Precision Measurements of D + s → ηe + ν e and D + s → η ′ e + ν e

Precision measurements of the semileptonic decays D + s → ηe + ν e and D + s → η ′ e + ν e are performed with 7.33 fb − 1 of e + e − collision data collected at center-of-mass energies between 4.128 and 4.226 GeV with the BESIII detector. The branching fractions obtained are B ( D + s → ηe + ν e ) = (2 . 255 ± 0 . 039 stat ± 0 . 051 syst )% and B ( D + s → η ′ e + ν e ) = (0 . 810 ± 0 . 038 stat ± 0 . 024 syst )%. Combining these results with the B ( D + → ηe + ν e ) and B ( D + → η ′ e + ν e ) obtained from previous BESIII measurements, the η − η ′ mixing angle in the quark ﬂavor basis is determined to be φ P = (40 . 0 ± 2 . 0 stat ± 0 . 6 syst ) ◦ . Moreover, from the ﬁts to the partial decay rates of D + s → ηe + ν e and D + s → η ′ e + ν e , the products of the hadronic transition form factors f η ( ′ ) + (0) and the modulus of the c → s Cabibbo-Kobayashi-Maskawa matrix element | V cs | are determined by using diﬀerent hadronic transition form factor parametrizations. Based on the two-parameter series expansion, the products f η + (0) | V cs | = 0 . 4519 ± 0 . 0071 stat ± 0 . 0065 syst and f η ′ + (0) | V cs | = 0 . 525 ± 0 . 024 stat ± 0 . 009 syst are extracted. All results determined in this work supersede those measured in the previous BESIII analyses based on the 3.19 fb − 1 subsample of data at 4.178 GeV.


II. BESIII DETECTOR AND MONTE CARLO SIMULATIONS
The BESIII detector [25] records symmetric e + e − collisions provided by the BEPCII storage ring [26] in the center-of-mass energy range from 2.0 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 [27,28].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.The solenoid is supported by an octagonal flux-return yoke with resistive plate counter muon identification modules interleaved with steel.The charged-particle momentum resolution at 1 GeV/c is 0.5%, and the 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, while that in the end cap region was 110 ps.The end cap TOF system was upgraded in 2015 using multi-gap resistive plate chamber technology, providing a time resolution of 60 ps [29].
Approximately 83% of the data used here was collected after this upgrade.Simulated samples produced with a geant4-based [30] Monte Carlo (MC) package, which includes the geometric description [31] 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 [32].The input cross section of e + e − → D ± s D * ∓ s is taken from Ref. [33].The ISR production of vector charmonium(-like) states and the continuum processes are incorporated in kkmc [32].In the simulation, the production of open-charm final states directly via e + e − annihilations is modeled with the generator conexc [34], and their subsequent decays are modeled by evtgen [35] with known branching fractions from the Particle Data Group [36].
Final state radiation from charged final-state particles is incorporated using the photos package [38].

III. ANALYSIS METHOD
A double-tag (DT) measurement strategy, analogous to what is used in Refs.[23,39,40], is employed.At E CM between 4.128 and 4.226 GeV, D s mesons are produced mainly from the process e s meson is fully reconstructed in one of several hadronic decay modes, discussed in Sec.IV; this is referred to as a single-tag (ST) candidate.This includes the D s directly from e + e − annihilations and the D s from D * s decays.Then, the signal decay of the D + s meson and the transition γ(π 0 ) from the D * ± s decay are reconstructed from the remaining particles in the event; these are the DT candidates.The branching fraction of the semileptonic decay is determined by Here, ST are the total DT and ST yields in data summing over tag mode i and data set j; ǭγ(π 0 )SL is the efficiency of detecting the tran-sition γ(π 0 ) and the semileptonic decay in the presence of the ST D − s candidate, weighted by the ST yields in data.It is calculated by where ǫ ij DT and ǫ ij ST are the detection efficiencies of the DT and ST candidates, respectively.The efficiencies do not include the branching fractions of η (′) [13].The quantity B sub is the product of the branching fractions of the relevant intermediate decays.

IV. SINGLE-TAG EVENT SELECTION
The ST D − s candidates are reconstructed from the fourteen hadronic decay modes , and K 0 S K − , where the subscripts of η and η ′ represent the decay modes used to reconstruct these mesons.Throughout this paper, ρ denotes ρ(770).
The selection criteria of K ± , π ± , K 0 S , γ, π 0 , and η are the same as those used in previous works [23,41,42].All charged tracks must be within a polar angle range |cos θ| < 0.93.Except for those from K 0 S decays, they are required to satisfy |V xy | < 1 cm and |V z | < 10 cm.Here, θ is the polar angle with respect to the MDC axis, and |V xy | and |V z | are the distances of the closest approach in the transverse plane and along the MDC axis, respectively.The particle identification (PID) of the charged particles is performed with the combined dE/dx and TOF information.The combined likelihoods (L ′ ) under the pion and kaon hypotheses are obtained.Kaon and pion candidates are required to satisfy L K > L π and L π > L K , respectively.
Each K 0 S candidate is reconstructed from two oppositely charged tracks satisfying |V z | < 20 cm.The two charged tracks are assigned as π + π − without imposing PID criteria.They are constrained to originate from a common vertex and are required to have an invariant mass within is the K 0 S nominal mass [13] and 12 MeV/c 2 corresponds to about three times the fitted resolution around the K 0 S nominal mass.The decay length of the K 0 S candidate is required to be greater than twice the vertex resolution away from the interaction point.
The π 0 and η mesons are reconstructed from photon pairs.Photon candidates are identified as isolated showers in the EMC.The deposited energy of each shower must be more than 25 MeV in the barrel region (| cos θ| < 0.80) and more than 50 MeV in the end cap region (0.86 < | cos θ| < 0.92).The different energy thresholds for the barrel and end cap regions are due to different energy resolutions.To exclude showers that originate from charged tracks, we require the angle subtended by the EMC shower and the position of the closest charged track at the EMC must be greater than 10 • as measured from the IP.The difference between the EMC time and the event start time, which is the interval of the trigger start time to the real collision time [43], is required to be within (0, 700) ns to suppress electronic noise and showers unrelated to the event.
To form π 0 and η candidates, we require the invariant masses of the selected photon pairs, M γγ , to be within the intervals (0.115, 0.150) and (0.500, 0.570) GeV/c 2 , respectively.To improve momentum resolution and suppress background, a kinematic fit is imposed on the selected photon pairs by constraining their invariant mass to the nominal π 0 or η mass [13].
The momentum of any pion, which does not originate from a K 0 S , η, or η ′ decay, is required to be greater than 0.1 GeV/c to reject the soft pions from D * + decays.For the tag mode D − s → π + π − h − (h = K or π), the peaking background from D − s → K 0 S (→ π + π − )h − is rejected by requiring the invariant mass of any π + π − combination at least 30 MeV/c 2 away from the nominal K 0 S mass [13].To suppress non-D ± s D * ∓ s events, the beam-constrained mass of the ST D − s candidate is required to be within the intervals shown in If there are multiple candidates for any tag mode, for a given ST D s charge, in one event, the candidate with the D − s recoil mass (3) closest to the nominal D * + s mass [13] is kept.Here, m D − s is the nominal D − s mass [13].The probability of the best candidate selection for individual tag modes ranges in (82-99)%.Figure 1 shows the invariant mass (M tag ) spectra of the accepted ST candidates for the 14 tag modes.For each tag mode, the ST yield is obtained by a fit to the corresponding M tag spectrum.The signal is described by the simulated shape for events with the angle between the reconstructed and generated four-momentum less than 15 • , convolved with a Gaussian function representing the difference in resolution between data and simulation.For the tag mode D − s → K 0 S K − , the peaking background from D − → K 0 S π − is described by the simulated shape convolved with the same Gaussian function used in the signal shape and its yield is left as a free parameter.The non-peaking background is modeled by a secondorder Chebychev polynomial, which has been validated using the inclusive simulation sample.The fit results for the data sample combined from all energy points are shown in Fig. 1.The candidates in the signal regions, marked with black arrows in each sub-figure, are kept for further analyses.The background contributions from e + e − → (γ ISR )D + s D − s , whose contribution is (0.7-1.1)% in the fitted ST yields for 14 tag modes based on simulation, are subtracted in this analysis.The resulting ST yields (N ST ) for the different tag modes in data and the corresponding ST efficiencies (ǫ ST ) are summarized in the second and third columns of Table 3, respectively.

V. DOUBLE-TAG EVENT SELECTION
The transition photon or π 0 and the semileptonic D + s decay candidate are selected from the particles remaining after ST reconstruction.The photon or π 0 providing the lowest energy difference, ∆E, is selected.Here, , where the recoil energy is calculated from the momenta of γ(π 0 ) and The signal candidates are examined using the kinematic variable where E i and p i , with i = (tag, γ(π 0 ), e or η (′) ), are the energies and momenta of particle i.To improve the M 2 miss resolution, all the selected candidate tracks in the tag side, transition γ(π 0 ), and η (′) e + of the signal side, plus the missing neutrino, are subjected to a kinematic fit with a net three constraints: seven are applied and the neutrino four-vector is determined.The fit requires energy and momentum conservation, and in addition, the invariant masses of the two D s mesons are constrained to the nominal D s mass, the invariant mass of the D − s γ(π 0 ) or D + s γ(π 0 ) combination is constrained to the nominal D * s mass, and the combination with the smaller χ 2 is kept.To suppress the background contributions from non-D s D * s events in D + s → η ′ γρ 0 e + ν e , the χ 2 is required to satisfy χ 2 < 200.
In the signal side, the η meson is reconstructed by η → γγ or π 0 π + π − , and the η ′ meson is reconstructed by η ′ → η γγ π + π − or γρ 0 π + π − .The selection criteria of η (′) are the same as in the ST selection.The positron candidate is identified by using the dE/dx, TOF, and EMC information.Combined likelihoods for the pion, kaon and positron hypotheses, L ′ π , L ′ K and L ′ e , are calculated.Charged tracks satisfying L ′ e > 0.001 and 8 are assigned as positron candidates.To suppress background contributions from D + s hadronic decays, the maximum energy of the unused showers (E max γ extra ) must be less than 0.3 GeV and events with additional charged tracks (N extra char ) are removed.The invariant mass of the η (′) and e + is required to be M η (′) e + < 1.9 GeV/c 2 for D + s → η (′) e + ν e to further suppress the background contributions of D + s → η (′) π + .To suppress contributions of backgrounds to D + s → η ′ γρ 0 e + ν e where the photon is from a π 0 decay, the opening angle between the missing momentum and the most energetic unused shower (θ γ, miss ) is required to satisfy cos θ γ, miss < 0.85.

VI. BRANCHING FRACTIONS A. Results of branching fractions
After imposing all selection criteria, the M 2 miss distributions of the accepted candidates for the semileptonic decays D + s → η (′) e + ν e are obtained, as shown in Fig. 2. For the semileptonic D + s decays reconstructed via two different η (′) decays, a simultaneous unbinned maximumlikelihood fit is performed on the two M 2 miss distributions, where the branching fractions of D + s → η (′) e + ν e measured with the different η (′) decay modes are constrained to be equal.The signal and background components are modeled with shapes derived from MC simulation.The yields of the peaking backgrounds due to D + s → η (′) π + π 0 and η (′) µ + ν µ are fixed according to the MC simulation.For D + s → η ′ γρ 0 e + ν e , there is a remaining background contribution from D + s → φ(1020) π 0 π + π − e + ν e .The yield of D + s → φ(1020) π 0 π + π − e + ν e is left free in the fit.The remaining combinatorial backgrounds are dominated by open charm (more than 60%) and e + e − → q q (about 30%).The magnitude of this contribution is a free parameter in the fit.The branching fractions of the intermediate decays are Table 3.The obtained values of NST, ǫST, and ǫDT for various signal decays in the i-th tag mode, where the efficiencies do not include the branching fractions of the sub-resonant decays and the uncertainties are statistical only.The ǫ η (′) = ǫ DT,η (′) /ǫST are the efficiencies of detecting the transition γ(π 0 ) and signal channels in the presence of the ST D − s candidates.
Tag mode NST ǫST ǫDT,η γγ ǫη γγ ǫDT,η and B(π 0 → γγ) = (98.823± 0.034)% [13].The branching fractions of D + s → η (′) e + ν e , the yields of other background contributions and the parameters of the Gaussian functions convolved with the distributions from MC simulation are left free during the fit.The branching fractions are calculated from the signal yields with Eq. 1.The signal efficiencies, the signal yields, and the obtained branching fractions are summarized in Table 4.
Table 4. Signal efficiencies (ǫ γ(π 0 )SL ), signal yields (NDT), and obtained branching fractions (BSL) for various semi-electronic decays.Efficiencies include the branching fractions of D * ∓ s decays but do not include the branching fractions of the η (′) decays.Numbers in the first and second parentheses are the statistical and systematic uncertainties, respectively.

B. Systematic uncertainties
Table 5 summarizes the sources of the systematic uncertainties in the measurements of the branching fractions of D + s → η (′) e + ν e .They are assigned relative to the measured branching fractions and are discussed below.In this table, the contributions to the systematic uncertainties listed in the upper part are treated as correlated, while those in the lower part are treated as uncorrelated.
The total systematic uncertainties of the branching fractions of D + s → ηe + ν e and D + s → η ′ e + ν e are calculated to be 2.3% and 2.9%, respectively, after taking into account correlated and uncorrelated systematic uncertainties and using the method described in Ref. [44].
a. ST D − s yields The uncertainty of the fits to the D − s invariant mass spectra is estimated by varying the signal and background shapes and repeating the fits for both data and MC sample.A variation of the signal shape is obtained by modifying the matching requirement between generated and reconstructed angles from 15 • to 10 • or 20 • .The background shape is changed to a thirdorder Chebychev polynomial.The relative change of the ST yields in data over the ST efficiencies is considered as the systematic uncertainty.Moreover, an additional uncertainty due to the background fluctuation of the fitted ST yields is included.The quadrature sum of these three terms, 0.5%, is assigned as the associated systematic uncertainty.
b. π 0 and η reconstruction The systematic uncertainty in the π 0 reconstruction has been studied by using the control sample of e + e − → K + K − π + π − π 0 .The systematic uncertainty in the η reconstruction is taken as equal to that of the π 0 due to the limited η sample.After correcting differences of the π 0 or η reconstruction efficiencies between data and MC simulation, which are 0.991 − 1.024, the systematic uncertainties, due to statistical uncertainties on these corrections, are listed in Table 5.
c. π ± tracking and PID efficiencies The tracking and PID efficiencies of π ± are studied by using the control sample of e + e − → K + K − π + π − .The momentum weighted data-MC differences due to π ± tracking efficiencies range from 0.981 − 1.001 for different signal decays and the signal efficiencies are corrected by these factors.The systematic uncertainties due to π ± tracking and π ± PID are listed in Table 5.The uncertainties of π ± tracking for D + s → η ′ ηπ + π − e + ν e and D + s → η ′ γρ 0 e + ν e are partly correlated, the common uncertainty of 0.6% is considered as fully correlated, and the remaining quadratic difference of 1.7% for d. e + tracking and PID efficiencies The e + tracking and PID efficiencies are studied by using the control sample of e + e − → γe + e − .The ratios are 1.000 ± 0.005 for e + tracking and 0.988 ± 0.002 for e + PID efficiencies.After corrections, the systematic uncertainties, due to statistical uncertainties on these factors, are listed in Table 5.
f. Smallest |∆E| The systematic uncertainty of selecting the transition γ(π 0 ) with the smallest |∆E| method is studied by using two control samples of D + s → K + K − π + and D + s → ηπ 0 π + .The difference of the efficiency of selecting the transition γ(π 0 ) candidates in data versus the simulation is 1.0%, which is assigned as the systematic uncertainty.
g. Peaking background The systematic uncertainty due to the peaking backgrounds from and D + s → η (′) µ + ν µ is estimated by varying the quoted branching fractions [13] by ±1σ and correcting by the data-MC difference for the mis-identification of π + → e + and µ + → e + .The relative changes of signal yields are taken as the corresponding systematic uncertainties and listed in Table 5.
h. Hadronic transition form factors The detection efficiencies are estimated by using signal MC events generated with the hadronic transition form factors measured in this work.The corresponding systematic uncertainties are estimated by varying the parameters by ±1σ and listed in Table 5.
j. Tag bias Due to different reconstruction environments in the inclusive and signal MC samples, the ST efficiencies determined by the inclusive MC sample may be different from those by the signal MC sample.This may lead to incomplete cancellation of the systematic uncertainties associated with the ST selection, referred to as "tag bias".Inclusive and signal MC efficiencies are compared and the tracking and PID efficiencies for kaons and pions are studied for different track multiplicities.The resulting ST-average offsets are assigned as the systematic uncertainties from tag bias and listed in Table 5.
k. χ 2 requirement The systematic uncertainty due to the χ 2 requirement is estimated with a hadronic DT sample with D + s → η ′ γρ 0 π + replacing the semileptonic signal.The difference of the accepted efficiencies of the χ 2 requirement between data and MC simulation is 1.5%, which is assigned as the systematic uncertainty for D + s → η ′ γρ 0 e + ν e .l. M η (′) e + requirement The efficiencies of the M η (′) e + < 1.9 GeV/c 2 requirement are greater than 99% for all signal decays and the differences of these efficiencies between data and MC simulation are negligible.
m. cos θ γ, miss requirement The systematic uncertainty due to the cos θ γ, miss requirement is estimated by varying the requirement by ±0.05.The differences of the branching fractions are negligible.
o. M 2 miss fit The systematic uncertainty due to the M 2 miss fit is considered in two parts.Since a Gaussian function is convolved with the simulated signal shapes to account for the resolution difference between data and MC simulation, the systematic uncertainty from the signal shape is ignored.The systematic uncertainty due to the background shape is assigned by varying the relative fractions of major backgrounds from e + e − → q q and non-D * ± s D ∓ s open-charm processes within ±30% according to the uncertainty of its input cross section in the inclusive MC sample.The changes in the branching fractions are taken as the corresponding systematic uncertainties and listed in Table 5.
p. MC statistics The relative uncertainties of the signal efficiencies are assigned as the systematic uncertainties due to MC statistics, as listed in Table 5.

VII. HADRONIC TRANSITION FORM FACTORS
The differential decay width can be expressed as ) where q is the momentum transfer to the e + ν e system, | p η (′) | is the magnitude of the meson 3-momentum in the D + s rest frame and G F is the Fermi constant.In the modified pole model [46], where M pole is fixed to m D * + s and α is a free parameter.The simple pole model [47] is obtained by setting α = 0 and leaving M pole free.In the two-parameter (2-Par) series expansion [46], the hadronic transition form factor is given by Here, P (q 2 ) = z(q 2 , m 2 . Φ is given by s and m η (′) are the masses of D + s and η (′) particles, m D * s is the pole mass of the vector form factor accounting for the strong interaction between D + s and η (′) mesons and usually taken as the mass of the lowest lying cs vector meson D * s [13], and χ V is obtained from dispersion relations using perturbative QCD [48].

A. Differential decay rates
To extract the hadronic transition form factors of the semileptonic decays, the differential decay rates are measured in different q 2 intervals.For the D + s → ηe + ν e decay, the q 2 range (m 2 e , 2.02) GeV 2 /c 4 is subdivided in eight intervals of 0.2 GeV 2 /c 4 width (except for a wider final bin), while three regions, (m 2 e , 0.3), (0.3,0.6), and (0.6,1.02) GeV 2 /c 4 , are defined for D + s → η ′ e + ν e .The differential decay rates in the individual q 2 intervals i are determined as where ∆Γ i = •NST is the decay rate in the i-th q 2 interval, N i prd is the number of events produced in the i-th q 2 interval, τ D + s is the D + s lifetime [13] and N ST is the number of the ST D − s mesons.In the i-th q 2 interval, the number of events produced in data is calculated as where (ε −1 ) ij is the element of the inverse efficiency matrix, obtained by analyzing the signal MC events.The statistical uncertainty of N i prd is given by where σ stat (N j DT ) is the statistical uncertainty of N j DT .The efficiency matrix ε ij is given by where N rec ij is the number of events generated in the j-th q 2 interval and reconstructed in the i-th q 2 interval, N gen j is the total number of events generated in the j-th q 2 interval, and ε tag is the ST efficiency.f corr j is the efficiency correction factor for the events generated in the j-th q 2 interval, which is obtained with the same analysis procedure as that in the branching fraction measurement.The product of the efficiency correction factors in each q 2 is listed in Table 6.
Tables 7 and 8 give the elements of the efficiency matrices weighted by the ST yields in the data sample.
The number of events observed in each reconstructed q 2 interval is obtained from a fit to the M 2 miss distribution of the corresponding events.Figures 3 and 4 show the results of the fits to the M 2 miss distributions in the reconstructed q 2 intervals.Tables 9 and 10 summarize the q 2 ranges, the fitted numbers of observed DT events (N DT ), the numbers of generated events (N prd ) calculated by the weighted efficiency matrix and the decay rates of D + s → η (′) e + ν e (∆Γ) in the individual q 2 intervals.

B. χ 2 construction and statistical covariance matrices
To extract the hadronic transition form factor parameters and |V cs |, the smallest χ 2 method is used to fit the partial decay rates of the different signal decays.Considering the correlations of the measured partial decay rates (∆Γ msr i ) among different q 2 intervals, the χ 2 is given by where ∆Γ th i is the theoretically expected decay rate in channel i, C ij is the element of the covariance matrix  7. The efficiency matrices for D + s → ηe + νe averaged over all 14 ST modes, where εij represents the efficiency in % for events produced in the j-th q 2 interval and reconstructed in the i-th q 2 interval.Efficiencies do not include the branching fractions of η (′) decays.s → η ′ e + νe averaged over all 14 ST modes, where εij represents the efficiency in % for events produced in the j-th q 2 interval and reconstructed in the i-th q 2 interval.Efficiencies do not include the branching fractions of η (′) decays.
Here, C stat ij and C sys ij are elements of the statistical and systematic covariance matrices, respectively.The elements of the statistical covariance matrix are defined as Tables 13 and 14 give the elements of the statistical correction density matrices for , and D + s → η ′ γρ 0 e + ν e , respectively.

C. Systematic uncertainties
Several sources of systematic uncertainties are discussed below.
a. D + s lifetime The uncertainties associated with the D + s lifetime are fully correlated across the q 2 intervals.
The element of the related systematic covariance matrix is calculated by where σ(∆Γ i ) = στ D + s • ∆Γ i and στ D + s is the uncertainty of the D + s lifetime [13].b.MC statistics Systematic efficiency uncertainties in and correlations between the q 2 intervals due to the limited MC size are calculated by  where the covariances of the inverse efficiency matrix elements are given by [49] c. Hadronic transition form factor Systematic uncertainties associated with the hadronic transition form factor used to generate signal MC events are estimated by re-weighting the signal MC events so that the q 2 spectrum agrees with the measured spectrum.For each  signal MC event, the weight ω is given by where f η (′) default + (q 2 ) is the default hadronic transition form factor used to generate the signal MC events.The default hadronic transition form factor uses the modified pole model with the parameter α = 0.25 and f + (0) = 1.0.The f η (′) measured + (q 2 ) is the measured hadronic transition form factor for D + s → η (′) e + ν e using the 2-Par series expansion with parameters obtained from the fit with the statistical covariance matrix.
The partial decay rates are then calculated in different q 2 intervals with the newly weighted efficiency matrix.The element of the covariance matrix is defined as where δ(∆Γ i ) denotes the change of the partial decay rate in the i-th q 2 interval.
d. Tracking, PID, and γ, η, π 0 reconstruction The systematic uncertainties associated with the e + tracking and PID efficiencies, pion tracking and PID efficiencies, and γ, η, π 0 reconstruction are estimated by varying the corresponding correction factors for efficiencies within ±1σ.Using the new efficiency matrix, the element of the corresponding systematic covariance matrix is calculated by where δ(∆Γ i ) denotes the change of the partial decay rate in the i-th q 2 interval.e. M 2 miss fit The systematic covariance matrix arising from the uncertainty in the M 2 miss fit has elements (20) where σ fit α is the systematic uncertainty of the number of signal events observed in the interval α obtained by varying the background shape in the M 2 miss fit.f.Remaining uncertainties The remaining uncertainties are assumed to be fully correlated across q 2 intervals and the element of the corresponding systematic covariance matrix is calculated by where σ(∆Γ i ) = σ sys • ∆Γ i and σ sys is the corresponding uncertainty reported in Table 5.
Tables 11 and 12 give the systematic uncertainties for all sources in the different q 2 intervals, and Tables 13  and 14 give the elements of the systematic covariance density matrices for D + s → ηe + ν e and D + s → η ′ e + ν e , respectively.

D. Results based on individual fits
For each semileptonic decay, the product f η (′) + (0)|V cs | and one of the parameters M pole , α, or r 1 are determined by constructing and minimizing the χ 2 defined in Eq. 12.
The covariance matrices used in these fits are shown in Tables 13 and 14. Figure 5 shows individual fits to the differential decay rates of D + s → ηe + ν e and D + s → η ′ e + ν e and (second row) the hadronic transition form factors as a function of q 2 .The results obtained from individual fits are listed in Table 15.

E. Results based on simultaneous fits
Since the results for the hadronic transition form factors are consistent with each other, simultaneous fits to the differential decay rates of D + s → ηe + ν e and D + s → η ′ e + ν e are performed to improve the statistical precision.
The values of ∆Γ i msr measured by the two η (′) subdecays are fitted simultaneously, with results shown in Fig. 6.In the fits, the ∆Γ i msr becomes a vector of length 2m and C ij becomes a 2m × 2m covariance matrix.The uncorrelated and correlated systematic uncertainties are the same as shown in Table 5.
For fully correlated systematic uncertainties, the matrices are constructed in the same way as done for the individual fits.For the uncorrelated systematic uncertainties, the matrix takes the form where A and B are the matrices obtained from the individual decays.Table 16 summarizes the fit results obtained from the simultaneous, where the obtained values of f η (′) + (0)|V cs | with different hadronic transition form factor parameterizations are consistent with each other.

Statistical correlation matrix
1.000 -0.168 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 8 1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1 1.000 -0.174 0.012 -0.002 -0.001 0.000 -0.001 0.000 2 1.000 -0.212 0.011 -0.003 -0.001 -0.001 0.000 3  [23], which used a subset of the dataset of the present analysis, the precision of the branching fractions of D + s → ηe + ν e and D + s → η ′ e + ν e is improved by a factor of 1.3 and 1.7, respectively, and the precision of f η ′ + (0)|V cs | is improved by a factor of 2.2.The precision of f η + (0)|V cs | is not improved because the uncertainty in the previous paper [23] is underestimated by a factor of two due to incorrect construction of the χ 2 in the fits to the partial decay rates (see [62] for details).For simple comparison, we also present the results based on 3.   a standard model fit (CKMfitter, [13]) as input, the form factors at zero momentum transfer squared f η (′) + (0) are determined.The measured hadronic transition form factors provide important pieces of information to test the various theoretical calculations [2, 4-6, 9, 11, 12]. Figure 7 shows the comparisons of the f η (′) + (0) obtained in this paper and different theoretical predictions.Alternatively, we determine |V cs | with the D + s → η (′) e + ν e decays by taking the f η (′) + (0) given by theoretical calculations [4].These results on |V cs | together with those measured by D → Kℓ + ν ℓ and D + s → ℓ + ν ℓ are important to test the unitarity of the CKM matrix.The branching fractions, hadronic transition form factors and |V cs | reported in this work supersede the corresponding results in Ref. [23], based on the 3.19 fb −1 subset of data at E CM = 4.178 GeV.+ (0) measured by this work with the theoretical calculations.The first and second uncertainties are statistical and systematic, respectively.The green bands correspond to the ±1σ limits of the form factors measured by this work.For the predictions by LQCD, no systematic uncertainties have been considered.Table 14.Statistical and systematic uncertainty density matrices for the measured partial decay rates of D + s → η ′ e + νe in different q 2 intervals.[62] In Ref. [23], the χ 2 is constructed to be χ

Statistical correlation matrix
where there is an additional term of σ(∆Γ msr i )σ(∆Γ msr j ) used for test but not removed when obtaining nominal results.This mistake only changes f η + (0)|Vcs| by 0.2%, but underestimate its uncertainty by a factor of 2.   24.26 (10) Figure 2 shows the simultaneous fits to the partial decay rates of D + s → ηe + ν e or D + s → η ′ e + ν e reconstructed with two different decay modes and the hadronic transition form factors as function of q 2 .The parameters obtained for the hadronic transition form factors are summarized in Table 2.

APPENDICES
Tables 3 and 4 summarize the q 2 ranges, the fitted numbers of observed events (N DT ), the numbers of generated events (N prd ) calculated by the weighted efficiency matrices and the decay rates (∆Γ) of D + s → ηe + ν e and D + s → η ′ e + ν e in various q 2 intervals, respectively.

Fig. 1 .
Fig. 1.Fits to the Mtag distributions of the accepted ST candidates from the data sample with all data sets.Points with error bars are data.The blue solid curves are the total fit results.The red dashed curves are the fitted backgrounds.The blue dotted curve in the K 0 S K − mode is the D − → K 0 S π − component.In each sub-figure, the pair of arrows denotes the signal regions.

Fig. 2 .
Fig.2.Fits to the M 2 miss distributions of the candidate events for various semileptonic decays.The points with error bars represent data.The blue solid curves denote the total fits, and the red solid dotted curves show the fitted combinatorial background contributions.Differences between dashed and dotted curves are the backgrounds from D + s → η (′) π + π 0 , η (′) µ + νµ, and D + s → φ(1020) π 0 π + π − e + νe.

Fig. 3 .
Fig. 3. Fits to the M 2 miss distributions in various reconstructed q 2 intervals for (top two rows) D + s → ηγγ e + νe and (bottom two rows) D + s → η π 0 π + π − e + νe.The points with error bars represent data.The blue solid curves denote the total fits, and the red solid dotted curves show the fitted combinatorial background contributions.Differences between black dashed and red dotted curves show the backgrounds from D + s → ηπ + π 0 and ηµ + νµ.

Fig. 5 .
Fig. 5. (Top row) Individual fits to the differential decay rates of D + s → ηe + νe and D + s → η ′ e + νe and (bottom row) projection on the hadronic transition form factors as a function of q 2 .The points with error bars are (top row) the measured differential decay rates and (bottom row) the hadronic transition form factors.The black, red, and blue curves are the form factors parameterized by simple pole model, modified pole model, and 2-Par series expansion, respectively.
19 fb −1 of data at E CM = 4.178 GeV in the Appendices.After fixing this issue, the precision of f η + (0)|V cs | is improved by a factor of 1.4 as expected.Combining the new branching fractions with those of D + → ηe + ν e and D + → η ′ e + ν e measured by BESIII [61], the η − η ′ mixing angle φ P = (40.0± 2.0 stat ± 0.6 syst ) • is extracted, providing information related to the gluon component in the η ′ meson.By analyzing the partial decay rates of D + s → ηe + ν e and D + s → η ′ e + ν e , the products of f η (′) + (0)|V cs | are determined.Furthermore, taking the value of |V cs | from

Fig. 6 .
Fig. 6. (Top row) Simultaneous fits to the differential decay rates of (left) D + s → ηγγ e + νe and D + s → η π 0 π + π − e + νe and (right) D + s → η ′ ηπ + π − e + νe and D + s → η ′ ρ 0 γ e + νe, and (bottom row) projection on the hadronic transition form factors as function of q 2 .The red circles and blue triangles with error bars are (top row) the measured differential decay rates for two η (′) channels and (bottom row) the hadronic transition form factors.The black, red, and blue curves are the form factors parameterized by simple pole model, modified pole model, and 2-Par series expansion, respectively.

Fig. 7 .
Fig. 7. Comparisons of the form factors f η (′)+ (0) measured by this work with the theoretical calculations.The first and second uncertainties are statistical and systematic, respectively.The green bands correspond to the ±1σ limits of the form factors measured by this work.For the predictions by LQCD, no systematic uncertainties have been considered.

Fig. 1 .
Fig. 1.Fits to the M 2miss distributions of the candidate events for various semileptonic decays from data taken at ECM = 4.178 GeV.The points with error bars represent data.The blue solid curves denote the total fits, and the red solid dotted curves show the fitted combinatorial background contributions.Differences between dashed and dotted curves are due to the backgrounds from D + s → η (′) π + π 0 , η (′) µ + νµ, and D + s → φ(1020) π 0 π + π − e + νe.

Fig. 2 .
Fig. 2. (Top row) Simultaneous fits to the differential decay rates of (left) D + s → ηγγ e + νe and D + s → η π 0 π + π − e + νe and (right) D + s → η ′ ηπ + π − e + νe and D + s → η ′ ρ 0 γ e + νe, and (bottom row) projection on the hadronic transition form factors as function of q 2 for the data sample taken at ECM = 4.178 GeV.The red circles and blue triangles with error bars are (top row) the measured differential decay rates for two η (′) channels and (bottom row) the hadronic transition form factors.The black, red, and blue curves are the form factors parameterized by simple pole model, modified pole model, and 2-Par series expansion, respectively.

Table 2 .
The MBC requirements for various energy points.

Table 5 .
Relative systematic uncertainties (in %) on the measurements of the branching fractions of D + s → ηe + νe and D + s → η ′ e + νe.The top and the bottom sections are correlated and uncorrelated, respectively.The uncertainty in the uncorrelated π ± tracking is obtained as the square root of the quadratic difference of the total uncertainty in the π ± tracking and the correlated portion.Sourceη γγ e + ν e η π 0 π + π − e + ν e η ′ ηπ + π − e + ν e η ′ γρ 0 e + ν e

Table 8 .
The efficiency matrices for D +

Table 9 .
The partial decay rates of D + s → ηe + νe in various q 2 intervals.Numbers in the parentheses are the statistical uncertainties.

Table 10 .
The partial decay rates of D + s → η ′ e + νe in various q 2 intervals.Numbers in the parentheses are the statistical uncertainties.

Table 12 .
Systematic uncertainties (%) of the measured decay rates of D + s → η ′ e + νe in various q 2 intervals.

Table 15 .
The parameters obtained from individual fits to the partial decay rates of D + s → ηe + νe or D + s → η ′ e + νe.The first and second uncertainties are statistical and systematic, respectively.N d.o.f is the number of degrees of freedom.

Table 16 .
The parameters obtained from simultaneous fits to the partial decay rates of D + s → ηe + νe or D + s → η ′ e + νe.The first and second uncertainties are statistical and systematic, respectively.N d.o.f is the number of degrees of freedom.
Figure1shows the results of the fits to the M 2 miss distributions of the candidate events for D + s → ηe + ν e and D + s → η ′ e + ν e , based on the 3.19 fb −1 of e + e − collision data taken at E CM = 4.178 GeV.The obtained signal yields, signal efficiencies and branching fractions are summarized in Table 1.

Table 1 .
Signal efficiencies (ǫ γ(π 0 )SL ), signal yields (NDT), and obtained branching fractions (BSL) for various semi-electronic decays based on the data sample taken at ECM = 4.178 GeV.Efficiencies do not include the branching fractions of η (′) subdecays.Numbers in the first and second parentheses are the statistical and systematic uncertainties, respectively.

Table 4 .
The partial decay rates of D + s → η ′ e + νe in various q 2 intervals of data for the data sample taken at ECM = 4.178 GeV.Numbers in the parentheses are the statistical uncertainties.