Global Fit for D0-D0 Mixing
(allowing for CP violation)
(through 24 September 2023)

 

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 63 measurements of 18 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) CLEOc Ψ(3770) ,   (ACP) BESIII 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) ,   (ACPK) LHCb(D*)

Theoretical Parameters:
Allowing for all CP violation, there are 10 fitted 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; RD 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 phase φ corresponds to a basis in which the dominant U-spin-changing (ΔU=2) dispersive and absorptive mixing amplitudes are real; see Kagan and Silvestrini, Eq. 107. (Choosing a basis ensures that Arg(q/p) is physically meaningful.) 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.

In addition, we perform a fit for alternative mixing and CP violation parameters as described below: (x12, y12, φ12) for Fit #2, and (x12, y12, φM2, φΓ2) for Fit #3.

 

 

Measurements used:


Index Observable Value Source
1 y CP − y CP (Kπ) (0.697 ± 0.028)%
World average   of D0 → K+ K / π+ π / K+ K K0
Our calculation of the y CP (Kπ) correction is from arXiv:2207.11867, Eq. (29).
This correction was first pointed out by Pajero, Morello in JHEP 03 (2022) 162.
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)%
 
(0.400 ± 0.045 ± 0.020)%
(0.551 ± 0.116 ± 0.059)%
(−0.029 ± 0.018 ± 0.001)%
(0.031 ± 0.035 ± 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+, B → D0μ X flavor tags. Correlation coefficients (stat. + syst.):
1   (0.121 + 0.13)   (−0.018 + 0.01)   (−0.016 + 0.01)
    1   (−0.012 − 0.02)   (−0.058 + 0.01)
        1   (0.069 + 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, 0.82 fb−1. 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 ACPK (0.068 ± 0.054 ± 0.016)%
LHCb   5.7 fb−1 pp collisions at √s = 13 TeV
D*+ → D0π+ flavor tags
〈t〉KD = (701.5 ± 1.1)/(410.3 ± 1.0) = 1.7097 ± 0.0050
62 (x2 + y2)/4 (0.0048 ± 0.0018)%
LHCb   3.0 fb−1 pp collisions at √s = 7, 8 TeV
D0 → K+ π π + π
63 ACP 0.132 ± 0.011 ± 0.007
BESIII   Ψ(3770) results, 2.93 fb−1. D0 → K π+,
difference between CP-even tagged and CP-odd tagged

 

MINUIT fit results
Four types of fits are performed, as follows:

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

Fit #2:   One-parameter description of indirect CP violation
This parametrization results from two simplifications: (1) sub-leading amplitudes in CF and DCS decays are neglected; and (2) sub-leading amplitudes in SCS decays are neglected in indirect CP violation observables, as their contribution is suppressed by mixing parameters x and y. These simplifictions have two consequences: (a) no direct CP violation in CF or DCS decays (AD= 0); and (b) only short-distance dispersive amplitudes contribute to indirect CP violation. Thus Arg(Γ12) = 0 (in the ΔU=2 basis, see note on Arg(q/p) above), and all indirect CP violation can be parameterized in terms of a phase difference between M12 and Γ12. This phase difference is denoted φ12 ≡ Arg(M1212). The magnitudes of the mixing amplitudes are parameterized as x12 ≡ 2|M12|/Γ and y12 ≡ |Γ12|/Γ. The "baseline" mixing + CPV parameters (x, y, |q/p|, φ) can be expressed in terms of (x12, y12, φ12); since four parameters depend only on three, there must be an additional constraint among the four. This relation, first derived by Ciuchini et al. and later independently obtained by Kagan and Sokoloff, is tanφ = (1-|q/p|2)/(1+|q/p|2) × (x/y).   Alternatively, one can use the quadratic equation (15) of Grossman, Nir, and Perez to reduce four parameters to three (e.g., see here).

We thus perform three separate fits:
2a: float x, y, and φ, and use the Ciuchini/Kagan formula to derive |q/p|; this yields proper (MINOS) errors for φ.
2b: float x, y, and |q/p|, and use the Ciuchini/Kagan formula to derive φ; this yields proper (MINOS) errors for |q/p|.
2c: fit for parameters x12, y12, and φ12. The relationships between these parameters and (x, y, |q/p|, φ) are derived by Kagan and Sokoloff (Eqs. 14, 15, 48, 52), but a factor of 1/√2 is missing from Eqs. (14) and (15):




Fit #3:   Two-parameter description of indirect CP violation
In this parameterization, sub-leading amplitudes in CF and DCS decays are neglected as in Fit #2, and thus AD= 0. However, to accommodate the "high-precision era" in charm mixing/CP violation measurements, sub-leading amplitudes in SCS decays are taken into account in indirect CP violation observables. There is one simplification: D0→ K+K, π+π final-state-dependent effects are neglected in AΓ, as they cancel at leading-order in U-spin breaking. The sub-leading SCS amplitudes contribute to Γ12 in addition to M12, and thus Arg(Γ12) may be nonzero. To account for this, we fit for four parameters: (x12, y12, φM2, φΓ2), where φM2 and φΓ2 are the phases of M12 and Γ12, respectively, relative to that of the dominant ΔU=2 dispersive and absorptive amplitudes. This parameterization is discussed in Kagan and Silvestrini. The relationships between (x, y, |q/p|, φ) and (x12, y12, φM2, φΓ2) are:


The first three relations correspond to Eqs. (14), (15), and (48) of Kagan and Sokoloff; the last relation corresponds to Eq. (110) of Kagan and Silvestrini.

Fit #4:   Two-parameter description of indirect CP violation (|q/p|, φ), direct CP violation in DCS decays allowed
In this case all parameters are floated, and we fit for the baseline parameters (x, y, |q/p|, φ).

The MINUIT output for Fits #1 − #4 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. Correlation coefficients among parameters are listed at the end.

The final results are:



χ 2 contributions for the all-CPV-allowed fit #4:

_____________________________________________

 

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

 

           

 

CPV-allowed plot, no mixing (x,y) = (0,0) point:   Δχ2 = 2631,   excluded at ≫ 11.5σ (limit of CERNLIB PROB routine)

No CPV (|q/p|, φ) = (1,0) point:   Δχ2 = 6.487,   excluded at 2.1σ

 

           

 

           

 

No CPV (φ2M, φ2Γ) = (0,0) point:   Δχ2 = 3.399,   excluded at 1.3σ
(Note: compared to the (|q/p|, φ) fit, the constraint AD = 0 raises the χ2 of the best-fit point, lowering Δχ2 of the no-CPV point)

_____________________________________________

 

MNCONTOUR-like 1-d plots:
Dashed red horizontal line denotes Δχ2 = 3.84, corresponding to a 95% C.L. interval.
(click on for .pdf versions)

 

                         

                        x = 0 point:   Δχ2 = 82.83,   x ≤ 0 excluded at 9.1σ                         y = 0 point:   Δχ2 = 1227,   y ≤ 0 excluded at > 11.5σ (limit of CERNLIB PROB)

 

                         

 

                         

 


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