# Fit results for D0- D0 Mixing (allowing for CP violation) (updated 4 September 2011)

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 38 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 ,   (x2, y, R D , 2√RDcos δ, 2√RDsin δ) Ψ(3770) ,   (RD, AD, x'±, y'±)BaBar ,   (RD, AD, x'±, y'±)Belle ,   (RD, x', y')CDF , (ACPK, ACPπ)BaBar , (ACPK, ACPπ)Belle , (ACPK, ACPπ)CDF , (ACPK −ACPπ)LHCb

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 (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
15
 ACPK ACPπ
 (0.00 ± 0.34 ± 0.13)% (−0.24 ± 0.52 ± 0.22)%
BaBar   385.8 fb−1 near ϒ(4S) resonance
16
 ACPK ACPπ
 (−0.43 ± 0.30 ± 0.11)% (0.43 ± 0.52 ± 0.12)%
Belle   540 fb−1 near ϒ(4S) resonance
17
 ACPK ACPπ
 (−0.24 ± 0.22 ± 0.10)% (0.22 ± 0.24 ± 0.11)%
 CDF   5.94 fb−1 pp collisions at √s = 1.96 TeV 〈t〉K/ τD = 2.65 ± 0.03;     〈t〉π/ τD = 2.40 ± 0.03
18 ACPK − ACPπ (−0.28 ± 0.70 ± 0.25)%
 LHCb   0.037 fb−1 pp collisions at √s = 7 TeV ( 〈t〉K − 〈t〉π ) / τD = 0.10 ± 0.01

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,   AK= 0,   Aπ= 0,   |q/p| = 1,   φ = 0)

Fit #2a:   no direct CP violation   (AD= 0,   AK= 0,   Aπ= 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 (AD= 0,   AK= 0,   Aπ= 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 = 109.6,   CL = 1.61 x 10 −24 ,   no mixing excluded at 10.1σ

No CPV (|q/p|, φ) = (1,0) point:   Δ χ 2 = 1.147,   CL = 0.436 ,   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 = 8.47,   x ≤ 0 excluded at 2.7σ                   y = 0 point:   Δ χ 2 = 36.2,   y ≤ 0 excluded at 5.9σ