Dalitz plot analysis of
B
!
decays
B. Aubert,
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
M. Battaglia,
5
D. N. Brown,
5
L. T. Kerth,
5
Yu. G. Kolomensky,
5
G. Lynch,
5
I. L. Osipenkov,
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. J. Asgeirsson,
8
B. G. Fulsom,
8
C. Hearty,
8
T. S. Mattison,
8
J. A. McKenna,
8
M. Barrett,
9
A. Khan,
9
A. Randle-Conde,
9
V. E. Blinov,
10
A. D. Bukin,
10,
*
A. R. Buzykaev,
10
V. P. Druzhinin,
10
V. B. Golubev,
10
A. P. Onuchin,
10
S. I. Serednyakov,
10
Yu. I. Skovpen,
10
E. P. Solodov,
10
K. Yu. Todyshev,
10
M. Bondioli,
11
S. Curry,
11
I. Eschrich,
11
D. Kirkby,
11
A. J. Lankford,
11
P. Lund,
11
M. Mandelkern,
11
E. C. Martin,
11
D. P. Stoker,
11
S. Abachi,
a12
C. Buchanan,
a12
H. Atmacan,
13
J. W. Gary,
13
F. Liu,
13
O. Long,
13
G. M. Vitug,
13
Z. Yasin,
13
L. Zhang,
13
V. Sharma,
14
C. Campagnari,
15
T. M. Hong,
15
D. Kovalskyi,
15
M. A. Mazur,
15
J. D. Richman,
15
T. W. Beck,
16
A. M. Eisner,
16
C. A. Heusch,
16
J. Kroseberg,
16
W. S. Lockman,
16
A. J. Martinez,
16
T. Schalk,
16
B. A. Schumm,
16
A. Seiden,
16
L. O. Winstrom,
16
C. H. Cheng,
17
D. A. Doll,
17
B. Echenard,
17
F. Fang,
17
D. G. Hitlin,
17
I. Narsky,
17
T. Piatenko,
17
F. C. Porter,
17
R. Andreassen,
18
G. Mancinelli,
18
B. T. Meadows,
18
K. Mishra,
18
M. D. Sokoloff,
18
P. C. Bloom,
19
W. T. Ford,
19
A. Gaz,
19
J. F. Hirschauer,
19
M. Nagel,
19
U. Nauenberg,
19
J. G. Smith,
19
S. R. Wagner,
19
R. Ayad,
20,
†
A. Soffer,
20,
‡
W. H. Toki,
20
R. J. Wilson,
20
E. Feltresi,
21
A. Hauke,
21
H. Jasper,
21
M. Karbach,
21
J. Merkel,
21
A. Petzold,
21
B. Spaan,
21
K. Wacker,
21
M. J. Kobel,
22
R. Nogowski,
22
K. R. Schubert,
22
R. Schwierz,
22
A. Volk,
22
D. Bernard,
23
G. R. Bonneaud,
23
E. Latour,
23
M. Verderi,
23
P. J. Clark,
24
S. Playfer,
24
J. E. Watson,
24
M. Andreotti,
25a,25b
D. Bettoni,
25a
C. Bozzi,
25a
R. Calabrese,
25a,25b
A. Cecchi,
25a,25b
G. Cibinetto,
25a,25b
P. Franchini,
25a,25b
E. Luppi,
25a,25b
M. Negrini,
25a,25b
A. Petrella,
25a,25b
L. Piemontese,
25a
V. Santoro,
25a,25b
R. Baldini-Ferroli,
26
A. Calcaterra,
26
R. de Sangro,
26
G. Finocchiaro,
26
S. Pacetti,
26
P. Patteri,
26
I. M. Peruzzi,
26,
x
M. Piccolo,
26
M. Rama,
26
A. Zallo,
26
R. Contri,
27a,27b
E. Guido,
27a
M. Lo Vetere,
27a,27b
M. R. Monge,
27a,27b
S. Passaggio,
27a
C. Patrignani,
27a,27b
E. Robutti,
27a
S. Tosi,
27a,27b
K. S. Chaisanguanthum,
28
M. Morii,
28
A. Adametz,
29
J. Marks,
29
S. Schenk,
29
U. Uwer,
29
F. U. Bernlochner,
30
V. Klose,
30
H. M. Lacker,
30
D. J. Bard,
31
P. D. Dauncey,
31
M. Tibbetts,
31
P. K. Behera,
32
X. Chai,
32
M. J. Charles,
32
U. Mallik,
32
J. Cochran,
33
H. B. Crawley,
33
L. Dong,
33
W. T. Meyer,
33
S. Prell,
33
E. I. Rosenberg,
33
A. E. Rubin,
33
Y. Y. Gao,
34
A. V. Gritsan,
34
Z. J. Guo,
34
N. Arnaud,
35
J. Be
́
quilleux,
35
A. D’Orazio,
35
M. Davier,
35
J. Firmino da Costa,
35
G. Grosdidier,
35
F. Le Diberder,
35
V. Lepeltier,
35
A. M. Lutz,
35
S. Pruvot,
35
P. Roudeau,
35
M. H. Schune,
35
J. Serrano,
35
V. Sordini,
35,
k
A. Stocchi,
35
G. Wormser,
35
D. J. Lange,
36
D. M. Wright,
36
I. Bingham,
37
J. P. Burke,
37
C. A. Chavez,
37
J. R. Fry,
37
E. Gabathuler,
37
R. Gamet,
37
D. E. Hutchcroft,
37
D. J. Payne,
37
C. Touramanis,
37
A. J. Bevan,
38
C. K. Clarke,
38
F. Di Lodovico,
38
R. Sacco,
38
M. Sigamani,
38
G. Cowan,
39
S. Paramesvaran,
39
A. C. Wren,
39
D. N. Brown,
40
C. L. Davis,
40
A. G. Denig,
41
M. Fritsch,
41
W. Gradl,
41
A. Hafner,
41
K. E. Alwyn,
42
D. Bailey,
42
R. J. Barlow,
42
G. Jackson,
42
G. D. Lafferty,
42
T. J. West,
42
J. I. Yi,
42
J. Anderson,
43
C. Chen,
43
A. Jawahery,
43
D. A. Roberts,
43
G. Simi,
43
J. M. Tuggle,
43
C. Dallapiccola,
44
E. Salvati,
44
S. Saremi,
44
R. Cowan,
45
D. Dujmic,
45
P. H. Fisher,
45
S. W. Henderson,
45
G. Sciolla,
45
M. Spitznagel,
45
R. K. Yamamoto,
45
M. Zhao,
45
P. M. Patel,
46
S. H. Robertson,
46
M. Schram,
46
A. Lazzaro,
47a,47b
V. Lombardo,
47a
F. Palombo,
47a,47b
S. Stracka,
47a
J. M. Bauer,
48
L. Cremaldi,
48
R. Godang,
48,
{
R. Kroeger,
48
D. J. Summers,
48
H. W. Zhao,
48
M. Simard,
49
P. Taras,
49
H. Nicholson,
50
G. De Nardo,
51a,51b
L. Lista,
51a
D. Monorchio,
51a,51b
G. Onorato,
51a,51b
C. Sciacca,
51a,51b
G. Raven,
52
H. L. Snoek,
52
C. P. Jessop,
53
K. J. Knoepfel,
53
J. M. LoSecco,
53
W. F. Wang,
53
L. A. Corwin,
54
K. Honscheid,
54
H. Kagan,
54
R. Kass,
54
J. P. Morris,
54
A. M. Rahimi,
54
J. J. Regensburger,
54
S. J. Sekula,
54
Q. K. Wong,
54
N. L. Blount,
55
J. Brau,
55
R. Frey,
55
O. Igonkina,
55
J. A. Kolb,
55
M. Lu,
55
R. Rahmat,
55
N. B. Sinev,
55
D. Strom,
55
J. Strube,
55
E. Torrence,
55
G. Castelli,
56a,56b
N. Gagliardi,
56a,56b
M. Margoni,
56a,56b
M. Morandin,
56a
M. Posocco,
56a
M. Rotondo,
56a
F. Simonetto,
56a,56b
R. Stroili,
56a,56b
C. Voci,
56a,56b
P. del Amo Sanchez,
57
E. Ben-Haim,
57
H. Briand,
57
J. Chauveau,
57
O. Hamon,
57
Ph. Leruste,
57
J. Ocariz,
57
A. Perez,
57
J. Prendki,
57
S. Sitt,
57
L. Gladney,
58
M. Biasini,
59a,59b
E. Manoni,
59a,59b
C. Angelini,
60a,60b
G. Batignani,
60a,60b
S. Bettarini,
60a,60b
G. Calderini,
60a,60b,
**
M. Carpinelli,
60a,60b,
††
A. Cervelli,
60a,60b
F. Forti,
60a,60b
M. A. Giorgi,
60a,60b
A. Lusiani,
60a,60c
G. Marchiori,
60a,60b
M. Morganti,
60a,60b
N. Neri,
60a,60b
E. Paoloni,
60a,60b
G. Rizzo,
60a,60b
J. J. Walsh,
60a
D. Lopes Pegna,
61
C. Lu,
61
J. Olsen,
61
A. J. S. Smith,
61
A. V. Telnov,
61
F. Anulli,
62a
E. Baracchini,
62a,62b
G. Cavoto,
62a
R. Faccini,
62a,62b
F. Ferrarotto,
62a
F. Ferroni,
62a,62b
M. Gaspero,
62a,62b
P. D. Jackson,
62a
L. Li Gioi,
62a
M. A. Mazzoni,
62a
S. Morganti,
62a
G. Piredda,
62a
F. Renga,
62a,62b
C. Voena,
62a
M. Ebert,
63
T. Hartmann,
63
H. Schro
̈
der,
63
R. Waldi,
63
T. Adye,
64
B. Franek,
64
E. O. Olaiya,
64
F. F. Wilson,
64
S. Emery,
65
L. Esteve,
65
G. Hamel de Monchenault,
65
W. Kozanecki,
65
G. Vasseur,
65
Ch. Ye
`
che,
65
M. Zito,
65
X. R. Chen,
66
H. Liu,
66
W. Park,
66
M. V. Purohit,
66
R. M. White,
66
J. R. Wilson,
66
M. T. Allen,
67
D. Aston,
67
R. Bartoldus,
67
J. F. Benitez,
67
R. Cenci,
67
J. P. Coleman,
67
M. R. Convery,
67
PHYSICAL REVIEW D
79,
072006 (2009)
1550-7998
=
2009
=
79(7)
=
072006(15)
072006-1
Ó
2009 The American Physical Society
J. C. Dingfelder,
67
J. Dorfan,
67
G. P. Dubois-Felsmann,
67
W. Dunwoodie,
67
R. C. Field,
67
A. M. Gabareen,
67
M. T. Graham,
67
P. Grenier,
67
C. Hast,
67
W. R. Innes,
67
J. Kaminski,
67
M. H. Kelsey,
67
H. Kim,
67
P. Kim,
67
M. L. Kocian,
67
D. W. G. S. Leith,
67
S. Li,
67
B. Lindquist,
67
S. Luitz,
67
V. Luth,
67
H. L. Lynch,
67
D. B. MacFarlane,
67
H. Marsiske,
67
R. Messner,
67,
*
D. R. Muller,
67
H. Neal,
67
S. Nelson,
67
C. P. O’Grady,
67
I. Ofte,
67
M. Perl,
67
B. N. Ratcliff,
67
A. Roodman,
67
A. A. Salnikov,
67
R. H. Schindler,
67
J. Schwiening,
67
A. Snyder,
67
D. Su,
67
M. K. Sullivan,
67
K. Suzuki,
67
S. K. Swain,
67
J. M. Thompson,
67
J. Va’vra,
67
A. P. Wagner,
67
M. Weaver,
67
C. A. West,
67
W. J. Wisniewski,
67
M. Wittgen,
67
D. H. Wright,
67
H. W. Wulsin,
67
A. K. Yarritu,
67
K. Yi,
67
C. C. Young,
67
V. Ziegler,
67
P. R. Burchat,
68
A. J. Edwards,
68
T. S. Miyashita,
68
S. Ahmed,
69
M. S. Alam,
69
J. A. Ernst,
69
B. Pan,
69
M. A. Saeed,
69
S. B. Zain,
69
S. M. Spanier,
70
B. J. Wogsland,
70
R. Eckmann,
71
J. L. Ritchie,
71
A. M. Ruland,
71
C. J. Schilling,
71
R. F. Schwitters,
71
B. W. Drummond,
72
J. M. Izen,
72
X. C. Lou,
72
F. Bianchi,
73a,73B
D. Gamba,
73a,73B
M. Pelliccioni,
73a,73B
M. Bomben,
74A,74b
L. Bosisio,
74A,74b
C. Cartaro,
74A,74b
G. Della Ricca,
74A,74b
L. Lanceri,
74A,74b
L. Vitale,
74A,74b
V. Azzolini,
75
N. Lopez-March,
75
F. Martinez-Vidal,
75
D. A. Milanes,
75
A. Oyanguren,
75
J. Albert,
76
Sw. Banerjee,
76
B. Bhuyan,
76
H. H. F. Choi,
76
K. Hamano,
76
G. J. King,
76
R. Kowalewski,
76
M. J. Lewczuk,
76
I. M. Nugent,
76
J. M. Roney,
76
R. J. Sobie,
76
J. J. Back,
77
T. J. Gershon,
77
P. F. Harrison,
77
J. Ilic,
77
T. E. Latham,
77
G. B. Mohanty,
77
E. M. T. Puccio,
77
H. R. Band,
78
X. Chen,
78
S. Dasu,
78
K. T. Flood,
78
Y. Pan,
78
R. Prepost,
78
C. O. Vuosalo,
78
and S. L. Wu
78
(
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
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 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
a12
University of California at Los Angeles, Los Angeles, California 90024, USA
13
University of California at Riverside, Riverside, California 92521, USA
14
University of California at San Diego, La Jolla, California 92093, USA
15
University of California at Santa Barbara, Santa Barbara, California 93106, USA
16
University of California at Santa Cruz, Institute for Particle Physics, Santa Cruz, California 95064, USA
17
California Institute of Technology, Pasadena, California 91125, USA
18
University of Cincinnati, Cincinnati, Ohio 45221, USA
19
University of Colorado, Boulder, Colorado 80309, USA
20
Colorado State University, Fort Collins, Colorado 80523, USA
21
Technische Universita
̈
t Dortmund, Fakulta
̈
t Physik, D-44221 Dortmund, Germany
22
Technische Universita
̈
t Dresden, Institut fu
̈
r Kern-und Teilchenphysik, D-01062 Dresden, Germany
23
Laboratoire Leprince-Ringuet, CNRS/IN2P3, Ecole Polytechnique, F-91128 Palaiseau, France
24
University of Edinburgh, Edinburgh EH9 3JZ, United Kingdom
25a
INFN Sezione di Ferrara, I-44100 Ferrara, Italy;
25b
Dipartimento di Fisica, Universita
`
di Ferrara, I-44100 Ferrara, Italy
26
INFN Laboratori Nazionali di Frascati, I-00044 Frascati, Italy
27a
INFN Sezione di Genova, I-16146 Genova, Italy;
27b
Dipartimento di Fisica, Universita
`
di Genova, I-16146 Genova, Italy
28
Harvard University, Cambridge, Massachusetts 02138, USA
29
Universita
̈
t Heidelberg, Physikalisches Institut, Philosophenweg 12, D-69120 Heidelberg, Germany
30
Humboldt-Universita
̈
t zu Berlin, Institut fu
̈
r Physik, Newtonstr. 15, D-12489 Berlin, Germany
31
Imperial College London, London, SW7 2AZ, United Kingdom
32
University of Iowa, Iowa City, Iowa 52242, USA
33
Iowa State University, Ames, Iowa 50011-3160, USA
34
Johns Hopkins University, Baltimore, Maryland 21218, USA
B. AUBERT
et al.
PHYSICAL REVIEW D
79,
072006 (2009)
072006-2
35
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
36
Lawrence Livermore National Laboratory, Livermore, California 94550, USA
37
University of Liverpool, Liverpool L69 7ZE, United Kingdom
38
Queen Mary, University of London, London, E1 4NS, United Kingdom
39
University of London, Royal Holloway and Bedford New College, Egham, Surrey TW20 0EX, United Kingdom
40
University of Louisville, Louisville, Kentucky 40292, USA
41
Johannes Gutenberg-Universita
̈
t Mainz, Institut fu
̈
r Kernphysik, D-55099 Mainz, Germany
42
University of Manchester, Manchester M13 9PL, United Kingdom
43
University of Maryland, College Park, Maryland 20742, USA
44
University of Massachusetts, Amherst, Massachusetts 01003, USA
45
Massachusetts Institute of Technology, Laboratory for Nuclear Science, Cambridge, Massachusetts 02139, USA
46
McGill University, Montre
́
al, Que
́
bec, Canada H3A 2T8
47a
INFN Sezione di Milano, I-20133 Milano, Italy;
47b
Dipartimento di Fisica, Universita
`
di Milano, I-20133 Milano, Italy
48
University of Mississippi, University, Mississippi 38677, USA
49
Universite
́
de Montre
́
al, Physique des Particules, Montre
́
al, Que
́
bec, Canada H3C 3J7
50
Mount Holyoke College, South Hadley, Massachusetts 01075, USA
51a
INFN Sezione di Napoli, I-80126 Napoli, Italy;
51b
Dipartimento di Scienze Fisiche, Universita
`
di Napoli Federico II, I-80126 Napoli, Italy
52
NIKHEF, National Institute for Nuclear Physics and High Energy Physics, NL-1009 DB Amsterdam, The Netherlands
53
University of Notre Dame, Notre Dame, Indiana 46556, USA
54
Ohio State University, Columbus, Ohio 43210, USA
55
University of Oregon, Eugene, Oregon 97403, USA
56a
INFN Sezione di Padova, I-35131 Padova, Italy;
56b
Dipartimento di Fisica, Universita
`
di Padova, I-35131 Padova, Italy
57
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
58
University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
59a
INFN Sezione di Perugia, I-06100 Perugia, Italy;
59b
Dipartimento di Fisica, Universita
`
di Perugia, I-06100 Perugia, Italy
60a
INFN Sezione di Pisa, I-56127 Pisa, Italy;
60b
Dipartimento di Fisica, Universita
`
di Pisa, I-56127 Pisa, Italy;
60c
Scuola Normale Superiore di Pisa, I-56127 Pisa, Italy
61
Princeton University, Princeton, New Jersey 08544, USA
62a
INFN Sezione di Roma, I-00185 Roma, Italy;
62b
Dipartimento di Fisica, Universita
`
di Roma La Sapienza, I-00185 Roma, Italy
63
Universita
̈
t Rostock, D-18051 Rostock, Germany
64
Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, OX11 0QX, United Kingdom
65
CEA, Irfu, SPP, Centre de Saclay, F-91191 Gif-sur-Yvette, France
66
University of South Carolina, Columbia, South Carolina 29208, USA
67
SLAC National Accelerator Laboratory, Stanford, California 94309, USA
68
Stanford University, Stanford, California 94305-4060, USA
69
State University of New York, Albany, New York 12222, USA
70
University of Tennessee, Knoxville, Tennessee 37996, USA
71
University of Texas at Austin, Austin, Texas 78712, USA
72
University of Texas at Dallas, Richardson, Texas 75083, USA
73a
INFN Sezione di Torino, I-10125 Torino, Italy;
73B
Dipartimento di Fisica Sperimentale, Universita
`
di Torino, I-10125 Torino, Italy
74A
INFN Sezione di Trieste, I-34127 Trieste, Italy;
††
Also with Universita
`
di Sassari, Sassari, Italy.
**
Also with 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.
{
Now at University of South Alabama, Mobile, AL 36688, USA.
k
Also with Universita
`
di Roma La Sapienza, I-00185 Roma, Italy.
x
Also with Universita
`
di Perugia, Dipartimento di Fisica, Perugia, Italy.
‡
Now at Tel Aviv University, Tel Aviv, 69978, Israel.
†
Now at Temple University, Philadelphia, PA 19122, USA.
*
Deceased.
DALITZ PLOT ANALYSIS OF
...
PHYSICAL REVIEW D
79,
072006 (2009)
072006-3
74b
Dipartimento di Fisica, Universita
`
di Trieste, I-34127 Trieste, Italy
75
IFIC, Universitat de Valencia-CSIC, E-46071 Valencia, Spain
76
University of Victoria, Victoria, British Columbia, Canada V8W 3P6
77
Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom
78
University of Wisconsin, Madison, Wisconsin 53706, USA
(Received 13 February 2009; published 22 April 2009)
We present a Dalitz plot analysis of charmless
B
decays to the final state
using a sample of
ð
465
5
Þ
10
6
B
B
pairs collected by the
BABAR
experiment at
ffiffiffi
s
p
¼
10
:
58 GeV
. We measure the
branching fractions
B
ð
B
!
Þ¼ð
15
:
2
0
:
6
1
:
2
0
:
4
Þ
10
6
,
B
ð
B
!
0
ð
770
Þ
Þ¼
ð
8
:
1
0
:
7
1
:
2
þ
0
:
4
1
:
1
Þ
10
6
,
B
ð
B
!
f
2
ð
1270
Þ
Þ¼ð
1
:
57
0
:
42
0
:
16
þ
0
:
53
0
:
19
Þ
10
6
, and
B
ð
B
!
nonresonant
Þ¼ð
5
:
3
0
:
7
0
:
6
þ
1
:
1
0
:
5
Þ
10
6
, where the uncertainties are statistical, system-
atic, and model-dependent, respectively. Measurements of branching fractions for the modes
B
!
0
ð
1450
Þ
and
B
!
f
0
ð
1370
Þ
are also presented. We observe no significant direct
CP
asymmetries
for the above modes, and there is no evidence for the decays
B
!
f
0
ð
980
Þ
,
B
!
c
0
,or
B
!
c
2
.
DOI:
10.1103/PhysRevD.79.072006
PACS numbers: 13.25.Hw, 12.15.Hh, 11.30.Er
I. INTRODUCTION
Decays of
B
mesons to three-body charmless final states
probe the properties of the weak interaction through their
dependence on the complex quark couplings described in
the Cabibbo-Kobayashi-Maskawa (CKM) matrix [
1
,
2
].
Furthermore, these decays test dynamical models for had-
ronic
B
decays.
One can measure direct
CP
asymmetries and constrain
magnitudes and phases of the CKM matrix elements using
individual channels that appear as intermediate resonances
in the
B
!
decay. For example, the CKM
angle
could be extracted from the interference between
the decay
B
!
c
0
, which has no
CP
-violating phase
(in the standard parametrization), and other modes such as
B
!
0
ð
770
Þ
[
3
–
8
].
Studies of
B
!
can also be useful for a
precise measurement of the CKM angle
. A theoretically
clean determination of this angle can be obtained from the
decay-time dependence of the interference between
B
0
!
þ
,
B
0
!
þ
, and
B
0
!
0
0
via the analysis of
the Dalitz plot for
B
0
!
þ
0
decays [
9
] (recently
implemented by
BABAR
[
10
] and Belle [
11
,
12
]). Charged
B
decays offer a large statistics sample with which to
determine additional resonant or nonresonant contributions
to the three-pion Dalitz plot that can affect the measure-
ment of
. For example, the Dalitz plot analysis of
B
!
allows one to check for effects from
B
!
!
ð
782
Þ
, that could cause large direct
CP
violation
due to
!
mixing [
13
]. It is particularly important to
limit the possible effects of broad scalar structures [includ-
ing the so-called
f
0
ð
600
Þ
or
] and nonresonant contribu-
tions [
14
–
19
].
Furthermore, a number of unexplained structures have
been observed in charmless
B
decays to
K
[
20
–
25
],
KK
[
26
,
27
], and
KKK
[
23
,
28
,
29
] final states. Verifying
the presence of these structures in
B
!
decays
would help to determine their nature and involvement in
hadronic
B
decays.
In this paper we present an amplitude analysis of
B
!
decays based on a
424 fb
1
data sample con-
taining
ð
465
5
Þ
10
6
B
B
pairs (
N
B
B
). The data were
collected with the
BABAR
detector [
30
] at the PEP-II
asymmetric-energy
e
þ
e
storage rings [
31
] operating at
the
ð
4
S
Þ
resonance with center-of-mass (CM) energy of
ffiffiffi
s
p
¼
10
:
58 GeV
. An additional total integrated luminos-
ity of
44 fb
1
was recorded 40 MeV below the
ð
4
S
Þ
resonance (‘‘off-peak’’ data) and was used to study back-
grounds. Compared to our previous publication [
32
], in
addition to doubling the data sample we have included
several improvements in reconstruction algorithms that
enhance the signal efficiency, made numerous modifica-
tions to the analysis to increase the sensitivity to direct
CP
violation effects (for example, by including more discrimi-
nating variables in the maximum likelihood fit), and im-
proved our model of the Dalitz plot structure.
The remainder of the paper is organized as follows:
Sec.
II
describes the amplitude analysis formalism,
Secs.
III
and
IV
give details about the selection of signal
B
decays and how backgrounds are considered, Sec.
V
presents the results from the likelihood fit, Sec.
VI
gives
an account of the various sources of systematic uncertain-
ties, while Sec.
VII
summarizes the results.
II. AMPLITUDE ANALYSIS FORMALISM
A number of intermediate states contribute to the decay
B
!
. We determine their contributions with a
maximum likelihood fit to the distribution of events in the
Dalitz plot. This procedure has been described in detail in
our previous publications [
20
,
21
,
32
].
The
B
!
decay contains two same-sign
pions in the final state. We distinguish these particles
according to the invariant mass they make when combined
with the oppositely charged pion, and draw the Dalitz plot
B. AUBERT
et al.
PHYSICAL REVIEW D
79,
072006 (2009)
072006-4
in terms of heavy and light invariant masses-squared of the
systems (denoted
m
2
max
and
m
2
min
, respectively), so
that each candidate has a uniquely defined position.
Moreover, we explicitly enforce the symmetrization of
the total amplitude under exchange of identical bosons.
The total signal amplitudes for
B
þ
and
B
decays are
given by
A
A
ð
m
2
max
;m
2
min
Þ¼
X
j
c
j
F
j
ð
m
2
max
;m
2
min
Þ
;
A
A
ð
m
2
max
;m
2
min
Þ¼
X
j
c
j
F
j
ð
m
2
max
;m
2
min
Þ
:
(1)
The complex coefficients
c
j
and
c
j
for a given decay mode
j
contain all the weak phase dependence. Since the
F
j
terms contain only strong dynamics,
F
j
F
j
. We use the
following parametrization [
21
] for the amplitude coeffi-
cients:
c
j
¼ð
x
j
þ
x
j
Þþ
i
ð
y
j
þ
y
j
Þ
c
j
¼ð
x
j
x
j
Þþ
i
ð
y
j
y
j
Þ
:
(2)
In this approach,
x
j
and
y
j
(
x
j
and
y
j
) are the
CP
-conserving (-violating) components of the decay
amplitude.
The
F
j
distributions describe the dynamics of the decay
amplitudes and are written as the product of an invariant-
mass term
R
j
, two Blatt-Weisskopf barrier form factors
X
J
,
and an angular function
T
j
F
j
ð
m
2
max
;m
2
min
Þ
R
j
ð
m
Þ
X
J
ð
p
?
Þ
X
J
ð
q
Þ
T
j
ð
m
Þ
;
(3)
where
m
(
J
) is the mass (spin) of the resonance,
p
?
is the
momentum of the bachelor pion that is not part of the
resonance in the
B
meson rest frame, and
q
is the momen-
tum of either daughter in the rest frame of the resonance
(we use the
c
¼
1
convention for all equations in this
paper). The
F
j
are normalized over the entire Dalitz plot:
Z
Z
j
F
j
ð
m
2
max
;m
2
min
Þj
2
dm
2
max
dm
2
min
¼
1
:
(4)
The Blatt-Weisskopf barrier form factors [
33
] are given
by
X
J
¼
0
ð
z
Þ¼
1
;X
J
¼
1
ð
z
Þ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
=
½
1
þð
zr
BW
Þ
2
q
;
X
J
¼
2
ð
z
Þ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
=
½ð
zr
BW
Þ
4
þ
3
ð
zr
BW
Þ
2
þ
9
q
;
(5)
where the meson radius parameter
r
BW
is taken to be
4
:
0
1
:
0
ð
GeV
=c
Þ
1
[
34
].
For most resonances in this analysis the
R
j
are taken to
be relativistic Breit-Wigner line shapes
R
j
ð
m
Þ¼
1
ð
m
2
0
m
2
Þ
im
0
ð
m
Þ
;
(6)
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-
J
resonance, the latter can be expressed as
ð
m
Þ¼
0
q
q
0
2
J
þ
1
m
0
m
X
2
J
ð
q
Þ
X
2
J
ð
q
0
Þ
:
(7)
The symbol
0
denotes the nominal width of the reso-
nance. The values of
m
0
and
0
are obtained from standard
tables [
34
] when they are well known. The symbol
q
0
denotes the value of
q
when
m
¼
m
0
.
The angular distribution terms
T
j
in Eq. (
3
) follow the
Zemach tensor formalism [
35
,
36
]. For the decay of a spin
zero
B
meson into a spin
J
resonance and a spin zero
bachelor particle this gives [
37
]
T
J
¼
0
j
¼
1
;T
J
¼
1
j
¼
2
~
p
~
q;
T
J
¼
2
j
¼
4
3
½
3
ð
~
p
~
q
Þ
2
ðj
~
p
jj
~
q
jÞ
2
;
(8)
where
~
p
is the momentum of the bachelor particle and
~
q
is
the momentum of the resonance daughter with charge
opposite from that of the bachelor particle, both measured
in the rest frame of the resonance.
The Gounaris-Sakurai parametrization [
38
] of the
P
-wave scattering amplitude for a broad resonance decay-
ing to two pions is used for the
0
ð
770
Þ
and
0
ð
1450
Þ
line
shapes
R
j
ð
m
Þ¼
1
þ
0
d=m
0
ð
m
2
0
m
2
Þþ
f
ð
m
Þ
im
0
ð
m
Þ
;
(9)
where
f
ð
m
Þ¼
0
m
2
0
q
3
0
q
2
½
h
ð
m
Þ
h
ð
m
0
Þ
þð
m
2
0
m
2
Þ
q
2
0
dh
dm
m
0
;
(10)
and the function
h
ð
m
Þ
is defined as
h
ð
m
Þ¼
2
q
m
ln
m
þ
2
q
2
m
;
(11)
with
dh
dm
m
0
¼
h
ð
m
0
Þ½ð
8
q
2
0
Þ
1
ð
2
m
2
0
Þ
1
þð
2
m
2
0
Þ
1
:
(12)
The normalization condition at
R
j
ð
0
Þ
fixes the parameter
d
¼
f
ð
0
Þ
=
ð
0
m
0
Þ
. It is found to be
d
¼
3
m
2
q
2
0
ln
m
0
þ
2
q
0
2
m
þ
m
0
2
q
0
m
2
m
0
q
3
0
:
(13)
We model the nonresonant component using an empiri-
cal function that has been found to accurately describe
nonresonant contributions in other charmless three-body
B
decays [
23
–
25
,
28
,
29
]:
A
nr
¼
c
nr
ð
e
nr
m
2
max
þ
e
nr
m
2
min
Þ
:
(14)
DALITZ PLOT ANALYSIS OF
...
PHYSICAL REVIEW D
79,
072006 (2009)
072006-5
We include this term in the coherent sum given by Eq. (
1
)
when calculating the total signal amplitude over the Dalitz
plot.
To allow comparison among experiments we present
results also in terms of fit fractions (
FF
j
), defined as the
integral of a single decay amplitude squared divided by the
total matrix element squared for the complete Dalitz plot
FF
j
¼
R
R
ðj
c
j
F
j
j
2
þj
c
j
F
j
j
2
Þ
dm
2
max
dm
2
min
R
R
ðj
A
j
2
þj
A
j
2
Þ
dm
2
max
dm
2
min
:
(15)
Note that the sum of all the fit fractions is not necessarily
unity due to the possible presence of constructive or de-
structive interference. The
CP
asymmetry for each con-
tributing resonance is determined from the fitted
parameters
A
CP;j
¼
j
c
j
j
2
j
c
j
j
2
j
c
j
j
2
þj
c
j
j
2
¼
2
ð
x
j
x
j
þ
y
j
y
j
Þ
ð
x
j
Þ
2
þð
x
j
Þ
2
þð
y
j
Þ
2
þð
y
j
Þ
2
:
(16)
The signal Dalitz plot probability density function
(PDF) is formed from the total amplitude as follows:
P
sig
ð
m
2
max
;m
2
min
;q
B
Þ¼
1
þ
q
B
2
j
A
j
2
"
þ
1
q
B
2
j
A
j
2
"
R
R
ðj
A
j
2
"
þj
A
j
2
"
Þ
dm
2
max
dm
2
min
;
(17)
where
q
B
is the charge of the
B
-meson candidate, and
"
"
ð
m
2
max
;m
2
min
Þ
and
"
"
ð
m
2
max
;m
2
min
Þ
are the signal recon-
struction efficiencies for
B
þ
and
B
events, respectively,
defined for all points in the Dalitz plot.
III. CANDIDATE SELECTION
We reconstruct
B
candidates from events that have four
or more charged tracks. Each track is required to be well
measured and to originate from the beam spot. They must
have a minimum transverse momentum of
50 MeV
=c
, and
a distance of closest approach to the beam spot of less than
1.5 cm in the transverse plane and less than 2.5 cm along
the detector axis.
B
candidates are formed from combina-
tions of three charged tracks, and particle identification
(PID) criteria are applied to reject electrons and to separate
pions from kaons. In our final state, the average selection
efficiency for pions that have passed the tracking and PID
requirements is about 93% including geometrical accep-
tance, while the average misidentification probability of
kaons as pions is close to 8%.
Two kinematic variables are used to identify signal
B
decays. The first variable is
E
¼
E
B
ffiffiffi
s
p
=
2
;
(18)
the difference between the reconstructed CM energy of the
B
-meson candidate (
E
B
) and
ffiffiffi
s
p
=
2
, where
ffiffiffi
s
p
is the total
CM energy. The second is the beam-energy-substituted
mass
m
ES
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
s=
4
j
~
p
B
j
2
q
;
(19)
where
~
p
B
is the
B
momentum measured in the CM frame.
The
m
ES
distribution for signal events peaks near the
B
mass with a resolution of around
2
:
5 MeV
=c
2
, while the
E
distribution peaks at zero with a resolution of approxi-
mately 20 MeV. We initially require events to lie in the
region formed by the following selection criteria:
5
:
200
<
m
ES
<
5
:
286 GeV
=c
2
and
0
:
075
<
E<
0
:
300 GeV
.
The region of
E
below
0
:
075 GeV
is heavily contami-
nated by four-body
B
-decay backgrounds and is not useful
for studying the continuum background. The selected
region is then subdivided into three areas: the ‘‘left side-
band’’ (
5
:
20
<m
ES
<
5
:
26 GeV
=c
2
and
j
E
j
<
0
:
075 GeV
) used to study the background
E
and Dalitz
plot distributions; the ‘‘upper sideband’’ (
5
:
230
<m
ES
<
5
:
286 GeV
=c
2
and
0
:
1
<
E<
0
:
3 GeV
) used to study
the background
m
ES
distributions; and the ‘‘signal region’’
(
5
:
272
<m
ES
<
5
:
286 GeV
=c
2
and
j
E
j
<
0
:
075 GeV
)
with which the final fit to data is performed. Following
the calculation of these kinematic variables, each of the
B
candidates is refitted with its mass constrained to the
world-average value of the
B
meson mass [
34
] in order
to improve the Dalitz plot position resolution and to make
sure all events lie within the kinematic boundary of the
Dalitz plot.
The dominant source of background comes from light-
quark and charm continuum production (
e
þ
e
!
q
q
,
where
q
¼
u
,
d
,
s
,
c
). This background is suppressed by
requirements on event-shape variables calculated in the
CM frame. We compute a neural network (NN) from the
following five variables: the ratio of the second- and
zeroth-order angular moments (
L
2
=L
0
), with
L
j
¼
P
i
p
i
j
cos
i
j
j
, where
i
is the angle of the track or neutral
cluster
i
with respect to the signal
B
thrust axis,
p
i
is its
momentum, and the sum excludes the daughters of the
B
candidate; the absolute value of the cosine of the angle
between the direction of the
B
and the detector axis; the
magnitude of the cosine of the angle between the signal
B
thrust axis and the detector axis; the output of a multi-
variate
B
-flavor tagging algorithm [
39
] multiplied by the
charge of the
B
candidate; and the ratio of the measured
proper time difference of the two
B
decay vertices and its
statistical uncertainty. We train the NN using samples of
off-peak data and signal Monte Carlo (MC) events gener-
ated with the phase-space distribution. A selection require-
ment is imposed on the NN output that accepts about 48%
of signal events while rejecting 97% of continuum back-
ground events.
Dalitz plot distributions of the reconstruction efficiency
for
B
þ
and
B
events are modeled with two-dimensional
histograms formed from a sample of around
7
10
6
B
!
phase-space MC events. All selection criteria
are applied except for the exclusion of certain invariant-
B. AUBERT
et al.
PHYSICAL REVIEW D
79,
072006 (2009)
072006-6
mass regions described below. We take the ratio of two
histograms, the denominator containing the true Dalitz plot
distribution of all generated MC events and the numerator
containing the reconstructed MC events. The reconstructed
events are weighted in order to correct for differences
between data and MC simulations in the tracking and
PID efficiencies. In order to give better resolution near
the edges of the Dalitz plot, where most reconstructed
events lie, the histograms are formed in the ‘‘square
Dalitz plot’’ [
10
,
32
] coordinates. We use
50
50
bins
and smooth these histograms by applying linear interpola-
tion between neighboring bins. The efficiency shows very
little variation across most of the Dalitz plot but decreases
towards the corners where one of the particles has low
momentum. The effect of experimental resolution on the
signal model is neglected since the resonances under con-
sideration are sufficiently broad. The average reconstruc-
tion efficiency for events in the signal region for the phase-
space MC sample is about 15%. The fraction of misrecon-
structed signal events is only 5%, and MC studies indicate
that there is no need for any explicit treatment of these
events.
IV. BACKGROUNDS
In addition to the continuum (
q
q
) background we also
have backgrounds from
B
B
events. There are four main
sources: (i) combinatorial background from three unrelated
tracks; (ii) three- and four-body
B
decays involving an
intermediate
D
meson; (iii) charmless two- and four-
body decays with an extra or missing particle; and
(iv) three-body decays with one or more particles misiden-
tified. We reject background from two-body decays of
D
mesons and charmonium states by excluding invariant
masses (in units of
GeV
=c
2
) in the ranges:
1
:
660
<
m
þ
<
1
:
920
,
3
:
051
<m
þ
<
3
:
222
, and
3
:
660
<
m
þ
<
3
:
820
. These ranges reject decays from
D
0
!
K
þ
(or
þ
),
J=
c
!
‘
þ
‘
, and
c
ð
2
S
Þ!
‘
þ
‘
respectively, where
‘
is a lepton that has been misidentified
as a pion. We also employ a special requirement to reject
the decay process
B
!
K
0
S
;
K
0
S
!
þ
, by exclud-
ing candidates where the vertexed mass of two oppositely
charged pions lies in the range of
½
478
;
516
MeV
=c
2
.
We use a large sample of MC-simulated
B
B
decays,
equivalent to approximately 3 times the integrated lumi-
nosity of the data sample, to identify the important
B
backgrounds that survive the invariant-mass exclusion re-
quirements described above. In total, 53
B
-meson decay
modes are identified for which larger samples of exclusive
MC events are used for further study. We combine modes
that have similar behavior in the discriminating variables
m
ES
and
E
into a
B
-background category. There are four
such categories: the first contains the two-body decays
B
0
!
þ
and
B
0
!
K
þ
, the second is dominated
by
B
!
K
and contains other decays with similar
topologies, the third contains only
B
0
!
þ
0
, and the
fourth contains the remaining backgrounds from
B
decays
that are combinatorial in nature. For each
B
-background
category the combined
m
ES
,
E
, and Dalitz plot distribu-
tions are created where the relative contributions of various
decay modes in a specific category are calculated from the
reconstruction efficiencies from MC simulations and the
branching fractions listed by the Particle Data Group [
34
]
and the Heavy Flavor Averaging Group [
40
]. These distri-
butions are used in the likelihood fit described below.
Background Dalitz plot distributions are included in the
likelihood fit through the use of two-dimensional histo-
grams. For backgrounds from
B
decays these histograms
are formed from the various MC samples. For the contin-
uum background the left sideband data sample is used.
Since this data sideband also contains events from
B
decays, MC samples are used to subtract these events. To
these
B
-subtracted sideband events, we add off-peak data
events from across the whole range of
m
ES
and
E
in order
to enhance statistics. We have verified that the shapes of
various discriminating variables are compatible between
the sideband and off-peak events. As for the reconstruction
efficiency histograms, the background Dalitz plot distribu-
tions are formed in the square Dalitz plot coordinates and
are smoothed by linear interpolation applied between
neighboring bins. Separate histograms are constructed for
B
þ
and
B
events. The
q
q
- and
B
-background PDFs are
identical in their construction, and the
q
q
PDF is shown
here as an example:
P
q
q
ð
m
2
max
;m
2
min
;q
B
Þ
¼
1
2
ð
1
q
B
A
q
q
Þ
1
þ
q
B
2
Q
ð
m
2
max
;m
2
min
Þ
R
R
Q
ð
m
2
max
;m
2
min
Þ
dm
2
max
dm
2
min
þ
1
q
B
2
Q
ð
m
2
max
;m
2
min
Þ
R
R
Q
ð
m
2
max
;m
2
min
Þ
dm
2
max
dm
2
min
;
(20)
where
A
q
q
is the charge asymmetry in the background,
and
Q
ð
m
2
max
;m
2
min
Þ
and
Q
ð
m
2
max
;m
2
min
Þ
are the Dalitz plot
distributions of
q
q
events in selected
B
þ
and
B
samples,
respectively.
V. MAXIMUM LIKELIHOOD FIT
To provide further discrimination between the signal and
background hypotheses in the likelihood fit, we include
PDFs for the kinematic variables
m
ES
and
E
, which
multiply that of the Dalitz plot. The signal
m
ES
shape is
modeled with the sum of a Gaussian function and a
Crystal-Ball line shape [
41
], and the
E
shape is modeled
with a double Gaussian function. The parameters of these
functions are obtained from a sample of
B
!
MC events, modeled according to the Dalitz plot distribu-
tion from Ref. [
32
], and are appropriately adjusted to
account for possible differences between data and
MC simulations determined with a control sample of
DALITZ PLOT ANALYSIS OF
...
PHYSICAL REVIEW D
79,
072006 (2009)
072006-7
B
þ
!
D
0
þ
;
D
0
!
K
þ
decays. These parameters are
fixed in the fit to data.
The
q
qm
ES
distribution is modeled with the experimen-
tally motivated ARGUS function [
42
]. The end point for
the ARGUS function is fixed to
5
:
289 GeV
=c
2
, and the
parameter describing the shape is fixed to the value deter-
mined from the combined sample of upper sideband and
off-peak data. We model the continuum
E
shape using a
linear function, the slope of which is fixed to the value
determined from the left sideband and off-peak data. The
B
B
background distributions are modeled with histograms
obtained from the mixture of
B
B
MC samples. The yields
of signal and
q
q
events are allowed to vary in the final fit to
the data while the yields of
B
B
backgrounds are fixed to 11
(two-body decays), 195 (
B
!
K
type), 117
(
B
0
!
þ
0
), and 495 (combinatorial) events.
The complete likelihood function is given by
L
¼
e
N
Y
N
e
j
X
k
N
k
P
j
k
ð
m
2
max
;m
2
min
;m
ES
;
E; q
B
Þ
;
(21)
where
N
is equal to
P
k
N
k
,
N
k
is the yield for the event
category
k
,
N
e
is the total number of events in the data
sample, and
P
j
k
is the PDF for the category
k
for event
j
,
which consists of a product of the Dalitz plot,
m
ES
, and
E
PDFs. The function
2ln
L
is minimized in an unbinned
fit to the data.
Our nominal signal Dalitz plot model comprises a
momentum-dependent nonresonant component and four
intermediate resonance states:
0
ð
770
Þ
,
0
ð
1450
Þ
,
f
2
ð
1270
Þ
, and
f
0
ð
1370
Þ
. The parameters used to
describe these states are summarized in Table
I
.Wefit
4335
B
candidates in the signal region selected from the
data to obtain the central values of the
x
j
,
x
j
,
y
j
, and
y
j
parameters for each component, and use Eqs. (
15
) and (
16
)
to calculate the fit fractions and
CP
asymmetries. We use
0
ð
770
Þ
as the reference amplitude, fixing its
x
,
y
, and
y
parameters to unity, zero, and zero, respectively. The
signal yield,
q
q
background yield, and
q
q
background
asymmetry are also free parameters of the fit, giving a total
of 20 free parameters.
The Dalitz plot model was determined using the results
of our previous analysis [
32
] and the changes in the fit
likelihood and
2
values when omitting or adding reso-
nances. The latter is calculated from the projection of the
fit results onto the Dalitz plot using the formula
2
¼
X
n
b
i
¼
1
½
y
i
f
ð
x
i
Þ
2
f
ð
x
i
Þ
;
(22)
where
y
i
is the number of data events in bin
i
and
f
ð
x
i
Þ
is
the number of events in that bin as predicted by the fit
result. The number of degrees of freedom is calculated as
n
b
h
1
, where
n
b
is the total number of bins used and
h
is the number of free parameters in the fit. A minimum of
20 entries in each bin is required; if this requirement is not
met then the neighboring bins are combined. Typically,
n
b
takes values around 100.
In our previous study we found significant contributions
from
0
ð
770
Þ
and
f
2
ð
1270
Þ
; with
f
0
ð
980
Þ
,
0
ð
1450
Þ
, and a uniform nonresonant term also in-
cluded in the model. Because of the larger data sample
and many improvements to the analysis, we find it neces-
sary to include an additional contribution from
f
0
ð
1370
Þ
, and to use a momentum-dependent nonreso-
nant amplitude [see Eq. (
14
)] in order to achieve a reason-
able agreement of the fit with the data. We do not find any
significant signal from
f
0
ð
980
Þ
, so we exclude this
channel from our nominal model and calculate an upper
limit for its fit fraction. The statistical significance of the
presence of a component is estimated by evaluating the
difference
ln
L
between the negative log-likelihood of
the nominal fit and that of a fit where all of the
x
,
y
,
x
, and
y
parameters for the given component are fixed to zero.
This is then used to evaluate a
p
value
p
¼
Z
1
2 ln
L
f
ð
z
;
n
d
Þ
dz;
(23)
where
f
ð
z
;
n
d
Þ
is the PDF of the
2
distribution and
n
d
is
the number of degrees of freedom, four in this case. We
then determine the equivalent one-dimensional signifi-
cance from this
p
value. We find that the
f
2
ð
1270
Þ
con-
tribution has a statistical significance of
6
:
1
, the
0
ð
1450
Þ
4
:
6
, and the
f
0
ð
1370
Þ
3
:
9
.
Since the mass and width of the
f
0
ð
1370
Þ
state are not
well known [
34
], we determine the preferred values
from data by scanning the likelihood values obtained
with different parameters. The mass and width are deter-
mined to be
m
f
0
ð
1370
Þ
¼
1400
40 MeV
=c
2
and
f
0
ð
1370
Þ
¼
300
80 MeV
, with a correlation of
ð
39
4
Þ
%
, where the errors are statistical only, and are obtained
from a fit to the two-dimensional likelihood profile.
Similarly, we determine the parameter of the nonresonant
line shape to be
nr
¼
0
:
28
0
:
06 GeV
2
c
4
(statistical
uncertainties only).
Possible contributions from
c
0
and
c
2
are not
significant so we set upper limits on their branching frac-
tions. Furthermore, we do not find any evidence for a very
TABLE I. Parameters used to describe intermediate states in
our nominal model. GS and RBW refer to the Gounaris-Sakurai
and relativistic Breit-Wigner line shapes, respectively.
Resonance
Line
shape
Mass
(
MeV
=c
2
)
Width
(MeV)
Ref.
0
ð
770
Þ
GS
775
:
49
0
:
34 149
:
4
1
:
0
[
34
]
0
ð
1450
Þ
GS
1465
25
400
60
[
34
]
f
2
ð
1270
Þ
RBW
1275
:
1
1
:
2
185
:
0
þ
2
:
9
2
:
4
[
34
]
f
0
ð
1370
Þ
RBW
1400
40
300
80
See text
B. AUBERT
et al.
PHYSICAL REVIEW D
79,
072006 (2009)
072006-8
broad enhancement at low
þ
invariant mass such as
could be caused by the decay
B
!
.
Figure
1
shows the
m
ES
and
E
distributions of signal
and
q
q
background determined from the fit with event-by-
event signal and
q
q
background probabilities for each
candidate event [
43
]. The background-subtracted Dalitz
plot of the data in the signal region can be seen in Fig.
2
.
The
2
per number of degrees of freedom of the projection
of the fit result onto the Dalitz plot is
82
=
84
. Using the
fitted signal distribution we calculate the average recon-
struction efficiency for our signal sample to be 18%.
We generate a large number of MC experiments with the
fitted parameters, and from the spread of results of fits to
those experiments we determine the statistical uncertain-
ties on the parameters,
FF
j
, and
A
CP;j
. This procedure
takes into account correlations between the
x
j
,
x
j
,
y
j
, and
y
j
parameters. The linear correlation coefficients be-
tween the
FF
j
and
A
CP;j
parameters are also obtained
and are presented in Appendix
A
. In order to calculate the
branching fraction for an intermediate mode, we multiply
)
2
(GeV/c
ES
m
5.274 5.276 5.278 5.28 5.282 5.284
)
2
Events/(0.28 MeV/c
0
10
20
30
40
50
60
70
80
E (GeV)
∆
-0.06 -0.04 -0.02
0
0.02 0.04 0.06
Events/(3 MeV)
0
10
20
30
40
50
60
70
)
2
(GeV/c
ES
m
5.274 5.276 5.278 5.28 5.282 5.284
)
2
Events/(0.28 MeV/c
0
10
20
30
40
50
60
70
80
E (GeV)
∆
-0.06 -0.04 -0.02
0
0.02 0.04 0.06
Events/(3 MeV)
0
10
20
30
40
50
60
70
80
FIG. 1. (Top) signal and (bottom)
q
q
distributions of (left)
m
ES
and (right)
E
obtained from the fit to data using event-by-event
signal and
q
q
background probabilities [
43
]. The solid lines show the PDF shapes used in the fit.
)
4
/c
2
(GeV
min
2
m
0 2 4 6 8 101214
)
4
/c
2
(GeV
max
2
m
0
5
10
15
20
25
FIG. 2. Background-subtracted Dalitz plot of the combined
B
!
data sample in the signal region. The plot
shows bins with greater than zero entries. The area of the boxes
is proportional to the number of entries. The depleted bands are
the charm and charmonia exclusion regions.
DALITZ PLOT ANALYSIS OF
...
PHYSICAL REVIEW D
79,
072006 (2009)
072006-9