Measurements of absolute branching fractions for $D$ mesons decays into two pseudoscalar mesons

Using a data sample of $e^+e^-$ collision data with an integrated luminosity of 2.93 fb$^{-1}$ taken at the center-of-mass energy $\sqrt s= 3.773$~GeV with the BESIII detector operating at the BEPCII storage rings, we measure the absolute branching fractions of the two-body hadronic decays $D^+\to \pi^+\pi^0$, $K^+ \pi^0$, $\pi^+ \eta$, $K^+\eta$, $\pi^+\eta^\prime$, $K^+\eta^\prime$, $K_S^0 \pi^+$, $K_S^0 K^+$, and $D^0\to \pi^+ \pi^-$, $K^+ K^-$, $K^\mp \pi^\pm$, $K_S^0 \pi^0$, $K_S^0 \eta$, $K_S^0 \eta^\prime$. Our results are consistent with previous measurements within uncertainties. Among them, the branching fractions for $D^+\to\pi^+\pi^0$, $K^+\pi^0$, $\pi^+\eta$, $\pi^+\eta^\prime$, $K_S^0 \pi^+$, $K_S^0 K^+$ and $D^0 \to K_S^0 \pi^0$, $K_S^0 \eta$, $K_S^0 \eta^\prime$ are determined with improved precision compared to the world average values.


I. INTRODUCTION
The two-body hadronic decays D → P 1 P 2 (throughout the text, D represents the D + and D 0 mesons and P denotes one of the pseudoscalar mesons π ± , K ± , K 0 S , π 0 , η and η ) serve as an ideal testbed to improve the understanding of the weak and strong interactions in decays of charmed mesons. These reactions proceed via external W -emission, internal W -emission or W -exchange processes. Due to the relatively simple topology, the amplitude of D → P 1 P 2 decay can be theoretically derived as a sum of different diagrams based on SU(3)-flavor symmetry [1]. Comprehensive and improved experimental measurements of the branching fractions for these decays may help to validate the theoretical calculations and provide important and complementary data to explore the effect of SU(3)-flavor symmetry breaking in hadronic decays of the D mesons [2][3][4][5].
Historically, experimental studies of singly or doubly-Cabibbo-suppressed (DCS) decays of D → P 1 P 2 with branching fractions at the 10 −4 level were challenging due to limited statistics and high background. In recent years, the D → P 1 P 2 decays have been widely studied in various experiments [6][7][8][9][10]. The BESIII Collaboration has recently reported measurements of the branching fractions for some D +(0) → P 1 P 2 decays [11][12][13][14] by analyzing the data sample corresponding to an integrated luminosity of 2.93 fb −1 [15] taken at the centerof-mass energy √ s = 3.773 GeV. Single-tag or doubletag methods, in which one or two D mesons are fully reconstructed, have been used in previous works. Analyzing the same data sample with the single-tag method, we report in this paper the measurements of the absolute branching fractions of the two-body hadronic decays includes both the Cabbibofavored decay of D 0 → K − π + and the DCS decay of D 0 → K + π − . Throughout this paper, charge conjugated modes are implied.

II. BESIII DETECTOR AND MONTE CARLO SIMULATION
The BESIII detector is a cylindrical detector with a solid-angle coverage of 93% of 4π that operates at the BEPCII collider. 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 provides a measurement of ionization energy loss (dE/dx) that is used for charged particle identification (PID). An array of time-of-flight counters (TOF) is located outside the MDC and provides further information for PID. 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 yoke of the magnet is instrumented with about 1272 m 2 resistive plate muon counters (MUC), arranged in nine layers in the barrel and eight layers in the end-caps, that are used to identify muons with momenta greater than 0.5 GeV/c. More details about the BESIII detector are described in Ref. [16].
A GEANT4-based [17] Monte Carlo (MC) simulation software package, which includes the geometric description of the detector and its response, is used to determine the detection efficiency and to estimate the potential background. An inclusive MC sample, which includes D 0D0 , D + D − and non-DD decays of the ψ(3770), Initial State Radiation (ISR) production of the ψ(3686) and J/ψ, e + e − → qq (q = u, d, s) continuum processes, Bhabha scattering events, di-muon events and di-tau events, is produced at √ s = 3.773 GeV. The ψ(3770) production is simulated by the MC generator KKMC [18], in which the effects of ISR [19] and Final State Radiation (FSR) [20] are considered. The known decay modes are generated using EvtGen [21] with the branching fractions taken from the Particle Data Group (PDG) [22], and unknown decay modes are generated using LundCharm [23].
All charged tracks, except for those from a K 0 S decay, are required to originate from the interaction region defined as V xy < 1 cm and |V z | < 10 cm, where V xy and |V z | denote the distances of the closest approach of the reconstructed track to the Interaction Point (IP) in the xy plane and in the z direction (along the beam direction), respectively. The polar angles of the charged tracks θ is required to satisfy |cosθ| < 0.93. Charged tracks are identified using confidence levels for the kaon (pion) hypothesis CL K(π) , calculated with both dE/dx and TOF information. The kaon (pion) candidates are required to satisfy CL K(π) > CL π(K) and CL K(π) > 0.
The K 0 S candidates are formed from two oppositely charged tracks with |V z | < 20 cm and |cosθ| < 0.93. The two charged tracks are assumed to be a π + π − pair without PID and are constrained to originate from a common decay vertex. To suppress the π + π − combinatorial background, the reconstructed decay length of the K 0 S candidate is required to be greater than twice its uncertainty. The π + π − invariant mass must be within the signal region, defined as ±0.012 GeV/c 2 around the K 0 S nominal mass [10].
The photon candidates are selected from isolated EMC clusters. To suppress the electronics noise and beam background, the clusters are required to start within 700 ns after the event start time and fall outside a cone angle of 10 • around the nearest extrapolated charged track. The minimum energy of each EMC cluster is required to be larger than 25 MeV in the barrel region (| cos θ| < 0.80) or 50 MeV in the end-cap region (0.86 < | cos θ| < 0.92) [16]. To select the π 0 and η meson candidates, the γγ invariant mass is required to be within (0.115, 0.150) GeV/c 2 and (0.515, 0.575) GeV/c 2 , respectively. The momentum resolution of π 0 and η is further improved with a kinematic fit that constrains the γγ invariant mass to the π 0 or η nominal mass [10]. For η mesons, the π + π − η invariant mass is required to be within the signal region, which is ±0.012 GeV/c 2 around the nominal η mass [10].
For D 0 decays to π + π − , K + K − and K ∓ π ± , the backgrounds arising from cosmic rays and Bhabha events are rejected with the same requirements as those used in Ref. [24], i.e., the two charged tracks must have a TOF time difference less than 5 ns and must not be consistent with the requirement for a muon pair or an electronpositron pair. Furthermore, at least one EMC cluster with an energy larger than 50 MeV or at least one additional charged track detected in the MDC is required.
At the ψ(3770) peak, the DD meson pairs are produced without additional particles, thus, the energies of the D mesons are equal to the beam energy E beam in the center-of-mass frame of the e + e − system. Two variables reflecting energy and momentum conservation are used to identify the D meson candidates. They are the energy difference and the beam-energy-constrained mass where E i and p i are the energy and momentum of the decay products of the D candidates in the center-of-mass frame of the e + e − system. For a given D decay mode, if there is more than one candidate per charm per event, the one with the least |∆E| is kept for further analysis. The combinatorial backgrounds are suppressed by mode dependent ∆E requirements, which correspond to ±3.0σ ∆E around the fitted ∆E peak, where σ ∆E is the resolution of the ∆E distribution. Figures 1 and 2 show the M BC distributions of the accepted single-tag D + and D 0 candidates, respectively. The signal yields of D mesons for the different processes are determined using unbinned maximum likelihood fits to the corresponding distributions, where the signal probability density function (PDF) is modeled by the MCsimulated shape convolved with a double Gaussian function that describes the resolution difference between data and MC simulation. The combinatorial background is described with an ARGUS function [25] with the endpoint fixed at E beam . For the DCS decay D + → K + π 0 , MC studies show that the sizeable peaking background from D + → K 0 S (→ π 0 π 0 )π + can not be ignored. Thus, in the M BC fit for this decay, the size and shape of the background D + → K 0 S (→ π 0 π 0 )π + are fixed based on MC simulation.
For the decays including K 0 S (η ) mesons in the final states, there are peaking backgrounds from non-K 0 S (nonη ) events in the K 0 S (η ) signal regions around the nominal D mass in the M BC distributions. To estimate these peaking backgrounds, the events in the K 0 S (η ) sideband regions, defined as 0.020 < |M π + π − (π + π − η) − M K 0 S (η ) | < 0.044 GeV/c 2 , are used. Figure 3 shows the distributions of M π + π − , M π + π − η as well as M π + π − versus M π + π − η for the D 0 → K 0 S η candidate events in data. In Fig. 3(a) and (b), the regions between the pair of solid (dashed) arrows denote the K 0 S and η signal (sideband) regions. To estimate the non-K 0 S and non-η peaking backgrounds in D 0 → K 0 S η decays, 2-dimensional (2D) signal and sideband regions, as shown in Fig. 3(c), are used. The solid box is the 2D signal region, where both of the π + π − and π + π − η combinations lie in the K 0 S and η signal regions, respectively. The dashed (dotted) boxes indicate the 2D sideband A (B) regions, in which one (both) of the π + π − and π + π − η combinations lie in the K 0 S (η ) sideband regions.
The yields of peaking backgrounds in the K 0 S (η ) sideband regions in data are obtained with similar fits to the corresponding M BC distributions. For the decays with a K 0 S or η alone in the final status, the net signal yields N net are obtained according to where N sig and N sb are the observed number of events in the signal and sideband regions, respectively, as obtained in the fit. In the decay D 0 → K 0 S η , the net signal yield  is estimated by where N sbA and N sbB denote the peaking background yield in the sideband regions A and B, respectively.

IV. BRANCHING FRACTION
The branching fraction of the D → P 1 P 2 decay is determined according to where N net is the background-subtracted signal yields of the data; N tot DD is the total number of DD pairs, which  Distributions of (a) M π + π − , (b) M π + π − η and (c) M π + π − versus M π + π − η of the D 0 → K 0 S η candidate events in data, where the regions between the pair of the solid (dashed) arrows denote the K 0 S (η ) signal (sideband) regions, the solid, dashed and dotted boxes denote the signal, sideband A and sideband B regions (see text), respectively. is (8, 296 ± 31 ± 64) × 10 3 for D + D − and (10, 597 ± 28 ± 87) × 10 3 for D 0D0 [26]; ε is the detection efficiency obtained by the MC simulation, and B denotes the product branching fractions of the intermediate resonances π 0 , η, K 0 S and η in the cascade decays. The detection efficiency ε is determined by analyzing the inclusive MC sample with the same analysis procedure as applied to the data, including the M BC fit and the background estimation. Because of the relatively high backgrounds in the DCS decays of D + → K + π 0 , K + η and K + η , their detection efficiencies are determined from MC samples of ψ(3770) → D + D − in which one D is forced to decay into a signal mode and the other decays generically. By fitting the M BC distributions we obtain the net signal yield from the MC samples for each decay. The detection efficiency is obtained by dividing the net signal yield by the total number of the produced signal events. To better describe the data, the MC simulated efficiencies are corrected by the differences between data and MC simulation as discussed in Sect. V.
Inserting the values of N net , N tot DD , ε and B i in Eq.
(5), we obtain the branching fractions of the decays of interest, as listed in Table I. For the branching fractions measured in this work, the first uncertainty is statistical and the second one is systematic. By subtracting the branching fraction of DCS decay D 0 → K + π − [10] from that of D 0 → K ∓ π ± , we obtain the branching fraction of D 0 → K − π + to be (3.882 ± 0.006 ± 0.051)%. Table II summarizes the sources of the systematic uncertainties in the branching fraction measurements. The uncertainties are estimated relative to the measured branching fractions and are described below.

V. SYSTEMATIC UNCERTAINTY
• N tot DD : The total number of DD pairs produced in data are cited from our previous work [26]. They are determined with a combined analysis in which both single-tag and double-tag events are used. Their uncertainties are included in our measure-ment.
• Tracking (PID) of K + (π + ): The tracking (PID) efficiencies for K + (π + ) are studied by using doubletag DD hadronic events. Small differences in the tracking (PID) efficiencies of K + (π + ) between data and MC simulation (denoted as data-MC differences) have been observed. To better describe the data, the MC simulated efficiencies are corrected by the momentum dependent data-MC differences for the K + or π + . Afterwards, the systematic uncertainty for tracking (PID) is assigned as 1.0% (0.6%) for each pion from η decays, and 0.3% (0.3%) per track for the others.
• K 0 S reconstruction: The K 0 S reconstruction efficiency, including the tracking efficiency for charged pions, is studied with control samples of J/ψ → K * (892) ∓ K ± with K * (892) ± → K 0 S π ± and J/ψ → φK 0 S K ± π ∓ . Small data-MC differences are found, as presented in Ref. [27]. We correct the MC efficiencies for these differences and assign a systematic uncertainty of 1.5% for each K 0 S . • π 0 and η reconstruction: The π 0 reconstruction efficiency is verified with double-tag hadronic events D 0 → K − π + and K − π + π + π − versusD 0 → K − π + π 0 and K 0 S (π + π − )π 0 . Small data-MC differences for the π 0 reconstruction efficiencies are found and are corrected to the MC simulation efficiencies. After corrections, the uncertainty for the π 0 reconstruction efficiency is taken as 1.0%. The uncertainty for the η reconstruction efficiency is assigned as 1.0%, too. • Background estimation: The uncertainty from the K 0 S (η ) sideband region is examined by changing the scale factors based on MC simulations and by shifting the K 0 S (η ) signal or sideband regions by ±2 MeV/c 2 . The maximum changes of the branching fractions with respect to the nominal results are assigned as the systematic uncertainties due to background estimation.
For the decay of D + → K + π 0 , we also examine the effect of the fixed peaking background of D + → K 0 S (→ π 0 π 0 )π + by considering the uncertainties of its world average branching fraction [10], the tracking and PID for π + and the π 0 selection. The effect is found to be negligible.
• MC statistics: The uncertainty in the efficiencies due to limited MC statistics is taken into account.
• Quantum coherence (QC) effects: Since D 0 andD 0 are coherently produced in the process e + e − → ψ(3770) → D 0D0 , quantum correlation is considered with a method introduced in Ref. [28] when measuring the signal yields. The correction factors are included in the signal yields listed in Table I. The parameters are quoted from the PDG [10] and Heavy Flavor Averaging Group [29] and their uncertainties propagate to the branching fractions as systematic uncertainties.
Assuming all the uncertainty sources are independent, the quadratic sum of these uncertainties gives the total systematic uncertainty in the measurement of the branching fraction for each decay.