Work supported in part by US Department of Energy contract DE-AC02-76SF00515

hep-ex:arxiv:0901.1291BABAR-PUB-08/048, SLAC-PUB-13508

Dalitz Plot Analysis of B− → D+π−π−

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. Tanabe

Lawrence 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. Todyshev

Budker 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. Winstrom

University 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. Porter

January 2009

Published in the Physical Review D

2

California 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. 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, 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

3

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. Touramanis

University 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. Wang

4

University 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. Wong

Ohio 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. Torrence

University 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. Sitt

Laboratoire 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,

5

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, 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. 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, 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 about383 million 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± 0.01± 0.05)× 10−3. The masses and widths of D∗0

2 and D∗00 , the 2+ and 0+ cu

P-wave states decaying to D+π−, are measured: 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 tothe 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

I. INTRODUCTION

Orbitally excited states of the D meson, denoted hereas DJ , where J is the spin of the meson, provide aunique opportunity to test Heavy Quark Effective The-

††Also with Universita di Sassari, Sassari, Italy

6

ory (HQET) [1, 2]. The simplest DJ meson consists of acharm quark and a light anti-quark in an orbital angu-lar momentum L = 1 (P-wave) state. Four such statesare expected with spin-parity JP = 0+ (j = 1/2), 1+

(j = 1/2), 1+ (j = 3/2) and 2+ (j = 3/2), which arelabeled here as D∗

0 , D′1, D1 and D∗

2 , respectively, wherej is a quantum number corresponding to the sum of thelight quark spin and the orbital angular momentum �L.

The conservation of parity and angular momentum instrong 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 summarized inFig. 1. Due to the finite c-quark mass, the two JP = 1+

states may be a mixture of the j = 1/2 and j = 3/2states. Thus the broad D′

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

J mesons [5] are given in Table 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] ele-ments |Vcb| and |Vub|. The Belle experiment has reportedthe first observation of the broad D∗0

0 and D′01 mesons in

B decay [12]. The FOCUS experiment has found evi-dence for broad structures in D+π− final states [13] withmass and width in agreement with the D∗0

0 found byBelle experiment. However, the Particle Data Group [5]considers that the D∗0

0 and D′01 states still need confir-

mation.

In this analysis, we fully reconstruct the decays B− →D+π−π− [26] and measure its 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 D0

J mesons. The decayB− → 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.

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

1.8

2.0

2.2

2.4

2.6

2.8

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

S-waveπ D-waveπ

)2M

ass

(GeV

/c

D∗2D1D′

1D∗

0

D∗

D

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

1

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

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-ple consists of 347.2 fb−1 corresponding to 382.9 ± 4.2million BB pairs (NBB) taken on the peak of the Υ(4S)resonance. Monte Carlo (MC) simulation is used tostudy the detector response, its acceptance, background,and to validate the analysis. The MC uses the Evt-Gen generator [27] and GEANT4 [28] to simulate res-onant (e+e− → Υ(4S) → BB) and continuum events(e+e− → qq, where q = u, d, s or c).

A detailed description of the BABAR detector is givenin Ref. [29]. 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.

7

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 100MeV/c and at least twelve hitsin the DCH. A K− candidate must be identified as akaon using a likelihood-based particle identification algo-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.

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 ΔE:

mES =√

(s/2 + �p0 · �pB)2/E20 − p2

B , (1)

ΔE = 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 ΔE 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. Toenhance discrimination between signal and continuum,restrictions are placed on the magnitude of the cosineof the thrust angle, cos Θth, defined as the angle be-tween the thrust axis of the selected B candidate andthe thrust axis of the remaining tracks and neutral clus-ters in the event. The distribution of | cos Θth| is stronglypeaked towards unity for continuum background but isuniform for signal events. We accept the events with| cos Θth| < 0.85. Additionally, we also require the ratioof the second to the zeroth Fox-Wolfram moment [30] tobe less than 0.30 in order to further reduce the continuumbackground.

To suppress backgrounds, restriction are placed onmES: 5.2754 < mES < 5.2820 GeV/c2, and ΔE: −130 <ΔE < 130 MeV. The selected samples of B candidates

) ) (GeV/c+π+π-m(K

1.84 1.85 1.86 1.87 1.88 1.89 1.9

)2E

vent

s/(0

.002

GeV

/c

0

200

400

600

800

1000

) ) (GeV/c+π+π-m(K

1.84 1.85 1.86 1.87 1.88 1.89 1.9

)2E

vent

s/(0

.002

GeV

/c

0

200

400

600

800

1000

2

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 shadedregion marks the D+ signal region.

are used as input to an unbinned extended maximumlikelihood fit to the ΔE 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 simula-tion, we determine that the correct candidate is selectedat least 65% of the time. We fit the ΔE distribution ofthe selected B− → D+π−π− candidates with a sum oftwo Gaussian functions with a common mean for the sig-nal and a linear function for the background. The signalparameters (mean, width of wide Gaussian, width andfraction of narrow Gaussian) and the slope of the back-ground function are free parameters of the fit. The dataand the result of the fit are shown in Fig. 3. A clear signalis evident in the ΔE distribution. The resulting signalyield is 3496 ± 74 events, where the error is statisticalonly.

To distinguish signal and background in the Dalitz plotstudies, we divide the candidates into three subsamples:the signal region, −21 < ΔE < 15MeV, the left side-band, −109 < ΔE < −73MeV, and the right sideband,67 < ΔE < 103MeV. 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 ΔE dis-tribution, we have generated a background MC sampleof 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. TheΔE distribution of the selected events from the back-ground MC sample is shown as the histogram in Fig. 3. 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 is

8

E (GeV)Δ-0.1 -0.05 0 0.05 0.1

Eve

nts

/ ( 0

.01

GeV

)

0

500

1000

1500

E (GeV)Δ-0.1 -0.05 0 0.05 0.1

Eve

nts

/ ( 0

.01

GeV

)

0

500

1000

1500

FIG. 3: ΔE distribution for D+π−π− candidates. Data(points with statistical errors) are compared to the results ofthe fit (solid curve), with the background distribution markedas a dashed line. The histogram is the ΔE distribution of thebackground MC sample as described in the text. The threeshaded regions mark the signal in the center and the two side-bands.

misidentified as a signal π−. The background histogramin Fig. 3 is fitted with a sum of two Gaussian functionswith a common mean for the peaking background, withparameters fixed to those obtained from the fit to data,and a linear function to describe the combinatorial back-ground. The amount of peaking background is estimatedat 82 ± 41 events. After peaking background subtrac-tion, the number of signal events above background isNsignal = 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

dΓdxdy

=|M|2

256π3m3B

, (3)

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] ε(x, y)∫

DP [S(x, y) ⊗R] ε(x, y)dxdy,

(4)

where the integral is performed over the whole Dalitzplot, B(x, y) and S(x, y) are the background and sig-nal terms, respectively, fbg is the fraction of backgroundevents, ε is the reconstruction efficiency, and R is thesignal resolution function.

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

L =Nevent∏

i=1

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

The goodness-of-fit is estimated using a two-dimensional histogram of cos θ (range from -1 to 1) versusm2

min(Dπ) (range from 4.04 to 15.23 GeV/c2), where theparameter θ is the helicity angle of the Dπ system andm2

min(Dπ) is the lesser of x and y. The helicity angle θis defined as the angle between the momentum vector ofthe bachelor pion and that of the pion of the Dπ systemin the Dπ rest-frame. The χ2 value is calculated usingthe formula

χ2 =∑

i

χ2i =

ntotal∑i=1

(Ncelli − Nfiti)2

Nfiti

, (7)

9

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 predicted bythe fit results. The number of degrees of freedom (NDF)is calculated as ntotal − k − 1, where k is the numberof free parameters in the fit. A minimum of ten entriesin each cell is required; if this requirement is not metthen neighboring cells are combined until ten events areaccumulated.

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)]

+∑

k

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 andphases 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 [31], 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]. The forms of FL(z), where z = p′r′or qr, for L = 0, 1, 2 are:

F0(z) = 1, (13)

F1(z) =

√1 + z2

0

1 + z2, (14)

F2(z) =

√9 + 3z2

0 + z40

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, though any value of pv between 0.015and 1.5 GeV/c gives comparable results.

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

(q

q0

)2L+1 (m0

m

)F 2

L(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 [32]. 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

2

10

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.

Since the choice of normalization, phase conventionand amplitude formalism may not always be the samefor different experiments, we use fit fractions instead ofamplitudes to allow for a more meaningful comparison ofresults. The fit fraction for the kth decay mode is definedas the integral of the resonance decay amplitudes dividedby the coherent matrix element squared for the completeDalitz plot:

fk =

∫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:

fL =

∫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. In order to obtain thesignal resolution on m2(Dπ), we study a sample of MCgenerated B− → Xπ− → D+π−π− decays, with themass and width of X set to 2.460 GeV/c2 (D∗

2 mass re-gion) and 0 MeV, respectively, and subject these eventsto the same analysis reconstruction chain. The recon-structed events are then classified into two categories:truth-matched (TM) events, where the B and the daugh-ters are correctly reconstructed, and self-crossfeed (SCF)events, where one or more of the daughters is not cor-rectly associated with the 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 polynomial

function. 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′0)

2

2σ21

) if q′ < q′0;2√

2π(σ1+σ2)exp(− (q′−q′

0)2

2σ22

) 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, convolution isperformed 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 be con-stant during convolution. The quantity q′max is computedusing the kinematics of three-body decay with qmin, q′and cos θ.

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 ε(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

εTM(x, y) = ε(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 inthe Dalitz plot is generated. Some events are gener-ated with one or more additional final-state photons toaccount for radiative corrections [33]. As a result, the

11

0

50

100

150

200

250

300

350

)4/c2) (GeVπ(D2 m5.95 6 6.05 6.1 6.15

θco

s

-1

-0.5

0

0.5

1

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

0

20

40

60

80

100

120

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

θco

s

-1

-0.5

0

0.5

1

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

SCF

frac

tion

(%)

0

1

2

3

4

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

)4/c2) (GeVπ(D 2m5 10 15 20 25

Eff

icie

ncy

(%)

0

2

4

6

8

10

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).

generated 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:

εTM(x, y) =PDFacc(x, y) × Nacc

PDFgen(x, y) × Ngen. (29)

Fig. 7 shows the MC and the fit result for the efficiencyas a function of m2(Dπ).

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 ΔE region defined in Section III. Eventsin the data ΔE 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 to be consistent, we are confident that the MC canaccurately represent the background distribution in thesignal region. Fig. 8a, 8b and Fig. 9a show the Dalitzplot distributions of sideband events in data, sidebandevents in MC and background events in the ΔE signalregion of MC, respectively. Fig. 8c, 8d and 8e show the

12

0

2

4

6

8

10

)4 /c2) (GeV1 π(D 2 m5 10 15 20 25

)4 /c2

) (G

eV 2

π

(D

2 m

5

10

15

20

25

0

2

4

6

8

)4 /c2) (GeV1 π(D2 m5 10 15 20 25

)4 /c2

) (G

eV 2

π

(D

2 m

5

10

15

20

25

)4 /c2) (GeVπ(D min2m

4 6 8 10 12 14

) 4

/c

2E

vent

s/(0

.15

GeV

0

50

100

150

)4 /c2) (GeVπ(D max2m

16 18 20 22 24 26

) 4

/c 2

Eve

nts/

(0.1

5 G

eV

0

50

100

150

)4 /c2 ) (GeVπ π(2m0 2 4 6 8 10 12

) 4

/c 2

Eve

nts/

(0.1

5 G

eV

0

50

100

150

200

(a) (b)

(c)

(d)

(e)

FIG. 8: Comparison of events in the ΔE sideband: Dalitz plotfor (a) data and (b) MC, and projections on (c) m2

min(Dπ), (d)m2

max(Dπ) and (e) m2(ππ) with data (points with statisticalerrors) and MC predictions (histograms).

comparisons of ΔE sideband events between data andMC in m2

min(Dπ), m2max(Dπ) and m2(ππ) projections,

respectively. Here m2min(Dπ) (m2

max(Dπ)) is the lesser(greater) of x and y.

The parameterization used to describe the backgroundis:

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)2)+ c15BGaussian(qmax − c12, c13, c14)+ c19BGaussian(z − c16, c17, c18), (30)

where the coefficients c0 to c19 are free parameters tobe 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 of

0

2

4

)4 /c2) (GeV1 π(D 2 m5 10 15 20 25

)4 /c2

) (G

eV2

π

(D

2 m

5

10

15

20

25

)4 /c2) (GeVπ(D min2m

4 6 8 10 12 14

) 4

/c

2E

vent

s/(0

.15

GeV

0

20

40

60

80

)4 /c2) (GeVπ(D max2m

16 18 20 22 24 26

) 4

/c 2

Eve

nts/

(0.1

5 G

eV

0

20

40

60

80

)4 /c2 ) (GeVπ π(2m0 2 4 6 8 10 12

) 4

/c 2

Eve

nts/

(0.1

5 G

eV

0

50

100

(a)

(b)

(c)

(d)

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

min(Dπ), (c)m2

max(Dπ) and (d) m2(ππ) with MC (points with statisticalerrors) and the fits (solid curves).

m2(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 MC are shown in Fig. 9b, 9c and 9d. Theχ2/NDF for the fit is 72/64.

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

(ε · B(D+)) · 2N(B+B−), (31)

where Nsignal = 3414 ± 85 is the fitted signal yield givenin Section III, ε is the average efficiency, B(D+) = (9.22±

13

0.21)% is the branching fraction for D+ → K−π+π+ [5,34], and the total number of B+B− events, N(B+B−) =(197.2±3.1)×106, is determined using NBB and the ratioof 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):

ε =∑Nacc

i=1 S(xi, yi) × wi∑Ngenj=1 S(xj , yj)

, (32)

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

The measured total branching fraction is B(B− →D+π−π−) = (1.08± 0.01)× 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 of

)4

/c2

) (GeV 1 π(D 2

m5 10 15 20 25

)4

/c2

) (G

eV 2

π

(D

2 m

5

10

15

20

25

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

the 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 nominalfit model. The components considered in this fit modelare D∗0

2 , D∗00 , D∗

v , B∗v and P-wave nonresonant. The

)4 /c2) (GeVπ(Dmin2m

4 4.5 5 5.5 6 6.5 7 7.5 8

)4 /c2

Ent

ries

/(0.

125

GeV

0

100

200

300

400

500

)4 /c2) (GeVπ(Dmax2m

16 18 20 22 24 26

)4 /c2

Ent

ries

/(0.

125

GeV

0

50

100

150

)4 /c2) (GeVπ π(2m0 2 4 6 8 10 12

)4 /c2

Eve

nts/

(0.1

5 G

eV

0

50

100

150

200

(a)

(b)

(c)

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.

14

θcos -1 -0.5 0 0.5 1

Eve

nts/

bin

20

40

60

80

100

θcos -1 -0.5 0 0.5 1

Eve

nts/

bin

0

50

100

150

(a)

(b)

FIG. 12: Result of the nominal fit to the data: the cos θdistributions for (a) 4.5 < m2(Dπ) < 5.5 GeV/c2 region and(b) 5.9 < m2(Dπ) < 6.2 GeV/c2 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 ΔE sidebandsin data.

P-wave nonresonant component is in addition to the fitmodel used in the previous measurement from Belle [12].The sum of the fractions (115 ± 5)% for the nominalfit differs from 100% because of destructive interferencesamong the amplitudes. The χ2/NDF for the nominal fitis 215/150. There are four cells having χ2 > 7 in the two-dimensional histogram of cos θ versus m2

min(Dπ), whichinflate the overall χ2. The central points of these cells are:(6.83,−0.722), (6.83,−0.611), (6.83, 0.5), (8.08,−0.722),where the first value is m2

min(Dπ), and the second valueis cos θ. In order to test the effect of these cells, we re-peat the nominal fit with these cells excluded from thefit and determine that the effect on the fitted parametersis much smaller than their statistical uncertainties. Thefit gives χ2/NDF = 176/146, correspond to a probabilityof 4.6%. The differences in the fitted parameters, whenincluding and excluding these cells, are included in thesystematic uncertainties.

The nominal fit model results in the following branch-

ing fractions: B(B− → D∗02 π−) × B(D∗0

2 → D+π−) =(3.5 ± 0.1) × 10−4 and B(B− → D∗0

0 π−) × B(D∗00 →

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 isclearly 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

0

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. Weconclude that a broad spin-0 state D∗0

0 is required in thefit to the data.

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 ΔE background shape, the uncertaintyin the average efficiency due to the fit models and a pos-sible fit bias also contribute to the systematic error.

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 [34]. The uncertainty in the ΔEbackground shape is estimated by comparing the signalyields between fitting the ΔE 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.

15

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

fD∗v

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

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

fB∗v

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

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 215/150 236/152 231/152 236/150 214/148 317/152 431/156

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 216/150

1 D∗02 , D∗

v , B∗v , P-NR 23761 1142/152

2 D∗02 , D∗

v , B∗v , P-NR, (2+) 23699 968/150

3 D∗02 , D∗

v , B∗v , P-NR, (1−) 23427 645/150

4 D∗02 , D∗

v , B∗v , P-NR, S-NR 23339 617/150

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

Systematic Source ΔB(B−→D+π−π−)

B(B−→D+π−π−)(%)

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

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, as

they 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. Wevary the set of cuts on ΔE, mES, R2, cos Θth and massof D+, which increase the number of signal events by25% and the background fraction to 36.5%, and repeatthe fits. The difference in the fit results is taken as anestimate of the systematic uncertainty due to the eventselection.

Fit biases are studied using 1248 parameterized MCsamples without event reconstruction and 262 MC sam-ples with full event reconstruction. Small biases are ob-served for some of the parameters. We combine thesebiases with those coming from high χ2 cells, as discussedin the previous section, in quadrature to obtain the totalsystematic contribution from the fit bias. The uncertain-ties in PID are obtained by comparing the nominal fitresults with those obtained when the PID corrections tothe reconstruction efficiency are varied according to theiruncertainties. The uncertainties in the efficiency and sig-nal resolution parametrization are found to be negligibleusing the fits to the reconstructed MC samples.

In addition to the above systematic uncertainties, wealso estimate a model-dependent uncertainty that comesfrom the uncertainty in the composition of the signal

16

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

0 .

Systematic Source ΔmD∗02

ΔΓD∗02

ΔmD∗00

ΔΓD∗00

ΔfD∗02

ΔfD∗00

ΔφD∗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

model 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)−1.

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 ± 0.01 ± 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 [35–37].

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 ± 0.1 ± 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.

17

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