Amplitude analysis and branching-fraction measurement of \boldmath $D_{s}^{+} \to K^0_{S}\pi^{+}\pi^{0}$

Utilizing a data set corresponding to an integrated luminosity of 6.32~$\rm fb^{-1}$, recorded by the BESIII detector at center-of-mass energies between 4.178 and 4.226~GeV, we perform an amplitude analysis of the decay $D_{s}^{+} \to K_{S}^{0}\pi^{+}\pi^{0}$ and determine the relative fractions and phase differences of different intermediate processes, which include $K_{S}^{0}\rho(770)^{+}$, $K_{S}^{0}\rho(1450)^{+}$, $K^{*}(892)^{0}\pi^{+}$, $K^{*}(892)^{+}\pi^{0}$, and $K^{*}(1410)^{0}\pi^{+}$. Using a double-tag technique, and making an efficiency correction that relies on our knowledge of the phase-space distribution of the decays coming from the amplitude analysis, the absolute branching fraction is measured to be $\mathcal{B}(D_{s}^{+} \to K_{S}^{0}\pi^{+}\pi^{0})=(5.43\pm0.30_{\text{stat}}\pm 0.15_{\text{syst}})\times 10^{-3}$.

Abstract: Utilizing a data set corresponding to an integrated luminosity of 6.32 fb −1 , recorded by the BESIII detector at center-of-mass energies between 4.178 and 4.226 GeV, we perform an amplitude analysis of the decay D + s → K 0 S π + π 0 and determine the relative fractions and phase differences of different intermediate processes, which include K 0 S ρ(770) + , K 0 S ρ(1450) + , K * (892) 0 π + , K * (892) + π 0 , and K * (1410) 0 π + . Using a double-tag technique, and making an efficiency correction that relies on our knowledge of the phase-space distribution of the decays coming from the amplitude analysis, the absolute branching fraction is measured to be B(D + s → K 0 S π + π 0 ) = (5.43 ± 0.30 stat ± 0.15 syst ) × 10 −3 .

Introduction
Knowledge of D ± s decay properties are vital input for studies of the B 0 s hadron, whose decay channels are dominated by the final states involving D ± s mesons [1]. Furthermore, hadronic D ± s decays probe the interplay of short-distance weak-decay matrix elements and longdistance QCD interactions, and the measured branching fractions (BFs) provide valuable information concerning the amplitudes and phases that the strong force induces in the decay process [2][3][4]. The singly Cabibbo-suppressed (SCS) decay D + s → K 0 π + π 0 has a large BF of the order of 10 −2 [1]. This decay, therefore, is often used as a reference channel for the other decays of D ± s mesons. Accurate knowledge of its substructure is essential to reduce the systematic uncertainties in those analyses using this channel. To date, there have been few measurements of charge-parity asymmetries A CP in SCS D ± s decay modes in general and none for the mode discussed here.
An amplitude analysis of the D + s decay to a three-body pseudoscalar meson final state is a powerful tool for studying the vector-pseudoscalar channels of the SCS D + s decay. Table 1 shows the current measured values and theoretical predictions, in various models, for the BFs of D + s → K 0 S ρ + , K * (892) 0 π + , and K * (892) + π 0 (ρ + denotes ρ(770) + throughout this paper). References [5] and [6] took into account quark flavor SU(3) symmetry and its breaking effects. Reference [4] used a generalized factorization method considering the resonance effects in the pole model for the annihilation contributions and introducing large strong phases between different topological diagrams. More precise experimental results are required to validate or falsify these theoretical predictions.
The CLEO collaboration has reported a measurement of B(D + s → K 0 π + π 0 ) = (1.00 ± 0.18)% [7], using 600 pb −1 of e + e − collisions recorded at a center-of-mass energy ( √ s) of 4.17 GeV. In this paper, by using 6.32 fb −1 of data collected with the BESIII detector at √ s = 4.178-4.226 GeV, we perform the first amplitude analysis of D + s → K 0 S π + π 0 and improve the measurement of its absolute BF.

Detector and data sets
The BESIII detector is a magnetic spectrometer [8,9] located at the Beijing Electron Positron Collider (BEPCII) [10]. The cylindrical core of the BESIII detector consists of a helium-based multilayer drift chamber (MDC), a plastic scintillator time-of-flight system (TOF), and a CsI(Tl) electromagnetic calorimeter (EMC), which are all enclosed in a superconducting solenoidal magnet providing a 1.0 T magnetic field. The solenoid is supported by an octagonal flux-return yoke with resistive plate counter muon identifier modules interleaved with steel. The acceptance of charged particles and photons is 93% over a 4π solid angle. The charged-particle momenta resolution at 1.0 GeV/c is 0.5%, and the specific energy loss (dE/dx) resolution is 6% for the electrons from Bhabha scattering. The EMC measures photon energies with a resolution of 2.5% (5%) at 1 GeV in the barrel (end-cap) region. The time resolution of the TOF barrel part is 68 ps, while that of the end-cap part is 110 ps. The end-cap TOF was upgraded in 2015 with multi-gap resistive plate chamber technology, providing a time resolution of 60 ps. The data samples used in this analysis are listed in Table 2. For some aspects of the analysis, these samples are organized into three sample groups, 4.178 GeV, 4.189-4.219 GeV, and 4.226 GeV, that were acquired during the same year under consistent running conditions. Since the cross section of D * ± s D ∓ s production in e + e − annihilation is about a factor of twenty larger than that of D + s D − s [11], and the D * ± s meson decays to γD ± s with a dominant BF of (93.5 ± 0.7)% [1], the signal events discussed in this paper are selected from the process e + e − → D * ± s D ∓ s → γD + s D − s . Simulated samples are produced with the geant4-based [12] Monte Carlo (MC) package, which includes the geometric description of the BESIII detector and the detector response. These samples are used to determine the detection efficiency and to estimate the background. The simulation includes the beam-energy spread and initial-state radiation (ISR) in e + e − annihilations modeled with the generator kkmc [13]. The generic MC samples consist of the production of DD pairs with consideration of quantum coherence for all neutral D modes, the non-DD decays of the ψ(3770), the ISR production of the J/ψ and ψ(3686) states, and the continuum processes. The known decay modes are modeled with evtgen [14,15] using the BFs taken from the Particle Data Group (PDG) [1], and the remaining unknown decays from the charmonium states with lundcharm [16,17]. Final-state radiation from charged particles is incorporated with the photos [18] package.

Event selection
The data samples were collected just above the D * ± s D ∓ s threshold. The tag method allows clean signal samples to be selected, which provide an opportunity to perform amplitude analyses and to measure the absolute BFs of the hadronic D + s meson decays. In the tag method, a single-tag (ST) candidate requires only one of the D ± s mesons to be reconstructed via a hadronic decay; a double-tag (DT) candidate has both D + s D − s mesons reconstructed via hadronic decays. The DT candidates are required to have the D + s meson decaying to the signal mode D + s → K 0 S π + π 0 and the D − s meson decaying to a tag mode. (Charge conjugation is implied throughout this paper.) Nine tag modes are reconstructed and the corresponding mass windows on the tagging D − s mass (M tag ) are listed in Table 3. The D ± s candidates are constructed from individual π ± , K ± , η, η , K 0 S and π 0 particles. Charged track candidates from the MDC must satisfy |cos θ| < 0.93, where θ is the polar angle with respect to the direction of the positron beam. The closest approach to the interaction point is required to be less than 10 cm along the beam direction and less than 1 cm in the plane perpendicular to the beam. Particle identification (PID) of charged particles is implemented by combining the dE/dx information in the MDC and the time-of-flight information from the TOF system. For charged kaon (pion) candidates, the probability for the kaon (pion) hypothesis is required to be larger than that for the pion (kaon) hypothesis. Table 3. Requirements on M tag for various tag modes, where the η and η subscripts denote the decay modes used to reconstruct these particles.

Tag mode
Mass window (GeV/c 2 ) The K 0 S mesons are reconstructed with pairs of two oppositely charged tracks, which satisfy |cos θ| < 0.93 and the distances of closest approach along the beam direction must be less than 20 cm. The decay length of the reconstructed K 0 S in the signal side decay is required to be more than twice that of the vertex resolution away from the interaction point. The invariant masses of these charged track pairs are required to be in the range [0.487, 0.511] GeV/c 2 .
Photons are reconstructed from the clusters of deposited energy in the EMC. The shower time is required to be within [0, 700] ns of the event start time in order to suppress electronics noise or e + e − beam background. Photon candidates within |cos θ| < 0.80 (barrel) are required to have an energy deposition larger than 25 MeV and those with 0.86 < |cos θ| < 0.92 (end-cap) must have an energy deposition larger than 50 MeV. To suppress the noise from hadronic shower splitoffs, the calorimeter positions of photon candidates must lie outside a cone of 10°from all charged tracks. The π 0 (η) candidates are reconstructed through π 0 → γγ (η → γγ) decays, with at least one barrel photon. The invariant mass of the photon pair for π 0 and η candidates must be in the ranges [0.115, 0.150] GeV/c 2 and [0.490, 0.580] GeV/c 2 , respectively. A kinematic fit that constrains the γγ invariant mass to the π 0 or η nominal mass [1] is performed to improve the mass resolution. The χ 2 of the kinematic fit is required to be less than 30. The η candidates are formed from the π + π − η combinations with an invariant mass within a range of [0.946, 0.970] GeV/c 2 .
D ± s candidates with M rec lying with the mass windows listed in Table 2 are retained for further study. The quantity M rec is defined as where E cm is the initial energy of the e + e − center-of-mass system, p Ds is the threemomentum of the D ± s candidate in the e + e − center-of-mass frame, and m Ds is the D ± s nominal mass [1].
A "K 0 S K" veto and a "D 0 " veto are applied on the signal D + s candidates. The Cabibbofavored D + s → K 0 S K + decay contributes background when the K + is misreconstructed as a π + . This background is reduced by a veto on the signal D + s , with S K + is the invariant mass of the K 0 S and reconstructed π + track, when it is assumed to be a kaon. There is also track-swap background where events through the exchange of K 0 S and K + , or π 0 and K − tracks. Events which simultaneously satisfy

Event selection
An eight-constraint kinematic fit is performed assuming the process e + e − → D * ± s D ∓ s → γD + s D − s , with D − s decaying to one of the tag modes and D + s decaying to the signal mode. The combination with the minimum χ 2 is chosen, assuming that a D * + s meson decays to D + s γ or a D * − s meson decays to D − s γ. In addition to the constraints of four-momentum conservation in the e + e − center-of-mass system, the invariant masses of (γγ) π 0 , (π + π − ) K 0 S , tag D − s , and D * ± s candidates are constrained to the corresponding nominal masses [1]. In order to ensure that all candidates fall within the phase-space boundary, the constraint of the signal D + s mass is added to the kinematic fit and the updated four-momenta are used for the amplitude analysis.
Furthermore, it is required that the energy of the transition photon from D * ± s → γD ± s is smaller than 0.18 GeV and the mass recoiling against this photon and the signal D + s candidate lies within the range [1.952, 1.995] GeV/c 2 . Finally, a mass window, [1.930, 1.990] GeV/c 2 , is applied on the signal D + s candidates. Fig. 1 shows the fits to the invariant-mass distributions of the accepted signal D + s candidates, M sig , for various data samples. In the fits, the signal is described by an MC-simulated shape convolved with a Gaussian resolution function, and the background is described by a second-order Chebyshev function. There are 352, 193, and 64 events retained for the amplitude analysis with purities, w sig , of 88.9 ± 6.8%, 84.6 ± 8.3%, and 75.9 ± 14.3% for the data samples at √ s = 4.178 GeV, 4.189-4.219 GeV, and 4.226 GeV, respectively.

Fit method
The intermediate-resonance composition in the decay D + s → K 0 S π + π 0 is determined by an unbinned maximum-likelihood fit to data. The likelihood function is constructed with a probability density function (PDF), which depends on the momenta of the three daughter particles. The amplitude of the nth intermediate state (A n ) is  The total amplitude M is then the coherent sum of the amplitudes of intermediate processes, M = c n A n , where c n = ρ n e iφn is the corresponding complex coefficient. The magnitude ρ n and phase φ n are free parameters in the fit, and are defined relative to those of a reference mode, for which they are fixed. The signal PDF f S (p j ) is given by where (p j ) is the detection efficiency parameterized in terms of the final four-momenta p j . The index j refers to the different particles in the final states, and R 3 (p j ) is the standard element of three-body phase space. The normalization integral is determined by an MC integration, where k is the index of the kth event and N M is the number of the selected MC events.
Here M g (p j ) is the PDF used to generate the MC samples in MC integration. To account for any bias caused by differences in PID and tracking efficiency between data and MC simulation, each signal MC event is weighted with a ratio, γ (p), of the efficiency of data to that of MC simulation and the MC integration then becomes Finally, the log-likelihood is written as 5) where N D is the number of candidate events in data.
Equation (4.5), however, is only appropriate for a data sample with negligible background. A signal-background combined PDF is introduced to account for the approximate 15% of background in this analysis. The background PDF is given by The background shape B(p j ) is derived from about 3500 background events selected from generic MC samples using RooNDKeysPdf [19] with the Dalitz distribution of M 2 S π 0 as input. This background PDF is added to the signal PDF incoherently. Then, the combined PDF is written as (4.7) Thus, (p j ) is factorized from the background PDF to make it a comparable equation with the signal PDF. In this way, the (p j ) term, which is independent of the fitted variables, is regarded as a constant and can be dropped during the log-likelihood fit. The background shape is extracted from the selected events, hence one has to manually divide the background PDF by the efficiency, B ≡ B/ . As a consequence, the combined PDF becomes (4.8) Next, the integration in the denominator of the background term can also be handled by the MC integration method in the same way as for the signal only sample: Eventually, for the case with a non-negligible background, Eq. (4.5) is extended to become where i indicate the data sample and w i sig are fixed during the log-likelihood fit.

Blatt-Weisskopf barrier factors
For the process a → bc, the Blatt-Weisskopf barrier F L (p j ) is parameterized as a function of the angular momenta L and the momenta q of the daughter b or c in the rest system of a,

Propagator
The intermediate resonances K * (892) 0,+ and K * (1410) 0 are parameterized as relativistic Breit-Wigner functions, where s a denotes the invariant-mass squared of the parent particle; m 0 and Γ 0 are the rest masses and the widths of the intermediate resonances, respectively, and are fixed to the PDG values [1]. We parameterize the ρ + and ρ(1450) + resonances by the Gounaris-Sakurai lineshape [20], which is given by . (4.13) The function f (m) is given by The normalization condition at P GS (0) fixes the parameter d = f (0)/(Γ 0 m 0 ). It is found to be (4.17)

Spin factors
The spin-projection operators are defined as [21] P  The quantities p a , p b , and p c are the momenta of particles a, b, and c, respectively, and r a = p b − p c . The covariant tensors are given bỹ The spin factors for S, P , and D wave decays are where theT (l) factors have the same definition ast (l) . The tensor describing the D + s decay is denoted byT and that of the a decay is denoted byt.

Fit results
The Dalitz plot of M 2 K 0 S π 0 versus M 2 K 0 S π + summed over all the data samples is shown in Fig. 2. We can see an anti-diagonal band corresponding to K 0 S ρ + . In the fit, the magnitude and phase of the reference amplitude D + s → K 0 S ρ + are fixed to 1.0 and 0.0, respectively, and the masses and widths of all resonances are fixed to the corresponding PDG averages [1]. In addition to the dominating amplitude D + s → K 0 S ρ + , we have tested for the contribution of all possible intermediate resonances including K * (892) 0 , K * (892) + , K * (1410), K * 0 (1430), K * 2 (1430), ρ(1450), K * (1680), ρ(1700), etc. We find that D + s → K 0 S ρ(1450) + , D + s → K * + (892)π 0 , and K * (1410) 0 π + have a statistical significances greater than three standard deviations and retain these amplitudes in the final model.
The calculation of the fit fractions (FFs) for individual amplitudes, involves the phasespace MC truth information without detector acceptance or resolution effects. The FF for the nth amplitude is defined as where N gen is the number of phase-space MC events at generator level. These FFs will not sum to unity if there is net constructive or destructive interference. Interference IN between the nth and the n th amplitudes is defined as (for n < n only) In order to determine the statistical uncertainties of FFs the amplitude coefficients are randomly sampled by a Gaussian-distributed amount set by the fit uncertainty and the covariance matrix. Then the distribution of each FF is fitted with a Gaussian function and the width of the Gaussian function is defined as the uncertainty of the FF. The magnitudes, phases, FFs, and significances for the amplitudes are listed in Table 4. The interference between amplitudes is listed in Table 5. The Dalitz plot projections are shown in Fig. 3. The assignment of systematic uncertainties is discussed in next section.  Table 4. Magnitudes, phases, FFs, and significances for the amplitudes. The uncertainties in the magnitudes are statistical only. The first and the second uncertainties in the phases and FFs are statistical and systematic, respectively. The total FF is 86.9%.

Amplitude
Magnitude Table 5. Interference between amplitudes, in % of the total amplitude. A denotes D +

Systematic uncertainties for amplitude analysis
The systematic uncertainties for the amplitude analysis are summarized in Table 6, with their assignment described below. The other source of potential bias arised from the knowledge of the background distributions. We follow an alternative procedure by determining the background shape with another two variables, M 2 K 0 S π + versus M 2 π + π 0 , and change the smooth parameters in RooNDKeysPdf [19]. This resulting change in results is small enough to be ignored and so we assign no uncertainty from this source. Table 6. Systematic uncertainties on the φ and FFs for each amplitude in units of the corresponding statistical uncertainties. The sources are: (i) Fixed parameters in the amplitudes, (ii) The R values, (iii) Background, (iv) Experimental effects, (v) Fit bias.

Amplitude
Source iv Experimental effects. To estimate the systematic uncertainty related to the difference in acceptance between MC and data associated with the PID and tracking efficiencies, that is γ in Eq. (4.4), the amplitude fit is performed varying the PID and tracking efficiencies according to their uncertainties.
v Fit bias. The amplitude analysis is performed on three-hundred data-sized signal MC samples and the pulls, which are the normalized-residual distributions of the fit, are inspected to look for biases or significant excursions from a normal distribution.
These studies indicate that the FFs of D + s → K * (892) 0 π + and D + s → K * (1410) 0 π + are slightly biased. In addition, the statistical uncertainties of the FF and the phase of D + s → K * (892) 0 π + are underestimated. Therefore, we correct the biased FFs by the mean values of the pull distributions and scale the underestimated uncertainties by the widths of the pulls. The systematic uncertainty due to the correction is assigned as the uncertainty of the mean value. An additional systematic uncertainty due to the normalization is taken into account by √ 2f ∆f , where f is the fitted width and ∆f is its uncertainty [22,23].

Branching fraction measurement of
With the selection criteria described in Sec. 3, the best tag candidate with M rec closest to the D ± s nominal mass [1] is chosen if there are multiple ST candidates. The yields for various tag modes are listed in Table 7 and obtained by fitting the corresponding M tag distributions. As an example, the fits to the data sample at √ s = 4.178 GeV are shown in Fig. 4. In the fits, the signal is modeled by an MC-simulated shape convolved with a Gaussian function to take into account the data-MC resolution difference. The background is described by a second-order Chebyshev function. MC studies show that there is no significant peaking background in any tag mode, except for D − → K 0 S π − and D − s → ηπ + π − π − faking the D − s → K 0 S K − and D − s → π − η tags, respectively. Therefore, in the fits, the MC-simulated shapes of these two peaking background sources are added to the background polynomial functions. Tag mode Once a tag mode is identified, we search for the signal decay D + s → K 0 S π + π 0 . In the case of multiple candidates, the DT candidate with the average mass, (M sig + M tag )/2, closest to the D ± s nominal mass is retained. To measure the BF, we start from the following equations for a single tag mode: 2) where N D + s D − s is the total number of D * ± s D ∓ s pairs produced from the e + e − collisions; N ST tag is the ST yield for the tag mode; N DT tag,sig is the DT yield; B tag and B sig are the BFs of the tag and signal modes, respectively; ST tag is the ST efficiency to reconstruct the tag mode; and DT tag,sig is the DT efficiency to reconstruct both the tag and the signal decay modes. In the case of more than one tag modes and sample groups, where α represents tag modes in the ith sample group. By isolating B sig , we find where N ST α,i and ST α,i are obtained from the data and generic MC samples, respectively, while DT α,sig,i is determined with signal MC samples, where D + s → K 0 S π + π 0 events are generated according to the results of the amplitude analysis. The two branching ratios B K 0 S →π + π − and B π 0 →γγ have been introduced to account for the fact that the signal is reconstructed through these decays.
The DT yield N DT total is found to be 666 ± 37 from the fit to the M sig distribution of the selected D + s → K 0 S π + π 0 candidates. The fit result is shown in Fig. 5. In the fit, the signal shape is described by an MC-simulated shape convolved with a Gaussian function to take into account the data-MC resolution difference. The background shape is described by an MC-simulated shape, which includes the small peaking background (2.1%) that is mainly from D + s → π + π + π − π 0 decays. The width of the Gaussian function is fixed to be 1.9 ± 1.1 MeV/c 2 , which is extracted from the control sample of D + s → K 0 S K + π 0 decays. Note that the DT yield is larger than the fit yields of Fig. 1 since the selection for the BF measurement is looser than that for the amplitude analysis and no kinematic fit is applied in the BF measurement.
We take the differences in pion tracking efficiency between data and MC simulation into account, and apply a correction to the MC signal efficiency of +0.3%. The differences in PID efficiency are negligible. The BF is determined to be B(D + s → K 0 S π + π 0 ) = (5.43 ± 0.30 stat ± 0.15 syst ) × 10 −3 .
In order to test CP conservation in the decay, the BFs are measured separately for the -14 - charge-conjugated modes. The BFs of D + s → K 0 S π + π 0 and D − s → K 0 S π − π 0 are measured to be (5.33±0.41 stat ±0.15 syst )×10 −3 and (5.63±0.44 stat ±0.16 syst )×10 −3 , respectively. From these measurements the asymmetry of the BFs is determined to be (2.7 ± 5.5 stat ± 0.9 syst )% Hence, no CP violation is observed. Note that the systematic uncertainties related to K 0 S and π 0 reconstructions cancel in the A CP calculation.
The following sources of the systematic uncertainties are taken into account for the BF measurement.
• Signal shape. The systematic uncertainty due to the signal shape is studied by repeating the fit with an alternative width of the convolved Gaussian. This width is varied according to the uncertainty of the control sample.
• Background shape. Since qq or non-D * ± s D ∓ s open charm are the major background sources, we alter the MC shapes by varying the relative fractions of the background from qq or non-D * + s D − s open charm by ±30%. The largest change is taken as the corresponding systematic uncertainty.
• π + tracking/PID efficiency. The π + tracking and PID efficiencies are studied with e + e − → K + K − π + π − events. The data-MC efficiency ratios of the π + tracking and PID efficiencies are 1.003 ± 0.002 and 1.000 ± 0.002, respectively. After multiplying the signal efficiencies by the factor 1.003, we assign 0.2% and 0.2% as the systematic uncertainties arising from π + PID and tracking, respectively.
• K 0 S reconstruction. The systematic uncertainty from the K 0 S reconstruction efficiency is assigned to be 1.5%, determined from studying a control sample of ψ(3770) → DD events containing hadronic D decays.
• π 0 reconstruction. A control sample of the process e + e − → K + K − π + π − π 0 is used to study the uncertainty due to π 0 reconstruction, which is assigned as 2.0%.
• MC statistics. The uncertainty due to the limited MC statistics is obtained by where f i is the tag yield fraction, and i and δ i are the signal efficiency and the corresponding uncertainty of tag mode i, respectively.
• Dalitz model. The uncertainty from the Dalitz model is estimated by varying the Dalitz model parameters based on their error matrix. The distribution of 600 efficiencies resulting from this variation is fitted by a Gaussian function and the deviation from the nominal mean value is taken as an uncertainty.
• Peaking background. The uncertainty of the peaking background is about 8% and corresponds to only one event in the D + s → K 0 S π + π 0 decay. Therefore the associated uncertainty in the BF measurement is negligible.
All of the systematic uncertainties are summarized in Table 8. Adding them in quadrature gives a total systematic uncertainty in the BF measurement of 2.8%.
These results can be compared to the current theoretical predictions [4][5][6]. The predictions in Ref [5] are consistent with our results, but their large uncertainties make the comparisons less conclusive. The calculations in Ref [6] have small uncertainties, while the predicted B(D + s → K 0 ρ + ) is over five standard deviations off the measured one. The predictions in Ref [4] have moderate uncertainties and match our measurements in principle, but the predicted B(D + s → K * (892) + π 0 ) is only marginally consistent with our measurement. Based on the current experimental and theoretical precisions, it is difficult to draw a definite conclusion to discriminate between models yet.
The asymmetry for the BFs of the decays D + s → K 0 S π + π 0 and D − s → K 0 S π − π 0 is determined to be (2.7 ± 5.5 stat ± 0.9 syst )%. No evidence for CP violation is found. S π + π 0 . The first and second uncertainties are statistical and systematic, respectively.