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

People working on this:   David Asner, Bostjan Golob, Milind Purohit, Alan Schwartz

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 δKππ 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 = (mCP+ − mCP−)/Γ, y = (ΓCP+ − ΓCP−)/(2Γ), and δ = δ D0 → Kn(π) − δ D0 → Kn(π) .

Allowing for CP violation, there are a total of 8 underlying parameters:   x, y, δ, δKππ, 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 (=δKππ), and φ are in radians.

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

Fit #2, no direct CP violation (AD=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]°

χ 2 contributions:

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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

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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