Dalitz plot analysis of B-→D+π-π

transcript

BABAR-PUB-08/048, SLAC-PUB-13508

Dalitz Plot Analysis of B−

→ D+

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. 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. 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

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, 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. ZalloINFN 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, 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

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. Wormser

Laboratoire 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, F. Di Lodovico, R. Sacco, and M. SigamaniQueen Mary, University of London, London, E1 4NS, United Kingdom

G. Cowan, S. Paramesvaran, 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, 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. Zhao

Massachusetts 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. J. Summers, and H. W. ZhaoUniversity of Mississippi, University, Mississippi 38677, USA

M. Simard and P. TarasUniversite 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

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

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,

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 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, R. Facciniab, F. Ferrarottoa, F. Ferroniab, M. Gasperoab,

P. D. Jacksona, L. Li Gioia, M. A. Mazzonia, S. Morgantia, G. Pireddaa, 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, 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, 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,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, 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. Ziegler

Stanford Linear Accelerator Center, Stanford, California 94309, USA

P. R. Burchat, A. J. Edwards, and T. S. MiyashitaStanford 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. SobieUniversity 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, R. Prepost, C. O. Vuosalo, and S. L. WuUniversity of Wisconsin, Madison, Wisconsin 53706, USA

We report on a Dalitz plot analysis of B−→ D+π−π− decays, based on a sample of about

383 × 106 Υ(4S) → BB decays collected with the BABAR detector at the PEP-II asymmetric-energy B Factory at SLAC. We find the total branching fraction of the three-body decay: B(B−

D+π−π−) = (1.08± 0.03± 0.05)× 10−3. We observe the established D∗02 and confirm the existence

of D∗00 in their decays to D+π−, where the D∗0

2 and D∗00 are the 2+ and 0+ cu P-wave states,

respectively. We measure the masses and widths of D∗02 and D∗0

0 to be: mD∗02

= (2460.4 ± 1.2 ±

1.2 ± 1.9) MeV/c2, ΓD∗02

= (41.8 ± 2.5 ± 2.1 ± 2.0) MeV, mD∗00

= (2297 ± 8 ± 5 ± 19) MeV/c2 and

ΓD∗00

= (273±12±17±45) MeV. The stated errors reflect the statistical and systematic uncertainties,

and the uncertainty related to the assumed composition of signal events and the theoretical model.

PACS numbers: 13.25.Hw, 14.40.Lb, 14.40.Nd

∗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

I. INTRODUCTION

Orbitally excited states of the D meson, denoted hereas DJ , where J is the spin of the meson, provide a uniqueopportunity to test the Heavy Quark Effective Theory(HQET) [1, 2]. The simplest DJ meson consists of acharm quark and a light anti-quark in an orbital angularmomentum L = 1 (P-wave) state. Four such states areexpected with spin-parity JP = 0+ (j = 1/2), 1+ (j =1/2), 1+ (j = 3/2) and 2+ (j = 3/2), which are labeledhere as D∗

0 , D′1, D1 and D∗

2 , respectively, where j is aquantum number corresponding to the sum of the light

quark spin and the orbital angular momentum ~L.The conservation of parity and angular momentum in

strong interactions imposes constraints on the strong de-cays of DJ states to Dπ and D∗π. The j = 1/2 statesare predicted to decay exclusively through an S-wave:D∗

0 → Dπ and D′1 → D∗π. The j = 3/2 states are

expected to decay through a D-wave: D1 → D∗π andD∗

2 → Dπ and D∗π. These transitions are summarizedin Fig. 1. Because of the finite c-quark mass, the twoJP = 1+ states may be mixtures of the j = 1/2 andj = 3/2 states. Thus the broad D′

1 state may decayvia a D-wave and the narrow D1 state may decay via anS-wave. The j = 1/2 states with L = 1, which decaythrough an S-wave, are expected to be wide (hundreds ofMeV/c2), while the j = 3/2 states that decay through aD-wave are expected to be narrow (tens of MeV/c2) [2–4]. Properties of the L = 1 D0

J mesons [5] are given inTable I.

The narrow DJ mesons have been previously observedand studied by a number of experiments [6–16]. DJ

mesons have also been studied in semileptonic B de-cays [17–24]. Precise knowledge of the properties of theDJ mesons is important to reduce uncertainties in themeasurements of semileptonic decays, and thus the de-termination of the Cabibbo-Kobayashi-Maskawa [25] ma-trix elements |Vcb| and |Vub|. The Belle Collaborationhas reported the first observation of the broad D∗0

0 andD′0

1 mesons in B decay [12]. The FOCUS Collaborationhas found evidence for broad structures in D+π− finalstates [13] with mass and width in agreement with theD∗0

0 found by Belle Collaboration. However, the Parti-cle Data Group [5] considers that the J and P quantumnumbers of the D∗0

0 and D′01 states still need confirma-

tion.In this analysis, we fully reconstruct the decays B− →

D+π−π− [26] and measure their branching fraction. Wealso perform an analysis of the Dalitz plot (DP) to mea-sure the exclusive branching fractions of B− → D0

Jπ−

and study the properties of the D0J mesons. The decay

B− → D+π−π− is expected to be dominated by the in-termediate states D∗0

2 π− and D∗00 π−, and has a possible

contribution from B− → D+π−π− nonresonant (NR) de-cay. The D0

1 and D′01 states can not decay strongly into

Dπ because of parity and angular momentum conserva-tion. However, the D∗(2007)0 (labeled as D∗

v here) massis close to the Dπ production threshold and it may con-tribute as a virtual intermediate state. The B∗ (labeledas B∗

v here) produced in a virtual process B− → B∗vπ−

may also contribute via the decay B∗v → D+π−. Possible

contributions from these virtual states are also studied inthis analysis.

II. THE BABAR DETECTOR AND DATASET

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

storage rings at SLAC between 1999 and 2006. The sam-

-0 -1 +0 +1 +1 +2 PJ

L= 0 L= 1j=1/2 j=3/2

S-waveπ D-waveπ

1’D 1D 2

FIG. 1: Mass spectrum for cu states. The vertical bars showthe widths. Masses and widths are from Ref. [5]. The dottedand dashed lines between the levels show the dominant piontransitions. Although it is not indicated in the figure, the two1+ states may be mixtures of j = 1/2 and j = 3/2, and D′

may decay via a D-wave and D1 may decay via an S-wave.

TABLE I: Properties of L = 1 D0J mesons [5].

JP Mass Width Decays Partial(MeV/c2) ( MeV) seen [5] waves

D∗00 0+ 2352 ± 50 261 ± 50 Dπ S

D′01 1+ 2427 ± 36 384+130

−105 D∗π S, DD0

1 1+ 2422.3 ± 1.3 20.4 ± 1.7 D∗π, D0π+π− S, DD∗0

2 2+ 2461.1 ± 1.6 43 ± 4 D∗π, Dπ D

ple consists of 347.2 fb−1 corresponding to (382.9±4.2)×106 BB pairs (NBB) taken on the peak of the Υ(4S) res-onance. Monte Carlo (MC) simulation is used to studythe detector response, its acceptance, background, andto validate the analysis. We use GEANT4 [27] to sim-ulate resonant e+e− → Υ(4S) → BB events (generatedby EvtGen [28]) and e+e− → qq (where q = u, d, s or c)continuum events (generated by JETSET [29]).

A detailed description of the BABAR detector is givenin Ref. [30]. Charged particle trajectories are measuredby a five-layer, double-sided silicon vertex tracker (SVT)and a 40-layer drift chamber (DCH) immersed in a 1.5 Tmagnetic field. Charged particle identification (PID) isachieved by combining information from a ring-imagingCherenkov device with ionization energy loss (dE/dx)measurements in the DCH and SVT.

III. EVENT SELECTION

Five charged particles are selected to reconstruct de-cays of B− → D+π−π− with D+ → K−π+π+. Thecharged particle candidates are required to have trans-verse momenta above 100 MeV/c and at least twelve hitsin the DCH. A K− candidate must be identified as akaon using a likelihood-based particle identification algo-

) ) (GeV/c+π+π-m(K

1.84 1.85 1.86 1.87 1.88 1.89 1.9

) ) (GeV/c+π+π-m(K

1.84 1.85 1.86 1.87 1.88 1.89 1.9

FIG. 2: K+π−π− invariant mass distribution for D+ candi-dates for the selected B−

→ D+π−π− decays without thecut on the mass of D+. Data (points with statistical errors)are compared to the results of the fit (solid curve), with thebackground distribution marked as a dashed line. The shadedarea marks the D+ signal region.

rithm (with an average efficiency of ∼85% and an averagemisidentification probability of ∼3%). Any combinationof K−π+π+ candidates with a common vertex and aninvariant mass between 1.8625 and 1.8745 GeV/c2 is ac-cepted as a D+ candidate. We fit the invariant massdistribution of the K−π+π+ candidates with a functionthat includes a Gaussian component for the signal and alinear term for the background. The signal parameters(mean and width of Gaussian) and slope of the back-ground function are free parameters of the fit. The dataand the result of the fit are shown in Fig. 2. The invariantmass resolution for this D+ decay is about 5.2 MeV/c2.The B− candidates are reconstructed by combining a D+

candidate and two charged tracks. The trajectories ofthe three daughters of the B− meson candidate are con-strained to originate from a common decay vertex. TheD+ and B− vertex fits are required to have converged.

At the Υ(4S) resonance, B mesons can be character-ized by two nearly independent kinematic variables, thebeam-energy substituted mass mES and the energy dif-ference ∆E:

mES =√

(s/2 + ~p0 · ~pB)2/E20 − p2

B, (1)

∆E = E∗B −

√s/2, (2)

where E and p are energy and momentum, the subscripts0 and B refer to the e+e−-beam system and the B can-didate, respectively; s is the square of the center-of-massenergy and the asterisk labels the center-of-mass frame.For B− → D+π−π− signal events, the mES distributionis well described by a Gaussian resolution function witha width of 2.6 MeV/c2 centered at the B− meson mass,while the ∆E distribution can be represented by a sum oftwo Gaussian functions with a common mean near zeroand different widths with a combined RMS of 20 MeV.

Continuum events are the dominant background. Sup-

pression of background from continuum events is pro-vided by two topological requirements. In particular, weemploy restrictions on the magnitude of the cosine of thethrust angle, cosΘth, defined as the angle between thethrust axis of the selected B candidate and the thrustaxis of the remaining tracks and neutral clusters in theevent. The distribution of | cosΘth| is strongly peaked to-wards unity for continuum background but is uniform forsignal events. We also select on the ratio of the second tothe zeroth Fox-Wolfram moment [31], R2, to further re-duce the continuum background. The value of R2 rangesfrom 0 to 1. Small values of R2 indicate a more spher-ical event shape (typical for a BB event) while valuesclose to 1 indicate a 2-jet event topology (typical for aqq event). We accept the events with | cosΘth| < 0.85and R2 < 0.30. The | cosΘth| (R2) cut eliminates about68% (71%) of the continuum background while retainingabout 90% (83%) of signal events.

To suppress backgrounds, restrictions are placed onmES: 5.2754 < mES < 5.2820 GeV/c2, and ∆E: −130 <∆E < 130 MeV. The selected samples of B candidatesare used as input to an unbinned extended maximumlikelihood fit to the ∆E distribution. The result of thefit is used to determine the fractions of signal and back-ground events in the selected data sample. For eventswith multiple candidates (∼ 3.5% of the selected events)satisfying the selection criteria, we choose the one withbest χ2 from the B vertex fit. Based on MC simulation,we determine that the correct candidate is selected atleast 65% of the time. We fit the mES distribution ofthe selected B− → D+π−π− candidates with a sum of aGaussian function for the signal and a background func-tion for the background having the probability density,P (x) ∝ x

√1 − x2 exp(−ξ(1 − x2)), where x = mES/m0

with m0 fixed at 5.29 GeV/c2 and ξ is a shape param-eter [32]. The signal parameters (mean, width of Gaus-sian) and the shape parameter of the background func-tion are free parameters of the fit. The data and theresult of the fit are shown in Fig. 3a. We fit the ∆E dis-tribution of the selected B− → D+π−π− candidates witha sum of two Gaussian functions with a common meanfor the signal and a linear function for the background.The signal parameters (mean, width of wide Gaussian,width and fraction of narrow Gaussian) and the slope ofthe background function are free parameters of the fit.The data and the result of the fit are shown in Fig. 3b.The resulting signal yield is 3496 ± 74 events, where theerror is statistical only. A clear signal is evident in bothmES and ∆E distributions.

To distinguish signal and background in the Dalitz plotstudies, we divide the candidates into three subsamples:the signal region, −21 < ∆E < 15 MeV, the left side-band, −109 < ∆E < −73 MeV, and the right sideband,67 < ∆E < 103 MeV. The events in the signal region areused in the Dalitz plot analysis, while the events in thesideband regions are used to study the background.

In order to check the shape of the background ∆E dis-tribution, we have generated a background MC sample

E (GeV)∆-0.1 -0.05 0 0.05 0.1

1500(b)

)2 (GeV/cESm5.23 5.24 5.25 5.26 5.27 5.28 5.29

FIG. 3: (a) mES and (b) ∆E distributions for D+π−π− can-didates. Data (points with statistical errors) are comparedto the results of the fits (solid curves), with the backgroundcontributions marked as dashed lines. The histograms are thecorresponding distributions of the background MC sample asdescribed in the text. The shaded area in (a) shows the sig-nal region, while the three shaded areas in (b) mark the signalregion in the center and the two sidebands.

of resonant and continuum events with B− → D+π−π−

signal events removed. The background MC sample hasbeen scaled to the same luminosity as the data. The ∆Edistribution of the selected events from the backgroundMC sample is shown as the histogram in Fig. 3b. Asmall amount of peaking background is found from mis-reconstructed decays of B0 → D+ρ− with ρ− → π−π0,where a π0 is missed and a random track in the event ismisidentified as a signal π−. The background histogramin Fig. 3b is fitted with a sum of two Gaussian func-tions with a common mean for the peaking background,with parameters fixed to those obtained from the fit todata, and a linear function to describe the combinatorialbackground. The amount of peaking background is esti-mated at 82± 41 events. After peaking background sub-traction, the number of signal events above backgroundis Nsignal = 3414 ± 85. The background fraction in thesignal region is (30.4 ± 1.1)%.

IV. DALITZ PLOT ANALYSIS

We refit the D+ and B− candidate momenta by con-straining the trajectories of the three daughters of the B−

meson candidate to originate from a common decay ver-tex while constraining the invariant masses of K−π+π+

and D+π−π− to the D+ and B− masses [5], respectively.The mass-constraints ensure that all events fall withinthe Dalitz plot boundary.

In the decay of a B− into a final state composedof three pseudo-scalar particles (D+π−π−), two degreesof freedom are required to describe the decay kinemat-ics. In this analysis we choose the two Dπ invariantmass-squared combinations x = m2(D+π−

1 ) and y =m2(D+π−

2 ) as the independent variables, where the twolike-sign pions π−

1 and π−2 are randomly assigned to x and

y. This has no effect on our analysis since the likelihoodfunction (described below) is explicitly symmetrized withrespect to interchange of the two identical particles.

The differential decay rate is generally given in termsof the Lorentz-invariant matrix element M by

|M|2256π3m3

where mB is the B meson mass. The Dalitz plot givesa graphical representation of the variation of the squareof the matrix element, |M|2, over the kinematically ac-cessible phase space (x,y) of the process. Non-uniformityin the Dalitz plot can indicate presence of intermediateresonances, and their masses and spin quantum numberscan be determined.

A. Probability Density Function

We describe the distribution of candidate events in theDalitz plot in terms of a probability density function(PDF). The PDF is the sum of signal and backgroundcomponents and has the form:

PDF(x, y) = fbgB(x, y)

DP B(x, y)dxdy

+ (1 − fbg)[S(x, y) ⊗R] ǫ(x, y)

DP [S(x, y) ⊗R] ǫ(x, y)dxdy,

where the integral is performed over the whole Dalitzplot, the S(x, y) ⊗ R is the signal term convolved withthe signal resolution function, B(x, y) is the backgroundterm, fbg is the fraction of background events, and ǫ isthe reconstruction efficiency.

An unbinned maximum likelihood fit to the Dalitz plotis performed in order to maximize the value of

Nevent∏

PDF(xi, yi) (5)

with respect to the parameters used to describe S, wherexi and yi are the values of x and y for event i respectively,and Nevent is the number of events in the Dalitz plot. Inpractice, the negative-log-likelihood (NLL) value

NLL = − lnL (6)

is minimized in the fit.

B. Goodness-of-fit

It is difficult to find a proper binning at the kinematicboundaries in the x-y plane of the Dalitz plot. For thisreason, we choose to estimate the goodness-of-fit χ2 inthe cos θ (range from -1 to 1) and m2

min(Dπ) (range from4.04 to 15.23 GeV2/c4) plane, which is a rectangular rep-resentation of the Dalitz plot. The parameter θ is thehelicity angle of the Dπ system and m2

min(Dπ) is thelesser of x and y. The helicity angle θ is defined as theangle between the momentum vector of the pion from theB decay (bachelor pion) and that of the pion of the Dπsystem in the Dπ rest-frame.

The χ2 value is calculated using the formula

χ2 =∑

χ2i =

ntotal∑

(Ncelli − Nfiti)2

for cells in a 18 × 18 grid of the two-dimensional his-togram. In Eq. (7), ntotal is the total number of cellsused, Ncelli is the number of events in each cell, Nfiti isthe expected number of events in that cell as predictedby the fit results. The number of degrees of freedom(NDF) is calculated as ntotal−k−1, where k is the num-ber of free parameters in the fit. We require Nfit ≥ 10;if this requirement is not met then neighboring cells arecombined until ten events are accumulated.

C. Matrix element M and Fit Parameters

This analysis uses an isobar model formulation inwhich the signal decays are described by a coherent sumof a number of two-body (Dπ system + bachelor pion)amplitudes. The orbital angular momentum between theDπ system and the bachelor pion is denoted here as L.The total decay matrix element M for B− → D+π−π−

is given by:

M =∑

L=(0,1,2)

ρLeiφL [NL(x, y) + NL(y, x)]

ρkeiφk [Ak(x, y) + Ak(y, x)] , (8)

where the first term represents the S-wave (L = 0), P-wave (L = 1) and D-wave (L = 2) nonresonant contribu-tions, the second term stands for the resonant contribu-tions, the parameters ρk and φk are the magnitudes and

phases of kth resonance, while ρL and φL correspond tothe magnitudes and phases of the nonresonant contribu-tions with angular momentum L. The functions NL(x, y)and Ak(x, y) are the amplitudes for nonresonant and res-onant terms, respectively.

The resonant amplitudes Ak(x, y) are expressed as:

Ak(x, y) = Rk(m)FL(p′r′)FL(qr)TL(p, q, cos θ), (9)

where Rk(m) is the kth resonance lineshape, FL(p′r′) andFL(qr) are the Blatt-Weisskopf barrier factors [33], andTL(p, q, cos θ) gives the angular distribution. The param-eter m (=

√x) is the invariant mass of the Dπ system.

The parameter p′ is the magnitude of the three momen-tum of the bachelor pion evaluated in the B-meson restframe. The parameters p and q are the magnitudes of thethree momenta of the bachelor pion and the pion of theDπ system, both in the Dπ rest frame. The parametersp′, p, q and θ are functions of x and y.

The nonresonant amplitudes NL(x, y) with L = 0, 1, 2are similar to Ak(x, y) but do not contain resonant massterms:

N0(x, y) = 1, (10)

N1(x, y) = F1(p′r′)F1(qr)T1(p, q, cos θ), (11)

N2(x, y) = F2(p′r′)F2(qr)T2(p, q, cos θ). (12)

The Blatt-Weisskopf barrier factors FL(p′r′) andFL(qr) depend on a single parameter, r′ or r, the ra-

dius of the barrier, which we take to be 1.6 (GeV/c)−1,similarly to Ref. [12]. A discussion of the systematic un-certainty associated with the choice of the values of r andr′ follows below. The forms of FL(z), where z = p′r′ orqr, for L = 0, 1, 2 are:

F0(z) = 1, (13)

F1(z) =

1 + z20

1 + z2, (14)

F2(z) =

9 + 3z20 + z4

9 + 3z2 + z4, (15)

where z0 = p′0r′ or q0r. Here p′0 and q0 represent the

values of p′ and q, respectively, when the invariant mass isequal to the pole mass of the resonance. For nonresonantterms, the fit results are not affected by the choice ofinvariant mass (we use the sum of mD and mπ) usedfor the calculations of p′0 and q0. For virtual D∗

v decay,D∗

v → D+π−, and virtual B∗v production in B− → B∗

vπ−,we use an exponential form factor in place of the Blatt-Weisskopf barrier factor, as discussed in Ref. [12]:

F (z) = exp (−(z − z′)), (16)

where z′ = rpv for D∗v → D+π− and z′ = r′pv for

B− → B∗vπ−. Here, we set pv = 0.038 GeV/c, which

gives the best fit, although any value of pv between 0.015and 1.5 GeV/c gives negligible effect on the fitted param-eters compared to their statistical errors.

The resonance mass term Rk(m) describes the inter-mediate resonance. All resonances in this analysis areparametrized with relativistic Breit-Wigner functions:

Rk(m) =1

(m20 − m2) − im0Γ(m)

, (17)

where the decay width of the resonance depends on m:

Γ(m) = Γ0

)2L+1(m0

F 2L(qr), (18)

where m0 and Γ0 are the values of the resonance polemass and decay width, respectively.

The terms TL(p, q, cos θ) describe the angular distribu-tion of final state particles and are based on the Zemachtensor formalism [34]. The definitions of TL(p, q, cos θ)for L = 0, 1, 2 are:

T0(p, q, cos θ) = 1, (19)

T1(p, q, cos θ) = −2pq cos θ, (20)

T2(p, q, cos θ) = 4p2q2(cos2 θ − 1/3). (21)

The signal function is then given by:

S(x, y) = |M|2. (22)

In this analysis, the masses of D∗v and B∗

v are takenfrom the world averages [5] while their widths are fixedat 0.1 MeV; the magnitude ρk and phase φk of the D∗0

amplitude are fixed to 1 and 0, respectively, while themasses and widths of D0

J resonances and other magni-tudes and phases are free parameters to be determinedin the fit. The effect of varying the masses of D∗

v and B∗v

within their errors [5] and widths of D∗v and B∗

v between0.001 and 0.3 MeV is negligible compared to the othermodel-dependent systematic uncertainties given below.

Since the choice of normalization, phase conventionand amplitude formalism may not always be the samefor different experiments, we use fit fractions and relativephases instead of amplitudes to allow for a more mean-ingful comparison of results. The fit fraction for the kth

decay mode is defined as the integral of the resonance de-cay amplitudes divided by the coherent matrix elementsquared for the complete Dalitz plot:

DP |ρk(Ak(x, y) + Ak(y, x))|2dxdy∫

DP |M|2dxdy. (23)

The fit fraction for nonresonant term with angular mo-mentum L has a similar form:

DP |ρL(NL(x, y) + NL(y, x))|2dxdy∫

DP |M|2dxdy. (24)

The fit fractions do not necessarily add up to unity be-cause of interference among the amplitudes.

To estimate the statistical uncertainties on the fit frac-tions, the fit results are randomly modified according tothe covariance matrix of the fit and the new fractions arecomputed using Eq. (23) or (24). The resulting fit frac-tion distribution is fitted with a Gaussian whose widthgives the error on the given fraction.

D. Signal Resolution Function

The detector has finite resolution, thus measured quan-tities differ from their true values. For the narrow res-onance D∗

2 with the expected width of about 40 MeV,the signal resolution needs to be taken into account. Inorder to obtain the signal resolution on m2(Dπ) aroundthe D∗

2 mass region, we study a sample of MC generatedB− → Xπ− → D+π−π− decays, with the mass andwidth of X set to 2.460 GeV/c2 (D∗

2 mass region) and 0MeV, respectively, and subject these events to the sameanalysis reconstruction chain. The reconstructed eventsare then classified into two categories: truth-matched(TM) events, where the B and the daughters are correctlyreconstructed, and self-crossfeed (SCF) events, where oneor more of the daughters is not correctly associated withthe generated particle.

The two-dimensional distribution of cos θ versusm2(Dπ) for truth-matched events is shown in Fig. 4.Since the resolution is independent of cos θ, we fit thedistribution of the quantity q′ = m2(Dπ)−m2

true using asum of two Gaussian functions with a common mean toobtain the resolution function for truth-matched events(RTM). The signal resolution for an invariant mass ofthe Dπ combination around the D∗0

2 region is about 3MeV/c2.

The two-dimensional distribution of cos θ versusm2(Dπ) for self-crossfeed events is shown in Fig. 5. TheSCF fraction, fSCF, varies from 0.5% to 4.0% with cos θ.We fit the fSCF distribution with a 4th-order polynomialfunction. The fSCF distribution and the result of thefit are shown in Fig. 6. The resolution for self-crossfeedevents varies between 5 MeV/c2 and 100 MeV/c2 withcos θ. We divide the cos θ interval into 40 bins of equalwidth and use these bins to describe the resolution func-tion (RSCF) in terms of a sum of two bifurcated Gaus-sian (BGaussian) functions with different means. TheBGaussian is a Gaussian as a function of q′ with threeparameters, q′0 the mean, and the two widths, σ1 on theleft and σ2 on the right side of the mean. The form ofBGaussian is:

BGaussian(q′ − q′0, σ1, σ2) =

2√2π(σ1+σ2)

exp(− (q′−q′

) if q′ < q′0;

2√2π(σ1+σ2)

exp(− (q′−q′

) if q′ ≥ q′0,(25)

where q′0, σ1 and σ2 are free parameters.The signal resolution function is then given by:

R(q′, cos θ) = (1 − fSCF(cos θ)) ×RTM(q′)

+ fSCF(cos θ) ×RSCF(q′, cos θ). (26)

The function R(q′, cos θ) represents the probability den-sity for an event having the true mass squared m2

true tobe reconstructed at m2(Dπ) for different cos θ regions.

The signal term S in Eq. (4) is convoluted with theabove resolution function. For each event, the convolu-

)4/c2) (GeVπ(D2 m5.95 6 6.05 6.1 6.15

FIG. 4: Two-dimensional histogram cos θ versus m2(Dπ) ofthe truth-matched events as defined in the text.

tion is performed using numerical integration:

S(x, y) ⊗R =

S(qmin + q′, q′max) ×R(q′, cos θ)dq′, (27)

where S is the signal function in Eq. (22), and qmin (qmax)is the lesser (greater) of x and y. The quantity cos θ isdetermined from qmin and qmax and is assumed to beconstant during convolution. The resolution in cos θ hasa negligible effect on the fitted parameters. The quan-tity q′max is computed using the kinematics of three-bodydecay with qmin, q′ and cos θ.

The resolution function and the integration methodin Eq. (27) have been fully tested using 262 MC sam-ples with full event reconstruction given below. We havecompared D invariant mass resolutions for D0 → K−π+,K−π+π−π+ and D+ → K−π+π+ between data andMC-simulated events and find that they agree withintheir statistical uncertainties. Estimated biases in thefitted parameters due to uncertainties in the signal reso-lution function are small and have been included into thesystematic errors.

E. Efficiency

The signal term S defined above is modified in orderto take into account experimental particle detection andevent reconstruction efficiency. Since different regionsof the Dalitz plot correspond to different event topolo-gies, the efficiency is not expected to be uniform over theDalitz plot. The term ǫ(x, y) in Eq. (4) is the overall effi-ciency for truth-matched and self-crossfeed signal events,hence the efficiency for truth-matched signal events is

ǫTM(x, y) = ǫ(x, y)(1 − fSCF(cos θ)). (28)

In order to determine the efficiency across the Dalitzplot, a sample of simulated B− → D+π−π− events in

)4 /c2) (GeVπ(D2 m4.5 5 5.5 6 6.5 7 7.5 8

FIG. 5: Two-dimensional histogram cos θ versus m2(Dπ) ofthe self-crossfeed events as defined in the text.

θcos -1 -0.5 0 0.5 1

FIG. 6: fSCF(cos θ) distribution. The observed self-crossfeedfractions (points with statistical errors) are compared to theresults of the fit (solid curve).

the Dalitz plot is generated. Some events are gener-ated with one or more additional final-state photons toaccount for radiative corrections [35]. As a result, thegenerated Dalitz plot is slightly distorted from the uni-form distribution. The number of generated events isNgen =1252k. Each event is subjected to the standardreconstruction and selection, described in Section III. Inaddition, we require that the candidate decay is truthmatched. After correcting for data/MC efficiency dif-ferences in particle identification, which are momentumdependent and thus vary over the Dalitz plot, the totalnumber of accepted events is Nacc = 121, 390. We em-ploy an unbinned likelihood method to fit the Dalitz plotdistributions for generated and accepted event samples.The PDF for generated events (PDFgen) is a fourth-ordertwo-dimensional polynomial while the PDF for acceptedevents (PDFacc) is a seventh-order two-dimensional poly-nomial. The efficiency function is then given by:

ǫTM(x, y) =PDFacc(x, y) × Nacc

PDFgen(x, y) × Ngen. (29)

)4/c2) (GeVπ(D 2m5 10 15 20 25

FIG. 7: The efficiency for signal decays as a function ofm2(Dπ), as determined by MC simulation (points with sta-tistical errors) and the results of the fit to the accepted andgenerated distributions (solid curve).

Fig. 7 shows the efficiency as a function of m2(Dπ) andthe fit result for MC-simulated events.

F. Background

The background distribution is modeled using MCbackground events, selected with the same criteria ap-plied to the data and requiring the B candidate to fallinto the signal ∆E region defined in Section III. Eventsin the data ∆E sidebands could also be used to model thebackground, however in MC studies we find differencesbetween the Dalitz plot distributions of the backgroundin the signal and sideband regions. Since we find theDalitz plot distributions of sideband events in data andin MC simulation to be consistent within their statistics,we are confident that the MC simulation can accuratelyrepresent the background distribution in the signal re-gion. Fig. 8a, 8b and Fig. 9a show the Dalitz plot dis-tributions of sideband events in data, sideband events inthe MC sample and background events in the ∆E sig-nal region of the MC sample, respectively. Fig. 8c, 8dand 8e show the comparisons of ∆E sideband events be-tween data and MC simulation in m2

min(Dπ), m2max(Dπ)

and m2(ππ) projections, respectively. Here m2min(Dπ)

(m2max(Dπ)) is the lesser (greater) of x and y.The parameterization used to describe the background

B(x, y) = c0(qmin − q1)c1 ×

exp (c2(qmin − q1) + c3(qmin − q1)2)

+ c4(q2 − qmax)c5 ×

exp (c6(q2 − qmax) + c7(q2 − qmax)2)

+ c8(z − z1))c9 exp (c10(z − z1) + c11(z − z1)

+ c15BGaussian(qmax − c12, c13, c14)

+ c19BGaussian(z − c16, c17, c18), (30)

where the coefficients c0 to c19 are free parameters to

)4 /c2) (GeV1 π(D 2 m5 10 15 20 25

)4 /c2

25 (a)

)4 /c2) (GeV1 π(D2 m5 10 15 20 25

)4 /c2

25 (b)

)4 /c2) (GeVπ(D min2m

4 6 8 10 12 14

150(c)

)4 /c2) (GeVπ(D max2m

16 18 20 22 24 26

150(d)

)4 /c2 ) (GeVπ π(2m0 2 4 6 8 10 12

200(e)

FIG. 8: Comparison of events in the ∆E sideband: Dalitzplot for (a) data and (b) MC-simulated events, and projec-tions on (c) m2

min(Dπ), (d) m2max(Dπ) and (e) m2(ππ) with

data (points with statistical errors) and MC predictions (his-tograms).

be determined from the fit, q1 = (mD + mπ)2 = 4.04GeV2/c4 and q2 = (mB − mπ)2 = 26.41 GeV2/c4 arethe lower and upper limits of the Dalitz plot, respec-tively, z1 = (2mπ)2 = 0.077 GeV2/c4 is the lower limit ofm2(ππ), qmin is the lesser of x and y, qmax is the greaterof x and y, z is the invariant m2(ππ), and BGaussian isgiven in Eq. (25).

The projections on m2min(Dπ), m2

max(Dπ) and m2(ππ)and the result of the fit for the background events in thesignal region of the MC sample are shown in Fig. 9b, 9cand 9d. The χ2/NDF for the fit is 72/64.

)4 /c2) (GeV1 π(D 2 m5 10 15 20 25

)4 /c2

25 (a)

)4 /c2) (GeVπ(D min2m

4 6 8 10 12 14

80 (b)

)4 /c2) (GeVπ(D max2m

16 18 20 22 24 26

)4 /c2 ) (GeVπ π(2m0 2 4 6 8 10 12

100 (d)

FIG. 9: Fit to background events in the ∆E signal regionof the MC sample: (a) Dalitz plot and projections on (b)m2

min(Dπ), (c) m2max(Dπ) and (d) m2(ππ) with MC predic-

tions (points with statistical errors) and the fits (solid curves).

V. RESULTS

A. Branching Fraction B(B−→ D+π−π−)

The total B− → D+π−π− branching fraction is calcu-lated using the relation:

B =Nsignal

(ǫ · B(D+)) · 2N(B+B−), (31)

where Nsignal = 3414 ± 85 is the fitted signal yield givenin Section III, ǫ is the average efficiency, B(D+) = (9.22±0.21)% is the branching fraction for D+ → K−π+π+ [5,36], and the total number of B+B− events, N(B+B−) =(197.2±3.1)×106, is determined using NBB and the ratio

of Γ(Υ(4S) → B+B−)/Γ(Υ(4S) → B0B0) (= 1.065 ±0.026) [5].

Since the reconstruction efficiencies vary slightly fordifferent resonances, the average efficiency is calculatedby weighing the accepted and generated events by S(x, y)

with the values for the parameters of our nominal Dalitzplot model (discussed below):

∑Nacc

i=1 S(xi, yi) × wi∑Ngen

j=1 S(xj , yj), (32)

where wi is the correction factor which depends on xand y due to particle identification efficiency. The valueǫ = (8.72 ± 0.05)% is obtained using this method.

The measured total branching fraction is B(B− →D+π−π−) = (1.08± 0.03)× 10−3, where the stated errorrefers to the statistical uncertainty only. A full discussionof the systematic uncertainties follows below.

B. Dalitz plot analysis results

The Dalitz plot distribution for data is shown in Fig.10. Since the composition of events in the Dalitz plotand their distributions are not known a priori, we havetried a variety of different assumptions. In particular, wetest the inclusion of various components, such as the vir-tual D∗

v and B∗v as well as S-, P- and D-wave modeling of

the nonresonant component, in addition to the expectedcomponents of D∗0

2 , D∗00 and background. The D-wave

nonresonant term does not improve the goodness-of-fitand the fraction of D-wave nonresonant contribution isclose to 0. The results of these tests with variations ofthe models are summarized in Table II. Of these models,model 1 produces the best fit quality with the smallestnumber of components, and we choose it as the nom-inal fit model. The components considered in this fitmodel are D∗0

2 , D∗00 , D∗

v, B∗v and P-wave nonresonant.

The P-wave nonresonant component is an addition tothe fit model used in the previous measurement fromBelle [12]. The sum of the fractions (115 ± 5)% for thenominal fit differs from 100% because of destructive in-terferences among the amplitudes. The χ2/NDF for thenominal fit is 220/153. To better understand the largeχ2/NDF, we look at the contributions to the total χ2

from individual cells. We find four cells with χ2 > 7,which inflate the total χ2. The central points in thesecells are at: (6.83,−0.722), (6.83,−0.611), (6.83, 0.5) and(8.08,−0.722), where the first value is m2

min(Dπ), and thesecond is cos θ. In order to determine the effect on thefitted parameters from these cells, we repeat the nomi-nal fit with these cells excluded. The resulting χ2/NDFis 182/149, corresponding to a probability of 3.4%. As-suming these large χ2 contributions are caused by anunknown systematic problem, removing them from thefit is reasonable. However, under the assumption thatthese high χ2 contributions have a statistical origin, theχ2 probability is 0.04% [37]. The low probability indi-cates that a model more complex than the isobar modelmay be necessary to describe the characteristics of thedata. The differences in the fitted D∗

2 and D∗0 parame-

ters, when these cells are included or excluded, are as-signed to systematic uncertainties, and are much smaller

TABLE II: Fit results for the masses, widths, fit fractions and phases from the Dalitz plot analysis of B−→ D+π−π−

for different models. The errors are statistical only. The magnitude and phase of the D∗02 amplitude are fixed to 1 and 0,

respectively. The background fraction is fixed to 30.4% as described in Section III. The nominal fit corresponds to model 1.The labels, S-NR and P-NR, denote the S-wave nonresonant and P-wave nonresonant contributions, respectively.

Parameter Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 Model 7mD∗0

2(MeV/c2) 2460.4 ± 1.2 2460.2 ± 1.0 2459.1 ± 1.0 2460.1 ± 1.1 2461.5 ± 1.2 2458.1 ± 1.1 2457.4 ± 1.0

ΓD∗02

( MeV) 41.8 ± 2.5 41.7 ± 2.4 41.1 ± 2.4 41.8 ± 2.4 42.0 ± 2.5 41.8 ± 2.4 41.7 ± 2.4

mD∗00

(MeV/c2) 2297 ± 8 2309 ± 7 2297 ± 7 2312 ± 10 2307 ± 11 2270 ± 8 2273 ± 5

ΓD∗00

( MeV) 273 ± 12 285 ± 11 288 ± 12 289 ± 20 313 ± 21 262 ± 12 276 ± 10

fD∗02

(%) 32.2 ± 1.3 30.8 ± 1.2 31.5 ± 1.1 30.7 ± 1.5 32.6 ± 1.3 32.9 ± 1.3 30.9 ± 1.1

φD∗02

(rad) 0.0 (fixed) 0.0 (fixed) 0.0 (fixed) 0.0 (fixed) 0.0 (fixed) 0.0 (fixed) 0.0 (fixed)

fD∗00

(%) 62.8 ± 2.5 59.0 ± 2.1 57.5 ± 1.7 57.0 ± 4.5 88.0 ± 8.1 64.8 ± 2.2 69.7 ± 1.1

φD∗00

(rad) −2.07 ± 0.06 −2.06 ± 0.05 −2.01 ± 0.05 −2.00 ± 0.12 −2.14 ± 0.10 −1.96 ± 0.06 −2.00 ± 0.05

v(%) 10.1 ± 1.4 11.3 ± 1.5 9.0 ± 1.2 11.0 ± 1.5 9.6 ± 1.3

φD∗

v(rad) 3.00 ± 0.12 2.99 ± 0.08 3.17 ± 0.10 3.05 ± 0.12 2.82 ± 0.17

v(%) 4.6 ± 2.6 1.4 ± 0.5 1.7 ± 0.8 12.2 ± 5.4 2.2 ± 1.4

φB∗

v(rad) 2.80 ± 0.21 −2.43 ± 0.28 −2.33 ± 0.28 2.52 ± 0.25 2.28 ± 0.38

fP-NR (%) 5.4 ± 2.4 1.6 ± 0.4 12.6 ± 4.0 12.7 ± 3.1φP-NR (rad) −0.89 ± 0.18 −1.46 ± 0.20 −0.84 ± 0.12 −0.71 ± 0.10fS-NR (%) 0.3 ± 0.3 5.2 ± 3.8φS-NR (rad) −0.77 ± 0.49 3.30 ± 0.23fbg (%) 30.4(fixed) 30.4(fixed) 30.4(fixed) 30.4(fixed) 30.4(fixed) 30.4(fixed) 30.4(fixed)NLL 22970 22982 22977 22982 22964 23046 23125χ2/NDF 220/153 240/152 236/154 239/153 216/150 328/160 454/161

than the statistical uncertainties. The removal of thesecells does not affect the choice of model 1 as the nominalfit from Table II.

Ref. [38] argues for an addition of a Dπ S-wave statenear the Dπ system threshold to the model of the Dππfinal state. We have performed tests using the models1-4 in Table II with the D∗

v replaced by a Dπ S-wavestate. Two different parametrizations for Dπ S-wavestate amplitude are used: one is the function given byEq.(8) of Ref. [38] with the numerator set to constant,the other function is the relativistic Breit-Wigner givenby Eq. (17). Among the tests we have performed withthese parameterizations, the model with D∗

2 , D∗0 , Dπ S-

wave (using Eq.(8) of Ref. [38]), B∗v and P-wave nonres-

onant gives the best fit with NLL and χ2/NDF values of22997 and 271/151, respectively, which are worse thanthose of the nominal fit even when allowing the Dπ S-wave’s parameters to vary. Each of these models alsorequires large fractions of D∗

The nominal fit model results in the following branch-ing fractions: B(B− → D∗0

2 π−) × B(D∗02 → D+π−) =

(3.5 ± 0.2) × 10−4 and B(B− → D∗00 π−) × B(D∗0

0 →D+π−) = (6.8±0.3)×10−4, where the errors are statisti-cal only. A full discussion of the systematic uncertaintiesfollows below.

Fig. 11a, 11b and 11c show the m2min(Dπ), m2

max(Dπ)and m2(ππ) projections respectively, while Fig. 12a and12b show the cos θ distributions for the D∗0

0 and D∗02

mass regions, respectively. The distributions in Fig. 11and 12 show good agreement between the data and thefit. The angular distribution in the D∗0

2 mass region is

clearly visible and is consistent with the expected D-wavedistribution of | cos2 θ − 1/3 |2 for a spin-2 state. Inaddition, the D∗0

0 signal and the reflection of D∗02 can

be easily distinguished in the m2min(Dπ) and m2

max(Dπ)projection, respectively. The lower edge of m2

min(Dπ)is better described with D∗

v component included thanwithout.

Table III shows the NLL and χ2/NDF values for thenominal fit and for the fits with the broad resonance D∗0

) (GeV 1 π(D 2

m5 10 15 20 25

FIG. 10: Data Dalitz Plot for B−→ D+π−π−.

)4 /c2) (GeVπ(Dmin2m

4 5 6 7 8 9 10

)4 /c2

)4 /c2) (GeVπ(Dmax2m

14 16 18 20 22 24 26

)4 /c2

)4 /c2) (GeVπ π(2m0 2 4 6 8 10 12

)4 /c2

FIG. 11: Result of the nominal fit to the data: projectionson (a) m2

min(Dπ), (b) m2max(Dπ) and (c) m2(ππ). The points

with error bars are data, the solid curves represent the nom-inal fit. The shaded areas show the D∗0

2 contribution, thedashed curves show the D∗0

0 signal, the dash-dotted curvesshow the D∗

v and B∗v signals, and the dotted curves show the

background.

excluded or with the JP of the broad resonance replacedby other quantum numbers. In all cases, the NLL andχ2/NDF values are significantly worse than that of thenominal fit. Fig. 12a illustrates the helicity distributionsin the D∗0

0 mass region from hypotheses 2-4; clearly thenominal fit gives the best description of the data. We

θcos -1 -0.5 0 0.5 1

100(a)

θcos -1 -0.5 0 0.5 1

FIG. 12: Result of the nominal fit to the data: the cos θdistributions for (a) 4.5 < m2(Dπ) < 5.5 GeV2/c4 region and(b) 5.9 < m2(Dπ) < 6.2 GeV2/c4 region. The points witherror bars are data, the solid curves represent the nominal fit.The dashed, dash-dotted and dotted curves in (a) show thefit of hypotheses 2-4 in Table III, respectively. The shadedhistograms show the cos θ distributions from ∆E sidebandsin data.

conclude that a broad spin-0 state D∗00 is required in the

fit to the data. The same conclusion is obtained whenperforming the same test on Models 2-5.

TABLE III: Comparison of the models with different reso-nances composition. The labels, S-NR and P-NR, denote theS-wave nonresonant and P-wave nonresonant contributions,respectively.

Hypothesis Model NLL χ2/NDFModel 1 (nominal fit) 22970 220/153

1 D∗02 , D∗

v , B∗v , P-NR 23761 1171/143

2 D∗02 , D∗

v , B∗v , P-NR, (2+) 23699 991/144

3 D∗02 , D∗

v , B∗v , P-NR, (1−) 23427 638/135

4 D∗02 , D∗

v , B∗v , P-NR, S-NR 23339 652/157

VI. SYSTEMATIC UNCERTAINTIES

A. Uncertainties on B(B−→ D+π−π−)

As listed in Table IV, the systematic error on the mea-surement of the total B− → D+π−π− branching fractionis due to the uncertainties on the following quantities:the number of B+B− events in the initial sample, thecharged track reconstruction and identification efficien-cies, and the D+ → K−π+π+ branching fraction. Theuncertainty in the ∆E background shape, the uncertaintyin the average efficiency due to the fit models and a pos-sible fit bias also contribute to the systematic error.

TABLE IV: Summary of systematic uncertainties (relative er-rors in %) in the measurement of the total branching fraction.

Systematic Source ∆B(B−→D+π−π−)

B(B−→D+π−π−)(%)

Number of B+B− events 1.6Tracking efficiencies 2.5PID 1.5∆E background shape 1.3D+ branching fraction 2.3Fit models 0.7Fit bias 1.0Total Systematics 4.4

The uncertainty on the number of B+B− eventsis determined using the uncertainties on Γ(Υ(4S) →B+B−)/Γ(Υ(4S) → B0B0) [5] and integrated luminos-ity (1.1%). The uncertainty on the input D+ branchingfraction is taken from [36]. The uncertainty in the ∆Ebackground shape is estimated by comparing the signalyields between fitting the ∆E distribution with a lin-ear background shape and with higher-order (second andthird-order) polynomials. The uncertainty in the fit mod-els is estimated by comparing the average efficiencies inEq. (32) using Models 2-5 of Table II. The fit bias is es-timated to be less than 1% by comparing the generatedand the fitted value of B(B− → D+π−π−) from resonantand continuum MC samples.

B. Uncertainties on Dalitz plot analysis results

The sources of systematic uncertainties that affect theresults of the Dalitz Plot analysis are summarized in Ta-ble V. These uncertainties are added in quadrature, asthey are uncorrelated, to obtain the total systematic er-ror.

The uncertainties due to the background parameteri-zation are estimated by comparing the results from thenominal fit with those obtained when the backgroundshape parameters are allowed to float in the fit. The er-rors from the uncertainty in the background fraction areestimated by comparing the fit results when the back-ground fraction is changed by its statistical error. We

vary the set of cuts on ∆E, mES, R2, cosΘth and massof D+, which increase the number of signal events by 25%and the background fraction to 36.5%, and repeat the fits.The difference in the fit results is taken as an estimateof the systematic uncertainty due to the event selection.Fit biases are studied using 1248 parameterized MC sam-ples and 262 MC samples with full event reconstruction.Small biases are observed for some of the parameters.We combine these biases with those coming from high χ2

cells, as discussed in the previous section, in quadratureto obtain the total systematic contribution from the fitbias. The uncertainties in PID are obtained by compar-ing the nominal fit results with those obtained when thePID corrections to the reconstruction efficiency are var-ied according to their uncertainties. The uncertainties inthe efficiency and signal resolution parametrization arefound to be negligible using the fits to the reconstructedMC samples.

In addition to the above systematic uncertainties, wealso estimate a model-dependent uncertainty that comesfrom the uncertainty in the composition of the signalmodel and the uncertainty in the Blatt-Weisskopf bar-rier factors. The model-dependent uncertainties are es-timated by comparing the fit results with Models 2-5 inTable II and by varying the radius of the barrier, r′ andr in Eqs. (14)-(16) from 0 to 5 (GeV/c)

VII. SUMMARY

In conclusion, we measure the total branching fractionof the B− → D+π−π− decay to be

B(B− → D+π−π−) = (1.08 ± 0.03 ± 0.05)× 10−3,

where the first error is statistical and the second is sys-tematic.

Analysis of the B− → D+π−π− Dalitz plot using theisobar model confirms the existence of a narrow D∗0

2 anda broad D∗0

0 resonance as predicted by Heavy Quark Ef-fective Theory. The mass and width of D∗0

2 are deter-mined to be:

mD∗02

= (2460.4 ± 1.2 ± 1.2 ± 1.9)MeV/c2 and

ΓD∗02

= (41.8 ± 2.5 ± 2.1 ± 2.0)MeV,

respectively, while for the D∗00 they are:

mD∗00

= (2297± 8 ± 5 ± 19)MeV/c2 and

ΓD∗00

= (273 ± 12 ± 17 ± 45)MeV,

where the first and second errors reflect the statisticaland systematic uncertainties, respectively, the third oneis the uncertainty related to the assumed compositionof signal events and the Blatt-Weisskopf barrier factors.The measured masses and widths of both states are con-sistent with the world averages [5] and the predictions ofsome theoretical models [39–41].

TABLE V: Summary of systematic uncertainties in the masses, widths and fit fractions of the D∗02 and D∗0

0 and the phase ofD∗0

Systematic Source ∆mD∗02

∆ΓD∗02

∆mD∗00

∆ΓD∗00

∆fD∗02

∆fD∗00

∆φD∗00

( MeV/c2) ( MeV) (MeV/c2) (MeV) (%) (%) (rad)Background parameterization 1.0 1.1 3 5 1.2 0.0 0.04Background fraction 0.1 0.4 2 1 0.4 0.4 0.00Event selection 0.6 1.6 1 14 0.3 0.8 0.08Fit bias 0.3 0.7 4 8 0.7 1.4 0.02PID efficiency 0.0 0.1 0 0 0.0 0.1 0.01Total systematic error 1.2 2.1 5 17 1.5 1.7 0.09Fit models 1.3 0.7 15 40 1.5 17.2 0.07r constant 1.4 1.9 12 21 3.8 7.8 0.17Total model-dependent error 1.9 2.0 19 45 4.1 18.9 0.18

We have also obtained exclusive branching fractionsfor D∗0

2 and D∗00 production:

B(B− → D∗02 π−) × B(D∗0

2 → D+π−)

= (3.5 ± 0.2 ± 0.2 ± 0.4)× 10−4 and

B(B− → D∗00 π−) × B(D∗0

0 → D+π−)

= (6.8 ± 0.3 ± 0.4 ± 2.0)× 10−4.

Our results for the masses, widths and branching frac-tions are consistent with but more precise than previousmeasurements performed by Belle [12].

The relative phase of the scalar and tensor amplitudeis measured to be

φD∗00

= −2.07 ± 0.06 ± 0.09 ± 0.18 rad.

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.

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Dalitz plot analysis of B-→D+π-π

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