Fit results for D⁰- D⁰ Mixing
(allowing for CP violation)

People working on this: David Asner, Bostjan Golob, Brian Petersen, Milind Purohit, Alan Schwartz
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>; R _D is the ratio of magnitudes of D⁰ → f and D⁰ → f amplitudes; δ and δ₂ are strong phase differences between D⁰ → f and D⁰ → f amplitudes, and φ is a weak phase difference. The mixing parameters are defined as x ≡ (m₂ − m₁)/Γ and y ≡ (Γ₂ − Γ₁)/(2Γ), where Γ = (Γ₁ + Γ₂)/2. In the absence of CP violation, our convention is x = (m_CP+ − m_CP−)/Γ, y = (Γ_CP+ − Γ_CP−)/(2Γ), and δ = δ_{D⁰ → K⁻n(π)} − δ_{D⁰
→ K⁻n(π)} .

Allowing for CP violation, there are a total of 8 underlying parameters: x, y, δ, δ₂, R _D, A _D, |q/p|, and φ From all experiments, there are currently 26 observables: y_CP, A_Γ, x _{K⁰_S π⁺ π ⁻}, y _{K⁰_S π⁺ π ⁻}, |q/p| _{K⁰_S π⁺ π ⁻}, Arg(q/p) = φ _{K⁰_S π⁺ π ⁻} (R_M)_semileptonic, (x) _{K⁺ π⁻ π ⁰}, (y) _{K⁺ π⁻ π ⁰}, (R_M)_{K⁺ π⁻ π
⁺ π ⁻} , (R_M)_Ψ(3770) , (y)_Ψ(3770) , (R_D)_Ψ(3770) , (R_D^1/2 cos δ)_Ψ(3770) , (R _D)_BaBar, (A _D)_BaBar, (x'^±, y'^±)_BaBar, (R _D)_Belle, (A _D)_Belle, (x'^±, y'^±)_Belle, The relationships between these observables and the underlying parameters 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.

Observables used:

Index Observable Value Source

1 y_CP (1.132 ± 0.266)% World average (COMBOS combination) of D⁰ → K⁺ K⁻ / π⁺ π ⁻ results

2 A_Γ (0.123 ± 0.248)% 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

(0.811 ± 0.334)%

(0.309 ± 0.281)%

0.95 ± 0.22 ^+0.10 _−0.09

(−0.035 ± 0.19 ± 0.09) radians

(0.81 ± 0.30 ^+0.13 _−0.17)%

(0.37 ± 0.25 ^+0.10 _−0.15)%

0.86 ± 0.30 ^+0.10 _−0.09

(−0.244 ± 0.31 ± 0.09) radians

No CPV: World average (COMBOS combination) of D⁰ → K⁰ π⁺π⁻ results
CPV-allowed: Belle D⁰ → K⁰ _S π⁺ π ⁻ results; correlation coefficients:

1 −0.007 −0.255α +0.216

−0.007 1 −0.019α −0.280

−0.255α −0.019α 1 −0.128α

+0.216 −0.280 −0.128α 1

(Note: α = (|q/p|+1)²/2 is a variable transformation factor)

7 R_M (0.0173 ± 0.0387)% World average (COMBOS combination) of D⁰ → K⁺l⁻ ν results

8

x"

y"

(2.39 ± 0.61 ± 0.32)%

(−0.14 ± 0.60 ± 0.40)%

BaBar K⁺ π ⁻ π ⁰ result; correlation coefficient = −0.34.

Note: x" = x cos δ₂ + y sin δ₂, y" = y cos δ₂ − x sin δ₂.

9 R_M (0.019 ± 0.0161)% BaBar K⁺ π ⁻ π ⁺ π ⁻ result

10

R_M

y

R _D

√R_D cos δ

(0.199 ± 0.173 ± 0)%

(−5.207 ± 5.571 ± 2.737)%

(−2.395 ± 1.739 ± 0.938)%

(8.878 ± 3.369 ± 1.579)%

CLEOc Ψ(3770) results; correlation coefficients:

1 −0.0644 0.0072 0.0607

−0.0644 1 −0.3172 −0.8331

0.0072 −0.3172 1 +0.3893

0.0607 −0.8331 +0.3893 1

11

R_D

x'²⁺

y'⁺

(0.303 ± 0.0189)%

(−0.024 ± 0.052)%

(0.98 ± 0.782)%

BaBar K⁺ π ⁻ results; correlation coefficients:

1 +0.77 −0.87

+0.77 1 −0.94

−0.87 −0.94 1

12

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.

13

R_D

x'²⁺

y'⁺

(0.364 ± 0.018)%

(0.032 ± 0.037)%

(−0.12 ± 0.58)%

Belle K⁺ π ⁻ results; correlation coefficients:

1 +0.655 −0.834

+0.655 1 −0.909

−0.834 −0.909 1

14

A _D

x'^{2 −}

y'⁻

(2.3 ± 4.7)%

(0.006 ± 0.034)%

(0.20 ± 0.54)%

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