with a model-independent Dalitz plot analysis of B

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First measurement of

₃

with a model-independent Dalitz plot analysis of B

! DK

, D ! K

⁰_S

^þ

decay

H. Aihara,⁵⁰K. Arinstein,²D. M. Asner,³⁸V. Aulchenko,²T. Aushev,¹⁵A. M. Bakich,⁴⁴K. Belous,¹⁴B. Bhuyan,¹⁰ M. Bischofberger,³⁰A. Bondar,²G. Bonvicini,⁵⁵A. Bozek,³³M. Bracˇko,^25,16T. E. Browder,⁷M.-C. Chang,⁵P. Chang,³² B. G. Cheon,⁶K. Chilikin,¹⁵R. Chistov,¹⁵K. Cho,¹⁹Y. Choi,⁴³J. Dalseno,^26,46Z. Dolezˇal,³A. Drutskoy,¹⁵S. Eidelman,²

D. Epifanov,²J. E. Fast,³⁸M. Feindt,¹⁸V. Gaur,⁴⁵N. Gabyshev,²A. Garmash,²Y. M. Goh,⁶B. Golob,^23,16J. Haba,⁸ H. Hayashii,³⁰Y. Horii,²⁹Y. Hoshi,⁴⁸W.-S. Hou,³²Y. B. Hsiung,³²H. J. Hyun,²¹T. Iijima,^29,28A. Ishikawa,⁴⁹R. Itoh,⁸ M. Iwabuchi,⁵⁷T. Julius,²⁷J. H. Kang,⁵⁷T. Kawasaki,³⁵C. Kiesling,²⁶H. J. Kim,²¹H. O. Kim,²¹J. B. Kim,²⁰J. H. Kim,¹⁹ K. T. Kim,²⁰M. J. Kim,²¹Y. J. Kim,¹⁹K. Kinoshita,⁴B. R. Ko,²⁰S. Koblitz,²⁶P. Kodysˇ,³S. Korpar,^25,16P. Krizˇan,^23,16 P. Krokovny,²B. Kronenbitter,¹⁸T. Kuhr,¹⁸T. Kumita,⁵²A. Kuzmin,²Y.-J. Kwon,⁵⁷S.-H. Lee,²⁰J. Li,⁴²Y. Li,⁵⁴J. Libby,¹¹ C. Liu,⁴¹Y. Liu,⁴Z. Q. Liu,¹²D. Liventsev,¹⁵R. Louvot,²²D. Matvienko,²K. Miyabayashi,³⁰H. Miyata,³⁵R. Mizuk,¹⁵

G. B. Mohanty,⁴⁵A. Moll,^26,46T. Mori,²⁸N. Muramatsu,⁴⁰Y. Nagasaka,⁹E. Nakano,³⁷M. Nakao,⁸Z. Natkaniec,³³ S. Nishida,⁸O. Nitoh,⁵³S. Ogawa,⁴⁷T. Ohshima,²⁸S. Okuno,¹⁷S. L. Olsen,^42,7G. Pakhlova,¹⁵C. W. Park,⁴³H. Park,²¹ H. K. Park,²¹K. S. Park,⁴³T. K. Pedlar,²⁴R. Pestotnik,¹⁶M. Petricˇ,¹⁶L. E. Piilonen,⁵⁴A. Poluektov,²K. Prothmann,^26,46

M. Ritter,²⁶M. Ro¨hrken,¹⁸M. Rozanska,³³S. Ryu,⁴²H. Sahoo,⁷Y. Sakai,⁸T. Sanuki,⁴⁹Y. Sato,⁴⁹O. Schneider,²² C. Schwanda,¹³A. J. Schwartz,⁴K. Senyo,⁵⁶O. Seon,²⁸M. E. Sevior,²⁷M. Shapkin,¹⁴T.-A. Shibata,⁵¹J.-G. Shiu,³²

B. Shwartz,²A. Sibidanov,⁴⁴F. Simon,^26,46J. B. Singh,³⁹P. Smerkol,¹⁶Y.-S. Sohn,⁵⁷A. Sokolov,¹⁴E. Solovieva,¹⁵ S. Stanicˇ,³⁶M. Staricˇ,¹⁶K. Sumisawa,⁸T. Sumiyoshi,⁵²G. Tatishvili,³⁸K. Trabelsi,⁸M. Uchida,⁵¹S. Uehara,⁸Y. Unno,⁶

S. Uno,⁸P. Urquijo,¹P. Vanhoefer,²⁶G. Varner,⁷K. E. Varvell,⁴⁴A. Vinokurova,²V. Vorobyev,²C. H. Wang,³¹ M.-Z. Wang,³²P. Wang,¹²Y. Watanabe,¹⁷K. M. Williams,⁵⁴E. Won,²⁰B. D. Yabsley,⁴⁴H. Yamamoto,⁴⁹J. Yamaoka,⁷

Y. Yamashita,³⁴C. Z. Yuan,¹²Z. P. Zhang,⁴¹V. Zhilich,²V. Zhulanov,²and A. Zupanc¹⁸ (Belle Collaboration)

1University of Bonn, Bonn

2Budker Institute of Nuclear Physics SB RAS and Novosibirsk State University, Novosibirsk 630090

3Faculty of Mathematics and Physics, Charles University, Prague

4University of Cincinnati, Cincinnati, Ohio 45221

5Department of Physics, Fu Jen Catholic University, Taipei

6Hanyang University, Seoul

7University of Hawaii, Honolulu, Hawaii 96822

8High Energy Accelerator Research Organization (KEK), Tsukuba

9Hiroshima Institute of Technology, Hiroshima

10Indian Institute of Technology Guwahati, Guwahati

11Indian Institute of Technology Madras, Madras

12Institute of High Energy Physics, Chinese Academy of Sciences, Beijing

13Institute of High Energy Physics, Vienna

14Institute of High Energy Physics, Protvino

15Institute for Theoretical and Experimental Physics, Moscow

16J. Stefan Institute, Ljubljana

17Kanagawa University, Yokohama

18Institut fu¨r Experimentelle Kernphysik, Karlsruher Institut fu¨r Technologie, Karlsruhe

19Korea Institute of Science and Technology Information, Daejeon

20Korea University, Seoul

21Kyungpook National University, Taegu

22E´ cole Polytechnique Fe´de´rale de Lausanne (EPFL), Lausanne

23Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana

24Luther College, Decorah, Iowa 52101

25University of Maribor, Maribor

26Max-Planck-Institut fu¨r Physik, Mu¨nchen

27University of Melbourne, School of Physics, Victoria 3010

28Graduate School of Science, Nagoya University, Nagoya

29Kobayashi-Maskawa Institute, Nagoya University, Nagoya

30Nara Women’s University, Nara

31National United University, Miao Li PHYSICAL REVIEW D85,112014 (2012)

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32Department of Physics, National Taiwan University, Taipei

33H. Niewodniczanski Institute of Nuclear Physics, Krakow

34Nippon Dental University, Niigata

35Niigata University, Niigata

36University of Nova Gorica, Nova Gorica

37Osaka City University, Osaka

38Pacific Northwest National Laboratory, Richland, Washington 99352

39Panjab University, Chandigarh

40Research Center for Nuclear Physics, Osaka University, Osaka

41University of Science and Technology of China, Hefei

42Seoul National University, Seoul

43Sungkyunkwan University, Suwon

44School of Physics, University of Sydney, NSW 2006

45Tata Institute of Fundamental Research, Mumbai

46Excellence Cluster Universe, Technische Universita¨t Mu¨nchen, Garching

47Toho University, Funabashi

48Tohoku Gakuin University, Tagajo

49Tohoku University, Sendai

50Department of Physics, University of Tokyo, Tokyo

51Tokyo Institute of Technology, Tokyo

52Tokyo Metropolitan University, Tokyo

53Tokyo University of Agriculture and Technology, Tokyo

54CNP, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061

55Wayne State University, Detroit, Michigan 48202

56Yamagata University, Yamagata

57Yonsei University, Seoul

(Received 28 April 2012; published 26 June 2012)

We present the first measurement of the angle3of the unitarity triangle using a model-independent Dalitz plot analysis ofB!DK,D!K⁰_S^þdecays. The method uses, as input, measurements of the strong phase of theD!K_S⁰^þamplitude from the CLEO Collaboration. The result is based on the full data set of77210⁶BBpairs collected by the Belle experiment at theð4SÞresonance. We obtain 3¼ ð77:3^þ15:1_14:94:14:3Þand the suppressed amplitude ratiorB¼0:1450:0300:0100:011.

Here the first error is statistical, the second is the experimental systematic uncertainty, and the third is the error due to the precision of the strong-phase parameters obtained by CLEO.

DOI:10.1103/PhysRevD.85.112014 PACS numbers: 12.15.Hh, 13.25.Hw, 14.40.Nd

I. INTRODUCTION

The angle3(also denoted as) is one of the least well- constrained parameters of the Unitarity Triangle. The measurement that currently dominates sensitivity to 3 uses B!DKdecays with the neutralDmeson decaying to a three-body final state such asK_S⁰^þ [1,2]. The weak phase3appears in the interference betweenb!cus and b!ucs transitions. The value of 3 is determined by exploiting differences between theK_S⁰^þ Dalitz plots forDmesons fromB^þ andB decay. Theoretical uncer- tainties in the3 determination inB!DKdecays are expected to be negligible [3], and the main difficulty in its measurement is the very low probability of the decays that are involved. However, the method based on Dalitz plot analysis requires the knowledge of the amplitude of the D⁰ !K_S⁰^þ decay, including its complex phase. The amplitude can be obtained from a model that involves isobar andK-matrix [4] descriptions of the decay dynam- ics, and thus results in a model uncertainty for the 3

measurement. In the latest model-dependent Dalitz plot

analyses performed byBABARand Belle, this uncertainty ranges from 3to 9[5–10].

A method to eliminate the model uncertainty using a binned Dalitz plot analysis has been proposed by Giri et al.

[1]. Information about the strong phase in the D⁰! K⁰_S^þ decay is obtained from the decays of quantum- correlated D⁰ pairs produced in the cð3770Þ !D⁰D⁰ process. As a result, the model uncertainty is replaced by a statistical error related to the precision of the strong- phase parameters. This method has been further developed in Refs. [11,12], where its experimental feasibility has been shown along with a proposed analysis procedure to optimally use the available B decays and correlated D⁰ pairs. In this paper, we report the first measurement of3

using a model-independent Dalitz plot analysis of theD! K⁰_S^þ decay from the modeB !DK, based on a 711 fb¹ data sample (corresponding to 77210⁶ BB pairs) collected by the Belle detector at the KEKB asymmetric-energy e^þe collider. This analysis uses the recent measurement of the strong phase inD⁰ !K⁰_S^þ

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and D⁰ !K_S⁰K^þK decays performed by the CLEO Collaboration [13,14].

II. THE MODEL-INDEPENDENT DALITZ PLOT ANALYSIS TECHNIQUE

The amplitude of theB^þ!DK^þ, D!K⁰_S^þ decay is a superposition of the B^þ!D⁰K^þ and B^þ ! D⁰K^þamplitudes

ABðm²_þ; m²Þ ¼AþrBe^ið^B^þ³^ÞA; (1) where m²_þ and m² are the Dalitz plot variables—the squared invariant masses of K⁰_S^þ and K⁰_S combinations, respectively,A ¼Aðm ²_þ; m²Þis the amplitude of the D⁰ !K_S⁰^þ decay, A¼Aðm²_þ; m²Þ is the amplitude of theD⁰ !K_S⁰^þdecay,rBis the ratio of the absolute values of theB^þ!D⁰K^þ andB^þ!D⁰K^þ amplitudes, andBis the strong-phase difference between them. In the case of CP conservation in the D decay Aðm²_þ; m²Þ ¼ Aðm ²; m²_þÞ. The Dalitz plot density of theD decay from B^þ!DK^þis given by

PB¼ jA_Bj² ¼ jAþrBe^ið^B^þ³^ÞAj²

¼Pþr²_BPþ2 ffiffiffiffiffiffiffi PP

p ðx_þCþyþSÞ; (2)

wherePðm²_þ; m²Þ ¼ jAj²,Pðm ²_þ; m²Þ ¼ jAj ²; while xþ¼rBcosðBþ3Þ; yþ¼rBsinðBþ3Þ; (3) and the functions C¼Cðm²_þ; m²Þ and S¼Sðm²_þ; m²Þ are the cosine and sine of the strong-phase difference Dðm²_þ; m²Þ ¼arg AargA between the D⁰ ! K_S⁰^þandD⁰!K⁰_S^þamplitudes.¹The equations for the charge-conjugate mode B!DK are obtained with the substitution 3 ! 3 and A$A; the corresponding parameters that depend on theB decay amplitude are

x¼r_Bcosð_B3Þ; y¼r_Bsinð_B3Þ: (4) Using bothBcharges, one can obtain3andBseparately.

Up to this point, the description of the model-dependent and model-independent techniques is the same. The model-dependent analysis deals directly with the Dalitz plot density, and the functionsCandSare obtained from model assumptions in the fit to theD⁰ !K_S⁰^þampli- tude. In the model-independent approach, the Dalitz plot is divided into 2N bins symmetric under the exchange m²$m²_þ. The bin index i ranges from N to N (excluding 0); the exchange m²_þ$m² corresponds to the exchange i$ i. The expected number of events in biniof the Dalitz plot of theDmeson fromB !DKis

N_i¼hB½Kiþr²_BKiþ2 ffiffiffiffiffiffiffiffiffiffiffiffiffi KiKi

p ðxciysiÞ; (5)

whereh_Bis a normalization constant andK_iis the number of events in theith bin of theK⁰_S^þDalitz plot of theD meson in a flavor eigenstate. A sample of flavor-taggedD⁰ mesons is obtained by reconstructingD!Ddecays (note that charge conjugation is assumed throughout this paper unless otherwise stated). The termsciandsiinclude information about the functionsCandSaveraged over the bin region:

c_i¼ R

D_ijAjjAjCdD ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R

DijAj²dDR

DijAj ²dD

q : (6)

HereD represents the Dalitz plot phase space andDi is the bin region over which the integration is performed. The termssiare defined similarly withCsubstituted byS. The absence of CPviolation in theDdecay impliesc_i¼c_i ands_i¼ s_i.

The values of the c_i and s_i terms are measured in the quantum correlations of D pairs by charm-factory experiments operating at the threshold ofDD pair production [13,14]. The measurement involves studies of the four- dimensional (4D) density of two correlatedD!K⁰_S^þ Dalitz plots, as well as decays of aDmeson tagged in aCP eigenstate decaying toK⁰_S^þ. The wave function of the two mesons is antisymmetric, thus the four-dimensional density of two correlatedD!K⁰_S^þDalitz plots is

jAcorrðm²_þ; m²; m⁰²_þ; m⁰²Þj²

¼ jAA⁰AA ⁰j²

¼PP⁰þPP ⁰2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PP⁰PP ⁰

p ðCC⁰þSS⁰Þ; (7)

where the primed and unprimed quantities correspond to the two decaying Dmesons. Similarly, the density of the decay D!K⁰_S^þ, where the D meson is in a CP eigenstate, is

jA_CPðm²_þ; m²Þj² ¼ jAAj² ¼PþP2 ffiffiffiffiffiffiffi PP

p C: (8)

CLEO uses these relations to obtainciandsivalues. Once they are measured, the system of equations (5) contains only three free parameters (x,y, andhB) for eachBcharge, and can be solved using a maximum likelihood method to extract the value of3.

We have neglected charm-mixing effects in D decays from both the B!DK process and in quantum- correlated DD production. It has been shown [15] that although the charm-mixing correction is of first order in the mixing parameters x_D,y_D, it is numerically small (of the order 0.2 for x_D, y_D0:01) and can be neglected at the current level of precision. Future precision measurements of3 can account for charm mixing andCPviola- tion (both in mixing and decay) using the measurement of the corresponding parameters.

In principle, the set of relations defined by Eq. (5) can be solved without external constraints on ci and si for N 2. However, due to the small value of rB, there is

1This paper follows the convention for strong phases in D decay amplitudes introduced in Ref. [12].

FIRST MEASUREMENT OF3WITH A MODEL-. . . PHYSICAL REVIEW D85,112014 (2012)

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very little sensitivity to thec_iands_iparameters inB ! DKdecays, which results in a reduction in the precision on3 that can be obtained [11].

III. CLEO INPUT

The procedure for a binned Dalitz plot analysis should give the correct results for any binning. However, the statistical accuracy depends strongly on the amplitude behavior across the bins. Large variations of the amplitude within a bin result in loss of coherence in the interference term. This effect becomes especially significant with lim- ited statistics when a small number of bins must be used to ensure a stable fit. Greater statistical precision is obtained for the binning in which the phase difference between the D⁰ andD⁰ amplitudes varies as little as possible within a bin [12]. For optimal precision, one also has to take the variations of the absolute value of the amplitude into account, along with contributions from background events.

The procedure to optimize the binning for the maximal statistical precision of3 has been proposed in Ref. [12]

and generalized to the case with background in Ref. [14]. It has been shown that as few as 16 bins are enough to reach a statistical precision that is only 10–20% worse than in the unbinned case.

The optimization of binning sensitivity uses the amplitude of the D!K⁰_S^þ decay. It should be noted, however, that although the choice of binning is model- dependent, a poor choice of model results only in a loss of precision, not bias, of the measured parameters [12].

CLEO measured ci and si parameters for four different binnings withN ¼8:

(1) Bins equally distributed in the phase differenceD

between theD⁰ andD⁰ decay amplitudes, with the amplitude from theBABARmeasurement [6].

(2) Same as option 1, but with the amplitude from the Belle analysis [10].

(3) Optimized for statistical precision according to the procedure from [12] (see Fig.1). The effect of the

background inBdata is not taken into account in the optimization. The amplitude is taken from the BABARmeasurement [6].

(4) Same as option 3, but optimized for an analysis with high background inBdata (e.g., at LHCb).

Our analysis uses the optimal binning shown in Fig.1 (option 3) as the baseline since it offers better statistical accuracy. In addition, we use the equal phase difference binning (D binning, option 1) as a cross-check.

The results of the CLEO measurement ofc_i ands_i for the optimal binning are presented in Table I. The same results in graphical form are shown in Fig.2. The values of c_i and s_i calculated from the Belle model [10] are compared to the measurements and are found to be in reasonable agreement with ²¼18:6 for the number of

4)

2/c (GeV

−2

m

0.5 1 1.5 2 2.5 3

)4 /c2 (GeV+2m

0.5 1 1.5 2 2.5 3

|Bin index i|

1 2 3 4 5 6 7 8

i>0 i<0

FIG. 1 (color online). Optimal binning of theD!K⁰_S^þ Dalitz plot. The color scale indicated corresponds to the absolute value of the bin index,jij.

TABLE I. Values ofciandsifor the optimal binning measured by CLEO [14], and calculated from the Belle D!K_S⁰^þ amplitude model.

CLEO measurement Belle model

c1 0:0090:0880:094 0:039 c2 þ0:9000:1060:082 þ0:771 c3 þ0:2920:1680:139 þ0:242 c4 0:8900:0410:044 0:867 c5 0:2080:0850:080 0:246 c6 þ0:2580:1550:108 þ0:023 c7 þ0:8690:0340:033 þ0:851 c8 þ0:7980:0700:047 þ0:662 s1 0:4380:1840:045 0:706 s2 0:4900:2950:261 þ0:124 s3 1:2430:3410:123 0:687 s4 0:1190:1410:038 0:108 s5 þ0:8530:1230:035 þ0:851 s6 þ0:9840:3570:165 þ0:930 s7 0:0410:1320:034 þ0:169 s8 0:1070:2400:080 0:596

Ci

iS

-1.5 -1 -0.5 0 0.5 1 1.5

1 1

2

3 2

3

4 4

5 5

6 6

7

8 8

CLEO Belle

model

-1 0 1

FIG. 2 (color online). Comparison of phase terms ðc_i; siÞ for the optimal binning measured by CLEO, and calculated from the BelleD!K⁰_S^þ amplitude model.

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degrees of freedomndf¼16(the correspondingp-value is p¼29%).

As is apparent from Fig.2, the chosen binning contains bins where the strong-phase difference betweenD⁰andD⁰ amplitudes is close to zero (bins with jij ¼2, 7, 8) and 180(bin withjij ¼4) which provide sensitivity tox, as well as bins with the strong-phase difference close to 90 and 270(bins withjij ¼1, 3, 5, 6), more sensitive toy. This ensures that the method is sensitive to 3 for any combination of3 andB values.

IV. ANALYSIS PROCEDURE

Equation (5) is the key relation used in the analysis, but it only holds if there is no background, a uniform Dalitz plot acceptance and no cross feed between bins. (Cross feed is due to invariant-mass resolution and radiative cor- rections.) In this section we outline the procedures that account for these experimental effects.

A. Efficiency profile

We note that the Eqs. (2), (7), and (8) do not change after the transformation P!P when the efficiency profile ðm²_þ; m²Þ is symmetric: ðm²_þ; m²Þ ¼ðm²; m²_þÞ. This implies that if the efficiency profile is the same in all of the three modes involved in the measurement (flavorD, correlated cð3770Þ !DD, and D from B!DK), the result will be unbiased even if no efficiency correction is applied.

The effect of nonuniform efficiency over the Dalitz plot cancels out when using a flavor-tagged D sample with kinematic properties that are similar to the sample for the signalB decay. This approach allows for the removal of systematic error associated with the possible inaccuracy in the description of the detector acceptance in the Monte Carlo (MC) simulation. The center-of-mass (CM) D momentum distribution for B!DK decays is practi- cally uniform in the narrow range 2:10 GeV=c < pD<

2:45 GeV=c. We assume that the efficiency profile depends mostly on the D momentum and take the flavor-tagged sample with an average momentum of pD¼2:3 GeV=c (we use a wider range ofDmomenta than inB!DKto increase the statistics). The assumption that the efficiency profile depends only on the Dmomentum is tested using MC simulation, and the remaining difference is treated as a systematic uncertainty.

While calculatingciandsi, CLEO applies an efficiency correction, therefore the values reported in their analysis correspond to a flat efficiency profile. To use theciandsi

values in the3 analysis, they have to be corrected for the Belle efficiency profile. This correction cannot be performed in a completely model-independent way, since the correction terms include the phase variation inside the bin. Fortunately, the calculations using the BelleD! K_S⁰^þ model show that this correction is negligible even for very large nonuniformity of the efficiency profile.

The difference between the uncorrectedciandsiterms and

those corrected for the efficiency, calculated using the efficiency profile parameterization used in the 605 fb¹ analysis [10], does not exceed 0.01, which is negligible compared to the statistical error.

B. Momentum resolution

Momentum resolution leads to migration of events between the bins. In the binned approach, this effect can be corrected in a nonparametric way. The migration can be described by a linear transformation of the number of events in bins:

N⁰_i¼X

_ikN_k; (9) whereNkis the number of events that binkwould contain without the cross feed, andN_i⁰is the reconstructed number of events in bini. The cross-feed matrixikis nearly a unit matrix:ik1forik. It is obtained from a signal MC simulation generated with the amplitude model reported in Ref. [10]. In the case of aD!K_S⁰^þdecay from aB, the cross feed depends on the parametersxandy. However, this is a minor correction to an already small effect due to cross feed; therefore it is neglected.

Migration of events between the bins also occurs due to final state radiation (FSR). Theciandsiterms in the CLEO measurement are not corrected for FSR; we therefore do not simulate FSR to obtain the cross-feed matrix to mini- mize the bias due to this effect. Comparison of the cross feed with and without FSR shows that this effect is negligible.

C. Fit procedure

The background contribution has to be accounted for in the calculation of the values Ni and Ki. Statistically the most effective way of calculating the number of signal events (especially in the case ofNi, where the statistics is a limiting factor) is to perform, in each biniof the Dalitz plot, an unbinned fit in the variables used to distinguish the signal from the background.

Two different approaches are used in this analysis to obtain theCPviolating parameters from the data: separate fits in bins, and a combined fit.

In the first, we fit the data distribution in each bin separately, with the number of events for signal and backgrounds as free parameters. Once the numbers of events in bins N_i are found, we use them in Eq. (5) to obtain the parametersðx; yÞ. This is accomplished by minimizing a negative logarithmic likelihood of the form

2logLðx;yÞ ¼ 2X

i

logpðhNiiðx;yÞ;Ni;N_iÞ; (10) where hN_iiðx; yÞis the expected number of events in the bin i obtained from Eq. (5). Here, N_i and _N_i are the observed number of events in data and the uncertainty on Ni, respectively. If the probability density function

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(PDF) p is Gaussian, this procedure is equivalent to a² fit; however, the assumption of the Gaussian distribution may introduce a bias in the case of low statistics in certain bins.

The procedure described above does not make any assumptions on the Dalitz distribution of the background events, since the fits in each bin are independent. Thus there is no associated systematic uncertainty. However, in the case of a small number of events and many background components this can be a limiting factor. Our second approach is to use the combined fit with a common likelihood for all bins. The relative numbers of background events in bins in such a fit can be constrained externally from MC and control samples. In addition, for the case of the combined fit, the two-step procedure of first extracting the numbers of signal events, and then using them to obtain ðx; yÞis not needed—the expected numbers of eventshN_ii as functions of ðx; yÞ can be included in the likelihood.

Thus the variables ðx; yÞ become free parameters of the combined likelihood fit, and the assumption that the number of signal events has a Gaussian distribution is not needed.

Both approaches are tested with the control samples and MC simulation. We choose the combined fit approach as the baseline, but the procedure with separate fits in bins is also used: it allows a clear demonstration of theCPasym- metry in each bin.

V. EVENT SELECTION

We use a data sample of77210⁶ BB pairs collected by the Belle detector. The decaysB!DK andB ! Dare selected for the analysis. The neutralDmeson is reconstructed in theK⁰_S^þ final state in all cases. We also selectD!Ddecays produced via thee^þe ! cccontinuum process as a high-statistics sample to deter- mine the K_i parameters related to the flavor-tagged D⁰ !K_S⁰^þ decay.

The Belle detector is described in detail elsewhere [16,17]. It is a large-solid-angle magnetic spectrometer consisting of a silicon vertex detector, a 50-layer central drift chamber for charged particle tracking and specific ionization measurement (dE=dx), an array of aerogel threshold Cherenkov counters, time-of-flight scintillation counters, and an array of CsI(Tl) crystals for electromag- netic calorimetry located inside a superconducting sole- noid coil that provides a 1.5 T magnetic field. An iron flux return located outside the coil is instrumented to detectK_L mesons and identify muons.

Charged tracks are required to satisfy criteria based on the quality of the track fit and the distance from the interaction point of the beams (IP). We require each track to have a transverse momentum greater than100 MeV=c, and the impact parameter relative to the IP to be less than 2 mm in the transverse and less than 10 mm in longitudinal projections. Separation of kaons and pions is accomplished

by combining the responses of the aerogel threshold Cherenkov counters and the time-of-flight scintillation counters with the dE=dx measurement from the central drift chamber. Neutral kaons are reconstructed from pairs of oppositely charged tracks with an invariant mass M within7 MeV=c² of the nominalK⁰_Smass, flight distance from the IP in the plane transverse to the beam axis greater than 0.1 mm, and the cosine of the angle between the projections of K_S⁰ flight direction and its momentum greater than 0.95.

The flavor of the neutralDmesons used forK_idetermi- nation is tagged by the charge of the slow pion in the decay D!D. The slow pion track is required to originate from theD⁰ decay vertex to improve the momentum and angular resolution. The selection of signal candidates is based on two variables, the invariant mass of the neutralD candidates MD¼M_K⁰

S^þ and the difference of the invariant masses of the D and the neutral D candidates M¼M_ðK⁰

S^þÞDM_K⁰

S^þ. We retain the events sat- isfying the following criteria: 1800 MeV=c²< M_D<

1920 MeV=c² and M <150 MeV=c². We also require the momentum of theD⁰candidate in the CM framep_Dto be in the range 1:8 GeV=c < p_D<2:8 GeV=cto reduce the effect of the efficiency profile on the3 measurement (see Sec.IVA). About 15% of selected events contain more than one D candidate that satisfies the requirements above; in this case we keep only one randomly selected candidate.

Selection of B!DK and B !D samples is based on the CM-energy difference E¼P

EiEbeam

and the beam-constrained B meson mass Mbc ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

E²_beam ðP

~ p_iÞ² q

, where Ebeam is the CM beam energy, andEiandp~iare the CM energies and momenta of theB candidate decay products. We select events with Mbc>

5:2 GeV=c²andjEj<0:18 GeVfor further analysis. We also impose a requirement on the invariant mass of the neutralDcandidatejM_K⁰

S^þM_D⁰j<11 MeV=c². Further separation of the background frome^þe !qq (q¼u,d,s, c) continuum events is done by calculating two variables that characterize the event shape. One is the cosine of the thrust angle cos _thr, where _thr is the angle between the thrust axis of the Bcandidate daughters and that of the rest of the event, calculated in the CM frame.

The other is a Fisher discriminant F composed of 11 parameters [18]: the production angle of the Bcandidate, the angle of theBthrust axis relative to the beam axis, and nine parameters representing the momentum flow in the event relative to theBthrust axis in the CM frame. We use the E, Mbc, cos _thr, and F variables in the maximum likelihood fit.

In both flavor D⁰ and B!DK (B!D) samples, the momenta of the tracks forming aD⁰candidate are constrained to give the nominalD⁰ mass in the calculation of the Dalitz plot variables.

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VI. FLAVOR-TAGGED SAMPLE D !D,D!K⁰_S^þ

The number of eventsKi in bin i of the flavor-tagged D!K⁰_S^þdecay is obtained from a two-dimensional unbinned fit to the distribution ofM_D andMvariables.

The fits in each Dalitz plot bin are performed indepen- dently. The fit uses a signal PDF and two background components: purely random combinatorial background and background with a real D⁰ and random slow pion track. The signal distribution is a product of the PDFs for MD(triple Gaussian) andM(sum of bifurcated Student’s t distribution and bifurcated Gaussian distribution). The combinatorial background is parameterized by a linear function inMDand by a function with a kinematic threshold at the^þmass inM:

pcombðMÞ ¼ ffiffiffi py

ð1þAy½1þBðMDm_D⁰ÞÞe^yC (11) where y¼Mm^þ, m^þ and m_D⁰ are the nominal masses of ^þ and D⁰, respectively, andA, B, and Care free parameters. A small correlation between theMD and M distributions is introduced that is controlled by the parameterB. The random slow pion background is parame-

terized as a product of the signal MD distribution and combinatorialMbackground shape.

The parameters of the signal and background distributions are obtained from the fit to data. The parameters of the signal PDF are constrained to be the same in all bins.

The free parameters in each bin are the number of signal eventsKi, the parameters of the background distribution, and fractions of the background components.

The fit results from the flavor-tagged D sample integrated over the whole Dalitz plot are shown in Fig.3. The number of signal events calculated from the integral of the signal distribution is 426 938825, the background fraction in the signal region jM_Dm_D⁰j<11 MeV=c², 144:5 MeV=c²<M <146:5 MeV=c² is 10:10:1%.

The signal yield in bins is shown in TableII.

VII. SELECTION OFB!D ANDB!DK SAMPLES

The decays B!DK andB!D have similar topology and background sources and their selection is performed in a similar way. The modeB!Dhas an order of magnitude larger branching ratio and a smaller amplitude ratior_B0:01due to the ratio of weak coeffi- cientsjVubV_cd j=jVcbV_ud j 0:02and the color suppression factor. This results in the smallCPviolation in this mode, therefore it is used as a control sample to test the procedures of the background extraction and Dalitz plot fit. In addition, signal resolutions inEandMbcand the Dalitz plot struc- ture of some background components are constrained from the control sample and used in the signal fit.

The number of signal events is obtained by fitting the 4D distribution of variablesMbc,E,cos _thr andF. The fits to the B!D and B!DK samples use the following three background components in addition to the signal PDF:

(i) Combinatorial background from the process e^þe!qq, whereq¼ ðu; d; s; cÞ.

(ii) RandomBB background, in which the tracks forming theB!Dcandidate come from decays of

2) (GeV/c MD

Events / (0.001)

0 5000 10000 15000 20000 25000 30000

Signal True D0

Fake D0

2) M (GeV/c

∆

Events / (8e-05)

0 5000 10000 15000 20000 25000 30000 35000 40000

1.8 1.85 1.9 0.14 0.142 0.144 0.146 0.148 0.15

FIG. 3 (color online). Projections of the flavor-tagged D!D, D!K⁰_S^þ data with 1:8 GeV=c < pD<2:8 GeV=c. (a) MD distribution for 144:5 MeV=c²<M <146:5 MeV=c². (b) M distribution for 1854 MeV=c²< MD<1876 MeV=c². Histograms show the fitted signal and background contributions, points with the error bars are the data. The fullD!K_S⁰^þDalitz plot is used.

TABLE II. Signal yields in Dalitz plot bins for the flavor- taggedD!D,D!K⁰_S^þsample with1:8 GeV=c <

pD<2:8 GeV=c.

Bini Ki Ki

1 43 261255 8770124

2 58 005268 182763

3 62 808274 160158

4 44 513253 26 482202

5 21 886177 13 146143

6 28 876197 176568

7 48 001265 22 476196

8 9279125 26 450181

Total 426 938825

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bothBmesons in the event. The number of possible Bdecay combinations that contribute to this background is large, therefore both the Dalitz distribution andðMbc;EÞdistribution are quite smooth.

(iii) PeakingBB background, in which all tracks forming theB!Dcandidate come from the same Bmeson. This kind of background is dominated by B!Ddecays reconstructed without theor from theD decay.

In addition, theB !DK fit includes a fourth component that modelsB!Ddecays in which the pion is misidentified as a kaon.

The PDF for the signal parameterization (as well as for each of the background components) is a product of the ðMbc;EÞandðcos _thr;FÞPDFs. TheðMbc;EÞPDF is a 2D double-Gaussian function, which has a correlation betweenMbcandE. The double-Gaussian function mod- els both the core and tails of the distribution. The ðcos _thr;FÞ distribution is parameterized by the sum of two functions (with different coefficients) of the form

pðx;FÞ ¼expðC1xþC2x²þC3x³Þ

GðF; F0ðxÞ; FLðxÞ; FRðxÞÞ; (12) where x¼cos _thr, GðF; F; L; RÞ is the bifurcated Gaussian distribution with the meanF and the widths _L

and_R, and functionsF0,_FLand_FRare polynomials that contain only even powers ofx. The parameters of the signal PDF are obtained from the signal MC simulation. However, to account for the possible imperfection of the simulation, we allow all the width parameters to scale by a common factor, which is obtained from theB!Dsample.

The combinatorial background from continuum e^þe!qq production is obtained from the experimental sample collected at a CM energy below the ð4SÞ resonance (off-resonance data). The parameterization in varia- blesðcos _thr;FÞfollows Eq. (12). The parameterization in ðMbc;EÞis the product of an exponential distribution in E and the empirical shape proposed by the ARGUS Collaboration [19] inMbc:

pcombðMbc;EÞ ¼expðEÞMbc ffiffiffi py

expðcyÞ; (13) wherey¼1Mbc=Ebeam,Ebeamis the CM beam energy, andandcare empirical parameters.

The parameters for random and peaking BB backgrounds are obtained from a generic MC sample.

Generator information is used to distinguish between the two: the latter contains only the events in which the candidate is formed of tracks coming from bothBmesons. The ðMbc;EÞdistributions for each of these backgrounds are parameterized by the sum of three components:

(i) the product of an exponential (inE) and Argus (in Mbc) functions, as for continuum background (as

2) (GeV/c Mbc

Events / ( 0.002 GeV/c2) 0 500 1000 1500 2000 2500 3000 3500

Signal BB peaking BB random u,d,s,c

∆E (GeV)

Events / (0.006 GeV)

0 500 1000 1500 2000 2500

θthr

cos

Events / (0.02)

0 200 400 600 800 1000 1200

Events / (0.2)

0 200 400 600 800 1000 1200 1400 1600

F

5.2 5.22 5.24 5.26 5.28 5.3 -0.1 0 0.1

0 0.2 0.4 0.6 0.8 1 -4 -2 0 2 4

FIG. 4 (color online). Projections of theB!Ddata. (a)Mbcdistribution withjEj<30 MeVandcos _thr<0:8requirements.

(b)Edistribution withMbc>5270 MeV=c²andcos _thr<0:8requirements. (c)cos _thrand (d)Fdistributions withjEj<30 MeV andMbc>5270 MeV=c²requirements. Histograms show the fitted signal and background contributions, points with error bars are the data. The entireD!K_S⁰^þDalitz plot is used.

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expected, this component dominates the randomBB background);

(ii) the product of an exponential in theEand bifurcated Gaussian distribution inMbc, where the mean of the Gaussian distribution is linear as a function of E; and

(iii) a two-dimensional Gaussian distribution inEand Mbc, which includes a correlation and is asymmetric inMbc. This component is small compared to the random BB contribution, but dominates the peakingBB background, which mostly consists of partially reconstructedBdecays.

The peaking background coming fromB^þB andB⁰B⁰ decays is treated separately inðMbc;EÞvariables, while a commonðcos _thr;FÞdistribution is used. In the case of the B!DKfit,B!Devents with the pion misidentified as a kaon are treated as a separate background category. The distributions of Mbc, E and cos _thr, F variables are parameterized in the same way as for the signal events and are obtained from MC simulation.

The Dalitz plot distributions of the background components are discussed in the next section. Note that the Dalitz distribution is described by the relative number of events in each bin. The numbers of events in bins can be free parameters in the fit, thus there will be no uncertainty due to the modeling of the background distribution over the Dalitz plot in such an approach. This procedure is

justified for background that is either well separated from the signal (such as peakingBB background in the case of B!D), or is constrained by a much larger number of events than the signal (such as the continuum background).

The results of the fit to B !D and B!DK data with the full Dalitz plot taken are shown in Figs.4and5, respectively. We obtain a total of 19 106 147signalB !Devents and117643signalB! DK events—55% more than in the 605 fb¹ model- dependent analysis [10]. The improvement partially comes from the larger integrated luminosity of the sample, and partially from the larger selection efficiency due to im- proved track reconstruction.

VIII. DATA FITS IN BINS

The data fits in bins for both B !D and B! DK samples are performed with two different procedures: separate fits for the number of events in bins and the combined fit with the free parameters ðx; yÞ as discussed in Sec. IV C. The combined fit is used to obtain the final values forðx; yÞ, while the separate fits provide a crosscheck of the fit procedure and a way to visualize the extent ofCPviolation within the sample. A study with MC pseudoexperiments is performed to check that the observed difference in the fit results between the two approaches agrees with expectation.

2) (GeV/c Mbc

Events / (0.002 GeV/c2) 0 50 100 150 200 250

300 _Signal

π

→D B BB peaking BB random u,d,s,c

∆E (GeV)

Events / (0.006 GeV)

0 20 40 60 80 100 120 140 160 180 200

θthr

cos

Events / (0.02)

0 100 200 300 400 500

F

Events / (0.2)

0 20 40 60 80 100 120 140 160 180 200 220 240

5.2 5.22 5.24 5.26 5.28 5.3 -0.1 0 0.1

0 0.2 0.4 0.6 0.8 1 -4 -2 0 2 4

FIG. 5 (color online). Projections of theB!DKdata. (a)Mbcdistribution withjEj<30 MeVandcos _thr<0:8requirements.

(b) E distribution with Mbc>5270 MeV=c² and cos _thr<0:8 requirements. (c) cos _thr and (d) F distributions with jEj<

30 MeVandMbc>5270 MeV=c²requirements. Histograms show the fitted signal and background contributions, points with error bars are the data. The entireD!K_S⁰^þ Dalitz plot is used.

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In the case of separate fits in bins, we first perform the fit to all events in the Dalitz plot. The fit uses background shapes fixed to those obtained from fits to the generic MC samples of continuum and BB decays. The signal shape parameters are fixed to those obtained from a fit to the signal MC sample except for the mean value and width scale factors ofEandMbcPDFs. As a next step, we fit the

4D ðMbc;E;cos _thr;FÞ distributions in each bin separately, with the signal peak positions and width scale factors fixed to the values obtained from the fit to all events.

The free parameters of each fit are the number of signal events, and the number of events in each background category.

The numbers of signal events in bins for theB !D sample extracted from the fits are given in TableIII. These numbers are used in the fit to extract ðx; yÞusing Eq. (5) after the cross feed and efficiency correction for both Ni

and K_i. Figure 6 illustrates the results of this fit. The numbers of signal events in each bin for B^þ andB are shown in Fig.6(a)together with the numbers of events in the flavor-tagged D⁰ sample (appropriately scaled). The difference in the number of signal events shown in Fig.6 (b) does not reveal CP violation. Figures 6(c) and 6(d) show the difference between the numbers of signal events for B^þ [B] data and scaled flavor-tagged D⁰ sample, both for the data and after the ðx; yÞ fit. The ²=ndf is reasonable for both the ðx; yÞfit and the comparison with the flavor-specificCPconserving amplitude.

UnlikeB!D, theB!DKsample has signifi- cantly different signal yields in bins ofB^þandBdata [see Fig. 7(b) and Table IV). The probability to obtain this difference as a result of a statistical fluctuation is 0.42%.

This value can be taken as the model-independent measure of the CP violation significance. The significance of 3

being nonzero is in general smaller since30results in a specific pattern of charge asymmetry. The fit of the signal TABLE III. Signal yields in Dalitz plot bins for the B!

D,D!K⁰_S^þ sample with the optimal binning.

Bini N_i N_i^þ

8 564:225:3 587:025:7 7 462:323:8 462:823:9 6 47:97:7 39:27:2 5 314:119:0 286:218:2 4 592:626:5 645:727:8 3 22:26:2 27:26:3 2 42:77:6 54:08:7 1 190:815:4 210:816:3

1 959:232:6 980:233:1

2 1288:737:0 1295:937:1

3 1395:838:4 1352:237:9

4 1045:534:7 1065:134:9

5 479:323:3 532:224:5

6 623:726:0 663:526:7

7 1081:035:3 1049:234:8

8 210:016:1 212:116:3

Total 9467:1103:6 9639:1104:7

Bin

-8 -6 -4 -2 0 2 4 6 8

Number of events

0 200 400 600 800 1000 1200

1400 -

B+

B

Bin

-8 -6 -4 -2 0 2 4 6 8 )- )-N(B+ N(B

-100 -50 0 50 100

/ndf=10.3/15 p=80%

χ2

Bin

-8 -6 -4 -2 0 2 4 6 8 )-N(flavor)- N(B

-100 -50 0 50

100 χ²/ndf(fit)=5.1/13 p=97%

/ndf(flavor)=14.3/15 p=50%

χ2

Bin

-8 -6 -4 -2 0 2 4 6 8 )-N(flavor)+ N(B

-100 -50 0 50

100 χ²/ndf(fit)=12.8/13 p=46%

/ndf(flavor)=21.0/15 p=14%

χ2

FIG. 6 (color online). Results of the fit to theB!Dsample. (a) Signal yield in bins of theD!K⁰_S^þ Dalitz plot: from B!D (red),B^þ!D^þ (blue) and flavor sample (histogram). (b) Difference of signal yields between the B^þ!D^þ and B!Ddecays. (c) Difference of signal yields between theB!Dand flavor samples (normalized to the totalB!D yield): yield from the separate fits (points with error bars), and as a result of the combinedðx; yÞfit (horizontal bars). (d) Same as (c) for B^þ!D^þdata.