Dalitz plot analysis of
B
!
D
þ
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. 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. J. Asgeirsson,
8
B. G. Fulsom,
8
C. Hearty,
8
T. S. Mattison,
8
J. A. McKenna,
8
M. Barrett,
9
A. Khan,
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,
12
C. Buchanan,
12
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. J. Flacco,
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
M. G. Wilson,
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
A. Buzzo,
27a
R. Contri,
27a,27b
M. Lo Vetere,
27a,27b
M. M. Macri,
27a
M. R. Monge,
27a,27b
S. Passaggio,
27a
C. Patrignani,
27a,27b
E. Robutti,
27a
A. Santroni,
27a,27b
S. Tosi,
27a,27b
K. S. Chaisanguanthum,
28
M. Morii,
28
A. Adametz,
29
J. Marks,
29
S. Schenk,
29
U. Uwer,
29
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
C. K. Lae,
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
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
X. Li,
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
F. Taylor,
45
R. K. Yamamoto,
45
M. Zhao,
45
P. M. Patel,
46
S. H. Robertson,
46
A. Lazzaro,
47a,47b
V. Lombardo,
47a
F. Palombo,
47a,47b
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
G. Calderini,
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
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,
61
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
M. Escalier,
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
PHYSICAL REVIEW D
79,
112004 (2009)
1550-7998
=
2009
=
79(11)
=
112004(16)
112004-1
Ó
2009 The American Physical Society
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
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
R. Kowalewski,
76
M. J. Lewczuk,
76
I. M. Nugent,
76
J. M. Roney,
76
R. J. Sobie,
76
T. J. Gershon,
77
P. F. Harrison,
77
J. Ilic,
77
T. E. Latham,
77
G. B. Mohanty,
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 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 British Columbia, Vancouver, British Columbia, Canada V6T 1Z1
9
Brunel University, Uxbridge, Middlesex UB8 3PH, United Kingdom
10
Budker Institute of Nuclear Physics, Novosibirsk 630090, Russia
11
University of California at Irvine, Irvine, California 92697, USA
12
University of California at 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,
112004 (2009)
112004-2
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
Stanford Linear Accelerator Center, 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
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
{
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.
**
Also with Universita
`
di Sassari, Sassari, Italy.
*
Deceased.
DALITZ PLOT ANALYSIS OF
B
!
D
þ
PHYSICAL REVIEW D
79,
112004 (2009)
112004-3
77
Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom
78
University of Wisconsin, Madison, Wisconsin 53706, USA
(Received 11 January 2009; published 11 June 2009)
We report on a Dalitz plot analysis of
B
!
D
þ
decays, based on a sample of about
383
10
6
ð
4S
Þ!
B
B
decays collected with the
BABAR
detector at the PEP-II asymmetric-energy
B
Factory at
SLAC. We find the total branching fraction of the three-body decay:
B
ð
B
!
D
þ
Þ¼ð
1
:
08
0
:
03
0
:
05
Þ
10
3
. We observe the established
D
0
2
and confirm the existence of
D
0
0
in their decays to
D
þ
, where the
D
0
2
and
D
0
0
are the
2
þ
and
0
þ
c
uP
-wave states, respectively. We measure the masses
and widths of
D
0
2
and
D
0
0
to be:
m
D
0
2
¼ð
2460
:
4
1
:
2
1
:
2
1
:
9
Þ
MeV
=c
2
,
D
0
2
¼ð
41
:
8
2
:
5
2
:
1
2
:
0
Þ
MeV
,
m
D
0
0
¼ð
2297
8
5
19
Þ
MeV
=c
2
, and
D
0
0
¼ð
273
12
17
45
Þ
MeV
. The
stated errors reflect the statistical and systematic uncertainties, and the uncertainty related to the assumed
composition of signal events and the theoretical model.
DOI:
10.1103/PhysRevD.79.112004
PACS numbers: 13.25.Hw, 14.40.Lb, 14.40.Nd
I. INTRODUCTION
Orbitally excited states of the
D
meson, denoted here as
D
J
, where
J
is the spin of the meson, provide a unique
opportunity to test the heavy quark effective theory
(HQET) [
1
,
2
]. The simplest
D
J
meson consists of a charm
quark and a light antiquark in an orbital angular momen-
tum
L
¼
1
(
P
-wave) state. Four such states are expected
with spin-parity
J
P
¼
0
þ
(
j
¼
1
=
2
),
1
þ
(
j
¼
1
=
2
),
1
þ
(
j
¼
3
=
2
), and
2
þ
(
j
¼
3
=
2
), which are labeled here as
D
0
,
D
0
1
,
D
1
, and
D
2
, respectively, where
j
is a quantum
number corresponding to the sum of the light quark spin
and the orbital angular momentum
~
L
.
The conservation of parity and angular momentum in
strong interactions imposes constraints on the strong de-
cays of
D
J
states to
D
and
D
. The
j
¼
1
=
2
states are
predicted to decay exclusively through an
S
-wave:
D
0
!
D
and
D
0
1
!
D
. The
j
¼
3
=
2
states are expected to
decay through a
D
-wave:
D
1
!
D
and
D
2
!
D
and
D
. These transitions are summarized in Fig.
1
. Because
of the finite
c
-quark mass, the two
J
P
¼
1
þ
states may be
mixtures of the
j
¼
1
=
2
and
j
¼
3
=
2
states. Thus the broad
D
0
1
state may decay via a
D
-wave and the narrow
D
1
state
may decay via an
S
-wave. The
j
¼
1
=
2
states with
L
¼
1
,
which decay through an
S
-wave, are expected to be wide
(hundreds of
MeV
=c
2
), while the
j
¼
3
=
2
states that decay
through a
D
-wave are expected to be narrow (tens of
MeV
=c
2
)[
2
–
4
]. Properties of the
L
¼
1
D
0
J
mesons [
5
]
are given in Table
I
.
The narrow
D
J
mesons have been previously observed
and studied by a number of experiments [
6
–
16
].
D
J
mesons
have also been studied in semileptonic
B
decays [
17
–
24
].
Precise knowledge of the properties of the
D
J
mesons is
important to reduce uncertainties in the measurements of
semileptonic decays, and thus the determination of the
Cabibbo-Kobayashi-Maskawa [
25
] matrix elements
j
V
cb
j
and
j
V
ub
j
. The Belle Collaboration has reported the first
observation of the broad
D
0
0
and
D
0
0
1
mesons in
B
decay
[
12
]. The FOCUS Collaboration has found evidence for
broad structures in
D
þ
final states [
13
] with mass and
width in agreement with the
D
0
0
found by the Belle
Collaboration. However, the Particle Data Group [
5
] con-
siders that the
J
and
P
quantum numbers of the
D
0
0
and
D
0
0
1
states still need confirmation.
In this analysis, we fully reconstruct the decays
B
!
D
þ
[
26
] and measure their branching fraction. We
also perform an analysis of the Dalitz plot (DP) to measure
the exclusive branching fractions of
B
!
D
0
J
and
study the properties of the
D
0
J
mesons. The decay
B
!
D
þ
is expected to be dominated by the intermediate
-
0
-
1
+
0
+
1
+
1
+
2
P
J
1.8
2.0
2.2
2.4
2.6
2.8
L= 0
L= 1
j=1/2
j=3/2
S-wave
π
D-wave
π
)
2
Mass (GeV/c
D
*
D
0
*
D
1
’
D
1
D
2
*
D
FIG. 1 (color online). Mass spectrum for
c
u
states. The verti-
cal bars show the widths. Masses and widths are from Ref. [
5
].
The dotted and dashed lines between the levels show the domi-
nant pion transitions. Although it is not indicated in the figure,
the two
1
þ
states may be mixtures of
j
¼
1
=
2
and
j
¼
3
=
2
, and
D
0
1
may decay via a
D
-wave and
D
1
may decay via an
S
-wave.
TABLE I. Properties of
L
¼
1
D
0
J
mesons [
5
].
J
P
Mass
Width
Decays
Partial
(
MeV
=c
2
)
(MeV)
seen [
5
]
waves
D
0
0
0
þ
2352
50 261
50
D
S
D
0
0
1
1
þ
2427
36 384
þ
130
105
D
S
,
D
D
0
1
1
þ
2422
:
3
1
:
320
:
4
1
:
7
D
,
D
0
þ
S
,
D
D
0
2
2
þ
2461
:
1
1
:
643
4
D
,
D
D
B. AUBERT
et al.
PHYSICAL REVIEW D
79,
112004 (2009)
112004-4
states
D
0
2
and
D
0
0
and has a possible contribution
from
B
!
D
þ
nonresonant (NR) decay. The
D
0
1
and
D
0
0
1
states cannot decay strongly into
D
because of
parity and angular momentum conservation. However, the
D
ð
2007
Þ
0
(labeled as
D
v
here) mass is close to the
D
production threshold and it may contribute as a virtual
intermediate state. The
B
(labeled as
B
v
here) produced
in a virtual process
B
!
B
v
may also contribute via
the decay
B
v
!
D
þ
. Possible contributions from these
virtual states are also studied in this analysis.
II. 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 at SLAC between 1999 and 2006. The sample
consists of
347
:
2fb
1
corresponding to
ð
382
:
9
4
:
2
Þ
10
6
B
B
pairs (
N
B
B
) taken on the peak of the
ð
4S
Þ
reso-
nance. Monte Carlo (MC) simulation is used to study the
detector response, its acceptance, background, and to vali-
date the analysis. We use GEANT4 [
27
] to simulate reso-
nant
e
þ
e
!
ð
4S
Þ!
B
B
events (generated by EvtGen
[
28
]) and
e
þ
e
!
q
q
(where
q
¼
u
,
d
,
s
,or
c
) continuum
events (generated by JETSET [
29
]).
A detailed description of the
BABAR
detector is given in
Ref. [
30
]. Charged particle trajectories are measured by a
five-layer, double-sided silicon vertex tracker (SVT) and a
40-layer drift chamber (DCH) immersed in a 1.5 T mag-
netic field. Charged particle identification (PID) is
achieved by combining information from a ring-imaging
Cherenkov device with ionization energy loss (
dE=dx
)
measurements in the DCH and SVT.
III. EVENT SELECTION
Five charged particles are selected to reconstruct decays
of
B
!
D
þ
with
D
þ
!
K
þ
þ
. The charged
particle candidates are required to have transverse mo-
menta above
100 MeV
=c
and at least 12 hits in the DCH.
A
K
candidate must be identified as a kaon using a
likelihood-based particle identification algorithm (with an
average efficiency of
85%
and an average misidentifica-
tion probability of
3%
). Any combination of
K
þ
þ
candidates with a common vertex and an invariant mass
between 1.8625 and
1
:
8745 GeV
=c
2
is accepted as a
D
þ
candidate. We fit the invariant mass distribution of the
K
þ
þ
candidates with a function that includes a
Gaussian component for the signal and a linear term for
the background. The signal parameters (mean and width of
Gaussian) and slope of the background function are free
parameters of the fit. The data and the result of the fit are
shown in Fig.
2
. The invariant mass resolution for this
D
þ
decay is about
5
:
2 MeV
=c
2
. The
B
candidates are recon-
structed by combining a
D
þ
candidate and two charged
tracks. The trajectories of the three daughters of the
B
meson candidate are constrained to originate from a com-
mon decay vertex. The
D
þ
and
B
vertex fits are required
to have converged.
At the
ð
4S
Þ
resonance,
B
mesons can be characterized
by two nearly independent kinematic variables, the beam-
energy substituted mass
m
ES
and the energy difference
E
:
m
ES
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð
s=
2
þ
~
p
0
~
p
B
Þ
2
=E
2
0
p
2
B
q
;
(1)
E
¼
E
B
ffiffiffi
s
p
=
2
;
(2)
where
E
and
p
are energy and momentum, the subscripts 0
and
B
refer to the
e
þ
e
-beam system and the
B
candidate,
respectively;
s
is the square of the center-of-mass energy
and the asterisk labels the center-of-mass frame. For
B
!
D
þ
signal events, the
m
ES
distribution is well de-
scribed by a Gaussian resolution function with a width of
2
:
6 MeV
=c
2
centered at the
B
meson mass, while the
E
distribution can be represented by a sum of two Gaussian
functions with a common mean near zero and different
widths with a combined root-mean-square (RMS) of
20 MeV.
Continuum events are the dominant background.
Suppression of background from continuum events is pro-
vided by two topological requirements. In particular, we
employ restrictions on the magnitude of the cosine of the
thrust angle,
cos
th
, defined as the angle between the
thrust axis of the selected
B
candidate and the thrust axis
of the remaining tracks and neutral clusters in the event.
The distribution of
j
cos
th
j
is strongly peaked towards
unity for continuum background but is uniform for signal
events. We also select on the ratio of the second to the
zeroth Fox-Wolfram moment [
31
],
R
2
, to further reduce the
continuum background. The value of
R
2
ranges from 0 to 1.
Small values of
R
2
indicate a more spherical event shape
(typical for a
B
B
event) while values close to 1 indicate a 2-
jet event topology (typical for a
q
q
event). We accept the
)
) (GeV/c
+
π
+
π
-
m(K
1.84
1.85
1.86
1.87
1.88
1.89
1.9
)
2
Events/(0.002 GeV/c
0
200
400
600
800
1000
)
) (GeV/c
+
π
+
π
-
m(K
1.84
1.85
1.86
1.87
1.88
1.89
1.9
)
2
Events/(0.002 GeV/c
0
200
400
600
800
1000
2
FIG. 2 (color online).
K
þ
invariant mass distribution for
D
þ
candidates for the selected
B
!
D
þ
decays without
the cut on the mass of
D
þ
. Data (points with statistical errors)
are compared to the results of the fit (solid curve), with the
background distribution marked as a dashed line. The shaded
area marks the
D
þ
signal region.
DALITZ PLOT ANALYSIS OF
B
!
D
þ
PHYSICAL REVIEW D
79,
112004 (2009)
112004-5
events with
j
cos
th
j
<
0
:
85
and
R
2
<
0
:
30
. The
j
cos
th
j
(
R
2
) cut eliminates about 68% (71%) of the continuum
background while retaining about 90% (83%) of signal
events.
To suppress backgrounds, restrictions are placed on
m
ES
:
5
:
2754
<
m
ES
<
5
:
2820 GeV
=c
2
, and
E
:
130
<
E
<
130 MeV
. The selected samples of
B
candidates are used
as input to an unbinned extended maximum likelihood fit
to the
E
distribution. The result of the fit is used to
determine the fractions of signal and background events
in the selected data sample. For events with multiple
candidates (
3
:
5%
of the selected events) satisfying the
selection criteria, we choose the one with best
2
from the
B
vertex fit. Based on MC simulation, we determine that
the correct candidate is selected at least 65% of the time.
We fit the
m
ES
distribution of the selected
B
!
D
þ
candidates with a sum of a Gaussian function for the signal
and a background function for the background having the
probability density,
P
ð
x
Þ/
x
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
x
2
p
exp
ð
ð
1
x
2
ÞÞ
,
where
x
¼
m
ES
=m
0
with
m
0
fixed at
5
:
29 GeV
=c
2
and
is a shape parameter [
32
]. The signal parameters (mean,
width of Gaussian) and the shape parameter of the back-
ground function are free parameters of the fit. The data and
the result of the fit are shown in Fig.
3(a)
. We fit the
E
distribution of the selected
B
!
D
þ
candidates
with a sum of two Gaussian functions with a common
mean for the signal and a linear function for the back-
ground. The signal parameters (mean, width of wide
Gaussian, width and fraction of narrow Gaussian) and the
slope of the background function are free parameters of the
fit. The data and the result of the fit are shown in Fig.
3(b)
.
The resulting signal yield is
3496
74
events, where the
error is statistical only. A clear signal is evident in both
m
ES
and
E
distributions.
To distinguish signal and background in the Dalitz plot
studies, we divide the candidates into three subsamples: the
signal region,
21
<
E
<
15 MeV
, the left sideband,
109
<
E
<
73 MeV
, and the right sideband,
67
<
E
<
103 MeV
. The events in the signal region are used
in the Dalitz plot analysis, while the events in the sideband
regions are used to study the background.
In order to check the shape of the background
E
distribution, we have generated a background MC sample
of resonant and continuum events with
B
!
D
þ
signal events removed. The background MC sample has
been scaled to the same luminosity as the data. The
E
distribution of the selected events from the background
MC sample is shown as the histogram in Fig.
3(b)
. A small
amount of peaking background is found from misrecon-
structed decays of
B
0
!
D
þ
with
!
0
, where
a
0
is missed and a random track in the event is mis-
identified as a signal
. The background histogram in
Fig.
3(b)
is fitted with a sum of two Gaussian functions
with a common mean for the peaking background, with
parameters fixed to those obtained from the fit to data, and
a linear function to describe the combinatorial background.
The amount of peaking background is estimated at
82
41
events. After peaking background subtraction, the number
of signal events above background is
N
signal
¼
3414
85
.
The background fraction in the signal region is
ð
30
:
4
1
:
1
Þ
%
.
IV. DALITZ PLOT ANALYSIS
We refit the
D
þ
and
B
candidate momenta by con-
straining the trajectories of the three daughters of the
B
meson candidate to originate from a common decay vertex
while constraining the invariant masses of
K
þ
þ
and
D
þ
to the
D
þ
and
B
masses [
5
], respectively. The
mass-constraints ensure that all events fall within the
Dalitz plot boundary.
In the decay of a
B
into a final state composed of three
pseudoscalar particles
ð
D
þ
Þ
, 2 degrees of freedom
are required to describe the decay kinematics. In this
analysis we choose the two
D
invariant mass-squared
combinations
x
¼
m
2
ð
D
þ
1
Þ
and
y
¼
m
2
ð
D
þ
2
Þ
as the
E (GeV)
∆
-0.1
-0.05
0
0.05
0.1
Events / ( 0.01 GeV )
0
500
1000
1500
E (GeV)
∆
-0.1
-0.05
0
0.05
0.1
Events / ( 0.01 GeV )
0
500
1000
1500
(b)
)
2
(GeV/c
ES
m
5.23
5.24
5.25
5.26
5.27
5.28
5.29
)
Events / ( 0.002 GeV/c
0
500
1000
1500
2000
)
2
(GeV/c
ES
m
5.23
5.24
5.25
5.26
5.27
5.28
5.29
)
Events / ( 0.002 GeV/c
0
500
1000
1500
2000
(a)
2
FIG. 3 (color online). (a)
m
ES
and (b)
E
distributions for
D
þ
candidates. Data (points with statistical errors) are
compared to the results of the fits (solid curves), with the
background contributions marked as dashed lines. The histo-
grams are the corresponding distributions of the background MC
sample as described in the text. The shaded area in (a) shows the
signal region, while the three shaded areas in (b) mark the signal
region in the center and the two sidebands.
B. AUBERT
et al.
PHYSICAL REVIEW D
79,
112004 (2009)
112004-6
independent variables, where the two like-sign pions
1
and
2
are randomly assigned to
x
and
y
. This has no
effect on our analysis since the likelihood function (de-
scribed below) is explicitly symmetrized with respect to
interchange of the two identical particles.
The differential decay rate is generally given in terms of
the Lorentz-invariant matrix element
M
by
d
2
dxdy
¼
j
M
j
2
256
3
m
3
B
;
(3)
where
m
B
is the
B
meson mass. The Dalitz plot gives a
graphical representation of the variation of the square of
the matrix element,
j
M
j
2
, over the kinematically acces-
sible phase space
ð
x; y
Þ
of the process. Nonuniformity in
the Dalitz plot can indicate presence of intermediate reso-
nances, and their masses and spin quantum numbers can be
determined.
A. Probability density function
We describe the distribution of candidate events in the
Dalitz plot in terms of a probability density function
(PDF). The PDF is the sum of signal and background
components and has the form
PDF
ð
x; y
Þ¼
f
bg
B
ð
x; y
Þ
R
DP
B
ð
x; y
Þ
dxdy
þð
1
f
bg
Þ
½
S
ð
x; y
Þ
R
ð
x; y
Þ
R
DP
½
S
ð
x; y
Þ
R
ð
x; y
Þ
dxdy
;
(4)
where the integral is performed over the whole Dalitz plot,
the
S
ð
x; y
Þ
R
is the signal term convolved with the signal
resolution function,
B
ð
x; y
Þ
is the background term,
f
bg
is
the fraction of background events, and
is the reconstruc-
tion efficiency.
An unbinned maximum likelihood fit to the Dalitz plot is
performed in order to maximize the value of
L
¼
Y
N
event
i
¼
1
PDF
ð
x
i
;y
i
Þ
(5)
with respect to the parameters used to describe
S
, where
x
i
and
y
i
are the values of
x
and
y
for event
i
respectively, and
N
event
is the number of events in the Dalitz plot. In practice,
the negative-log-likelihood (NLL) value
NLL
¼
ln
L
(6)
is minimized in the fit.
B. Goodness-of-fit
It is difficult to find a proper binning at the kinematic
boundaries in the
x
-
y
plane of the Dalitz plot. For this
reason, we choose to estimate the goodness-of-fit
2
in the
cos
(range from
1
to 1) and
m
2
min
ð
D
Þ
(range from 4.04
to
15
:
23 GeV
2
=c
4
) plane, which is a rectangular represen-
tation of the Dalitz plot. The parameter
is the helicity
angle of the
D
system and
m
2
min
ð
D
Þ
is the lesser of
x
and
y
. The helicity angle
is defined as the angle between the
momentum vector of the pion from the
B
decay (bachelor
pion) and that of the pion of the
D
system in the
D
rest-
frame.
The
2
value is calculated using the formula
2
¼
X
i
2
i
¼
X
n
total
i
¼
1
ð
N
cell
i
N
fit
i
Þ
2
N
fit
i
;
(7)
for cells in a
18
18
grid of the two-dimensional histo-
gram. In Eq. (
7
),
n
total
is the total number of cells used,
N
cell
i
is the number of events in each cell, and
N
fit
i
is the
expected number of events in that cell as predicted by the
fit results. The number of degrees of freedom (NDF) is
calculated as
n
total
k
1
, where
k
is the number of free
parameters in the fit. We require
N
fit
10
; if this require-
ment is not met then neighboring cells are combined until
ten events are accumulated.
C. Matrix element
M
and fit parameters
This analysis uses an isobar model formulation in which
the signal decays are described by a coherent sum of a
number of two-body (
D
system
þ
bachelor pion) ampli-
tudes. The orbital angular momentum between the
D
system and the bachelor pion is denoted here as
L
. The
total decay matrix element
M
for
B
!
D
þ
is
given by
M
¼
X
L
¼ð
0
;
1
;
2
Þ
L
e
i
L
½
N
L
ð
x; y
Þþ
N
L
ð
y; x
Þ
þ
X
k
k
e
i
k
½
A
k
ð
x; y
Þþ
A
k
ð
y; x
Þ
;
(8)
where the first term represents the
S
-wave (
L
¼
0
),
P
-wave (
L
¼
1
), and
D
-wave (
L
¼
2
) nonresonant con-
tributions, the second term stands for the resonant contri-
butions, the parameters
k
and
k
are the magnitudes and
phases of the
k
th resonance, while
L
and
L
correspond
to the magnitudes and phases of the nonresonant contribu-
tions with angular momentum
L
. The functions
N
L
ð
x; y
Þ
and
A
k
ð
x; y
Þ
are the amplitudes for nonresonant and reso-
nant terms, respectively.
The resonant amplitudes
A
k
ð
x; y
Þ
are expressed as
A
k
ð
x; y
Þ¼
R
k
ð
m
Þ
F
L
ð
p
0
r
0
Þ
F
L
ð
qr
Þ
T
L
ð
p; q;
cos
Þ
;
(9)
where
R
k
ð
m
Þ
is the
k
th resonance line shape,
F
L
ð
p
0
r
0
Þ
and
F
L
ð
qr
Þ
are the Blatt-Weisskopf barrier factors [
33
], and
T
L
ð
p; q;
cos
Þ
gives the angular distribution. The parame-
ter
m
ð¼
ffiffiffi
x
p
Þ
is the invariant mass of the
D
system. The
parameter
p
0
is the magnitude of the three momentum of
the bachelor pion evaluated in the
B
-meson rest frame. The
parameters
p
and
q
are the magnitudes of the three mo-
menta of the bachelor pion and the pion of the
D
system,
both in the
D
rest frame. The parameters
p
0
,
p
,
q
, and
are functions of
x
and
y
.
DALITZ PLOT ANALYSIS OF
B
!
D
þ
PHYSICAL REVIEW D
79,
112004 (2009)
112004-7
The nonresonant amplitudes
N
L
ð
x; y
Þ
with
L
¼
0
,1,2
are similar to
A
k
ð
x; y
Þ
but do not contain resonant mass
terms:
N
0
ð
x; y
Þ¼
1
;
(10)
N
1
ð
x; y
Þ¼
F
1
ð
p
0
r
0
Þ
F
1
ð
qr
Þ
T
1
ð
p; q;
cos
Þ
;
(11)
N
2
ð
x; y
Þ¼
F
2
ð
p
0
r
0
Þ
F
2
ð
qr
Þ
T
2
ð
p; q;
cos
Þ
:
(12)
The Blatt-Weisskopf barrier factors
F
L
ð
p
0
r
0
Þ
and
F
L
ð
qr
Þ
depend on a single parameter,
r
0
or
r
, the radius of the
barrier, which we take to be
1
:
6
ð
GeV
=c
Þ
1
, similarly to
Ref. [
12
]. A discussion of the systematic uncertainty asso-
ciated with the choice of the values of
r
and
r
0
follows
below. The forms of
F
L
ð
z
Þ
, where
z
¼
p
0
r
0
or
qr
, for
L
¼
0
, 1, 2 are
F
0
ð
z
Þ¼
1
;
(13)
F
1
ð
z
Þ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
þ
z
2
0
1
þ
z
2
s
;
(14)
F
2
ð
z
Þ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
9
þ
3
z
2
0
þ
z
4
0
9
þ
3
z
2
þ
z
4
s
;
(15)
where
z
0
¼
p
0
0
r
0
or
q
0
r
. Here
p
0
0
and
q
0
represent the
values of
p
0
and
q
, respectively, when the invariant mass
is equal to the pole mass of the resonance. For nonresonant
terms, the fit results are not affected by the choice of
invariant mass (we use the sum of
m
D
and
m
) used for
the calculations of
p
0
0
and
q
0
. For virtual
D
v
decay,
D
v
!
D
þ
, and virtual
B
v
production in
B
!
B
v
, we use
an exponential form factor in place of the Blatt-Weisskopf
barrier factor, as discussed in Ref. [
12
]:
F
ð
z
Þ¼
exp
ðð
z
z
0
ÞÞ
;
(16)
where
z
0
¼
rp
v
for
D
v
!
D
þ
and
z
0
¼
r
0
p
v
for
B
!
B
v
. Here, we set
p
v
¼
0
:
038 GeV
=c
, which gives the
best fit, although any value of
p
v
between 0.015 and
1
:
5 GeV
=c
gives a negligible effect on the fitted parame-
ters compared to their statistical errors.
The resonance mass term
R
k
ð
m
Þ
describes the intermedi-
ate resonance. All resonances in this analysis are parame-
trized with relativistic Breit-Wigner functions:
R
k
ð
m
Þ¼
1
ð
m
2
0
m
2
Þ
im
0
ð
m
Þ
;
(17)
where the decay width of the resonance depends on
m
:
ð
m
Þ¼
0
q
q
0
2
L
þ
1
m
0
m
F
2
L
ð
qr
Þ
;
(18)
where
m
0
and
0
are the values of the resonance pole mass
and decay width, respectively.
The terms
T
L
ð
p; q;
cos
Þ
describe the angular distribu-
tion of final-state particles and are based on the Zemach
tensor formalism [
34
]. The definitions of
T
L
ð
p; q;
cos
Þ
for
L
¼
0
, 1, 2 are
T
0
ð
p; q;
cos
Þ¼
1
;
(19)
T
1
ð
p; q;
cos
Þ¼
2
pq
cos
;
(20)
T
2
ð
p; q;
cos
Þ¼
4
p
2
q
2
ð
cos
2
1
=
3
Þ
:
(21)
The signal function is then given by
S
ð
x; y
Þ¼j
M
j
2
:
(22)
In this analysis, the masses of
D
v
and
B
v
are taken from
the world averages [
5
] while their widths are fixed at
0.1 MeV; the magnitude
k
and phase
k
of the
D
0
2
amplitude are fixed to 1 and 0, respectively, while the
masses and widths of
D
0
J
resonances and other magnitudes
and phases are free parameters to be determined in the fit.
The effect of varying the masses of
D
v
and
B
v
within their
errors [
5
] and widths of
D
v
and
B
v
between 0.001 and
0.3 MeV is negligible compared to the other model-
dependent systematic uncertainties given below.
Since the choice of normalization, phase convention,
and amplitude formalism may not always be the same for
different experiments, we use fit fractions and relative
phases instead of amplitudes to allow for a more mean-
ingful comparison of results. The fit fraction for the
k
th
decay mode is defined as the integral of the resonance
decay amplitudes divided by the coherent matrix element
squared for the complete Dalitz plot:
f
k
¼
R
DP
j
k
ð
A
k
ð
x; y
Þþ
A
k
ð
y; x
ÞÞj
2
dxdy
R
DP
j
M
j
2
dxdy
:
(23)
The fit fraction for nonresonant term with angular momen-
tum
L
has a similar form:
f
L
¼
R
DP
j
L
ð
N
L
ð
x; y
Þþ
N
L
ð
y; x
ÞÞj
2
dxdy
R
DP
j
M
j
2
dxdy
:
(24)
The fit fractions do not necessarily add up to unity because
of interference among the amplitudes.
To estimate the statistical uncertainties on the fit frac-
tions, the fit results are randomly modified according to the
covariance matrix of the fit and the new fractions are
computed using Eq. (
23
)or(
24
). The resulting fit fraction
distribution is fitted with a Gaussian whose width gives the
error on the given fraction.
D. Signal resolution function
The detector has finite resolution, thus measured quan-
tities differ from their true values. For the narrow reso-
nance
D
2
with the expected width of about 40 MeV, the
signal resolution needs to be taken into account. In order to
obtain the signal resolution on
m
2
ð
D
Þ
around the
D
2
mass
region, we study a sample of MC generated
B
!
X
!
D
þ
decays, with the mass and width of
X
set to
2
:
460 GeV
=c
2
(
D
2
mass region) and 0 MeV, respectively,
B. AUBERT
et al.
PHYSICAL REVIEW D
79,
112004 (2009)
112004-8
and subject these events to the same analysis reconstruc-
tion chain. The reconstructed events are then classified into
two categories: truth-matched (TM) events, where the
B
and the daughters are correctly reconstructed, and self-
crossfeed (SCF) events, where one or more of the daugh-
ters is not correctly associated with the generated particle.
The two-dimensional distribution of
cos
versus
m
2
ð
D
Þ
for truth-matched events is shown in Fig.
4
.
Since the resolution is independent of
cos
, we fit the
distribution of the quantity
q
0
¼
m
2
ð
D
Þ
m
2
true
using a
sum of two Gaussian functions with a common mean to
obtain the resolution function for truth-matched events
(
R
TM
). The signal resolution for an invariant mass of the
D
combination around the
D
0
2
region is about
3 MeV
=c
2
.
The two-dimensional distribution of
cos
versus
m
2
ð
D
Þ
for self-crossfeed events is shown in Fig.
5
. The
SCF fraction,
f
SCF
, varies from 0.5% to 4.0% with
cos
.
We fit the
f
SCF
distribution with a fourth-order polynomial
function. The
f
SCF
distribution and the result of the fit are
shown in Fig.
6
. The resolution for self-crossfeed events
varies between
5 MeV
=c
2
and
100 MeV
=c
2
with
cos
.We
divide the
cos
interval into 40 bins of equal width and use
these bins to describe the resolution function (
R
SCF
)in
terms of a sum of two bifurcated Gaussian (BGaussian)
functions with different means. The BGaussian is a
Gaussian as a function of
q
0
with three parameters,
q
0
0
the mean, and the two widths,
1
on the left and
2
on
the right side of the mean. The form of BGaussian is
BGaussian
ð
q
0
q
0
0
;
1
;
2
Þ
¼
8
>
>
<
>
>
:
2
ffiffiffiffiffi
2
p
ð
1
þ
2
Þ
exp
ð
q
0
q
0
0
Þ
2
2
2
1
if
q
0
<q
0
0
;
2
ffiffiffiffiffi
2
p
ð
1
þ
2
Þ
exp
ð
q
0
q
0
0
Þ
2
2
2
2
if
q
0
q
0
0
;
(25)
where
q
0
0
,
1
, and
2
are free parameters.
The signal resolution function is then given by
R
ð
q
0
;
cos
Þ¼ð
1
f
SCF
ð
cos
ÞÞ
R
TM
ð
q
0
Þ
þ
f
SCF
ð
cos
Þ
R
SCF
ð
q
0
;
cos
Þ
:
(26)
The function
R
ð
q
0
;
cos
Þ
represents the probability density
for an event having the true mass-squared
m
2
true
to be
reconstructed at
m
2
ð
D
Þ
for different
cos
regions.
The signal term
S
in Eq. (
4
) is convoluted with the above
resolution function. For each event, the convolution is
performed using numerical integration:
S
ð
x; y
Þ
R
¼
Z
S
ð
q
min
þ
q
0
;q
0
max
Þ
R
ð
q
0
;
cos
Þ
dq
0
;
(27)
where
S
is the signal function in Eq. (
22
) and
q
min
(
q
max
)is
the lesser (greater) of
x
and
y
. The quantity
cos
is deter-
mined from
q
min
and
q
max
and is assumed to be constant
during convolution. The resolution in
cos
has a negligible
effect on the fitted parameters. The quantity
q
0
max
is com-
puted using the kinematics of three-body decay with
q
min
,
q
0
, and
cos
.
FIG. 4. Two-dimensional histogram
cos
versus
m
2
ð
D
Þ
of
the truth-matched events as defined in the text.
0
20
40
60
80
100
120
)
4
/c
2
) (GeV
π
(D
2
m
4.5
5
5.5
6
6.5
7
7.5
8
θ
cos
-1
-0.5
0
0.5
1
FIG. 5. Two-dimensional histogram
cos
versus
m
2
ð
D
Þ
of
the self-crossfeed events as defined in the text.
θ
cos
-1
-0.5
0
0.5
1
SCF fraction (%)
0
1
2
3
4
FIG. 6 (color online).
f
SCF
ð
cos
Þ
distribution. The observed
self-crossfeed fractions (points with statistical errors) are com-
pared to the results of the fit (solid curve).
DALITZ PLOT ANALYSIS OF
B
!
D
þ
PHYSICAL REVIEW D
79,
112004 (2009)
112004-9