Global Fit for D0- D0 Mixing
(allowing for CP violation)
(through 1 July 2021)

 

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

Experimental Observables:
From all experiments there are 61 observables:   y CP ,   A Γ ,   (x, y, |q/p|, φ) Belle K0S π+ π ,   (xCP, yCP, Δx, Δy) 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) ,   (RD, AD, x', y'±)BaBar ,   (RD, AD, x', y'±)Belle ,   (RD, x'2, y')CDF ,   (RD±, x', y'±)LHCb , (ACPK, ACPπ)BaBar , (ACPK, ACPπ)Belle , (ACPK − ACPπ)CDF , (ACPK −ACPπ) LHCb(D*) , (ACPK −ACPπ) LHCb(B →D0μX)

Theoretical Parameters:
Allowing for CP violation, there are 10 underlying parameters:   x, y, δ, δKππ, RD, AD, Aπ, AK, |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 (0.719 ± 0.113)% World average (COMBOS combination)   of D0 → K+ K / π+ π / K+ K K0
2 A Γ (0.0089 ± 0.0113)% World average (COMBOS combination)   of D0 → K+ K / π+ π results
3-6
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
7-10
 
 
 
 
 
11-14
x (no CPV)
y (no CPV)
 
xCP
yCP
Δx
Δy
 
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)%
 
(0.397 ± 0.046 ± 0.029)%
(0.459 ± 0.120 ± 0.085)%
(−0.027 ± 0.018 ± 0.001)%
(0.020 ± 0.036 ± 0.013)%
LHCb   D0 → K0S π+π results using 1 fb-1 (√s = 7 TeV)
D*+ → D0π+ flavor tag. Correlation coefficient = +0.37, no CPV.
 
3 fb-1 results (√s = 7, 8 TeV) allowing for CPV.
D*+ → D0π+, B → D0μ X flavor tags. Correlation coefficients (stat. + syst.):
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)
 
5.4 fb-1 results (√s = 13 TeV) allowing for CPV.
D*+ → D0π+ flavor tags. Correlation coefficients (stat. + syst.):
1   (0.11 + 0.13)   (−0.02 + 0.01)   (−0.01 + 0.01)
    1   (−0.01 − 0.02)   (−0.05 + 0.01)
        1   (0.08 + 0.31)
For (x, y, |q/p|, φ) → (xCP, yCP, Δx, Δy) mapping, see PRD 99, 012007 (2019)
15-16
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.
17-18
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.
19 (x2 + y2)/2 (0.0130 ± 0.0269)% World average (COMBOS combination)   of D0 → K+l ν results
20-21
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.
22-26
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
27-29
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
30-32
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.
33-35
(no CPV)
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
33-35
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
36-38
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.
39-41
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
42-44
RD+
x' 2+
y' +
(0.338 ± 0.0161)%
(−0.0019 ± 0.0447)%
(0.581 ± 0.526)%
LHCb   K+ π results for 3.0 fb−1 (√s = 7, 8 TeV)
B → D*+μ X, D*+ → D0π+ flavor tags. Correlation coefficients:
1   0.823   −0.920
0.823   1   −0.962
−0.920   −0.962   1
45-47
RD
x' 2 −
y'
(0.360 ± 0.0166)%
(0.0079 ± 0.0433)%
(0.332 ± 0.523)%
LHCb   K+ π results for 3.0 fb−1 (√s = 7, 8 TeV)
B → D*+μ X, D*+ → D0π+ flavor tags. Correlation coefficients:
1   0.812   −0.918
0.812   1   −0.956
−0.918   −0.956   1
48-50
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)
D*+ → D0π+ flavor tag. Correlation coefficients:
1   0.843   −0.935
0.843   1   −0.963
−0.935   −0.963   1
51-53
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)
D*+ → D0π+ flavor tag. Correlation coefficients:
1   0.846   −0.935
0.846   1   −0.964
−0.935   −0.964   1
54-55
ACPK
ACPπ
(0.00 ± 0.34 ± 0.13)%
(−0.24 ± 0.52 ± 0.22)%
BaBar   385.8 fb−1 near ϒ(4S) resonance
56-57
ACPK
ACPπ
(−0.43 ± 0.30 ± 0.11)%
(0.43 ± 0.52 ± 0.12)%
Belle   540 fb−1 near ϒ(4S) resonance
58-59
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
60 ACPK − ACPπ (−0.154 ± 0.029)%
LHCb   8.9 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〉D = 1.71 ± 0.10
61 (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
Five separate fits are performed as follows:

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

Fits #2a and #2b:   no direct CP violation in doubly-Cabibbo-suppressed amplitudes   (AD= 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 #2c:   no direct CP violation in doubly-Cabibbo-suppressed amplitudes   (AD= 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(M1212). 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 MINUIT output for all five fits (in order) are given here.
(Note that x, y, R D, A D, A π and A K are in percent; δ, δ2 (=δKππ), and φ are in radians.)

The final results are:



Note that for the No-direct-CPV results, the values listed for (δ, δKππ, RD) are from Fit 2a rather than Fit 2b (but they are almost identical).

χ 2 contributions for CPV-allowed fit:

_____________________________________________

 

MNCONTOUR-like 2-d plots:
(click on for .eps versions)

 

           

 

CPV-allowed plot, no mixing (x,y) = (0,0) point:   Δ χ 2 = 2099,   excluded at ≫ 11.5σ (limit of CERNLIB PROB routine)

No CPV (|q/p|, φ) = (1,0) point:   Δ χ 2 = 5.633,   excluded at 1.56σ

 

           

 

           

 

_____________________________________________

MNCONTOUR-like 1-d plots:
Dashed red horizontal line denotes Δχ 2 = 3.84, corresponding to a 95% C.L. interval.
(click on for .eps versions)

 

           

                        x = 0 point:   Δ χ 2 = 68.30,   x ≤ 0 excluded at 8.18σ                         y = 0 point:   Δ χ 2 = 477.2,   y ≤ 0 excluded at > 11.4σ (limit of CERNLIB PROB)

 

           

 

           

_____________________________________________

 

Comparison with results before LHCb's 5.4 fb−1 D0 → K0S π+π measurement, using identical binning:
Left-most plots include all results except for LHCb's 5.4 fb−1 D0 → K0S π+π measurement of (xCP, yCP, Δx, Δy); right-most plots include that result too (all results).
(click on for .eps versions)

 

           

 

           

 

 


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