Measurement of
CP
-violating asymmetries in
B
0
!ð
Þ
0
decays using
a time-dependent Dalitz plot analysis
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,
6
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
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,
§
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,
42
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,56b
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,
75
S. Dasu,
76
Y. Pan,
76
R. Prepost,
76
and S. L. Wu
76
(
B
A
B
AR
Collaboration)
PHYSICAL REVIEW D
88,
012003 (2013)
1550-7998
=
2013
=
88(1)
=
012003(26)
012003-1
Ó
2013 American Physical Society
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
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
University of California at Santa Cruz, Institute for Particle Physics, 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, Newtonstraße 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, 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
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
J. P. LEES
et al.
PHYSICAL REVIEW D
88,
012003 (2013)
012003-2
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
Tel Aviv University, School of Physics and Astronomy, 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 15 April 2013; published 19 July 2013)
We present results for a time-dependent Dalitz plot measurement of
CP
-violating asymmetries in the
mode
B
0
!
þ
0
. The data set is derived from the complete sample of
471
10
6
B
B
meson pairs
collected with the
BABAR
detector at the PEP-II asymmetric-energy
e
þ
e
collider at the SLAC National
Accelerator Laboratory operating on the
ð
4
S
Þ
resonance. We extract parameters describing the time-
dependent
B
0
!
decay probabilities and
CP
asymmetries, including
C
¼
0
:
016
0
:
059
0
:
036
,
C
¼
0
:
234
0
:
061
0
:
048
,
S
¼
0
:
053
0
:
081
0
:
034
, and
S
¼
0
:
054
0
:
082
0
:
039
, where
the uncertainties are statistical and systematic, respectively. We perform a two-dimensional likelihood
scan of the direct
CP
-violation asymmetry parameters for
B
0
!
decays, finding the change in
2
between the minimum and the origin (corresponding to no direct
CP
violation) to be
2
¼
6
:
42
.We
present information on the
CP
-violating parameter
in a likelihood scan that incorporates
B
!
measurements. To aid in the interpretation of our results, statistical robustness studies are performed to
assess the reliability with which the true values of the physics parameters can be extracted. Significantly,
these studies indicate that
cannot be reliably extracted with our current sample size, though the other
physics parameters are robustly extracted.
DOI:
10.1103/PhysRevD.88.012003
PACS numbers: 11.30.Er, 12.15.Ff
I. INTRODUCTION
Within the standard model (SM) of particle physics,
CP
violation in the quark sector is described by the Cabibbo-
Kobayashi-Maskawa (CKM) quark-mixing matrix.
Physics beyond the SM may result in measured values of
observables that deviate from the values expected based on
other CKM parameter measurements and the SM.
The decay
B
0
!
þ
0
[
1
] is well suited to the study
of
CP
violation and has been previously explored by both
*
Present address: University of Tabuk, Tabuk 71491, Saudi
Arabia.
†
Also at Universita
`
di Perugia, Dipartimento di Fisica,
Perugia, Italy.
‡
Present address: University of Huddersfield, Huddersfield
HD1 3DH, United Kingdom.
§
Deceased.
∥
Present address: University of South Alabama, Mobile,
Alabama 36688, USA.
¶
Also at Universita
`
di Sassari, Sassari, Italy.
MEASUREMENT OF
CP
-VIOLATING ASYMMETRIES IN
...
PHYSICAL REVIEW D
88,
012003 (2013)
012003-3
the
BABAR
[
2
] and Belle [
3
] Collaborations. Early studies
of this mode were ‘‘quasi-two-body’’ (Q2B) analyses
that treated each
resonance separately in the decays
B
0
!
0
0
ð
0
!
þ
Þ
and
B
0
!
ð
!
0
Þ
.
However, as first noted by Snyder and Quinn [
4
], a com-
plete time-dependent Dalitz plot (DP) analysis is sensitive
to the interference between the strong and weak amplitudes
in the regions where the
þ
,
, and
0
resonances over-
lap. This interference allows the unambiguous extraction
of the strong and weak relative phases, and of the
CP
-violating parameter
arg
½
V
td
V
tb
=
ð
V
ud
V
ub
Þ
,
where
V
qq
0
are components of the CKM matrix. A preci-
sion measurement of
is of interest because it serves to
further test the SM and constrain new physics that may
contribute to loops in Feynman diagrams.
In this paper, we present an update of an earlier
BABAR
analysis. We use the full
BABAR
data set collected at the
ð
4
S
Þ
resonance, corresponding to an increase of 25% in
the number of
B
meson decays, and include a number of
improvements to both the reconstruction and selection
procedures. Among these are improved charged-particle
tracking and particle identification (PID), and a reopti-
mized multivariate discriminator, used both for event se-
lection and as a variable in the final fit.
Another new feature of this analysis is a series of studies
of the statistical robustness with which the true values of
physics parameters can be extracted. These studies, de-
scribed in an appendix, reveal that
cannot be reliably
extracted with our current sample size, though the other
physics parameters are robustly extracted.
Section
II
contains an introduction to the theory behind
this analysis and the formalism used. We proceed to de-
scriptions of the detector (Sec.
III
), the data sets (Sec.
IV
),
and the event selection procedures (Sec.
V
). This is fol-
lowed by a presentation of the fitting procedure (Sec.
VI
)
and of the systematic studies (Sec.
VII
). Finally, we present
the fit results (Sec.
VIII
) and a conclusion (Sec.
IX
). An
overview of robustness studies is provided in an Appendix.
II. THEORY OVERVIEW
A. Time-independent probability distribution
The time-independent amplitudes for
B
0
and
B
0
decays
to
þ
0
are given by
A
3
¼
f
þ
A
þ
þ
f
A
þ
f
0
A
0
;
A
3
¼
f
þ
A
þ
þ
f
A
þ
f
0
A
0
;
(1)
respectively, where
A
and
A
with
2fþ
;
;
0
g
are
complex amplitudes associated with the
þ
,
, and
0
resonances, respectively, and
f
¼
f
ð
m;
Þ
are defined
in terms of modified relativistic Breit-Wigner resonances
[
5
] modeling the three
resonances. The angle
is the
helicity angle for the resonance, defined as the angle
between the
0
(
) momentum and the negative of the
momentum of the recoiling
(
þ
) for the
þ
(
), and
as the angle between the
þ
momentum and the negative
of the momentum of the recoiling
0
for the
0
. All
helicity angles are calculated in the
rest frame. In the
fit, we include the
ð
770
Þ
as well as its radial excitation, the
ð
1450
Þ
; therefore, each
f
is a sum of modified relativis-
tic Breit-Wigner resonances,
F
, for the
ð
770
Þ
and
ð
1450
Þ
,
f
ð
m;
Þ/
F
ð
770
Þ
ð
m;
Þþ
a
0
e
i
0
F
ð
1450
Þ
ð
m;
Þ
;
(2)
where
a
0
and
0
are the magnitude and phase of the
ð
1450
Þ
resonance relative to the
ð
770
Þ
. We include
systematic uncertainties, described in Sec.
VII A
, to ac-
count for possible contributions from the
ð
1700
Þ
.
B. Time-dependent probability distribution
Using the time-independent amplitudes
A
3
and
A
3
,we
can express the full time-dependent probability for a me-
son that is a
B
0
(
A
3
)or
B
0
(
A
þ
3
) at the time the other
B
meson decays, to decay to
þ
0
as
j
A
3
ð
t
Þj
2
¼
e
j
t
j
=
B
0
4
B
0
j
A
3
j
2
þj
A
3
j
2
ðj
A
3
j
2
j
A
3
j
2
Þ
cos
ð
m
d
t
Þ
2Im
q
p
A
3
A
3
sin
ð
m
d
t
Þ
;
(3)
where
B
0
is the mean neutral
B
lifetime,
m
d
is the mass
difference between the heavy and light neutral
B
mass
eigenstates,
p
and
q
are the complex parameters in the
definitions of the neutral mass eigenstates
p
j
B
0
i
q
j
B
0
i
,
and
t
is the time difference between the decays of the
fully reconstructed
B
meson (
B
3
) and the
B
meson used to
determine the
B
flavor (
B
tag
). In Eq. (
3
), as in the fit, we
assume that the heavy and light mass eigenstates have the
same lifetime, that there is no
CP
violation in
B
0
B
0
mixing
(
j
q=p
j¼
1
), and that
CPT
is conserved.
C. Square Dalitz plot formalism
While nonresonant phase-space decays uniformly popu-
late the kinematically allowed region of a DP, signal
events populate the boundaries of this region due to the low
mass of the
resonances relative to the
B
mass. In par-
ticular, the interference regions of the signal DP, which
provide sensitivity to the relative phases of the
reso-
nances, are confined to small regions in the three corners of
the DP. In order to expand these regions of interest and
avoid the use of bins of variable size, we perform a trans-
formation of the DP that maps the kinematically allowed
region onto a dimensionless unit square. The transforma-
tion is described by
dm
þ
dm
!j
det
J
j
dm
0
d
0
;
(4)
with the square Dalitz plot (SDP) coordinates,
J. P. LEES
et al.
PHYSICAL REVIEW D
88,
012003 (2013)
012003-4
m
0
1
arccos
2
m
0
m
min
0
m
max
0
m
min
0
1
;
(5)
0
1
0
;
(6)
where
m
is the invariant mass of the
0
system,
m
0
is
the invariant mass of the two charged pion candidates,
0
is
the
0
helicity angle defined earlier,
m
max
0
¼
m
B
0
m
0
and
m
min
0
¼
2
m
þ
are the kinematic limits of the
m
0
mass,
and
J
is the Jacobian of the transformation. The determi-
nant of the Jacobian is given by
j
det
J
j¼
4
j
p
þ
jj
p
0
j
m
0
@m
0
@m
0
@
cos
0
@
0
;
(7)
where
j
p
þ
j¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð
E
þ
Þ
2
m
2
þ
q
;
(8)
j
p
0
j¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð
E
0
Þ
2
m
2
0
q
;
(9)
and the energies
E
þ
and
E
0
of the
þ
and
0
are defined in
the
þ
center-of-mass (CM) frame. Figure
1
shows an
example of a standard DP (left) and its transformed SDP
counterpart (right), plotted using simulated
B
0
!
de-
cays, where the three
resonances are assumed to have the
same amplitude.
D.
U=I
formalism
If one explicitly inserts Eq. (
1
) into Eq. (
3
), the full time-
dependent amplitude for a
B
0
or
B
0
meson to decay to
þ
0
can be written in terms of
j
A
3
j
2
j
A
3
j
2
¼
X
2½þ
;
;
0
j
f
j
2
U
þ
2
X
<
2½þ
;
;
0
ð
Re
½
f
f
U
;
Re
Im
½
f
f
U
;
Im
Þ
;
(10)
and
Im
q
p
A
3
A
3
¼
X
2½þ
;
;
0
j
f
j
2
I
þ
X
<
2½þ
;
;
0
ð
Re
½
f
f
I
Im
þ
Im
½
f
f
I
Re
Þ
;
(11)
with
U
¼j
A
j
2
j
A
j
2
;
(12)
U
;
Re
ð
Im
Þ
¼
Re
ð
Im
Þ½
A
A
A
A
;
(13)
I
¼
Im
½
A
A
;
(14)
I
Re
¼
Re
½
A
A
A
A
;
(15)
I
Im
¼
Im
½
A
A
þ
A
A
:
(16)
The 27 real-valued
U
and
I
coefficients provide an alter-
native parametrization to tree and penguin amplitudes (as
well as
) or to the amplitudes
A
and
A
[
6
]. The
U
and
I
parameters can also be directly related to the Q2B
C
and
S
parameters often used in
CP
-violation analyses [
7
], where
C
parametrizes direct
CP
violation, and
S
parametrizes
mixing-induced
CP
violation (involving the angle
in
this analysis). The related parameter
C
describes
the asymmetry between the rates
ð
B
0
!
þ
Þþ
ð
B
0
!
þ
Þ
and
ð
B
0
!
þ
Þþ
ð
B
0
!
þ
Þ
,
0
5
10
15
20
25
30
0
5
10
15
20
25
30
m
2
(
π
+
π
0
) (GeV/c
2
)
2
m
2
(
π
–
π
0
) (GeV/c
2
)
2
0
1
2
3
4
5
22
23
24
25
26
27
B
0
→
π
+
π
–
π
0
(kin.)
interference regs.
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
100
200
300
400
θ
'
m
'
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
m
'
θ
'
interference regions
m
(
ρ
0
)
=
1.5 GeV/c
2
m
(
ρ
+
)
=
1.5 GeV/c
2
m
(
ρ
–
)
=
1.5 GeV/c
2
FIG. 1 (color online). Nominal (left) and square (right)
B
0
!
Dalitz plots obtained from Monte Carlo generated events without
detector simulation [
2
]. The amplitudes in Eq. (
1
) are generated with values
A
þ
¼
A
¼
A
0
¼
1
so that they interfere destructively for
equal
masses. The hatched areas indicate the main overlap regions between the different
bands. The dashed lines in the square
Dalitz plot correspond to
m
ð
þ
;
;
0
Þ¼
1
:
5 GeV
=c
2
. The middle plot depicts the Jacobian determinant of the transformation and shows
the distribution in the square Dalitz plot for uniformly distributed events in the nominal Dalitz plot.
MEASUREMENT OF
CP
-VIOLATING ASYMMETRIES IN
...
PHYSICAL REVIEW D
88,
012003 (2013)
012003-5
while
S
is related to the strong phase difference between
the different amplitudes describing the decay
B
0
!
.
The
U
and
I
parameters are related to the
C
and
S
parame-
ters through the relations
C
þ
¼
U
þ
U
þ
þ
;
C
¼
U
U
þ
;
S
þ
¼
2
I
þ
U
þ
þ
;
S
¼
2
I
U
þ
;
(17)
and
A
¼
U
þ
þ
U
þ
U
þ
þ
þ
U
þ
;
(18)
where
C
¼ð
C
þ
þ
C
Þ
=
2
;
(19)
C
¼ð
C
þ
C
Þ
=
2
;
(20)
S
¼ð
S
þ
þ
S
Þ
=
2
;
(21)
S
¼ð
S
þ
S
Þ
=
2
:
(22)
Note that while
C
,
S
,
C
, and
S
do not depend on
interference effects between the
resonances, the
U
and
I
parameter formalism accounts for these features and is
thus appropriate for a full DP analysis. While some degree
of physical intuition is lost when using the
U
and
I
pa-
rameters instead of the standard complex amplitudes and
phases, there are several practical motivating factors for
their adoption in the fit:
(i) Whereas there is a twofold ambiguity in
(
versus
90
), there is a unique solution in a fit to the
U
and
I
parameters, which encompasses both solutions
for
).
(ii) The
U
and
I
parametrization results in uncertainties
that are more Gaussian than those in a standard
amplitude and phase fit.
(iii) It is simpler to average
U
and
I
fit results from
different measurements or experiments that publish
the full covariance matrix.
For physical solutions, there are constraints between
the
U
and
I
parameters. In the case of three
resonances
(
þ
,
, and
0
), a fit to the complex tree and penguin
amplitudes as well as the weak phase
involves 11 un-
known parameters, which reduce to 10 parameters when the
arbitrary global phase is removed. A
U
and
I
fit is equiva-
lent to such a fit, but involves many more parameters.
However, when the
0
0
amplitude is small, as is observed
in nature, the values of 11 of the 27
U
and
I
parameters
become unimportant. Due to the high degree of correlation
between the various
U
and
I
fit parameters, the 27 parame-
ters actually represent only 12 independent parameters.
Neglecting the arbitrary phase and the overall normaliza-
tion, this reduces to 10, and once isospin relations are taken
into account, only 9 independent parameters remain.
Because the
U
and
I
formalism is used in the final fit
without any constraints on the parameters (aside from
fixing
U
þ
þ
¼
1
to set the overall normalization), it is
possible for the free parameters to take on unphysical
values that do not correspond to any physical set of
amplitudes. The final fit values from the 2007
BABAR
analysis [
2
] are one such unphysical set. We determined
that no biases are introduced due to the fitted values of the
parameters being unphysical.
III. THE
BABAR
DETECTOR AND EXPERIMENT
The data used in this analysis were collected with the
BABAR
detector at the PEP-II asymmetric-energy
e
þ
e
storage ring at SLAC. A detailed description of the
BABAR
detector is presented in Ref. [
8
]. The tracking system used
for track and vertex reconstruction has two components: a
5-layer silicon vertex tracker and a drift chamber, both
operating within a 1.5 T magnetic field generated by a
superconducting solenoidal magnet. A detector of inter-
nally reflected Cherenkov light associates Cherenkov pho-
tons with tracks for particle identification. The energies of
photons and electrons are determined from the measured
light produced in electromagnetic showers inside a CsI(Tl)
crystal electromagnetic calorimeter. Muon candidates are
identified with the use of the instrumented flux return of the
solenoid. The flux return instrumentation initially con-
sisted of resistive plate chambers and was later modified
to consist of a mixture of resistive plate chambers and
limited streamer tubes.
IV. DATA SAMPLE AND MC SIMULATION
A. Data samples
For the final fit, we use the full ‘‘on-resonance’’
BABAR
data set of
431
:
0fb
1
[
9
] collected at the
ð
4
S
Þ
resonance
energy (
ffiffiffi
s
p
¼
10
:
58 GeV
=c
2
). When optimizing back-
ground suppression criteria,
44
:
6fb
1
of ‘‘off-resonance’’
data, collected 40 MeV below the
ð
4
S
Þ
resonance, are
used to model ‘‘continuum’’
e
þ
e
!
q
q
(
q
¼
u
,
d
,
s
,
c
)
background.
B. Monte Carlo samples
Event simulation based on the Monte Carlo (MC)
method is used to evaluate backgrounds and to determine
signal-event reconstruction efficiencies. For all MC
samples, detector response is accounted for using the
GEANT4
package [
10
] in a full
BABAR
detector simulation.
B
0
!
þ
and
B
0
!
0
0
signal decays are simu-
lated in separate MC data sets, but the
B
0
!
0
0
sample
is not used to determine background selection criteria or to
model signal distributions since the nominal branching
fraction for
B
0
!
is
11
:
5
3
:
1
times larger than
the branching fraction for
B
0
!
0
0
[
11
].
B
-decay backgrounds are modeled using MC samples
consisting of
B
decays to specific final states as well as
‘‘generic’’ MC samples consisting of charged and neutral
B
decays to unconstrained final states. In the generic MC,
dominant branching fractions are fixed to the results of
J. P. LEES
et al.
PHYSICAL REVIEW D
88,
012003 (2013)
012003-6
recent measurements [
12
]. Due to the uncertainty on
branching fractions for charmless decays, all charmless
events are removed from the generic MC and charmless
modes of interest are explicitly included among the
B
background samples consisting of decays to specific final
states. We use a total of 24
B
-decay MC samples corre-
sponding to 29 different final states (Table
I
) as well as MC
samples of generic charged and neutral
B
decays.
The expected number of events for each charmless
B
background is calculated according to
n
exp
¼
2
n
BB
BB
mode
;
(23)
where
n
BB
is the number of produced
B
B
pairs,
B
is the
branching fraction [
12
] (approximately
1
=
2
) for an
ð
4
S
Þ
to decay to a charged or neutral
B
B
pair (whichever is
appropriate for the mode in question),
B
mode
is the branch-
ing fraction for the
B
decay mode, and
is the efficiency
for reconstructing events in the mode, determined from
MC. The factor of 2 is included because either of the
B
mesons in a given event may decay to the mode of interest.
In the case of the charged and neutral generic
B
back-
grounds, the number of events expected for each mode is
n
exp
¼
n
MC
n
BB
B
n
gen
;
(24)
where
n
gen
is the number of generated charged (neutral)
MC events and
n
MC
is the number of charged (neutral)
generic MC events that remain after all selection criteria
have been applied.
An additional simulated data set, which is used for valida-
tion studies, consists of DP-parametrized
B
0
!
decays.
This data set is used to verify flavor-tagging conventions.
V. EVENT SELECTION AND BACKGROUND
SUPPRESSION
A. Event preselection
The kinematics of
B
meson decays that are fully recon-
structed at
BABAR
can be characterized by two variables:
TABLE I.
B
background decay modes included in the final fit. Modes generated taking into account interference effects in the Dalitz
plot are indicated by the ‘‘Dalitz’’ label. Longitudinal polarization is indicated by ‘‘[longitudinal].’’
Class
Decay mode
B
[
10
6
]
# of expected events
0
B
þ
!
þ
0
[longitudinal]
24
:
0
1
:
9
129
10
0
B
þ
!
a
þ
1
ð! ð
Þ
þ
Þ
0
26
753
14
0
B
þ
!
a
0
1
ð!
Þ
þ
20
637
11
1
B
þ
!
þ
K
0
s
ð!
þ
Þ
7
:
99
0
:
35
6
:
96
0
:
31
2
B
þ
!
K
þ
þ
Dalitz
51
:
0
2
:
934
:
8
2
:
0
2
B
þ
!
þ
þ
Dalitz
16
:
2
1
:
5
203
19
3
B
þ
!
0
þ
10
:
9
1
:
4
120
15
3
B
þ
!
þ
K
0
s
ð!
0
0
Þ
3
:
54
0
:
15
24
:
2
1
:
1
4
B
þ
!
þ
0
5
:
7
0
:
538
:
6
3
:
4
4
B
þ
!
K
þ
0
12
:
9
0
:
618
:
6
0
:
9
5
B
0
!
K
0
s
þ
Dalitz
44
:
8
2
:
615
:
7
0
:
9
6
B
0
!
þ
[longitudinal]
24
:
2
3
:
1
122
16
6
B
0
!
a
1
þ
33
561
9
6
B
0
!
a
0
1
0
11
:
0
1
:
722
:
8
3
:
5
7
B
0
!
K
þ
19
:
4
0
:
621
:
6
0
:
7
8
B
0
!
K
þ
0
Dalitz
35
:
9
2
:
6
398
29
9
B
0
!
K
0
ð
892
Þð!
K
þ
Þ
40
:
1
2
:
031
:
8
1
:
6
9
B
0
!
K
0
ð
1430
Þð!
K
þ
Þ
12
:
4
2
:
43
:
2
0
:
6
9
B
0
!
0
ð!
0
Þ
0
0
:
35
0
:
18
4
:
2
2
:
1
10
B
0
!
0
K
0
s
ð!
þ
Þ
3
:
39
0
:
21
21
:
8
1
:
4
11
B
0
!
D
ð!
0
Þ
þ
3
:
35
0
:
27
399
32
12
B
0
!
D
0
ð!
K
þ
;K
þ
0
Þ
0
46
:
7
4
:
5
124
12
13
B
0
!
D
0
ð!
þ
Þ
0
0
:
367
0
:
034
48
5
14
B
0
!
J=
c
ð!
e
þ
e
;
þ
Þ
0
2
:
09
0
:
19
153
14
15
B
0
!
neutral generic
b
!
c
decays
466
14
16
B
þ
!
charged generic
b
!
c
decays
921
21
Total
3478
65
MEASUREMENT OF
CP
-VIOLATING ASYMMETRIES IN
...
PHYSICAL REVIEW D
88,
012003 (2013)
012003-7