Measurement of the absolute branching fraction of inclusive semielectronic $D_s^+$ decays

We measure the inclusive semielectronic decay branching fraction of the $D_s^+$ meson. A double-tag technique is applied to $e^+e^-$ annihilation data collected by the BESIII experiment at the BEPCII collider, operating in the center-of-mass energy range $4.178 - 4.230$ GeV. We select positrons from $D_s^+\rightarrow Xe^{+}\nu_e$ with momenta greater than 200 MeV/$c$, and determine the laboratory momentum spectrum, accounting for the effects of detector efficiency and resolution. The total positron yield and semielectronic branching fraction are determined by extrapolating this spectrum below the momentum cutoff. We measure the $D_s^+$ semielectronic branching fraction to be $\mathcal{B}\left(D_s^+\rightarrow Xe^{+}\nu_e\right)=\left(6.30\pm0.13\;(\text{stat.})\pm 0.10\;(\text{syst.})\right)\%$, showing no evidence for unobserved exclusive semielectronic modes. We combine this result with external data taken from literature to determine the ratio of the $D_s^+$ and $D^0$ semielectronic widths, $\frac{\Gamma(D_{s}^{+}\rightarrow Xe^+\nu_e)}{\Gamma(D^0\rightarrow Xe^+\nu_e)}=0.790\pm 0.016\;(\text{stat.})\pm0.020\;(\text{syst.})$. Our results are consistent with and more precise than previous measurements.

k Also at Frontiers Science Center for Rare Isotopes, Lanzhou University, Lanzhou 730000, People's Republic of China l Also at Lanzhou Center for Theoretical Physics, Lanzhou University, Lanzhou 730000, People's Republic of China m Currently at Istinye University, 34010 Istanbul, Turkey (Dated: April 30, 2021) We measure the inclusive semielectronic decay branching fraction of the D + s meson. A double-tag technique is applied to e + e − annihilation data collected by the BESIII experiment at the BEPCII collider, operating in the center-of-mass energy range 4.178 − 4.230 GeV. We select positrons from D + s → Xe + νe with momenta greater than 200 MeV/c, and determine the laboratory momentum spectrum, accounting for the effects of detector efficiency and resolution. The total positron yield and semielectronic branching fraction are determined by extrapolating this spectrum below the momentum cutoff. We measure the D + s semielectronic branching fraction to be B D + s → Xe + νe = (6.30 ± 0.13 (stat.) ± 0.10 (syst.)) %, showing no evidence for unobserved exclusive semielectronic modes. We combine this result with external data taken from literature to determine the ratio of the D + s and D 0 semielectronic widths, Γ(D 0 →Xe + νe) = 0.790 ± 0.016 (stat.) ± 0.020 (syst.). Our results are consistent with and more precise than previous measurements. The D + s meson is the ground state of charmedstrange mesons, and precise measurements of its semileptonic decays allow for crucial tests of Standard Model predictions of flavor-changing interactions. This article presents a new measurement of the D + smeson inclusive semielectronic branching fraction and positron momentum spectrum.
(Here and throughout this article charge conjugate modes are implied.) A previous measurement by the CLEO-c experiment reported the first results for these quantities, including the measurement of B (D + s → Xe + ν e ) = (6.52 ± 0.39 (stat.) ± 0.15 (syst.)) % [1]. Measuring this branching fraction with improved precision contributes to a comprehensive understanding of D + s decays and is an important component of the overall experimental and theoretical heavy-flavor physics program. Table I lists the six exclusive D + s semielectronic modes that have been observed to date and their branching fractions, as well as the branching fraction of D + s → τ + ν τ → e + ν e ν τ ν τ and the previously measured D + s → Xe + ν e branching fraction [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. By comparing the inclusive semielectronic branching fraction with the sum of all measured exclusive semielectronic branching fractions, we can estimate the branching fraction for further unobserved D + s semielectronic decays. Measurements of the semielectronic branching fractions for different charmed mesons can be combined to probe for non-spectator effects in heavy-meson decays [16,17]. The CLEO-c measurement of the ratio of the D + s and D 0 semielectronic widths, Γ(D + s →Xe + νe) Γ(D 0 →Xe + νe) = 0.815 ± 0.052, is in agreement with predictions employing an effective quark model [18] and shows that non-spectator effects are present in semielectronic charmed-meson decays. It has also been demonstrated with CLEOc data that the inclusive semielectronic momentum spectrum can be used to make sensitive tests for specific non-spectator processes, such as Weak Annihilation (WA) [19,20]. Strong understanding of these processes are required for application of heavy-quark-expansion in extracting CKM elements from inclusive semileptonic B meson decays [21]. Thus, the improved precision of both the inclusive branching fraction and the momentum spectrum of D + s semielectronic decays reported in this article have potential to contribute to reducing theoretical uncertainties in determining CKM parameters with heavy-meson decays.
The remainder of the article is organized in seven sections. The BESIII detector, the analyzed data, and the Monte Carlo (MC) simulation samples are described in Sec. II. An overview of the measurement technique is presented in Sec. III. Event-selection requirements based on full reconstruction of hadronic D − s decays are discussed in Sec. IV. Semielectronic decay selection requirements and further analysis of candidate signal events are presented in Sec. V. The systematic uncertainties of our measurement are evaluated in Sec. VI. We conclude with a summary of our results in Sec. VII and acknowledgements in Sec. VIII.

II. DETECTOR AND DATA SAMPLES
The BESIII detector records the results of symmetric e + e − collisions provided by the BEPCII collider [22]. BEPCII produces collisions at center-of-mass energies (E cm ) between 2 and 4.9 GeV, and BESIII has collected the world's largest data samples near a number of pairproduction threshold energies for charmed hadrons. The BESIII detector is composed of the following sub-systems for particle detection and identification: a helium-based multilayer drift chamber (MDC), a plastic scintillator time-of-flight system (TOF), a CsI(Tl) electromagnetic calorimeter (EMC), a 1.0 T superconducting solenoid, and a set of resistive plate chambers for muon I: Branching fractions for observed D + s semielectronic decays and for D + s → τ + ν τ → e + ν e ν τ ν τ . The listed uncertainties are the total uncertainties. The D + s → f 0 (980)e + ν e , f 0 (980) → ππ branching fraction is calculated based on the measurements of D + s → π + π − e + ν e from [2] and corrected for the corresponding π 0 π 0 branching fraction by an isospin correction factor of 3 2 .
Inclusive MC samples with the equivalent of five times the luminosity of data and exclusive MC samples with the equivalent of thirty-five times the luminosity of data are used to optimize selection criteria, investigate distributions of signal and background processes, and determine the efficiencies of our selection criteria. Hadronic simulation samples are produced using the event generators KKMC [25] and EvtGen [26,27]. BabayagaNLO [28] is used to produce radiative Bhabha (e + e − → γe + e − ) samples. Final-state radiation (FSR) of particles is simulated with PHOTOS [29]. Interaction of the simulated particles with the detector material is handled by GEANT4 [30], which uses a detailed XMLbased description of the detector geometry [31].
In 2016, BESIII collected 3.19 fb −1 of data at E cm = 4.178 GeV, which provides approximately 6.4 × 10 6 D + s mesons primarily through the process e + e − → D * + s D − s , with a small contribution from e + e − → D + s D − s . We analyze the entirety of this sample, as well as 2.08 fb −1 of data collected at center-of-mass energies in the range E cm = 4.189 − 4.219 GeV in 2017, and 1.05 fb −1 of data collected in the range E cm = 4.225 − 4.230 GeV in 2013. These three samples are analyzed separately due to differing detector and running conditions. A summary of the data sets with their E cm [32], integrated luminosities [33], and estimated number of D + s mesons produced is shown in Table II.   TABLE II: E cm , integrated luminosities, and estimated number of D s mesons (N Ds ) for the analyzed data samples. In each case, the first listed uncertainty is statistical and the second is systematic.

III. MEASUREMENT TECHNIQUE
Our analysis procedure employs the double-tag technique pioneered by the MARKIII collaboration [34]. We fully reconstruct hadronic D − s mesons (the tag or tag-side), and determine the number of signal events by analyzing the remaining charged tracks unused in the reconstruction of the tags (the recoil or recoil-side). We refer to events where a tag meson is found as singletag events, and events where a semielectronic decay is identified on the recoil side in addition to the tag meson as double-tag events. Our single-tag selection is described in more detail in Sec. IV.
Since identification and reconstruction of positron tracks is only possible above a certain momentum threshold with the BESIII detector, double-tag candidate tracks are only accepted with p > 200 MeV/c. The differential inclusive semielectronic branching fraction as a function of the electron momentum p e ( dBSL dpe ) is given as (1) In this equation n ST is the number of the observed single-tag events, ∆n DT is the number of the observed (semielectronic) double-tag events in a particular p e momentum bin, and ST and DT are the single-tag and double-tag reconstruction efficiencies, respectively. We define DT = ST × sig , where sig is the momentumdependent efficiency of reconstructing only the signal side, and ST is the tag-side reconstruction efficiency given that a signal event is present on the recoil-side. We define b tag ≡ ST ST as the tag bias, which accounts for the difference in the single-tag detection efficiency given that a semielectronic decay is present on the recoil side, and ∆N DT as the true number of double-tag signal events in our single-tag sample for a particular p e momentum bin.
We begin our signal selection by sorting recoil-side tracks with p > 200 MeV/c into momentum bins. For a given momentum bin, we further sort these tracks by their charge: tracks with charge opposite to the tag (right-sign or RS) and tracks with the same charge as the tag (wrong-sign or WS). The WS sample is used to estimate charge-symmetric backgrounds in the RS distributions, which include the true signal. For both charge categories in each momentum bin, we sort tracks into three mutually exclusive particle identification hypotheses: electron/positron (e ID), pion (π ID), and kaon (K ID). The similar detector response to muons and pions allows us to treat muons as identical to pions with negligible uncertainty. For each of these six categories in a given momentum bin, we determine the number of tracks that originate from a true D + s meson by fitting to the invariant-mass distribution of the tag D − s candidate.
We relate the true number of double-tag signal events in our single-tag sample and the observed number of signal-candidate tracks from double-tag events through detector response matrices for charged particle tracking (A trk ) and PID (A PID ): (2) Here, A trk (a|p i , p j ) gives the probability of a particle of type a (a ∈ {e, π, K}) tracked in momentum bin p i to have true momentum in bin p j . A PID (b|a, p i ) gives the probability of a particle of type a passing the PID requirements of particle type b (b ∈ {e, π, K}) in track momentum bin p i .
We determine the observed double-tag yields n b DT (p i ) by fitting the invariant-mass distribution of tag-side mesons. For both RS and WS tracks in each momentum bin, we account for e ID efficiencies and particles faking e ID with the PID unfolding method introduced in Eq. (2): Here, a is the efficiency of our PID selection requirements for the particles of species a and P a→b is the probability of a particle of species a passing the selection requirements for a particle of species b. We apply this PID unfolding to the fitted RS and WS yields for each momentum bin and take the e solution to determine the number of tracked positrons originating from a D + s meson. We then take the difference of the number of tracked RS positrons and WS positrons to subtract the contributions of positrons from Dalitz decays of light mesons in the final state of D + s decay such as π 0 → γe + e − and any other charge-symmetric backgrounds.
This leaves us with the momentum spectrum of tracked positrons from D + s → Xe + ν e events and a smaller contribution from D + s → τ + ν τ → e + ν e ν τ ν τ . To account for tracking inefficiency and mismeasurement, we execute the second unfolding from Eq. (2): This provides the true sum of D + s → Xe + ν e and D + s → τ + ν τ → e + ν e ν τ ν τ positron momentum spectra. We then subtract the contribution of D + s → τ + ν τ → e + ν e ν τ ν τ based on the branching fraction in Table I to obtain the D + s → Xe + ν e momentum spectrum for p e > 200 MeV/c. To account for positrons with p e ≤ 200 MeV/c, we fit the expected D + s → Xe + ν e momentum spectrum with an assumed spectrum, described in more detail in Sec. V, and add the fitted yields in the region p e ≤ 200 MeV/c to the measured yields with p e > 200 MeV/c. This gives us the signal-efficiency-corrected number of signal events which allows us to calculate the branching fraction similarly to Eq. (1) through with the MC simulation prediction of b tag and the determined number of single-tag events n ST .

IV. SINGLE-TAG EVENT SELECTION
We select single-tag candidates using only the D − s → K + K − π − hadronic decay mode because it is unique in having sufficient statistics and well-known backgrounds. All tag candidate daughters are required to satisfy the following track-quality requirements: the track must originate from a region within 1 (10) cm of the e + e − interaction point perpendicular (parallel) to the z axis, which is the symmetry axis of the MDC, and must be tracked with an angle θ with respect to the z axis that satisfies | cos θ| < 0.93. We apply π/K ID based on dE/dx and TOF measurements to all tag-candidate daughters to maximize the purity of the tag sample. Multiple candidates in a single event are allowed, both to increase the single-tag selection efficiency and to minimize the tag bias, b tag . We calculate the recoil mass against the tag D − s candidate, where p Ds is the momentum of the reconstructed tag D − s candidate, and m Ds is the known D s mass [35]. We require the recoil mass to be consistent with a D * + s D − s event hypothesis by imposing requirements for each data set: 2057 < M Rec < 2177 MeV/c 2 for E cm = 4.178 GeV, 2145 < M Rec < 2190 MeV/c 2 for E cm = 4.189 − 4.219 GeV, and 2150 < M Rec < 2200 MeV/c 2 for E cm = 4.225 − 4.230 GeV. For about 20% of cases, our event selection identifies more than one candidate per event after these recoil-mass requirements. Background due to extra candidates is subtracted as part of the fitting procedure described later in this section. The M Rec distributions for each data set can be seen in Fig. 1. While differences between the MC simulation and data can be seen in the figure, our measurement is not sensitive to the MC simulation of this distribution.
We determine the number of true single-tag candidates by performing an unbinned fit to the distribution of the invariant mass of the tag D − s candidates, M Inv . The signal shape is based on MC simulation and convolved with a Gaussian function whose width and mean are left free in the fit to account for a possible difference in resolution and calibration between data and the MC simulation. The distribution of backgrounds in this variable is modeled using a second-order Chebyshev polynomial. The fit range is chosen to be within ±40 MeV/c 2 of the known D + s mass for the E cm = 4.178 GeV distribution corresponding to approximately 5σ of the simulated M Inv distribution at E cm = 4.178 GeV. In the E cm = 4.189 − 4.219 GeV and E cm = 4.225 − 4.230 GeV data sets, the fit range is reduced to be within ±35 MeV/c 2 of the known D s mass due to non-polynomial background structures appearing at the edges of the fit range. The single-tag fits to each data set are shown in Fig. 2   The single-tag reconstruction efficiency depends on the topology of the recoil-side decay, which we account for through b tag . We determine the tag bias from MC samples for double-tag events from each of the observed semielectronic modes listed in Table I and for D + s → τ + ν τ → e + ν e ν τ ν τ . The D + s → Xe + ν e tag bias is determined by averaging the tag biases for the six observed modes weighted by the branching fractions in Table I. The single-tag efficiencies with no specification on the signal-side and the determined tag biases for D + s → Xe + ν e and D + s → τ + ν τ → e + ν e ν τ ν τ for each data set are shown in Table IV. After a single-tag candidate is found, we begin searching for positron candidates among the recoil-side tracks. We sort recoil-side tracks that satisfy track requirements defined in Sec. IV into eighteen 50-MeV/c momentum bins between 200 MeV/c and 1100 MeV/c. Mutually exclusive PID hypotheses are assigned to each track based on information from the MDC, TOF, and EMC.
Based on our PID assigment, we fill tag-side invariantmass distributions in each momentum bin for each of the six categories of recoil-side tracks defined in Sec. III. We determine the number of tracks in each category originating from true D + s events by performing an unbinned fit to the tag-side invariant mass distribution with the same signal shape used for the single-tag fit, with the Gaussian parameters fixed to those determined in the single-tag fit. We employ a first-order Chebyshev polynomial to model the distribution of backgrounds. Two examples of the 172 fits we perform are shown in Fig. 3.
The full set of fits is made available as supplemental material [36]. Measured yields as a function of momentum bin for each category are shown in Fig. 4.
With the yields determined from the fits, we perform the matrix unfolding procedure described in Sec. III to correct for the inefficiencies of our electron/positron identification and for the misidentification of pions and kaons as electrons in each momentum bin for both RS and WS tracks, as described in Eq. (3). The elements of A PID for each data set are determined by applying our PID requirements to MC samples of particles originating from D + s decays. Differing detector conditions of the three data sets introduce deviations among the PID rates from the data sets. The rates that populate the A PID   1930 1940 1950 1960 1970 1980 1990 2000  results of the unfolding procedure for both RS and WS tracks are shown in Fig. 6, as well as the difference of the PID-unfolded right-sign and wrong-sign yields, which gives the PID-unfolded momentum spectra for tracked positrons originating from D + s → Xe + ν e and D + s → τ + ν τ → e + ν e ν τ ν τ events. The assessment of systematic uncertainties related to the two rates to which we are most sensitive (the electron identification efficiency and the pion-faking-electron rate) are discussed in Sec. VI.
After taking the difference of the RS and WS PID-corrected yields, we apply the tracking unfolding matrix to correct for tracking reconstruction efficiencies and momentum bin mis-assignment (Eq. (4)). The momentum bin mis-assignment is caused not only by imperfect detector resolution, but also by FSR from electrons and positrons, which increases the likelihood for the track momentum to be less than the momentum produced in the D + s decay. The tracking unfolding matrices are consistent among the MC samples.
We determine the D + s → Xe + ν e positron momentum spectrum for each data sample by subtracting the contribution of positrons from D + s → τ + ν τ → e + ν e ν τ ν τ events from the spectra obtained from the tracking unfolding procedure. We take the D + s → τ + ν τ → e + ν e ν τ ν τ positron momentum spectra from MC samples. We fix the normalization according to where the values of B(D + s → τ + ν τ → e + ν e ν τ ν τ ) and b tag,τ are given in Table I and Table IV, respectively. The subtracted τ component is shown for each data set in Fig. 7.
By determining the signal-efficiency-corrected number of D + s → Xe + ν e events in each data set with p e > 200 MeV/c, all effects of the detector response, except for small effects in b tag , have been accounted for. As such, we sum the determined yields from each data set in each momentum bin to produce a combined D + s → Xe + ν e momentum spectrum. A table with the total summed yields as a function of momentum along with their statistical uncertainties is provided in the supplemental material [36]. We determine the number of D + s → Xe + ν e events with p e ≤ 200 MeV/c by fitting a shape based on MC simulation to the combined yields with p e > 200 MeV/c. The shape is constructed by adding the momentum spectra predicted by MC simulation of the six observed exclusive modes (φe + ν e , ηe + ν e , η e + ν e , K 0 e + ν e , K * (892) 0 e + ν e , and f 0 (980)e + ν e ) in proportion to the branching fractions listed in Table I. The momentum spectra for the η e + ν, K 0 e + ν e , K * (892) 0 e + ν e , and f 0 (980)e + ν e modes are taken from MC samples generated with the ISGW2 model [37] for the decay of the D + s meson. We generate RS e yields WS e yields separate MC samples for the two largest modes, φe + ν e and ηe + ν e , using simple-pole parameterizations of the respective decay form factors as functions of the leptonneutrino system squared four-momentum q 2 . The formfactor parameters are taken from measurements of the BABAR collaboration for φe + ν e [3] and the BESIII collaboration for ηe + ν e [7].
The integral of the fitted spectrum from p e = 0 MeV/c to p e = 200 MeV/c is added to the number of signalefficiency-corrected D + s → Xe + ν e events with p e > 200 MeV/c as described in Eq. 5, and the statistical uncertainty is scaled by the ratio by which the total yield increases due to this correction. The fit to data with the assumed momentum spectrum can be found in Fig. 8. As a crosscheck, we also fit to the data sets separately, and find consistent results. The yields in data both without and with the correction for the data with p e < 200 MeV/c are shown in Table V. Using Eq. (6), we determine B (D + s → Xe + ν e ) with the momentum-extrapolated number of signalefficiency-corrected double-tag events from Table V, the  observed number of single-tag events from Table III  and the tag bias from Table IV. The branching fractions determined from each data set independently, their average, and the branching fraction determined from their combination are shown along with the associated statistical uncertainties in Table VI. The difference of the "Combined" result from Table VI, B (D + s → Xe + ν e ) = (6.30 ± 0.13(stat.)) %, and the sum of the observed exclusive semielectronic branching fractions from Table I gives the unobserved D + s semielectronic branching fraction as (−0.04 ± 0.21) %, where the stated uncertainty includes the total uncertainty from the exclusive measurements, but only the statistical uncertainty from the inclusive measurement presented in this article.

VI. SYSTEMATIC UNCERTAINTY
Our methods to determine the relative systematic uncertainty on our measured B (D + s → Xe + ν e ) are described below.  we perform the matrix inversion and reperform the analysis with the new inverted matrix. All variations produce a negligible change in our final result, which indicates a negligible systematic uncertainty from the statistical uncertainty of the MC samples as well as the stability of the algorithm for inverting our efficiency matrices.

B. Tracking
Simulation of our tracking efficiency is studied with a control sample of radiative Bhabha events. Tracking efficiencies as a function of momentum are measured in each data set, as well as in MC samples produced with the BabayagaNLO package [28]. The ratios of the measured efficiencies in data and MC samples are weighted by the predicted momentum distribution from signal MC simulation and the number of single-tag events in each data set to determine the systematic uncertainty. This results in a relative systematic uncertainty of 0.7%.
In addition, we investigate the systematic uncertainty in the individual tracking efficiency matrix entries. As we assign a systematic uncertainty for the total tracking efficiency, we probe the uncertainty in the individual entries by keeping the sum of a row of the matrix constant while varying the individual entries. The specific variation is as follows: This variation is chosen as a conservative estimation of the uncertainties from FSR and detector resolution. We see negligible change when we perform such a variation, so we only assign the previously stated uncertainty for tracking.

C. PID
Similar to our procedure in assessing the systematic uncertainty in our tracking efficiencies, we measure e ID efficiencies as a function of momentum and track angle in radiative Bhabha control samples for each data set.
We also probe the accuracy of the pion-faking-electron rates from MC simulation via a control sample of pions collected in each data sample through the decay chains D * + → π + D 0 , D 0 → K − π + , K − π + π + π − . To determine the total uncertainty from PID rates, we simultaneously vary the e ID efficiencies and the pion-faking-electron rates using the central values of the measured data-to-MC efficiency ratios and reperform our analysis. This yields a 0.8% change in the final branching fraction, which we assign as the relative systematic uncertainty due to PID.
As our sensitivity to kaon-faking-electron rates is small due to the relatively few number of kaons, the systematic uncertainty in kaon-faking-electron rates is neglected.

D. Tag Bias
We follow the procedure laid out in [38], which assigns a fraction of 1 − b tag as the systematic uncertainty based on the particles in the final state of the single-tag D − s decay. The specific guidelines for variation of detectorresponse parameters are as follows: 1.0% per kaon for tracking, 0.5% per pion for tracking, and 0.5% per kaon or pion for PID. For D − s → K + K − π − , with two kaons and one pion, the quadrature sum is 2.9%. With b tag from Table IV (including the contribution from D + s → τ + ν τ → e + ν e ν τ ν τ ), taking 2.9% of 1 − b tag yields a 0.03% relative systematic uncertainty. We additionally propagate the uncertainties in the branching fractions (Table I) through the calculation of the weighted-average b tag . This yields a 0.07% relative systematic uncertainty in the bias. We add these in quadrature and assign the relative systematic uncertainty due to tag bias as 0.1%.

E. Number of Single Tags
We investigate the systematic uncertainty in the invariant-mass fitting procedure used to determine the number of single tags by varying the choice of background distribution from the nominal second-order Chebyshev polynomial. We use both first-order and third-order Chebyshev polynomials as variations in fitting to each data set. Using the first-order Chebyshev polynomial gives a larger difference in the yields in all cases, while not significantly degrading the quality of the fit. We take an average of the changes for each data set weighted by the single-tag yields to determine the systematic uncertainty, which results in a 0.6% relative systematic uncertainty in the number of single tags.

F. Background Shapes
To assess the uncertainty due to our chosen background shapes in our signal-side fits of the tag invariant mass, we use background distributions based on MC simulation instead of the nominal first-order Chebyshev polynomial to model backgrounds in each of the signal-side fits. We then reperform the analysis with the yields determined from these alternative fits. The relative difference in N DT is 0.4%, which we assign as the relative systematic uncertainty due to this source.

H. Spectrum Extrapolation
We assess the uncertainty due to the momentum spectrum extrapolation by generating an ensemble of alternative momentum spectra and fitting these to the data. Each spectrum in this ensemble is created by Gaussian sampling the branching fractions of the six observed exclusive semielectronic modes and adding spectra for unobserved decay modes. We then add these spectra in proportion based on the sampled branching fractions.
We consider effects from the combination of three unobserved decay modes: D + s → h 1 (1415) e + ν e , D + s → f 1 (1510) e + ν e , and D + s → γe + ν e . MC samples for D + s → h 1 (1415) e + ν e and D + s → f 1 (1510) e + ν e are generated based on the ISGW2 model's predictions [37]. We generate MC samples for D + s → γe + ν e based on the model of Yang and Yang [39]. We determine the normalization of the D + s → h 1 (1415) e + ν e and D + s → f 1 (1510) e + ν e spectra by fixing the relative branching fraction of these decays to the ISGW2 predictions and fitting them in addition to our nominal momentum spectrum. The D + s → γe + ν e spectrum is fixed to its measured 90% confidence level upper limit, 1.3 × 10 −4 [40].
From this fit, we determine an upper limit at the 90% confidence-level for the sum of the branching fractions of D + s → h 1 (1415) e + ν e and D + s → f 1 (1510) e + ν e . In our toy ensemble, their summed spectra are fixed based on this upper limit.
We performed this same procedure excluding the D + s → γe + ν e spectrum, but found the combination of modes to produce the largest variation.
The resulting systematic uncertainty is determined by filling a distribution of the relative change in B (D + s → Xe + ν e ) between the alternative momentum spectra and our nominal spectrum. The linear sum of the mean (0.33%) and RMS (0.34%) of this distribution is taken as the uncertainty, which gives a 0.7% relative systematic uncertainty.
We also probe uncertainty in the models we employ by using the ISGW2-predicted momentum spectra for the D + s → φe + ν e and D + s → ηe + ν e modes. Using these alternative spectra instead of the nominal spectra gives a less than 0.1% difference in the measured branching fraction, so we conclude that any uncertainty due to model dependence in our analysis is negligible.

I. Summary of Systematic Uncertainties
The assigned relative systematic uncertainties for all sources are listed in Table VII. Our systematic uncertainty is not dominated by any single source, but the largest contributions come from the momentum spectrum extrapolation and imperfect simulation of PID and tracking efficiencies. The total relative systematic uncertainty is obtained from the quadrature sum of the assigned relative uncertainties. This gives a total relative systematic uncertainty of 1.6%.

VII. SUMMARY AND DISCUSSION
Using data collected by the BESIII detector in the center-of-mass-energy range of E cm = 4.178 − 4.230 GeV, we measure the inclusive semielectronic branching fraction of the D + s meson to be B D + s → Xe + ν e = (6.30 ± 0.13 (stat.) ± 0.10 (syst.)) %.
We also measure the lab-frame momentum spectrum of the positrons produced in this decay, which can be seen in Fig. 8.
Our result is consistent with the measurement from the CLEO-c experiment [1], with a factor of 3 reduction of the statistical uncertainty and a factor of 1.5 reduction of the systematic uncertainty. The total precision of our measurement is 2.6%, which corresponds to approximately a 2.5 times improvement in the total precision compared to the measurement from CLEO-c.
By taking the difference between our measurement of B (D + s → Xe + ν e ) and the sum of the best available measurements for the exclusive semielectronic modes in Table I, we calculate the unobserved semielectronic branching fraction to be where the systematic uncertainty includes the total uncertainty on the measured exclusive branching fractions. Our measurement provides no evidence for the existence of unobserved D + s semileptonic decay modes and constrains the branching fractions of all unobserved decay modes. In addition, the measured momentum spectrum can be used to further constrain the decay rates of modes with characteristic momentum spectra. The spectrum is included in tabular form in the supplemental material [36].
With our updated measurement of the D + s semielectronic branching fraction, the CLEO-c measurement of the D 0 semielectronic branching fraction [1], and the 2020 PDG values for the D + s and D 0 lifetimes [35], we find where the systematic uncertainty includes the total uncertainty from B D 0 → Xe + ν e . This result is in agreement with the prediction of Γ(D 0 →Xe + ν) = 0.813 from [18], supporting the conclusion that the difference in the semileptonic decay widths of D + s and D 0 mesons can be accounted for within the Standard Model by nonspectator interactions. Further theoretical analysis of our measured spectrum, similar to those of [19] and [20], can constrain specific processes like WA of the constituent c and s quarks of the D + s , with potential extensions to determinations of |V ub | in semileptonic B decays [21].

VIII. ACKNOWLEDGMENTS
The BESIII collaboration thanks the staff of BEPCII and the IHEP computing center for their strong support.

IX. DOUBLE-TAG FITS
The following section contains the results of the double-tag fits in which the observed D + s → Xe + ,D + s → Xπ + , and D + s → XK + yields are determined for each sign hypothesis and momentum bin. In each plot, the solid blue line is the result of the total fit, the dashed red line is the fitted distribution of non-D − s → K + K − π − backgrounds, the dotted black line is the fitted signal distribution, the filled red histogram is the MC simulation-predicted contributions from backgrounds, and the black points are data. Binning is arbitrary and used solely for display.