a K0

S, the K0S flight length with

respect to this vertex must have > +3 standard deviationsignificance in order to reduce combinatoric background.The K0

S candidate is not mass constrained, since this wasfound to have a negligible effect on resolution.We further define B meson decay candidates using

the energy difference ∆E = E∗B − √

s/2 in the centerof mass (c.m.) frame, and the beam-energy substituted

mass mES =√

((s/2 + ~pi · ~pB)/Ei)2 − ~p 2B, where (Ei, ~pi)

is the initial state four-momentum vector in the labora-tory frame and

√s is the c.m. energy; E∗

B is the B mesonenergy in the c.m. and ~pB is its laboratory frame mo-mentum.We require that B decay signal events satisfy 5.272 <

mES < 5.286 GeV/c2 and |∆E| < 0.020 GeV. In orderto correct for background events in the signal region, wedefine a ∆E sideband region by 0.030 < |∆E| < 0.050GeV; we have verified through MC studies that sidebandevents in the mES signal range correctly represent back-ground in the B meson signal region. We refer to theprocedure by which we correct for background in the sig-nal region by subtracting the ∆E sideband events in themES signal range by the term “sideband subtraction”.In Figs. 1(a)-(d) we show the mES distributions in the

∆E signal region for the decay processes of Eqs. (1)- (4),where the filled histograms show the sideband distribu-tions. We fit each distribution with a signal Gaussianfunction with mass and width as free parameters, andan ARGUS background function [12] with a free expo-nential slope parameter. In each figure, the solid curverepresents the total function and the dashed curve showsthe background contribution. Clear mES signals are ob-served in Figs. 1(a)-(d), and in each figure the sidebanddistribution is consistent with the fitted background.The ∆E distributions for the mES signal region

(Figs. 1(e)-(h)) exhibit clear signal peaks. We fit eachdistribution with a linear background function and a sig-nal function consisting of two Gaussian functions with acommon center; all parameters are free in the fits. Ineach case, the filled histogram is from the mES sidebandregion defined by 5.250 < mES < 5.264 GeV/c2, and is

8

500

1000

1500

500

1000

1500

500

1000

1500

S0K-πψ J/→-B

Data

Sideband

Fit result

Background

(a)

2000

4000

2000

4000

2000

4000+K-πψ J/→0B

(b)

100

200

100

200

100

200 S0K-π(2S)ψ →-B

(c)

)2 (GeV/cESm5.2 5.22 5.24 5.26 5.28 5.30

200

400

600

800

)2 (GeV/cESm5.2 5.22 5.24 5.26 5.28 5.30

200

400

600

800

)2 (GeV/cESm5.2 5.22 5.24 5.26 5.28 5.30

200

400

600

800 +K-π(2S)ψ →0B(d)

2E

vent

s/2

MeV

/c

200

400

200

400

200

400

200

400S0K-πψ J/→-B(e)

500

1000

1500

500

1000

1500

500

1000

1500

500

1000

1500+K-πψ J/→0B(f)

50

100

50

100

50

100

50

100S0K-π(2S)ψ →-B(g)

E (GeV)∆-0.1 -0.05 0 0.05 0.1

100

200

300

E (GeV)∆-0.1 -0.05 0 0.05 0.1

100

200

300

E (GeV)∆-0.1 -0.05 0 0.05 0.1

100

200

300

E (GeV)∆-0.1 -0.05 0 0.05 0.1

100

200

300+K-π(2S)ψ →0B(h)

Eve

nts/

2 M

eV

FIG. 1: The mES distributions, (a)-(d), and (∆E) distributions, (e)-(h), for the decay modes B− → J/ψπ−K0S, B

0 →J/ψπ−K+, B− → ψ(2S)π−K0

S , and B0 → ψ(2S)π−K+. The points show the data, and the solid curves represent thefit functions. The dashed curves indicate the background contributions, and the filled histograms show the correspondingdistributions for the sideband regions.

in good agreement with the fitted background. For thedecay modes of Eqs. (1)-(4), the fraction of events withmore than one B meson signal candidate ranges from0.5 to 1.1 %. For such events, the candidate with thesmallest value of |∆E| is selected.We summarize the principal selection criteria in Ta-

ble I, and in Table II provide an overview of the datasamples in the B-meson signal region used in the analy-sis described in this paper.

IV. THE DALITZ PLOTS AND INVARIANT

MASS PROJECTIONS

The Dalitz plots of m2ψπ−

versus m2Kπ−

are shown in

Fig. 2 for the signal regions defined in Table I for the Bmeson decay modes specified in Eqs. (1)-(4). The corre-sponding mKπ− , mψπ− , and mψK mass projections are

represented by the data points in Figs. 3, 4, and 5, re-spectively. In each figure the filled histogram is obtainedfrom the relevant ∆E sideband region.

In Fig. 3, the contributions due to the K∗(892) domi-nate the mass distributions. Small, but clear, K∗

2 (1430)signals are evident for the J/ψ decay modes, and theseseem to be present for the ψ(2S) modes also. Previousanalyses [13, 14] have shown that, for the J/ψ modes,the region between the K∗(892) and K∗

2 (1430) signals(∼ 1.1− 1.3 GeV/c2) contains a significant Kπ− S-wavecontribution. In the K∗(892) region, the presence of theS-wave amplitude has been demonstrated through its in-terference with the K∗(892) P -wave amplitude [14]. Thisinterference yields a strong forward-backward asymmetryin the Kπ− angular distribution, as is seen in the verti-cal K∗(892) bands of Fig. 2(b) and Fig. 2(d), and as isshown in Sec. VII, Figs. 13(a) and Fig. 13(c). These fea-tures of the Kπ− mass and angular distributions will be

9

)4/c2 (GeV2-πS

0Km

0 1 2 3 4 5

)4/c2

(G

eV2

- πψJ/

m

16

18

20

22

24

1

10

210(a)

)4/c2 (GeV2-πS

0Km

0 1 2 3

)4/c2

(G

eV2

- π(2

S)ψ

m

15

20

1

10

210(c)

)4/c2 (GeV2-π+K

m0 1 2 3 4 5

)4/c2

(G

eV2

- πψJ/

m

16

18

20

22

24

1

10

210(b)

)4/c2 (GeV2-π+K

m0 1 2 3

)4/c2

(G

eV2

- π(2

S)ψ

m

15

20

1

10

210(d)

FIG. 2: The m2ψπ−

versus m2Kπ−

Dalitz plot distributions for the signal regions for the decay modes (a) B− → J/ψπ−K0S , (b)

B0 → J/ψπ−K+, (c) B− → ψ(2S)π−K0S, (d) B

0 → ψ(2S)π−K+. The intensity scale is logarithmic.

analyzed in detail in Secs. VII and VIII below.

The mψπ− distributions of Fig. 4 show no peakingstructure at the mass reported for the Z(4430)− [5] (in-dicated by the dashed vertical line in each figure). InFig. 4(b) there seems to be a peak at ∼ 4.65 GeV/c2 andperhaps a weaker one just below 4.4 GeV/c2, while inFigs. 4(c) and (d) there seems to be a peak just below∼ 4.5 GeV/c2. These features are discussed in Secs. Xand XI in conjunction with reflections resulting from theKπ− mass and angular structures.

Similarly, the mψK distributions of Fig. 5 show noevidence of narrow structure. In fact, in overall shapethese distributions approximate mirror images of thosein Fig. 4. This is not unexpected if both result primarilyfrom Kπ− reflection, since then the high-mass region ofone distribution would be correlated strongly with thelow mass region of the other, and vice versa. Since Fig. 5

shows no evidence of interesting features, and since ouremphasis in this paper is on the search for the Z(4430)−,we do not discuss the ψK systems any further in thepresent analysis.We note that in Figs. 3-5, and in other invariant mass

distributions to follow, the J/ψ -ψ(2S) mass differencecauses significant differences in the range spanned in therespective decay modes. This should be kept in mindwhen making J/ψ -ψ(2S) comparisons.

V. THE ψπ− MASS RESOLUTION

In Ref. [5], the width of the Z(4430)− is given as 45+35−18

MeV, where we have combined statistical and system-atic errors in quadrature. This value is very similar tothat of the K∗(892) [3], although with larger uncertain-

10

TABLE I: Summary of the principal criteria used to select Bcandidates.

Selection category criterion

J/ψ → e+e− 2.95 < mee < 3.14 GeV/c2

J/ψ → µ+µ− 3.06 < mµµ < 3.14 GeV/c2

ψ(2S) → e+e− 3.44 < mee < 3.74 GeV/c2

ψ(2S) → J/ψπ+π− 3.655 < mJ/ψππ < 3.715 GeV/c2

(J/ψ → e+e−)

ψ(2S) → µ+µ− 3.64 < mµµ < 3.74 GeV/c2

ψ(2S) → J/ψπ+π− 3.655 < mJ/ψππ < 3.715 GeV/c2

(J/ψ → µ+µ−)

K0S → π+π− 0.472 < mππ < 0.522 GeV/c2

Flight length significance > +3σ

mES signal region 5.272 < mES < 5.286 GeV/c2

∆E signal region |∆E| < 0.020 GeV

100

200

300

400

100

200

300

400 S0K-πψ J/→-B(a)

20

40

60

80

20

40

60

80S0K-π(2S)ψ →-B(c)

) 2 (GeV/c-πKm1 1.5 2

500

1000

1500

DataE sideband∆

) 2 (GeV/c-πKm1 1.5 2

500

1000

1500+K-πψ J/→0B(b)

) 2 (GeV/c-πKm0.8 1 1.2 1.4 1.6

100

200

300

) 2 (GeV/c-πKm0.8 1 1.2 1.4 1.6

100

200

300 +K-π(2S)ψ →0B(d)

2E

vent

s/10

MeV

/c 2

Eve

nts/

10 M

eV/c

FIG. 3: The mKπ− mass projections for the Dalitz plots ofFig. 2. The data points are for the mES-∆E signal regions,and the filled histograms are for the ∆E sideband regions.

20

40

60

20

40

60 S0K-πψ J/→-B (a)

10

20

10

20S0K-π(2S)ψ →-B (c)

) 2 (GeV/c-πψJ/m3.5 4 4.5

50

100

150

200Data

E sideband ∆

) 2 (GeV/c-πψJ/m3.5 4 4.5

50

100

150

200+K-πψ J/→0B (b)

) 2 (GeV/c-π(2S)ψm3.8 4 4.2 4.4 4.6 4.8

20

40

60

) 2 (GeV/c-π(2S)ψm3.8 4 4.2 4.4 4.6 4.8

20

40

60+K-π(2S)ψ →0B (d)

2E

vent

s/10

MeV

/c

FIG. 4: The mψπ− mass projections for the Dalitz plots ofFig. 2. The data points are for the mES-∆E signal regions,and the filled histograms are for the ∆E sideband regions.The dashed vertical lines indicate mψπ− = 4.433 GeV/c2.

20

40

60

20

40

60 S0K-πψ J/→-B(a)

5

10

15

20

5

10

15

20 S0K-π(2S)ψ →-B(c)

)2 (GeV/cKψJ/m3.5 4 4.5 5

50

100

150

200

)2 (GeV/cKψJ/m3.5 4 4.5 5

50

100

150

200 +K-πψ J/→0B(b)

)2 (GeV/c(2S)Kψm4.5 5

20

40

60

DataE sideband∆

)2 (GeV/c(2S)Kψm4.5 5

20

40

60 +K-π(2S)ψ →0B(d)

2E

vent

s/10

MeV

/c 2

Eve

nts/

10 M

eV/c

2E

vent

s/10

MeV

/c

FIG. 5: The mψK mass projections for the Dalitz plots ofFig. 2. The data points are for the mES-∆E signal regions,and the filled histograms are for the ∆E sideband regions.

ties, which we have no difficulty observing, as shown inFig. 3. However, mass resolution degrades with increas-ing Q-value, where Q-value is the difference between theinvariant mass value in question and the correspondingthreshold mass value. Since the Q-value for the K∗(892)is only ∼ 260 MeV/c2, while that for the Z(4430)− is∼ 600 MeV/c2 in the ψ(2S)π mode, and would be ∼ 1200MeV/c2 in the J/ψπ mode, the mass resolution should beworse for the Z(4430)− than for the K∗(892), and henceshould be systematically investigated.

We do this by using MC simulated data for the B me-son decay modes of Eqs. (1)-(4). For each mode, thereconstructed events were divided into 50 MeV/c2 inter-vals of ψπ− mass, and within each interval it was found

11

TABLE II: The data samples used in the analysis.

Decay mode Signal region events ∆E sideband events Net analysis sample

B− → J/ψπ−K0S 4229 ± 65 485± 22 3744 ± 68

B0 → J/ψπ−K+ 14251 ± 119 1269 ± 36 12982 ± 124

B− → ψ(2S)π−K0S 703± 26 161± 13 542± 29

B0 → ψ(2S)π−K+ 2405 ± 49 384± 20 2021 ± 53

that the distribution of (reconstructed - generated) ψπ−

mass could be described by a double Gaussian functionwith a common mean at zero. The local ψπ− mass resolu-tion is characterized by the half-width-at-half-maximum(HWHM) value for this line shape, and the dependenceof this quantity on ψπ− mass is shown in Fig. 6 for theindividual decay modes. For both J/ψ modes, the res-olution varies from ∼ 2 MeV/c2 at threshold to ∼ 9MeV/c2 at the maximum mass value, while for bothψ(2S)π modes, the variation is from ∼ 2 − 6 MeV/c2.At the Z(4430)− mass value, indicated by the dashedvertical lines in Fig. 6, the resolution is ∼ 4 MeV/c2 forψ(2S)π, and ∼ 7 MeV/c2 for J/ψπ. We note that, forJ/ψπ, the resolution at a Q-value of ∼ 600 MeV/c2 isessentially the same as for ψ(2S)π at the Z(4430)−. Itfollows that failure to observe the Z(4430)− in its J/ψπ−

or ψ(2S)π− decay mode in the present experiment shouldnot be attributed to inadequate mass resolution.

5

10 (a) S0K-πψ J/→-B

5

10 (c) S0K-π(2S)ψ →-B

)2 (GeV/c-πψJ/m3.5 4 4.5

5

10 (b) +K-πψ J/→0B

)2 (GeV/c-π(2S)ψm4 4.5

5

10 (d) +K-π(2S)ψ →0B

Res

olut

ion

(M

eV)

FIG. 6: The ψπ− mass resolution in 50 MeV/c2 intervalsas a function of ψπ− mass for the decay modes (a) B− →J/ψπ−K0

S , (b) B0 → J/ψπ−K+, (c) B− → ψ(2S)π−K0

S, and(d) B0 → ψ(2S)π−K+. The dashed vertical lines indicatemψπ− = 4.433 GeV/c2.

VI. EFFICIENCY CORRECTION

In the search for the Z(4430)−, a detailed understand-ing of event reconstruction efficiency over the entire finalstate Dalitz plot for each of the B meson decay processesof Eqs. (1)-(4) is necessary. This is because efficiencyvariation can, in principle, lead to the creation of spuri-ous signals or to the distortion of real effects such thattheir significance is reduced. Even when the process ofefficiency correction leads to no significant change in theinterpretation of the data, it is important to demonstrateclearly that this is in fact the case.

The efficiency correction procedure which we follow isdescribed in detail in Appendix A. For reasons discussedthere, we use a “rectangular Dalitz plot” for which thevariables are chosen to be mKπ− and cos θK , the nor-malized dot-product between the Kπ− three-momentumvector in the parent-B rest frame and the kaon three-momentum vector after a Lorentz transformation fromthe B rest frame to the Kπ− rest frame. For the Dalitzplots shown in Fig. 2, the y-axis variable, m2

ψπ−, varies

linearly with cos θK .

The average efficiency E0 depends on ψ decay mode,as shown in Fig. 7. The individual fitted curves in thisfigure are used to calculate the value of E0 for an event ata particular value ofmKπ− which has been reconstructedin the relevant ψ decay mode.

As explained in Appendix A, E0 is then modulatedby a linear combination of 12 Legendre polynomials incos θK , whose multiplicative coefficients E1−E12 are ob-tained from curves representing their individual mKπ−

dependence. In this way, an efficiency value can be cal-culated for each reconstructed event in our data sam-ple. The inverse of this efficiency then provides a weight-factor which is associated with this event in any distribu-tion to which it contributes. This enables us to correctany distribution under study for efficiency-loss effects.

A specific example is provided by the Kπ− mass dis-tributions of Fig. 8. A weight value for each event is cal-culated according to its particular ψ decay mode, as de-scribed above. The histograms of Fig. 8 are then formed

12

0.1

0.2

0.3 S0K

-πψ J/→-B(a)

ee→ψ(2S),J/ψµµ →ψ(2S),J/ψ

ππ(ee)ψ J/→(2S)ψππ)µµ(ψ J/→(2S)ψ

0.1

0.2

0.3 S0K

-π(2S)ψ →-B(c)

) 2 (GeV/c-πKm1 1.5 2

0.1

0.2

0.3+K

-πψ J/→0B(b)

) 2 (GeV/c-πKm0.8 1 1.2 1.4

0.1

0.2

0.3+K

-π(2S)ψ →0B(d)

2E

ffic

ienc

y/50

MeV

/c

FIG. 7: The mKπ− dependence of the average efficiency,E0, for the B meson decay processes (a) B− → J/ψπ−K0

S ,(b) B0 → J/ψπ−K+, (c) B− → ψ(2S)π−K0

S , and (d)B0 → ψ(2S)π−K+. The key in (a) applies to all four fig-ures, and, as indicated, the individual J/ψ and ψ(2S) decaymodes are treated separately. The curves result from fifth-order polynomial fits to the data points.

by summing these weights in each mass interval of eachplot. Sideband subtraction is accomplished by assigningsideband events negative weight. The contributions fromthe different ψ decay modes are distinguished by shad-ing, and the final histograms represent the sum of thesecontributions.In Appendix A it is pointed out that the use of

high-order Legendre polynomials is necessary because ofsignificant decrease in efficiency for cos θK ∼ +1 and0.72 < mKπ− < 0.92 GeV/c2, and for cos θK ∼ −1 and0.97 < mKπ− < 1.27 GeV/c2. The former loss is dueto the failure to reconstruct low-momentum pions in thelaboratory frame, and the latter is due to a similar failureto reconstruct low-momentum kaons. The dependence ofefficiency on laboratory frame momentum is shown inFig. 9. In Figs. 9(a)-(d), there is a significant decrease inefficiency for pions of momentum < 0.1 GeV/c, while inFigs. 9(f),(h) there is similar decrease for charged kaonsbelow ∼ 0.25 GeV/c. For K0

S, the effect is similar to thatfor charged pions (Figs. 9(e),(g)).The Lorentz boost from the laboratory frame to the

Kπ− rest frame translates the laboratory frame lossesinto the losses localized in mKπ− and cos θK which ourefficiency study reveals. The effect of these regions oflow efficiency on the uncorrected ψπ mass distributionsis discussed in Sec. IX.

VII. FITS TO THE Kπ− MASS DISTRIBUTIONS

The most striking aspects of the Dalitz plots of Fig. 2pertain to the Kπ− system, as shown explicitly in Fig. 3.In order to investigate the features of the ψπ− distribu-

tions of Fig. 4, it is necessary to understand how structurein theKπ− mass and cos θK distributions reflects into theψπ− system. We begin this process by making detailedfits to the sideband-subtracted and efficiency-correctedKπ− mass distributions of Fig. 8.As discussed in Sec. IV, we expect that the Kπ− sys-

tem can be described in terms of a superposition of S-,P -, and D-wave amplitudes. Since we correct for cos θKdependence of the efficiency, it follows that the correctedKπ− mass distributions of Fig. 8 can be described bya sum of S-, P -, and D-wave intensity contributions,since any interference terms vanish when integrated overcos θK . Consequently, we describe the mKπ− mass pro-jections as follows,

dN

dmKπ= N× (5)

[

fS

(

GSR

GSdmKπ

)

+ fP

(

GPR

GP dmKπ

)

+ fD

(

GDR

GDdmKπ

)]

,

where the integrals are over the full mKπ range and thefractions f are such that

fS + fP + fD = 1 . (6)

The P - and D-wave intensities, GP and GD, are ex-pressed in terms of the squared moduli of Breit-Wigner(BW) amplitudes. For the P -wave,

GP (mKπ) =BP (mKπ)(p · q) q2

DP (qRP )

(m2P −m2

Kπ)2 +m2

PΓ2P (q, RP )

, (7)

where

• BP (mKπ) describes the B-decay vertex;

• p is the momentum of the ψ in the B rest frame;

• q is the momentum of theK in theKπ− rest frame;

• DP is the P -wave Blatt-Weisskopf barrier factorwith radius RP [15];

• the mass-dependent total width is

ΓP = Γ0P

(

q2

q2P

)

DP (qPRP )

DP (qRP )

(

q

qP

)(

mP

mKπ

)

, (8)

with qP = q evaluated at mP ; mP is the mass, and Γ0P

the width, of the K∗(892). We leave the mass and widthfree in the fits and choose RP = 3.0 GeV−1 [18].Similarly, for the D-wave,

GD(mKπ) =BD(mKπ)(p · q) q4

DD(qRD)

(m2D −m2

Kπ)2 +m2

DΓ2D(q, RD)

, (9)

where

• BD(mKπ) describes the B-decay vertex;

13

500

1000

1500

2000

500

1000

1500

2000 2

Eve

nts/

10 M

eV/c

500

1000

1500

2000 S0K

-πψ J/→-B(a)

ee→ψJ/

µµ →ψJ/

200

400

600

200

400

600

2E

vent

s/10

MeV

/c

200

400

600S0K

-π(2S)ψ →-B(c)

ee→(2S)ψ

µµ →(2S)ψ

ππ(ee)ψ J/→(2S)ψ

ππ)µµ(ψ J/→(2S)ψ

1 1.5 2

2000

4000

6000

1 1.5 2

2000

4000

6000

) 2 (GeV/c-πKm1 1.5 2

2E

vent

s/10

MeV

/c

2000

4000

6000+K

-πψ J/→0B(b)

0.8 1 1.2 1.4 1.6

500

1000

1500

0.8 1 1.2 1.4 1.6

500

1000

1500

) 2 (GeV/c-πKm0.8 1 1.2 1.4 1.6

2E

vent

s/10

MeV

/c

500

1000

1500

+K-π(2S)ψ →0B(d)

FIG. 8: The Kπ− mass distributions, after sideband subtraction and efficiency correction, for the decay modes (a) B− →J/ψπ−K0

S , (b) B0 → J/ψπ−K+, (c) B− → ψ(2S)π−K0

S, and (d) B0 → ψ(2S)π−K+. The contributions from the individualψ decay modes are obtained separately, as described in the text, and are accumulated as indicated by the keys in (a) and (c),to form the final histograms.

• DD is the D-wave Blatt-Weisskopf barrier factorwith radius RD [15];

• the mass-dependence of the total width is approx-imated by the Kπ− contribution as follows,

ΓD = Γ0D

(

q4

q4D

)

DD(qDRD)

DD(qRD)

(

q

qD

)(

mD

mKπ

)

, (10)

with qD = q evaluated at mD; mD is the mass, and Γ0D

the width of the K∗2 (1430). We fix mD and Γ0

D to theirnominal values [3] and choose RD = 1.5 GeV−1 [16].The S-wave contribution is described using the I = 1/2

amplitude for S-wave K−π+ elastic scattering [18]. We

write

GS(mKπ) = BS(mKπ)(p · q)|TS |2 , (11)

where BS(mKπ) describes the B-decay vertex; TS is theinvariant amplitude, which is related to AS , the complexKπ− scattering amplitude, by

|TS | =(

mKπ

q

)

|AS | . (12)

For mKπ− > 1.5 GeV/c2, |AS | is obtained by interpo-lation from the measured values [16]. For lower massvalues, AS is a pure-elastic amplitude (within error) and

14

0.1

0.2

0.3S0K

-πψ J/→-B(a)

0.1

0.2

0.3S0K

-πψ J/→-B(e)

0.1

0.2

0.3 +K-πψ J/→0B(b)

0.1

0.2

0.3 +K-πψ J/→0B(f)

0.1

0.2

0.3S0K

-π(2S)ψ →-B(c)

0.1

0.2

0.3S0K

-π(2S)ψ →-B(g)

) (GeV/c) -πP(0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3+K

-π(2S)ψ →0B(d)

P(K) (GeV/c)0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3+K

-π(2S)ψ →0B(h)

Eff

icie

ncy/

10 M

eV/c

FIG. 9: (a)-(d): The dependence of efficiency on pion momen-tum in the laboratory frame for the B meson decay modes ofEqs. (1)-(4). (e)-(h): The corresponding efficiency depen-dence on kaon laboratory frame momentum.

is parameterized as

AS =1

cot δB − i+ e2iδB

(

1

cot δR − i

)

, (13)

where the first term is non-resonant, and the second is aresonant term rotated by 2δB in order to maintain elasticunitarity. In Eq. 13,

q cot δB =1

a+

1

2rq2 , (14)

with a = 1.94 GeV−1 and r = 1.76 GeV−1 [16];

cot δR =m2S −m2

Kπ

mSΓS, (15)

with

ΓS = Γ0S

(

q

qS

)

mS

mKπ; (16)

mS (= 1.435 GeV/c2) is the mass of the K∗0 (1430) reso-

nance and Γ0S (= 0.279 GeV) is its width [16]. We choose

to fix the S-wave parameters to the indicated values.Although the vertex functions BS , BP , and BD de-

pend, in principle, on mKπ, we find that our best fitsto the mass distributions are obtained when each ver-tex function is set to one (BS = BP = BD = 1). Inthis regard, we emphasize that our goal is not to ob-tain a precise amplitude decomposition of the Kπ− massspectrum. This would require taking account of the an-gular correlations between the ψ decay products and the

Kπ− system. For Kπ− S-, P -, and D-wave this is ex-tremely complicated [17], and is far beyond the scope ofthe present analysis. Our aim is to obtain an accuratedescription of the Kπ− mass distributions in terms ofthe expected angular momentum contributions so thatwe can reliably project the observed structures onto therelated ψπ− mass distributions.The results of the fits to the mKπ− distributions of

Fig. 8 are shown by the curves in Fig. 10. Good descrip-tions are obtained, even though each fit has only fivefree parameters and the fit function has exactly the samestructure in each case.Table III summarizes the output from the fits. The

χ2/NDF (NDF = Number of Degrees of Freedom) val-ues are satisfactory, and the mass and width values forthe K∗(892)0 and K∗(892)− modes are internally consis-tent and agree with their nominal values [3]; the widthvalues for the ψ(2S) modes are slightly low, but the un-certainties are quite large because of the smaller datasample involved.The fractional contributions for the two J/ψ modes

agree well with each other, and there is similar agree-ment for the ψ(2S) modes. This demonstrates that thecharged and neutral Kπ− distributions are very similarin shape, and so we combine them for the remainder ofthe analysis, unless we explicitly indicate otherwise.The results of repeating the fits for the combined distri-

butions are shown in Fig. 11, where we use a logarithmicy-axis scale in order to display the individual contribu-tions more clearly. The solid curves describe the distri-butions very well. For mKπ ∼ 0.7 GeV/c2, the curves inboth plots are slightly below the data, and we believe thatthis results from the parameterization of the S-wave am-plitude at low Kπ− mass values. If we normalize the S-wave amplitudes of Ref. [16] and Ref. [18] at 1.2 GeV/c2

and average them from threshold to 1.2 GeV/c2, the dis-crepancy is removed. However, the fits become slightlyworse in the region between the two K∗’s. Since the lat-ter region is very important to the present analysis, whilethe region around 0.7 GeV/c2 is much less so, we do notmake use of the modified S-wave amplitude.Based on Table III, the following branching fractions

can be calculated:

B(B− → J/ψπ−K0) = (1.101± 0.021)× 10−3 ,

(17)

B(B0 → J/ψπ−K+) = (1.079± 0.011)× 10−3 ,

(18)

B(B− → ψ(2S)π−K0) = (0.588± 0.034)× 10−3 ,

(19)

B(B0 → ψ(2S)π−K+) = (0.557± 0.016)× 10−3 ,

(20)

where we have corrected for the relevant J/ψ , ψ(2S),K0S → π0π0, and K0

L branching fractions [3]. Thequoted errors result from the statistical uncertainties ofTable III. The published value corresponding to Eq. (18)

15

1000

2000 S0K

-πψ J/→-B(a)

200

400

600 S0K

-π(2S)ψ →-B(c)

) 2 (GeV/c-πKm1 1.5 2

2000

4000

6000+K

-πψ J/→0B(b)

) 2 (GeV/c-πKm0.8 1 1.2 1.4 1.6

500

1000

1500

+K-π(2S)ψ →0B(d)

2E

vent

s/10

MeV

/c

FIG. 10: The efficiency-corrected and sideband-subtracted Kπ− mass distributions of Fig. 8 for the decay modes (a) B− →J/ψπ−K0

S , (b) B0 → J/ψπ−K+, (c) B− → ψ(2S)π−K0

S, and (d) B0 → ψ(2S)π−K+. The data are shown as open dots, andthe curves correspond to the fits described in the text.

TABLE III: Summary of the fit results corresponding to Fig. 10. For each decay mode, we list the total number of Kπevents after efficiency-correction and sideband-subtraction, χ2/NDF , K∗(892) mass and width, and the percentage S-, P -, andD-wave intensity contributions. Only statistical uncertainties are given.

Mode Corrected Events χ2/NDF m(K∗(892)) Γ(K∗(892)) S-wave P -wave D-wave(MeV/c2) (MeV) (%) (%) (%)

B− → J/ψπ−K0S 20985±393 117.6/149 892.9±0.8 49.0±1.9 17.0±1.6 72.5±1.3 10.5±1.0

B0 → J/ψπ−K+ 57231±561 171.4/149 895.5±0.4 48.9±1.0 15.7±0.8 73.5±0.7 10.8±0.5

B− → ψ(2S)π−K0S 5016±292 98.1/90 891.6±2.1 44.8±6.0 23.4±4.5 71.3±4.4 5.3±2.7

B0 → ψ(2S)π−K+ 13237±377 81.5/90 895.8±1.0 43.8±3.0 25.4±2.2 68.2±2.0 6.4±1.2

is (1.2 ± 0.6) × 10−3 [3], and for the mode of Eq. (20)an upper limit of 1 × 10−3 (90% confidence level (c.l.))is quoted [3]. No other information on these branch-ing fraction values exists; the present measurements thusrepresent significant improvements, even in the absenceof systematic error studies.

We note that the charged and neutral B meson decayrates to J/ψπ−K agree very well, and that this is truealso for decay to ψ(2S)π−K; also, the latter decays occurat slightly more than half the rate of the former.

VIII. THE Kπ− LEGENDRE POLYNOMIAL

MOMENTS

At the beginning of Sec. VII we pointed out the needto understand the Kπ− mass dependence of the angularstructure in the Kπ− system. In order to do this, wechoose to represent the Kπ− angular distribution at agiven mKπ− in terms of a Legendre polynomial expan-sion, following much the same procedure as described inAppendix A for our efficiency studies. In the notation of

16

) 2 (GeV/c-πKm1 1.5 2

-110

1

10

210

310

410 0,+K

-πψ J/→-,0B(a)

DataFit resultS-waveP-waveD-wave

) 2 (GeV/c-πKm0.8 1 1.2 1.4 1.6

-110

1

10

210

310

410 0,+K

-π(2S)ψ →-,0B(b) 2

Eve

nts/

10 M

eV/c

FIG. 11: The results of the fits to the Kπ− mass distributions for the combined Kπ− charge configurations (a) B−,0 →J/ψπ−K0,+ and (b) B−,0 → ψ(2S)π−K0,+. The data are shown as open dots, and the individual fit contributions are asindicated.

Eq. (A5), we write

dN

d cos θK= N

L∑

i=0

〈Pi〉Pi(cos θK) , (21)

where N is the number of events (after correction) in asmall mass interval centered at mKπ− , and L = 2ℓmax,where ℓmax is the maximum orbital angular momentumrequired to describe the Kπ− system at mKπ− . We canre-write Eq. (21) as

dN

d cos θK=N

2+

L∑

i=1

(N〈Pi〉)Pi(cos θK) , (22)

and extract the coefficients from the data using

N〈Pi〉 ≈N∑

j=1

Pi(cos θKj) , (23)

as in Appendix A. We refer to this coefficient as “theunnormalized Pi moment” in the course of our discus-sion. In order to incorporate the efficiency weighting andsideband subtraction procedures, we extend Eq. (23) asfollows:

N〈Pi〉 ≈N∑

j=1

(

1

ǫj

)

Pi(cos θKj)

+

NSB∑

k=1

(−1

ǫk

)

Pi(cos θKk) , (24)

where NSB is the number of sideband events falling inthis mKπ− interval, and the efficiency values, ǫ, are ob-tained as described in Appendix A. For convenience,we introduce the notation 〈PUi 〉 = N〈Pi〉, where the su-perscript U indicates that we refer to the unnormalizedmoment, and re-write Eq. (22) as

dN

d cos θK=N

2+

L∑

i=1

〈PUi 〉Pi(cos θK) . (25)

An overview of the (mKπ− , cos θK) structure is pro-vided by the rectangular Dalitz plots (see Appendix A)of Fig. 12. Data for the B decay modes of Eqs. (1) and (2)have been combined in Fig. 12(a), and data for those ofEqs. (3) and (4) have been combined in Fig. 12(b). InFig. 12(a), the intensity for cos θK < 0 is stronger thanfor cos θK > 0 in the region mKπ− < 0.85 GeV/c2. Asimilar asymmetry is present in the K∗(892) band, wherein addition there is a clear decrease in intensity aroundcos θK = 0. Above ∼ 1.2 GeV/c2 the backward region isenhanced, especially in the increased intensity region attheK∗

2 (1430). FormKπ− > 1.5 GeV/c2 the overall inten-sity is significantly decreased, but the backward region ofcos θK continues to be favored. Despite the smaller datasample of Fig. 12(b), the backward region seems againto be favored in the K∗(892) and K∗

2 (1430) regions, butlittle more can be said.For both Fig. 12(a) and 12(b) it is necessary to examine

the mKπ− dependence of the unnormalized moments inorder to quantify these qualitative features.

17

)2 (GeV/c-πKm1 1.5 2-1

-0.5

0

0.5

1

1

10

210(a)

)2 (GeV/c-πKm0.8 1 1.2 1.4 1.6-1

-0.5

0

0.5

1

1

10

210(b)

Kθco

s

FIG. 12: The cos θK versus mKπ− rectangular Dalitz plots for the combined decay modes (a) B−,0 → J/ψπ−K0,+, (b)B−,0 → ψ(2S)π−K0,+. The plots are obtained after efficiency weighting, but without ∆E sideband subtraction. The intensityscale is logarithmic, and is the same for both plots.

In order to facilitate discussion of the mKπ− depen-dence of the unnormalized moments, we first expressthem in terms of S-, P - and D-wave Kπ− amplitudes.These expressions have been obtained from the B →J/ψπK analysis of Ref. [17], after integration over theJ/ψ decay angles; they apply equally to the B mesondecay to ψ(2S)πK.

For an interval of mKπ− containing N events:

N = S2

0+ P 2

0+ D2

0+ P 2

+1 + P 2−1 +D2

+1 +D2−1 (26)

〈PU1 〉 = S0P0 cos(δS0− δP0

) (27)

+ 2

√

2

5P0D0 cos(δP0

− δD0)

+

√

6

5[P+1D+1 cos(δP+1

− δD+1)

+ P−1D−1 cos(δP−1

− δD−1)] ,

〈PU2 〉 =

√

2

5P

2

0+

√

10

7D

2

0(28)

+√

2S0D0 cos(δS0− δD0

)

−(

1√10

(

P 2+1 + P 2

−1

)

+5√10

28

(

D2+1 +D2

−1

)

)

,

〈PU3 〉 = 3

√

6

35P0D0 cos(δP0

− δD0) (29)

− 3

√

2

35(P+1D+1 cos(δP+1

− δD+1)

+ P−1D−1 cos(δP−1

− δD−1)) ,

〈PU4 〉 = 3√

2

7D

2

0− 2

√2

7

(

D2+1 +D2

−1

)

. (30)

The Si, Pi, and Di are amplitude magnitudes and idenotes the relevant helicity; the corresponding phaseangles are denoted by δ with the appropriate subscript.The helicity-zero terms, denoted in bold-face font, pro-vide the corresponding description of K−π+ elastic scat-tering. Equations (26)-(30) define the five measurablequantities accessible to the present analysis if we restrictourselves to S-, P -, and D-wave Kπ− amplitudes. How-ever, the equations involve seven amplitude magnitudesand six relative phase values, and so they cannot besolved in each mKπ− interval. For this reason, we canonly measure the mass dependence of the moments ofthe Kπ− system, and then use the results to understandhow the mKπ− and cos θK structure reflects into the ob-served ψπ− mass distributions, as will be discussed inSec. IX. In the following, allKπ− moments are sideband-subtracted and efficiency-corrected.

18

The dependence of 〈PU1 〉, and of 〈PU2 〉, on mKπ−

is shown in Fig. 13. For each moment, the behaviorfor ψ(2S) (Fig. 13(c),(d)) is very similar to that forJ/ψ (Fig. 13(a),(b)), but the statistical uncertainties aresignificantly larger because the net analysis sample issmaller by a factor of approximately six (Table II).

2>

/10

MeV

/cU 1

U

1(a) U

1(c) 

/10

MeV

/cU 2

U

2(b) U

2(d) <P

FIG. 13: The mKπ− dependence of (a) 〈PU1 〉 and (b) 〈PU2 〉for B0,− → J/ψπ−K+,0; the mKπ− dependence of (c) 〈PU1 〉and (d) 〈PU2 〉 for B0,− → ψ(2S)π−K+,0.

The distributions of Fig. 10 and Figs. 13-15 can becompared to those observed in Ref. [16], Fig. 6, for theK−π+ system produced in the reaction K−p→ K−π+nat 11 GeV/c K− beam momentum; the latter are rep-resentative of the moments structure of K−π+ elasticscattering. A striking overall feature is the strong sup-pression of the mass structure for mKπ− > 1 GeV/c2 rel-ative to the K∗(892) which is observed for the B mesondecay processes of Eqs. (1)-(4) in comparison to K−π+

elastic scattering.With this in mind, the 〈PU1 〉 distribution of Fig. 13(a)

bears a remarkable similarity to that in Ref. [16], exceptthat the sign is reversed. This is entirely consistent withthe analysis of Ref. [14], performed with a much smallerBABAR data sample (∼ 81 fb−1), which showed that theS0−P0 relative phase of Eq. (27) differed by π from thatobtained for K−π+ elastic scattering. In the K∗(892)region, the D-wave terms in Eq. (27) should be negli-gible, so that the relative phase offset should yield theobserved sign reversal w.r.t. K−π+ scattering. This signreversal continues through the K∗

2 (1430) region, whichsuggests that S0 −P0 interference remains the dominantcontribution to 〈PU1 〉, especially since 〈PU3 〉 is systemat-ically positive in this region (Figs. 14(a),(c)) (Note thatthe first term in Eq. (29) differs by only 1.8% from thesecond term in Eq. (27)).

The behavior of 〈PU2 〉 in the K∗(892) region is verysimilar to that observed in Ref. [16], and, ignoring D-wave contributions, agrees in magnitude and sign witha calculation using the values of P0, P+1, and P−1 fromRef. [14]. In Ref. [16], 〈PU2 〉 is positive and much largerat the K∗

2 (1430) than at the K∗(892). Clearly, this is notthe case in Figs. 13(b) and 13(d), where 〈PU2 〉 is small andnegative at the K∗

2 (1430). From Eq. (28), this could oc-cur if S0 is also shifted in phase by π relative to D0, whilethe D2

0 and (D2+1+D

2−1) contributions to Eq. (28) essen-

tially cancel. The latter is suggested by the observationof only a small 〈PU4 〉 signal at the K∗

2 (1430) in Fig. 14(b),and the absence of signal in this region of Fig. 14(d). InEq. (30), D2

0 is favored 3:2 over (D2+1+D

2−1), whereas in

Eq. (28) the ratio is 4:5, hence the conjecture that thesecontributions may in effect be canceling in Figs. 13(b)and 13(d).

2>

/10

MeV

/cU 3

U

3(a) U

3(c) 

/10

MeV

/cU 4

U

4(b) U

4(d) 1.2 GeV/c2)of (a) 〈PU3 〉 and (b) 〈PU4 〉 for B0,− → J/ψπ−K+,0; themKπ− dependence of (c) 〈PU3 〉 and (d) 〈PU4 〉 for B0,− →ψ(2S)π−K+,0.

The 〈PU3 〉 and 〈PU4 〉 moments require that D-wavecontributions be present. At the statistical level of thepresent analysis, we do not expect to observe such con-tributions for mKπ− < 1.2 GeV/c2, and so we beginthe distributions of Fig. 14 at this value. In Figs. 14(a)and 14(c), 〈PU3 〉 is systematically positive for all mKπ−

mass values above 1.3 GeV/c2. This results from the en-hancement observed in Fig. 12 for cos θK < 0, and is sim-ilar to the behavior in Ref. [16], but at a much-reducedintensity level, as mentioned previously. The 〈PU4 〉 mo-ments of Fig. 14(b) and 14(d) have been discussed al-ready. The dip to negative values atmKπ− ∼ 1.3 GeV/c2

in Fig. 14(b) is interesting, since it may indicate that the

19

relative strength of the D20 and (D2

+1 + D2−1) contribu-

tions to Eq. (30) varies with mKπ− . The detailed am-plitude analyses of the K∗(892) region [14, 19] do notconsider such a possibility for the P0, P+1, and P−1 am-plitudes, but a mass-independent approach to these anal-yses would require a much larger data sample.Finally, the extended mKπ− range available for the

combined B−,0 → J/ψπ−K0,+ data samples allows us tosearch for evidence of F -wave amplitude contributionsassociated with the K∗

3 (1780), which has mass ∼ 1.78GeV/c2 and width ∼ 0.20 GeV [16]. In Fig. 15(a) weshow the mKπ− dependence of 〈PU5 〉 for mKπ− > 1.2GeV/c2, and in Fig. 15(b), the dependence of 〈PU6 〉 formKπ− > 1.5 GeV/c2. Interference between D- and F -wave amplitudes could yield a 〈PU5 〉 distribution char-acterized by the underlying D-wave BW amplitude. Forzero relative phase, the mKπ− dependence resulting fromthe overlap of the leading edge of the F -wave BW ampli-tude with the entire D-wave BW amplitude would resem-ble the real part of theD-wave BW. Figure 15(a) exhibitsjust such behavior; the intensity increases from near zeroto a maximum below the K∗

2 (1430), passes through zeronear the nominal mass value, reaches a minimum justbelow 1.5 GeV/c2, returns to zero near 1.6 GeV/c2, andhas no clear structure thereafter. A 〈PU6 〉 moment wouldinvolve F 2

0 and (F 2+1 + F 2

−1) intensity contributions ofopposite sign, just as for the P - and D-waves. We seeno clear signal in Fig.15(b), although the distribution issystematically negative in the region 1.7 − 1.9 GeV/c2,so it could be that these contributions almost cancel, asmay be the case for the D-wave amplitudes. In the over-all mKπ− distributions, these contributions add, and it isinteresting to note that in Fig. 11(a) there is a small ex-cess of events above the fitted curve in the region of theK∗

3 (1780). Our fit to the mass distribution could pos-sibly be improved slightly in this region by including aK∗

3 (1780) contribution, but we do not do this at present.It seems reasonable to interpret the mKπ− dependence of〈PU5 〉 observed in Fig. 15(a) as indicating the presence ofa K∗

3 (1780) amplitude, in which case this would be thefirst evidence for B meson decay to a final state includinga spin three resonance.In summary, the angular structures observed for the

Kπ− systems produced in B−,0 → J/ψπ−K0,+ andB−,0 → ψ(2S)π−K0,+ show interesting features, whichcan be well understood on the basis of the expected Kπ−

amplitude contributions. Moreover, the main featuresagree well between the J/ψ and ψ(2S) modes, takingaccount of the statistical limitations of the latter datasample.

IX. REFLECTION OF Kπ− STRUCTURE INTO

THE ψπ− MASS DISTRIBUTIONS

We now investigate the extent to which reflection of theKπ− mass and angular structures described in Secs. VIIand VIII are able to reproduce the efficiency-corrected

) 2 (GeV/c-πKm1.2 1.4 1.6 1.8 2 2.2

2>

/10

MeV

/cU 5

U

5(a) 

/10

MeV

/cU 6

U

6(b) 1.2GeV/c2 for B0,− → J/ψπ−K+,0; (b) the mKπ− dependenceof 〈PU6 〉 for mKπ− > 1.5 GeV/c2 for B0,− → J/ψπ−K+,0.

and sideband-subtracted ψπ− mass distributions.We do this by using a MC generator which initially

creates large samples of unit weight B−,0 → ψπ−K0,+

events with the correct B meson production angular dis-tribution in the overall c.m. frame, and distributed inKπ− mass according to the fit functions obtained as de-scribed in Sec. VII. The distribution in cos θK is uniformat each value of mKπ− , and so is described by

dN

d cos θK=N

2, (31)

which is just Eq. (25) with no angular structure.Since we are now dealing with efficiency-corrected dis-

tributions, and since mass resolution cannot generate lo-cal structure (Fig. 6), we need not subject the generatedevents to detector response simulation and subsequentevent reconstruction. Consequently we can create verylarge MC samples.The Kπ− mass distributions generated in this way

for B−,0 → J/ψπ−K0,+ and B−,0 → ψ(2S)π−K0,+ areshown in Figs. 16(a) and 16(b), respectively. Each distri-bution contains ten million events, and has been normal-ized to the total number of events in the correspondingcorrected data sample; each represents the relevant fit

20

function of Fig. 11 very well. We can use these events tocreate ψπ− mass distributions according to any desiredselection criteria, and we provide several examples of thislater.

) 2 (GeV/c-πKm1 1.5 2

2E

vent

s/10

MeV

/c

0

2000

4000

6000

80000,+K-πψ J/→-,0B(a)

) 2 (GeV/c-πKm0.8 1 1.2 1.4 1.60

500

1000

1500

20000,+K-π(2S)ψ →-,0B(b)

FIG. 16: The Kπ− mass distributions generated according tothe Kπ− fit functions of Fig. 11 for (a) B−,0 → J/ψπ−K0,+

and (b) B−,0 → ψ(2S)π−K0,+. Each distribution containsten million events, normalized to the total number of eventsobserved in the corrected data sample (Fig. 11).

However, in Sec. VIII we showed that there is a greatdeal of angular structure in the Kπ− system, as repre-sented by the mKπ− dependence of the 〈PUi 〉 moments,and this affects the reflection from the Kπ− system intothe ψπ− mass distribution.We take this Kπ− angular structure into account by

returning to Eq. (25), removing a factor of N/2 on theright side, and so obtaining

dN

d cos θK=N

2

(

1 +

L∑

i=1

(2

N)〈PUi 〉Pi(cos θK)

)

, (32)

i.e.

dN

d cos θK=N

2

(

1 +

L∑

i=1

〈PNi 〉Pi(cos θK)

)

, (33)

where

〈PNi 〉 = 2

N〈PUi 〉 (34)

is defined to be the “the normalized Pi moment”. We canthus incorporate the measured Kπ− angular structureinto our generator by giving weight wj to the jth eventgenerated, where

wj = 1 +L∑

i=1

〈PNi 〉Pi(cos θKj) . (35)

The 〈PNi 〉 are evaluated for the mKπ− value of the jth

event by linear interpolation of the values obtained bynormalizing the 〈PUi 〉 of Figs. 13-15 according to Eq. (34).The results are shown in Figs. 17-19, where we have

used modified mass intervals in order to reduce statisti-cal fluctuation. We interpolate linearly using the lines

connecting the measured values. In order to take intoaccount the statistical uncertainties, we also interpolateusing lines connecting the +1σ error values, and linesconnecting the −1σ error values, as shown by the shadedregions in each plot.

>

N 1N

1(a) N

1(c) 

N 2<

P-2

-1

0

1

>N

2(b) N

2(d) <P

FIG. 17: The normalized moments corresponding to Fig. 13obtained by using Eq. (34). Mass intervals have been com-bined in order to reduce statistical fluctuations. The linesindicate the linear interpolations used in weighting the MCevents. The shaded regions indicate the ±1σ variations usedto account of statistical uncertainties.

The ψπ− mass distributions for the entire Kπ− massrange for the decay modes B−,0 → J/ψπ−K0,+ andB−,0 → ψ(2S)π−K0,+ are shown in Fig. 20(a) andFig. 20(b), respectively. The points represent the dataafter correcting for efficiency and subtracting the eventsin the ∆E sideband. The dashed curves show the reflec-tion from Kπ− assuming a flat cos θK distribution. Thesolid curves are obtained by weighting each event accord-ing to Eq. (35). The shaded bands associated with thesolid curves indicate the effect of interpolation using ±1σnormalized moment values, as described above.We emphasize that the absolute normalization of the

curves shown in Figs. 20(a) and 20(b) is established bythe scale factor used to normalize each of our ten-million-event MC samples to the corresponding corrected numberof events, as shown in Figs. 16(a) and 16(b), respectively.Comparison of the dashed and solid curves of Fig. 20

shows that it is important to modulate the cos θK distri-butions using the normalized moment weights of Eq. (35).Since the individual Pi(cos θK) functions integrate to zeroover cos θK , the incorporation of the wj weights does notaffect the distributions of Fig. 16. This also means thatthe associated dashed and solid curves of Fig. 20 inte-grate to the same total number of events.

21

>

N 3N

3(a) N

3(c) 

N 4N

4(b) N

4(d) 

N 5N

5(a) 

N 6N

6(b) <P

0,+K-πψ J/→-,0B


The reasons for the enhancement of the solid curves athigh ψπ− mass values, and their suppression at lowermass values are made clear by the rectangular Dalitzplots of Figs. 21(a) and 21(b). We plot cos θψ againstmψπ− , where cos θψ is the normalized dot-product of theψπ− three-momentum vector in the parent B meson restframe and the ψ three-momentum vector in the ψπ− restframe. A guide to the structures observed in these plots isprovided by Figs. 21(c) and 21(d), where we indicate thelocus of the K∗(892) band (0.795 − 0.995 GeV/c2) andthat of the K∗

2 (1430) band (1.330 − 1.530 GeV/c2), aschosen in Ref. [5]; in addition, we label regions (A)-(E),defined in Sec. X. The dashed vertical lines indicate theZ(4430)− region from Ref. [5]. The high-mψπ− region ofeach of these bands corresponds to cos θK < 0, and inFig. 21(a) this region is clearly populated preferentiallyfor both K∗ bands, and even for the Kπ− mass rangein between. This corresponds to the backward-forwardasymmetry observed in Fig. 12, and to the negative val-ues observed for 〈PU1 〉 in Fig. 13(a), and to the positivevalues of 〈PU3 〉 in Fig. 14(a). These high-mψπ− enhance-ments are compensated by the low-mψπ− suppression ofthe solid curve relative to the dashed curve in Fig. 20(a),since the integral along anymKπ− locus is independent ofthe cos θK distribution. Similar behavior is observed forFig. 20(b), but at a reduced statistical level. Backward-forward asymmetry is observed in Fig. 12(b), where 〈PU1 〉is primarily negative in Fig. 13(c) and 〈PU3 〉 is positive inFig. 14(c), so that the net effect on the ψπ− mass distri-bution of Fig. 20(b) is much the same as in Fig. 20(a).

In Figs. 20(c) and 20(d), we show the residuals (data- solid curves) for the distributions in Figs. 20(a) and20(b), respectively. The dashed vertical lines indicatemψπ− = 4.433 GeV/c2 [5].

There is an excess of events in Fig. 20(c) for mJ/ψπ−

∼ 4.61 GeV/c2, as well as in Fig. 25(b) and Fig. 26(c) be-low. The effect is associated with the K∗(892) region ofKπ− mass (Fig. 24(b)), for which the mJ/ψπ− distribu-tion decreases steeply in the high-mass region. The massresolution there is ∼ 9 MeV/c2 (Fig. 6(a),(c)) and wehave not incorporated this into our calculations. For thisreason, the calculated curve falls systematically belowthe data, hence yielding a spurious peak in the residualdistribution. For the corresponding mψ(2S)π− distribu-

tions, the mass resolution is ∼ 4 MeV/c2 (Fig. 6(b),(d))at ∼ 4.6 GeV/c2, the statistical fluctuations are larger,and no similar effect is observed (Figs. 20(d), 25(g), and26(d)). Apart from this, the distribution of the residualsshows no evidence of statistically significant departurefrom zero at any J/ψπ− mass value.

In Fig. 20(b), and correspondingly in Fig. 20(d), thesmall excess of events at ∼ 4.48 GeV/c2 provides the onlyindication of a narrow signal. As shown in Sec. XI B, thisyields a 2.7σ enhancement with mass ∼ 4.476 GeV/c2

and width consistent with that reported in Ref. [5]. Thedot-dashed curve in Fig. 20(b) was obtained from thedashed curve by modulating the Kπ− angular distribu-tion using instead the normalized Kπ− moments from

22

2E

vent

s/10

MeV

/c

0

500

1000 KθFlat cos

K moments-πψJ/(a)

0

200

400K

θFlat cosK moments-π(2S)ψ

K moments-πψJ/(b)

)2 (GeV/c-πψJ/m3.5 4 4.5

2R

esid

ual/1

0 M

eV/c

-200

0

2000,+K-πψ J/→-,0B (c)

)2 (GeV/c-π(2S)ψm3.8 4 4.2 4.4 4.6 4.8

-200

0

2000,+K-π(2S)ψ →-,0B (d)

FIG. 20: The ψπ− mass distributions for the combined decay modes (a) B−,0 → J/ψπ−K0,+ and (b) B−,0 → ψ(2S)π−K0,+.The points show the data after efficiency correction and ∆E sideband subtraction. The dashed curves show the Kπ− reflectionfor a flat cos θK distribution, while the solid curves show the result of cos θK weighting. The shaded bands represent the effectof statistical uncertainty on the normalized moments. In (b), the dot-dashed curve indicates the effect of weighting with thenormalized J/ψπ−K moments. The dashed vertical lines indicate the value of mψπ− = 4.433 GeV/c2. In (c) and (d), we showthe residuals (data-solid curve) for (a) and (b), respectively.

the B−,0 → J/ψπ−K0,+ data, which show no evidenceof a Z(4430)− signal. This curve and the solid curve differonly slightly in the range∼ 4.2 GeV/c2 to ∼ 4.55 GeV/c2,so that the Kπ− background function at ∼ 4.48 GeV/c2

is not very sensitive to the modulation procedure, nor tothe presence of a small, narrow mψ(2S)π− enhancement(see Sec. XIB for a quantitative discussion).We conclude that the mψ(2S)π− distribution of

Fig. 20(b), and the residual distribution of Fig. 20(d),do not provide confirmation of the Z(4430)− signal re-ported in Ref. [5].

X. COMPARISON TO THE BELLE RESULTS

We now compare our results to those obtained by Bellefor B → ψ(2S)π−K [5].

A. The ψπ− mass resolution

In Sec. V we showed (Fig. 6) our mass resolution de-pendence on Q-value, and obtained HWHM ∼ 4 MeV/c2

for the ψ(2S)π− system at the Z(4430)−. In Ref. [5],it is stated only that the mass resolution is 2.5 MeV/c2.Since the width of the Z(4430)− is ∼ 45 MeV [5], mass

resolution should not be an issue for the comparison ofsimilar data samples (see Sec. XE).

B. Efficiency

We have made a detailed study of efficiency over eachDalitz plot for each J/ψ and ψ(2S) decay mode sepa-rately (Sec. VI), and have identified efficiency losses as-sociated with low-momentum pions and kaons in the lab-oratory frame (Fig. 9). We illustrate the effect of suchlosses on the mψπ− distributions using our ten-million-event B−,0 → ψπ−K0,+ samples weighted to take ac-count of the Kπ− angular structure (Sec. IX). In Fig. 22we show the ψπ− distributions obtained as for Fig. 20(solid curves). We then require that the momentum ofthe π be less than 100 MeV/c in the laboratory frame(Fig. 9) and obtain the shaded distributions in the mψπ−

threshold regions. Similarly, the requirement that thekaon momentum be less than 250 MeV/c in the labora-tory frame (Fig. 9) yields the cross-hatched regions nearmaximum ψπ− mass [20]. It follows that the regions oflower efficiency discussed in Appendix A should have nosignificant effect on the region of the Z(4430)−.As a direct check of the effect of our efficiency-

correction procedure, we show our mψπ− distribu-tions before and after correction in Figs. 23(a),(b)

23

1

10

210

ψ

J/θco

s

-0.5

0

0.5

1

(a)1

10

210

(2

S)ψθ

cos

-0.5

0

0.5

1

(b)

) 2 (GeV/c-πψJ/m3.5 4 4.5

ψ

J/θco

s

-1

-0.5

0

0.5

1

(c)

(1430)2*

K

(892)*

K

A

B

C

D

E

) 2 (GeV/c-π(2S)ψm4 4.2 4.4 4.6 4.8

(2

S)ψθ

cos

-1

-0.5

0

0.5

1

(d)

(1430)2*

K

(892)*

K

A

B

C

D

E

FIG. 21: The cos θψ versus mψπ− rectangular Dalitz plots for (a) B−,0 → J/ψπ−K0,+, and (b) B−,0 → ψ(2S)π−K0,+; (c)and (d), the corresponding plots indicating the loci of the K∗(892) and K∗

2 (1430) resonance bands defined in the text; regionsA-E, defined by Eqs. (36)-(40), are indicated. The dashed vertical lines show the mass range 4.400 < mψπ− < 4.460 GeV/c2.

) 2 (GeV/c-πψJ/m3.5 4 4.5

2E

vent

s/10

MeV

/c

0

500

1000

10M evts normalizedSlow kaons 1.2 %Slow pions 1.6 %

(a)0,+K-πψ J/→-,0B

) 2 (GeV/c-π(2S)ψm3.8 4 4.2 4.4 4.6 4.80

100

200

30010M evts normalizedSlow kaons 2.9 %Slow pions 1.6 %

(b)0,+K-π(2S)ψ →-,0B

FIG. 22: (a) The curve of Fig. 20(a), and (b) that ofFig. 20(b), obtained by cos θK weighting; the shaded regionsnear threshold correspond to P (π−) < 0.1 GeV/c in the lab-oratory frame, while the cross-hatched regions near maxi-mum ψπ− mass represent P (K) < 0.25 GeV/c in the labora-tory frame; the dashed vertical lines indicate mψπ− = 4.433

GeV/c2.

and 23(d),(e) for B−,0 → J/ψπ−K0,+ and B−,0 →ψ(2S)π−K0,+, respectively. Then in Fig. 23(c) and 23(f)we show the ratio of the uncorrected and corrected distri-butions as a measure of average efficiency. As expected,the average value decreases rapidly near threshold, andnear the maximum value for both distributions. Awayfrom these regions, the efficiency increases slowly withincreasing mass. We conclude that our event reconstruc-tion efficiency should have no effect on any Z(4430)−

signal in our data.

In Fig. 23(f) we show the result of a linear fit, excludingthe regionsmψ(2S)π− < 3.9 GeV/c2 and mψ(2S)π− > 4.71

GeV/c2, which are seriously affected by the loss of lowmomentum pions and kaons, respectively. The fitted effi-ciency value increases from 13.7 % to 15.0 % over thefitted region. The low-efficiency regions are excludedwhen we compare our uncorrected mψ(2S)π− distributionto that from Belle (Sec. XE). This is due to the factthat for both experiments the reconstruction efficiencyfor very low momentum charged-particle tracks in thelaboratory frame decreases rapidly to zero (cf. Fig. 9).

24

There is no detailed discussion of efficiency in Ref. [5].

100

200

100

200

100

200

No efficiency correction(a)

S0K-πψ J/→-B+K-πψ J/→0B

20

40

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60

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40

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60

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S0K-π(2S)ψ →-B+K-π(2S)ψ →0B

500

1000

500

1000

500

1000Corrected for efficiency(b)

S0K-πψ J/→-B+K-πψ J/→0B

100

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400

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100

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400Corrected for efficiency(e)

S0K-π(2S)ψ →-B+K-π(2S)ψ →0B

) 2 (GeV/c-πψJ/m3.5 4 4.5

0.1

0.2

0.3(c)

) 2 (GeV/c-π(2S)ψm3.8 4 4.2 4.4 4.6

0.1

0.2

0.3(f)

2E

vent

s/10

MeV

/c2

Rat

io/1

0 M

eV/c

FIG. 23: The cumulative ψπ− mass distributions for the K0S

andK+ decay modes (a) and (d) before, and (b) and (e) after,efficiency correction. The ratio of uncorrected to correcteddata is shown in (c) and (f). The fitted line in (f) is describedin the text.

C. The Kπ− mass and cos θK structures

In Ref. [5], the K∗(892) and K∗2 (1430) regions

of the Dalitz plot are removed, and the remain-ing non-Z(4430)− contribution to the mψ(2S)π− massdistribution is described by a second-order polyno-mial in mψ(2S)π− multiplied by the momentum of the

ψ(2S) in the ψ(2S)π− rest frame and by the factor√

mmax −mψ(2S)π− ; for two-out-of-three phase space,this latter factor should be the momentum of the recoilK in the B rest frame. Also, the ∆E sideband contri-butions are not subtracted prior to their fit. There is nodiscussion of a possible description of the background interms of Kπ− mass and angular structures.

In our analysis, we have considered the effect of theKπ− mass and cos θK structures (Sec. VII and Sec. VIII),and have shown how the observed features affect theψπ− mass distributions in Sec. IX. The resulting dashedcurves of Fig. 22 cannot be described in terms of second-order polynomials. However, each corresponds to a pro-jection of the entire Dalitz plot. In order to make a directcomparison to the Belle data, we investigate the relevantregions of Kπ− mass in the following section.

D. Regions of Kπ− mass

The ψ(2S)π mass distribution shown in Fig. 2 of Ref.[5]has a “K∗ veto” applied. This means that events within100 MeV/c2 of the K∗(892) or the K∗

2 (1430) have beenremoved, and hence that the Kπ− mass range has, ineffect, been divided into five regions, as follows:

regionA : mKπ− < 0.795 GeV/c2 , (36)

regionB : 0.795 < mKπ− < 0.995 GeV/c2 , (37)

regionC : 0.995 < mKπ− < 1.332 GeV/c2 , (38)

regionD : 1.332 < mKπ− < 1.532 GeV/c2 , (39)

regionE : mKπ− > 1.532 GeV/c2 . (40)

These regions are labeled in Fig. 21(c) and Fig. 21(d).The ψ(2S) mass distribution of Ref. [5] thus containsevents (with sideband contribution) from regions A, C,and E.In Fig. 24 we show the corrected mψπ− distributions

for regions A-E of Kπ− mass. The solid curves andshaded bands correspond to those in Fig. 20, with thesame overall normalization constants as obtained forFig. 16, i.e., there is no renormalization in the separateKπ− mass regions. In Figs. 24(f)-(j), the dot-dashedcurves were obtained using the normalized moments fromB−,0 → J/ψπ−K0,+ in conjunction with Eq. (35). Forregions A and B, there is almost no difference betweenthe solid and dot-dashed curves, while in the other re-gions the differences are less than, or of the order of, thestatistical fluctuations in the associated data. The resid-uals (obtained by subtracting the solid curves from thedata) corresponding to Fig. 24 are shown in Fig. 25, andshow no evidence of structure. In Figs. 26 and Fig. 27we make similar comparisons for the combined data inthe K∗ regions (B and D), and for the K∗-veto region(A, C, and E). Again the residuals reveal no significantstructure.

E. Direct comparison

The ψ(2S)π− mass distribution of Fig. 28(a) is a re-production of that in Ref. [5], except for the additionof error bars [21]. In Fig. 28(b) we show the equiva-lent distribution for our combined analysis samples forthe B meson decay processes of Eqs. (3) and (4) for theK∗-veto region (A, C, and E). The mass intervals arethe same as for Fig. 28(a), but no efficiency-correctionhas been performed. As mentioned in Sec. XC, whenwe make quantitative comparisons between Fig. 28(a)and Fig. 28(b) we exclude the low-efficiency regions nearthreshold and at high mass, and use only the region3.9 < mψ(2S)π− < 4.71 GeV/c2. We make a global com-parison of the data samples in Table IV. The BABAR

sample contains ∼ 8% more background than does theBelle sample. The net signal ratio is 1.18± 0.09 in favorof Belle, although the corresponding integrated luminos-ity ratio is 1.46. It follows that for BABAR, net signal per

25

-50

0

50

1002<0.795 GeV/c-πKm

0,+K-πψ J/→-,0B

K moments-πψJ/

(a)

0

1002<0.795 GeV/c-πKm

0,+K-π(2S)ψ →-,0B

K moments-π(2S)ψK moments-πψJ/

(f)

0

500

10002<0.995 GeV/c-πK0.795<m

(b)

100

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4002<0.995 GeV/c-πK0.795<m

(g)

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2002<1.332 GeV/c-πK0.995<m

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(h)

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)2 (GeV/c-πψJ/m3.5 4 4.5

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50100

2>1.532 GeV/c-πKm (e)

)2 (GeV/c-π(2S)ψm3.8 4 4.2 4.4 4.6 4.8

-200

2040 2>1.532 GeV/c-πKm (j)

2E

vent

s/10

MeV

/c

FIG. 24: The ψπ− mass distributions in regions A-E of Kπ− mass for the combined decay modes (a-e) B−,0 → J/ψπ−K0,+,and (f-j) B−,0 → ψ(2S)π−K0,+; the open dots represent the data, and the solid curves and shaded bands are as in Fig. 20,but calculated for the relevant Kπ− mass region using the same overall normalization constant as for Fig. 16. In (f-j), thedot-dashed curves are obtained using Kπ− normalized moments for B−,0 → J/ψπ−K0,+ in Eq. (35), instead of those fromB−,0 → ψ(2S)π−K0,+; the dot-dashed vertical lines indicate mψπ− = 4.433 GeV/c2.

-50

0

500,+K

-πψ J/→-,0B2<0.795 GeV/c-πKm(a)

-50

0

50

100 0,+K-π(2S)ψ →-,0B2<0.795 GeV/c-πKm

(f)

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200 2<0.995 GeV/c-πK0.795<m(b)

-100

0

100

200 2<0.995 GeV/c-πK0.795<m(g)

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100 2<1.332 GeV/c-πK0.995<m(c)

-100

0

1002<1.332 GeV/c-πK0.995<m

(h)

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50

1002<1.532 GeV/c-πK1.332<m

(d)

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0

100

2002<1.532 GeV/c-πK1.332<m

(i)

)2 (GeV/c-πψJ/m3.5 4 4.5

-50

0

502>1.532 GeV/c-πKm (e)

)2 (GeV/c-π(2S)ψm3.8 4 4.2 4.4 4.6 4.8

-200

2040 2>1.532 GeV/c-πKm (j)

2

Res

idua

l/10

MeV

/c

FIG. 25: The residuals (data - solid curve) corresponding to Fig. 24; the dot-dashed vertical lines indicate mψπ− = 4.433

GeV/c2.

26

2E

vent

s/10

MeV

/c

500

1000 0,+K-πψ J/→-,0B(1430)

2

*(892) + K

*K

(a)

100

200

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400

0,+K-π(2S)ψ →-,0B(1430)

2

*(892) + K

*K

(b)

)2 (GeV/c-πψJ/m3.5 4 4.5

2R

esid

ual/1

0 M

eV/c

-200

0

200(c)

)2 (GeV/c-π(2S)ψm3.8 4 4.2 4.4 4.6 4.8

-200

0

200(d)

FIG. 26: The ψπ− mass distributions for (a) B−,0 → J/ψπ−K0,+, and (b) B−,0 → ψ(2S)π−K0,+, for mKπ− regions B andD combined; the open dots represent the data, the solid, dashed, and dot-dashed curves, and the shaded bands, correspond tothose of Fig. 20(a),(b); (c) and (d), the corresponding residual distributions. The dashed vertical lines indicate mψπ− = 4.433

GeV/c2.

2E

vent

s/10

MeV

/c

0

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300 0,+K-πψ J/→-,0B veto

*K

(a)

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100

2000,+K-π(2S)ψ →-,0B

veto*

K

(b)

)2 (GeV/c-πψJ/m3.5 4 4.5

2R

esid

ual/1

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eV/c

-100

0

100(c)

)2 (GeV/c-π(2S)ψm3.8 4 4.2 4.4 4.6 4.8

-100

0

100

(d)

FIG. 27: The ψπ− mass distributions for (a) B−,0 → J/ψπ−K0,+, and (b) B−,0 → ψ(2S)π−K0,+, for mKπ− regions A, C, andE, combined; the open dots represent the data, the solid, dashed, and dot-dashed curves, and the shaded bands, correspond tothose of Fig. 20(a),(b); (c) and (d), the corresponding residual distributions. The dashed vertical lines indicate mψπ− = 4.433

GeV/c2.

27

unit luminosity is ∼ 1.34, while for Belle it is 1.08, andthat this significant increase in signal yield comes at thecost of only a modest increase in background level. Forboth experiments the background distribution increasestoward threshold, and differs markedly in ψ(2S)π− massdependence from the signal.

10

20

30

10

20

30

(a) Belle

data in signal region

data in sideband region

)2 (GeV/c-π(2S)ψm3.8 4 4.2 4.4 4.6 4.8

10

20

30

)2 (GeV/c-π(2S)ψm3.8 4 4.2 4.4 4.6 4.8

10

20

30

(b)BABAR

data in signal region

data in sideband region

2E

vent

s/10

MeV

/c

FIG. 28: a) The ψ(2S)π− mass distribution after K∗ vetofrom Ref. [5]; the data points represent the signal region

(we have assigned√N errors at present), and the shaded

histogram represents the background contribution estimatedfrom the ∆E sideband regions; b) shows the correspondingdistribution from the BABAR analysis. The dashed verticalline indicates mψ(2S)π−= 4.433 GeV/c2.

We have shown our efficiency-corrected and sideband-subtracted mass distribution corresponding to Fig. 28(b)in Fig. 27(b), and should compare the latter to theequivalent Belle distribution. However, this is not avail-able, and so we make do with the distributions ofFig. 28 instead. In order to justify the use of

√N er-

ror assignments, we combine adjacent mass intervals formψ(2S)π− < 4.18 GeV/c2 and mψ(2S)π− > 4.55 GeV/c2

so that we obtain at least ten events (signal + sideband)in each mass interval. We then create the sideband-subtracted distributions of Fig. 29(a) and Fig. 29(b) fromthe data of Fig. 28(a) and Fig. 28(b), respectively. InFig. 29(b), we have scaled our data by the factor 1.18 tocompensate for the statistical difference between the ex-periments. In Fig. 29(c) we show the result of subtractingthe distribution of Fig. 29(b) from that of Fig. 29(a), witherrors combined in quadrature. There is no evidence ofany statistically significant difference, in particular nearmψ(2S)π− = 4.433 GeV/c2, indicated by the dashed ver-tical line. If the low-efficiency (Fig. 23(f)) cross-hatchedregions of Fig. 29(c) are excluded, the χ2−value for the

0

10

20

30 (a)Belle

0

10

20

30 (b)BABAR (*1.18)

) 2 (GeV/c-π(2S)ψm3.8 4 4.2 4.4 4.6 4.8

2D

iff.

/10

MeV

/c

-20

-10

0

10 (c)

/NDF = 54.7/582χ

2E

vent

s/10

MeV

/c

FIG. 29: a) The distribution of Fig. 28(a) after combiningmass intervals for mψ(2S)π− < 4.18 GeV/c2 and mψ(2S)π− >

4.55 GeV/c2, and carrying out sideband subtraction. b) Thedistribution of Fig. 28(b) after following the same procedure,and in addition scaling by 1.18, as described in the text. c)The difference, (a)-(b), where the errors have been combinedin quadrature; χ2/NDF = 54.7/58 (Probability=59.9%) ex-cluding the low-efficiency regions (cross-hatched).

remaining region is found to be 54.7 for 59 mass inter-vals. There is one normalization constant (1.18), so thatthe comparison yields χ2/NDF = 54.7/58, with corre-sponding probability 59.9%.

We conclude that the Belle and BABAR distributionsof Fig. 29 are statistically consistent. We have shownin Fig. 20 and Figs. 24-27 that all of our corrected ψπ−

distributions are well-described by reflections of the massand angular structures of the Kπ− system. We refer tothis as our Kπ− background, and in Sec. XI we quan-tify the extent to which an additional Z(4430)− signalis required to describe the corrected BABAR ψπ− massdistributions.

We have mentioned previously that it is the backward-forward asymmetry in the Kπ− angular distribution asa function of mKπ− which yields the high mass enhance-ments seen in our ψπ− mass distributions. We show thiseffect explicitly in Fig. 30, where we plot the distributionof cos θπ = − cos θK for regions A, C, and E ofKπ− mass.For cos θπ ∼ 1, mψ(2S)π− is near its maximum value,

and m2ψπ is related linearly to cos θπ, and so it is not

surprising that the sideband subtracted distributions ofFig. 30(b) and Fig. 30(d) bear a strong shape resemblanceto the correspondingmψ(2S)π− distributions of Fig. 27(a)and Fig. 27(b), respectively. We note also that the in-crease in the sideband distribution of Fig. 30(c) towards

28

TABLE IV: Summary of mψ(2S)π− data from regions A, C, and E, combined (Fig. 28) for Belle and BABAR in the mass range

3.9-4.71 GeV/c2.

Category Belle [Fig. 28(a)] BABAR [Fig. 28(b)]

Total signal region events (N) 824± 29 786± 28

Sideband contribution (B) 172± 9 234± 15

B/N 21.5% 29.8%

Net signal 652± 30 552± 32

cos θπ ∼ −1 corresponds to the increase in Fig. 30(b)toward ψπ− threshold. Figure 30 illustrates how impor-tant it is to take into account the angular structure inthe Kπ− system in creating the shape of the associatedψπ− mass distribution, even after the removal of the K∗

regions.

100

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(a) 0,+K-πψ J/→-,0B

Signal regionE sideband∆ 20

40

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80

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80 (c) 0,+K-π(2S)ψ →-,0B

-πθcos-1 -0.5 0 0.5 10

100

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300(b) Sideband-subtracted

-πθcos-1 -0.5 0 0.5 1

20

40

60

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Eve

nts/

0.1

FIG. 30: The cos θπ distributions for the combined Kπ− massregions A, C, and E, (a), (b) for the decay modes B−,0 →J/ψπ−K0,+, and (c), (d) for B−,0 → ψ(2S)π−K0,+. In (a)and (c), the points represent the data in the signal region,and the shaded histograms show the ∆E sideband events. In(b) and (d), the signal region distributions are shown aftersideband subtraction. No efficiency correction is applied tothe data.

XI. FITS TO THE CORRECTED ψπ− MASS

DISTRIBUTIONS

A. The fit results

In Fig. 31 we show the results of the fits to the cor-rected BABAR data of Figs. 20(a),(b), Figs. 26(a),(b), andFigs. 27(a),(b). In each fit, the Kπ− background distri-bution has been multiplied by a free normalization pa-rameter; an S-wave BW line shape, with free normaliza-tion, mass and width parameters, has been added inco-herently in order to quantify the search for a Z(4430)−

signal. In each figure, the solid curve represents thefit result; the parameter values for the correspondingZ(4430)− signal are summarized in Table V. Table Valso contains the results obtained when the mass of theZ(4430)− is fixed (4.433 GeV/c2), and when both themass and width (45 MeV) are fixed [5]. For all fits,χ2/NDF is acceptable, but deteriorates slightly, or failsto improve, as first the mass is fixed and then the massand width are fixed. We start each fit at the mass andwidth values of m = 4.433 GeV/c2 and Γ = 45 MeV.

For the fit to the J/ψπ− mass distribution using theK∗-veto sample with mass and width free, the obtainedmass is ∼ 100 MeV/c2 larger than that of the Z(4430)−,the width is essentially undetermined, the signal is morethan 2σ negative, and it remains at least 1σ negative forthe other fits.

For the K∗ region, the fitted mass value is closer tothat of the Z(4430)−, but otherwise the results are verysimilar to those for the K∗-veto region in that negativesignal values are obtained.

For the total sample, the results are no better. Thesignal is negative by ∼ 2.1σ, and remains negative by atleast 2σ as the constraints are applied.

We conclude that there is no evidence forZ(4430)− production via the decay sequenceB−,0 → Z(4430)−K0,+, Z(4430)−→ J/ψπ−.

For the fit to the ψ(2S)π− K∗-veto sample (which isequivalent to the Belle analysis sample, but sideband-

29

0

500

1000 (a) 0,+K-πψ J/→-,0B

Events-πAll K

0

200

400(d) 0,+K

-π(2S)ψ →-,0B Events-πAll K

0

500

(b)(1430)

2

*(892) + K

*K

100

200

300

400 (e)(1430)

2

*(892) + K

*K

) 2 (GeV/c-πψJ/m3.5 4 4.5

0

100

200

(c) veto

*K

) 2 (GeV/c-π(2S)ψm3.8 4 4.2 4.4 4.6 4.8

0

100

200 (f) veto

*K

2E

vent

s/10

MeV

/c

FIG. 31: The results of the fits to the corrected mass distributions, (a)-(c) for J/ψπ−, and (d)-(f) for ψ(2S)π−. In eachfigure, the open dots represent the data, and the solid curve represents the fit function, which consists of the Kπ− backgrounddistribution with normalization free, and a relativistic S-wave BW line shape, with mass, width, and normalization free; thedotted curves indicate the Kπ− background functions; the dashed vertical lines indicate mψπ− = 4.433 GeV/c2.

subtracted and efficiency-corrected), we obtain mass andwidth values which are consistent with theirs, but a pos-itive signal which is only ∼ 1.9σ from zero; fixing massand width increases this to only ∼ 3.1σ. Since our ef-ficiency in the Z(4430)− region is almost constant (cf.Fig. 23(f)), our corrected signal size with mass and widthfree (426 ± 229 events) corresponds to ∼ 61 ± 33 ob-served events. This converts to a signal of ∼ 70 events inFig. 29(b), and we estimate a similar value for the Belledistribution of Fig. 29(a). The signal size reported inRef. [5] is ∼ 120 events, obtained on the basis of a back-ground description which ignores Kπ− mass and angularstructures. It is interesting that we find a small posi-tive signal with mass and width consistent with the Bellevalues. However, with the present small data sample,it seems impossible to decide whether this is due to theproduction of a real state, or to the imprecision of thenormalized Kπ− moments, primarily in the region be-tween the two K∗’s.

For the K∗ region, our fitted mass value is ∼ 50MeV/c2 higher than the Belle value, and the signal de-viates from zero only by ∼ 2.5σ; the imposition of Bellemass and width values yields a signal which is less than

∼ 0.5σ from zero.

We note that each of these regions corresponds to ap-proximately half of the cos θψ(2S) range (cf. Fig. 21(d)),

so that for a flat Z(4430)− angular distribution, we wouldexpect naively that the signal size for each region wouldbe the same. The values in Table V are consistent withthis, however the central mass values differ by ∼ 5 stan-dard deviations. Although there could be significant in-terference effects associated with the K∗ regions, it doesnot seem possible that these could cause the signal tobe displaced by approximately one full-width. This, to-gether with the fact that both signals are in the 2-3 stan-dard deviation range, suggests that a more likely inter-pretation is that both are simply statistical fluctuations.

Finally, for the complete sample, our fitted mass valueis ∼ 40 MeV/c2 higher than the Belle value; the widthis consistent with Belle’s, but the signal size is only ∼2.7σ from zero; this is reduced to ∼ 2.1σ when the Belleparameters are imposed.

We conclude that there is no convincing evidence forproduction of the Z(4430)− state via the decay sequenceB−,0 → Z(4430)−K0,+, Z(4430)−→ψ(2S)π−, especiallysince the fits take no account of the Kπ− background un-

30

certainty associated with the normalized Kπ− moments.

B. Can the Kπ− background absorb a Z(4430)−

signal?

Since our evidence for the existence of a Z(4430)− sig-nal in the B−,0 → ψ(2S)π−K0,+ data sample is less thancompelling, it is reasonable to ask whether our use ofthe normalized Kπ− moments to modulate the ψ(2S)π−

background shape might have removed part, or all, of areal Z(4430)− signal.Firstly, our Kπ− background curves in Figs. 20, 26,

and 27 show no tendency to peak in a narrow regionaround the Z(4430)− mass position, indicated by thedashed vertical line in each figure. Secondly, for theψ(2S)π− distributions in these figures, the dot-dashedcurves obtained using the normalized Kπ− momentsfrom the B−,0 → J/ψπ−K0,+ modes, for which there isno Z(4430)− signal, do not differ significantly in shape inthe signal region from the solid curves from the B−,0 →ψ(2S)π−K0,+ modes. Finally, in Ref. [5] it is stated thatit is not possible to create a narrow peak from the S-,P -, and D-wave amplitude structure of the Kπ− system.We agree with this statement and provide a quantita-tive demonstration below. However, we first show howa Z(4430)− signal would affect the cos θK versus mKπ−

Dalitz plot.To this end, we have generated a MC sample of events

corresponding to B−,0 → Z(4430)−K0,+, Z(4430)− →ψ(2S)π− where the Z(4430)− has the Belle central massand width values, and decays isotropically. Figure 32shows the Dalitz plot which results. The Z(4430)− eventsyield a narrow locus confined almost entirely to the regioncos θK < 0.For the Legendre polynomials which we have used to

modulate the Kπ− angular distribution, such a distribu-tion yields the following behavior:

• P1(cos θK) is almost always negative;

• P2(cos θK), which is negative for | cos θK | < 0.58,is negative for ∼ 0.75 < mKπ < 1.55 GeV/c2 i.e.

almost always;

• P3(cos θK), which is positive for −0.78 < cos θK <0, is positive over almost the entire mKπ− rangeabove 1.2 GeV/c2, which is where we make use ofP3(cos θK) in the data;

• P4(cos θK), which is positive for | cos θK | < 0.33and | cos θK | > 0.88, is mainly positive for mKπ− >1.2 GeV/c2.

We estimate the effect of such a Z(4430)− signal on ourKπ− background by adding a MC-generated Z(4430)−

signal incoherently to the corrected ψ(2S)π− data ofFig. 20(b), and subjecting this new sample to the pro-cedure for creating our Kπ− background contribution to

1

10

210

)2 (GeV/c-πKm0.8 1 1.2 1.4 1.6

Kθ

cos

-1

-0.5

0

0.5

1

FIG. 32: The cos θK versus mKπ− rectangular Dalitz plot forthe B−,0 → Z(4430)−K0,+, Z(4430)−→ ψ(2S)π− MC eventsgenerated using mZ = 4.433 GeV/c2, ΓZ = 0.045 GeV, andwith isotropic Z(4430)− decay.

the ψ(2S)π− mass distribution. As for Fig. 32, we usemZ = 4433 MeV/c2 and ΓZ = 45 MeV in the simulation,and generate a flat cos θψ angular distribution. On thebasis of the Belle result, we would expect an observed sig-nal of ∼ 200 events for the full cos θψ angular range (cf.Fig. 21(d)), and this would yield an efficiency-correctedsignal of ∼ 1500 events (cf. Fig. 23(f)). Consequently,we generate 1500 events, of which 1493 events are withinthe (mES, ∆E) signal region. The mψ(2S)π− distribu-tion for these events is shown as the shaded histogramof Fig. 33(a); the points with error bars show the re-sult of combining these MC events with our correctedψ(2S)π− data distribution (Fig. 20(b)). The mKπ− dis-tribution for this combined sample is shown in Fig. 33(b),where the solid curve shows the result of a fit to the com-bined data, and the shaded histogram indicates the re-flection of the Z(4430)− MC events. We use this newfit curve to generate 10 million events corresponding toB−,0 → ψ(2S)π−K0,+ as before, and normalize this sam-ple to the combined data and MC-generated Z(4430)−

sample. We add the unnormalizedKπ− moments for thesimulated Z(4430)− events to the corrected Kπ− datamoments, and use the distribution of Fig. 33(b) to createnew normalized Kπ− moments as in Sec. IX. Finally, wefollow the weighting procedure described in Sec. IX tocreate the new Kπ− background description shown bythe dashed curve of Fig. 34 in comparison to the com-bined mψ(2S)π− distribution of Fig. 33(a).

Clearly, this dashed curve does not describe theZ(4430)− signal region. However, the shape of the curve

31

TABLE V: Results of the fits to the corrected BABAR ψπ− mass distributions; all fits use the relevant BABAR Kπ− backgroundshape.

Type of fit total K∗ region K∗ veto

J/ψπ− sample; m = 4455± 8 MeV/c2 m = 4454± 4 MeV/c2 m = 4545 ± 30 MeV/c2

Z(4430)− mass and width free Γ = 42± 27 MeV Γ = 17± 12 MeV Γ = 100± 96 MeV

NZ = −901± 420 NZ = −514± 223 NZ = −411± 181

χ2/NDF = 131/154 χ2/NDF = 135/154 χ2/NDF = 159/154

J/ψπ− sample; Γ = 82± 77 MeV Γ = 79± 39 MeV Γ = 10± 12 MeV

Z(4430)− fixed mass and free width NZ = −1098 ± 490 NZ = −810± 393 NZ = −86± 64

χ2/NDF = 136/155 χ2/NDF = 143/155 χ2/NDF = 162/155

J/ψπ− sample; NZ = −704± 249 NZ = −540± 225 NZ = −147± 105

Z(4430)− fixed mass and width χ2/NDF = 137/156 χ2/NDF = 144/156 χ2/NDF = 164/156

ψ(2S)π− sample; m = 4476± 8 MeV/c2 m = 4483± 3 MeV/c2 m = 4439 ± 8 MeV/c2

Z(4430)− mass and width free Γ = 32± 16 MeV Γ = 17± 12 MeV Γ = 41± 33 MeV

NZ = 703± 260 NZ = 447± 177 NZ = 426± 229

χ2/NDF = 93/96 χ2/NDF = 91/96 χ2/NDF = 106/96

ψ(2S)π− sample; Γ = 97± 77 MeV Γ = 100± 82 MeV Γ = 36± 26 MeV

Z(4430)− fixed mass and free width NZ = 710± 440 NZ = 246± 247 NZ = 414± 194

χ2/NDF = 101/97 χ2/NDF = 102/97 χ2/NDF = 107/97

ψ(2S)π− sample; NZ = 440± 212 NZ = 89± 162 NZ = 431± 137

Z(4430)− fixed mass and width χ2/NDF = 101/98 χ2/NDF = 101/98 χ2/NDF = 107/98

has been changed compared to that of Fig. 20(b) be-cause of the effect of the Z(4430)− signal events on thelow-order Legendre polynomial Kπ− moments. This isshown by the shaded histogram in Fig. 34, which rep-resents the difference between the dashed curve in thisfigure and the solid curve in Fig. 20(b).

A complete representation of the highly localizedZ(4430)− distribution in Fig. 32 requires the use of Leg-endre polynomials to order more than 30, and so ourlow-order representation of the Kπ− angular structure is

unable to do this. The shaded histogram does reach amaximum at about the Z(4430)− mass value, but corre-sponds to a width of ∼ 160 MeV, which is almost fourtimes larger than the input signal value.

A fit to the distribution of Fig. 34 with the normal-ization of the new Kπ− background, and the normaliza-tion, mass and width of the Z(4430)− signal, free yieldsmZ(4430)− = 4433±3 MeV/c2, ΓZ(4430)− = 34±12 MeV,and NZ(4430)− = 1402± 315 events so that the width isreduced to ∼ 75%, and the signal to ∼ 94%, of the input

32

value, while the mass is essentially unchanged. The solidcurve in Fig. 34 represents the fit result, and the dot-ted curve shows the reduced level of Kπ− background.If the Z(4430)− width is fixed to 45 MeV, we obtainNZ(4430)− = 1558 ± 220 events. This is consistent withthe input value, and presumably more properly reflectsthe effect of the statistical uncertainties in the underlyingdata distribution. We therefore use this to estimate themagnitude of the systematic signal reduction factor, andso obtain a value of ∼ 90%.

)2 (GeV/c-π(2S)ψm3.8 4 4.2 4.4 4.6 4.8

2E

vent

s/10

MeV

/c

0

200

400

600 (a)

)2 (GeV/c-πKm0.8 1 1.2 1.4 1.6

2E

vent

s/10

MeV

/c

1

10

210

310

)2 (GeV/c-πKm0.8 1 1.2 1.4 1.6

2E

vent

s/10

MeV

/c

1

10

210

310(b)

FIG. 33: (a): The combined mψ(2S)π− mass distribution for

the data of Fig. 20(b) and the MC Z(4430)− sample generatedas described in the text; the shaded histogram represents theMC sample; (b) the Kπ− mass distribution corresponding toFig. 33(a); the shaded histogram represents the reflection ofthe MC Z(4430)− events, and the curve shows the result ofthe fit.

This is a direct demonstration in support of the state-ment in the Belle letter [5] that a narrow peak in theψπ− mass distribution cannot be generated by only S-,P -, and D-wave amplitudes in the Kπ− system, and in-dicates that our use of low-order Legendre polynomials increating our Kπ− background could lead to an approxi-mately 10% systematic reduction of a narrow Z(4430)−

signal of the reported magnitude.

C. Branching fractions

In Table VI, we summarize the branching fraction val-ues and their 95% c.l. upper limits, obtained for theindividual B decay modes studied in the present analysisby repeating the fits of Figs. 31(a) and 31(d), but withZ(4430)− mass and width fixed to the central values ob-tained by Belle. The errors quoted are statistical, and

)2 (GeV/c-π(2S)ψm3.8 4 4.2 4.4 4.6 4.8

2E

vent

s/10

MeV

/c

0

200

400

600

FIG. 34: The mψ(2S)π− distribution of Fig. 33(a) for the dataand MC samples combined as described in the text. Thedashed curve represents the Kπ− background distribution,obtained as described in the text, and the shaded histogramrepresents the difference between this dashed curve and thesolid curve of Fig. 20(b), i.e. it represents the impact of theMC Z(4430)− signal. The solid curve is the result of a fitusing the Kπ− background function and a relativistic S-waveBW, and the dotted curve shows the resulting renormalizeddashed curve.

the upper limits were obtained using these values. Forthe ψ(2S)π− modes, the branching-fraction and upper-limit values have been increased by 10% in order to takeaccount of possible reduction of Z(4430)− signal size, asdescribed in Sec. XIB.The branching fraction for the decay mode B0 →

Z(4430)−K+, Z(4430)− → ψ(2S)π− from Belle is (4.1±1.0 ± 1.4)× 10−5, to be compared to our upper limit of3.1× 10−5 at 95% c.l.

XII. SUMMARY AND CONCLUSIONS

We have searched for evidence supporting the existenceof the Z(4430)− in the ψπ− mass distributions result-ing from the decays B−,0 → ψπ−K0,+ in a large datasample recorded by the BABAR detector at the PEP-IIe+e− collider at SLAC. Since the relevant Dalitz plots aredominated by mass and angular distribution structuresin the Kπ− system, we decided to investigate the extentto which the reflections of these features might describethe associated ψπ− mass distributions. To this end, weobtained a detailed description of the mass and angu-lar structures of the Kπ− system based on the expectedunderlying S-, P -, and D-wave Kπ− amplitude contri-

33

TABLE VI: The Z(4430)− signal size, the branching fraction value, and its 95% c.l. upper limit for each decay mode; theerrors quoted are statistical, and the upper limits were obtained using these values; the Z(4430)− mass and width have beenfixed to the central values obtained by Belle [5].

Decay mode Z(4430)− signal Branching fraction Upper limit(×10−5) (×10−5 at 95% c.l.)

B− → Z(4430)−K0, Z(4430)− → J/ψπ− −17± 140 −0.1± 0.8 1.5

B0 → Z(4430)−K+, Z(4430)− → J/ψπ− −670± 203 −1.2± 0.4 0.4

B− → Z(4430)−K0, Z(4430)− → ψ(2S)π− 148 ± 117 2.0± 1.7 4.7

B0 → Z(4430)−K+, Z(4430)− → ψ(2S)π− 415 ± 170 1.9± 0.8 3.1

butions, and in the process even found evidence of D−Fwave interference in the J/ψ decay modes. The frac-tional S-, P -, and D-wave intensity contributions to thecorrected Kπ− mass distributions for B− → J/ψπ−K0

Sand B0 → J/ψπ−K+ were found to be the same withinerror, as are the branching fraction values after all cor-rections, and so we combined these data in our analysisof the J/ψπ− mass distributions. We observed similarfeatures for the corresponding ψ(2S) decay modes, eventhough the analysis sample is ∼ 6.6 times smaller thanthat for J/ψ (Table II), and we combined the chargedand neutral B-meson samples for the ψ(2S) analysis also.

We next investigated the ψπ− mass distributions onthe basis of our detailed analysis of the Kπ− system.We used a MC generator to create large event samplesfor B−,0 → ψπ−K0,+ with Kπ− mass distribution gen-erated according to the overall fit function obtained fromthe corrected data, but with a uniform cos θK distribu-tion. The cos θK dependence was then modulated usingnormalized Legendre polynomial moments whose valueswere obtained from our corrected data by linear interpo-lation.

The total corrected J/ψπ− mass distribution(Fig. 20(a)) is well described by this Kπ− back-ground, whose form can be seen more clearly inFig. 22(a). The residuals (Fig. 20(c)) show no evidenceof a Z(4430)− signal, and this is true also for thevarious regions of Kπ− mass shown in Fig. 24(a)-(e),Fig. 26(a),(c), and Fig. 27(a),(c). When we fit thedata using a function which allows the presence of aZ(4430)− signal, we obtain only negative Z(4430)−

signal intensities (Fig. 31(a)-(c), Table V). We find thisto be the case also for the B− and B0 modes separately,and summarize the corresponding branching fractionupper limits in Table VI for Z(4430)− mass and widthfixed at the central values obtained by Belle [5].

We conclude that there is no evidence to support theexistence of a narrow resonant structure in the J/ψπ−

mass distributions for our data on the decay modes

B0,− → J/ψπ−K+,0.

The corresponding corrected ψ(2S)π− distributionsof Figs. 20(b),(d), Figs. 24(f)-(j) and Figs. 25(f)-(j),Figs. 26(b) and (d), and Figs. 27(b) and (d) likewise showno clear evidence of a narrow signal at the Z(4430)− massposition.

We have directly compared our uncorrected ψ(2S)π−

data with K∗ veto, to those from Ref. [5]; there is noevidence of statistically significant difference (Fig. 29(c)).

In order to quantify our Z(4430)− production rate es-timates, we fit our total corrected ψπ− mass distribu-tions of Figs. 20(a) and 20(b), using the Kπ− back-ground shapes shown in these figures, together with aZ(4430)− line shape. For J/ψπ− we obtain a negativesignal, while for ψ(2S)π− we obtain a 2.7 standard devia-tion signal with fitted width consistent with the value ob-tained by Belle, but with central mass value∼ 43 MeV/c2

higher than that reported by Belle, which correspondsto a +4.7 standard deviation difference. These fit re-sults are shown in Figs. 31(a) and 31(d), respectively,and are summarized in Table V. We repeated thesefits for the individual decay modes of Eqs.(1)-(4) withZ(4430)− mass and width fixed to the central values re-ported by Belle, and obtained the branching fraction andupper limit values summarized in Table VI. In particular,we find a branching fraction upper limit for the processB(B0 → Z(4430)−K+, Z− → ψ(2S)π−) < 3.1 × 10−5

at 95% c.l., and a corresponding value for the reactionB(B− → Z(4430)−K0, Z− → ψ(2S)π−) < 4.7 × 10−5

at 95% c.l. We conclude that our analyses provide nosignificant evidence for the existence of the Z(4430)−.

It will be of great interest to see whether or not theZ(4430)− is confirmed by a future analysis based upona significantly larger data sample than is available atpresent.

34

XIII. ACKNOWLEDGMENTS

We are grateful for the extraordinary contributions ofour PEP-II colleagues in achieving the excellent luminos-ity and machine conditions that have made this work pos-sible. The success of this project also relies critically onthe expertise and dedication of the computing organiza-tions that support BABAR. The collaborating institutionswish to thank SLAC for its support and the kind hospi-tality extended to them. This work is supported by theUS Department of Energy and National Science Foun-dation, the Natural Sciences and Engineering ResearchCouncil (Canada), the Commissariat a l’Energie Atom-ique and Institut National de Physique Nucleaire et dePhysique des Particules (France), the Bundesministeriumfur Bildung und Forschung and Deutsche Forschungsge-meinschaft (Germany), the Istituto Nazionale di FisicaNucleare (Italy), the Foundation for Fundamental Re-search on Matter (The Netherlands), the Research Coun-cil of Norway, the Ministry of Education and Science ofthe Russian Federation, Ministerio de Educacion y Cien-cia (Spain), and the Science and Technology FacilitiesCouncil (United Kingdom). Individuals have receivedsupport from the Marie-Curie IEF program (EuropeanUnion) and the A. P. Sloan Foundation.

APPENDIX A: THE DALITZ PLOT

EFFICIENCY-CORRECTION PROCEDURE

The efficiency is obtained using samples of Monte Carloevents corresponding to the decay processes of Eqs. (1)-(4) generated uniformly over the final state Dalitz plot.In general, the phase space volume element in the

Dalitz plot corresponding to the decayB → ψπK is givenby:

dρ ∼ d(m2Kπ) · d(m2

ψπ) , (A1)

where mKπ− (mψπ−) is the invariant mass of the Kπ−

(ψπ−) system.However, when the efficiency is studied in such recti-

linear area elements, those elements at the plot boundaryare partially outside the plot, and this leads to a rathercumbersome efficiency treatment. The phase space vol-ume element of Eq. (A1) may be transformed to

dρ′ ∼ p · q

mKπ·mKπ d(mKπ) d(cosθK) , (A2)

i.e.

dρ′ ∼ p · q d(mKπ)d(cosθK) , (A3)

where p is the momentum of the ψ daughter of the Bin the B rest frame, and q is the momentum of the Kin the rest frame of the Kπ− system. This expression issuch that the phase space density is uniform in cos θK ata given value of mKπ− .

The range of cos θK is [-1,1], and that of mKπ− is fromthreshold tomB−mψ, so that the resultant “Dalitz Plot”is rectangular in shape, with the factor p · q representingthe Jacobian of the variable transformation. A plot ofthis kind can then be used readily to study efficiency be-havior over the entire phase space region without theproblems incurred at the boundary of a conventionalDalitz plot (see, for example, Appendix B of Ref. [22]).The reconstruction efficiency calculated using the

Monte Carlo simulated events is parametrized as a func-tion of mKπ− and cos θK , and then used to correctthe data by weighting each event by the inverse of itsparametrized efficiency value. For a given mass inter-val I = [mKπ− ,mKπ− + dmKπ− ], let N be the numberof generated events, and let nreco, represent the numberof reconstructed events. The generated cos θK distribu-tion is flat, but in general efficiency effects will causethe reconstructed cos θK distribution to have structure.Writing the angular distribution in terms of appropri-ately normalized Legendre polynomials,

dN

d cos θK= N〈P0〉P0(cos θK) (A4)

and,

dnrecod cos θK

= nreco

L∑

i=0

〈Pi〉Pi(cos θK) (A5)

where the normalizations are such that,

∫ 1

−1

Pi(cos θK)Pj (cos θK)d(cos θK) = δij , (A6)

where Pi =√2πY 0

i , and Y0i is a spherical harmonic func-

tion. The value of L is obtained empirically.Using this orthogonality condition, the coefficients in

the expansion are obtained from

〈Pj〉 =1

nreco

∫ 1

−1

Pj(cos θK)dnrecod cos θK

d(cos θK) , (A7)

where the integral is given, to a good approximation fora large enough MC sample, by

∑nreco

i=1 Pj (cos θKi). The

index i runs over the reconstructed events in mass in-terval I, such that nreco〈Pj 〉 ∼ ∑nreco

i=1 Pj (cos θKi), and

the effect of efficiency loss on the angular distribution isrepresented through these coefficients. The absolute effi-ciency, calculated as a function of cos θK and mKπ− , inmass interval I, is then given by

E(cos θK ,mKπ−) =nreco

(

∑Li=0〈Pi〉Pi(cos θK)

)

N〈P0〉P0(cos θK).

(A8)

With

E0 =nrecoN

(A9)

35

and

Ej = 2nreco〈Pj 〉

N= 2

∑nreco

i=1 Pj (cos θKi)

N, (A10)

for a large enough sample (note that the factor 2 enterssince 〈P0〉P0(cos θK) = 1/2), Eq. (A8) becomes,

E(cos θK ,mKπ−) = E0

+E1P1(cos θK) + ...+ ELPL(cos θK) . (A11)

The mean value of the Pj (cos θKi), with i = 1, ..., nreco,

corresponding to mass interval I, is written as 〈Pj 〉. Ther.m.s. deviation of the Pj (cos θKi

) w.r.t. 〈Pj 〉, σ, is givenby

σ2 =

nreco∑

i=1

(Pj (cos θKi)− 〈Pj 〉)2

nreco − 1. (A12)

The error on the mean is then δ〈Pj 〉 = σ√nreco

and from

Eq. (A12),

δ〈Pj〉 =

√

∑nreco

i=1 (Pj(cos θKi)− 〈Pj〉)2

nreco(nreco − 1)

=

√

√

√

√

√

[

∑nreco

i=1(Pj (cos θKi

))2

nreco

]

− 〈Pj〉2

nreco − 1.

(A13)

The uncertainty in the parameter E0 = nreco

N is givenby:

δ(E0) = E0

√

1

nreco+

1

N, (A14)

and the uncertainty in the coefficient Ej is given by:

δ(Ej) =2

N·

√

∑nreco

i=1 (Pj(cos θKi))2 +

(Pnreco

i=1 (Pj(cos θKi)))

2

N .

(A15)

For each of the processes represented by Eqs.(1)-(4),the efficiency analysis is carried out in 50 MeV/c2 Kπ−

mass intervals from threshold to the maximum value ac-cessible. As shown in Sec. VI, Fig. 7, the Kπ− massdependence of the average efficiency parameter, E0, de-pends on the decay mode of the ψ involved, and sois obtained by using the MC sample for that particu-lar decay mode. The angular dependence representedby E1, E2,...etc. does not depend on the individual ψdecay mode, and so these coefficients are calculated bycombining the MC samples for the individual ψ modes.For the B meson decay processes of Eqs. (1)- (4), themain features of the angular dependence of the efficiencyare very similar, and so we present the results only forB0 → ψ(2S)π−K+ by way of illustration.Simulated MC events are subjected to the same re-

construction and event-selection procedures as those ap-plied to the data. For the process of Eq. (4), the cos θKdistributions for the surviving MC events are shown foreach Kπ− mass interval in Fig. 35. We chose a smallinterval size (0.02) in order to investigate the signifi-cant decrease in efficiency observed for cos θK ∼ +1 and0.720 < mKπ− < 0.920 GeV/c2 (Figs. 35(c)-(f)) andfor cos θK ∼ −1 and 0.970 < mKπ− < 1.270 GeV/c2

(Figs. 35(h)-(m)). Representation of such localized lossesrequires the use of high-order Legendre polynomials; wefind that L = 12 yields a satisfactory description, asdemonstrated by the curves in Fig. 35. The Kπ− massdependence of the resulting values of E1 - E12 is shownin Fig. 36, and is parametrized in each case by the fifth-order polynomial curve shown. These parameterizations,together with those describing theKπ− mass dependenceof E0 for the individual ψ decay modes (Sec. VI) enableus to calculate the efficiency at any point in the relevantrectangular Dalitz plot, and hence for each event in thecorresponding data sample. We then assign to each eventa weight given by the inverse of this efficiency value, andby using this weight are able to create efficiency-correcteddistributions.

As discussed in Sec. VI, the efficiency loss for cos θK ∼+1 is due to the failure to reconstruct low momentumcharged pions in the laboratory frame, while that forcos θK ∼ −1 is due to the similar loss of low momen-tum kaons (Fig. 9).

[1] M. Gell-Mann, Baryons And Mesons,” Phys. Lett. 8, 214(1964).

[2] G. Trilling, J. Phys. G 33, 1019 (2006).[3] C. Amsler et al. [Particle Data Group], Phys. Lett. B667,

1 (2008).[4] L. Maiani et al., Phys. Rev. D 71, 014028 (2005); L.

Maiani et al., Phys. Rev. D 72, 031502(R) (2005); L.Maiani, A.D. Polosa, and V. Riquer, Phys. Rev. Lett.

99, 182003 (2007).[5] S.-K. Choi et al. [Belle Collaboration], Phys. Rev. Lett.

100, 142001 (2008).[6] The use of charge conjugate reactions is implied through-

out this paper.[7] L. Maiani, A.D. Polosa, and V. Riquer, New Journal of

Physics, 10, 073004 (2008).[8] M. Karliner and H. J. Lipkin, Phys. Lett. B 638, 221

36

10

20

30) < 0.670π0.620 < m(K

2GeV/c

(a)20

40

60 ) < 0.720π0.670 < m(K2GeV/c

(b)20

40

60

80) < 0.770π0.720 < m(K

2GeV/c

(c)

50

100 ) < 0.820π0.770 < m(K2GeV/c

(d)

50

100) < 0.870π0.820 < m(K

2GeV/c

(e)

50

100) < 0.920π0.870 < m(K

2GeV/c

(f)

50

100

) < 0.970π0.920 < m(K2GeV/c

(g)

50

100

) < 1.020π0.970 < m(K2GeV/c

(h)

50

100

) < 1.070π1.020 < m(K2GeV/c

(i)

50

100

150

) < 1.120π1.070 < m(K2GeV/c

(j)

50

100

150) < 1.170π1.120 < m(K

2GeV/c

(k)

50

100

150) < 1.220π1.170 < m(K

2GeV/c

(l)

50

100

150

) < 1.270π1.220 < m(K2GeV/c

(m)

50

100

150

) < 1.320π1.270 < m(K2GeV/c

(n)50

100

150 ) < 1.370π1.320 < m(K2GeV/c

(o)

Kθcos-1 -0.5 0 0.5 10

50

100

150) < 1.420π1.370 < m(K

2GeV/c

(p)

Kθcos-1 -0.5 0 0.5 10

50

100

150) < 1.470π1.420 < m(K

2GeV/c

(q)

Kθcos-1 -0.5 0 0.5 10

50

100

) < 1.520π1.470 < m(K2GeV/c

(r)

Kθcos-1 -0.5 0 0.5 10

20

40

60

80 ) < 1.570π1.520 < m(K2GeV/c

(s)

Kθcos-1 -0.5 0 0.5 1

Eve

nts/

0.02

FIG. 35: The cos θK distributions in 50 MeV/c2 Kπ− mass intervals for the decay mode B0 → ψ(2S)π−K+. The pointsrepresent the data, and the curves show the functions calculated from the moments.

(2006); Ce Meng and Kuang-Ta Chao, arXiv:0708.4222[hep-ph]; J. L. Rosner, Phys. Rev. D 76, 114002 (2007);M. Karliner and H. J. Lipkin, arXiv:0802.0649 [hep-ph].

[9] We use “ψ” to represent “J/ψ or ψ(2S)” unless these arerequired explicitly.

[10] B. Aubert et al. [BABAR Collaboration], Nucl. Instrum.Meth. A 479, 1 (2002).

[11] W. Menges, Nuclear Science Symposium ConferenceRecord, 2005 IEEE, 3, 1470 (2005).

[12] H. Albrecht et al. [ARGUS Collaboration], Z. Phys. C

48, 543 (1990).[13] B. Aubert et al. [BABAR Collaboration], Phys. Rev. Lett.

87, 241801 (2001).[14] B. Aubert et al. [BABAR Collaboration], Phys. Rev. D

71, 032005 (2005).[15] J. M. Blatt and V. F. Weisskopf, Theoretical Nuclear

Physics, John Wiley and Sons (1952).[16] D. Aston et al. [LASS Collaboration], Nucl. Phys. B296,

493 (1988); W. Dunwoodie, private communication.[17] S. T’Jampens, Ph.D. Thesis, Universite Paris XI (2002);

http://arxiv.org/abs/0708.4222

http://arxiv.org/abs/0802.0649

37

-0.04

-0.02

0

0.02

0.04

2

/50

MeV

/c1

E(a)

-0.04

-0.02

0

0.02

0.04

2/5

0 M

eV/c

2E

(b)

-0.04

-0.02

0

0.02

0.04

2/5

0 M

eV/c

3E

(c)

-0.04

-0.02

0

0.02

0.04

2/5

0 M

eV/c

4E

(d)

-0.04

-0.02

0

0.02

0.04

2/5

0 M

eV/c

5E

(e)

-0.04

-0.02

0

0.02

0.04

2/5

0 M

eV/c

6E

(f)

-0.04

-0.02

0

0.02

0.04

2/5

0 M

eV/c

7E

(g)

-0.04

-0.02

0

0.02

0.04

2/5

0 M

eV/c

8E

(h)

) 2 (GeV/c-πKm0.8 1 1.2 1.4

-0.04

-0.02

0

0.02

0.04

2/5

0 M

eV/c

9E

(i)

) 2 (GeV/c-πKm0.8 1 1.2 1.4

-0.04

-0.02

0

0.02

0.04

2/5

0 M

eV/c

10E

(j)

) 2 (GeV/c-πKm0.8 1 1.2 1.4

-0.04

-0.02

0

0.02

0.04

2/5

0 M

eV/c

11E

(k)

) 2 (GeV/c-πKm0.8 1 1.2 1.4

-0.04

-0.02

0

0.02

0.04

2/5

0 M

eV/c

12E

(l)

FIG. 36: The Kπ− mass dependence of the coefficients E1 through E12 for the decay mode B0 → ψ(2S)π−K+. The pointsshow the calculated values, and the curves result from fits to the data using a fifth-order polynomial.

SLAC-R-836, Appendix D.[18] E. M. Aitala et al. [E791 Collaboration], Phys. Rev. D

73, 032004 (2006) [Erratum-ibid. D 74, 059901 (2006)].[19] B. Aubert et al. [BABAR Collaboration], Phys. Rev. D

76, 031102(R) (2007).[20] We have verified that the difference in the Lorentz boost

between the BABAR and Belle experiments does not sig-nificantly affect these losses of low-momentum particles.

[21] We thank the Belle Collaboration for allowing us to re-produce the histograms in Fig. 2 of Ref. [5].

[22] V. Ziegler, Ph.D. Thesis, SLAC-R-868 (2007).

arxiv:0811.0564v1 [hep-ex] 4 nov 20084 g. raven and h. l. snoek nikhef, national institute for...

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