70. D0 -D0 Mixing

1

70. D0 -D0 Mixing
Revised September 2023 by D.M. Asner (BNL) and A.J. Schwartz (U. of Cincinnati).
The formalism for D0 -D0 mixing is closely related to that for CP violation; for further details
on the latter, see the note “CP Violation in the Quark Sector” in this Review. The time evolutionof the D0 –D0 system is described by the Schrödinger equation!

∂ D0 (t)i∂t D0 (t)i= M− Γ2

!

D0 (t),
D0 (t)(70.1)

where the M and Γ matrices are Hermitian, and CP T invariance requires that M11 = M22 ≡ M
and Γ11 = Γ22 ≡ Γ . The off-diagonal elements of M and Γ are referred to as the dispersive andabsorptive parts, respectively, of the mixing. The mass eigenstates D1 and D2 of the HamiltonianM - iΓ/2 are defined as|D1,2 i ≡ p|D0 i ± q|D0 i ,(70.2)
where normalization imposes |p|2 + |q|2 = 1. If p = ±q, then the mass eigenstates are CP
eigenstates and CP is conserved in mixing. Our phase convention is CP |D0 i = −|D0 i andArg(q/p) ∈ [−π/2, π/2], which imply that, in the absence of CP violation in mixing, D2 is CP -evenand D1 is CP -odd.
The eigenvalues of M - iΓ/2 areω1,2 =

i
M− Γ
2



q
i
±
M12 − Γ12
p
2




i
≡ m1,2 − Γ1,2 ,
2(70.3)

where m1,2 and Γ1,2 are real and correspond to the masses and decay widths, respectively, of theD1,2 mass eigenstates. As the trace Γ11 + Γ22 = 2Γ is unchanged by diagonalizing Γ, Γ must equal(Γ1 + Γ2 )/2, the mean decay width. Solving for the eigenstates of the eigenvalues yields
 2

q
p

=

∗ − iΓ∗M12
2 12.
M12 − 2i Γ12(70.4)

If CP is conserved in mixing, then (q/p)2 = 1 and M12 and Γ12 must be real. In this case, thedifference in eigenvalues is ∆m ≡ m2 − m1 = 2M12 and ∆Γ ≡ Γ2 − Γ1 = 2Γ12 . The signs of ∆mand ∆Γ are difficult to predict from theory and thus must be determined experimentally.
We define dimensionless mixing parameters x and y as
x≡

∆m
Γ

(70.5)

y≡

∆Γ
.
2Γ

(70.6)

These parameters are measured in several ways. The most precise values are obtained by measuring
the time dependence of D0 decays. For all methods, the initial flavor of the D0 or D0 when produced
must be determined. The most common method used for this is to reconstruct D∗+ → D0 π + orD∗− → D0 π − decays; the charge of the accompanying pion (which has low momentum in the labframe and is often referred to as the “soft” pion) determines the flavor of the neutral D. BABAR
and LHCb have also identified the flavor of the neutral D by reconstructing the semileptonic decays
B + → D0 `+ ν, B 0 → D∗− `+ ν, B − → D0 `− ν, and B 0 → D∗+ `− ν; in this case the charge of the
S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
31st May, 2024 10:13am

70. D0 -D0 Mixing

2

accompanying lepton determines the D flavor. Both experiments have used both tags together
to select “double-tagged” B → D∗± `∓ ν, D∗± → (D0 , D0 )π ± decays, which have especially highpurity. At e+ e− collider experiments such as Belle, BABAR, and BESIII, the D flavor can also be
determined by fully reconstructing a flavor-specific D decay on the “opposite side” of an event, i.e.,
recoiling against the signal-side D decay.
At BESIII, where DD pairs are produced near their threshold via e+ e− → ψ(3770) → D0 D0 ,there is relatively little background and the purity of opposite-side tagging is equivalent to that
achieved using D∗± decays. However, BESIII operates at a symmetric e+ e− collider, and the DD
pairs are produced almost at rest in the lab frame. As a consequence, the D’s do not travel any
appreciable distance before decaying, and time-dependent analyses are not possible. To overcome
this, measurements of mixing at BESIII utilize the quantum coherence of the initial ψ(3770) →D0 D0 state and time-integrated measurements [1–5].

70.1 Time-Dependent Analyses
Our notation is as follows: Cabibbo-favored (CF) decay amplitudes are denoted Af ≡ hf |H|D0 i
and Af ≡ hf |H|D0 i; i.e, the final state is f = K + `− ν, K + π −, K + π − π 0, etc. Doubly-Cabibbosuppressed (DCS) decay amplitudes are denoted Af ≡ hf |H|D0 i and Af ≡ hf |H|D0 i. When
discussing decay rates, we neglect constant phase-space factors, which divide out in ratios of rates
to common final states.
Starting from a pure |D0 i or |D0 i state at t = 0, the time-dependent decay rates to “wrong-sign”
final states can be writtenr(t) ≡ hf |H|D0 (t)ir(t) ≡ hf |H|D0 (t)i

2

= Af
2

= Af

2

2
q 2
g+ (t) λ−1
f + g− (t)p(70.7)

22
p 2
g+ (t) λf + g− (t) ,q(70.8)

where
λf ≡

q Af
,
p Af

(70.9)

1  −iω t
e 1 ± e−iω2 t .
2(70.10)

q Af
,
p Af

and
g± (t) =λf ≡A change in convention for the relative phase of D0 and D0 would cancel between q/p and Af /Af or
Af /Af , leaving λf and λf unchanged. For multibody final states, these equations apply separately
to each point in phase-space. Integrating over regions of phase-space can lead to enhanced sensitivity
to CP violation; see the discussion below on multibody decays and the note “Review of Multibody
Charm Analyses" in this Review. As the mixing parameters x and y are very small, r(t) and r(t)are usually expanded to second order in x and y.

70.2 Semileptonic decays
Consider the final state f = K + `− ν̄` , where Af = Af = 0 is an excellent approximation in theStandard Model. The final state f is accessible from a D0 only via mixing,1 and the decay rate isr(t) = Af

2

2 q 2
q 2
|g− (t)|2 ≈ Afpp

1x2 + y 24!

(Γ t)2 e−Γ t .

(70.11)

There exists a doubly Cabibbo-suppressed amplitude in which the c and ū quarks exchange a W , and then the
resulting d quark (from c) decays semileptonically. We neglect this second-order process.

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70. D0 -D0 Mixing

3

For r(t), q/p is replaced by p/q. In the Standard Model, CP violation in charm mixing is small
and |q/p| ≈ 1. In the limit of CP conservation, r(t) = r(t), and the time-integrated mixed decayrate relative to the time-integrated unmixed decay rate for semileptonic decays isZ ∞r(t) dt
Z ∞ 0

Af

2

=
e−Γ t dt

x2 + y 2≡ RM .
2(70.12)0

Table 70.1 summarizes results for RM from semileptonic decays. The world average from the Heavy
Flavor Averaging Group (HFLAV) [6] is RM = (1.30 ± 2.69) × 10−4 .
Table 70.1: Results for RM = (x2 + y 2 )/2 in D0 semileptonic decays. The HFLAV averageassumes statistical and systematic uncertainties are uncorrelated. When a single uncertainty islisted, that corresponds to statistical and systematic uncertainties combined. The measurementswith an asterisk (∗ ) have been superseded and thus are not included in the HFLAV average.
Year Experiment
Final state(s)RM (×10−3 )90% C.L. (×10−3 )
−1
(∗)+
−
2008 Belle (492 fb ) [7]Ke νe
0.13±0.22±0.20< 0.61
+0.70
−1
(∗)+
−
2007 BABAR (344 fb ) [8]Ke νe
0.04 −0.60(−1.3, 1.2)
2005 CLEO (9.0 fb−1 ) [9]K (∗)+ e− ν e
1.6±2.9±2.9< 7.8+3.0 +0.0
10+−
1.1 −2.7 −0.1
1996 E791 (2 × 10 evts) [10] K ` ν `< 5.0
HFLAV Average [6]0.130 ± 0.269
2005∗ Belle (253 fb−1 ) [11]K (∗)+ e− ν e0.02±0.47±0.14
< 1.0
∗
−1
2004 BABAR (87 fb ) [12]K (∗)+ e− ν e
2.3±1.2±0.4< 4.2

70.3 Wrong-sign decays to hadronic non-CP eigenstates
Consider the final state f = K + π −, i.e., Af¯ and Af are CF, Af and Af are DCS. Because CF
and DCS decays proceed via tree-level amplitudes, and such amplitudes involve only the first two
quark generations, direct CP violation is negligible2 . The ratios of decay amplitudes can be writtenqAf+ −iδf= − RD
e
Af

Af
Af

q

− −iδfe,= − RD(70.13)where δf is the strong phase difference between the DCS and CF amplitudes. The minus signoriginates from the weak phase difference between the amplitudes, specifically, the relative minus∗ V ). The
sign between Vus and Vcd (which produces a relative minus sign between Vcs∗ Vud and Vus
cd−+
parameters RD and RD are the ratios of the DCS decay rate to the CF decay rate. From the−+relevant CKM matrix elements, one estimates RD, RD∼ tan4 θc , where θc is the Cabibbo angle.
With the parameterization of Eq. (70.13), Eq. (70.9) becomes
q
p Af
+ p
=
−
RD
λ−1
=
e−i(δf +φ)(70.14)
f
q Af
q
λf

q
q Af
− q
=
= − RDe−i(δf −φ) ,p Afp(70.15)where φ = Arg(q/p). The weak phase φ is independent of the final state f and is often referred to−+
as “universal.” For convenience, we define the mean decay rate RD ≡ (RD+ RD)/2, and the decay−−++rate asymmetry AD ≡ (RD − RD )/(RD + RD ).
2

For two quark generations, the weak phases can be defined to eliminate all weak-phase differences.

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70. D0 -D0 Mixing

4

With these definitions, we expand the decay rates in Eqs. (70.7) and (70.8) to second order inthe small mixing parameters x and y to obtain [13, 14]:r(t) = Af

2"

RD (1 + AD ) +qe−Γ t RD (1 − AD ) +

q

e

−Γ t

0 2)
q 2 (x0+2 + y+
q 0
y+ (Γ t) +RD (1 + AD )(Γ t)2pp
4#(70.16)andr(t) = Af

2

"

#

0 2)
p 0
p 2 (x0−2 + y−
RD (1 − AD )y− (Γ t) +(Γ t)2 , (70.17)
q
q
4

where
x0± = x cos(δf ± φ) + y sin(δf ± φ)(70.18)0y±= y cos(δf ± φ) − x sin(δf ± φ) .

(70.19)Defining the “strong-phase-rotated” mixing parametersx0 ≡ x cos δf + y sin δf(70.20)0y ≡ y cos δf − x sin δf(70.21)x0± = x0 cos φ ± y 0 sin φ(70.22)0y±= y 0 cos φ(70.23)gives
0∓ x sin φ ,0 are obtained from x0 , y 0 via an additional “weak-phase rotation.” To summarize,i.e., x0± and y±
parameters (x0 , y 0 ) are the mixing parameters (x, y) rotated by the strong phase δf , and parameters0 ) are the parameters (x0 , y 0 ) rotated by the weak phase +φ for D 0 decays and −φ for D 0(x0± , y±
0 2 = x0 2 + y 0 2 = x0 2 + y 0 2 = x2 + y 2 . In Eqs. (70.16) and (70.17), a
decays. Note that x0+2 + y+−−
2 )(Γ t)2 /4 has been dropped, as it is negligible relative to the otherfourth term RD (1 ± AD )(x2± − y±terms for the range of decay times measured by experiments.
Comparing Eqs. (70.16) and (70.17), one sees that r(t) 6= r(t) and CP is violated if either
AD 6= 0, |q/p| =6 1, or φ 6= 0. These three inequalities correspond, respectively, to the three types of−+CP violation: in the decay amplitudes (RD
6= RD); in the mixing; and due to interference between
a mixed decay amplitude (i.e., mixing is followed by decay) and an unmixed decay amplitude.
Whereas CP violation in the decay amplitudes is parameterized by AD , CP violation in mixing isparameterized by AM ≡ (|q/p| − |p/q|)/(|q/p| + |p/q|).
In the limit of CP conservation, AD = 0, |q/p| = 1, and φ = 0. In this caser(t) = r(t) = Af

2

"

e

−Γ t

RD +

p

x0 2 + y 0 2
RD y (Γ t) +(Γ t)24
0#,(70.24)and the total number of D0 → f decays divided by the total number of D0 → f¯ decays isZ ∞r(t) dtpx0 2 + y 0 2= RD + RD y 0 +.
R = Z ∞ 0 2
2
−Γ t
Af e
dt

(70.25)0

The ratio R is more straightforward to measure than r(t) or r̄(t), as there is no decay-timedependence. In Table 70.2 we report measurements of R, RD , and AD in D0 → K + π − decays
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70. D0 -D0 Mixing

5

normalized to D0 → K − π + decays, and results from HFLAV [15] obtained from a global fit to all
relevant data that allows for both mixing and CP violation (see Section 70.7). The experiments
typically perform a single fit for parameters RD , x0 2 , and y 0 ; results for x0 2 and y 0 are listed in+0 ) and
Table 70.3. Allowing for CP violation, the experiments measure parameters (RD, x0+2 , y+
−
−
+
0
2
0
0
(RD , x− , y− ) [or equivalently (RD , AD ) instead of (RD , RD )] by separately fitting the D → K + π −and D0 → K − π + event samples.
Table 70.2: Results for R, RD , and AD as measured using D0 → K ± π ∓ decays. When a single
uncertainty is listed, that corresponds to statistical and systematic uncertainties combined. The
measurements with an asterisk (∗ ) have been superseded and thus are not included in the HFLAV
global fit (Section 70.7). The measurements with a dagger († ) are not included in the HFLAV
global fit due to much poorer precision.
Year
2018
2017
2014
2013
2007
2013b∗
2013a∗
2008∗
2006∗
2005†
2000†
1998†

Experiment
LHCb (5.0 fb−1 D∗ tag) [16]LHCb (3.0 fb−1 B +D∗
double tag) [17]Belle (976 fb−1 ) [18]
CDF (9.6 fb−1 ) [19]BABAR (384 fb−1 ) [20]
HFLAV Fit Result [15]LHCb (3.0 fb−1 D∗ tag) [21]LHCb (1.0 fb−1 ) [22]
CDF (1.5 fb−1 ) [23]Belle (400 fb−1 ) [24]
FOCUS (234 evts) [25]
CLEO (9.0 fb−1 ) [26]E791 (5643 evts) [27]R (×10−3 )—RD (×10−3 )3.454±0.031

AD (%)−0.01±0.91—
3.86±0.06
4.30±0.05
3.53±0.08±0.04

3.48±0.10
3.53±0.13
3.51±0.35
3.03±0.16±0.10
3.434 ± 0.019
3.568±0.066
3.52±0.15
3.04±0.55
3.64±0.18
5.17 +1.47−1.58 ±0.76
4.8±1.2±0.4—−3.15±3.31——−2.1±5.2±1.5−0.70 ± 0.36
−0.7±1.9——
2.3±4.7
13 +33−25 ±10−1 +16−17 ±1——4.25±0.04
4.15±0.10
3.77±0.08±0.05+0.63
4.29 −0.61
±0.27
+0.63
3.32 −0.65 ±0.40
6.8 +3.4−3.3 ±0.7

Extraction of the mixing parameters x and y from measurements of x0 and y 0 requires knowledgeof the strong phase difference δKπ . This can be determined from the decay rates of D± →√K + π − ,where D+ (D− ) denotes the CP -even (CP -odd) eigenstate. Since |D± i = (|D0 i ∓ |D0 i)/ 2,√
(70.26)2 A(D± → K + π − ) = A(D0 → K + π − ) ∓ A D0 → K + π − .
Squaring this amplitude and using Eq. (70.13) yields the relationcos δKπ =|A(D+ → K + π − )|2 − |A(D− → K + π − )|2.
2 |A(D0 → K + π − )| |A(D0 → K + π − )|(70.27)

Measuring the right-hand side is possible if one can identify pure D+ , D− , D0 , and D0 initialstates. This is accomplished at CLEOc and BESIII utilizing the processes e+ e− → ψ(3770) →D0 D0 → (fCP )(K + π − ), or ψ(3770) → D0 D0 → (fD0 )(K + π − ), where fCP denotes a CP -specificfinal state, and fD0 denotes a D0 -flavor-specific final state. In the first case, quantum coherence
and CP symmetry ensures that the K + π − state originates from a neutral D with CP oppositethat of fCP . In the second case, at the time when the D0 decays, the opposite side is D0 . However,it can potentially mix to D0 before decaying to K + π − , and this introduces some dependence on
the mixing parameters x and y. This dependence is seen explicitly in the observable
ACP
Kπ ≡

|A(D− → K − π + )|2 − |A(D+ → K − π + )|2.
|A(D− → K − π + )|2 + |A(D+ → K − π + )|2
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(70.28)

70. D0 -D0 Mixing

6

Table 70.3: Results for x02 and y 0 , as measured using D0 → K ± π ∓ decays. When a single
uncertainty is listed, that corresponds to statistical and systematic uncertainties combined. The
measurements with an asterisk (∗ ) have been superseded and thus are not included in the HFLAV
global fit. The measurements with a dagger († ) are not included in the HFLAV global fit due tomuch poorer precision. All confidence limits and intervals correspond to 95% C.L. The Belle 2006
results restrict x02 to the physical region. The BABAR confidence intervals are obtained from thefit, whereas Belle uses a Feldman-Cousins method, and CDF uses a Bayesian method.
Year

No CP violationy 0 (%)

Experiment

Allowing for CP violation
x02 (×10−3 )y 0 (%)(
D0 : 0.061±0.037
0.501±0.074
D0 : 0.016±0.039
0.554±0.074

x02 (×10−3 )
!

2018
2017
2014
2013
2007
2006

5.0 fb−1
LHCb[16] 0.039±0.027 0.528±0.052
D∗ tag
(
3.0 fb−1
D0 : −0.019±0.447

∗
LHCb B +D
0.46±0.37
[17] 0.028±0.310
D0 : 0.079±0.433
double tag
Belle (976 fb−1 ) [18]0.09±0.22
0.46±0.34—−1
CDF (9.6 fb ) [19]0.08±0.18
0.43±0.43—(
0 : −0.24±0.52
D
BABAR (384 fb−1 ) [20] −0.22±0.37 0.97±0.54
D0 : −0.20±0.50
Belle (400 fb−1 ) [24]

0.18 +0.21
−0.23

∗ 

0.06 +0.40
−0.39

!

∗

3.0 fb−1[21] 0.055±0.049 0.48±0.10
D∗ tag2013a∗ LHCb (1.0 fb−1 ) [22]−0.09±0.13
0.72±0.24
2008∗ CDF (1.5 fb−1 ) [23]−0.12±0.35
0.85±0.76
2005† FOCUS (234 evts) [25]< 8.3
−7.2 < y 0 < 4.1+1.3
2000† CLEO (9.0 fb−1 ) [26]0.00±0.23−2.3 −1.4†
1998 E791 (5643 evts) [27]
< 17
< 13
2013b∗ LHCb(

0.581±0.526
0.332±0.523——
0.98±0.78
0.96±0.75< 0.72−2.8 < y 0 < 2.1

D0 : 0.049±0.070
D0 : 0.060±0.068
—
—
< 8.0
0.00±0.23—

0.51±0.14
0.45±0.14——−11.2 < y 0 < 6.7−2.5 +1.4−1.6—

To lowest order in the mixing parameters [28],√2 RD cos δKπ + y
CP
AKπ =
,
1+R

(70.29)where R is defined in Eq. (70.25). Such measurements are discussed in Section 70.5.
70.3.1 Wrong-sign decays to multibody final states
For multibody final states, Eqs. (70.13)-(70.25) apply to each point in phase-space.3 Although x
and y do not vary across phase-space, knowledge of the resonant substructure is needed to determine
the strong phase difference δ from point to point to extract x and y. Alternatively, experimental
knowledge of the strong phase difference between D0 and D0 decay amplitudes across phase space[29] allow one to determine x and y independent of a decay model of resonant substructure. This
phase information can be measured at the charm threshold, where CLEO-c and BESIII took data.
A time-dependent analysis at BABAR [30, 31] of D0 → K + π − π 0 decays, relative to CF D0 →+K π − π 0 decays, determined the strong phase variation across the Dalitz plot and reported x00 =3
However, if the decay amplitudes involve the third generation, then Af /Af and Af /Af in Eq. (70.13) also haveweak phase differences. However, this phase difference is estimated to be negligible for the sensitivity of current andforeseeable future experimental measurements.

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70. D0 -D0 Mixing

7

+0.55
00
(2.61 +0.57−0.68 ± 0.39)% and y = (−0.06 −0.64 ± 0.34)% . These mixing parameters are defined as

x00 = x cos δKππ0 + y sin δKππ0(70.30)00

y = y cos δKππ0 − x sin δKππ0 ,(70.31)in analogy with x0 , y 0 , and δKπ of Eqs. (70.20) and (70.21). Here, δKππ0 is the strong phase difference
between the amplitudes A(D0 → K + ρ− ) and A(D0 → K + ρ− ). The phase difference δKππ0 can be
determined in a manner similar to that for δKπ : by using Eq. (70.27) and quantum-correlated
measurements of the branching fractions B(D+ → K + ρ− ), B(D− → K + ρ− ), B(D0 → K + ρ− ), andTable 70.4: Results from time-dependent multibody analyses. The errors are statistical, systematic, and, when a third error is listed, due to the decay-model, respectively. The measurementwith an asterisk (∗ ) has been superseded and thus is not included in the HFLAV global fit. The
measurement with a dagger († ) is not included in the HFLAV global fit due to poorer precision.
The 2019 LHCb result utilizes strong-phase measurements from CLEO-c [32] and thus is decaymodel independent. The 2019–2023 LHCb fits determine CP -violating parameters ∆x and ∆y; thetranslation of these parameters to |q/p| and φ is given in Ref. [33].
No CP Violation
Year
Experiment
Final state(s)x (×10−3 )
−1
0
+
−
2023
LHCb (5.4 fb B tag) [34]KS π π4.29 ± 1.48 ± 0.26
−1
∗
0
+
−
2021
LHCb (5.4 fb D tag) [35]KS π π3.97 ± 0.46 ± 0.29
2019 LHCb (3.0 fb−1 B, D∗ tags) [36] KS0 π + π −2.7 ± 1.6 ± 0.4
2016
LHCb (1.0 fb−1 D∗ tag) [37]KS0 π + π −−8.6 ± 5.3 ± 1.7
2016
BABAR (468 fb−1 ) [38]π+π−π015 ± 12 ± 6+0.6−10 π+π−
2014
Belle (921 fb ) [39]K
5.6
± 1.9 +0.3
−0.9 −0.9( S
KS0 π + π −1.6±2.3±1.2±0.8
2010
BABAR (469 fb−1 ) [40]KS0 K + K −
+1.0
2007∗
Belle (540 fb−1 ) [41]KS0 π + π −8.0 ± 2.9 +0.9
−0.7 −1.4+32
2005†
CLEO (9.0 fb−1 ) [42]KS0 π + π −19 −33 ± 4 ± 4

Year
2023
2021

2019

Experiment
LHCb (5.4 fb−1 ) [34]
LHCb (5.4 fb−1 ) [35]

LHCb (3.0 fb−1 ) [36]2014

Belle (921 fb−1 ) [39]2007∗‡

Belle (540 fb−1 ) [41]With CP Violation
Final state(s) (|q/p|∆x × 103 =KS0 π + π −−0.77 ± 0.93 ± 0.28

 0.996 ± 0.052∆x × 103 =KS0 π + π − −0.27 ± 0.18 ± 0.01
+0.22

 1.05 −0.17
KS0 π + π −∆x × 103 = −0.53 ± 0.70 ± 0.22
+0.05 +0.06
KS0 π + π −0.90 +0.16−0.15 −0.04 −0.05
KS0 π + π −

‡

+0.06
0.86 +0.30−0.29 −0.03 ± 0.08

y (×10−3 )12.61 ± 3.12 ± 0.83
4.59 ± 1.20 ± 0.85
7.4 ± 3.6 ± 1.1
0.3 ± 4.6 ± 1.3
2±9±5+0.4 +0.3
3.0 ± 1.5 −0.5−0.6
5.7±2.0±1.3±0.7+0.8 +0.6
3.3 ± 2.4 −1.2
−0.8
−14 ± 24 ± 8 ± 4

φ(∆y × 103 =3.01 ± 1.92 ± 0.26


+2.7 ◦
 (3.2 −2.9 )∆y × 103 = 0.20 ± 0.36 ± 0.13

+6.3 ◦
 (−5.2 −9.2 )∆y × 103 = 0.6 ± 1.6 ± 0.3◦(−6 ± 11 ± 3 +3
−4 )
+5 +2 ◦(−14 +16−18 −3 −4 )This result allows for all types of CP violation and is superseded by Ref. [39], which assumes no direct CP
violation in CF or DCS decays.

31st May, 2024

70. D0 -D0 Mixing

8

B(D0 → K + ρ− ) in e+ e− → ψ(3770) events.
For the decay modes D0 and D0 → K + π − π + π − , Belle measured R = (0.324 ± 0.008 ±0.007)% [43]. Subsequently, a phase-space-integrated analysis from LHCb [44] measured the prodK3π and the strong-phase-rotated mixing parameter y 00
uct of a “coherence factor” RD
K3π . Thismeasurement resulted in an observation of charm mixing with 8.2σ significance.
Both the sign and magnitude of x and y without strong phases entering or sign ambiguity canbe determined by measuring the time-dependent resonant substructure of multibody D0 decays
to self-conjugate final states [41, 42]. For such decays, e.g., D0 → KS0 π + π − , the DCS and CF
decay amplitudes populate the same Dalitz plot, which allows for direct measurement of the strong
phase difference. Belle [39, 41], BABAR [40], and CLEO [45] have measured the overall phasedifference between D0 → K ∗ (892)− π + and D0 → K ∗ (892)+ π − to be [173.9 ± 0.7 (stat. only)]◦ ,◦[177.6 ± 1.1 (stat. only)]◦ , and [189 ± 10 ± 3 +15− 5 ] , respectively. These results are close to the◦∗ V ; thus
180 expected from Cabibbo factors, i.e., the relative minus sign between Vcs∗ Vud and Vus
cd
they indicate a small strong phase. Four LHCb measurements [34–37] of x, y using D0 → KS0 π + π −decays are decay-model independent, as the model of resonances in the intermediate state is replaced
by strong-phase measurements from CLEO-c [32] and BESIII [46]. Table 70.4 summarizes resultsfrom time-dependent analyses of self-conjugate multibody final states. World average values forthe measurements listed are given later, as a result of the HFLAV global fit.
With regard to resonant substructure in D0 → KS0 π + π − decays, Belle [39, 41] measured therelative strong phase (statistical errors only) and the ratio R (central values only) of the DCS
fit fraction relative to the CF fit fraction for five excited K states: K ∗ (892)+ π − , K0∗ (1430)+ π − ,K2∗ (1430)+ π − , K ∗ (1410)+ π − , and K ∗ (1680)+ π − . Similarly, BABAR [40, 47, 48] reported central
values of R for K ∗ (892)+ π − , K0∗ (1430)+ π − , and K2∗ (1430)+ π − . The systematic uncertainties on
R are not evaluated. Large differences in R are observed among these final states, which indicatessignificant hadronic effects.

70.4 Decays to CP Eigenstates
When the final state f is a CP eigenstate, there is no distinction between f and f . Thus
Af = Af and Af = Af . We denote final states with CP eigenvalues ±1 by f± . Decays toCP eigenstates proceed mainly via singly Cabibbo-suppressed amplitudes. Such amplitudes cancontain internal loops and thus involve the third quark generation; in this manner a weak phasewould appear in the decay amplitude, leading to direct CP violation. However, such internal loop
amplitudes are suppressed, and the presence of a weak phase is often neglected.
The mixing parameter y may be measured by comparing the rate for D0 decays to CP eigenstates such as K + K − with the rate to flavor eigenstates such as K − π + [14]. If decays to K + K −have a shorter effective lifetime than those to K − π + , then Γ+ > Γ− , or, since CP violation is verysmall, Γ2 > Γ1 and y is positive. For small mixing (x, y  1), the decay rates for D0 → f± and
D0 → f± have an approximately exponential time dependence:r± (t) ∝ exp(−Γ± t)(70.32)r± (t) ∝ exp(−Γ± t) ,(70.33)where the effective decay widths are given byq(y cos φ − x sin φ)ppΓ± = Γ 1 ±(y cos φ + x sin φ) .
q

Γ± = Γ 1 ±31st May, 2024

(70.34)(70.35)

70. D0 -D0 Mixing

9

Table 70.5: Results for yCP and AΓ from D0 decays to CP eigenstates. When a single uncertainty
is listed, that corresponds to statistical and systematic uncertainties combined. The measurementswith an asterisk (∗ ) have been superseded.
Year Experiment
Final state(s)yCP (%)AΓ (×10−3 )
‡
−1
∗
+
−
+
−
2022 LHCb (6 fb D tag) [49]K K + π π 0.696±0.026± 0.013
—
−1
∗
+
−
+
−
2021 LHCb (8.4 fb B, D tags) [50] K K + π π—
0.10±0.11± 0.03
2021 LHCb (6 fb−1 D∗ tag) [50]K +K − + π+π−—
0.27±0.13± 0.03
2021 LHCb (6 fb−1 D∗ tag) [50]K +K −—
0.23±0.15± 0.03
−1∗+−2021 LHCb (6 fb D tag) [50]π π—0.40±0.28± 0.04
2020 Belle (976 fb−1 ) [51]KS0 ω0.96±0.91+0.64
—
−0.62
−1
+
−
+
−
2019 LHCb (3 fb B tag) [52]K K +π π0.57±0.13± 0.09—
2017 LHCb (3 fb−1 D∗ tag) [53]K +K − + π+π−—−0.13±0.28± 0.10
2017 LHCb (3 fb−1 D∗ tag) [53]K +K −—−0.30±0.32±0.10−1∗+−2017 LHCb (3 fb D tag) [53]π π—0.46±0.58±0.122016 Belle (976 fb−1 ) [54]K + K − + π + π − 1.11±0.22±0.09−0.3±2.0±0.7
−1
+
−
+
−
2015 LHCb (3 fb B tag) [55]
K K +π π—−1.25±0.73+0.262015 LHCb (3 fb−1 B tag) [55]
K +K −—−1.34±0.77 −0.34
+0.25
−1
+
−
2015 LHCb (3 fb B tag) [55]π π—−0.92±1.45 −0.33

+
−
+
−

K K , π π
KS0 π 0, KS0 π 0 π 0 −2.0 ± 1.3 ± 0.7—
2015 BES III (2.9 fb−1 ) [56]
 K 0 η, K 0 ω
SS2014 CDF (9.7 fb−1 ) [57]K +K − + π+π−—
−1.2±1.2−1+−2014 CDF (9.7 fb ) [57]K K—
−1.9±1.5±0.42014 CDF (9.7 fb−1 ) [57]π+π−—−0.1±1.8±0.3
2012 BABAR (468 fb−1 ) [58]K + K − + π + π − 0.72±0.18±0.12
0.9±2.6±0.6−10+−2009 Belle (673 fb ) [59]
KS K K
0.11±0.61±0.52—
2002 CLEO (9.0 fb−1 ) [60]K +K − + π+π−
−1.2±2.5±1.4—
6+−2000 FOCUS (1 × 10 evts) [61]K K
3.42±1.39±0.74—
1999 E791 (2 × 1010 evts) [62]K +K −
0.73±2.89±1.03—
HFLAV Average [63]0.719 ± 0.113
0.089 ± 0.113∗−1+−2020 LHCb (5.4 fb B tag) [64]K K—
−0.43±0.36± 0.05
2020∗ LHCb (5.4 fb−1 B tag) [64]π+π−—0.22±0.70± 0.08
∗
−1
∗
+
−
2013 LHCb (1.0 fb D tag) [65]K K—
−0.35±0.62±0.122013∗ LHCb (1.0 fb−1 D∗ tag) [65]π+π−—0.33±1.06±0.14
∗§
−1
∗
+
−
2011 LHCb (29 pb D tag) [66]K K
0.55±0.63±0.41−5.9±5.9±2.1
2009∗ BABAR (384 fb−1 ) [67]K +K −
1.16±0.22±0.18—
2008∗ BABAR (384 fb−1 ) [68]K + K − + π + π − 1.03±0.33±0.19
2.6±3.6±0.8
2007∗ Belle (540 fb−1 ) [69]K + K − + π + π − 1.31±0.32±0.25
0.1±3.0±1.5∗−12003 BABAR (91 fb ) [70]K +K − + π+π−
0.8±0.4 +0.5
—
−0.4
+0.7
∗
−1
+
−
2001 Belle (23.4 fb ) [71]K K
−0.5±1.0 −0.8—‡

This measurement has sufficient precision that yCP for the normalization channel D0 → K − π + must be accountedfor. Thus, the measurement is yCP (h+ h− )−yCP (K − π + ). HFLAV accounts for this small correction in their global
fit.
§This result for yCP is not superseded, but it is not included in the HFLAV average due to having some correlations
with the result of Ref. [52] but much worse precision.

31st May, 2024

70. D0 -D0 Mixing

10

Thus, the effective decay rate to a CP eigenstate combining equal numbers of D0 and D0 decays(e.g., an untagged sample with no production asymmetry) is
r± (t) + r± (t) ∝ e−(1± yCP )Γ t ,(70.36)

where
1
yCP =
2



q
p
+
p
q



1
y cos φ −
2



q
p
−
p
q



x sin φ(70.37)≈ y cos φ − AM x sin φ .

(70.38)If CP is conserved, yCP = y. Most measurements of yCP have used D0 → K + K − and D0 →π + π − decays, which are CP -even, measured relative to D0 → K − π + . Belle measured yCP also using
D0 → KS0 ω decays [51], which are CP -odd, and D0 → KS0 K + K − decays [59], which are dominatedby the CP -odd final state KS0 φ. Table 70.5 summarizes the current status of measurements.
In addition to yCP , Belle [54], BABAR [58], CDF [57], and LHCb [50, 66] have reported measurements of the decay-rate asymmetry for CP -even final states:AΓ ≡≈Γ+ − Γ +(1/τ+ ) − (1/τ + )τ + − τ+==(1/τ+ ) + (1/τ + )τ + + τ+Γ+ + Γ +
1
2



q
p
−
p
q



y cos φ −

1
2



q
p
+
p
q

(70.39)

x sin φ(70.40)≈ AM y cos φ − x sin φ .

(70.41)

If CP is conserved, AΓ = 0.
There is a contribution to Eq. (70.41) from direct CP violation, i.e., |Af /Af | =6 1 [72, 73]. For+−+−
f = K K and π π , this contribution can be estimated from measurements of ACP (K + K − ) andACP (π + π − ) (see below) and is much smaller than the current uncertainty on AM ; thus we neglect
it here. We note that, when averaging AΓ measurements over K + K − and π + π − final states, thecontribution from direct CP violation cancels, as it has the same magnitude but opposite signs for
K + K − and π + π − due to U -spin symmetry [73].
The asymmetry AΓ is an asymmetry in the full decay widths. An asymmetry in partial widths
is referred to as ACP and is final-state dependent:
ACP ≡Γ (D0 → f ) − Γ (D0 → f ).
Γ (D0 → f ) + Γ (D0 → f )(70.42)

Unlike AΓ , which is measured by fitting decay time distributions, ACP is measured by fitting forsignal yields and (aside from acceptance effects) does not require measuring decay times. For neutral
D decays, ACP receives contributions from both direct (in the decay amplitudes) and indirect (dueto mixing) processes: ACP (D0 → f ) = AfCP + Aindirect. The latter indirect contribution depends
CPon the mixing parameters x and y:
Aindirect
=
CP

1
2



q
p
+
p
q



x sin φ −

1
2



q
p
−
p
q



y cos φ = −AΓ .

(70.43)

Numerous measurements of ACP for decays to CP eigenstates are listed in this Review [74].
Table 70.6 summarizes the current status of measurements of the difference in ACP for D0 → K + K −31st May, 2024

70. D0 -D0 Mixing

11

and D0 → π + π − decays: ∆ACP ≡ ACP (K + K − ) − ACP (π + π − ). Within the Standard Model,
ππ
AKK
CP ≈ −ACP [75], and ∆ACP essentially doubles any direct CP violation present. The differenceis also advantageous experimentally, as several systematic uncertainties cancel. As AindirectisCP
independent of final state, it subtracts out of ∆ACP . However, at hadron experiments such as
LHCb, there is a difference in efficiencies between K + K − and π + π − such that htiKK 6= htiππ , i.e.,the mean decay times slightly differ. This difference leads to a small contribution to ∆ACP from
Aindirect[72]. The most recent ∆ACP result from LHCb [76], based on 8.9 fb−1 of data, differs fromCPzero with a statistical significance of 5.3σ. Thus, this measurement constitutes the first observationof CP violation in charm decays. These CP asymmetries are included in HFLAV’s global fit forcharm mixing parameters discussed below.
Table 70.6: Results for the difference in time-integrated CP asymmetries ∆ACP between D0 →K + K − and D0 → π + π − decays. When a single uncertainty is listed, that corresponds to statisticaland systematic uncertainties combined. The measurements with an asterisk (∗ ) have been either
superseded or combined with subsequent results and thus are not included in the HFLAV global
fit.
Year
2019
2013
2008
2008
2016∗
2014∗
2013∗
2012∗
2012‡
‡

Experiment
LHCb (8.9 fb−1 B, D∗ tags) [76]
CDF (9.7 fb−1 D∗ tag) [77]
BABAR (386 fb−1 ) [78]Belle (540 fb−1 ) [79]
LHCb (3.0 fb−1 D∗ tag) [80]
LHCb (3.0 fb−1 B tag) [81]LHCb (1.0 fb−1 B tag) [82]
LHCb (0.62 fb−1 D∗ tag) [83]Belle (976 fb−1 ) [84]

∆ACP (×10−3 )−1.54 ± 0.29
−6.2±2.1±1.0
2.4±6.2±2.6−8.6±6.0±0.7−1.0 ± 0.8 ± 0.3
1.4±1.6±0.8
4.9±3.0±1.4−8.2±2.1±1.1−8.7±4.1±0.6This preliminary result was not published and thus is not included in the HFLAV global fit.

70.5 Quantum-correlated D0 D0 Analyses
Measurements of RD , cos δKπ , sin δKπ , x, and y can be obtained from a combined fit to timeintegrated yields of single-tagged (ST) and double-tagged (DT) D0 D0 events produced at theψ(3770) resonance. Single-tagged events are those in which either the D0 or D0 decay is reconstructed (identified), and the other neutral D decays generically. Double-tagged events are thosein which both the D0 and D0 decays are identified. Due to quantum correlations, the decay of a
D0 , D0 , D+ , or D− projects the other neutral D into a state D0 , D0 , D− , or D+ , respectively.
The CP -specific D− and D+ decays (or, neglecting CP violation, D1 and D2 decays) include interference between D0 and D0 amplitudes, and this provides sensitivity to RD and cos δKπ . The
flavor-specific D0 and D0 decays include interference between D1 and D2 amplitudes, and thisprovides sensitivity to x and y. For details of this method, see Refs. [1–5].
BESIII has reported results using 2.9 fb−1 of e+ e− → ψ(3770) data, where the quantumcorrelated D0 D0 pairs are produced in a C = −1 state. They measure yCP = (−2.0±1.3±0.7)% [56]from DT yields using a CP -eigenstate tag for one D and a flavor-specific semileptonic tag for theother; and they measure ACPKπ = (13.2 ± 1.1 ± 0.7)% [85] from DT yields using a CP tag for one±∓
D and a K π tag for the other. For yCP , the CP eigenstates used are K − K + (f+ ), π + π − (f+ ),KS0 π 0 π 0 (f+ ), KS0 π 0 (f− ), KS0 η (f− ), and KS0 ω (f− ). For ACP
Kπ , seven additional CP eigenstates
31st May, 2024

1270. D0 -D0 Mixing

are included: π 0 π 0 (f+ ), KS0 η 0 (f− ), KS0 φ (f− ), KL0 π 0 (f+ ), KL0 ω (f+ ), KL0 π 0 π 0 (f− ), and π + π − π 0(mixed CP ). Using Eq. (70.29) and external inputs for the CP -even fraction of D0 → π + π − π 0◦(from Ref. [86]) and values of RD and y (from HFLAV [87]), BESIII obtains δKπ = (7.6 +10.4−11.6 ) [85].
CLEO-c has reported results using 0.82 fb−1 of e+ e− → ψ(3770) data [88–90]. The values fory, RM , cos δKπ , and sin δKπ are obtained from a combined fit to the ST (hadronic only) and DT
yields. The DT yields include events in which one D is reconstructed in a hadronic mode and theother D is partially reconstructed in flavor-specific D → K ∓ e± ν and D → K ∓ µ± ν modes. The+0.07
CLEO-c analysis obtains cos δKπ = 0.81 +0.22−0.18 −0.05 and sin δKπ = −0.01 ± 0.41 ± 0.04. These fits
allow cos δKπ and sin δKπ (and also x2 ) to be unphysical. Constraining cos δKπ and sin δKπ to thephysical range [−1, +1] (i.e., interpreting δKπ as an angle) and also using external inputs for x, y,◦
and yCP from HFLAV 2012 [91], CLEO-c obtains δKπ = (18 +11−17 ) [90].

70.6 Summary of Experimental Results
The first evidence for D0 -D0 mixing was obtained in 2007 by Belle [69] and BABAR [20]. These
results were confirmed by CDF [92] and, much later, by LHCb [22]. There are now numerous
measurements of D0 -D0 mixing with various levels of sensitivity. For D0 → K + π − decays, LHCb
[21,22], CDF [19], and Belle [18] each exclude the no-mixing hypothesis by more than five standard
deviations. LHCb [44] reported the observation of charm mixing in D0 → K + π − π + π − decays with
8.2σ significance. However, the strong phase difference for this decay is not known, and thus the
mixing parameters x and y cannot be extracted. The most precise measurements of x and y are
obtained from a time-dependent Dalitz plot analysis of D0 → KS0 π + π − decays. This method wasdeveloped at CLEO [93] and subsequently used at Belle [39,41] and BABAR [40] with much higher
statistics. BABAR has applied this method also to D0 → KS0 K + K − decays [40]. It has recently
been used by LHCb with very high statistics [34,35] to obtain the most precise values of x and y todate. This measurement resulted in the first observation (> 5σ significance) of dispersive mixing(x 6= 0).
The experimental measurements establish that D0 and D0 mesons mix. This mixing is presumably dominated by long-distance amplitudes, which are difficult to calculate. Under the assumptionthat the observed mixing is due entirely to non-Standard Model processes, significant constraints
on new physics models can be obtained [94]. A significant limitation to interpreting charm mixing
in terms of new physics is the theoretical uncertainty on Standard Model predictions [95, 96]. We
note that the HFLAV global fit results for x and y (see below) indicate that charm mixing is at theupper end of the range of these predictions. The current situation would benefit from knowledge ofthe strong phase difference δKππ between D0 → K + π − π 0 and D0 → K + π − π 0 decays, and similarly
the strong phase difference δKπππ between D0 → K + π − π + π − and D0 → K + π − π + π − decays. Such00
knowledge would allow one to extract x and y from measurements of (x002, y 00 ) and (x002
K3π , yK3π ) for
these three- and four-body decays.
With regard to CP violation, by combining four separate measurements from two data sets
totalling 8.9 fb−1 of data, LHCb observed CP violation in D decays for the first time [76]. The
observable measured is the difference ∆ACP = ACP (K + K − ) − ACP (π + π − ), which is dominatedby direct CP violation, i.e., the potential contributions from indirect CP violation mostly cancel.
The amount of direct CP violation observed is small: ∆ACP = (−0.154 ± 0.029)%. A theory
calculation indicates this value is consistent with Standard Model expectations [97]; however, newphysics contributions cannot be excluded. A subsequent LHCb measurement of ACP (K + K − ) =(0.068 ± 0.056)% differs from zero by only 1.2σ [98]; thus, the ∆ACP result is interpreted asindicating direct CP violation in D0 → π + π − decays.
31st May, 2024

70. D0 -D0 Mixing

13

70.7 HFLAV Global Fit for Charm Mixing Parameters
The Heavy Flavor Averaging Group (HFLAV) performs global fits to all relevant mixing measurements to obtain world average values for the following parameters: mixing parameters x andy; strong phase differences δKπ and δKππ0 ; the ratio RD of D0 → K + π − and D0 → K + π − partial
ππ
widths; direct CP -violating parameters AD (K + π − ), AKK
CP ≡ AK , and ACP ≡ Aπ ; and indirect CP Γ
violating parameters |q/p|, Arg(q/p) ≡ φ, φ12 , φM
2 , and φ2 . Four separate fits are performed: (a)assuming no CP violation; (b) assuming no subleading amplitudes for indirect CP violation; (c) assuming no subleading amplitudes in Cabibbo-favored and doubly Cabibbo-suppressed decays; and(d) allowing for all CP violation. Fits (b) and (c) correspond to theory expectations [73, 99, 100],with the latter being less restrictive. For fit (b), four fitted parameters are reduced to three using
the relationship tan φ = (x/y) · (1 − |q/p|2 )/(1 + |q/p|2 ) [99–101]; alternatively, one can fit for thethree parameters x12 ≡ 2|M12 |/Γ , y12 ≡ |Γ12 |/Γ , and φ12 ≡ Arg(M12 /Γ12 ), from which x, y, |q/p|,and φ can be derived. Details of these fits are given in Ref. [87].
Table 70.7: HFLAV global fit results (see text) [15].
Parameter

x (%)
y (%)
δKπ (◦ )RD (%)
AD (%)|q/p|
φ (◦ )
δKππ (◦ )Aπ (%)AK (%)x12 (%)y12 (%)
φ12 (◦ )
◦
φM
2 ( )
Γ
◦
φ2 ( )
2
χ /d.o.f.

No CP
violation
Fit (a)0.434 +0.126−0.1390.646 ± 0.024
11.5 +3.6−3.7
0.344 ± 0.002−−−
23.1 +22.8
−23.2
−
−
−
−
−
−
−
100.6/(63−5)= 1.74

No subleading No subleading
amplitudes for amplitudes inindirect CP V CF+DCS decays
Fit (b)Fit (c)0.407 ± 0.044−+0.024−0.643 −0.023
+3.6
10.9
11.3 +3.6−3.8−3.8
0.344 ± 0.002 0.344 ± 0.002−−−
1.005 ± 0.007−−0.19 ± 0.26
+21.4
24.4 +21.5
24.5 −22.5−22.5
0.202 ± 0.059 0.200 ± 0.059
0.045 ± 0.053 0.043 ± 0.053
0.407 ± 0.044 0.407 ± 0.044
0.641 +0.024
0.643 +0.024−0.023−0.023+0.92−
0.65 −0.90
0.48 ± 0.92−
2.40 +1.55−−1.54
68.9/(63−8)
66.5/(63−9)= 1.25= 1.23

All CP
95% C.L. Intervalviolation(CP V allowed)
allowedFit (d)0.407 ± 0.044[0.320, 0.493]
0.645 +0.024[0.600,0.692]−0.02311.4 +3.5[3.7, 18.2]−3.8
0.344 ± 0.002[0.340, 0.347]−0.77 ± 0.35[−1.46, −0.08]+0.016[0.96, 1.02]
0.994 −0.015[−4.88,−0.37]−2.6 +1.1
−1.2
+21.4
[−20.1,
65.7]
24.6 −22.5[0.10, 0.34]
0.218 ± 0.060[−0.04, 0.17]
0.062 ± 0.054[0.321, 0.494]−[0.595, 0.687]−[−1.13, 2.49]−[−1.35, 2.32]−[−0.61, 5.45]−
65.4/(63−10)= 1.23

The fits use Belle, BABAR, CDF, and LHCb measurements of D0 → K (∗)+ `− ν, K + K − , π + π − ,K + π − , K + π − π 0 , K + π − π + π − , KS0 π + π − , KS0 K + K − , and π + π − π 0 decays, as well as CLEO-c and
BESIII measurements of cos δ, sin δ, and ACP (K + π−) obtained from quantum-correlated branching
fractions measured in e+ e− → ψ(3770) → D0 D0 reactions. Correlations among observables aretaken into account by using the error matrices provided by the experiments. Three observables
input to the fit are themselves world average values calculated by HFLAV: RM = (x2 + y 2 )/2 from
D0 → K (∗)+ `− ν decays (Table 70.1), and yCP and AΓ from D0 → fCP decays (Table 70.5). A
measurement by LHCb of RM using D0 → K + π − π + π − decays is input separately.
The results of the fit as of September, 2023 are listed in Table 70.7. Confidence contours in thetwo dimensions (x, y) and (|q/p|, φ) resulting from the fit are plotted in Fig. 70.1. These contours
31st May, 2024

70. D0 -D0 Mixing

14

0.8

CPV allowed

0.7
0.6
0.5

Arg(q/p) [degrees]y (%)

are obtained by allowing, for any point in the two-dimensional plane, all other fitted parameters
to take their preferred values. The 1σ–5σ boundaries drawn are the loci of points in which theχ2 has risen above the minimum by 2.30, 6.18, 11.83, 19.33, and 28.67 units. The fit excludes
the no-mixing point x = y = 0 at more than 11.5σ. The fit is consistent with CP conservation(|q/p| = 1, φ = 0) at the 2σ level. The χ2 of the all-CP -violation-allowed fit is 65.4 for 63−10 = 53
degrees of freedom, which is satisfactory. One-dimensional likelihood functions for parameters are
obtained by allowing, for any value of the parameter, all other fitted parameters to take theirpreferred values. The resulting likelihood functions give central values, 68.3% C.L. intervals, and
95% C.L. intervals as listed in Table 70.7.
1σ
2 σ
3 σ
4σ
5 σ

!"#$%
!"#$%&%'

10

5

0
0.4

−5

0.3
1σ
2 σ
3 σ
4σ
5 σ

0.2

!"#$%
0.1
0.1

!"#$%&%'

0.2

0.3

0.4

0.5

0.6

0.7

0.8

−10

−0.1

x (%)

−0.05

0

0.05

0.1
|q/p|-1

Figure 70.1: Two-dimensional 1σ-5σ contours for (x, y) (left) and for (|q/p|, Arg(q/p)) (right) as
obtained by HFLAV [15]. In the right plot, the black dot denotes the no-CP -violation point (0, 0).
From the results of the HFLAV global fit, we conclude the following: (1) Since CP violation issmall and yCP is positive, the CP -even state is shorter-lived, as in the K 0 K 0 system. (2) Since x is
positive, the CP -even state is heavier, unlike in the K 0 K 0 system. (3) The strong phase differenceδKπ is small but probably nonzero: the 95% C.L. interval is 3.7◦ < δKπ < 18.2◦ . (4) While direct
CP violation has been observed in D decays, there is still no evidence for indirect CP violation,i.e., |q/p| 6= 1 or φ 6= 0. Observing such CP violation at the current level of sensitivity would
indicate new physics.

70.8 Future Data
Current results are based primarily upon CLEO-c (0.82 fb−1 of e+ e− → ψ(3770) data), Belle and√
BABAR (∼1.4 ab−1 of e+ e− → Υ (4S) data), CDF (9.6 fb−1 of pp̄ collision data at s = 1.96 TeV),√
and LHCb Runs 1 and 2 (3.0 fb−1 + 5.9 fb−1 of pp collision data at s = 7, 8, 13 TeV).
BESIII has accumulated 8 fb−1 of e+ e− → ψ(3770) data and plans to collect a total of 20 fb−1
by 2025. These data should provide strong phase measurements that would enable improved modelindependent determinations of mixing parameters from Belle II and LHCb. In 2019, Belle II began
accumulating 50 ab−1 of e+ e− → Υ (4S) data [102], which is expected to take approximately ten
years to collect. At LHCb, Run 2 was completed in 2018, Run 3 is now in progress, and Run 4 isplanned for 2027-30 [103]. The goal for Runs 3 + 4 is to accumulate an additional 50 fb−1 of pp√
data at s ≈ 14 TeV [104]. These data, along with the large e+ e− dataset from Belle II, should
provide more precise measurements of D0 -D0 mixing and direct CP violation, and might possibly
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70. D0 -D0 Mixing

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