# Combination of Direct and Indirect CP Violation (updated 12 April 2012)

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(K+K) − ACP+π),

where ACP are time-integrated CP asymmetries. The underlying theoretical parameters are:
(e.g., see Grossman, Kagan, and Nir, Phys. Rev. D75 (2007) 036008 )

aCPdir   ≡   (|AD0→f |2 − |A D0→f |2) / (|AD0→f |2 + |A D0→f |2) ,

aCPind   ≡   (1/2) [ (|q/p| + |p/q|) x sin φ   −   (|q/p| - |p/q|) y cos φ ] .

We use the following relations between the observables and underlying parameters:
(from Gersabeck et al., J. Phys. G 39 (2012) 045005)

AΓ   =   − aCPind − aCPdir yCP

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

≈   ΔaCPdir (1 + yCP 〈t〉/τ )   +   aCPind Δ〈t〉/τ .

The first relation constrains mostly aCPind (indirect CP violation), and aCPdir (the direct CP violation contribution) can differ for different final states. In the second relation, 〈t〉/τ denotes the mean decay time in units of the D0 lifetime; ΔX denotes the difference in quantity X between K+K and π+π final states; and X denotes the average for quantity X. We neglect the last term in this relation as all three factors are O(10−2) or smaller, and thus this term is negligible with respect to the other two terms. [Note that Δ〈t〉/τ « 〈t〉/τ, and it is expected that aCPdir < ΔaCPdir because aCPdir(K+K) and aCPdir+π) are expected to have opposite signs.]

A χ2 fit is performed in the plane ΔaCPdir vs. aCPind. For the BaBar result the difference of the quoted values for ACP(K+K) and ACP+π) 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. We use the current world average value yCP = (1.064 ± 0.209)% and the measurements listed in the table below.

Year Experiment Results Δ〈t〉/τ 〈t〉 Comment Reference
2007 Belle AΓ = (0.01 ±0.30 (stat.) ±0.15 (syst.))% - - 540 fb−1 near Υ(4S) resonance M. Staric et al. (Belle Collab.), Phys. Rev. Lett. 98, 211803 (2007).
2008 BaBar AΓ = (0.26 ±0.36 (stat.) ±0.08 (syst.))% - - 384 fb−1 near Υ(4S) resonance B. Aubert et al. (BaBar Collab.), Phys. Rev. D 78, 011105(R) (2008).
2011 LHCb AΓ = (−0.59 ±0.59 (stat.) ±0.21 (syst.))% - - 28 pb−1 s  = 7 TeV pp collisions R. Aaij et al. (LHCb Collab.), arXiv:1112.4698 (submitted to JHEP).
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).
2008 Belle ΔACP = (−0.86 ±0.60 (stat.) ±0.07 (syst.))% 0.00 1.00 540 fb−1 near Υ(4S) resonance M. Staric et al. (BELLE Collab.), Phys. Lett. B 670, 190 2008).
2011 LHCb ΔACP = (−0.82 ±0.21 (stat.) ±0.11 (syst.))% 0.10 2.08 0.62 fb−1 s  = 7 TeV pp collisions R. Aaij et al. (LHCb Collab.), arXiv:1112.0938 (accepted by PRL).
2012 CDF Prelim. ΔACP = (−0.62 ±0.21 (stat.) ±0.10 (syst.))% 0.25 2.58 9.7 fb−1 s  = 1.96 TeV p p  collisions The CDF Collaboration, CDF Note 10784.
Fit Result
 Agreement with no CP violation CL = 6.1x10−5

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, 95% CL, and 99.7% CL regions are plotted as ellipses with the best fit value as a cross indicating the one-dimensional uncertainties in their center.

From the fit, the change in χ2 from the minimum value for the no-CPV point (0,0) is 19.4; this corresponds to a CL of 6.1x10−5 for two degrees of freedom. Thus the data is consistent with no CP violation at 0.006% CL. The central values and ± 1σ errors for the individual parameters are:

aCPind = (-0.025 ± 0.231 )%

ΔaCPdir = (−0.656 ± 0.154 )%

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