Combination of Direct and Indirect CP Violation
(updated 16 April 2025)

 

People working on this:   Marco Gersabeck


Notation: This combination uses measurements of direct and indirect CP violation to extract the level of agreement for a no-CP-violation hypothesis. The observables are:

A Γ ≡ [τ(D0 → h+ h−) − τ(D0 → h+ h− )] / [τ(D0 → h+ h−) + τ(D0 → h+ h− )]

where h+ h− can be K+ K− or π+ π−, and

ΔACP ≡ ACP(KK) - ACP(ππ).

Following Gersabeck et al., J. Phys. G 39 (2012) 045005, the following relations between observables and underlying parameters are obtained:

AΓ = -aCPind - aCPdir yCP

which thus constrains mostly indirect CP violation (aCPind), and where the direct CP violation contribution (aCPdir) can differ for different final states. The latest LHCb measurement measures ΔY, which is approximately equal to AΓ with the relative difference being yCP. Further,

ΔACP = ΔaCPdir (1 + yCP 〈t〉/τ) - Δ〈t〉/τ AΓ,

where Δ〈X〉 is the difference between quantity X for the KK and ππ final state and 〈X〉 is their average. 〈t〉/τ is the mean decay time relative to the true lifetime of the D0 meson. The contribution of the decay-dependent direct CP violation to AΓ is known to be ≤10-4. Therefore, all AΓ measurements are combined, irrespective of whether they are based on KK or ππ decays, and thus aCPind = -AΓ is assumed, leading to

ΔACP = ΔaCPdir (1 + yCP 〈t〉/τ) + Δ〈t〉/τ aCPind,


A χ2 fit is performed in the plane ΔaCPdir vs. aCPind. For the BaBar result the difference of the quoted values for ACP(KK) and ACP(ππ) is calculated, adding all uncertainties in quadrature. This may overestimate the systematic uncertainty for the difference as it neglects correlated errors; however, the result is conservative and the effect is small as all measurements are statistically limited. For all measurements, statistical and systematic uncertainties are added in quadrature when calculating the χ2.

For world average values of mixing parameters, in particular yCP = (0.655 ± 0.028)% which is used in this fit, click here. For tables of measured CP asymmetries, click here.


Year Experiment Results Δ〈t〉/τ 〈t〉/τ Comment Reference
2012 BaBar AΓ = (0.09 ±0.26 (stat.) ±0.06 (syst.))% - - 468 fb−1 near Υ(4S) resonance J.P. Lees et al. (BaBar Collab.), Phys.Rev. D87 (2013) 012004.
2014 CDF AΓ = (−0.12 ±0.12)% - - 9.7 fb−1 √ s  = 1.96 TeV p p  collisions T.A. Aaltonen et al. (CDF Collab.), Phys.Rev. D90 (2014) 111103.
2015 Belle AΓ = (−0.03 ±0.20 (stat.) ±0.07 (syst.))% - - 976 fb−1 near Υ(4S) resonance M. Staric et al. (Belle Collab.), Phys.Lett. B753 (2016) 412.
2021 LHCb ΔY(KK) = (−0.003 ±0.013 (stat.) ±0.003 (syst.))%
ΔY(ππ) = (-0.036 ±0.024 (stat.) ±0.004 (syst.))%
 - - 9 fb−1 √ s  = 7+8+13 TeV pp collisions, prompt D* and B→D0μX R. Aaij et al. (LHCb Collab.), Phys.Rev.Lett. 118 (2017) 261803.
2008 BaBar ACP(KK) = (0.00 ±0.34 (stat.) ±0.13 (syst.))%
ACP(ππ) = (−0.24 ±0.52 (stat.) ±0.22 (syst.))% 		0.00 1.00 385.8 fb−1 near Υ(4S) resonance B. Aubert et al. (BABAR Collab.), Phys. Rev. Lett. 100, 061803 (2008).
2012 CDF ΔACP = (−0.62 ±0.21 (stat.) ±0.10 (syst.))% 0.25 2.58 9.7 fb−1 √ s  = 1.96 TeV p p  collisions T. Aaltonen et al. (CDF Collab.), Phys.Rev.Lett. 109 (2012) 111801.
2014 LHCb ΔACP = (0.14 ±0.16 (stat.) ±0.08 (syst.))% 0.01 1.07 3 fb−1 √ s  = 7+8 TeV pp collisions, B→D0μX R. Aaij et al. (LHCb Collab.), JHEP 1407 (2014) 041.
2016 LHCb ΔACP = (−0.10 ±0.08 (stat.) ±0.03 (syst.))% 0.12 2.10 3 fb−1 √ s  = 7+8 TeV pp collisions, prompt D* R. Aaij et al. (LHCb Collab.), Phys.Rev.Lett. 116 (2016) 191601.
2019 LHCb ΔACP = (-0.09 ±0.08 (stat.) ±0.05 (syst.))% 0.00 1.21 6 fb−1 √ s  = 13 TeV pp collisions, B→D0μX R. Aaij et al. (LHCb Collab.), arXiv:1903.08726.
2019 LHCb ΔACP = (−0.182 ±0.032 (stat.) ±0.009 (syst.))% 0.13 1.74 6 fb−1 √ s  = 13 TeV pp collisions, prompt D* R. Aaij et al. (LHCb Collab.), arXiv:1903.08726.
Fit Result
Agreement with no CP violation
CL = 6.9x10−8

 

Combination Plot: The combination plot shows the measurements listed in the Table above for ΔACP and AΓ, where the bands represent ±1σ intervals. The point of no CP violation (0,0) is shown as a filled circle, and two-dimensional 68% CL, 99.7% CL, and 99.99997% CL regions are plotted as ellipses with the best fit value as a cross indicating the one-dimensional uncertainties in their center. The second plot provides a zoomed-in view of the fit region.

 

 

 

From the fit, the change in χ2 from the minimum value for the no-CPV point (0,0) is 32.3; this corresponds to a CL of 9.7x10−8 for two degrees of freedom or 5.3 standard deviations. The central values and ± 1σ errors for the individual parameters are:

aCPind = (-0.010 ± 0.012 )%

ΔaCPdir = (−0.159 ± 0.029 )%

 

With the current level of measurements, an average solely of measurements from KK final states becomes feasible. This fit avoids the assumption of universal indirect CP violation. As not all measurements were published with a separate result for the KK final state, only the results in the table below are used. Compared to the fits above, the term proportional to aCPdir yCP is no longer neglected, resulting in the time-dependent measurements being represented by sloped bands, although the smallness of yCP suppresses this effect so they still appear as vertical bands in the figure below. The value for yCP is also extracted from the KK final state alone, for which the only available measurement is that of the LHCb collaboration [PRD105(2022)092013], yCP(KK)−yCP(Kπ) = (0.708 ±0.033)%, corrected for yCP(Kπ) as described above to yield yCP(KK) = (0.666 ±0.033)%. The time integrated asymmetry enters the fit according to the equation

ACP = aCPdir+aCPind〈t〉/τ


Year Experiment Results 〈t〉/τ Comment Reference
2014 CDF AΓ(KK) = (−0.19 ±0.15 (stat.) ±0.04 (syst.))% - 9.7 fb−1 √ s  = 1.96 TeV p p  collisions T.A. Aaltonen et al. (CDF Collab.), Phys.Rev. D90 (2014) 111103.
2021 LHCb ΔY(KK) = (−0.003 ±0.013 (stat.) ±0.003 (syst.))% - 9 fb−1 √ s  = 7+8+13 TeV pp collisions, prompt D* and B→D0μX R. Aaij et al. (LHCb Collab.), Phys.Rev.Lett. 118 (2017) 261803.
2008 BaBar ACP(KK) = (0.00 ±0.34 (stat.) ±0.13 (syst.))% 1.00 385.8 fb−1 near Υ(4S) resonance B. Aubert et al. (BABAR Collab.), Phys. Rev. Lett. 100, 061803 (2008).
2008 Belle ACP(KK) = (-0.43 ±0.30 (stat.) ±0.11 (syst.))% 1.00 540 fb−1 near Υ(4S) resonance M. Staric et al. (Belle Collab.), Phys. Lett. B670 (2008) 190.
2023 LHCb ACP(KK) = (+0.068 ±0.054 (stat.) ±0.016 (syst.))% 1.75 5.7 fb−1 √ s  = 13 TeV pp collisions, prompt D* R. Aaij et al. (LHCb Collab.), Phys. Rev. Lett. 131 (2023) 091802.
Fit Result
Agreement with no CP violation
CL = 0.49


Combination Plot: The combination plot shows the measurements listed in the Table above for ΔACP and AΓ, where the bands represent ±1σ intervals. The point of no CP violation (0,0) is shown as a filled circle, and two-dimensional 68% CL, 99.7% CL, and 99.99997% CL regions are plotted as ellipses with the best fit value as a cross indicating the one-dimensional uncertainties in their center. The second plot provides a zoomed-in view of the fit region.

 

 

From the fit, the change in χ2 from the minimum value for the no-CPV point (0,0) is 1.44; this corresponds to a CL of 0.49 for two degrees of freedom or 0.7 standard deviations. The central values and ± 1σ errors for the individual parameters are:

aCPind(KK) = (-0.001 ± 0.013 )%

aCPdir(KK) = (0.054 ± 0.060 )%
This page is maintained by M. Gersabeck and A. Schwartz and was last updated