Global Fit for D⁰- D⁰ Mixing
(allowing for CP violation)
(through 26 July 2020)

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:
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:
From all experiments there are 57 observables: y_CP, A_Γ, (x, y, |q/p|, φ) _{Belle
K⁰_S π⁺ π ⁻}, (x, y) _{LHCb
K⁰_S π⁺ π ⁻}, (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") _{K⁺ π⁻ π ⁰}, (R_D, x², y, cos δ, sin δ) _Ψ(3770) , (R_D, A_D, x'^2±, y'^±)_BaBar, (R_D, A_D, x'^2±, y'^±)_Belle, (R_D, x'², y')_CDF, (R_D^±, x'^2±, y'^±)_LHCb, (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^*), (A_CP^K −A_CP^π) _{LHCb(B →D⁰μX)}

Theoretical Parameters:
Allowing for CP violation, there are 10 underlying parameters: x, y, δ, δ_Kππ, R_D, A_D, A_π, A_K, |q/p|, and Arg(q/p) = φ. The first two parameters govern mixing; the next two are strong phases; R _D is the ratio Γ(D⁰→ f)/Γ(D⁰ → f); the next three are direct CP-violating asymmetries for D⁰ → K⁺ π⁻, D⁰ → π⁺ π⁻, and D⁰ → K⁺ K⁻, respectively; and the last two are indirect CP-violating parameters. The relationships between these parameters and the measured observables are given below. The observables appear in blue (on the left sides of the equations), the underlying parameters in magenta (on the right sides), and intermediate variables in black.

Measurements used:

Index Observable Value Source

1 y_CP (0.719 ± 0.113)% World average (COMBOS combination) of D⁰ → K⁺ K⁻ / π⁺ π ⁻ / K⁺ K⁻ K⁰

2 A_Γ (−0.0309 ± 0.0204)% World average (COMBOS combination) of D⁰ → K⁺ K⁻ / π⁺ π ⁻ results

(3-5)

6

x (no CPV)

y (no CPV)

|q/p| (no dCPV)

Arg(q/p)=φ (no dCPV)

x

y

|q/p|

φ

0.56 ± 0.19 ^+0.067 _−0.127

0.30 ± 0.15 ^+0.050 _−0.078

0.90 ^+0.16 _−0.15 ^+0.078 _−0.064

(−6 ± 11 ^+4.2 ₋₅) degrees

(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 D⁰ → K⁰ _S π⁺ π ⁻ results using 921 fb⁻¹.

Correlation coefficient is +0.012 for no-CPV; for CPV-allowed they are:

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

(7-9)

10

x (no CPV)

y (no CPV)

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)%

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)

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

11-12

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.

13-14

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.

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

16-17

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.

18

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; 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

19

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

+0.77 1 −0.94

−0.87 −0.94 1

20

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.

21a

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

+0.737 1 −0.948

−0.865 −0.948 1

21b

R_D

x'²⁺

y'⁺

(0.364 ± 0.018)%

(0.032 ± 0.037)%

(−0.12 ± 0.58)%

Belle K⁺ π ⁻ CPV-allowed results using 400 fb⁻¹. Correlation coefficients:

1 +0.655 −0.834

+0.655 1 −0.909

−0.834 −0.909 1

22

A _D

x'^{2 −}

y'⁻

(2.3 ± 4.7)%

(0.006 ± 0.034)%

(0.20 ± 0.54)%

Belle K⁺ π ⁻ CPV-allowed results using 400 fb⁻¹;

correlation coefficients same as above.

23

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

0.90 1 −0.98

−0.97 −0.98 1

24

R_D⁺

x'²⁺

y'⁺

(0.338 ± 0.0161)%

(−0.0019 ± 0.0447)%

(0.581 ± 0.526)%

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

1 0.823 −0.920

0.823 1 −0.962

−0.920 −0.962 1

25

R_D⁻

x'^{2 −}

y'⁻

(0.360 ± 0.0166)%

(0.0079 ± 0.0433)%

(0.332 ± 0.523)%

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

1 0.812 −0.918

0.812 1 −0.956

−0.918 −0.956 1

26

R_D⁺

x'²⁺

y'⁺

(0.3454 ± 0.0045)%

(0.0061 ± 0.0037)%

(0.501 ± 0.074)%

LHCb K⁺ π ⁻ results for 5.0 fb⁻¹ (√s = 7, 8 TeV)
D^*+ → D⁰π⁺ flavor tag. Correlation coefficients:

1 0.843 −0.935

0.843 1 −0.963

−0.935 −0.963 1

27

R_D⁻

x'^{2 −}

y'⁻

(0.3454 ± 0.0045)%

(0.0016 ± 0.0039)%

(0.554 ± 0.074)%

LHCb K⁺ π ⁻ results for 5.0 fb⁻¹ (√s = 7, 8 TeV)
D^*+ → D⁰π⁺ flavor tag. Correlation coefficients:

1 0.846 −0.935

0.846 1 −0.964

−0.935 −0.964 1

28

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

29

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

30

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

31 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

32 (x² + y²)/4 (0.0048 ± 0.0018)%

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

D⁰ → K⁺ π ⁻ π ⁺ π ⁻

MINUIT fit results
Five separate fits are performed as follows:

Fit #1: no CP violation (A_D= 0, A_K= 0, A_π= 0, |q/p| = 1, φ = 0)

Fits #2a and #2b: no direct CP violation in doubly-Cabibbo-suppressed amplitudes (A_D= 0)
In addition, we impose the relation tanφ = (1-|q/p|²)/(1+|q/p|²) × (x/y) to reduce four independent parameters to three. This relation was first derived by Ciuchini et al. and was later independently obtained by Kagan and Sokoloff. Alternatively, one can use the quadratic equation (15) of Grossman, Nir, and Perez to reduce four parameters to three (e.g., see here). We use the Ciuchini/Kagan formula to perform two separate fits: first we float x, y, and φ and from them derive |q/p| (this yields proper errors for φ). Then we float x, y, and |q/p| and from them derive φ (this yields proper errors for |q/p|).

Fit #2c: no direct CP violation in doubly-Cabibbo-suppressed amplitudes (A_D= 0) fit for theory parameters x₁₂, y₁₂, and φ₁₂
Here we fit for the underlying theory parameters x₁₂ ≡ 2|M₁₂|/Γ, y₁₂ ≡ |Γ₁₂|/Γ, and φ₁₂ ≡ Arg(M₁₂/Γ₁₂). The relationships between these parameters and our nominal parameters (x, y, |q/p|, φ) are given by Kagan and Sokoloff Eqs. (14, 15, 48, 52), but a factor of 2^-1/2 is missing from Eqs. (14) and (15). An alternative derivation (our own) is here; these differ from Kagan and Sokoloff but give identical results.

Fit #3: allowing all CP violation (all parameters floated)

The MINUIT output for all five fits (in order) are given here.
(Note that x, y, R _D, A _D, A _π and A _K are in percent; δ, δ₂ (=δ_Kππ), and φ are in radians.)

The final results are:

Note that for the No-direct-CPV results, the values listed for (δ, δ_Kππ, R_D) are from Fit 2a rather than Fit 2b (but they are almost identical).

χ ² contributions for CPV-allowed fit: