Global Fit for D⁰-D⁰ Mixing and CP Violation
(through 14 September 2025)

People working on this: Alan Schwartz, Marco Gersabeck
For a complete list of references click here
For world average values of measured observables (used below) click here

Notation and Phase Convention:
The mass eigenstates are denoted D₁ ≡ p|D⁰> + q|D⁰> and D₂ ≡ p|D⁰> − q|D⁰>; δ and δ_Kππ are strong phase differences between D⁰ → f and D⁰ → f amplitudes, and φ is the weak phase difference Arg(q/p). We define δ ≡ δ_{D⁰ → K⁻n(π)} − δ_{D⁰
→ K⁻n(π)}. The mixing parameters are defined as x ≡ (m₂ − m₁)/Γ and y ≡ (Γ₂ − Γ₁)/(2Γ), where Γ = (Γ₁ + Γ₂)/2. Our convention is (CP)|D⁰> = −|D⁰> and (CP)|D⁰> = −|D⁰>; thus, in the absence of CP violation, x = (m_CP+ − m_CP−)/Γ and y = (Γ_CP+ − Γ_CP−)/(2Γ).

Experimental Observables:
For this version of the global fit, there are input 63 experimental measurements: y_CP, A_Γ, (x, y, |q/p|, φ) _{Belle
K⁰_S π⁺ π ⁻}, (x_CP, y_CP, Δx, Δy) _{LHCb
K⁰_S π⁺ π ⁻ (Runs 1, 2)}, (x, y) _{BaBar
K⁰_S h⁺ h⁻}, (x, y) _{BaBar
π⁰ π⁺ π⁻}, (R_M)/2 _{LHCb
K⁺ π⁻ π⁺ π ⁻}, (R_M)_semileptonic, (x", y") _{BaBar K⁺ π⁻ π ⁰}, (R_D, x², y, cos δ, sin δ) _{CLEOc Ψ(3770)} , (A_CP^Kπ) _{BESIII Ψ(3770)} , (R_D, A_D, x'^2±, y'^±)_{BaBar K⁺ π ⁻}, (R_D, x'², y') _{Belle K⁺ π ⁻}, (R_D, x'², y') _{CDF K⁺ π ⁻}, (R_D^±, x'^2±, y'^±) _{LHCb K⁺ π ⁻ (D^*+B tagged)}, (R_Kπ, c_Kπ, c′_Kπ, A_Kπ, Δc_Kπ, Δc′_Kπ, Ã_Kπ, Δc̃_Kπ, Δc̃′ _Kπ) _{LHCb K⁺ π ⁻ (D* tagged)}, (A_CP^K, A_CP^π) _BaBar, (A_CP^K, A_CP^π) _Belle, (A_CP^K, A_CP^π) _CDF, (A_CP^K −A_CP^π) _{LHCb (D^*+B tagged)}, (A_CP^K) _{LHCb (D^* tagged)}

Theoretical Parameters:
There are two equivalent sets of parameters governing mixing and indirect CP violation: x, y, |q/p|, Arg(q/p) ≡ φ and x₁₂, y₁₂, Arg(M₁₂) ≡ φ₂^M, Arg(Γ₁₂) ≡ φ₂^Γ. For both sets, the first two parameters govern mixing, and the latter two parameters govern indirect CP violation. In addition to these four parameters, there are six additional parameters fitted: δ, δ_Kππ are strong phases; R_D is the ratio Γ(D⁰→ f)/Γ(D⁰ → f); and A_D, A_π, A_K are direct CP-violating parameters for D⁰ → K⁺ π⁻, D⁰ → π⁺ π⁻, and D⁰ → K⁺ K⁻, respectively. The subscript 2 for φ₂^M and φ₂^Γ denotes the basis in which the dominant U-spin-changing (ΔU=2) dispersive and absorptive mixing amplitudes are real; see Kagan and Silvestrini Eq. 107. Choosing a basis ensures that the phases φ, φ₂^M, and φ₂^Γ are physically meaningful.

The relationships between x, y, |q/p|, φ, δ, δ_Kππ, R_D, A_D, A_π, A_K and the measured observables are given below. The observables appear in blue (left side), and the fitted parameters appear in magenta (right side).

The relationships between (x, y, |q/p|, φ) and (x₁₂, y₁₂, φ₂^M, φ₂^Γ) are:

The first three relations correspond to Eqs. (14), (15), and (48) of Kagan and Sokoloff; the last relation corresponds to Eq. (110) of Kagan and Silvestrini.
Note that if φ₂^M = φ₂^Γ, then |x| = x₁₂, |y| = y₁₂, |q/p| = 1, and φ = −φ₂^M. In this case, there is no CP violation in mixing.

Measurements used:

Index Observable Value Source

1 y_CP − y_CP(Kπ) (0.697 ± 0.028)%

World average of D⁰ → K⁺ K⁻ / π⁺ π ⁻ / K⁺ K⁻ K⁰

Our calculation of the y_CP(Kπ) correction is from arXiv:2207.11867, Eq. (29).

This correction was first pointed out by Pajero, Morello in JHEP 03 (2022) 162.

2 A_Γ (0.0089 ± 0.0113)% World average (COMBOS combination) of D⁰ → K⁺ K⁻ / π⁺ π ⁻ results

3-6

x

y

x

y

|q/p|

φ

0.40 ± 0.17 ± 0.04

0.29 ± 0.14 ± 0.03

(0.58 ± 0.19 ^+0.0734 _−0.1177)%

(0.27 ± 0.16 ^+0.0546 _−0.0854)%

0.82 ^+0.20 _−0.18 ^+0.0807 _−0.0645

(−13 ⁺¹² ₋₁₃ ^+4.15 _−4.77) degrees

Belle + Belle II D⁰ → K⁰ _S π⁺ π ⁻ results using

951 fb⁻¹ and 408 fb⁻¹, respectively. The correlation is negligible.

Belle D⁰ → K⁰ _S π⁺ π ⁻ results using 921 fb⁻¹. Correlation coefficients:

1 0.054 −0.074 −0.031

0.054 1 0.034 −0.019

−0.074 0.034 1 0.044

−0.031 −0.019 0.044 1

7-10

11-14

x (no CPV)

y (no CPV)

x_CP

y_CP

Δx

Δy

x_CP

y_CP

Δx

Δy

(−0.86 ± 0.53 ± 0.17)%

(0.03 ± 0.46 ± 0.13)%

(0.27 ± 0.16 ± 0.04)%

(0.74 ± 0.36 ± 0.11)%

(−0.053 ± 0.070 ± 0.022)%

(0.06 ± 0.16 ± 0.03)%

(0.400 ± 0.045 ± 0.020)%

(0.551 ± 0.116 ± 0.059)%

(−0.029 ± 0.018 ± 0.001)%

(0.031 ± 0.035 ± 0.013)%

LHCb D⁰ → K⁰_S π⁺π⁻ results using 1 fb^-1 (√s = 7 TeV)

D^*+ → D⁰π⁺ flavor tag. Correlation coefficient = +0.37, no CPV.

3 fb^-1 results (√s = 7, 8 TeV) allowing for CPV.

D^*+ → D⁰π⁺, B → D⁰μ⁻ X flavor tags. Correlation coefficients (stat. + syst.):

1 (−0.17 + 0.15) (0.04 + 0.01) (−0.02 − 0.02)

1 (−0.03 − 0.05) (0.01 − 0.03)

1 (−0.13 + 0.14)

5.4 fb^-1 results (√s = 13 TeV) allowing for CPV.

D^*+ → D⁰π⁺, B → D⁰μ⁻ X flavor tags. Correlation coefficients (stat. + syst.):

1 (0.121 + 0.13) (−0.018 + 0.01) (−0.016 + 0.01)

1 (−0.012 − 0.02) (−0.058 + 0.01)

1 (0.069 + 0.31)

For (x, y, |q/p|, φ) → (x_CP, y_CP, Δx, Δy) mapping, see PRD 99, 012007 (2019)

15-16

x

y

(0.16 ± 0.23 ± 0.12 ± 0.08)%

(0.57 ± 0.20 ± 0.13 ± 0.07)%

BaBar D⁰ → K⁰_S π⁺π⁻ and D⁰ → K⁰_S K⁺ K⁻ combined;

Correlation coefficient = +0.0615, no CPV.

17-18

x

y

(1.5 ± 1.2 ± 0.6)%

(0.2 ± 0.9 ± 0.5)%

BaBar D⁰ → π⁰ π⁺π⁻

Correlation coefficient = −0.006, no CPV.

19 (x² + y²)/2 (0.0130 ± 0.0269)% World average (COMBOS combination) of D⁰ → K⁺l⁻ ν results

20-21

x"

y"

(2.61 ^+0.57 _−0.68 ± 0.39)%

(−0.06 ^+0.55 _−0.64 ± 0.34)%

BaBar K⁺ π ⁻ π ⁰ result; correlation coefficient = −0.75.

Note: x" = x cos δ_Kππ + y sin δ_Kππ, y" = y cos δ_Kππ − x sin δ_Kππ.

22-26

R_D

x²

y

cos δ

sin δ

(0.533 ± 0.107 ± 0.045)%

(0.06 ± 0.23 ± 0.11)%

(4.2 ± 2.0 ± 1.0)%

0.81 ^+0.22_−0.18 ^+0.07_−0.05

−0.01 ± 0.41 ± 0.04

CLEO-c Ψ(3770) results, 0.82 fb⁻¹. Correlation coefficients:

1 0 0 −0.42 0.01

1 −0.73 0.39 0.02

1 −0.53 −0.03

1 0.04

1

27-29

R_D

x'²⁺

y'⁺

(0.303 ± 0.0189)%

(−0.024 ± 0.052)%

(0.98 ± 0.78)%

BaBar K⁺ π ⁻ results; correlation coefficients:

1 +0.77 −0.87

1 −0.94

1

30-32

A _D

x'^{2 −}

y'⁻

(−2.1 ± 5.4)%

(−0.020 ± 0.050)%

(0.96 ± 0.75)%

BaBar K⁺ π ⁻ results; correlation coefficients same as above.

33-35

R_D

x'²

y'

(0.353 ± 0.013)%

(0.009 ± 0.022)%

(0.46 ± 0.34)%

Belle K⁺ π ⁻ no-CPV results using 976 fb⁻¹. Correlation coefficients:

1 +0.737 −0.865

1 −0.948

1

36-38

R_D

x'²

y'

(0.351 ± 0.035)%

(0.008 ± 0.018)%

(0.43 ± 0.43)%

CDF K⁺ π ⁻ results for 9.6 fb⁻¹. Correlation coefficients:

1 0.90 −0.97

1 −0.98

1

39-47

R_Kπ

c _Kπ

c′_Kπ

Ã_Kπ

Δc̃ _Kπ

Δc̃′_Kπ

A_Kπ

Δc _Kπ

Δc′_Kπ

(0.3427 ± 0.0019)%

(0.528 ± 0.033)%

(12.0 ± 3.5) × 10⁻⁶

(−0.82 ± 0.59)%

(0.032 ± 0.036)%

(−2.0 ± 3.8) × 10⁻⁶

(−0.9 ± 2.0)%

(−0.01 ± 0.10)%

(4.6 ± 9.8) × 10⁻⁶

LHCb K⁺ π ⁻ results for 3.0 + 5.4 = 8.4 fb⁻¹ (√s = 7, 8, 13 TeV)
D^*+ → D⁰π⁺ flavor tag. Correlation coefficients:

1 −0.927 0.803 0.008 −0.007 0.000 0.003 −0.002 0.002

1 −0.943 −0.014 0.013 −0.006 −0.005 0.004 −0.004

1 0.007 −0.006 0.000 0.003 −0.003 0.003

1 −0.934 0.810 0.000 0.000 0.000

1 −0.943 0.000 0.000 0.000

1 0.000 0.000 0.000

1 −0.938 0.811

1 −0.943

Note: above values in red (<1.5%) are neglected in the fit. Definitions:

^{dN^±}⁄_dt = R_Kπ^± + √(R_Kπ^±) (c_Kπ ± Δc_Kπ) t + (c′_Kπ ± Δc′_Kπ) t²

where R_Kπ^± ≡ R_Kπ (1 ± A_Kπ). Also:

Ã_Kπ = A_Kπ − 2a^d_KK

Δc̃_Kπ = Δc_Kπ − c_Kπ a^d_KK + 2A_Γ √(R_Kπ)

Δc̃′_Kπ = Δc′_Kπ − 2c′_Kπ a^d_KK + 2c_Kπ A_Γ √(R_Kπ)

48-50

R_D⁺

x'²⁺

y'⁺

(0.3500 ± 0.0073)%

(0.008 ± 0.015)%

(0.41 ± 0.20)%

LHCb K⁺ π ⁻ results for 3.0 + 5.4 = 8.4 fb⁻¹ (√s = 7, 8, 13 TeV)
B → D^*+μ⁻ X, D^*+ → D⁰π⁺ flavor tags. Correlation coefficients:

1 0.624 −0.749

1 −0.943

1

51-53

R_D⁻

x'^{2 −}

y'⁻

(0.3440 ± 0.0074)%

(−0.005 ± 0.017)%

(0.68 ± 0.21)%

LHCb K⁺ π ⁻ results for 3.0 + 5.4 = 8.4 fb⁻¹ (√s = 7, 8, 13 TeV)
B → D^*+μ⁻ X, D^*+ → D⁰π⁺ flavor tags. Correlation coefficients:

1 0.629 −0.745

1 −0.946

1

54-55

A_CP^K

A_CP^π

(0.00 ± 0.34 ± 0.13)%

(−0.24 ± 0.52 ± 0.22)%

BaBar 385.8 fb⁻¹ near ϒ(4S) resonance

56-57

A_CP^K

A_CP^π

(−0.43 ± 0.30 ± 0.11)%

(0.43 ± 0.52 ± 0.12)%

Belle 540 fb⁻¹ near ϒ(4S) resonance

58-59

A_CP^K

A_CP^π

(−0.32 ± 0.21)%

(0.31 ± 0.22)%

CDF 9.7 fb⁻¹ pp collisions at √s = 1.96 TeV

( 〈t〉_K − 〈t〉_π ) / τ_D = 0.27 ± 0.01

60 A_CP^K − A_CP^π (−0.154 ± 0.029)%

LHCb 8.9 fb⁻¹ pp collisions at √s = 7, 8, 13 TeV

D^*+ → D⁰π⁺ and B → D⁰μ⁻ X flavor tags

( 〈t〉_K − 〈t〉_π )/τ_D = 0.115 ± 0.002; 〈t〉/τ_D = 1.71 ± 0.10

61 A_CP^K (0.068 ± 0.054 ± 0.016)%

LHCb 5.7 fb⁻¹ pp collisions at √s = 13 TeV

D^*+ → D⁰π⁺ flavor tags

〈t〉_K /τ_D = (701.5 ± 1.1)/(410.3 ± 1.0) = 1.7097 ± 0.0050

62 (x² + y²)/4 (0.0048 ± 0.0018)%

LHCb 3.0 fb⁻¹ pp collisions at √s = 7, 8 TeV

D⁰ → K⁺ π ⁻ π ⁺ π ⁻

63 A_CP^Kπ 0.132 ± 0.011 ± 0.007

BESIII Ψ(3770) results, 2.93 fb⁻¹. D⁰ → K⁻ π⁺,

difference between CP-even tagged and CP-odd tagged

MINUIT fit results
Three main fits are performed, as follows:

Fit #1: no indirect CP violation
This fit takes |q/p| = 1 and φ = 0. In addition, we neglect sub-leading amplitudes in Cabibbo-favored (CF) and doubly Cabibbo-suppressed (DCS) decays, i.e., they proceed via tree diagrams. This implies no direct CP violation in these decays, and A_D = 0.

Fit #2: One-parameter description of indirect CP violation
This parametrization results from two assumptions: (1) sub-leading amplitudes in CF and DCS decays are negligible (A_D = 0); (2) sub-leading amplitudes in singly Cabibbo-suppressed (SCS) decays (i.e., loop amplitudes) are neglected in indirect CP violation observables, as their contribution is suppressed by the small mixing parameters x and y. As a consequence, only short-distance dispersive amplitudes contribute to indirect CP violation. Thus Arg(Γ₁₂) = 0 (in the ΔU=2 basis, see note on Arg(q/p) above), and all indirect CP violation is due to Arg(M₁₂) ≠ 0, or more formally Arg(M₁₂) ≠ Arg(Γ₁₂). This difference is denoted φ₁₂ ≡ Arg(M₁₂/Γ₁₂). This reduces four mixing + CP violation parameters to three: x₁₂, y₁₂, φ₁₂. For alternative parameters (x, y, |q/p|, φ), this introduces a constraint among them, first derived by Ciuchini et al. and later independently by Kagan and Sokoloff:

tanφ = ^(1-|q/p|²)⁄_(1+|q/p|²) × (^x⁄_y).
Alternatively, one can use the quadratic equation (15) of Grossman, Nir, and Perez to reduce four parameters to three. We perform this fit three times: once for parameters x, y, φ (giving MINOS errors for φ); once for parameters x, y, |q/p| (giving MINOS errors for |q/p|); and once for parameters x₁₂, y₁₂, φ₁₂.

Fit #3: Two-parameter description of indirect CP violation
In this parameterization, sub-leading amplitudes in SCS decays are accounted for in indirect CP violation observables. Such sub-leading amplitudes contribute to Γ₁₂ in addition to M₁₂, and thus Arg(Γ₁₂) can be nonzero. We thus fit for all four parameters: (x₁₂, y₁₂, φ₂^M, φ₂^Γ) or equivalently (x, y, |q/p|, φ). We perform this fit twice: (a) neglecting sub-leading amplitudes in CF and DCS decays, i.e., A_D = 0; and (b) allowing all sub-leading amplitudes, i.e., floating all ten parameters. The results are essentially identical except for a slight shift in the φ central value: (−1.46 ± 1.04)° → (−1.51 ± 1.04)°.

The MINUIT output for Fits #1 − #3 are given here. In this output, x, y, R _D, A _D, A _π and A _K are in percent; and δ, δ₂ (= δ_Kππ), and φ are in radians. Correlation coefficients among parameters are listed at the end. The final results are:

χ ² contributions for the all-CPV-allowed fit #3:

_____________________________________________

MNCONTOUR-like 2-d plots:
(click on for .eps versions)

CPV-allowed plot, no mixing (x,y) = (0,0) point: Δχ² = 4328, excluded at ≫ 11.5σ (limit of CERNLIB PROB routine)

No indirect CP violation point (|q/p|, φ) = (1,0): Δχ² = 2.16, consistent with CP conservation (0.95σ)

No indirect CP violation point (φ₂^M, φ₂^Γ) = (0,0): Δχ² = 1.97, consistent with CP conservation (0.89σ)
Note: the slight difference from the (|q/p|, φ) plot is due to the constraint A_D = 0. (This raises the χ² of the best-fit point, slightly lowering Δχ² of the no-CPV point.)