First measurement of the form factors in $D^+_{s}\rightarrow K^0 e^+\nu_e$ and $D^+_{s}\rightarrow K^{*0} e^+\nu_e$ decays

We report on new measurements of Cabibbo-suppressed semileptonic $D_s^+$ decays using $3.19~\mathrm{fb}^{-1}$ of $e^+e^-$ annihilation data sample collected at a center-of-mass energy of 4.178~GeV with the BESIII detector at the BEPCII collider. Our results include branching fractions $\mathcal B({D^+_s\rightarrow K^0 e^+\nu_{e}})=(3.25\pm0.38({\rm stat.})\pm0.16({\rm syst.}))\times10^{-3}$ and $\mathcal B({D^+_s\rightarrow K^{*0} e^+\nu_{e}})=(2.37\pm0.26({\rm stat.})\pm0.20({\rm syst.}))\times10^{-3}$ which are much improved relative to previous measurements, and the first measurements of the hadronic form-factor parameters for these decays. For $D^+_s\rightarrow K^0 e^+\nu_{e}$, we obtain $f_+(0)=0.720\pm0.084({\rm stat.})\pm0.013({\rm syst.})$, and for $D^+_s\rightarrow K^{*0} e^+\nu_{e}$, we find form-factor ratios $r_V=V(0)/A_1(0)=1.67\pm0.34({\rm stat.})\pm0.16({\rm syst.})$ and $r_2=A_2(0)/A_1(0)=0.77\pm0.28({\rm stat.})\pm0.07({\rm syst.})$.

We report on new measurements of Cabibbo-suppressed semileptonic D + s decays using 3.19 fb −1 of e + e − annihilation data sample collected at a center-of-mass energy of 4.178 GeV with the BESIII detector at the BEPCII collider. Our results include branching fractions B(D + s → K 0 e + νe) = (3.25 ± 0.38(stat.) ± 0.16(syst.)) × 10 −3 and B(D + s → K * 0 e + νe) = (2.37 ± 0.26(stat.) ± 0.20(syst.)) × 10 −3 which are much improved relative to previous measurements, and the first measurements of the hadronic form-factor parameters for these decays. For D + s → K 0 e + νe, we obtain f+(0) = 0.720 ± 0.084(stat.) ± 0.013(syst.), and for D + s → K * 0 e + νe, we find form-factor ratios rV = V (0)/A1(0) = 1.67 ± 0.34(stat.) ± 0.16(syst.) and r2 = A2(0)/A1(0) = 0.77 ± 0.28(stat.) ± 0.07(syst.). The study of D + s semileptonic (SL) decays provides valuable information about weak and strong interactions in mesons composed of heavy quarks. (Throughout this Letter, charge-conjugate modes are implied unless explicitly noted.) Measurement of the total SL decay width of the D + s , and comparison with that of the D mesons, can help elucidate the role of nonperturbative effects in heavy-meson decays [1,2]. The Cabibbo-suppressed (CS) SL decays, including the branching fractions (BFs) for D + s → K 0 e + ν e and D + s → K * 0 e + ν e [3], are especially poorly measured. Detailed investigations of the dynamics of these decays allow measurements of SL decay partial widths, which depend on the hadronic form factors (FFs) describing the interaction between the final-state quarks. Measurements of these FFs provide experimental tests of theoretical predictions of Lattice QCD (LQCD). Reference [4] predicts that the FFs have minimal dependence on the spectator-quark mass, with values for D + s → K 0 ℓ + ν ℓ and D + → π 0 ℓ + ν ℓ differing by less than 5%. Experimental verification of this predicted instance of U -spin (d ↔ s) symmetry would be a significant success for LQCD. A complementary LQCD test is provided by comparing measured and predicted FF parameters for D + s → K * 0 ℓ + ν ℓ and D + → ρ 0 ℓ + ν ℓ . The combination of these measurements has the potential to verify LQCD FF predictions for SL charm decays to both pseudoscalar and vector mesons, useful for further applying the LQCD to SL B decays for precise determination of Cabibbo-Kobayashi-Maskawa (CKM) parameters [4,5].
In this Letter, we report on improved measurements of the absolute BFs and first measurements of the FFs for the decays D + s → K 0 e + ν e and D + s → K * 0 e + ν e . Our measurements have been made with 3.19 fb −1 e + e − annihilation data recorded with the BESIII detector at the BEPCII collider. The center-of-mass energy for our data is √ s = 4.178 GeV. The cross section is ∼ 1 nb for the production of D * + s D − s + c.c. at this energy. Our data sample is the largest collected by any experiment for D + s studies in the clean near-threshold environment.
Details about the BESIII detector design and performance are provided in Ref. [6]. A GEANT4-based [7] Monte Carlo (MC) simulation package, which includes the geometric description of the detector and the detector response, is used to determine signal detection efficiencies and to estimate potential backgrounds. Signal MC samples of e + e − → D * + s D − s with a D + s meson decaying to K ( * )0 e + ν e together with a D − s decaying to the studied decay modes used for this analysis are generated with CONEXC [8] using EVTGEN [10], with initial-state radiation (ISR) [8,9] and final-state radiation (FSR) effects [11] included. The simulation of the SL decay D + s → K * (0) e + ν e is matched with the FFs measured in this work. To study the backgrounds, inclusive MC samples consisting of open-charm states, radiative return to J/ψ and ψ(2S) and continuum processes are generated. All known decay modes of open-charm and ψ states are simulated as specified by the Particle Data Group (PDG) [12], while the remaining unknown decays are modeled with LUNDCHARM [13].
As described above, D + s mesons are produced at √ s = 4.178 GeV predominantly through D * + s D − s [14], with 94% of the D * + s decaying to γD + s . The first step of our analysis is to select "single-tag" (ST) events with a fully reconstructed D − s candidate. The D − s hadronic decay tag modes that are used for this analysis are listed in Table I. In this ST sample, we select the SL decay D + s → K ( * )0 e + ν e plus an isolated photon consistent with being from the D * s → γD s transition. The selected events are referred to as the double-tag (DT) sample. For a specific tag mode i, the ST and DT event yields can be expressed as where N DsD * s is the number of D s D * s pairs; B i ST and B i SL are the BFs of the D − s tag mode and the D + s SL decay mode, respectively; ǫ i ST is the efficiency for finding the tag candidate; and ǫ i DT is the efficiency for simultaneously finding the tag D − s and the SL decay. The DT efficiency ǫ i DT includes the BF for D * + s → γD + s . The BF for the SL decay is given by where N DT is the total yield of DT events, N ST is the total ST yield, and ǫ SL = is the average efficiency for finding SL decay weighted by the measured yields of tag modes in data.
Selection criteria for photons, charged pions and charged kaons are the same as those used in Ref. [15]. To reconstruct π 0 and η candidates, the invariant masses of the accepted photon pairs must be within (0.115, 0.150) GeV/c 2 and (0.50, 0.57) GeV/c 2 , respectively. To improve the momentum resolution, a kinematic fit is performed to constrain the γγ invariant mass to the nominal π 0 or η mass [3], and

ST mode
the χ 2 of the kinematic fit is required to be less than 20. The fitted π 0 and η momenta are used for reconstruction of the D − s tag candidates. K 0 S mesons are reconstructed from two oppositely charged tracks with its invariant mass within (0.485, 0.510) GeV/c 2 . A fit is applied to constrain these two charged tracks to a common vertex, and this K 0 S decay vertex is required to be separated from the interaction point by more than twice the standard deviation of the measured flight distance. We select ρ − → π − π 0 by requiring the invariant mass M π − π 0 to be within (0.626, 0.924) GeV/c 2 [3]. The decay modes η ′ → π + π − η and η ′ → γπ + π − are used to select η ′ mesons, with the invariant masses of the π + π − η and γπ + π − required to be within (0.940, 0.976) GeV/c 2 and (0.940, 0.970) GeV/c 2 , respectively. Additionally, to suppress backgrounds from D * decays, the momenta of the photons from η ′ → γπ + π − and all pions are required to be greater than 0.1 GeV/c. For all events passing the ST selection criteria, we calculate the recoil mass against the tag with the following formula: where m D − s and p D − s are the known mass [3] and measured momentum of the tag  Fig. 1. Signals are modeled with the MC-simulated signal shape convoluted with Gaussians to account for the resolution differences between data and MC, while the combinatorial backgrounds are parameterized with second-or third-order polynomial functions. Due to misidentification of π − as K − , the backgrounds from In the fit, the shape of this background is described by using the MC simulation and its size is set as a  Table I, which also includes the ST yields for all tag modes. The total reconstructed ST yield in our data sample is N ST = 341, 325 ± 1, 764.
In signal events, the system recoiling against the D − s tag consists of the SL decay D + s → K 0 e + ν e or D + s → K * 0 e + ν e . 2 We select these from the additional tracks accompanying the tag, that is a K 0 → K 0 S → π + π − with the ST criteria al-4 ready described, and K * 0 → K + π − therefore requiring that there be exactly three tracks in the event and with the invariant candidates must satisfy L e > 0.001 and L e /(L e + L π + L K ) > 0.8. Energy loss due to bremsstrahlung is partial- 14 ly recovered by adding the energy of EMC showers that are within 5 • of the electron direction and not matched to oth- 16 er particles [16,17]. Backgrounds from D + s → K 0 π + reconstructed as D + s → K 0 e + ν e and D + s → K + π + π − re- constructed as D + s → K * 0 e + ν e are rejected by requiring the K 0 e + or K * 0 e + invariant mass to be less than 1.78 GeV/c 2 .

20
Backgrounds associated with fake photons are suppressed by requiring E γmax , the largest energy of any unused photon, to be less than 0.20 GeV.
To identify a photon produced directly from D * ± s , we per- 24 form two kinematic fits for each γ candidate, one assuming that the γ combines with the tag to form a D * − s and the other 26 assuming that the SL decay comes from a D * + s parent. We require the D ∓ s D * ± s pair to conserve energy and momentum 28 in the center-of-mass frame, and the D ± s candidates are constrained to the known mass. The neutrino is treated as a miss- 30 ing particle. When we assume the tag to be the daughter of a D * − s , we constrain the mass of the photon plus tag candi-32 date to be consistent with the expected D * − s mass; otherwise we constrain the mass of the photon plus SL decay to be consistent with the D * + s mass. Finally, we select the photon and hypothesis with the smallest kinematic fit χ 2 . 36 We obtain information about the undetected neutrino with the missing-mass squared of the event, calculated from the 38 energies and momenta of the tag (E D − s , p D − s ), the transition photon (E γ , p γ ), and the detected SL decay products (E SL = 40 E K ( * )0 + E e + , p SL = p K ( * )0 + p e + ) as follows: Figure 2 shows the MM 2 distributions of the accepted candidate events for D + s → K 0 e + ν e and D + s → K * 0 e + ν e in data. The signal DT yield N DT is obtained by performing an unbinned maximum likelihood fit to MM 2 . In the fit, the signal is described with an MC-derived signal shape convolved with a Gaussian, and the background is described by a shape obtained from the inclusive MC sample, in which no peaking backgrounds are observed. We obtain 117.2 ± 13.9 and 155.0±17.2 events for D + s → K 0 e + ν e and D + s → K * 0 e + ν e , respectively, where the uncertainties are statistical only. No peaking backgrounds are observed in K ( * )0 mass sideband.
With the DT technique, the BF measurements are insensitive to the systematic uncertainties of the ST selection. The uncertainties of the e + tracking and PID efficiencies have all been determined to be 1.0% [17], while the uncertainty of the K ( * )0 reconstruction is 1.5 (2.3)%. The uncertainty associated with the MM 2 fit is estimated to be 3.5 (3.8)% by varying the fitting ranges and the signal and background shapes. The uncertainty due to the selection of the γ is estimated to be 2.0% based on selecting the best photon candidate in a control sample of e + e − → D + * s D − s events with two hadronic tags, D + s → K 0 S K + and D − s → K + K − π − . The uncertainties due to the E γmax and M K ( * )0 e + requirements are estimated to be 1.7 (1.7)% and 0.7 (0.9)% by comparing the nominal BF with that measured with alternative requirements. The uncertainty due to the MC signal modeling is estimated to be 0.9 (1.8)% by varying the input FF parameters by ±1σ as determined in this work. We also consider the systematic uncertainties of N ST (0.5%), evaluated by using alternative signal shapes when fitting the M D − s spectra, and of the MC statistics (0.4 [0.3]%). The uncertainty due to different tag dependencies between data and MC simulation is estimated to be 0.8 (0.3)%. Additionally, for D + s → K * 0 e + ν e decay, the systematic uncertainty for the possible S-wave component in Kπ system is estimated to be 6.0% according to Refs. [18,19]. Adding these contributions in quadrature gives total systematic uncertainties of 5.1% and 8.3% for B(D + s → K 0 e + ν e ) and B(D + s → K * 0 e + ν e ), respectively. The D + s → K 0 e + ν e differential decay width with respect to the mass squared (q 2 ) of the e + ν e system is expressed as [20]: In this equation p K 0 is the K 0 momentum in the rest frame of the D + s , G F is the Fermi constant [3], |V cd | is the CKM matrix element, and f K + (q 2 ) is the hadronic FF. To extract the FF parameters, we fit to the differential decay rates  Table II. A least-χ 2 fit is performed accounting for correlations among q 2 bins. We fix the pole mass m pole at the D * + nominal mass [3]. The fits to the differential decay rate and projections of the fits onto f + (q 2 ) for D + s → K 0 e + ν e are shown in Figs. 3(a) and (b), and the FF fit results are summarized in the third column of Table II. The systematic uncertainties in the extracted parameters are estimated as in Ref. [21]. These include the same systematic effects as the BF measurements, along with the D + s -lifetime uncertainty. Using |V cd | = 0.22492 ± 0.00050 [3], we obtain f K + (0) as shown in the last column of Table II.  TABLE II. FF results from fits to D + s → K 0 e + νe, where the first errors are statistical and the second systematic. [22] 0.172 ± 0.010 ± 0.001 0.765 ± 0.044 ± 0.004 Modified pole [22] 0.163 ± 0.017 ± 0.003 0.725 ± 0.076 ± 0.013 z series (2 par.) [23] 0.162 ± 0.019 ± 0.003 0.720 ± 0.084 ± 0.013 The differential decay rate of D + s → K * 0 e + ν e depends on five variables: Kπ mass-squared (m 2 Kπ ), e + ν e mass-squared (q 2 ), the angle between the K + and D + s momenta in the Kπ rest frame (θ K ), the angle between the ν e and D + s momenta in the e + ν e system (θ e ), and the acoplanarity angle between the Kπ and e + ν e decay planes (χ). The differential decay rate can be expressed in terms of three helicity amplitudes [24,25]: is the momentum of the Kπ system in the rest frame of the D + s , and V (q 2 ) and A 1/2 (q 2 ) are the vector and axial FFs, respectively. Because A 1 (q 2 ) is common to all three helicity amplitudes, it is natural to define the FF ratios r V = V (0)/A 1 (0) and r 2 = A 2 (0)/A 1 (0). The A 1/2 (q 2 ) and V (q 2 ) are assumed to have simple pole forms, We perform a five-dimensional maximum likelihood fit in the space of M 2 K + π − , q 2 , cos θ e , cos θ K , and χ for the D + s → K * 0 e + ν e events within −0.15 < MM 2 < 0.15 GeV 2 /c 4 in a similar manner to Refs. [24,25]. We ignored the possible S-wave component in Kπ system due to limited statistics. The projections of the fit onto M 2 K + π − , q 2 , cos θ e , cos θ K , and χ are shown in Figs. 3 (c-g). In this fit, the K * 0 Breit-Wigner function follows Ref. [24], with a mass and width fixed to those reported in Ref. [3]. We obtain r V = 1.67 ± 0.34(stat.) and r 2 = 0.77 ± 0.28(stat.). The fit procedure has been validated by analyzing a large inclusive MC sample, and the pull distribution of each fitted parameter was consistent with a normal distribution. The systematic uncertainties in the FF ratio measurements are estimated by comparing the nominal values with those obtained after varying one source of uncertainty, as described in Ref. [19]. The systematic uncertainties in measuring r V (r 2 ) arise mainly from the uncertainties related to tracking, PID and photon detection (1. . Combining all of these in quadrature, we find the systematic uncertainties in r V and r 2 of D + s → K * 0 e + ν e to be 9.3% and 8.7%, respectively. In summary, using a data sample corresponding to an integrated luminosity of 3.19 fb −1 that was collected at √ s = 4.178 GeV by the BESIII detector, we measure the absolute BFs of the CS SL decays D + s → K 0 e + ν e and D + s → K * 0 e + ν e to be B(D + s → K 0 e + ν e ) = (3.25 ± 0.38(stat.) ± 0.16(syst.)) × 10 −3 and B(D + s → K * 0 e + ν e ) = (2.37 ± 0.26(stat.) ± 0.20(syst.)) × 10 −3 . These are the most precise measurements to date. Theoretical predictions of these BFs range from 2.0 × 10 −3 to 3.9 × 10 −3 [20,[26][27][28][29] for D + s → K 0 e + ν e and 1.7 × 10 −3 to 2.3 × 10 −3 [20,[27][28][29][30] for D + s → K * 0 e + ν e , respectively. Since the predicated BF 2.0 × 10 −3 based on a double-pole model in Ref. [26] is more than 2 standard deviations away from the mean value of our measured B(D + s → K 0 e + ν e ), thus at a confidence level of 95%, our measurement disfavors this prediction.
By analyzing the dynamics of D + s → K 0 e + ν e and D + s → K * 0 e + ν e decays for the first time, we determine the FF of D + s → K 0 e + ν e to be f K + (0) = 0.720 ± 0.084(stat.) ± 0.013(syst.) and the FF ratios of D + s → K * 0 e + ν e to be r V = 1.67±0.34(stat.)±0.16(syst.) and r 2 = 0.77±0.28(stat.)± 0.07(syst.). With the FF of D + → π 0 e + ν e measured by BESIII [21] and that of D + → ρ 0 e + ν e by CLEO [24],  we calculate the ratios of the FFs of D + s → K 0 e + ν e to D + → π 0 e + ν e and D + s → K * 0 e + ν e to D + → ρ 0 e + ν e decays, as shown in Table III, which are consistent with LQCD predictions [4]. These measurements provide a first test of the LQCD prediction that the FFs are insensitive to spectator quarks, which has important implications when considering the corresponding B and B s decays [4,5].