Measurement of D0-anti-D0 Mixing using the Ratio of Lifetimes for the Decays D0 -->K-pi+, K-K+, and pi-pi+

We present a measurement of $D^0$-$\bar{D^0}$ mixing parameters using the ratios of lifetimes extracted from a sample of $D^0$ mesons produced through the process $D^{*+}\to\D^0\pi^+$, that decay to $K^{+}\pi^{-}$, K^{-}K^{+}$, or $\pi^-\pi^+$. The Cabibbo-suppressed modes $K^{-}K^{+}$ and $\pi^-\pi^+$ are compared to the Cabibbo-favored mode $K^{+}\pi^{-}$ to obtain a measurement of $y_{CP}$, which in the limit of CP conservation corresponds to the mixing parameter $y$. The analysis is based on a data sample of 384 $fb^{-1}$ collected by the $BaBar$ detector at the PEP-II asymmetric-energy $e^+e^-$ collider. We obtain $y_{CP} = [1.24\pm 0.39\mathrm{(stat)} \pm 0.13\mathrm{(syst)}]%$, which is evidence of $D^0$-$\bar{D^0}$ mixing at the $3\sigma$ level, and $\Delta Y = [-0.26\pm 0.36\mathrm{(stat)} \pm 0.08 \mathrm{(syst)}]%$, where $\Delta Y$ constrains possible CP violation. Combining this result with a previous $BaBar$ measurement of $y_{CP}$ obtained from a separate sample of $D^0\to K^{-}K^{+}$ events, we obtain $y_{CP} = [1.03\pm 0.33 \mathrm{(stat)} \pm 0.19 \mathrm{(syst)}]%$.

Several recent studies have shown evidence for mixing in the D 0 -D 0 system at the 1% level [1,2,3]. This is consistent with Standard Model (SM) expectations [4] and provides strong constraints for new physics models [5]. One consequence of D 0 -D 0 mixing is that the D 0 decay time distribution can be different for decays to different CP eigenstates [6]. An observation of CP violation in D 0 -D 0 mixing with the present experimental sensitivity would provide evidence for physics beyond the SM [7]. We present a measurement of this lifetime difference and results of a search for evidence of CP violation in D 0 -D 0 mixing.
The two neutral D mass eigenstates |D 1 and |D 2 can be represented as where |p| 2 + |q| 2 = 1. We characterize the rate of D 0 -D 0 mixing with the parameters x ≡ ∆m/Γ and y ≡ ∆Γ/2Γ, where ∆m = m 1 − m 2 and ∆Γ = Γ 1 − Γ 2 are the differences between the mass and width eigenvalues of the states in Eq. (1), respectively, and Γ = (Γ 1 + Γ 2 )/2 is the average width. If either x or y is non-zero, mixing will occur.
The effects of CP violation in D 0 -D 0 mixing can be parameterized in terms of the quantities where A f ≡ f |H D |D 0 (A f ≡ f |H D |D 0 ) is the amplitude for D 0 (D 0 ) to decay into a final state f , and H D is the Hamiltonian for the decay. A value of r m = 1 would indicate CP violation in mixing. A non-zero value of ϕ f would indicate CP violation in the interference between mixing and decay. Within the SM, CP violation in decay is expected to be small in the D 0 -D 0 system [8] and is considered elsewhere [9]. D 0 -D 0 mixing will alter the decay time distribution of D 0 and D 0 mesons that decay into final states of specific CP . To a good approximation, these decay time distributions can be treated as exponential with effective lifetimes τ + hh and τ − hh , given by [8] where τ Kπ is the lifetime for the Cabibbo-favored decays D 0 → K − π + and D 0 → K + π − , and τ + hh (τ − hh ) is the lifetime for the Cabibbo-suppressed decays of the D 0 (D 0 ) into CP -even final states (such as K − K + and π − π + ). These effective lifetimes can be combined into the quantities y CP and ‡ Now at Tel Aviv University, Tel Aviv, 69978, Israel § Also with Università di Perugia, Dipartimento di Fisica, Perugia, Italy ¶ Also with Universita' di Sassari, Sassari, Italy ∆Y : where τ hh = (τ + hh + τ − hh )/2 and A τ = (τ + hh − τ − hh )/(τ + hh + τ − hh ). Both y CP and ∆Y are zero if there is no D 0 -D 0 mixing. In the limit of CP conservation, y CP = y and ∆Y = 0, with the convention that cos ϕ f > 0.
We measure the D 0 lifetime in the three different D 0 decay modes, K − π + , K − K + , and π − π + . We use D 0 mesons coming from D * + → D 0 π + decays [10]; the requirement of a D * + parent strongly suppresses the backgrounds. We use the charge of the D * ± to split the K − K + and π − π + samples into those originating from D 0 and from D 0 mesons for measuring the CP -violating parameters. To avoid potential bias, we finalize our data selection criteria, the procedures for fitting and for extracting the statistical limits, and determine the systematic errors, prior to examining the mixing results.
Most systematic errors related to signal events are expected to cancel in the lifetime ratios. Background events can contain effects that differ in each decay mode, making them difficult to characterize. Therefore, the event selection is chosen to produce very pure samples. The decay time distribution of signal candidates is fit to an exponential convolved with a resolution function that uses event-by-event decay time errors. The decay-time resolution parameters are allowed to vary in the fit. Residual background components are modeled using Monte Carlo (MC) simulated events and control samples obtained from the data.
We use 384 fb −1 of e + e − colliding-beam data recorded near √ s = 10.6 GeV with the BABAR detector [11] at the PEP-II asymmetric-energy storage rings. We begin by reconstructing candidate D 0 decays into the final states K − π + , π − π + , and K − K + . We require tracks to satisfy particle identification criteria based upon dE/dx ionization energy loss and Cherenkov angle measurements. We fit pairs of tracks with the appropriate mass hypotheses to a common vertex. We require the invariant mass of a candidate track pair to be within the range 1.78-1.94 GeV/c 2 . To further reduce backgrounds, we require the helicity angle θ H , defined as the angle between the positively charged track in the D 0 rest frame and the D 0 direction in the lab frame, to satisfy | cos θ H | < 0.7. This is particularly helpful for rejecting combinatorial background, especially in the π − π + mode.
We reconstruct D * + candidates by combining a D 0 candidate with a slow pion track (denoted π + s ), requiring them to originate from a common vertex constrained to the e + e − interaction region. We require the π + s momentum to be greater than 0.1 GeV/c in the laboratory frame and less than 0.45 GeV/c in the e + e − center-of-mass (CM) frame. We perform a vertex-constrained combined fit to the D 0 production and decay vertices, requiring the χ 2 -based probability, P (χ 2 ), to be at least 0.1%. The decay time t and its estimated uncertainty σ t for each D 0 candidate are determined by this fit. We reject slow electrons that fake π + s candidates using dE/dx measurements in the tracking volume and further veto any π + s candidate that may have originated from a reconstructed gamma conversion or π 0 Dalitz decay. To reduce combinatorial backgrounds from D 0 's produced via B-meson decay we require each D 0 to have a momentum in the CM frame greater than 2.5 GeV/c. We also require −2 < t < 4 ps and σ t < 0.5 ps. The most probable value of σ t for signal events is 0.16 ps. For cases where multiple D * ± candidates in an event share one or more tracks, we retain only the candidate with the highest P (χ 2 ).
The distribution of the difference in the reconstructed D * + and D 0 masses (δm) peaks near 145.4 MeV/c 2 . Backgrounds are suppressed by discarding D * + candidates with a value of δm deviating more than 0.8 MeV/c 2 from the peak. Invariant mass distributions for the selected D 0 candidates are shown in Fig. 1. For the lifetime fit, we only use events within 15 MeV/c 2 of the D 0 signal peak (shaded regions in Fig 1); the event yields and purity within this signal region are also given. GeV/c 2 in the π − π + decay mode, are mainly due to candidates with misidentified kaons or pions.) Also shown are the yield and purity of the three D 0 samples as calculated inside the ±15 MeV/c 2 mass window.
The D 0 lifetime is determined from an unbinned maximum likelihood fit to the reconstructed decay time and its estimated error for events in the signal region. The fit is performed simultaneously to all five decay samples ( . The D 0 candidates in the signal region can be divided into three components: D 0 signal events, combinatorial background, and mis-reconstructed charm events. Each component is described by its own probability density function (PDF) which also depends upon the D 0 or D 0 decay mode.
The measured decay-time distribution of signal events is described by an exponential convolved with a resolution function. The resolution function is the sum of three Gaussian functions with widths proportional to σ t . The three Gaussian functions share a common mean which is allowed to be offset from zero in order to take detector misalignment effects into account. The effect of the offset is studied as part of the crosschecks and taken into account as a systematic uncertainty. The resolution function parameters are all permitted to vary in the fit. Up to an overall scale factor in the width, the resolution function is observed to have the same shape for all modes, including the offset. To account for the small (1.5%) differences in width, we introduce two parameters S K − K + and S π − π + to scale the overall width of the K − K + and π − π + resolution functions relative to the width of the K − π + resolution function. All other resolution function parameters are shared among the different modes and are determined by a simultaneous fit to all modes together.
About 0.4% of the D 0 signal in the K − K + and π − π + modes consists of a correctly reconstructed D 0 combined with an unrelated π s ; this is estimated from MC and verified in data. These candidates have the same resolution and lifetime behavior as those from correctly reconstructed D * + decays, but about half of them will be tagged as the wrong flavor. We therefore include a 0.2% component in the signal PDF that uses the lifetime of the opposite flavor state.
The decay-time distribution of the combinatorial background is described by a sum of a Gaussian and a modified Gaussian with a power-law tail to account for a small number of events with large reconstructed lifetimes. The means of these functions are allowed to float in the fit. Each of the three decay modes has its own shape for the combinatorial background. These shapes are determined from fits to the events in the sideband region defined by 1.89 < M hh < 1.92 GeV/c 2 and 0.151 < δm < 0.159 GeV/c 2 . We determine the amount of combinatorial background using MC samples scaled to the same luminosity as the data, modeling all known, relevant physics processes. The fraction of combinatorial background in the K − π + mode is estimated to be (0.032 ± 0.003)%, in the K − K + mode (0.16 ± 0.02)%, and in the π − π + mode (1.8 ± 0.2)%. The uncertainties are determined by comparing data and MC events in the (M hh , δm) sideband where the combinatorial background is dominant.
Mis-reconstructed charm background events have one or more of the charm decay products either not reconstructed or reconstructed with the wrong particle hypothesis. Most are D 0 mesons from a D * + → D 0 π s decay with a correctly reconstructed π s . For the K − π + mode, most of the charm background is semileptonic decays D 0 → K − ℓ + ν with the charged lepton misidentified as a pion. The semileptonic decays also contribute to the K − K + final state, but the dominant contribution is from D 0 → K − π + π 0 in which the π 0 is not reconstructed and the π + is misidentified as a kaon. There is also a small contribution from D + → K − π + π + decays. In the π − π + mode, the charm background is almost exclusively due to mis-reconstructed D 0 → K − π + decays in which the kaon has been misidentified as a pion. The decaytime distributions of the charm backgrounds are described by an exponential convolved with a Gaussian. The parameters are fixed to values obtained in a fit to MC events. The fraction of charm background events in the signal region is estimated from MC simulation and crosschecked by comparing data and MC events in a (M hh , δm) sideband region defined by 1.78 < M hh < 1.80 GeV/c 2 and 0.14 < δm < 0.16 GeV/c 2 , where the charm background is the dominant contribution. We estimate the charm background to be (0.009 ± 0.002)% of events in the signal region for K − π + , (0.2 ± 0.1)% for K − K + , and (0.15 ± 0.15)% for π − π + .
The results of the lifetime fits are shown in Fig. 2. The fitted D 0 lifetime τ Kπ is found to be 409.33 ± 0.70 (stat) fs, consistent with the world-average lifetime [12]. From the fit results we calculate y CP and ∆Y for the K − K + mode, the π − π + mode, and the two modes combined, taking into account any correlations between the fitted lifetimes. The dominant correlation of 11% arises primarily because the decaytime resolution offset is shared between the decay modes. The y CP and ∆Y results are listed in Table I. The combined result is obtained by fitting the data with common lifetimes for the K − K + and π − π + modes, and assuming the same value of ϕ f for the K − K + and π − π + decay modes. Sample yCP ∆Y K − K + (1.60 ± 0.46 ± 0.17)% (−0.40 ± 0.44 ± 0.12)% π − π + (0.46 ± 0.65 ± 0.25)% ( 0.05 ± 0.64 ± 0.32)% Combined (1.24 ± 0.39 ± 0.13)% (−0.26 ± 0.36 ± 0.08)% Various cross-checks have been performed to ensure that the fit is unbiased and the assumptions in the fit model are well-founded. An offset in the resolution function is measured in the fit to be −4.75 ± 0.51 fs. This offset was seen in our recent K − π + mixing analysis [1] and has also been observed in other BABAR measurements of charm decays. Because we measure ratios of lifetimes, the presence of a common offset has minimal impact on the values y CP and ∆Y . However, differences in the offset between the three decay modes, or between the D 0 and D 0 , could introduce a bias. No resolution offset is found in the MC samples. However, we are able to introduce offsets in the fits to the MC sample of up to twice the size of the offset in data by misaligning the Silicon Vertex Tracker (SVT). In all cases the offsets are found to be consistent between all modes.
The fitting procedure has been validated with generic MC samples weighted to the luminosity of the data sample and with dedicated signal MC samples. The signal efficiency is found to be independent of the true decay time and the fitted lifetimes are consistent with the generated value.
The assumption that the resolution function is the same for all decay modes except for a scale factor is tested by fitting each sample independently. This gives mixing parameters and resolution offsets consistent with the nominal fit, but with significantly larger statistical uncertainties. The lifetime has also been extracted in independent fits to the flavor-separated samples of D 0 → K − π + and D 0 → K + π − decays. The fitted lifetimes and resolution functions in these two samples are consistent with each other. To cross-check the effect of the resolution offset, we performed further studies by dividing the data sample into subsamples with different sensitivities to detector effects and fitting each subsample independently. Besides the D * tagged samples used for this mixing measurement, we also use a control sample of D 0 → K − π + decays where the D 0 is not required to come from a D * decay. This untagged sample has about five times as many D 0 decays as the D * tagged samples combined, allowing us to divide the sample more finely. The quantities used to divide the data into subsamples for these tests include the run period, the azimuthal and polar angle of the D 0 meson, and the orientation of the D 0 decay plane with respect to the X-Y (bending) plane of the detector. In all of the variables mentioned, the resolution offset is observed to have a large variation (typically between −10 fs and 0 fs), but the fitted lifetimes are consistent among samples. Furthermore, the weighted average of the mixing parameters from the sub-divided data samples is in almost all cases nearly identical to that obtained by fitting the full data sample with one common lifetime and resolution function as described previously. The largest variation is observed with the polar angle of the D 0 meson in the laboratory frame, where decays perpendicular to the beam line are found to have almost no resolution offset, while decays into the forward region of the detector have a large offset. Since the acceptance for D 0 → K − K + decays is lower in the forward region than for D 0 → K − π + or D 0 → π − π + decays, the polar angle dependence in the offset could potentially introduce a different average offset for each of the three modes. This is accounted for in the systematic errors.
The systematic uncertainties on the mixing parameters are small since most uncertainties in the lifetimes cancel in the ratios. We have considered variations in the signal and background fit models, changes to the event selection and detector effects that could introduce biases in the lifetime. Table II summarizes the various systematic uncertainties. The evaluation of each of these is described below. The systematic uncertainty on y CP and ∆Y averaged over the two CP modes is occasionally smaller than the individual uncertainties because of anti-correlations.
We vary the signal PDF shape, and the size and position of the signal region. As part of the PDF shape variations, we perform a fit without a resolution offset. The effect of the polar angle dependence in the resolution offset is evaluated by performing the fit with separate, floating offsets in seven bins of polar angle, but sharing all other resolution parameters and lifetimes across all polar angle bins. The difference in the mixing parameters between this fit and the nominal fit is found to be small (< 0.02%). The largest systematic contribution to y KK CP (0.12%) is due to widening the signal region mass interval from 15 to 25 MeV/c 2 . The choice of signal region determines the level of mis-reconstructed signal events included in the fit.
The mis-reconstructed charm background is a very small component in the lifetime fit and is determined using MC events. Varying the charm background fraction (depending on the mode) and the effective lifetime, both within their associated uncertainties, yields a minor contribution to the systematic uncertainty.
Because of the high purity, the results have little sensitivity to the modeling of the combinatorial background, except in the π − π + mode where varying the fraction of combinatorial background by 10% yields a systematic uncertainty in y ππ CP of 0.14%. We also alter the fit procedure by using a different sideband region and by substituting the MC decay time distribution for that obtained from fitting the data. Neither variation contributes a large systematic uncertainty.
We have studied the effect of varying the event selection criteria, which could potentially affect the lifetime measurement. Changing the treatment of events where multiple D * + candidates share one or more tracks (either keeping all of them or throwing them all out) has little effect, while changing the upper bound on the decay time uncertainty from 0.5 to 0.4 ps yields the largest systematic uncertainty on y ππ CP of 0.172%. As with the D 0 mass window, the choice of the σ t range af- II: Summary of systematic uncertainties on yCP and ∆Y , separately for K − K + and π − π + and averaged over the two CP modes, in percent. To evaluate the effect of possible misalignments in the SVT on the mixing parameters, signal MC events are reconstructed with different alignment parameters, and the analysis is repeated. The misalignments introduce resolution offsets in the MC of up to 10 fs and the corresponding fitted lifetimes change by up to 3 fs. However, since the lifetimes of all decay modes change by similar amounts, the effect on y CP and ∆Y is small. We also changed the energy loss correction applied in the tracking by 20% since a previous analysis has shown that the energy loss is underestimated in the reconstruction of data events [13]. This changes the fitted lifetimes by about 0.5 fs but has little effect on the mixing parameters.
We combine the results shown in table I, with those from a previous BABAR study [14], based on 91 fb −1 of data, that does not require a D * + parent to identify the D 0 decays. While use of these untagged D 0 decays increases the sensitivity to y CP through a factor of five increase in statistics, it also introduces different background behavior and therefore different systematic errors. We have not used these untagged events in the current analysis, and thus the untagged data sample of the earlier analysis is essentially disjoint and its results statistically independent. Systematic uncertainties in the previous analysis were dominated by the limited number of simulated events. Since the MC samples in the present study are presented here are entirely independent, this uncertainty is not correlated with those on the new results. Conservatively assuming the remaining systematic uncertainties to be 100% correlated, we combine the two results using the BLUE method [15] and obtain y CP = [1.03 ± 0.33(stat) ± 0.19(syst))]%.
In summary, we have obtained a value of y CP = [1.24 ± 0.39(stat) ± 0.13(syst))]% which is evidence of D 0 -D 0 mixing at the 3σ level. It is compatible with our previous result [14] and the recent lifetime ratio measurement from Belle of y CP = [1.31 ± 0.32(stat) ± 0.25(syst)]% [2]. We find no evidence for CP violation and determine ∆Y to be [−0.26 ± 0.36(stat) ± 0.08(syst)]%. The result is consistent with SM estimates for mixing.
We are grateful for the extraordinary contributions of our PEP-II colleagues in achieving the excellent luminosity and machine conditions that have made this work possible. The success of this project also relies critically on the expertise and dedication of the computing organizations that support BABAR. The collaborating institutions wish to thank SLAC for its support and the kind hospitality extended to them. This work is supported by the US Department of Energy and National Science Foundation, the Natural Sciences and Engineering Research Council (Canada), the Commissariatà l'Energie Atomique and Institut National de Physique Nucléaire et de Physique des Particules (France), the Bundesministerium für Bildung und Forschung and Deutsche Forschungsgemeinschaft (Germany), the Istituto Nazionale di Fisica Nucleare (Italy), the Foundation for Fundamental Research on Matter (The Netherlands), the Research Council of Norway, the Ministry of Education and Science of the Russian Federation, Ministerio de Educación y Ciencia (Spain), and the Science and Technology Facilities Council (United Kingdom). Individuals have received support from the Marie-Curie IEF program (European Union) and the A. P. Sloan Foundation.