Global Fit for D⁰- D⁰ Mixing
(allowing for CP violation)
(updated 31 March 2019)

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 D ₁ ≡ p|D⁰> + q|D⁰> and D ₂ ≡ 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 49 observables:   y _CP ,   A _Γ ,   (x, y, |q/p|, φ) _{Belle K⁰S π⁺ π ⁻} ,   (x, y) _{LHCb K⁰S π⁺ π ⁻} ,   (x, y) _{BaBar K⁰S 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	(0.715 ± 0.111)%	World average (COMBOS combination)   of D0 → K+ K− / π+ π − / K+ K− K0
2	A Γ	(−0.032 ± 0.026)%	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)   xCP yCP 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 → K0S π+π − results using 1 fb-1 (√s = 7 TeV) Correlation coefficient = +0.37, no CPV.   3 fb-1 results (√s = 7, 8 TeV) allowing for CPV. Correlation coefficients (statistical + systematic): 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) → (xCP, yCP, 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 → K0S π+π − and D0 → K0S 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 δKππ + y sin δKππ,   y" = y cos δKππ − x sin δKππ.
18	R D x 2 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	RD 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	RD 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	RD 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	RD 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	RD+ 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) Correlation coefficients: 1   0.843   −0.935 0.843   1   −0.963 −0.935   −0.963   1
25	RD− 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) Correlation coefficients: 1   0.846   −0.935 0.846   1   −0.964 −0.935   −0.964   1
26	ACPK ACPπ	(0.00 ± 0.34 ± 0.13)% (−0.24 ± 0.52 ± 0.22)%	BaBar   385.8 fb−1 near ϒ(4S) resonance
27	ACPK ACPπ	(−0.32 ± 0.21)% (0.31 ± 0.22)%	CDF   9.7 fb−1 pp collisions at √s = 1.96 TeV ( 〈t〉K − 〈t〉π ) / τD = 0.27 ± 0.01
28	ACPK − ACPπ	(−0.154 ± 0.029)%	LHCb   9.0 fb−1 pp collisions at √s = 7, 8, 13 TeV D*+ → D0π+ and B → D0μ− X flavor tags ( 〈t〉K − 〈t〉π )/τD = 0.115 ± 0.002;   〈t〉/τD = 1.71 ± 0.10
29	(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: 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.

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

Fit #2a:   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 #2b:   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 final results are:

MIGRAD correlation coefficients:

χ ² contributions:

MNCONTOUR-like 2-d plots (click on for high resolution):

           

 

           

 

           

 

Old scales (for comparison):

 

       

 

CPV-allowed plot, no mixing (x,y) = (0,0) point:   Δ χ ² = 2028,   excluded at ≫ 11.5σ (limit of PROB routine)

No CPV (|q/p|, φ) = (1,0) point:   Δ χ ² = 0.935,   CL = 0.63

MNCONTOUR-like 1-d plots (click on for .eps versions):   red dashed horizontal line denotes Δχ ² = 3.84, corresponding to 95% C.L. interval. Cusp points result from multiple solutions.

           

                        x = 0 point:   Δ χ ² = 10.68,   x ≤ 0 excluded at 3.1σ                                   y = 0 point:   Δ χ ² = 142.0,   y ≤ 0 excluded at > 11.4σ (limit of PROB routine)