of 12
Observation of direct
CP
violation in the measurement of the Cabibbo-Kobayashi-Maskawa
angle

with
B

!
D
ð

Þ
K
ð

Þ
decays
J. P. Lees,
1
V. Poireau,
1
V. Tisserand,
1
E. Grauges,
2
A. Palano,
3a,3b
G. Eigen,
4
B. Stugu,
4
D. N. Brown,
5
L. T. Kerth,
5
Yu. G. Kolomensky,
5
G. Lynch,
5
H. Koch,
6
T. Schroeder,
6
D. J. Asgeirsson,
7
C. Hearty,
7
T. S. Mattison,
7
J. A. McKenna,
7
R. Y. So,
7
A. Khan,
8
V. E. Blinov,
9
A. R. Buzykaev,
9
V. P. Druzhinin,
9
V. B. Golubev,
9
E. A. Kravchenko,
9
A. P. Onuchin,
9
S. I. Serednyakov,
9
Yu. I. Skovpen,
9
E. P. Solodov,
9
K. Yu. Todyshev,
9
A. N. Yushkov,
9
D. Kirkby,
10
A. J. Lankford,
10
M. Mandelkern,
10
H. Atmacan,
11
J. W. Gary,
11
O. Long,
11
G. M. Vitug,
11
C. Campagnari,
12
T. M. Hong,
12
D. Kovalskyi,
12
J. D. Richman,
12
C. A. West,
12
A. M. Eisner,
13
J. Kroseberg,
13
W. S. Lockman,
13
A. J. Martinez,
13
B. A. Schumm,
13
A. Seiden,
13
D. S. Chao,
14
C. H. Cheng,
14
B. Echenard,
14
K. T. Flood,
14
D. G. Hitlin,
14
P. Ongmongkolkul,
14
F. C. Porter,
14
A. Y. Rakitin,
14
R. Andreassen,
15
Z. Huard,
15
B. T. Meadows,
15
M. D. Sokoloff,
15
L. Sun,
15
P. C. Bloom,
16
W. T. Ford,
16
A. Gaz,
16
U. Nauenberg,
16
J. G. Smith,
16
S. R. Wagner,
16
R. Ayad,
17,
*
W. H. Toki,
17
T. M. Karbach,
18,
B. Spaan,
18
K. R. Schubert,
19
R. Schwierz,
19
D. Bernard,
20
M. Verderi,
20
P. J. Clark,
21
S. Playfer,
21
D. Bettoni,
22a
C. Bozzi,
22a
R. Calabrese,
22a,22b
G. Cibinetto,
22a,22b
E. Fioravanti,
22a,22b
I. Garzia,
22a,22b
E. Luppi,
22a,22b
L. Piemontese,
22a
V. Santoro,
22a
R. Baldini-Ferroli,
23
A. Calcaterra,
23
R. de Sangro,
23
G. Finocchiaro,
23
P. Patteri,
23
I. M. Peruzzi,
23,
M. Piccolo,
23
M. Rama,
23
A. Zallo,
23
R. Contri,
24a,24b
E. Guido,
24a,24b
M. Lo Vetere,
24a,24b
M. R. Monge,
24a,24b
S. Passaggio,
24a
C. Patrignani,
24a,24b
E. Robutti,
24a
B. Bhuyan,
25
V. Prasad,
25
M. Morii,
26
A. Adametz,
27
U. Uwer,
27
H. M. Lacker,
28
T. Lueck,
28
P. D. Dauncey,
29
U. Mallik,
30
C. Chen,
31
J. Cochran,
31
W. T. Meyer,
31
S. Prell,
31
A. E. Rubin,
31
A. V. Gritsan,
32
N. Arnaud,
33
M. Davier,
33
D. Derkach,
33
G. Grosdidier,
33
F. Le Diberder,
33
A. M. Lutz,
33
B. Malaescu,
33
P. Roudeau,
33
M. H. Schune,
33
A. Stocchi,
33
G. Wormser,
33
D. J. Lange,
34
D. M. Wright,
34
C. A. Chavez,
35
J. P. Coleman,
35
J. R. Fry,
35
E. Gabathuler,
35
D. E. Hutchcroft,
35
D. J. Payne,
35
C. Touramanis,
35
A. J. Bevan,
36
F. Di Lodovico,
36
R. Sacco,
36
M. Sigamani,
36
G. Cowan,
37
D. N. Brown,
38
C. L. Davis,
38
A. G. Denig,
39
M. Fritsch,
39
W. Gradl,
39
K. Griessinger,
39
A. Hafner,
39
E. Prencipe,
39
R. J. Barlow,
40,
§
G. Jackson,
40
G. D. Lafferty,
40
E. Behn,
41
R. Cenci,
41
B. Hamilton,
41
A. Jawahery,
41
D. A. Roberts,
41
C. Dallapiccola,
42
R. Cowan,
43
D. Dujmic,
43
G. Sciolla,
43
R. Cheaib,
44
D. Lindemann,
44
P. M. Patel,
44,
k
S. H. Robertson,
44
P. Biassoni,
45a,45b
N. Neri,
45a
F. Palombo,
45a,45b
S. Stracka,
45a,45b
L. Cremaldi,
46
R. Godang,
46,
{
R. Kroeger,
46
P. Sonnek,
46
D. J. Summers,
46
X. Nguyen,
47
M. Simard,
47
P. Taras,
47
G. De Nardo,
48a,48b
D. Monorchio,
48a,48b
G. Onorato,
48a,48b
C. Sciacca,
48a,48b
M. Martinelli,
49
G. Raven,
49
C. P. Jessop,
50
J. M. LoSecco,
50
W. F. Wang,
50
K. Honscheid,
51
R. Kass,
51
J. Brau,
52
R. Frey,
52
N. B. Sinev,
52
D. Strom,
52
E. Torrence,
52
E. Feltresi,
53a,53b
N. Gagliardi,
53a,53b
M. Margoni,
53a,53b
M. Morandin,
53a
M. Posocco,
53a
M. Rotondo,
53a
G. Simi,
53a
F. Simonetto,
53a,53b
R. Stroili,
53a,53b
S. Akar,
54
E. Ben-Haim,
54
M. Bomben,
54
G. R. Bonneaud,
54
H. Briand,
54
G. Calderini,
54
J. Chauveau,
54
O. Hamon,
54
Ph. Leruste,
54
G. Marchiori,
54
J. Ocariz,
54
S. Sitt,
54
M. Biasini,
55a,55b
E. Manoni,
55a,55b
S. Pacetti,
55a,55b
A. Rossi,
55a,55b
C. Angelini,
56a,56b
G. Batignani,
56a,56b
S. Bettarini,
56a,56b
M. Carpinelli,
56a,56b,
**
G. Casarosa,
56a,56b
A. Cervelli,
56a,56b
F. Forti,
56a,56b
M. A. Giorgi,
56a,56b
A. Lusiani,
56a,56c
B. Oberhof,
56a,56b
A. Perez,
56a
G. Rizzo,
56a,56b
J. J. Walsh,
56a
D. Lopes Pegna,
57
J. Olsen,
57
A. J. S. Smith,
57
F. Anulli,
58a
R. Faccini,
58a,58b
F. Ferrarotto,
58a
F. Ferroni,
58a,58b
M. Gaspero,
58a,58b
L. Li Gioi,
58a
M. A. Mazzoni,
58a
G. Piredda,
58a
C. Bu
̈
nger,
59
O. Gru
̈
nberg,
59
T. Hartmann,
59
T. Leddig,
59
C. Voß,
59
R. Waldi,
59
T. Adye,
60
E. O. Olaiya,
60
F. F. Wilson,
60
S. Emery,
61
G. Hamel de Monchenault,
61
G. Vasseur,
61
Ch. Ye
`
che,
61
D. Aston,
62
R. Bartoldus,
62
J. F. Benitez,
62
C. Cartaro,
62
M. R. Convery,
62
J. Dorfan,
62
G. P. Dubois-Felsmann,
62
W. Dunwoodie,
62
M. Ebert,
62
R. C. Field,
62
M. Franco Sevilla,
62
B. G. Fulsom,
62
A. M. Gabareen,
62
M. T. Graham,
62
P. Grenier,
62
C. Hast,
62
W. R. Innes,
62
M. H. Kelsey,
62
P. Kim,
62
M. L. Kocian,
62
D. W. G. S. Leith,
62
P. Lewis,
62
B. Lindquist,
62
S. Luitz,
62
V. Luth,
62
H. L. Lynch,
62
D. B. MacFarlane,
62
D. R. Muller,
62
H. Neal,
62
S. Nelson,
62
M. Perl,
62
T. Pulliam,
62
B. N. Ratcliff,
62
A. Roodman,
62
A. A. Salnikov,
62
R. H. Schindler,
62
A. Snyder,
62
D. Su,
62
M. K. Sullivan,
62
J. Va’vra,
62
A. P. Wagner,
62
W. J. Wisniewski,
62
M. Wittgen,
62
D. H. Wright,
62
H. W. Wulsin,
62
C. C. Young,
62
V. Ziegler,
62
W. Park,
63
M. V. Purohit,
63
R. M. White,
63
J. R. Wilson,
63
A. Randle-Conde,
64
S. J. Sekula,
64
M. Bellis,
65
P. R. Burchat,
65
T. S. Miyashita,
65
E. M. T. Puccio,
65
M. S. Alam,
66
J. A. Ernst,
66
R. Gorodeisky,
67
N. Guttman,
67
D. R. Peimer,
67
A. Soffer,
67
S. M. Spanier,
68
J. L. Ritchie,
69
A. M. Ruland,
69
R. F. Schwitters,
69
B. C. Wray,
69
J. M. Izen,
70
X. C. Lou,
70
F. Bianchi,
71a,71b
D. Gamba,
71a,71b
S. Zambito,
71a,71b
L. Lanceri,
72a,72b
L. Vitale,
72a,72b
F. Martinez-Vidal,
73
A. Oyanguren,
73
P. Villanueva-Perez,
73
H. Ahmed,
74
J. Albert,
74
Sw. Banerjee,
74
F. U. Bernlochner,
74
H. H. F. Choi,
74
G. J. King,
74
R. Kowalewski,
74
M. J. Lewczuk,
74
I. M. Nugent,
74
J. M. Roney,
74
R. J. Sobie,
74
N. Tasneem,
74
T. J. Gershon,
75
P. F. Harrison,
75
T. E. Latham,
75
H. R. Band,
76
S. Dasu,
76
Y. Pan,
76
R. Prepost,
76
and S. L. Wu
76
PHYSICAL REVIEW D
87,
052015 (2013)
1550-7998
=
2013
=
87(5)
=
052015(12)
052015-1
Ó
2013 American Physical Society
(
B
A
B
AR
Collaboration)
1
Laboratoired’Annecy-le-VieuxdePhysiquedesParticules(LAPP),Universite
́
deSavoie,CNRS/IN2P3,F-74941Annecy-Le-Vieux,France
2
Departament ECM, Facultat de Fisica, Universitat de Barcelona, 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
Institute of Physics, University of Bergen, N-5007 Bergen, Norway
5
Lawrence Berkeley National Laboratory and University of California, Berkeley, California 94720, USA
6
Ruhr Universita
̈
t Bochum, Institut fu
̈
r Experimentalphysik 1, D-44780 Bochum, Germany
7
University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z1
8
Brunel University, Uxbridge, Middlesex UB8 3PH, United Kingdom
9
Budker Institute of Nuclear Physics, Novosibirsk 630090, Russia
10
University of California at Irvine, Irvine, California 92697, USA
11
University of California at Riverside, Riverside, California 92521, USA
12
University of California at Santa Barbara, Santa Barbara, California 93106, USA
13
Institute for Particle Physics, University of California at Santa Cruz, Santa Cruz, California 95064, USA
14
California Institute of Technology, Pasadena, California 91125, USA
15
University of Cincinnati, Cincinnati, Ohio 45221, USA
16
University of Colorado, Boulder, Colorado 80309, USA
17
Colorado State University, Fort Collins, Colorado 80523, USA
18
Technische Universita
̈
t Dortmund, Fakulta
̈
t Physik, D-44221 Dortmund, Germany
19
Technische Universita
̈
t Dresden, Institut fu
̈
r Kern- und Teilchenphysik, D-01062 Dresden, Germany
20
Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS/IN2P3, F-91128 Palaiseau, France
21
University of Edinburgh, Edinburgh EH9 3JZ, United Kingdom
22a
INFN Sezione di Ferrara, I-44100 Ferrara, Italy
22b
Dipartimento di Fisica, Universita
`
di Ferrara, I-44100 Ferrara, Italy
23
INFN Laboratori Nazionali di Frascati, I-00044 Frascati, Italy
24a
INFN Sezione di Genova, I-16146 Genova, Italy
24b
Dipartimento di Fisica, Universita
`
di Genova, I-16146 Genova, Italy
25
Indian Institute of Technology Guwahati, Guwahati, Assam, 781 039, India
26
Harvard University, Cambridge, Massachusetts 02138, USA
27
Universita
̈
t Heidelberg, Physikalisches Institut, Philosophenweg 12, D-69120 Heidelberg, Germany
28
Humboldt-Universita
̈
t zu Berlin, Institut fu
̈
r Physik, Newtonstr. 15, D-12489 Berlin, Germany
29
Imperial College London, London, SW7 2AZ, United Kingdom
30
University of Iowa, Iowa City, Iowa 52242, USA
31
Iowa State University, Ames, Iowa 50011-3160, USA
32
Johns Hopkins University, Baltimore, Maryland 21218, USA
33
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
34
Lawrence Livermore National Laboratory, Livermore, California 94550, USA
35
University of Liverpool, Liverpool L69 7ZE, United Kingdom
36
Queen Mary, University of London, London, E1 4NS, United Kingdom
37
University of London, Royal Holloway and Bedford New College, Egham, Surrey TW20 0EX, United Kingdom
38
University of Louisville, Louisville, Kentucky 40292, USA
39
Johannes Gutenberg-Universita
̈
t Mainz, Institut fu
̈
r Kernphysik, D-55099 Mainz, Germany
40
University of Manchester, Manchester M13 9PL, United Kingdom
41
University of Maryland, College Park, Maryland 20742, USA
42
University of Massachusetts, Amherst, Massachusetts 01003, USA
43
Massachusetts Institute of Technology, Laboratory for Nuclear Science, Cambridge, Massachusetts 02139, USA
44
McGill University, Montre
́
al, Que
́
bec, Canada H3A 2T8
45a
INFN Sezione di Milano, I-20133 Milano, Italy
45b
Dipartimento di Fisica, Universita
`
di Milano, I-20133 Milano, Italy
46
University of Mississippi, University, Mississippi 38677, USA
47
Universite
́
de Montre
́
al, Physique des Particules, Montre
́
al, Que
́
bec, Canada H3C 3J7
48a
INFN Sezione di Napoli, I-80126 Napoli, Italy
48b
Dipartimento di Scienze Fisiche, Universita
`
di Napoli Federico II, I-80126 Napoli, Italy
49
NIKHEF, National Institute for Nuclear Physics and High Energy Physics, NL-1009 DB Amsterdam, The Netherlands
50
University of Notre Dame, Notre Dame, Indiana 46556, USA
51
Ohio State University, Columbus, Ohio 43210, USA
J. P. LEES
et al.
PHYSICAL REVIEW D
87,
052015 (2013)
052015-2
52
University of Oregon, Eugene, Oregon 97403, USA
53a
INFN Sezione di Padova, I-35131 Padova, Italy
53b
Dipartimento di Fisica, Universita
`
di Padova, I-35131 Padova, Italy
54
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
55a
INFN Sezione di Perugia, I-06100 Perugia, Italy
55b
Dipartimento di Fisica, Universita
`
di Perugia, I-06100 Perugia, Italy
56a
INFN Sezione di Pisa, I-56127 Pisa, Italy
56b
Dipartimento di Fisica, Universita
`
di Pisa, I-56127 Pisa, Italy
56c
Scuola Normale Superiore di Pisa, I-56127 Pisa, Italy
57
Princeton University, Princeton, New Jersey 08544, USA
58a
INFN Sezione di Roma, I-00185 Roma, Italy
58b
Dipartimento di Fisica, Universita
`
di Roma La Sapienza, I-00185 Roma, Italy
59
Universita
̈
t Rostock, D-18051 Rostock, Germany
60
Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 0QX, United Kingdom
61
CEA, Irfu, SPP, Centre de Saclay, F-91191 Gif-sur-Yvette, France
62
SLAC National Accelerator Laboratory, Stanford, California 94309, USA
63
University of South Carolina, Columbia, South Carolina 29208, USA
64
Southern Methodist University, Dallas, Texas 75275, USA
65
Stanford University, Stanford, California 94305-4060, USA
66
State University of New York, Albany, New York 12222, USA
67
School of Physics and Astronomy, Tel Aviv University, Tel Aviv, 69978, Israel
68
University of Tennessee, Knoxville, Tennessee 37996, USA
69
University of Texas at Austin, Austin, Texas 78712, USA
70
University of Texas at Dallas, Richardson, Texas 75083, USA
71a
INFN Sezione di Torino, I-10125 Torino, Italy
71b
Dipartimento di Fisica Sperimentale, Universita
`
di Torino, I-10125 Torino, Italy
72a
INFN Sezione di Trieste, I-34127 Trieste, Italy
72b
Dipartimento di Fisica, Universita
`
di Trieste, I-34127 Trieste, Italy
73
IFIC, Universitat de Valencia-CSIC, E-46071 Valencia, Spain
74
University of Victoria, Victoria, British Columbia, Canada V8W 3P6
75
Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom
76
University of Wisconsin, Madison, Wisconsin 53706, USA
(Received 7 January 2013; published 22 March 2013)
We report the determination of the Cabibbo-Kobayashi-Maskawa
CP
-violating angle

through the
combination of various measurements involving
B

!
DK

,
B

!
D

K

, and
B

!
DK

decays
performed by the
BABAR
experiment at the PEP-II
e
þ
e

collider at SLAC National Accelerator
Laboratory. Using up to 474 million
B

B
pairs, we obtain

¼ð
69
þ
17

16
Þ

modulo 180

. The total uncertainty
is dominated by the statistical component, with the experimental and amplitude-model systematic
uncertainties amounting to

4

. The corresponding two-standard-deviation region is
41

<<
102

.
This result is inconsistent with

¼
0
with a significance of 5.9 standard deviations.
DOI:
10.1103/PhysRevD.87.052015
PACS numbers: 13.25.Hw, 11.30.Er, 12.15.Hh, 14.40.Nd
I. INTRODUCTION AND OVERVIEW
In the Standard Model (SM), the mechanism of
CP
violation in weak interactions arises from the joint effect
of three mixing angles and the single irreducible phase in
the three-family Cabibbo-Kobayashi-Maskawa (CKM)
quark-mixing matrix [
1
]. The unitarity of the CKM matrix
V
implies a set of relations among its elements,
V
ij
, with
i
¼
u
,
c
,
t
and
j
¼
d
,
s
,
b
. In particular,
V
ud
V

ub
þ
V
cd
V

cb
þ
V
td
V

tb
¼
0
, which can be depicted in the com-
plex plane as a unitarity triangle whose sides and angles are
related to the magnitudes and phases of the six elements of
the first and third columns of the matrix,
V
id
and
V
ib
. The
parameter

, defined as
arg
½
V
ud
V

ub
=V
cd
V

cb

, is one of
the three angles of this triangle. From measurements of the
sides and angles of the unitarity triangle from many decay
processes, it is possible to overconstrain our knowledge
of the CKM mechanism, probing dynamics beyond the
*
Now at the University of Tabuk, Tabuk 71491, Saudi Arabia.
Present address: the European Organization for Nuclear
Research (CERN), Geneva, Switzerland.
Also at Universita
`
di Perugia, Dipartimento di Fisica,
Perugia, Italy.
§
Present address: the University of Huddersfield, Huddersfield
HD1 3DH, United Kingdom.
k
Deceased.
{
Present address: the University of South Alabama, Mobile,
Alabama 36688, USA.
**
Also at Universita
`
di Sassari, Sassari, Italy.
OBSERVATION OF DIRECT
CP
VIOLATION IN THE
...
PHYSICAL REVIEW D
87,
052015 (2013)
052015-3
SM [
2
]. In this context, the angle

is particularly relevant
since it is the only
CP
-violating parameter that can be
cleanly determined using tree-level
B
meson decays [
3
].
In spite of a decade of successful operation and experi-
mental efforts by the
B
factory experiments,
BABAR
and
Belle,

is poorly known due to its large statistical uncer-
tainty. Its precise determination is an important goal of
present and future flavor-physics experiments.
Several methods have been pursued to extract

[
4
9
].
Those using charged
B
meson decays into
D
ðÞ
K

and
DK

final states, denoted generically as
D
ðÞ
K
ðÞ
, yield
low theoretical uncertainties since the decays involved do
not receive contributions from penguin diagrams (see
Fig.
1
). This is a very important distinction from most
other measurements of the angles. Here, the symbol
D
ðÞ
indicates either a
D
0
(
D

0
)ora

D
0
(

D

0
) meson, and
K

refers to
K

ð
892
Þ

states. The methods to measure

based
on
B

!
D
ðÞ
K
ðÞ
decays rely on the interference be-
tween the CKM- and color-favored
b
!
c

us
and the sup-
pressed
b
!
u

cs
amplitudes, which arises when the
D
0
from a
B

!
D
0
K

decay [
10
] (and similarly for the other
related
B
decays) is reconstructed in a final state which can
be produced also in the decay of a

D
0
originating from
B

!

D
0
K

(see Fig.
1
). The interference between the
b
!
c

us
and
b
!
u

cs
tree amplitudes results in observ-
ables that depend on their relative weak phase

, on the
magnitude ratio
r
B
j
A
ð
b
!
u

cs
Þ
=
A
ð
b
!
c

us
Þj
, and
on the relative strong phase

B
between the two amplitudes.
In the case of a nonzero weak phase

and a nonzero strong
phase

B
, the
B

and
B
þ
decay rates are different, which is
a manifestation of direct
CP
violation. The hadronic pa-
rameters
r
B
and

B
are not precisely known from theory,
and may have different values for
DK

,
D

K

, and
DK

final states. They can be measured directly from data by
simultaneously reconstructing several
D
-decay final states.
The three main approaches employed by the
B
factory
experiments are:
(i) the Dalitz plot or Giri-Grossman-Soffer-Zupan
(GGSZ) method, based on three-body, self-
conjugate final states, such as
K
0
S

þ


[
7
];
(ii) the Gronau-London-Wyler (GLW) method, based
on decays to
CP
-eigenstate final states, such as
K
þ
K

and
K
0
S

0
[
8
];
(iii) the Atwood-Dunietz-Soni (ADS) method, based on
D
decays to doubly-Cabibbo-suppressed final
states, such as
D
0
!
K
þ


[
9
].
To date, the GGSZ method has provided the highest
statistical power in measuring

. The other two methods
provide additional information that can further constrain
the hadronic parameters and thus allow for a more robust
determination of

. The primary issue with all these meth-
ods is the small product branching fraction of the decays
involved, which range from
5

10

6
to
5

10

9
, and
the small size of the interference, proportional to
r
B

c
F
j
V
cs
V

ub
j
=
j
V
us
V

cb
j
0
:
1
, where
c
F

0
:
2
is a
color suppression factor [
11
13
]. Therefore a precise
determination of

requires a very large data sample and
the combination of all available methods involving differ-
ent
D
decay modes.
Recently, Belle [
14
] and LHCb [
15
] have presented the
preliminary results of the combination of their measure-
ments related to

, yielding

to be
ð
68
þ
15

14
Þ

and
ð
71
þ
17

16
Þ

,
respectively. Attempts to combine the results by
BABAR
,
Belle, CDF, and LHCb have been performed by the
CKMfitter and UTfit groups [
2
]. Their most recent results
are
ð
66

12
Þ

and
ð
72

9
Þ

, respectively.
The
BABAR
experiment [
16
] at the PEP-II asymmetric-
energy
e
þ
e

collider at SLAC has analyzed charged
B
decays into
DK

,
D

K

, and
DK

final states using the
GGSZ [
17
19
], GLW [
20
22
], and ADS [
22
24
] methods,
providing a variety of measurements and constraints on

.
The results are based on a data set collected at a center-of-
mass energy equal to the mass of the

ð
4
S
Þ
resonance, and
about 10% of data collected 40 MeV below. We present
herein the combination of published
BABAR
measurements
using detailed information on correlations between pa-
rameters that we have not previously published. This com-
bination represents the most complete study of the data
sample collected by
BABAR
and benefits from the possi-
bility to access and reanalyze the data sample (see Sec.
II
for details).
Other analyses related to

[
25
27
]or
2

þ

[
28
,
29
]
have not been included, because the errors on the experi-
mental measurements are too large.
II. INPUT MEASUREMENTS
In the GGSZ approach, where
D
mesons are recon-
structed into the
K
0
S

þ


and
K
0
S
K
þ
K

final states
[
17
19
], the signal rates for
B

!
D
ðÞ
K

and
B

!
DK

decays are analyzed as a function of the position
in the Dalitz plot of squared invariant masses
m
2


m
2
ð
K
0
S
h

Þ
,
m
2
þ

m
2
ð
K
0
S
h
þ
Þ
, where
h
is either a charged
pion or kaon (
h
¼

,
K
). We assume no
CP
violation in
the neutral
D
and
K
meson systems and neglect small
D
0


D
0
mixing effects [
30
,
31
], leading to

A
ð
m
2

;m
2
þ
Þ¼
A
ð
m
2
þ
;m
2

Þ
, where

A
(
A
) is the

D
0
(
D
0
) decay
amplitude. In this case, the signal decay rates can be
written as [
32
]
FIG. 1. Dominant Feynman diagrams for the decays
B

!
D
0
K

(left) and
B

!

D
0
K

(right). The left diagram proceeds
via
b
!
c

us
transition, while the right diagram proceeds via
b
!
u

cs
transition and is both CKM- and color-suppressed.
J. P. LEES
et al.
PHYSICAL REVIEW D
87,
052015 (2013)
052015-4

ðÞ

ð
m
2

;m
2
þ
Þ/j
A

j
2
þ
r
ðÞ
B

2
j
A
j
2
þ
2

Re
½
z
ðÞ

A
y

A

;

s

ð
m
2

;m
2
þ
Þ/j
A

j
2
þ
r
2
s

j
A
j
2
þ
2Re
½
z
s

A
y

A

;
(1)
with
A


A
ð
m
2

;m
2
Þ
and
A
y

is the complex conju-
gate of
A

. The symbol

for
B

!
D

K

accounts for
the different
CP
parity of the
D

when it is reconstructed
into
D
0
(

¼þ
1
) and
D
(

¼
1
) final states, as a
consequence of the opposite
CP
eigenvalue of the

0
and
the photon [
33
]. Here,
r
ðÞ
B

and
r
s

are the magnitude ratios
between the
b
!
u

cs
and
b
!
c

us
amplitudes for
B

!
D
ðÞ
K

and
B

!
DK

decays, respectively, and

ðÞ
B
,

s
are their relative strong phases. The analysis extracts the
CP
-violating observables [
19
]
z
ðÞ


x
ðÞ

þ
iy
ðÞ

;
z
s


x
s

þ
iy
s

;
(2)
defined as the suppressed-to-favored complex amplitude
ratios
z
ðÞ

¼
r
ðÞ
B

e
i
ð

ðÞ
B


Þ
and
z
s

¼
r
s

e
i
ð

s


Þ
, for
B

!
D
ðÞ
K

and
B

!
DK

decays, respectively.
The hadronic parameter

is defined as
e
i
s

R
A
c
ð
p
Þ
A
u
ð
p
Þ
e
i
ð
p
Þ
d
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
R
A
2
c
ð
p
Þ
d
p
R
A
2
u
ð
p
Þ
d
p
q
;
(3)
where
A
c
ð
p
Þ
and
A
u
ð
p
Þ
are the magnitudes of the
b
!
c

us
and
b
!
u

cs
amplitudes as a function of the
B

!
DK
0
S


phase-space position
p
, and

ð
p
Þ
is their relative
strong phase. This coherence factor, with
0
<<
1
in the
most general case and

¼
1
for two-body
B
decays,
accounts for the interference between
B

!
DK

and
other
B

!
DK
0
S


decays, as a consequence of the
K

natural width [
12
]. In our analysis,

has been fixed
to 0.9 and a systematic uncertainty has been assigned, vary-
ing its value by

0
:
1
, as estimated using a Monte Carlo
simulation based on the Dalitz plot model of
B

!
DK
0
S


decays [
18
]. Thus the parameter

s
is an effective
strong-phase difference averaged over the phase space.
Table
I
summarizes our experimental results for the
CP
-violating parameters
z
ðÞ

and
z
s

. Complete
12

12
covariance matrices for statistical, experimental system-
atic, and amplitude-model uncertainties are reported in
Ref. [
17
]. The
z
ðÞ

and
z
s

observables are unbiased and
have Gaussian behavior with small correlations, even for
low values of
r
ðÞ
B
,
r
s
and relatively low-statistics samples.
Furthermore, their uncertainties have a minimal depen-
dence on their central values and are free of physical
bounds [
19
]. These good statistical properties allow for
easier combination of several measurements into a single
result. For example, the rather complex experimental
GGSZ likelihood function can be parametrized by a
12-dimensional (correlated) Gaussian probability density
function (P.D.F.), defined in the space of the
z
ðÞ

and
z
s

measurements from Table
I
. After this combination has
been performed, the values of

and of the hadronic
parameters
r
ðÞ
B
,
r
s
,

ðÞ
B
, and

s
can be obtained.
The
D
decay amplitudes
A

have been determined
from Dalitz plot analyses of tagged
D
0
mesons from
D
!
D
0

þ
decays produced in
e
þ
e

!
c

c
events
[
18
,
34
], assuming an empirical model to describe the
variation of the amplitude phase as a function of the
Dalitz plot variables. A model-independent, binned ap-
proach also exists [
7
,
35
], which optimally extracts infor-
mation on

for higher-statistics samples than the ones
available. This type of analysis has been performed as a
proof of principle by the Belle collaboration [
36
], giving
consistent results to the model-dependent approach [
37
].
The LHCb collaboration has also released results of a
model-independent GGSZ analysis [
38
].
In order to determine

with the GLW method, the
analyses measure the direct
CP
-violating partial decay
rate asymmetries
A
ðÞ
CP



ð
B

!
D
ðÞ
CP

K

Þ

ð
B
þ
!
D
ðÞ
CP

K
þ
Þ

ð
B

!
D
ðÞ
CP

K

Þþ

ð
B
þ
!
D
ðÞ
CP

K
þ
Þ
;
(4)
and the ratios of charge-averaged partial rates using
D
decays to
CP
and flavor eigenstates,
R
ðÞ
CP


2

ð
B

!
D
ðÞ
CP

K

Þþ

ð
B
þ
!
D
ðÞ
CP

K
þ
Þ

ð
B

!
D
ðÞ
0
K

Þþ

ð
B
þ
!

D
ðÞ
0
K
þ
Þ
;
(5)
where
D
ðÞ
CP

refers to the
CP
eigenstates of the
D
ðÞ
meson
system. We select
D
mesons 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
eigen-
state
K


þ
(
D
0
from
B

!
D
0
h

)or
K
þ


(

D
0
from
B
þ
!

D
0
h
þ
). We recontruct
D

mesons in the states
D
0
and
D
. The observables
A
s
CP

and
R
s
CP

for
B

!
DK

decays are defined similarly.
TABLE I.
CP
-violating complex parameters
z
ðÞ


x
ðÞ

þ
iy
ðÞ

and
z
s


x
s

þ
iy
s

, measured using the GGSZ technique
[
17
]. The first uncertainty is statistical, the second is the experi-
mental systematic uncertainty and the third is the systematic
uncertainty associated with the
D
0
decay amplitude models. The
sample analyzed contains 468 million
B

B
pairs.
Real part (%)
Imaginary part (%)
z

6
:
0

3
:
9

0
:
7

0
:
66
:
2

4
:
5

0
:
4

0
:
6
z
þ

10
:
3

3
:
7

0
:
6

0
:
7

2
:
1

4
:
8

0
:
4

0
:
9
z



10
:
4

5
:
1

1
:
9

0
:
2

5
:
2

6
:
3

0
:
9

0
:
7
z

þ
14
:
7

5
:
3

1
:
7

0
:
3

3
:
2

7
:
7

0
:
8

0
:
6
z
s

7
:
5

9
:
6

2
:
9

0
:
712
:
7

9
:
5

2
:
7

0
:
6
z
s
þ

15
:
1

8
:
3

2
:
9

0
:
6
4
:
5

10
:
6

3
:
6

0
:
8
OBSERVATION OF DIRECT
CP
VIOLATION IN THE
...
PHYSICAL REVIEW D
87,
052015 (2013)
052015-5
For later convenience, the GLW observables can be
related to
z
ðÞ

and
z
s

(neglecting mixing and
CP
violation
in neutral
D
decays) as
A
ðÞ
CP

¼
x
ðÞ


x
ðÞ
þ
1
þj
z
ðÞ
j
2
x
ðÞ

þ
x
ðÞ
þ
Þ
(6)
and
R
ðÞ
CP

¼
1
þj
z
ðÞ
j
2
x
ðÞ

þ
x
ðÞ
þ
Þ
;
(7)
where
j
z
ðÞ
j
2
is the average value of
j
z
ðÞ
þ
j
2
and
j
z
ðÞ

j
2
.For
B

!
DK

decays, similar relations to Eqs. (
6
) and (
7
)
hold, with

¼
1
, since the effects of the non-
K

B
!
DK
events and the width of the
K

are incorporated
into the systematic uncertainties of the
A
s
CP

and
R
s
CP

measurements [
22
].
Table
II
summarizes the results obtained for the GLW
observables. In order to avoid overlaps with the samples
selected in the Dalitz plot analysis, the results for
B

!
D
CP

K

decays are corrected by removing the
contribution from
D
CP

!
K
0
S

,

!
K
þ
K

candidates
[
20
]. For the decays
B

!
D

CP

½
D
CP


0

K

,
B

!
D

CP
þ
½
D
CP



K

, and
B

!
D
CP

K

, such informa-
tion is not available. In this case, the overlap is accounted
for by increasing the uncertainties quoted in Refs. [
21
,
22
]
by 10% while keeping the central values unchanged. The
10% increase in the experimental uncertainties is approxi-
mately the change observed in
B

!
D
CP

K

decays
when excluding or including
D
!
K
0
S

in the measure-
ment. The impact on the combination has been found to be
negligible.
As in the case of the GGSZ observables,
A
ðÞ
CP

,
A
s
CP

,
R
ðÞ
CP

, and
R
s
CP

have Gaussian uncertainties near the best
solution, with small statistical and systematic correlations,
as given in Ref. [
20
] for
B

!
D
CP

K

decays. The
GLW method has also been exploited by the Belle [
39
],
CDF [
40
], and LHCb collaborations [
41
], with consistent
results.
In the ADS method, the
D
0
meson from the
favored
b
!
c

us
amplitude is reconstructed in the
doubly-Cabibbo-suppressed decay
K
þ


, while the

D
0
from the
b
!
u

cs
suppressed amplitude is reconstructed
in the favored decay
K
þ


[
22
,
23
]. The product branching
fractions for these final states, which we denote as
B

!
½
K
þ



D
K

,
B

K
þ




D
K

,
B

K
þ



D
K

,
and their
CP
conjugates, are small (
10

7
). However,
the two interfering amplitudes are of the same order of
magnitude, allowing for possible large
CP
asymmetries.
We measure charge-specific ratios for
B
þ
and
B

decay
rates to the ADS final states, which are defined as
R
ðÞ



ð
B

K



ðÞ
D
K

Þ

ð
B

K



ðÞ
D
K

Þ
;
(8)
and similarly for
R
s

, where the favored decays
B

K


þ

D
K

,
B

K


þ


D
K

, and
B

!
½
K


þ

D
K

serve as normalization so that many system-
atic uncertainties cancel. The rates in Eq. (
8
) depend on

and the
B
-decay hadronic parameters. They are related to
z
ðÞ

and
z
s

through
R
ðÞ

¼
r
ðÞ
B

2
þ
r
2
D
þ
2
r
D
h
x
ðÞ

cos

D

y
ðÞ

sin

D
i
;
(9)
where
r
D
¼j
A
ð
D
0
!
K
þ


Þ
=
A
ð
D
0
!
K


þ
Þj
and

D
are the ratio between magnitudes of the suppressed
and favored
D
-decay amplitudes and their relative strong
phase, respectively. As in Eq. (
1
), the symbol

for
B

!
D

K

decays accounts for the different
CP
parity
of
D

!
D
0
and
D

!
D
. The values of
r
D
and

D
are
taken as external constraints in our analysis. As for the
GLW method, the effects of other
B

!
DK
0
S


events,
not going through
K

, and the
K

width, are incorpo-
rated in the systematic uncertainties on
R
s

. Thus similar
relations hold for these observables with

¼
1
.
The choice of the observables
R

(and similarly for
R


and
R
s

) rather than the original ADS observables
R
ADS

ð
R
þ
þ
R

Þ
=
2
and
A
ADS
R


R
þ
Þ
=
2
R
ADS
[
9
] is moti-
vated by the fact that the set of variables
ð
R
ADS
;A
ADS
Þ
is
not well-behaved since the uncertainty on
A
ADS
depends on
the central value of
R
ADS
, while
R
þ
and
R

are statistically
independent observables. Although systematic uncertain-
ties are largely correlated, the measurements of
R
þ
and
R

are effectively uncorrelated since the total uncertainties are
dominated by the statistical component.
We have also reconstructed
B

K



0

D
K

de-
cays [
24
] from which the observables
R
K
0

have been
measured, which are related to the GGSZ observables as
R
K
0

¼
r
2
B

þ
r
2
K
0
þ
2

K
0
r
K
0
½
x

cos

K
0

y

sin

K
0

;
(10)
where

K
0
is a
D
decay coherence factor similar to
that defined in Eq. (
3
) for the
B

!
DK
0
S


decay, and
where
r
K
0
and

K
0
are hadronic parameters for
D
0
!
K



0
decays analogous to
r
D
and

D
.
TABLE II. GLW observables measured for the
B

!
DK

(based on 467 million
B

B
pairs) [
20
],
B

!
D

K

(383 million
B

B
pairs) [
21
], and
B

!
DK

(379 million
B

B
pairs) [
22
]
decays, corrected by removing the contribution from
D
CP

!
K
0
S

,

!
K
þ
K

candidates. The first uncertainty is statistical,
the second is systematic.
CP
-even
CP
-odd
R
CP

1
:
18

0
:
09

0
:
05
1
:
03

0
:
09

0
:
04
A
CP

0
:
25

0
:
06

0
:
02

0
:
08

0
:
07

0
:
02
R

CP

1
:
31

0
:
13

0
:
04
1
:
10

0
:
13

0
:
04
A

CP


0
:
11

0
:
09

0
:
01
0
:
06

0
:
11

0
:
02
R
s
CP

2
:
17

0
:
35

0
:
09
1
:
03

0
:
30

0
:
14
A
s
CP

0
:
09

0
:
13

0
:
06

0
:
23

0
:
23

0
:
08
J. P. LEES
et al.
PHYSICAL REVIEW D
87,
052015 (2013)
052015-6
Table
III
summarizes the measurements of the ADS
charge-specific ratios for the different final states. Contrary
to the case of the GGSZ and GLWobservables,
R
ðÞ

,
R
s

, and
R
K
0

do not have Gaussian behavior. The experimental
likelihood function for each of the four decay modes, shown
in Fig.
2
for
B

!
DK

and
B

!
D

K

decays, is well
described around the best solution by an analytical P.D.F.
composed of the sum of two asymmetric Gaussian func-
tions. For the
B

!
DK

channel, we use instead a simple
Gaussian approximation since in this case the experimental
likelihood scans are not available. The effect of this approxi-
mation has been verified to be negligible, given the small
statistical weight of this sample in the combination.
Measurements using the ADS technique have also been
performed by the Belle [
42
,
43
], CDF [
44
], and LHCb col-
laborations [
41
], with consistent results.
III. OTHER MEASUREMENTS
Similar analyses related to

measurement have
been carried out using the decay
B

!
DK

with the
D
!

þ



0
final state [
25
], and the neutral
B
decay

B
0
!
D

K

ð
892
Þ
0
,

K

ð
892
Þ
0
!
K


þ
,with
D
!
K
0
S

þ


[
26
] and
D
!
K


,
K



0
,
K





[
27
]. For neutral
B
decays,
r
B
is naively expected to be larger
(

0
:
3
) because both interfering amplitudes are color sup-
pressed and thus
c
F

1
. However, the overall rate of events
is smaller than for
B

!
DK

decays. The flavor of the
neutral
B
meson is tagged by the charge of the kaon
produced in the

K

0
decay,

K

ð
892
Þ
0
!
K


þ
or
K

ð
892
Þ
0
!
K
þ


.
Experimental analyses of the time-dependent decay rates
of
B
!
D
ðÞ


and
B
!
D

ð
770
Þ

decays have also
been used to constrain

[
28
,
29
]. In these decays, the
interference occurs between the favored
b
!
c

ud
and the
suppressed
b
!
u

cd
tree amplitudes with and without
B
0


B
0
mixing, resulting in a total weak-phase difference
2

þ

[
45
], where

is the angle of the unitarity triangle
defined as
arg
½
V
cd
V

cb
=V
td
V

tb

. The magnitude ratios
between the suppressed and favored amplitudes
r
D
ðÞ

and
r
D
are expected to be

2%
, and have to be estimated
either by analyzing suppressed charged
B
decays
(e.g.,
B
þ
!
D
þ

0
) with an isospin assumption or from
self-tagging neutral
B
decays to charmed-strange mesons
(e.g.,
B
0
!
D
þ
s


) assuming SU(3) flavor symmetry and
neglecting contributions from
W
-exchange diagrams [
45
].
Performing a time-dependent Dalitz plot analysis of
B
!
D
K
0


decays [
46
] could in principle avoid the
problem of the smallness of
r
. In these decays the two
interfering amplitudes are color suppressed, and it is ex-
pected to be

0
:
3
, but the overall rate of events is too small
with the current data sample.
In both cases, the errors on the experimental measure-
ments are too large for a meaningful determination of

,and
have not been included in the combined determination of

reported in this paper. However, these decay channels
might provide important information in future experiments.
TABLE III. ADS observables included into the combination
for
B

!
DK

with
D
!
K
(based on 467 million
B

B
pairs)
and
D
!
K
0
(based on 474 million
B

B
pairs),
B

!
D

K

(467 million
B

B
pairs), and
B

!
DK

(379 million
B

B
pairs)
decays [
22
24
]. The first uncertainty is statistical, the second is
systematic.
B
þ
B

R

0
:
022

0
:
009

0
:
003 0
:
002

0
:
006

0
:
002
R


[
D
0
]
0
:
005

0
:
008

0
:
003 0
:
037

0
:
018

0
:
009
R


[
D
]
0
:
009

0
:
016

0
:
007 0
:
019

0
:
023

0
:
012
R
s

0
:
076

0
:
042

0
:
011 0
:
054

0
:
049

0
:
011
R
K
0

0
:
005
þ
0
:
012
þ
0
:
001

0
:
010

0
:
004
0
:
012
þ
0
:
012
þ
0
:
002

0
:
010

0
:
004
+
R
Likelihood
0
0.2
0.4
0.6
0.8
1
(a)
-
R
Likelihood
0
0.2
0.4
0.6
0.8
1
(b)
0
π
*,
+
R
Likelihood
0
0.2
0.4
0.6
0.8
1
(c)
0
π
*,
-
R
Likelihood
0
0.2
0.4
0.6
0.8
1
(d)
γ
*,
+
R
Likelihood
0
0.2
0.4
0.6
0.8
1
(e)
γ
*,
-
R
Likelihood
0.2
0.4
0.6
0.8
1
(f)
0
π
π
K
+
R
Likelihood
0
0.2
0.4
0.6
0.8
1
(g)
0
π
π
K
-
R
0
0.05
0
0.05
0
0.1
0
0.1
0
0.05
0
0.05
0
0.05
0
0.05
Likelihood
0
0.2
0.4
0.6
0.8
1
(h)
FIG. 2. Experimental likelihoods as functions of the ADS
charge-specific ratios
R

(a,b),
R


[
D
0
] (c,d),
R


[
D
]
(e,f), and
R
K
0

(g,h), from Refs. [
23
,
24
], including systematic
uncertainties. The P.D.F.s are normalized so that their maximum
values are equal to 1. These distributions are well parametrized
by sums of two asymmetric Gaussian functions with mean values
as given in Table
III
.
OBSERVATION OF DIRECT
CP
VIOLATION IN THE
...
PHYSICAL REVIEW D
87,
052015 (2013)
052015-7
IV. COMBINATION PROCEDURE
We combine all the GGSZ, GLW, and ADS observables
(34 in total) to extract

in two different stages. First, we
extract the best-fit values for the
CP
-violating quantities

z
ðÞ

and

z
s

, whose definitions correspond to those for the
quantities
z
ðÞ

and
z
s

of the GGSZ analysis given in Eq. (
2
).
Their best-fit values are obtained by maximizing a
combined likelihood function constructed as the product
of partial likelihood P.D.F.s for GGSZ, GLW, and ADS
measurements. The GGSZ likelihood function uses a 12-
dimensional Gaussian P.D.F. with measurements
z
ðÞ

and
z
s

and their covariance matrices for statistical, experi-
mental systematic and amplitude-model uncertainties, and
mean (expected) values

z
ðÞ

and

z
s

. Similarly, the GLW
likelihood is formed as the product of four-dimensional
Gaussian P.D.F.s for each
B
decay with measurements
A
ðÞ
CP

,
A
s
CP

,
R
ðÞ
CP

,
R
s
CP

, and their covariance matrices,
and expected values given by Eqs. (
6
) and (
7
) after replac-
ing the
z
ðÞ

and
z
s

observables by the

z
ðÞ

and

z
s

pa-
rameters. Finally, the ADS P.D.F. is built from the product
of experimental likelihoods shown in Fig.
2
. With this
construction, GGSZ, GLW, and ADS observables are taken
as uncorrelated. Similarly, the individual measurements
are considered uncorrelated as the experimental uncertain-
ties are dominated by the statistical component.
The combination requires external inputs for the
D
hadronic parameters
r
D
,

D
,
r
K
0
,

K
0
, and

K
0
.
We assume Gaussian P.D.F.s for
r
D
¼
0
:
0575

0
:
0007
[
30
] and
r
K
0
¼
0
:
0469

0
:
0011
[
47
], while for the
other three we adopt asymmetric Gaussian parametriza-
tions based on the experimental likelihoods available either
from world averages for

D
¼ð
202
:
0
þ
9
:
9

11
:
2
Þ

[
30
] or from
the CLEOc collaboration for

K
0
¼ð
47
þ
14

17
Þ

and

K
0
¼
0
:
84

0
:
07
[
48
]. The values of

D
and

K
0
have been corrected for a shift of 180

in the definition of
the phases between Refs. [
23
,
24
] and Refs. [
30
,
48
]. The
correlations between
r
D
and

D
, and between

K
0
and

K
0
, are small and have been neglected. All five external
observables are assumed to be uncorrelated with the rest of
the input observables.
The results for the combined
CP
-violating parameters

z
ðÞ

and

z
s

are summarized in Table
IV
. Figure
3
shows
comparisons of two-dimensional regions corresponding to
one-, two-, and three-standard-deviation regions in the

z

,

z


, and

z
s

planes, including statistical and systematic
uncertainties for GGSZ only, GGSZ and GLW methods
combined, and the overall combination. These contours
have been obtained using the likelihood ratio method,

2 ln
L
¼
s
2
, where
s
is the number of standard
deviations, where
2 ln
L
represents the variation of the
combined log-likelihood with respect to its maximum
value [
47
]. With this construction, the approximate
confidence level (C.L.) in two dimensions for each pair
of variables is 39.3%, 86.5%, and 98.9%. In these
two-dimensional regions, the separation of the
B

and
B
þ
positions is equal to
2
r
B
j
sin

j
,
2
r

B
j
sin

j
,
2
r
s
j
sin

j
and is a measurement of direct
CP
violation,
while the angle between the lines connecting the
B

and
B
þ
centers with the origin (0, 0) is equal to
2

. Therefore,
the net difference between

x
þ
and

x

observed in Table
IV
and Fig.
3
is clear evidence for direct
CP
violation in
B

!
DK

decays.
In Fig.
3
, we observe that when the information from the
GLW measurements isincludedtheconstraintson the best-fit
values of the parameters are improved. However, the con-
straints on

y

are poor due to the quadratic dependence and
the fact that
r
B
1
. This is the reason why the GLW method
alone can hardly constrain

. Similarly, Eq. (
9
) for the ADS
method represents two circles in the
ð

x

;

y

Þ
plane centered
at
ð
r
B
cos

D
;r
D
sin

D
Þ
and with radii
ffiffiffiffiffiffiffi
R

p
.Itisnotpos-
sible to determine

with only ADS observables because the
true
ð

x

;

y

Þ
points are distributed over two circles [
49
].
Therefore, while the GLW and ADS methods alone can
hardly determine

, when combined with the GGSZ mea-
surements they help to improve significantly the constraints
on the
CP
-violating parameters

z

,

z


,and

z
s

.
V. INTERPRETATION OF RESULTS
In a second stage, we transform the combined
ð

x

;

y

Þ
,
ð

x


;

y


Þ
, and
ð

x
s

;

y
s

Þ
measurements into the physically
relevant quantities

and the set of hadronic parameters
u
r
B
;r

B
;r
s
;
B
;

B
;
s
Þ
. We adopt a frequentist pro-
cedure [
50
] to obtain one-dimensional confidence intervals
of well-defined C.L. that takes into account non-Gaussian
effects due to the nonlinearity of the relations between the
observables and physical quantities. This procedure is
identical to that used in Refs. [
17
,
18
,
20
,
22
,
23
].
We define a
2
function as
2
ð
;
u
Þ
2 ln
L
ð
;
u
Þ
2
½
ln
L
ð
;
u
Þ
ln
L
max

;
(11)
where
2 ln
L
ð
;
u
Þ
is the variation of the combined log-
likelihood with respect to its maximum value, with the

z
ðÞ

TABLE IV.
CP
-violating complex parameters

z
ðÞ

¼

x
ðÞ

þ
i

y
ðÞ

and

z
s

¼

x
s

þ
i

y
s

obtained from the combination of GGSZ,
GLW, and ADS measurements. The first error is statistical
(corresponding to

2 ln
L
¼
1
), the second is the experimen-
tal systematic uncertainty including the systematic uncertainty
associated to the GGSZ decay amplitude models.
Real part (%)
Imaginary part (%)

z

8
:
1

2
:
3

0
:
74
:
4

3
:
4

0
:
5

z
þ

9
:
3

2
:
2

0
:
3

1
:
7

4
:
6

0
:
4

z



7
:
0

3
:
6

1
:
1

10
:
6

5
:
4

2
:
0

z

þ
10
:
3

2
:
9

0
:
8

1
:
4

8
:
3

2
:
5

z
s

13
:
3

8
:
1

2
:
613
:
9

8
:
8

3
:
6

z
s
þ

9
:
8

6
:
9

1
:
2
11
:
0

11
:
0

6
:
1
J. P. LEES
et al.
PHYSICAL REVIEW D
87,
052015 (2013)
052015-8