of 4
1
Measurement of
D
0
-
D
0
mixing parameters using
D
0
K
0
S
π
+
π
and
D
0
K
0
S
K
+
K
decays
The
B
A
B
AR
Collaboration
The following includes supplementary material for the Elec
tronic Physics Auxiliary Publication Service.
TABLE I:
D
0
K
0
S
π
+
π
complex amplitudes,
ππ
P-vector and
S-wave parameters, and fit fractions, as obtained from
the mixing fit. The
ππ
S-wave parameters
β
5
,
f
prod
14
, and
f
prod
15
are fixed to zero due to the lack of sensitivity. We also report
the mass and the width of the
K
(892)
resonance. Errors are statistical only. The fit fraction is d
efined as the integral over
the entire DP of a single component divided by the coherent su
m of all components. The sum of fit fractions is 103
.
3%. A
detailed description of the parameters can be found elsewhe
re [1]. Equations (14) and (15) in [1] have been corrected as f
ollows,
A
Kπ L
=0
(
s
) =
T
Kπ L
=0
(
s
)
(
s
), where
ρ
(
s
) =
q/
s
is the phase-space factor and
T
Kπ L
=0
(
s
) =
F
sin(
δ
F
+
φ
F
)
e
i
(
δ
F
+
φ
F
)
+
R
sin
δ
R
e
i
(
δ
R
+
φ
R
)
e
i
2(
δ
F
+
φ
F
)
, with tan
δ
R
=
M
K
0
(1430)
Γ
K
0
(1430)
(
s
)
/
(
M
2
K
0
(1430)
s
), cot
δ
F
= 1
/
(
aq
)+
rq/
2,
s
the invariant mass
squared of the
system, and
q
the momentum of the kaon (or pion) in the
rest frame [2]. The symbol
indicates the
parameters fixed in the mixing fit to the values extracted from
a time-integrated DP fit to the same data. The results from
this time-integrated DP fit for the amplitude model paramete
rs agree within statistical errors with the results reporte
d here.
Component
Amplitude
Phase (rad)
Fit fraction (%)
K
(892)
1
.
735
±
0
.
005
2
.
331
±
0
.
004
57
.
0
ρ
(770)
0
1
0
21
.
1
K
0
(1430)
2
.
650
±
0
.
015
1
.
497
±
0
.
007
6
.
1
K
2
(1430)
1
.
303
±
0
.
013
2
.
498
±
0
.
012
1
.
9
ω
(782)
0
.
0420
±
0
.
0006
2
.
046
±
0
.
014
0
.
6
K
(892)
+
0
.
164
±
0
.
003
0
.
768
±
0
.
019
0
.
6
K
(1680)
0
.
90
±
0
.
03
2
.
97
±
0
.
04
0
.
3
f
2
(1270)
0
.
410
±
0
.
013
2
.
88
±
0
.
03
0
.
3
K
0
(1430)
+
0
.
145
±
0
.
014
1
.
78
±
0
.
10
<
0
.
1
K
2
(1430)
+
0
.
115
±
0
.
013
2
.
69
±
0
.
11
<
0
.
1
ππ
S-wave
15
.
4
β
1
5
.
54
±
0
.
06
0
.
054
±
0
.
007
β
2
15
.
64
±
0
.
06
3
.
125
±
0
.
005
β
3
44
.
6
±
1
.
2
2
.
731
±
0
.
015
β
4
9
.
3
±
0
.
2
2
.
30
±
0
.
02
f
prod
11
11
.
43
±
0
.
11
0
.
005
±
0
.
009
f
prod
12
15
.
5
±
0
.
4
1
.
13
±
0
.
02
f
prod
13
7
.
0
±
0
.
7
0
.
99
±
0
.
11
Parameter value
s
prod
0
3
.
92637
S-wave parameters
M
K
0
(1430)
( MeV
/c
2
)
1421
.
5
±
1
.
6
Γ
K
0
(1430)
( MeV
/c
2
)
247
±
3
F
0
.
62
±
0
.
04
φ
F
(rad)
0
.
100
±
0
.
010
R
1
φ
R
(rad)
1
.
10
±
0
.
02
a
( GeV
/c
1
)
0
.
224
±
0
.
003
r
( GeV
/c
1
)
15
.
01
±
0
.
13
K
(892) parameters
M
K
(892)
( MeV
/c
2
)
893
.
70
±
0
.
07
Γ
K
(892)
( MeV
/c
2
)
46
.
74
±
0
.
15
2
TABLE II:
D
0
K
0
S
K
+
K
complex amplitudes and fit fractions, as obtained from the mi
xing fit. We also report the mass
and the width of the
φ
(1020) resonance, and the
a
0
(980) coupling constant to
K
K
as determined from the fit. Errors are
statistical only. The fit fraction is defined as the integral o
ver the entire DP of a single component divided by the coheren
t sum
of all components. The sum of fit fractions is 163
.
4%. A detailed description of the parameters can be found els
ewhere [1].
The symbol
indicates the parameters fixed in the mixing fit to the values e
xtracted from a time-integrated DP fit to the same
data. The results from this time-integrated DP fit for the amp
litude model parameters agree within statistical errors wi
th the
results reported here.
Component
Amplitude
Phase (rad)
Fit fraction (%)
a
0
(980)
0
1
0
51
.
8
φ
(1020)
0
.
2313
±
0
.
0011
0
.
977
±
0
.
008
44
.
1
a
0
(1450)
+
0
.
93
±
0
.
03
1
.
66
±
0
.
07
25
.
6
a
0
(980)
+
0
.
635
±
0
.
006
2
.
91
±
0
.
02
19
.
5
a
0
(1450)
0
0
.
83
±
0
.
10
1
.
93
±
0
.
12
19
.
3
f
0
(1370)
0
.
16
±
0
.
05
0
.
2
±
0
.
2
1
.
7
f
2
(1270)
0
.
385
±
0
.
015
0
.
06
±
0
.
04
0
.
7
a
0
(980)
0
.
125
±
0
.
008
2
.
47
±
0
.
04
0
.
7
φ
(1020) and
a
0
(980) parameters
Value
M
φ
(1020)
( MeV
/c
2
)
1019
.
55
±
0
.
02
Γ
φ
(1020)
( MeV
/c
2
)
4
.
60
±
0
.
04
g
K
K
( MeV
/c
2
)
537
±
9
)
2
(GeV/c
0
D
m
1.84
1.86
1.88
1.9
2
Events / 0.8 MeV/c
3
10
4
10
a)
)
2
m (GeV/c
0.144
0.146
0.148
2
Events / 0.8 MeV/c
3
10
4
10
Data
Signal
s
π
Random
0
Misrecon. D
0
S
K
0
S
K
0
D
Combinatorial
b)
)
2
(GeV/c
0
D
m
1.84
1.86
1.88
1.9
2
Events / 60 KeV/c
2
10
3
10
4
10
c)
)
2
m (GeV/c
0.144
0.146
0.148
2
Events / 60 KeV/c
2
10
3
10
4
10
d)
FIG. 1: (color online) Distributions of
m
D
0
and ∆
m
for (a,b)
K
0
S
π
+
π
and (c,d)
K
0
S
K
+
K
data after all selection criteria
(points). The curves superimposed represent the fit project
ions for signal plus background (solid lines) and for differe
nt
background components (shaded regions). The arrows indica
te the definition of the signal region.
[1] B. Aubert
et al.
(
B
A
B
AR
Collaboration), Phys. Rev. D
78
, 034023 (2008).
3
)
4
/c
2
(GeV
-
s
1
2
3
)
4
/c
2
(GeV
+
s
1
2
3
1
10
2
10
3
10
a)
)
4
/c
2
(GeV
+
s
1
1.2
1.4
1.6
1.8
)
4
/c
2
(GeV
0
s
1
1.2
1.4
1.6
1.8
1
10
2
10
b)
FIG. 2: DP distributions for (a)
D
0
K
0
S
π
+
π
and (b)
D
0
K
0
S
K
+
K
data after all selection criteria, in the signal region.
The gray scale indicates the number of events per bin. The sol
id lines show the kinematic limits of the
D
0
decay. The
s
0
DP
variable is defined as
s
0
=
m
2
(
h
+
h
). For
D
0
decays the variables
s
and
s
+
are interchanged.
)
4
/c
2
(GeV
-
s
1
2
3
4
/c
2
Events / 0.035 GeV
0
20000
40000
60000
80000
)
4
/c
2
(GeV
-
s
1
2
3
4
/c
2
Events / 0.035 GeV
0
20000
40000
60000
80000
a)
)
4
/c
2
(GeV
+
s
1
2
3
4
/c
2
Events / 0.035 GeV
0
5000
10000
15000
)
4
/c
2
(GeV
+
s
1
2
3
4
/c
2
Events / 0.035 GeV
0
5000
10000
15000
b)
)
4
/c
2
(GeV
0
s
0
0.5
1
1.5
2
4
/c
2
Events / 0.025 GeV
0
5000
10000
15000
)
4
/c
2
(GeV
0
s
0
0.5
1
1.5
2
4
/c
2
Events / 0.025 GeV
0
5000
10000
15000
c)
)
4
/c
2
(GeV
-
s
1
1.2
1.4
1.6
1.8
4
/c
2
Events / 0.024 GeV
0
2000
4000
6000
)
4
/c
2
(GeV
-
s
1
1.2
1.4
1.6
1.8
4
/c
2
Events / 0.024 GeV
0
2000
4000
6000
d)
)
4
/c
2
(GeV
+
s
1
1.2
1.4
1.6
1.8
4
/c
2
Events / 0.024 GeV
0
1000
2000
3000
4000
5000
)
4
/c
2
(GeV
+
s
1
1.2
1.4
1.6
1.8
4
/c
2
Events / 0.024 GeV
0
1000
2000
3000
4000
5000
e)
)
4
/c
2
(GeV
0
s
1
1.2
1.4
1.6
1.8
4
/c
2
Events / 0.024 GeV
0
5000
10000
15000
20000
25000
)
4
/c
2
(GeV
0
s
1
1.2
1.4
1.6
1.8
4
/c
2
Events / 0.024 GeV
0
5000
10000
15000
20000
25000
f)
FIG. 3: DP projections for (a,b,c)
D
0
K
0
S
π
+
π
and (d,e,f)
D
0
K
0
S
K
+
K
data after all selection criteria, in the signal
region (points). The histograms represent the mixing fit pro
jections. For
D
0
decays the variables
s
and
s
+
are interchanged.
[2] D. Aston
et al.
(LASS Collaboration), Nucl. Phys. B
296
, 493 (1988); W. Dunwoodie, private communication.
4
TABLE III: Summary of the contributions to the experimental
systematic uncertainty on the mixing parameters.
Source
x/
10
3
y/
10
3
Analysis biases and fitting procedure (Monte Carlo statisti
cs)
0.75
0.66
Selection criteria
0.47
0.57
Signal and background yields
0.11
0.07
Efficiency variations across the DP
0.37
0.18
Modeling of the DP distributions for misreconstructed
D
0
decays
0.33
0.14
Modeling of the proper-time distributions for signal and mi
sreconstructed
D
0
decays
0.13
0.13
Modeling of the proper-time error distributions for signal
and misreconstructed
D
0
decays
0.06
0.09
Misidentification of the
D
0
flavor for signal and random
π
+
s
events
0.49
0.40
Mixing in the random
π
+
s
background component
0.10
0.08
PDF normalization
0.11
0.05
Misalignment of the detector
0.28
0.83
Total experimental systematic uncertainty
1.18
1.30
TABLE IV: Summary of the contributions to the
D
0
decay amplitude model systematic uncertainty on the mixing
parameters.
Source
x/
10
3
y/
10
3
Breit-Wigner parameters and alternative GS lineshapes
0.3
5
0.12
Alternative K-matrix solutions and P-vector parameteriza
tion
0.13
0.19
S- and P-waves, and
ππ
S-wave parameters
0.68
0.53
Form factors
0.25
0.23
Angular dependence
0.05
0.17
Add/remove resonances
0.17
0.23
Total amplitude model systematic uncertainty
0.83
0.69