Search for CP Violation in the Decay $D^+\rightarrow K^0_S K^+$

We search for CP violation in the decay $D^+\rightarrow K^0_S K^+$ using a data sample with an integrated luminosity of 977 fb$^{-1}$ collected with the Belle detector at the KEKB $e^+e^-$ asymmetric-energy collider. No CP violation has been observed and the CP asymmetry in $D^+\rightarrow K^0_S K^+$ decay is measured to be $(-0.25\pm0.28\pm0.14)%$, which is the most sensitive measurement to date. After subtracting CP violation due to $K^0-\bar{K}^0$ mixing, the CP asymmetry in $D^+\rightarrow\bar{K}^0 K^+$ decay is found to be $(+0.08\pm0.28\pm0.14)%$.


Introduction
Studies of CP violation in charmed meson decays provide a promising opportunity to search for new physics beyond the standard model (SM) [1] in the absence of disagreement between experimental measurements and the SM interpretation of CP violation in K and B meson decays [2][3][4]. Recently, the LHCb collaboration has reported ∆A CP = (−0.82 ± 0.21 ± 0.11)% [5] where ∆A CP is the CP asymmetry difference between D 0 → K + K − 1 and D 0 → π + π − decays. Thereafter, the CDF collaboration has also announced ∆A CP = (−0.62 ± 0.21 ± 0.10)% [6], which strongly supports the non-zero ∆A CP measured from the LHCb collaboration. Together with results from the BaBar and Belle collaborations, the value of ∆A CP is significantly different from zero [7]. Taking into account that the indirect CP asymmetries in the two decays are approximately equal [8], ∆A CP can be expressed as ∆A CP = ∆a dir CP + a ind CP ∆ t /τ, where a dir CP and a ind CP denote direct and indirect CP violation, respectively, and t /τ is the mean proper decay time of the selected signal sample in units of the D 0 lifetime [9]. The factor ∆ t /τ in eq. (1.1) depends on the experimental conditions and the largest value reported to date is 0.26 ± 0.01 from the CDF measurement [6]. Therefore, ∆A CP reveals a significant direct CP violation difference between the two decays. Within the SM, direct CP violation in the charm sector is expected to be present only in singly Cabibbosuppressed (SCS) decays, and even there is expected to be small, O(0.1%) [10]. Hence, the current ∆A CP measurements engender questions of whether the origin of the asymmetry lies within [11][12][13][14] or beyond [15][16][17][18] the SM. The origin of ∆A CP calls for the precise measurements of A CP in D 0 → K + K − and D 0 → π + π − . A complementary test is a precise measurement of A CP in another SCS charmed hadron decay, D + →K 0 K + , as suggested in ref. [13]. As shown in figures 1(a) and 1(b), the decay D + →K 0 K + shares the same decay diagrams with D 0 → K + K − by exchanging the spectator quarks, d ↔ u. Although there are additional contributions to the two decays as shown in figures 1(c) and 1(d), these are expected to be small due to helicity-and color-suppression considerations 2 . Therefore, neglecting the latter contributions in D + →K 0 K + and D 0 → K + K − decays, the direct CP asymmetries in the two decays are expected to be the same.
In this paper, we report results from a search for CP violation in the decay D + → K 0 S K + that originates from D + →K 0 K + decay, where K 0 S decays to π + π − . The CP asymmetry in the decay, A CP , is then defined as where Γ is the partial decay width. In eq. (1.2), A D + →K 0 K + CP is the CP asymmetry in the decay D + →K 0 K + and AK 0 CP is that inK 0 → π + π − decay induced by K 0 −K 0 mixing in the SM [19][20][21] in which the decayK 0 → π + π − arises from K 0 S → π + π − together with a small contribution from K 0 L → π + π − , where the latter is known precisely from K 0 L semileptonic decays, AK 0 CP = (−0.332 ± 0.006)% [2]. As shown in eq. (1.2), the product of the two small asymmetries is neglected. The D + decaying to the final state K 0 S K + proceeds from D + →K 0 K + decay, which is SCS. In the SM, direct CP violation in SCS charmed meson decays is predicted to occur with a non-vanishing phase that enters the diagram shown in figure 1(b) in the Kobayashi-Maskawa ansatz [22]. The current average of ∆A CP favors a negative value of direct CP violation in D 0 → K + K − decay. Correspondingly, the CP asymmetry in D + → K 0 S K + decays is more likely to have a negative value since the two CP asymmetry terms shown in eq. (1.2) are negative.

Methodology
We determine A D + →K 0 S K + CP by measuring the asymmetry in the signal yield where N rec is the number of reconstructed decays. The asymmetry in eq. (2.1) includes the forward-backward asymmetry (A F B ) due to γ * -Z 0 interference and higher order QED effects in e + e − → cc [23][24][25], and the detection efficiency asymmetry between K + and K − (A K + ) as well as A CP . In addition, ref. [26] calculates another asymmetry source, denoted A D , due to the differences in interactions ofK 0 and K 0 mesons with the material of the detector. Since we reconstruct the K 0 S with π + π − combinations, the π + π − detection asymmetry cancels out for K 0 S . The asymmetry of eq. (2.1) can be written as by neglecting the terms involving the product of asymmetries. In eq. (2.2), A D + →K 0 S K + CP is the sum of A D + →K 0 K + CP and AK 0 CP as stated in eq. (1.2), where the former is independent of all kinematic variables while the latter is known to depend on the K 0 S decay time according to ref. [27], and A D + F B is an odd function of the cosine of the polar angle θ c.m.s.

D +
of the D + momentum in the center-of-mass system (c.m.s.). A K + depends on the transverse momentum p lab T K + and the polar angle θ lab K + of the K + in the laboratory frame (lab). Here, A D is a function of the lab momentum p lab K 0 S of the K 0 S . To correct for A K + in eq. (2.2), we use the technique developed in our previous publication [28]. We use D 0 → K − π + and D + s → φπ + decays where the φ is reconstructed with K + K − combinations and hence the K + K − detection asymmetry nearly cancels out [29] (the residual small effect is included in the systematic error). Since these are Cabibbo-favored decays for which the direct CP asymmetry is expected to be negligible, in analogy to eq. (2.2), can be written as , assuming the same A F B for D 0 and D + s mesons. We also obtain A D according to ref. [26]. After these A K + and A D corrections 3 , we obtain We subsequently extract A CP and A F B as a function of cos θ c.m.s.
by taking sums and differences: Note that extracting A CP in eq. (2.5) using eq. (2.6a) is crucial here to cancel out the Belle detector's asymmetric acceptance in cos θ c.m.s. D + .

Data and event selections
The data used in this analysis were recorded at the Υ(nS) resonances (n = 1, 2, 3, 4, 5) or near the Υ(4S) resonance with the Belle detector at the e + e − asymmetric-energy collider KEKB [30]. The data sample corresponds to an integrated luminosity of 977 fb −1 . The Belle detector is a large solid angle magnetic spectrometer that consists of a silicon vertex detector (SVD), a 50-layer central drift chamber (CDC), an array of aerogel threshold Cherenkov counters (ACC), a barrel-like arrangement of time-of-flight scintillation counters (TOF), and an electromagnetic calorimeter comprising CsI(Tl) crystals located inside a superconducting solenoid coil that provides a 1.5 T magnetic field. An iron flux return located outside the coil is instrumented to detect K 0 L mesons and to identify muons. A detailed description of the Belle detector can be found in ref. [31].
Except for the tracks from K 0 S decays we require charged tracks to originate from the vicinity of the interaction point (IP) by limiting the impact parameters along the beam direction (z-axis) and perpendicular to it to less than 4 cm and 2 cm, respectively. All charged tracks other than those from K 0 S decays are identified as pions or kaons by requiring the ratio of particle identification likelihoods, L K /(L K +L π ), constructed using information from the CDC, TOF, and ACC, to be larger or smaller than 0.6, respectively [32]. For both kaons and pions, the efficiencies and misidentification probabilities are about 90% and 5%, respectively.
We form K 0 S candidates adopting the standard Belle K 0 S criteria [33], requiring the invariant mass of the charged track pair to be within [0.4826, 0.5126] GeV/c 2 . The "loose" K 0 S candidates not satisfying these standard selections are also used in this analysis with additional requirements described later.
The K 0 S and K + candidates are combined to form a D + candidate by fitting their tracks to a common vertex; the D + candidate is fitted to the independently measured IP 3 We profile to give the production vertex. To remove combinatorial background as well as D + mesons that are produced in possibly CP -violating B meson decays, we require the D + meson momentum calculated in the c.m.s. (p * D + ) to be greater than 2.5 and 3.0 GeV/c for the data taken at the Υ(4S) and Υ(5S) resonances, respectively. For the data taken below Υ(4S), where no B mesons are produced, we apply the requirement p * D + >2.0 GeV/c. In addition to the selections described above, we further optimize the signal sensitivity with four variables: the goodness-of-fit values of the D + decay-and production-vertex fits χ 2 D and χ 2 P , the transverse momentum of the K + in the lab p lab T K + , and the angle ξ between the D + momentum vector (as reconstructed from its daughters) and the vector joining the D + production and decay vertices. We optimize the requirement on these four variables with the standard and loose The optimal set of (χ 2 D , χ 2 P , p lab T K + , ξ) requirements are found to be (<100, <10, >0.30 GeV/c, <40 • ), (<100, <10, >0.25 GeV/c, <115 • ), and (<100, <10, >0.20 GeV/c, <125 • ) for the data taken below the Υ(4S), at the Υ(4S), and at the Υ(5S), respectively. Note that p * D + is highly correlated with p lab T K + and ξ; hence, a tighter p * D + requirement on the Υ(5S) sample results in looser p lab T K + and ξ requirements and vice versa for the data taken below the Υ(4S). The D + candidates with the loose K 0 S requirement are further optimized with two additional variables: the χ 2 of the fit of tracks from the K 0 S decay and the kaon from the D + meson decay to a single vertex (χ 2 Khh ) and the angle ζ between the K 0 S momentum vector (as reconstructed from its daughters) and the vector joining the D + and K 0 S decay vertices. The two variables are again varied simultaneously and the optimum is found to be χ 2 Khh >6 and ζ<3 • for all data. The inclusion of D + candidates with the loose K 0 S requirement improves the statistical sensitivity by approximately 5%. After the final selections described above, we find no significant peaking backgrounds-for example, D + → π + π − K + decays-in the Monte Carlo (MC) simulated events [34]. Figure 2 shows the distributions of M (K 0 S K + ) and M (K 0 S K − ) together with the results of the fits described below. Each D ± → K 0 S K ± signal is parameterized as two Gaussian distributions with a common mean. The combinatorial background is parameterized with the unnormalized form e α+βM (K 0 S K ± ) , where α and β are fit parameters. The asymmetry and the sum of the D + and D − yields are directly obtained from a simultaneous fit to the D + and D − candidate distributions. Besides the asymmetry and the sum of the D + and D − yields, the common parameters in the simultaneous fit are the widths of the two Gaussians and the ratio of their amplitudes. The asymmetry and the sum of the D + → K 0 S K + and D − → K 0 S K − yields from the fit are (+0.048 ± 0.275)% and 276812 ± 1156, respectively, where the errors are statistical.
In order to measure the CP asymmetry in D + → K 0 S K + decays, we must also reconstruct D 0 → K − π + and D + s → φπ + decays: see eqs. (2.2), (2.3), and (2.4). For the reconstruction of the D 0 → K − π + and D + s → φπ + decays, we require the same track quality, particle identification, vertex fit quality, and p * D requirements as used for the reconstruction of the D + → K 0 S K + decays, where the mass window for the φ is ±16 MeV/c 2 [29] of the nominal φ mass [2]. 4 Extraction of A CP in the decay D + → K 0 S K + To obtain A K + , we first extract A D + s →φπ + rec from a simultaneous fit to the mass distributions of D + s and D − s candidates with similar parameterizations as for D ± → K 0 S K ± decays except that, for the D ± s → φπ ± signal description, a single Gaussian is used. The values of A D + s →φπ + rec are evaluated in 10×10×10 bins of the three-dimensional (3D) phase space (p lab T π + , cos θ lab π + , cos θ c.m.s. ). Each D 0 → K − π + andD 0 → K + π − candidate is then weighted with a factor of 1 − A D + s →φπ + rec and 1 + A D + s →φπ + rec , respectively, in the corresponding bin of this space. After this weighting, the asymmetry in the D 0 → K − π + decay sample becomes A K − . The detector asymmetry, A K − , is measured from simultaneous fits to the weighted M (K ∓ π ± ) distributions in 10×10 bins of the 2D phase space (p lab T K − , cos θ lab K − ) with similar parameterizations as used for D + → K 0 S K + decays except that, for the D 0 → K − π + signal description, a sum of a Gaussian and bifurcated Gaussian is used. Figure 3 shows the measured A K − in bins of p lab T K − and cos θ lab K − together with A D 0 →K − π + rec for comparison; we observe that A D 0 →K − π + rec shows a cos θ lab K − dependency that is inherited from A D 0 F B while A K − does not. The average of A K − over the phase space is (−0.150 ± 0.029)%, where the error is due to the limited statistics of the D 0 → K − π + sample.
Based on a recent study of A D [26], we obtain the dilution asymmetry in bins of K 0 S lab momentum. For the present analysis, A D is approximately 0.1% after integrating over the phase space of the two-body decay. The data samples shown in figure 2 are divided into 10×10×16 bins of the 3D phase space (p lab T K + , cos θ lab . Each D ± → K 0 S K ± candidate is then weighted with a factor of (1 ∓ A K + )(1 ∓ A D ) in this space. The weighted M (K 0 S K ± ) distributions in bins of cos θ c.m.s.

Systematic uncertainty
The entire analysis procedure is validated with fully simulated MC events [34] and the result is consistent with null input asymmetry. We also consider other sources of systematic uncertainty. The dominant one in the A CP measurement is the A K + determination, the uncertainty of which is mainly due to the statistical uncertainties in the D 0 → K − π + and D + s → φπ + samples. These are found to be 0.029% and 0.119%, respectively, from a simplified simulation study. A possible A CP in the D 0 → K − π + final state is estimated using A CP = −y sin δ sin φ √ R [35]. A calculation with 95% upper and lower limits on D 0 −D 0 mixing and CP violation parameters y, φ, and strong phase difference δ and Cabibbo suppression factor R from ref. [3], A CP in the D 0 → K − π + final state is estimated to be less than 0.005% and this is included as one of systematic uncertainties in the A K + determination. As reported in our previous publication [29], the magnitude of A KK rec for  the φ reconstruction in D + s → φπ + decays is 0.051%, which is also added to the systematic uncertainty in the A K + measurement. By adding the contributions in quadrature, the systematic uncertainty in the A K + determination is estimated to be 0.133%. We estimate 0.008% and 0.021% systematic uncertainties due to the choice of the fitting method and that of the cos θ c.m.s. D + binning, respectively. Finally, we add the systematic uncertainty in the A D correction, which is 0.010% based on ref. [26]. The quadratic sum of the above uncertainties, 0.135%, is taken as the total systematic uncertainty.

Results
We find A D + →K 0 S K + CP = (−0.246 ± 0.275 ± 0.135)%. This measurement supersedes our previous determination of A D + →K 0 S K + CP [28]. In Table 1, we compare all the available measurements and give their weighted average.
According to Grossman and Nir [27], we can estimate the experimentally measured CP asymmetry induced by SM K 0 −K 0 mixing, AK 0 CP . The efficiency as a function of K 0 S decay time in our detector is obtained from MC simulated events. The efficiency is then used in eq. (2.10) of ref. [27] to obtain the correction factor that takes into account, for AK 0 CP , the dependence on the kaon decay time. The result is 0.987 ± 0.007. By multiplying the correction factor 0.987 ± 0.007 and the asymmetry due to the neutral kaons [2], we find the experimentally measured AK 0 CP to be (−0.328 ± 0.006)%. Experiment A D + →K 0 S K + CP (%) FOCUS [36] +7.1 ± 6.1 ± 1.2 CLEO [37] −0.2 ± 1.5 ± 0.9 Belle (this measurement) −0.246 ± 0.275 ± 0.135 New world average −0.23 ± 0.30 Table 1. Summary of A D + →K 0 S K + CP measurements (where the first uncertainties are statistical and the second systematic), together with their average (assuming the uncertainties to be uncorrelated, the error on the average represents the total uncertainty).

Conclusion
We report the most sensitive CP asymmetry measurement to date for the decay D + → K 0 S K + using a data sample corresponding to an integrated luminosity of 977 fb −1 collected with the Belle detector. The CP asymmetry in the decay is measured to be (−0.25 ± 0.28 ± 0.14)%. After subtracting the contribution due to K 0 −K 0 mixing (AK 0 CP ), the CP asymmetry in the charm decay (A D + →K 0 K + CP ) is measured to be (+0.08 ± 0.28 ± 0.14)%, which can be compared with direct CP violation in D 0 → K + K − . For the latter the current averages of ∆A CP and CP asymmetry in D 0 → K + K − favor a negative value [3]. Our result, on the other hand, does not show this tendency for D + →K 0 K + decays, albeit with a significant statistical uncertainty.