Search for CP Violation in Neutral D Meson Cabibbo-suppressed Three-body Decays
aa r X i v : . [ h e p - e x ] S e p B A B AR -PUB-07/074SLAC-PUB-13058arXiv:0802.4035 [hep-ex] Phys. Rev. D78 , 051102(R) (2008)
Search for CP Violation in Neutral D Meson Cabibbo-suppressed Three-body Decays
B. Aubert, M. Bona, Y. Karyotakis, J. P. Lees, V. Poireau, X. Prudent, V. Tisserand, A. Zghiche, J. Garra Tico, E. Grauges, L. Lopez, A. Palano, M. Pappagallo, G. Eigen, B. Stugu, L. Sun, G. S. Abrams, M. Battaglia, D. N. Brown, J. Button-Shafer, R. N. Cahn, R. G. Jacobsen, J. A. Kadyk, L. T. Kerth, Yu. G. Kolomensky, G. Kukartsev, G. Lynch, I. L. Osipenkov, M. T. Ronan, ∗ K. Tackmann, T. Tanabe, W. A. Wenzel, C. M. Hawkes, N. Soni, A. T. Watson, H. Koch, T. Schroeder, D. Walker, D. J. Asgeirsson, T. Cuhadar-Donszelmann, B. G. Fulsom, C. Hearty, T. S. Mattison, J. A. McKenna, M. Barrett, A. Khan, M. Saleem, L. Teodorescu, 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, K. Yu. Todyshev, M. Bondioli, S. Curry, I. Eschrich, D. Kirkby, A. J. Lankford, P. Lund, M. Mandelkern, E. C. Martin, D. P. Stoker, S. Abachi, C. Buchanan, J. W. Gary, F. Liu, O. Long, B. C. Shen, ∗ G. M. Vitug, Z. Yasin, L. Zhang, H. P. Paar, S. Rahatlou, V. Sharma, C. Campagnari, T. M. Hong, D. Kovalskyi, M. A. Mazur, J. D. Richman, T. W. Beck, A. M. Eisner, C. J. Flacco, C. A. Heusch, J. Kroseberg, W. S. Lockman, T. Schalk, B. A. Schumm, A. Seiden, M. G. Wilson, L. O. Winstrom, E. Chen, C. H. Cheng, D. A. Doll, B. Echenard, F. Fang, D. G. Hitlin, I. Narsky, T. Piatenko, F. C. Porter, R. Andreassen, G. Mancinelli, B. T. Meadows, K. Mishra, M. D. Sokoloff, F. Blanc, P. C. Bloom, W. T. Ford, J. F. Hirschauer, A. Kreisel, M. Nagel, U. Nauenberg, A. Olivas, J. G. Smith, K. A. Ulmer, S. R. Wagner, R. Ayad, † A. M. Gabareen, A. Soffer, ‡ W. H. Toki, R. J. Wilson, D. D. Altenburg, E. Feltresi, A. Hauke, H. Jasper, M. Karbach, J. Merkel, A. Petzold, B. Spaan, K. Wacker, V. Klose, M. J. Kobel, H. M. Lacker, W. F. Mader, R. Nogowski, J. Schubert, K. R. Schubert, R. Schwierz, J. E. Sundermann, A. Volk, D. Bernard, G. R. Bonneaud, E. Latour, Ch. Thiebaux, M. Verderi, P. J. Clark, W. Gradl, S. Playfer, A. I. Robertson, J. E. Watson, M. Andreotti, D. Bettoni, C. Bozzi, R. Calabrese, A. Cecchi, G. Cibinetto, P. Franchini, E. Luppi, M. Negrini, A. Petrella, L. Piemontese, E. Prencipe, V. Santoro, F. Anulli, R. Baldini-Ferroli, A. Calcaterra, R. de Sangro, G. Finocchiaro, S. Pacetti, P. Patteri, I. M. Peruzzi, § M. Piccolo, M. Rama, A. Zallo, A. Buzzo, R. Contri, M. Lo Vetere, M. M. Macri, M. R. Monge, S. Passaggio, C. Patrignani, E. Robutti, A. Santroni, S. Tosi, K. S. Chaisanguanthum, M. Morii, R. S. Dubitzky, J. Marks, S. Schenk, U. Uwer, D. J. Bard, P. D. Dauncey, J. A. Nash, W. Panduro Vazquez, M. Tibbetts, P. K. Behera, X. Chai, M. J. Charles, U. Mallik, J. Cochran, H. B. Crawley, L. Dong, V. Eyges, W. T. Meyer, S. Prell, E. I. Rosenberg, A. E. Rubin, Y. Y. Gao, A. V. Gritsan, Z. J. Guo, C. K. Lae, A. G. Denig, M. Fritsch, G. Schott, N. Arnaud, J. B´equilleux, A. D’Orazio, M. Davier, J. Firmino da Costa, G. Grosdidier, A. H¨ocker, V. Lepeltier, F. Le Diberder, A. M. Lutz, S. Pruvot, P. Roudeau, M. H. Schune, J. Serrano, V. Sordini, A. Stocchi, W. F. Wang, G. Wormser, D. J. Lange, D. M. Wright, I. Bingham, J. P. Burke, C. A. Chavez, J. R. Fry, E. Gabathuler, R. Gamet, D. E. Hutchcroft, D. J. Payne, C. Touramanis, A. J. Bevan, K. A. George, F. Di Lodovico, R. Sacco, M. Sigamani, G. Cowan, H. U. Flaecher, D. A. Hopkins, S. Paramesvaran, F. Salvatore, A. C. Wren, D. N. Brown, C. L. Davis, K. E. Alwyn, N. R. Barlow, R. J. Barlow, Y. M. Chia, C. L. Edgar, G. D. Lafferty, T. J. West, J. I. Yi, J. Anderson, C. Chen, A. Jawahery, D. A. Roberts, G. Simi, J. M. Tuggle, C. Dallapiccola, S. S. Hertzbach, X. Li, E. Salvati, S. Saremi, R. Cowan, D. Dujmic, P. H. Fisher, K. Koeneke, G. Sciolla, M. Spitznagel, F. Taylor, R. K. Yamamoto, M. Zhao, S. E. Mclachlin, ∗ P. M. Patel, S. H. Robertson, A. Lazzaro, V. Lombardo, F. Palombo, J. M. Bauer, L. Cremaldi, V. Eschenburg, R. Godang, R. Kroeger, D. A. Sanders, D. J. Summers, H. W. Zhao, S. Brunet, D. Cˆot´e, M. Simard, P. Taras, F. B. Viaud, H. Nicholson, G. De Nardo, L. Lista, D. Monorchio, C. Sciacca, M. A. Baak, G. Raven, H. L. Snoek, C. P. Jessop, K. J. Knoepfel, J. M. LoSecco, G. Benelli, L. A. Corwin, K. Honscheid, H. Kagan, R. Kass, J. P. Morris, A. M. Rahimi, J. J. Regensburger, S. J. Sekula, Q. K. Wong, N. L. Blount, J. Brau, R. Frey, O. Igonkina, J. A. Kolb, M. Lu, R. Rahmat, N. B. Sinev, D. Strom, J. Strube, E. Torrence, G. Castelli, N. Gagliardi, A. Gaz, M. Margoni, M. Morandin, M. Posocco, M. Rotondo, F. Simonetto, R. Stroili, C. Voci, P. del Amo Sanchez, E. Ben-Haim, H. Briand, G. Calderini, J. Chauveau, P. David, L. Del Buono, O. Hamon, Ph. Leruste, J. Malcl`es, J. Ocariz, A. Perez, J. Prendki, L. Gladney, M. Biasini, R. Covarelli, E. Manoni, C. Angelini, G. Batignani, S. Bettarini, M. Carpinelli, ¶ A. Cervelli, F. Forti, M. A. Giorgi, A. Lusiani, G. Marchiori, M. Morganti, N. Neri, E. Paoloni, G. Rizzo, J. J. Walsh, J. Biesiada, Y. P. Lau, D. Lopes Pegna, C. Lu, J. Olsen, A. J. S. Smith, A. V. Telnov, E. Baracchini, G. Cavoto, D. del Re, E. Di Marco, R. Faccini, F. Ferrarotto, F. Ferroni, M. Gaspero, P. D. Jackson, M. A. Mazzoni, S. Morganti, G. Piredda, F. Polci, F. Renga, C. Voena, M. Ebert, T. Hartmann, H. Schr¨oder, R. Waldi, T. Adye, B. Franek, E. O. Olaiya, W. Roethel, F. F. Wilson, S. Emery, M. Escalier, A. Gaidot, S. F. Ganzhur, G. Hamel de Monchenault, W. Kozanecki, G. Vasseur, Ch. Y`eche, M. Zito, X. R. Chen, H. Liu, W. Park, M. V. Purohit, R. M. White, J. R. Wilson, M. T. Allen, D. Aston, R. Bartoldus, P. Bechtle, J. F. Benitez, R. Cenci, J. P. Coleman, M. R. Convery, J. C. Dingfelder, J. Dorfan, G. P. Dubois-Felsmann, W. Dunwoodie, R. C. Field, T. Glanzman, S. J. Gowdy, M. T. Graham, P. Grenier, C. Hast, W. R. Innes, J. Kaminski, M. H. Kelsey, H. Kim, P. Kim, M. L. Kocian, D. W. G. S. Leith, S. Li, B. Lindquist, S. Luitz, V. Luth, H. L. Lynch, D. B. MacFarlane, H. Marsiske, R. Messner, D. R. Muller, H. Neal, S. Nelson, C. P. O’Grady, I. Ofte, A. Perazzo, M. Perl, B. N. Ratcliff, A. Roodman, A. A. Salnikov, R. H. Schindler, J. Schwiening, A. Snyder, D. Su, M. K. Sullivan, K. Suzuki, S. K. Swain, J. M. Thompson, J. Va’vra, A. P. Wagner, M. Weaver, W. J. Wisniewski, M. Wittgen, D. H. Wright, H. W. Wulsin, A. K. Yarritu, K. Yi, C. C. Young, V. Ziegler, P. R. Burchat, A. J. Edwards, S. A. Majewski, T. S. Miyashita, B. A. Petersen, L. Wilden, S. Ahmed, M. S. Alam, R. Bula, J. A. Ernst, B. Pan, M. A. Saeed, S. B. Zain, S. M. Spanier, B. J. Wogsland, R. Eckmann, J. L. Ritchie, A. M. Ruland, C. J. Schilling, R. F. Schwitters, J. M. Izen, X. C. Lou, S. Ye, F. Bianchi, D. Gamba, M. Pelliccioni, M. Bomben, L. Bosisio, C. Cartaro, F. Cossutti, G. Della Ricca, L. Lanceri, L. Vitale, V. Azzolini, N. Lopez-March, F. Martinez-Vidal, D. A. Milanes, A. Oyanguren, J. Albert, Sw. Banerjee, B. Bhuyan, K. Hamano, R. Kowalewski, I. M. Nugent, J. M. Roney, R. J. Sobie, T. J. Gershon, P. F. Harrison, J. Ilic, T. E. Latham, G. B. Mohanty, H. R. Band, X. Chen, S. Dasu, K. T. Flood, P. E. Kutter, Y. Pan, M. Pierini, R. Prepost, C. O. Vuosalo, and S. L. Wu (The B A B AR Collaboration) Laboratoire de Physique des Particules, IN2P3/CNRS et Universit´e de Savoie, F-74941 Annecy-Le-Vieux, France Universitat de Barcelona, Facultat de Fisica, Departament ECM, E-08028 Barcelona, Spain Universit`a di Bari, Dipartimento di Fisica and INFN, I-70126 Bari, Italy University of Bergen, Institute of Physics, N-5007 Bergen, Norway Lawrence Berkeley National Laboratory and University of California, Berkeley, California 94720, USA University of Birmingham, Birmingham, B15 2TT, United Kingdom Ruhr Universit¨at Bochum, Institut f¨ur Experimentalphysik 1, D-44780 Bochum, Germany University of Bristol, Bristol BS8 1TL, United Kingdom University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z1 Brunel University, Uxbridge, Middlesex UB8 3PH, United Kingdom Budker Institute of Nuclear Physics, Novosibirsk 630090, Russia University of California at Irvine, Irvine, California 92697, USA University of California at Los Angeles, Los Angeles, California 90024, USA University of California at Riverside, Riverside, California 92521, USA University of California at San Diego, La Jolla, California 92093, USA University of California at Santa Barbara, Santa Barbara, California 93106, USA University of California at Santa Cruz, Institute for Particle Physics, Santa Cruz, California 95064, USA California Institute of Technology, Pasadena, California 91125, USA University of Cincinnati, Cincinnati, Ohio 45221, USA University of Colorado, Boulder, Colorado 80309, USA Colorado State University, Fort Collins, Colorado 80523, USA Universit¨at Dortmund, Institut f¨ur Physik, D-44221 Dortmund, Germany Technische Universit¨at Dresden, Institut f¨ur Kern- und Teilchenphysik, D-01062 Dresden, Germany Laboratoire Leprince-Ringuet, CNRS/IN2P3, Ecole Polytechnique, F-91128 Palaiseau, France University of Edinburgh, Edinburgh EH9 3JZ, United Kingdom Universit`a di Ferrara, Dipartimento di Fisica and INFN, I-44100 Ferrara, Italy Laboratori Nazionali di Frascati dell’INFN, I-00044 Frascati, Italy Universit`a di Genova, Dipartimento di Fisica and INFN, I-16146 Genova, Italy Harvard University, Cambridge, Massachusetts 02138, USA Universit¨at Heidelberg, Physikalisches Institut, Philosophenweg 12, D-69120 Heidelberg, Germany Imperial College London, London, SW7 2AZ, United Kingdom University of Iowa, Iowa City, Iowa 52242, USA Iowa State University, Ames, Iowa 50011-3160, USA Johns Hopkins University, Baltimore, Maryland 21218, USA Universit¨at Karlsruhe, Institut f¨ur Experimentelle Kernphysik, D-76021 Karlsruhe, Germany Laboratoire de l’Acc´el´erateur Lin´eaire, IN2P3/CNRS et Universit´e Paris-Sud 11,Centre Scientifique d’Orsay, B. P. 34, F-91898 ORSAY Cedex, France Lawrence Livermore National Laboratory, Livermore, California 94550, USA University of Liverpool, Liverpool L69 7ZE, United Kingdom Queen Mary, University of London, E1 4NS, United Kingdom University of London, Royal Holloway and Bedford New College, Egham, Surrey TW20 0EX, United Kingdom University of Louisville, Louisville, Kentucky 40292, USA University of Manchester, Manchester M13 9PL, United Kingdom University of Maryland, College Park, Maryland 20742, USA University of Massachusetts, Amherst, Massachusetts 01003, USA Massachusetts Institute of Technology, Laboratory for Nuclear Science, Cambridge, Massachusetts 02139, USA McGill University, Montr´eal, Qu´ebec, Canada H3A 2T8 Universit`a di Milano, Dipartimento di Fisica and INFN, I-20133 Milano, Italy University of Mississippi, University, Mississippi 38677, USA Universit´e de Montr´eal, Physique des Particules, Montr´eal, Qu´ebec, Canada H3C 3J7 Mount Holyoke College, South Hadley, Massachusetts 01075, USA Universit`a di Napoli Federico II, Dipartimento di Scienze Fisiche and INFN, I-80126, Napoli, Italy NIKHEF, National Institute for Nuclear Physics and High Energy Physics, NL-1009 DB Amsterdam, The Netherlands University of Notre Dame, Notre Dame, Indiana 46556, USA Ohio State University, Columbus, Ohio 43210, USA University of Oregon, Eugene, Oregon 97403, USA Universit`a di Padova, Dipartimento di Fisica and INFN, I-35131 Padova, Italy Laboratoire de Physique Nucl´eaire et de Hautes Energies,IN2P3/CNRS, Universit´e Pierre et Marie Curie-Paris6,Universit´e Denis Diderot-Paris7, F-75252 Paris, France University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA Universit`a di Perugia, Dipartimento di Fisica and INFN, I-06100 Perugia, Italy Universit`a di Pisa, Dipartimento di Fisica, Scuola Normale Superiore and INFN, I-56127 Pisa, Italy Princeton University, Princeton, New Jersey 08544, USA Universit`a di Roma La Sapienza, Dipartimento di Fisica and INFN, I-00185 Roma, Italy Universit¨at Rostock, D-18051 Rostock, Germany Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, OX11 0QX, United Kingdom DSM/Dapnia, CEA/Saclay, F-91191 Gif-sur-Yvette, France University of South Carolina, Columbia, South Carolina 29208, USA Stanford Linear Accelerator Center, Stanford, California 94309, USA Stanford University, Stanford, California 94305-4060, USA State University of New York, Albany, New York 12222, USA University of Tennessee, Knoxville, Tennessee 37996, USA University of Texas at Austin, Austin, Texas 78712, USA University of Texas at Dallas, Richardson, Texas 75083, USA Universit`a di Torino, Dipartimento di Fisica Sperimentale and INFN, I-10125 Torino, Italy Universit`a di Trieste, Dipartimento di Fisica and INFN, I-34127 Trieste, Italy IFIC, Universitat de Valencia-CSIC, E-46071 Valencia, Spain University of Victoria, Victoria, British Columbia, Canada V8W 3P6 Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom University of Wisconsin, Madison, Wisconsin 53706, USA (Dated: August 3, 2018)Using 385 fb − of e + e − collision data collected at center-of-mass energies around 10.6 GeV, wesearch for time-integrated CP violation in the Cabibbo-suppressed decays D /D → π − π + π and D /D → K − K + π with both model-independent and model-dependent methods. Measurementsof the asymmetries in amplitudes of flavor states and CP eigenstates provide constraints on theoriesbeyond the standard model, some of which predict CP violation in amplitudes at the 1% level orhigher. We find no evidence of CP violation and hence no conflict with the standard model. PACS numbers: 14.40.Lb, 13.25.Ft, 11.30.Er
Charge-parity violation (
CP V ) [1], manifested as anasymmetry between the decay rates of a particle andits CP -conjugate antiparticle, requires at least two in-terfering complex quantum mechanical amplitudes withdifferent phases. The strong phase of each amplituderespects CP symmetry while the weak phase changessign under charge-conjugation. In the standard model(SM), direct CP V is due to relative weak phases thattypically enter as a difference in phase between “treelevel” and “penguin” [2] SM amplitudes. The pen-guin amplitudes in charm decays are, however, toosmall ( O (0 . CP V . Ex-tensions of the SM introduce additional amplitudes of O (1%) [3, 4, 5] with relative weak phases that can pro-duce CP V in charmed particle decays [6]. Current ex-perimental searches [7, 8, 9, 10, 11, 12] are approachingthis level of sensitivity. Observation of
CP V with currentexperimental sensitivities would provide strong evidenceof new physics.A recent theory paper [3] argues that singly Cabibbo-suppressed (SCS) D (meaning either D or D ) decaysare uniquely sensitive to CP V in c → u ¯ dd, u ¯ ss transitionsand probe contributions from supersymmetric gluonicpenguins. Such transitions do not affect the Cabibbo-favored ( c → s ¯ du ) or doubly Cabibbo-suppressed ( c → d ¯ su ) decays. Time-integrated CP asymmetries in D de-cays can have three components: direct CP V in decaysto specific states, indirect
CP V in D – D mixing, andindirect CP V in interference of decays with and withoutmixing. Indirect
CP V is predicted to be universal foramplitudes with final CP eigenstates, but direct CP V can be non-universal depending on the specifics of thenew physics.We search for time-integrated
CP V in the three-bodySCS decays D → π − π + π , K − K + π . These decays pro-ceed via CP eigenstates ( e.g., ρ π , φπ ) and also viaflavor states ( e.g., ρ ± π ∓ , K ∗± K ∓ ), thus making it pos-sible to probe CP V in both types of amplitudes and inthe interference between them. Measuring interferenceeffects in a Dalitz plot (DP) probes asymmetries in boththe magnitudes and phases of the amplitudes, not sim-ply in the overall decay rates. We adopt four approachesin our search for evidence of
CP V , three of which aremodel-independent. First, we quantify differences be-tween the D and D DPs in two dimensions. Second,we look for differences in the angular moments of the D and D intensity distributions. Third, in a model-dependent approach, we look for CP V in the amplitudesdescribing intermediate states in the D and D decays.Finally, we look for a phase-space-integrated asymmetry.The first two methods are sensitive to differences in theshapes of the D and D DPs, allowing regions of phasespace with
CP V to be identified. The third method as- sociates any
CP V observed using the first two methodswith specific intermediate amplitudes. The last methodis insensitive to differences in the DP shapes, so comple-ments the other methods. To minimize bias, we finalizethe analysis procedure without looking at the data.We perform the present analysis using 385 fb − of e + e − collision data collected at 10.58 GeV and 10.54 GeVcenter-of-mass (CM) energies with the B A B AR detec-tor [13] at the PEP-II storage rings. The event se-lection criteria are those used in our measurement ofthe branching ratios of the decays D → π − π + π and D → K − K + π [14]. In particular, we study D mesonsproduced in D ∗ + → D π + and D ∗− → D π − de-cays that distinguish between D and D . We requirethe D candidate CM momentum > .
77 GeV /c and | m D ∗± − m D − . /c | < . /c . Here, m refers to a reconstructed invariant mass. Around ± D mass, we find82468 ± π − π + π and 11278 ± K − K + π signalevents with purities of about 98%. We determine the sig-nal reconstruction efficiency as a function of the positionin the DP using simulated D and D decays [14] from e + e − → cc events, subjected to the same selection pro-cedure that is applied to the data.A direct comparison of the efficiency-corrected andbackground-subtracted DPs for D and D events is thesimplest way to look for CP V . Figure 1 shows the nor-malized residuals ∆ in DP area elements, where∆ = ( n D − R · n D ) / q σ n D + R · σ n D , (1)and n denotes the number of events in a DP element and σ its uncertainty. The factor R , equal to 0 . ± .
006 for π − π + π and 1 . ± .
016 for K − K + π , is the ratio ofthe number of efficiency-corrected D to D events. Thisis introduced to allow for any asymmetry in the produc-tion cross section due to higher order QED correctionsor in the branching fractions for D and D decay to thesame final state.We calculate χ /ν = ( P νi =1 ∆ i ) /ν , where ν is thenumber of DP elements: 1429 for π − π + π and 726 for K − K + π . In an ensemble of simulated experiments withno CP V , we find the distribution of χ /ν values to havea mean of 1.012 ± ± π − π + π ( K − K + π ). The measuredvalue in the data is 1.020 for π − π + π and 1.056 for K − K + π , so we obtain a one-sided Gaussian confidencelevel (CL) for consistency with no CP V of 32 .
8% for π − π + π and 16 .
6% for K − K + π . The same analysisprocedure, when applied to simulated samples with ei-ther 1% fractional change in magnitude or 1 ◦ changein phase between the D and D amplitudes for decayto any of the main resonant states, gives a χ /ν that isabout 2 σ away from the no CP V hypothesis. Systematic ) /c ) (GeV p - p ( m ) / c ) ( G e V p + p ( m -4-3-2-101234 (a) ) /c ) (GeV p - (K m ) / c ) ( G e V p + ( K m -4-3-2-101234 (b) FIG. 1: (color). Normalized residuals in Dalitz plot elements, defined in Eq. 1, for (a) D → π − π + π and (b) D → K − K + π . uncertainties are small (as will be clear from the model-dependent results of Tables I–II) and have not been in-cluded in the CL calculation.The angular moments of the cosine of the helicity an-gle of the D decay products reflect the spin and massstructure of intermediate resonant and nonresonant am-plitudes [15]. We define the helicity angle θ H for decaysof the type D → r ( AB ) C as the angle between the mo-mentum of A in the AB rest frame and the direction op-posite to the D momentum in that same frame. The an-gular moments [16] of order l are defined as the efficiency-corrected invariant mass distributions of events weightedby spherical harmonics Y l ( θ H ) = p / π P l (cos θ H ).Here P l are the Legendre polynomials of order l . Tostudy differences between the D and D amplitudes, wecalculate the quantities X l for l = 0 −
7, where X l = (cid:0) P l − R · P l (cid:1)q σ P l + R · σ P l , (2)and P l ( P l ) are obtained from D ( D ) events. Highermoments are zero within errors in both data and simu-lation. For illustration, we show the X l distributions for l = 0 −
2, in Fig. 2.We then define χ /ν of the angular moment distribu-tions of a two-body channel summed over all intervals ininvariant mass as χ /ν = k X X i =0 7 X j =0 X i ρ ij X j /ν, (3)where ν = 8 k , k is the number of intervals, and ρ ij is the correlation coefficient between X i , X j : ρ ij = h X i X j i − h X i i h X j i q h X i i − h X i i · q(cid:10) X j (cid:11) − h X j i . (4)We determine the ρ ij in each mass interval by simulat-ing experiments with no CP V . We test the method onreal data by randomly assigning events as D or D , andthen calculating χ /ν for the difference in their angularmoments. We repeat this experiment 500 times and findthe resulting χ /ν distribution to be consistent with no CP V , validating our calculation of ρ ij . We then look atthe D flavor in the data and calculate the χ /ν valuesfor the two-body channels with charge combinations + , − and + ,
0. Finally, we obtain a one-sided Gaussian CLfor consistency with no
CP V using the reference valueand r.m.s. deviation from simulation. We find the CLfor no
CP V to be 28 .
2% for the π + π − , 28 .
4% for the π + π , 63 .
1% for the K + K − , and 23 .
8% for the K + π sub-systems. Again, a 1% fractional change in magni-tude or 1 ◦ change in phase of any of the main resonantamplitudes gives a χ /ν that is about 2 σ away from theno CP V hypothesis.The Dalitz plot amplitude A can be parametrized asa sum of amplitudes A r ( s + , s − ) for all relevant interme-diate states r , each with a complex coefficient, i.e., A = P r a r e iφ r A r ( s + , s − ), where a r and φ r are real. Here s + and s − are the squared invariant masses of the pairof final state particles with charge combinations + , − ,
0. The fit fraction for each process r is defined as f r ≡ R | a r A r | ds + ds − / R |A| ds + ds − . We model incoherent, CP -symmetric background empirically [15, 17]. In theabsence of CP V , we expect the values of a r and φ r (and ) ) (GeV/c - p + p m(0.26 1.01 1.76 X -202 ) ) (GeV/c - p + p m(0.26 1.01 1.76 X -202 ) ) (GeV/c - p + p m(0.26 1.01 1.76 X -202 ) ) (GeV/c p + p m(0.26 1.01 1.76 X -202 ) ) (GeV/c p + p m(0.26 1.01 1.76 X -202 ) ) (GeV/c p + p m(0.26 1.01 1.76 X -202) ) (GeV/c - K + m(K1 1.5 X -202 ) ) (GeV/c - K + m(K1 1.5 X -202 ) ) (GeV/c - K + m(K1 1.5 X -202 ) ) (GeV/c p + m(K0.6 1 1.4 X -202 ) ) (GeV/c p + m(K0.6 1 1.4 X -202 ) ) (GeV/c p + m(K0.6 1 1.4 X -202 FIG. 2: (color online). Normalized residuals for the first three Legendre polynomial moments of the π − π + (row 1), π + π (row2), K − K + (row 3), and K + π (row 4) sub-systems. The confidence level for no CP violation (dashed line) is obtained fromthe first eight moments. The error bars represent ± σ . hence f r ) to be identical for D and D decay. The re-sults obtained with this assumption are listed in Ref. [17]for D → π − π + π and in Ref. [15] for D → K − K + π .To allow the possibility of CP V in the present analysis,we let a second process – not necessarily of SM origin– contribute to each of the amplitudes A r , thus permit-ting the a r , φ r , f r for D and D to differ. We sum-marize the results of the fit to the data in terms of thedifferences ∆ a r = a D r − a D r , ∆ φ r = φ D r − φ D r , and∆ f r = f D r − f D r in Table I for π − π + π and in Table IIfor K − K + π . The CP asymmetry in any amplitude, rel-ative to that of the whole decay, is no larger than a fewpercent.Systematic uncertainties in the quantities describ-ing CP asymmetries, reported in Tables I–II, arise fromexperimental effects, and also from uncertainties in themodels used to describe the data. We determine theseseparately, as described in Refs. [15, 17], and add themin quadrature. For all variations described below, weassign the maximum deviation from the central valueas a systematic uncertainty, accounting for correlationsamong parameters. For resonance lineshapes and form-factors, we vary the parameters [18] by ± σ . Similarly,we vary the signal efficiency parameters for separatelyfor D and D events by ± σ , the ratios of particle-identification rates in data and simulation by ± σ , andthe background shapes by using simulation rather thandata sidebands. We include uncertainties from D – D misidentification, estimated from simulation, in the ex-perimental systematic uncertainty.To this point, we have described the investigation oftime-integrated CP asymmetry in neutral D meson de-cays using information from the DP distributions. Dif-ferences in the overall branching fractions for the D and D decays to π − π + π , K − K + π would also indi-cate time-integrated CP V . This information is not cap-tured by the differential comparisons of the DP struc- tures already described, and is complementary to them.To correct for any production asymmetry in D -flavorassignment, we weight each event by the relative effi-ciency for flavor assignment, as described in Ref. [7].Since there is an asymmetry [7] between the number ofevents reconstructed at forward and backward polar an-gles ( θ CM D ) of the D candidate CM momentum, we extractthe CP asymmetry value, a CP ≡ N D − N D N D + N D , in intervals of | cos θ CM D | . Here, N denotes the number of signal events.Any forward-backward asymmetry is canceled by aver-aging over symmetric intervals in cos θ CM D , as shown inEqs. 3–5 of Ref. [7]. In Fig. 3 we show the a CP for eventsin the D mass window used in the DP analysis. Weperform χ minimization to obtain the central values:[ − . ± .
41 (stat) ± .
17 (syst)] % for π − π + π and[1 . ± .
67 (stat) ± .
25 (syst)] % for K − K + π finalstates. The systematic uncertainties result from signalefficiency, particle-identification, background treatment,and D − D misidentification. As a consistency check, werepeat the analysis with a larger D mass window ( ± . σ )and find consistent results: [ − . ± .
34 (stat) ± . π − π + π and [0 . ± .
24 (stat) ± . K − K + π .In summary, our model-independent and model-dependent analyses show no evidence of CP V in the SCSdecays D → π − π + π and D → K − K + π . The interme-diate amplitudes include well-defined flavor states ( e.g., ρ ± π ∓ , K ∗± K ∓ ) and CP -odd eigenstates ( e.g., ρ π , φπ ). With the null results of Ref. [7, 8, 9, 10] for CP -even eigenstates D → K + K − and D → π + π − , we con-clude that any CP V in the SCS charm decays occurs ata rate which is not larger than a few percent. These re-sults are in accord with the SM predictions, and provideconstraints on some models beyond the SM [3].We are grateful for the excellent luminosity and ma-chine conditions provided by our PEP-II colleagues, andfor the substantial dedicated effort from the comput- | DCM q | cos C P ppp a -0.0100.01 (a) | DCM q | cos C P p KK a -0.0200.020.04 (b) FIG. 3: (color online). Phase-space-integrated CP asymmetry as a function of the cosine of the polar angle of the reconstructed D candidate CM momentum for (a) D → π − π + π and (b) D → K − K + π decays. The dashed lines represent the centralvalues, and the shaded regions the 1 σ intervals.TABLE I: Model-dependent CP asymmetry in the D → π − π + π Dalitz plots. The first and second errors are statistical andsystematic, respectively. For details on the Dalitz plot parametrization and the a r , φ r , and f r values, see Ref. [17]. As explainedin text, ∆ f r is closely related to ∆ a r and ∆ φ r State f r (%) ∆ a r (%) ∆ φ r ( ◦ ) ∆ f r (%) ρ + (770) 68 -3.2 ± ± ± ± ± ± ρ (770) 26 2.1 ± ± ± ± ± ± ρ − (770) 35 2.0 ± ± ± ± ± ± ρ + (1450) 0.1 2 ± ± ± ± ± ± ρ (1450) 0.3 13 ± ± ± ± ± ± ρ − (1450) 1.8 -3 ± ± ± ± ± ± ρ + (1700) 4 19 ± ± ± ± ± ± ρ (1700) 5 -31 ± ±
12 -7 ± ± ± ± ρ − (1700) 3 -3 ± ±
11 -3 ± ± ± ± f (980) 0.2 0.0 ± ± ± ± ± ± f (1370) 0.4 -0.3 ± ± ± ± ± ± f (1500) 0.4 0.4 ± ± ± ± ± ± f (1710) 0.3 -3 ± ± ± ±
11 0.0 ± ± f (1270) 1.3 8 ± ± ± ± ± ± σ (400) 0.8 -0.3 ± ± ± ± ± ± ± ± ± ± ± ± ing organizations that support B A B AR . The collaborat-ing institutions wish to thank SLAC for its support andkind hospitality. This work is supported by DOE andNSF (USA), NSERC (Canada), CEA and CNRS-IN2P3(France), BMBF and DFG (Germany), INFN (Italy),FOM (The Netherlands), NFR (Norway), MES (Russia),MEC (Spain), and STFC (United Kingdom). Individu-als have received support from the University ResearchCouncil (University of Cincinnati), the Marie Curie EIF(European Union), and the A. P. Sloan Foundation. ∗ Deceased † Now at Temple University, Philadelphia, Pennsylvania19122, USA ‡ Now at Tel Aviv University, Tel Aviv, 69978, Israel § Also with Universit`a di Perugia, Dipartimento di Fisica,Perugia, Italy ¶ Also with Universita’ di Sassari, Sassari, Italy[1] J.H. Christenson, J.W. Cronin, V.L. Fitch, andR. Turlay, Phys. Rev. Lett. , 138 (1964).[2] M.A. Shifman, “ITEP Lectures in Particle Physics”, TABLE II: Model-dependent CP asymmetry in the D → K − K + π Dalitz plots. The errors are statistical and systematic,respectively. We show the a (980) contribution, when it is included in place of the f (980), in square brackets. For details onthe Dalitz plot parametrization and the a r , φ r , and f r values, see Ref. [15]. We use Model-I of Ref. [15] to obtain central valuesand Model-II for study of systematic errors.State f r (%) ∆ a r (%) ∆ φ r ( ◦ ) ∆ f r (%) K ∗ (892) +
45 2 ± ± ± ± ± ± K ∗ (1410) + ± ±
37 1 ± ± ± ± K + π ( S ) 16 -130 ± ±
51 -9 ± ± ± ± φ (1020) 19 -1 ± ± ± ± ± ± f (980) 7 14 ± ± ± ± ± ± ˆ a (980) ˜ [6] [19 ± ±
6] [-7 ± ±
8] [0.6 ± ± f ′ (1525) 0.1 -38 ± ± ± ±
12 0.0 ± ± K ∗ (892) −
16 1 ± ± ± ± ± ± K ∗ (1410) − ± ±
68 -23 ± ± ± ± K − π ( S ) 3 8 ± ±
36 32 ± ±
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