arXiv:1007.0504v2 [hep-ex] 15 Feb 2011 B A B AR-PUB-10/008 SLAC-PUB-14187 arXiv:1007.0504 [hep-ex] Measurement of CP observables in B ± → D CP K ± decays and constraints on the CKM angle γ P. del Amo Sanchez, J. P. Lees, V. Poireau, E. Prencipe, and V. Tisserand Laboratoire d’Annecy-le-Vieux de Physique des Particules (LAPP), Universit´ e de Savoie, CNRS/IN2P3, F-74941 Annecy-Le-Vieux, France J. Garra Tico and E. Grauges Universitat de Barcelona, Facultat de Fisica, Departament ECM, E-08028 Barcelona, Spain M. Martinelli ab , A. Palano ab , and M. Pappagallo ab INFN Sezione di Bari a ; Dipartimento di Fisica, Universit` a di Bari b , I-70126 Bari, Italy G. Eigen, B. Stugu, and L. Sun University of Bergen, Institute of Physics, N-5007 Bergen, Norway M. Battaglia, D. N. Brown, B. Hooberman, L. T. Kerth, Yu. G. Kolomensky, G. Lynch, I. L. Osipenkov, and T. Tanabe Lawrence Berkeley National Laboratory and University of California, Berkeley, California 94720, USA C. M. Hawkes and A. T. Watson University of Birmingham, Birmingham, B15 2TT, United Kingdom H. Koch and T. Schroeder Ruhr Universit¨ at Bochum, Institut f¨ ur Experimentalphysik 1, D-44780 Bochum, Germany D. J. Asgeirsson, C. Hearty, T. S. Mattison, and J. A. McKenna University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z1 A. Khan and A. Randle-Conde Brunel University, Uxbridge, Middlesex UB8 3PH, United Kingdom V. E. Blinov, A. R. Buzykaev, V. P. Druzhinin, V. B. Golubev, A. P. Onuchin, S. I. Serednyakov, Yu. I. Skovpen, E. P. Solodov, K. Yu. Todyshev, and A. N. Yushkov Budker Institute of Nuclear Physics, Novosibirsk 630090, Russia M. Bondioli, S. Curry, D. Kirkby, A. J. Lankford, M. Mandelkern, E. C. Martin, and D. P. Stoker University of California at Irvine, Irvine, California 92697, USA H. Atmacan, J. W. Gary, F. Liu, O. Long, and G. M. Vitug University of California at Riverside, Riverside, California 92521, USA C. Campagnari, T. M. Hong, D. Kovalskyi, and J. D. Richman University of California at Santa Barbara, Santa Barbara, California 93106, USA A. M. Eisner, C. A. Heusch, J. Kroseberg, W. S. Lockman, A. J. Martinez, T. Schalk, B. A. Schumm, A. Seiden, 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, D. G. Hitlin, P. Ongmongkolkul, F. C. Porter, and A. Y. Rakitin
P. del Amo Sanchez, J. P. Lees, V. Poireau, E. Prencipe, and V. TisserandLaboratoire d’Annecy-le-Vieux de Physique des Particules (LAPP),
Universite de Savoie, CNRS/IN2P3, F-74941 Annecy-Le-Vieux, France
J. Garra Tico and E. GraugesUniversitat de Barcelona, Facultat de Fisica, Departament ECM, E-08028 Barcelona, Spain
M. Martinelliab, A. Palanoab, and M. Pappagalloab
INFN Sezione di Baria; Dipartimento 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
M. Battaglia, D. N. Brown, B. Hooberman, L. T. Kerth,
Yu. G. Kolomensky, G. Lynch, I. L. Osipenkov, and T. TanabeLawrence Berkeley National Laboratory and University of California, Berkeley, California 94720, USA
C. M. Hawkes 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, C. Hearty, T. S. Mattison, and J. A. McKennaUniversity of British Columbia, Vancouver, British Columbia, Canada V6T 1Z1
A. Khan and A. Randle-CondeBrunel University, Uxbridge, Middlesex UB8 3PH, United Kingdom
V. E. Blinov, A. R. Buzykaev, V. P. Druzhinin, V. B. Golubev, A. P. Onuchin,
S. I. Serednyakov, Yu. I. Skovpen, E. P. Solodov, K. Yu. Todyshev, and A. N. YushkovBudker Institute of Nuclear Physics, Novosibirsk 630090, Russia
M. Bondioli, S. Curry, D. Kirkby, A. J. Lankford, M. Mandelkern, E. C. Martin, and D. P. StokerUniversity of California at Irvine, Irvine, California 92697, USA
H. Atmacan, J. W. Gary, F. Liu, O. Long, and G. M. VitugUniversity of California at Riverside, Riverside, California 92521, USA
C. Campagnari, T. M. Hong, D. Kovalskyi, and J. D. RichmanUniversity of California at Santa Barbara, Santa Barbara, California 93106, USA
A. M. Eisner, C. A. Heusch, J. Kroseberg, W. S. Lockman,
A. J. Martinez, T. Schalk, B. A. Schumm, A. Seiden, 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, D. G. Hitlin, P. Ongmongkolkul, F. C. Porter, and A. Y. Rakitin
California Institute of Technology, Pasadena, California 91125, USA
R. Andreassen, M. S. Dubrovin, G. Mancinelli, B. T. Meadows, and M. D. SokoloffUniversity of Cincinnati, Cincinnati, Ohio 45221, USA
P. C. Bloom, W. T. Ford, A. Gaz, M. Nagel, U. Nauenberg, J. G. Smith, and S. R. WagnerUniversity of Colorado, Boulder, Colorado 80309, USA
R. Ayad∗ and W. H. TokiColorado State University, Fort Collins, Colorado 80523, USA
H. Jasper, T. M. Karbach, J. Merkel, A. Petzold, B. Spaan, and K. WackerTechnische Universitat Dortmund, Fakultat Physik, D-44221 Dortmund, Germany
M. J. Kobel, K. R. Schubert, and R. SchwierzTechnische Universitat Dresden, Institut fur Kern- und Teilchenphysik, D-01062 Dresden, Germany
D. Bernard 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, E. Fioravantiab,
P. Franchiniab, E. Luppiab, M. Muneratoab, M. Negriniab, A. Petrellaab, and L. Piemontesea
INFN Sezione di Ferraraa; Dipartimento di Fisica, Universita di Ferrarab, I-44100 Ferrara, Italy
R. Baldini-Ferroli, A. Calcaterra, R. de Sangro, G. Finocchiaro, M. Nicolaci,
S. Pacetti, P. Patteri, I. M. Peruzzi,† M. Piccolo, M. Rama, and A. ZalloINFN Laboratori Nazionali di Frascati, I-00044 Frascati, Italy
R. Contriab, E. Guidoab, M. Lo Vetereab, M. R. Mongeab, S. Passaggioa, C. Patrignaniab, E. Robuttia, and S. Tosiab
INFN Sezione di Genovaa; Dipartimento di Fisica, Universita di Genovab, I-16146 Genova, Italy
B. Bhuyan and V. PrasadIndian Institute of Technology Guwahati, Guwahati, Assam, 781 039, India
C. L. Lee 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
F. U. Bernlochner, M. Ebert, H. M. Lacker, T. Lueck, and A. VolkHumboldt-Universitat zu Berlin, Institut fur Physik, Newtonstr. 15, D-12489 Berlin, Germany
P. D. Dauncey and M. TibbettsImperial College London, London, SW7 2AZ, United Kingdom
P. K. Behera and U. MallikUniversity of Iowa, Iowa City, Iowa 52242, USA
C. Chen, 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, and Z. J. GuoJohns Hopkins University, Baltimore, Maryland 21218, USA
3
N. Arnaud, M. Davier, D. Derkach, J. Firmino da Costa, G. Grosdidier, F. Le Diberder, A. M. Lutz, B. Malaescu,
A. Perez, P. Roudeau, M. H. Schune, J. Serrano, V. Sordini,‡ A. Stocchi, L. Wang, and G. WormserLaboratoire de l’Accelerateur Lineaire, IN2P3/CNRS et Universite Paris-Sud 11,
Centre Scientifique d’Orsay, B. P. 34, F-91898 Orsay Cedex, France
D. J. Lange and D. M. WrightLawrence Livermore National Laboratory, Livermore, California 94550, USA
I. Bingham, C. A. Chavez, J. P. Coleman, 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, 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, W. Gradl, and A. HafnerJohannes Gutenberg-Universitat Mainz, Institut fur Kernphysik, D-55099 Mainz, Germany
K. E. Alwyn, D. Bailey, R. J. Barlow, G. Jackson, G. D. Lafferty, and T. J. WestUniversity of Manchester, Manchester M13 9PL, United Kingdom
J. Anderson, R. Cenci, A. Jawahery, D. A. Roberts, G. Simi, and J. M. TuggleUniversity of Maryland, College Park, Maryland 20742, USA
C. Dallapiccola and E. SalvatiUniversity of Massachusetts, Amherst, Massachusetts 01003, USA
R. Cowan, D. Dujmic, P. H. Fisher, G. Sciolla, and M. ZhaoMassachusetts Institute of Technology, Laboratory for Nuclear Science, Cambridge, Massachusetts 02139, USA
D. Lindemann, P. M. Patel, S. H. Robertson, and M. SchramMcGill University, Montreal, Quebec, Canada H3A 2T8
P. Biassoniab, A. Lazzaroab, V. Lombardoa, F. Palomboab, and S. Strackaab
INFN Sezione di Milanoa; Dipartimento di Fisica, Universita di Milanob, I-20133 Milano, Italy
L. Cremaldi, R. Godang,§ R. Kroeger, P. Sonnek, and D. J. SummersUniversity of Mississippi, University, Mississippi 38677, USA
X. Nguyen, M. Simard, and P. TarasUniversite de Montreal, Physique des Particules, Montreal, Quebec, Canada H3C 3J7
G. De Nardoab, 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
4
L. A. Corwin, K. Honscheid, R. Kass, J. P. Morris, and A. M. RahimiOhio State University, Columbus, Ohio 43210, USA
N. L. Blount, J. Brau, R. Frey, O. Igonkina, J. A. Kolb, R. Rahmat, N. B. Sinev, D. Strom, J. Strube, and E. TorrenceUniversity of Oregon, Eugene, Oregon 97403, USA
G. Castelliab, E. Feltresiab, N. Gagliardiab, M. Margoniab, M. Morandina,
M. Posoccoa, M. Rotondoa, F. Simonettoab, and R. Stroiliab
INFN Sezione di Padovaa; Dipartimento di Fisica, Universita di Padovab, I-35131 Padova, Italy
E. Ben-Haim, G. R. Bonneaud, H. Briand, G. Calderini, J. Chauveau,
O. Hamon, Ph. Leruste, G. Marchiori, J. Ocariz, 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
M. Biasiniab, E. Manoniab, and A. Rossiab
INFN Sezione di Perugiaa; Dipartimento di Fisica, Universita di Perugiab, I-06100 Perugia, Italy
C. Angeliniab, G. Batignaniab, S. Bettariniab, M. Carpinelliab,¶ G. Casarosaab, A. Cervelliab,
F. Fortiab, M. A. Giorgiab, A. Lusianiac, 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, L. Li Gioia, M. A. Mazzonia, G. Pireddaa, and F. Rengaab
INFN Sezione di Romaa; Dipartimento di Fisica,Universita di Roma La Sapienzab, I-00185 Roma, Italy
T. Hartmann, T. Leddig, 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, G. Hamel de Monchenault, G. Vasseur, Ch. Yeche, and M. ZitoCEA, Irfu, SPP, Centre de Saclay, F-91191 Gif-sur-Yvette, France
M. T. Allen, D. Aston, D. J. Bard, R. Bartoldus, J. F. Benitez, C. Cartaro, M. R. Convery, J. Dorfan,
G. P. Dubois-Felsmann, W. Dunwoodie, R. C. Field, M. Franco Sevilla, B. G. Fulsom, A. M. Gabareen,
M. T. Graham, P. Grenier, C. Hast, W. R. Innes, 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,D. R. Muller, H. Neal, S. Nelson, C. P. O’Grady, I. Ofte, M. Perl, T. Pulliam, B. N. Ratcliff,
A. Roodman, A. A. Salnikov, V. Santoro, R. H. Schindler, J. Schwiening, A. Snyder, D. Su,
M. K. Sullivan, S. Sun, K. Suzuki, 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, C. C. Young, and V. ZieglerSLAC National Accelerator Laboratory, Stanford, California 94309 USA
X. R. Chen, W. Park, M. V. Purohit, R. M. White, and J. R. WilsonUniversity of South Carolina, Columbia, South Carolina 29208, USA
S. J. SekulaSouthern Methodist University, Dallas, Texas 75275, USA
M. Bellis, P. R. Burchat, A. J. Edwards, and T. S. Miyashita
5
Stanford 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
N. Guttman and A. SofferTel Aviv University, School of Physics and Astronomy, Tel Aviv, 69978, Israel
P. Lund and S. M. SpanierUniversity of Tennessee, Knoxville, Tennessee 37996, USA
R. Eckmann, J. L. Ritchie, A. M. Ruland, C. J. Schilling, R. F. Schwitters, and B. C. WrayUniversity of Texas at Austin, Austin, Texas 78712, USA
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. Lanceriab, and L. Vitaleab
INFN Sezione di Triestea; Dipartimento di Fisica, Universita di Triesteb, I-34127 Trieste, Italy
N. Lopez-March, F. Martinez-Vidal, D. A. Milanes, and A. OyangurenIFIC, Universitat de Valencia-CSIC, E-46071 Valencia, Spain
J. Albert, Sw. Banerjee, H. H. F. Choi, K. Hamano, G. J. King,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, T. E. Latham, and E. M. T. PuccioDepartment of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom
H. R. Band, S. Dasu, K. T. Flood, Y. Pan, R. Prepost, C. O. Vuosalo, and S. L. WuUniversity of Wisconsin, Madison, Wisconsin 53706, USA
(Dated: February 16, 2011)
Using the entire sample of 467 million Υ (4S) → BB decays collected with the BABAR detectorat the PEP-II asymmetric-energy B factory at SLAC, we perform an analysis of B±
→ DK±
decays, using decay modes in which the neutral D meson decays to either CP -eigenstates or non-CP -eigenstates. We measure the partial decay rate charge asymmetries for CP -even and CP -odd Dfinal states to be ACP+ = 0.25 ± 0.06 ± 0.02 and ACP− = −0.09 ± 0.07 ± 0.02, respectively, wherethe first error is the statistical and the second is the systematic uncertainty. The parameter ACP+
is different from zero with a significance of 3.6 standard deviations, constituting evidence for directCP violation. We also measure the ratios of the charged-averaged B partial decay rates in CP andnon-CP decays, RCP+ = 1.18 ± 0.09 ± 0.05 and RCP− = 1.07 ± 0.08 ± 0.04. We infer frequentistconfidence intervals for the angle γ of the unitarity triangle, for the strong phase difference δB , andfor the amplitude ratio rB, which are related to the B−
→ DK− decay amplitude by rBei(δB−γ) =
A(B−→ D0K−)/A(B−
→ D0K−). Including statistical and systematic uncertainties, we obtain0.24 < rB < 0.45 (0.06 < rB < 0.51) and, modulo 180◦, 11.3◦ < γ < 22.7◦ or 80.8◦ < γ < 99.2◦ or157.3◦ < γ < 168.7◦ (7.0◦ < γ < 173.0◦) at the 68% (95%) confidence level.
PACS numbers: 13.25.Hw, 12.15.Hh, 11.30.Er
∗Now at Temple University, Philadelphia, Pennsylvania 19122,
USA†Also with Universita di Perugia, Dipartimento di Fisica, Perugia,
Italy‡Also with Universita di Roma La Sapienza, I-00185 Roma, Italy
6
I. INTRODUCTION
In the standard model (SM) of fundamental particles,CP violation in weak interactions is allowed by a sin-gle, irreducible phase in the 3 × 3 Cabibbo-Kobayashi-Maskawa (CKM) quark flavor-mixing matrix [1, 2]. Theunitarity of the CKM matrix, V , implies a set of rela-tions among its elements Vij , in particular the conditionVudV
∗ub + VcdV
∗cb + VtdV
∗tb = 0, which can be depicted in
the complex plane as a “unitarity” triangle, whose sidesand angles are related to the magnitudes and phases ofthe six elements Vid and Vib, where i = u, c, t. Over-constraining the unitarity triangle by means of precisemeasurements of all its sides and angles allows tests ofwhether the CKM mechanism is the correct descriptionof CP violation. Any inconsistencies among the variousexperimental constraints would reveal effects of physicsbeyond the standard model.
After a decade of successful operation and a total ofabout 1.3 billion BB pairs collected by the BABAR andBelle experiments, the three CKM angles have been mea-sured with varied precision. The angle β has been mea-sured with the highest precision, to around 1◦, usingB0 → (cc)K(∗)0 decays. Using a variety of two-body Bdecays (B → ππ, ρπ, ρρ and a1(1260)π) the angle α hasbeen measured to a precision of around 4◦. The angle γhas a relatively large uncertainty, around 14◦, comparedwith α and β. The lack of precision in our knowledge ofγ reflects the difficulty in measuring this angle. The un-certainties of the CKM angles quoted in this paragraphare taken from [3].
Several techniques for measuring γ in a theoreticallyclean way are based on B meson decays to open-charmfinal states, D(∗)0Xs and D(∗)0Xs (Xs = K(∗)±, K(∗)0).In these decays, the interference between the b → cusand b → ucs tree amplitudes, when the D0 and D0 de-cay to a common final state, leads to observables thatdepend on the relative weak phase γ. The size of theinterference also depends on the magnitude of the ratiorB and the relative phase strong phase δB of the twoamplitudes, which can not be precisely calculated fromtheory. They can be extracted directly from data bysimultaneously reconstructing several related B → DKdecays. Many methods have been proposed to extractγ from B decays using D(∗)K(∗)± and D(∗)K(∗)0 finalstates (here and in the following D refers to any admix-ture of the neutral D0 meson and its CP -conjugate D0).The three methods that have been used most produc-tively to date are the “GLW” method [4, 5], based onCabibbo-suppressed D decays to CP -eigenstates, suchas K+K− or K0
Sπ0; the “ADS” method [6, 7], where
the D is reconstructed in Cabibbo-favored and doubly-
§Now at University of South Alabama, Mobile, Alabama 36688,
USA¶Also with Universita di Sassari, Sassari, Italy
Cabibbo-suppressed final states such as K±π∓; and the“GGSZ” method [8], which studies the Dalitz-plot distri-bution of the products of D decays to multi-body self-conjugate final states, such as K0
Sπ+π−. A common
problem with these methods is the small overall branch-ing fraction of these decays ranging from 5 × 10−6 to5×10−9. Therefore a precise determination of γ requiresa very large data sample. BABAR has published several γrelated measurements: GLW analyses of B± → DK± [9],D∗K± [10] and DK∗± [11] decays; ADS analyses ofB± → D(∗)K± [12, 13], DK∗± [11] and B0 → DK∗0
[14]; and GGSZ analyses of B± → D(∗)K(∗)± [15, 16]and B0 → DK∗0 decays [17]. To date, the single mostprecise experimental determination of γ from BABAR isγ = (68± 14± 4± 3)◦ and 39◦ < γ < 98◦, obtained fromthe GGSZ analysis of B± → D(∗)K(∗)± decays [16]. Inthis measurement, the first error represents the statisti-cal uncertainty, the second is the experimental systematicuncertainty, and the third reflects the uncertainty on thedescription of the D Dalitz-plot distributions.
II. GLW ANALYSIS OF B → DK DECAYS
In this paper we present the update of the GLW analy-sis ofB± → DK± decays based on the full BABAR datasetcollected near the Υ (4S) resonance. In addition to a 22%increase in statistics of the data sample, this study bene-fits from other significant improvements compared to ourprevious result [9]:
• More refined charged track reconstruction and par-ticle identification algorithms, with higher purityand efficiency, have been employed;
• The event shape variable F , used to discriminatethe signal from the continuum e+e− → qq back-ground (described in detail in Section IV) has beenremoved from the selection criteria and has in-stead been included in the final fit to the selectedB candidates. This allows us to increase the sig-nal efficiency by about 30%. At the same time itprovides a larger sample of continuum backgroundevents, thus allowing for the determination of thebackground properties directly from data (see Sec-tion V);
• Better kaon/pion separation, which is needed todistinguish B± → DK± candidates from the twelvetimes more abundant B± → Dπ± decays, isachieved through the use of a global likelihoodbased not only on the Cherenkov angle θC recon-structed by the Cherenkov detector, but also on thespecific energy loss dE/dx measured by the track-ing devices. The inclusion of dE/dx in the like-lihood increases the kaon identification efficiencyand decreases the pion misidentification both atlow momentum and outside of the geometrical ac-ceptance of the Cherenkov detector (which is 10%lower than the acceptance of the tracking devices).
7
In order to determine γ from B± → DK± decayswith the GLW method, we measure the two direct-CP -violating partial decay rate asymmetries,
ACP± ≡ Γ(B− → DCP±K−)− Γ(B+ → DCP±K
+)
Γ(B− → DCP±K−) + Γ(B+ → DCP±K+),
(1)and the two ratios of charge averaged partial rates usingD decays to CP and flavor eigenstates,
RCP± ≡ 2Γ(B− → DCP±K
−) + Γ(B+ → DCP±K+)
Γ(B− → D0K−) + Γ(B+ → D0K+),
(2)where DCP± refer to the CP eigenstates of the D mesonsystem. We then extract γ, together with the other twounknowns rB and δB, by means of a frequentist proce-dure, which exploits the following relations [4, 5], neglect-ing D0–D0 mixing [18]:
RCP± = 1 + r2B ± 2rB cos δB cos γ , (3)
ACP± =±2rB sin δB sin γ
1 + r2B ± 2rB cos δB cos γ. (4)
Here, rB≡∣
∣A(B−→D0K−)/A(B−→D0K−)∣
∣ is the mag-
nitude of the ratio of the amplitudes for B−→D0K− andB−→D0K− and δB the difference of their strong phases.Taking into account the CKM factor (|VubVcs/VcbVus| ≈0.4) and color-suppression of the B−→D0K− amplitude,rB is expected to be around 0.1. The current worldaverages for the B± → DK± GLW observables fromthe measurements in [9, 19, 20] are summarized in Ta-ble I. The world averages for the parameters rB and δB
TABLE I: World averages at 68% confidence level [21] for theGLW observables in B → DK decays.
level (CL) [3].To reduce the systematic uncertainties from branch-
ing fractions and reconstruction efficiencies of differentD channels appearing in the numerator and denomina-tor of Eq. 2, we approximate RCP± with the double ratios
RCP± ≈R±
K/π
RK/π, (5)
where
R±K/π ≡ Γ(B− → DCP±K
−) + Γ(B+ → DCP±K+)
Γ(B− → DCP±π−) + Γ(B+ → DCP±π+),
(6)and
RK/π ≡ Γ(B− → D0K−) + Γ(B+ → D0K+)
Γ(B− → D0π−) + Γ(B+ → D0π+). (7)
Equation 5 would be exact in the limit in which theCabibbo-suppressed contributions to the B± → Dπ±
amplitudes vanish, as well as terms proportional torBrD ≈ 5× 10−3, as we will discuss in Section VII. Thisapproximation results in a systematic uncertainty on thefinal values of RCP±.The paper is organized as follows. In Section III we
describe the data sample used for these measurementsand the main features of the BABAR detector and of thePEP-II storage rings. In Section IV we summarize theprocedure adopted to select B± → Dh± candidates andsuppress the main backgrounds. In Section V we intro-duce the simultaneous extended maximum likelihood fitused to extract the observables RCP± and ACP±. In Sec-tion VI we explain how, by applying the same fit pro-cedure to selected control samples, we estimate the ir-reducible background present in the final samples. Adiscussion of the sources of systematic uncertainties andthe evaluation of the uncertainties is presented in Sec-tion VII. Section VIII lists the final results on the GLWobservables RCP± and ACP±, including statistical andsystematic uncertainties. It also contains a descriptionof the statistical method used to construct frequentistconfidence intervals for the parameters γ, δB, and rB .Section IX gives a summary of our results.
III. DATA SAMPLE AND DETECTOR
The measurements presented in this paper use the en-tire BB data sample collected with the BABAR detectorat the PEP-II asymmetric-energy B factory at the SLACNational Accelerator Laboratory. The BB pairs are pro-duced from the decays of Υ (4S) mesons that originatein collisions of 9.0GeV electrons and 3.1GeV positrons(√s=10.58GeV=MΥ (4S)c
2). In total, (467 ± 5) × 106
BB pairs, approximately equally divided into B0B0 andB+B−, have been collected in the years from 1999 un-til early 2008. The B meson pairs are produced almostat rest in the Υ (4S) center-of-mass (CM) frame, but theasymmetric beam energies boost them in the laboratoryframe by (βγ)CM ≈ 0.56.The BABAR detector is described in detail else-
where [22]. Primary and secondary vertex reconstruc-tion and charged-particle tracking are provided by a fivelayer double-sided silicon vertex tracker and a 40 layerdrift chamber. Charged particle identification (PID) isprovided by measurement of specific ionization energyloss in the tracking devices and of the Cherenkov radia-tion cone in a ring-imaging detector. Photons and elec-trons are identified by combining the information fromthe tracking devices and the energy deposits in the elec-tromagnetic calorimeter, which consists of 6580 thallium-doped CsI crystals. These systems are located inside a1.5T solenoidal superconducting magnet. Finally, theflux return of the magnet is instrumented with resistiveplate chambers and limited streamer tubes in order todiscriminate muons from pions. We use the GEANT4 [23]
8
software toolkit to simulate interactions of particles inthe detector, taking into account the varying acceleratorand detector conditions.
IV. EVENT SELECTION
We reconstruct B± → Dh± decays, where the chargedtrack h is either a kaon or a pion. Neutral D mesonsare reconstructed in the CP -even eigenstates π−π+ andK−K+ (DCP+), in the CP -odd eigenstates K0
Sπ0, K0
Sφ
and K0Sω (DCP−), and in the non-CP -eigenstate K−π+
(D0 from B− → D0h−) or K+π− (D0 from B+ →D0h+). CP violation in the K0–K0 system is neglected,i.e. the K0
Sis assumed to be a pure CP = +1 eigenstate.
The DCP daughters are reconstructed in the decay modesK0
S→ π+π−, φ → K+K− and ω → π−π+π0.
We optimize all our event selection requirements bymaximizing the significance of the expected B± → DK±
signal yield, defined as Nsig/√
Nsig +Nbkg, where Nsig
(Nbkg) is the expected signal (background) yield. Theoptimization is done for each D decay channel using sim-ulated signal and background events, which are generatedwith the EVTGEN software package [24].Neutral pions are reconstructed by combining pairs
of photon candidates with energy deposits larger than30MeV that are not matched to charged tracks andwhose energy deposition profile is consistent with thatexpected from a photon. The photon pair invariant massis required to differ from the nominal π0 mass [25] by lessthan 2.5 times its resolution (σ ≈ 6MeV/c2) and the totalπ0 energy in the laboratory frame must be greater than240MeV for D → K0
Sπ0 and 210MeV for ω → π+π−π0.
Neutral kaons are reconstructed from pairs of oppo-sitely charged tracks with invariant mass within 2.5σ(σ ≈ 2.1MeV/c2) of the nominal K0
Smass [25]. The ratio
between theK0Ssigned 3-dimensional flight length and its
uncertainty, determined from the position of the K0Sand
the D decay vertices and the K0Smomentum direction,
must be greater than 1.9, 2.0, and 2.2 for D → K0Sπ0,
D → K0Sφ, and D → K0
Sω, respectively.
The φ candidates are reconstructed from pairs of oppo-sitely charged tracks passing kaon identification criteriawith typical kaon selection efficiency of ≈ 98% and pionmisidentification of ≈ 15%. The two tracks are assignedthe kaon mass hypothesis and their invariant mass is re-quired to be within 6.5MeV/c2 of the nominal φmass [25](the resolution is σ = 1.0MeV/c2 and the natural width isΓφ = 4.3MeV). We also require that the helicity angle θHbetween the flight direction of one of the two kaons andthe D flight direction, in the φ rest frame, satisfies thecondition | cos θH | > 0.4. This requirement exploits thefact that in D → K0
Sφ decays the φ is produced in a lon-
gitudinally polarized state, thus cos θH follows a cos2 θHdistribution, while in φ candidates not from D → K0
Sφ
decays, cos θH is approximately uniformly distributed.The ω candidates are reconstructed from π+π−π0 com-
binations with invariant mass within 17MeV/c2 (2Γω)
of the nominal ω mass [25] (the resolution is σ =6.9MeV/c2). The charged pion candidates are requiredto pass pion identification criteria with pion selectionefficiency around 98% and kaon misidentification ratearound 12%. To improve the ω momentum resolution,the invariant mass of the two photons forming the π0 can-didate is constrained to the nominal π0 mass. We defineθN as the angle between the normal to the ω decay planeand the D momentum in the ω rest frame, and θππ asthe angle between the flight direction of one of the threepions in the ω rest frame and the flight direction of one ofthe other two pions in the two-pion rest frame. The quan-tities cos θN and cos θππ follow cos2 θN and (1−cos2 θππ)distributions, respectively, for the signal, and are almostuniformly distributed for wrongly reconstructed ω candi-dates. We require the product cos2 θN sin2 θππ > 0.046.
Neutral D candidates are formed from two-body com-binations of K±, π±, K0
S, π0, φ and ω candidates consis-
tent with one of the six D decay channels under study.To improve the DCP− momentum resolution, the invari-ant masses of the π0 and K0
Sdaughters are constrained
to the nominal π0 and K0Smasses. To suppress poorly
reconstructed D candidates and candidates from ran-dom combinations, we perform a geometric fit of the Ddaughters to a common origin, and reject D candidatesfor which the χ2 probability of the vertex fit is lowerthan 0.01%. The invariant mass of a D candidate MD
must be within a range that corresponds to slightly morethan twice the MD resolution, which varies from about6MeV/c2 for the K0
Sφ channel to about 44MeV/c2 for the
K0Sπ0 channel. We apply the following particle identifica-
tion criteria to the charged daughters of the D meson: inD → π+π−, the two pion candidates must pass the samepion identification criteria adopted in the reconstructionof ω → π−π+π0; in D → K+K−, the two kaon can-didates are required to pass tighter kaon identificationcriteria than those applied to the φ daughters (typicalkaon selection efficiency around 94%, and pion misiden-tification rate around 6%); in D → K−π+, the kaoncandidate must pass the same kaon identification criteriarequired for the φ daughters. In order to reduce the largecombinatorial background from random combinations oftracks and photons in e+e− → qq events (q = u, d, s, c),we put requirements on the cosine of the D decay angle,| cos θD|. We define θD as the angle between one of the Ddaughters in the D rest frame, and the direction of the Dmeson in the B rest frame. Due to angular momentumconservation we expect the distribution of cos θD to beuniform for B± → Dh±, D → π+π− and D → K0
Sπ0 sig-
nal events, while for qq events the distribution is stronglypeaked at ±1. We require | cos θD| < 0.74 (0.99) for theB± → Dh±, D → π+π− (D → K0
Sπ0) channel.
The invariant mass distributions of the reconstructedD candidates, after all the other selection criteria de-scribed in this section have been applied, are shown inFig. 1.
We reconstruct B± meson candidates by combining aneutral D candidate with a track h±. For the D→Kπ
9
)2 (GeV/cDM∆-0.06 -0.04 -0.02 0 0.02 0.04 0.06
)2E
vent
s / (
0.00
3 G
eV/c
0
200
400
600
800
1000 Dh→B -K+ K→D
)2 (GeV/cDM∆-0.06 -0.04 -0.02 0 0.02 0.04 0.06
)2E
vent
s / (
0.00
3 G
eV/c
0
50
100
150
200
250
300
350
400 Dh→B
-π+π →D
)2 (GeV/cDM∆-0.05 0 0.05
)2E
vent
s / (
0.00
4 G
eV/c
0
100
200
300
400
500 Dh→B
0πS0 K→D
)2 (GeV/cDM∆-0.06 -0.04 -0.02 0 0.02 0.04 0.06
)2E
vent
s / (
0.00
3 G
eV/c
0
50
100
150
200
250
300 Dh→B
ωS0 K→D
)2 (GeV/cDM∆-0.03 -0.02 -0.01 0 0.01 0.02 0.03
)2E
vent
s / (
0.00
1 G
eV/c
0
20
40
60
80
100
120
140
160 Dh→B
φS0 K→D
)2 (GeV/cDM∆-0.04 -0.02 0 0.02 0.04
)2E
vent
s / (
0.00
2 G
eV/c
0
1000
2000
3000
4000
5000 Dh→B
π K→D
FIG. 1: Distributions of the difference between the D candidate’s invariant mass and the nominal D0 mass [25], as measuredin the B±
→ Dh± samples. All selection criteria described in Section IV, except that on the D invariant mass MD, have beenapplied, including the χ2-based candidate selection. In addition we reduce the background by requiring the fit variables tosatisfy mES > 5.27GeV/c2, ∆E > −0.05GeV, and F > −0.25. The ∆MD selection requirements are depicted by the verticallines.
mode, the charge of the track h must match that of thekaon from the D meson decay. This selects b → c medi-ated B decays B− → D0h− and B+ → D0h+. The con-tamination from b → u mediated B decays followed bydoubly-Cabibbo-suppressed D decay, i.e. B− → D0K−,D0 → K−π+, and from D0–D0 mixing is negligible.In the B± → Dh±, D → π+π− channel we requirethat the invariant mass of the (h±π∓) system is greaterthan 1.9GeV/c2 to reject background from B−→D0π−,D0→K−π+ and B−→K∗0π−, K∗0→K−π+ decays andtheir CP conjugates. Here π is the pion from the D andh is the track from the B candidate taken with the kaonmass hypothesis. To improve the B momentum resolu-
tion, the neutral D invariant mass is constrained to thenominal D0 mass [25] for all D decay channels.We identify signal B → DK and B → Dπ candidates
using two kinematic variables: the difference between theCM energy of the B meson (E∗
B) and the beam energy,
∆E = E∗B −
√s/2 , (8)
and the beam-energy-substituted mass,
mES =√
(s/2 + pee · pB)2/E2ee − p2B, (9)
where (EB ,pB) and (Eee,pee) are the four-momenta ofthe B meson and of the initial e+e− system, respectively,
10
measured in the laboratory frame. ThemES distributionsfor B± → Dh± signals are centered at the B mass [25],have a root-mean-square of approximately 2.6MeV/c2,and do not depend strongly on either the D decay modeor the nature of the track h. In contrast, the ∆E dis-tributions depend on the mass assigned to the track h.We evaluate ∆E with the kaon mass hypothesis so thatthe peaks of the distributions are centered near zero forB± → DK± events and are shifted by approximately+50MeV for B± → Dπ± events. The ∆E resolutiondepends on the kinematics of the decay, and is typically16MeV for all D decay modes under study after the Dinvariant mass is constrained to its nominal value. We re-tain B candidates with mES and ∆E within the intervals5.20 < mES < 5.29GeV/c2 and −80 < ∆E < 120MeV,which define the region for the fit described later.In order to discriminate the signal from e+e− → qq
background events, denoted qq in the following, we con-struct a Fisher discriminant F based on the four event-shape quantities LROE
20 , | cos θ∗T |, | cos θ∗B| and HROE20 .
These quantities, evaluated in the CM frame, are definedas:
• LROE20 = L2/L0 is the ratio of the second and ze-
roth event shape moments of the energy flow in therest of event (ROE), i.e. considering all the chargedtracks and neutral clusters in the event that are notused to reconstruct the B candidate. They are de-fined as L2 =
∑
i pi cos2 θi and L0 =
∑
i pi, wherepi are the momenta and θi the angles of the chargedand neutral particles in the ROE, with respect tothe thrust axis of the B candidate’s decay products.The thrust axis is defined as the direction that max-imizes the sum of the longitudinal momenta of theparticles used to define it;
• θ∗T is the angle between the thrust axis of the Bcandidate’s decay products and the beam axis;
• θ∗B is the angle between the B candidate momen-tum and the beam axis;
• HROE20 = H2/H0 is the ratio of the second and ze-
roth Fox-Wolfram moments H2 and H0 [26], com-puted using charged tracks and photons in theROE.
The quantity F is a linear combination of the four afore-mentioned event-shape variables:
F = c1LROE20 + c2| cos θ∗T |+ c3| cos θ∗B|+ c4H
ROE20 . (10)
The values of the coefficients ci are the ones which max-imize the separation between simulated signal eventsand a continuum background sample provided by off-resonance data, taken ≈ 40MeV below the Υ (4S) res-onance. The maximum likelihood fit described in Sec-tion V is restricted to events with F within the interval−1.5 < F < 1.5, to remove poorly reconstructed candi-dates.
For events with multiple B± → Dh± candidates(about 16% of the selected events), we choose the B can-didate with the smallest χ2 =
∑
c(Mc−MPDGc )2/(σ2
Mc+
Γ2c) formed from the measured and true masses, Mc and
MPDGc , of all the unstable particles c produced in the
B decay tree (D, π0, K0S, φ, ω), scaled by the sum in
quadrature of the resolution σMcof the reconstructed
mass and the intrinsic width Γc. From simulated signalevents, we find that this algorithm has a probability toselect the correct candidate between 98.2% and 99.9%depending on the D decay mode. We also find that thealgorithm has negligible effect on the MD distributions.We compare the distribution of each selection variable
in data and simulated events after the requirements onall other variables have been applied. In order not tointroduce biases that may artificially enhance the signalyield, we perform a blind study by explicitly removing, inthis comparison, events consistent with the B± → DK±
signal, i.e. those with |mES−mB| < 10 MeV/c2, |∆E| <40 MeV, F > −0.8 and track h passing kaon identi-fication criteria. We find excellent agreement betweendata and simulated events, both for events consistentwith the B± → Dπ± signal (|mES −mB| < 10 MeV/c2,|∆E− 50 MeV| < 40 MeV, F > −0.8 and track h failingthe kaon identification criteria) and for background-likeevents. We correct for small differences in the means andwidths of the distributions of the invariant masses of theunstable particles and of mES and ∆E both when apply-ing to data the selection criteria obtained from simulatedevents and in the final fit described in the next section.The total reconstruction efficiencies, based on simu-
lated B± → DK± events, are summarized in the sec-ond column of Table II. For the reasons explained in
TABLE II: Reconstruction efficiency for B → DK from sim-ulated events. We also quote the efficiency and purity in asignal-enriched subsample (see text for details).
D0 mode Efficiency after Efficiency in Purity infull selection signal-enriched signal-enriched
Section II, the efficiencies are 40% to 60% higher thanin our previous study of the same decay channels [9].The efficiencies obtained for B± → Dπ± events fromthe simulation are statistically consistent with those forB± → DK±, where the D meson is reconstructed inthe same final state. For illustration purposes we definea signal-enriched sample for each D decay mode, con-taining all B± → Dh± candidates satisfying the criteria−40 < ∆E < 100MeV, 0.2 < F < 1.5, 5.275 < mES <5.285GeV/c2, and whose daughter track h passes charged
11
kaon identification criteria. The typical kaon efficiency is≈ 77% and the pion misidentification rate is ≈ 2%. Thereconstruction efficiencies and the expected purities forthe signal-enriched subsamples, determined on simulateddata, are listed in Table II.
V. MAXIMUM LIKELIHOOD FIT
We measure R(±)K/π and ACP± using simultaneous ex-
tended and unbinned maximum likelihood fits to the dis-tributions of the three variables ∆E, mES, and F of Bcandidates selected in data. The dataset is split into24 subgroups by means of three discrete variables: thecharge η = ±1 of the reconstructed B meson (×2 sub-groups); the two-body D decay final state X (×6), allow-ing for a more accurate description of the correspondingprobability density functions compared to the larger CP±subgroups; and a PID variable denoting whether or notthe track h from the B passes (p) or fails (f) chargedkaon identification criteria (×2). The pion misidentifi-cation rate of these criteria is determined directly fromdata as described later, and is expected from simulationto be around 2%. The corresponding kaon identificationefficiency is (77 ± 1)%, as determined from the signalMC samples after weighting the bidimensional distribu-tion of the momentum and polar angle of the track h bythe ratio of the analogous distributions observed in MCand data kaon control samples. The uncertainty on thekaon identification efficiency is dominated by the system-atic contribution from the uncertainties on the weights.We perform in total three simultaneous fits to these 24subgroups: one fit for the two CP -even D final states(8 subgroups), one for the three CP -odd D final states(12 subgroups), and one for the D → Kπ decay (4 sub-groups).The likelihood function L for each of these simultane-
ous fits has the form
L(~ν) = e−NNn
n!
∏
s
Ns∏
i=1
Ps(mES,i,∆Ei,Fi;~ν), (11)
where s ranges over the subgroups under consideration,Ns is the number of events in subgroup s, n is the totalnumber of events in the fit n =
∑
s Ns, and N is theexpected number of events. We minimize − lnL withrespect to the set of fit parameters ~ν specified later.The probability Ps,i ≡ Ps(mESi,∆Ei,Fi) for an eventi is the sum of six signal and background components:B± → DK± signal, B± → Dπ± signal, background can-didates from e+e− → qq events, irreducible backgroundarising from charmless B± → XK± and B± → Xπ± de-cays, and background candidates from other BB events(reducible BB background):
NsPs,i = NDπs PDπ
s,i +NDKs PDK
s,i +
N qqs Pqq
s,i +NBBs PBB
s,i +
NXπs PXπ
s,i +NXKs PXK
s,i , (12)
where the N js are the expected yields in each component
j. In case of negligible correlations among the fit vari-ables, each probability density function (PDF) P factor-izes as:
P(mES,∆E,F) = P(mES)P(∆E)P(F). (13)
The irreducible BB background originates from eventswhere a B meson decays to the same final state Xh asthe signal, but without the production of an intermediatecharmed meson in the decay chain. When exploiting the∆E, mES, and F variables, this background is thereforeindistinguishable from the signal. As an example, thedecay B± → K+K−K± (X = K+K−) is an irreduciblebackground for B± → DCP+K
±, DCP+ → K+K−.As described later in Section VI, the irreducible back-ground yield can be estimated by studying sideband re-gions of the D candidate invariant mass distribution,and can then be fixed in the final fit, where we assumePDhi = PXh
of B± → Dπ±, D → X signal events, the true kaon iden-tification efficiency ε of the PID selector, and the pionmisidentification rate m of the PID selector:
NDKη,p,X =
1
2
(
1− ηAXDK
)
NDπtot,X RX
K/π ε , (14)
NDKη,f,X =
1
2
(
1− ηAXDK
)
NDπtot,X RX
K/π (1− ε) , (15)
NDπη,p,X =
1
2
(
1− ηAXDπ
)
NDπtot,X m, (16)
NDπη,f,X =
1
2
(
1− ηAXDπ
)
NDπtot,X (1−m) . (17)
Because the ratios RXK/π are small, the fit is not able
to determine the value of ε. Therefore we fix it to theaforementioned value of ε = (77± 1)%. The reconstruc-tion and selection efficiencies for true B± → DK± andB± → Dπ± candidates, where theD meson decays to thesame final state, are assumed to be identical. A system-atic uncertainty is assigned due to this assumption (seeSection VII). The simultaneous fit to the two CP -evenmodes constrains
Aπ+π−
DK = AK+K−
DK ≡ ACP+ , (18)
Rπ+π−
K/π = RK+K−
K/π ≡ R+K/π , (19)
while the simultaneous fit to the three CP -odd modesconstrains
AK0
Sπ0
DK = AK0
Sφ
DK = AK0
Sω
DK ≡ ACP− , (20)
RK0
Sπ0
K/π = RK0
Sφ
K/π = RK0
Sω
K/π ≡ R−K/π . (21)
ThemES distributions of the signal components are pa-rameterized using an asymmetric Gaussian shape, i.e. a
12
Gaussian with different widths on both sides of the peak.We use the same shape for B± → DK± and B± → Dπ±,so the mES B± → DK± signal shape (whose parame-ters are floating in the fit) will mostly be determined bythe much more abundant B± → Dπ± control sample.Since the selection efficiencies for the two channels arethe same, we expect the number of reconstructed candi-dates from B± → Dπ± to be about twelve times higherthan for B± → DK±. We have checked that the mES
shapes for B± → DK± and B± → Dπ± are consistent,and that the assumption that they are identical does notbias the parameters of interest.The ∆E distribution of the B± → DK± signal compo-
nent is parameterized with a double Gaussian shape. Thecore Gaussian has a mean close to zero, a width around16MeV and, according to the simulation, accounts forabout 90% of the true B± → DK± candidates. The sec-ond Gaussian accounts for the remaining 10% of candi-dates whose energy has been poorly measured. The meanand width of the core Gaussian are directly determinedfrom data, while the remaining three parameters (the dif-ference between the two means, the ratio between the twowidths and the ratio of the integrals of the two Gaussianfunctions) are fixed from the simulation. In contrast tothe mES case, the B± → Dπ± ∆E shape is not the sameas for B± → DK±. This is due to the fact that we alwaysassign the kaon mass hypothesis to the track: the wrongmass assignment, in the case of B± → Dπ±, introducesa shift to the reconstructed energy of the pion and thusto ∆E, since ∆E = E∗
B −√s/2 = E∗
D +E∗h−
√s/2. The
shift depends on the magnitude of the momentum p ofthe track h in the laboratory frame,
∆Eshift(p) = γCM
(
√
m2K + p2 −
√
m2π + p2
)
. (22)
Therefore we parameterize the B± → Dπ± ∆E signalcomponent with the sum of two Gaussians whose meansare computed event-per-event by adding ∆Eshift(p) tothe means of the Gaussian functions used to describe theB± → DK± ∆E signal. The other parameters of theB± → Dπ± and B± → DK± ∆E distributions (the twowidths and the ratio of the integrals) are identical. Again,we exploit the B± → Dπ± control sample to determinethe shape of the B± → DK± signal. In the case of thehigh statistics flavor mode D → Kπ, we add a linearbackground component to the double Gaussian shape toaccount for misreconstructed events, which peak in mES
but not in ∆E. The ratio between the integral of thelinear component and that of the two Gaussian functionsis fixed from simulated signal events.For the reducible BB background, Eq. 13 does not hold
because of significant correlations between the ∆E andmES distributions. This reflects the fact that this back-ground is composed of two categories of B candidateswith different mES and ∆E distribution:
• B candidates formed from random combinations ofcharged tracks and neutral objects in the event,which populate the whole mES-∆E plane;
• B candidates from B± → Dρ±, B± → DK∗±,B± → D∗h± (D∗ → Dπ), where a pion from theρ, K∗ or D∗ decay is not reconstructed. Thesecandidates peak in mES close to the B mass, butwith broader resolution compared to the signal, andare shifted towards negative ∆E values, typicallypeaking at ∆E ≈ −mπc
2, therefore outside of the∆E fit region; however, the tail on the positive sideof the distribution extends into the ∆E fit region.
We parametrize the mES-∆E distribution of the BBbackground by means of two factorizing components:
PBB(mES,∆E) = f × gpeak(mES)hpeak(∆E) +
(1 − f) × gcont(mES)hcont(∆E). (23)
The mES component of the peaking part, gpeak(mES),is parameterized with a Gaussian function for X =π+π−, K0
Sω, K0
Sφ. For X = K+K−, K0
Sπ0 we use the
“Crystal Ball” lineshape [27], an empirical smooth func-tion that better describes the non-Gaussian tail on thenegative side of the distribution,
C(x) =
{
nn
|α|n e− |α|2
2
(
n|α| − |α| − x
)−n
x < −|α| ,exp
(
− 12 x
2)
x ≥ −|α| ,(24)
with x = (x−µ)/σ and x → −x for α < 0. For X = Kπwe use an empirical function of the form:
N(x) = exp
(
− 1
2τ2{
ln2[1 + Λτ(x− µ)] + τ4}
)
, (25)
with Λ = sinh(τ√ln 4)/(στ
√ln 4). Here µ is the posi-
tion of the peak, while σ and τ are parameters relatedto the width of the distribution on the two sides of thepeak. The ∆E component of the peaking part hpeak(∆E)is described with a simple exponential function for thefive CP self-conjugate D final states, and with a Landaufunction for the non-CP -eigenstate final state. The BBpurely combinatorial background component is describedby the 2-dimensional product of a linear background,hcont(∆E), and an empirical function introduced by theARGUS collaboration [28], gcont(mES) = A(mES/m0):
A(x) = x(1 − x2)p exp(
−ζ[
1− x2])
, (26)
where m0 =√s/(2c2) = 5.29GeV/c2 is the kinematic
endpoint of the mES distribution. All the parameters ofthe BB background mES-∆E distribution are fixed fromsimulated BB events. The only exception is the width ofthe Landau function used for hpeak(∆E) in X = K−π+.This parameter controls the behaviour at low ∆E values,∆E ≈ −80MeV, where we find the simulation not to besufficiently precise given the high statistics of this chan-nel. We note that the shape parameters differ across thesix final states, but are similar across the charge and PIDselector subgroups belonging to one final state.In qq events, B candidates arise from random combina-
tions of charged tracks and neutral particles produced in
13
the hadronization of the light quark-antiquark pairs pro-duced in e+e− collisions. Similarly to the combinatorialcomponent of the BB background, the qq backgrounddistribution in the mES-∆E plane is parameterized bythe product of an ARGUS function in mES and a linearbackground in ∆E. We float the slope of the linear com-ponents, while the parameters of the ARGUS functionare fixed, in each D final state, from simulated qq events.They are in good agreement across the final states andother subgroups.
The F distributions are parameterized in a similar wayfor all fit components. We find that the distributions ofB± → DK± and B± → Dπ± signal events are consis-tent with each other, as expected since their kinematicsare very similar, and choose to parameterize them withthe same shape. For this we use the sum of two asym-metric Gaussian functions. Some channels with lowerstatistics don’t require the full complexity of this param-eterization: in those cases we use a single asymmetricGaussian, a double Gaussian, or a single Gaussian. Inparticular we use: for the signal components a doubleasymmetric Gaussian, except for X = K0
Sφ, where a
double Gaussian function is adopted; for the BB back-ground components a double asymmetric Gaussian incase of X = K0
Sπ0,K−π+, a double Gaussian in case
of X = K0Sω, and a single Gaussian otherwise; for the qq
background components a double asymmetric Gaussian,except for X = K0
Sφ, where we use a single Gaussian.
In summary, the floating parameters of the fits are:all parameters related to the signal yields, and thereforeto the GLW parameters, as given in Eqns. 14-17, exceptε; all background yields and CP -asymmetries except theirreducible background yields and asymmetries, the BBasymmetries for CP− modes and for the (CP+, p) sub-groups (B → DCP+h candidates where the track h passesthe kaon identification criteria), and the BB yield in the(K0
the overall width and mean of the ∆E signal, the mES
signal shape, and the ∆E and F shape for qq background.A full list of the floating parameters can be found in Ta-ble VIII. The non-floating parameters are fixed to theirexpectations obtained from simulation or, in case of theirreducible background yields, to values obtained fromdata control samples (see next section). Non-floating CPasymmetries are fixed to zero. We assign systematic un-certainties due to the fixed parameters.
We check that the fitter is correctly implemented bygenerating and fitting a large number of test datasets us-ing the final PDFs. In this study, we include an analyticdescription for the conditional variable ∆Eshift. Theresiduals for a given parameter, divided by the measuredparameter error, should follow a Gaussian distributionwith zero mean (µ) and unitary width (σ). We observe nosignificant deviations from the expected distribution. Inparticular, R+
K/π shows the largest shift from zero mean
(µ = −0.06±0.07) and ACP+ shows the largest deviationfrom unity width (σ = 1.13±0.06) among the parametersof interest.
We investigate fit biases, arising from possible discrep-ancies between the true signal distribution and the cho-sen fit model, by fitting a large number of test datasets,in which the B± → DK± and B± → Dπ± signal com-ponents are taken from simulated samples of sufficientstatistics, while the background components are ran-domly generated according to their PDFs. Of all floatingparameters, only RK/π acquires a significant bias, result-ing in corrections of 0.5 and 1.0 times the expected sta-tistical uncertainties on these parameters in the CP andflavor modes, respectively. This bias is caused by smalldifferences in the ∆E distributions of the signal compo-nents across the kaon PID subgroups (p and f), whichthe final PDF does not account for. A second, smallercontribution to this bias is a small discrepancy betweenthe ∆E(π) signal shape of B± → Dπ± events and the∆E(K) shape of B± → DK± events. The biases inthe RK/π parameters are correlated, and partly cancelin the ratio, resulting in a smaller bias on the GLW pa-rameters RCP±. The largest (smallest) remaining bias is0.12 (0.05) times the expected statistical uncertainty forRCP+ (ACP−). We correct the final values of the param-eters ACP and RK/π for the observed biases, and assignsystematic uncertainties to these corrections.
VI. IRREDUCIBLE BACKGROUND
DETERMINATION
As discussed in the previous section, the irreduciblebackground arises from charmless B± → Xh± decays,which have the same final states as the B± → D(→X)h± signal and therefore the same distribution of thethree fit variables ∆E, mES, and F .In the D0 → K−π+ flavor mode, the irreducible back-
ground – taking into account the measured branchingfractions for B± → K±π∓K± and B± → K±π∓π± [25]and a selection efficiency of ≈ 1%, estimated from sim-ulated events – is negligible compared to the expectedsignal yields (about 3400 B± → DK± and 45000 B± →Dπ± expected signal events). On the other hand, in theCP modes, where the signal yields are expected to be anorder of magnitude lower than in K−π+, and the upperlimits for the branching ratios of B± → Xh± decays areat the 10−5 level, we cannot a priori exclude a relevantirreducible background contribution.We estimate the irreducible background yields in our
sample by exploiting the fact that the D invariant massdistribution for this background is approximately uni-form, while for the signal it is peaked around the nom-inal D mass. Therefore we can select a control samplecontaining irreducible background candidates, but withthe signal strongly suppressed, by applying the same se-lection as for the signal, with the only difference that theD invariant mass is required to lie in a region (D invari-ant mass sidebands) which is separated by at least a fewσMD
from the nominal D mass (see Table III). We thenperform an extended maximum likelihood fit to the mES,
14
∆E, and F distributions of the control sample in orderto measure the irreducible background yields in the D in-variant mass sidebands. The fit is similar to the nominalone described in the previous section. However, due tothe limited statistics available in the sidebands, we areforced to fix more parameters compared to the nominalfit; in particular, we fix any possible charge asymmetry ofthe B± → Xh± decays to zero (a systematic uncertaintyis assigned to this assumption). Finally, since the D can-didate invariant mass distribution of the irreducible back-ground is approximately uniform, we scale the obtainedyields by the ratio of the widths of the D signal and con-trol sideband mass regions to obtain the irreducible back-ground yield NXh (scale factor in Table III). Table IVshows the scaled irreducible background yields that enterthe final fit.
TABLE III: D mass sideband definitions, the scale factor de-fined as the ratio of the widths of the D mass signal andsideband regions.
TABLE IV: Irreducible background yields estimated from MD
sidebands in data.
D decay mode NXK NXπ
K+K− 93 ± 10 −5 ± 8π+π− 4 ± 6 0 ± 9K0
Sπ0
−4 ± 9 65 ± 23K0
Sω 3 ± 6 0 ± 8K0
Sφ 0.5 ± 0.7 1.4 ± 1.0
VII. SYSTEMATIC UNCERTAINTIES
We consider nine sources of systematic uncertaintythat may affect the GLW parameters ACP± and RCP±.Their contributions are summarized in Table V.First, we estimate the influence of fixed parameters of
the nominal PDF. We perform a large number of testfits to the data, similar to the nominal fit. In each ofthese test fits the fixed parameters are varied accordingto their covariance matrices. From the resulting distri-butions we calculate the systematic covariances of the fitparameters ACP± and RK/π. The parameters responsi-ble for the largest uncertainty are the mES endpoint m0,and parameters related to the measured yields, e.g. bbbackground asymmetries and the efficiency of the kaonselector.
The uncertainties in the irreducible background eventyields introduce a systematic uncertainty in the B± →DCPh
± yields and therefore in RCP±. Likewise, anycharge asymmetry in this background would affect themeasured values of ACP±. We again perform a series oftest fits to on-peak data, where we vary the B± → Xh±
yields and asymmetries by their uncertainties. For thelatter, we take the uncertainties to be ±10% for X =K+K− and ±20% for the other modes, which are conser-vative estimates consistent with the existing upper limitson the CP asymmetries in those decays [21]. For ACP+,the possible CP asymmetries in the peaking backgrounddominate the systematic error.As explained in Section V, we correct the fit results
for biases observed in Monte Carlo studies. We take theassociated systematic uncertainties to be half the sizeof the bias corrections, summed in quadrature with thestatistical uncertainties on the biases. The latter are dueto the limited number of test fits used to estimate thecorrections.We investigate a potential charge asymmetry of the
BABAR detector, due to a possible charge bias in trackingefficiency (e.g. K+ vs K−) and/or particle identification.Our analysis includes a number of control samples, inwhich the CP asymmetry is expected to be negligible:the six B± → Dπ± samples and the B± → DK± flavormode (D → Kπ). The weighted average of the chargeasymmetry in the control samples is (−0.95 ± 0.44)%,from which we assign uncertainties of 1.4% to both ACP+
and ACP−. We consider these uncertainties to be 100%correlated.The measured CP asymmetry in B± → DK±, D →
K0Sφ, can be diluted by the presence of B± → DK±
decays followed by D decays to the same final stateK0
SK+K− as the signal but with opposite CP content,
such as D → K0Sa0, a0 → K+K−. The same can hap-
pen in the B± → DK±, D → K0Sω analysis with back-
grounds fromB± → DK±, D → K0Sπ−π+π0. This back-
ground can also affect the ratios RCP−. It is possible toobtain correction factors to both ACP− and RCP− froma fit to the distributions of the relevant helicity angles,cos θN and cos θH forK0
Sω and K0
Sφ, respectively. The fit
is performed on dedicated B± → Dπ± samples, in whichthe selection requirements on the helicity angles have notbeen applied. It can be shown [29] that for these two finalstates the observed charge asymmetries and ratios shouldbe corrected by a factor
AtrueCP = Aobs
CP · 1 + fǫ|z|2R′
1− fǫ|z|2, (27)
RtrueK/π = Robs
K/π · 1 + fǫ|z|21 + fǫ|z|2R′
. (28)
Here, R′ is the ratio of the R±K/π values, where R−
K/π
is taken from a single fit to the D0 → K0Sπ0 final state
only (as opposed to using all three CP− final states un-
der study), R′ = R+K/π/R
K0Sπ0
K/π , and fǫ = ǫsig/ǫbkg is the
ratio of the efficiencies of the selection criterion on the
15
helicity angles: fǫ,K0Sω = 0.71 and fǫ,K0
Sφ = 0.64. To ap-
ply these corrections, we first perform a fit of the K0Sπ0
final state alone to obtain RK0
Sπ0
K/π . We then perform the
simultaneous fit of the CP+ final states, from which wetake the value of R+
K/π . Finally, we include the correc-
tion factors into the CP− final PDF, which will allow thelikelihood fitter to correctly estimate their influence. Theparameter |z|2 in Eqns. 27 and 28 is extracted from fitsof the helicity angle distributions in the D0 → K0
Sω and
D0 → K0Sφ subsamples to the function |z|2+3 cos2 θ [29].
We subtract the background expected from the MonteCarlo simulation, which has been rescaled to match thedata. We find |z|2 = 0.065 ± 0.033 in the case of K0
Sω,
and |z|2 = 0.217 ± 0.063 in the case of K0Sφ. The un-
certainties contain propagated uncertainties due to thebackground subtraction. The resulting corrections are:
AtrueCP (K0
Sω) = Aobs
CP (K0Sω) × (1.105± 0.056) , (29)
AtrueCP (K0
Sφ) = Aobs
CP (K0Sφ) × (1.35± 0.12) , (30)
RtrueK/π(K0
Sω) = Robs
K/π(K0Sω) × (0.9929± 0.0066) , (31)
RtrueK/π(K0
Sφ) = Robs
K/π(K0Sφ) × (0.981± 0.016) . (32)
In order to assign systematic uncertainties, we propagatethe uncertainties on the correction factors into the finalresult.When calculating RCP through Eq. 5 one has to take
into account that this equation is an approximation. Wedefine the double ratios used to approximate RCP± asR±. They are given by
R± =Γ(B− → DCP±K
−) + Γ(B+ → DCP±K+)
Γ(B− → DfK−) + Γ(B+ → DfK+)
× Γ(B− → Dfπ−) + Γ(B+ → Dfπ
+)
Γ(B− → DCP±π−) + Γ(B+ → DCP±π+),(33)
where Df denotes the K−π+ final state. These can bewritten as
R± =1 + r2B ± 2rB cos δB cos γ
1 + r2Br2D + 2rBrD cos(δB−δD) cos γ
× 1 + r2Bπr2D + 2rBπrD cos(δBπ−δD) cos γ
1 + r2Bπ ± 2rBπ cos δBπ cos γ,(34)
where rBπ and δBπ are defined, in analogy to rB and δB,as rBπe
i(δBπ−γ) = A(B− → D0π−)/A(B− → D0π−),while rD and δD are defined as rDeiδD = A(D0 →K−π+)/A(D0 → K−π+). We write Eq. 34 in theform R± = RCP± × (1 + Rc), and we assign a relativesystematic uncertainty based on the value of the cor-rection Rc. Taking sin θC = 0.2257 ± 0.0010 (whereθC is the Cabibbo angle) and rB=0.104+0.015
−0.025 from [3],
and expressing rD = |VcdVus|/|VudVcs| = tan2 θC , andrBπ = rB tan2 θC , we find Rc ≈ 4rB tan2 θC ≈ 2.2%.Here, we have conservatively assumed values for the co-sine terms which maximize Rc. We thus assign a relative
uncertainty of 2.2% to the values of RCP , fully correlatedbetween RCP+ and RCP−.
We also consider the influence on the measured value ofACP of misreconstructed signal B candidates, i.e. candi-dates reconstructed, in events containing a true B → DKdecay with D decaying to the same final state X as thereconstructed candidate, from random combinations ofparticles produced in the true B → DK decay and theparticles of the ROE. The fraction of these candidatesranges from 0.3% to 12% in simulated B± → DCPK
±
events, depending on the channel. Since we treat thiscomponent as signal, we implicitly assume that its chargeasymmetry is equal to the asymmetry in the signal com-ponent. We use simulated signal events to estimate theratio between misreconstructed and true B+ → DK+
candidates and the ratio between misreconstructed andtrue B− → DK− candidates, and find these two quanti-ties to differ by less than 0.1%, from which we derive anupper limit on the difference between the observed andthe true value of ACP .
The yield double ratios RCP± should be corrected bythe corresponding double ratio of selection efficiencies.We find from simulated events that the efficiency doubleratios are compatible with each other, and their averagevalue is very close to unity, (99.46± 0.23)%. Thus we donot correct the central values but conservatively assigna relative uncertainty equal to 1 − (0.9946 − 0.0023) =0.0077.
The final PDF doesn’t contain an explicit descriptionof the conditional parameter ∆Eshift, assuming implic-itly that the distribution of ∆Eshift observed in data isthe same for all the components of the fit. However, thedistributions are found to be slightly different across thecomponents, thus introducing a possible bias in the fit re-sults. To estimate the size of this bias, we use simulatedevents to obtain parameterizations of the ∆Eshift distri-butions of all the fit components and repeat the fits todata. We assign the differences compared to the resultsof the nominal fits as the systematic uncertainty. We ex-pect this effect to be highly correlated between ACP pa-rameters, because the PDFs are similar in each D decaychannel. Thus they are affected by non-uniform ∆Eshift
distributions in a similar way. The same argument holdsfor the RK/π parameters. We studied the effect of assign-ing a 0%, 50%, and 100% correlation. The uncorrelatedcase gave the largest deviations from the nominal results,the fully correlated case gave the smallest. However, thevariation was found to be at the 10% level. We assign thesystematic uncertainty corresponding to a correlation of50%.
Table V lists the contributions of the effects discussedabove. Compared to our previous analysis [9], the sys-tematic uncertainty on ACP+ is reduced due to betterunderstanding of the detector intrinsic charge asymme-try (the determination of which benefits from the largerdataset) and due to improved evaluation of the correla-tions among the different sources of systematic uncertain-ties. The uncertainty on ACP− is only slightly reduced.
16
By contrast, the systematic uncertainties on RCP± are in-creased due to two additional sources of uncertainty thatwere not considered previously: the bias correction andthe differences of the ∆Eshift distributions among thefit components. The systematic correlations between theGLW parameters ~y = (ACP+, ACP−, RCP+, RCP−)
The final values of the GLW parameters that we mea-sure are:
ACP+ = 0.25± 0.06(stat)± 0.02(syst) , (36)
ACP− = −0.09± 0.07(stat)± 0.02(syst) , (37)
RCP+ = 1.18± 0.09(stat)± 0.05(syst) , (38)
RCP− = 1.07± 0.08(stat)± 0.04(syst) . (39)
The statistical correlations among these four quantitiesare:
C(stat)[~y] =
1 0.0 −0.08 0.01 0.0 0.03
1 0.101
. (40)
The results are in good agreement with those from ourprevious analysis [9] and the current world averages [21].Figure 2 shows the ∆E projections of the final fits tothe CP subsamples and Figures 3-5 show mES and Fprojections as well as projections of the fit to the D0 →K−π+ flavor mode.The statistical significance of a non-zero ACP+ value
is determined from the maximum value of the likelihoodfunction of the nominal fit and that of a dedicated null-hypothesis fit, where ACP+ was fixed to zero,
Sstat =√
2 ln(Lnom/Lnull) = 3.7. (41)
Taking into account systematic uncertainties, the statis-tical significance of ACP+ is slightly decreased to:
Sstat+syst =Sstat
√
1 +σ2syst
σ2stat
= 3.6. (42)
This constitutes evidence for direct CP violation incharged B decays and the first evidence of direct CPviolation in B → DK.We constrain the CKM angle γ, the strong phase
δB, and the amplitude ratio rB from the present mea-surement by adopting the frequentist procedure also ex-ploited in [15]. We define a multivariate Gaussian likeli-hood function
L(γ, δB, rB) =1
Nexp
(
−1
2(~y − ~yt)
TV −1cov (~y − ~yt)
)
(43)
relating the experimentally measured observables ~yand their statistical and systematic covariance matri-ces Vcov = Vstat + Vsyst with the corresponding truthparameters ~yt = ~yt(γ, δB, rB) calculated using Eqns. 3and 4. The matrices Vstat and Vsyst are constructed from
Eqns. 35-40. The normalization is N = (2π)2√
|Vcov|.We then define a χ2-function as
χ2(γ, δB, rB) = −2 lnL(γ, δB, rB). (44)
Due to the inherent eight-fold ambiguity of the GLWmethod there are eight equivalent minima of the χ2-function, χ2
min, which correspond to the same value ofrB and to eight alternative solutions for (γ, δB). To eval-uate the confidence level of a certain truth parameter (forexample γ) at a certain value (γ0) we consider the valueof the χ2-function at the new minimum, χ2
min(γ0, δ′B, r
′B),
satisfying ∆χ2 = χ2min(γ0, δ
′B, r
′B)−χ2
min ≥ 0. In a purelyGaussian situation for the truth parameters the CL is
17
E (GeV)∆-0.05 0 0.05 0.1
Eve
nts
/ ( 0
.01
GeV
)
0
10
20
30
40
50
60
E (GeV)∆-0.05 0 0.05 0.1
Eve
nts
/ ( 0
.01
GeV
)
0
10
20
30
40
50
60- DK→ -B
,-K+ K→D -π+π →D
a)
E (GeV)∆-0.05 0 0.05 0.1
Eve
nts
/ ( 0
.01
GeV
)
0
10
20
30
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60
E (GeV)∆-0.05 0 0.05 0.1
Eve
nts
/ ( 0
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GeV
)
0
10
20
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50
60+ DK→ +B
,-K+ K→D -π+π →D
b)
E (GeV)∆-0.05 0 0.05 0.1
Eve
nts
/ ( 0
.01
GeV
)
0
5
10
15
20
25
30
35
40
45
E (GeV)∆-0.05 0 0.05 0.1
Eve
nts
/ ( 0
.01
GeV
)
0
5
10
15
20
25
30
35
40
45 - DK→ -B
,0πS0 K→D
,ωS0 K→D
φS0 K→D
c)
E (GeV)∆-0.05 0 0.05 0.1
Eve
nts
/ ( 0
.01
GeV
)
0
5
10
15
20
25
30
35
40
45
E (GeV)∆-0.05 0 0.05 0.1
Eve
nts
/ ( 0
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GeV
)
0
5
10
15
20
25
30
35
40
45+ DK→ +B
,0πS0 K→D
,ωS0 K→D
φS0 K→D
d)
FIG. 2: ∆E projections of the fits to the data, split intosubsets of definite CP of the D candidate and charge ofthe B candidate: a) B−
→DCP+K−, b) B+
→DCP+K+, c)
B−→DCP−K
−, d) B+→DCP−K
+. The curves are the fullPDF (solid, blue), and B→Dπ (dash-dotted, green) stackedon the remaining backgrounds (dotted, purple). The regionbetween the solid and the dash-dotted lines represents theB→DK contribution. We show the subsets of the data sam-ple in which the track h from the B decay is identified as akaon. We require candidates to lie inside the signal-enrichedregion defined in Sec. IV, except for the plotted variable.
)2 (GeV/cESm5.25 5.26 5.27 5.28
)2E
vent
s / (
0.0
008
GeV
/c
0
5
10
15
20
25
30
35
40
)2 (GeV/cESm5.25 5.26 5.27 5.28
)2E
vent
s / (
0.0
008
GeV
/c
0
5
10
15
20
25
30
35
40- DK→ -B
,-K+ K→D -π+π →D
a)
)2 (GeV/cESm5.25 5.26 5.27 5.28
)2E
vent
s / (
0.0
008
GeV
/c
0
5
10
15
20
25
30
35
40
)2 (GeV/cESm5.25 5.26 5.27 5.28
)2E
vent
s / (
0.0
008
GeV
/c
0
5
10
15
20
25
30
35
40+ DK→ +B
,-K+ K→D -π+π →D
b)
)2 (GeV/cESm5.25 5.26 5.27 5.28
)2E
vent
s / (
0.0
008
GeV
/c
0
5
10
15
20
25
30
35
40
)2 (GeV/cESm5.25 5.26 5.27 5.28
)2E
vent
s / (
0.0
008
GeV
/c
0
5
10
15
20
25
30
35
40- DK→ -B
,0πS0 K→D
,ωS0 K→D
φS0 K→D
c)
)2 (GeV/cESm5.25 5.26 5.27 5.28
)2E
vent
s / (
0.0
008
GeV
/c
0
5
10
15
20
25
30
35
40
)2 (GeV/cESm5.25 5.26 5.27 5.28
)2E
vent
s / (
0.0
008
GeV
/c
0
5
10
15
20
25
30
35
40+ DK→ +B
,0πS0 K→D
,ωS0 K→D
φS0 K→D
d)
FIG. 3: mES projections of the fits to the data, split intosubsets of definite CP of the D candidate and charge of theB candidate: a) B−
→ DCP+K−, b) B+
→ DCP+K+, c)
B−→ DCP−K
−, d) B+→ DCP−K+. We show the subsets
of the data sample in which the track h from the B decay isidentified as a kaon. See caption of Fig. 2 for line definitions.Only a subrange of the whole fit range is shown in order toprovide a closer view of the signal peak.
given by the probability that ∆χ2 is exceeded for a χ2-
18
F-1.5 -1 -0.5 0 0.5 1 1.5
Eve
nts
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.06
)
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50
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)
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50 - DK→ -B
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a)
F-1.5 -1 -0.5 0 0.5 1 1.5
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nts
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F-1.5 -1 -0.5 0 0.5 1 1.5
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50 + DK→ +B
,-K+ K→D -π+π →D
b)
F-1.5 -1 -0.5 0 0.5 1 1.5
Eve
nts
/ ( 0
.06
)
0
10
20
30
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60
F-1.5 -1 -0.5 0 0.5 1 1.5
Eve
nts
/ ( 0
.06
)
0
10
20
30
40
50
60- DK→ -B
,0πS0 K→D
,ωS0 K→D
φS0 K→D
c)
F-1.5 -1 -0.5 0 0.5 1 1.5
Eve
nts
/ ( 0
.06
)
0
10
20
30
40
50
60
F-1.5 -1 -0.5 0 0.5 1 1.5
Eve
nts
/ ( 0
.06
)
0
10
20
30
40
50
60+ DK→ +B
,0πS0 K→D
,ωS0 K→D
φS0 K→D
d)
FIG. 4: F projections of the fits to the data, split into sub-sets of definite CP of the D candidate and charge of theB candidate: a) B−
→ DCP+K−, b) B+
→ DCP+K+, c)
B−→ DCP−K
−, d) B+→ DCP−K
+. We show the subsetsof the data sample in which the track h from the B decay isidentified as a kaon. See caption of Fig. 2 for line definitions.
distribution with one degree of freedom:
1− CL =1√
2Γ(1/2)
∫ ∞
∆χ2
e−t/2t−1/2 dt. (45)
E (GeV)∆-0.05 0 0.05 0.1
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π K→D
a)
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1000
2000
3000
4000
5000
6000
7000
8000
9000± Dh→ ±B
π K→D
b)
F-1.5 -1 -0.5 0 0.5 1 1.5
Eve
nts
/ ( 0
.06
)
0
1000
2000
3000
4000
5000
F-1.5 -1 -0.5 0 0.5 1 1.5
Eve
nts
/ ( 0
.06
)
0
1000
2000
3000
4000
5000± Dh→ ±B
π K→D
c)
FIG. 5: Projections of (a) ∆E, (b) mES, and (c) F variablesof the fit to the B±
→ Dh±, D0→ K−π+ flavor mode. No
requirements are put on the PID of the track h from the Bdecay and on the fit variables not plotted. See caption ofFig. 2 for line definitions.
A more accurate approach is to take into account thenon-linearity of the GLW relations, Eqns. 3 and 4. Inthis case one should consider ∆χ2 as a test statistic, andcalculate (1−CL) by means of a Monte Carlo procedure,described in the following. For a certain value of interest(γ0), we:
1. calculate ∆χ2 = χ2min(γ0, δ
′B, r
′B)− χ2
min as before;
2. generate a “toy” result A′CP±, R
′CP±, using Eq. 43
with values γ0, δ′B, r
′B as the PDF;
3. calculate ∆χ2′ of the toy result as in the first step,i.e. minimize again with respect to δB and rB ;
19
4. calculate (1 − CL) as the fraction of toy resultswhich perform better than the measured data, i.e.1− CL = N(∆χ2 > ∆χ2′)/Ntoy.
Figures 6 and 7 illustrate 1-CL as a function of γ and rBas obtained from this study. From these distributions weextract 68% and 95% CL confidence intervals for γ andrB , as summarized in Table VII. Due to the γ ↔ δB am-biguity of the GLW method, the 1D CL intervals for δBare identical to those for γ. At the 68% CL we are able todistinguish six out of eight solutions for γ (and δB), twoof which are in good agreement with the current worldaverages [25]. At the 95% CL we are able to exclude theintervals [0◦, 7.0◦], [173.0◦, 187.0◦] and [353.0◦, 360◦] forγ and δB. For rB we deduce at 68% CL:
rB = 0.35+0.10−0.11(stat + syst). (46)
TABLE VII: 68% and 95% CL intervals for the parameters γ,and rB , taking into account both statistical and systematicuncertainties. The confidence intervals for δB are identical tothose for γ due to the intrinsic γ ↔ δB ambiguity of the GLWmethod.
γ mod 180 [◦] rB68% CL [11.3, 22.7] [0.24, 0.45]
[80.8, 99.2][157.3, 168.7]
95% CL [7.0, 173.0] [0.06, 0.51]
In order to facilitate the future combination of thesemeasurements with the results of the Dalitz plot analysisof B± → DK±, D → K0
Sh+h− decays (h = π,K) [16],
we recompute the GLW parameters after excluding fromthe nominal fit the DCP− → K0
Sφ (φ → K+K−) sub-
sample. The sample obtained in this way is statisticallyindependent of that selected in [16]. The final values ofthe GLW parameters that we measure in this case are:
ACP+ = 0.25± 0.06(stat)± 0.02(syst) , (47)
ACP− = −0.08± 0.07(stat)± 0.02(syst) , (48)
RCP+ = 1.18± 0.09(stat)± 0.05(syst) , (49)
RCP− = 1.03± 0.09(stat)± 0.04(syst) . (50)
The statistical correlations among these four quantitiesare:
C(stat)[~y] =
1 0 −0.08 01 0 0.04
1 0.091
, (51)
and the systematic correlations are
C(syst)[~y] =
1 0.56 −0.06 01 0 0
1 0.121
. (52)
]° [γ
1-C
L
00.1
0.20.30.40.50.60.7
0.80.9
1
0 20 40 60 80 100 120 140 160 180
68%
95%
Br0.1 0.2 0.3 0.4 0.5 0.6
1-C
L
00.1
0.20.30.40.50.60.7
0.80.9
1
68%
95%
FIG. 6: 1-CL as a function of γ (top) and rB (bottom).Both statistical and systematic uncertainties are taken intoaccount. For the angle γ, the plot is identical in the range[180◦, 360◦]. The horizontal lines show the 68% CL (dashed)and the 95% CL (dotted). Due to the symmetry of Eqns. 3and 4 the plot for the strong phase δB is identical to the onefor γ.
To compare the results obtained after removing theDCP− → K0
Sφ subsample with those from the B± →
DK±, D → K0Sh+h− analyses, which are expressed in
terms of the variables x± = rB cos(δB ± γ) and y± =rB sin(δB ± γ), we use the GLW parameters measuredin this way to determine the quantities x± through therelations:
x± =1
4[RCP+(1∓ACP+)−RCP−(1∓ACP−)] . (53)
We obtain
x+ = −0.057± 0.039(stat)± 0.015(syst) , (54)
x− = 0.132± 0.042(stat)± 0.018(syst) . (55)
These results are in good agreement with the currentworld averages [21] and have precision close to the singlemost precise measurements [16]. We also measure r2B ,which provides a constraint on x± and y± via r2B = x2
±+y2±, from
r2B =1
2(RCP+ +RCP− − 2) . (56)
We determine:
r2B = 0.105± 0.067(stat)± 0.035(syst) . (57)
20
]° [γ
Br
0.1
0.2
0.3
0.4
0.5
0.6
0 20 40 60 80 100 120 140 160 180
95% CL68% CL
]° [γ
]° [
Bδ
0 20 40 60 80 100 120 140 160 1800
20
40
60
80
100
120
140
160
180
95% CL68% CL
FIG. 7: Contours at 68% (dotted, red) and 95% (solid, green)2-dimensional CL in the (γ, rB) and (γ, δB) planes. See alsothe caption of Fig. 6 regarding symmetries.
The constraints that could be placed on the quantitiesy± from these measurements, by exploiting the relationr2B = x2
± ± y2±, are much weaker than those provided bythe B± → DK±, D → K0
Sh+h− analysis.
As a final check of consistency we consider the quantitya,
a = ACP+RCP+ +ACP−RCP− . (58)
From Eqns. 3 and 4 one expects a to satisfy a = 0. Wemeasure a = 0.19±0.11(stat + syst), which is compatiblewith 0.
IX. SUMMARY
Using the entire dataset collected by BABAR at thee+e− center-of-mass energy close to the Υ (4S) mass, wehave reconstructed B± → DK± decays, with D mesonsdecaying to non-CP (Kπ), CP -even (K+K−, π+π−) andCP -odd (K0
Sπ0, K0
Sφ, K0
Sω) eigenstates.
Through an improved analysis method compared tothe previous BABAR measurement [9] and through an en-larged dataset, corresponding to an increase in integrated
luminosity at the Υ (4S) peak from 348 fb−1 to 426 fb−1,we obtain the most precise measurements of the GLWparameters ACP± and RCP± to date:
ACP+ = 0.25± 0.06(stat)± 0.02(syst) ,
ACP− = −0.09± 0.07(stat)± 0.02(syst) ,
RCP+ = 1.18± 0.09(stat)± 0.05(syst) ,
RCP− = 1.07± 0.08(stat)± 0.04(syst) .
We measure a value of ACP+ which is 3.6 standarddeviations from zero, which constitutes the first evidencefor direct CP violation in B → DK decays.From the measured values of the GLW parameters, we
extract confidence intervals for the CKM angle γ, thestrong phase δB , and the amplitude ratio rB, using a fre-quentist approach, taking into account both statisticaland systematic uncertainties. At the 68% CL we find thatboth γ and δB (modulo 180◦) belong to one of the threeintervals [11.3◦, 22.7◦], [80.8◦, 99.2◦] or [157.3◦, 168.7◦],and that
rB ∈ [0.24, 0.45].
At 95% CL, we exclude the intervals [0◦, 7.0◦],[173.0◦, 187.0◦] and [353.0◦, 360◦] for γ and δB, and mea-sure
rB ∈ [0.06, 0.51].
Our results are in agreement with the current world av-erages [25].To facilitate the combination of these measurements
with the results of our Dalitz plot analysis of B± →DK±, D → K0
Sh+h− (h = K, π) [16], we exclude the
D → K0Sφ, φ → K+K− channel from this analysis – thus
removing events selected also in [16] – and then deter-mine
ACP− = −0.08± 0.07(stat)± 0.02(syst) ,
RCP− = 1.03± 0.09(stat)± 0.04(syst) .
For comparison with the results of the B± → DK±, D →K0
Sh+h− analyses, which are expressed in terms of the
variables x± = rB cos(δB ± γ) and y± = rB sin(δB ± γ),we express our results for the GLW observables in termsof x+ and x−. We measure
x+ = −0.057± 0.039(stat)± 0.015(syst) ,
x− = 0.132± 0.042(stat)± 0.018(syst) ,
at 68% CL. These results are in good agreement with thecurrent world averages [21] and have precision compara-ble to the single most precise measurements [16]. We alsoevaluate rB after the exclusion of the D → K0
Sφ channel,
and obtain a weak constraint on r2B = x2± ± y2±:
r2B = 0.105± 0.067(stat)± 0.035(syst)
at 68% CL.
21
X. ACKNOWLEDGEMENTS
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 hospital-ity extended to them. This work is supported by the USDepartment of Energy and National Science Foundation,the Natural Sciences and Engineering Research Council(Canada), the Commissariat a l’Energie Atomique and
Institut National de Physique Nucleaire et de Physiquedes Particules (France), the Bundesministerium fur Bil-dung und Forschung and Deutsche Forschungsgemein-schaft (Germany), the Istituto Nazionale di Fisica Nu-cleare (Italy), the Foundation for Fundamental Researchon Matter (The Netherlands), the Research Council ofNorway, the Ministry of Education and Science of theRussian Federation, Ministerio de Ciencia e Innovacion(Spain), and the Science and Technology Facilities Coun-cil (United Kingdom). Individuals have received supportfrom the Marie-Curie IEF program (European Union),the A. P. Sloan Foundation (USA) and the BinationalScience Foundation (USA-Israel).
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TABLE VIII: Fit result of the three final fits to data, before correcting for fit biases (see Section VII).
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