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

People working on this:   David Asner, Bostjan Golob, 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 D ₁ ≡ p|D⁰> + q|D⁰> and D ₂ ≡ p|D⁰> − q|D⁰>; R _D is the ratio of magnitudes of D⁰ → f and D⁰ → f amplitudes; δ and δ_Kππ 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, δ_Kππ, R _D, A _D, |q/p|, and φ From all experiments, there are currently 28 observables:   y _CP ,   A _Γ ,   x _{K⁰S π⁺ π ⁻} ,   y _{K⁰S π⁺ π ⁻} ,   |q/p| _{K⁰S π⁺ π ⁻} ,   Arg(q/p) = φ _{K⁰S π⁺ π ⁻}   (R _M ) _semileptonic ,   (x") _{K⁺ π⁻ π ⁰} ,   (y") _{K⁺ π⁻ π ⁰} ,   (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 ,   (R _D, x', y') _CDF . 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	(1.072 ± 0.257)%	World average (COMBOS combination)   of D0 → K+ K− / π+ π − / K+ K− K0
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	(0.0130 ± 0.0269)%	World average (COMBOS combination)   of D0 → K+l− ν results
8	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.
9	R M y R D √RD 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
10	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
11	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.
12	RD 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
13	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.
14	RD x' 2 y'	(0.304 ± 0.055)% (−0.012 ± 0.035)% (0.85 ± 0.76)%	CDF   K+ π − results; correlation coefficients: 1   0.923   −0.971 0.923   1   −0.984 −0.971   −0.984   1

MINUIT fit results: note that x, y, R _D, and A _D are in percent; δ, δ₂ (=δ_Kππ), and φ are in radians.

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

Fit #2, no direct CP violation (A_D=0, fixed):

Fit #3, allowing direct CP violation (i.e., all parameters floated):

The final results allowing for CPV are:

Parameter	Central value (with 1σ error)	95% C.L. Interval
x	(1.00 +0.24 −0.26 )%	[0.48,   1.46]%
y	(0.76 +0.17 −0.18 )%	[0.40,   1.10]%
δ	(22.5 +10.4 −11.0 )°	[−0.6,   43.2]°
δKππ	(11.2 +20.6 −22.5 )°	[−34.5,   51.1]°
RD	(0.336 ± 0.009)%	[0.320,   0.353]%
AD	(−2.1 ± 2.4)%	[−6.8,   2.7]%
|q/p|	0.86 +0.17 −0.15	[0.59,   1.22]
φ	(−8.8 +7.6 −7.2 )°	[−23.0,   6.3]°

MIGRAD correlation coefficients:

χ ² contributions:

MNCONTOUR-like 2-d plots:

           

 

 

CPV-allowed plot, no mixing (x,y) = (0,0) point:   Δ χ ² = 102.6,   CL = 5.3 x 10 ⁻²³ ,   no mixing excluded at 9.8σ

No CPV (|q/p|, φ) = (1,0) point:   Δ χ ² = 1.33,   CL = 0.486 ,   consistent with CP conservation

MNCONTOUR-like 1-d plots:   red dashed horizontal line denotes Δχ ² = 3.84, corresponding to 95% C.L. Cusp points result from multiple solutions.

           

x = 0 point:   Δ χ ² = 11.6,   x ≤ 0 excluded at 3.4σ                   y = 0 point:   Δ χ ² = 16.5,   y ≤ 0 excluded at 4.1σ