Measurement of the charm-mixing parameter y CP in D 0 → K 0 S ω decays at Belle

We report the first measurement of the charm-mixing parameter y CP in D 0 decays to the CP -odd final state K 0 S ω . The study uses the full Belle e þ e − annihilation data sample of 976 fb − 1 taken at or near the ϒ ð 4 S Þ centre-of-mass energy. We find y CP ¼ ð 0 . 96 (cid:2) 0 . 91 (cid:2) 0 . 62 þ 0 . 17 − 0 . 00 Þ % , where the first uncertainty is statistical, the second is systematic due to event selection and background, and the last is due to possible presence of CP -even decays in the data sample.

We report the first measurement of the charm-mixing parameter y CP in D 0 decays to the CP-odd final state K 0 S ω. The study uses the full Belle e þ e − annihilation data sample of 976 fb −1 taken at or near the ϒð4SÞ centre-of-mass energy. We find y CP ¼ ð0.96 AE 0.91 AE 0.62 þ0.17 −0.00 Þ%, where the first uncertainty is statistical, the second is systematic due to event selection and background, and the last is due to possible presence of CP-even decays in the data sample. DOI: 10.1103/PhysRevD.102.071102 In systems of neutral mesons and antimesons, flavorchanging weak interactions induce mixing. The mixing phenomenon originates due to the difference between mass and flavor eigenstates and has been observed in the K 0 −K 0 , B 0 ðd;sÞ −B 0 ðd;sÞ , and D 0 −D 0 systems [1]. In the latter case, the mass eigenstates jD 1;2 i with masses m 1;2 and widths Γ 1;2 can be expressed as linear combinations of the flavor eigenstates, with jpj 2 þ jqj 2 ¼ 1. The mixing rate is characterized by two parameters: x ¼ Δm=Γ and y ¼ ΔΓ=2Γ. Here Δm ¼ m 2 − m 1 and ΔΓ ¼ Γ 2 − Γ 1 are the differences in mass and decay width, respectively, and Γ ¼ ðΓ 2 þ Γ 1 Þ=2 is the average decay width of the two mass eigenstates. If CP is conserved, p ¼ q ¼ 1= ffiffi ffi 2 p , and the mass eigenstates jD 1;2 i coincide with CP-odd ðD − Þ and -even ðD þ Þ states, respectively. Here the phase convention is chosen such that For small values of the mixing parameters, jxj, jyj ≪ 1, the decay-time dependence of initially produced D 0 andD 0 mesons decaying to a CP eigenstate is approximately exponential. The effective lifetime here differs from that in decays to flavor eigenstates such as D 0 → K − π þ [2]. Summing D 0 andD 0 decays, the time-dependent decay rate to a CP eigenstate can be written as where η f ¼ þ1ð−1Þ for CP-even (-odd) final states. Neglecting possible CP violation in decays, y CP is related to x and y as where ϕ ¼ argðq=pÞ. In the limit of CP conservation ðjq=pj ¼ 1; ϕ ¼ 0Þ, y CP ¼ y. Note that y CP also depends on CP violation in decay, making the difference in y CP between CP-even and -odd final states sensitive to CP violation in decay [3]. The most precise measurement of y CP has been performed with decays to CP-even final states K þ K − and π þ π − [4][5][6]. A mixing search in CP-odd decays was also performed by Belle using 673 fb −1 data in D 0 → K 0 S K þ K − [7] by comparing the effective lifetimes in CP-even and -odd components of this final state and assuming jq=pj ¼ 1. The current world average value of y CP is ð0.715 AE 0.111Þ% [8].
In this paper, we search for D-mixing in the CP-odd decay D 0 → K 0 S ω with ω → π þ π − π 0 . This decay is favorable as it has a relatively large branching fraction of ð0.99 AE 0.05Þ% [1], nearly 5 times that of D 0 → K 0 S ϕ, and the two charged tracks from the D 0 decay vertex allow for an accurate measurement of the D 0 decay time. The narrowness of the ω peak leads to small contamination by other resonant or nonresonant decays to the D 0 → K 0 S π þ π − π 0 final state. We extract y CP by comparing the lifetimes of K 0 S ω and K − π þ . Since dΓðD 0 → K − π þ Þ=dt ∝ e −Γt , Eq. (2) implies Our study is based on the full data sample of 976 fb −1 recorded with the Belle [9] detector at the KEKB asymmetric-energy e þ e − collider [10] at a center-of-mass energy near the ϒð4SÞ resonance. The detector components relevant for this work are a silicon vertex detector (SVD), a 50-layer central drift chamber (CDC), and an electromagnetic calorimeter (ECL) composed of CsI(Tl) crystals, all located inside a superconducting solenoid coil that provides a 1.5 T magnetic field. Two inner detector configurations were used. A 2.0 cm radius beam pipe with a three-layer SVD was used for the initial 16% of the sample and a 1.5 cm radius beam pipe with a four-layer SVD for the rest. Charged particle identification is accomplished by combining specific ionization measurements in the CDC with the information from an array of aerogel threshold Cherenkov counters and a barrellike arrangement of timeof-flight scintillation counters. The analysis procedure is established using Monte Carlo (MC) simulated samples. Particle decays are modeled by the EvtGen package [11], with the simulation of detector response performed with GEANT3 [12].
We select charged tracks originating from the collision region with jdrj < 0.5 cm and jdzj < 2.0 cm, where dr and dz are the impact parameters with respect to the nominal interaction point in the plane transverse and parallel to the e þ beam, respectively. We require these charged tracks to have at least two associated hits in the SVD, in both the z and azimuthal projections. Charged hadrons are identified with a likelihood ratio LðK=πÞ ¼ L K =ðL K þ L π Þ, where L π and L K are the individual likelihood values for the π AE and K AE hypothesis based on all the available particle identification information. We require LðK=πÞ > 0.6 and LðK=πÞ < 0.4 for K AE and π AE candidates, respectively. The K 0 S candidates are reconstructed from pairs of oppositely charged tracks (assumed to be pions) that form a common vertex and are identified with an artificial neural network [13] that combines seven kinematic variables of the K 0 S including the finite flight length for K 0 S vertex from the e þ e − interaction point. More details on K 0 S identification can be found in Ref. [14]. The invariant mass of the selected candidates is required to satisfy 487 MeV=c 2 < M K 0 S < 508 MeV=c 2 that corresponds to approximately 3 standard deviations (σ) in mass resolution. The K 0 S purity is 96% after all the K 0 S selections are applied. π 0 meson candidates are reconstructed from photon pairs. Photons are contiguous regions of energy deposit in the ECL without any associated charged tracks. The ratio of the energy deposited in the central 3 × 3 array of crystals relative to that in the central 5 × 5 array of crystals is required to be greater than 0.75. The energy of each photon must be greater than 50, 100, and 150 MeV in the barrel region, forward, and backward end cap, respectively. The π 0 momentum is required to be greater than 300 MeV=c, and its invariant mass is required to be in the range 120 MeV=c 2 < M γγ < 148 MeV=c 2 , which corresponds to approximately AE3σ around the nominal π 0 mass [1].
As the ω lifetime is negligible, we determine the D 0 decay vertex from a kinematic fit constraining the K 0 S , π þ , π − , and π 0 candidates to come from a common vertex. We constrain the π 0 mass in this fit by introducing a large uncertainty of 1.0 cm on its vertex position. We select D 0 → K 0 S π þ π − π 0 candidates in the ω mass region by requiring 750 MeV=c 2 < M πππ 0 < 810 MeV=c 2 that corresponds to approximately AE3σ in resolution around the nominal ω mass [1]. The purity of the ω sample after all selection criteria is 91.4%. We retain a D 0 → K 0 S π þ π − π 0 candidate if its invariant mass is in the range 1.80 GeV=c 2 < M D < 1.92 GeV=c 2 and a D 0 → K − π þ candidate if its invariant mass is in the range 1.83 GeV=c 2 < M D < 1.90 GeV=c 2 . The tighter requirement in the latter case is due to better mass resolution. The D Ãþ candidates are reconstructed from the selected D 0 and π þ slow candidates requiring the mass difference between D Ãþ and D 0 to lie in the range m π þ < ΔM < 150 MeV=c 2 . Here, π þ slow is the charged pion whose momentum tends to be low compared to the final-state particles originating from the D 0 decay, and m π þ is the charged pion nominal mass [1]. In order to suppress combinatorial background further and veto D 0 mesons coming from B decays, the D Ãþ momentum in the center-of-mass frame is required to be greater than 2.55 GeV=c.
The production vertex of the D 0 , i.e., the D Ãþ vertex is obtained by constraining the D 0 momentum to the interaction region (IR). The π þ slow candidate is refitted to the D Ãþ vertex to improve resolution of ΔM. As the IR position varies with changing accelerator conditions, we update the mean position every 10,000 hadronic events. The IR position resolution is determined by comparing the mean IR position with the true production vertex position using MC. The mean width of the IR is 3.34 mm along the z axis and 82 μm in the horizontal and 4.3 μm in the vertical directions. To further improve vertex resolutions, we require confidence levels to exceed 10 −3 for both fits. After applying all selection criteria, there are on average 1.40 (1.01) candidates per event in the D 0 → K 0 S ω (Kπ) decay. We retain the one having the minimum χ 2 value determined from the π slow vertex fit.
The proper decay time of D 0 candidates is calculated by projecting the flight length vector connecting the D Ãþ and D 0 decay vertices along the direction of the momentum vector ⃗p and then dividing by the magnitude of ⃗p and multiplying by the D 0 mass. The error on the proper decay time, σ t , is calculated from the error matrix of the production vertex position, the decay vertex position, and the momentum ⃗p. The diagonal elements correspond to the variances in these quantities, whereas the offdiagonal elements give the correlations among their uncertainties. The resolution on the decay time is 310 fs for D 0 → K 0 S ω decays and 162 fs for D 0 → Kπ decays. For both samples, a loose requirement σ t < 900 fs is imposed.
The worsening in resolution in the D 0 → K 0 S ω case is due to the presence of π 0 and K 0 S in the final state. According to MC simulation, the selected events can be grouped into the following four categories: signal, random π slow background composed of correctly reconstructed D 0 mesons combined with a misreconstructed π slow , combinatorial background, and background due to partially reconstructed multibody charm decays. We first perform a two-dimensional (2D) unbinned maximum-likelihood fit to the variables ðM D ; ΔMÞ in order to extract signal and background fractions. These are then used in the lifetime fits to normalize different lifetime components.
The probability density functions (PDFs) of different event categories are parametrized as follows. For the D 0 → K 0 S ω decay mode, the signal distribution in M D is modeled with the sum of a Crystal Ball (CB) function [15] and three Gaussian functions all constrained to a common mean, while the distribution in ΔM is parametrized with the sum of two Gaussian functions constrained to a common mean (double Gaussian function) to describe the core, and the sum of an asymmetric Gaussian function and a CB function to model the tails. To account for a correlation between the core widths of ΔM and M D , we parametrize the former with a second-order polynomial of jM D − m D 0 j, where m D 0 is the nominal mass [1] of the D 0 meson.
The signal distribution of the D 0 → K − π þ decay mode is parametrized in M D with a sum of a CB function, a double Gaussian function, and an asymmetric Gaussian function, while in ΔM it is modeled with a double Gaussian function to describe the core, and with a sum of a CB function and two asymmetric Gaussian functions to describe the tails. The correlation between the core widths of ΔM and M D is parametrized as for the D 0 → K 0 S ω mode. The distribution of random π slow background is peaking in M D and smooth in ΔM. The former is parametrized with the signal PDF and the latter with a threshold function, where Q ≡ ΔM − m π þ , and α and β are two shape parameters. The distribution of combinatorial background is smooth in both variables. We parametrize it in M D with either a first-order polynomial (K − π þ ) or a second-order polynomial (K 0 S ω); and in ΔM with the threshold function as in Eq. (5).
The background due to partially reconstructed multibody charm decays is smooth in M D but exhibits a broad peak in ΔM. In the case of K 0 S ω, this background is small (about 3% of the total background) and its shape in M D is very similar to that of the combinatorial background. We decide to combine this background with the combinatorial background by adding an additional Gaussian term to the parametrization in ΔM. The parameters of this additional function and its fraction are fixed from the fit to MC simulation. In the case of K − π þ , we treat this background separately. The distribution is parametrized with an exponential function in M D and with a double Gaussian function in ΔM whose parameters are fixed to values obtained from MC simulation.
The robustness of our fitting model is tested with MC samples that correspond to the Belle data set in integrated luminosity. The obtained signal and background fractions  S ω (top) and D 0 → K − π þ (bottom). Points with error bars represent the data. The curves show projections of fitted PDF: total PDF projection in solid black, signal contribution in double dot-dashed red, combinatorial background in dashed black, random π slow background in dotted magenta, and multibody background as dash-dotted green. (The total PDF is hard to see as it closely follows the data points.) in the signal region, defined in Table I, are consistent with the ones determined with MC "truth matching"; the difference between the two is, in all cases, within 1 standard deviation.
After validating the fitting model, we proceed to fit the data sample. The results are shown in Fig. 1 and are listed in Table II. We measure the signal fractions of 96.3% (K 0 S ω) and 99.6% (K − π þ ) by integrating events in the signal region.
Finally, we perform unbinned maximum-likelihood fits for lifetime using the events in the signal region. We parametrize the proper decay-time distribution as where the first term represents signal and the second term background, f sig , is the fraction of signal events determined with the 2D fit described earlier, τ is the effective signal lifetime, and Rðt − t 0 Þ is the resolution function. The resolution function is parametrized with the sum of three (K 0 S ω) or four (K − π þ ) Gaussian functions constrained to the common mean. Besides the effective lifetime τ, the free parameters of the fit are the resolution function mean, the widths, and the fraction of each Gaussian function.
The background term BðtÞ is parametrized with two lifetime components: a zero-lifetime component corresponding to combinatorial background and a component with an effective lifetime τ b corresponding to multibody charm background, where f 0 is the fraction of zero-lifetime component and R b ðt − t 0 Þ is the resolution function for background, parametrized with a sum of three Gaussian functions constrained to the common mean. The parameters of BðtÞ are obtained by fitting the proper-time distribution of events in the sidebands as defined in Table I. The sidebands are chosen such that they contain negligible amounts of signal.
The lifetime fitting model is tested with four statistically independent MC samples, each corresponding to the integrated luminosity in data. The resulting fitted lifetimes are found to be consistent with the generated value, and y CP determined from the fitted lifetimes of D 0 → K 0 S ω and D 0 → K − π þ is compatible with zero within 1 standard deviation.
Lifetime fits on the data are shown in Fig. 2. The χ 2 per number of degrees of freedom of the D 0 → K 0 S ω and  D 0 → K − π þ lifetime fits are 0.90 and 1.10, respectively. We measure τ K 0 S ω ¼ ð410.47 AE 3.73Þ fs and τ Kπ ¼ ð406.53 AE 0.57Þ fs, and y CP ¼ ð0.96 AE 0.91Þ%, where the uncertainties are statistical.
Besides D 0 → K 0 S ω decay, the reconstructed final state K 0 S π þ π − π 0 might include contributions from other intermediate resonances, or no resonance at all. Depending on orbital angular momenta, some of these decay modes might be CP-even. The presence of CP-even component in the signal reduces the measured y CP by a factor of 1 − 2f CPþ , where f CPþ is the fraction of CP-even decays in the signal component. Since this fraction is not well known in the selected mass region of ω, we assign a systematic uncertainty to the measured y CP by conservatively assuming that all non-ω decays are CP-even. The fraction of non-ω decays is determined from a fit to the M πππ 0 distribution in which the M πππ 0 requirement is loosened, but events are still required to be in the signal region. The fraction of events under the ω peak obtained from the fit and corrected for a small amount of random combinations of ω and K 0 S (2.5%) is 88.0%, while the signal fraction from the 2D fit is 96.3%. From the ratio of the two (91.4%), we find the upper limit f CPþ ¼ 8.6%. The systematic uncertainty in y CP due to the possible presence of CP-even decays in the sample is therefore at most 2f CPþ · y CP ¼ þ0.17%.
Other sources of systematic uncertainties are listed in Table III. We vary the requirement on the K 0 S flight length in steps of 0.1 mm up to 1.0 mm; we find no significant bias in the D 0 lifetime and assign the maximum variation observed of 0.01% as the systematic uncertainty in y CP . To assign systematics due to different energy thresholds used for different barrel regions, we divide the whole barrel region into three equal bins and assign a maximum energy threshold of each photon of 70 MeV to each bin. We observe an average bias of 0.1% which we assign as the systematic due to π 0 reconstruction. We vary our selection criteria on σ t by AE50 fs and find a 0.21% variation in y CP . Variation of D mass window position and size by AE2.5 MeV=c 2 leads to a 0.13% change in y CP . We vary the signal fraction by its statistical and systematic uncertainties; we find a 0.14% variation due to statistics and, from MC simulation, 0.10% due to the fixed shape parameters in the ðM D ; ΔMÞ fit. These two contributions are combined in quadrature, and the result is assigned as the systematic uncertainty due to the signal fraction. Note that difference between the data and fit visible in Fig. 1 for the D 0 → Kπ mode has a negligible effect on the extracted lifetime.
By choosing different sidebands to obtain the decay-time dependence of background BðtÞ, we find a variation of 0.32% in y CP . We also vary the background lifetime by the lifetime difference obtained in simulation between background events in the signal region and those in the sidebands; we find a variation of 0.03% in y CP . We vary each fixed background shape parameter by its uncertainty; by taking into account correlations among the parameters, we obtain a variation of 0.43% in y CP . By summing the above contributions in quadrature, we obtain a total systematic uncertainty of 0.62%; the systematic uncertainty due to the possible presence of CP-even decays in the data sample (discussed earlier) is treated separately.
In summary, we have measured for the first time the mixing parameter y CP in the CP-odd decay D 0 → K 0 S ω. We obtain y CP ¼ ð0.96 AE 0.91 AE 0.62 þ0.17 −0.00 Þ%; where the first uncertainty is statistical, the second is systematic due to event selection and background, and the last is due to the possible presence of CP-even decays in the final state. The result is consistent with our previous measurement in the CP-odd decay D 0 → K 0 S ϕ [7], as well as with measurements in the CP-even decays D 0 → K þ K − and D 0 → π þ π − [4][5][6]. The result also agrees with the world average of y CP [8]. In the future, comparing more precise measurements of y CP with that of y may reveal new physics effects in the charm system.