Measurement of D0-D0bar mixing and search for CP violation in D0 ->K+K-, pi+pi- decays with the full Belle data set

We report an improved measurement of D0-D0bar mixing and a search for CP violation in D0 decays to CP-even final states K+K- and pi+pi-. The measurement is based on the final Belle data sample of 976/fb. The results are y_CP=(1.11 +- 0.22 +- 0.09)% and A_Gamma=(-0.03 +- 0.20 +- 0.07)%, where the first uncertainty is statistical and the second is systematic.


Introduction
Mixing of neutral mesons originates from a difference between mass and flavor eigenstates of the meson-antimeson system. For D 0 mesons, the mass eigenstates are usually expressed as |D 0 1,2 = p|D 0 ± q|D 0 (the sum for D 0 1 and the difference for D 0 2 ), with |p| 2 + |q| 2 = 1. The D 0 -D 0 mixing rate is characterized by two parameters: x = ∆m/Γ and y = ∆Γ/2Γ, where ∆m = m 2 − m 1 and ∆Γ = Γ 2 − Γ 1 are the differences in mass and decay width, respectively, between the mass eigenstates D 0 2 and D 0 1 , and Γ is the average D 0 decay width. If p = q, the mass eigenstates are also CP eigenstates; otherwise, D 0 1,2 are not CP eigenstates and CP violation arises in decays of D 0 mesons [1].
Mixing in D 0 decays to CP eigenstates, such as D 0 → K + K − , gives rise to an effective lifetime τ that differs from that in decays to flavor eigenstates such as D 0 → K − π + [2]. The observable is equal to the mixing parameter y if CP is conserved 1 . Otherwise, the effective lifetimes of D 0 and D 0 decaying to the same CP eigenstate differ and the asymmetry is non-zero. The observables y CP and A Γ are, in the absence of direct CP violation, related to the mixing parameters x and y as [2,3] y CP = 1 using phase convention CP |D 0 = −|D 0 1 2 (|q/p| + |p/q|)y cos φ − 1 2 (|q/p| − |p/q|)x sin φ and A Γ = 1 2 (|q/p|−|p/q|)y cos φ− 1 2 (|q/p|+|p/q|)x sin φ, where φ = arg(q/p).
The first evidence for D 0 -D 0 mixing was obtained in 2007 by Belle using D 0 → K + K − and D 0 → π + π − [4] and by BaBar using "wrong-sign" D 0 → K + π − decays [5]. These results were later confirmed with high precision by LHCb [6] and CDF [7]. The asymmetry A Γ has been measured by Belle [4], BaBar [8], CDF [9] and LHCb [10,11]. The measurements of y CP have been reported also by BaBar [8], LHCb [12] and BESIII [13]. Here, we report a new measurement of D 0 → K + K − , π + π − decays using almost twice as much data as in Ref. [4] and an improved analysis method. The resolution function now accounts for a dependence upon polar angle and different configurations of the silicon vertex detector (see below).

Event selection
The measurement is based on the final data set of 976 fb −1 recorded by the Belle detector [14] at the KEKB asymmetric-energy e + e − collider [15], which operated primarily at the center-of-mass energy of the Υ(4S) resonance, and 60 MeV below. A fraction of the data was recorded at the Υ(1S), Υ(2S), Υ(3S), and Υ(5S) resonances; these data are included in the measurement. The Belle detector is described in detail elsewhere [14]. It includes a silicon vertex detector (SVD), a central drift chamber (CDC), an array of aerogel Cherenkov counters, and time-of-flight scintillation counters. Two different SVD configurations were used: a 3-layer configuration for the first 153 fb −1 of data and a 4-layer configuration [16] for the remaining 823 fb −1 of data.
The decays D 0 → K + K − , D 0 → π + π − and D 0 → K − π + are reconstructed in the decay chain D * + → D 0 π + , where the charge of the D * -daughter pion (which has low momentum and thus is referred to as "slow") is used to tag the initial flavor of the D 0 meson. 2 Each final-state charged particle is required to have at least two associated SVD hits in each of the longitudinal and azimuthal measuring coordinates. To select pion and kaon candidates, we impose particle identification criteria based on energy deposition in the CDC, the track time of flight, and information from the aerogel Cherenkov counters [17]. The identification efficiencies and the misidentification probabilities are about 85% and 9%, respectively, for the D 0 daughters, and about 99% and 2%, respectively, for the slow pion from D * + decay. The D 0 daughters are refitted to a common vertex. The D 0 production vertex is determined as the intersection of the D 0 trajectory with that of the slow pion, subject to the constraint that they both originate from the e + e − interaction region. Confidence levels exceeding 10 −3 are required for both fits. To reject D mesons produced in B-meson decays and also to suppress combinatorial background, the D * + momentum in the e + e − center-of-mass system (CMS) is required to satisfy p * D > 2.5 GeV/c for the data taken below the Υ(5S) resonance and p * D > 3.1 GeV/c for the Υ(5S) data. We select D 0 candidates using two kinematic variables: the invariant mass M of the D 0 and the energy released in the D * + decay q = (M D * − M − m π )c 2 , where M D * is the invariant mass of the D * + decay products and m π is the mass of the charged pion. The proper decay time of the D 0 candidate is calculated from the projection of the vector joining the two vertices, L, onto the D 0 momentum vector p: [18]. The proper decay time uncertainty σ t of the candidate D 0 is evaluated from the error matrices of the production and decay vertices.
The samples of events for the lifetime measurements are selected using variables ∆M ≡ M − m D 0 , ∆q = q − q 0 , and σ t , where q 0 is the nominal energy released in the D * + decay (5.86 MeV). These selection criteria are optimized using Monte Carlo (MC) simulation by minimizing the statistical uncertainty on y CP . The simulation is based on EvtGen [19] and Pythia generators [20]; simulated events were processed through a full Belle detector simulation us-ing Geant 3 [21] and Fluka [22] to simulate hadronic interactions. The optimization gives the following selection criteria: |∆M | < 2.25σ M for all events, where σ M is the r.m.s. width of the D 0 invariant mass peak; |∆q| < 0.66 MeV and σ t < 440 fs for the 3-layer SVD configuration; and |∆q| < 0.82 MeV and σ t < 370 fs for the 4-layer SVD configuration. The D 0 peak, shown in Fig. 1, is not purely Gaussian in shape. In addition, the width σ M depends on the decay mode and on the SVD configuration.
Background is estimated from sidebands in M . The sideband position is optimized using MC simulation in order to minimize systematic uncertainties arising from small differences between the decay time distribution of events in the sideband and that of background events in the signal region. The sideband windows are shown in Fig. 1. The yields of selected events are 242 × 10 3 K + K − , 114 × 10 3 π + π − , and 2.61 × 10 6 K − π + , with signal purities of 98.0%, 92.9% and 99.7%, respectively. The dominant background is combinatorial.

Lifetime fit
The measurement is performed by doing a simultaneous binned maximum likelihood fit to five data samples: and the sum of D 0 → K − π + and D 0 → K + π − . The proper decay time distribution is parameterized as where τ is the effective lifetime, N is the signal yield, R(t) is a resolution function, and B(t) is the background contribution that is fixed from a fit to the sideband distribution. The decay time acceptance is studied with MC simulations and found to be constant to good precision within the selected range. The construction of the resolution function is similar to that of our previous analysis [4] but improved to take into account a possible shape asymmetry and D 0 polar angle dependence. It is constructed using a normalized distribution of σ t : for each σ t bin, a common-mean double-or triple-Gaussian probability density function is constructed. The fractions w k and widths σ pull constructed and generated proper decay times, respectively, of simulated D 0 decays. The resolution function is where G(t; µ i , σ ik ) is a Gaussian distribution of mean µ i and width σ ik ; f i is the fraction of events in the i-th bin of the σ t distribution; the index k runs over the number of Gaussians n g used for bin i; and the index i runs over the number of σ t bins. The means and widths of the Gaussians are parameterized as where t 0 is a resolution function offset, a is a parameter to model a possible asymmetry of the resolution function, σ i is the bin central value, σ t is the mean of the σ t distribution, and s k is a width-scaling factor. The parameters s k , t 0 and a, in addition to N and τ , are free parameters in the fit. To construct R(t) with Eq. 4, a sideband-subtracted σ t distribution is used.
From studies of the proper decay time distribution of D 0 → K − π + decays, we observe a significant dependence of its mean value on cos θ * (see Fig. 2), where θ * is the polar angle of D 0 in CMS with respect to the direction of e + . From MC studies, we find that this effect is due to a small misalignment of the SVD detector. The effect can be corrected for when fitting for the lifetime by allowing the resolution function offset t 0 to vary with cos θ * . We thus measure y CP and A Γ in bins of cos θ * , with the resolution function calculated separately for each bin. An additional requirement | cos θ * | < 0.9 is imposed to suppress events with large offsets (about 1% of events).
The background term in Eq. 3 is parameterized as the sum of a component with zero lifetime and a component with an effective lifetime τ b : The resolution function R b (t) is also parameterized with Eq. 4 except that, for each σ t bin, the function is taken to be symmetric (a = 0) and always composed of three Gaussians, with the second and third scaling factors being equal (s 2 = s 3 ). The σ t distribution is taken from an M sideband. The fraction p of the zero-lifetime component is found to be cos θ * -dependent; its value is fixed in each bin using MC simulation. The parameters t 0 , s 1 , s 2 and τ b are determined separately for each decay mode and SVD configuration from a fit to sideband distributions summed over cos θ * bins. However, the background shape is still cos θ * dependent, because the σ t distribution, the zero-lifetime fraction p and the yield N b all depend on cos θ * . The quality of these fits exceeds 15% confidence level (CL).
To extract y CP and A Γ , the decay modes are fitted simultaneously in each cos θ * bin and separately for each of the two SVD configurations. The parameters shared among the decay modes are y CP and A Γ (between KK and ππ), t 0 and a (among all decay modes), and parameters s 1 , s 2 and s 3 , up to an overall scaling factor. Results for individual cos θ * bins and for the two data sets are combined into an overall result via a least-squares fit to a constant.
The fitting procedure is tested with a generic MC sample equivalent to six times the data statistics. The fitted y CP and A Γ are consistent with the input zero value, and the fitted Kπ lifetime is consistent with the generated value. Linearity tests performed with MC-simulated events re-weighted to reflect different y CP and A Γ values show no bias.
The fitting procedure is then applied to the measured data. The fitted proper decay time distributions summed over cos θ * bins and running periods with the two SVD configurations are shown in Fig. 3. The pulls, plotted beneath each fitted distribution, show no significant structure. The normalized χ 2 is 1.13. 3 The confidence levels of individual fits in bins of cos θ * are above 5%, except for one with CL=3.3%, and are distributed uniformly.
The fitted values of y CP and A Γ in bins of cos θ * are shown in Figs. 4 and 5. The values obtained with a least-squares fit to a constant are y CP = (1.11 ± 0.22)% and A Γ = (−0.03 ± 0.20)%, where the uncertainties are statistical only; the confidence levels are 32% and 40%, respectively. The fitted D 0 lifetime is (408.46 ± 0.54) fs (statistical uncertainty only), which is consistent with the current world average of (410.1 ± 1.5) fs [18].

Systematic uncertainties
The estimated systematic uncertainties are listed in Table 1. The impact of imperfect SVD alignment is studied with a dedicated signal MC simulation in which different local and global SVD misalignments are modeled. Local misalignment refers to a 3 We use Pearson's definition of χ 2 and take only the bins with the fitted function greater than one. random translation and rotation of each individual silicon strip detector according to the alignment precision, while global misalignment refers to a translation and rotation of the entire SVD with respect to the CDC. The systematic uncertainties are taken to be the r.m.s. of the differences between these results and the nominal result that assumes perfect SVD alignment. We obtain 0.060% for y CP and 0.041% for A Γ .
The uncertainty due to the position of the mass window is estimated by varying the position of the window by the small differences found between MC simulation and data in the position of the D 0 mass peak, about ±1 MeV/c 2 . This resulting uncertainty is relatively small: 0.007% for y CP and 0.009% for A Γ . Background contributes to the systematic uncertainty in two ways: statistical fluctuations of sideband distributions and modeling. The former is found to contribute 0.051% for y CP and 0.050% for A Γ . The latter arises from modeling the background distribution with that of sideband events; this uncertainty is estimated from MC simulation to be 0.029% for y CP and 0.007% for A Γ . Combining the two contributions in quadrature gives total uncertainties of 0.059% for y CP and 0.050% for A Γ .
Systematics due to the resolution function are estimated using two alternative parameterizations in the fit: one in which the parameter a in Eq. 5 is fixed to zero, and the other in which this parameter is floated but not shared among different decay modes. We find variations of 0.030% for y CP and 0.002% for A Γ . Systematics due to binning are estimated by varying the number of bins in cos θ * and t. This contribution is found to be 0.021% for y CP and 0.010% for A Γ .
Possible acceptance variations with decay time are tested by fitting decay time distributions of MC events that pass the selection criteria. We always recover the generated lifetimes, for all decay modes, indicating uniform acceptance. We conclude that this effect is negligible. All individual contributions are added in quadrature to obtain overall systematic uncertainties of 0.09% for y CP and 0.07% for A Γ .

Conclusions
Using the final Belle data set, we measure the difference from unity of the ratio of lifetimes of D 0 mesons decaying to CP -even eigenstates K + K − , π + π − and to the flavor eigenstate K − π + . Our result is y CP = [+1.11 ± 0.22 (stat.) ± 0.09 (syst.)]% . (7) The significance of this measurement is 4.7σ when both statistical and systematic uncertainties are combined in quadrature. We also search for CP violation, measuring a CP asymmetry A Γ = [−0.03 ± 0.20 (stat.) ± 0.07 (syst.)]% .
This value is consistent with zero. These results are significantly more precise than our previous results [4] and supersede them. They are compatible with results from other experiments [8][9][10][11][12][13] and the world average values [23].