Measurement of the Dynamics of the Decays ${ D_s^+ \rightarrow \eta^{(\prime)} e^{+} \nu_e}$

Using $e^+e^-$ annihilation data corresponding to an integrated luminosity of 3.19\,fb$^{-1}$ collected at a center-of-mass energy of 4.178~GeV with the BESIII detector, we measure the absolute branching fractions $\mathcal{B}_{D_s^+ \rightarrow \eta e^{+} \nu_e }$ = $(2.323\pm0.063_{\rm stat}\pm0.063_{\rm syst})\%$ and $\mathcal{B}_{D_s^+ \rightarrow \eta^{\prime} e^{+} \nu_e}$ = $(0.824\pm0.073_{\rm stat}\pm0.027_{\rm syst})\%$ via a tagged analysis technique, where one $D_s$ is fully reconstructed in a hadronic mode. Combining these measurements with previous BESIII measurements of $\mathcal{B}_{D^+\to\eta^{(\prime)} e^{+} \nu_e}$, the $\eta-\eta^\prime$ mixing angle in the quark flavour basis is determined to be $\phi_{\rm P} = (40.1\pm2.1_{\rm stat}\pm0.7_{\rm syst})^\circ$. From the first measurements of the dynamics of $D^+_s\to \eta^{(\prime)}e^+\nu_e$ decays, the products of the hadronic form factors $f_+^{\eta^{(\prime)}}(0)$ and the Cabibbo-Kobayashi-Maskawa matrix element $|V_{cs}|$ are determined with different form factor parameterizations. For the two-parameter series expansion, the results are $f^{\eta}_+(0)|V_{cs}| = 0.4455\pm0.0053_{\rm stat}\pm0.0044_{\rm syst}$ and $f^{\eta^{\prime}}_+(0)|V_{cs}| = 0.477\pm0.049_{\rm stat}\pm0.011_{\rm syst}$.

Exclusive D semi-leptonic (SL) decays provide a powerful way to extract the weak and strong interaction couplings of quarks due to simple theoretical treatment [1][2][3]. In the Standard Model, the rate of D + s → ηe + ν e and D + s → η ′ e + ν e depends not only on V cs , an element of the Cabibbo-Kobayashi-Maskawa (CKM) matrix describing weak transitions between the charm and strange quarks, but also on the dynamics of strong interaction, parameterized by the form factor (FF) f η (′) + (q 2 ), where q is the momentum transfer to the e + ν e system. Unlike the final-state hadrons K and π, the mesons η (′) are especially intriguing because the spectator quark plays an important role in forming the final state. This gives access to the singlet-octet mixing of η − η ′ -gluon [4,5], whose mixing parameter can be determined from the SL decays, and, consequently, gives a deeper understanding of non-perturbative QCD confinement.
Measurements of f η (′) + (0) are crucial to calibrate these theoretical calculations. Once the predicted f η (′) + (0) pass these experimental tests, they will help determine |V cs |, and, in return, help test the unitarity of the CKM quark mixing matrix. Additionally, measurements of the branching fractions (BFs) of D + s → η (′) e + ν e can shed light on η − η ′ -gluon mixing. The η − η ′ mixing angle in the quark flavour basis, φ P , can be related to the BFs of the D and , in which a possible gluon component cancels [9]. Determination of φ P gives a complementary constraint on the role of gluonium in the η ′ , thus helping to improve our understanding of nonperturbative QCD dynamics and benefiting theoretical calculations of D and B decays involving the η (′) . Previous measurements of the BFs of D + s → η (′) e + ν e were made by CLEO [12][13][14] and BESIII [15], but these measurements include large uncertainties. This Letter reports improved measurements of the BFs and the first experimental studies of the dynamics of D + s → η (′) e + ν e [16]. Based on these, the first measurements of f η (′) + (0) are made, and measurements of |V cs | and φ P are presented.
This analysis is performed using e + e − collision data corresponding to an integrated luminosity of 3.19 fb −1 taken at a center-of-mass energy E CM = 4.178 GeV with the BESIII detector. A description of the design and performance of the BESIII detector can be found in Ref. [17]. For the data used in this Letter, the end cap time-of-flight system was upgraded with multi-gap resistive plate chambers with a time resolution of 60 ps [18,19]. Monte Carlo (MC) simulated events are generated with a geant4-based [20] detector simulation software package, which includes the geometric description and a simulation of the response of the detector. An inclusive MC sample with equivalent luminosity 35 times that of data is produced at E CM = 4.178 GeV. It includes open charm processes, initial state radiation (ISR) production of ψ(3770), ψ(3686) and J/ψ, qq (q = u, d, s) continuum processes, along with Bhabha scattering, µ + µ − , τ + τ − and γγ events. The open charm processes are generated using conexc [21]. The effects of ISR and final state radiation (FSR) are considered. The known particle decays are generated with the BFs taken from the Particle Data Group (PDG) [22] by evtgen [23], and the other modes are generated using lundcharm [24].
The SL decays D + s → η (′) e + ν e are simulated with the modified pole model [25].
At E CM = 4.178 GeV, D + s mesons are produced mainly from the processes e + e − → D + s D * − s + c.c. → D + s γ(π 0 )D − s . We first fully reconstruct one D − s in one of several hadronic decay modes [called the single-tag (ST) D − s ]. We then examine the SL decays of the D + s and the γ(π 0 ) from the D * s [called double-tag (DT) D + s ]. The BF of the SL decay is determined by where N tot ST and N tot DT are the ST and DT yields in data, ǫ γ(π 0 )SL is the efficiency of finding γ(π 0 )η (′) e + ν e determined by k ST and ǫ k DT are the efficiencies of selecting ST and DT candidates in the k-th tag mode, and estimated by analyzing the inclusive MC sample and the independent signal MC events of various DT modes, respectively.
The ST D − s candidates are reconstructed using fourteen hadronic decay modes as shown in Fig. 1. The selection criteria for charged tracks and K 0 S , and the particle identification (PID) requirements for π ± and K ± , are the same as those used in Ref. [26]. Positron PID is performed by using the specific ionization energy loss in the main drift chamber, the time of flight, and the energy deposited in the electromagnetic calorimeter (EMC). Confidence levels for the pion, kaon and positron hypotheses (L π , L K and L e ) are formed. Positron candidates must satisfy L e > 0.001 and L e /(L e +L π +L K ) > 0.8. The energy loss of the positron due to bremsstrahlung is partially recovered by adding the energies of the EMC showers that are within 10 • of the positron direction and not matched to other particles (FSR recovery).
To remove soft pions originating from D * transitions, the momenta of pions from the ST D − s are required to be larger than 0.1 GeV/c. For the tag modes D − s → π + π − π − and K − π + π − , the contributions of D − s → K 0 S π − and K 0 S K − are removed by requiring M π + π − outside ±0.03 GeV/c 2 around the K 0 S nominal mass [22].
The nonpeaking background is modeled by a second-or third-order Chebychev polynomial. To account for the resolution difference between data and MC simulation, the MC simulated shape(s) is convolved with a Gaussian for each tag mode. The reliability of the fitted nonpeaking background has been verified using the inclusive MC sample. Events in the signal regions, denoted by the boundaries in each subfigure of Fig. 1, are kept for further analysis. The total ST yield is N tot ST = 395142 ± 1923. Once the D − s tag has been found, the photon or π 0 from the D * + s transition is selected. We define the energy or tag] are the energy and momentum of γ(π 0 ) or D − s tag, respectively. All unused γ or π 0 candidates are looped over and that with the minimum |∆E| is chosen. Candidates with ∆E ∈ (−0.04, 0.04) GeV are accepted. The signal candidates are examined by the where E i and p i (i = e or η (′) ) are the energy and momentum of e + or η (′) . To suppress backgrounds 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. We require M η ′ e + < 1.9 GeV/c 2 for D + s → η ′ e + ν e and cos θ hel ∈ (−0.85, 0.85) for D + s → η ′ γρ 0 e + ν e to further suppress the D + s → η ′ π + and D + s → φe + ν e backgrounds, where θ hel is the helicity angle between the momentum directions of the π + and the η ′ in the ρ 0 rest frame. Figure 2 shows the MM 2 distribution after all selection criteria have been applied. The signal yields are determined from a simultaneous unbinned maximum likelihood fit to these spectra, where B D + s →η (′) e + νe measured using two different η (′) subdecays are constrained to be the same after considering the different efficiencies and subdecay BFs. The signal and background components in the fit are described by shapes derived from MC simulation. For the decay D + s → η ′ γρ 0 e + ν e , some peaking background from D + s → φe + ν e still remains. This background is modeled by a separate component in the fit; its size and shape are fixed based on MC simulation. Table I summarizes the efficiencies for finding SL decays, the observed signal yields, and the obtained BFs.
With the DT method, the BF measurements are insensitive to the ST selection. The following relative systematic uncertainties in the BF measurements are assigned. The uncertainty in the ST yield is estimated to be 0.6% by alternative fits to the M tag spectra with different signal shapes, background parameters, and fit ranges. The uncertainties in the tracking or PID efficiencies are assigned as 0.5% per π ± by studying e + e − → K + K − π + π − , and 0.5% per e + by radiative Bhabha process, respectively. The uncertainties of the E max γ extra and N extra char requirements are estimated to be 0.5% and 0.9% by analyzing DT hadronic events. The uncertainties of the ∆E requirement, FSR recovery and θ hel requirement are estimated with and without each requirement, and the BF changes are 0.8%, 0.8%, and 0.1%, respectively, which are taken as the individual uncertainties. The uncertainties of the selection of neutral particles are assigned as 1.0% per photon by studying J/ψ → π + π − π 0 [27] and 1.0% per π 0 or η by studying e + e − → K + K − π + π − π 0 .
The uncertainty due to the signal model is estimated to be 0.5% by comparing the DT efficiencies before and after re-weighting the q 2 distribution of the signal MC events to data. The uncertainty of the MM 2 fit is assigned as 0.9%, 1.3%, 1.2% and 1.2% for D + s → η γγ e + ν e , η π 0 π + π − e + ν e , η ′ ηπ + π − e + ν e and η ′ γρ 0 e + ν e (the same sequence later), respectively, by repeating fits with different fit ranges and different signal and background shapes. The ST efficiencies may be different due to the different multiplicities in the tag environments, leading to incomplete cancelation of the systematic uncertainties associated with the ST selection.
The associated uncertainty is assigned as 0.4%, 0.3%, 0.3%, 0.3%, from studies of the efficiency differences for tracking and PID of K ± and π ± as well as the selection of neutral particles between data and MC simulation in different environments.
The uncertainty due to the M η ′ e + requirement is found to be negligible. The uncertainty due to peaking background is assigned to be 1.4% by varying its size by ±1σ of the corresponding BF. The uncertainties due to the quoted BFs, 0.9%, 1.4%, 1.8% and 1.9% of η (′) decays [22] are also considered. For each decay, the total systematic uncertainty is determined to be 2.7%, 3.3%, 3.4% and 4.0% by adding all these uncertainties in quadrature.
To study the D + s → η (′) e + ν e dynamics, the candidate events are divided into various q 2 intervals. The measured partial decay width ∆Γ i msr in the ith q 2 interval is determined by ∆Γ i is the lifetime of the D + s meson [22,29], and N i pro is the DT yield produced in the ith q 2 interval, calculated by Here m is the number of q 2 intervals, N j obs is the observed DT yield obtained from similar fits to the MM 2 distribution as described previously, and ǫ ij is the efficiency matrix determined from signal MC events and is given by is the DT yield generated in the jth q 2 interval and reconstructed in the ith q 2 interval, N j gen is the total signal yield generated in the jth q 2 interval, and k sums over all tag modes. See Tables 1 and 2 of Ref. [30] for details about the range, N i obs , N i prd , and ∆Γ i msr of each q 2 interval for D + s → ηe + ν e and D + s → η ′ e + ν e , respectively. In theory, the differential decay width can be expressed where |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 [31], where M pole is fixed to M D * + s and α is a free parameter. Setting α = 0 and leaving M pole free, it is the simple pole model [32]. In the two-parameter (2 Par.) series expansion [31] f + (q 2 ) = 1 Here, A(q 2 ) = P (q 2 )Φ(q 2 , t 0 ), B(q 2 ) = r 1 (t 0 )z(q 2 , t 0 ), , and r k is a free parameter. The functions P (q 2 ), Φ(q 2 , t 0 ), and z(q 2 , t 0 ) are defined following Ref. [31].
For each SL decay, the product f + (0)|V cs | and one other parameter, M pole , α, or r 1 , are determined by constructing and minimizing with ∆Γ i msr and the theoretically expected value ∆Γ i exp , where C ij = C stat ij + C syst ij is the covariance matrix of ∆Γ i msr among q 2 intervals, as shown in Tables 3 and  4 in Ref. [30]. For each η (′) subdecay, the statistical covariance matrix is constructed with the statistical uncertainty in each q 2 interval (σ(N α obs )) as C stat The systematic covariance matrix is obtained by summing all the covariance matrices for all systematic uncertainties, which are all constructed with the systematic uncertainty in each q 2 interval (δ(∆Γ i msr )) as C syst ij = δ(∆Γ i msr )δ(∆Γ j msr ). Here, an additional systematic uncertainty in τ D + s (0.8%) [22,29] is involved besides those in the BF measurements.
The ∆Γ i msr measured by the two η (′) subdecays are fitted simultaneously, with results shown in Fig 3. In the fits, the ∆Γ i msr becomes a vector of length 2m. Uncorrelated systematic uncertainties are from tag bias, quoted BFs, η (and π 0 ) reconstruction, and FF parametrization, while other systematic uncertainties are fully correlated. Table II summarizes the fit results, where the obtained f η (′) + (0)|V cs | with different FF parameterizations are consistent with each other.
The BESIII collaboration thanks the staff of BEPCII and the IHEP computing center for their strong support.