of 31
Study of
CP
violation in Dalitz-plot analyses of
B
0
!
K
þ
K

K
0
S
,
B
þ
!
K
þ
K

K
þ
, and
B
þ
!
K
0
S
K
0
S
K
þ
J. P. Lees,
1
V. Poireau,
1
V. Tisserand,
1
J. Garra Tico,
2
E. Grauges,
2
D. A. Milanes,
3
A. Palano,
3,4
M. Pappagallo,
3,4
G. Eigen,
5
B. Stugu,
5
D. N. Brown,
6
L. T. Kerth,
6
Yu. G. Kolomensky,
6
G. Lynch,
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
V. E. Blinov,
10
A. R. Buzykaev,
10
V. P. Druzhinin,
10
V. B. Golubev,
10
E. A. Kravchenko,
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
D. Kirkby,
11
A. J. Lankford,
11
M. Mandelkern,
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
C. A. West,
13
A. M. Eisner,
14
J. Kroseberg,
14
W. S. Lockman,
14
A. J. Martinez,
14
T. Schalk,
14
B. A. Schumm,
14
A. Seiden,
14
D. S. Chao,
15
C. H. Cheng,
15
D. A. Doll,
15
B. Echenard,
15
K. T. Flood,
15
D. G. Hitlin,
15
P. Ongmongkolkul,
15
F. C. Porter,
15
A. Y. Rakitin,
15
R. Andreassen,
16
Z. Huard,
16
B. T. Meadows,
16
M. D. Sokoloff,
16
L. Sun,
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
B. Spaan,
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
D. Bettoni,
23
C. Bozzi,
23
R. Calabrese,
23,24
G. Cibinetto,
23,24
E. Fioravanti,
23,24
I. Garzia,
23,24
E. Luppi,
23,24
M. Munerato,
23,24
M. Negrini,
23,24
L. Piemontese,
23
V. Santoro,
23
R. Baldini-Ferroli,
25
A. Calcaterra,
25
R. de Sangro,
25
G. Finocchiaro,
25
P. Patteri,
25
I. M. Peruzzi,
25,
M. Piccolo,
25
M. Rama,
25
A. Zallo,
25
R. Contri,
26,27
E. Guido,
26,27
M. Lo Vetere,
26,27
M. R. Monge,
26,27
S. Passaggio,
26
C. Patrignani,
26,27
E. Robutti,
26
B. Bhuyan,
28
V. Prasad,
28
C. L. Lee,
29
M. Morii,
29
A. J. Edwards,
30
A. Adametz,
31
J. Marks,
31
U. Uwer,
31
H. M. Lacker,
32
T. Lueck,
32
P. D. Dauncey,
33
P. K. Behera,
34
U. Mallik,
34
C. Chen,
35
J. Cochran,
35
W. T. Meyer,
35
S. Prell,
35
A. E. Rubin,
35
A. V. Gritsan,
36
Z. J. Guo,
36
N. Arnaud,
37
M. Davier,
37
D. Derkach,
37
G. Grosdidier,
37
F. Le Diberder,
37
A. M. Lutz,
37
B. Malaescu,
37
P. Roudeau,
37
M. H. Schune,
37
A. Stocchi,
37
G. Wormser,
37
D. J. Lange,
38
D. M. Wright,
38
I. Bingham,
39
C. A. Chavez,
39
J. P. Coleman,
39
J. R. Fry,
39
E. Gabathuler,
39
D. E. Hutchcroft,
39
D. J. Payne,
39
C. Touramanis,
39
A. J. Bevan,
40
F. Di Lodovico,
40
R. Sacco,
40
M. Sigamani,
40
G. Cowan,
41
D. N. Brown,
42
C. L. Davis,
42
A. G. Denig,
43
M. Fritsch,
43
W. Gradl,
43
A. Hafner,
43
E. Prencipe,
43
D. Bailey,
44
R. J. Barlow,
44,
G. Jackson,
44
G. D. Lafferty,
44
E. Behn,
45
R. Cenci,
45
B. Hamilton,
45
A. Jawahery,
45
D. A. Roberts,
45
G. Simi,
45
C. Dallapiccola,
46
R. Cowan,
47
D. Dujmic,
47
G. Sciolla,
47
R. Cheaib,
48
D. Lindemann,
48
P. M. Patel,
48
S. H. Robertson,
48
M. Schram,
48
P. Biassoni,
49,50
N. Neri,
49
F. Palombo,
49,50
S. Stracka,
49,50
L. Cremaldi,
51
R. Godang,
51,
§
R. Kroeger,
51
P. Sonnek,
51
D. J. Summers,
51
X. Nguyen,
52
M. Simard,
52
P. Taras,
52
G. De Nardo,
53,54
D. Monorchio,
53,54
G. Onorato,
53,54
C. Sciacca,
53,54
M. Martinelli,
55
G. Raven,
55
C. P. Jessop,
56
K. J. Knoepfel,
56
J. M. LoSecco,
56
W. F. Wang,
56
K. Honscheid,
57
R. Kass,
57
J. Brau,
58
R. Frey,
58
N. B. Sinev,
58
D. Strom,
58
E. Torrence,
58
E. Feltresi,
59,60
N. Gagliardi,
59,60
M. Margoni,
59,60
M. Morandin,
59
M. Posocco,
59
M. Rotondo,
59
F. Simonetto,
59,60
R. Stroili,
59,60
S. Akar,
61
E. Ben-Haim,
61
M. Bomben,
61
G. R. Bonneaud,
61
H. Briand,
61
G. Calderini,
61
J. Chauveau,
61
O. Hamon,
61
Ph. Leruste,
61
G. Marchiori,
61
J. Ocariz,
61
S. Sitt,
61
M. Biasini,
62,63
E. Manoni,
62,63
S. Pacetti,
62,63
A. Rossi,
62,63
C. Angelini,
64,65
G. Batignani,
64,65
S. Bettarini,
64,65
M. Carpinelli,
64,65,
k
G. Casarosa,
64,65
A. Cervelli,
64,65
F. Forti,
64,65
M. A. Giorgi,
64,65
A. Lusiani,
64,65
B. Oberhof,
64,65
E. Paoloni,
64,65
A. Perez,
64
G. Rizzo,
64,65
J. J. Walsh,
64
D. Lopes Pegna,
67
J. Olsen,
67
A. J. S. Smith,
67
A. V. Telnov,
67
F. Anulli,
68
G. Cavoto,
68
R. Faccini,
68,69
F. Ferrarotto,
68
F. Ferroni,
68
L. Li Gioi,
68
M. A. Mazzoni,
68
G. Piredda,
68
C. Bu
̈
nger,
70
O. Gru
̈
nberg,
70
T. Hartmann,
70
T. Leddig,
70
H. Schro
̈
der,
70
C. Voss,
70
R. Waldi,
70
T. Adye,
71
E. O. Olaiya,
71
F. F. Wilson,
71
S. Emery,
72
G. Hamel de Monchenault,
72
G. Vasseur,
72
Ch. Ye
`
che,
72
D. Aston,
73
D. J. Bard,
73
R. Bartoldus,
73
C. Cartaro,
73
M. R. Convery,
73
J. Dorfan,
73
G. P. Dubois-Felsmann,
73
W. Dunwoodie,
73
M. Ebert,
73
R. C. Field,
73
M. Franco Sevilla,
73
B. G. Fulsom,
73
A. M. Gabareen,
73
M. T. Graham,
73
P. Grenier,
73
C. Hast,
73
W. R. Innes,
73
M. H. Kelsey,
73
P. Kim,
73
M. L. Kocian,
73
D. W. G. S. Leith,
73
P. Lewis,
73
B. Lindquist,
73
S. Luitz,
73
V. Luth,
73
H. L. Lynch,
73
D. B. MacFarlane,
73
D. R. Muller,
73
H. Neal,
73
S. Nelson,
73
M. Perl,
73
T. Pulliam,
73
B. N. Ratcliff,
73
A. Roodman,
73
A. A. Salnikov,
73
R. H. Schindler,
73
A. Snyder,
73
D. Su,
73
M. K. Sullivan,
73
J. Va’vra,
73
A. P. Wagner,
73
M. Weaver,
73
W. J. Wisniewski,
73
M. Wittgen,
73
D. H. Wright,
73
H. W. Wulsin,
73
C. C. Young,
73
V. Ziegler,
73
W. Park,
74
M. V. Purohit,
74
R. M. White,
74
J. R. Wilson,
74
A. Randle-Conde,
74
S. J. Sekula,
75
M. Bellis,
76
J. F. Benitez,
76
P. R. Burchat,
76
T. S. Miyashita,
76
M. S. Alam,
77
J. A. Ernst,
77
R. Gorodeisky,
78
N. Guttman,
78
D. R. Peimer,
78
A. Soffer,
78
P. Lund,
79
S. M. Spanier,
79
R. Eckmann,
80
J. L. Ritchie,
80
A. M. Ruland,
80
C. J. Schilling,
80
R. F. Schwitters,
80
B. C. Wray,
80
J. M. Izen,
81
X. C. Lou,
81
F. Bianchi,
82,83
D. Gamba,
82,83
L. Lanceri,
84,85
L. Vitale,
84,85
F. Martinez-Vidal,
86
A. Oyanguren,
86
H. Ahmed,
87
J. Albert,
87
Sw. Banerjee,
87
F. U. Bernlochner,
87
H. H. F. Choi,
87
PHYSICAL REVIEW D
85,
112010 (2012)
1550-7998
=
2012
=
85(11)
=
112010(31)
112010-1
Ó
2012 American Physical Society
G. J. King,
87
R. Kowalewski,
87
M. J. Lewczuk,
87
I. M. Nugent,
87
J. M. Roney,
87
R. J. Sobie,
87
N. Tasneem,
87
T. J. Gershon,
88
P. F. Harrison,
88
T. E. Latham,
88
E. M. T. Puccio,
88
H. R. Band,
89
S. Dasu,
89
Y. Pan,
89
R. Prepost,
89
and S. L. Wu
89
(
B
A
B
AR
Collaboration)
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
3
INFN Sezione di Bari, I-70126 Bari, Italy
4
Dipartimento di Fisica, Universita
`
di Bari, I-70126 Bari, Italy
5
University of Bergen, Institute of Physics, N-5007 Bergen, Norway
6
Lawrence Berkeley National Laboratory and University of California, Berkeley, California 94720, USA
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, Ecole Polytechnique, CNRS/IN2P3, F-91128 Palaiseau, France
22
University of Edinburgh, Edinburgh EH9 3JZ, United Kingdom
23
INFN Sezione di Ferrara, I-44100 Ferrara, Italy
24
Dipartimento di Fisica, Universita
`
di Ferrara, I-44100 Ferrara, Italy
25
INFN Laboratori Nazionali di Frascati, I-00044 Frascati, Italy
26
INFN Sezione di Genova, I-16146 Genova, Italy
27
Dipartimento di Fisica, Universita
`
di Genova, I-16146 Genova, Italy
28
Indian Institute of Technology Guwahati, Guwahati, Assam, 781 039, India
29
Harvard University, Cambridge, Massachusetts 02138, USA
30
Harvey Mudd College, Claremont, California 91711
31
Universita
̈
t Heidelberg, Physikalisches Institut, Philosophenweg 12, D-69120 Heidelberg, Germany
32
Humboldt-Universita
̈
t zu Berlin, Institut fu
̈
r Physik, Newtonstrasse 15, D-12489 Berlin, Germany
33
Imperial College London, London, SW7 2AZ, United Kingdom
34
University of Iowa, Iowa City, Iowa 52242, USA
35
Iowa State University, Ames, Iowa 50011-3160, USA
36
Johns Hopkins University, Baltimore, Maryland 21218, USA
37
Laboratoire de l’Acce
́
le
́
rateur Line
́
aire, IN2P3/CNRS et Universite
́
Paris-Sud 11,
Centre Scientifique d’Orsay, B. P. 34, F-91898 Orsay Cedex, France
38
Lawrence Livermore National Laboratory, Livermore, California 94550, USA
39
University of Liverpool, Liverpool L69 7ZE, United Kingdom
40
Queen Mary, University of London, London, E1 4NS, United Kingdom
41
University of London, Royal Holloway and Bedford New College, Egham, Surrey TW20 0EX, United Kingdom
42
University of Louisville, Louisville, Kentucky 40292, USA
43
Johannes Gutenberg-Universita
̈
t Mainz, Institut fu
̈
r Kernphysik, D-55099 Mainz, Germany
44
University of Manchester, Manchester M13 9PL, United Kingdom
45
University of Maryland, College Park, Maryland 20742, USA
46
University of Massachusetts, Amherst, Massachusetts 01003, USA
47
Massachusetts Institute of Technology, Laboratory for Nuclear Science, Cambridge, Massachusetts 02139, USA
48
McGill University, Montre
́
al, Que
́
bec, Canada H3A 2T8
49
INFN Sezione di Milano, I-20133 Milano, Italy
50
Dipartimento di Fisica, Universita
`
di Milano, I-20133 Milano, Italy
51
University of Mississippi, University, Mississippi 38677, USA
J. P. LEES
et al.
PHYSICAL REVIEW D
85,
112010 (2012)
112010-2
52
Universite
́
de Montre
́
al, Physique des Particules, Montre
́
al, Que
́
bec, Canada H3C 3J7
53
INFN Sezione di Napoli, I-80126 Napoli, Italy
54
Dipartimento di Scienze Fisiche, Universita
`
di Napoli Federico II, I-80126 Napoli, Italy
55
NIKHEF, National Institute for Nuclear Physics and High Energy Physics, NL-1009 DB Amsterdam, The Netherlands
56
University of Notre Dame, Notre Dame, Indiana 46556, USA
57
The Ohio State University, Columbus, Ohio 43210, USA
58
University of Oregon, Eugene, Oregon 97403, USA
59
INFN Sezione di Padova, I-35131 Padova, Italy
60
Dipartimento di Fisica, Universita
`
di Padova, I-35131 Padova, Italy
61
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
62
INFN Sezione di Perugia, I-06100 Perugia, Italy
63
Dipartimento di Fisica, Universita
`
di Perugia, I-06100 Perugia, Italy
64
INFN Sezione di Pisa, I-56127 Pisa, Italy
65
Dipartimento di Fisica, Universita
`
di Pisa, I-56127 Pisa, Italy
66
Scuola Normale Superiore di Pisa, I-56127 Pisa, Italy
67
Princeton University, Princeton, New Jersey 08544, USA
68
INFN Sezione di Roma, I-00185 Roma, Italy
69
Dipartimento di Fisica, Universita
`
di Roma La Sapienza, I-00185 Roma, Italy
70
Universita
̈
t Rostock, D-18051 Rostock, Germany
71
Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, OX11 0QX, United Kingdom
72
CEA, Irfu, SPP, Centre de Saclay, F-91191 Gif-sur-Yvette, France
73
SLAC National Accelerator Laboratory, Stanford, California 94309 USA
74
University of South Carolina, Columbia, South Carolina 29208, USA
75
Southern Methodist University, Dallas, Texas 75275, USA
76
Stanford University, Stanford, California 94305-4060, USA
77
State University of New York, Albany, New York 12222, USA
78
Tel Aviv University, School of Physics and Astronomy, Tel Aviv, 69978, Israel
79
University of Tennessee, Knoxville, Tennessee 37996, USA
80
University of Texas at Austin, Austin, Texas 78712, USA
81
University of Texas at Dallas, Richardson, Texas 75083, USA
82
INFN Sezione di Torino, I-10125 Torino, Italy
83
Dipartimento di Fisica Sperimentale, Universita
`
di Torino, I-10125 Torino, Italy
84
INFN Sezione di Trieste, I-34127 Trieste, Italy
85
Dipartimento di Fisica, Universita
`
di Trieste, I-34127 Trieste, Italy
86
IFIC, Universitat de Valencia-CSIC, E-46071 Valencia, Spain
87
University of Victoria, Victoria, British Columbia, Canada V8W 3P6
88
Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom
89
University of Wisconsin, Madison, Wisconsin 53706, USA
(Received 31 January 2012; published 22 June 2012)
We perform amplitude analyses of the decays
B
0
!
K
þ
K

K
0
S
,
B
þ
!
K
þ
K

K
þ
, and
B
þ
!
K
0
S
K
0
S
K
þ
,
and measure
CP
-violating parameters and partial branching fractions. The results are based on a data
sample of approximately
470

10
6
B

B
decays, collected with the
BABAR
detector at the PEP-II
asymmetric-energy
B
factory at the SLAC National Accelerator Laboratory. For
B
þ
!
K
þ
K

K
þ
,we
find a direct
CP
asymmetry in
B
þ
!

ð
1020
Þ
K
þ
of
A
CP
¼ð
12
:
8

4
:
4

1
:
3
Þ
%
, which differs from zero
by
2
:
8

. For
B
0
!
K
þ
K

K
0
S
, we measure the
CP
-violating phase

eff
ð

ð
1020
Þ
K
0
S
Þ¼ð
21

6

2
Þ

. For
B
þ
!
K
0
S
K
0
S
K
þ
, we measure an overall direct
CP
asymmetry of
A
CP
¼ð
4
þ
4

5

2
Þ
%
. We also perform an
angular-moment analysis of the three channels and determine that the
f
X
ð
1500
Þ
state can be described
well by the sum of the resonances
f
0
ð
1500
Þ
,
f
0
2
ð
1525
Þ
, and
f
0
ð
1710
Þ
.
DOI:
10.1103/PhysRevD.85.112010
PACS numbers: 13.25.Hw, 13.25.Jx, 13.66.Bc, 14.40.Nd
*
Now at the University of Tabuk, Tabuk 71491, Saudi Arabia.
Also with Universita
`
di Perugia, Dipartimento di Fisica, Perugia, Italy.
Now at the University of Huddersfield, Huddersfield HD1 3DH, UK.
§
Now at University of South Alabama, Mobile, AL 36688, USA.
k
Also with Universita
`
di Sassari, Sassari, Italy.
STUDY OF
CP
VIOLATION IN DALITZ-PLOT
...
PHYSICAL REVIEW D
85,
112010 (2012)
112010-3
I. INTRODUCTION
In the standard model (SM),
CP
violation in the quark
sector is entirely described by a single weak phase in the
Cabibbo-Kobayashi-Maskawa (CKM) quark-mixing ma-
trix. Studies of time-dependent
CP
violation in
B
0
!
ð
c

c
Þ
K
0
decay
1
have yielded precise measurements [
1
,
2
]of
sin2

, where


arg
½ð
V

cb
V
cd
Þ
=
ð
V

tb
V
td
Þ
and
V
ij
are the
elements of the CKM matrix. Measurements of time-
dependent
CP
violation in
b
!
q

qs
(
q
¼
u
,
d
,
s
) decays
offer an alternative method for measuring

. Such decays
are dominated by
b
!
s
loop diagrams, and therefore are
sensitive to possible new physics (NP) contributions appear-
ingin the loops of these diagrams. As a result, the effective

(

eff
) measured in such decays could differ from the

measured in
B
0
c

c
Þ
K
0
. Deviations of

eff
from

are
also possible in the SM, due to additional amplitudes from
b
!
u
tree diagrams, loop diagrams containing different
CKM factors (‘‘
u
penguins’’), and final-state interactions.
The decay mode
B
0
!
K
0
S
is particularly suited for a
NP search, as

eff
for this mode is expected to be very near
in value to

in the SM, with
sin2

eff

sin2

in the range
(

0
:
01
, 0.04) [
3
5
]. However, the measurement of

eff
is
complicated due to other
B
0
!
K
þ
K

K
0
S
decays that in-
terfere with
B
0
!
K
0
S
. In general,
K
þ
K

K
0
S
is not a
CP
eigenstate: the
K
þ
K

K
0
S
system is
CP
even (odd) if the
K
þ
K

system has even (odd) angular momentum. Thus,
one must account for the (mostly
S
-wave)
K
þ
K

K
0
S
states
that interfere with
K
0
S
. This can be done by measuring

eff
using a Dalitz-plot (DP) analysis of
B
0
!
K
þ
K

K
0
S
.
A further benefit of a DP analysis is that it allows both
sin2

eff
and
cos2

eff
to be determined, through the inter-
ference of odd and even partial waves, which eliminates a
trigonometric ambiguity between

eff
and
90



eff
.
The related decay mode
B
þ
!
K
þ
is another interest-
ing channel in which to search for NP. This decay is also
dominated by a
b
!
s
penguin amplitude, and its direct
CP
asymmetry,
A
CP
, is predicted to be small in the SM,
(0.0–4.7)% [
5
,
6
], so a significant deviation from zero could
be a signal of NP.
In addition to measuring

eff
in
B
0
!
K
0
S
, it is possible
to measure it for the other resonant and nonresonant
B
0
!
K
þ
K

K
0
S
decays. However, these decays may contain a
mixture of even and odd partial waves, so the final state is
not guaranteed to be a
CP
eigenstate, thus posing a chal-
lenge to a measurement of

eff
. A DP analysis can reveal
which partial waves are present, thus eliminating a source
of systematic uncertainty affecting the extraction of

eff
,
without having to rely on theoretical predictions.
Previous analyses of
B
þ
!
K
þ
K

K
þ
[
7
,
8
] and
B
0
!
K
þ
K

K
0
S
[
9
,
10
] have revealed a complex DP structure that
is poorly understood. Both modes exhibit a large peak
around
m
ð
K
þ
K

Þ
1500 MeV
=c
2
, which has been
dubbed the
f
X
ð
1500
Þ
. Both
BABAR
and Belle have mod-
eled it as a scalar resonance. The recent DP analysis of
B
0
!
K
0
S
K
0
S
K
0
S
by
BABAR
[
11
] does not yield evidence for
this resonance. It is important to clarify the properties of
the
f
X
ð
1500
Þ
with a larger data sample, and, in particular,
to determine its spin, as that affects the

eff
measurement
in
B
0
!
K
þ
K

K
0
S
.
An additional feature seen in
B
0
!
K
þ
K

K
0
S
and
B
þ
!
K
þ
K

K
þ
decays is a large broad ‘‘nonresonant’’ (NR)
contribution. Previous analyses have found that a uniform-
phase-space model is insufficient to describe the NR term,
and have instead parametrized it with an empirical model.
The NR term has been taken to be purely
K
þ
K

S
wave in
B
þ
!
K
þ
K

K
þ
[
7
,
8
], while smaller
K
þ
K
0
S
and
K

K
0
S
S
-wave terms have been seen in
B
0
!
K
þ
K

K
0
S
[
9
,
10
],
which correspond effectively to higher-order
K
þ
K

partial
waves. Because the NR contribution dominates much of
the available phase space, it is crucial to study its angular
distribution if one wishes to accurately measure

eff
over
the entire
B
0
!
K
þ
K

K
0
S
DP.
Because of the importance of understanding the DP
structure in
B
0
!
K
þ
K

K
0
S
, we study the related modes
B
þ
!
K
þ
K

K
þ
and
B
þ
!
K
0
S
K
0
S
K
þ
along with
B
0
!
K
þ
K

K
0
S
. The mode
B
þ
!
K
þ
K

K
þ
is valuable because
it has the most signal events by far of any
B
!
KKK
mode.
Far fewer events are expected in
B
þ
!
K
0
S
K
0
S
K
þ
, but its
DP has a simplified spin structure due to the fact that the
two
K
0
S
mesons in the final state are forbidden (by Bose-
Einstein statistics) to be in an odd angular momentum
configuration. This implies that the
f
X
ð
1500
Þ
can decay
to
K
0
S
K
0
S
only if it has even spin, and it also ensures that the
nonresonant component in
B
þ
!
K
0
S
K
0
S
K
þ
does not con-
tain any
K
0
S
K
0
S
P
-wave contribution.
In this paper we report the results of DP analyses of
B
þ
!
K
þ
K

K
þ
and
B
þ
!
K
0
S
K
0
S
K
þ
, and a time-
dependent DP analysis of
B
0
!
K
þ
K

K
0
S
. In Sec.
II
,we
introduce the formalism used for the DP amplitude analy-
ses. In Sec.
III
, we briefly describe the
BABAR
detector and
data sets used, and Sec.
IV
describes the event selection
and backgrounds. Section
V
describes the maximum-
likelihood (ML) fit parametrization and implementation.
In Sec.
VI
, we present studies of the DP structure in the
three modes, which enable us to determine the nominal DP
models. In Sec.
VII
, we then present the final fit results
including measurements of
CP
violation. We discuss sys-
tematic uncertainties in Sec.
VIII
and summarize our re-
sults in Sec.
IX
.
II. DECAY MODEL FORMALISM
Taking advantage of the interference pattern in the DP,
we measure the magnitudes and phases of the different
resonant decay modes using an unbinned maximum-
likelihood fit.
1
Charge-conjugate decays are implied throughout, unless oth-
erwise indicated.
J. P. LEES
et al.
PHYSICAL REVIEW D
85,
112010 (2012)
112010-4
We consider the decay of a
B
meson with four-
momentum
p
B
into the three daughters
K
1
,
K
2
, and
K
3
,
with corresponding four-momenta
p
1
,
p
2
, and
p
3
. The
squares of the invariant masses are given by
s
ij
¼
m
2
ij
¼
ð
p
i
þ
p
j
Þ
2
.
We will use the following convention for the
K
indices:
(i) For
B

!
K

K
K

,
K
1

K

,
K
2

K
, and
K
3

K

. The indices for the two like-sign kaons
are defined such that
s
12
s
23
.
(ii) For
B

!
K
0
S
K
0
S
K

,
K
1

K
0
S
,
K
2

K
0
S
, and
K
3

K

. The indices for the two
K
0
S
are defined such that
s
13
s
23
.
(iii) For
B
ðÞ
0
!
K
þ
K

K
0
S
,
K
1

K
þ
,
K
2

K

, and
K
3

K
0
S
.
The
s
ij
obey the relation
s
12
þ
s
13
þ
s
23
¼
m
2
B
þ
m
2
K
1
þ
m
2
K
2
þ
m
2
K
3
:
(1)
The DP distribution of the
B

decays is given by
d

ds
12
ds
23
¼
1
ð
2

Þ
3
1
32
m
3
B
þ
j
A
ðÞ
j
2
;
(2)
where
A
(

A
) is the Lorentz-invariant amplitude of the
B
þ
(
B

) three-body decay, and is a function of
s
12
and
s
23
.
For
B
0
!
K
þ
K

K
0
S
, the time dependence of the decay
rate is a function of DP location. With

t

t
sig

t
tag
defined as the proper time interval between the decay of
the fully reconstructed
B
0
!
K
þ
K

K
0
S
(
B
0
sig
) and that of
the other meson (
B
0
tag
) from the

ð
4
S
Þ
, the time-dependent
decay rate over the DP is given by
d

ds
12
ds
23
d

t
¼
1
ð
2

Þ
3
1
32
m
3
B
0
e
j

t
j
=
B
0
4

B
0
½j
A
j
2
þj

A
j
2

Q
ð
1

2
w
Þðj
A
j
2
j

A
j
2
Þ

cos
m
d

t
þ
Q
ð
1

2
w
Þ

2Im
½
e

2
i

AA


sin
m
d

t

;
(3)
where

B
0
is the neutral
B
meson lifetime and

m
d
is the
B
0


B
0
mixing frequency.
A
(

A
) is the amplitude of the
B
0
sig
(

B
0
sig
) decay and
Q
¼þ
1
ð
1
Þ
when the
B
0
tag
is iden-
tified as a
B
0
(

B
0
). The parameter
w
is the fraction of events
in which the
B
0
tag
is tagged with the incorrect flavor.
We describe the distribution of signal events in the DP
using an isobar approximation, which models the total
amplitude as a coherent sum of amplitudes from
N
indi-
vidual decay channels (‘‘isobars’’):
A
ðÞ
¼
X
N
j
¼
1
A
ðÞ
j
;
(4)
where
A
j

a
j
F
j
ð
s
12
;s
23
Þ
;

A
j


a
j

F
j
ð
s
12
;s
23
Þ
:
(5)
The
F
j
are DP-dependent dynamical amplitudes described
below, and
a
j
are complex coefficients describing the
relative magnitude and phase of the different decay chan-
nels. All the weak phase dependence is contained in
a
j
, and
F
j
contains strong dynamics only.
The amplitudes must be symmetric under exchange of
identical bosons, so for
B
þ
!
K
þ
K

K
þ
,
F
j
ð
s
12
;s
23
Þ
is replaced by
F
j
ð
s
12
;s
23
Þþ
F
j
ð
s
23
;s
12
Þ
. Similarly, in
B
þ
!
K
0
S
K
0
S
K
þ
,
F
j
ð
s
12
;s
23
Þ
is replaced by
F
j
ð
s
12
;s
23
Þþ
F
j
ð
s
12
;s
13
Þ
.
We parametrize the complex coefficients as
a
j
¼
c
j
ð
1
þ
b
j
Þ
e
i
ð

j
þ

j
Þ
;

a
j
¼
c
j
ð
1

b
j
Þ
e
i
ð

j


j
Þ
;
(6)
where
c
j
,
b
j
,

j
, and

j
are real numbers. We define the fit
fraction (
FF
j
) for an intermediate state as
FF
j

R
R
ðj
A
j
j
2
þj

A
j
j
2
Þ
ds
12
ds
23
R
R
ðj
A
j
2
þj

A
j
2
Þ
ds
12
ds
23
:
(7)
Note that the sum of the fit fractions is not necessarily
unity, due to interference between states. This interference
can be quantified by the interference fit fractions
FF
jk
,
defined as
FF
jk

2Re
R
R
ð
A
j
A

k
þ

A
j

A

k
Þ
ds
12
ds
23
R
R
ðj
A
j
2
þj

A
j
2
Þ
ds
12
ds
23
:
(8)
With this definition,
X
j
FF
j
þ
X
j<k
FF
jk
¼
1
:
(9)
In the
B
þ
modes, the direct
CP
asymmetry
A
CP
ð
j
Þ
for a
particular intermediate state is given by
A
CP
ð
j
Þ
R
R
ðj

A
j
j
2
j
A
j
j
2
Þ
ds
12
ds
23
R
R
ðj

A
j
j
2
þj
A
j
j
2
Þ
ds
12
ds
23
¼

2
b
j
1
þ
b
2
j
;
(10)
while there can also be a
CP
asymmetry in the interference
between two intermediate states, which depends on both
the
b
’s and

’s of the interfering states. We define the
CP
-violating phase difference as


j

arg
ð
a
j

a

j
Þ¼
2

j
:
(11)
For
B
0
!
K
þ
K

K
0
S
, we can define the direct
CP
asym-
metry as in Eq. (
10
), while we can also compute the
effective

for an intermediate state as

eff
;j

1
2
arg
ð
e
2
i
a
j

a

j
Þ¼

þ

j
;
(12)
which quantifies the
CP
violation due to the interference
between mixing and decay.
The resonance dynamics are contained within the
F
j
terms, which are the product of the invariant mass and
angular distributions,
F
L
j
ð
s
12
;s
23
Þ¼
R
j
ð
m
Þ
X
L
ðj
~
p
?
j
r
0
Þ
X
L
ðj
~
q
j
r
Þ
T
j
ð
L;
~
p;
~
q
Þ
;
(13)
where
(i)
L
is the spin of the resonance.
STUDY OF
CP
VIOLATION IN DALITZ-PLOT
...
PHYSICAL REVIEW D
85,
112010 (2012)
112010-5
(ii)
m
is the invariant mass of the decay products of the
resonance.
(iii)
R
j
ð
m
Þ
is the resonance mass term or ‘‘line shape’’
(e.g. Breit-Wigner).
(iv)
~
p
?
is the momentum of the ‘‘bachelor’’ particle,
i.e., the particle not belonging to the resonance,
evaluated in the rest frame of the
B
.
(v)
~
p
and
~
q
are the momenta of the bachelor particle and
one of the resonance daughters, respectively, both
evaluated in the rest frame of the resonance. For
K
þ
K

resonances,
~
q
is assigned to the momentum
of the
K
þ
, except for
B

!
K

K
þ
K

decays, in
which case
~
q
is assigned to the momentum of the
K

.For
K
0
S
K
0
S
resonances, it is irrelevant to which
K
0
S
we assign
~
q
, so we arbitrarily assign
~
q
to which-
ever
K
0
S
forms the smaller angle with the
K
þ
.
(vi)
X
L
are Blatt-Weisskopf angular momentum barrier
factors [
12
]:
L
¼
0
:
X
L
ð
z
Þ¼
1
;
(14)
L
¼
1
:
X
L
ð
z
Þ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
þ
z
2
0
1
þ
z
2
s
;
(15)
L
¼
2
:
X
L
ð
z
Þ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
9
þ
3
z
2
0
þ
z
4
0
9
þ
3
z
2
þ
z
4
s
;
(16)
where
z
equals
j
~
q
j
r
or
j
~
p
?
j
r
0
, and
z
0
is the value of
z
when the invariant mass of the pair of daughter
particles equals the mass of the parent resonance.
r
and
r
0
are effective meson radii. We take
r
0
as
zero, while
r
is taken to be
4

2
:
5
ð
GeV
=c
Þ

1
for
each resonance.
(vii)
T
j
ð
L;
~
p;
~
q
Þ
are the Zemach tensors [
13
], which
describe the angular distributions:
L
¼
0
:
T
j
¼
1
;
(17)
L
¼
1
:
T
j
¼
4
~
p
~
q;
(18)
L
¼
2
:
T
j
¼
16
3
½
3
ð
~
p
~
q
Þ
2
ðj
~
p
jj
~
q
2

:
(19)
The helicity angle of a resonance is defined as the angle
between
~
p
and
~
q
, measured in the rest frame of the reso-
nance. For a
K
1
K
2
resonance, the helicity angle will be
called

3
, and is the angle between
K
3
and
K
1
.In
B
0
!
K
þ
K

K
0
S
, because
~
q
is defined as the
K
þ
momentum for
both
B
0
and

B
0
decays, there is a sign flip between
B
0
and

B
0
amplitudes for odd-
LK
þ
K

resonances:

F
j
ð
s
12
;s
23
Þ¼
F
j
ð
s
12
;s
13
Þ¼ð
1
Þ
L
F
j
ð
s
12
;s
23
Þ
:
(20)
In contrast, for
B
þ
!
K
þ
K

K
þ
and
B
þ
!
K
0
S
K
0
S
K
þ
,

F
j
ð
s
12
;s
23
Þ
always equals
F
j
ð
s
12
;s
23
Þ
.
For most resonances in this analysis the
R
j
are taken to
be relativistic Breit-Wigner (RBW) [
14
] line shapes:
R
j
ð
m
Þ¼
1
ð
m
2
0

m
2
Þ
im
0

ð
m
Þ
;
(21)
where
m
0
is the nominal mass of the resonance and

ð
m
Þ
is
the mass-dependent width. In the general case of a spin-
L
resonance, the latter can be expressed as

ð
m
Þ¼

0

j
~
q
j
j
~
q
0
j

2
L
þ
1

m
0
m

X
2
L
ðj
~
q
j
r
Þ
:
(22)
The symbol

0
denotes the nominal width of the reso-
nance. The values of
m
0
and

0
are listed in Table
I
. The
symbol
j
~
q
0
j
denotes the value of
j
~
q
j
when
m
¼
m
0
.
For the
f
0
ð
980
Þ
line shape the Flatte
́
form [
16
] is used. In
this case
R
j
ð
m
Þ¼
1
ð
m
2
0

m
2
Þ
i
ð
g



ð
m
Þþ
g
K

KK
ð
m
ÞÞ
;
(23)
where


ð
m
Þ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1

4
m
2


=m
2
q
;
(24)

KK
ð
m
Þ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1

4
m
2
K
=m
2
q
:
(25)
Here,
m
K
is the average of the
K

and
K
0
S
masses, and
g

and
g
K
are coupling constants for which the values are
given in Table
I
.
In this paper, we test several different models to account
for NR
B
!
KKK
decays.
BABAR’s
previous analysis [
7
]
of
B
þ
!
K
þ
K

K
þ
modeled the NR decays with an ex-
ponential model given by
F
NR
ð
s
12
;s
23
Þ¼
e
s
12
þ
e
s
23
;
(26)
TABLE I. Parameters of the DP model used in the fit. Values
are given in
MeV
ð
=c
2
Þ
unless specified otherwise. All parame-
ters are taken from Ref. [
14
], except for the
f
0
ð
980
Þ
parameters,
which are taken from Ref. [
15
].
Resonance
Parameters
Line shape

ð
1020
Þ
m
0
¼
1019
:
455

0
:
020
RBW

0
¼
4
:
26

0
:
04
f
0
ð
980
Þ
m
0
¼
965

10
Flatte
́
g

¼ð
0
:
165

0
:
018
Þ
GeV
2
=c
4
g
K
=g

¼
4
:
21

0
:
33
f
0
ð
1500
Þ
m
0
¼
1505

6
RBW

0
¼
109

7
f
0
ð
1710
Þ
m
0
¼
1720

6
RBW

0
¼
135

8
f
0
2
ð
1525
Þ
m
0
¼
1525

5
RBW

0
¼
73
þ
6

5
NR decays
See text
c
0
m
0
¼
3414
:
75

0
:
31
RBW

0
¼
10
:
3

0
:
6
J. P. LEES
et al.
PHYSICAL REVIEW D
85,
112010 (2012)
112010-6
where the symmetrization is explicit.
is a parameter to be
determined empirically. This model consists purely of
K
þ
K

S
-wave decays.
The most recently published
B
0
!
K
þ
K

K
0
S
analyses
by Belle [
9
] and
BABAR
[
10
] both used what we will call
an
extended exponential model
. This model adds
K
þ
K
0
S
and
K

K
0
S
exponential terms:
A
NR
ð
s
12
;s
23
Þ¼
a
12
e
s
12
þ
a
13
e
s
13
þ
a
23
e
s
23
;

A
NR
ð
s
12
;s
23
Þ¼
a
12
e
s
12
þ
a
13
e
s
23
þ
a
23
e
s
13
:
(27)
We also test a
polynomial model
, consisting of explicit
S
-wave and
P
-wave terms, each of which has a quadratic
dependence on
m
12
:
A
NR
ð
s
12
;s
23
Þ¼ð
a
S
0
þ
a
S
1
x
þ
a
S
2
x
2
Þ
þð
a
P
0
þ
a
P
1
x
þ
a
P
2
x
2
Þ
P
1
ð
cos

3
Þ
;
(28)
where
x

m
12


, and

is an offset that we define as


1
2
ð
m
B
þ
1
3
ð
m
K
1
þ
m
K
2
þ
m
K
3
ÞÞ
;
(29)
and
P
1
is the first Legendre polynomial. In this paper, we
normalize the
P
such that
Z
1

1
P
ð
x
Þ
P
k
ð
x
Þ
dx
¼

‘k
:
(30)
Note that in the
B
þ
!
K
þ
K

K
þ
channel, we symmetrize
all terms in Eq. (
28
):
A
NR
;
total
¼
A
NR
ð
s
12
;s
23
Þþ
A
NR
ð
s
23
;s
12
Þ
:
(31)
This results in
S
-wave and
P
-wave terms for both the
(
K
1
K
2
) and (
K
2
K
3
) pairs. In the
B
þ
!
K
0
S
K
0
S
K
þ
channel,
the
P
-wave term is forbidden by Bose-Einstein symmetry.
In Sec.
VI
, we present studies that allow us to determine
the nominal DP model. The components of the nominal
model are summarized in Table
I
. Other components, taken
into account only to estimate the systematic uncertainties
due to the DP model, are discussed in Sec.
VIII
.
III. THE
BABAR
DETECTOR AND DATA SET
The data used in this analysis were collected with the
BABAR
detector at the PEP-II asymmetric energy
e
þ
e

storage rings. The
B
0
!
K
þ
K

K
0
S
and
B
þ
!
K
0
S
K
0
S
K
þ
modes use an integrated luminosity of
429 fb

1
or
ð
471

3
Þ
10
6
B

B
pairs collected at the
ð
4
S
Þ
resonance
(‘‘on resonance’’). The
B
þ
!
K
þ
K

K
þ
mode uses
426 fb

1
or
ð
467

5
Þ
10
6
B

B
pairs collected on reso-
nance. We also use approximately
44 fb

1
collected
40 MeV below the
ð
4
S
Þ
(‘‘off resonance’’) to study
backgrounds.
A detailed description of the
BABAR
detector is given in
Ref. [
17
]. Charged-particle trajectories are measured with
a five-layer, double-sided silicon vertex tracker (SVT) and
a 40-layer drift chamber (DCH), both operating inside a
1.5-T magnetic field. Charged-particle identification (PID)
is achieved by combining information from a ring-imaging
Cherenkov device and ionization energy loss (
dE=dx
)
measurements from the DCH and SVT. Photons are de-
tected and their energies measured in a CsI(Tl) electro-
magnetic calorimeter inside the magnet coil. Muon
candidates are identified in the instrumented flux return
of the solenoid.
We use
GEANT4
-based [
18
] software to simulate the
detector response and account for the varying beam and
environmental conditions. Using this software, we gener-
ate signal and background Monte Carlo (MC) event
samples in order to estimate the efficiencies and expected
backgrounds.
IV. EVENT SELECTION AND BACKGROUNDS
A.
B
þ
!
K
þ
K

K
þ
The
B
þ
!
K
þ
K

K
þ
candidates are reconstructed from
three charged tracks that are each consistent with a kaon
hypothesis. The PID requirement is about 85% efficient for
kaons, with a pion misidentification rate of around 2%. The
tracks are required to form a good-quality vertex. Also, the
total energy in the event must be less than 20 GeV.
Most backgrounds arise from random track combina-
tions in
e
þ
e

!
q

q
ð
q
¼
u;d;s;c
Þ
events (hereafter referred
to as
continuum
events). These backgrounds peak at
cos

T
¼
1
, where

T
is the angle in the
e
þ
e

center-
of-mass (c.m.) frame between the thrust axis of the
B
-candidate decay products and the thrust axis of the rest
of the event. To reduce these backgrounds, we require
j
cos

T
j
<
0
:
95
. Additional continuum suppression is
achieved by using a neural network (NN) classifier with
five input variables:
j
cos

T
j
,
j
cos

B
j
,
j

t=

t
j
,
L
2
=
L
0
,
and the output of a
B
-flavor tagging algorithm. Here,

B
is
the angle in the
e
þ
e

c.m. frame between the
B
-candidate
momentum and the beam axis,

t
is the difference between
the decay times of the
B
þ
and
B

candidates with


t
its
uncertainty, and
L
k
¼
P
j
j
p
j
j
P
k
ð
cos

j
Þ
. The sum includes
every track and neutral cluster not used to form the
B
candidate, and

j
is the angle in the
e
þ
e

c.m. frame
between the momentum
p
j
and the
B
-candidate thrust
axis.
P
k
is the
k
th Legendre polynomial. The NN is trained
on signal MC events and off-resonance data. We place a
requirement on the NN output that removes 65% of con-
tinuum events while removing only 6% of signal events.
Further discrimination is achieved with the energy-
substituted mass
m
ES

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð
s=
2
þ
p
i
p
B
Þ
2
=E
2
i

p
2
B
p
and energy
difference

E

E

B

1
2
ffiffiffi
s
p
, where (
E
B
,
p
B
) and (
E
i
,
p
i
)
are the four-vectors of the
B
candidate and the initial
electron-positron system measured in the laboratory
frame, respectively. The asterisk denotes the
e
þ
e

c.m.
frame, and
s
is the invariant mass squared of the electron-
positron system. Signal events peak at the
B
mass
(
5
:
279 GeV
=c
2
) for
m
ES
, and at zero for

E
. We require
5
:
27
<m
ES
<
5
:
29 GeV
=c
2
and
j

E
j
<
0
:
1 GeV
.An
m
ES
STUDY OF
CP
VIOLATION IN DALITZ-PLOT
...
PHYSICAL REVIEW D
85,
112010 (2012)
112010-7
sideband region with
m
ES
<
5
:
27 GeV
=c
2
is used for back-
ground characterization. After the calculation of
m
ES
and

E
, we refit each
B
candidate with the invariant mass of
the candidate constrained to agree with the nominal
B
mass
[
14
], in order to improve the resolution on the DP position
and to ensure that Eq. (
1
) is satisfied. About 8% of signal
events have multiple
B
candidates that pass the selection
criteria. If an event has multiple
B
candidates, we select the
one with the best vertex
2
. To avoid having events that
have candidates in both the
m
ES
sideband and in the signal
region, the best-candidate selection is performed prior to
the
m
ES
and

E
selection. The overall selection efficiency
for
B
þ
!
K
þ
K

K
þ
is 33%.
We use MC simulation to study backgrounds from
B
decays (
B

B
background). In this paper, we treat
B
!
KKK
decays containing intermediate charm decays as back-
ground, except for
B
!
c
0
K
ð
c
0
!
KK
Þ
, which we treat
as signal. Most of the
B

B
backgrounds come from
B
!
D
ðÞ
X
decays. We study 20 of the most prominent
B
þ
B

background modes using simulated exclusive samples, and
split these modes into six classes, summarized in Table
II
.
These classes have distinct kinematic distributions, and so
will be handled separately in the ML fit, as described in
Sec.
V
. Class 1 contains various charmless
B
þ
decays, the
largest of which is
B
þ
!
K
þ
K


þ
. Class 2 includes a
number of decays containing
D
0
!
K
þ
K

in the decay
chain. Class 3 includes various decays containing

D
0
!
K
þ


. Class 4 consists of
B
þ
!

D
0
K
þ
ð

D
0
!
K
þ


Þ
decays. We also include classes for signal-like
B
þ
!
K
þ
K

K
þ
decays coming from
B
þ
!

D
0
K
þ
(class 5)
and
B
þ
!
J=
c
K
þ
(class 6). These decays have the
same
m
ES
and

E
distributions as signal, but can be
distinguished from charmless signal by their location on
the DP. We include a seventh
B

B
background class, which
contains the remaining inclusive
B
þ
B

and
B
0

B
0
decays.
B.
B
þ
!
K
0
S
K
0
S
K
þ
The
B
þ
!
K
0
S
K
0
S
K
þ
candidates are reconstructed by
combining a charged track with two
K
0
S
!

þ


candi-
dates. The charged track is required to satisfy a kaon-PID
requirement that is about 95% efficient for kaons, with a
pion misidentification rate of around 4%. The
K
0
S
candi-
dates are each required to have a mass within
12 MeV
=c
2
of the nominal
K
0
S
mass and a lifetime significance exceed-
ing 3 standard deviations. We also require that
cos
K
S
>
0
:
999
, where
K
S
is the angle between the momentum
vector of the
K
0
S
candidate and the vector connecting the
decay vertices of the
B
þ
and
K
0
S
candidates in the labora-
tory frame. The total energy in the event must be less than
20 GeV.
To reduce continuum backgrounds, we require
j
cos

T
j
<
0
:
9
. We also use the same NN as for
B
þ
!
K
þ
K

K
þ
, and place a requirement on the NN output that
removes 49% of continuum events while removing 4% of
signal events. Finally, the
B
candidates are required to
satisfy
5
:
26
<m
ES
<
5
:
29 GeV
=c
2
and
j

E
j
<
0
:
1GeV
.
An
m
ES
sideband region with
m
ES
<
5
:
26 GeV
=c
2
is used
for background characterization. After the calculation of
m
ES
and

E
, the
B
candidates are refitted with a
B
mass
constraint. The overall selection efficiency for
B
þ
!
K
0
S
K
0
S
K
þ
(with both
K
0
S
!

þ


)is27%.
About 2% of signal events have multiple
B
candidates
that pass the selection criteria. In such cases, we choose the
B
candidate whose
K
0
S
candidates have invariant masses
closest to the nominal
K
0
S
mass. Because there can be
multiple
B
candidates that share one or more of the same
kaon candidates, multiple
B
candidates may still remain
after this step. In this case, we select the
B
candidate whose
K
þ
candidate has PID information most consistent with the
kaon hypothesis. If multiple
B
candidates still remain, we
select the one with the best vertex
2
. The best candidate
selection is performed prior to the
m
ES
and

E
selection.
B

B
backgrounds are studied with MC events. We study
10 of the most prominent background decay modes using
simulated exclusive samples, and group them into three
classes, summarized in Table
III
. Class 1 contains
B
þ
!
D
0

þ
ð
D
0
!
K
0
S
K
0
S
Þ
and
B
þ
!
K
0
S
K
ð
K
!
K
0
S

þ
Þ
decays. Class 2 contains various
B
þ
B

and
B
0

B
0
decays,
dominated by the charmless decays
B
0
!
K
0
S
K
0
S
K
0
S
and
B
0
!
K
ðÞþ
K

K
0
S
. Signal-like
B
þ
!
K
0
S
K
0
S
K
þ
decays
coming from
B
þ
!
D
0
K
þ
make up class 3. The remaining
B

B
backgrounds are grouped into a fourth class.
TABLE II. Summary of the
B

B
backgrounds in
B
þ
!
K
þ
K

K
þ
. The ‘‘Expected yields’’
column gives the expected number of events for
467

10
6
B

B
pairs, based on MC simulation.
The ‘‘Fitted yields’’ column gives the fitted number of events from the best solution of the fit on
the data (see Sec.
VII A
).
Class
Decay
Expected yields
Fitted yields
1
B
þ
!
charmless
42

5
Fixed
2
B
þ
!

D
ðÞ
0
X
,

D
0
!
K
þ
K

195

7
170

21
3
B
þ
!

D
ðÞ
0
X
,

D
0
!
K
þ


117

5
133

34
4
B
þ
!

D
0
K
þ
ð

D
0
!
K
þ


Þ
92

523

9
5
B
þ
!

D
0
K
þ
ð

D
0
!
K
þ
K

Þ
233

13
238

22
6
B
þ
!
J=
c
K
þ
ð
J=
c
!
K
þ
K

Þ
38

545

10
7
B
þ
B

=B
0

B
0
remaining
386

12
261

56
J. P. LEES
et al.
PHYSICAL REVIEW D
85,
112010 (2012)
112010-8