Measurement of the branching fraction of D + s

: Utilizing 7 . 33 fb − 1 of e + e − collision data taken at the center-of-mass energies of 4.128, 4.157, 4.178, 4.189, 4.199, 4.209, 4.219, and 4.226 GeV with the BESIII detector, the branching fraction of the leptonic decay D + s → τ + ν τ via τ + → µ + ν µ ¯ ν τ is measured to be B D + s → τ + ν τ = (5 . 34 ± 0 . 16 stat ± 0 . 10 syst )% . Combining this branching fraction with the world averages of the measurements of the masses of τ + and

Beijing 100871, People's Republic of China h Also at School of Physics and Electronics, Hunan University, Changsha 410082, China i Also at Guangdong Provincial Key Laboratory of Nuclear Science, Institute of Quantum Matter, South China Normal University, Guangzhou 510006, China j Also at Frontiers Science Center for Rare Isotopes, Lanzhou University, Lanzhou 730000, People's Republic of China k Also at Lanzhou Center for Theoretical Physics, Lanzhou University, Lanzhou 730000, People's Republic of China l Also at the Department of Mathematical Sciences, IBA, Karachi , Pakistan Abstract: Utilizing 7.33 fb −1 of e + e − collision data taken at the center-of-mass energies of 4.128, 4.157, 4.178, 4.189, 4.199, 4.209, 4.219, and 4.226 GeV with the BESIII detector, the branching fraction of the leptonic decay D + s → τ + ν τ via τ + → µ + ν µ ντ is measured to be B D + s →τ + ντ = (5.37 ± 0.17 stat ± 0.15 syst )%.Combining this branching fraction with the world averages of the measurements of the masses of τ + and D + s as well as the lifetime of D + s , we extract the product of the decay constant of D + s and the c → s Cabibbo-Kobayashi-Maskawa matrix element to be f

INTRODUCTION
Leptonic decays offer an ideal laboratory for studying strong and weak interaction effects in the charmed meson system.In the standard model (SM) of particle physics, the D + s meson decays into ℓ + ν ℓ (ℓ = e, µ or τ ) via annihilation mediated by a virtual W + boson.Throughout this paper, the inclusion of charge conjugate channels is always implied.The partial width of D + s → ℓ + ν ℓ at lowest order can be related to the D + s decay constant where G F is the Fermi coupling constant, |V cs | is the c → s Cabibbo-Kobayashi-Maskawa (CKM) matrix element, m ℓ is the mass of the lepton, and m D + s is the mass of the D + s meson.Extraction of f D + s in experiments is important for testing various theoretical calculations based on different approaches [2][3][4][5][6][7][8][9][10].In recent years, the precision of calculations of f D + s based on Lattice Quantum Chromodynamics (LQCD) has reached a level of 0.2% [7], and much progress has been achieved in the experimental studies of D + s → ℓ + ν ℓ decays by the CLEO [11-13], BaBar [14], Belle [15], and BESIII [16][17][18][19][20][21] collaborations.Based on the average of the branching fractions (BFs) reported by these experiments, one can derive f D + s with a precision of 1.0%.Precise and intensive estimations of f D + s are still desirable to test theoretical calculations with higher precision.Improved measurements of f Ds × |V cs | are therefore important for testing the unitarity of the CKM matrix [22] with higher sensitivity.
In the SM, the ratio of the BFs of D + s → τ + ν τ and D + s → µ + ν µ can be written as which only depends on the charged lepton and D + s meson masses.Inserting the world averages of m τ , m µ , and m Ds [23] in the above equation gives R τ /µ = 9.75 ± 0.01.Measurements of the BFs of D + s → ℓ + ν ℓ allow this ratio to be determined experimentally and provide an important test of τ − µ lepton flavor universality.
In this paper, we present a measurement of the BF of D + s → τ + ν τ via the decay of τ + → µ + ν µ ντ , by analyzing 7.  [24][25][26] with the BESIII detector [27].Following previous measurements, we have not corrected the BF of D + s → τ + ν τ by the effect of radiative photons since their uncertainties can be considered individually later, details of which are reviewed in "Leptonic Decays of Charged Pseudoscalar Mesons" by the Particle Data Group (PDG) [23].Based on this measurement, we determine f D + s × |V cs | with an improved accuracy, and test τ − µ lepton flavor universality with D + s → ℓ + ν ℓ decays.

BESIII DETECTOR AND MONTE CARLO SIMULATION
The BESIII detector [27] records symmetric e + e − collisions provided by the BEPCII storage ring [28] in the center-of-mass energy range from 2.00 to 4.95 GeV, with a peak luminosity of 1×10 33 cm −2 s −1 achieved at √ s = 3.77 GeV.BESIII has collected large data samples in this energy region [29].The cylindrical core of the BESIII detector covers 93% of the full solid angle and 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 [30].The solenoid is supported by an octagonal flux-return yoke with modules of resistive plate muon counters (MUC) interleaved with steel.The charged-particle momentum resolution at 1 GeV/c is 0.5%, and specific ionization energy loss dE/dx resolution is 6% for 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 in the TOF barrel region is 68 ps.The end-cap TOF system was upgraded in 2015 using multi-gap resistive plate chamber technology, providing a time resolution of 60 ps [31][32][33].Approximately 83% of the data used here was collected after this upgrade.Simulated data samples, namely inclusive MC samples, produced with a geant4based [34] Monte Carlo (MC) package, which includes the geometric description of the BESIII detector and the detector response, are used to determine detection efficiencies and to estimate backgrounds.The simulation models the beam-energy spread and initial-state radiation (ISR) in the e + e − annihilations with the generator kkmc [35,36].In the simulation, the production of open-charm processes directly produced via e + e − annihilations are modeled with the generator conexc [37], and their subsequent decays are modeled by evtgen [38,39] with known BFs from the Particle Data Group [23].The ISR production of vector charmonium (-like) states and the continuum processes are incorporated in kkmc [35,36].The remaining unknown charmonium decays are modeled with lundcharm [40,41].Final-state radiation from charged final-state particles is incorporated using the photos package [42].The input cross section line shape of e + e − → D * ± s D ∓ s is based on the cross section measurement in the energy range from threshold to 4.7 GeV.

ANALYSIS METHOD
In e + e − collisions with data taken at the center-of-mass energies between 4.128 and 4.226 GeV, the D ± s mesons are produced mainly via the e + e − → D * ± s D ∓ s → γ(π 0 )D + s D − s process.For our analysis we adopt the double-tag (DT) method pioneered by the MARK III Collaboration [43].The D − s meson, when fully reconstructed via any hadronic decay mode, is referred to as the single-tag (ST) D − s meson.Events in which the transition γ(π 0 ) from the D * + s meson and the leptonic decay of D + s → τ + ν τ are reconstructed, in addition to the ST D − s meson, are denoted as DT events.The BF of D + s → τ + ν τ is determined by where N j DT and N j ST are the yields of the DT events and ST D − s mesons in data, respectively; and ϵ j DT and ϵ j ST are the efficiencies of the DT events and ST D − s mesons estimated with MC simulation, respectively.Here, ϵ j DT , which includes the efficiency of simultaneously finding the tag side, the transition γ(π 0 ) and D + s → τ + ν τ as well as the BF of D * + s → γ(π 0 )D + s , B τ + →µ + νµ ντ is the BF of τ + → µ + ν µ ντ and j denotes the ST mode.The weighted mean method [44] is utilized to calculate the final BF, taking into account the statistical and tag mode dependent uncertainty as discussed later.
In selecting K ± , π ± , K 0 S , γ, π 0 , and η candidates, we use the same selection criteria as those adopted in our previous studies [17,45,46].For each good charged track, the polar angle (θ) with respect to the beam direction is required to be within the MDC acceptance |cos θ| < 0.93, where θ is defined with respect to the z axis, which is the symmetry axis of the MDC.The distance of its closest approach relative to the interaction point is required to be within 10.0 cm along the beam direction (|V z |) and within 1.0 cm in the plane transverse to the beam direction (|V xy |).Particle identification (PID) for good charged tracks combines the measurements of the dE/dx in the MDC and the flight time in the TOF to form probabilities L(h)(h = K, π) for each hadron (h) hypothesis.The charged tracks are assigned as kaons or pions if their probabilities satisfy L(K) > L(π) and L(π) > L(K), respectively.
K 0 S candidates are reconstructed via K 0 S → π + π − decays.The two charged pions are required to satisfy |V z | < 20 cm and |cos θ| < 0.93.They are assumed to be π + π − without particle identification (PID) requirements and their invariant mass is required to be within (0.486, 0.510) GeV/c 2 .The distance from the K 0 S decay vertex to the interaction point is required to be greater than twice the vertex resolution.
Photon candidates are selected by using the information measured by the EMC and are required to satisfy the following criteria.The energy of each shower in the barrel (end-cap) region of the EMC [27] is required to be greater than 25 (50) MeV.To suppress backgrounds associated with charged tracks, the angle between the shower position and the closest intersection point of any charged track with the EMC inner surface, projected from the interaction point, must be greater than 10 degrees.To suppress electronic noise and energy deposits unrelated to the event of interest, any candidate shower is required to start within [0, 700] ns from the event start time.
The backgrounds from non-D ± s D * ∓ s processes are suppressed by using the beam-constrained mass of the ST D − s candidate defined as where E beam is the beam energy ( √ s/2) and ⃗ p ST is the momentum of the ST D − s candidate in the e + e − rest frame.Figure 1 shows the M BC distribution of the ST candidates at 4.178 GeV.The M BC value is required to be within (2.010, 2.061 + i × 0.003) GeV/c 2 , where i takes the value 0, 3, 4, 5, 6, 7, 8, 9 for the energy points 4.128, 4.157, 4.178, 4.189, 4.199, 4.209, 4.219, 4.226 If there are multiple candidates present per tag mode per charge, only the one with the D − s recoil mass closest to the D * + s nominal mass [23] is kept for further analysis.The distributions of the invariant masses (M ST ) of the accepted ST candidates from data for each tag mode are shown in Fig. 2. The yields of ST D − s mesons reconstructed in each tag mode are determined from fits to their individual M ST distributions.In the fits, the signal is described by the simulated shape convolved with a Gaussian function that represents the resolution difference between data and simulation.In the fit to the D − s → K 0 S K − tag mode, the shape of the peaking background D − → K 0 S π − is modeled by the simulated shape convolved with the same Gaussian resolution function as used for the signal shape and its size is left free.The fraction of the . The combinatorial background is described by a first to third order Chebychev function, which is validated by analyzing the inclusive MC sample.The second and third columns of Table 2 summarize the yields of ST D − s mesons (N ST ) for each tag mode obtained in data and the corresponding detection efficiencies (ϵ ST ), respectively.In this table, the N ST quantities are obtained by summing over all energy points, and the ϵ ST quantities are obtained by weighting the corresponding yields of ST D − s mesons in data at each energy points.

DOUBLE-TAG CANDIDATES
The D + s → τ + ν τ candidates are selected in the system recoiling against the ST D − s mesons via the decay of τ + → µ + ν µ ντ by using the residual neutral showers and charged tracks which have not been used in the ST selection.As the detection efficiencies and background levels do not vary greatly with √ s, the analysis combines the samples over all the energy points.
Excluding the daughter particles originating from the tag side, only one good charged track is allowed in each DT candidate and its charge must be opposite to that of the tagside decay.The deposited energy of muon candidates in the EMC is required to be within (0.0, 0.3) GeV.To separate muons from hadrons, the muon candidates must have momenta greater than 0.5 GeV/c, and fulfill requirements on the muon travelling length in the MUC (d µ ) with dependence of momentum (p µ ) and flight direction (cos θ µ ) in the MUC [17] as shown in Table 1 and Fig. 3 based on the control sample of e + e − → γµ + µ − .To select the D + s → τ + ν τ signals and the transition γ(π 0 ) from D * + s , we define two kinematic variables: the energy difference where E miss is defined as , and the missing mass squared of the neutrinos in which E k and ⃗ p k are the energy and momentum of ST D − s , transition γ(π 0 ), or µ + , respectively.All γ and π 0 candidates that have not been used in tag selection are looped over.If there are multiple γ or π 0 combinations satisfying the selection criteria, we choose the one leading to the minimum |∆E|.To suppress the backgrounds from D + s → µ + ν µ and D + s → ηπ + decays, which peak in the M 2 3ν distribution around 0 and 0.3 GeV 2 /c 4 , respectively, the value of M 2 3ν is required to be within (0.5, 2.0) GeV 2 /c 4 as shown in Fig. 4.

BRANCHING FRACTION DETERMINATION
Following Refs.[20,47,48], we discriminate signal from background by using the variable E tot extra γ .It is defined as the total energy of the good isolated EMC showers which have not been used in tag selection.The distributions of E tot extra γ of the accepted DT candidates in data are shown in Fig. 5.
Study of the inclusive MC sample shows that the background events can be divided into three categories: BKGI, BKGII, and BKGIII.The BKGI component corresponds to events with an incorrectly reconstructed ST D − s .The BKGII component corresponds to events with a correctly reconstructed ST D − s and D + s → K 0 L µ + ν µ , in which the K 0 L meson passes through the detector without undergoing decay or significant interaction.The BKGIII component consists of events with a correctly reconstructed ST D − s and a D + s decaying to any other background final state apart from K 0 L µ + ν µ , The DT signal yield is extracted by analyzing the E tot extra γ distribution as shown in Fig. 5.To minimize the effect of the imperfect signal shape, we adopt an extrapolation technique following Refs.[20,47,48].A bin maximum likelihood fit is performed on the events with E tot extra γ > 0.6 GeV, where the signal is negligible, and the sizes and shapes of BKGI and BKGII are fixed.The signal DT yield is obtained by subtracting the yields of BKGI, BKGII, and BKGIII from the yield of all events (N j tot ) in the E tot extraγ signal region.In the D * s rest frame, the transition photon has a monochromatic energy of 139 MeV.When evaluated in the laboratory rest frame, the D * s momentum causes a smearing of ±15 MeV on the photon energy.After further considering the resolution effect, we define E tot extraγ < 0.4 GeV as the signal region.Details of BKGI, BKGII, and BKGIII are given below.
The shape of the BKGI component is derived using the data DT events situated in the corresponding M ST sideband regions.The M ST sideband regions are indicated inside the brown line segments in Fig. 2. For tag modes with neutrals, the remaining contamination from signal in sideband regions is subtracted.The size of this component is fixed at Class , where f j 1 is the sideband scale factor, defined as the ratio of the numbers of background events in the M ST sideband and signal ranges.The f j 1 value is obtained by fitting the M ST distribution from the inclusive MC sample after imposing the DT requirements.

N I j
Class is obtained by counting events in the E tot extra γ signal region in data.The shape of the BKGII component is modeled by the simulated events corrected by a 2D data-MC difference for the K 0 L detector response.The correction factors are obtained by using a control sample of D 0 → K 0 L π + π − decays from 2.93 fb −1 of e + e − collision data collected at √ s = 3.773 GeV [49,50].The yield of this component is fixed at N II j Class , which is calculated by taking the probability not to reconstruct the K 0 L meson from MC simulation and assuming the BF of D + s → K 0 µ + ν µ decays to be the same as the corresponding decay mode involving electrons [23].The shape of the BKGIII component is estimated from the inclusive MC sample.The MC simulation shows that the leading six D + s non-peaking background components are Finally, the signal DT yield in data is obtained by The efficiencies of detecting DT events (ϵ j DT ) are estimated by using the signal MC samples of e + e − → D ∓ s D * ± s with the D − s meson decaying to the tag mode and D + s → τ + ν τ with τ + → µ + ν µ ντ .All numbers discussed above are summarized in Table 2.For each tag mode, inserting the individual values of N j ST , ϵ j ST , N j DT , and ϵ j DT in Eq. 3.1 gives the corresponding BF.The systematic uncertainties in the BF measurement are estimated in the next section.The obtained BFs are summarized in the last column of Table 2.

SYSTEMATIC UNCERTAINTIES
Sources of the relative systematic uncertainties in the measurement of the BF of D + s → τ + ν τ are summarized in Table 3 and discussed below.Note that the DT method means that most uncertainties due to the selection of ST D − s candidates cancel.

TAG-MODE DEPENDENT SYSTEMATIC UNCERTAINTIES
Several sources of potential systematic bias are associated with the tag mode, and are hence classified as tag-mode dependent.
The systematic uncertainties on the fitted yields of the ST D − s mesons are assessed by using alternative signal and background shapes.The alternative signal shapes are obtained by changing the baseline choices derived from inclusive MC sample to those from the signal MC sample.The alternative background shapes are obtained by varying the order of the nominal Chebychev function by ±1.For a given ST mode, the differences in the ratio of the yields of ST D − s mesons over the corresponding efficiency for all variations, and the background fluctuation of the fitted yield of ST D − s are re-weighted by the yields of ST D − s mesons in various data samples and are added in quadrature.An additional component to this uncertainty is statistical in nature, and accounts for the contribution of background fluctuations to the fitted yields of ST D − s mesons.The effects due to the signal shape, the background shape, and the background fluctuation are 0.08%, 0.12%, and 0.46%, respectively.The corresponding overall systematic uncertainty from all these sources is assigned to be 0.48%, which is the quadrature sum of these three terms.
The ST efficiencies obtained from the inclusive MC sample may differ from those estimated with the signal MC events generated with events containing the ST D − s and D + s → τ + ν τ decays, thereby causing possible tag bias.The size of this bias is estimated by measuring for each tag ε D + s →τ + ντ ST , the efficiency in the signal MC sample, and ε inclusive D + s ST , the efficiency in the inclusive MC sample, and multiplying (ε −1) by the estimated data-MC differences in the tracking and PID efficiencies without any correction, which are 1.0% for charged pions and kaons, and 2.0% for π 0 , η(γγ) and K 0 S decays.The resulting numbers are weighted by the ST yields in each tag to yield an overall systematic uncertainty of 0.37%.
After weighting by the yields of ST D − s mesons in each data sample, the uncertainty from the limited MC sample sizes is assigned to be 0.29%.ντ , the first, second, and third uncertainties are the statistical, tag-mode dependent systematic and tag-mode independent systematic, respectively.The listed efficiencies do not include the BFs of the sub decays.
The index j from 1 to 14 represents the tag modes π 0 , respectively.The ϵ j DT /ϵ j ST varies within 46% for different tag modes; this is mainly due to the significantly different signal environments for some tag modes containing low momentum photon and pions in the signal and inclusive MC samples.The systematic uncertainties related to the µ + tracking and PID efficiencies are investigated by using a control sample of e + e − → γµ + µ − decays.By considering the dependencies of the µ + efficiencies on the µ momentum, polar angle, and different energy points, the difference of µ + tracking efficiencies between data and MC simulation is (−0.32±0.18)%.After correcting the signal efficiencies to data, the associated systematic uncertainty is assigned to be 0.18%.The difference of the µ + PID efficiencies between data and MC simulation is found to be −(11.86± 0.33)%.A similar large difference in the µ + PID efficiency between data and simulation was observed for D + s → µ + ν µ events in previous analyses at BESIII and is understood to arise from imperfections in the simulation of the length of the muon traveling in the MUC [17].After correcting the signal efficiencies to data, the uncertainty 0.33% is assigned as the corresponding systematic uncertainty.
The efficiency of the γ selection is studied by using a control sample of J/ψ → π + π − π 0 decays [51], while the π 0 reconstruction efficiency is studied with a sample of e + e − → K + K − π + π − π 0 events [52].The systematic uncertainty of selecting the transition γ or π 0 is estimated to be 1.00%, accounting for the relative BFs of The systematic uncertainty associated with the M 2 3ν requirement is assessed by reperforming the measurement with enlarging or shrinking this requirement by ±1 or ±2 bin sizes, resulting in 24 variations.Among all variations, the maximum change of BF, 1.75%, is taken as the corresponding systematic uncertainty.
The systematic uncertainty associated with the requirement of no extra charged tracks (N charge extra ) is studied with the DT sample of The difference of the acceptance efficiencies between data and MC simulation, 0.41%, is taken as the systematic uncertainty.
The systematic uncertainty in the E tot extra γ fit has contributions associated with the three classes of background.The systematic uncertainty arising from the BKGI is estimated by varying the sideband scale factor by ±1σ and the corresponding change of 0.10% in the fitted signal yield is taken as the systematic uncertainty.The systematic uncertainty arising from the shape of BKGII is assessed by replacing the corrected shape of E tot extra γ with the uncorrected one and is found to be negligible.We also change the level of BKGII background by varying the misidentification rate by ±1σ and the BF of The relative difference of the fitted signal yield, 1.39%, is assigned as the associated systematic uncertainty.The uncertainty due to the non-peaking shape of BKGIII is estimated by varying the f 2 by ±1σ and the relative components of the leading six background modes [23], and is assigned to be 0.69%.After adding these contributions in quadrature, the uncertainty associated with the E tot extra γ fit is assigned to be 1.56%.The uncertainty on the BF of τ + → µ + ν τ ντ contributes a systematic uncertainty of 0.23% [23].

TOTAL SYSTEMATIC UNCERTAINTIES
By adding the individual components in quarature, we determine the total tag-mode dependent and independent systematic uncertainties to be 0.67% and 2.62%, respectively, and the total relative systematic uncertainty to be 2.70%.

RESULTS
The measured values B D + s →τ + ντ are listed in Table 2 for each tag mode.Weighting each measurement by the inverse squares of the combined statistical and tag-mode dependent systematic uncertainties yields B D + s →τ + ντ = (5.37 ± 0.17 stat ± 0.15 syst )%.
Here, the first uncertainty is statistical, and the second is the quadrature sum of the tagmode dependent and independent systematic uncertainties.

SUMMARY
By analyzing e + e − collision data collected with a total integrated luminosity of 7.33 fb −1 at the center-of-mass energies between 4.128 GeV and 4.226 GeV, we determine the BF of D + s → τ + ν τ via τ + → µ + ν µ ντ to be (5.37 ± 0.17 stat ± 0.15 syst )%.This result is consistent with the previous measurements [23] [23], we obtain R τ /µ = 9.89 ± 0.50, which is consistent with the expectation based on lepton flavor universality.
Improved measurements of B(D + s → τ + ν τ ) are foreseen with the larger data sets that BESIII is expected to accumulate in the coming years [29].

ACKNOWLEDGEMENT
The BESIII Collaboration thanks the staff of BEPCII and the IHEP computing center for their strong support.

Figure 1 .
Figure 1.The M BC distributions of the ST D − s candidates in data and inclusive MC samples at 4.178 GeV.The candidates between the two red arrows are retained for further analysis.
Figure 2 shows the fit results for the data sample at √ s = 4.178 GeV.In each sub-figure, the red arrows show the chosen M ST signal regions.The candidates located in these signal regions are retained for further analysis.Based on simulation, the e + e − → (γ ISR )D + s D − s process is found to contribute about (0.7-1.1)% in the fitted number of ST D − s mesons for each tag mode.The reported yields have this contribution subtracted.The efficiencies of reconstructing ST D − s mesons (N ST ) are estimated by analyzing the inclusive MC sample in the same way as real data.

Figure 2 .
Figure 2. The fits to the M ST distributions of the surviving ST D − s candidates for each tag mode.The points with error bars denote the data sample at √ s = 4.178 GeV.The blue solid curves represent the best fit results.The red dashed curves represent the fitted backgrounds.For the D − s → K 0 S K − tag mode, the blue dotted curve is the peaking background from D − → K 0 S π − .In each figure, the range within the two arrows indicate the chosen M ST signal regions and the brown line segments indicate the sideband regions.

Figure 3 .
Figure 3.The distributions of d µ vs. p µ in different |cos θ µ | regions of e + e − → γµ + µ − candidates in data.The regions above the red line are retained for further analysis.

Figure 4 .
Figure 4.The M 2 3ν distributions of accepted candidates in data and the inclusive MC samples with the E tot extra γ <0.4 GeV requirement.Candidates with M 2 3ν within the two red arrows are retained for further analysis.

Figure 5 .
Figure 5.The distributions of E tot extra γ of the DT candidates for D + s → τ + ν τ with τ + → µ + ν µ ντ .Black points with error bars are the combined data sample.Solid blue histograms denote the resutlts.Filled pink shadows, open circles with error bars, filled green histograms, and dashed blue histograms are Signal, BKGI, BKGII, and BKGIII, respectively.The area to the left of the red arrow denotes the signal region.

Figure 8 .
Figure 8.Comparison of |V cs | values in this with previous work.For experimental results, the inner error bar is the statistical uncertainty and the outer is the combined statistical and systematic uncertainty.The green band denotes the CKM Fitter average and the yellow one denotes the experimental average.The last line is the BESIII combined result which does not include the BESIII result in Ref. [18].

Table 1 .
Identification criteria for muon candidates.

Table 2 .
The fitted yields of ST D − Systematic uncertainties which do not depend on tag modes are classified as tag-mode independent.

Table 3 .
Systematic uncertainties in the BF measurement.
Using this BF and the world average values of G F , m µ , m D + s , and τ D + s [23] with Γ D + s →τ + ντ = B D + s →τ + ντ /τ D + s , we determine the product of f D + s and |V cs | to be f D + s |V cs | = (246.7 ± 3.9 stat ± 3.6 syst ) MeV, where the systematic uncertainty is dominated by that of the measured BF (1.86%) and the lifetime of D + s (0.8%).Making use of |V cs | = 0.97349 ± 0.00016 from the global fit in the SM [23, 53], we obtain f D + s = (253.4± 4.0 stat ± 3.7 syst ) MeV. = 0.987 ± 0.016 stat ± 0.014 syst .In the calculation of |V cs |, one additional uncertainty (0.2%) for the input value of f D + s is included.In the determination of f D + s , however, the uncertainty from the input value |V cs | has negligible effect.Our value |V cs | agrees with our previous results obtained via D → Kℓ + ν ℓ [54-57], D + s → µ + ν µ [17], and D + s → η (′) e + ν e decays [45].

-1 BESIII 6.32 fb ν
This work is supported in part by National Key R&D Program of Comparison of the BFs measured in this work with previous measurements, where the inner error bar is the statistical uncertainty and the outer is the combined statistical and systematic uncertainty.The last line is the BESIII combined result which does not include the BESIII result in Ref. [18].Figure 7. Comparison of f D +s values in this with previous work and LQCD calculations.For experimental results, the inner error bar is the statistical uncertainty and the outer is the combined statistical and systematic uncertainty.The green band denotes the FLAG average and the yellow one denotes the experimental average.The last line is the BESIII combined result which does not include the BESIII result in Ref.[18].