Measurement of CP Violation Parameters with a Dalitz Plot Analysis of B
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B. Aubert,
1
M. Bona,
1
D. Boutigny,
1
Y. Karyotakis,
1
J. P. Lees,
1
V. Poireau,
1
X. Prudent,
1
V. Tisserand,
1
A. Zghiche,
1
E. Grauges,
2
L. Lopez,
3
A. Palano,
3
J. C. Chen,
4
N. D. Qi,
4
G. Rong,
4
P. Wang,
4
Y. S. Zhu,
4
G. Eigen,
5
I. Ofte,
5
B. Stugu,
5
G. S. Abrams,
6
M. Battaglia,
6
D. N. Brown,
6
J. Button-Shafer,
6
R. N. Cahn,
6
Y. Groysman,
6
R. G. Jacobsen,
6
J. A. Kadyk,
6
L. T. Kerth,
6
Yu. G. Kolomensky,
6
G. Kukartsev,
6
D. Lopes Pegna,
6
G. Lynch,
6
L. M. Mir,
6
T. J. Orimoto,
6
M. Pripstein,
6
N. A. Roe,
6
M. T. Ronan,
6,
*
K. Tackmann,
6
W. A. Wenzel,
6
P. del Amo Sanchez,
7
M. Barrett,
7
T. J. Harrison,
7
A. J. Hart,
7
C. M. Hawkes,
7
A. T. Watson,
7
T. Held,
8
H. Koch,
8
B. Lewandowski,
8
M. Pelizaeus,
8
T. Schroeder,
8
M. Steinke,
8
J. T. Boyd,
9
J. P. Burke,
9
W. N. Cottingham,
9
D. Walker,
9
D. J. Asgeirsson,
10
T. Cuhadar-Donszelmann,
10
B. G. Fulsom,
10
C. Hearty,
10
N. S. Knecht,
10
T. S. Mattison,
10
J. A. McKenna,
10
A. Khan,
11
P. Kyberd,
11
M. Saleem,
11
D. J. Sherwood,
11
L. Teodorescu,
11
V. E. Blinov,
12
A. D. Bukin,
12
V. P. Druzhinin,
12
V. B. Golubev,
12
A. P. Onuchin,
12
S. I. Serednyakov,
12
Yu. I. Skovpen,
12
E. P. Solodov,
12
K. Yu Todyshev,
12
M. Bondioli,
13
M. Bruinsma,
13
M. Chao,
13
S. Curry,
13
I. Eschrich,
13
D. Kirkby,
13
A. J. Lankford,
13
P. Lund,
13
M. Mandelkern,
13
E. C. Martin,
13
D. P. Stoker,
13
S. Abachi,
14
C. Buchanan,
14
S. D. Foulkes,
15
J. W. Gary,
15
F. Liu,
15
O. Long,
15
B. C. Shen,
15
L. Zhang,
15
E. J. Hill,
16
H. P. Paar,
16
S. Rahatlou,
16
V. Sharma,
16
J. W. Berryhill,
17
C. Campagnari,
17
A. Cunha,
17
B. Dahmes,
17
T. M. Hong,
17
D. Kovalskyi,
17
J. D. Richman,
17
T. W. Beck,
18
A. M. Eisner,
18
C. J. Flacco,
18
C. A. Heusch,
18
J. Kroseberg,
18
W. S. Lockman,
18
T. Schalk,
18
B. A. Schumm,
18
A. Seiden,
18
D. C. Williams,
18
M. G. Wilson,
18
L. O. Winstrom,
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E. Chen,
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C. H. Cheng,
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A. Dvoretskii,
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F. Fang,
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D. G. Hitlin,
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I. Narsky,
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T. Piatenko,
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F. C. Porter,
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G. Mancinelli,
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B. T. Meadows,
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K. Mishra,
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M. D. Sokoloff,
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F. Blanc,
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P. C. Bloom,
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S. Chen,
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W. T. Ford,
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J. F. Hirschauer,
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A. Kreisel,
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M. Nagel,
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U. Nauenberg,
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J. G. Smith,
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J. Zhang,
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E. A. Eckhart,
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A. Soffer,
22,†
W. H. Toki,
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F. Winklmeier,
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Q. Zeng,
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D. D. Altenburg,
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E. Feltresi,
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A. Hauke,
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H. Jasper,
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J. Merkel,
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A. Petzold,
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B. Spaan,
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K. Wacker,
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T. Brandt,
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V. Klose,
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H. M. Lacker,
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W. F. Mader,
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J. Schubert,
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D. Bernard,
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Ch. Thiebaux,
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M. Verderi,
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P. J. Clark,
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M. Andreotti,
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D. Bettoni,
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I. M. Peruzzi,
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M. Piccolo,
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M. Lo Vetere,
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M. M. Macri,
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D. J. Bard,
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M. B. Nikolich,
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N. Arnaud,
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quilleux,
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S. Rodier,
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M. H. Schune,
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D. J. Lange,
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C. A. Chavez,
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I. J. Forster,
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J. R. Fry,
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E. Gabathuler,
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R. Gamet,
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D. E. Hutchcroft,
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D. J. Payne,
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K. C. Schofield,
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C. Touramanis,
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A. J. Bevan,
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K. A. George,
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F. Di Lodovico,
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W. Menges,
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R. Sacco,
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G. Cowan,
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H. U. Flaecher,
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D. A. Hopkins,
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P. S. Jackson,
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T. R. McMahon,
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F. Salvatore,
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A. C. Wren,
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D. N. Brown,
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C. L. Davis,
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J. Allison,
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N. R. Barlow,
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Y. M. Chia,
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W. D. Hulsbergen,
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A. Jawahery,
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D. A. Roberts,
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G. Simi,
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G. Blaylock,
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C. Dallapiccola,
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S. S. Hertzbach,
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X. Li,
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T. B. Moore,
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E. Salvati,
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R. Cowan,
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G. Sciolla,
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S. J. Sekula,
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M. Spitznagel,
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F. Taylor,
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R. K. Yamamoto,
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H. Kim,
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S. E. Mclachlin,
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P. M. Patel,
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S. H. Robertson,
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A. Lazzaro,
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V. Lombardo,
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F. Palombo,
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J. M. Bauer,
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L. Cremaldi,
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V. Eschenburg,
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R. Godang,
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R. Kroeger,
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D. A. Sanders,
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D. J. Summers,
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H. W. Zhao,
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S. Brunet,
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D. Co
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M. Simard,
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P. Taras,
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F. B. Viaud,
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H. Nicholson,
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N. Cavallo,
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G. De Nardo,
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F. Fabozzi,
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C. Gatto,
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L. Lista,
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D. Monorchio,
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P. Paolucci,
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D. Piccolo,
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C. Sciacca,
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M. A. Baak,
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G. Raven,
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H. L. Snoek,
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C. P. Jessop,
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J. M. LoSecco,
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G. Benelli,
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L. A. Corwin,
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K. K. Gan,
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K. Honscheid,
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D. Hufnagel,
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H. Kagan,
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R. Kass,
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J. P. Morris,
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A. M. Rahimi,
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J. J. Regensburger,
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R. Ter-Antonyan,
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Q. K. Wong,
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N. L. Blount,
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J. Brau,
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R. Frey,
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O. Igonkina,
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J. A. Kolb,
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M. Lu,
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R. Rahmat,
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N. B. Sinev,
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D. Strom,
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J. Strube,
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E. Torrence,
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A. Gaz,
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M. Margoni,
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M. Morandin,
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A. Pompili,
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M. Posocco,
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M. Rotondo,
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F. Simonetto,
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R. Stroili,
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C. Voci,
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E. Ben-Haim,
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H. Briand,
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J. Chauveau,
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P. David,
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L. Del Buono,
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Ch. de la Vaissie
`
re,
58
O. Hamon,
58
PRL 99, 251801 (2007)
PHYSICAL REVIEW LETTERS
week ending
21 DECEMBER 2007
0031-9007=07=99(25)=251801(7) 251801-1 © 2007 The American Physical Society
B. L. Hartfiel,
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Ph. Leruste,
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J. Malcle
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J. Ocariz,
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A. Perez,
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J. Prendki,
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R. Kowalewski,
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I. M. Nugent,
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J. M. Roney,
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R. J. Sobie,
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J. J. Back,
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P. F. Harrison,
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T. E. Latham,
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G. B. Mohanty,
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H. R. Band,
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K. T. Flood,
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J. J. Hollar,
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P. E. Kutter,
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B. Mellado,
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Y. Pan,
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M. Pierini,
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R. Prepost,
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S. L. Wu,
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Z. Yu,
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and H. Neal
81
(BABAR Collaboration)
1
Laboratoire de Physique des Particules, IN2P3/CNRS et Universite
´
de Savoie, F-74941 Annecy-Le-Vieux, France
2
Facultat de Fisica, Departament ECM, Universitat de Barcelona, E-08028 Barcelona, Spain
3
Dipartimento di Fisica and INFN, Universita
`
di Bari, I-70126 Bari, Italy
4
Institute of High Energy Physics, Beijing 100039, China
5
Institute of Physics, University of Bergen, N-5007 Bergen, Norway
6
Lawrence Berkeley National Laboratory and University of California, Berkeley, California 94720, USA
7
University of Birmingham, Birmingham, B15 2TT, United Kingdom
8
Institut fu
¨
r Experimentalphysik 1, Ruhr Universita
¨
t Bochum, D-44780 Bochum, Germany
9
University of Bristol, Bristol BS8 1TL, United Kingdom
10
University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z1
11
Brunel University, Uxbridge, Middlesex UB8 3PH, United Kingdom
12
Budker Institute of Nuclear Physics, Novosibirsk 630090, Russia
13
University of California at Irvine, Irvine, California 92697, USA
14
University of California at Los Angeles, Los Angeles, California 90024, USA
15
University of California at Riverside, Riverside, California 92521, USA
16
University of California at San Diego, La Jolla, California 92093, USA
17
University of California at Santa Barbara, Santa Barbara, California 93106, USA
18
Institute for Particle Physics, University of California at Santa Cruz, Santa Cruz, California 95064, USA
19
California Institute of Technology, Pasadena, California 91125, USA
20
University of Cincinnati, Cincinnati, Ohio 45221, USA
21
University of Colorado, Boulder, Colorado 80309, USA
22
Colorado State University, Fort Collins, Colorado 80523, USA
23
Institut fu
¨
r Physik, Universita
¨
t Dortmund, D-44221 Dortmund, Germany
24
Institut fu
¨
r Kern- und Teilchenphysik, Technische Universita
¨
t Dresden, D-01062 Dresden, Germany
25
Laboratoire Leprince-Ringuet, CNRS/IN2P3, Ecole Polytechnique, F-91128 Palaiseau, France
26
University of Edinburgh, Edinburgh EH9 3JZ, United Kingdom
PRL 99, 251801 (2007)
PHYSICAL REVIEW LETTERS
week ending
21 DECEMBER 2007
251801-2
27
Dipartimento di Fisica and INFN, Universita
`
di Ferrara, I-44100 Ferrara, Italy
28
Laboratori Nazionali di Frascati dell’INFN, I-00044 Frascati, Italy
29
Dipartimento di Fisica and INFN, Universita
`
di Genova, I-16146 Genova, Italy
30
Harvard University, Cambridge, Massachusetts 02138, USA
31
Physikalisches Institut, Universita
¨
t Heidelberg, Philosophenweg 12, D-69120 Heidelberg, Germany
32
Imperial College London, London, SW7 2AZ, United Kingdom
33
University of Iowa, Iowa City, Iowa 52242, USA
34
Iowa State University, Ames, Iowa 50011-3160, USA
35
Johns Hopkins University, Baltimore, Maryland 21218, USA
36
Institut fu
¨
r Experimentelle Kernphysik, Universita
¨
t Karlsruhe, D-76021 Karlsruhe, Germany
37
Laboratoire de l’Acce
´
le
´
rateur Line
´
aire, IN2P3/CNRS et Universite
´
Paris-Sud 11, Centre Scientifique d’Orsay, B. P. 34, F-91898
ORSAY Cedex, France
38
Lawrence Livermore National Laboratory, Livermore, California 94550, USA
39
University of Liverpool, Liverpool L69 7ZE, United Kingdom
40
Queen Mary, University of London, E1 4NS, United Kingdom
41
University of London, Royal Holloway and Bedford New College, Egham, Surrey TW20 0EX, United Kingdom
42
University of Louisville, Louisville, Kentucky 40292, USA
43
University of Manchester, Manchester M13 9PL, United Kingdom
44
University of Maryland, College Park, Maryland 20742, USA
45
University of Massachusetts, Amherst, Massachusetts 01003, USA
46
Laboratory for Nuclear Science, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
47
McGill University, Montre
´
al, Que
´
bec, Canada H3A 2T8
48
Dipartimento di Fisica and INFN, Universita
`
di Milano, I-20133 Milano, Italy
49
University of Mississippi, University, Mississippi 38677, USA
50
Physique des Particules, Universite
´
de Montre
´
al, Montre
´
al, Que
´
bec, Canada H3C 3J7
51
Mount Holyoke College, South Hadley, Massachusetts 01075, USA
52
Dipartimento di Scienze Fisiche and INFN, Universita
`
di Napoli Federico II, I-80126, Napoli, Italy
53
NIKHEF, National Institute for Nuclear Physics and High Energy Physics, NL-1009 DB Amsterdam, The Netherlands
54
University of Notre Dame, Notre Dame, Indiana 46556, USA
55
Ohio State University, Columbus, Ohio 43210, USA
56
University of Oregon, Eugene, Oregon 97403, USA
57
Dipartimento di Fisica and INFN, Universita
`
di Padova, I-35131 Padova, Italy
58
Laboratoire de Physique Nucle
´
aire et de Hautes Energies, IN2P3/CNRS, Universite
´
Pierre et Marie Curie-Paris6,
Universite
´
Denis Diderot-Paris7, F-75252 Paris, France
59
University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
60
Dipartimento di Fisica and INFN, Universita
`
di Perugia, I-06100 Perugia, Italy
61
Dipartimento di Fisica, Scuola Normale Superiore, and INFN, Universita
`
di Pisa, I-56127 Pisa, Italy
62
Prairie View A&M University, Prairie View, Texas 77446, USA
63
Princeton University, Princeton, New Jersey 08544, USA
64
Dipartimento di Fisica and INFN, Universita
`
di Roma La Sapienza, I-00185 Roma, Italy
65
Universita
¨
t Rostock, D-18051 Rostock, Germany
66
Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, OX11 0QX, United Kingdom
67
DSM/Dapnia, CEA/Saclay, F-91191 Gif-sur-Yvette, France
68
University of South Carolina, Columbia, South Carolina 29208, USA
69
Stanford Linear Accelerator Center, Stanford, California 94309, USA
70
Stanford University, Stanford, California 94305-4060, USA
71
State University of New York, Albany, New York 12222, USA
72
University of Tennessee, Knoxville, Tennessee 37996, USA
73
University of Texas at Austin, Austin, Texas 78712, USA
74
University of Texas at Dallas, Richardson, Texas 75083, USA
75
Dipartimento di Fisica Sperimentale and INFN, Universita
`
di Torino, I-10125 Torino, Italy
76
Dipartimento di Fisica and INFN, Universita
`
di Trieste, I-34127 Trieste, Italy
77
IFIC, Universitat de Valencia-CSIC, E-46071 Valencia, Spain
78
University of Victoria, Victoria, British Columbia, Canada V8W 3P6
79
Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom
80
University of Wisconsin, Madison, Wisconsin 53706, USA
81
Yale University, New Haven, Connecticut 06511, USA
(Received 23 March 2007; published 17 December 2007)
We report the results of a CP violation analysis of the decay B
! D
0
K
, where D
0
indicates a neutral D meson detected in the final state
0
, excluding K
0
S
0
. The analysis makes use
PRL 99, 251801 (2007)
PHYSICAL REVIEW LETTERS
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251801-3
of 324 10
6
e
e
! B
B events recorded by the BABAR experiment at the PEP-II e
e
storage ring.
Analyzing the
0
Dalitz plot distribution and the B
! D
0
K
branching fraction and decay
rate asymmetry, we find the following one-standard-deviation constraints on the amplitude ratio and on
the weak and strong phases: 0:06 <r
B
< 0:78, 30
<<76
, 27
<<78
. We also measure the
magnitudes and phases of the components of the D
0
!
0
decay amplitude.
DOI: 10.1103/PhysRevLett.99.251801 PACS numbers: 13.25.Hw, 12.15.Hh, 11.30.Er
An important component of the program to study CP
violation is the measurement of the angle
argV
ud
V
ub
=V
cd
V
cb
of the unitarity triangle related to
the Cabibbo-Kobayashi-Maskawa quark mixing matrix
[1]. The decays B ! D
0
K
can be used to measure
with essentially no hadronic uncertainties, exploiting in-
terference between b ! u
cs and b ! c
us decay ampli-
tudes [2]. In one of the measurement methods [3], is
extracted by analyzing the D-decay Dalitz plot distribution
in B
! DK
with multibody D decays [4]. This method
has only been used with the Cabibbo-favored decay D !
K
0
S
[5,6], and Cabibbo-suppressed decays are ex-
pected to be similarly sensitive to [7]. We present here
the first CP-violation study of B
! DK
with a multi-
body, Cabibbo-suppressed D decay, D !
0
.
The data used in this analysis were collected with the
BABAR detector at the PEP-II e
e
storage ring, and they
include 288 fb
1
taken on the 4S resonance and
27 fb
1
collected 40 MeV below the resonance. Samples
of simulated Monte Carlo (MC) events were analyzed with
the same reconstruction and analysis procedures. These
samples include an e
e
! B
B sample 5 times larger
than the data, a continuum e
e
! q
q sample, where q
is a u, d, s,orc quark, with luminosity equivalent to the
data, and a signal sample 300 times larger than the data,
with both phase space D decays and decays generated
according to the amplitudes measured by CLEO [8]. The
BABAR detector and the methods used for particle recon-
struction and identification are described in Ref. [9].
We use event-shape variables [10] to suppress the con-
tinuum background, and we identify kaon and pion
candidates using specific ionization and Cherenkov radia-
tion. The invariant mass of D candidates must satisfy
1830 <M
D
< 1895 MeV=c
2
. We require 5272 <m
ES
<
5300 MeV=c
2
, where m
ES
E
2
c:m:
=4 jp
B
p
j
2
, E
c:m:
is
the total e
e
center of mass (c.m.) energy, and p
B
is
the B candidate c.m. momentum. Events must satisfy
70 < E<60 MeV, where E E
B
E
c:m:
=2 and
E
B
is the B candidate c.m. energy. We exclude the decay
mode D ! K
0
S
0
, which is a previously studied CP eigen-
state not related to the method of Ref. [3], by rejecting
candidates with 489 <M
< 508 MeV=c
2
or for
which the distance between the
vertex and the B
candidate decay vertex is more than 1.5 cm. We reject
B
! D
0
K
candidates in which the K
invari-
ant mass satisfies 1840 <MK
< 1890 MeV=c
2
,to
suppress B
! D
0
K
decays. We require d>0:25,
where d [10] is a neural net variable that separates signal
candidates (which peak toward d 1) from those with a
misreconstructed D (peaking toward d 0). In events with
multiple candidates (9% of the sample), we keep the
candidate whose m
ES
value is closest to the nominal B
mass [11]. The final signal reconstruction efficiency is
11:4%.
For each B
! D
0
K
candidate, we compute the
neural net variable q [10]. The q distribution of B
B events
peaks toward q 1, while that of continuum peaks at q
0.For 2fq; dg, we define the variables
0
tanh
1
f
1
2
max
min
=
1
2
max
min
g, where
q
max
d
max
1, q
min
0:1, and d
min
0:25 are the al-
lowed ranges for q and d. The
0
variables can be conven-
iently fit with Gaussians, as described later.
As in Ref. [10], we identify in the MC samples ten event
types, one signal, and nine different backgrounds. We list
them here with the labels used to refer to them throughout
the Letter. DK
sig
: B
! D
0
K
events that are cor-
rectly reconstructed; these are the only events considered
to be signal. DK
bgd
: B
! D
0
K
events that are
misreconstructed; namely, some of the particles used to
form the final state do not originate from the B
!
D
0
K
decay. D
D
(D
D6
): B
! D
0
, D
0
!
0
decays, where the decay D
0
!
0
is cor-
rectly reconstructed (misreconstructed). DKX: B !
D
K
events not containing the decay D !
0
. DX: B ! D
and B ! D
decays,
excluding D !
0
. BBC
D
(BBC
D6
): all other B
B
events with a correctly reconstructed (misreconstructed) D
candidate. qq
D
(qq
D6
): continuum e
e
! q
q events with
a correctly reconstructed (misreconstructed) D candidate.
The measurement of the CP parameters proceeds in
three steps, each involving an unbinned maximum like-
lihood fit. In step 1, we measure the complex Dalitz plot
amplitude fs
;s
for the decay D
0
!
0
, where
s
m
2
0
are the squared invariant masses of the
0
pairs. In step 2, we extract the numbers of B
and
B
signal events and background yields. We obtain the CP
parameters in step 3.
We parametrize fs
;s
using the isobar model,
fs
;s
a
NR
e
iNR
P
r
a
r
e
i
r
A
r
s
;s
=N
f
, where
the first term represents a nonresonant contribution, the
sum is over all intermediate two-body resonances r, and N
f
is such that
R
ds
ds
jfs
;s
j
2
1. The amplitude for
the decay chain D
0
! rC, r ! AB is A
r
s
;s
F
r
F
s
m
2
r
M
2
AB
im
r
r
M
AB
1
, where m
r
is the peak
PRL 99, 251801 (2007)
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mass of the resonance [11], M
2
AB
is the squared invariant
mass of the AB pair, F
r
is a spin-dependent form factor
[12], and
r
M
AB
is the mass-dependent width for the
resonance r [12]. The spin factors F
s
are F
0
m
2
D
, F
1
M
2
BC
M
2
AC
m
2
D
m
2
C
m
2
A
m
2
B
M
2
AB
, and F
2
F
2
1
1
3
2
CD
2
AB
m
2
D
, where
2
jk
M
2
AB
2m
2
j
2m
2
k
m
2
j
m
2
k
2
M
2
jk
, and m
i
is the mass of particle i
[11].
In step 1, we determine the parameters a
NR
, a
r
,
NR
, and
r
by fitting a large sample of D
0
and
D
0
mesons, flavor
tagged through their production in the decay D
!
D
0
[13]. To select this sample, we require the c.m.
momentum of the D
candidate to be greater than
2770 MeV=c, and jM
D
M
D
145:4 MeV=c
2
j <
0:6 MeV=c
2
, where M
D
is the invariant mass of the D
candidate. The signal and background yields are obtained
from a fit to the M
D
distribution, modeling the signal as a
Gaussian and the background as an exponential. The signal
Gaussian peaks at 1863:7 0:4 MeV=c
2
and has a width
of 17:4 0:8 MeV=c
2
.
Of the D
0
candidates in the signal region 1848 <M
D
<
1880 MeV=c
2
, we obtain from the fit N
S
44 780 250
signal and N
B
830 70 background events. To
obtain the parameters of fs
;s
, we fit these candidates
with the probability distribution function (PDF)
N
S
jfs
;s
j
2
s
;s
N
B
jf
B
s
;s
j
2
, where the
background PDF f
B
s
;s
is a binned distribution ob-
tained from events in the sideband 1930 <M
D
<
1990 MeV=c
2
, and s
;s
is an efficiency function,
parametrized as a two-dimensional third-order polynomial
determined from MC. To within the MC-signal statistical
uncertainty, s
;s
s
;s
. The region M
D
<
1848 MeV=c
2
, which contains D
0
! K
0
events
that are absent from the signal region, is not used.
Table I summarizes the results of this fit, with systematic
errors obtained by varying the masses and widths of the
1700 and resonances, setting F
r
1, and varying
s
;s
to account for uncertainties in reconstruction and
particle identification. The Dalitz plot distribution of the
data is shown in Fig. 1(a). The distribution is marked by
three destructively interfering amplitudes, suggesting
an I 0-dominated final state [14].
The fit for step i 2f2; 3g uses the PDF
P
C
i
X
t
N
t
2
1 CA
t
P
C
i;t
i
Z
P
C
i;t
0
i
d
n
i
0
i
; (1)
where
i
is the set of n
i
event variables
2
fE; q
0
;d
0
g,
3
fE; q
0
;s
;s
g, t corresponds to one of the ten event
types listed above, N
t
N
t
N
t
is the number of events
of type t, A
t
N
t
N
t
=N
t
is their charge asymmetry,
C 1 is the electric charge of the B candidate, and
P
t
N
t
. Using MC, we verify that the variables in each set
i
are uncorrelated for each event type. Therefore, the PDFs
P
C
i;t
are the products
P
2;t
E; q
0
;d
0
E
t
EQ
t
q
0
C
t
d
0
;
P
C
3;t
E; q
0
;s
;s
E
t
EQ
t
q
0
D
0C
t
s
;s
:
(2)
The parameters of the Dalitz plot PDF D
0
C
DK
sig
s
;s
are
obtained from the data as described below. Those of all
other functions in Eq. (2) are obtained from the MC
samples. The functions E
t
E are parametrized as the
TABLE I. Result of the fit to the D
! D
0
sample, showing the amplitudes ratios R
r
a
r
=a
770
, phase differences
r
r
770
, and fit fractions f
r
R
ja
r
A
r
s
;s
j
2
ds
ds
. The first (second) errors are statistical (systematic). We take the
mass (width) of the meson to be 400600 MeV=c
2
.
State R
r
(%)
r
(
) f
r
(%)
770 100 0 67:8 0:0 0:6
0
770 58:8 0:6 0:216:2 0:6 0:426:2 0:5 1:1
770 71:4 0:8 0:3 2:0 0:6 0:634:6 0:8 0:3
1450 21 6 13 146 18 24 0:11 0:07 0:12
0
1450 33 6 410 8 13 0:30 0:11 0:07
1450 82 5 416 3 31:79 0:22 0:12
1700 225 18 14 17 2 34:1 0:7 0:7
0
1700 251 15 13 17 2 25:0 0:6 1:0
1700 200 11 7 50 3 33:2 0:4 0:6
f
0
980 1:50 0:12 0:17 59 5 40:25 0:04 0:04
f
0
1370 6:3 0:9 0:9 156 9 60:37 0:11 0:09
f
0
1500 5:8 0:6 0:612 9 40:39 0:08 0:07
f
0
1710 11:2 1:4 1:751 8 70:31 0:07 0:08
f
2
1270 104 3 21 171 3 41:32 0:08 0:10
400 6:9 0:6 1:28 4 80:82 0:10 0:10
Nonresonant 57 7 8 11 4 20:84 0:21 0:12
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