^{31,28}

^{,B}

^{106}

^{,A}

^{114}

^{,A}

^{53}

^{,B}

^{130}

^{,B}

^{97}

^{,A}

^{116}

^{,A}

^{136b}

^{,A}

^{104a}

^{,A}

^{53}

^{,A}

^{10}

^{,B}

^{111}

^{,A}

^{112}

^{,B}

^{76}

^{,B}

^{119}

^{,B}

^{119,50}

^{,B}

^{118}

^{,B}

^{64}

^{,A}

^{94}

^{,B}

^{68}

^{,*}

^{,A}

^{100a,100b}

^{,A}

^{136b}

^{,A}

^{39}

^{,B}

^{105}

^{,A}

^{97}

^{,A}

^{59}

^{,A}

^{136b}

^{,A}

^{100a,100b}

^{,A}

^{24a}

^{,A}

^{62}

^{,A}

^{35}

^{,B}

^{37}

^{,A}

^{133a,133b}

^{,A}

^{99a,99b}

^{,A}

^{46}

^{,B}

^{11,12,13}

^{,A}

^{97}

^{,A}

^{11,12}

^{,B}

^{97}

^{,A}

^{88}

^{,B}

^{24a}

^{,A}

^{69,46}

^{,B}

^{30}

^{,B}

^{7}

^{,A}

^{64}

^{,A}

^{105}

^{,A}

^{105}

^{,A}

^{11}

^{,A}

^{24a,24b}

^{,A}

^{25}

^{,A}

^{97}

^{,A}

^{55}

^{,B}

^{100a,100b}

^{,†}

^{,A}

^{111}

^{,A}

^{100a,100b}

^{,A}

^{70}

^{,A}

^{17}

^{,A}

^{97}

^{,A}

^{76}

^{,A}

^{84}

^{,B}

^{44}

^{,A}

^{17}

^{,A}

^{29}

^{,B}

^{58}

^{,B}

^{52}

^{,B}

^{117}

^{,B}

^{38}

^{,B}

^{100a}

^{,A}

^{24a,24b}

^{,A}

^{139}

^{,B}

^{44}

^{,A}

^{60}

^{,A}

^{111}

^{,A}

^{63}

^{,A}

^{71}

^{,A}

^{76}

^{,A}

^{22}

^{,B}

^{36}

^{,B}

^{55}

^{,A}

^{64}

^{,A}

^{133a,133b}

^{,A}

^{82a,82b}

^{,A}

^{67}

^{,A}

^{25}

^{,A}

^{75a}

^{,A}

^{62}

^{,A}

^{105}

^{,A}

^{17}

^{,B}

^{111}

^{,A}

^{17}

^{,B}

^{11,12}

^{,A}

^{111}

^{,A}

^{111}

^{,A}

^{17}

^{,A}

^{11,12}

^{,B}

^{6}

^{,A}

^{16}

^{,A}

^{107}

^{,A}

^{11,12}

^{,B}

^{116}

^{,A}

^{104a,104b}

^{,A}

^{94}

^{,B}

^{48}

^{,B}

^{104a}

^{,A}

^{104a,104b}

^{,A}

^{111}

^{,A}

^{133a}

^{,A}

^{25}

^{,A}

^{24a,24b}

^{,A}

^{17}

^{,A}

^{100a,100b}

^{,A}

^{8}

^{,A}

^{111,94}

^{60}

^{,‡}

^{,A}

^{133b,133b}

^{,A}

^{96}

^{,B}

^{11,12}

^{,B}

^{15}

^{,A}

^{24a,24b}

^{,A}

^{137}

^{,B}

^{95a}

^{,A}

^{48}

^{,B}

^{138}

^{,A}

^{72}

^{,B}

^{100a,100b}

^{,A}

^{38}

^{,B}

^{76}

^{,§}

^{,A}

^{48}

^{,B}

^{61,46}

^{,B}

^{11,12}

^{,A}

^{123}

^{,A}

^{67}

^{,A}

^{111}

^{,A}

^{2}

^{,A}

^{67}

^{,A}

^{47}

^{,A}

^{105}

^{,A}

^{133a}

^{,B}

^{123}

^{,A}

^{67}

^{,A}

^{31,28}

^{,B}

^{138}

^{,A}

^{111}

^{,A}

^{90}

^{,B}

^{9a,9b}

^{,A}

^{48}

^{,B}

^{30}

^{,B}

^{105}

^{,A}

^{81}

^{,B}

^{17}

^{,A}

^{92}

^{,A}

^{87}

^{,B}

^{20}

^{,A}

^{4}

^{,B}

^{60}

^{,A}

^{81}

^{,B}

^{22}

^{,B}

^{111}

^{,‡}

^{,A}

^{128}

^{,B}

^{31,28}

^{,B}

^{93}

^{31}

^{,B}

^{126}

^{,A}

^{40}

^{,B}

^{70}

^{,A}

^{91}

^{,A}

^{5}

^{,B}

^{130}

^{,B}

^{19}

^{,B}

^{74}

^{,B}

^{39}

^{,B}

^{54}

^{,B}

^{22}

^{,B}

^{92}

^{,A}

^{31}

^{,B}

^{110}

^{,B}

^{53}

^{,B}

^{55}

^{,B}

^{29}

^{,B}

^{17}

^{,A}

^{111}

^{,A}

^{136b}

^{,A}

^{20}

^{,B}

^{8}

^{,A}

^{18}

^{,B}

^{7}

^{,A}

^{69,46}

^{,B}

^{30}

^{,B}

^{136b}

^{,A}

^{11,12}

^{,A}

^{61,46}

^{,B}

^{76}

^{,B}

^{11,12}

^{,B}

^{65}

^{,B}

^{49}

^{,B}

^{11,12}

^{,B}

^{142}

^{,B}

^{33}

^{,A}

^{68}

^{,A}

^{134}

^{,A}

^{27}

^{,B}

^{57}

^{,A}

^{14}

^{,A}

^{138}

^{,A}

^{105}

^{,A}

^{55}

^{,A}

^{29}

^{,B}

^{54}

^{,B}

^{1}

^{,A}

^{111}

^{,A}

^{41}

^{,B}

^{98}

^{,B}

^{17}

^{,A}

^{39}

^{,B}

^{137,31}

^{,B}

^{16}

^{,A}

^{15}

^{,A}

^{91}

^{,A}

^{126}

^{,A}

^{46}

^{,B}

^{136b}

^{,A}

^{111}

^{,A}

^{23}

^{,B}

^{24a,24b}

^{,A}

^{100a,100c}

^{,A}

^{55}

^{,A}

^{111}

^{,A}

^{31}

^{,B}

^{43}

^{,A}

^{99a}

^{,A}

^{97}

^{,A}

^{95a 95b}

^{,A}

^{25}

^{,A}

^{135}

^{,A}

^{129}

^{,B}

^{77}

^{,B}

^{9b}

^{,A}

^{11,12}

^{,B}

^{9b}

^{,A}

^{20}

^{,A}

^{82a,82b}

^{,B}

^{83}

^{,B}

^{17}

^{,A}

^{90}

^{,B}

^{58,79,80}

^{,B}

^{120}

^{,B}

^{81}

^{,B}

^{111}

^{,A}

^{48}

^{,B}

^{133a}

^{,B}

^{93}

^{,B}

^{31,28}

^{,B}

^{46}

^{,B}

^{37}

^{,B}

^{139,31}

^{,B}

^{111}

^{,A}

^{75a}

^{,A}

^{101}

^{,B}

^{31,28}

^{,B}

^{136b}

^{,A}

^{100a,100b}

^{,A}

^{97}

^{,A}

^{127}

^{,B}

^{17}

^{,A}

^{89,90}

^{,B}

^{11,12,13}

^{,A}

^{135}

^{,A}

^{58,79}

^{,B}

^{58,80}

^{,B}

^{20}

^{,B}

^{3}

^{,A}

^{75a,75b}

^{,A}

^{16}

^{100a,100b}

^{,A}

^{82a}

^{,B}

^{54}

^{,B}

^{26}

^{,A}

^{26}

^{,¶}

^{,A}

^{25}

^{,A}

^{121}

^{,B}

^{80}

^{,B}

^{60}

^{,A}

^{66}

^{,B}

^{123}

^{,A}

^{25}

^{,A}

^{25}

^{,A}

^{137}

^{,B}

^{104a,104b}

^{,A}

^{104a}

^{,†}

^{,A}

^{1}

^{,A}

^{17}

^{,A}

^{95a}

^{,A}

^{44}

^{,A}

^{140}

^{,A}

^{115}

^{,A}

^{112}

^{,A}

^{20}

^{,A}

^{100a}

^{,A}

^{113}

^{,A}

^{111}

^{,A}

^{86}

^{,A}

^{39}

^{,B}

^{125}

^{,A}

^{65}

^{,B}

^{100a,100b}

^{,A}

^{70}

^{,A}

^{73a,73b}

^{,A}

^{17,48}

^{,**}

^{136b}

^{,A}

^{111}

^{,A}

^{99a}

^{,A}

^{25}

^{,A}

^{88}

^{,B}

^{82a}

^{,B}

^{62}

^{,A}

^{119,51}

^{,B}

^{31,28}

^{,B}

^{20}

^{,B}

^{31}

^{,B}

^{24a}

^{,A}

^{128}

^{,B}

^{101}

^{,B}

^{56}

^{,B}

^{4,34}

^{,B}

^{8}

^{,A}

^{67}

^{,A}

^{42}

^{,B}

^{20}

^{,B}

^{125}

^{,A}

^{82a,82b}

^{,A}

^{73b}

^{,A}

^{90}

^{,B}

^{113}

^{,A}

^{141}

^{,B}

^{81}

^{,B}

^{11,12}

^{,A}

^{74}

^{,B}

^{11,12}

^{,B}

^{131}

^{,B}

^{130}

^{,B}

^{87}

^{,B}

^{95a,95b}

^{,A}

^{72,122}

^{,B}

^{95a,95b}

^{,A}

^{11,12}

^{21}

^{,A}

^{102}

^{,A}

^{9b}

^{,A}

^{136a,136b}

^{,A}

^{123}

^{,A}

^{20}

^{,A}

^{11,12}

^{,A}

^{58,80}

^{,B}

^{124}

^{,A}

^{46}

^{,B}

^{95a,95b}

^{,A}

^{111}

^{,A}

^{31,28}

^{,B}

^{132}

^{,B}

^{76}

^{,A}

^{20}

^{,††}

^{,A}

^{109,32,103}

^{,B}

^{133a}

^{,B}

^{45}

^{,B}

^{78}

^{,A}

^{136b}

^{,A}

^{74}

^{,B}

^{1}

^{,A}

^{11,12}

^{60}

^{,A}

^{131}

^{,B}

^{58,80}

^{,B}

^{29}

^{,B}

^{31,28}

^{,B}

^{30}

^{,B}

^{30}

^{,B}

^{107}

^{,A}

^{111}

^{,A}

^{18}

^{,B}

^{59}

^{,A}

^{134}

^{,A}

^{11,12}

^{,B}

^{105}

^{,A}

^{21}

^{,A}

^{74}

^{,B}

^{105}

^{,A}

^{100a}

^{,A}

^{20}

^{,B}

^{85}

^{,B}

^{87}

^{,B}

^{41}

^{,B}

^{106}

^{,A}

^{112}

^{,A}

^{111}

^{,A}

^{53}

^{,B}

^{55}

^{,A}

^{57}

^{,A}

^{140}

^{,A}

^{41}

^{,B}

^{90}

^{,B}

^{58,80}

^{,B}

^{25}

^{,A}

^{100a,100b}

^{,A}

^{108}

^{,B}

^{11,12}

^{,B}

^{58,79}

^{,B}

^{11,12}

^{,B}

^{61,46}

^{,B}

^{A}

^{B}

Deceased.

Present address: University of Huddersfield, Huddersfield HD1 3DH, United Kingdom.

Also at Università di Sassari, I-07100 Sassari, Italy.

Present address: University of South Alabama, Mobile, Alabama 36688, USA.

Present address: Università di Bologna and INFN Sezione di Bologna, I-47921 Rimini, Italy.

Present address: European Organization for Nuclear Research (CERN), Geneva, Switzerland.

Present address: Wuhan University, Wuhan 430072, China.

^{3}.

We report measurements of

The breaking of

In particular,

In this article the inclusion of charge-conjugated decay modes is implied unless otherwise stated.

The trigonometric ambiguity can be resolved experimentally by measuring

An experimentally elegant and powerful approach to accessing

Feynman diagrams describing

In an

Equation

The last term in Eq.

Although elegant and appealing, the measurements of

Time-dependent Dalitz plot analyses of

In this article, we present measurements of

The approach of combining the existing data of the

The paper is structured as follows: Section

The results presented herein utilize data collected with the

The

The

The Belle detector consists of a silicon vertex detector (SVD), a 50-layer central drift chamber (CDC), an array of aerogel threshold Cherenkov counters (ACC), a barrel-like arrangement of time-of-flight scintillation counters (TOF), and an electromagnetic calorimeter comprised of CsI(Tl) crystals (ECL) located inside a superconducting solenoid coil that provides a 1.5 T magnetic field. A steel flux return located outside of the coil is instrumented to detect

The Monte Carlo event generators used at

The first part of the analysis, described in Sec.

The second part of the analysis, described in Sec.

The

The reconstructed charmed meson decays are characterized by two observables: the

Two-dimensional

The signal and background yields are estimated using a two-dimensional unbinned maximum-likelihood (ML) fit to the

The following four separate categories are considered for the background.

The first source of background arises from the combination of correctly reconstructed

The second background category is composed of real

The third background category contains background from

The fourth background category accounts for the remaining combinatorial background originating from random combinations of tracks. This “combinatorial background” is parametrized by a first-order polynomial function in

In the two-dimensional fit of the

Data distributions of

Signal and background yields determined by a two-dimensional fit to the

The

Dalitz plot data distributions for all three combinations of

The

The form factors

The Zemach formalism

The propagator term

The isobar ansatz parametrizes the

In this parametrization, the decay amplitude

The production vector

In this analysis, the

The

Experimental effects, such as detector acceptance variations, reconstruction algorithms, or the event selection criteria, can produce nonuniformities in the reconstruction efficiency as a function of the Dalitz plot phase space,

The simulated detector response then undergoes the same reconstruction algorithms and event selection requirements as the data. The generated MC sample contains

The efficiency map is constructed using an approach

In the first step, the angular variations of the efficiency are estimated by expanding the

The chosen order

The reconstruction efficiency is nearly flat over large parts of the Dalitz plot phase space. The efficiency decreases slightly at larger values of

Variation of the Dalitz plot reconstruction efficiency as a function of

The Dalitz plot distributions of the background are estimated from the data using two

The

The fraction of “wrong

The background p.d.f.

Due to the high statistics of the Belle

The

The results for the estimated

Projections of the Dalitz plot data distributions (points with error bars) for

Results for the amplitude magnitudes

The quality of the fit is estimated by a two-dimensional

The fit fractions (

To further test the agreement of the Dalitz plot amplitude model with the data, we follow an approach employed by

Dalitz plot data distributions (points with error bars) for

The chosen parametrization and composition of the

The Dalitz plot amplitude analysis of

The addition of further resonances [e.g., the

The inclusion of a term constant in phase space to the baseline model to account for possible additional direct nonresonant three-body decays results in negligibly small fit fractions for this component.

Instead of using the

The CLEO experiment made a model-independent determination of the relative strong phase between

The performance of the

Charged pion candidates are formed from tracks that are reconstructed from hits in the tracking detectors that meet charged particle quality criteria

Neutral pions are reconstructed by combining two photon candidates. The invariant mass of a

Neutral kaons are reconstructed in the decay mode

Neutral

Neutral

The beam-energy-constrained mass is defined as

The neural network combines information characterizing the shape of the events and is based on 16 modified Fox-Wolfram moments

The following requirements are made on

The

For

For

In addition to the contributions from the signal and the signal-like cross-feed, the fit model accounts for the following three separate sources of background. The first source originates from partially reconstructed

There is a small “combinatorial

The third source of background originates from

In total, we obtain

Data distributions for

Summary of the

At

The experimental conditions and the instrumentation of the detectors are different for

The time-dependent Dalitz plot analysis to measure the

For the signal, the p.d.f.s are constructed from Eqs.

Neutral

To account for the signal-like cross-feed from partially reconstructed

In the fit, the parameters

A separate fit is performed to directly measure the

A rich variety of intermediate

Distributions of the proper-time interval (data points with error bars) and the corresponding asymmetries for

In Figs.

In Figs.

Cross-checks are performed to validate the measurement procedure. The

This analysis accounts for two classes of systematic uncertainties on the measured

The estimation of the experimental systematic uncertainty on the

Experimental systematic uncertainties on the

The systematic uncertainty due to vertex reconstruction accounts for the applied vertex reconstruction algorithms, the requirements applied to select

Experiment-specific resolution models are applied to account for effects due to the finite experimental

The parameters of the

The signal purity is estimated by the three-dimensional unbinned ML fit to the

The

The neutral

The systematic uncertainty due to a possible small fit bias in

The effect due to the applied Dalitz plot reconstruction efficiency correction for neutral

Most systematic uncertainties are independent for

Additional contributions to the systematic uncertainty from possible sources of peaking background and the tag-side interference have been considered and can be neglected.

The total experimental systematic uncertainty is the quadratic sum of all these contributions.

The model uncertainty accounts for the dependence of the

Uncertainties on the

For the masses and widths of resonances fixed to the world averages, each resonance parameter is varied within its uncertainty to estimate the associated model uncertainty.

The model uncertainty due to the chosen

The LASS parametrization is used to model the

The model uncertainty due to the chosen Blatt-Weisskopf barrier factors for

The fraction of wrong

The model uncertainty due to the applied Dalitz plot reconstruction efficiency correction is estimated by replacing the parametrized efficiency map by the corresponding two-dimensional binned distributions.

In the Dalitz plot amplitude analysis, the background is described by a parametrized model taken from the

The signal and background fractions used in the Dalitz plot amplitude analysis are determined by the fit of the two-dimensional

The statistical uncertainties on the Dalitz plot amplitude model parameters that are summarized in Table

The dependence of the model on resonances with very small contributions is estimated by removing resonances with fit fractions of 0.1% or lower. The doubly Cabibbo-suppressed

As a further cross-check and estimate of the possible model dependence, a pure isobar

The total model uncertainty is the quadratic sum of all contributions. Overall, the uncertainty due to the Dalitz plot amplitude model is small compared to the statistical uncertainty and the experimental systematic uncertainty.

The statistical significance of the results is determined by a likelihood-ratio approach. The change in

Obtained

In summary, we have measured

We measure

Due to the absence of penguin amplitudes, the

The combined

We thank the PEP-II and KEKB groups for the excellent operation of the accelerators. The

The CKM angle