Strong Phase and D0 − D0 mixing at BES-III
Xiao-Dong Cheng1,2,∗ Kang-Lin He1,† Hai-Bo Li1,‡ Yi-Fang Wang1,§ and Mao-Zhi Yang1¶
Institute of High Energy Physics, P.O.Box 918, Beijing 100049, China
Department of Physics, Henan Normal University, XinXiang, Henan 453007, China
(Dated: October 25, 2018)
Most recently, both BaBar and Belle experiments found evidences of neutral D mixing. In this
paper, we discuss the constraints on the strong phase difference in D0 → Kπ decay from the
measurements of the mixing parameters, y′, yCP and x at the B factories. With CP tag technique at
ψ(3770) peak, the extraction of the strong phase difference at BES-III are discussed. The sensitivity
of the measurement of the mixing parameter y is estimated in BES-III experiment at ψ(3770) peak.
Finally, we also make an estimate on the measurements of the mixing rate RM .
PACS numbers: 13.25.Ft, 12.15.Ff, 13.20.Fc, 11.30.Er
Due to the smallness of ∆C = 0 amplitude in the
Standard Model (SM), D0 − D0 mixing offers a unique
opportunity to probe flavor-changing interactions which
may be generated by new physics. The recent measure-
ments from BaBar and Belle experiments indicate that
the D0 − D0 mixing may exist [1, 2]. At the B fac-
tories, the decay time information can be used to ex-
tract the neutral D mixing parameters. At t = 0 the
only term in the amplitude is the direct doubly-Cabibbo-
suppressed (DCS) mode D0 → K+π−, but for t > 0
D0 − D0 mixing may contribute through the sequence
D0 → D0 → K+π− , where the second stage is Cabibbo
favored (CF). The interference of this term with the DCS
contribution involves the lifetime and mass differences
of the neutral D mass eigenstates, as well as the final-
state strong phase difference δKπ between the CF and
the DCS decay amplitudes. This interference plays a
key role in the measurement of the mixing parameters at
time-dependent measurements.
With the assumption of CPT invariance, the mass
eigenstates of D0 −D0 system are |D1〉 = p|D0〉+ q|D0〉
and |D2〉 = p|D0〉−q|D0〉 with eigenvalues µ1 = m1−
and µ2 = m2 −
Γ2, respectively, where the m1 and Γ1
(m2 and Γ2) are the mass and width of D1 (D2). For
the method of detecting D0 − D0 mixing involving the
D0 → Kπ decay mentioned above, in order to separate
the DCS decay from the mixing signal, one must study
the time-dependent decay rate. The proper-time evolu-
tion of the particle states |D0
(t)〉 and |D0
(t)〉 are
given by
|D0phys(t)〉 = g+(t)|D
0〉 − q
g−(t)|D0〉,
|D0phys(t)〉 = g+(t)|D
0〉 − p
g−(t)|D0〉, (1)
where
(e−im2t−
Γ2t ± e−im1t− 12Γ1t), (2)
with definitions
m ≡ m1 +m2
, ∆m ≡ m2 −m1,
Γ ≡ Γ1 + Γ2
, ∆Γ ≡ Γ2 − Γ1, (3)
Note the sign of ∆m and ∆Γ is to be determined by
experiments.
In practice, one define the following mixing parameters
x ≡ ∆m
, y ≡ ∆Γ
. (4)
The time-dependent decay amplitudes for D0
(t) →
K+π− and D0
(t) → K−π+ are described as
〈K+π−|H|D0phys(t)〉 = g+(t)AK+π− −
g−(t)AK+π−
AK+π− [λg+(t)− g−(t)], (5)
〈K−π+|H|D0phys(t)〉 = g+(t)AK−π+ −
g−(t)AK−π+
AK−π+ [λg+(t)− g−(t)], (6)
where AK+π− = 〈K+π−|H|D0〉, AK+π− =
〈K+π−|H|D0〉, AK−π+ = 〈K−π+|H|D0〉, and
AK−π+ = 〈K−π+|H|D0〉. Here, λ and λ are de-
fined as:
λ ≡ p
AK+π−
AK+π−
, (7)
λ ≡ q
AK−π+
AK−π+
. (8)
From Eqs. (5) and (6), one can derive the general ex-
pression for the time-dependent decay rate, in agreement
http://arxiv.org/abs/0704.0120v3
with [3, 4]:
dΓ(D0
(t) → K+π−)
dtN = |AK
+π− |2
e−Γt ×
[(|λ|2 + 1)cosh(yΓt) +
(|λ|2 − 1)cos(xΓt) +
2Re(λ)sinh(yΓt) +
2Im(λ)sin(xΓt)] (9)
dΓ(D0
(t) → K−π+)
dtN = |AK−π+ |
e−Γt ×
[(|λ|2 + 1)cosh(yΓt) +
(|λ|2 − 1)cos(xΓt) +
2Re(λ)sinh(yΓt) +
2Im(λ)sin(xΓt)] (10)
where N is a common normalization factor. In order to
simplify the above formula, we make the following defi-
nition:
≡ (1 +AM )e−iβ , (11)
where β is the weak phase in mixing and AM is a real-
valued parameter which indicates the magnitude of CP
violation in the mixing. For f = K−π+ final state, we
define
AK+π−
AK+π−
r′e−iα
AK−π+
AK−π+
re−iα, (12)
where r′ and α′ (r and α) are the ratio and relative phase
of the DCS decay rate and the CF decay rate. Then, λ
and λ can be parameterized as
λ = −
1 +AM
e−i(α
′−β) , (13)
λ = −
r(1 +AM )e
−i(α+β). (14)
In order to demonstrate the CP violation in decay, we
define
RD(1 + AD) and
1 +AD
Thus, Eqs. (13) and (14) can be expressed as
λ = −
1 +AD
1 +AM
e−i(δ−φ) , (15)
λ = −
1 +AM
1 +AD
e−i(δ+φ) , (16)
where δ =
α′ + α
is the averaged phase difference be-
tween DCS and CF processes, and φ =
α− α′
We can characterize the CP violation in the mixing
amplitude, the decay amplitude, and the interference
between amplitudes with and without mixing, by real-
valued parameters AM , AD, and φ as in Ref [5, 6].
In the limit of CP conservation, AM , AD and φ are
all zero. AM = 0 means no CP violation in mixing,
namely, |q/p| = 1; AD = 0 means no CP violation in
decay, for this case, r = r′ = RD = |AK−π+/AK−π+ |2 =
|AK+π−/AK+π− |2; φ = 0 means no CP violation in the
interference between decay and mixing.
In experimental searches, one can define CF decay as
right-sign (RS) and DCS decay or via mixing followed
by a CF decay as wrong-sign (WS). Here, we define the
ratio of WS to RS decays as for D0:
R(t) =
dΓ(D0
(t) → K+π−)
dtN × e−Γ|t| × 2|AK+π− |2
, (17)
and for D0:
R(t) =
dΓ(D0
(t) → K−π+)
dtN × e−Γ|t| × 2|AK−π+ |2
, (18)
Taking into account that |λ|, |λ| ≪ 1 and x, y ≪ 1,
keeping terms up to order x2, y2 and RD in the ex-
pressions, neglecting CP violation in mixing, decay and
the interference between decay with and without mixing
(AM = 0, AD = 0, and φ = 0), expanding the time-
dependent for xt, yt <∼ Γ−1, combing Eqs. (9) and (10),
we can write Eqs. (17) and (18) as
R(t) = R(t) = RD +
(Γt)2, (19)
where
x′ = xcosδ + ysinδ, (20)
y′ = −xsinδ + ycosδ. (21)
In the limit of SU(3) symmetry, AK+π− and AK+π−
(AK−π+ and AK−π+) are simply related by CKM fac-
tors, AK+π− = (VcdV
us/VcsV
ud)AK+π− [7]. In particular,
AK+π− and AK+π− have the same strong phase, leading
to α′ = α = 0 in Eq. (12). But the SU(3) symmetry
is broken according to the recent precise measurements
from the B factories, the ratio [5]:
R = BR(D
0 → K+π−)
BR(D0 → K+π−)
, (22)
is unity in the SU(3) symmetry limit. But, the world
average for this ratio is
Rexp = 1.21± 0.03, (23)
computed from the individual measurements using the
standard method of Ref. [4]. Since the SU(3) is bro-
ken in D → Kπ decays at the level of 20%, in which
case the strong phase δ should be non-zero. Recently, a
time-dependent analysis in D → Kπ has been performed
based on 384 fb−1 luminosity at Υ (4S) [1]. By assuming
CP conservation, they obtained the following neutral D
mixing results
RD = (3.03± 0.16± 0.10)× 10−3,
= (−0.22± 0.30± 0.21)× 10−3,
y′ = (9.7± 4.4± 3.1)× 10−3. (24)
TABLE I: Experimental results used in the paper. Only one
error is quoted, we have combined in quadrature statistical
and systematic contributions.
Parameter BaBar (×10−3) Belle(×10−3) Technique
-0.22± 0.37 [1] 0.18+0.21−0.23 [8] Kπ
′ 9.7± 5.4 [1] 0.6+4.0−3.9 [8] Kπ
RD 3.03± 0.19 [1] 3.64 ± 0.17 [8] Kπ
yCP - 13.1 ± 4.1 [2] K
−, π+π−
x - 8.0± 3.4 [9] KSπ
y - 3.3± 2.8 [9] KSπ
The result is inconsistent with the no-mixing hypoth-
esis with a significance of 3.9 standard deviations. The
results from BaBar and Belle are in agreement within 2
standard deviation on the exact analysis of y′ measure-
ment by using D → Kπ as listed in Table I. As indicated
in Eq. (23), the strong phase δ should be non-zero due to
the SU(3) violation. One has to know the strong phase
difference exactly in order to extract the direct mixing
parameters, x and y as defined in Eqs. (4). However,
at the B factory, it is hard to do that with a model-
independent way [7, 10]. In order to extract the strong
phase δ we need data near the DD threshold to do a CP
tag as discussed in Ref. [7]. Here, we would like to figure
out the possible physics solution of the strong phase δ by
using the recent results from the B factories with differ-
ent decay modes, so that we can have an idea about the
sensitivity to measure the strong phase at the BES-III
project.
In Ref [2], Belle collaboration also reported the result
of yCP =
τ(D0→K+π−)
τ(D0→fCP )
− 1, where fCP = K+K− and
yCP = (13.1± 3.2± 2.5)× 10−3. (25)
The result is about 3.2σ significant deviation from zero
(non-mixing). In the limit of CP symmetry, yCP = y [11,
12]. In the decay of D0 → KSπ+π−, Belle experiment
has done a Dalitz plot (DP) analysis [9], they obtained
the direct mixing parameters x and y as
x = (8.0± 3.4)× 10−3, y = (3.3± 2.8)× 10−3, (26)
where the error includes both statistic and systematic un-
certainties. Since the parameterizations of the resonances
on the DP are model-dependent, the results suffer from
large uncertainties from the DP model. In this analysis,
they see a significance of 2.4 standard deviations from
non-mixing. Here, we will use the value of x measured in
the DP analysis for further discussion. As shown in Eq.
(21), once y, y′ and x are known, it is straightforward to
extract the strong phase difference between DCS and CF
decay in D0 → Kπ decay. If taking the measured central
values of x, yCP (≈ y) , and y′ as input parameters, we
found two-fold solutions for tanδ as below:
tanδ = 0.35± 0.63, or − 7.14± 29.13, (27)
which are corresponding to (19± 32)0 and (−820 ± 30)0,
respectively.
At ψ(3770) peak, to extract the mixing parameter y,
one can make use of rates for exclusive D0D0 combina-
tion, where both the D0 final states are specified (known
as double tags or DT), as well as inclusive rates, where
either the D0 or D0 is identified and the other D0 de-
cays generically (known as single tags or ST) [13]. With
the DT tag technique [14, 15], one can fully consider
the quantum correlation in C = −1 and C = +1 D0D0
pairs produced in the reaction e+e− → D0D0(nπ0) and
e+e− → D0D0γ(nπ0) [13, 16, 17], respectively.
For the ST, in the limit of CP conservation, the rate
of D0 decays into a CP eigenstate is given as [13]:
Γfη ≡ Γ(D0 → fη) = 2A2fη [1− ηy] , (28)
where fη is a CP eigenstate with eigenvalue η = ±1, and
Afη = |〈fη|H|D0〉| is the real-valued decay amplitude.
For the DT case, Gronau et. al. [7] and Xing [18]
have considered time-integrated decays into correlated
pairs of states, including the effects of non-zero final state
phase difference. As discussed in Ref. [7], the rate of
(D0D0)C=−1 → (l±X)(fη) is described as [7]:
Γl;fη ≡ Γ[(l±X)(fη)] = A2l±XA
(1 + y2)
≈ A2l±XA
, (29)
where Al±X = |〈l±X |H|D0〉| is real-valued amplitude for
semileptonic decays, here, we neglect y2 term since y ≪
For C = −1 initial D0D0 state, y can be expressed in
term of the ratios of DT rates and the double ratios of
ST rates to DT rates [13]:
Γl;f+Γf−
Γl;f−Γf+
Γl;f−Γf+
Γl;f+Γf−
. (30)
For a small y, its error, ∆(y), is approximately 1/
Nl±X ,
where Nl±X is the total number of (l
±X) events tagged
with CP -even and CP -odd eigenstates. The num-
ber Nl±X of CP tagged events is related to the to-
tal number of D0D0 pairs N(D0D0) through Nl±X ≈
N(D0D0)[BR(D0 → l± +X)× BR(D0 → f±)× ǫtag] ≈
1.5 × 10−3N(D0D0), here we take the branching ratio-
times-efficiency factor (BR(D0 → f±)× ǫtag) for tagging
CP eigenstates is about 1.1% (the total branching ratio
into CP eigenstates is larger than about 5% [4]). We find
∆(y) =
N(D0D0)
= ±0.003. (31)
If we take the central value of y from the measurement
of yCP at Belle experiment [2], thus, at BES-III exper-
iment [19], with 20fb−1 data at ψ(3770) peak, the sig-
nificance of the measurement of y could be around 4.3 σ
deviation from zero.
We can also take advantage of the coherence of the
D0 mesons produced at the ψ(3770) peak to extract the
strong phase difference δ between DCS and CF decay am-
plitudes that appears in the time-dependent mixing mea-
surement in Eq. (19) [7, 13]. Because the CP properties
of the final states produced in the decay of the ψ(3770)
are anti-correlated [16, 17], one D0 state decaying into a
final state with definite CP properties immediately iden-
tifies or tags the CP properties of the other side. As
discussed in Ref. [7], the process of one D0 decaying to
K−π+, while the other D0 decaying to a CP eigenstate
fη can be described as
ΓKπ;fη ≡ Γ[(K−π+)(fη)] ≈ A2A2fη |1 + η
−iδ|2
≈ A2A2fη (1 + 2η
RDcosδ),
where A = |〈K−π+|H|D0〉| and Afη = |〈fη|H|D0〉| are
the real-valued decay amplitudes, and we have neglected
the y2 terms in Eq. (32). In order to estimate the total
sample of events needed to perform a useful measurement
of δ, one defined [7, 10] an asymmetry
ΓKπ;f+ − ΓKπ;f−
ΓKπ;f+ + ΓKπ;f−
, (33)
where ΓKπ;f± is defined in Eq. (32), which is the rates for
the ψ(3770) → D0D0 configuration to decay into flavor
eigenstates and a CP -eigenstates f±. Eq. (32) implies a
small asymmetry, A = 2
RDcosδ. For a small asymme-
try, a general result is that its error ∆A is approximately
NK−π+ , where NK−π+ is the total number of events
tagged with CP -even and CP -odd eigenstates. Thus one
obtained
∆(cosδ) ≈ 1
NK−π+
. (34)
The expected number NK−π+ of CP -tagged events
can be connected to the total number of D0D0 pairs
N(D0D0) through NK−π+ ≈ N(D0D0)BR(D0 →
K−π+)×BR(D0 → f±)×ǫtag ≈ 4.2×10−4N(D0D0) [7],
here, as in Ref [7], we take the branching ratio-times-
efficiency factor BR(D0 → f±) × ǫtag = 1.1%. With
0 0.2 0.4 0.6 0.8 1
FIG. 1: Illustrative plot of the expected error (∆δ) of the
strong phase with various central values of cosδ. The expected
error of cosδ is 0.04 by ssuming 20fb−1 data at ψ(3770) peak
at BES-III. The two asterisks correspond to δ = 190 and
−820, respectively.
the measured RD = (3.03± 0.19)× 10−3 and BR(D0 →
K−π+) = 3.8% [4], one found [7]
∆(cosδ) ≈ ±444√
N(D0D0)
. (35)
At BESIII, about 72 × 106 D0D0 pairs can be collected
with 4 years’ running. If considering both K−π+ and
K+π− final states, we thus estimate that one may be
able to reach an accuracy of about 0.04 for cosδ. Fig-
ure 1 shows the expected error of the strong phase δ
with various central values of cosδ. With the expected
∆(cosδ) = ±0.04, the sensitivity of the strong phase
varies with the physical value of cosδ. For δ = 190 and
−820, the expected error could be ∆(δ) = ±8.70 and
±2.90, respectively.
By combing the measurements of x inD0 → KSππ and
yCP from Belle, one can obtain RM = (1.18±0.6)×10−4.
At the ψ(3770) peak, D0D0 pair are produced in a state
that is quantum-mechanically coherent [16, 17]. This en-
ables simple new method to measure D0 mixing param-
eters in a way similar proposed in Ref. [7]. At BES-III,
the measurement of RM can be performed unambigu-
ously with the following reactions [16]:
(i) e+e− → ψ(3770) → D0D0 → (K±π∓)(K±π∓),
(ii) e+e− → ψ(3770) → D0D0 → (K−e+ν)(K−e+ν),
(iii) e+e− → D−D∗+ → (K+π−π−)(π+
[K+e−ν]).
Reaction (i) in Eq. (36) can be normalized to D0D0 →
(K−π+)(K+π−), the following time-integrated ratio is
obtained by neglecting CP violation:
N [(K−π+)(K−π+)]
N [(K−π+)(K+π−)]
2 + y2
= RM . (37)
For the case of semileptonic decay, as (ii) in Eq. (36), we
N(l±l±)
N(l±l∓)
x2 + y2
= RM , (38)
The observation of reaction (i) would be definite evi-
dence for the existence of D0 −D0 mixing since the final
state (K±π∓)(K±π∓) can not be produced from DCS
decay due to quantum statistics [16, 17]. In particular,
the initial D0D0 pair is in an odd eigenstate of C which
will preclude, in the absence of mixing between the D0
and D0 over time, the formation of the symmetric state
required by Bose statistics if the decays are to be the
same final state. This final state is also very appealing
experimentally, because it involves a two-body decay of
both charm mesons, with energetic charged particles in
the final state that form an overconstrained system. Par-
ticle identification is crucial in this measurement because
if both the kaon and pion are misidentified in one of the
two D-meson decays in the event, it becomes impossi-
ble to discern whether mixing has occurred. At BESIII,
where the data sample is expected to be 20 fb−1 inte-
grated luminosity at ψ(3770) peak, the limit will be 10−4
at 95% C.L. for RM , but only if the particle identification
capabilities are adequate.
Reactions (ii) and (iii) offer unambiguous evidence
for the mixing because the mixing is searched for in the
semileptonic decays for which there are no DCS decays.
Of course since the time-evolution is not measured, obser-
vation of Reactions (ii) and (iii) actually would indicate
the violation of the selection rule relating the change in
charm to the change in leptonic charge which holds true
in the standard model [16].
In Table II, the sensitivity for RM measurements in
different decay modes are estimated with 4 years’ run at
BEPCII.
TABLE II: The sensitivity for RM measurements at BES-III
with different decay modes with 4 years’ run at BESPCII
0 Mixing
Reaction Events Sensitivity
RS(×104) RM (×10
ψ(3770) → (K−π+)(K−π+) 10.4 1.0
ψ(3770) → (K−e+ν)(K−e+ν) 8.9
ψ(3770) → (K−e+ν)(K−µ+ν) 8.1 3.7
ψ(3770) → (K−µ+ν)(K−µ+ν) 7.3
In the limit of CP conservation, by combing the mea-
surements of x in D0 → KSππ and yCP from Belle, one
can obtain RM = (1.18± 0.6)× 10−4. With 20fb−1 data
at BES-III, about 12 events for the precess D0D0 →
(K±π∓)(K±π∓) can be produced. One can observe 3.0
events after considering the selection efficiency at BE-
SIII, which could be about 25% for the four charged
particles. The background contamination due to double
particle misidentification is about 0.6 event with 20fb−1
data at BES-III [20]. Table III lists the expected mixing
signal for Nsig = N(K
±π∓)(K±π∓), background Nbkg ,
and the Poisson probability P (n), where n is the possible
number of observed events in experiment. In Table III,
we assume the RM = 1.18× 10−4, the expected number
of mixing signal events are estimated with 10fb−1 and
20fb−1, respectively.
TABLE III: The expected mixing signal for Nsig =
N(K±π∓)(K±π∓), background Nbkg , and the Poisson prob-
ability P (n) in 10 fb−1 and 20 fb−1 at BES-III at ψ(3770)
peak, respectively. Here, we take the mixing rate RM =
1.18× 10−4.
10 fb−1 (ψ(3770)) 20 fb−1 (ψ(3770))
36 million D0D0 72 million D0D0
Nsig 1.5 3.0
Nbkg 0.3 0.6
P (n = 0) 15.7% 2.5%
P (n = 1) 29.1% 9.1%
P (n = 2) 26.9% 16.9%
P (n = 3) 16.6% 20.9%
P (n = 4) 7.7% 19.3%
P (n = 5) 2.8% 14.3%
P (n = 6) 0.9% 8.8%
P (n = 7) 0.2% 4.7%
P (n = 8) 0.1% 2.2%
P (n = 9) 0.01% 0.9%
In conclusion, we discuss the constraints on the strong
phase difference in D0 → Kπ decay according to the
most recent measurements of y′, yCP and x from B fac-
tories. We estimate the sensitivity of the measurement of
mixing parameter y at ψ(3770) peak in BES-III experi-
ment. With 20 fb−1 data, the uncertainty ∆(y) could be
0.003. Thus, assuming y at a percent level, we can make
a measurement of y at a significance of 4.3σ deviation
from zero. The sensitivity of the strong phase differ-
ence at BES-III are obtained by using data near the DD
threshold with CP tag technique at BES-III experiment.
Finally, we estimated the sensitivity of the measurements
of the mixing rate RM , and find that BES-III experiment
may not be able to make a significant measurement of
RM with current luminosity by using coherent DD state
at ψ(3770) peak.
One of the authors (H. B. Li) would like to thank
David Asner and Zhi-Zhong Xing for stimulating dis-
cussion, Chang-Zheng Yuan for useful discussion on the
statistics used in this paper, and also thank Stephen
L. Olsen and Yang-Heng Zheng for commenting on this
manuscript. We thank BES-III collaboration for provid-
ing us many numerical results based on GEANT4 simula-
tion. This work is supported in part by the National Nat-
ural Science Foundation of China under contracts Nos.
10205017, 10575108,10521003, and the Knowledge Inno-
vation Project of CAS under contract Nos. U-612 and
U-530 (IHEP).
∗ Electronic address: chengxd@ihep.ac.cn
† Electronic address: hekl@ihep.ac.cn
‡ Electronic address: lihb@ihep.ac.cn
§ Electronic address: yfwang@ihep.ac.cn
¶ Electronic address: yangmz@ihep.ac.cn
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mailto:chengxd@ihep.ac.cn
mailto:hekl@ihep.ac.cn
mailto:lihb@ihep.ac.cn
mailto:yfwang@ihep.ac.cn
mailto:yangmz@ihep.ac.cn
http://arxiv.org/abs/hep-ex/0703020
http://arxiv.org/abs/hep-ex/0703036
http://arxiv.org/abs/hep-ph/0703235
http://arxiv.org/abs/hep-ex/0603031

  Most recently, both BaBar and Belle experiments found evidences of neutral
$D$ mixing. In this paper, we discuss the constraints on the strong phase
difference in $D^0 \to K\pi$ decay from the measurements of the mixing
parameters, $y^\prime$, $y_{CP}$ and $x$ at the $B$ factories. The sensitivity
of the measurement of the mixing parameter $y$ is estimated in BES-III
experiment at $\psi(3770)$ peak. We also make an estimate on the measurements
of the mixing rate $R_M$. Finally, the sensitivity of the strong phase
difference at BES-III are obtained by using data near the $D\bar{D}$ threshold
with CP tag technique at BES-III experiment.

Strong Phase and D0 − D0 mixing at BES-III
Xiao-Dong Cheng1,2,∗ Kang-Lin He1,† Hai-Bo Li1,‡ Yi-Fang Wang1,§ and Mao-Zhi Yang1¶
Institute of High Energy Physics, P.O.Box 918, Beijing 100049, China
Department of Physics, Henan Normal University, XinXiang, Henan 453007, China
(Dated: October 25, 2018)
Most recently, both BaBar and Belle experiments found evidences of neutral D mixing. In this
paper, we discuss the constraints on the strong phase difference in D0 → Kπ decay from the
measurements of the mixing parameters, y′, yCP and x at the B factories. With CP tag technique at
ψ(3770) peak, the extraction of the strong phase difference at BES-III are discussed. The sensitivity
of the measurement of the mixing parameter y is estimated in BES-III experiment at ψ(3770) peak.
Finally, we also make an estimate on the measurements of the mixing rate RM .
PACS numbers: 13.25.Ft, 12.15.Ff, 13.20.Fc, 11.30.Er
Due to the smallness of ∆C = 0 amplitude in the
Standard Model (SM), D0 − D0 mixing offers a unique
opportunity to probe flavor-changing interactions which
may be generated by new physics. The recent measure-
ments from BaBar and Belle experiments indicate that
the D0 − D0 mixing may exist [1, 2]. At the B fac-
tories, the decay time information can be used to ex-
tract the neutral D mixing parameters. At t = 0 the
only term in the amplitude is the direct doubly-Cabibbo-
suppressed (DCS) mode D0 → K+π−, but for t > 0
D0 − D0 mixing may contribute through the sequence
D0 → D0 → K+π− , where the second stage is Cabibbo
favored (CF). The interference of this term with the DCS
contribution involves the lifetime and mass differences
of the neutral D mass eigenstates, as well as the final-
state strong phase difference δKπ between the CF and
the DCS decay amplitudes. This interference plays a
key role in the measurement of the mixing parameters at
time-dependent measurements.
With the assumption of CPT invariance, the mass
eigenstates of D0 −D0 system are |D1〉 = p|D0〉+ q|D0〉
and |D2〉 = p|D0〉−q|D0〉 with eigenvalues µ1 = m1−
and µ2 = m2 −
Γ2, respectively, where the m1 and Γ1
(m2 and Γ2) are the mass and width of D1 (D2). For
the method of detecting D0 − D0 mixing involving the
D0 → Kπ decay mentioned above, in order to separate
the DCS decay from the mixing signal, one must study
the time-dependent decay rate. The proper-time evolu-
tion of the particle states |D0
(t)〉 and |D0
(t)〉 are
given by
|D0phys(t)〉 = g+(t)|D
0〉 − q
g−(t)|D0〉,
|D0phys(t)〉 = g+(t)|D
0〉 − p
g−(t)|D0〉, (1)
where
(e−im2t−
Γ2t ± e−im1t− 12Γ1t), (2)
with definitions
m ≡ m1 +m2
, ∆m ≡ m2 −m1,
Γ ≡ Γ1 + Γ2
, ∆Γ ≡ Γ2 − Γ1, (3)
Note the sign of ∆m and ∆Γ is to be determined by
experiments.
In practice, one define the following mixing parameters
x ≡ ∆m
, y ≡ ∆Γ
. (4)
The time-dependent decay amplitudes for D0
(t) →
K+π− and D0
(t) → K−π+ are described as
〈K+π−|H|D0phys(t)〉 = g+(t)AK+π− −
g−(t)AK+π−
AK+π− [λg+(t)− g−(t)], (5)
〈K−π+|H|D0phys(t)〉 = g+(t)AK−π+ −
g−(t)AK−π+
AK−π+ [λg+(t)− g−(t)], (6)
where AK+π− = 〈K+π−|H|D0〉, AK+π− =
〈K+π−|H|D0〉, AK−π+ = 〈K−π+|H|D0〉, and
AK−π+ = 〈K−π+|H|D0〉. Here, λ and λ are de-
fined as:
λ ≡ p
AK+π−
AK+π−
, (7)
λ ≡ q
AK−π+
AK−π+
. (8)
From Eqs. (5) and (6), one can derive the general ex-
pression for the time-dependent decay rate, in agreement
http://arxiv.org/abs/0704.0120v3
with [3, 4]:
dΓ(D0
(t) → K+π−)
dtN = |AK
+π− |2
e−Γt ×
[(|λ|2 + 1)cosh(yΓt) +
(|λ|2 − 1)cos(xΓt) +
2Re(λ)sinh(yΓt) +
2Im(λ)sin(xΓt)] (9)
dΓ(D0
(t) → K−π+)
dtN = |AK−π+ |
e−Γt ×
[(|λ|2 + 1)cosh(yΓt) +
(|λ|2 − 1)cos(xΓt) +
2Re(λ)sinh(yΓt) +
2Im(λ)sin(xΓt)] (10)
where N is a common normalization factor. In order to
simplify the above formula, we make the following defi-
nition:
≡ (1 +AM )e−iβ , (11)
where β is the weak phase in mixing and AM is a real-
valued parameter which indicates the magnitude of CP
violation in the mixing. For f = K−π+ final state, we
define
AK+π−
AK+π−
r′e−iα
AK−π+
AK−π+
re−iα, (12)
where r′ and α′ (r and α) are the ratio and relative phase
of the DCS decay rate and the CF decay rate. Then, λ
and λ can be parameterized as
λ = −
1 +AM
e−i(α
′−β) , (13)
λ = −
r(1 +AM )e
−i(α+β). (14)
In order to demonstrate the CP violation in decay, we
define
RD(1 + AD) and
1 +AD
Thus, Eqs. (13) and (14) can be expressed as
λ = −
1 +AD
1 +AM
e−i(δ−φ) , (15)
λ = −
1 +AM
1 +AD
e−i(δ+φ) , (16)
where δ =
α′ + α
is the averaged phase difference be-
tween DCS and CF processes, and φ =
α− α′
We can characterize the CP violation in the mixing
amplitude, the decay amplitude, and the interference
between amplitudes with and without mixing, by real-
valued parameters AM , AD, and φ as in Ref [5, 6].
In the limit of CP conservation, AM , AD and φ are
all zero. AM = 0 means no CP violation in mixing,
namely, |q/p| = 1; AD = 0 means no CP violation in
decay, for this case, r = r′ = RD = |AK−π+/AK−π+ |2 =
|AK+π−/AK+π− |2; φ = 0 means no CP violation in the
interference between decay and mixing.
In experimental searches, one can define CF decay as
right-sign (RS) and DCS decay or via mixing followed
by a CF decay as wrong-sign (WS). Here, we define the
ratio of WS to RS decays as for D0:
R(t) =
dΓ(D0
(t) → K+π−)
dtN × e−Γ|t| × 2|AK+π− |2
, (17)
and for D0:
R(t) =
dΓ(D0
(t) → K−π+)
dtN × e−Γ|t| × 2|AK−π+ |2
, (18)
Taking into account that |λ|, |λ| ≪ 1 and x, y ≪ 1,
keeping terms up to order x2, y2 and RD in the ex-
pressions, neglecting CP violation in mixing, decay and
the interference between decay with and without mixing
(AM = 0, AD = 0, and φ = 0), expanding the time-
dependent for xt, yt <∼ Γ−1, combing Eqs. (9) and (10),
we can write Eqs. (17) and (18) as
R(t) = R(t) = RD +
(Γt)2, (19)
where
x′ = xcosδ + ysinδ, (20)
y′ = −xsinδ + ycosδ. (21)
In the limit of SU(3) symmetry, AK+π− and AK+π−
(AK−π+ and AK−π+) are simply related by CKM fac-
tors, AK+π− = (VcdV
us/VcsV
ud)AK+π− [7]. In particular,
AK+π− and AK+π− have the same strong phase, leading
to α′ = α = 0 in Eq. (12). But the SU(3) symmetry
is broken according to the recent precise measurements
from the B factories, the ratio [5]:
R = BR(D
0 → K+π−)
BR(D0 → K+π−)
, (22)
is unity in the SU(3) symmetry limit. But, the world
average for this ratio is
Rexp = 1.21± 0.03, (23)
computed from the individual measurements using the
standard method of Ref. [4]. Since the SU(3) is bro-
ken in D → Kπ decays at the level of 20%, in which
case the strong phase δ should be non-zero. Recently, a
time-dependent analysis in D → Kπ has been performed
based on 384 fb−1 luminosity at Υ (4S) [1]. By assuming
CP conservation, they obtained the following neutral D
mixing results
RD = (3.03± 0.16± 0.10)× 10−3,
= (−0.22± 0.30± 0.21)× 10−3,
y′ = (9.7± 4.4± 3.1)× 10−3. (24)
TABLE I: Experimental results used in the paper. Only one
error is quoted, we have combined in quadrature statistical
and systematic contributions.
Parameter BaBar (×10−3) Belle(×10−3) Technique
-0.22± 0.37 [1] 0.18+0.21−0.23 [8] Kπ
′ 9.7± 5.4 [1] 0.6+4.0−3.9 [8] Kπ
RD 3.03± 0.19 [1] 3.64 ± 0.17 [8] Kπ
yCP - 13.1 ± 4.1 [2] K
−, π+π−
x - 8.0± 3.4 [9] KSπ
y - 3.3± 2.8 [9] KSπ
The result is inconsistent with the no-mixing hypoth-
esis with a significance of 3.9 standard deviations. The
results from BaBar and Belle are in agreement within 2
standard deviation on the exact analysis of y′ measure-
ment by using D → Kπ as listed in Table I. As indicated
in Eq. (23), the strong phase δ should be non-zero due to
the SU(3) violation. One has to know the strong phase
difference exactly in order to extract the direct mixing
parameters, x and y as defined in Eqs. (4). However,
at the B factory, it is hard to do that with a model-
independent way [7, 10]. In order to extract the strong
phase δ we need data near the DD threshold to do a CP
tag as discussed in Ref. [7]. Here, we would like to figure
out the possible physics solution of the strong phase δ by
using the recent results from the B factories with differ-
ent decay modes, so that we can have an idea about the
sensitivity to measure the strong phase at the BES-III
project.
In Ref [2], Belle collaboration also reported the result
of yCP =
τ(D0→K+π−)
τ(D0→fCP )
− 1, where fCP = K+K− and
yCP = (13.1± 3.2± 2.5)× 10−3. (25)
The result is about 3.2σ significant deviation from zero
(non-mixing). In the limit of CP symmetry, yCP = y [11,
12]. In the decay of D0 → KSπ+π−, Belle experiment
has done a Dalitz plot (DP) analysis [9], they obtained
the direct mixing parameters x and y as
x = (8.0± 3.4)× 10−3, y = (3.3± 2.8)× 10−3, (26)
where the error includes both statistic and systematic un-
certainties. Since the parameterizations of the resonances
on the DP are model-dependent, the results suffer from
large uncertainties from the DP model. In this analysis,
they see a significance of 2.4 standard deviations from
non-mixing. Here, we will use the value of x measured in
the DP analysis for further discussion. As shown in Eq.
(21), once y, y′ and x are known, it is straightforward to
extract the strong phase difference between DCS and CF
decay in D0 → Kπ decay. If taking the measured central
values of x, yCP (≈ y) , and y′ as input parameters, we
found two-fold solutions for tanδ as below:
tanδ = 0.35± 0.63, or − 7.14± 29.13, (27)
which are corresponding to (19± 32)0 and (−820 ± 30)0,
respectively.
At ψ(3770) peak, to extract the mixing parameter y,
one can make use of rates for exclusive D0D0 combina-
tion, where both the D0 final states are specified (known
as double tags or DT), as well as inclusive rates, where
either the D0 or D0 is identified and the other D0 de-
cays generically (known as single tags or ST) [13]. With
the DT tag technique [14, 15], one can fully consider
the quantum correlation in C = −1 and C = +1 D0D0
pairs produced in the reaction e+e− → D0D0(nπ0) and
e+e− → D0D0γ(nπ0) [13, 16, 17], respectively.
For the ST, in the limit of CP conservation, the rate
of D0 decays into a CP eigenstate is given as [13]:
Γfη ≡ Γ(D0 → fη) = 2A2fη [1− ηy] , (28)
where fη is a CP eigenstate with eigenvalue η = ±1, and
Afη = |〈fη|H|D0〉| is the real-valued decay amplitude.
For the DT case, Gronau et. al. [7] and Xing [18]
have considered time-integrated decays into correlated
pairs of states, including the effects of non-zero final state
phase difference. As discussed in Ref. [7], the rate of
(D0D0)C=−1 → (l±X)(fη) is described as [7]:
Γl;fη ≡ Γ[(l±X)(fη)] = A2l±XA
(1 + y2)
≈ A2l±XA
, (29)
where Al±X = |〈l±X |H|D0〉| is real-valued amplitude for
semileptonic decays, here, we neglect y2 term since y ≪
For C = −1 initial D0D0 state, y can be expressed in
term of the ratios of DT rates and the double ratios of
ST rates to DT rates [13]:
Γl;f+Γf−
Γl;f−Γf+
Γl;f−Γf+
Γl;f+Γf−
. (30)
For a small y, its error, ∆(y), is approximately 1/
Nl±X ,
where Nl±X is the total number of (l
±X) events tagged
with CP -even and CP -odd eigenstates. The num-
ber Nl±X of CP tagged events is related to the to-
tal number of D0D0 pairs N(D0D0) through Nl±X ≈
N(D0D0)[BR(D0 → l± +X)× BR(D0 → f±)× ǫtag] ≈
1.5 × 10−3N(D0D0), here we take the branching ratio-
times-efficiency factor (BR(D0 → f±)× ǫtag) for tagging
CP eigenstates is about 1.1% (the total branching ratio
into CP eigenstates is larger than about 5% [4]). We find
∆(y) =
N(D0D0)
= ±0.003. (31)
If we take the central value of y from the measurement
of yCP at Belle experiment [2], thus, at BES-III exper-
iment [19], with 20fb−1 data at ψ(3770) peak, the sig-
nificance of the measurement of y could be around 4.3 σ
deviation from zero.
We can also take advantage of the coherence of the
D0 mesons produced at the ψ(3770) peak to extract the
strong phase difference δ between DCS and CF decay am-
plitudes that appears in the time-dependent mixing mea-
surement in Eq. (19) [7, 13]. Because the CP properties
of the final states produced in the decay of the ψ(3770)
are anti-correlated [16, 17], one D0 state decaying into a
final state with definite CP properties immediately iden-
tifies or tags the CP properties of the other side. As
discussed in Ref. [7], the process of one D0 decaying to
K−π+, while the other D0 decaying to a CP eigenstate
fη can be described as
ΓKπ;fη ≡ Γ[(K−π+)(fη)] ≈ A2A2fη |1 + η
−iδ|2
≈ A2A2fη (1 + 2η
RDcosδ),
where A = |〈K−π+|H|D0〉| and Afη = |〈fη|H|D0〉| are
the real-valued decay amplitudes, and we have neglected
the y2 terms in Eq. (32). In order to estimate the total
sample of events needed to perform a useful measurement
of δ, one defined [7, 10] an asymmetry
ΓKπ;f+ − ΓKπ;f−
ΓKπ;f+ + ΓKπ;f−
, (33)
where ΓKπ;f± is defined in Eq. (32), which is the rates for
the ψ(3770) → D0D0 configuration to decay into flavor
eigenstates and a CP -eigenstates f±. Eq. (32) implies a
small asymmetry, A = 2
RDcosδ. For a small asymme-
try, a general result is that its error ∆A is approximately
NK−π+ , where NK−π+ is the total number of events
tagged with CP -even and CP -odd eigenstates. Thus one
obtained
∆(cosδ) ≈ 1
NK−π+
. (34)
The expected number NK−π+ of CP -tagged events
can be connected to the total number of D0D0 pairs
N(D0D0) through NK−π+ ≈ N(D0D0)BR(D0 →
K−π+)×BR(D0 → f±)×ǫtag ≈ 4.2×10−4N(D0D0) [7],
here, as in Ref [7], we take the branching ratio-times-
efficiency factor BR(D0 → f±) × ǫtag = 1.1%. With
0 0.2 0.4 0.6 0.8 1
FIG. 1: Illustrative plot of the expected error (∆δ) of the
strong phase with various central values of cosδ. The expected
error of cosδ is 0.04 by ssuming 20fb−1 data at ψ(3770) peak
at BES-III. The two asterisks correspond to δ = 190 and
−820, respectively.
the measured RD = (3.03± 0.19)× 10−3 and BR(D0 →
K−π+) = 3.8% [4], one found [7]
∆(cosδ) ≈ ±444√
N(D0D0)
. (35)
At BESIII, about 72 × 106 D0D0 pairs can be collected
with 4 years’ running. If considering both K−π+ and
K+π− final states, we thus estimate that one may be
able to reach an accuracy of about 0.04 for cosδ. Fig-
ure 1 shows the expected error of the strong phase δ
with various central values of cosδ. With the expected
∆(cosδ) = ±0.04, the sensitivity of the strong phase
varies with the physical value of cosδ. For δ = 190 and
−820, the expected error could be ∆(δ) = ±8.70 and
±2.90, respectively.
By combing the measurements of x inD0 → KSππ and
yCP from Belle, one can obtain RM = (1.18±0.6)×10−4.
At the ψ(3770) peak, D0D0 pair are produced in a state
that is quantum-mechanically coherent [16, 17]. This en-
ables simple new method to measure D0 mixing param-
eters in a way similar proposed in Ref. [7]. At BES-III,
the measurement of RM can be performed unambigu-
ously with the following reactions [16]:
(i) e+e− → ψ(3770) → D0D0 → (K±π∓)(K±π∓),
(ii) e+e− → ψ(3770) → D0D0 → (K−e+ν)(K−e+ν),
(iii) e+e− → D−D∗+ → (K+π−π−)(π+
[K+e−ν]).
Reaction (i) in Eq. (36) can be normalized to D0D0 →
(K−π+)(K+π−), the following time-integrated ratio is
obtained by neglecting CP violation:
N [(K−π+)(K−π+)]
N [(K−π+)(K+π−)]
2 + y2
= RM . (37)
For the case of semileptonic decay, as (ii) in Eq. (36), we
N(l±l±)
N(l±l∓)
x2 + y2
= RM , (38)
The observation of reaction (i) would be definite evi-
dence for the existence of D0 −D0 mixing since the final
state (K±π∓)(K±π∓) can not be produced from DCS
decay due to quantum statistics [16, 17]. In particular,
the initial D0D0 pair is in an odd eigenstate of C which
will preclude, in the absence of mixing between the D0
and D0 over time, the formation of the symmetric state
required by Bose statistics if the decays are to be the
same final state. This final state is also very appealing
experimentally, because it involves a two-body decay of
both charm mesons, with energetic charged particles in
the final state that form an overconstrained system. Par-
ticle identification is crucial in this measurement because
if both the kaon and pion are misidentified in one of the
two D-meson decays in the event, it becomes impossi-
ble to discern whether mixing has occurred. At BESIII,
where the data sample is expected to be 20 fb−1 inte-
grated luminosity at ψ(3770) peak, the limit will be 10−4
at 95% C.L. for RM , but only if the particle identification
capabilities are adequate.
Reactions (ii) and (iii) offer unambiguous evidence
for the mixing because the mixing is searched for in the
semileptonic decays for which there are no DCS decays.
Of course since the time-evolution is not measured, obser-
vation of Reactions (ii) and (iii) actually would indicate
the violation of the selection rule relating the change in
charm to the change in leptonic charge which holds true
in the standard model [16].
In Table II, the sensitivity for RM measurements in
different decay modes are estimated with 4 years’ run at
BEPCII.
TABLE II: The sensitivity for RM measurements at BES-III
with different decay modes with 4 years’ run at BESPCII
0 Mixing
Reaction Events Sensitivity
RS(×104) RM (×10
ψ(3770) → (K−π+)(K−π+) 10.4 1.0
ψ(3770) → (K−e+ν)(K−e+ν) 8.9
ψ(3770) → (K−e+ν)(K−µ+ν) 8.1 3.7
ψ(3770) → (K−µ+ν)(K−µ+ν) 7.3
In the limit of CP conservation, by combing the mea-
surements of x in D0 → KSππ and yCP from Belle, one
can obtain RM = (1.18± 0.6)× 10−4. With 20fb−1 data
at BES-III, about 12 events for the precess D0D0 →
(K±π∓)(K±π∓) can be produced. One can observe 3.0
events after considering the selection efficiency at BE-
SIII, which could be about 25% for the four charged
particles. The background contamination due to double
particle misidentification is about 0.6 event with 20fb−1
data at BES-III [20]. Table III lists the expected mixing
signal for Nsig = N(K
±π∓)(K±π∓), background Nbkg ,
and the Poisson probability P (n), where n is the possible
number of observed events in experiment. In Table III,
we assume the RM = 1.18× 10−4, the expected number
of mixing signal events are estimated with 10fb−1 and
20fb−1, respectively.
TABLE III: The expected mixing signal for Nsig =
N(K±π∓)(K±π∓), background Nbkg , and the Poisson prob-
ability P (n) in 10 fb−1 and 20 fb−1 at BES-III at ψ(3770)
peak, respectively. Here, we take the mixing rate RM =
1.18× 10−4.
10 fb−1 (ψ(3770)) 20 fb−1 (ψ(3770))
36 million D0D0 72 million D0D0
Nsig 1.5 3.0
Nbkg 0.3 0.6
P (n = 0) 15.7% 2.5%
P (n = 1) 29.1% 9.1%
P (n = 2) 26.9% 16.9%
P (n = 3) 16.6% 20.9%
P (n = 4) 7.7% 19.3%
P (n = 5) 2.8% 14.3%
P (n = 6) 0.9% 8.8%
P (n = 7) 0.2% 4.7%
P (n = 8) 0.1% 2.2%
P (n = 9) 0.01% 0.9%
In conclusion, we discuss the constraints on the strong
phase difference in D0 → Kπ decay according to the
most recent measurements of y′, yCP and x from B fac-
tories. We estimate the sensitivity of the measurement of
mixing parameter y at ψ(3770) peak in BES-III experi-
ment. With 20 fb−1 data, the uncertainty ∆(y) could be
0.003. Thus, assuming y at a percent level, we can make
a measurement of y at a significance of 4.3σ deviation
from zero. The sensitivity of the strong phase differ-
ence at BES-III are obtained by using data near the DD
threshold with CP tag technique at BES-III experiment.
Finally, we estimated the sensitivity of the measurements
of the mixing rate RM , and find that BES-III experiment
may not be able to make a significant measurement of
RM with current luminosity by using coherent DD state
at ψ(3770) peak.
One of the authors (H. B. Li) would like to thank
David Asner and Zhi-Zhong Xing for stimulating dis-
cussion, Chang-Zheng Yuan for useful discussion on the
statistics used in this paper, and also thank Stephen
L. Olsen and Yang-Heng Zheng for commenting on this
manuscript. We thank BES-III collaboration for provid-
ing us many numerical results based on GEANT4 simula-
tion. This work is supported in part by the National Nat-
ural Science Foundation of China under contracts Nos.
10205017, 10575108,10521003, and the Knowledge Inno-
vation Project of CAS under contract Nos. U-612 and
U-530 (IHEP).
∗ Electronic address: chengxd@ihep.ac.cn
† Electronic address: hekl@ihep.ac.cn
‡ Electronic address: lihb@ihep.ac.cn
§ Electronic address: yfwang@ihep.ac.cn
¶ Electronic address: yangmz@ihep.ac.cn
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mailto:chengxd@ihep.ac.cn
mailto:hekl@ihep.ac.cn
mailto:lihb@ihep.ac.cn
mailto:yfwang@ihep.ac.cn
mailto:yangmz@ihep.ac.cn
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