Measurements of the branching fractions for $D^+\to K^0_SK^0_SK^+$, $K^0_SK^0_S\pi^+$ and $D^0\to K^0_SK^0_S$, $K^0_SK^0_SK^0_S$

By analyzing $2.93\ \rm fb^{-1}$ of data taken at the $\psi(3770)$ resonance peak with the BESIII detector, we measure the branching fractions for the hadronic decays $D^+\to K^0_SK^0_S K^+$, $D^+\to K^0_SK^0_S \pi^+$, $D^0\to K^0_S K^0_S$ and $D^0\to K^0_SK^0_SK^0_S$. They are determined to be ${\mathcal B}(D^+\to K^0_SK^0_SK^+)=(2.54 \pm 0.05_{\rm stat.} \pm 0.12_{\rm sys.})\times 10^{-3}$, ${\mathcal B}(D^+\to K^0_SK^0_S\pi^+)=(2.70 \pm 0.05_{\rm stat.} \pm 0.12_{\rm sys.})\times 10^{-3}$, ${\mathcal B}(D^0\to K^0_SK^0_S)=(1.67 \pm 0.11_{\rm stat.} \pm 0.11_{\rm sys.})\times 10^{-4}$ and ${\mathcal B}(D^0\to K^0_SK^0_SK^0_S)=(7.21 \pm 0.33_{\rm stat.} \pm 0.44_{\rm sys.})\times 10^{-4}$, where the second one is measured for the first time and the others are measured with significantly improved precision over the previous measurements.


Introduction
Hadronic decays of D mesons open a window to probe for the physics mechanisms in charmed meson decays, e.g., CP violation, D 0 D0 mixing and SU(3) symmetry breaking effects.Since the discovery of D mesons in 1976, the hadronic decays of D mesons have been extensively investigated [1].However, the existing measurements of the D hadronic decays containing at least two K 0 S mesons in the final state are still very poor due to limited statistics [1].
In this Letter, we report the measurements of the branching fractions for the hadronic decays Throughout this Letter, charged conjugate modes are implied.These decays have simpler event topologies and suffer less from combinatorial backgrounds than other decay modes containing two K 0 S in the final state.The comprehensive or improved measurements of three-body decays will benefit the understanding of the interplay between the weak and strong interactions in multibody decays where theoretical predictions are poorer than two-body decays.The improved measurements of two-body decays can serve to better explore the contributions of W-exchange diagrams and final-state interactions [2,3,4,5], as well as SU(3)-flavor symmetry breaking effects [6,7,8,9,10] in D meson decays.In addition, these measurements will also help to improve background estimations in the precision measurements of D and B meson decays.
The data sample used for this analysis, which has an integrated luminosity of 2.93 fb −1 [11], was taken at the ψ(3770) resonance peak with the BESIII detector [12].The D 0 D0 and D + D − pairs produced in ψ(3770) decay provide cleaner D 0 and D + meson samples than those used in previous studies at ARGUS [13,14], CLEO [15,16] and FOCUS [17].To optimize the precision for these measurements, we use a single-tag method, in which either a D or D is reconstructed in an event.We combine the yields measured with previously reported values of the cross sections for e + e − → D 0 D0 and D + D − at the ψ(3770) resonance peak [18].

BESIII detector and Monte Carlo simulation
The BESIII detector is a magnetic spectrometer that operates at the BEPCII collider.It has a cylindrical geometry with a solid-angle coverage of 93% of 4π.It consists of several main components.A 43-layer main drift chamber (MDC) surrounding the beam pipe performs precise determinations of charged particle trajectories and measures the specific ionization (dE/dx) for charged particle identification (PID).An array of timeof-flight counters (TOF) is located outside the MDC and provides additional PID information.A CsI(Tl) electromagnetic calorimeter (EMC) surrounds the TOF and is used to measure the energies of photons and electrons.A solenoidal superconducting magnet outside the EMC provides a 1 T magnetic field in the central tracking region of the detector.The iron flux return of the magnet is instrumented with 1272 m 2 of resistive plate muon counters (MUC) arranged in nine layers in the barrel and eight layers in the endcaps for identification of muons with momentum greater than 0.5 GeV/c.More details about the BESIII detector are described in Ref. [12].
A GEANT4-based [19] Monte Carlo (MC) simulation software package, which includes the geometric description and response of the detector, is used to determine the detection efficiency and to estimate background for each decay mode.An inclusive MC sample, which includes the D 0 D0 , D + D − and non-D D decays of the ψ(3770), initial-state-radiation (ISR) production of the ψ(3686) and J/ψ, the e + e − → q q (q = u, d, s) continuum process, the Bhabha scattering events, the di-muon events and the di-tau events, is produced at √ s = 3.773 GeV.The equivalent luminosity of the MC sample is ten times of data.The ψ(3770) decays are generated by the MC generator KKMC [20], which incorporates both ISR effects [21] and final-stateradiation (FSR) effects [22].Known decay modes are generated using EvtGen [23] with input branching fractions from the Particle Data Group (PDG) [1].Unmeasured decays are generated using LundCharm [24].

Data analysis
All charged tracks used in this analysis are required to be within a polar-angle (θ) range of |cos θ| < 0.93.The good charged tracks, except when used to reconstruct K 0 S mesons, are required to originate within an interaction region defined by V xy < 1.0 cm and V z < 10.0 cm, where V xy and V z are the distances of closest approach of the reconstructed track to the interaction point (IP) perpendicular to (xy) and along (z) the beam direction.
The charged kaons and pions are identified by the dE/dx and TOF measurements.The combined confidence levels for pion and kaon hypotheses (CL π and CL K ) are calculated, respectively.The charged track is identified as kaon (pion K 0 S candidate mesons are reconstructed through the π + π − decay mode.Charged pions used in K 0 S candidates mesons are required to satisfy V z < 20.0 cm.The two oppositely charged tracks are assumed to be a π + π − pair without PID requirements.To reconstruct K 0 S , the π + π − combination is constrained to have a common vertex.The candidate is accepted if it has an invariant mass M π + π − within 12 MeV/c 2 of the K 0 S nominal mass [1] and satisfies L/σ L > 2, where L is the measured flight distance and σ L is its uncertainty.
To identify D candidates, we use two selection variables, the energy difference ∆E ≡ E D − E beam and the beam-energy-constrained mass , where E beam is the beam energy and E D and p D are the energy and momentum of the D candidate in the e + e − center-of-mass system.For each signal decay mode, only the combination with the minimum |∆E| is kept in events where more than one candidate passes the selection requirements.Modedependent ∆E cuts are determined separately for data and MC based on fits to the respective ∆E distributions.These are set at ±3σ, where σ is the ∆E resolution (Table 1).
The combinatorial π + π − | non−K 0 S pairs with invariant mass in K 0 S signal region may also satisfy the K 0 S selection criteria and contribute peaking background around the D mass in the M BC distribution.This peaking background is estimated with events in the K 0 S sideband region, defined as 0.020 S candidates in data with the corresponding distribution for the inclusive MC.In the figure, the solid (dashed) arrows delineate the K 0 S signal (sideband) regions.In the analyses of the events in data.The solid box, in which both of the π + π − combinations lie in the K 0 S signal regions, denotes the 2D signal region.The dot-dashed (dashed) boxes indicate the 2D sideband 1 (2) regions, in which one (two) of the π + π − combinations lie in the K 0 S sideband regions and the others are in the K 0 S signal region.For the of the candidate events in data is shown in Fig. 1 (c).The region in which all three π + π − combinations lie in the K 0 S signal regions is taken as the three-dimensional (3D) signal region.The 3D sideband i (i = 1, 2, 3) regions denote those in which i of the three π + π − pairs lie in the K 0 S sideband regions and the rest are located in the K 0 S signal regions.The resulting M BC distributions of the accepted candidate events in the 2D or 3D signal region, sideband 1 region and sideband 2 region are shown in the subfigures of the first, second and third rows of Fig. 2, re-

Decay modes
candidate events in data.In these figures, all selection criteria have been imposed except for the K 0 S mass requirement and M BC is required to be within 5 MeV/c 2 around the D nominal mass [1].
candidate events.The dots with error bars are data, the solid curves are the total fits, and the dashed curves are the fitted backgrounds.The first, second and third rows correspond to the fits to the candidate events in the 2D or 3D signal region, sideband 1 region and sideband 2 region, respectively.
spectively.By fitting these M BC distributions as shown in Fig. 2, we obtain the fitted yields of D signal in the 2D or 3D signal region, sideband 1 region and sideband 2 region, N K 0 S sig , N sb1 , N sb2 , which are given in Table 2.In the fits, the D signal is modeled by a MCsimulated shape convoluted with a Gaussian function with free parameters accounting for the difference of detector resolution between data and MC.The combinatorial backgrounds are described by an ARGUS function [25] with an endpoint of 1.8865 GeV/c 2 .In the M BC fits for the 2D or 3D sideband events, the parameters of the convoluted Gaussian function are fixed at the values determined for the signal region.For the D 0 → K 0 S K 0 S K 0 S decays, the peaking backgrounds from sideband 3 region are negligible since few events survive.
In this analysis, the combinatorial background in the M π + π − distribution are assumed to be flat, which implies that the ratio of background yields between the K 0 S signal and sideband regions is 0.5.Thus, the net numbers of the D 0 → K 0 S K 0 S , D + → K 0 S K 0 S K + and K 0 S K 0 S π + decays can be calculated by and the net number of the D 0 → K 0 S K 0 S K 0 S decays can be calculated by where N K 0 S sig and N sbi are D signal yields from the fit in the 2D or 3D signal regions and sideband i regions, respectively.N b other is the normalized number of residual peaking background.For the any possible particle combination).This kind of background peaks around the nominal D mass [1] when the K 0 S from a D − ( D0 ) decay has momentum similar to that of a K 0 L produced in D + (D 0 ) decay.These peaking backgrounds cannot be modeled by the events from the 2D or 3D sideband region and are estimated by analyzing the inclusive MC sample.The measured values of N b other and N net are given in Table 2.

Branching fractions
The branching fraction for the hadronic decay D +(0) → f is determined by where N net is the net number of D +(0) → f decays in data, ǫ is the detection efficiency including the branching fraction of K 0 S → π + π − , L is the integrated luminosity of data [11] and ) cross section at the ψ(3770) resonance peak.
The detection efficiencies are determined by analyzing the inclusive MC sample.In this sample, the signal MC events for D + → K 0 S K 0 S π + are produced as a mixed sample containing 90% of the D + → K 0 S K * (892) + , K * (892) + → K 0 S π + decays and 10% of the direct three-body decay in phase space S are produced using a phase-space model.Detailed studies show that the momentum and polar-angle distributions of the daughter particles in data are well modeled by the MC simulation for each decay mode.By analyzing the inclusive MC sample with the same analysis procedure applied to the data (including the M BC fits and the calculation of the net signal yields), we obtain the net number of D mesons observed for each decay.The detection efficiency ǫ is obtained by dividing the net D signal by the total number of signal events, taking into account the efficiency correction discussed in Sect. 5.

Systematic uncertainty
Table 3 shows the systematic uncertainties in the branching fraction measurements.Each of them, estimated relative to the measured branching fraction, is discussed below.
• MC statistics: The uncertainties due to the limited MC statistics are 0.5%, 0.4%, 1.8% and 1.3% for

respectively. • Luminosity of data:
The uncertainty in the quoted integrated luminosity of data is 0.5% [11].• D D cross section: The uncertainties of the quoted D + D − and D 0 D0 cross sections are 1.6% [18].
• K 0 S reconstruction: The K 0 S reconstruction efficiency has been studied as a function of momentum by using the control samples J/ψ → K * (892) ∓ K ± and J/ψ → φK 0 S K ± π ∓ .Small data-MC efficiency differences are found and presented in Ref. [26].To correct the K 0 S reconstruction efficiency, a piecewise fit to these differences as a function of K 0 S momentum is performed.For the efficiencies of detecting the decays , the momentum weighted differences associated with K 0 S reconstruction between data and MC are determined to be (+3.9± 1.9)%, (+3.0 ± 1.4)%, (+1.8 ± 0.8)% and (+5.9 ± 2.8)%, respectively, where the uncertainties are statistical.These corrections are applied to the detection efficiencies, after which only the statistical uncertainties of the differences are retained.On average, the residual uncertainty for each K 0 S is no more than 1.0%.Furthermore, the difference of the momentumweighted efficiencies between data and MC from the different fits, which is 1.0% per K 0 S , is included as an additional uncertainty.Finally, we assign 1.5% per K 0 S as the systematic uncertainty for the reconstruction efficiency .
• Tracking [PID] for K + (π + ): The tracking [PID] efficiencies for K + and π + are investigated using doubly tagged D D hadronic events.The difference of momentum weighted efficiencies between data and MC of the tracking [PID] are determined to be (+2.1 ± 0.4)% [(−0.3 ± 0.1)%] for the K + in the D + → K 0 S K 0 S K + decay and (+0.4 ± 0.3)% [(−0.3 ± 0.1)%] for the π + in the D + → K 0 S K 0 S π + decay, where the uncertainties are statistical.After correcting the detection efficiencies by these differences, we take 0.5% [0.5%] as the systematic uncertainties in tracking [PID] for the K + and π + , respectively.
which are assigned as the relevant systematic uncertainties.
• ∆E requirement: To investigate the systematic uncertainty associated with the ∆E requirement, we repeat the measurements using alternative ∆E requirements of ±(4, 5, 6) times the resolution around the ∆E peaks.The maximum changes of the branching fractions, 2.0%, 1.5%, 2.0% and 1.5% for , are taken as the associated systematic uncertainties.
• Normalization of peaking backgrounds: In the nominal analysis, the normalization factor for the peaking backgrounds, which is the ratio of background yields between the K 0 S signal and sideband regions, has been assumed to be 0.5.The branching fractions are recalculated with alternative normalization factors determined by MC simulation.The corresponding changes on the branching fractions, 0.5%, 1.4%, 2.4% and 0.7% for , are assigned as the systematic uncertainties associated with the peaking background (PBKG) normalization.On the other hand, the uncertainties of the residual peaking backgrounds are dominated by the uncertainties of the input branching fractions for D − ( D0 ) → K 0 S X, which contribute additional uncertainties of 0.1%, 0.1% and 0.4% for the measured branching fractions for • K 0 S sideband: To evaluate the systematic uncertainty due to the choice of K 0 S sideband region, we remeasure the branching fractions after shifting the K 0 S sideband by ±2 MeV/c 2 .The corresponding maximum changes in the branching fraction, which are 0.5%, 0.5%, 2.0% and 1.0% for , respectively, are taken as the systematic uncertainties.
• MC modeling: For the three-body decays, we examine the reweighted detection efficiencies by including the possible sub-resonances a 0 (980) and f 0 (980) in the signal MC samples.The maximum change of the reweighted detection efficiencies, 1.0%, is taken as the systematic uncertainty in MC modeling.
Adding all of above systematic uncertainties in quadrature, we obtain the total systematic uncertainties of 4.7%, 4.4%, 6.8% and 6.1% for

Summary
In summary, by analyzing 2.93 fb −1 of data collected at √ s = 3.773 GeV with the BESIII detector, we measure the branching fractions for the hadronic decays S using a singletag method.Table 4 presents the comparisons of the measured branching fractions with the PDG values [1].The branching fraction for D + → K 0 S K 0 S π + is measured for the first time and the others are consistent with previous measurements, but with much improved precision.We also determine the branching fraction ratios B(D + → K 0 S K 0 S K + )/B(D + → K 0 S K 0 S π + ) = 0.941 ± 0.025 stat.± 0.040 sys.and B(D 0 → K 0 S K 0 S )/B(D 0 → K 0 S K 0 S K 0 S ) = 0.232 ± 0.019 stat.± 0.016 sys., in which the systematic uncertainties in the D + D − (or D 0 D0 ) cross section, the integrated luminosity of data, as well as the reconstruction efficiencies and the branching fractions of the two K 0 S mesons cancel.The results in this analysis provide helpful experimental data to probe for the interplay between the weak and strong interactions in charmed meson decay [2,3,4,5].In addition, the measured branching fraction for the two-body decay D 0 → K 0 S K 0 S can also help to understand SU(3)-flavor symmetry breaking effects in D meson decays [6,7,8,9,10].

TABLE 1 :
∆E requirements (in MeV) for data and MC samples.

TABLE 2 :
Input quantities and results for the determination of the branching fractions as described in the text.The uncertainties are statistical only.

•
M BC fit: In order to estimate the systematic uncertainty associated with the M BC fit, we repeat the measurements by varying the fit range ((1.8415, 1.8865) GeV/c 2 ), signal shape (with different MC matching requirements) and endpoint of the ARGUS function (±0.2 MeV/c 2 ).