Measurement of
CP
observables in
B
!
D
CP
K
decays and constraints on the CKM angle
P. del Amo Sanchez,
1
J. P. Lees,
1
V. Poireau,
1
E. Prencipe,
1
V. Tisserand,
1
J. Garra Tico,
2
E. Grauges,
2
M. Martinelli,
3a,3b
A. Palano,
3b,3b
M. Pappagallo,
3a,3b
G. Eigen,
4
B. Stugu,
4
L. Sun,
4
M. Battaglia,
5
D. N. Brown,
5
B. Hooberman,
5
L. T. Kerth,
5
Yu. G. Kolomensky,
5
G. Lynch,
5
I. L. Osipenkov,
5
T. Tanabe,
5
C. M. Hawkes,
6
A. T. Watson,
6
H. Koch,
7
T. Schroeder,
7
D. J. Asgeirsson,
8
C. Hearty,
8
T. S. Mattison,
8
J. A. McKenna,
8
A. Khan,
9
A. Randle-Conde,
9
V. E. Blinov,
10
A. R. Buzykaev,
10
V. P. Druzhinin,
10
V. B. Golubev,
10
A. P. Onuchin,
10
S. I. Serednyakov,
10
Yu. I. Skovpen,
10
E. P. Solodov,
10
K. Yu. Todyshev,
10
A. N. Yushkov,
10
M. Bondioli,
11
S. Curry,
11
D. Kirkby,
11
A. J. Lankford,
11
M. Mandelkern,
11
E. C. Martin,
11
D. P. Stoker,
11
H. Atmacan,
12
J. W. Gary,
12
F. Liu,
12
O. Long,
12
G. M. Vitug,
12
C. Campagnari,
13
T. M. Hong,
13
D. Kovalskyi,
13
J. D. Richman,
13
A. M. Eisner,
14
C. A. Heusch,
14
J. Kroseberg,
14
W. S. Lockman,
14
A. J. Martinez,
14
T. Schalk,
14
B. A. Schumm,
14
A. Seiden,
14
L. O. Winstrom,
14
C. H. Cheng,
15
D. A. Doll,
15
B. Echenard,
15
D. G. Hitlin,
15
P. Ongmongkolkul,
15
F. C. Porter,
15
A. Y. Rakitin,
15
R. Andreassen,
16
M. S. Dubrovin,
16
G. Mancinelli,
16
B. T. Meadows,
16
M. D. Sokoloff,
16
P. C. Bloom,
17
W. T. Ford,
17
A. Gaz,
17
M. Nagel,
17
U. Nauenberg,
17
J. G. Smith,
17
S. R. Wagner,
17
R. Ayad,
18,
*
W. H. Toki,
18
H. Jasper,
19
T. M. Karbach,
19
J. Merkel,
19
A. Petzold,
19
B. Spaan,
19
K. Wacker,
19
M. J. Kobel,
20
K. R. Schubert,
20
R. Schwierz,
20
D. Bernard,
21
M. Verderi,
21
P. J. Clark,
22
S. Playfer,
22
J. E. Watson,
22
M. Andreotti,
23a,23b
D. Bettoni,
23a
C. Bozzi,
23a
R. Calabrese,
23a,23b
A. Cecchi,
23a,23b
G. Cibinetto,
23a,23b
E. Fioravanti,
23a,23b
P. Franchini,
23a,23b
E. Luppi,
23a,23b
M. Munerato,
23a,23b
M. Negrini,
23a,23b
A. Petrella,
23a,23b
L. Piemontese,
23a
R. Baldini-Ferroli,
24
A. Calcaterra,
24
R. de Sangro,
24
G. Finocchiaro,
24
M. Nicolaci,
24
S. Pacetti,
24
P. Patteri,
24
I. M. Peruzzi,
24,
†
M. Piccolo,
24
M. Rama,
24
A. Zallo,
24
R. Contri,
25a,25b
E. Guido,
25a,25b
M. Lo Vetere,
25a,25b
M. R. Monge,
25a,25b
S. Passaggio,
25a
C. Patrignani,
25a,25b
E. Robutti,
25a
S. Tosi,
25a,25b
B. Bhuyan,
26
V. Prasad,
26
C. L. Lee,
27
M. Morii,
27
A. Adametz,
28
J. Marks,
28
S. Schenk,
28
U. Uwer,
28
F. U. Bernlochner,
29
M. Ebert,
29
H. M. Lacker,
29
T. Lueck,
29
A. Volk,
29
P. D. Dauncey,
30
M. Tibbetts,
30
P. K. Behera,
31
U. Mallik,
31
C. Chen,
32
J. Cochran,
32
H. B. Crawley,
32
L. Dong,
32
W. T. Meyer,
32
S. Prell,
32
E. I. Rosenberg,
32
A. E. Rubin,
32
Y. Y. Gao,
33
A. V. Gritsan,
33
Z. J. Guo,
33
N. Arnaud,
34
M. Davier,
34
D. Derkach,
34
J. Firmino da Costa,
34
G. Grosdidier,
34
F. Le Diberder,
34
A. M. Lutz,
34
B. Malaescu,
34
A. Perez,
34
P. Roudeau,
34
M. H. Schune,
34
J. Serrano,
34
V. Sordini,
34,
‡
A. Stocchi,
34
L. Wang,
34
G. Wormser,
34
D. J. Lange,
35
D. M. Wright,
35
I. Bingham,
36
C. A. Chavez,
36
J. P. Coleman,
36
J. R. Fry,
36
E. Gabathuler,
36
R. Gamet,
36
D. E. Hutchcroft,
36
D. J. Payne,
36
C. Touramanis,
36
A. J. Bevan,
37
F. Di Lodovico,
37
R. Sacco,
37
M. Sigamani,
37
G. Cowan,
38
S. Paramesvaran,
38
A. C. Wren,
38
D. N. Brown,
39
C. L. Davis,
39
A. G. Denig,
40
M. Fritsch,
40
W. Gradl,
40
A. Hafner,
40
K. E. Alwyn,
41
D. Bailey,
41
R. J. Barlow,
41
G. Jackson,
41
G. D. Lafferty,
41
T. J. West,
41
J. Anderson,
42
R. Cenci,
42
A. Jawahery,
42
D. A. Roberts,
42
G. Simi,
42
J. M. Tuggle,
42
C. Dallapiccola,
43
E. Salvati,
43
R. Cowan,
44
D. Dujmic,
44
P. H. Fisher,
44
G. Sciolla,
44
M. Zhao,
44
D. Lindemann,
45
P. M. Patel,
45
S. H. Robertson,
45
M. Schram,
45
P. Biassoni,
46a,46b
A. Lazzaro,
46a,46b
V. Lombardo,
46a
F. Palombo,
46a,46b
S. Stracka,
46a,46b
L. Cremaldi,
47
R. Godang,
47,
x
R. Kroeger,
47
P. Sonnek,
47
D. J. Summers,
47
X. Nguyen,
48
M. Simard,
48
P. Taras,
48
G. De Nardo,
49a,49b
D. Monorchio,
49a,49b
G. Onorato,
49a,49b
C. Sciacca,
49a,49b
G. Raven,
50
H. L. Snoek,
50
C. P. Jessop,
51
K. J. Knoepfel,
51
J. M. LoSecco,
51
W. F. Wang,
51
L. A. Corwin,
52
K. Honscheid,
52
R. Kass,
52
J. P. Morris,
52
A. M. Rahimi,
52
N. L. Blount,
53
J. Brau,
53
R. Frey,
53
O. Igonkina,
53
J. A. Kolb,
53
R. Rahmat,
53
N. B. Sinev,
53
D. Strom,
53
J. Strube,
53
E. Torrence,
53
G. Castelli,
54a,54b
E. Feltresi,
54a,54b
N. Gagliardi,
54a,54b
M. Margoni,
54a,54b
M. Morandin,
54a
M. Posocco,
54a
M. Rotondo,
54a
F. Simonetto,
54a,54b
R. Stroili,
54a,54b
E. Ben-Haim,
55
G. R. Bonneaud,
55
H. Briand,
55
G. Calderini,
55
J. Chauveau,
55
O. Hamon,
55
Ph. Leruste,
55
G. Marchiori,
55
J. Ocariz,
55
J. Prendki,
55
S. Sitt,
55
M. Biasini,
56a,56b
E. Manoni,
56a,56b
A. Rossi,
56a,56b
C. Angelini,
57a,57b
G. Batignani,
57a,57b
S. Bettarini,
57a,57b
M. Carpinelli,
57a,57b,
k
G. Casarosa,
57a,57b
A. Cervelli,
57a,57b
F. Forti,
57a,57b
M. A. Giorgi,
57a,57b
A. Lusiani,
57a,57c
N. Neri,
57a,57b
E. Paoloni,
57a,57b
G. Rizzo,
57a,57b
J. J. Walsh,
57a
D. Lopes Pegna,
58
C. Lu,
58
J. Olsen,
58
A. J. S. Smith,
58
A. V. Telnov,
58
F. Anulli,
59a
E. Baracchini,
59a,59b
G. Cavoto,
59a
R. Faccini,
59a,59b
F. Ferrarotto,
59a
F. Ferroni,
59a,59b
M. Gaspero,
59a,59b
L. Li Gioi,
59a
M. A. Mazzoni,
59a
G. Piredda,
59a
F. Renga,
59a,59b
T. Hartmann,
60
T. Leddig,
60
H. Schro
̈
der,
60
R. Waldi,
60
T. Adye,
61
B. Franek,
61
E. O. Olaiya,
61
F. F. Wilson,
61
S. Emery,
62
G. Hamel de Monchenault,
62
G. Vasseur,
62
Ch. Ye
`
che,
62
M. Zito,
62
M. T. Allen,
63
D. Aston,
63
D. J. Bard,
63
R. Bartoldus,
63
J. F. Benitez,
63
C. Cartaro,
63
M. R. Convery,
63
J. Dorfan,
63
G. P. Dubois-Felsmann,
63
W. Dunwoodie,
63
R. C. Field,
63
M. Franco Sevilla,
63
B. G. Fulsom,
63
A. M. Gabareen,
63
M. T. Graham,
63
P. Grenier,
63
C. Hast,
63
W. R. Innes,
63
M. H. Kelsey,
63
H. Kim,
63
P. Kim,
63
M. L. Kocian,
63
D. W. G. S. Leith,
63
S. Li,
63
B. Lindquist,
63
S. Luitz,
63
V. Luth,
63
H. L. Lynch,
63
D. B. MacFarlane,
63
H. Marsiske,
63
D. R. Muller,
63
H. Neal,
63
S. Nelson,
63
C. P. O’Grady,
63
I. Ofte,
63
M. Perl,
63
T. Pulliam,
63
B. N. Ratcliff,
63
A. Roodman,
63
PHYSICAL REVIEW D
82,
072004 (2010)
1550-7998
=
2010
=
82(7)
=
072004(20)
072004-1
Ó
2010 The American Physical Society
A. A. Salnikov,
63
V. Santoro,
63
R. H. Schindler,
63
J. Schwiening,
63
A. Snyder,
63
D. Su,
63
M. K. Sullivan,
63
S. Sun,
63
K. Suzuki,
63
J. M. Thompson,
63
J. Va’vra,
63
A. P. Wagner,
63
M. Weaver,
63
C. A. West,
63
W. J. Wisniewski,
63
M. Wittgen,
63
D. H. Wright,
63
H. W. Wulsin,
63
A. K. Yarritu,
63
C. C. Young,
63
V. Ziegler,
63
X. R. Chen,
64
W. Park,
64
M. V. Purohit,
64
R. M. White,
64
J. R. Wilson,
64
S. J. Sekula,
65
M. Bellis,
66
P. R. Burchat,
66
A. J. Edwards,
66
T. S. Miyashita,
66
S. Ahmed,
67
M. S. Alam,
67
J. A. Ernst,
67
B. Pan,
67
M. A. Saeed,
67
S. B. Zain,
67
N. Guttman,
68
A. Soffer,
68
P. Lund,
69
S. M. Spanier,
69
R. Eckmann,
70
J. L. Ritchie,
70
A. M. Ruland,
70
C. J. Schilling,
70
R. F. Schwitters,
70
B. C. Wray,
70
J. M. Izen,
71
X. C. Lou,
71
F. Bianchi,
72a,72b
D. Gamba,
72a,72b
M. Pelliccioni,
72a,72b
M. Bomben,
73a,73b
L. Lanceri,
73a,73b
L. Vitale,
73a,73b
N. Lopez-March,
74
F. Martinez-Vidal,
74
D. A. Milanes,
74
A. Oyanguren,
74
J. Albert,
75
Sw. Banerjee,
75
H. H. F. Choi,
75
K. Hamano,
75
G. J. King,
75
R. Kowalewski,
75
M. J. Lewczuk,
75
I. M. Nugent,
75
J. M. Roney,
75
R. J. Sobie,
75
T. J. Gershon,
76
P. F. Harrison,
76
T. E. Latham,
76
E. M. T. Puccio,
76
H. R. Band,
77
S. Dasu,
77
K. T. Flood,
77
Y. Pan,
77
R. Prepost,
77
C. O. Vuosalo,
77
and S. L. Wu
77
1
Laboratoire d’Annecy-le-Vieux de Physique des Particules (LAPP),
Universite
́
de Savoie, CNRS/IN2P3, F-74941 Annecy-Le-Vieux, France
2
Universitat de Barcelona, Facultat de Fisica, Departament ECM, E-08028 Barcelona, Spain
3a
INFN Sezione di Bari, I-70126 Bari, Italy;
3b
Dipartimento di Fisica, Universita
`
di Bari, I-70126 Bari, Italy
4
University of Bergen, Institute of Physics, N-5007 Bergen, Norway
5
Lawrence Berkeley National Laboratory and University of California, Berkeley, California 94720, USA
6
University of Birmingham, Birmingham, B15 2TT, United Kingdom
7
Ruhr Universita
̈
t Bochum, Institut fu
̈
r Experimentalphysik 1, D-44780 Bochum, Germany
8
University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z1
9
Brunel University, Uxbridge, Middlesex UB8 3PH, United Kingdom
10
Budker Institute of Nuclear Physics, Novosibirsk 630090, Russia
11
University of California at Irvine, Irvine, California 92697, USA
12
University of California at Riverside, Riverside, California 92521, USA
13
University of California at Santa Barbara, Santa Barbara, California 93106, USA
14
University of California at Santa Cruz, Institute for Particle Physics, Santa Cruz, California 95064, USA
15
California Institute of Technology, Pasadena, California 91125, USA
16
University of Cincinnati, Cincinnati, Ohio 45221, USA
17
University of Colorado, Boulder, Colorado 80309, USA
18
Colorado State University, Fort Collins, Colorado 80523, USA
19
Technische Universita
̈
t Dortmund, Fakulta
̈
t Physik, D-44221 Dortmund, Germany
20
Technische Universita
̈
t Dresden, Institut fu
̈
r Kern- und Teilchenphysik, D-01062 Dresden, Germany
21
Laboratoire Leprince-Ringuet, CNRS/IN2P3, Ecole Polytechnique, F-91128 Palaiseau, France
22
University of Edinburgh, Edinburgh EH9 3JZ, United Kingdom
23a
INFN Sezione di Ferrara, I-44100 Ferrara, Italy;
23b
Dipartimento di Fisica, Universita
`
di Ferrara, I-44100 Ferrara, Italy
24
INFN Laboratori Nazionali di Frascati, I-00044 Frascati, Italy
25a
INFN Sezione di Genova, I-16146 Genova, Italy;
25b
Dipartimento di Fisica, Universita
`
di Genova, I-16146 Genova, Italy
26
Indian Institute of Technology Guwahati, Guwahati, Assam, 781 039, India
27
Harvard University, Cambridge, Massachusetts 02138, USA
28
Universita
̈
t Heidelberg, Physikalisches Institut, Philosophenweg 12, D-69120 Heidelberg, Germany
29
Humboldt-Universita
̈
t zu Berlin, Institut fu
̈
r Physik, Newtonstr. 15, D-12489 Berlin, Germany
30
Imperial College London, London, SW7 2AZ, United Kingdom
31
University of Iowa, Iowa City, Iowa 52242, USA
32
Iowa State University, Ames, Iowa 50011-3160, USA
33
Johns Hopkins University, Baltimore, Maryland 21218, USA
34
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
35
Lawrence Livermore National Laboratory, Livermore, California 94550, USA
36
University of Liverpool, Liverpool L69 7ZE, United Kingdom
37
Queen Mary, University of London, London, E1 4NS, United Kingdom
38
University of London, Royal Holloway and Bedford New College, Egham, Surrey TW20 0EX, United Kingdom
39
University of Louisville, Louisville, Kentucky 40292, USA
40
Johannes Gutenberg-Universita
̈
t Mainz, Institut fu
̈
r Kernphysik, D-55099 Mainz, Germany
41
University of Manchester, Manchester M13 9PL, United Kingdom
42
University of Maryland, College Park, Maryland 20742, USA
P. DEL AMO SANCHEZ
et al.
PHYSICAL REVIEW D
82,
072004 (2010)
072004-2
43
University of Massachusetts, Amherst, Massachusetts 01003, USA
44
Massachusetts Institute of Technology, Laboratory for Nuclear Science, Cambridge, Massachusetts 02139, USA
45
McGill University, Montre
́
al, Que
́
bec, Canada H3A 2T8
46a
INFN Sezione di Milano, I-20133 Milano, Italy;
46b
Dipartimento di Fisica, Universita
`
di Milano, I-20133 Milano, Italy
47
University of Mississippi, University, Mississippi 38677, USA
48
Universite
́
de Montre
́
al, Physique des Particules, Montre
́
al, Que
́
bec, Canada H3C 3J7
49a
INFN Sezione di Napoli, I-80126 Napoli, Italy;
49b
Dipartimento di Scienze Fisiche, Universita
`
di Napoli Federico II, I-80126 Napoli, Italy
50
NIKHEF, National Institute for Nuclear Physics and High Energy Physics, NL-1009 DB Amsterdam, The Netherlands
51
University of Notre Dame, Notre Dame, Indiana 46556, USA
52
Ohio State University, Columbus, Ohio 43210, USA
53
University of Oregon, Eugene, Oregon 97403, USA
54a
INFN Sezione di Padova, I-35131 Padova, Italy;
54b
Dipartimento di Fisica, Universita
`
di Padova, I-35131 Padova, Italy
55
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
56a
INFN Sezione di Perugia, I-06100 Perugia, Italy;
56b
Dipartimento di Fisica, Universita
`
di Perugia, I-06100 Perugia, Italy
57a
INFN Sezione di Pisa, I-56127 Pisa, Italy;
57b
Dipartimento di Fisica, Universita
`
di Pisa, I-56127 Pisa, Italy;
57c
Scuola Normale Superiore di Pisa, I-56127 Pisa, Italy
58
Princeton University, Princeton, New Jersey 08544, USA
59a
INFN Sezione di Roma, I-00185 Roma, Italy;
59b
Dipartimento di Fisica, Universita
`
di Roma La Sapienza, I-00185 Roma, Italy
60
Universita
̈
t Rostock, D-18051 Rostock, Germany
61
Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, OX11 0QX, United Kingdom
62
CEA, Irfu, SPP, Centre de Saclay, F-91191 Gif-sur-Yvette, France
63
SLAC National Accelerator Laboratory, Stanford, California 94309 USA
64
University of South Carolina, Columbia, South Carolina 29208, USA
65
Southern Methodist University, Dallas, Texas 75275, USA
66
Stanford University, Stanford, California 94305-4060, USA
67
State University of New York, Albany, New York 12222, USA
68
Tel Aviv University, School of Physics and Astronomy, Tel Aviv, 69978, Israel
69
University of Tennessee, Knoxville, Tennessee 37996, USA
70
University of Texas at Austin, Austin, Texas 78712, USA
71
University of Texas at Dallas, Richardson, Texas 75083, USA
72a
INFN Sezione di Torino, I-10125 Torino, Italy;
72b
Dipartimento di Fisica Sperimentale, Universita
`
di Torino, I-10125 Torino, Italy
73a
INFN Sezione di Trieste, I-34127 Trieste, Italy;
73b
Dipartimento di Fisica, Universita
`
di Trieste, I-34127 Trieste, Italy
74
IFIC, Universitat de Valencia-CSIC, E-46071 Valencia, Spain
75
University of Victoria, Victoria, British Columbia, Canada V8W 3P6
76
Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom
77
University of Wisconsin, Madison, Wisconsin 53706, USA
(Received 6 July 2010; published 8 October 2010)
Using the entire sample of
467
10
6
ð
4
S
Þ!
B
B
decays collected with the
BABAR
detector at the
PEP-II asymmetric-energy
B
factory at the SLAC National Accelerator Laboratory, we perform an analysis
of
B
!
DK
decays, using decay modes in which the neutral
D
meson decays to either
CP
-eigenstates
or non-
CP
-eigenstates. We measure the partial decay rate charge asymmetries for
CP
-even and
CP
-odd
D
final states to be
A
CP
þ
¼
0
:
25
0
:
06
0
:
02
and
A
CP
¼
0
:
09
0
:
07
0
:
02
, respectively, where the
first error is the statistical and the second is the systematic uncertainty. The parameter
A
CP
þ
is different
from zero with a significance of 3.6 standard deviations, constituting evidence for direct
CP
violation. We
*
Now at Temple University, Philadelphia, Pennsylvania 19122, USA
†
Also with Universita
`
di Perugia, Dipartimento di Fisica, Perugia, Italy
‡
Also with Universita
`
di Roma La Sapienza, I-00185 Roma, Italy
x
Now at University of South Alabama, Mobile, Alabama 36688, USA
k
Also with Universita
`
di Sassari, Sassari, Italy
MEASUREMENT OF
CP
OBSERVABLES IN
...
PHYSICAL REVIEW D
82,
072004 (2010)
072004-3
also measure the ratios of the charged-averaged
B
partial decay rates in
CP
and non-
CP
decays,
R
CP
þ
¼
1
:
18
0
:
09
0
:
05
and
R
CP
¼
1
:
07
0
:
08
0
:
04
. We infer frequentist confidence intervals for the
angle
of the unitarity triangle, for the strong phase difference
B
, and for the amplitude ratio
r
B
, which
are related to the
B
!
DK
decay amplitude by
r
B
e
i
ð
B
Þ
¼
A
ð
B
!
D
0
K
Þ
=A
ð
B
!
D
0
K
Þ
.
Including statistical and systematic uncertainties, we obtain
0
:
24
<r
B
<
0
:
45
(
0
:
06
<r
B
<
0
:
51
) and,
modulo 180
,
11
:
3
<<
22
:
7
or
80
:
8
<<
99
:
2
or
157
:
3
<<
168
:
7
(
7
:
0
<<
173
:
0
)at
the 68% (95%) confidence level.
DOI:
10.1103/PhysRevD.82.072004
PACS numbers: 13.25.Hw, 12.15.Hh, 11.30.Er
I. INTRODUCTION
In the standard model (SM) of fundamental particles,
CP
violation in weak interactions is allowed by a single,
irreducible phase in the
3
3
Cabibbo-Kobayashi-
Maskawa (CKM) quark flavor-mixing matrix [
1
,
2
]. The
unitarity of the CKM matrix,
V
, implies a set of relations
among its elements
V
ij
, in particular, the condition
V
ud
V
ub
þ
V
cd
V
cb
þ
V
td
V
tb
¼
0
, which can be depicted
in the complex plane as a ‘‘unitarity’’ triangle, whose sides
and angles are related to the magnitudes and phases of the
six elements
V
id
and
V
ib
, where
i
¼
u
,
c
,
t
. Overcon-
straining the unitarity triangle by means of precise mea-
surements of all its sides and angles allows tests of whether
the CKM mechanism is the correct description of
CP
violation. Any inconsistencies among the various experi-
mental constraints would reveal effects of physics beyond
the standard model.
After a decade of successful operation and a total of
about
1
:
3
10
9
B
B
pairs collected by the
BABAR
and
Belle experiments, the three CKM angles have been mea-
sured with varied precision. The angle
has been
measured with the highest precision, to around 1
, using
B
0
!ð
c
c
Þ
K
ðÞ
0
decays. Using a variety of two-body
B
decays (
B
!
,
,
and
a
1
ð
1260
Þ
) the angle
has been measured to a precision of around 4
. The angle
has a relatively large uncertainty, around 14
, compared
with
and
. The lack of precision in our knowledge of
reflects the difficulty in measuring this angle. The uncer-
tainties of the CKM angles quoted in this paragraph are
taken from [
3
].
Several techniques for measuring
in a theoretically
clean way are based on
B
meson decays to open-charm
final states,
D
ðÞ
0
X
s
and
D
ðÞ
0
X
s
(
X
s
¼
K
ðÞ
,
K
ðÞ
0
). In
these decays, the interference between the
b
!
c
us
and
b
!
u
cs
tree amplitudes, when the
D
0
and
D
0
decay to a
common final state, leads to observables that depend on the
relative weak phase
. The size of the interference also
depends on the magnitude of the ratio
r
B
and the relative
strong phase
B
of the two amplitudes, which cannot be
precisely calculated from theory. They can be extracted
directly from data by simultaneously reconstructing sev-
eral related
B
!
DK
decays. Many methods have been
proposed to extract
from
B
decays using
D
ðÞ
K
ðÞ
and
D
ðÞ
K
ðÞ
0
final states (here and in the following
D
refers to
any admixture of the neutral
D
0
meson and its
CP
-conjugate
D
0
). The three methods that have been
used most productively to date are the ‘‘GLW’’ method
[
4
,
5
], based on Cabibbo-suppressed
D
decays to
CP
-eigenstates, such as
K
þ
K
or
K
0
S
0
; the ‘‘ADS’’
method [
6
,
7
], where the
D
is reconstructed in Cabibbo-
favored and doubly-Cabibbo-suppressed final states such
as
K
; and the ‘‘GGSZ’’ method [
8
], which studies the
Dalitz-plot distribution of the products of
D
decays to
multibody self-conjugate final states, such as
K
0
S
þ
.
A common problem with these methods is the small overall
branching fraction of these decays ranging from
5
10
6
to
5
10
9
. Therefore a precise determination of
re-
quires a very large data sample.
BABAR
has published
several
related measurements: GLW analyses of
B
!
DK
[
9
],
D
K
[
10
] and
DK
[
11
] decays; ADS analy-
ses of
B
!
D
ðÞ
K
[
12
,
13
],
DK
[
11
] and
B
0
!
DK
0
[
14
]; and GGSZ analyses of
B
!
D
ðÞ
K
ðÞ
[
15
,
16
] and
B
0
!
DK
0
decays [
17
]. To date, the single most precise
experimental determination of
from
BABAR
is
¼
ð
68
14
4
3
Þ
and
39
<<
98
, obtained from
the GGSZ analysis of
B
!
D
ðÞ
K
ðÞ
decays [
16
]. In
this measurement, the first error represents the statistical
uncertainty, the second is the experimental systematic
uncertainty, and the third reflects the uncertainty on the
description of the
D
Dalitz-plot distributions.
II. GLW ANALYSIS OF
B
!
DK
DECAYS
In this paper we present the update of the GLW analysis
of
B
!
DK
decays based on the full
BABAR
data set
collected near the
ð
4
S
Þ
resonance. In addition to a 22%
increase in statistics of the data sample, this study benefits
from other significant improvements compared to our pre-
vious result [
9
]:
(i) More refined charged track reconstruction and par-
ticle identification algorithms, with higher purity and
efficiency, have been employed;
(ii) The event shape variable
F
, used to discriminate
the signal from the continuum
e
þ
e
!
q
q
back-
ground (described in detail in Sec.
IV
) has been
removed from the selection criteria and has instead
been included in the final fit to the selected
B
candidates. This allows us to increase the signal
efficiency by 40% to 60%. At the same time it
provides a larger sample of continuum background
P. DEL AMO SANCHEZ
et al.
PHYSICAL REVIEW D
82,
072004 (2010)
072004-4
events, thus allowing for the determination of the
background properties directly from data (see
Sec.
V
);
(iii) Better kaon/pion separation, which is needed to
distinguish
B
!
DK
candidates from the 12
times more abundant
B
!
D
decays, is
achieved through the use of a global likelihood
based not only on the Cherenkov angle
C
recon-
structed by the Cherenkov detector, but also on the
specific energy loss
dE=dx
measured by the track-
ing devices. The inclusion of
dE=dx
in the like-
lihood increases the kaon identification efficiency
and decreases the pion misidentification both at low
momentum and outside of the geometrical accep-
tance of the Cherenkov detector (which is 10%
lower than the acceptance of the tracking devices).
In order to determine
from
B
!
DK
decays with
the GLW method, we measure the two direct-
CP
-violating
partial decay rate asymmetries,
A
CP
ð
B
!
D
CP
K
Þ
ð
B
þ
!
D
CP
K
þ
Þ
ð
B
!
D
CP
K
Þþ
ð
B
þ
!
D
CP
K
þ
Þ
;
(1)
and the two ratios of charge averaged partial rates using
D
decays to
CP
and flavor eigenstates,
R
CP
2
ð
B
!
D
CP
K
Þþ
ð
B
þ
!
D
CP
K
þ
Þ
ð
B
!
D
0
K
Þþ
ð
B
þ
!
D
0
K
þ
Þ
;
(2)
where
D
CP
refer to the
CP
eigenstates of the
D
meson
system. We then extract
, together with the other two
unknowns
r
B
and
B
, by means of a frequentist procedure,
which exploits the following relations [
4
,
5
], neglecting
D
0
-
D
0
mixing [
18
]:
R
CP
¼
1
þ
r
2
B
2
r
B
cos
B
cos
;
(3)
A
CP
¼
2
r
B
sin
B
sin
1
þ
r
2
B
2
r
B
cos
B
cos
:
(4)
Here,
r
B
j
A
ð
B
!
D
0
K
Þ
=A
ð
B
!
D
0
K
Þj
is the
magnitude of the ratio of the amplitudes for
B
!
D
0
K
and
B
!
D
0
K
and
B
the difference of their strong
phases. Taking into account the CKM factor (
j
V
ub
V
cs
=
V
cb
V
us
j
0
:
4
) and color-suppression of the
B
!
D
0
K
amplitude,
r
B
is expected to be around 0.1. The
current world averages for the
B
!
DK
GLW observ-
ables from the measurements in [
9
,
19
,
20
] are summarized
in Table
I
. The world averages for the parameters
r
B
and
B
are
r
B
¼
0
:
104
þ
0
:
015
0
:
025
and
B
¼ð
117
þ
17
24
Þ
at 68% confi-
dence level (CL) [
3
].
To reduce the systematic uncertainties from branching
fractions and reconstruction efficiencies of different
D
channels appearing in the numerator and denominator of
Eq. (
2
), we approximate
R
CP
with the double ratios
R
CP
R
K=
R
K=
;
(5)
where
R
K=
ð
B
!
D
CP
K
Þþ
ð
B
þ
!
D
CP
K
þ
Þ
ð
B
!
D
CP
Þþ
ð
B
þ
!
D
CP
þ
Þ
;
(6)
and
R
K=
ð
B
!
D
0
K
Þþ
ð
B
þ
!
D
0
K
þ
Þ
ð
B
!
D
0
Þþ
ð
B
þ
!
D
0
þ
Þ
:
(7)
Equation (
5
) would be exact in the limit in which the
Cabibbo-suppressed contributions to the
B
!
D
am-
plitudes vanish, as well as terms proportional to
r
B
r
D
5
10
3
, as we will discuss in Sec.
VII
. This approxima-
tion results in a systematic uncertainty on the final values
of
R
CP
.
The paper is organized as follows. In Sec.
III
we de-
scribe the data sample used for these measurements and the
main features of the
BABAR
detector and of the PEP-II
storage rings. In Sec.
IV
we summarize the procedure
adopted to select
B
!
Dh
candidates and suppress the
main backgrounds. In Sec.
V
we introduce the simulta-
neous extended maximum likelihood fit used to extract the
observables
R
CP
and
A
CP
. In Sec.
VI
we explain how, by
applying the same fit procedure to selected control
samples, we estimate the irreducible background present
in the final samples. A discussion of the sources of system-
atic uncertainties and the evaluation of the uncertainties is
presented in Sec.
VII
. Section
VIII
lists the final results on
the GLW observables
R
CP
and
A
CP
, including statistical
and systematic uncertainties. It also contains a description
of the statistical method used to construct frequentist
confidence intervals for the parameters
,
B
, and
r
B
.
Section
IX
gives a summary of our results.
III. DATA SAMPLE AND DETECTOR
The measurements presented in this paper use the entire
B
B
data sample collected with the
BABAR
detector at the
PEP-II asymmetric-energy
B
factory at the SLAC National
Accelerator Laboratory. The
B
B
pairs are produced from
the decays of
ð
4
S
Þ
mesons that originate in collisions
of 9.0 GeV electrons and 3.1 GeV positrons (
ffiffiffi
s
p
¼
10
:
58 GeV
¼
M
ð
4
S
Þ
c
2
Þ
. In total,
ð
467
5
Þ
10
6
B
B
pairs, approximately equally divided into
B
0
B
0
and
B
þ
B
, have been collected in the years from 1999 until
early 2008. The
B
meson pairs are produced almost at
rest in the
ð
4
S
Þ
center-of-mass (CM) frame, but the
TABLE I. World averages at 68% confidence level [
21
] for the
GLW observables in
B
!
DK
decays.
CP
of the
DR
CP
A
CP
þ
11
:
10
0
:
09
0
:
24
0
:
07
11
:
06
0
:
10
0
:
10
0
:
08
MEASUREMENT OF
CP
OBSERVABLES IN
...
PHYSICAL REVIEW D
82,
072004 (2010)
072004-5
asymmetric beam energies boost them in the laboratory
frame by
ð
Þ
CM
0
:
56
.
The
BABAR
detector is described in detail elsewhere
[
22
]. Primary and secondary vertex reconstruction and
charged-particle tracking are provided by a five-layer
double-sided silicon vertex tracker and a 40-layer drift
chamber. Charged particle identification (PID) is provided
by measurement of specific ionization energy loss in the
tracking devices and of the Cherenkov radiation cone in a
ring-imaging detector. Photons and electrons are identified
by combining the information from the tracking devices
and the energy deposits in the electromagnetic calorimeter,
which consists of 6580 thallium-doped CsI crystals. These
systems are located inside a 1.5 T solenoidal superconduct-
ing magnet. Finally, the flux return of the magnet is in-
strumented with resistive plate chambers and limited
streamer tubes in order to discriminate muons from pions.
We use the
GEANT4
[
23
] software toolkit to simulate inter-
actions of particles in the detector, taking into account the
varying accelerator and detector conditions.
IV. EVENT SELECTION
We reconstruct
B
!
Dh
decays, where the charged
track
h
is either a kaon or a pion. Neutral
D
mesons are
reconstructed in the
CP
-even eigenstates
þ
and
K
K
þ
(
D
CP
þ
), in the
CP
-odd eigenstates
K
0
S
0
,
K
0
S
and
K
0
S
!
(
D
CP
), and in the non-
CP
-eigenstate
K
þ
(
D
0
from
B
!
D
0
h
)or
K
þ
(
D
0
from
B
þ
!
D
0
h
þ
).
CP
violation in the
K
0
-
K
0
system is neglected, i.e. the
K
0
S
is assumed to be a pure
CP
¼þ
1
eigenstate. The
D
CP
daughters are reconstructed in the decay modes
K
0
S
!
þ
,
!
K
þ
K
and
!
!
þ
0
.
We optimize all our event selection requirements by
maximizing the significance of the expected
B
!
DK
signal yield, defined as
N
sig
=
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
N
sig
þ
N
bkg
p
, where
N
sig
(
N
bkg
) is the expected signal (background) yield. The
optimization is done for each
D
decay channel using
simulated signal and background events, which are gener-
ated with the
EVTGEN
software package [
24
].
Neutral pions are reconstructed by combining pairs
of photon candidates with energy deposits larger than
30 MeV that are not matched to charged tracks and whose
energy deposition profile is consistent with that expected
from a photon. The photon pair invariant mass is required
to differ from the nominal
0
mass [
25
] by less than
2.5 times its resolution (
6 MeV
=c
2
) and the total
0
energy in the laboratory frame must be greater than
240 MeV for
D
!
K
0
S
0
and 210 MeV for
!
!
þ
0
.
Neutral kaons are reconstructed from pairs of oppositely
charged tracks with invariant mass within
2
:
5
(
2
:
1 MeV
=c
2
) of the nominal
K
0
S
mass [
25
]. The ratio
between the
K
0
S
signed 3-dimensional flight length and its
uncertainty, determined from the position of the
K
0
S
and the
D
decay vertices and the
K
0
S
momentum direction, must be
greater than 1.9, 2.0, and 2.2 for
D
!
K
0
S
0
,
D
!
K
0
S
,
and
D
!
K
0
S
!
, respectively.
The
candidates are reconstructed from pairs of oppo-
sitely charged tracks passing kaon identification criteria
with typical kaon selection efficiency of
98%
and pion
misidentification of
15%
. The two tracks are assigned
the kaon mass hypothesis and their invariant mass is re-
quired to be within
6
:
5 MeV
=c
2
of the nominal
mass
[
25
] (the resolution is
¼
1
:
0 MeV
=c
2
and the natural
width is
¼
4
:
3 MeV
). We also require that the helicity
angle
H
between the flight direction of one of the two
kaons and the
D
flight direction, in the
rest frame,
satisfies the condition
j
cos
H
j
>
0
:
4
. This requirement
exploits the fact that in
D
!
K
0
S
decays the
is pro-
duced in a longitudinally polarized state, thus
cos
H
follows a
cos
2
H
distribution, while in
candidates
not from
D
!
K
0
S
decays,
cos
H
is approximately uni-
formly distributed.
The
!
candidates are reconstructed from
þ
0
combinations with invariant mass within
17 MeV
=c
2
(
2
!
) of the nominal
!
mass [
25
] (the resolution is
¼
6
:
9 MeV
=c
2
). The charged pion candidates are required to
pass pion identification criteria with pion selection effi-
ciency around 98% and kaon misidentification rate around
12%. To improve the
!
momentum resolution, the invari-
ant mass of the two photons forming the
0
candidate is
constrained to the nominal
0
mass. We define
N
as the
angle between the normal to the
!
decay plane and the
D
momentum in the
!
rest frame, and
as the angle
between the flight direction of one of the three pions in
the
!
rest frame and the flight direction of one of the other
two pions in the two-pion rest frame. The quantities
cos
N
and
cos
follow
cos
2
N
and
ð
1
cos
2
Þ
distribu-
tions, respectively, for the signal, and are almost uniformly
distributed for wrongly reconstructed
!
candidates. We
require the product
cos
2
N
sin
2
>
0
:
046
.
Neutral
D
candidates are formed from two-body combi-
nations of
K
,
,
K
0
S
,
0
,
and
!
candidates consistent
with one of the six
D
decay channels under study. To
improve the
D
CP
momentum resolution, the invariant
masses of the
0
and
K
0
S
daughters are constrained to the
nominal
0
and
K
0
S
masses. To suppress poorly recon-
structed
D
candidates and candidates from random combi-
nations, we perform a geometric fit of the
D
daughters to a
common origin, and reject
D
candidates for which the
2
probability of the vertex fit is lower than 0.01%. The
invariant mass of a
D
candidate
M
D
must be within a range
that corresponds to slightly more than twice the
M
D
reso-
lution: the range varies from about
6 MeV
=c
2
for the
K
0
S
channel to about
44 MeV
=c
2
for the
K
0
S
0
channel.
We apply the following particle identification criteria to the
charged daughters of the
D
meson: in
D
!
þ
, the
two-pion candidates must pass the same pion identification
criteria adopted in the reconstruction of
!
!
þ
0
;in
D
!
K
þ
K
, the two kaon candidates are required to pass
P. DEL AMO SANCHEZ
et al.
PHYSICAL REVIEW D
82,
072004 (2010)
072004-6
tighter kaon identification criteria than those applied to the
daughters (typical kaon selection efficiency around 94%,
and pion misidentification rate around 6%); in
D
!
K
þ
, the kaon candidate must pass the same kaon iden-
tification criteria required for the
daughters. In order to
reduce the large combinatorial background from random
combinations of tracks and photons in
e
þ
e
!
q
q
events
(
q
¼
u
,
d
,
s
,
c
), we put requirements on the cosine of the
D
decay angle,
j
cos
D
j
. We define
D
as the angle between
one of the
D
daughters in the
D
rest frame, and the
direction of the
D
meson in the
B
rest frame. Because of
angular momentum conservation we expect the distribu-
tion of
cos
D
to be uniform for
B
!
Dh
,
D
!
þ
and
D
!
K
0
S
0
signal events, while for
q
q
events the
distribution is strongly peaked at
1
. We require
j
cos
D
j
<
0
:
74
(0.99) for the
B
!
Dh
,
D
!
þ
(
D
!
K
0
S
0
) channel.
The invariant mass distributions of the reconstructed
D
candidates, after all the other selection criteria described in
this section have been applied, are shown in Fig.
1
.
We reconstruct
B
meson candidates by combining a
neutral
D
candidate with a track
h
. For the
D
!
K
mode, the charge of the track
h
must match that of the
kaon from the
D
meson decay. This selects
b
!
c
medi-
ated
B
decays
B
!
D
0
h
and
B
þ
!
D
0
h
þ
. The con-
tamination from
b
!
u
mediated
B
decays followed by
doubly-Cabibbo-suppressed
D
decay, i.e.
B
!
D
0
K
,
D
0
!
K
þ
, and from
D
0
-
D
0
mixing is negligible. In
the
B
!
Dh
,
D
!
þ
channel we require that the
invariant mass of the
ð
h
Þ
system is greater than
1
:
9 GeV
=c
2
to reject background from
B
!
D
0
,
D
0
!
K
þ
and
B
!
K
0
,
K
0
!
K
þ
decays
and their
CP
conjugates. Here
is the pion from the
D
and
h
is the track from the
B
candidate taken with the kaon
mass hypothesis. To improve the
B
momentum resolution,
the neutral
D
invariant mass is constrained to the nominal
D
0
mass [
25
] for all
D
decay channels.
We identify signal
B
!
DK
and
B
!
D
candidates
using two kinematic variables: the difference between the
CM energy of the
B
meson (
E
B
) and the beam energy,
)
2
(GeV/c
D
M
∆
-0.06 -0.04 -0.02
0
0.02
0.04
0.06
)
2
Events / (0.003 GeV/c
0
200
400
600
800
1000
Dh
→
B
-
K
+
K
→
D
)
2
(GeV/c
D
M
∆
-0.06 -0.04 -0.02
0
0.02
0.04
0.06
)
2
Events / (0.003 GeV/c
0
50
100
150
200
250
300
350
400
Dh
→
B
-
π
+
π
→
D
)
2
(GeV/c
D
M
∆
-0.05
0
0.05
)
2
Events / (0.004 GeV/c
0
100
200
300
400
500
Dh
→
B
0
π
S
0
K
→
D
)
2
(GeV/c
D
M
∆
-0.06 -0.04 -0.02
0
0.02
0.04
0.06
)
2
Events / (0.003 GeV/c
0
50
100
150
200
250
300
Dh
→
B
ω
S
0
K
→
D
)
2
(GeV/c
D
M
∆
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
)
2
Events / (0.001 GeV/c
0
20
40
60
80
100
120
140
160
Dh
→
B
φ
S
0
K
→
D
)
2
(GeV/c
D
M
∆
-0.04
-0.02
0
0.02
0.04
)
2
Events / (0.002 GeV/c
0
1000
2000
3000
4000
5000
Dh
→
B
π
K
→
D
FIG. 1 (color online). Distributions of the difference between the
D
candidate’s invariant mass and the nominal
D
0
mass [
25
], as
measured in the
B
!
Dh
samples. All selection criteria described in Sec.
IV
, except that on the
D
invariant mass
M
D
, have been
applied, including the
2
-based candidate selection. In addition we reduce the background by requiring the fit variables to satisfy
m
ES
>
5
:
27 GeV
=c
2
,
E>
0
:
05 GeV
, and
F
>
0
:
25
. The
M
D
selection requirements are depicted by the vertical lines.
MEASUREMENT OF
CP
OBSERVABLES IN
...
PHYSICAL REVIEW D
82,
072004 (2010)
072004-7
E
¼
E
B
ffiffiffi
s
p
=
2
;
(8)
and the beam-energy-substituted mass,
m
ES
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð
s=
2
þ
p
ee
p
B
Þ
2
=E
2
ee
p
2
B
q
;
(9)
where
ð
E
B
;
p
B
Þ
and
ð
E
ee
;
p
ee
Þ
are the four-momenta of the
B
meson and of the initial
e
þ
e
system, respectively,
measured in the laboratory frame. The
m
ES
distributions
for
B
!
Dh
signals are centered at the
B
mass [
25
],
have a root-mean-square of approximately
2
:
6 MeV
=c
2
,
and do not depend strongly on either the
D
decay mode or
the nature of the track
h
. In contrast, the
E
distributions
depend on the mass assigned to the track
h
. We evaluate
E
with the kaon mass hypothesis so that the peaks of the
distributions are centered near zero for
B
!
DK
events
and are shifted by approximately
þ
50 MeV
for
B
!
D
events. The
E
resolution depends on the kinematics
of the decay, and is typically 16 MeV for all
D
decay
modes under study after the
D
invariant mass is con-
strained to its nominal value. We retain
B
candidates
with
m
ES
and
E
within the intervals
5
:
20
<m
ES
<
5
:
29 GeV
=c
2
and
80
<
E<
120 MeV
, which define
the region for the fit described later.
In order to discriminate the signal from
e
þ
e
!
q
q
background events, denoted
q
q
in the following, we con-
struct a Fisher discriminant
F
based on the four event-
shape quantities
L
ROE
20
,
j
cos
T
j
,
j
cos
B
j
and
H
ROE
20
. These
quantities, evaluated in the CM frame, are defined as
(i)
L
ROE
20
¼
L
2
=L
0
is the ratio of the second and zeroth
event shape moments of the energy flow in the rest of
event (ROE), i.e. considering all the charged tracks
and neutral clusters in the event that are not used to
reconstruct the
B
candidate. They are defined as
L
2
¼
P
i
p
i
cos
2
i
and
L
0
¼
P
i
p
i
, where
p
i
are the
momenta and
i
the angles of the charged and neu-
tral particles in the ROE, with respect to the thrust
axis of the
B
candidate’s decay products. The thrust
axis is defined as the direction that maximizes the
sum of the longitudinal momenta of the particles
used to define it;
(ii)
T
is the angle between the thrust axis of the
B
candidate’s decay products and the beam axis;
(iii)
B
is the angle between the
B
candidate momentum
and the beam axis;
(iv)
H
ROE
20
¼
H
2
=H
0
is the ratio of the second and ze-
roth Fox-Wolfram moments
H
2
and
H
0
[
26
], com-
puted using charged tracks and photons in the ROE.
The quantity
F
is a linear combination of the four afore-
mentioned event-shape variables:
F
¼
c
1
L
ROE
20
þ
c
2
j
cos
T
jþ
c
3
j
cos
B
jþ
c
4
H
ROE
20
:
(10)
The values of the coefficients
c
i
are the ones which max-
imize the separation between simulated signal events and a
continuum background sample provided by off-resonance
data, taken
40 MeV
below the
ð
4
S
Þ
resonance. The
maximum likelihood fit described in Sec.
V
is restricted to
events with
F
within the interval
1
:
5
<
F
<
1
:
5
,to
remove poorly reconstructed candidates.
For events with multiple
B
!
Dh
candidates (about
16% of the selected events), we choose the
B
candidate
with the smallest
2
¼
P
c
ð
M
c
M
PDG
c
Þ
2
=
ð
2
M
c
þ
2
c
Þ
formed from the measured and true masses,
M
c
and
M
PDG
c
, of all the unstable particles
c
produced in the
B
decay tree (
D
,
0
,
K
0
S
,
,
!
), scaled by the sum in quad-
rature of the resolution
M
c
of the reconstructed mass and
the intrinsic width
c
. From simulated signal events, we
find that this algorithm has a probability to select the
correct candidate between 98.2% and 99.9% depending
on the
D
decay mode. We also find that the algorithm
has negligible effect on the
M
D
distributions.
We compare the distribution of each selection variable in
data and simulated events after the requirements on all
other variables have been applied. In order not to introduce
biases that may artificially enhance the signal yield, we
perform a blind study by explicitly removing, in this
comparison, events consistent with the
B
!
DK
signal,
i.e. those with
j
m
ES
m
B
j
<
10 MeV
=c
2
,
j
E
j
<
40 MeV
,
F
>
0
:
8
and track
h
passing kaon identifica-
tion criteria. We find excellent agreement between data and
simulated events, both for events consistent with the
B
!
D
signal (
j
m
ES
m
B
j
<
10 MeV
=c
2
,
j
E
50 MeV
j
<
40 MeV
,
F
>
0
:
8
and track
h
failing the
kaon identification criteria) and for backgroundlike events.
We correct for small differences in the means and widths of
the distributions of the invariant masses of the unstable
particles and of
m
ES
and
E
both when applying to data
the selection criteria obtained from simulated events and in
the final fit described in the next section.
The total reconstruction efficiencies, based on simulated
B
!
DK
events, are summarized in the second column
of Table
II
. For the reasons explained in Sec.
II
, the
efficiencies are 40% to 60% higher than in our previous
study of the same decay channels [
9
]. The efficiencies
obtained for
B
!
D
events from the simulation are
TABLE II. Reconstruction efficiency for
B
!
DK
from simu-
lated events. We also quote the efficiency and purity in a signal-
enriched subsample (see text for details).
D
0
mode
Efficiency after
full selection
Efficiency in
signal-enriched
subsample
Purity in
signal-enriched
subsample
K
þ
52%
22%
96%
K
þ
K
44%
18%
85%
þ
38%
17%
68%
K
0
S
0
24%
10%
83%
K
0
S
20%
9%
91%
K
0
S
!
10%
4%
71%
P. DEL AMO SANCHEZ
et al.
PHYSICAL REVIEW D
82,
072004 (2010)
072004-8
statistically consistent with those for
B
!
DK
, where
the
D
meson is reconstructed in the same final state. For
illustration purposes we define a signal-enriched sample
for each
D
decay mode, containing all
B
!
Dh
candi-
dates satisfying the criteria
40
<
E<
100 MeV
,
0
:
2
<
F
<
1
:
5
,
5
:
275
<m
ES
<
5
:
285 GeV
=c
2
, and whose
daughter track
h
passes charged kaon identification crite-
ria. The typical kaon efficiency is
77%
and the pion
misidentification rate is
2%
. The reconstruction efficien-
cies and the expected purities for the signal-enriched
subsamples, determined on simulated data, are listed in
Table
II
.
V. MAXIMUM LIKELIHOOD FIT
We measure
R
ðÞ
K=
and
A
CP
using simultaneous ex-
tended and unbinned maximum likelihood fits to the dis-
tributions of the three variables
E
,
m
ES
, and
F
of
B
candidates selected in data. The data set is split into 24
subgroups by means of three discrete variables: the charge
¼
1
of the reconstructed
B
meson (
2
subgroups);
the two-body
D
decay final state
X
(
6
), allowing for a
more accurate description of the corresponding probability
density functions compared to the larger
CP
subgroups;
and a PID variable denoting whether or not the track
h
from the
B
passes (
p
) or fails (
f
) charged kaon identifica-
tion criteria (
2
). The pion misidentification rate of these
criteria is determined directly from data as described later,
and is expected from simulation to be around 2%. The
corresponding kaon identification efficiency is
ð
77
1
Þ
%
,
as determined from the signal MC samples after weighting
the bidimensional distribution of the momentum and polar
angle of the track
h
by the ratio of the analogous distribu-
tions observed in MC and data kaon control samples. The
uncertainty on the kaon identification efficiency is domi-
nated by the systematic contribution from the uncertainties
on the weights. We perform in total three simultaneous fits
to these 24 subgroups: one fit for the two
CP
-even
D
final
states (8 subgroups), one for the three
CP
-odd
D
final
states (12 subgroups), and one for the
D
!
K
decay
(4 subgroups).
The likelihood function
L
for each of these simulta-
neous fits has the form
L
ð
~
Þ¼
e
N
N
n
n
!
Y
s
Y
N
s
i
¼
1
P
s
ð
m
ES
;i
;
E
i
;
F
i
;
~
Þ
;
(11)
where
s
ranges over the subgroups under consideration,
N
s
is the number of events in subgroup
s
,
n
is the total number
of events in the fit
n
¼
P
s
N
s
, and
N
is the expected number
of events. We minimize
ln
L
with respect to the set of fit
parameters
~
specified later. The probability
P
s;i
P
s
ð
m
ES
i
;
E
i
;
F
i
Þ
for an event
i
is the sum of six signal
and background components:
B
!
DK
signal,
B
!
D
signal, background candidates from
e
þ
e
!
q
q
events, irreducible background arising from charmless
B
!
XK
and
B
!
X
decays, and background can-
didates from other
B
B
events (reducible
B
B
background):
N
s
P
s;i
¼
N
D
s
P
D
s;i
þ
N
DK
s
P
DK
s;i
þ
N
q
q
s
P
q
q
s;i
þ
N
B
B
s
P
B
B
s;i
þ
N
X
s
P
X
s;i
þ
N
XK
s
P
XK
s;i
;
(12)
where the
N
j
s
are the expected yields in each component
j
.
In case of negligible correlations among the fit variables,
each probability density function (PDF)
P
factorizes as
P
ð
m
ES
;
E;
F
Þ¼
P
ð
m
ES
Þ
P
ð
E
Þ
P
ð
F
Þ
:
(13)
The irreducible
B
B
background originates from events
where a
B
meson decays to the same final state
Xh
as
the signal, but without the production of an intermediate
charmed meson in the decay chain. When exploiting
the
E
,
m
ES
, and
F
variables, this background is therefore
indistinguishable from the signal. As an example, the
decay
B
!
K
þ
K
K
(
X
¼
K
þ
K
) is an irreducible
background for
B
!
D
CP
þ
K
,
D
CP
þ
!
K
þ
K
.As
described later in Sec.
VI
, the irreducible background yield
can be estimated by studying sideband regions of the
D
candidate invariant mass distribution, and can then be fixed
in the final fit, where we assume
P
Dh
i
¼
P
Xh
i
.
We express the signal yield parameters
N
DK
s
and
N
D
s
through the
CP
asymmetries
A
X
DK
and
A
X
D
of
B
!
DK
,
D
!
X
and
B
!
D
,
D
!
X
, their branching fraction
ratios,
R
X
K=
, the total number
N
D
tot
;X
of
B
!
D
,
D
!
X
signal events, the true kaon identification efficiency
"
of
the PID selector, and the pion misidentification rate
m
of
the PID selector:
N
DK
;p;X
¼
1
2
ð
1
A
X
DK
Þ
N
D
tot
;X
R
X
K=
";
(14)
N
DK
;f;X
¼
1
2
ð
1
A
X
DK
Þ
N
D
tot
;X
R
X
K=
ð
1
"
Þ
;
(15)
N
D
;p;X
¼
1
2
ð
1
A
X
D
Þ
N
D
tot
;X
m;
(16)
N
D
;f;X
¼
1
2
ð
1
A
X
D
Þ
N
D
tot
;X
ð
1
m
Þ
:
(17)
Because the ratios
R
X
K=
are small, the fit is not able to
determine the value of
"
. Therefore we fix it to the afore-
mentioned value of
"
¼ð
77
1
Þ
%
. The reconstruction and
selection efficiencies for true
B
!
DK
and
B
!
D
candidates, where the
D
meson decays to the same final
state, are assumed to be identical. A systematic uncertainty
is assigned due to this assumption (see Sec.
VII
). The simul-
taneous fit to the two
CP
-even modes constrains
A
þ
DK
¼
A
K
þ
K
DK
A
CP
þ
;
(18)
MEASUREMENT OF
CP
OBSERVABLES IN
...
PHYSICAL REVIEW D
82,
072004 (2010)
072004-9