# Global Fit for D0- D0 Mixing (allowing for CP violation) (updated 30 June 2014)

People working on this:   Alan Schwartz, Bostjan Golob, Marco Gersabeck

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 41 observables:   y CP ,   A Γ ,   (x, y, |q/p|, φ) Belle K0S π+ π ,   (x, y) BaBar K0S h+ h ,   (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', 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 (0.866 ± 0.155)% World average (COMBOS combination)   of D0 → K+ K / π+ π / K+ K K0
2 A Γ (−0.014 ± 0.052)% 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
7-8
 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
 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
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.
12a
 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
12
 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
13
 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.
14
 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
15
 RD+ x' 2+ y' +
 (0.3545 ± 0.0095)% (0.0049 ± 0.0070)% (0.51 ± 0.14)%
LHCb   K+ π results for 3.0 fb−1 (√s = 7, 8 TeV)
Correlation coefficients:
 1 0.862 −0.942 0.862 1 −0.968 −0.942 −0.968 1
16
 RD− x' 2 − y' −
 (0.3591 ± 0.0094)% (0.0060 ± 0.0068)% (0.45 ± 0.14)%
LHCb   K+ π results for 3.0 fb−1 (√s = 7, 8 TeV)
Correlation coefficients:
 1 0.858 −0.941 0.858 1 −0.966 −0.941 −0.966 1
17
 ACPK ACPπ
 (0.00 ± 0.34 ± 0.13)% (−0.24 ± 0.52 ± 0.22)%
BaBar   385.8 fb−1 near ϒ(4S) resonance
18
 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
19
 ACPK ACPπ
 (−0.32 ± 0.21 ± 0.09)% (0.55 ± 0.36 ± 0.09)%
Belle preliminary   976 fb−1 in e+e collisions
20 ACPK − ACPπ (−0.34 ± 0.15 ± 0.10)%
 LHCb   1.0 fb−1 pp collisions at √s = 7 TeV D*+ → D0π+ flavor tag ( 〈t〉K − 〈t〉π ) / τD = 0.1119 ± 0.0013 ± 0.0017
21 ACPK − ACPπ (0.14 ± 0.16 ± 0.08)%
 LHCb   3 fb−1 pp collisions at √s = 7 TeV B → D0μ− X flavor tag ( 〈t〉K − 〈t〉π ) / τD = 0.014 ± 0.004

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

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

Fit #2a:   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 #2b:   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 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:

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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 = 421.0,   excluded at ≫ 11.5σ (limit of PROB routine)

No CPV (|q/p|, φ) = (1,0) point:   Δ χ 2 = 1.32,   CL = 0.48

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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 = 6.96,   x ≤ 0 excluded at 2.4σ                   y = 0 point:   Δ χ 2 = 89.6,   y ≤ 0 excluded at 9.4σ