of 10
Measurement of the neutral
D
meson mixing parameters in a
time-dependent amplitude analysis of the
D
0
π
þ
π
π
0
decay
J. P. Lees,
1
V. Poireau,
1
V. Tisserand,
1
E. Grauges,
2
A. Palano,
3
G. Eigen,
4
D. N. Brown,
5
Yu. G. Kolomensky,
5
H. Koch,
6
T. Schroeder,
6
C. Hearty,
7
T. S. Mattison,
7
J. A. McKenna,
7
R. Y. So,
7
V. E. Blinov,
8a,8b,8c
A. R. Buzykaev,
8a
V. P. Druzhinin,
8a,8b
V. B. Golubev,
8a,8b
E. A. Kravchenko,
8a,8b
A. P. Onuchin,
8a,8b,8c
S. I. Serednyakov,
8a,8b
Yu. I. Skovpen,
8a,8b
E. P. Solodov,
8a,8b
K. Yu. Todyshev,
8a,8b
A. J. Lankford,
9
J. W. Gary,
10
O. Long,
10
A. M. Eisner,
11
W. S. Lockman,
11
W. Panduro Vazquez,
11
D. S. Chao,
12
C. H. Cheng,
12
B. Echenard,
12
K. T. Flood,
12
D. G. Hitlin,
12
J. Kim,
12
T. S. Miyashita,
12
P. Ongmongkolkul,
12
F. C. Porter,
12
M. Röhrken,
12
Z. Huard,
13
B. T. Meadows,
13
B. G. Pushpawela,
13
M. D. Sokoloff,
13
L. Sun,
13
*
J. G. Smith,
14
S. R. Wagner,
14
D. Bernard,
15
M. Verderi,
15
D. Bettoni,
16a
C. Bozzi,
16a
R. Calabrese,
16a,16b
G. Cibinetto,
16a,16b
E. Fioravanti,
16a,16b
I. Garzia,
16a,16b
E. Luppi,
16a,16b
V. Santoro,
16a
A. Calcaterra,
17
R. de Sangro,
17
G. Finocchiaro,
17
S. Martellotti,
17
P. Patteri,
17
I. M. Peruzzi,
17
M. Piccolo,
17
A. Zallo,
17
S. Passaggio,
18
C. Patrignani,
18
B. Bhuyan,
19
U. Mallik,
20
C. Chen,
21
J. Cochran,
21
S. Prell,
21
H. Ahmed,
22
A. V. Gritsan,
23
N. Arnaud,
24
M. Davier,
24
F. Le Diberder,
24
A. M. Lutz,
24
G. Wormser,
24
D. J. Lange,
25
D. M. Wright,
25
J. P. Coleman,
26
E. Gabathuler,
26
D. E. Hutchcroft,
26
D. J. Payne,
26
C. Touramanis,
26
A. J. Bevan,
27
F. Di Lodovico,
27
R. Sacco,
27
G. Cowan,
28
Sw. Banerjee,
29
D. N. Brown,
29
C. L. Davis,
29
A. G. Denig,
30
M. Fritsch,
30
W. Gradl,
30
K. Griessinger,
30
A. Hafner,
30
K. R. Schubert,
30
R. J. Barlow,
31
G. D. Lafferty,
31
R. Cenci,
32
A. Jawahery,
32
D. A. Roberts,
32
R. Cowan,
33
R. Cheaib,
34
S. H. Robertson,
34
B. Dey,
35a
N. Neri,
35a
F. Palombo,
35a,35b
L. Cremaldi,
36
R. Godang,
36
§
D. J. Summers,
36
P. Taras,
37
G. De Nardo,
38
C. Sciacca,
38
G. Raven,
39
C. P. Jessop,
40
J. M. LoSecco,
40
K. Honscheid,
41
R. Kass,
41
A. Gaz,
42a
M. Margoni,
42a,42b
M. Posocco,
42a
M. Rotondo,
42a
G. Simi,
42a,42b
F. Simonetto,
42a,42b
R. Stroili,
42a,42b
S. Akar,
43
E. Ben-Haim,
43
M. Bomben,
43
G. R. Bonneaud,
43
G. Calderini,
43
J. Chauveau,
43
G. Marchiori,
43
J. Ocariz,
43
M. Biasini,
44a,44b
E. Manoni,
44a
A. Rossi,
44a
G. Batignani,
45a,45b
S. Bettarini,
45a,45b
M. Carpinelli,
45a,45b
G. Casarosa,
45a,45b
M. Chrzaszcz,
45a
F. Forti,
45a,45b
M. A. Giorgi,
45a,45b
A. Lusiani,
45a,45c
B. Oberhof,
45a,45b
E. Paoloni,
45a,45b
M. Rama,
45a
G. Rizzo,
45a,45b
J. J. Walsh,
45a
A. J. S. Smith,
46
F. Anulli,
47a
R. Faccini,
47a,47b
F. Ferrarotto,
47a
F. Ferroni,
47a,47b
A. Pilloni,
47a,47b
G. Piredda,
47a
C. Bünger,
48
S. Dittrich,
48
O. Grünberg,
48
M. Heß,
48
T. Leddig,
48
C. Voß,
48
R. Waldi,
48
T. Adye,
49
F. F. Wilson,
49
S. Emery,
50
G. Vasseur,
50
D. Aston,
51
C. Cartaro,
51
M. R. Convery,
51
J. Dorfan,
51
W. Dunwoodie,
51
M. Ebert,
51
R. C. Field,
51
B. G. Fulsom,
51
M. T. Graham,
51
C. Hast,
51
W. R. Innes,
51
P. Kim,
51
D. W. G. S. Leith,
51
S. Luitz,
51
V. Luth,
51
D. B. MacFarlane,
51
D. R. Muller,
51
H. Neal,
51
B. N. Ratcliff,
51
A. Roodman,
51
M. K. Sullivan,
51
J. Va
vra,
51
W. J. Wisniewski,
51
M. V. Purohit,
52
J. R. Wilson,
52
A. Randle-Conde,
53
S. J. Sekula,
53
M. Bellis,
54
P. R. Burchat,
54
E. M. T. Puccio,
54
M. S. Alam,
55
J. A. Ernst,
55
R. Gorodeisky,
56
N. Guttman,
56
D. R. Peimer,
56
A. Soffer,
56
S. M. Spanier,
57
J. L. Ritchie,
58
R. F. Schwitters,
58
J. M. Izen,
59
X. C. Lou,
59
F. Bianchi,
60a,60b
F. De Mori,
60a,60b
A. Filippi,
60a
D. Gamba,
60a,60b
L. Lanceri,
61
L. Vitale,
61
F. Martinez-Vidal,
62
A. Oyanguren,
62
J. Albert,
63
A. Beaulieu,
63
F. U. Bernlochner,
63
G. J. King,
63
R. Kowalewski,
63
T. Lueck,
63
I. M. Nugent,
63
J. M. Roney,
63
N. Tasneem,
63
T. J. Gershon,
64
P. F. Harrison,
64
T. E. Latham,
64
R. Prepost,
65
and S. L. Wu
65
(
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
3
INFN Sezione di Bari and 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 V6T 1Z1, Canada
8a
Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090, Russia
8b
Novosibirsk State University, Novosibirsk 630090, Russia
8c
Novosibirsk State Technical University, Novosibirsk 630092, Russia
9
University of California at Irvine, Irvine, California 92697, USA
10
University of California at Riverside, Riverside, California 92521, USA
11
University of California at Santa Cruz, Institute for Particle Physics, Santa Cruz, California 95064, USA
12
California Institute of Technology, Pasadena, California 91125, USA
13
University of Cincinnati, Cincinnati, Ohio 45221, USA
14
University of Colorado, Boulder, Colorado 80309, USA
15
Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS/IN2P3, F-91128 Palaiseau, France
16a
INFN Sezione di Ferrara, I-44122 Ferrara, Italy
16b
Dipartimento di Fisica e Scienze della Terra, Università di Ferrara, I-44122 Ferrara, Italy
PHYSICAL REVIEW D
93,
112014 (2016)
2470-0010
=
2016
=
93(11)
=
112014(10)
112014-1
© 2016 American Physical Society
17
INFN Laboratori Nazionali di Frascati, I-00044 Frascati, Italy
18
INFN Sezione di Genova, I-16146 Genova, Italy
19
Indian Institute of Technology Guwahati, Guwahati, Assam 781 039, India
20
University of Iowa, Iowa City, Iowa 52242, USA
21
Iowa State University, Ames, Iowa 50011, USA
22
Physics Department, Jazan University, Jazan 22822, Kingdom of Saudi Arabia
23
Johns Hopkins University, Baltimore, Maryland 21218, USA
24
Laboratoire de l
Accélérateur Linéaire, IN2P3/CNRS et Université Paris-Sud 11,
Centre Scientifique d
Orsay, F-91898 Orsay Cedex, France
25
Lawrence Livermore National Laboratory, Livermore, California 94550, USA
26
University of Liverpool, Liverpool L69 7ZE, United Kingdom
27
Queen Mary, University of London, London E1 4NS, United Kingdom
28
University of London, Royal Holloway and Bedford New College, Egham,
Surrey TW20 0EX, United Kingdom
29
University of Louisville, Louisville, Kentucky 40292, USA
30
Johannes Gutenberg-Universität Mainz, Institut für Kernphysik, D-55099 Mainz, Germany
31
University of Manchester, Manchester M13 9PL, United Kingdom
32
University of Maryland, College Park, Maryland 20742, USA
33
Massachusetts Institute of Technology, Laboratory for Nuclear Science, Cambridge,
Massachusetts 02139, USA
34
McGill University, Montréal, Québec H3A 2T8, Canada
35a
INFN Sezione di Milano, I-20133 Milano, Italy
35b
Dipartimento di Fisica, Università di Milano, I-20133 Milano, Italy
36
University of Mississippi, University, Mississippi 38677, USA
37
Université de Montréal, Physique des Particules, Montréal, Québec H3C 3J7, Canada
38
INFN Sezione di Napoli and Dipartimento di Scienze Fisiche, Università di Napoli Federico II,
I-80126 Napoli, Italy
39
NIKHEF, National Institute for Nuclear Physics and High Energy Physics, NL-1009 DB Amsterdam,
The Netherlands
40
University of Notre Dame, Notre Dame, Indiana 46556, USA
41
Ohio State University, Columbus, Ohio 43210, USA
42a
INFN Sezione di Padova, I-35131 Padova, Italy
42b
Dipartimento di Fisica, Università di Padova, I-35131 Padova, Italy
43
Laboratoire de Physique Nucléaire et de Hautes Energies, IN2P3/CNRS, Université Pierre et Marie
Curie-Paris6, Université Denis Diderot-Paris7, F-75252 Paris, France
44a
INFN Sezione di Perugia, I-06123 Perugia, Italy
44b
Dipartimento di Fisica, Università di Perugia, I-06123 Perugia, Italy
45a
INFN Sezione di Pisa, I-56127 Pisa, Italy
45b
Dipartimento di Fisica, Università di Pisa, I-56127 Pisa, Italy
45c
Scuola Normale Superiore di Pisa, I-56127 Pisa, Italy
46
Princeton University, Princeton, New Jersey 08544, USA
47a
INFN Sezione di Roma, I-00185 Roma, Italy
47b
Dipartimento di Fisica, Università di Roma La Sapienza, I-00185 Roma, Italy
48
Universität Rostock, D-18051 Rostock, Germany
49
Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 0QX, United Kingdom
50
CEA, Irfu, SPP, Centre de Saclay, F-91191 Gif-sur-Yvette, France
51
SLAC National Accelerator Laboratory, Stanford, California 94309, USA
52
University of South Carolina, Columbia, South Carolina 29208, USA
53
Southern Methodist University, Dallas, Texas 75275, USA
54
Stanford University, Stanford, California 94305, USA
55
State University of New York, Albany, New York 12222, USA
56
School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel
57
University of Tennessee, Knoxville, Tennessee 37996, USA
58
University of Texas at Austin, Austin, Texas 78712, USA
59
University of Texas at Dallas, Richardson, Texas 75083, USA
60a
INFN Sezione di Torino, I-10125 Torino, Italy
60b
Dipartimento di Fisica, Università di Torino, I-10125 Torino, Italy
61
INFN Sezione di Trieste and Dipartimento di Fisica, Università di Trieste, I-34127 Trieste, Italy
62
IFIC, Universitat de Valencia-CSIC, E-46071 Valencia, Spain
63
University of Victoria, Victoria, British Columbia V8W 3P6, Canada
J. P. LEES
et al.
PHYSICAL REVIEW D
93,
112014 (2016)
112014-2
64
Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom
65
University of Wisconsin, Madison, Wisconsin 53706, USA
(Received 4 April 2016; published 28 June 2016)
We perform the first measurement on the
D
0
̄
D
0
mixing parameters using a time-dependent amplitude
analysis of the decay
D
0
π
þ
π
π
0
. The data were recorded with the
BABAR
detector at center-of-mass
energies at and near the
Υ
ð
4
S
Þ
resonance, and correspond to an integrated luminosity of approximately
468
.
1
fb
1
. The neutral
D
meson candidates are selected from
D

ð
2010
Þ
þ
D
0
π
þ
s
decays where the
flavor at the production is identified by the charge of the low-momentum pion,
π
þ
s
. The measured mixing
parameters are
x
¼ð
1
.
5

1
.
2

0
.
6
Þ
%
and
y
¼ð
0
.
2

0
.
9

0
.
5
Þ
%
, where the quoted uncertainties are
statistical and systematic, respectively.
DOI:
10.1103/PhysRevD.93.112014
I. INTRODUCTION
The first evidence for
D
0
-
̄
D
0
mixing, which had been
sought for more than two decades since it was first
predicted
[1]
, was obtained by
BABAR
[2]
and Belle
[3]
in 2007. These results were rapidly confirmed by CDF
[4]
.
The techniques utilized in those analyses and more recent,
much higher statistics LHCb analyses
[5
7]
do
not
directly
measure the normalized mass and the width differences
of the neutral
D
eigenstates,
x
and
y
. In contrast, a time-
dependent amplitude analysis of the Dalitz plot (DP) of
neutral
D
mesons decaying into self-conjugate final states
provides direct measurements of both these parameters. This
technique was introduced using
D
0
K
0
S
π
π
þ
decays by
the CLEO Collaboration
[8]
, and the first measurement by
the Belle Collaboration
[9]
provided stringent constraints on
the mixing parameters. More recent measurements with this
final state by the
BABAR
and Belle Collaborations
[10,11]
contribute significantly to the Heavy Flavor Averaging
Group (HFAG) global fits that determine world average
mixing and
CP
violation parameter values
[12]
.
This paper reports the first measurement of mixing
parameters from a time-dependent amplitude analysis of
the singly Cabibbo-suppressed decay
D
0
π
þ
π
π
0
. The
inclusion of charge conjugate reactions is implied through-
out this paper. No measurement of
CP
violation is attempted
as the data set lacks sufficient sensitivity to be interesting.
The
D
0
candidates are selected from
D

ð
2010
Þ
þ
D
0
π
þ
s
decays where the
D
0
flavor at production is identified by the
charge of the slow pion,
π
þ
s
.
The
D
0
and
̄
D
0
meson flavor eigenstates evolveand decay
as mixtures of the weak Hamiltonian eigenstates
D
1
and
D
2
with masses and widths
m
1
,
Γ
1
and
m
2
,
Γ
2
, respectively. The
mass eigenstates can be expressed as superpositions of the
flavor eigenstates,
j
D
1
;
2
p
j
D
0
i
q
j
̄
D
0
i
where the com-
plex coefficients
p
and
q
satisfy
j
p
j
2
þj
q
j
2
¼
1
. The mixing
parameters are defined as normalized mass and width
differences,
x
ð
m
1
m
2
Þ
=
Γ
D
and
y
ð
Γ
1
Γ
2
Þ
=
2
Γ
D
.
Here,
Γ
D
is the average decay width,
Γ
D
ð
Γ
1
þ
Γ
2
Þ
=
2
.
These mixing parameters appear in the expression for
the decay rate at each point
ð
s
þ
;s
Þ
in the
D
0
decay
Dalitz plot at the decay time
t
, where
s

m
2
ð
π

π
0
Þ
.
For a charm meson tagged at
t
¼
0
as a
D
0
, the decay rate is
proportional to
j
M
ð
D
0
Þj
2
1
2
e
Γ
D
t

j
A
f
j
2
½
cosh
ð
y
Γ
D
t
Þþ
cos
ð
x
Γ
D
t
Þ
þ




q
p
̄
A
f




2
½
cosh
ð
y
Γ
D
t
Þ
cos
ð
x
Γ
D
t
Þ
2

Re

q
p
A

f
̄
A
f

sinh
ð
y
Γ
D
t
Þ
Im

q
p
A

f
̄
A
f

sin
ð
x
Γ
D
t
Þ

;
ð
1
Þ
where
f
represents the
π
þ
π
π
0
final state that is commonly
accessible to decays of both flavor eigenstates, and
A
f
and
̄
A
f
are the decay amplitudes for
D
0
and
̄
D
0
to final state
f
.
The amplitudes are functions of position in the DP and are
defined in our description of the fitting model in Sec.
IVA
Eq.
(3)
. In Eq.
(1)
, the first term is the direct decay rate to the
final state
f
and is always the dominant term for sufficiently
small decay times. The second term corresponds to mixing.
Initially, the cosh
ð
y
Γ
D
t
Þ
and cos
ð
x
Γ
D
t
Þ
contributions to this
term cancel, but over time the cosh
ð
y
Γ
D
t
Þ
contribution can
become dominant. The third term is the interference term. It
depends explicitly on the real and imaginary parts of
A

f
̄
A
f
and on the real and imaginary parts of
q=p
. As for the mixing
rate, the interference rate is initially zero, but it can become
important at later decay times. The variation of the total
decay rate from purely exponential depends on the relative
strengths of the direct and mixing amplitudes, their relative
phases, the mixing parameters
x
and
y
, and on the magnitude
*
Present address: Wuhan University, Wuhan 43072, China.
Present address: Università di Bologna and INFN Sezione di
Bologna, I-47921 Rimini, Italy.
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.
MEASUREMENT OF THE NEUTRAL
D
MESON MIXING
...
PHYSICAL REVIEW D
93,
112014 (2016)
112014-3
and phase of
q=p
. HFAG reports the world averages to be
x
¼ð
0
.
49
þ
0
.
14
0
.
15
Þ
%
and
y
¼ð
0
.
61

0
.
08
Þ
%
assumingno
CP
violation
[12]
.
In this time-dependent amplitude analysis of the DP, we
measure
x
,
y
,
τ
D
1
=
Γ
D
, and resonance parameters of the
decay model. At the level of precision of this measurement,
CP
violation can be neglected. Direct
CP
violation in this
channel is well constrained
[13]
, and indirect
CP
violation
due to
q=p
1
is also very small, as reported by HFAG
[12]
. We assume no
CP
violation, i.e.,
q=p
¼
1
, and
̄
A
f
ð
s
þ
;s
Þ¼
A
f
ð
s
;s
þ
Þ
.
This paper is organized as follows: Section
II
discusses
the
BABAR
detector and the data used in this analysis.
Section
III
describes the event selection. Section
IV
presents the model used to describe the amplitudes in
the DP and the fit to the data. Section
V
discusses and
quantifies the sources of systematic uncertainty. Finally, the
results are summarized in Sec.
VI
.
II. THE
BABAR
DETECTOR AND DATA
This analysis is based on a data sample corresponding to
an integrated luminosity of approximately
468
.
1
fb
1
recorded at, and 40 MeV below, the
Υ
ð
4
S
Þ
resonance
by the
BABAR
detector at the PEP-II2 asymmetric energy
e
þ
e
collider
[14]
. The
BABAR
detector is described in
detail elsewhere
[15,16]
. Charged particles are measured
with a combination of a 40-layer cylindrical drift chamber
(DCH) and a 5-layer double-sided silicon vertex tracker
(SVT), both operating within the 1.5 T magnetic field
of a superconducting solenoid. Information from a ring-
imaging Cherenkov detector is combined with specific
ionization (
dE=dx
) measurements from the SVT and DCH
to identify charged kaon and pion candidates. Electrons are
identified, and photons measured, with a CsI(Tl) electro-
magnetic calorimeter. The return yoke of the superconduct-
ing coil is instrumented with tracking chambers for the
identification of muons.
III. EVENT SELECTION
We reconstruct
D
D
0
π
þ
s
decays coming from
e
þ
e
c
̄
c
in the channel
D
0
π
þ
π
π
0
.
D
candidates
from
B
-meson decays are disregarded due to high back-
ground level. The pion from the
D
decay is called the
slow pion
(denoted
π
þ
s
) because of the limited phase space
available. The mass difference of the reconstructed
D
and
D
0
is defined as
Δ
m
m
ð
π
þ
π
π
0
π
þ
s
Þ
m
ð
π
þ
π
π
0
Þ
. Many
of the selection criteria and background veto algorithms
discussed below are based upon previous
BABAR
analy-
ses
[17,18]
.
To select well-measured slow pions, we require that the
π
þ
s
tracks have at least 10 hits measured in the DCH; and
we reduce backgrounds from other nonpion tracks by
requiring that the
dE=dx
values reported by the SVT
and DCH be consistent with the pion hypothesis. The
Dalitz decay
π
0
γ
e
þ
e
produces background when we
misidentify the
e
þ
as a
π
þ
s
. We reduce such background by
trying to reconstruct an
e
þ
e
pair using the candidate
π
þ
s
track as the
e
þ
and combine it with a
γ
. If the
e
þ
e
vertex is
within the SVT volume and the invariant mass is in the
range
115
<m
ð
γ
e
þ
e
Þ
<
155
MeV, then the event is
rejected. Real photon conversions in the detector material
are another source of background in which electrons can be
misidentified as slow pions. To identify such conversions,
we first create a candidate
e
þ
e
pair using the slow pion
candidate and an identified electron, and perform a least-
squares fit. The event is rejected if the invariant mass of the
putative pair is less than 60 MeVand the constrained vertex
position is within the SVT tracking volume.
We require that the
D
0
and
π
þ
s
candidates originate from
a common vertex, and that the
D
candidate originates
from the
e
þ
e
interaction region (beam spot). A kinematic
fit to the entire decay chain is performed with geometric
constraints at each decay vertex. In addition, the
γγ
and
π
þ
π
π
0
invariant masses are constrained to be the nominal
π
0
and
D
0
masses, respectively
[13]
. The
χ
2
probability of
the
D
fit must be at least 0.1%. About 15% of events with
at least one candidate satisfying all selection criteria (other
than the final
D
0
mass and
Deltam
cuts described below)
have at least two such candidates. In these events, we select
the candidate with the smallest
χ
2
value.
To suppress misidentifications from low-momentum
neutral pions, we require the laboratory momentum of
the
π
0
candidate to be greater than 350 MeV. The
reconstructed
D
0
proper decay time
t
, obtained from our
kinematic fit, must be within the time window
2
<t<
3
ps
and have an uncertainty
σ
t
<
0
.
8
ps. Combinatorial and
B
meson decay background is removed by requiring
p

ð
D
0
Þ
>
2
.
8
GeV, where
p

is the momentum measured
in the
e
þ
e
center-of-mass frame for the event. The
reconstructed
D
0
mass must be within 15 MeV of the
nominal
D
0
mass
[13]
and the reconstructed
Δ
m
must be
within 0.6 MeV of the nominal
D
D
0
mass difference
[13]
. After imposing all other event selection requirements
as mentioned earlier, these
p

ð
D
0
Þ
,
σ
t
,
m
ð
π
þ
π
π
0
Þ
, and
Δ
m
criteria were chosen to maximize the significance of
the signal yield obtained from a 2D fit to the
m
,
Δ
m
plane
of data, where the significance was calculated as
S=
ffiffiffiffiffiffiffiffiffiffiffiffi
S
þ
B
p
with
S
and
B
as the numbers of signal and background
events, respectively.
The signal probability density functions (PDFs) in both
m
and
Δ
m
are each defined as the sum of two Gaussian
functions. The
m
ð
π
þ
π
π
0
Þ
background distribution is
parametrized by the sum of a linear function and a single
Gaussian, which is used to model the
D
0
K
π
þ
π
0
contribution when we misidentify the kaon track as a pion.
We use a threshold-like function
[19]
to model the
Δ
m
background as a combination of real
D
0
mesons with
random slow pion candidates near kinematic threshold.
J. P. LEES
et al.
PHYSICAL REVIEW D
93,
112014 (2016)
112014-4
For many purposes, we use
full
Monte Carlo (MC)
simulations in which each data set is roughly the same size
as that observed in the real data and the background is a
mixture of
b
̄
b
,
c
̄
c
,
τ
þ
τ
and
u
̄
u=d
̄
d=s
̄
s
events scaled to the
data luminosity. The signal MC component is generated
with four combinations of
x
¼
1%
,
y
¼
1%
. We create
four samples for each set of mixing values except
x
¼
y
¼
þ
1%
which has ten samples.
Based upon detailed study of full MC events, we have
identified four specific misreconstructions of the
D
0
candidate that we can safely remove from the signal region
without biasing the measured parameters. The first mis-
reconstruction creates a peaking background in the corner
of the DP when the
K
daughter of a
D
0
K
π
þ
decay is
misidentified as a pion. To veto these events, we assign
the kaon mass hypothesis for the
π
þ
π
candidates and
calculate the
m
ð
K
π
þ
Þ
invariant mass. We remove more
than 95% of these misreconstructions by requiring
j
m
ð
K
π
þ
Þ
m
ð
D
0
Þj
>
20
MeV.
The second misreconstruction occurs when the
D
0
signal candidate shares one or more tracks with a
D
0
K
π
þ
π
0
decay. To veto these decays, we create a list of all
D
0
K
π
þ
π
0
candidates in the event that satisfy
j
m
ð
K
π
þ
π
0
Þ
m
ð
D
0
Þj
<
20
MeV,
j
Δ
m
Δ
m
PDG
j
<
3
MeV,
and
χ
2
veto
<
1000
, where
χ
2
veto
ð
m;
Δ
m
Þ¼

m
ð
K
π
þ
π
0
Þ
m
PDG
ð
D
0
Þ
σ
m

2
þ

Δ
m
Δ
m
PDG
σ
Δ
m

2
;
ð
2
Þ
where
m
PDG
denotes the nominal value for the mass taken
from Ref.
[13]
and
σ
m
(
σ
Δ
m
) is the
m
(
Δ
m
) uncertainty
reported by the fit. Such an additional veto is applied for the
specific case when the
π
þ
π
0
from a
D
0
K
π
þ
π
0
decay is
paired with a random
π
to form a signal candidate. We
can eliminate more than 95% of these misreconstructions
by finding the
K
candidate in the event that yields a
m
ð
K
π
þ
π
0
Þ
invariant mass closest to the nominal
D
0
mass
and requiring
j
m
ð
K
π
þ
π
0
Þ
m
ð
D
0
Þj
>
40
MeV. The
background from
D
0
K
π
þ
π
0
due to misidentifying
the kaon track as a pion falls outside the signal region
mass window and is negligible.
The third misreconstruction is the peaking background
when the
π
þ
π
pair from a
D
0
K
0
S
π
þ
π
decay is
combined with a random
π
0
to form a signal candidate.
To veto these events, we combine the
π
þ
π
from a
D
0
π
þ
π
π
0
candidate with
K
0
S
π
þ
π
candidates in the
same event and require
j
m
ð
K
0
S
π
þ
π
Þ
m
ð
D
0
Þj
>
20
MeV
for each.
The fourth misreconstruction is pollution from
D
0
K
0
S
π
0
ð
π
þ
π
Þ
π
0
decay. Although a real
D
0
decay, its
amplitude does not interfere with those for
prompt
D
0
π
þ
π
π
0
. We eliminate
99%
of these events by
removing candidates with
475
<m
ð
π
þ
π
Þ
<
505
MeV.
The
K
0
S
veto also removes other potential backgrounds
associated with
K
0
S
decays.
Figure
1
shows the
m
ð
π
þ
π
π
0
Þ
and
Δ
m
distributions of
D
0
candidates passing all the above requirements except for
the requirement on the shown variable. We relax the
requirements on
Δ
m
and
m
ð
π
þ
π
π
0
Þ
to perform a 2D-fit
in the
m
ð
π
þ
π
π
0
Þ
Δ
m
plane, whose projections are also
shown in Fig.
1
. The fit determines that about 91% of the
138
;
000
candidates satisfying all selection requirements
(those between the dashed lines in Fig.
1
), including those
for
m
ð
π
þ
π
π
0
Þ
and
Δ
m
cuts, are signal.
IV. MEASUREMENT OF THE
MIXING PARAMETERS
A. Fit model
The mixing parameters are extracted through a fit to
the DP distribution of the selected events as a function of
FIG. 1. (a) The reconstructed
D
0
mass distribution of data (dots)
with its fit projection (blue line), requiring
j
Δ
m
Δ
m
PDG
j
<
0
.
6
MeV; (b) The
Δ
m
distribution of data (dots) with its fit
projection (blue line), requiring
j
m
ð
π
þ
π
π
0
Þ
m
D
0
j
<
15
MeV.
The underlying histograms shown in shaded bands represent
contributions from different background categories defined in
Sec.
IV
. The vertical dashed lines mark the actual
m
ð
π
þ
π
π
0
Þ
or
Δ
m
requirement for the DP analysis.
MEASUREMENT OF THE NEUTRAL
D
MESON MIXING
...
PHYSICAL REVIEW D
93,
112014 (2016)
112014-5
time
t
. The data is fit with a total PDF which is the sum of
three component PDFs describing the signal,
broken-
charm
backgrounds, and combinatorial background.
The signal DP distribution is parametrized in terms of an
isobar model
[20
22]
. The total amplitude is a coherent
sum of partial waves
W
k
with complex weights
c
k
,
̄
A
f
ð
s
;s
þ
Þ¼
A
f
ð
s
þ
;s
Þ¼
X
k
c
k
W
k
ð
s
þ
;s
Þ
;
ð
3
Þ
where
A
f
and
̄
A
f
are the final state amplitudes introduced in
Eq.
(1)
. Our model uses relativistic Breit-Wigner functions
each multiplied by a real spin-dependent angular factor
using the same formalism with the Zemach variation as
described in Ref.
[23]
for
W
k
, and constant
W
NR
¼
1
for
the nonresonant term. As in Ref.
[23]
,
W
k
also includes the
Blatt-Weisskopf form factors with the radii of
D
0
and
intermediate resonances set at
5
GeV
1
and
1
.
5
GeV
1
,
respectively. The CLEO Collaboration modeled the decay
as a coherent combination of four amplitudes: those with
intermediate
ρ
þ
,
ρ
0
,
ρ
resonances and a uniform non-
resonant term
[24]
. This form works well to describe lower
statistics samples. In this analysis we use the model we
developed for our higher statistics search for time-
integrated
CP
violation
[18]
, which also includes other
resonances as listed in Table
I
. The partial wave with a
ρ
þ
resonance is the reference amplitude. The true decay time
distribution at any point in the DP depends on the
amplitude model and the mixing parameters. We model
the observed decay time distribution at each point in the DP
as an exponential with average decay time coming from the
mixing formalism [Eq.
(1)
] convolved with the decay time
resolution, modeled as the sum of three Gaussians with
widths proportional to
σ
t
and determined from simulation.
As the ability to reconstruct
t
varies with the position in the
DP, our parametrization of the signal PDF includes
σ
t
functions that depend on
m
2
ð
π
þ
π
Þ
, defined separately in
six ranges, each as an exponential convolved with a
Gaussian. Efficiency variations across the Dalitz plot are
modeled by a histogram obtained from simulated decays
generated with a uniformly populated phase space.
In addition to correctly reconstructed signal decay
chains, a small fraction of the events,
<
1%
, contain
D
0
π
þ
π
π
0
(
̄
D
0
π
þ
π
π
0
) decays which are correctly recon-
structed, but then paired with false slow pion candidates to
create fake
D
(
D

) candidates. As these are real
D
0
decays, their DP and decay time distributions are described in
the fit assuming a randomly tagged flavor. The total ampli-
tude for this contribution is
A
0
f
ð
s
þ
;s
Þ¼
f
RS
A
f
ð
s
þ
;s
Þþ
ð
1
f
RS
Þ
A
f
ð
s
;s
þÞ
, where
f
RS
is the
lucky fraction
that
we have a fake slow pion with the correct charge. As roughly
half of these events are assigned the wrong
D
flavor, we set
f
RS
¼
50%
in the nominal fit. We later vary this fraction to
determine a corresponding systematic uncertainty.
Backgrounds from misreconstructed signal decays and
other
D
0
decays are referred to as broken-charm. In the fit,
the Dalitz-plot distribution for this category is described by
histograms taken from the simulations. The decay time
distributions are described by the sum of two exponentials
convolved with Gaussians whose parameters are taken
from fits to the simulations.
We use sideband data to estimate combinatorial back-
ground. The data are taken from the sidebands with
TABLE I. Results of the fit to the
D
0
π
þ
π
π
0
sample showing each resonance amplitude magnitude, phase, and fit fraction
f
r
R
j
c
k
A
k
ð
s
þ
;s
Þj
2
ds
ds
þ
. The uncertainties are statistical only. We take the mass (width) of the
f
0
ð
500
Þ
to be 500 (400) MeV. In
the fit, all resonance masses and widths are fixed to the listed values, which are taken from earlier world averages produced by the
Particle Data Group
[13]
.
Resonance parameters
Fit to data results
State
J
PC
Mass (MeV)
Width (MeV)
Magnitude
Phase (°)
Fraction
f
r
(%)
ρ
ð
770
Þ
þ
1
−−
775.8
150.3
1
0
66
.
4

0
.
5
ρ
ð
770
Þ
0
1
−−
775.8
150.3
0
.
55

0
.
01
16
.
1

0
.
423
.
9

0
.
3
ρ
ð
770
Þ
1
−−
775.8
150.3
0
.
73

0
.
01
1
.
6

0
.
535
.
6

0
.
4
ρ
ð
1450
Þ
þ
1
−−
1465
400
0
.
55

0
.
07
7
.
7

8
.
21
.
1

0
.
3
ρ
ð
1450
Þ
0
1
−−
1465
400
0
.
19

0
.
07
70
.
4

15
.
90
.
1

0
.
1
ρ
ð
1450
Þ
1
−−
1465
400
0
.
53

0
.
06
8
.
2

6
.
71
.
0

0
.
2
ρ
ð
1700
Þ
þ
1
−−
1720
250
0
.
91

0
.
15
23
.
3

10
.
31
.
5

0
.
5
ρ
ð
1700
Þ
0
1
−−
1720
250
0
.
60

0
.
13
56
.
3

16
.
00
.
7

0
.
3
ρ
ð
1700
Þ
1
−−
1720
250
0
.
98

0
.
17
78
.
9

8
.
51
.
7

0
.
6
f
0
ð
980
Þ
0
þþ
980
44
0
.
06

0
.
01
58
.
8

2
.
90
.
3

0
.
1
f
0
ð
1370
Þ
0
þþ
1434
173
0
.
20

0
.
03
19
.
6

9
.
50
.
3

0
.
1
f
0
ð
1500
Þ
0
þþ
1507
109
0
.
18

0
.
02
7
.
4

7
.
40
.
3

0
.
1
f
0
ð
1710
Þ
0
þþ
1714
140
0
.
40

0
.
08
42
.
9

8
.
80
.
3

0
.
1
f
2
ð
1270
Þ
2
þþ
1275.4
185.1
0
.
25

0
.
01
8
.
8

2
.
60
.
9

0
.
1
f
0
ð
500
Þ
0
þþ
500
400
0
.
26

0
.
01
4
.
1

3
.
70
.
9

0
.
1
NR
0
.
43

0
.
07
22
.
1

11
.
70
.
4

0
.
1
J. P. LEES
et al.
PHYSICAL REVIEW D
93,
112014 (2016)
112014-6
m
ð
π
þ
π
π
0
Þ
<
1
.
80
GeV or
m
ð
π
þ
π
π
0
Þ
>
1
.
92
GeV, and
outside of the region
0
.
144
<
Δ
m<
0
.
147
GeV, where
most of the broken-charm background events reside. The
weighted sum of the two sideband regions is used to
describe the combinatorial background in the signal region.
The sideband weights and their uncertainties are deter-
mined from full MC simulation. We model these events in
t
similarly to the broken-charm category. The decay time is
described by the sum of two exponentials convolved with
Gaussians. As an
ad hoc
description of
σ
t
between 0 and
0.8 ps, the
σ
t
function for the combinatorial background is
an exponential convolved with a Gaussian, but we use
different values in six ranges of
j
t
j
.
The best-fit parameters are determined by an unbinned
maximum-likelihood fit. The central values for
x
and
y
were
blinded until the systematic uncertainties were estimated.
Because of the high statistics and the complexity of the
model,the fit iscomputationallyintensive.We have therefore
developed an open-source framework called
GooFit
[25]
to
exploit the parallel processing power of graphical processing
units.Boththeframeworkand the specificanalysiscode used
in this analysis are publicly available
[26]
.
B. Fit results
The time-integrated Dalitz plot for the signal region data
is shown in Fig.
2(a)
. The amplitude parameters determined
by the fit described above are listed in Table
I
.Our
amplitude parameters and the associated fractions are
generally consistent with the previous
BABAR
results based
on a subset of our data
[18]
. The normalized difference
between the signal DP and the model is shown in Fig.
2(b)
.
FIG. 2. The (a) Dalitz plot and (b) difference between the Dalitz plot and fit model prediction normalized by the associated statistical
uncertainty in each bin, both time integrated for the data. Also shown underneath are the projections of (c)
m
2
π
þ
π
0
, (d)
m
2
π
π
0
, and
(e)
m
2
π
þ
π
for our data (points) and fit model (blue solid lines), together with the fit residuals normalized by the associated statistical
uncertainties. The PDF components for signal (red dotted) and background (green dashed) events are shown. Note the narrow gap in
(e) due to the
K
0
S
veto.
MEASUREMENT OF THE NEUTRAL
D
MESON MIXING
...
PHYSICAL REVIEW D
93,
112014 (2016)
112014-7
The
m
2
ð
π

π
0
Þ
and
m
2
ð
π
þ
π
Þ
projections of the data and
model are shown in Figs.
2(c)
2(e)
. Differences between
the data and the fit model are apparent in both the Dalitz plot
itself and the projections. Large pull values are observed
predominantly near low and high values of
m
2
in all
projections. However, we understand the origin of these
discrepancies, and the systematic uncertainties induced on
the mixing parameters are small, as discussed below. Our fit
reports the raw mixing parameters as
x
¼ð
2
.
08

1
.
17
Þ
%
and
y
¼ð
0
.
14

0
.
89
Þ
%
. The correlation coefficient
between
x
and
y
is
0
.
6%
. The measured
D
0
lifetime is
τ
D
¼ð
410
.
2

3
.
8
Þ
fs, and agrees with the world average
of
ð
410
.
1

1
.
5
Þ
fs
[13]
. The central values of
x
and
y
are
later corrected by the estimated fit biases as discussed
in Sec.
V
.
V. SYSTEMATIC UNCERTAINTIES
Most sources of systematic uncertainty are studied by
varying some aspect of the fit, measuring the resulting
x
and
y
values, and taking the full differences between the
nominal and the varied results as the corresponding
systematic uncertainty.
To study instrumental effects that may not be well
simulated and are not covered in other studies, we divide
the data into four groups of disjoint bins and calculate
χ
2
with respect to the overall average for each group for both
x
and
y
. Within a group, each bin has roughly the same
statistics. Four bins of
m
ð
π
þ
π
π
0
Þ
give
χ
2
¼
3
.
9
(0.2) for
x
(
y
); five bins of each of
D
0
laboratory momentum
p
lab
,
cos
θ
, and
φ
give
χ
2
values of 1.5, 1.2, and 3.2 (5.9, 5.1, and
6.9) for
x
(
y
), respectively. Altogether, the summed
χ
2
is
27.9 for
ν
¼
37
degrees of freedom. Ignoring possible
correlations, the
p
-value for the hypothesis that the varia-
tions are consistent with being purely statistical fluctuations
around a common mean value is
85%
. Therefore, we
assign no additional systematic uncertainties.
Table
II
summarizes the systematic uncertainties
described in detail below. Combining them in quadrature,
we find total systematic uncertainties of 0.56% for
x
and
0.46% for
y
.
As mentioned earlier, one source of background comes
from events in which the
D
0
is correctly reconstructed, but
is paired with a random slow pion. We assume the lucky
fraction
f
RS
to be exactly 50% in the nominal fit. To
estimate the uncertainty associated with this assumption,
we vary the fraction from 40% to 60% and take the largest
variations as an estimate of the uncertainty.
The detector resolution leads to correlations between
reconstructed
D
0
mass and the decay time,
t
. We divide the
sample into four ranges of
D
0
mass with approximately
equal statistics and fit them separately; we find the
variations consistent with statistical fluctuations. Because
the average decay time is correlated with the reconstructed
D
0
mass, we refit the data by introducing separate time
resolution functions for each range, allowing the sets of
parameters to vary independently. The associated system-
atic uncertainties are taken as the differences from the
nominal values.
The DP distribution of the signal is modeled as a
coherent sum of quasi-two-body decays, involving several
resonances. To study the sensitivity to the choice of the
model, we remove some resonances from the coherent sum.
To decide if removing a resonance provides a
reasonable
description of the data, we calculate the
χ
2
of a fit using an
adaptive binning process where each bin contains at least a
reasonable number of events so that its statistical uncer-
tainty is well determined. With 1762 bins, the nominal fit
has
χ
2
¼
2794
. We separately drop the four partial waves
that individually increase
χ
2
by less than 80 units:
f
0
ð
1370
Þ
,
f
0
ð
1500
Þ
,
f
0
ð
1710
Þ
, and
ρ
ð
1700
Þ
. We take
the largest variations as the systematic uncertainties. The
other partial waves individually when removed produce
Δ
χ
2
>
165
. Additional uncertainties from our amplitude
model due to poor knowledge of the mass and width of
f
0
ð
500
Þ
are accounted for by floating the mass and width of
f
0
ð
500
Þ
in the fit to data and taking the variations in
x
and
y
.
The default resonance radius used in the Breit-Wigner
resonances in the isobar components is
1
.
5
GeV
1
,as
mentioned earlier. We vary it in steps of
0
.
5
GeV
1
from a
radius of 0 to
2
.
5
GeV
1
and again take the largest variations.
The efficiency as a function of position in the DP in the
nominal fit is modeled using a histogram taken from events
generated with a uniform phase space distribution. As a
variation, we parametrize the efficiency using a third-
degree polynomial in
s
þ
,
s
and take the difference in
mixing parameters as the uncertainty in the efficiency
TABLE II. Summary of systematic uncertainties. The various
sources are added in quadrature to find the total systematic
uncertainty.
Source
x
[%]
y
[%]
Lucky
false slow pion fraction
0.01
0.01
Time resolution dependence
0.03
0.02
On reconstructed
D
0
mass
Amplitude-model variations
0.31
0.12
Resonance radius
0.02
0.10
DP efficiency parametrization
0.03
0.03
DP normalization granularity
0.03
0.04
Background DP distribution
0.21
0.11
Decay time window
0.18
0.19
σ
t
cutoff
0.01
0.01
Number of
σ
t
ranges
0.11
0.26
σ
t
parametrization
0.05
0.03
Background-model MC time
0.06
0.11
Distribution parameters
Fit bias correction
0.29
0.02
SVT misalignment
0.20
0.23
Total
0.56
0.46
J. P. LEES
et al.
PHYSICAL REVIEW D
93,
112014 (2016)
112014-8