Fit results for D0- D0 Mixing
(allowing for CP violation)
(updated 14 November 2010)

 

People working on this:   David Asner, Bostjan Golob, 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 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 → K−n(π) − δ D0 → K−n(π) .


Allowing for CP violation, there are a total of 8 underlying parameters:   x, y, δ, δKππ, R D, A D, |q/p|, and Arg(q/p) = φ. From all experiments, there are currently 30 observables:   y CP ,   A Γ ,   (x, y, |q/p|, φ) Belle K0S π+ π − ,   (x, y) BaBar K0S h+ h− ,   (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 (1.107 ± 0.217)% 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:   Belle + CLEO (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)
7-8
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.
9 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
x 2
y
R D
2√RD cos δ
2√RD sin δ
(0.1549 ± 0.2223)%
(2.997 ± 2.293)%
(0.4118 ± 0.0948)%
(12.64 ± 2.86)%
(−0.5242 ± 6.426)%
CLEOc   Ψ(3770) results; correlation coefficients:
1   −0.6217   −0.00224   0.3698   0.01567
    1   0.00414   −0.5756   −0.0243  
        1   0.00350   0.00978
            1   0.0471
                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 #3, allowing all (direct and indirect) CP violation, i.e., all parameters floated:

 

The final results are:

 

 

MIGRAD correlation coefficients:

 

χ 2 contributions:

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

 

           

 

 

CPV-allowed plot, no mixing (x,y) = (0,0) point:   Δ χ 2 = 109.6,   CL = 1.56 x 10 −24 ,   no mixing excluded at 10.2σ

No CPV (|q/p|, φ) = (1,0) point:   Δ χ 2 = 1.218,   CL = 0.456 ,   consistent with CP conservation

 

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MNCONTOUR-like 1-d plots (click on for eps.gz 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 = 8.53,   x ≤ 0 excluded at 2.7σ                   y = 0 point:   Δ χ 2 = 36.2,   y ≤ 0 excluded at 5.9σ

 

           

 

           

 

 

This page is maintained by A. Schwartz and was last updated