of 29
Time-dependent analysis of
B
0
K
0
S
π
π
þ
γ
decays and studies of the
K
þ
π
π
þ
system in
B
þ
K
þ
π
π
þ
γ
decays
P. del Amo Sanchez,
1
J. P. Lees,
1
V. Poireau,
1
V. Tisserand,
1
E. Grauges,
2
A. Palano,
3a,3b
G. Eigen,
4
B. Stugu,
4
D. N. Brown,
5
L. T. Kerth,
5
Yu. G. Kolomensky,
5
M. J. Lee,
5
G. Lynch,
5
H. Koch,
6
T. Schroeder,
6
C. Hearty,
7
T. S. Mattison,
7
J. A. McKenna,
7
R. Y. So,
7
A. Khan,
8
V. E. Blinov,
9a,9b,9c
A. R. Buzykaev,
9a
V. P. Druzhinin,
9a,9b
V. B. Golubev,
9a,9b
E. A. Kravchenko,
9a,9b
A. P. Onuchin,
9a,9b,9c
S. I. Serednyakov,
9a,9b
Yu. I. Skovpen,
9a,9b
E. P. Solodov,
9a,9b
K. Yu. Todyshev,
9a,9b
A. J. Lankford,
10
J. W. Gary,
11
O. Long,
11
M. Franco Sevilla,
12
T. M. Hong,
12
D. Kovalskyi,
12
J. D. Richman,
12
C. A. West,
12
A. M. Eisner,
13
W. S. Lockman,
13
W. Panduro Vazquez,
13
B. A. Schumm,
13
A. Seiden,
13
D. S. Chao,
14
C. H. Cheng,
14
B. Echenard,
14
K. T. Flood,
14
D. G. Hitlin,
14
J. Kim,
14
T. S. Miyashita,
14
P. Ongmongkolkul,
14
F. C. Porter,
14
M. Röhrken,
14
R. Andreassen,
15
Z. Huard,
15
B. T. Meadows,
15
B. G. Pushpawela,
15
M. D. Sokoloff,
15
L. Sun,
15
W. T. Ford,
16
J. G. Smith,
16
S. R. Wagner,
16
R. Ayad,
17
,
W. H. Toki,
17
B. Spaan,
18
D. Bernard,
19
M. Verderi,
19
S. Playfer,
20
D. Bettoni,
21a
C. Bozzi,
21a
R. Calabrese,
21a,21b
G. Cibinetto,
21a,21b
E. Fioravanti,
21a,21b
I. Garzia,
21a,21b
E. Luppi,
21a,21b
V. Santoro,
21a
A. Calcaterra,
22
R. de Sangro,
22
G. Finocchiaro,
22
S. Martellotti,
22
P. Patteri,
22
I. M. Peruzzi,
22
M. Piccolo,
22
A. Zallo,
22
R. Contri,
23a,23b
M. R. Monge,
23a,23b
S. Passaggio,
23a
C. Patrignani,
23a,23b
B. Bhuyan,
24
V. Prasad,
24
A. Adametz,
25
U. Uwer,
25
H. M. Lacker,
26
U. Mallik,
27
C. Chen,
28
J. Cochran,
28
S. Prell,
28
H. Ahmed,
29
A. V. Gritsan,
30
N. Arnaud,
31
M. Davier,
31
D. Derkach,
31
G. Grosdidier,
31
F. Le Diberder,
31
A. M. Lutz,
31
B. Malaescu,
31
,
P. Roudeau,
31
A. Stocchi,
31
G. Wormser,
31
D. J. Lange,
32
D. M. Wright,
32
J. P. Coleman,
33
J. R. Fry,
33
E. Gabathuler,
33
D. E. Hutchcroft,
33
D. J. Payne,
33
C. Touramanis,
33
A. J. Bevan,
34
F. Di Lodovico,
34
R. Sacco,
34
G. Cowan,
35
D. N. Brown,
36
C. L. Davis,
36
A. G. Denig,
37
M. Fritsch,
37
W. Gradl,
37
K. Griessinger,
37
A. Hafner,
37
K. R. Schubert,
37
R. J. Barlow,
38
G. D. Lafferty,
38
R. Cenci,
39
B. Hamilton,
39
A. Jawahery,
39
D. A. Roberts,
39
R. Cowan,
40
R. Cheaib,
41
P. M. Patel,
41
,*
S. H. Robertson,
41
B. Dey,
42a
N. Neri,
42a
F. Palombo,
42a,42b
L. Cremaldi,
43
R. Godang,
43
,
D. J. Summers,
43
M. Simard,
44
P. Taras,
44
G. De Nardo,
45a,45b
G. Onorato,
45a,45b
C. Sciacca,
45a,45b
G. Raven,
46
C. P. Jessop,
47
J. M. LoSecco,
47
K. Honscheid,
48
R. Kass,
48
M. Margoni,
49a,49b
M. Morandin,
49a
M. Posocco,
49a
M. Rotondo,
49a
G. Simi,
49a,49b
F. Simonetto,
49a,49b
R. Stroili,
49a,49b
S. Akar,
50
E. Ben-Haim,
50
M. Bomben,
50
G. R. Bonneaud,
50
H. Briand,
50
G. Calderini,
50
J. Chauveau,
50
Ph. Leruste,
50
G. Marchiori,
50
J. Ocariz,
50
M. Biasini,
51a,51b
E. Manoni,
51a
A. Rossi,
51a
C. Angelini,
52a,52b
G. Batignani,
52a,52b
S. Bettarini,
52a,52b
M. Carpinelli,
52a,52b
G. Casarosa,
52a,52b
M. Chrzaszcz,
52a
F. Forti,
52a,52b
M. A. Giorgi,
52a,52b
A. Lusiani,
52a,52c
B. Oberhof,
52a,52b
E. Paoloni,
52a,52b
M. Rama,
52a
G. Rizzo,
52a,52b
J. J. Walsh,
52a
D. Lopes Pegna,
53
J. Olsen,
53
A. J. S. Smith,
53
F. Anulli,
54a
R. Faccini,
54a,54b
F. Ferrarotto,
54a
F. Ferroni,
54a,54b
M. Gaspero,
54a,54b
A. Pilloni,
54a,54b
G. Piredda,
54a
C. Bünger,
55
S. Dittrich,
55
O. Grünberg,
55
M. Hess,
55
T. Leddig,
55
C. Voß,
55
R. Waldi,
55
T. Adye,
56
E. O. Olaiya,
56
F. F. Wilson,
56
S. Emery,
57
G. Vasseur,
57
D. Aston,
58
D. J. Bard,
58
C. Cartaro,
58
M. R. Convery,
58
J. Dorfan,
58
G. P. Dubois-Felsmann,
58
W. Dunwoodie,
58
M. Ebert,
58
R. C. Field,
58
B. G. Fulsom,
58
M. T. Graham,
58
C. Hast,
58
W. R. Innes,
58
P. Kim,
58
D. W. G. S. Leith,
58
S. Luitz,
58
V. Luth,
58
D. B. MacFarlane,
58
D. R. Muller,
58
H. Neal,
58
T. Pulliam,
58
B. N. Ratcliff,
58
A. Roodman,
58
R. H. Schindler,
58
A. Snyder,
58
D. Su,
58
M. K. Sullivan,
58
J. Va
vra,
58
W. J. Wisniewski,
58
H. W. Wulsin,
58
M. V. Purohit,
59
J. R. Wilson,
59
A. Randle-Conde,
60
S. J. Sekula,
60
M. Bellis,
61
P. R. Burchat,
61
E. M. T. Puccio,
61
M. S. Alam,
62
J. A. Ernst,
62
R. Gorodeisky,
63
N. Guttman,
63
D. R. Peimer,
63
A. Soffer,
63
S. M. Spanier,
64
J. L. Ritchie,
65
R. F. Schwitters,
65
J. M. Izen,
66
X. C. Lou,
66
F. Bianchi,
67a,67b
F. De Mori,
67a,67b
A. Filippi,
67a
D. Gamba,
67a,67b
L. Lanceri,
68a,68b
L. Vitale,
68a,68b
F. Martinez-Vidal,
69
A. Oyanguren,
69
J. Albert,
70
Sw. Banerjee,
70
A. Beaulieu,
70
F. U. Bernlochner,
70
H. H. F. Choi,
70
G. J. King,
70
R. Kowalewski,
70
M. J. Lewczuk,
70
T. Lueck,
70
I. M. Nugent,
70
J. M. Roney,
70
R. J. Sobie,
70
N. Tasneem,
70
T. J. Gershon,
71
P. F. Harrison,
71
T. E. Latham,
71
H. R. Band,
72
S. Dasu,
72
Y. Pan,
72
R. Prepost,
72
and S. L. Wu
72
(
B
A
B
AR
Collaboration)
1
Laboratoire d
Annecy-le-Vieux de Physique des Particules (LAPP), Université de Savoie, CNRS/IN2P3,
F-74941 Annecy-Le-Vieux, France
2
Universitat de Barcelona, Facultat de Fisica, Departament ECM, E-08028 Barcelona, Spain
3a
INFN Sezione di Bari, I-70126 Bari, Italy
3b
Dipartimento di Fisica, Università di Bari, I-70126 Bari, Italy
4
University of Bergen, Institute of Physics, N-5007 Bergen, Norway
5
Lawrence Berkeley National Laboratory and University of California, Berkeley, California 94720, USA
6
Ruhr Universität Bochum, Institut für Experimentalphysik 1, D-44780 Bochum, Germany
7
University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z1
8
Brunel University, Uxbridge, Middlesex UB8 3PH, United Kingdom
9a
Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090, Russia
9b
Novosibirsk State University, Novosibirsk 630090, Russia
9c
Novosibirsk State Technical University, Novosibirsk 630092, Russia
PHYSICAL REVIEW D
93,
052013 (2016)
2470-0010
=
2016
=
93(5)
=
052013(29)
052013-1
© 2016 American Physical Society
10
University of California at Irvine, Irvine, California 92697, USA
11
University of California at Riverside, Riverside, California 92521, USA
12
University of California at Santa Barbara, Santa Barbara, California 93106, USA
13
University of California at Santa Cruz, Institute for Particle Physics, Santa Cruz, California 95064, USA
14
California Institute of Technology, Pasadena, California 91125, USA
15
University of Cincinnati, Cincinnati, Ohio 45221, USA
16
University of Colorado, Boulder, Colorado 80309, USA
17
Colorado State University, Fort Collins, Colorado 80523, USA
18
Technische Universität Dortmund, Fakultät Physik, D-44221 Dortmund, Germany
19
Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS/IN2P3, F-91128 Palaiseau, France
20
University of Edinburgh, Edinburgh EH9 3JZ, United Kingdom
21a
INFN Sezione di Ferrara, I-44122 Ferrara, Italy
21b
Dipartimento di Fisica e Scienze della Terra, Università di Ferrara, I-44122 Ferrara, Italy
22
INFN Laboratori Nazionali di Frascati, I-00044 Frascati, Italy
23a
INFN Sezione di Genova, I-16146 Genova, Italy
23b
Dipartimento di Fisica, Università di Genova, I-16146 Genova, Italy
24
Indian Institute of Technology Guwahati, Guwahati, Assam, 781 039, India
25
Universität Heidelberg, Physikalisches Institut, D-69120 Heidelberg, Germany
26
Humboldt-Universität zu Berlin, Institut für Physik, D-12489 Berlin, Germany
27
University of Iowa, Iowa City, Iowa 52242, USA
28
Iowa State University, Ames, Iowa 50011-3160, USA
29
Physics Department, Jazan University, Jazan 22822, Kingdom of Saudi Arabia
30
Johns Hopkins University, Baltimore, Maryland 21218, USA
31
Laboratoire de l
Accélérateur Linéaire, IN2P3/CNRS et Université Paris-Sud 11, Centre Scientifique
d
Orsay, F-91898 Orsay Cedex, France
32
Lawrence Livermore National Laboratory, Livermore, California 94550, USA
33
University of Liverpool, Liverpool L69 7ZE, United Kingdom
34
Queen Mary, University of London, London, E1 4NS, United Kingdom
35
University of London, Royal Holloway and Bedford New College, Egham,
Surrey TW20 0EX, United Kingdom
36
University of Louisville, Louisville, Kentucky 40292, USA
37
Johannes Gutenberg-Universität Mainz, Institut für Kernphysik, D-55099 Mainz, Germany
38
University of Manchester, Manchester M13 9PL, United Kingdom
39
University of Maryland, College Park, Maryland 20742, USA
40
Massachusetts Institute of Technology, Laboratory for Nuclear Science, Cambridge,
Massachusetts 02139, USA
41
McGill University, Montréal, Québec, Canada H3A 2T8
42a
INFN Sezione di Milano, I-20133 Milano, Italy
42b
Dipartimento di Fisica, Università di Milano, I-20133 Milano, Italy
43
University of Mississippi, University, Mississippi 38677, USA
44
Université de Montréal, Physique des Particules, Montréal, Québec, Canada H3C 3J7
45a
INFN Sezione di Napoli, I-80126 Napoli, Italy
45b
Dipartimento di Scienze Fisiche, Università di Napoli Federico II, I-80126 Napoli, Italy
46
NIKHEF, National Institute for Nuclear Physics and High Energy Physics,
NL-1009 DB Amsterdam, Netherlands
47
University of Notre Dame, Notre Dame, Indiana 46556, USA
48
Ohio State University, Columbus, Ohio 43210, USA
49a
INFN Sezione di Padova, I-35131 Padova, Italy
49b
Dipartimento di Fisica, Università di Padova, I-35131 Padova, Italy
50
Laboratoire de Physique Nucléaire et de Hautes Energies, IN2P3/CNRS, Université Pierre et Marie
Curie-Paris6, Université Denis Diderot-Paris7, F-75252 Paris, France
51a
INFN Sezione di Perugia, I-06123 Perugia, Italy
51b
Dipartimento di Fisica, Università di Perugia, I-06123 Perugia, Italy
52a
INFN Sezione di Pisa, I-56127 Pisa, Italy
52b
Dipartimento di Fisica, Università di Pisa, I-56127 Pisa, Italy
52c
Scuola Normale Superiore di Pisa, I-56127 Pisa, Italy
53
Princeton University, Princeton, New Jersey 08544, USA
54a
INFN Sezione di Roma, I-00185 Roma, Italy
54b
Dipartimento di Fisica, Università di Roma La Sapienza, I-00185 Roma, Italy
55
Universität Rostock, D-18051 Rostock, Germany
P. DEL AMO SANCHEZ
et al.
PHYSICAL REVIEW D
93,
052013 (2016)
052013-2
56
Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, OX11 0QX, United Kingdom
57
CEA, Irfu, SPP, Centre de Saclay, F-91191 Gif-sur-Yvette, France
58
SLAC National Accelerator Laboratory, Stanford, California 94309 USA
59
University of South Carolina, Columbia, South Carolina 29208, USA
60
Southern Methodist University, Dallas, Texas 75275, USA
61
Stanford University, Stanford, California 94305-4060, USA
62
State University of New York, Albany, New York 12222, USA
63
Tel Aviv University, School of Physics and Astronomy, Tel Aviv, 69978, Israel
64
University of Tennessee, Knoxville, Tennessee 37996, USA
65
University of Texas at Austin, Austin, Texas 78712, USA
66
University of Texas at Dallas, Richardson, Texas 75083, USA
67a
INFN Sezione di Torino, I-10125 Torino, Italy
67b
Dipartimento di Fisica, Università di Torino, I-10125 Torino, Italy
68a
INFN Sezione di Trieste, I-34127 Trieste, Italy
68b
Dipartimento di Fisica, Università di Trieste, I-34127 Trieste, Italy
69
IFIC, Universitat de Valencia-CSIC, E-46071 Valencia, Spain
70
University of Victoria, Victoria, British Columbia, Canada V8W 3P6
71
Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom
72
University of Wisconsin, Madison, Wisconsin 53706, USA
(Received 19 December 2015; published 29 March 2016)
We measure the time-dependent
CP
asymmetry in the radiative-penguin decay
B
0
K
0
S
π
π
þ
γ
, using a
sample of
471
×
10
6
Υ
ð
4
S
Þ
B
̄
B
events recorded with the
BABAR
detector at the PEP-II
e
þ
e
storage
ring at SLAC. Using events with
m
K
ππ
<
1
.
8
GeV
=c
2
, we measure the branching fractions of
B
þ
K
þ
π
π
þ
γ
and
B
0
K
0
π
π
þ
γ
, the branching fractions of the kaonic resonances decaying to
K
þ
π
π
þ
,as
well as the overall branching fractions of the
B
þ
ρ
0
K
þ
γ
,
B
þ
K

0
π
þ
γ
and
S
-wave
B
þ
ð
K
π
Þ

0
0
π
þ
γ
components. For events from the
ρ
mass band, we measure the
CP
-violating parameters
S
K
0
S
π
þ
π
γ
¼
0
.
14

0
.
25

0
.
03
and
C
K
0
S
π
þ
π
γ
¼
0
.
39

0
.
20
þ
0
.
03
0
.
02
, where the first uncertainties are statistical and the
second are systematic. We extract from this measurement the time-dependent
CP
asymmetry related to the
CP
eigenstate
ρ
0
K
0
S
and obtain
S
K
0
S
ργ
¼
0
.
18

0
.
32
þ
0
.
06
0
.
05
, which provides information on the photon
polarization in the underlying
b
s
γ
transition.
DOI:
10.1103/PhysRevD.93.052013
I. INTRODUCTION
The
V
-
A
structure of the Standard Model (SM) weak
interaction implies that the circular polarization of the
photon emitted in
b
s
γ
transitions is predominantly left-
handed, with contamination by oppositely polarized pho-
tons suppressed by a factor
m
s
=m
b
[1,2]
. Thus,
B
0
mesons
decay mostly to right-handed photons while decays of
̄
B
0
mesons produce mainly left-handed photons. Therefore, the
mixing-induced
CP
asymmetry in
B
f
CP
γ
decays, where
f
CP
is a
CP
eigenstate, is expected to be small. This
prediction may be altered by new-physics (NP) processes in
which opposite helicity photons are involved. Especially, in
some NP models
[3
5]
, the right-handed component may
be comparable in magnitude to the left-handed component,
without affecting the SM prediction for the inclusive
radiative decay rate. The present branching fraction meas-
urement of (
B
ð
B
X
s
γ
Þ
exp
¼ð
3
.
43

0
.
21

0
.
07
Þ
×
10
4
[6]
) agrees with the SM prediction of (
B
ð
B
X
s
γ
Þ
th
¼
ð
3
.
36

0
.
23
Þ
×
10
4
[7]
) calculated at next-to-next-to-
leading order. Further information on right-handed photon
could be obtained by measuring
CP
asymmetries in different
exclusive radiative decay modes. Furthermore,
B
meson
decays to
K
ππγ
can display an interesting hadronic structure,
since several resonances decay to three-body
K
ππ
final state
(referred to as
kaonic resonances
throughout the article).
The decays of these resonances themselves exhibit a
resonant structure, with contributions from
K

π
,
K
ρ
,and
a
ð
K
π
Þ

0
0
π
S
-wave component.
In the present analysis, we extract information about the
K
ππ
resonant structure by means of an amplitude analysis
of the
m
K
ππ
and
m
K
π
spectra in
B
þ
K
þ
π
π
þ
γ
decays.
Assuming isospin symmetry, we use these results to extract
*
Deceased.
Present address: University of Tabuk, Tabuk 71491, Saudi
Arabia.
Present address: Laboratoire de Physique Nucléaire et de
Hautes Energies, IN2P3/CNRS, F-75252 Paris, France.
§
Present address: University of Huddersfield, Huddersfield
HD1 3DH, United Kingdom.
Present address: University of South Alabama, Mobile,
Alabama 36688, USA.
Also at Università di Sassari, I-07100 Sassari, Italy.
TIME-DEPENDENT ANALYSIS OF
...
PHYSICAL REVIEW D
93,
052013 (2016)
052013-3
the mixing-induced
CP
parameters of the process
B
0
K
0
S
ρ
0
γ
from the time-dependent analysis of
B
0
K
0
S
π
π
þ
γ
decays without an explicit amplitude analysis of this mode.
Charge conjugation is implicit throughout the document.
The Belle Collaboration has previously reported a time-
dependent
CP
asymmetry measurement of
B
0
K
0
S
ρ
0
γ
decays
[8]
. Similar measurements with
B
0
K
0
S
π
0
γ
decays
have been reported by
BABAR
[9]
and Belle
[10]
.No
evidence for NP was found in these measurements. The
observed
CP
asymmetry parameters are compatible with the
SM predictions. LHCb has recently reported a nonzerovalue
of the photon polarization in
B
þ
K
þ
π
π
þ
γ
decays via the
distribution of the angle of the photon with respect to the
plane defined by the final state hadrons
[11]
. Studies of
the processes
B
þ
K
þ
π
π
þ
γ
and
B
0
K
0
S
π
π
þ
γ
includ-
ing measurements of the branching fractions have been
performedbyboth
BABAR
[12]
andBelle
[13]
usingsamples
of
232
×
10
6
and
152
×
10
6
B
̄
B
pairs,respectively.Thelatter
analysis also determined the branching fraction of the
resonant decay
B
þ
K
1
ð
1270
Þ
þ
γ
.
The article is organized as follows. In Sec.
II
we briefly
describe the
BABAR
detector and the data set. In Sec.
III
we
describe the analysis strategy. The amplitude analysis of
B
þ
K
þ
π
π
þ
γ
decays and the time-dependent analysis
of
B
0
K
0
S
π
π
þ
γ
decays are described in Secs.
IV
and
V
,
respectively. Finally, we summarize the results in Sec.
VI
.
II. THE
BABAR
DETECTOR AND DATA SET
The data used in this analysis were collected with the
BABAR
detector at the PEP-II2 asymmetric-energy
e
þ
e
storage ring at SLAC. The sample consists of an integrated
luminosity of
426
.
0
fb
1
[14]
, corresponding to
ð
470
.
9

2
.
8
Þ
×
10
6
B
̄
B
pairs collected at the
Υ
ð
4
S
Þ
resonance (
on
resonance
) and
44
.
5
fb
1
collected about 40 MeV below
the
Υ
ð
4
S
Þ
(
off resonance
). A detailed description of the
BABAR
detector is presented in Refs.
[15,16]
. The tracking
system used for track and vertex reconstruction has two
components: a silicon vertex tracker and a drift chamber,
both operating within a 1.5 T magnetic field generated by a
superconducting solenoidal magnet. A detector of inter-
nally reflected
Č
erenkov light (DIRC) is used for charged
particle identification. The energies of photons and elec-
trons are determined from the measured light produced in
electromagnetic showers inside a CsI(Tl) crystal electro-
magnetic calorimeter (EMC). Muon candidates are iden-
tified with the use of the instrumented flux return of the
solenoid.
III. ANALYSIS STRATEGY
The main goal of the present study is to perform a time-
dependent analysis of
B
0
K
0
S
π
π
þ
γ
decays to extract the
decay and mixing-induced
CP
asymmetry parameters,
C
K
0
S
ργ
and
S
K
0
S
ργ
, in the
B
0
K
0
S
ρ
0
γ
mode. However,
due to the large natural width of the
ρ
ð
770
Þ
0
, a non-
negligible number of
B
0
K

ð
K
0
S
π

Þ
π
γ
events, which
do not contribute to
S
K
0
S
ργ
, are expected to lie under the
ρ
ð
770
Þ
0
resonance and modify
S
K
0
S
ργ
. Using the formalism
developed in Ref.
[17]
, which assumes the SM, the
so-
called
dilution factor
D
K
0
S
ργ
can be expressed as
D
K
0
S
ργ
S
K
0
S
π
þ
π
γ
S
K
0
S
ργ
¼
R
½j
A
ρ
K
0
S
j
2
j
A
K
π
j
2
j
A
ð
K
π
Þ
0
π
j
2
þ
2
ð
A

ρ
K
0
S
A
K
π
Þþ
2
ð
A

ρ
K
0
S
A
ð
K
π
Þ
0
π
Þ
dm
2
R
½j
A
ρ
K
0
S
j
2
þj
A
K
π
j
2
þj
A
ð
K
π
Þ
0
π
j
2
þ
2
ð
A

ρ
K
0
S
A
K
π
Þþ
2
ð
A

ρ
K
0
S
A
ð
K
π
Þ
0
π
Þ
dm
2
;
ð
1
Þ
where
S
K
0
S
π
þ
π
γ
is the effective value of the mixing-induced
CP
asymmetry measured for the whole
B
0
K
0
S
π
π
þ
γ
data set and
A
RP
is the (complex) amplitude of the mode
RP
, where
R
represents a hadronic resonance and
P
a
pseudoscalar particle. Here,
ð
A
Þ
denotes the real part of
the complex number
A
. We assume the final state
K
0
S
π
þ
π
to originate from a few resonant decay modes where
R
corresponds to
ρ
0
,
K
,
K

,
ð
K
π
Þ
0
or
ð
K
π
Þ

0
S
wave.
Since a small number of events is expected in this sample,
the extraction of the
A
RP
amplitudes from the
B
0
K
0
S
π
π
þ
γ
sample is not feasible. Instead, the amplitudes
of the resonant modes are extracted from a fit to the
m
K
π
spectrum in the decay channel
B
þ
K
þ
π
π
þ
γ
, which has
more signal events and is related to
B
0
K
0
S
π
π
þ
γ
by isospin symmetry. Assuming that the resonant ampli-
tudes are the same in both modes, the dilution factor is
calculated from those of
B
þ
K
þ
π
π
þ
γ
. While the entire
phase-space region is used to extract the amplitudes in the
charged decay channel, the integration region over the plane
of the
K
þ
π
and
π
þ
π
invariant masses in the calculation of
D
K
0
S
ργ
is optimized in order to maximize the sensitivity on
S
K
0
S
ργ
. Note that the expression of
D
K
0
S
ργ
used in the present
analysis slightly differs from the one used in the previous
analysis performed by the Belle Collaboration
[8]
.
Moreover, the decay to the
K
þ
π
π
þ
γ
final state proceeds
in general through resonances with a three-body
K
þ
π
π
þ
final state. Although the contributions of some of these
states to the
B
þ
K
þ
π
π
þ
γ
decay, such as
K
1
ð
1270
Þ
or
K

2
ð
1430
Þ
have been measured, not all the contributions
have been identified
[18]
. Since each of these resonances
has different
K

π
and
K
ρ
mass spectra (see Sec.
IV B 3
), it
is necessary to first determine the three-body resonance
content of the
m
K
ππ
spectrum by fitting the charged
B
þ
K
þ
π
π
þ
γ
sample.
P. DEL AMO SANCHEZ
et al.
PHYSICAL REVIEW D
93,
052013 (2016)
052013-4
Two types of Monte Carlo (MC) samples are used to
characterize signal and background and to optimize the
selection in both analyses of
B
þ
K
þ
π
π
þ
γ
and
B
0
K
0
S
π
π
þ
γ
. Generic
B
̄
B
MC and MC samples for specific
exclusive final states are used to study backgrounds from
B
-meson decays, whereas only MC samples for specific
exclusive final states are used to study signal events. The
size of the generic
B
̄
B
MC sample approximately corre-
sponds to 3 times that of the data sample.
IV. AMPLITUDE ANALYSIS OF
B
þ
K
þ
π
π
þ
γ
DECAYS
In Sec.
IVA
, we describe the selection requirements used
to obtain the signal candidates and to suppress back-
grounds. In Sec.
IV B 1
, we describe the unbinned extended
maximum-likelihood fit method used to extract the yield of
B
þ
K
þ
π
π
þ
γ
correctly reconstructed (CR) signal events
from the data. Using information from this fit, the
K
þ
π
π
þ
,
K
þ
π
, and
π
þ
π
invariant-mass spectra (
m
K
ππ
,
m
K
π
, and
m
ππ
) for CR signal events are extracted by means of the
s
P
lot
technique
[19]
. In the second step, we perform a
binned maximum-likelihood fit to the CR signal
s
P
lot
of
m
K
ππ
to determine from data the branching fractions of the
various kaonic resonances decaying to
K
þ
π
π
þ
. We finally
perform a binned maximum-likelihood fit to the CR signal
s
P
lot
of
m
K
π
to extract from data the amplitudes and the
branching fractions of the two-body resonances decaying to
K
þ
π
and
π
þ
π
. The use of a one-dimensional fit in this
final step is purely pragmatic; with a larger sample size, a
two-dimensional approach would have been possible. The
m
K
π
projection is used due to the narrower width of the
K

ð
892
Þ
resonance compared with that of the
ρ
0
ð
770
Þ
.
The
m
K
ππ
and the
m
K
π
fit models are described in
Secs.
IV B 2
and
IV B 3
, respectively.
In Sec.
IV C
, we present the results of the three fits
described above, and finally, we discuss systematic uncer-
tainties on the results in Sec.
IV D
.
A. Event selection and backgrounds
We reconstruct
B
þ
K
þ
π
π
þ
γ
candidates from a
high-energy photon, a pair of oppositely charged tracks
consistent with pion hypotheses and one charged track
consistent with a kaon hypothesis, based on information
from the tracking system, from the EMC and from the
DIRC. The center-of-mass energy of the photon is required
to be between 1.5 and 3.5 GeV, as expected in a
B
radiative
decay. The system formed by the final state particles is
required to have a good-quality vertex.
A
B
-meson candidate is characterized kine-
matically by the energy-substituted mass
m
ES
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð
s=
2
þ
p
i
·
p
B
Þ
2
=E
2
i
p
2
B
p
and energy difference
Δ
E
¼
E

B
ffiffiffi
s
p
=
2
, where
ð
E
B
;
p
B
Þ
and
ð
E
i
;
p
i
Þ
are the
four-vectors of the
B
candidate and of the initial electron-
positron system, respectively, in the laboratory frame. The
asterisk denotes the center-of-mass frame, and
s
is the
square of the invariant mass of the electron-positron
system. We require
5
.
200
<m
ES
<
5
.
292
GeV
=c
2
and
j
Δ
E
j
<
0
.
200
GeV.
Since the
Υ
ð
4
S
Þ
is only just above the threshold for
B
̄
B
production, the decay products from such events are
approximately spherical in the center-of-mass frame,
whereas
e
þ
e
q
̄
q
(
q
¼
u
,
d
,
s
,
c
) continuum back-
ground events have a dijetlike structure. To enhance
discrimination between signal and the continuum back-
ground we use a Fisher discriminant
[20]
to combine six
discriminating variables: the angle between the momentum
of the
B
candidate and the beam (
z
) axis in the center-of-
mass frame, the angles between the
B
thrust axis
[21,22]
and the
z
axis and between the
B
thrust axis and that of the
rest of the event, the zeroth-order momentum-weighted
Legendre polynomial
L
0
and the second-to-zeroth-order
Legendre polynomials ratio
L
2
=L
0
of the energy flow about
the
B
thrust axis, and the second-to-zeroth-order Fox-
Wolfram moments
[23]
ratio. The momentum-weighted
Legendre polynomials are defined by
L
0
¼
P
i
j
p
i
j
and
L
2
¼
P
i
j
p
i
j
1
2
ð
3
cos
2
θ
i
1
Þ
, where
θ
i
is the angle with
respect to the
B
thrust axis of track or neutral cluster
i
and
p
i
is its momentum. The sums exclude the
B
candidate and
all quantities are calculated in the
Υ
ð
4
S
Þ
frame. The Fisher
discriminant is trained using off-resonance data for the
continuum and a mixture of simulated exclusive decays for
the signal. The final sample of candidates is selected with a
requirement on the Fisher discriminant output value (
F
)
that retains 90% of the signal and rejects 73% of the
continuum background.
We use simulated events to study the background from
B
decays other than our signal (
B
background). In prelimi-
nary studies, a large number of channels were considered,
of which only those with at least one event expected after
selection are considered here. The main
B
backgrounds
originate from
b
s
γ
processes.
B
background decays are
grouped into classes of modes with similar kinematic and
topological properties.
In order to reduce backgrounds from photons coming
from
π
0
and
η
mesons, we construct
π
0
and
η
likelihood
ratios,
L
R
, for which the photon candidate
γ
1
is associated
with all other photons in the event,
γ
2
, such that
L
R
;h
0
¼
p
ð
m
γ
1
γ
2
;E
γ
2
j
h
0
Þ
p
ð
m
γ
1
γ
2
;E
γ
2
j
K
þ
π
þ
π
γ
Þþ
p
ð
m
γ
1
γ
2
;E
γ
2
j
h
0
Þ
;
ð
2
Þ
where
h
0
is either
π
0
or
η
, and
p
is a probability density
function in terms of
m
γ
1
γ
2
and the energy of
γ
2
in the
laboratory frame,
E
γ
2
. The value of
L
R
;
ð
π
0
=
η
Þ
corresponds to
the probability for a photon candidate to originate from a
π
0
=
η
decay. We require
L
R
;
π
0
<
0
.
860
ð
L
R
;
η
<
0
.
957
Þ
,
resulting, if applied before any other selection cut, in a
signal efficiency of
93%
ð
95%
Þ
and in background
TIME-DEPENDENT ANALYSIS OF
...
PHYSICAL REVIEW D
93,
052013 (2016)
052013-5
rejection factors of
83%
ð
87%
Þ
for continuum events and
63%
ð
10%
Þ
for
B
-background events.
The optimization of the selection criteria was done using
the BumpHunter algorithm
[24]
. We optimized the
S=
ffiffiffiffiffiffiffiffiffiffiffiffi
S
þ
B
p
figure of merit using several selection variables
from which the kaon and pion particle identification levels,
the
π
0
and
η
likelihood ratios and the vertex
χ
2
of the
system formed by the final state particles. In the optimi-
zation, we used CR signal events from simulation, off-
resonance data for combinatorial background and generic
B
̄
B
simulated events (filtered to remove signal) for
B
backgrounds.
Table
I
summarizes the six mutually exclusive
B
-
background classes that are considered in the present
analysis.
B. The maximum-likelihood fit and extraction of the
physical observables
1. The m
ES
,
Δ
E, and
F
probability density
functions (PDFs)
We perform an unbinned extended maximum-likelihood
fit to extract the
B
þ
K
þ
π
π
þ
γ
event yield. We further
obtain the signal
m
K
ππ
,
m
K
π
and
m
ππ
spectra, where the
background is statistically subtracted using the
s
P
lot
technique. Note that this technique may produce bins with
negative entries. The fit is performed using the L
AURA
++
package
[25]
. The fit uses the variables
m
ES
,
Δ
E
, and the
Fisher-discriminant output
F
, to discriminate CR signal
events from other event categories. The likelihood function
L
i
for the event
i
is the sum
L
i
¼
X
j
N
j
P
i
j
ð
m
ES
;
Δ
E;
F
Þ
;
ð
3
Þ
where
j
stands for the event species (signal, continuum and
the various
B
backgrounds) and
N
j
is the corresponding
yield. The CR yield is a free parameter in the fit to the data,
while the misreconstructed signal yield is fixed, defined as
the product of the misreconstructed signal ratio obtained
from simulation and the signal branching fraction taken
from Ref.
[18]
. If no correlation is seen among the fitting
variables, the PDF
P
i
j
is the product of three individual
PDFs:
P
i
j
¼
P
i
j
ð
m
ES
Þ
P
i
j
ð
Δ
E
Þ
P
i
j
ð
F
Þ
:
ð
4
Þ
Otherwise, the correlations are taken into account through
multidimensional PDFs that depend on the correlated
variables. The total likelihood is given by
L
¼
exp

X
j
N
j

Y
i
L
i
:
ð
5
Þ
The
m
ES
distribution of CR signal events is parametrized
by a Crystal Ball (CB) function
[26
28]
defined as
CB
ð
x
;
μ
;
σ
;
α
;n
Þ¼
(
ð
n
α
Þ
n
exp
ð
α
2
=
2
Þ
ðð
μ
x
Þ
=
σ
þ
n=
α
α
Þ
n
x
μ
ασ
;
exp
½
1
2
ð
x
μ
σ
Þ
2

x>
μ
ασ
;
ð
6
Þ
where the parameters
μ
and
σ
designate the mean and
width, respectively, of a Gaussian distribution that is joined
at
μ
ασ
to a power law tail. The
Δ
E
distribution of CR
signal events is parametrized by a modified Gaussian (
~
G)
defined as
TABLE I. Summary of
B
-background classes included in the fit model to
B
þ
K
þ
π
π
þ
γ
decays. If the yield is a free parameter in
the fit, the listed values correspond to the fit result. Otherwise the expected value is given, which is computed from the branching fraction
and selection efficiency. The terms
X
su
ð
sd
Þ
ð
K
π
Þ
designate all
X
su
ð
sd
Þ
decays but the
K
π
final state. The functions used to parametrize
the
B
-background probability density functions of
m
ES
,
Δ
E
and
F
are also given. The notations
Exp
,
CB
and
~
G
correspond to the
exponential function, the Crystal Ball function [given in Eq.
(6)
] and the modified Gaussian function [given in Eq.
(7)
], respectively.
Class
Probability density functions
Varied
Number of events
m
ES
Δ
E
F
B
0
X
sd
ð
K
π
Þ
γ
~
G
þ
Exp
Gaussian
No
2872

242
B
þ
X
su
ð
K
π
Þ
γ
ARGUS
B
0
K

0
ð
K
π
Þ
γ
Two-dimensional
~
G
Yes
1529

116
B
0
X
sd
ð
K
π
Þ
γ
Nonparametric
B
þ
K
ð
K
π
Þ
γ
Linear
þ
Exp
~
G
No
442

50
B
þ
X
su
ð
K
π
Þ
γ
ARGUS
B
0
K

0
η
~
G
þ
Gaussian
þ
~
G
No
56

21
ARGUS
Constant
B
þ
a
þ
1
ð
ρ
0
π
þ
Þ
π
0
γ
CB
Asymmetric
Asymmetric
No
17

9
B
þ
K

0
ð
K
π
Þ
π
þ
π
0
γ
Gaussian
Gaussian
B
f
charged and neutral generic decays
g
ARGUS
Exp
Gaussian
Yes
3270

385
P. DEL AMO SANCHEZ
et al.
PHYSICAL REVIEW D
93,
052013 (2016)
052013-6
~
G
ð
x
;
μ
;
σ
l
;
σ
r
;
α
l
;
α
r
Þ
¼
exp

ð
x
μ
Þ
2
2
σ
2
k
þ
α
k
ð
x
μ
Þ
2

x
μ
<
0
k
¼
l;
x
μ
0
k
¼
r:
ð
7
Þ
The
μ
and
σ
l
parameters are free in the fit to the data, while
the other parameters are fixed to values determined from
simulations. Correlations between
m
ES
and
Δ
E
in CR
signal are taken into account through a two-dimensional
PDF. It is constructed as the product of a conditional PDF
(CB for
m
ES
) by a marginal PDF (
~
G for
Δ
E
). The
dependences on
Δ
E
of the CB parameters
μ
and
σ
are
parametrized by two second-order polynomials, while
those of the parameters
α
and
n
are parametrized by two
first-order polynomials. The three parameters of both
second-order polynomials are determined by the fit, while
the parameters of the first-order polynomials are fixed in
the fit to the values determined from simulations. The
F
PDF of CR signal events is parametrized by a Gaussian, for
which the mean and variance are left free in the fit to the
data. No significant correlations were found between
F
and
either
m
ES
or
Δ
E
.
The shape parameters of the PDFs of misreconstructed
signal events are fixed to values determined from simu-
lations. The
m
ES
PDF is parametrized by the sum of an
asymmetric Gaussian and of an ARGUS shape function
[29]
, while the
Δ
E
and
F
PDFs are parametrized by a first-
order polynomial and a Gaussian, respectively.
The
m
ES
,
Δ
E
and
F
PDFs for continuum events are
parametrized by an ARGUS shape function, a second-order
Chebychev polynomial and an exponential function,
respectively, with parameters determined by the fit, except
for the exponential shape parameter, which is fixed to the
value determined from a fit to off-resonance data.
The
m
ES
,
Δ
E
and
F
PDFs for all the classes of
B
-
background events are described by parametric functions,
given in Table
I
, except for the
B
0
K
πγ
background
m
ES
and
Δ
E
PDFs, for which significant correlations are
present. These are taken into account through a nonpara-
metric two-dimensional PDF, defined as a histogram
constructed from a mixture of simulated events. No
significant correlations were found among the fit variables
for the other species in the fit. The distributions of the
combined
B
0
̄
B
0
and
B
þ
B
generic
B
backgrounds were
studied using generic
B
̄
B
MC from which all other
B
-
background class contributions were filtered out. The shape
parameters of the
B
-background PDFs are fixed to values
determined from simulated events. If the yield of a class is
allowed to vary in the fit, the number of events listed in
Table
I
corresponds to the fit results. For the other classes,
the expected numbers of events are computed by multi-
plying the selection efficiencies estimated from simulations
by the world average branching fractions
[6,18]
, scaled to
the data set luminosity. The yield of the
B
0
K
πγ
class,
which has a clear signature in
m
ES
, and that of the generic
B
-background class are left free in the fit to the data. The
remaining background yields are fixed.
2. The m
K
ππ
spectrum
We model the
m
K
ππ
distribution as a coherent sum of five
resonances described by relativistic Breit-Wigner (
R
k
) line
shapes
[18]
, with widths that are taken to be constant. The
total decay amplitude is then defined as
j
A
ð
m
;
c
k
Þj
2
¼
X
J




X
k
c
k
R
J
k
ð
m
Þ




2
;
ð
8
Þ
with
R
J
k
ð
m
Þ¼
1
ð
m
0
k
Þ
2
m
2
im
0
k
Γ
0
k
;
ð
9
Þ
and where
c
k
¼
α
k
e
i
φ
k
and
m
¼
m
K
ππ
. In Eq.
(8)
, the index
J
runs over the different spin parities (
J
P
) and the index
k
runs over the
K
þ
π
π
þ
resonances of the same
J
P
. The
coefficients
α
k
and
φ
k
are the magnitude and the phase of
the complex coefficients,
c
k
, corresponding to a given
resonance. Due to the fact that helicity angles are not
explicitly taken into account in the fit model, it only has to
account for interference between resonances with the same
spin parity
J
P
. Table
II
details the resonances in the
m
K
ππ
fit
model. The
K
1
ð
1270
Þ
magnitude is fixed to 1, and the
K
1
ð
1270
Þ
,
K

ð
1680
Þ
, and
K

2
ð
1430
Þ
phases are fixed to 0.
It has been checked that the choice of reference does not
affect the results. The remaining parameters of the complex
coefficients are left free in the fit: namely the
K
1
ð
1400
Þ
,
K

ð
1410
Þ
,
K

ð
1680
Þ
, and
K

2
ð
1430
Þ
magnitudes as well as
the two relative phases, that between the two
J
P
¼
1
þ
resonances and that between the two
J
P
¼
1
resonances.
In addition to the complex coefficients, the widths of the
two resonances,
K
1
ð
1270
Þ
and
K

ð
1680
Þ
, are left free in
the fit. In the case of the
K
1
ð
1270
Þ
, this is motivated by the
fact that the width quoted in Ref.
[18]
might be under-
estimated according to the measurements reported in
Ref.
[30]
. In the case of the
K

ð
1680
Þ
, the uncertainty
on the width quoted in Ref.
[18]
is large. In total, eight
parameters are kept free in the fit.
TABLE II. The five kaonic resonances decaying to
K
þ
π
π
þ
included in the model used to fit the
m
K
ππ
spectrum. The pole
mass
m
0
and the width
Γ
0
are taken from Ref.
[18]
.
J
P
K
res
Mass
m
0
(MeV
=c
2
) Width
Γ
0
(MeV
=c
2
)
1
þ
K
1
ð
1270
Þ
1272

790

20
K
1
ð
1400
Þ
1403

7
174

13
1
K

ð
1410
Þ
1414

15
232

21
K

ð
1680
Þ
1717

27
322

110
2
þ
K

2
ð
1430
Þ
1425
.
6

1
.
598
.
5

2
.
7
TIME-DEPENDENT ANALYSIS OF
...
PHYSICAL REVIEW D
93,
052013 (2016)
052013-7
Note that we do not take phase-space effects into account
here. However, distortions of line shapes of the
K
þ
π
π
þ
resonances may occur from two sources: the available
energy in the production process (i.e.
B
K
res
γ
), and a
mass of intermediate-state particles close to threshold, as
for instance in the case of
K
1
ð
1270
Þ
K
ρ
ð
770
Þ
0
. For each
K
þ
π
π
þ
resonance, the first source of distortion is studied
by comparing the invariant-mass distribution generated by
E
VT
G
EN
[31]
to the
R
k
mass used as an input to the
generator. We see no significant distortion. For each
resonance, the second source of distortion is estimated
from the known properties of all decaying processes.
Ideally, one should perform an iterative procedure in which
the input values of the decaying processes are compared
to the results of the fit to the data repeating the procedure until
the fit results converge to values compatible with the inputs.
However, due to the limited size of the data sample, we use
the effective model described in Eqs.
(8)
and
(9)
where no
correction is applied to the line shapes. As described in
Sec.
IV C 1
, this approach describes the data well.
The fit fractions FF
ð
k
Þ
extracted for each resonance, as
well as the interference fit fractions FF
ð
k; l
Þ
between the
same
J
P
resonances, are calculated as
FF
ð
k
Þ¼
j
c
k
j
2
h
R
k
R

k
i
P
μν
ð
c
μ
c

ν
Þh
R
μ
R

ν
i
;
ð
10
Þ
FF
ð
k; l
Þ¼
2
c
k
c

l
Þh
R
k
R

l
ig
P
μν
ð
c
μ
c

ν
Þh
R
μ
R

ν
i
;
ð
11
Þ
where the terms
h
R
μ
R

ν
i
are
h
R
μ
R

ν
Z
R
μ
R

ν
dm:
ð
12
Þ
The sum of fit fractions is defined as the algebraic sum of
all fit fractions. This quantity is not necessarily unity due to
the possible presence of net constructive or destructive
interference.
The branching fraction to the
K
þ
π
þ
π
γ
final state is
determined from the fitted yield of the CR signal event
category,
N
CR
sig
, the weighted CR signal efficiency
h
ε
þ
i
, and
the number of charged
B
events,
N
B

:
B
ð
B
þ
K
þ
π
þ
π
γ
Þ¼
N
CR
sig
h
ε
þ
i
×
N
B

;
ð
13
Þ
with
h
ε
þ
X
k
ε
þ
k
FF
ð
k
Þ
P
l
FF
ð
l
Þ
:
ð
14
Þ
Here,
k
and
l
run over the kaonic resonances,
ε
þ
k
represents
the efficiency without requirement on
m
K
ππ
for resonance
k
listed in Table
III
and FF are the fit fractions extracted from
a binned maximum-likelihood fit to the CR signal
s
P
lot
of
m
K
ππ
plotted in 80 bins. The term
N
B

is obtained from the
total number of
B
̄
B
pairs in the full
BABAR
data set,
N
B
̄
B
,
and the corresponding
Υ
ð
4
S
Þ
branching fraction taken
from Ref.
[18]
:
N
B

¼
2
×
N
B
̄
B
×
B
ð
Υ
ð
4
S
Þ
B
þ
B
Þ
¼ð
483
.
2

6
.
4
Þ
×
10
6
:
ð
15
Þ
The branching fraction of each kaonic resonance,
B
ð
B
þ
K
res
ð
K
þ
π
þ
π
Þ
γ
Þ¼
FF
ð
k
Þ
N
CR
sig
ε
þ0
k
×
N
B

;
ð
16
Þ
is computed using the corresponding fit fraction FF
ð
k
Þ
and
efficiency accounting for the requirement on
m
K
ππ
,
ε
þ0
k
,
listed in Table
III
.
3. The m
K
π
spectrum
In a third step, we perform a binned maximum-
likelihood fit to the efficiency-corrected CR signal
m
K
π
s
P
lot
with 90 bins to extract amplitudes and branching
fractions of the intermediate resonances decaying to
K
þ
π
and
π
þ
π
. The branching fractions of the intermediate state
resonances are obtained using the averaged efficiency
h
ε
þ
i
such that
B
ð
B
þ
Rh
γ
Þ¼
FF
ð
R
Þ
N
CR
sig
h
ε
þ
i
N
B

;
ð
17
Þ
where
R
denotes an intermediate state resonance and
h
is
either a kaon or a pion, and FF
ð
R
Þ
is the corresponding fit
fraction. The resonance
R
is decaying either to
K
þ
π
when
h
¼
π
þ
or to
π
þ
π
when
h
¼
K
þ
. To correct for efficiency
effects, we construct efficiency maps in the
m
K
π
-
m
ππ
plane for each kaonic resonance in the fit model. For
each exclusive decay, the efficiency map is determined
from the phase-space decay of that resonance [
B
þ
K
res
ð
K
þ
π
π
þ
Þ
γ
]. The efficiency map of the combined
sample shown in Fig.
1
is obtained by applying weights
to the individual maps, which were extracted from the fit to
the
m
K
ππ
spectrum. The
m
K
π
spectrum is corrected for
TABLE III. Efficiencies
ε
þ
k
ð
ε
þ0
k
Þ
for correctly reconstructed
signal candidates for each kaonic resonance from simulations
without(with) the applied requirement
m
K
ππ
<
1
.
8
GeV
=c
2
.
K
res
ε
þ
k
ε
þ0
k
K
1
ð
1270
Þ
þ
0
.
2190

0
.
0006
0
.
2130

0
.
0006
K
1
ð
1400
Þ
þ
0
.
2250

0
.
0013
0
.
2110

0
.
0013
K

ð
1410
Þ
þ
0
.
2056

0
.
0012
0
.
1926

0
.
0013
K

2
ð
1430
Þ
þ
0
.
2130

0
.
0015
0
.
2092

0
.
0016
K

ð
1680
Þ
þ
0
.
1878

0
.
0022
0
.
1276

0
.
0020
P. DEL AMO SANCHEZ
et al.
PHYSICAL REVIEW D
93,
052013 (2016)
052013-8
efficiency effects by dividing the (
m
K
π
;m
ππ
)
s
P
lot
dis-
tribution by the combined efficiency map and integrating
over the
m
ππ
dimension. The approach of projecting the
m
ππ
-
m
K
π
phase space of
B
þ
K
þ
π
π
þ
γ
onto the
m
K
π
axis was chosen since the sample size was too small for a
two-dimensional fit. This is further complicated by the
four-body nature of the decay: since the value of
m
K
ππ
can
vary from event to event, the kinematic boundaries for the
m
ππ
-
m
K
π
plane vary as well. We model the
m
K
π
spectrum
as the projection of two
1
P
-wave and one
0
þ
S
-wave
components. The two
P
-wave components, namely the
K

ð
892
Þ
0
and the
ρ
ð
770
Þ
0
resonances, are described by
relativistic Breit-Wigner (RBW) and Gounaris-Sakurai
(GS)
[32]
line shapes, respectively. The
0
þ
(
S
-wave)
component of the
K
π
spectrum, designated by
ð
K
π
Þ

0
0
,is
modeled by the LASS parametrization
[33]
, which consists
of the
K

0
ð
1430
Þ
0
resonance together with an effective
range nonresonant (NR) component.
Due to the relatively low mass of the
K
þ
π
π
þ
reso-
nances, the line shapes of the two-body resonances are
distorted; the phase space is noticeably different for events
below and above the resonance pole mass. To account for
this effect, we model the invariant-mass-dependent magni-
tude of each resonance
R
j
by:
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
H
R
j
ð
m
K
π
;m
ππ
Þ
q
, where H
is a two-dimensional histogram. The
K

ð
892
Þ
0
and
ρ
ð
770
Þ
0
histograms are directly generated from the Monte Carlo
event generator
[31]
, while the LASS parametrized
S
-wave
histogram is obtained by applying weights to the sample of
phase-space-generated events, as described below. To take
into account the interference between the components,
invariant-mass-dependent phases
Φ
R
j
ð
m
Þ
are required. We
make the hypothesis that the phases can be directly taken
from the analytical expression of the corresponding line
shape:
Φ
R
j
ð
m
Þ¼
arccos

½
R
j
ð
m
Þ
j
R
j
ð
m
Þj

8
>
>
>
>
>
>
<
>
>
>
>
>
>
:
m
¼
m
K
π
R
j
ð
m
K
π
Þ
is taken as
RBW for
K

ð
892
Þ
0
and
as LASS for
S
-wave
;
m
¼
m
ππ
R
j
ð
m
ππ
Þ
is taken as a GS
line shape for
ρ
ð
770
Þ
0
;
ð
18
Þ
where the line shapes are taken from the following
expressions.
The RBW parametrization used to determine the corre-
sponding invariant-mass-dependent phase,
Φ
K

ð
m
K
π
Þ
,is
defined as
R
j
ð
m
Þ¼
1
ð
m
2
0
m
2
Þ
im
0
Γ
ð
m
Þ
;
ð
19
Þ
where
m
0
is the nominal mass of the resonance and
Γ
ð
m
Þ
is
the mass-dependent width. In the general case of a spin-
J
resonance, the latter can be expressed as
Γ
ð
m
Þ¼
Γ
0

j
q
j
j
q
j
0

2
J
þ
1

m
0
m

X
2
J
ðj
q
j
r
Þ
X
2
J
ðj
q
j
0
r
Þ
:
ð
20
Þ
The symbol
Γ
0
denotes the nominal width of the resonance.
The values of
m
0
and
Γ
0
are listed in Table
IV
. The symbol
q
is the momentum of one of the resonance daughters,
evaluated in the resonance rest frame. The modulus of
q
is a
function of
m
and the resonance daughter masses
m
a
and
m
b
, given by
j
q
m
2

1
ð
m
a
þ
m
b
Þ
2
m
2

1
=
2

1
ð
m
a
m
b
Þ
2
m
2

1
=
2
:
ð
21
Þ
The symbol
j
q
j
0
denotes the value of
j
q
j
when
m
¼
m
0
.
The
X
J
ðj
q
j
r
Þ
function describes the Blatt-Weisskopf barrier
factor
[34]
with a barrier radius of
r
. Defining the quantity
z
¼j
q
j
r
, the Blatt-Weisskopf barrier function for a spin-1
resonance is given by
X
J
¼
1
ð
z
Þ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffi
1
þ
z
2
0
1
þ
z
2
s
;
ð
22
Þ
where
z
0
represents the value of
z
when
m
¼
m
0
.
For the
ρ
ð
770
Þ
0
we use the GS parametrization, which
describes the
P
-wave scattering amplitude for a broad
resonance decaying to two pions
R
j
ð
m
Þ¼
1
þ
C
·
Γ
0
=m
0
ð
m
2
0
m
2
Þþ
f
ð
m
Þ
im
0
Γ
ð
m
Þ
;
ð
23
Þ
)
2
(GeV/c
π
K
m
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
)
2
(GeV/c
π
π
m
0.4
0.6
0.8
1
1.2
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
FIG. 1. Combination of the efficiency maps for each kaonic
resonance. The relative weights used for the combination are
extracted from a fit to the
m
K
ππ
spectrum (Sec.
IV C
). Large
fluctuations at high
m
K
π
or
m
ππ
are due to the small number of
events.
TIME-DEPENDENT ANALYSIS OF
...
PHYSICAL REVIEW D
93,
052013 (2016)
052013-9