Measurement of the absolute branching fractions for purely leptonic $D_s^+$ decays

We report new measurements of the branching fraction $\cal B(D_s^+\to \ell^+\nu)$, where $\ell^+$ is either $\mu^+$ or $\tau^+(\to\pi^+\bar{\nu}_\tau)$, based on $6.32$ fb$^{-1}$ of electron-positron annihilation data collected by the BESIII experiment at six center-of-mass energy points between $4.178$ and $4.226$ GeV. Simultaneously floating the $D_s^+\to\mu^+\nu_\mu$ and $D_s^+\to\tau^+\nu_\tau$ components yields $\cal B(D_s^+\to \tau^+\nu_\tau) = (5.21\pm0.25\pm0.17)\times10^{-2}$, $\cal B(D_s^+\to \mu^+\nu_\mu) = (5.35\pm0.13\pm0.16)\times10^{-3}$, and the ratio of decay widths $R=\frac{\Gamma(D_s^+\to \tau^+\nu_\tau)}{\Gamma(D_s^+\to \mu^+\nu_\mu)} = 9.73^{+0.61}_{-0.58}\pm 0.36$, where the first uncertainties are statistical and the second systematic. No evidence of ${\it CP}$ asymmetry is observed in the decay rates $D_s^\pm\to\mu^\pm\nu_\mu$ and $D_s^\pm\to\tau^\pm\nu_\tau$: $A_{\it CP}(\mu^\pm\nu) = (-1.2\pm2.5\pm1.0)\%$ and $A_{\it CP}(\tau^\pm\nu) = (+2.9\pm4.8\pm1.0)\%$. Constraining our measurement to the Standard Model expectation of lepton universality ($R=9.75$), we find the more precise results $\cal B(D_s^+\to \tau^+\nu_\tau) = (5.22\pm0.10\pm 0.14)\times10^{-2}$ and $A_{\it CP}(\tau^\pm\nu_\tau) = (-0.1\pm1.9\pm1.0)\%$. Combining our results with inputs external to our analysis, we determine the $c\to \bar{s}$ quark mixing matrix element, $D_s^+$ decay constant, and ratio of the decay constants to be $|V_{cs}| = 0.973\pm0.009\pm0.014$, $f_{D^+_s} = 249.9\pm2.4\pm3.5~\text{MeV}$, and $f_{D^+_s}/f_{D^+} = 1.232\pm0.035$, respectively.


I. INTRODUCTION
Purely leptonic decays of heavy mesons are the subject of great experimental and theoretical interest because of their potential for precise tests of the Standard Model (SM), including determination of the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements and sensitivity to non-SM physics.Leptonic decays of charmed mesons play an important role in this, with clean experimental signatures and the opportunity for rigorous tests of strong-interaction theory, especially lattice-QCD (LQCD) calculations.In the decay process D + s → + ν , the charm quark (c) and antistrange quark (s) annihilate through a virtual W boson to a charged and neutral lepton pair.(Throughout this article, charge conjugate modes are implied unless otherwise noted.)According to the SM, the branching fraction for this process (ignoring radiative corrections) is given as follows: where τ D + s is the D + s lifetime, m D + s is the mass of D + s , m is the mass of the charged lepton (e + , µ + , or τ + ), and G F is the Fermi coupling constant, all of which are known to precision [1].The remaining two factors, |V cs | and f D + s , must be determined experimentally and are of great interest.(1) V cs is a fundamental SM parameter, the CKM matrix element describing the coupling between the c and s quarks.(2) f D + s is the D + s decay constant, the amplitude for quark-antiquark annihilation inside the meson, which can be thought of as the overlap of the wave functions of c and s at zero spatial separation.
It follows from Eq. ( 1) that measurement of the branching fraction B(D + s → + ν ) is essentially a determination of f 2 In practice, we can determine f D + s by combining a measurement of B(D + s → + ν ) with an independent determination of |V cs |, thereby testing theoretical predictions, primarily from LQCD.Testing the LQCD calculations in D and D + s decays is especially important to validate their application to the B-meson sector, in which the precision of experimental determination of f B + is very limited due to the small value of |V ub |, 0.00361 +0.00011  −0.00009 [1].It is also possible to reverse this approach, determining |V cs | from B(D + s → + ν ) with a theoretical estimate of f D + s from LQCD and comparing the result with other experimental determinations of |V cs |.
The ratio of decay widths for leptonic decays to µ and τ is also an interesting quantity to measure:

Ds
) 2 . ( In this ratio, the decay constant and the CKM element cancel, giving a very precise SM prediction of R = 9.75 ± 0.01.Any deviation from this value potentially indicates the existence of non-SM physics. Using the known values of the lepton masses and other constants [1], the measured D + s lifetime [1] (including recent improvements in precision by the LHCb Collaboration [2]), the weighted average of recent four-flavor LQCD calculations [3] (f D + s = 249.9± 0.5 MeV), and the latest determination of the c → s coupling from the global fit of CKM parameters [1] (|V cs | = 0.97320 ± 0.00011), one arrives at the following SM predictions of the D + s leptonic branching fractions: In this article, we report new measurements of the branching fractions , and of the CP-violating asymmetries A CP ( ± ν ).These measurements have been made with 6.32 fb −1 of e + e − annihilation data collected at center-of-mass energies between 4178 and 4226 MeV with the BESIII detector [4] at the Beijing Electron Positron Collider (BEPCII) [5].This work, which uses a larger data sample and a procedure that is simultaneously sensitive to both D + s → µ + ν µ and D + s → τ + ν τ decays, is distinct from and supersedes our previous measurement of B(D + s → µ + ν µ ) [6].

II. THE BESIII EXPERIMENT AND DATA SETS
BESIII [4] is a cylindrical spectrometer with a geometrical acceptance of 93% of 4π.It consists of a small-celled, helium-based main drift chamber (MDC), a plastic scintillator time-of-flight system (TOF), a CsI(Tl) electromagnetic calorimeter (EMC), a superconducting solenoid providing a 1.0-T magnetic field, and a muon counter (MUC).The charged particle momentum resolution is 0.5% at a transverse momentum of 1 GeV/c.The specific ionization (dE/dx) measurement provided by the MDC has a resolution of 6%, and provides 3σ separation of charged pions and kaons.The time resolution of the TOF is 80 ps (110 ps) in the central barrel (end-cap) region of the detector.The end-cap TOF system was upgraded with multigap resistive plate chamber technology in 2015 [7], improving its time resolution to 60 ps.Approximately 83% of the sample employed in this work was taken with the improved configuration.The energy resolution for photons is 2.5% (5%) at 1 GeV in the barrel (end-cap) region of the EMC.A more detailed description of the BESIII detector is given in Ref. [4].
The data samples employed in this work were taken at six e + e − center-of-mass energies (E cm = 4178, 4189, 4199, 4209, 4219, and around 4226 MeV [8]).The integrated luminosities for these subsamples are 3189.0,526.7, 526.0, 517.1, 514.6, and 1047.3 pb −1 [9], respectively.For simplicity, we refer to these datasets as 4180, 4190, 4200, 4210, 4220, and 4230 in the rest of this article.For some aspects of our analysis, especially in assessing systematic uncertainties, we organize the samples into three groups that were acquired during the same year under consistent running conditions.The 4180 sample was taken in 2016, while the group 4190-4220 was taken in 2017 and 4230 was taken in 2013.
To assess background processes and determine detection efficiencies, we produce and analyze geant4based [10] Monte Carlo (MC) simulation samples for all six datasets, with sizes that are 40 times the integrated luminosity of data ("40×").The MC samples are produced using known decay rates [1] and correct angular distributions by two event generators, EvtGen [11] for charm and charmonium decays and KKMC [12] for continuum processes.The samples consist of e + e − → D D, s , DD * π, DDπ, q q(q = u, d, s), γJ/ψ, γψ(3686), and τ + τ − .Charmonium decays that are not accounted for by exclusive measurements are simulated by Lundcharm [13].Additionally we generate separate samples consisting only of signal events, with size 300 times larger than is expected in our data.All MC simulations include the effects of initialstate radiation (ISR) and final-state radiation (FSR).We simulate ISR with ConExc [14] for e + e − → cc events within the framework of EvtGen, and with KKMC for noncharm continuum processes.FSR is simulated with PHOTOS [15].

III. ANALYSIS METHOD
We employ the double-tag technique pioneered by the MARK III Collaboration [16] in our selection of D ± s → ± ν decays in e + e − → D * ± s D ∓ s events.In this method, a D − s is fully reconstructed through one of several hadronic decay modes (tag side), while we reconstruct only one charged track from the D + s (signal side).Note that the reconstructed tag-side D s can either be directly produced from the e + e − collision (direct) or be the daughter of a D * s (indirect).We reconstruct the radiative photon from D * s → γD s in the tag-side selection and use it in the analysis of the signal side.(Further explanation is provided in Sec.III A.) The absolute branching fraction determined by this method does not depend on the integrated lumi-nosity or the produced number of D * ± s D ∓ s pairs: where N DT = i N i DT is the summed yield over tag modes of double-tag (DT ) events in which the tag side and signal side are simultaneously reconstructed, N i ST is the single-tag (ST ) yield of reconstructed D − s for tag mode i, and µν,i DT and i ST are the corresponding reconstruction efficiencies.Similarly for B(D + s → τ + ν τ ) we have where τ ν,i DT does not include B(τ + → π + ντ ) but ν,i DT does include B(D * s → γD s ).In the ratios of N ν DT /N i ST and ν,i DT / i ST , systematic uncertainty associated with the tag-side analysis mostly cancels, except for a possible uncertainty due to variations in ST reconstruction efficiencies (tag bias, discussed in Sec.IV A).

A. Selection of tagged D − s candidates
The ST D − s is reconstructed using tracks and EMC showers that pass several quality requirements.The selection criteria for D − s daughters and the reconstruction procedures are the same as those described in Ref. [17].Tracks must be within the fiducial region (| cos θ| < 0.93, where θ is the polar angle relative to the positron beam direction) and originate within 1 cm (10 cm) of the interaction point in the plane transverse to the beam direction (along the beam direction).This requirement on the primary vertex is not applied for the reconstruction of K 0 S → π + π − , for which we constrain the charged pion pair to have a common vertex with a loose fitquality requirement of χ 2 < 100 for 1 degree of freedom.To be selected as a photon candidate, an EMC shower must not be associated with any charged track [18], must have an EMC hit time between 0 and 700 ns to suppress activity that is not consistent with originating from the collision event, and must have an energy of at least 25 MeV if it is in the barrel region of the detector (| cos θ| < 0.8) and 50 MeV if it is in the end-cap region (0.86 < | cos θ| < 0.92) to suppress noise in the EMC.
We apply K/π particle identification (PID) based on TOF and dE/dx measurements, with the identity as a pion or kaon assigned based on which hypothesis has the higher likelihood.Pions from the intermediate states K 0 S → π + π − , η → π + π − π 0 and η → π + π − η are not required to satisfy the K/π-identification requirement.We also require that the reconstructed momentum for any charged or neutral pion have a magnitude of at least 100 MeV/c to suppress events from D * → Dπ.
We select tag modes for this analysis to maximize signal sensitivity and minimize tag bias by performing the entire analysis procedure on our cocktail MC sample with various combinations of tag modes.The following thirteen D − s hadronic decay modes are used: , and η γρ → γρ 0 .Requirements on invariant-mass ranges for the intermediate states π 0 , K 0 S , η, η 3π , ρ, η ππη , and η γρ are chosen to cover ±(3 − 4)σ of the signal mass resolutions, with the exception of ρ, for which 520 (500) < M ππ < 1000 MeV/c 2 is selected for the charged (neutral) case.We also require an isolation criterion for photons used in the reconstruction of η → γρ 0 ; the reconstructed photons must be separated from the extrapolated positions of all charged particles by more than 20 in the center-ofmass system of the initial e + e − , where 2050 < M rec < 2195 MeV/c 2 for the 4180 data to en-sure that events are consistent with e + e − → D * ± s D ∓ s .For other data samples taken at higher E cm , the tails of the indirect D s events extend more widely.We expand the selected M rec range to maintain roughly constant KKπ tag efficiencies for different values of E cm , except for the 4230 data, whose energy is above the threshold for production of D * s D * s .In this case we require 2040 < M rec < 2220 MeV/c 2 to suppress D s from D * s D * s events, which mostly have M rec > 2230 MeV/c 2 .When multiple reconstructed candidates are found for a given D s tag mode and electric charge, we keep only the one with the best D * s → γD s photon candidate.In the D * s rest frame, the emitted photon energy is monochromatic, with energy (m 2 Ds c 4 )/(2m D * s c 2 ) = 138.9MeV.Once the tag-side D − s is reconstructed, we loop over the remaining photon candidates to construct the four-momentum of the D * s candidate based on two hypotheses.For the first hypothesis, we assume that the tag is direct, the photon is on the signal side, and p(D * + s ) = p e + e − − p tag .For the second, we assume that the tag is indirect, the photon is on the tag side, and p(D * − s ) = p γ + p tag .We then choose between these the combination that gives E γ closest to 138.9 MeV in the D * s rest frame and use this in the rest of our analysis.Additionally, we require 119 < E γ < 149 MeV to suppress backgrounds further.This range is selected based on MC studies of the signal side, as is described in Sec.III B. The resulting optimized photon selection efficiency is ∼ 90%.While we do not use information about the transition photon in determining the tag-side yields, we perform the photon reconstruction and apply the additional selection on E γ at this point to minimize the systematic uncertainty due to the best-tag selection.The effect on the signal efficiency of selecting the transition photon is minimal due to the simple decay topology of the signal side, with only one charged track.
To determine the tag yields, we perform an unbinned maximum likelihood fit to M inv (D − s ) in the range 1900 < M inv (D − s ) < 2030 MeV/c 2 .The signal functions are based on distributions from MC simulations, obtained by the Gaussian kernel estimation method [19], and convolved with a Gaussian function to account for differing resolution between data and MC samples.In practice it is difficult to float the width of the Gaussian function in fits for tag modes with larger backgrounds in smaller samples (e.g., 4220 data).Because of this, we assume that the relative difference between data and MC samples is consistent among the different datasets and simultaneously fit to all six samples, sharing a single convolved Gaussian function for a given tag mode.We estimate possible systematic uncertainty associated with this assumption in Sec.IV B.
Backgrounds in the invariant-mass fits are represented by low-degree Chebyshev polynomials (first to third, depending on the tag mode).Figure 2 shows fits to the M inv (D − s ) distributions of the thirteen tag modes for the 4180 dataset.In the fits of tag modes , which are taken into account in the fits.Table I shows the ST efficiencies, ST .The corresponding ST yields from data are also shown in Table II.
Signal selection begins with the requirement that there be only one additional track that is unused in the reconstruction of the tag (N tk = 1).The corresponding particle must have electric charge opposite to the tag and satisfy the pion PID criteria described in Sec.III A. (The pion PID response closely approximates that for a muon because pions and muons are charged particles with similar masses.)It is significant that the systematic uncertainty arising from the PID requirement in the signal selection does not cancel, as it does for tag reconstruction.We study control samples of all datasets and observe differences in PID efficiencies between data and MC samples of about 1%.Here the D 0 sample is obtained via e + e − → D * + (→ π + D 0 )D ( * )− .Corrections are applied to MC-determined efficiencies.
We split the signal-track sample into two parts based on the energy-deposit properties of muons and pions in the EMC, as was done previously in similar analyses [20,21].Candidates with signal-track energy deposit satisfying E EMC ≤ 300 MeV are classified as µ-like and the remainder as π-like.Based on MC simulation, we estimate that the µ-like sample includes ∼ 99% of D + s → µ + ν µ events and ∼ 58% of As mentioned in Sec.III A, we impose the additional requirement on signal candidates from D * s → γD s that the selected photon have an energy in the D * s rest frame that satisfies 119 < E γ < 149 MeV.This criterion is optimized based on a detailed MC study.Distributions of E γ for MC and data are shown in Fig. 3.Here the input B(D + s → µ + ν µ ) and B(D + s → τ + ν τ ) in our MC samples are 5.38 × 10 −3 and 5.54 × 10 −2 , respectively.
We suppress candidate signal events that are not true D + s → + ν by considering three additional variables that are sensitive to unreconstructed particles.The first is cos θ miss , where θ miss is the polar angle of p miss = − p tag − p γ − p µ/π in the e + e − center-of-mass frame.We require | cos θ miss | < 0.9 to ensure that p miss points into the fiducial volume of our detector.The second variable is AN G, the opening angle between p miss and the most energetic unused EMC shower.Events with a K 0 L or an energetically asymmetric decay from π 0 → γγ or η → γγ tend to leave detectable EMC energy deposits near the p miss direction.We require AN G > 40 • to suppress such events.The third variable is E max neu , the maximum unused EMC shower energy.Requiring E max neu < 300 MeV helps to ensure that nothing energetic is unaccounted for in selected events.
C. Fit

Fitting to simulated samples
We infer the presence of neutrinos in the final states from the event missing mass-squared, is computed in the e + e − center-of-mass frame, and m Ds is fixed to the known value [1].We determine the B(D + s → + ν) signal yields with a simultaneous unbinned maximum likelihood fit to the two-dimensional distributions of the tag-side invariant mass M inv (D − s ) versus M 2 miss for the six E cm samples.The fit range on the M inv (D − s ) axis is the same as we use to fit to the tagside s → τ + ν τ , but not from τ + → π + ντ , and everything else is correct, with peaks in M inv but smooth distributions in M 2 miss .(Note that, while such events are not analyzed correctly, they are not background and we float this contribution with the τ + → π + ντ signal.)(5) All remaining background, both charm and noncharm, which has a smooth continuous shape.Two-dimensional background distributions for fitting are constructed based on products of onedimensional probability distribution functions (PDFs) for M inv (D − s ) and M 2 miss .Fits are performed with MC-based shapes for signals, similarly extracted by the kernel estimation method [19], and the following distributions for the five background components.( 1 and (m) K − π + π − .Points are data, while red-solid lines represent the total fits, blue-dotted lines are the fitted background shapes, and the orange-dashed lines correspond to the fitted signal shapes.The magenta-dotted lines only seen in the two tag modes, (a) K 0 S K − and (b) K − K + π − , are the fitted nonsmooth background shapes representing Tag mode 4180 4190 4200 4210 4220 4230  ing that the data/MC differences do not depend on E cm and running conditions.
Normalizations of the five background components are fixed whenever an absolute estimate is possible and otherwise are allowed to float freely in fitting, as follows.
(1) Wrong tag: fixed based on the MC-estimated ratio of the correctly reconstructed yields to the wrongly reconstructed yields.(We float this component in assessing associated systematic uncertainties.)(2) Wrong photon: ratio to the signal component is fixed according to the MC estimation.(3) Wrong track: fixed according to a MC estimation for which the MC sample is scaled to the N ST observed in data.(4) τ decays to final states other than π + ντ : constrained to the signal yield, (5) Remainder: floated freely.Normalizations of the two signal components (D + s → µν µ and τ + ν τ ) are floated freely, except for the constraints introduced by the simultaneous fit and the ratio of the yields between the µ-like and π-like samples.This ratio is fixed for each of the six datasets based on MC estimations.
The means of the convolved Gaussian functions in fitting M inv (D − s ) and M 2 miss are floated for both data and MC simulations.The widths are floated in fitting to data and are fixed to a negligibly small value in fitting to MC samples.Because the M inv (D − s ) resolution depends on the D − s decay mode, the M inv (D − s ) PDF is obtained as a sum of PDFs for all tag modes, weighted by the observed N ST .We obtain the M 2 miss PDFs in a similar way, as MC simulation predicts weak tag-mode dependence.The coefficients of the Chebyshev polynomials for backgrounds (1) and ( 5) are shared because MC simulations demonstrate that the statistical sensitivity of our data is insufficient to distinguish their slopes.
We determine ν DT for each of the thirteen tag modes by counting the reconstructed candidates that match the MC-generated true signal particles.The resultant efficiencies are shown in Tables III and IV.
To validate these DT efficiencies and our overall fitting procedure, we perform tests on ten independent data-size MC samples and compare the fitted signal yields and the corresponding branching fractions.Table V shows the differences between the fitted signal yields and the MCpredicted yields for the data-size samples and for a 40× sample.We see reasonable agreement between the fitted and generated yields.
Figure 4 shows projections of the selected data sample onto the M 2 miss and M inv (D − s ) axes, summed over the six data samples.Scaled MC distributions are overlaid, with a breakdown of the components of the MCsimulated background.There is agreement between data and MC simulation for both signal and background.Figures 5 and 6 show comparisons between data and MC samples for signal-track momentum and the cosine of the polar angle, again demonstrating excellent agreement.

Fitting to data
The fitting procedure described in Sec.III B accounts for all signal and background processes that have been observed experimentally to date.We now introduce two additional physics processes that are expected theoretically but have not yet been observed, D + s → γµ + ν µ and The radiative leptonic decay D + s → γµ + ν µ is not helicity suppressed, but its experimental detection is difficult and has not yet been achieved.Past measurements of D + s leptonic decays [20,21] have relied on theoretical predictions of the branching fraction for this mode, with a 1% (relative) systematic uncertainty.For our analysis we note that the M 2 miss distribution for D + s → γµ + ν µ must have a high-side tail and could therefore affect our signal fits significantly.We generate a signal MC sample for this process following the procedure of previous BE-SIII studies of D + → γe + ν e [22] and D + s → γe + ν e [23].This adopts a factorization approach [24] in modeling Tag mode 4180 4190 4200 4210 4220 4230 D + s → γµ + ν µ events, requiring a minimum photon energy of 10 MeV. Figure 7 shows the predicted M 2 miss distribution for D + s → γµ + ν µ , along with distributions for other processes.Because of the high-side tail of the distribution for D + s → γµ + ν µ , we explicitly include this additional PDF in fitting our data.

The suppressed hadronic decay D +
s → π + π 0 has not been observed, with only an upper limit on the branching fraction of B(D + s → π + π 0 ) < 3.4 × 10 −4 at 90% confidence level [1].Events of this type exhibit a clear peak in M 2 miss , as is demonstrated by MC simulation and shown in Fig. 7.We do not include D + s → π + π 0 in our nominal data fit, but introduce it with a branching fraction equal to the upper limit as a systematic variation.s → µ + νµ and τ + (→ π + ντ )ντ signal on top (blue).Background components are stacked below this in the following order from the top to the bottom: tag-side misreconstructed (cyan), misreconstructed transition photon (light brown), signal side misreconstructed (dark brown), D + s → τ + ντ , where the τ + decays to a final state other than π + ντ (green), other charm background (gray), and noncharm sources (black, at bottom).
An alternative procedure that provides a more statistically precise but model-dependent determination of ) is to fit our data with the ratio R fixed to the SM prediction of 9.75.This fit yields a D + s → τ + (→ π + ντ )ν τ signal of 946 ± 18 events and B(D + s → τ + ν τ ) = (5.22 ± 0.10)%.
Because of this anticorrelation between the D + s → µ + ν µ and D + s → τ + ν τ signal processes, caution is necessary in extracting the decay constant f D + s and CKM matrix element |V cs | from an average of the measured branching fractions.We circumvent this difficulty by requiring lepton flavor universality (LFU), which requires that values of f D + s |V cs | extracted from B(D + s → µ + ν µ ) and B(D + s → τ + ν τ ) be identical.From Eqs. ( 1) and ( 2), it can be seen that this LFU constraint is equivalent to the condition R = 9.75.Thus, in Sec.V we present our combined average value of f D + s |V cs | by using the measurement obtained with the constraint R = 9.75.

CP-violating Asymmetries
We also measure the CP-violating asymmetries, where = µ or τ .Procedures are identical to those applied to the full sample, except that we determine the branching fractions separately for D + s and D − s .To search for systematic effects specific to this measurement, we look at ∆N , where i runs over the thirteen tag modes and N ± tag,i = N ± ST,i / ± ST,i for D ± s → i mode, and we combine the six datasets.We obtain ∆N ± tag = (+0.6 ± 0.8)%, consistent with zero CP asymmetry, which involves the simulations of charge-dependent tracking and PID efficiencies.We conservatively assign 1.0% (0.6% and 0.8% combined in quadrature) as a possible systematic uncertainty due to charge dependence in particle reconstruction and assign no additional systematic uncertainty to our measurements.Potential systematic effects that are associated with our DT fitting procedure are canceled in the determination of A CP .
The nominal fit yields 1123 ± 40 events and 463 +33 −32 events for D − s → µ − ν µ and D − s → τ − ν τ candidates, respectively, and 1077 ± 38 events and 487 +33 −32 events for D + s → µ + ν µ and D + s → τ + ν τ candidates, respectively.Table VI shows the resultant branching fractions for D + s and D − s , as well as A CP , based on the two fitting methods: our principal method, which yields both ), and the alternative method imposing the SM constraint, labeled as B SM τ ν .The first uncertainties quoted in A CP are statistical and the second systematic.All three A CP values show no evidence of CP violation.This is the first measurement of A CP (τ ν) and the most precise determinations to date of both A CP (µν) and A CP (τ ν) SM .ACP (τ ± ν) +2.9 ± 4.8 ± 1.0 ACP (τ ± ν) SM −0.1 ± 1.9 ± 1.0

IV. SYSTEMATIC UNCERTAINTIES
We consider a wide variety of potential sources of systematic uncertainty in our measurements of the branching fractions B(D + s → µ + ν µ ) and B(D + s → τ + ν τ ), and their ratio R. Procedures are described in the following two subsections and the resulting estimates are listed in Table VII.The sources of systematic uncertainty subdivide into two categories.Sources associated with the two-dimensional simultaneous DT fitting procedure affect all measurements, while those that are not related to fitting largely cancel in measuring the ratio R.

A. Nonfitting systematic uncertainties
We directly estimate the systematic uncertainties associated with the two input branching fractions, B(D * s → γD s ) and B(τ + → π + ντ ), by propagating the uncertain- ties from Ref. [1].
We estimate the systematic uncertainty associated with the reconstruction of the signal track, µ or π, by reconstructing events from the continuum process e + e − → K + K − π + π − .By comparing the pion reconstruction efficiency over the relevant momentum range for 4180 data and MC samples, the reliability of the simulation is found to be better than 1%.The stability of tracking over our six data samples is demonstrated by consistent performance on control samples of radiative µ-pair events.On this basis we assign a 1% systematic uncertainty for our branching fraction measurements.
The systematic uncertainty associated with the reconstruction of the photon from D * s → γD s is estimated by reconstructing J/ψ → π + π − π 0 events [28].Comparison of the photon-reconstruction efficiency in data and MC miss distributions for various types of MC events for the µ-like case.(Shapes are very similar for the π-like case.)The dotted-blue line represents samples gives a 1% systematic uncertainty for this source.
In selecting our signal-side sample, we require that there be only one charged track in addition to the daughters of the reconstructed tag, as described in Sec.III B. We estimate the systematic uncertainty for this requirement based on double-hadronic-tag (DHT) events in which a D − s is tagged in one of our thirteen modes, while the D + s decays into either K 0 S π or KKπ.The uncertainty is 0.2% for all three branching fraction measurements.
As described in Sec.III A, we handle events with multiple ST candidates for a given D s mode and charge by choosing the one with E γ , the D * s → γD s photon energy in the D * s rest frame, closest to the expected value.We investigate systematic effects in this selection by comparing efficiencies for DHT events in data and MC samples.The agreement is found to have some dependence on event complexity (charged and neutral particle multiplicity), but is no worse than 1%, so we assign this as the systematic uncertainty in the best-photon selection for all branching fraction measurements.
Our signal-selection procedure (Sec.III B) includes three additional requirements that are designed to suppress events with unreconstructed particles.We study systematic uncertainties associated with these using the same DHT events.For the requirement E max neu < 300 MeV, we compare the efficiencies in data and MC samples for the standard requirement and probe the stability of the data/MC agreement by also testing with requirements less or more restrictive than this by 50 MeV.We find an uncertainty of 0.3% for all branching fractions.We similarly probe the AN G > 40 • and | cos θ miss | < 0.90 efficiencies, although in this case the results with the D s DHT sample are limited by sample size.We augment with data collected at E cm = 3773 MeV, with an integrated luminosity of 2.93 fb −1 , and copious production of ψ(3770) → D D. In this sample we measure data and MC efficiencies for D 0 decays into the three hadronic modes, K − π + , K − π + π 0 , and K − π + π − π + , and for the semileptonic decay D0 → K + e − νe .Based on these studies, we assign a 1% systematic uncertainty for the AN G and | cos θ miss | requirements for all branching fraction measurements.
The determination of the D s leptonic branching fractions with Eqs. ( 3) and ( 4) depends on the efficiency ratios ν,i DT / i ST .Both ST and DT selection involve reconstructing a hadronic D − s decay, and we expect the efficiency for this tag reconstruction to depend on the event environment.The different topologies of leptonic D + s decays (only one track) and generic D + s decays (most with multiple tracks and showers) produce a mode-dependent bias in reconstructing the D − s tag that may be imperfectly modeled in the MC simulation.We estimate the systematic uncertainty associated with this effect by studying the BESIII detector's tracking and PID efficiencies for events with different particle multiplicities using the large E cm = 3773 MeV data sample mentioned earlier.The size of this uncertainty varies among the three branching fraction and R measurements, as is shown in Table VII.

B. Fitting systematic uncertainties
To assess the systematic uncertainties associated with our fitting procedure, we generate toy Monte Carlo samples based on the observed data distributions.We fit to these toy samples while varying an analysis selection requirement (or fitting procedure, PID requirement, etc.) and take the difference between the averages of these ensembles with the nominal fit procedure and with the alternative procedure and assign it as a systematic uncertainty.Table VII shows that these estimated systematic uncertainties vary significantly among the measurements of the three branching fraction and R.
The uncertainty in the determination of the denominators in Eqs. ( 3) and ( 4) arises mainly from fitting to M inv (D − s ) for ST candidates.The dominant effect comes from background and signal shapes (including the convolved Gaussian functions, which are independently determined for 4180, 4190 − 4220 and 4230).We also investigate the contamination from e + e − → γ ISR D + s D − s , and find the uncertainty associated with this to be negligible.
For a conservative estimate of the uncertainty due to the assumption of a fixed ratio R γ = B(D + (s) → γµ + ν µ )/B(D + (s) → µ + ν µ ) = 0.1, we vary R γ by ±0.1.To allow for the possible effect of the unobserved mode D + s → π + π 0 , which is excluded from our nominal fit, we include the PDF for this mode in an alternative fit, with the normalization set to the experimental upper limit, B(D + s → π + π 0 ) < 3.4 × 10 −4 .We consider a possible systematic uncertainty due to π-ID efficiency, measuring the effect with the D s and D 0  the other five datasets are provided as supplemental material [27].The black points are data, the shaded histograms correspond to the 40× background MC sample scaled to the integrated luminosity of data, and the lines represent the fitted signal and background shapes.The red-solid, orange-dashed, and blue-dotted lines represent the total, D + s → µ + νµ, and D + s → τ + ντ , while black-dot-dashed and green-long-dashed lines correspond to the total background and the case when both tag and signal sides are misreconstructed, respectively.data samples mentioned in Sec.III B. The uncertainty due to the rate for misidentification of µ as π is estimated by comparing the rate between data and our MC simulation in the D + s → µ + ν µ events, in which we heavily suppress the contribution from D + s → τ + (τ + → π + ντ )ν τ by requiring the signal track to penetrate deep into our MUC, as is done in Ref. [6].
In the nominal fitting procedure, we share two convolved Gaussian functions [one each for M 2 miss and M inv (D − s )] over the six data samples, effectively assuming that any data/MC differences are independent of E cm and changes in running conditions.We estimate a possible uncertainty due to this assumption with an alternative fit using independent Gaussian functions for each of the 4180, 4190 − 4220 and 4230 data groups.
The relative size of the background component arising from misreconstruction on the tag side is fixed according to MC simulation in our nominal fit procedure.We estimate the potential systematic uncertainty introduced by this constraint with an alternative fit allowing this component to float freely.
The relative size of the wrong-photon background component is also fixed in our nominal fitting procedure.For a systematic test, we vary this by ±1.4% (relative), the quadrature sum of a 1.0% uncertainty for photon reconstruction and a second 1.0% uncertainty for the bestphoton selection method.
The size of the background component in which the signal track is misreconstructed is fixed in our nominal procedure to MC simulation, normalized to the ST yields in data.The primary source is the decay D + s → K 0 π + , and we use the the uncertainties in B(D + s → K 0 S π + ) [1] and in our N ST determination to estimate the systematic uncertainty in the estimate of this background component.
Background events in which both the tag side and the signal track are misreconstructed are parametrized in the nominal fit with a first-order Chebyshev polynominal for M inv (D − s ) and a first-order exponential polynomial for M 2 miss .We use a MC-based shape for M inv (D − s ) and a first-order Chebyshev polynominal for M 2 miss in an alternative fit to estimate a possible systematic uncertainty due to the assumed background shape.
MC studies show that D + s → τ + ν τ events with τ + decays into final states other than π + ντ that are counted as signal are dominated by τ + → µ + ν µ ντ and τ + → π + π 0 ντ for the µ-like sample and by τ + → π + π 0 ντ and τ + → e + ν e ντ for the π-like sample.We estimate the uncertainty in the estimate of these events with variations based on the uncertainties in the measured branching fractions [1].
In our nominal fitting procedure, we fix the relative yields of signal between the µ-like and π-like samples according to MC simulation.The µ-like and π-like samples are defined by E EMC ≤ 300 MeV and E EMC > 300 MeV, respectively.Thus, to assess the systematic uncertainty associated with this criterion, we look at distributions of E EMC and see how well our MC agrees with data.We look at distributions for muons from D + s → µ + ν µ and for pions from D * + D − and D * + D * − , with D * + → D 0 π + and D 0 → K − π + , where the π + coming from the D 0 decay deposits E EMC , while requiring 800 < | p π | < 1100 MeV/c to match our signal pion and muon tracks.We observe a 4% (relative) difference in partitioning rates between data and MC samples.We vary the rate by ±4% to estimate this systematic uncertainty.
As an alternative fitting procedure, we constrain the yields of D + s → µ + ν µ and D + s → τ + ν τ to the SM expectation for the ratio R = 9.75, derived from Eq. ( 2).The uncertainty in this prediction arises from the input particle masses, which are precisely known.We estimate a possible systematic uncertainty due to this constraint by varying R by ±0.01.
Our measurements are summarized in Table VIII, along with previously published experimental results.In this section, the first uncertainty quoted is statistical and the second is systematic.We measure the absolute branching fraction B(D + s → τ + ν τ ) = (5.21± 0.25 ± 0.17)%, which is the most precise measurement to date and is in agreement with the SM prediction of B(D + s → τ + ν τ ) = (5.221± 0.018)% (see Sec.I for the predicted branching fractions).For the ratio of the two decay widths, we obtain R = Γ(D + s → τ + ν τ )/Γ(D + s → µ + ν µ ) = 9.73 +0.61 −0.58 ± 0.36, which is also consistent with the SM prediction of 9.75.The precision of these measurements is limited by the sample size.
We also obtain B(D + s → µ + ν µ ) = (5.35±0.13±0.16)× 10 −3 , which is again the most precise to date.It is con- sistent with the previously published BESIII result of (5.49 ± 0.16 ± 0.15) × 10 −3 [6], which only analyzed the 4180 data, and supersedes that result.Note that the analysis methods and background compositions for these two BESIII analyses are very different.In Ref. [6], we require µ identification with the MUC subdetector to suppress π from D + s → τ + ν τ , resulting in a smaller systematic uncertainty.
The measured B(D + s → τ + ν τ ) with the SM constraint leads to f D + s |V cs | = (243.2± 2.3 ± 3.3 ± 1.0) MeV, where the last uncertainty comes from the external inputs (lepton masses, m Ds , and the D + s lifetime [1]).Table VIII presents results based on other fitting schemes.By taking |V cs | = 0.97320 ± 0.00011 from the constrained global fit [1] and the average of the recent four-flavor LQCD predictions, f D + s = (249.9± 0.5) MeV [3], one finds an expected product of these of (243.2 ± 0.5) MeV, in excellent agreement with our result.
By taking |V cs | = 0.97320 ± 0.00011 [1] as an input, we    s → µ + νµ, and D + s → τ + ντ , while black-dot-dashed and green-long-dashed lines correspond to the total background and the case when both tag and signal sides are misreconstructed, respectively.

g
Also at Key Laboratory for Particle Physics, Astrophysics and Cosmology, Ministry of Education; Shanghai Key Laboratory for Particle Physics and Cosmology; Institute of Nuclear and Particle Physics, Shanghai 200240, People's Republic of China h Also at Key Laboratory of Nuclear Physics and Ion-beam Application (MOE) and Institute of Modern Physics, Fudan University, Shanghai 200443, People's Republic of China i Also at Harvard University, Department of Physics, Cambridge, MA, 02138, USA j Currently at: Institute of Physics and Technology, Peace Ave.54B, Ulaanbaatar 13330, Mongolia k Also at State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, People's Republic of China l School of Physics and Electronics, Hunan University, Changsha 410082, China m Also at Guangdong Provincial Key Laboratory of Nuclear Science, Institute of Quantum Matter, South China Normal University, Guangzhou 510006, China (Dated: September 23, 2021)

sFIG. 1 .
FIG. 1. Recoil mass distributions against D −s → K − K + π − for 4180 data and MC-simulated background, which mostly consists of charm decays and continuum processes.

FIG. 3 .
FIG. 3. Eγ spectra in the D * s rest frame for µ-like (top) and π-like (bottom) samples summed over the six Ecm values (weighted by integrated luminosity).The gray-shaded histograms represent the scaled 40× MC predictions, while the crosshatched histograms show the signals (red, bottom-left to top-right) and the backgrounds (blue, top-left to bottomright).The black points with error bars are the corresponding measurements from data, and the dashed vertical lines show our nominal selection requirement of 119 < Eγ < 149 MeV.

FIG. 4 .
FIG. 4. Projections onto the M 2 miss (left) and Minv(D − s ) (right) axes, for µ-like (top) and π-like (bottom) candidates.Black points are data, summed over all six samples.The filled histograms represent the prediction of the 40× MC sample, normalized to the integrated luminosity of data, with the D +s → µ + νµ and τ + (→ π + ντ )ντ signal on top (blue).Background components are stacked below this in the following order from the top to the bottom: tag-side misreconstructed (cyan), misreconstructed transition photon (light brown), signal side misreconstructed (dark brown), D + s → τ + ντ , where the τ + decays to a final state other than π + ντ (green), other charm background (gray), and noncharm sources (black, at bottom).

FIG. 5 .
FIG. 5. Distributions of the momentum of signal candidate tracks in the D + s rest frame for the DT sample with the additional requirement 1900 < Minv(D − s ) < 2030 MeV/c 2 for the µ-like (top) and the π-like samples (bottom).The black points are data and the overlaid histograms represent the 40× MC sample (normalized to the integrated luminosity of data), with green for the total, gray filled for the total background, and crosshatched for the signals, D + s → µ + νµ (blue, top-left to bottomright) and D + s → τ + (→ π + ντ )ντ (red, bottom-left to top-right).The distributions shown on the left correspond to only −0.20 < M 2 miss < 0.20 (GeV/c 2 ) 2 , while we show the entire momentum spectra on the right.

FIG. 6 .
FIG. 6. Distributions of cosines of polar angles in the D + s rest frame of µ-like (top) and π-like (bottom) signal-candidate tracks, based on the nominal DT selection and the additional requirements 1930 < Minv(D − s ) < 1990 MeV/c 2 and −0.1 < M 2 miss < +0.2 (GeV/c 2 ) 2 .Black points are data and the overlaid histograms correspond to the 40× MC sample scaled to the integrated luminosity of data, with gray shading for the total, and crosshatched for the signals, D + s → µ + νµ (blue, top-left to bottom-right) and D + s → τ + (→ π + ντ )ντ (red, bottom-left to top-right), respectively.The difference between gray and (the sum of red and blue) is the background.

FIG. 8 .
FIG. 8. Projections onto the M 2 miss (left) and Minv(D − s ) (right) axes of the two-dimensional fit to 4180 data for the µ-like (top) and the π-like (bottom) samples.Figures showing fit resultsfor the other five datasets are provided as supplemental material[27].The black points are data, the shaded histograms correspond to the 40× background MC sample scaled to the integrated luminosity of data, and the lines represent the fitted signal and background shapes.The red-solid, orange-dashed, and blue-dotted lines represent the total, D + s → µ + νµ, and D + s → τ + ντ , while black-dot-dashed and green-long-dashed lines correspond to the total background and the case when both tag and signal sides are misreconstructed, respectively.
FIG. 8. Projections onto the M 2 miss (left) and Minv(D − s ) (right) axes of the two-dimensional fit to 4180 data for the µ-like (top) and the π-like (bottom) samples.Figures showing fit resultsfor the other five datasets are provided as supplemental material[27].The black points are data, the shaded histograms correspond to the 40× background MC sample scaled to the integrated luminosity of data, and the lines represent the fitted signal and background shapes.The red-solid, orange-dashed, and blue-dotted lines represent the total, D + s → µ + νµ, and D + s → τ + ντ , while black-dot-dashed and green-long-dashed lines correspond to the total background and the case when both tag and signal sides are misreconstructed, respectively.

2 FIG. 9 .
FIG. 9. Projections onto the M 2miss axis of the two-dimensional fit to 4180 data (top left), 4190 data (top right), 4200 data left), 4210 data (middle right), 4220 data (bottom left), and 4230 data (bottom right) for the µ-like sample.The black points are data, the shaded histograms correspond to the 40× background MC sample scaled to the integrated luminosity of data, and the lines represent the fitted signal and background shapes.The red-solid, orange-dashed, and blue-dotted lines represent the total, D + s → µ + νµ, and D + s → τ + ντ , while black-dot-dashed and green-long-dashed lines correspond to the total background and the case when both tag and signal sides are misreconstructed, respectively.

FIG. 10 .
FIG. 10.Projections onto the Minv(D −s ) axis of the two-dimensional fit to 4180 data (top left), 4190 data (top right), 4200 data (middle left), 4210 data (middle right), 4220 data (bottom left), and 4230 data (bottom right) for the µ-like sample.The black points are data, the shaded histograms correspond to the 40× background MC sample scaled to the integrated luminosity of data, and the lines represent the fitted signal and background shapes.The red-solid, orange-dashed, and blue-dotted lines represent the total, D + s → µ + νµ, and D + s → τ + ντ , while black-dot-dashed and green-long-dashed lines correspond to the total background and the case when both tag and signal sides are misreconstructed, respectively.

2 FIG. 11 .
FIG. 11.Projections onto the M 2miss axis of the two-dimensional fit to 4180 data (top left), 4190 data (top right), 4200 data (middle left), 4210 data (middle right), 4220 data (bottom left), and 4230 data (bottom right) for the π-like sample.The black points are data, the shaded histograms correspond to the 40× background MC sample scaled to the integrated luminosity of data, and the lines represent the fitted signal and background shapes.The red-solid, orange-dashed, and blue-dotted lines represent the total, D + s → µ + νµ, and D + s → τ + ντ , while black-dot-dashed and green-long-dashed lines correspond to the total background and the case when both tag and signal sides are misreconstructed, respectively.

FIG. 12 .
FIG. 12. Projections onto the Minv(D −s ) axis of the two-dimensional fit to 4180 data (top left), 4190 data (top 4200 data (middle left), 4210 data (middle right), 4220 data (bottom left), and 4230 data (bottom right) for the π-like sample.The black points are data, the shaded histograms correspond to the 40× background MC sample scaled to the integrated luminosity of data, and the lines represent the fitted signal and background shapes.The red-solid, orange-dashed, and blue-dotted lines represent the total, D + s → µ + νµ, and D + s → τ + ντ , while black-dot-dashed and green-long-dashed lines correspond to the total background and the case when both tag and signal sides are misreconstructed, respectively.
• .For the decay mode D − s → K − π − π + we exclude the dipion mass range 486 < M ππ < 510 MeV/c 2 while we choose −0.2 < M 2 miss < 0.2 (GeV/c 2 ) 2 on the other axis to avoid peaking backgrounds in M 2 miss > 0.2 (GeV/c 2 ) 2 coming from D + s → K 0 π + when the K 0 is undetected.Background processes contributing D + s → + ν candidate signal events can be classified in five major categories.(1)The tag-side D − s is misreconstructed, but everything else is correct, producing smooth distributions in M inv , but peaks in M 2 miss .(2)The photon from D * s → γD s is misreconstructed, but everything else is correct, leading to peaking sharply in M inv and broadly in M 2 miss .(3) The signal-side track is not from D + s → (µ + /τ + )ν, but everything else is correct, giving peaks in M inv and smooth distributions in M 2 miss .(4) The reconstructed track is from D +

TABLE I .
ST reconstruction efficiencies ( ST in %) with their statistical uncertainties for the thirteen tag modes and six data samples.Efficiencies do not include the following intermediate-state branching fractions:

TABLE III .
DT reconstruction efficiencies (in %) with statistical uncertainties for D + s → µ + νµ signal events in each of the thirteen tag modes.These efficiencies do not include the following intermediate-state branching fractions: K and ρ → ππ.

TABLE IV .
DT reconstruction efficiencies (in %) with statistical uncertainties for D + s → τ + (→ πντ )ντ signal events in each of the thirteen tag modes.These efficiencies do not include the following intermediate-state branching fractions:

TABLE V .
Relative difference in % between the measured signal yields and generated numbers for ten "sets" of data-sized MC samples, the average of these, and a 40× MC sample.

TABLE VI .
Summary of charge-dependent branching fractions and ACP (in %) for B(D + s → µ + νµ) and B(D + s → τ + ντ ).The uncertainties reported in branching fractions are only statistical.The first uncertainties quoted in ACP are statistical and the second systematic.

TABLE VII .
Systematic uncertainties on Bµν , Bτν , B SM τ ν , and R. The notation "cncl."indicates that a systematic uncertainty cancels in the calculation of the branching fraction ratio R, and "neg."signifies that an uncertainty is negligible.