of 28
SLAC-PUB-15425
B
A
B
AR
-PUB-12/026
Measurement of
CP
-violating asymmetries in
B
0
(
ρπ
)
0
decays using a
time-dependent Dalitz plot analysis
J. P. Lees, V. Poireau, and V. Tisserand
Laboratoire d’Annecy-le-Vieux de Physique des Particules (LAPP),
Universit ́e de Savoie, CNRS/IN2P3, F-74941 Annecy-Le-Vieux, France
E. Grauges
Universitat de Barcelona, Facultat de Fisica, Departament ECM, E-08028 Barcelona, Spain
A. Palano
ab
INFN Sezione di Bari
a
; Dipartimento di Fisica, Universit`a di Bari
b
, I-70126 Bari, Italy
G. Eigen and B. Stugu
University of Bergen, Institute of Physics, N-5007 Bergen, Norway
D. N. Brown, L. T. Kerth, Yu. G. Kolomensky, and G. Lynch
Lawrence Berkeley National Laboratory and University of California, Berkeley, California 94720, USA
H. Koch and T. Schroeder
Ruhr Universit ̈at Bochum, Institut f ̈ur Experimentalphysik 1, D-44780 Bochum, Germany
D. J. Asgeirsson, C. Hearty, T. S. Mattison, J. A. McKenna, and R. Y. So
University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z1
A. Khan
Brunel University, Uxbridge, Middlesex UB8 3PH, United Kingdom
V. E. Blinov, A. R. Buzykaev, V. P. Druzhinin, V. B. Golubev, E. A. Kravchenko, A. P. Onuchin,
S. I. Serednyakov, Yu. I. Skovpen, E. P. Solodov, K. Yu. Todyshev, and A. N. Yushkov
Budker Institute of Nuclear Physics, Novosibirsk 630090, Russia
D. Kirkby, A. J. Lankford, and M. Mandelkern
University of California at Irvine, Irvine, California 92697, USA
H. Atmacan, J. W. Gary, O. Long, and G. M. Vitug
University of California at Riverside, Riverside, California 92521, USA
C. Campagnari, T. M. Hong, D. Kovalskyi, J. D. Richman, and C. A. West
University of California at Santa Barbara, Santa Barbara, California 93106, USA
A. M. Eisner, J. Kroseberg, W. S. Lockman, A. J. Martinez, B. A. Schumm, and A. Seiden
University of California at Santa Cruz, Institute for Particle Physics, Santa Cruz, California 95064, USA
D. S. Chao, C. H. Cheng, B. Echenard, K. T. Flood, D. G. Hitlin, P. Ongmongkolkul, F. C. Porter, and A. Y. Rakitin
California Institute of Technology, Pasadena, California 91125, USA
R. Andreassen, Z. Huard, B. T. Meadows, M. D. Sokoloff, and L. Sun
University of Cincinnati, Cincinnati, Ohio 45221, USA
P. C. Bloom, W. T. Ford, A. Gaz, U. Nauenberg, J. G. Smith, and S. R. Wagner
University of Colorado, Boulder, Colorado 80309, USA
R. Ayad
and W. H. Toki
Colorado State University, Fort Collins, Colorado 80523, USA
arXiv:1304.3503v1 [hep-ex] 11 Apr 2013
2
B. Spaan
Technische Universit ̈at Dortmund, Fakult ̈at Physik, D-44221 Dortmund, Germany
K. R. Schubert and R. Schwierz
Technische Universit ̈at Dresden, Institut f ̈ur Kern- und Teilchenphysik, D-01062 Dresden, Germany
D. Bernard and M. Verderi
Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS/IN2P3, F-91128 Palaiseau, France
P. J. Clark and S. Playfer
University of Edinburgh, Edinburgh EH9 3JZ, United Kingdom
D. Bettoni
a
, C. Bozzi
a
, R. Calabrese
ab
, G. Cibinetto
ab
, E. Fioravanti
ab
,
I. Garzia
ab
, E. Luppi
ab
, L. Piemontese
a
, and V. Santoro
a
INFN Sezione di Ferrara
a
; Dipartimento di Fisica, Universit`a di Ferrara
b
, I-44100 Ferrara, Italy
R. Baldini-Ferroli, A. Calcaterra, R. de Sangro, G. Finocchiaro,
P. Patteri, I. M. Peruzzi,
M. Piccolo, M. Rama, and A. Zallo
INFN Laboratori Nazionali di Frascati, I-00044 Frascati, Italy
R. Contri
ab
, E. Guido
ab
, M. Lo Vetere
ab
, M. R. Monge
ab
, S. Passaggio
a
, C. Patrignani
ab
, and E. Robutti
a
INFN Sezione di Genova
a
; Dipartimento di Fisica, Universit`a di Genova
b
, I-16146 Genova, Italy
B. Bhuyan and V. Prasad
Indian Institute of Technology Guwahati, Guwahati, Assam, 781 039, India
M. Morii
Harvard University, Cambridge, Massachusetts 02138, USA
A. Adametz and U. Uwer
Universit ̈at Heidelberg, Physikalisches Institut, Philosophenweg 12, D-69120 Heidelberg, Germany
H. M. Lacker and T. Lueck
Humboldt-Universit ̈at zu Berlin, Institut f ̈ur Physik, Newtonstr. 15, D-12489 Berlin, Germany
P. D. Dauncey
Imperial College London, London, SW7 2AZ, United Kingdom
U. Mallik
University of Iowa, Iowa City, Iowa 52242, USA
C. Chen, J. Cochran, W. T. Meyer, S. Prell, and A. E. Rubin
Iowa State University, Ames, Iowa 50011-3160, USA
A. V. Gritsan
Johns Hopkins University, Baltimore, Maryland 21218, USA
N. Arnaud, M. Davier, D. Derkach, G. Grosdidier, F. Le Diberder, A. M. Lutz,
B. Malaescu, P. Roudeau, M. H. Schune, A. Stocchi, and G. Wormser
Laboratoire de l’Acc ́el ́erateur Lin ́eaire, IN2P3/CNRS et Universit ́e Paris-Sud 11,
Centre Scientifique d’Orsay, B. P. 34, F-91898 Orsay Cedex, France
D. J. Lange and D. M. Wright
Lawrence Livermore National Laboratory, Livermore, California 94550, USA
C. A. Chavez, J. P. Coleman, J. R. Fry, E. Gabathuler, D. E. Hutchcroft, D. J. Payne, and C. Touramanis
University of Liverpool, Liverpool L69 7ZE, United Kingdom
3
A. J. Bevan, F. Di Lodovico, R. Sacco, and M. Sigamani
Queen Mary, University of London, London, E1 4NS, United Kingdom
G. Cowan
University of London, Royal Holloway and Bedford New College, Egham, Surrey TW20 0EX, United Kingdom
D. N. Brown and C. L. Davis
University of Louisville, Louisville, Kentucky 40292, USA
A. G. Denig, M. Fritsch, W. Gradl, K. Griessinger, A. Hafner, and E. Prencipe
Johannes Gutenberg-Universit ̈at Mainz, Institut f ̈ur Kernphysik, D-55099 Mainz, Germany
R. J. Barlow,
G. Jackson, and G. D. Lafferty
University of Manchester, Manchester M13 9PL, United Kingdom
E. Behn, R. Cenci, B. Hamilton, A. Jawahery, and D. A. Roberts
University of Maryland, College Park, Maryland 20742, USA
C. Dallapiccola
University of Massachusetts, Amherst, Massachusetts 01003, USA
R. Cowan, D. Dujmic, and G. Sciolla
Massachusetts Institute of Technology, Laboratory for Nuclear Science, Cambridge, Massachusetts 02139, USA
R. Cheaib, D. Lindemann, P. M. Patel,
§
and S. H. Robertson
McGill University, Montr ́eal, Qu ́ebec, Canada H3A 2T8
P. Biassoni
ab
, N. Neri
a
, F. Palombo
ab
, and S. Stracka
ab
INFN Sezione di Milano
a
; Dipartimento di Fisica, Universit`a di Milano
b
, I-20133 Milano, Italy
L. Cremaldi, R. Godang,
R. Kroeger, P. Sonnek, and D. J. Summers
University of Mississippi, University, Mississippi 38677, USA
X. Nguyen, M. Simard, and P. Taras
Universit ́e de Montr ́eal, Physique des Particules, Montr ́eal, Qu ́ebec, Canada H3C 3J7
G. De Nardo
ab
, D. Monorchio
ab
, G. Onorato
ab
, and C. Sciacca
ab
INFN Sezione di Napoli
a
; Dipartimento di Scienze Fisiche,
Universit`a di Napoli Federico II
b
, I-80126 Napoli, Italy
M. Martinelli and G. Raven
NIKHEF, National Institute for Nuclear Physics and High Energy Physics, NL-1009 DB Amsterdam, The Netherlands
C. P. Jessop, J. M. LoSecco, and W. F. Wang
University of Notre Dame, Notre Dame, Indiana 46556, USA
K. Honscheid and R. Kass
Ohio State University, Columbus, Ohio 43210, USA
J. Brau, R. Frey, N. B. Sinev, D. Strom, and E. Torrence
University of Oregon, Eugene, Oregon 97403, USA
E. Feltresi
ab
, N. Gagliardi
ab
, M. Margoni
ab
, M. Morandin
a
,
M. Posocco
a
, M. Rotondo
a
, G. Simi
a
, F. Simonetto
ab
, and R. Stroili
ab
INFN Sezione di Padova
a
; Dipartimento di Fisica, Universit`a di Padova
b
, I-35131 Padova, Italy
S. Akar, E. Ben-Haim, M. Bomben, G. R. Bonneaud, H. Briand, G. Calderini,
J. Chauveau, O. Hamon, Ph. Leruste, G. Marchiori, J. Ocariz, and S. Sitt
4
Laboratoire de Physique Nucl ́eaire et de Hautes Energies,
IN2P3/CNRS, Universit ́e Pierre et Marie Curie-Paris6,
Universit ́e Denis Diderot-Paris7, F-75252 Paris, France
M. Biasini
ab
, E. Manoni
ab
, S. Pacetti
ab
, and A. Rossi
ab
INFN Sezione di Perugia
a
; Dipartimento di Fisica, Universit`a di Perugia
b
, I-06100 Perugia, Italy
C. Angelini
ab
, G. Batignani
ab
, S. Bettarini
ab
, M. Carpinelli
ab
,
∗∗
G. Casarosa
ab
, A. Cervelli
ab
,
F. Forti
ab
, M. A. Giorgi
ab
, A. Lusiani
ac
, B. Oberhof
ab
, A. Perez
a
, G. Rizzo
ab
, and J. J. Walsh
a
INFN Sezione di Pisa
a
; Dipartimento di Fisica, Universit`a di Pisa
b
; Scuola Normale Superiore di Pisa
c
, I-56127 Pisa, Italy
D. Lopes Pegna, J. Olsen, and A. J. S. Smith
Princeton University, Princeton, New Jersey 08544, USA
F. Anulli
a
, R. Faccini
ab
, F. Ferrarotto
a
, F. Ferroni
ab
, M. Gaspero
ab
, L. Li Gioi
a
, M. A. Mazzoni
a
, and G. Piredda
a
INFN Sezione di Roma
a
; Dipartimento di Fisica,
Universit`a di Roma La Sapienza
b
, I-00185 Roma, Italy
C. B ̈unger, O. Gr ̈unberg, T. Hartmann, T. Leddig, C. Voß, and R. Waldi
Universit ̈at Rostock, D-18051 Rostock, Germany
T. Adye, E. O. Olaiya, and F. F. Wilson
Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, OX11 0QX, United Kingdom
S. Emery, G. Hamel de Monchenault, G. Vasseur, and Ch. Y`eche
CEA, Irfu, SPP, Centre de Saclay, F-91191 Gif-sur-Yvette, France
D. Aston, R. Bartoldus, J. F. Benitez, C. Cartaro, M. R. Convery, J. Dorfan, G. P. Dubois-Felsmann,
W. Dunwoodie, M. Ebert, R. C. Field, M. Franco Sevilla, B. G. Fulsom, A. M. Gabareen, M. T. Graham,
P. Grenier, C. Hast, W. R. Innes, M. H. Kelsey, P. Kim, M. L. Kocian, D. W. G. S. Leith, P. Lewis, B. Lindquist,
S. Luitz, V. Luth, H. L. Lynch, D. B. MacFarlane, D. R. Muller, H. Neal, S. Nelson, M. Perl, T. Pulliam,
B. N. Ratcliff, A. Roodman, A. A. Salnikov, R. H. Schindler, A. Snyder, D. Su, M. K. Sullivan, J. Va’vra,
A. P. Wagner, W. J. Wisniewski, M. Wittgen, D. H. Wright, H. W. Wulsin, C. C. Young, and V. Ziegler
SLAC National Accelerator Laboratory, Stanford, California 94309 USA
W. Park, M. V. Purohit, R. M. White, and J. R. Wilson
University of South Carolina, Columbia, South Carolina 29208, USA
A. Randle-Conde and S. J. Sekula
Southern Methodist University, Dallas, Texas 75275, USA
M. Bellis, P. R. Burchat, T. S. Miyashita, and E. M. T. Puccio
Stanford University, Stanford, California 94305-4060, USA
M. S. Alam and J. A. Ernst
State University of New York, Albany, New York 12222, USA
R. Gorodeisky, N. Guttman, D. R. Peimer, and A. Soffer
Tel Aviv University, School of Physics and Astronomy, Tel Aviv, 69978, Israel
S. M. Spanier
University of Tennessee, Knoxville, Tennessee 37996, USA
J. L. Ritchie, A. M. Ruland, R. F. Schwitters, and B. C. Wray
University of Texas at Austin, Austin, Texas 78712, USA
J. M. Izen and X. C. Lou
University of Texas at Dallas, Richardson, Texas 75083, USA
5
F. Bianchi
ab
, D. Gamba
ab
, and S. Zambito
ab
INFN Sezione di Torino
a
; Dipartimento di Fisica Sperimentale, Universit`a di Torino
b
, I-10125 Torino, Italy
L. Lanceri
ab
and L. Vitale
ab
INFN Sezione di Trieste
a
; Dipartimento di Fisica, Universit`a di Trieste
b
, I-34127 Trieste, Italy
F. Martinez-Vidal, A. Oyanguren, and P. Villanueva-Perez
IFIC, Universitat de Valencia-CSIC, E-46071 Valencia, Spain
H. Ahmed, J. Albert, Sw. Banerjee, F. U. Bernlochner, H. H. F. Choi, G. J. King,
R. Kowalewski, M. J. Lewczuk, I. M. Nugent, J. M. Roney, R. J. Sobie, and N. Tasneem
University of Victoria, Victoria, British Columbia, Canada V8W 3P6
T. J. Gershon, P. F. Harrison, and T. E. Latham
Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom
H. R. Band, S. Dasu, Y. Pan, R. Prepost, and S. L. Wu
University of Wisconsin, Madison, Wisconsin 53706, USA
We present results for a time-dependent Dalitz plot measurement of
CP
-violating asymmetries in
the mode
B
0
π
+
π
π
0
. The dataset is derived from the complete sample of 471
×
10
6
B
B
meson
pairs collected with the
B
A
B
AR
detector at the PEP-II asymmetric-energy
e
+
e
collider at the
SLAC National Accelerator Laboratory operating on the Υ(4
S
) resonance. We extract parameters
describing the time-dependent
B
0
ρπ
decay probabilities and
CP
asymmetries, including
C
=
0
.
016
±
0
.
059
±
0
.
036, ∆
C
= 0
.
234
±
0
.
061
±
0
.
048,
S
= 0
.
053
±
0
.
081
±
0
.
034, and ∆
S
= 0
.
054
±
0
.
082
±
0
.
039, where the uncertainties are statistical and systematic, respectively. We perform a
two-dimensional likelihood scan of the direct
CP
-violation asymmetry parameters for
B
0
ρ
±
π
decays, finding the change in
χ
2
between the minimum and the origin (corresponding to no direct
CP
violation) to be ∆
χ
2
= 6
.
42. We present information on the
CP
-violating parameter
α
in a
likelihood scan that incorporates
B
±
ρπ
measurements.
PACS numbers: 11.30.Rd, 11.30.Er, 12.15.Ff
I. INTRODUCTION
Within the standard model (SM) of particle physics,
CP
violation in the quark sector is described by the
Cabibbo-Kobayashi-Maskawa (CKM) quark-mixing ma-
trix. Physics beyond the SM may result in measured val-
ues of observables that deviate from the values expected
based on other CKM parameter measurements and the
SM.
The decay
B
0
π
+
π
π
0
[1] is well suited to the study
of
CP
violation and has been previously explored by both
the
B
A
B
AR
[2] and Belle [3] Collaborations. Early studies
of this mode were “quasi-two-body” (Q2B) analyses that
treated each
ρ
resonance separately in the decays
B
0
ρ
0
π
0
(
ρ
0
π
+
π
) and
B
0
ρ
±
π
(
ρ
±
π
±
π
0
). How-
Now at the University of Tabuk, Tabuk 71491, Saudi Arabia
Also with Universit`a di Perugia, Dipartimento di Fisica, Perugia,
Italy
Now at the University of Huddersfield, Huddersfield HD1 3DH,
UK
§
Deceased
Now at University of South Alabama, Mobile, Alabama 36688,
USA
∗∗
Also with Universit`a di Sassari, Sassari, Italy
ever, as first noted by Snyder and Quinn [4], a complete
time-dependent Dalitz plot (DP) analysis is sensitive to
the interference between the strong and weak amplitudes
in the regions where the
ρ
+
,
ρ
, and
ρ
0
resonances over-
lap. This interference allows the unambiguous extraction
of the strong and weak relative phases, and of the
CP
-
violating parameter
α
arg[
V
td
V
tb
/
(
V
ud
V
ub
)], where
V
qq
are components of the CKM matrix. A precision
measurement of
α
is of interest because it serves to fur-
ther test the SM and constrain new physics that may
contribute to loops in Feynman diagrams.
In this paper, we present an update of an earlier
B
A
B
AR
analysis. We use the full
B
A
B
AR
dataset collected at the
Υ(4
S
) resonance, corresponding to an increase of 25% in
the number of
B
meson decays, and include a number of
improvements to both the reconstruction and selection
procedures. Among these are improved charged-particle
tracking and particle identification (PID), and a reop-
timized multivariate discriminator, used both for event
selection and as a variable in the final fit.
Section II contains an introduction to the theory be-
hind this analysis and the formalism used. We proceed
to descriptions of the detector (Sec. III), the datasets
(Sec. IV), and the event selection procedures (Sec. V).
This is followed by a presentation of the fitting proce-
dure (Sec. VI) and of the systematic studies (Sec. VII).
6
Finally, we present the fit results (Sec. VIII) and a con-
clusion (Sec. IX). An overview of robustness studies is
provided in an Appendix.
II. THEORY OVERVIEW
A. Time-Independent Probability Distribution
The time-independent amplitudes for
B
0
and
B
0
de-
cays to
π
+
π
π
0
are given by
A
3
π
=
f
+
A
+
+
f
A
+
f
0
A
0
,
A
3
π
=
f
+
A
+
+
f
A
+
f
0
A
0
,
(1)
respectively, where
A
κ
and
A
κ
with
κ
∈ {
+
,
,
0
}
are
complex amplitudes associated with the
ρ
+
,
ρ
, and
ρ
0
resonances, respectively, and
f
κ
=
f
κ
(
m,θ
κ
) are de-
fined in terms of modified relativistic Breit-Wigner reso-
nances [5] modeling the three
ρ
resonances. The angle
θ
κ
is the helicity angle for the resonance, defined as the an-
gle between the
π
0
(
π
) momentum and the negative of
the momentum of the recoiling
π
(
π
+
) for the
ρ
+
(
ρ
),
and as the angle between the
π
+
momentum and the neg-
ative of the momentum of the recoiling
π
0
for the
ρ
0
. All
helicity angles are calculated in the
ρ
rest frame. In the
fit, we include the
ρ
(770) as well as its radial excitation,
the
ρ
(1450); therefore, each
f
κ
is a sum of modified rela-
tivistic Breit-Wigner resonances,
F
, for the
ρ
κ
(770) and
ρ
κ
(1450):
f
κ
(
m,θ
κ
)
F
ρ
(770)
(
m,θ
κ
) +
a
ρ
e
ρ
F
ρ
(1450)
(
m,θ
κ
)
,
(2)
where
a
ρ
and
φ
ρ
are the magnitude and phase of the
ρ
(1450) resonance relative to the
ρ
(770). We include sys-
tematic uncertainties, described in Sec. VII A, to account
for possible contributions from the
ρ
(1700).
B. Time-Dependent Probability Distribution
Using the time-independent amplitudes
A
3
π
and
A
3
π
,
we can express the full time-dependent probability for a
meson that is a
B
0
(
A
3
π
) or
B
0
(
A
+
3
π
) at the time the
other
B
meson decays, to decay to
π
+
π
π
0
as
|A
±
3
π
(∆
t
)
|
2
=
e
−|
t
|
B
0
4
τ
B
0
(
|
A
3
π
|
2
+
|
A
3
π
|
2
(
|
A
3
π
|
2
−|
A
3
π
|
2
)
cos(∆
m
d
t
)
±
2 Im
[
q
p
A
3
π
A
3
π
]
sin(∆
m
d
t
)
)
,
(3)
where
τ
B
0
is the mean neutral
B
lifetime, ∆
m
d
is the
mass difference between the heavy and light neutral
B
mass eigenstates,
p
and
q
are the complex param-
eters in the definitions of the neutral mass eigenstates
p
|
B
0
〉 ±
q
|
B
0
, and ∆
t
is the time difference between
the decays of the fully reconstructed
B
meson (
B
3
π
) and
the
B
meson used to determine the
B
flavor (
B
tag
). In
Eq. (3), as in the fit, we assume that the heavy and light
mass eigenstates have the same lifetime, that there is no
CP
violation in
B
0
B
0
mixing (
|
q/p
|
= 1), and that
CPT
is conserved.
C. Square Dalitz Plot Formalism
While nonresonant phase-space decays uniformly pop-
ulate the kinematically allowed region of a DP, signal
ρπ
events populate the boundaries of this region due to
the low mass of the
ρ
resonances relative to the
B
mass.
In particular, the interference regions of the signal DP,
which provide sensitivity to the relative phases of the
ρ
resonances, are confined to small regions in the three cor-
ners of the DP. In order to expand these regions of inter-
est and avoid the use of bins of variable size, we perform
a transformation of the DP that maps the kinematically
allowed region onto a dimensionless unit square. The
transformation is described by
dm
+
dm
→|
det
J
|
dm
,
(4)
with the square Dalitz plot (SDP) coordinates
m
1
π
arccos
(
2
m
0
m
min
0
m
max
0
m
min
0
1
)
,
(5)
θ
1
π
θ
0
,
(6)
where
m
±
is the invariant mass of the
π
±
π
0
system,
m
0
is
the invariant mass of the two charged pion candidates,
θ
0
is the
ρ
0
helicity angle defined earlier,
m
max
0
=
m
B
0
m
π
0
and
m
min
0
= 2
m
π
+
are the kinematic limits of the
m
0
mass, and
J
is the Jacobian of the transformation. The
determinant of the Jacobian is given by
|
det
J
|
= 4
|
p
+
||
p
0
|
m
0
∂m
0
∂m
cos
θ
0
∂θ
,
(7)
where
|
p
+
|
=
(
E
+
)
2
m
2
π
+
,
(8)
|
p
0
|
=
(
E
0
)
2
m
2
π
0
,
(9)
and the energies
E
+
and
E
0
of the
π
+
and
π
0
are defined
in the
π
+
π
center-of-mass (CM) frame. Figure 1 shows
an example of a standard DP (left) and its transformed
SDP counterpart (right), plotted using simulated
B
0
ρπ
decays, where the three
ρ
resonances are assumed to
have the same amplitude.
7
0
5
10
15
20
25
30
0
5
10
15
20
25
30
m
2
(
π
+
π
0
) (GeV/c
2
)
2
m
2
(
π
π
0
) (GeV/c
2
)
2
0
1
2
3
4
5
22
23
24
25
26
27
B
0
π
+
π
π
0
(kin.)
interference regs.
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
100
200
300
400
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
100
200
300
400
θ
'
m'
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
m'
θ
'
interference regions
m(
ρ
0
)
=
1.5 GeV/c
2
m(
ρ
+
)
=
1.5 GeV/c
2
m(
ρ
)
=
1.5 GeV/c
2
FIG. 1: (color online) Nominal (left) and square (right)
B
0
ρπ
Dalitz plots obtained from Monte Carlo generated events
without detector simulation [2]. The amplitudes in Eq. (1) are generated with values
A
+
=
A
=
A
0
= 1 so that they interfere
destructively for equal
ππ
masses. The hatched areas indicate the main overlap regions between the different
ρ
bands. The
dashed lines in the square Dalitz plot correspond to
m
(
ρ
+
,
,
0
) = 1
.
5 GeV
/c
2
. The middle plot depicts the Jacobian determinant
of the transformation and shows the distribution in the square Dalitz plot for uniformly distributed events in the nominal Dalitz
plot.
D.
U/I
Formalism
If one explicitly inserts Eq. (1) into Eq. (3), the full
time-dependent amplitude for a
B
0
or
B
0
meson to decay
to
π
+
π
π
0
can be written in terms of
|A
3
π
|
2
±|
A
3
π
|
2
=
κ
[+
,
,
0]
|
f
κ
|
2
U
±
κ
+2
κ<σ
[+
,
,
0]
(
Re [
f
κ
f
σ
]
U
±
,
Re
κσ
Im [
f
κ
f
σ
]
U
±
,
Im
κσ
)
,
(10)
and
Im
[
q
p
A
3
π
A
3
π
]
=
κ
[+
,
,
0]
|
f
κ
|
2
I
κ
+
κ<σ
[+
,
,
0]
(
Re [
f
κ
f
σ
]
I
Im
κσ
+Im [
f
κ
f
σ
]
I
Re
κσ
)
,
(11)
with
U
±
κ
=
|
A
κ
|
2
±|
A
κ
|
2
,
(12)
U
±
,
Re(Im)
κσ
= Re(Im)
[
A
κ
A
σ
±
A
κ
A
σ
]
,
(13)
I
κ
= Im
[
A
κ
A
κ
]
,
(14)
I
Re
κσ
= Re
[
A
κ
A
σ
A
σ
A
κ
]
,
(15)
I
Im
κσ
= Im
[
A
κ
A
σ
+
A
σ
A
κ
]
.
(16)
The 27 real-valued
U
and
I
coefficients provide an al-
ternative parameterization to tree and penguin ampli-
tudes (as well as
α
) or to the amplitudes
A
κ
and
A
κ
[6].
The
U
and
I
parameters can also be directly related
to the Q2B
C
and
S
parameters often used in
CP
-
violation analyses [7], where
C
parameterizes direct
CP
violation, and
S
parameterizes mixing-induced
CP
vio-
lation (involving the angle
α
in this analysis). The re-
lated parameter ∆
C
describes the asymmetry between
the rates Γ(
B
0
ρ
+
π
) + Γ(
B
0
ρ
π
+
) and Γ(
B
0
ρ
π
+
) + Γ(
B
0
ρ
+
π
), while ∆
S
is related to the
strong phase difference between the different amplitudes
describing the decay
B
0
ρπ
. The
U
and
I
parame-
ters are related to the
C
and
S
parameters through the
relations
C
+
=
U
+
U
+
+
,
C
=
U
U
+
,
S
+
=
2
I
+
U
+
+
,
S
=
2
I
U
+
,
(17)
and
A
ρπ
=
U
+
+
U
+
U
+
+
+
U
+
,
(18)
where
C
= (
C
+
+
C
)
/
2
,
(19)
C
= (
C
+
−C
)
/
2
,
(20)
S
= (
S
+
+
S
)
/
2
,
(21)
S
= (
S
+
−S
)
/
2
.
(22)
Note that while
C
,
S
, ∆
C
, and ∆
S
do not depend on
interference effects between the
ρ
resonances, the
U
and
I
parameter formalism accounts for these features and
is thus appropriate for a full DP analysis. While some
degree of physical intuition is lost when using the
U
and
I
parameters instead of the standard complex amplitudes