Time-dependent and time-integrated angular analysis of
B
!
’K
0
S
0
and
’K
B. Aubert,
1
M. Bona,
1
Y. Karyotakis,
1
J. P. Lees,
1
V. Poireau,
1
E. Prencipe,
1
X. Prudent,
1
V. Tisserand,
1
J. Garra Tico,
2
E. Grauges,
2
L. Lopez,
3a,3b
A. Palano,
3a,3b
M. Pappagallo,
3a,3b
G. Eigen,
4
B. Stugu,
4
L. Sun,
4
G. S. Abrams,
5
M. Battaglia,
5
D. N. Brown,
5
R. N. Cahn,
5
R. G. Jacobsen,
5
L. T. Kerth,
5
Yu. G. Kolomensky,
5
G. Lynch,
5
I. L. Osipenkov,
5
M. T. Ronan,
5,
*
K. Tackmann,
5
T. Tanabe,
5
C. M. Hawkes,
6
N. Soni,
6
A. T. Watson,
6
H. Koch,
7
T. Schroeder,
7
D. Walker,
8
D. J. Asgeirsson,
9
B. G. Fulsom,
9
C. Hearty,
9
T. S. Mattison,
9
J. A. McKenna,
9
M. Barrett,
10
A. Khan,
10
V. E. Blinov,
11
A. D. Bukin,
11
A. R. Buzykaev,
11
V. P. Druzhinin,
11
V. B. Golubev,
11
A. P. Onuchin,
11
S. I. Serednyakov,
11
Yu. I. Skovpen,
11
E. P. Solodov,
11
K. Yu. Todyshev,
11
M. Bondioli,
12
S. Curry,
12
I. Eschrich,
12
D. Kirkby,
12
A. J. Lankford,
12
P. Lund,
12
M. Mandelkern,
12
E. C. Martin,
12
D. P. Stoker,
12
S. Abachi,
13
C. Buchanan,
13
J. W. Gary,
14
F. Liu,
14
O. Long,
14
B. C. Shen,
14,
*
G. M. Vitug,
14
Z. Yasin,
14
L. Zhang,
14
V. Sharma,
15
C. Campagnari,
16
T. M. Hong,
16
D. Kovalskyi,
16
M. A. Mazur,
16
J. D. Richman,
16
T. W. Beck,
17
A. M. Eisner,
17
C. J. Flacco,
17
C. A. Heusch,
17
J. Kroseberg,
17
W. S. Lockman,
17
A. J. Martinez,
17
T. Schalk,
17
B. A. Schumm,
17
A. Seiden,
17
M. G. Wilson,
17
L. O. Winstrom,
17
C. H. Cheng,
18
D. A. Doll,
18
B. Echenard,
18
F. Fang,
18
D. G. Hitlin,
18
I. Narsky,
18
T. Piatenko,
18
F. C. Porter,
18
R. Andreassen,
19
G. Mancinelli,
19
B. T. Meadows,
19
K. Mishra,
19
M. D. Sokoloff,
19
P. C. Bloom,
20
W. T. Ford,
20
A. Gaz,
20
J. F. Hirschauer,
20
M. Nagel,
20
U. Nauenberg,
20
J. G. Smith,
20
K. A. Ulmer,
20
S. R. Wagner,
20
R. Ayad,
21,
†
A. Soffer,
21,
‡
W. H. Toki,
21
R. J. Wilson,
21
D. D. Altenburg,
22
E. Feltresi,
22
A. Hauke,
22
H. Jasper,
22
M. Karbach,
22
J. Merkel,
22
A. Petzold,
22
B. Spaan,
22
K. Wacker,
22
M. J. Kobel,
23
W. F. Mader,
23
R. Nogowski,
23
K. R. Schubert,
23
R. Schwierz,
23
A. Volk,
23
D. Bernard,
24
G. R. Bonneaud,
24
E. Latour,
24
M. Verderi,
24
P. J. Clark,
25
S. Playfer,
25
J. E. Watson,
25
M. Andreotti,
26a,26b
D. Bettoni,
26a
C. Bozzi,
26a
R. Calabrese,
26a,26b
A. Cecchi,
26a,26b
G. Cibinetto,
26a,26b
P. Franchini,
26a,26b
E. Luppi,
26a,26b
M. Negrini,
26a,26b
A. Petrella,
26a,26b
L. Piemontese,
26a
V. Santoro,
26a,26b
R. Baldini-Ferroli,
27
A. Calcaterra,
27
R. de Sangro,
27
G. Finocchiaro,
27
S. Pacetti,
27
P. Patteri,
27
I. M. Peruzzi,
27,
x
M. Piccolo,
27
M. Rama,
27
A. Zallo,
27
A. Buzzo,
28a
R. Contri,
28a,28b
M. Lo Vetere,
28a,28b
M. M. Macri,
28a
M. R. Monge,
28a,28b
S. Passaggio,
28a
C. Patrignani,
28a,28b
E. Robutti,
28a
A. Santroni,
28a,28b
S. Tosi,
28a,28b
K. S. Chaisanguanthum,
29
M. Morii,
29
A. Adametz,
30
J. Marks,
30
S. Schenk,
30
U. Uwer,
30
V. Klose,
31
H. M. Lacker,
31
D. J. Bard,
32
P. D. Dauncey,
32
J. A. Nash,
32
M. Tibbetts,
32
P. K. Behera,
33
X. Chai,
33
M. J. Charles,
33
U. Mallik,
33
J. Cochran,
34
H. B. Crawley,
34
L. Dong,
34
W. T. Meyer,
34
S. Prell,
34
E. I. Rosenberg,
34
A. E. Rubin,
34
Y. Y. Gao,
35
A. V. Gritsan,
35
Z. J. Guo,
35
C. K. Lae,
35
N. Arnaud,
36
J. Be
́
quilleux,
36
A. D’Orazio,
36
M. Davier,
36
J. Firmino da Costa,
36
G. Grosdidier,
36
A. Ho
̈
cker,
36
V. Lepeltier,
36
F. Le Diberder,
36
A. M. Lutz,
36
S. Pruvot,
36
P. Roudeau,
36
M. H. Schune,
36
J. Serrano,
36
V. Sordini,
36,
k
A. Stocchi,
36
G. Wormser,
36
D. J. Lange,
37
D. M. Wright,
37
I. Bingham,
38
J. P. Burke,
38
C. A. Chavez,
38
J. R. Fry,
38
E. Gabathuler,
38
R. Gamet,
38
D. E. Hutchcroft,
38
D. J. Payne,
38
C. Touramanis,
38
A. J. Bevan,
39
C. K. Clarke,
39
K. A. George,
39
F. Di Lodovico,
39
R. Sacco,
39
M. Sigamani,
39
G. Cowan,
40
H. U. Flaecher,
40
D. A. Hopkins,
40
S. Paramesvaran,
40
F. Salvatore,
40
A. C. Wren,
40
D. N. Brown,
41
C. L. Davis,
41
A. G. Denig,
42
M. Fritsch,
42
W. Gradl,
42
G. Schott,
42
K. E. Alwyn,
43
D. Bailey,
43
R. J. Barlow,
43
Y. M. Chia,
43
C. L. Edgar,
43
G. Jackson,
43
G. D. Lafferty,
43
T. J. West,
43
J. I. Yi,
43
J. Anderson,
44
C. Chen,
44
A. Jawahery,
44
D. A. Roberts,
44
G. Simi,
44
J. M. Tuggle,
44
C. Dallapiccola,
45
X. Li,
45
E. Salvati,
45
S. Saremi,
45
R. Cowan,
46
D. Dujmic,
46
P. H. Fisher,
46
G. Sciolla,
46
M. Spitznagel,
46
F. Taylor,
46
R. K. Yamamoto,
46
M. Zhao,
46
P. M. Patel,
47
S. H. Robertson,
47
A. Lazzaro,
48a,48b
V. Lombardo,
48a
F. Palombo,
48a,48b
J. M. Bauer,
49
L. Cremaldi,
49
R. Godang,
49,
{
R. Kroeger,
49
D. A. Sanders,
49
D. J. Summers,
49
H. W. Zhao,
49
M. Simard,
50
P. Taras,
50
F. B. Viaud,
50
H. Nicholson,
51
G. De Nardo,
52a,52b
L. Lista,
52a
D. Monorchio,
52a,52b
G. Onorato,
52a,52b
C. Sciacca,
52a,52b
G. Raven,
53
H. L. Snoek,
53
C. P. Jessop,
54
K. J. Knoepfel,
54
J. M. LoSecco,
54
W. F. Wang,
54
G. Benelli,
55
L. A. Corwin,
55
K. Honscheid,
55
H. Kagan,
55
R. Kass,
55
J. P. Morris,
55
A. M. Rahimi,
55
J. J. Regensburger,
55
S. J. Sekula,
55
Q. K. Wong,
55
N. L. Blount,
56
J. Brau,
56
R. Frey,
56
O. Igonkina,
56
J. A. Kolb,
56
M. Lu,
56
R. Rahmat,
56
N. B. Sinev,
56
D. Strom,
56
J. Strube,
56
E. Torrence,
56
G. Castelli,
57a,57b
N. Gagliardi,
57a,57b
M. Margoni,
57a,57b
M. Morandin,
57a
M. Posocco,
57a
M. Rotondo,
57a
F. Simonetto,
57a,57b
R. Stroili,
57a,57b
C. Voci,
57a,57b
P. del Amo Sanchez,
58
E. Ben-Haim,
58
H. Briand,
58
G. Calderini,
58
J. Chauveau,
58
P. David,
58
L. Del Buono,
58
O. Hamon,
58
Ph. Leruste,
58
J. Ocariz,
58
A. Perez,
58
J. Prendki,
58
S. Sitt,
58
L. Gladney,
59
M. Biasini,
60a,60b
R. Covarelli,
60a,60b
E. Manoni,
60a,60b
C. Angelini,
61a,61b
G. Batignani,
61a,61b
S. Bettarini,
61a,61b
M. Carpinelli,
61a,61b,
**
A. Cervelli,
61a,61b
F. Forti,
61a,61b
M. A. Giorgi,
61a,61b
A. Lusiani,
61a,61c
G. Marchiori,
61a,61b
M. Morganti,
61a,61b
N. Neri,
61a,61b
E. Paoloni,
61a,61b
G. Rizzo,
61a,61b
J. J. Walsh,
61a
D. Lopes Pegna,
62
C. Lu,
62
J. Olsen,
62
A. J. S. Smith,
62
A. V. Telnov,
62
F. Anulli,
63a
E. Baracchini,
63a,63b
G. Cavoto,
63a
D. del Re,
63a,63b
E. Di Marco,
63a,63b
R. Faccini,
63a,63b
F. Ferrarotto,
63a
F. Ferroni,
63a,63b
M. Gaspero,
63a,63b
P. D. Jackson,
63a
L. Li Gioi,
63a
M. A. Mazzoni,
63a
S. Morganti,
63a
PHYSICAL REVIEW D
78,
092008 (2008)
1550-7998
=
2008
=
78(9)
=
092008(27)
092008-1
Ó
2008 The American Physical Society
G. Piredda,
63a
F. Polci,
63a,63b
F. Renga,
63a,63b
C. Voena,
63a
M. Ebert,
64
T. Hartmann,
64
H. Schro
̈
der,
64
R. Waldi,
64
T. Adye,
65
B. Franek,
65
E. O. Olaiya,
65
F. F. Wilson,
65
S. Emery,
66
M. Escalier,
66
L. Esteve,
66
S. F. Ganzhur,
66
G. Hamel de Monchenault,
66
W. Kozanecki,
66
G. Vasseur,
66
Ch. Ye
`
che,
66
M. Zito,
66
X. R. Chen,
67
H. Liu,
67
W. Park,
67
M. V. Purohit,
67
R. M. White,
67
J. R. Wilson,
67
M. T. Allen,
68
D. Aston,
68
R. Bartoldus,
68
P. Bechtle,
68
J. F. Benitez,
68
R. Cenci,
68
J. P. Coleman,
68
M. R. Convery,
68
J. C. Dingfelder,
68
J. Dorfan,
68
G. P. Dubois-Felsmann,
68
W. Dunwoodie,
68
R. C. Field,
68
A. M. Gabareen,
68
S. J. Gowdy,
68
M. T. Graham,
68
P. Grenier,
68
C. Hast,
68
W. R. Innes,
68
J. Kaminski,
68
M. H. Kelsey,
68
H. Kim,
68
P. Kim,
68
M. L. Kocian,
68
D. W. G. S. Leith,
68
S. Li,
68
B. Lindquist,
68
S. Luitz,
68
V. Luth,
68
H. L. Lynch,
68
D. B. MacFarlane,
68
H. Marsiske,
68
R. Messner,
68
D. R. Muller,
68
H. Neal,
68
S. Nelson,
68
C. P. O’Grady,
68
I. Ofte,
68
A. Perazzo,
68
M. Perl,
68
B. N. Ratcliff,
68
A. Roodman,
68
A. A. Salnikov,
68
R. H. Schindler,
68
J. Schwiening,
68
A. Snyder,
68
D. Su,
68
M. K. Sullivan,
68
K. Suzuki,
68
S. K. Swain,
68
J. M. Thompson,
68
J. Va’vra,
68
A. P. Wagner,
68
M. Weaver,
68
C. A. West,
68
W. J. Wisniewski,
68
M. Wittgen,
68
D. H. Wright,
68
H. W. Wulsin,
68
A. K. Yarritu,
68
K. Yi,
68
C. C. Young,
68
V. Ziegler,
68
P. R. Burchat,
69
A. J. Edwards,
69
S. A. Majewski,
69
T. S. Miyashita,
69
B. A. Petersen,
69
L. Wilden,
69
S. Ahmed,
70
M. S. Alam,
70
J. A. Ernst,
70
B. Pan,
70
M. A. Saeed,
70
S. B. Zain,
70
S. M. Spanier,
71
B. J. Wogsland,
71
R. Eckmann,
72
J. L. Ritchie,
72
A. M. Ruland,
72
C. J. Schilling,
72
R. F. Schwitters,
72
B. W. Drummond,
73
J. M. Izen,
73
X. C. Lou,
73
F. Bianchi,
74a,74b
D. Gamba,
74a,74b
M. Pelliccioni,
74a,74b
M. Bomben,
75a,75b
L. Bosisio,
75a,75b
C. Cartaro,
75a,75b
G. Della Ricca,
75a,75b
L. Lanceri,
75a,75b
L. Vitale,
75a,75b
V. Azzolini,
76
N. Lopez-March,
76
F. Martinez-Vidal,
76
D. A. Milanes,
76
A. Oyanguren,
76
J. Albert,
77
Sw. Banerjee,
77
B. Bhuyan,
77
H. H. F. Choi,
77
K. Hamano,
77
R. Kowalewski,
77
M. J. Lewczuk,
77
I. M. Nugent,
77
J. M. Roney,
77
R. J. Sobie,
77
T. J. Gershon,
78
P. F. Harrison,
78
J. Ilic,
78
T. E. Latham,
78
G. B. Mohanty,
78
H. R. Band,
79
X. Chen,
79
S. Dasu,
79
K. T. Flood,
79
Y. Pan,
79
M. Pierini,
79
R. Prepost,
79
C. O. Vuosalo,
79
and S. L. Wu
79
(
B
A
B
AR
Collaboration)
1
Laboratoire de Physique des Particules, IN2P3/CNRS et Universite
́
de Savoie, 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
Dipartmento di Fisica, Universita
`
di Bari, I-70126 Bari, Italy
4
University of Bergen, Institute of Physics, N-5007 Bergen, Norway
5
Lawrence Berkeley National Laboratory and University of California, Berkeley, California 94720, USA
6
University of Birmingham, Birmingham, B15 2TT, United Kingdom
7
Ruhr Universita
̈
t Bochum, Institut fu
̈
r Experimentalphysik 1, D-44780 Bochum, Germany
8
University of Bristol, Bristol BS8 1TL, United Kingdom
9
University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z1
10
Brunel University, Uxbridge, Middlesex UB8 3PH, United Kingdom
11
Budker Institute of Nuclear Physics, Novosibirsk 630090, Russia
12
University of California at Irvine, Irvine, California 92697, USA
13
University of California at Los Angeles, Los Angeles, California 90024, USA
14
University of California at Riverside, Riverside, California 92521, USA
15
University of California at San Diego, La Jolla, California 92093, USA
16
University of California at Santa Barbara, Santa Barbara, California 93106, USA
17
University of California at Santa Cruz, Institute for Particle Physics, Santa Cruz, California 95064, USA
18
California Institute of Technology, Pasadena, California 91125, USA
19
University of Cincinnati, Cincinnati, Ohio 45221, USA
20
University of Colorado, Boulder, Colorado 80309, USA
21
Colorado State University, Fort Collins, Colorado 80523, USA
22
Technische Universita
̈
t Dortmund, Fakulta
̈
t Physik, D-44221 Dortmund, Germany
23
Technische Universita
̈
t Dresden, Institut fu
̈
r Kern- und Teilchenphysik, D-01062 Dresden, Germany
24
Laboratoire Leprince-Ringuet, CNRS/IN2P3, Ecole Polytechnique, F-91128 Palaiseau, France
25
University of Edinburgh, Edinburgh EH9 3JZ, United Kingdom
26a
INFN Sezione di Ferrara, I-44100 Ferrara, Italy
26b
Dipartimento di Fisica, Universita
`
di Ferrara, I-44100 Ferrara, Italy
27
INFN Laboratori Nazionali di Frascati, I-00044 Frascati, Italy
28a
INFN Sezione di Genova, I-16146 Genova, Italy
28b
Dipartimento di Fisica, Universita
`
di Genova, I-16146 Genova, Italy
29
Harvard University, Cambridge, Massachusetts 02138, USA
30
Universita
̈
t Heidelberg, Physikalisches Institut, Philosophenweg 12, D-69120 Heidelberg, Germany
B. AUBERT
et al.
PHYSICAL REVIEW D
78,
092008 (2008)
092008-2
31
Humboldt-Universita
̈
t zu Berlin, Institut fu
̈
r Physik, Newtonstr. 15, D-12489 Berlin, Germany
32
Imperial College London, London, SW7 2AZ, United Kingdom
33
University of Iowa, Iowa City, Iowa 52242, USA
34
Iowa State University, Ames, Iowa 50011-3160, USA
35
Johns Hopkins University, Baltimore, Maryland 21218, USA
36
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
37
Lawrence Livermore National Laboratory, Livermore, California 94550, USA
38
University of Liverpool, Liverpool L69 7ZE, United Kingdom
39
Queen Mary, University of London, London, E1 4NS, United Kingdom
40
University of London, Royal Holloway and Bedford New College, Egham, Surrey TW20 0EX, United Kingdom
41
University of Louisville, Louisville, Kentucky 40292, USA
42
Johannes Gutenberg-Universita
̈
t Mainz, Institut fu
̈
r Kernphysik, D-55099 Mainz, Germany
43
University of Manchester, Manchester M13 9PL, United Kingdom
44
University of Maryland, College Park, Maryland 20742, USA
45
University of Massachusetts, Amherst, Massachusetts 01003, USA
46
Massachusetts Institute of Technology, Laboratory for Nuclear Science, Cambridge, Massachusetts 02139, USA
47
McGill University, Montre
́
al, Que
́
bec, Canada H3A 2T8
48a
INFN Sezione di Milano, I-20133 Milano, Italy
48b
Dipartimento di Fisica, Universita
`
di Milano, I-20133 Milano, Italy
49
University of Mississippi, University, Mississippi 38677, USA
50
Universite
́
de Montre
́
al, Physique des Particules, Montre
́
al, Que
́
bec, Canada H3C 3J7
51
Mount Holyoke College, South Hadley, Massachusetts 01075, USA
52a
INFN Sezione di Napoli, I-80126 Napoli, Italy
52b
Dipartimento di Scienze Fisiche, Universita
`
di Napoli Federico II, I-80126 Napoli, Italy
53
NIKHEF, National Institute for Nuclear Physics and High Energy Physics, NL-1009 DB Amsterdam, The Netherlands
54
University of Notre Dame, Notre Dame, Indiana 46556, USA
55
Ohio State University, Columbus, Ohio 43210, USA
56
University of Oregon, Eugene, Oregon 97403, USA
57a
INFN Sezione di Padova, I-35131 Padova, Italy
57b
Dipartimento di Fisica, Universita
`
di Padova, I-35131 Padova, Italy
58
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
59
University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
60a
INFN Sezione di Perugia, I-06100 Perugia, Italy
60b
Dipartimento di Fisica, Universita
`
di Perugia, I-06100 Perugia, Italy
61a
INFN Sezione di Pisa, I-56127 Pisa, Italy
61b
Dipartimento di Fisica, Universita
`
di Pisa, I-56127 Pisa, Italy
61c
Scuola Normale Superiore di Pisa, I-56127 Pisa, Italy
62
Princeton University, Princeton, New Jersey 08544, USA
63a
INFN Sezione di Roma, I-00185 Roma, Italy
63b
Dipartimento di Fisica, Universita
`
di Roma La Sapienza, I-00185 Roma, Italy
64
Universita
̈
t Rostock, D-18051 Rostock, Germany
65
Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, OX11 0QX, United Kingdom
66
CEA, Irfu, SPP, Centre de Saclay, F-91191 Gif-sur-Yvette, France
67
University of South Carolina, Columbia, South Carolina 29208, USA
68
Stanford Linear Accelerator Center, Stanford, California 94309, USA
69
Stanford University, Stanford, California 94305-4060, USA
70
State University of New York, Albany, New York 12222, USA
71
University of Tennessee, Knoxville, Tennessee 37996, USA
72
University of Texas at Austin, Austin, Texas 78712, USA
73
University of Texas at Dallas, Richardson, Texas 75083, USA
**
Also at Universita
`
di Sassari, Sassari, Italy.
{
Present address: University of South Alabama, Mobile, AL 36688, USA.
k
Also at Universita
`
di Roma La Sapienza, I-00185 Roma, Italy.
x
Also at Universita
`
di Perugia, Dipartimento di Fisica, Perugia, Italy.
‡
Present address: Tel Aviv University, Tel Aviv, 69978, Israel.
†
Present address: Temple University, Philadelphia, PA 19122, USA.
*
Deceased.
TIME-DEPENDENT AND TIME-INTEGRATED ANGULAR
...
PHYSICAL REVIEW D
78,
092008 (2008)
092008-3
74a
INFN Sezione di Torino, I-10125 Torino, Italy
74b
Dipartimento di Fisica Sperimentale, Universita
`
di Torino, I-10125 Torino, Italy
75a
INFN Sezione di Trieste, I-34127 Trieste, Italy
75b
Dipartimento di Fisica, Universita
`
di Trieste, I-34127 Trieste, Italy
76
IFIC, Universitat de Valencia-CSIC, E-46071 Valencia, Spain
77
University of Victoria, Victoria, British Columbia, Canada V8W 3P6
78
Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom
79
University of Wisconsin, Madison, Wisconsin 53706, USA
(Received 26 August 2008; published 18 November 2008)
We perform a time-dependent and time-integrated angular analysis of the decays
B
0
!
’K
ð
892
Þ
0
,
’K
2
ð
1430
Þ
0
, and
’
ð
K
Þ
0
0
with the final sample of about
465
10
6
B
B
pairs recorded with the
BABAR
detector. Twenty-four parameters are investigated, including the branching fractions,
CP
-violation
parameters, and parameters sensitive to final-state interactions. We use the dependence on the
K
invariant mass of the interference between the scalar and vector or tensor components to resolve discrete
ambiguities of the strong and weak phases. We use the time evolution of the
B
!
’K
0
S
0
channel to
extract the
CP
-violation phase difference
00
¼
0
:
28
0
:
42
0
:
04
between the
B
and
B
decay
amplitudes. When the
B
!
’K
channel is included, the fractions of longitudinal polarization
f
L
of the vector-vector and vector-tensor decay modes are measured to be
0
:
494
0
:
034
0
:
013
and
0
:
901
þ
0
:
046
0
:
058
0
:
037
, respectively. This polarization pattern requires the presence of a positive-helicity
amplitude in the vector-vector decay from a currently unknown source.
DOI:
10.1103/PhysRevD.78.092008
PACS numbers: 13.25.Hw, 11.30.Er, 12.15.Hh, 13.88.+e
I. INTRODUCTION
Charge-parity (
CP
) symmetry violation has been recog-
nized as one of the fundamental requirements for produc-
ing a matter-dominated universe [
1
] and therefore it has
played an important role in understanding fundamental
physics since its initial discovery in the
K
meson system
in 1964 [
2
]. A significant
CP
-violating asymmetry has
been observed in decays of neutral
B
mesons to final states
containing charmonium, due to interference between
B
0
-
B
0
mixing and direct decay amplitudes [
3
]. It has
been established [
4
] that the
CP
-violating decays of the
K
0
L
meson are due to
CP
violation in decay amplitudes, as
well as in
K
0
-
K
0
mixing, and this kind of ‘‘direct’’
CP
asymmetry in
B
decays has also been observed recently
[
5
]. The
CP
asymmetries are generally much larger in
B
decays than in
K
decays [
6
] because they directly probe the
least flat unitarity triangle constructed from the Cabibbo-
Kobayashi-Maskawa (CKM) matrix elements
V
ij
, which
relate weak and flavor quark eigenstates [
7
]. This triangle
reflects the unitarity of the CKM matrix, and two of
its angles are the phase differences of its sides on the
complex plane:
arg
ð
V
td
V
tb
=V
ud
V
ub
Þ
and
arg
ð
V
cd
V
cb
=V
td
V
tb
Þ
. Because of the large
CP
-violating
effects,
B
decays provide an excellent testing ground of
fundamental interactions.
The
CP
-violating effects observed to date are self-
consistent within the standard model with a single complex
phase in the CKM mechanism [
7
]. However, this mecha-
nism alone is believed to be insufficient to produce the
present matter-dominated Universe. Therefore, it is impor-
tant to search for new sources of
CP
-violating interactions.
While direct access to new fundamental particles and
interactions may be beyond the energy reach of operating
accelerators, one can look for them in virtual transitions.
New particles in virtual transitions, including but not lim-
ited to supersymmetric particles [
8
], would provide addi-
tional amplitudes with different phases. Depending on the
model parameters, sizable
CP
-violating effects, either
B
0
-
B
0
mixing-induced or direct, could be observed in
pure penguin modes which involve virtual loops as in the
example shown in Fig.
1
. Some of the first observed
gluonic penguin decays,
B
ðÞ
!
0
K
ðÞ
[
9
] and
B
ðÞ
!
’K
ðÞ
ðÞ
[
10
], remain promising channels in which to look for
new physics. The latter type of decay is illustrated in
Fig.
1
and is the focus of this paper. For example, com-
parison of the value of
sin2
obtained from these modes
with that from charmonium modes such as
B
ðÞ
!
J=
c
K
ðÞ
ðÞ
[
3
,
11
], or measurement of direct
CP
violation, can probe
new physics participating in penguin loops [
12
].
FIG. 1. Penguin diagram describing the decay
B
!
’K
ðÞ
.
B. AUBERT
et al.
PHYSICAL REVIEW D
78,
092008 (2008)
092008-4
The
ð
V
-
A
Þ
nature of the weak interaction leads to left-
handed fermion couplings in interactions with
W
bosons,
such as those shown in Fig.
1
. Combined with helicity
conservation in strong interactions and spin-flip suppres-
sion of relativistic decay products, this leads to certain
expectations of the spin alignment in weak
B
meson decays
to light particles with spin, such as
B
ðÞ
!
’K
ðÞ
[
13
].
However, the large fraction of transverse polarization in
the
B
ðÞ
!
’K
ðÞ
ð
892
Þ
decay measured by
BABAR
[
14
] and by
Belle [
15
] indicates a significant departure from the naive
expectation of predominant longitudinal polarization. This
suggests the presence of other contributions to the decay
amplitude, previously neglected, either within or beyond
the standard model [
16
]. The presence of a substantial
transverse amplitude also allows the study of
CP
violation
in the angular distribution of
B
ðÞ
!
’K
ðÞ
decays, an ap-
proach complementary to either mixing-induced or yield
asymmetry studies. Polarization measurements in
B
decays
are discussed in a recent review [
17
,
18
]. In Table
I
, we list
BABAR
’s recent measurements of the branching fraction
and longitudinal polarization in the decays
B
ðÞ
!
’K
ðÞ
ðÞ
J
[
19
–
23
]. Measurements in
B
ðÞ
!
K
ðÞ
decays have also
revealed a large fraction of transverse polarization [
24
].
In this analysis, we use the final sample of about
465
10
6
ð
4
S
Þ!
B
B
pairs recorded with the
BABAR
detector
at the PEP-II asymmetric-energy
e
þ
e
storage rings at
Stanford Linear Accelerator Center (SLAC). We employ
all of these techniques for
CP
-violation and polarization
measurements in the study of a single
B
-decay topology
B
ðÞ
0
!
’
ð
K
ðÞ
Þ
. Overall, 27 independent parameters sensi-
tive to
CP
violation, spin alignment, or strong- or weak-
interaction phases describe three decay channels (twelve in
either vector-vector or vector-tensor and three in vector-
scalar decays), which leaves only one overall phase un-
measurable. The three channels in our amplitude analysis
are
B
ðÞ
0
!
’K
ðÞ
ð
892
Þ
0
,
’K
ðÞ
2
ð
1430
Þ
0
, and
’
ð
K
ðÞ
Þ
0
0
. The
latter contribution includes the
K
ðÞ
0
ð
1430
Þ
0
resonance to-
gether with a nonresonant component, as measured by the
LASS experiment [
25
]. While we describe the analysis of
these three neutral-
B
meson decays, this technique, with
the exception of time-dependent measurements, has also
been applied recently to the charged-
B
meson decays
[
21
,
22
].
We use the time evolution of the
B
ðÞ
0
!
’K
0
S
0
channel
to extract the mixing-induced
CP
-violating phase differ-
ence between the
B
and
B
decay amplitudes, which is
equivalent to a measurement of
sin2
to a good approxi-
mation. With the
B
ðÞ
0
!
’K
channel included, the
fractions of longitudinal and parity-odd transverse ampli-
tudes in the vector-vector and vector-tensor decay modes
are measured. We use the dependence on the
K
invariant
mass of the interference between the scalar and vector or
scalar and tensor components to resolve discrete ambigu-
ities of the strong and weak phases. Using either interfer-
ence between different channels or
B
0
-
B
0
mixing, we
measure essentially all 27 independent parameters except
for three quantities that characterize the parity-odd trans-
verse amplitude in the vector-tensor decay, which is found
to be consistent with zero.
TABLE I.
BABAR
’s recent measurements of the branching fraction
B
and longitudinal polarization fraction
f
L
in the decays
B
!
’K
ðÞ
J
. The spin
J
and parity
P
quantum numbers of the
K
ðÞ
J
mesons are quoted. The upper limits are shown at the 90% confidence
level. For a complete list of all observables in each analysis see the references listed. Results indicated with
y
are superseded by this
analysis.
Mode
J
P
Ref.
B
ð
10
6
Þ
f
L
’K
0
0
[
19
]
8
:
4
þ
1
:
5
1
:
3
0
:
5
1
’K
þ
0
[
19
]
10
:
0
þ
0
:
9
0
:
8
0
:
5
1
’K
0
ð
1430
Þ
0
0
þ
[
20
]
y
4
:
6
0
:
7
0
:
6
1
’K
0
ð
1430
Þ
þ
0
þ
[
21
]
7
:
0
1
:
3
0
:
9
1
’K
ð
892
Þ
0
1
[
20
]
y
9
:
2
0
:
7
0
:
60
:
51
0
:
04
0
:
02
’K
ð
892
Þ
þ
1
[
22
]
11
:
2
1
:
0
0
:
90
:
49
0
:
05
0
:
03
’K
ð
1410
Þ
þ
1
[
21
]
<
4
:
3
’K
ð
1680
Þ
0
1
[
23
]
<
3
:
5
’K
1
ð
1270
Þ
þ
1
þ
[
21
]
6
:
1
1
:
6
1
:
10
:
46
þ
0
:
12
þ
0
:
06
0
:
13
0
:
07
’K
1
ð
1400
Þ
þ
1
þ
[
21
]
<
3
:
2
’K
2
ð
1430
Þ
0
2
þ
[
20
]
y
7
:
8
1
:
1
0
:
60
:
85
þ
0
:
06
0
:
07
0
:
04
’K
2
ð
1430
Þ
þ
2
þ
[
21
]
8
:
4
1
:
8
1
:
00
:
80
þ
0
:
09
0
:
10
0
:
03
’K
2
ð
1770
Þ
þ
2
[
21
]
<
15
:
0
’K
2
ð
1820
Þ
þ
2
[
21
]
<
16
:
3
’K
3
ð
1780
Þ
0
3
[
23
]
<
2
:
7
’K
4
ð
2045
Þ
0
4
þ
[
23
]
<
15
:
3
TIME-DEPENDENT AND TIME-INTEGRATED ANGULAR
...
PHYSICAL REVIEW D
78,
092008 (2008)
092008-5
II. ANALYSIS STRATEGY
Earlier studies of
B
ðÞ
0
!
’K
decays by the
BABAR
Collaboration [
20
,
23
] indicate the presence of three sig-
nificant
K
partial waves:
ð
K
Þ
0
0
[spin
J
¼
0
, including
the resonance
K
0
ð
1430
Þ
0
],
K
ð
892
Þ
0
(
J
¼
1
), and
K
2
ð
1430
Þ
0
(
J
¼
2
). These correspond to the following
decays, with the number of independent amplitudes char-
acterizing different spin projections given in parentheses:
B
ðÞ
0
!
’
ð
K
ðÞ
Þ
0
0
(one),
’K
ðÞ
ð
892
Þ
0
(three), and
’K
ðÞ
2
ð
1430
Þ
0
(three). No significant contribution from other final states
has been found with
K
invariant mass
m
K
up to
2.15 GeV [
21
,
23
]. See Fig.
2
for an illustration of the
B
ðÞ
0
!
’K
contributions. Therefore, we limit our analysis to
the mass range
m
K
<
1
:
55 GeV
without any significant
loss of
B
ðÞ
0
!
’K
signal through charmless
K
reso-
nant or nonresonant production.
There has been no extensive study of the
B
ðÞ
0
!
’K
0
S
0
decay, except for the study of
B
ðÞ
0
!
’K
ðÞ
ð
892
Þ
0
[
14
].
However, due to isospin symmetry of the
K
0
0
and
K
systems, the same amplitude composition is ex-
pected in the
’K
and
’K
0
S
0
final states. We do not
expect any charmless resonance structure in the
’K
or
’
combinations, while we veto the charm resonance
states, such as
D
ð
s
Þ
!
’
.
It is instructive to do a simple counting of the amplitude
parameters in
B
ðÞ
0
!
’K
ðÞ
decays with the three
K
spin
contributions discussed above. With seven independent
A
J
complex amplitudes for
B
decays and seven
A
J
amplitudes for
B
decays, we could construct 28 indepen-
dent real parameters. Here
J
refers to the spin of the
K
system and
to the spin projection of the
’
meson onto the
direction opposite to the
B
meson flight direction in the
’
rest frame. However, one overall phase is not measurable
and we are left with 27 real measurable parameters.
Among these parameters, 26 parameters have been or can
be measured in the decay
B
ðÞ
0
!
’K
[
20
]. Those are
branching fractions, polarization parameters, strong
phases, and
CP
asymmetries. Some of the phases are
extracted from the interference effects between different
modes. However, due to limited statistics some of the
CP
asymmetries were not measured in earlier analyses and we
now extend those measurements.
TABLE II. Definitions of 27 real parameters measurable with the
B
0
!
’K
decays. Three resonance final states with spin
J
¼
0
;
1
;
2
are considered in the
K
spectrum. The branching fraction
B
is calculated as a ratio of the average partial decay widths for
B
0
ð
Þ
and
B
0
ð
Þ
and the total width
total
where we neglect any difference in the
B
0
and
B
0
widths. This definition allows for
differences between the
B
0
and
B
0
decay amplitudes,
A
J
and
A
J
, as discussed in the text.
’K
0
ð
1430
Þ
’K
ð
892
Þ
’K
2
ð
1430
Þ
Parameter
Definition
J
¼
0
J
¼
1
J
¼
2
B
J
1
2
ð
J
þ
J
Þ
=
total
B
0
B
1
B
2
f
LJ
1
2
ðj
A
J
0
j
2
=
j
A
J
j
2
þj
A
J
0
j
2
=
j
A
J
j
2
Þ
1
f
L
1
f
L
2
f
?
J
1
2
ðj
A
J
?
j
2
=
j
A
J
j
2
þj
A
J
?
j
2
=
j
A
J
j
2
Þ
None
f
?
1
f
?
2
k
J
1
2
ð
arg
ð
A
J
k
=
A
J
0
Þþ
arg
ð
A
J
k
=A
J
0
ÞÞ
None
k
1
k
2
?
J
1
2
ð
arg
ð
A
J
?
=
A
J
0
Þþ
arg
ð
A
J
?
=A
J
0
Þ
Þ
None
?
1
?
2
0
J
1
2
ð
arg
ð
A
00
=
A
J
0
Þþ
arg
ð
A
00
=A
J
0
ÞÞ
0
01
02
A
CPJ
ð
J
J
Þ
=
ð
J
þ
J
Þ
A
CP
0
A
CP
1
A
CP
2
A
0
CPJ
ðj
A
J
0
j
2
=
j
A
J
j
2
j
A
J
0
j
2
=
j
A
J
j
2
Þ
=
ðj
A
J
0
j
2
=
j
A
J
j
2
þj
A
J
0
j
2
=
j
A
J
j
2
Þ
0
A
0
CP
1
A
0
CP
2
A
?
CPJ
ðj
A
J
?
j
2
=
j
A
J
j
2
j
A
J
?
j
2
=
j
A
J
j
2
Þ
=
ðj
A
J
?
j
2
=
j
A
J
j
2
þj
A
J
?
j
2
=
j
A
J
j
2
Þ
None
A
?
CP
1
A
?
CP
2
k
J
1
2
ð
arg
ð
A
J
k
=
A
J
0
Þ
arg
ð
A
J
k
=A
J
0
ÞÞ
None
k
1
k
2
?
J
1
2
ð
arg
ð
A
J
?
=
A
J
0
Þ
arg
ð
A
J
?
=A
J
0
Þ
Þ
None
?
1
?
2
0
J
1
2
ð
arg
ð
A
00
=
A
J
0
Þ
arg
ð
A
00
=A
J
0
ÞÞ
0
01
02
00
1
2
arg
ð
A
00
=
A
00
Þ
00
None
None
0.75
1.45
2.15
Events / 20 MeV
0
50
100
(GeV)
π
K
m
FIG. 2. Invariant
K
mass distribution from the
B
!
’K
analysis from Refs. [
20
,
23
]. The solid (dashed) line is a projec-
tion of the signal-plus-background (background only) fit result.
The narrow charm background peak at 1.865 GeV comes from
D
0
decays to
K
and is not associated with
’K
production.
The arrow indicates the mass range considered in this analysis.
B. AUBERT
et al.
PHYSICAL REVIEW D
78,
092008 (2008)
092008-6
Finally, one parameter, which relates the phases of the
B
and
B
decay amplitudes, can be measured using only the
interference between decays with and without
B
0
-
B
0
mix-
ing, to final states which can be decomposed as
CP
eigen-
states, such as
’K
0
S
0
. In Table
II
all 27 real parameters
measurable with
B
ðÞ
0
!
’K
ðÞ
decays are summarized.
These parameters are expressed in terms of the
A
J
and
A
J
amplitudes for
B
0
!
’K
þ
or
’K
0
0
and
B
0
!
’K
þ
or
’
K
0
0
decays. We also refer to a transformed
set of amplitudes
A
J
0
and
A
J
1
¼ð
A
J
k
A
J
?
Þ
=
ffiffiffi
2
p
. The
parameters in Table
II
are expressed as six
CP
-averaged
and six
CP
-violating parameters for the vector-vector and
vector-tensor decays. The
in the definitions of
?
J
and
?
J
accounts for the sign flip
A
?
J
¼
A
?
J
if
CP
is
conserved. The parametrization in Table
II
is motivated by
the negligible
CP
violation expected in these decays.
Therefore, the polarization parameters specific to either
B
(superscript ‘‘
’’) or
B
(superscript ‘‘
þ
’’) are the
CP
-averaged parameters with small
CP
-violating correc-
tions which are either multiplicative (for rates) or additive
(for phases):
B
J
¼
B
J
ð
1
A
CPJ
Þ
=
2
;
(1)
f
LJ
¼
f
LJ
ð
1
A
0
CPJ
Þ
;
(2)
f
?
J
¼
f
?
J
ð
1
A
?
CPJ
Þ
;
(3)
k
J
¼
k
J
k
J
;
(4)
?
J
¼
?
J
?
J
þ
2
2
;
(5)
0
J
¼
0
J
0
J
:
(6)
In this section we discuss further the method for the
measurement of the relative phase, along with all of the
other parameters. First we review the angular distributions,
follow with a discussion of the
K
invariant mass distri-
butions critical to separating different partial waves, then
introduce interference effects between amplitudes from
different decays, and finally discuss time-dependent
distributions.
A. Angular distributions
We discuss here the angular distribution of the decay
products in the chain
B
!
’K
!ð
K
þ
K
Þð
K
Þ
inte-
grated over time. First we look at the decay of a
B
meson
only and leave the
B
for later discussion, which involves
CP
violation. Angular momentum conservation in the
decay of a spinless
B
meson leads to three possible spin
projections of the
’
meson onto its direction of flight, each
corresponding to a complex amplitude
A
J
with
¼
0
or
1
. The three
values are allowed with the
K
spin states
J
1
, but only
¼
0
contributes with a spin-zero
K
. The
angular distributions can be expressed as functions of
H
i
¼
cos
i
and
. Here
i
is the angle between the
direction of the
K
meson from the
K
!
K
(
1
)or
’
!
K
K
(
2
) and the direction opposite to the
B
in the
K
or
’
rest frame, and
is the angle between the decay planes of
the two systems, as shown in Fig.
3
. The differential decay
width is
d
3
d
H
1
d
H
2
d
/
X
A
J
Y
J
ð
H
1
;
Þ
Y
1
ð
H
2
;
0
Þ
2
;
(7)
where
Y
J
are the spherical harmonic functions, with
J
¼
2
for
K
2
ð
1430
Þ
,
J
¼
1
for
K
ð
892
Þ
, and
J
¼
0
for
ð
K
Þ
0
,
including
K
0
ð
1430
Þ
. We do not consider higher values of
J
because no significant contribution from those states is
expected. Only resonances with spin-parity combination
P
¼ð
1
Þ
J
are possible in the decay
K
!
K
due to
parity conservation in these strong-interaction decays.
If we ignore interference between modes with different
spins
J
of the
K
system in Eq. (
7
), then for each decay
mode we have three complex amplitudes
A
J
which appear
in the angular distribution. We discuss interference be-
tween different modes later in this section. The differential
decay rate for each decay mode involves six real quantities
iJ
, including terms that account for interference between
amplitudes of common
J
:
d
3
J
J
d
H
1
d
H
2
d
¼
X
i
iJ
f
iJ
ð
H
1
;
H
2
;
Þ
;
(8)
where the functions
f
iJ
ð
H
1
;
H
2
;
Þ
are given in
Table
III
. The
iJ
parameters are defined as
1
J
¼
j
A
J
0
j
2
j
A
J
j
2
¼
f
LJ
;
(9)
2
J
¼
j
A
J
k
j
2
þj
A
J
?
j
2
j
A
J
j
2
¼
j
A
J
þ
1
j
2
þj
A
J
1
j
2
j
A
J
j
2
¼ð
1
f
LJ
Þ
;
(10)
FIG. 3. Definition of decay angles given in the rest frames of
the decaying parents.
TIME-DEPENDENT AND TIME-INTEGRATED ANGULAR
...
PHYSICAL REVIEW D
78,
092008 (2008)
092008-7
3
J
¼
j
A
J
k
j
2
j
A
J
?
j
2
j
A
J
j
2
¼
2
<
e
ð
A
J
þ
1
A
J
1
Þ
j
A
J
j
2
¼ð
1
f
LJ
2
f
?
J
Þ
;
(11)
4
J
¼
=
m
ð
A
J
?
A
J
k
Þ
j
A
J
j
2
¼
=
m
ð
A
J
þ
1
A
J
1
Þ
j
A
J
j
2
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
f
?
J
ð
1
f
LJ
f
?
J
Þ
q
sin
ð
?
J
k
J
Þ
;
(12)
5
J
¼
<
e
ð
A
J
k
A
J
0
Þ
j
A
J
j
2
¼
<
e
ð
A
J
þ
1
A
J
0
þ
A
J
1
A
J
0
Þ
ffiffiffi
2
p
j
A
J
j
2
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
f
LJ
ð
1
f
LJ
f
?
J
Þ
q
cos
ð
k
J
Þ
;
(13)
6
J
¼
=
m
ð
A
J
?
A
J
0
Þ
j
A
J
j
2
¼
=
m
ð
A
J
þ
1
A
J
0
A
J
1
A
J
0
Þ
ffiffiffi
2
p
j
A
J
j
2
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
f
?
J
f
LJ
q
sin
ð
?
J
Þ
:
(14)
The above terms are specific to the
B
0
decays and are
denoted with the superscript ‘‘
,’’ as introduced in
Eqs. (
1
)–(
6
). The angular distributions for the
B
0
decays
are described by the same Eq. (
8
), but with
iJ
replaced by
þ
iJ
, and with definitions given by Eqs. (
9
)–(
14
), replacing
A
by
A
and superscript ‘‘
’’ by ‘‘
þ
.’’
B. Mass distributions
The differential decay width given in Eq. (
7
) is parame-
trized as a function of helicity angles. However, it also
depends on the invariant mass
m
K
of the
K
resonance,
and the amplitudes should be considered as functions of
m
K
. Without considering interference between different
modes, as shown in Eq. (
8
), this mass dependence decou-
ples from the angular dependence. Nonetheless, this de-
pendence is important for separating different
K
states.
The interference effects will be considered in the next
subsection. A relativistic spin-
J
Breit-Wigner (B-W) com-
plex amplitude
R
J
can be used to parametrize the reso-
nance masses with
J
¼
1
and
J
¼
2
[
18
]:
R
J
ð
m
Þ¼
m
J
J
ð
m
Þ
ð
m
2
J
m
2
Þ
im
J
J
ð
m
Þ
¼
sin
J
e
i
J
;
(15)
where we use the following convention:
cot
J
¼
m
2
J
m
2
m
J
J
ð
m
Þ
:
(16)
The mass-dependent widths are given by
1
ð
m
Þ¼
1
m
1
m
1
þ
r
2
q
2
1
1
þ
r
2
q
2
q
q
1
3
;
(17)
2
ð
m
Þ¼
2
m
2
m
9
þ
3
r
2
q
2
2
þ
r
4
q
4
2
3
þ
3
r
2
q
2
þ
r
4
q
4
q
q
2
5
;
(18)
where
J
is the resonance width,
m
J
is the resonance mass,
q
is the momentum of a daughter particle in the resonance
system after its two-body decay (
q
J
is evaluated at
m
¼
m
J
), and
r
is the interaction radius.
The parametrization of the scalar
ð
K
Þ
0
0
mass distribu-
tion requires more attention. Studies of
K
scattering were
performed by the LASS experiment [
25
]. It was found that
the scattering is elastic up to about 1.5 GeV and can be
parametrized with the amplitude:
R
0
ð
m
Þ¼
sin
0
e
i
0
;
(19)
where
0
¼
R
þ
B;
(20)
R
represents a resonant
K
0
ð
1430
Þ
0
contribution and
B
represents a nonresonant contribution. The mass depen-
dence of
B
is described by means of an effective range
parametrization of the usual type:
cot
B
¼
1
aq
þ
1
2
bq;
(21)
where
a
is the scattering length and
b
is the effective range.
The mass dependence of
R
is described by means of a
TABLE III. Parametrization of the angular distribution in Eq. (
8
) in the
B
0
!
’
ð
K
Þ
J
decays where three resonance final states with
spin
J
¼
0
;
1
;
2
are considered. The common constant is quoted for each decay mode and is omitted from each individual function
below. The three helicity angle parameters
ð
H
1
;
H
2
;
Þ
are discussed in the text.
’K
0
ð
1430
Þ
0
’K
ð
892
Þ
0
’K
2
ð
1430
Þ
0
J
¼
0
J
¼
1
J
¼
2
Common constant
3
=
4
9
=
8
15
=
32
f
1
J
ð
H
1
;
H
2
;
Þ
H
2
2
H
2
1
H
2
2
ð
3
H
2
1
1
Þ
2
H
2
2
f
2
J
ð
H
1
;
H
2
;
Þ
0
1
4
ð
1
H
2
1
Þð
1
H
2
2
Þ
3
H
2
1
ð
1
H
2
1
Þð
1
H
2
2
Þ
f
3
J
ð
H
1
;
H
2
;
Þ
0
1
4
ð
1
H
2
1
Þð
1
H
2
2
Þ
cos2
3
H
2
1
ð
1
H
2
1
Þð
1
H
2
2
Þ
cos2
f
4
J
ð
H
1
;
H
2
;
Þ
0
1
2
ð
1
H
2
1
Þð
1
H
2
2
Þ
sin2
6
H
2
1
ð
1
H
2
1
Þð
1
H
2
2
Þ
sin2
f
5
J
ð
H
1
;
H
2
;
Þ
0
ffiffiffi
2
p
H
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
H
2
2
q
H
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
H
2
2
q
cos
ffiffiffi
6
p
H
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
H
2
1
q
ð
3
H
2
1
1
Þ
H
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
H
2
2
q
cos
f
6
J
ð
H
1
;
H
2
;
Þ
0
ffiffiffi
2
p
H
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
H
2
1
q
H
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
H
2
2
q
sin
ffiffiffi
6
p
H
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
H
2
1
q
ð
3
H
2
1
1
Þ
H
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
H
2
2
q
sin
B. AUBERT
et al.
PHYSICAL REVIEW D
78,
092008 (2008)
092008-8
B-W parametrization of a form similar to Eq. (
16
):
cot
R
¼
m
2
0
m
2
m
0
0
ð
m
Þ
;
(22)
where
m
0
is the resonance mass, and
0
ð
m
Þ
is defined as
0
ð
m
Þ¼
0
m
0
m
q
q
0
:
(23)
The invariant amplitude
M
J
ð
m
Þ
is proportional to
R
J
ð
m
Þ
:
M
J
ð
m
Þ/
m
q
R
J
ð
m
Þ
(24)
and can be expressed, for example, for
J
¼
0
,as
M
0
ð
m
Þ/
m
q
cot
B
iq
þ
e
2
i
B
0
m
2
0
=q
0
ð
m
2
0
m
2
Þ
im
0
0
ð
m
Þ
:
(25)
The resulting
ð
K
Þ
0
0
invariant mass distribution is
shown in Fig.
4
, along with the phase and distributions
for the other resonances. The mass parameters describing
the three spin states in the
m
distribution are shown in
Table
IV
. Measurements of the LASS experiment are used
for the parameters of the
J
¼
0
contribution and for the
interaction radius [
25
,
26
]. The values of
m
0
,
0
,
a
, and
b
used in this analysis are different from those quoted in
Ref. [
25
] due to better handling of the fit to the LASS data
[
26
]. The two sets of values are consistent within errors and
lead to similar results.
To account for the three-body kinematics in the analysis
of
B
0
!
’K
decays, we multiply the amplitude squared
j
M
J
ð
m
Þj
2
by the phase-space factor
F
ð
m
Þ
:
F
ð
m
Þ¼
2
m
½
m
2
max
ð
m
Þ
m
2
min
ð
m
Þ
;
(26)
where
m
2
max
and
m
2
min
are the maximum and minimum
values of the Dalitz plot range of
m
2
’K
(
m
’K
is the
’K
invariant mass) at any given value of
m
K
; see the kine-
matics section of Ref. [
18
]. Because of slow dependence of
the factor in Eq. (
26
)on
m
in any small range of
m
, the
difference of this approach from the quasi-two-body ap-
proximation is small.
C. Interference effects
The differential decay width discussed in Eq. (
7
) in-
volves interference terms between resonances with differ-
ent spins
J
. These interference terms have unique angular
and mass dependences which cannot be factorized in the
full distribution. We can parametrize the mass and angular
amplitude for each spin state
J
as follows:
A
0
ð
m
K
;
1
;
2
;
Þ¼
Y
0
0
ð
H
1
;
Þ
Y
0
1
ð
H
2
;
0
Þ
M
0
ð
m
K
Þ
A
00
;
(27)
A
1
ð
m
K
;
1
;
2
;
Þ¼
X
¼
0
;
1
Y
1
ð
H
1
;
Þ
Y
1
ð
H
2
;
0
Þ
M
1
ð
m
K
Þ
A
1
;
(28)
A
2
ð
m
K
;
1
;
2
;
Þ¼
X
¼
0
;
1
Y
2
ð
H
1
;
Þ
Y
1
ð
H
2
;
0
Þ
M
2
ð
m
K
Þ
A
2
:
(29)
The interference will appear in the angular-mass distri-
butions as
2
<
e
ð
A
i
ð
m
K
;
1
;
2
;
Þ
A
j
ð
m
K
;
1
;
2
;
ÞÞ
.As
we can see from Fig.
4
, the overlap between the
P
- and
D
-wave
K
contributions is negligibly small, and we will
consider only the interference between the
J
¼
0
and
J
¼
1
,or
J
¼
0
and
J
¼
2
amplitudes. The resulting two
interference terms, properly normalized, are defined for
J
¼
1
and
J
¼
2
:
2
<
e
ð
A
J
A
0
Þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
j
A
J
j
2
p
j
A
00
j
¼
X
9
i
¼
7
iJ
ð
m
K
Þ
f
iJ
ð
H
1
;
H
2
;
Þ
;
(30)
where the angular dependence is defined in Table
V
, and
iJ
ð
m
K
Þ
are defined for
i
¼
7
;
8
;
9
as
(GeV)
π
K
m
11.5
Intensity
0
50
(a)
(GeV)
π
K
m
1
1.5
Phase (degrees)
0
100
(b)
FIG. 4 (color online). Intensity
j
M
J
ð
m
K
Þj
2
(a) and phase
arg
ð
M
J
ð
m
K
ÞÞ
(b) of the invariant amplitudes for
J
¼
0
(solid
line),
J
¼
1
(dashed line), and
J
¼
2
(long-dashed line)
K
contributions as a function of the invariant
K
mass
m
K
. The
taller two arrows indicate the low
m
K
region, while the shorter
two arrows indicate the high
m
K
region. The relative intensity
of the amplitudes is taken from Fig.
2
, while the absolute
intensity is shown in arbitrary units.
TABLE IV. Parametrization of the
K
invariant mass distri-
bution in the
B
0
!
’
ð
K
Þ
J
decays where three resonance final
states with spin
J
¼
0
;
1
;
2
are considered. The resonance mass
m
J
, width
J
[
18
,
25
,
26
], interaction radius
r
, scattering length
a
,
and effective range
b
are considered [
25
,
26
]. Combined errors
are quoted, except for
ð
K
Þ
0
0
where the systematic errors are
quoted last while the central values and statistical errors have
been updated [
26
] with respect to Ref. [
25
].
ð
K
Þ
0
0
K
ð
892
Þ
0
K
2
ð
1430
Þ
0
J
¼
0
J
¼
1
J
¼
2
m
J
(MeV)
1435
5
5 896
:
00
0
:
25 1432
:
4
1
:
3
J
(MeV)
279
6
21
50
:
3
0
:
6
109
5
r
(
GeV
1
)
3
:
4
0
:
72
:
7
1
:
3
a
(
GeV
1
)
1
:
95
0
:
09
0
:
06
b
(
GeV
1
)
1
:
76
0
:
36
0
:
67
TIME-DEPENDENT AND TIME-INTEGRATED ANGULAR
...
PHYSICAL REVIEW D
78,
092008 (2008)
092008-9
7
J
ð
m
K
Þ¼
ffiffiffiffiffiffiffiffi
f
LJ
q
<
e
ð
M
J
ð
m
K
Þ
M
0
ð
m
K
Þ
e
i
0
J
Þ
;
(31)
8
J
ð
m
K
Þ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
f
LJ
f
?
J
q
<
e
ð
M
J
ð
m
K
Þ
M
0
ð
m
K
Þ
e
i
k
J
e
i
0
J
Þ
;
(32)
9
J
ð
m
K
Þ¼
ffiffiffiffiffiffiffiffi
f
?
J
q
=
m
ð
M
J
ð
m
K
Þ
M
0
ð
m
K
Þ
e
i
?
J
e
i
0
J
Þ
:
(33)
The above terms are specific to the
B
0
decays and are
denoted with superscript ‘‘
,’’ as introduced in Eqs. (
1
)–
(
6
). The interference distributions for the
B
0
decays are
described by the same Eq. (
30
), but replacing
A
by
A
and
iJ
by
þ
iJ
and with definitions given by Eqs. (
31
)–(
33
),
replacing superscript ‘‘
’’ by ‘‘
þ
.’’
The main difference now is that the
iJ
ð
m
K
Þ
parame-
ters, as defined for
i
¼
7
;
8
;
9
, have a different dependence
on mass to those defined for
i
¼
1
–
6
in Eqs. (
9
)–(
14
). This
dependence now includes the phase of the resonance am-
plitude as a function of mass. This dependence becomes
crucial in resolving the phase ambiguities.
As can be seen from Eqs. (
8
)–(
14
), for any given set
of values
ð
k
J
;
?
J
;
k
J
;
?
J
Þ
a simple transformation
of phases, for example,
ð
2
k
J
;
?
J
;
k
J
;
?
J
Þ
, gives rise to another set of values that
satisfy the above equations in an identical manner. This
results in a fourfold ambiguity (twofold for each of
B
0
and
B
0
decays). At any given value of
m
K
the distributions,
including the interference terms in Eqs. (
32
) and (
33
), are
still invariant under the above transformations if we flip the
sign of the phase
arg
ð
M
J
ð
m
K
Þ
M
0
ð
m
K
Þ
e
i
0
J
Þ
. At a given
value of
m
K
this phase has to be determined from the data
and we cannot resolve the ambiguity. However, the mass
dependence of this phase is unique, given that the parame-
ters
0
J
are constant. Therefore, the two ambiguous solu-
tions for each
B
0
and
B
0
decay can be fully resolved from
the
m
K
dependence of the angular distributions in
Eq. (
30
).
This technique of resolving the two ambiguous solutions
in
B
!
VV
decays has been introduced in the analysis of
B
0
!
J=
c
K
0
decays [
27
] and has been used in
BABAR
’s
earlier analysis of both
B
!
’K
0
and
’K
decays
[
20
,
21
]. It is based on Wigner’s causality principle [
28
],
where the phase of a resonant amplitude increases with
increasing invariant mass; see Eq. (
15
). As a result, both
the
P
-wave and
D
-wave resonance phase shifts increase
rapidly in the vicinity of the resonance, while the corre-
sponding
S
-wave increases only gradually, as seen in
Fig.
4
.
D. Time-dependent distributions
Measurement of the time-dependent
CP
asymmetry
A
ð
t
Þ
in the decay of a neutral
B
meson to a
CP
eigen-
state, dominated by the tree-level
b
!
c
amplitude or by
the penguin
b
!
s
amplitude, such as
B
0
!ð
c
c
Þ
K
0
S
or
B
0
!ð
s
s
Þ
K
0
S
, where
ð
c
c
Þ
and
ð
s
s
Þ
are charmonium or
quarkonium states, respectively, gives an approximation
eff
to the CKM unitarity triangle angle
[
29
]. The
CP
asymmetry is defined by
A
ð
t
Þ¼
N
ð
t; B
0
tag
Þ
N
ð
t;
B
0
tag
Þ
N
ð
t; B
0
tag
Þþ
N
ð
t;
B
0
tag
Þ
¼
S
sin
ð
m
B
t
Þ
C
cos
ð
m
B
t
Þ
;
(34)
and
sin
ð
2
eff
Þ¼=
m
q
p
A
A
q
p
A
A
¼
CP
S=
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
C
2
p
;
(35)
where
N
ð
t; B
0
tag
Þ
or
N
ð
t;
B
0
tag
Þ
is the number of events
observed to decay at time
t
, in which the flavor of the
B
meson opposite to that of the decaying
B
at
t
¼
0
(re-
ferred to as the flavor ‘‘tag’’) is known to be
B
0
or
B
0
,
respectively,
CP
¼
1
is the
CP
eigenvalue of the final
state; amplitudes
A
and
A
describe the direct decays of
B
0
and
B
0
respectively to the final state; and
m
B
is the
mixing frequency due to the difference in masses between
the
B
meson eigenstates. We use a convention with
A
¼
CP
A
in the absence of
CP
violation. The above asym-
metry follows from the time evolution of each flavor:
TABLE V. Parametrization of the angular distribution in Eq. (
30
). Interference between either
J
¼
0
and
J
¼
1
,or
J
¼
0
and
J
¼
2
,
contributions in the
B
0
!
’
ð
K
Þ
J
decays is considered. The common constant is quoted for each decay mode and is omitted from
each individual function below. The three helicity angle parameters
ð
H
1
;
H
2
;
Þ
are discussed in the text.
’K
ð
892
Þ
=’
ð
K
Þ
0
’K
2
ð
1430
Þ
=’
ð
K
Þ
0
J
¼
1
J
¼
2
Common constant
3
ffiffiffi
3
p
=
4
3
ffiffiffi
5
p
=
8
f
7
J
ð
H
1
;
H
2
;
Þ
H
1
H
2
2
ð
3
H
2
1
1
Þ
H
2
2
f
8
J
ð
H
1
;
H
2
;
Þ
1
ffiffi
2
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
H
2
1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
H
2
2
q
H
2
cos
ffiffiffi
6
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
H
2
1
q
H
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
H
2
2
q
H
2
cos
f
9
J
ð
H
1
;
H
2
;
Þ
1
ffiffi
2
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
H
2
1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
H
2
2
q
H
2
sin
ffiffiffi
6
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
H
2
1
q
H
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
H
2
2
q
H
2
sin
B. AUBERT
et al.
PHYSICAL REVIEW D
78,
092008 (2008)
092008-10