of 21
Amplitude analysis of
B
0
!
K
þ



0
and evidence of direct
CP
violation in
B
!
K


decays
J. P. Lees,
1
V. Poireau,
1
E. Prencipe,
1
V. Tisserand,
1
J. Garra Tico,
2
E. Grauges,
2
M. Martinelli,
3a,3b
D. A. Milanes,
3a
A. Palano,
3a,3b
M. Pappagallo,
3a,3b
G. Eigen,
4
B. Stugu,
4
L. Sun,
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
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
M. Bondioli,
10
S. Curry,
10
D. Kirkby,
10
A. J. Lankford,
10
M. Mandelkern,
10
D. P. Stoker,
10
H. Atmacan,
11
J. W. Gary,
11
F. Liu,
11
O. Long,
11
G. M. Vitug,
11
C. Campagnari,
11
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
T. Schalk,
13
B. A. Schumm,
13
A. Seiden,
13
C. H. Cheng,
14
D. A. Doll,
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
M. S. Dubrovin,
15
B. T. Meadows,
15
M. D. Sokoloff,
15
P. C. Bloom,
16
W. T. Ford,
16
A. Gaz,
16
M. Nagel,
16
U. Nauenberg,
16
J. G. Smith,
16
S. R. Wagner,
16
R. Ayad,
17,
*
W. H. Toki,
17
B. Spaan,
18
M. J. Kobel,
19
K. R. Schubert,
19
R. Schwierz,
19
D. Bernard,
20
M. Verderi,
20
P. J. Clark,
21
S. Playfer,
21
J. E. Watson,
21
D. Bettoni,
22a,22b
C. Bozzi,
22a
R. Calabrese,
22a,22b
G. Cibinetto,
22a,22b
E. Fioravanti,
22a,22b
I. Garzia,
22a,22b
E. Luppi,
22a,22b
M. Munerato,
22a,22b
M. Negrini,
22a,22b
L. Piemontese,
22a
R. Baldini-Ferroli,
23
A. Calcaterra,
23
R. de Sangro,
23
G. Finocchiaro,
23
M. Nicolaci,
23
S. Pacetti,
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
C. L. Lee,
26
M. Morii,
26
A. J. Edwards,
27
A. Adametz,
28
J. Marks,
28
U. Uwer,
28
F. U. Bernlochner,
29
M. Ebert,
29
H. M. Lacker,
29
T. Lueck,
29
P. D. Dauncey,
30
M. Tibbetts,
30
P. K. Behera,
31
U. Mallik,
31
C. Chen,
32
J. Cochran,
32
H. B. Crawley,
32
W. T. Meyer,
32
S. Prell,
32
E. I. Rosenberg,
32
A. E. Rubin,
32
A. V. Gritsan,
33
Z. J. Guo,
33
N. Arnaud,
34
M. Davier,
34
D. Derkach,
34
G. Grosdidier,
34
F. Le Diberder,
34
A. M. Lutz,
34
B. Malaescu,
34
P. Roudeau,
34
M. H. Schune,
34
A. Stocchi,
34
G. Wormser,
34
D. J. Lange,
35
D. M. Wright,
35
I. Bingham,
36
C. A. Chavez,
36
J. P. Coleman,
36
J. R. Fry,
36
E. Gabathuler,
36
D. E. Hutchcroft,
36
D. J. Payne,
36
C. Touramanis,
36
A. J. Bevan,
37
F. Di Lodovico,
37
R. Sacco,
37
M. Sigamani,
37
G. Cowan,
38
S. Paramesvaran,
38
D. N. Brown,
39
C. L. Davis,
39
A. G. Denig,
40
M. Fritsch,
40
W. Gradl,
40
A. Hafner,
40
K. E. Alwyn,
41
D. Bailey,
41
R. J. Barlow,
41
G. Jackson,
41
G. D. Lafferty,
41
R. Cenci,
42
B. Hamilton,
42
A. Jawahery,
42
D. A. Roberts,
42
G. Simi,
42
C. Dallapiccola,
43
E. Salvati,
43
R. Cowan,
44
D. Dujmic,
44
G. Sciolla,
44
D. Lindemann,
45
P. M. Patel,
45
S. H. Robertson,
45
M. Schram,
45
P. Biassoni,
46a,46b
A. Lazzaro,
46a,46b
V. Lombardo,
46a
F. Palombo,
46a,46b
S. Stracka,
46a,46b
L. Cremaldi,
47
R. Godang,
47,
R. Kroeger,
47
P. Sonnek,
47
D. J. Summers,
47
X. Nguyen,
48
P. Taras,
48
G. De Nardo,
49a,49b
D. Monorchio,
49a,49b
G. Onorato,
49a,49b
C. Sciacca,
49a,49b
G. Raven,
50
H. L. Snoek,
50
C. P. Jessop,
51
K. J. Knoepfel,
51
J. M. LoSecco,
51
W. F. Wang,
51
K. Honscheid,
52
R. Kass,
52
J. Brau,
53
R. Frey,
53
N. B. Sinev,
53
D. Strom,
53
E. Torrence,
53
E. Feltresi,
54a,54b
N. Gagliardi,
54a,54b
M. Margoni,
54a,54b
M. Morandin,
54a
M. Posocco,
54a
M. Rotondo,
54a
F. Simonetto,
54a,54b
R. Stroili,
54a,54b
E. Ben-Haim,
55
M. Bomben,
55
G. R. Bonneaud,
55
H. Briand,
55
G. Calderini,
55
J. Chauveau,
55
O. Hamon,
55
Ph. Leruste,
55
G. Marchiori,
55
J. Ocariz,
55
S. Sitt,
55
M. Biasini,
56a,56b
E. Manoni,
56a,56b
A. Rossi,
56a,56b
C. Angelini,
57a,57b
G. Batignani,
57a,57b
S. Bettarini,
57a,57b
M. Carpinelli,
57a,57b,
x
G. Casarosa,
57a,57b
A. Cervelli,
57a,57b
F. Forti,
57a,57b
M. A. Giorgi,
57a,57b
A. Lusiani,
57a,57c
N. Neri,
57a,57b
B. Oberhof,
57a,57b
E. Paoloni,
57a,57b
A. Perez,
57a
G. Rizzo,
57a,57b
J. J. Walsh,
57a
D. Lopes Pegna,
58
C. Lu,
58
J. Olsen,
58
A. J. S. Smith,
58
A. V. Telnov,
58
F. Anulli,
59a
G. Cavoto,
59a
R. Faccini,
59a,59b
F. Ferrarotto,
59a
F. Ferroni,
59a,59b
M. Gaspero,
59a,59b
L. Li Gioi,
59a
M. A. Mazzoni,
59a
G. Piredda,
59a
C. Bu
̈
nger,
60
T. Hartmann,
60
T. Leddig,
60
H. Schro
̈
der,
60
R. Waldi,
60
T. Adye,
61
E. O. Olaiya,
61
F. F. Wilson,
61
S. Emery,
62
G. Hamel de Monchenault,
62
G. Vasseur,
62
Ch. Ye
`
che,
62
D. Aston,
63
D. J. Bard,
63
R. Bartoldus,
63
J. F. Benitez,
63
C. Cartaro,
63
M. R. Convery,
63
J. Dorfan,
63
G. P. Dubois-Felsmann,
63
W. Dunwoodie,
63
R. C. Field,
63
M. Franco Sevilla,
63
B. G. Fulsom,
63
A. M. Gabareen,
63
M. T. Graham,
63
P. Grenier,
63
C. Hast,
63
W. R. Innes,
63
M. H. Kelsey,
63
H. Kim,
63
P. Kim,
63
M. L. Kocian,
63
D. W. G. S. Leith,
63
P. Lewis,
63
S. Li,
63
B. Lindquist,
63
S. Luitz,
63
V. Luth,
63
H. L. Lynch,
63
D. B. MacFarlane,
63
D. R. Muller,
63
H. Neal,
63
S. Nelson,
63
I. Ofte,
63
M. Perl,
63
T. Pulliam,
63
B. N. Ratcliff,
63
A. Roodman,
63
A. A. Salnikov,
63
V. Santoro,
63
R. H. Schindler,
63
A. Snyder,
63
D. Su,
63
M. K. Sullivan,
63
J. Va’vra,
63
A. P. Wagner,
63
M. Weaver,
63
W. J. Wisniewski,
63
M. Wittgen,
63
D. H. Wright,
63
H. W. Wulsin,
63
A. K. Yarritu,
63
C. C. Young,
63
V. Ziegler,
63
W. Park,
64
M. V. Purohit,
64
R. M. White,
64
J. R. Wilson,
64
A. Randle-Conde,
65
S. J. Sekula,
65
M. Bellis,
66
P. R. Burchat,
66
T. S. Miyashita,
66
M. S. Alam,
67
J. A. Ernst,
67
R. Gorodeisky,
68
N. Guttman,
68
D. R. Peimer,
68
A. Soffer,
68
P. Lund,
69
S. M. Spanier,
69
R. Eckmann,
70
J. L. Ritchie,
70
A. M. Ruland,
70
C. J. Schilling,
70
R. F. Schwitters,
70
B. C. Wray,
70
J. M. Izen,
71
X. C. Lou,
71
F. Bianchi,
72a,72b
D. Gamba,
72a,72b
L. Lanceri,
73a,73b
L. Vitale,
73a,73b
N. Lopez-March,
74
F. Martinez-Vidal,
74
A. Oyanguren,
74
H. Ahmed,
75
PHYSICAL REVIEW D
83,
112010 (2011)
1550-7998
=
2011
=
83(11)
=
112010(21)
112010-1
Ó
2011 American Physical Society
J. Albert,
75
Sw. Banerjee,
75
H. H. F. Choi,
75
G. J. King,
75
R. Kowalewski,
75
M. J. Lewczuk,
75
C. Lindsay,
75
I. M. Nugent,
75
J. M. Roney,
75
R. J. Sobie,
75
T. J. Gershon,
76
P. F. Harrison,
76
T. E. Latham,
76
E. M. T. Puccio,
76
H. R. Band,
77
S. Dasu,
77
Y. Pan,
77
R. Prepost,
77
C. O. Vuosalo,
77
and S. L. Wu
77
(
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
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, CNRS/IN2P3, Ecole Polytechnique, 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
Harvey Mudd College, Claremont, California 91711
28
Universita
̈
t Heidelberg, Physikalisches Institut, Philosophenweg 12, D-69120 Heidelberg, Germany
29
Humboldt-Universita
̈
t zu Berlin, Institut fu
̈
r Physik, Newtonstr. 15, D-12489 Berlin, Germany
30
Imperial College London, London, SW7 2AZ, United Kingdom
31
University of Iowa, Iowa City, Iowa 52242, USA
32
Iowa State University, Ames, Iowa 50011-3160, USA
33
Johns Hopkins University, Baltimore, Maryland 21218, USA
34
Laboratoire de l’Acce
́
le
́
rateur Line
́
aire, IN2P3/CNRS et Universite
́
Paris-Sud 11, Centre Scientifique d’Orsay,
B. P. 34, F-91898 Orsay Cedex, France
35
Lawrence Livermore National Laboratory, Livermore, California 94550, USA
36
University of Liverpool, Liverpool L69 7ZE, United Kingdom
37
Queen Mary, University of London, London, E1 4NS, United Kingdom
38
University of London, Royal Holloway and Bedford New College, Egham, Surrey TW20 0EX, United Kingdom
39
University of Louisville, Louisville, Kentucky 40292, USA
40
Johannes Gutenberg-Universita
̈
t Mainz, Institut fu
̈
r Kernphysik, D-55099 Mainz, Germany
41
University of Manchester, Manchester M13 9PL, United Kingdom
42
University of Maryland, College Park, Maryland 20742, USA
43
University of Massachusetts, Amherst, Massachusetts 01003, USA
44
Massachusetts Institute of Technology, Laboratory for Nuclear Science, Cambridge, Massachusetts 02139, USA
45
McGill University, Montre
́
al, Que
́
bec, Canada H3A 2T8
46a
INFN Sezione di Milano, I-20133 Milano, Italy
46b
Dipartimento di Fisica, Universita
`
di Milano, I-20133 Milano, Italy
47
University of Mississippi, University, Mississippi 38677, USA
J. P. LEES
et al.
PHYSICAL REVIEW D
83,
112010 (2011)
112010-2
48
Universite
́
de Montre
́
al, Physique des Particules, Montre
́
al, Que
́
bec, Canada H3C 3J7
49a
INFN Sezione di Napoli, I-80126 Napoli, Italy
49b
Dipartimento di Scienze Fisiche, Universita
`
di Napoli Federico II, I-80126 Napoli, Italy
50
NIKHEF, National Institute for Nuclear Physics and High Energy Physics, NL-1009 DB Amsterdam, The Netherlands
51
University of Notre Dame, Notre Dame, Indiana 46556, USA
52
Ohio State University, Columbus, Ohio 43210, USA
53
University of Oregon, Eugene, Oregon 97403, USA
54a
INFN Sezione di Padova, I-35131 Padova, Italy
54b
Dipartimento di Fisica, Universita
`
di Padova, I-35131 Padova, Italy
55
Laboratoire de Physique Nucle
́
aire et de Hautes Energies, IN2P3/CNRS, Universite
́
Pierre et Marie Curie-Paris6,
Universite
́
Denis Diderot-Paris7, F-75252 Paris, France
56a
INFN Sezione di Perugia, I-06100 Perugia, Italy
56b
Dipartimento di Fisica, Universita
`
di Perugia, I-06100 Perugia, Italy
57a
INFN Sezione di Pisa, I-56127 Pisa, Italy
57b
Dipartimento di Fisica, Universita
`
di Pisa, I-56127 Pisa, Italy
57c
Scuola Normale Superiore di Pisa, I-56127 Pisa, Italy
58
Princeton University, Princeton, New Jersey 08544, USA
59a
INFN Sezione di Roma, I-00185 Roma, Italy
59b
Dipartimento di Fisica, Universita
`
di Roma La Sapienza, I-00185 Roma, Italy
60
Universita
̈
t Rostock, D-18051 Rostock, Germany
61
Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, OX11 0QX, United Kingdom
62
CEA, Irfu, SPP, Centre de Saclay, F-91191 Gif-sur-Yvette, France
63
SLAC National Accelerator Laboratory, Stanford, California 94309 USA
64
University of South Carolina, Columbia, South Carolina 29208, USA
65
Southern Methodist University, Dallas, Texas 75275, USA
66
Stanford University, Stanford, California 94305-4060, USA
67
State University of New York, Albany, New York 12222, USA
68
Tel Aviv University, School of Physics and Astronomy, Tel Aviv, 69978, Israel
69
University of Tennessee, Knoxville, Tennessee 37996, USA
70
University of Texas at Austin, Austin, Texas 78712, USA
71
University of Texas at Dallas, Richardson, Texas 75083, USA
72a
INFN Sezione di Torino, I-10125 Torino, Italy
72b
Dipartimento di Fisica Sperimentale, Universita
`
di Torino, I-10125 Torino, Italy
73a
INFN Sezione di Trieste, I-34127 Trieste, Italy
73b
Dipartimento di Fisica, Universita
`
di Trieste, I-34127 Trieste, Italy
74
IFIC, Universitat de Valencia-CSIC, E-46071 Valencia, Spain
75
University of Victoria, Victoria, British Columbia, Canada V8W 3P6
76
Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom
77
University of Wisconsin, Madison, Wisconsin 53706, USA
(Received 1 May 2011; published 29 June 2011)
We analyze the decay
B
0
!
K
þ



0
with a sample of
4
:
54

10
8
B

B
events collected by the
BABAR
detector at the PEP-II asymmetric-energy
B
factory at SLAC, and extract the complex amplitudes of seven
interfering resonances over the Dalitz plot. These results are combined with amplitudes measured in
B
0
!
K
0
S

þ


decays to construct isospin amplitudes from
B
0
!
K


and
B
0
!
K
decays. We measure the
phase of the isospin amplitude

3
=
2
, useful in constraining the Cabibbo-Kobayashi-Maskawa unitarity
triangle angle

and evaluate a
CP
rate asymmetry sum rule sensitive to the presence of new physics
operators. We measure direct
CP
violation in
B
0
!
K


decays at the level of
3

when measurements
from both
B
0
!
K
þ



0
and
B
0
!
K
0
S

þ


decays are combined.
DOI:
10.1103/PhysRevD.83.112010
PACS numbers: 11.30.Er, 11.30.Hv, 13.25.Hw
I. INTRODUCTION
In the standard model (SM),
CP
violation in weak
interactions is parametrized by an irreducible complex
phase in the Cabibbo-Kobayashi-Maskawa (CKM) quark
mixing matrix [
1
,
2
]. The unitarity of the CKM matrix is
typically expressed as a triangular relationship among its
parameters such that decay amplitudes are sensitive to the
*
Now at Temple University, Philadelphia, PA 19122, USA
Also with Universita
`
di Perugia, Dipartimento di Fisica,
Perugia, Italy
Now at University of South Alabama, Mobile, AL 36688,
USA
x
Also with Universita
`
di Sassari, Sassari, Italy
AMPLITUDE ANALYSIS OF
...
PHYSICAL REVIEW D
83,
112010 (2011)
112010-3
angles of the triangle denoted

,

,

. Redundant mea-
surements of the parameters of the CKM matrix provide an
important test of the SM, since violation of the unitarity
condition would be a signature of new physics. The angle

remains the least well measured of the CKM angles. Tree
amplitudes in
B
!
K


decays are sensitive to

but are
Cabibbo-suppressed relative to loop-order (penguin)
contributions involving radiation of either a gluon (QCD
penguins) or a photon (electroweak penguins or EWPs)
from the loop.
It has been shown that QCD penguin contributions can
be eliminated by constructing a linear combination of
B
0
!
K


and
B
0
!
K

0

0
weak decay amplitudes
that is pure (isospin)
I
¼
3
=
2
[
3
],
A
3
=
2
ð
K


Þ¼
1
ffiffiffi
2
p
A
ð
B
0
!
K


Þþ
A
ð
B
0
!
K

0

0
Þ
:
(1)
Since a transition from
I
¼
1
2
to
I
¼
3
=
2
is possible only
via

I
¼
1
operators,
A
3
=
2
must be free of

I
¼
0
,
namely, QCD contributions. The weak phase of
A
3
=
2
,
given by

3
=
2
¼
1
2
Arg
ð

A
3
=
2
=
A
3
=
2
Þ
, is equal to the
CKM angle

in the absence of EWP operators [
4
].
Here,

A
3
=
2
denotes the
CP
conjugate of the amplitude
in Eq. (
1
).
The relative magnitudes and phases of the
B
0
!
K


and
B
0
!
K

0

0
amplitudes in Eq. (
1
) are measured from
their interference over the available decay phase space
(Dalitz plot or DP) to the common final state
B
0
!
K
þ



0
. The phase difference between
B
0
!
K


and

B
0
!
K


þ
is measured in the DP analysis of the
self-conjugate final state
B
0
!
K
0
S

þ


[
5
] where the
strong phases cancel. This argument is extended to
B
0
!
K
decay amplitudes [
6
,
7
] where an isospin decomposi-
tion of amplitudes gives
A
3
=
2
ð
K
Þ¼
1
ffiffiffi
2
p
A
ð
B
0
!


K
þ
Þþ
A
ð
B
0
!

0
K
0
Þ
:
(2)
Here, the
B
0
!


K
þ
and
B
0
!

0
K
0
decays do not
decay to a common final state preventing a direct measure-
ment of their relative phase. The amplitudes in Eq. (
2
) do,
however, interfere with the
B
0
!
K


amplitude in
their decays to
B
0
!
K
þ



0
and
B
0
!
K
0
S

þ


final
states so that an indirect measurement of their relative
phase is possible.
The
CP
rate asymmetries of the isospin amplitudes
A
3
=
2
ð
K


Þ
and
A
3
=
2
ð
K
Þ
have been shown to obey a
sum rule [
8
],
j

A
3
=
2
ð
K


Þj
2
j
A
3
=
2
ð
K


Þj
2
¼j
A
3
=
2
ð
K
Þj
2
j

A
3
=
2
ð
K
Þj
2
:
(3)
This sum rule is exact in the limit of SU(3) symmetry
and large deviations could be an indication of new
strangeness violating operators. Measurements of
B
0
!
K


and
B
0
!
K
amplitudes are used to evaluate
Eq. (
3
).
We present an update of the DP analysis of the flavor-
specific
B
0
!
K
þ



0
decay from Ref. [
9
] with a sample
of
4
:
54

10
8
B

B
events. The isobar model used to pa-
rametrize the complex amplitudes of the intermediate
resonances contributing to the final state is presented in
Sec.
II
. The
BABAR
detector and data set are briefly
described in Sec.
III
. The efficient selection of signal
candidates is described in Sec.
IV
and the unbinned maxi-
mum likelihood (ML) fit performed with the selected
events is presented in Sec.
V
. The complex amplitudes of
the intermediate resonances contributing to the
B
0
!
K
þ



0
decay are extracted from the result of the ML
fit in Sec.
VI
together with the accounting of the systematic
uncertainties in Sec.
VII
. Several important results are
discussed in Sec.
VIII
. Measurements of
B
0
!
K
from
this article and Ref. [
5
] are used to produce a measurement
of

3
=
2
using Eq. (
2
). It is shown that the large phase
difference between
B
0
!
K
ð
892
Þ


and
B
0
!
K

0
ð
892
Þ

0
amplitudes makes a similar measurement us-
ing Eq. (
1
) impossible with the available data set. We find
that the sum rule in Eq. (
3
) holds within the experimental
uncertainty. Additionally, we find evidence for a direct
CP
asymmetry in
B
0
!
K


decays when the results of
Ref. [
5
] are combined with measurements in this article.
The conventions and results of Ref. [
5
] are summarized
where necessary. Finally in Sec.
IX
, we summarize our
results.
II. ANALYSIS OVERVIEW
We present a DP analysis of the
B
0
!
K
þ



0
decay
in which we measure the magnitudes and relative phases
of five resonant amplitudes:

ð
770
Þ

K
þ
,

ð
1450
Þ

K
þ
,

ð
1700
Þ

K
þ
,
K

ð
892
Þ
þ


,
K

ð
892
Þ
0

0
, two
K
S-waves:
ð
K
Þ

0
0
,
ð
K
Þ
0
, and a nonresonant (NR) con-
tribution, allowing for
CP
violation. The notation for the
S-waves denotes phenomenological amplitudes described
by coherent superpositions of an elastic effective-range
term and the
K

0
ð
1430
Þ
resonances [
10
]. Here, we describe
the decay amplitude formalism and conventions used in
this analysis.
The
B
0
!
K
þ



0
decay amplitude is a function of
two independent kinematic variables: we use the squares of
the invariant masses of the pairs of particles
K
þ


and
K
þ

0
,
x
¼
m
2
K
þ


and
y
¼
m
2
K
þ

0
. The total decay am-
plitude is a linear combination of
k
isobars, each having
amplitude
A
k
given by
A
ðÞ
k
¼
a
ðÞ
k
e
i

ðÞ
k
Z
DP
f
k
ð
J; x; y
Þ
dxdy;
(4)
where








Z
DP
f
k
ð
J; x; y
Þ
dxdy








¼
1
:
(5)
J. P. LEES
et al.
PHYSICAL REVIEW D
83,
112010 (2011)
112010-4
Here,

A
k
denotes the
CP
conjugate amplitude and
a
ðÞ
k
e
i

ðÞ
k
is the complex coefficient of the isobar. The
normalized decay dynamics of the intermediate state are
specified by the functions
f
k
that for a spin-
J
resonance in
the
K
þ


decay channel describe the angular dependence
T
k
ð
J; x; y
Þ
, Blatt-Weisskopf centrifugal barrier factor [
11
]
B
k
ð
J; x
Þ
, and mass distribution of the resonance
L
k
ð
J; x
Þ
,
f
k
ð
J; x; y
Þ¼
T
k
ð
J; x; y
Þ
B
k
ð
J; x
Þ
L
k
ð
J; x
Þ
:
(6)
The branching fractions
B
k
(
CP
averaged over
B
0
and

B
0
), and
CP
asymmetry,
A
CP
ð
k
Þ
, are given by
B
k
¼
j
A
k
j
2
þj

A
k
j
2
j
P
j
A
j
j
2
þj
P
j

A
j
j
2

N
sig
N
B

B
h

i
DP
;
(7)
A
CP
ð
k
Þ¼
j

A
k
j
2
j
A
k
j
2
j

A
k
j
2
þj
A
k
j
2
¼

a
2
k

a
2
k

a
2
k
þ
a
2
k
;
(8)
where
N
sig
is the number of
B
0
!
K
þ



0
events se-
lected from a sample of
N
B

B
B
-meson decays. The average
DP efficiency,
h

i
DP
, is given by
j
P
k
ð
a
k
e
i

k
þ

a
k
e
i


k
Þ
R
DP

ð
x; y
Þ
f
k
ð
J; x; y
Þ
dxdy
j
j
P
k
ð
a
k
e
i

k
þ

a
k
e
i


k
Þ
R
DP
f
k
ð
J; x; y
Þ
dxdy
j
;
(9)
where

ð
x; y
Þ
is the DP-dependent signal selection
efficiency.
We use the Zemach tensor formalism [
12
] for the angu-
lar distribution
T
ð
J; x; y
Þ
of a process by which a pseudo-
scalar
B
-meson produces a spin-
J
resonance in association
with a bachelor pseudoscalar meson. We define
~
p
and
~
q
as
the momentum vectors of the bachelor particle and reso-
nance decay product, respectively, in the rest frame of the
resonance
k
. The choice of the resonance decay product
defines the helicity convention for each resonance where
the cosine of the helicity angle is
cos

H
¼
~
p

~
q=
ðj
~
p
jj
~
q
.
We choose the resonance decay product with momentum
~
q
to be the


for
K
þ


resonances, the

0
for



0
resonances, and the
K
þ
for

0
K
þ
resonances (see Fig.
1
).
The decay of a spin-
J
resonance into two pseudoscalars
is damped by a Blatt-Weisskopf barrier factor, character-
ized by the phenomenological radius
R
of the resonance.
The Blatt-Weisskopf barrier factors
B
ð
J; x
Þ
are normalized
to 1 when
ffiffiffi
x
p
equals the pole mass
M
of the resonance. We
parametrize the barrier factors in terms of
z
¼j
~
q
j
R
and
z
0
¼j
~
q
0
j
R
, where
j
~
q
0
j
is the value of
j
~
q
j
when
ffiffiffi
x
p
¼
M
.
The angular distributions and Blatt-Weisskopf barrier
factors for the resonance spins used in this analysis are
summarized in Table
I
.
We use the relativistic Breit-Wigner (RBW) line shape
to describe the
K

ð
892
Þ
þ
;
0
resonances,
L
RBW
ð
J; x
Þ¼
1
M
2

x

iM

ð
J; x
Þ
:
(10)
Here, the mass-dependent width

ð
J; x
Þ
is defined by

ð
J; x
Þ¼

0
M
ffiffiffi
x
p

j
~
q
j
j
~
q
0
j

2
J
þ
1
B
ð
J; x
Þ
2
;
(11)
where

0
is the natural width of the resonance.
The Gounaris-Sakurai (GS) parametrization [
13
] is used
to describe the line shape of the broad

ð
770
Þ

,

ð
1450
Þ

and

ð
1700
Þ

resonances decaying to two pions,
L
GS
ð
J; x
Þ¼
1
þ
d

0
=M
M
2
þ
g
ð
x
Þ
x

iM

ð
J; x
Þ
;
(12)
where

ð
J; x
Þ
is defined in Eq. (
11
). Expressions for the
constant
d
and the function
g
ð
x
Þ
in terms of
M
and

0
are
given in Ref. [
13
]. The parameters of the

line shapes,
M
,
and

0
are taken from Ref. [
14
] using updated line-shape
fits with data from
e
þ
e

annihilation [
15
] and
decays
[
16
].
An effective-range parametrization was suggested [
17
]
for the
K
scalar amplitudes,
ð
K
Þ
0
and
ð
K
Þ

0
0
which
dominate for
m
K
<
2 GeV
=c
2
, to describe the slowly
increasing phase as a function of the
K
mass. We use
the parametrization chosen in the LASS experiment [
18
],
tuned for
B
-meson decays [
10
],
L
LASS
ð
x
Þ¼
ffiffiffi
x
p
=M
2
j
~
q
j
cot
B

i
j
~
q
j
þ
e
2
i
B

0
=
j
~
q
0
j
M
2

x

iM

ð
0
;x
Þ
;
(13)
with
FIG. 1. The helicity angle
ð

H
Þ
and momenta of particles
ð
~
q;
~
p
Þ
in the rest frame of a resonance
k
.
TABLE I. The angular distributions
T
ð
J; x; y
Þ
, and Blatt-
Weisskopf barrier factors
B
ð
J; x
Þ
, for a resonance of spin-
J
decaying to two pseudoscalar mesons.
Spin-
JT
ð
J; x; y
Þ
B
ð
J; x
Þ
011
1

2
j
~
p
jj
~
q
j
cos

H
ffiffiffiffiffiffiffiffi
1
þ
z
2
0
1
þ
z
2
r
2
4
3
j
~
p
j
2
j
~
q
j
2
ð
3cos
2

H

1
Þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
9
þ
3
z
2
0
þ
z
4
0
9
þ
3
z
2
þ
z
4
r
AMPLITUDE ANALYSIS OF
...
PHYSICAL REVIEW D
83,
112010 (2011)
112010-5
cot
B
¼
1
a
j
~
q
j
þ
1
2
r
j
~
q
j
;
(14)
where
a
is the scattering length and
r
the effective range
(see Table
II
). We impose a cutoff for the
K
S-waves so
that
L
LASS
is given only by the second term in Eq. (
13
) for
ffiffiffi
x
p
>
1
:
8 GeV
=c
2
. Finally, the NR
K
þ



0
amplitude is
taken to be constant across the DP.
In addition to the seven resonant amplitudes and the NR
component described above we model the contributions to
the
K
þ



0
final state from
B
0
!

D
0

0
and
B
0
!
D

K
þ
with a double Gaussian distribution given by
L
DG
ð
x
Þ¼
1

f

1
exp


ð
M
1

ffiffiffi
x
p
Þ
2
2

2
1

þ
f

2
exp


ð
M
2

ffiffiffi
x
p
Þ
2
2

2
2

:
(15)
The fraction
f
is the relative weight of the two Gaussian
distributions parametrized by the masses
M
1
,
M
2
and
widths

1
,

2
. The
D
-mesons are modeled as noninterfer-
ing isobars and are distinct from the charmless signal
events.
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 between October 1999 and September 2007.
This corresponds to an integrated luminosity of
413 fb

1
or approximately
N
B

B
¼
4
:
54

0
:
05

10
8
B

B
pairs
taken on the peak of the

ð
4
S
Þ
resonance (on resonance)
and
41 fb

1
recorded at a center-of-mass (CM) energy
40 MeV below (off resonance).
A detailed description of the
BABAR
detector is given in
Ref. [
20
]. Charged-particle trajectories are measured by a
five-layer, double-sided silicon vertex tracker (SVT) and
a 40-layer drift chamber (DCH) coaxial with a 1.5 T
magnetic field. Charged-particle identification is achieved
by combining the information from a ring-imaging
Cherenkov device (DIRC) and the ionization energy loss
(
dE=dx
) measurements from the DCH and SVT. Photons
are detected, and their energies are measured in a CsI(Tl)
electromagnetic calorimeter (EMC) inside the coil. Muon
candidates are identified in the instrumented flux return of
the solenoid. We use GEANT4-based [
21
] software to
simulate the detector response and account for the varying
beam and environmental conditions. Using this software,
we generate signal and background Monte Carlo (MC)
event samples in order to estimate the efficiencies and
expected backgrounds in this analysis.
IV. EVENT SELECTION AND BACKGROUNDS
We reconstruct
B
0
!
K
þ



0
candidates from a

0
candidate and pairs of oppositely charged tracks that are
required to form a good quality vertex. The charged-
particle candidates are required to have transverse mo-
menta above
100 MeV
=c
and at least 12 hits in the DCH.
We use information from the tracking system, EMC, and
DIRC to select charged tracks consistent with either a kaon
or pion hypothesis. The

0
candidate is built from a pair of
photons, each with an energy greater than 50 MeV in the
laboratory frame and a lateral energy deposition profile in
the EMC consistent with that expected for an electromag-
netic shower. The invariant mass of each

0
candidate,
m

0
, must be within 3 times the associated mass error,

ð
m

0
Þ
, of the nominal

0
mass
134
:
9766 MeV
=c
2
[
19
].
We also require
j
cos



0
j
, the modulus of the cosine of the
TABLE II. The model for the
B
0
!
K
þ



0
decay com-
prises a nonresonant (NR) amplitude and seven intermediate
states listed below. The three line shapes are described in the
text and the citations reference the parameters used in the fit. We
use the same LASS parameters [
18
] for both neutral and charged
K
systems.
Resonance
Line shape
Parameters
Spin-
J
¼
1

ð
770
Þ

GS [
14
]
M
¼
775
:
5

0
:
6 MeV
=c
2

0
¼
148
:
2

0
:
8 MeV
R
¼
0
þ
1
:
5

0
:
0
ð
GeV
=c
Þ

1

ð
1450
Þ

GS [
14
]
M
¼
1409

12 MeV
=c
2

0
¼
500

37 MeV
R
¼
0
þ
1
:
5

0
:
0
ð
GeV
=c
Þ

1

ð
1700
Þ

GS [
14
]
M
¼
1749

20 MeV
=c
2

0
¼
235

60 MeV
R
¼
0
þ
1
:
5

0
:
0
ð
GeV
=c
Þ

1
K

ð
892
Þ
þ
RBW [
19
]
M
¼
891
:
6

0
:
26 MeV
=c
2

0
¼
50

0
:
9 MeV
R
¼
3
:
4

0
:
7
ð
GeV
=c
Þ

1
K

ð
892
Þ
0
RBW [
19
]
M
¼
896
:
1

0
:
27 MeV
=c
2

0
¼
50
:
5

0
:
6 MeV
R
¼
3
:
4

0
:
7
ð
GeV
=c
Þ

1
Spin-
J
¼
0
ð
K
Þ
0
,
ð
K
Þ

0
0
LASS [
18
]
M
¼
1412

3 MeV
=c
2

0
¼
294

6 MeV
a
¼
2
:
07

0
:
10
ð
GeV
=c
Þ

1
r
¼
3
:
32

0
:
34
ð
GeV
=c
Þ

1
NR
Constant
Noninterfering components

D
0
DG
M
1
¼
1862
:
3 MeV
=c
2

1
¼
7
:
1 MeV
M
2
¼
1860
:
1 MeV
=c
2

2
¼
22
:
4 MeV
f
¼
0
:
12
D

DG
M
1
¼
1864
:
4 MeV
=c
2

1
¼
9
:
9 MeV
M
2
¼
1860
:
6 MeV
=c
2

2
¼
21
:
3 MeV
f
¼
0
:
32
J. P. LEES
et al.
PHYSICAL REVIEW D
83,
112010 (2011)
112010-6