Measurements of branching fractions of leptonic and hadronic Ds+ meson decays and extraction of the Ds+ meson decay constant

We present measurements of absolute branching fractions of hadronic and leptonic Ds+ decays to K- K+ pi+, anti-K0 K+, eta pi+, mu+ nu_mu and tau+ nu_tau and report a search for the leptonic Ds+ to e+ nu_e decays. The results are obtained from a data sample of 913 fb^-1 collected at or near the Y(4S) and Y(5S) resonances with the Belle detector at the KEKB asymmetric-energy e+e- collider. The branching fractions of hadronic decays are measured to be Br(Ds+ ->K- K+ pi+) = (5.06 +- 0.15 +- 0.21)%, Br(Ds+ ->anti-K0 K+) = (2.95 +- 0.11 +- 0.09)%, and Br(Ds+ ->eta pi+) = (1.82 +- 0.14 +- 0.07)%, where the first and second uncertainties are statistical and systematic, respectively. The branching fractions of leptonic decays are measured to be Br(Ds+ ->mu+ nu_mu) = (0.531 +- 0.028 +- 0.020)%, and Br(Ds+ ->tau+ nu_tau) = (5.70 +- 0.21 +0.31-0.30)%, which are combined to determine the Ds+ meson decay constant f_Ds = (255.5 +- 4.2 +- 5.1) MeV. We find no significant signal for Ds+ ->e+ nu_e decays and set an upper limit of Br(Ds+ ->e+ nu_e)<1.0 (0.83) x 10^-4 at 95% (90%) confidence level.


Introduction
Precise determination of the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix leads to a deeper understanding of the flavor structure in the Standard Model (SM) and provides a portal to New Physics (NP) processes at higher energy scales. Many of the constraints on the CKM unitarity triangle given by the precise experimental results on decays of B mesons (see ref. [1] for a review of results from the Belle collaboration) rely on lattice gauge theory (LQCD) calculations of quantities that parameterize nonperturbative QCD contributions to weak decays and mixing (see section 17 in ref. [2] for a review). Among these quantities, the pseudoscalar meson decay constants play an important role -without them, for example, an interpretation of measurements of purely leptonic decays B + → τ + ν τ [3,4] and B 0 s → µ + µ − [5] that are particularly sensitive to NP contributions is not possible. In some NP scenarios, the leptonic decay rates of the D + s mesons could also be modified although the expected effects are smaller than in the B meson sector [6][7][8][9]. Measurements of leptonic decays of charmed hadrons, D + s → + ν where + = e + , µ + or τ + , therefore enable precision tests of LQCD calculations of decay constants performed in the charm sector and can provide additional constraints on NP. 1 Purely leptonic decays of mesons are among the simplest and theoretically cleanest processes. The branching fraction of D + s meson leptonic decays that proceed via the mutual annihilation of the c and s-quarks into a virtual W + boson is given in the SM by (1.1) Here, m Ds is the D + s meson mass, τ Ds is its lifetime, m is the lepton mass, V cs is the relevant CKM matrix element, and G F is the Fermi coupling constant. The parameter f Ds is the D + s meson decay constant and is related to the wave-function overlap of the meson's constituent quark and anti-quark. The leptonic decays of pseudoscalar mesons are suppressed by helicity conservation and their decay rates are thus proportional to the square of the charged lepton mass. Leptonic D + s decays into electrons with B ∼ 10 −7 are not observable yet whereas decays to taus are favored over decays to muons. In particular, the ratio of the latter decays is equal to R Ds τ /µ ≡ B(D + s → τ + ν τ )/B(D + s → µ + ν µ ) = m 2 τ /m 2 µ ·(1−m 2 τ /m 2 Ds ) 2 /(1−m 2 µ /m 2 Ds ) 2 = 9.762±0.031, based on the world average masses of the muon, tau and D + s meson given in ref. [2]. Any deviation from this expectation could only be interpreted as violation of lepton universality in charged currents and would hence point to NP effects [10].
In the context of the SM, a measurement of B(D + s → + ν ) determines the D + s meson decay constant since the magnitude of the CKM matrix element |V cs | is precisely determined from other measurements and the assumption that the CKM matrix is unitary. Measurements of f Ds have been made previously by several groups: CLEO [11][12][13], Belle [14] and BaBar [15]. The current world average is f exp Ds = (260.0 ± 5.4) MeV [2]. Within the SM, f Ds has been predicted using several methods [16][17][18][19][20][21][22][23] and most calculations give values lower than the f Ds measurement although within theoretical and experimental uncertainties. The largest discrepancy is with an unquenched LQCD calculation that yields f LQCD Ds = (248.0 ± 2.5) MeV [16]. Measurements of f Ds with an accuracy that matches the precision of theoretical calculations are thus necessary to check and further constrain theoretical methods.
Hadronic decays, D + s → K − K + π + and D + s → K 0 K + , are the reference modes for the measurements of branching fractions of the D + s decays to any other final state [2]. In addition, precise measurements of the absolute hadronic D + s meson branching fractions improve our knowledge of the B (s) decays involving D + s , such as B 0 (s) → D ( * )− s D ( * )+ (s) [2], and of most of the other branching fraction measurements of B s mesons performed at LHCb, like B 0 s → µ + µ − [5]. For B s -decay branching fraction measurements performed at LHCb, the key systematic uncertainty [24] is the ratio of fragmentation fractions f s /f d , whose experimental systematic error is dominated by B(D + s → K − K + π + ) [25,26]. 2 Normalization branching fractions, B(D + s → K − K + π + ) and B(D + s → K 0 K + ), have been measured so far only by CLEO [27] (see also the very recent update in ref. [28]). It is therefore important to provide new and independent measurements.
In this paper, we present results of absolute branching fraction measurements of D + s → µ + ν µ and D + s → τ + ν τ decays and perform a search for D + s → e + ν e decays. The measurement of B(D + s → µ + ν µ ) presented here supersedes the previous Belle measurement [14]. The analysis described here has a number of significant improvements, including an increased data sample and significantly improved inclusive D + s reconstruction efficiency. The combined effect of these improvements and the accompanying change in the extraction of relevant signal yields results in a reduction of the expected error of B(D + s → µ + ν µ ) by more than a factor of two. The new analysis has improved systematic uncertainties. In addition, we present first measurements of absolute branching fractions of the D + s normalization decays, D + s → K − K + π + and D + s → K 0 K + , and of D + s → ηπ + decays. This analysis is based on a data sample of 913 fb −1 recorded at and near the Υ(4S) and Υ(5S) resonances -well above the open charm threshold -by the Belle detector at the KEKB asymmetric-energy collider [29,30].
The rest of the paper is structured as follows. We describe the Belle detector and the data sample in section 2. In section 3, we present the method of measuring the absolute branching fraction of D + s decays. The inclusive and exclusive event reconstruction steps are described in sections 4 and 5, respectively. Determination of the absolute branching fractions is discussed in section 6. Systematic uncertainties are itemized in section 7. We summarize our results in section 8 and conclude in section 9.

Belle detector and data sample
The data used in this analysis were collected with the Belle detector at the KEKB asymmetric energy e + e collider. 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 timeof-flight scintillation counters (TOF), and an electromagnetic calorimeter (ECL) comprised of CsI(Tl) crystals located inside a superconducting solenoid coil that provides a 1.5 T magnetic field. An iron flux-return located outside of the coil is instrumented to detect K 0 L mesons and to identify muons (KLM). The detector is described in detail elsewhere [1,31]. Two inner detector configurations were used. A 2.0 cm diameter beampipe and a 3-layer silicon vertex detector was used for the first sample of 156 fb −1 , while a 1.5 cm diameter beampipe, a 4-layer silicon detector and a small-cell inner drift chamber were used to record the remaining 757 fb −1 .
Charged particles are reconstructed with the CDC and the SVD. Each is required to have an impact parameter with respect to the interaction point (IP) of less than 1.5 cm along the beam direction (z) and less than 0.5 cm in the transverse (r − φ) plane. A likelihood ratio for a given track to be a kaon or pion, L (K,π) , is obtained by utilizing specific ionization energy loss measurements in the CDC, light yield measurements from the ACC, and time-of-flight information from the TOF. For electron identification, we use position, cluster energy, and shower shape in the ECL, combined with track momentum and dE/dx measurements in the CDC and hits in the ACC. For muon identification, we extrapolate the CDC track to the KLM and compare the measured range and transverse deviation in the KLM with the expected values. Photons are detected with the ECL and are required to have energies in the laboratory frame of at least 50 (100) MeV in the ECL barrel (endcaps). Neutral pion candidates are reconstructed using photon pairs with an invariant mass between 120 and 150 MeV/c 2 , which corresponds to ±3.2 σ around the nominal π 0 mass [2], where σ represents the invariant mass resolution. Neutral kaon candidates are reconstructed using charged pion pairs with an invariant mass within ±20 MeV/c 2 (±5 σ) of the nominal K 0 mass.
We use Monte Carlo (MC) events generated with EVTGEN [32] and JETSET [33] and then processed through the detailed detector simulation implemented in GEANT3 [34]. QED final state radiation from charged particles is added during generation using the PHOTOS package [35]. The simulated samples for e + e − annihilation to qq (q = u, d, s, c, and b) are equivalent to six times the integrated luminosity of the data and are used to develop methods to separate signal events from backgrounds, identify types of background events, determine reconstruction efficiencies and the distributions needed for the extraction of the signal decays.

Method overview
The method of absolute branching fraction measurement of D − s decays is similar to the one previously used by Belle [14,36] and BaBar [15]. In this method, the e + e − → cc events that contain D − s mesons produced through the reactions are fully reconstructed in two steps. In these events, one of the two charm quarks hadronizes into a D * − s meson while the other hadronizes into a tagging charm hadron, denoted D tag , that is one of D 0 , D + , Λ + c , D * + or D * 0 . The strangeness of the event is conserved by requiring an additional kaon, denoted K frag , be produced in the fragmentation process; K frag is either K + or K 0 S . In events where Λ + c is the tagging charm hadron, the baryon number of the event is conserved by requiring an anti-proton. Since Belle collected data at energies well above the D ( * ) tag K frag D * − s threshold, additional particles can be produced in the course of hadronization. These particles are denoted as X frag and consist of an even number of kaons plus any number of pions or photons. In this measurement, only pions are considered when reconstructing the fragmentation system X frag . 3 We require D − s mesons to be produced in a D * − s → D − s γ decay, which provides a powerful kinematic constraint that improves the resolution of the missing mass (defined below) and suppresses the combinatorial background.
In the first step of the measurement, no requirements are placed on the daughters of the signal D − s meson in order to obtain an inclusive sample of D − s events that is used for normalization in the calculation of the branching fractions. The number of inclusively reconstructed D − s mesons is extracted from the distribution of events in the missing mass, M miss (D tag K frag X frag γ), recoiling against the D tag K frag X frag γ system where p miss is the missing four-momentum in the event Here, p e + and p e − are the known four-momenta of the colliding positron and electron beams, respectively, and p Dtag , p K frag , p X frag , and p γ are the measured four-momenta of the reconstructed D tag , strangeness-conserving kaon, fragmentation system and the photon from D * − s → D − s γ, respectively. Correctly reconstructed events described in eq. (3.1) produce a peak in the M miss (D tag K frag X frag γ) at the nominal D − s meson mass. In the second step of the analysis, we search for the decay products of a specific D − s meson decay within the inclusive D − s meson sample reconstructed in the first step. In particular, we reconstruct purely leptonic D − s → e − ν e , D − s → µ − ν µ , and D − s → τ − ν τ decays within the inclusive D − s sample by requiring an additional charged track identified as an electron, muon or charged pion in the rest of the event. In the case of D − s → τ − ν τ decays, the electron, muon or pion identifies the subsequent tau decay to e − ν e ν τ , µ − ν µ ν τ or π − ν τ , respectively. Hadronic decays, D − s → K 0 K − and ηπ − , are reconstructed partially by explicitly requiring only the charged kaon or pion (originating directly from D − s meson decay) in the rest of the event but with no requirements on the neutral hadrons (K 0 or η) in order to increase the reconstruction efficiency. In the case of D − s → K − K + π − , all three charged tracks are required in the rest of the event. More details are given in section 5.

Inclusive D ± s reconstruction
The reconstruction of the inclusive D ± s sample starts with the reconstruction of the tagging charmed hadron, D tag . To maximize the reconstruction efficiency with reasonable purity, the ground-state D tag hadrons (D 0 , D + , Λ + c ) are reconstructed in the 18 hadronic decay modes listed in table 1. Only modes with up to one π 0 in the final state are used to avoid large backgrounds. If D tag is reconstructed as Λ + c baryon, an additional anti-proton is required in order to conserve the baryon number in the event.
The magnitude of the center-of-mass (CMS) momentum of the D tag candidates is required to be greater than 2.3 GeV/c (or 2.5 GeV/c for the less clean D tag modes) to reduce the combinatorial background and e + e − → BB events. The decay products of the D tag candidate are fitted to a common vertex; candidates with a poor fit quality are discarded by 3.6 Λπ + π + π − 2.6 Sum 16.8 Table 1. Summary of D tag = D 0 , D + and Λ + c decay modes used in this measurement. The branching fractions are taken from ref. [2].
requiring χ 2 /n.d.f. < 20, where n.d.f. is the number of degrees of freedom of the kinematic fit. The purity of the D tag sample, given as a fraction of correctly reconstructed D tag candidates, is rather low at this stage -around 17% in the signal region, defined as ±3 σ interval around the nominal D tag mass, where σ is the D tag decay-mode-dependent invariant mass resolution that ranges from 4 to 12 MeV/c 2 . To further purify the D tag sample, we train a NeuroBayes [37] neural network using a small sample of data (0.7% of the total sample). The network combines information from the following input variables into a single variable: the distance between the decay and the production vertices of the D tag candidate in the r − φ plane, where the D tag production vertex is defined by the intersection of its trajectory with the IP region; the χ 2 /n.d.f. of the vertex fit of the D tag candidate; the cosine of the angle between the D tag momentum and the vector joining its decay and production vertices in the r − φ plane; for two-body decays, the cosine of the angle between the momentum of either D tag daughter and the boost direction of the laboratory frame in the D tag rest frame; the particle identification likelihood ratios; and, for the D tag decay modes with a π 0 , the smaller of the two photon energies. To obtain the signal and background distributions of the network's input variables, a statistical tool to unfold the data distributions ( s Plot [38]) is applied. Network is then applied to the complementary subsample (again representing around 0.7% of the total sample) that we use to optimize the selection on the network output variable for each D tag mode individually by maximizing S/ √ S + B, where S (B) refers to the signal (background) yield in the signal window of D tag invariant mass determined by performing a fit to the D tag invariant mass distribution. 4 After the optimization, the purity of the correctly reconstructed D tag candidates increases from 17% to 42% while only 16% of signal D tag candidates is lost. We retain only D tag candidates from the signal region of the D tag invariant mass in the rest of the analysis.
Once the ground-state D tag hadrons have been reconstructed, D 0 and D + mesons originating from D * decays are identified by reconstructing the decays D * + → D 0 π + , D + π 0 , and D * 0 → D 0 π 0 , D 0 γ. We do this to purify the subsequent K frag X frag γ reconstruction: by absorbing one more particle into the tagging charm hadron, the subsequent combinato-rial background is reduced. In addition, by reconstructing D * + → D 0 π + decays, we can determine the flavor or charm quantum number of D 0 or D 0 candidates reconstructed in final states with a K 0 S . The pion from the D * decay is refitted to the D production vertex to improve the resolution of the mass difference, ∆M = M (Dπ) − M (D). The laboratory frame energy of the photon(s) originating from the π 0 produced in D * → Dπ 0 (produced directly in D * 0 → D 0 γ) is required to be larger than 50 (175) MeV. In the D 0 γ final state, the γ candidate is combined with each other photon and, if the two-photon invariant mass is within 10 MeV/c 2 around the nominal π 0 mass and their energy asymmetry ) is smaller than 0.5, the D * 0 candidate is rejected. For all D * decays, the mass difference, ∆M , is required to be within 3 σ of the corresponding nominal mass difference.
For strangeness conserving kaon candidate, K frag , all K ± or K 0 S candidates that do not overlap with the D tag candidate are considered.
From the remaining tracks and π 0 candidates in the event that do not overlap with the D tag K frag candidate, we form the X frag candidates. Only modes with up to three pions and up to one π 0 are used to suppress the combinatorial background. In addition, pions must have a momentum above 100 MeV/c in the laboratory frame. At this stage, no requirement is applied to the total charge of the X frag system.
The D tag , X frag and K frag candidates are combined to form D tag K frag X frag combinations. We keep only combinations with total charge ±1; these constitute the inclusive sample of D * ∓ s mesons. The charm and strange quark content of the D tag K frag X frag system is required to be consistent with that recoiling from a D * s : if D tag is reconstructed in a flavor-specific decay mode and K frag is charged, the kaon charge and the charm quantum number of D tag must be opposite the D * s charge; if K frag is neutral the charm quantum number of D tag must be opposite the D * s charge; and if D tag is reconstructed in a self-conjugate decay mode, the charge of K frag must be opposite the D * s charge. All other candidates are rejected. A kinematic fit to D tag K frag X frag candidates is performed in which the particles are required to originate from a common point within the IP region and the D tag mass is constrained to its nominal value. We select only one D tag K frag X frag candidate per event that has its missing mass, closest to the nominal D * + s mass and between 2.00 and 2.25 GeV/c 2 , which corresponds to a ±3 σ interval.
Finally, a photon candidate is identified that is consistent with the decay D * ± s → D ± s γ and does not overlap with the D tag K frag X frag system. We require that the energy of the photon candidate be larger than 120 MeV in the laboratory frame and that the cosine of the angle between the CMS momenta of the D tag hadron and the photon candidate be negative, since the signal photon should be in the hemisphere opposite the D tag hadron. We perform a similar kinematic fit with the signal photon included and with the missing mass recoiling against the D tag K frag X frag system constrained to the nominal D * + s mass. All D tag K frag X frag γ candidates are required to have a CMS momentum larger than 2.8 GeV/c and M miss (D tag K frag X frag γ) > 1.83 GeV/c 2 (see eq. (3.2)). After the final selections, there are an average of 2.1 D tag K frag X frag γ candidates per event; these are due solely to multiple γ candidates. Among these, we select the one with the highest NeuroBayes network output that is trained to separate signal photons from others based on photon energy, the detecting region of the ECL (forward, barrel or backward region), the ratio of the energies summed in 3 × 3 and 5 × 5 ECL crystals in the transverse plane around the crystal with the largest energy deposit, the invariant mass of the combination of the photon candidate with any other photon candidate that is closest to the π 0 nominal mass, the energy asymmetry of this two-photon combination, and the invariant mass and energy asymmetry of the twophoton combination whose invariant mass is second closest to the nominal π 0 mass. A relative gain of 23% in absolute reconstruction efficiency is obtained by applying the best D tag K frag X frag γ candidate selection instead of a completely random selection. Figure 1 shows the distributions of M miss (D tag K frag X frag γ) for each X frag mode.

Inclusive D + s yield extraction
The yield of inclusively reconstructed D + s mesons is determined by performing a χ 2 fit to the missing mass M miss (D tag K frag X frag γ) distribution for each X frag mode. The events fall into six categories: signal candidates; mis-reconstructed signal candidates, where either K frag or one of the pions forming the X frag system originates in reality from a D + s decay; background candidates where the signal γ candidate originates from D * 0 → D 0 γ decays; background candidates where the signal γ originates from the π 0 produced in D * (s) → D (s) π 0 decays; background candidates with a bad γ -the energy deposited in the ECL being produced by an unmatched charged track or by a beam-induced interaction; and background candidates where the signal γ originates from a π 0 that does not itself originate from a D * (s) decay. Each of the six categories is represented with a smoothed non-parametric histogram [39] probability density function (PDF), H(M miss (D tag K frag X frag γ)), taken from a large sample of MC events. The fit function for a given X frag mode is written as where N represents the yield of each component and the first (second) term describes the contribution of signal (the sum of the five background components). The histogram PDF of the signal, H , is numerically convolved with a Gaussian function, G(σ cal ), centered at zero and with width σ cal , which takes into account possible differences between M miss (D tag K frag X frag γ) resolutions in the data and MC samples. The calibration of σ cal is described in the next paragraph. The position of the signal peak in data relative to the position in the MC, δ M miss , is a free parameter of the fit. We also float all normalization parameters, N X frag i , except the normalization of the background component where the signal γ candidate originates from D * 0 → D 0 γ decays, which is fixed relative to the more abundant and similar background component where the signal γ candidate originates from the π 0 produced in D * 0 (s) → D (s) π 0 decays. The fraction f D 0 γ/D (s) π 0 is fixed to the value obtained in the MC sample.
Sum 94360 ± 1310 Table 2. Yields of inclusively reconstructed D ± s mesons per individual X frag mode. The uncertainties are statistical only.
We calibrate the M miss (D tag K frag X frag γ) resolution using the mass difference between In the exclusive reconstruction of D * + s mesons, the same requirements are used for the signal photon candidate as in the inclusive reconstruction. The dominant contribution to the ∆M and M miss (D tag K frag X frag γ) resolutions is the signal photon energy resolution. In the former case, the smearing of the D + s momentum cancels almost completely in the mass difference while, in the latter case, the impact of experimental smearing of p miss (D tag K frag X frag ) on M miss (D tag K frag X frag γ) is minimized by performing a mass constrained vertex fit of D tag K frag X frag candidates to the nominal D * + s mass. According to the MC study, the ∆M and M miss (D tag K frag X frag γ) resolutions are the same to within a few percent, which justifies the calibration of the M miss (D tag K frag X frag γ) resolution by comparing ∆M resolutions of exclusively reconstructed D * + s → D + s γ decays obtained from data and MC. We parameterize the contribution of correctly reconstructed D * + s → D + s γ decays in ∆M as in the case of the signal parameterization of M miss (D tag K frag X frag γ) as a histogram PDF convolved with a Gaussian function, H sig (∆M − δ ∆M ) ⊗ G(σ cal ), where δ ∆M is the difference between the peak positions in data and MC. The background shape is fitted by a second order polynomial. We achieve best agreement between data and simulated ∆M distributions when σ cal = 2.0 ± 0.2 MeV/c 2 .
The fitted inclusive D ± s yields for each X frag mode are given in table 2. The PDFs describe well the observed data distributions as can be seen from the normalized fit residuals in figure 1. The normalized χ 2 values of the fits are between 1.06 and 1.32.
In the fit to the M miss (D tag K frag X frag γ) distributions, we set σ cal = 2.0 MeV/c 2 . To estimate the systematic uncertainty, we repeat the fits to the M miss (D tag K frag X frag γ) distributions by changing σ cal by ±0.2 MeV/c 2 (one standard deviation) and assign the resulting difference of ±1.39% in the inclusive D ± s yield as the systematic error. In the nominal fit, the fraction f D 0 γ/D (s) π 0 is fixed to the MC-determined value. We vary this parameter by ±5% to conservatively estimate the possible differences between data and MC in the relative production rates of D * + and D * 0 mesons. The impact on the inclusive D ± s yields is found to be small (±0.41%). To account for the limited statistics of the MC sample used to determine the histogram PDFs, we repeatedly vary the contents of all bins of all histogram templates within their statistical uncertainties and refit. The systematic uncertainty is taken to be the root mean square (RMS) of the obtained distribution of the inclusive signal yield (±0.51%).
The total inclusive D ± s yield, including systematic uncertainties, is N inc Ds = 94360 ± 1310(stat.) ± 1450(syst.). 5 Reconstruction of D + s decays within the inclusive D + s sample Using the inclusive sample of D + s mesons, we reconstruct specific D + s meson decays to the following final states: K − K + π + , K 0 K + , ηπ + , e + ν e , µ + ν µ , and τ + ν τ ; for the last mode, the τ lepton is reconstructed via its decays to e + ν e ν τ , µ + ν µ ν τ , and π + ν τ . We keep only inclusive D + s candidates within the signal region of M miss (D tag K frag X frag γ) defined as 1.95 GeV/c 2 < M miss (D tag K frag X frag γ) < 1.99 GeV/c 2 ; the sideband regions are used to study background properties except in the case of the D + s → K − K + π + decay, where all inclusive D + s candidates are kept. We find that 88.7% of correctly reconstructed inclusive D + s candidates populate the M miss (D tag K frag X frag γ) signal region, within which the purity of the inclusive D + s sample ranges between 3.5% (X frag = π ± π ∓ π 0 ) and 41% (X frag = nothing).
In the following subsections, we describe the reconstruction procedure and signal yield extraction for the six decay modes.

D +
s → K − K + π + The reconstruction of D + s → K − K + π + decays requires exactly three charged tracks in the rest of the event with a net charge equal to the charge of the inclusively reconstructed D + s candidate. The track with charge opposite that of the inclusive D + s candidate is selected to be the K − candidate while the two same-sign tracks are identified as K + or π + based on their likelihood ratios, L K,π .
The exclusively reconstructed D + s → K − K + π + candidates within the inclusive D + s sample are identified as a peak at the nominal mass of the D * + s in the invariant mass distribution of the K − K + π + γ combination, M (K − K + π + γ). Here, γ stands for the signal photon candidate used to reconstruct the inclusive D + s candidate in the recoil against the D tag K frag X frag γ system. The M (K − K + π + γ) is chosen over the D + s invariant mass, M (K − K + π + ), because both sides -inclusive and exclusive -have to be correctly reconstructed to produce peaks in M (K − K + π + γ) and M miss (D tag K frag X frag γ). Correctly reconstructed D + s → K − K + π + events will peak in M (K − K + π + ) even if the inclusive reconstruction of D + s candidates fails, e.g., the photon candidate is incorrectly identified. The M (K − K + π + γ) and M miss (D tag K frag X frag γ) are correlated due to their common input: the photon four-momentum. Imposing the requirement that inclusive D + s candidates populate the signal M miss (D tag K frag X frag γ) region would distort the background distribution to have a peaking structure in M (K − K + π + γ). We avoid this by taking all inclusive D + s candidates into consideration rather than only those populating the signal region in M miss (D tag K frag X frag γ).
We parameterize the M (K − K + π + γ) distribution as where the three terms describe correctly reconstructed D * + s → D + s γ → K − K + π + γ decays (signal), mis-reconstructed D * + s → D + s π 0 → K − K + π + γγ decays where one of the photons from the π 0 decay is lost, and random combinations of charged tracks or photon (combinatorial background). The latter is parameterized as a third order polynomial while the first two contributions are represented using the non-parametric histogram PDFs taken from MC. We convolve the signal histogram PDF with a Gaussian function, G(σ excl cal ), to take into account the differences between the resolutions of M (K − K + π + γ) in real and MC samples. We estimate σ excl cal by the procedure described in section 4.1, the only difference being that the resolution on M (K − K + π + γ) is calibrated instead of the D * + s and D + s mass difference. We determine σ excl cal = 3.2 ± 0.2 MeV/c 2 . Free parameters of the fit are the normalization parameters, N i , the position of the signal peak relative to the peak position in the MC, δ M , and the combinatorial background shape parameters, c i .
The M (K − K + π + γ) distribution of exclusively reconstructed D + s → K − K + π + decays within the inclusive D + s sample is shown in figure 2 with the superimposed fit. The number of correctly reconstructed D + where the error is statistical only.
5.2 D + s → K 0 K + We reconstruct D + s → K 0 K + decays by requiring only one additional charged kaon in the rest of the event whose charge equals that of the inclusively reconstructed D + s candidate. The neutral kaon is not reconstructed; rather, it is identified as a peak at the nominal mass-squared of the neutral kaon in the missing-mass-squared distribution where the missing four-momentum is given by An explicit reconstruction of the K 0 meson to two oppositely charged pions would lead to a significant signal loss. The signal peak of D + s → K 0 K + in the M 2 miss (D tag K frag X frag γK) distribution is used to calibrate the M 2 miss resolution, which is important in the extraction of the signal yield of D + s → µ + ν µ decays. Since the flavor of the neutral kaon is not determined, the doubly Cabibbo suppressed decays, D + s → K 0 K + , also contribute to the peak in M 2 miss (D tag K frag X frag γK). Their relative contribution can be estimated naively to be equal to tan 4 θ C ≈ 0.29% (θ C being the Cabibbo mixing angle), which is an order of magnitude below the expected statistical uncertainty and thus safely neglected.
The signal yield of partially reconstructed D + s → K 0 K + decays is extracted by performing a binned extended maximum likelihood fit to the M 2 miss (D tag K frag X frag γK) distribution. The signal component is parameterized as a sum of three Gaussian functions with a common mean. In the fit, we fix the signal shape parameters to the values determined from the MC sample except for the mean and the resolution scaling factor, s, of the core and the second Gaussian function. In addition to the signal contribution, two-body D + s → π 0 K + and D + s → ηK + decays peak in the M 2 miss (D tag K frag X frag γK) distribution. We use the same parameterization for these peaking backgrounds as for signal except that we fix their means to the nominal masses squared of π 0 and η mesons. Other background sources that produce distinct structures in M 2 miss (D tag K frag X frag γK) are: D s → K * + K 0 → K + π 0 K 0 decays and D + s → ηπ + decays where the π + is misidentified as signal K + candidate. Both contributions are parameterized as the sum of a Gaussian and a bifurcated Gaussian function, where all parameters are fixed to the values determined from the MC sample. The combinatorial background is parameterized with a fourth-order polynomial whose coefficients are determined with the fit to the M 2 miss (D tag K frag X frag γK) distribution for inclusive D + are constrained to the expected values based on their known branching fractions and MC determined efficiencies. The M 2 miss (D tag K frag X frag γK) distribution of partially reconstructed D + s → K 0 K + decays within the inclusive D + s sample is shown in figure 3 with the superimposed fit. The number of correctly reconstructed D + s → K 0 K + decays is where the error is statistical only. The yield of Cabibbo suppressed D + s → π 0 K + decays, 108 ± 31 (statistical error only), is found to be consistent within uncertainties with the expectation, 52 ± 18, based on a measurement performed by CLEO [40].

D +
s → ηπ + As in the case of D + s → K 0 K + decays, we perform a partial reconstruction of D + s → ηπ + decays. We require only one charged track consistent with the pion hypothesis in the rest of the event. To avoid a significant signal loss, we do not perform an explicit reconstruction of the η meson but rather identify it as a peak at the nominal mass-squared of the η in the missing-mass-squared distribution where the missing four-momentum is given by The sample of events with inclusive D + s candidate plus one additional positively charged pion contains a significant contribution of D + s → τ + ν τ decays with the tau lepton decaying hadronically to a charged pion and a neutrino. The contribution of these events is suppressed by requiring that the extra neutral energy in the ECL (E ECL ) be larger than 1.0 GeV, where E ECL represents the sum over all energy deposits in the ECL that are not associated with the charged pion candidate and the tracks and neutrals used in the inclusive reconstruction of the D + s candidate [3]. The D + s → τ + ν τ → π + ν τ ν τ decays peak at zero in E ECL while D + s → ηπ + decays deposit a significant amount of energy in the ECL via the η decay products. (See section 5.5 for more details.) The signal yield of partially reconstructed D + s → ηπ + decays is extracted by performing a binned maximum likelihood fit to the M 2 miss (D tag K frag X frag γπ) distribution. We use the same parameterization for the signal component as in the case of D + s → K 0 K + decays. In addition, the resolution scaling factor of the core and the second Gaussian, s, is allowed to float within a Gaussian constraint in the fit, where the mean and width of the constraint are set to the value and uncertainty determined in the fit of the D + s → K 0 K + candidates (s = 1.177 ± 0.052), respectively. We identify a single source of peaking background to be the two-body D + s → K 0 π + decay that peaks at the neutral kaon mass-squared and is parameterized in the same way as the signal. Other background sources that produce distinct structures in M 2 miss (D tag K frag X frag γπ) are D + s → K 0 K + decays with a misidentified kaon and D + s → ρ 0 K + → π + π − K + decays; both are parameterized as bifurcated Gaussian functions. The combinatorial background is parameterized with a fourth order polynomial whose coefficients are fixed to the values determined with the fit to the M 2 miss (D tag K frag X frag γπ) distribution of inclusive D + s candidates from the M miss (D tag K frag X frag γ) < 1.95 GeV/c 2 sideband region. Yields of all but two event categories are free parameters of the fit; the D + s → K 0 π + and D + s → K 0 K + yields are constrained to expected values based on their known branching fractions and MC-determined efficiencies.
The M 2 miss (D tag K frag X frag γπ) distribution is shown in figure 4 with the superimposed fit. The number of correctly reconstructed D + s → ηπ + decays is where the error is statistical only.

D
The D + s → µ + ν µ decays are reconstructed by requiring one additional charged track consistent with the muon hypothesis in the rest of the event. The single missing neutrino is then identified as a peak at zero in the missing-mass-squared distribution where the missing four-momentum is given by The signal yield is extracted by performing a binned maximum likelihood fit to the M 2 miss (D tag K frag X frag γµ) distribution. The signal component is parameterized as the sum of three Gaussian functions with a common mean, where all parameters except the mean are fixed to their MC-determined values. As in the case of D + s → ηπ + decays, the resolution scaling factor of the first and second Gaussians, s, is allowed to float within a Gaussian constraint in the fit, except that the mean and width of the constraint here are set to the value and uncertainty determined in the fits of the D + s → K 0 K + and D + s → ηπ + candidates (s = 1.177 ± 0.049). The leptonic D + s → τ + ν τ → µ + ν µ ν τ ν τ decays produce three neutrinos in the final state and therefore do not peak at M 2 miss (D tag K frag X frag γµ). Their contribution is found to be very well described by an exponential function. In the fit, we include also contributions of hadronic D + s → ηπ + and D + s → K 0 K + decays where the muon candidate is a misidentified pion or kaon, respectively. The former is parameterized as the sum of two Gaussian functions and the latter as a bifurcated Gaussian function. In both cases, we fix the shape parameters to the MC-determined values. The combinatorial background is parameterized with an exponential function whose shape parameter is fixed to the value determined from the fit to the M 2 miss (D tag K frag X frag γµ) distribution for candidates in the M miss (D tag K frag X frag γ) < 1.95 GeV/c 2 sideband region. Free parameters of the fits are the yield parameters of all but two spectral components; the D + s → ηπ + and D + s → K 0 K + yields are constrained to the expected values based on their measured branching fractions and MC-determined efficiencies.
The distribution of M miss (D tag K frag X frag γµ) with superimposed fit is shown in figure  5. The number of reconstructed D + s → µ + ν µ decays is where the error is statistical only.

D +
s → τ + ν τ The reconstruction of D + s → τ + ν τ requires one charged track in the rest of the event that is identified as an electron, muon or a pion (denoted as D + s → τ + (X + )ν τ where X + = e + , µ + or π + ) indicating the subsequent decay of the τ + lepton to e + ν e ν τ , µ + ν µ ν τ or π + ν τ . 5 Due to the multiple neutrinos in the final state, these decays do not peak in the missing-masssquared distribution: where the missing four-momentum is given by The background in the D + s → τ + (π + )ν τ sample is much larger than in the leptonic modes, but is reduced significantly by requiring the magnitude of the missing momentum of the event, | p miss (D tag K frag X frag γπ)|, to be larger than 1.2 GeV/c in the laboratory frame. The background in this sample is further reduced by requiring 0.0 < M 2 miss (D tag K frag X frag γπ) < 0.6 GeV 2 /c 4 . 6 In the case of D + s → τ + ν τ → + ν ν τ ν τ (where + = e + or µ + ), we require M 2 miss (D tag K frag X frag γ ) > 0.3 GeV 2 /c 4 to veto D + s → + ν decays.
The signal yield of D + s → τ + ν τ decays is extracted from the simultaneous binned maximum likelihood fit to the E ECL distributions of the three tau submodes. The signal decay has either zero or a small value of E ECL , while background events tend to have larger values due to the contributions from additional neutral clusters. Signal components include the cross-feed contribution from other τ decays: in the case of leptonic τ decays, the crossfeed contribution is found to be small (around 3% of the signal contribution from leptonic modes) while, in the case of hadronic τ + → π + ν τ decays, a large cross-feed contribution originates from hadronic τ + → ρ + ν τ decays by missing the neutral pion from the ρ + decay (20% of the signal contribution from the pion mode). Backgrounds from several different D + s decays are found to contribute to D + s → τ + (X + )ν τ samples and are listed in table 3. The peaking background is dominated by D + s decays with K 0 L in the final state, e.g., D + s → K 0 + ν in the case of leptonic tau decays or D + s → K 0 K + in the case of hadronic tau decays. If the K 0 L deposits little or no energy in the ECL then these decay modes produce an E ECL distribution very similar to the signal. The non-peaking background, dominated by inclusive η decays, is much less problematic since it rises smoothly with increasing E ECL . The E ECL distributions of the above categories as well as of the combinatorial background are described with non-parametric histogram PDFs taken from MC samples. In the final fit, four parameters are allowed to vary: the total signal yield summed over three tau decay modes (we fix the relative contribution of the signal from each tau decay mode i to the total signal yield with the ratio of (B(τ → i) · ε i )/( 3 i=1 B(τ → i) · ε i )) and the yields of combinatorial background in each tau decay mode. The background contributions from D + s decays are fixed to the values given in table 3. Figure 6 shows the result of the simultaneous fit to the three E ECL distributions for D + s → τ + (e + )ν τ , D + s → τ + (µ + )ν τ , and D + s → τ + (π + )ν τ decays. The signal yield is where the error is statistical only. As a check, we fit the E ECL distributions while floating 6 Due to the lack of phase space and the fact that D + s → τ + ντ → π + ντ ντ decays have only two neutrinos in the final state, these decays populate a relatively narrow region in M 2 miss (DtagK frag X frag γπ). Table 3. Estimated background yields of various D + s decays contributing to the three D s → τ + ν τ samples. The uncertainties include the uncertainty of their branching fractions (taken from [2]), the error on the inclusive D + s yield, as well as the uncertainty due to limited MC sample size used to determine the efficiencies and systematic uncertainty related to particle identification.

Estimated background yields Background Source
the yield of each of the three tau decay modes. The resulting yields are where the errors are statistical only. The sum of individual yields is in good agreement with the total signal yield determined in the simultaneous fit.

D
We reconstruct D + s → e + ν e decays in the same way as D s → τ + (e + )ν τ decays, except that we focus on candidates populating the M 2 miss (D tag K frag X frag γe) region around zero. 7 To improve the purity, we require that E ECL < 0.5 GeV and p miss (D tag K frag X frag γe) > 0.7 GeV/c in the laboratory frame. These two requirements reject around 80% of the background candidates while keeping around 75% of the signal candidates in the search window defined as −0.10 GeV 2 /c 4 < M 2 miss (D tag K frag X frag γe) < 0.15 GeV 2 /c 4 . The background in this window is estimated by interpolating the observed yield from missingmass-squared sidebands: −0.30 GeV 2 /c 4 < M 2 miss (D tag K frag X frag γe) < −0.10 GeV 2 /c 4 and 0.15 GeV 2 /c 4 < M 2 miss (D tag K frag X frag γe) < 0.60 GeV 2 /c 4 . This is done by performing a binned maximum likelihood fit in these sidebands, using a model consisting of a sum of two exponential functions describing contributions of D + s → τ + (e + )ν τ decays and 7 The final states differ only in the number of neutrinos. combinatorial background. The shape and normalization of the former are fixed to MCbased expectations. Contributions of other D + s decays are found to be negligible. The shape parameter of the combinatorial background is fixed to the value determined in a binned maximum likelihood fit to the M 2 miss (D tag K frag X frag γe) distribution of events from M miss (D tag K frag X frag γ) sidebands. The normalization of the combinatorial background is a free parameter of the fit. Figure 7 shows the M 2 miss (D tag K frag X frag γe) distribution with superimposed fit. We observe n eν obs = 8 (5.8) in the M 2 miss (D tag K frag X frag γe) signal region (denoted by the two vertical lines in figure 7), which is in good agreement with the estimated background b eν = 8.7 ± 0.9(stat.) ± 0.8(syst.), (5.9) where the first error is statistical and the second is systematic. The latter is estimated by varying the yield of D + s → τ + (e + )ν τ decays and the combinatorial background parameters, changing both shape and normalization, and repeating the fits.  Table 4. Signal yields, tag bias corrected efficiencies and measured branching fractions for all five studied D + s decay modes. The first uncertainty is statistical and the second is systematic. In the case of D + s → τ + ν τ decays the efficiencies include the branching fractions of the τ + decay modes.

Determination of absolute branching fractions
For a given final state f , the absolute branching fraction of the D + s → f decay is given by . To take this effect into account, a ratio of D + s inclusive reconstruction efficiency for D + s → f decays, ε inc Ds→f , and the average D + s inclusive reconstruction efficiency, ε inc Ds = i B(D + s → i)ε inc Ds→i , is included in the denominator of eq. (6.1): f bias = ε inc Ds→f /ε inc Ds . The ratio f bias is taken from the MC sample.

Systematic uncertainties
Systematic uncertainties in the calculation of the branching fractions arise due to imperfect knowledge of the size of the inclusive D + s sample, the reconstruction efficiencies and the modeling of signal and background contributions in the fits from which the signal yields of D + s → f decays are extracted. The estimated systematic uncertainties are itemized in table 5 and described below.  Normalization The systematic error related to the normalization is assigned to be ±2.1% (where statistical and systematic errors given in eq. (4.2) are combined in quadrature) and is common for all studied D + s → f decays.
Tag bias Possible differences in relative rates of individual D + s decay modes between MC simulation and data that impact the f bias calculation are estimated by studying the distributions of the number of charged particles and π 0 's, N ch +N π 0 , produced in D + s decays. We obtain the N ch + N π 0 data distribution in the following way: first we count the number of remaining charged tracks, N reco ch , and π 0 candidates, N reco π 0 , that are not associated to the D tag K frag X frag γ candidate; in a second step, we determine, the inclusive D + s yields in bins of the N reco ch + N reco π 0 by performing fits to the M miss (D tag K frag X frag γ) distributions. The obtained distribution of N reco ch + N reco π 0 is proportional to the true distribution of N ch + N π 0 , but with a considerable amount of convolution. 8 We obtain the N ch +N π 0 distribution from the N reco ch +N reco π 0 distribution using the singular value decomposition algorithm [41]. Finally, we estimate the ratio between the inclusive D + s reconstruction efficiencies in the data and MC samples, ε inc Ds | DATA /ε inc Ds | MC = 0.9768 ± 0.0134, using the unfolded data and MC N ch + N π 0 distributions and the MC-determined dependence of the inclusive D + s reconstruction efficiency on N ch +N π 0 . The ratio is found to be consistent with unity within the uncertainty. Nevertheless, we correct the measured branching fractions by this factor and assign the error on this ratio as a source of systematic uncertainty (±1.4%) that is common for all studied D + s decay modes.
Tracking, particle ID, and efficiency The systematic errors in the D + s → f reconstruction efficiencies arise from several sources. First, we assign a 0.35% error per reconstructed charged track in the final state due to the uncertainty on the efficiency of the charged-track reconstruction estimated using partially reconstructed D * + → D 0 (K 0 S π + π − )π + decays. The e + e − → e + e − + − ( = e or µ) and D * + → D 0 (K − π + )π + samples are used to estimate the lepton, kaon and pion identification corrections and pion (kaon) to lepton misidentification probabilities. Finally, we include statistical uncertainties of the MC-determined efficiencies f bias · ε(D + s → f |inc. D + s ) as a source of systematic error. Since the branching fractions are determined relative to the number of inclusively reconstructed D + s mesons, the systematic uncertainties in the reconstruction of the inclusive D + s cancel.
Dalitz model In the case of D + s → K − K + π + decays, the reconstruction efficiency varies weakly across the K − K + π − Dalitz distribution. In calculating B(D + s → K − K + π + ), we use the Dalitz-plot-integrated MC efficiency. The decay amplitude in the MC is simply the incoherent sum of all known resonant two-body contributions (φπ + , K * 0 K + , f 0 (980)π + and others). We use a dedicated MC in which the D + s → K − K + π + decay dynamics are simulated according to results of the Dalitz plot analysis performed by the CLEO collaboration [42]. The ratio of efficiencies from both samples is found to be consistent with unity within the uncertainty (±1.1%) which is conservatively assigned as a source of systematic error.
Fit model The systematic error due to limited statistics of the MC sample used to construct the histogram PDFs is evaluated by varying repeatedly the contents of all bins of all histogram PDFs within their statistical uncertainties and refitting. The RMS of the distribution of fit results is assigned as a systematic uncertainty due to the fit model. To estimate the systematic error due to the possible signal E ECL shape difference between MC and data, we use the ratio of data to MC for the E ECL histograms of the backgroundsubtracted D + s → K − K + π + and D + s → K 0 K + samples to modify the signal PDF and repeat the fit. In a similar way, we estimate the systematic error due to the possible E ECL shape difference of combinatorial background in MC and data by using the ratio of E ECL histograms of D + s → τ + ν τ candidates populating the M miss (D tag K frag X frag γ) sidebands and modifying accordingly the combinatorial E ECL PDF. For D + s → µ + ν µ , D + s → K 0 K + , and D + s → ηπ + decays, we vary the values of all fixed parameters within their uncertainties (taking correlations into account) and assign the differences with respect to the nominal fits as the fit model systematic uncertainty. In the case of D + s → K − K + π + , the fits are repeated by changing σ excl cal within its uncertainty to assess this source of systematic error.
D + s background In the fits, we fix the contributions of certain background D + s decays to the expectations based on their measured branching fractions. We estimate the systematic error related to this by changing these branching fractions by their experimental errors [2]. In the case of D + s → τ + (e + )ν τ and D + s → τ + (µ + )ν τ decays, the systematic uncertainty from this source originates mainly from D + s → K 0 ν decays; in the case of D + s → τ + (π + )ν τ , from D + s → ηπ + and D + s → ρK + decays.
, and B(D + s → ηπ + ) (right) for each X frag mode. The measured branching fractions when all X frag modes are combined are indicated with vertical blue lines with statistical and total errors indicated as inner (dark blue) and outer (light blue) bands.
τ cross-feed In the nominal fit to the E ECL distribution of reconstructed D + s → τ + ν τ decays, we fix the cross-feed contributions relative to the signal contributions. We vary the ratios of cross-feed candidates within their uncertainties, repeat the fits, and take the differences from the nominal fits as the systematic uncertainty.
As a check, we determine the branching fractions of hadronic D + s decays for each X frag mode separately. Figure 8 shows the measured branching fractions of the three hadronic decay modes, subdivided by X frag mode. They are found to be in good agreement within uncertainties.
We find no evidence for D + s → e + ν e decays and set an upper limit on B(D + s → e + ν e ) using the modified frequentist approach (or CL s method) [43,44]. We construct test statistics from pseudo-experiments according to the signal plus background and backgroundonly hypotheses [45]. For each pseudo-experiment, a likelihood ratio is computed depending on the expected number of signal events for a given value of B(D + s → e + ν e ) (according to eq. (6.1)), incl. · f bias · ε(D + s → e + ν|incl. D + s ) · B(D + s → e + ν e ) = (59370 ± 3150(stat. + syst.)) · B(D + s → e + ν e ), the expected number of background events, b eν (see eq. (5.9)), and the observed number of events, n eν obs (see eq. (5.8)). The CL s is defined as the ratio of confidence levels, CL s+b /CL b , where the CL s+b and CL b are the probabilities for signal-plus-background or backgroundonly generated pseudo-experiments, respectively, to have a test-statistic value larger than or equal to that observed in the data. The observed distribution of CL s as a function of branching fraction of D + s → e + ν e decays is shown in figure 9. The systematic uncertainties of s eν and b eν are included in the CL s method using the techniques described in refs. [43,44]. The sources of systematic uncertainties that are considered in the evaluation of s eν are listed in table 5. The external inputs necessary in the extraction of f Ds from the measured branching fractions are given in table 6. The CKM matrix element |V cs | is obtained from the wellmeasured |V ud | = 0.97425 ± 0.00022 and |V cb | = (40.9 ± 1.1) × 10 −3 [2] by using the relation |V cs | = |V ud | − |V cb | 2 /2, following the prescription given in the section Decay constants of charged pseudoscalar mesons in ref. [2]. All but one of the external inputs are very precisely measured and do not introduce additional uncertainties; the exception is the D + s lifetime, τ Ds , which introduces an 0.70% relative uncertainty. Table 7 summarizes the obtained values of f Ds using the D + s → µ + ν µ and D + s → τ + ν τ decays. The error-weighted average is f Ds = (255.5 ± 4.2(stat.) ± 4.8(syst.) ± 1.8(τ Ds )) MeV, where the correlation of the systematic uncertainties between the µ + ν µ and τ + ν τ have been taken into account. This is the most precise measurement of f Ds to date.  Table 6. Numerical values of external parameters used in extraction of f Ds . All values are taken from ref. [2] except for |V cs | where we follow the prescription given in the section Decay constants of charged pseudoscalar mesons of ref. [2].

Conclusions
In conclusion, we measure the absolute branching fractions of hadronic decays to K − K + π + , K 0 K + , and ηπ + , and of leptonic D + s decays to µ + ν µ and τ + ν τ using the large data sample collected by the Belle experiment at the KEKB collider. The branching fractions of hadronic D + s decays are in agreement with the measurements performed by CLEO [27,28]. The measurements of branching fractions of leptonic D + s decays are consistent with the previous measurements performed by the CLEO and BaBar collaborations [11][12][13]15]. From the measured branching fractions of leptonic D + s decays, we determine the D + s meson decay constant, f Ds = (255.5 ± 4.2(stat.) ± 4.8(syst.) ± 1.8(τ Ds )) MeV, which represents the single most precise measurement to date and is found to be in agreement with the most precise lattice QCD calculation [16]. We find no evidence for D + s → e + ν e decays and we set the most stringent upper limit of B(D + s → e + ν e ) < 1.0 (0.83) × 10 −4 at 95 (90)% C.L.