Measurements of the branching fractions for the semi-leptonic decays $D^+_s\to\phi e^{+}\nu_{e}$, $\phi \mu^{+}\nu_{\mu}$, $\eta \mu^{+}\nu_{\mu}$ and $\eta'\mu^{+}\nu_{\mu}$

By analyzing 482 pb$^{-1}$ of $e^+e^-$ collision data collected at the center-of-mass energy $\sqrt s=4.009$ GeV with the BESIII detector, we measure the %absolute branching fractions for the semi-leptonic decays $D_{s}^{+}\to \phi e^{+}\nu_{e}$, $\phi \mu^{+}\nu_{\mu}$, $\eta \mu^{+}\nu_{\mu}$ and $\eta'\mu^{+}\nu_{\mu}$ to be ${\mathcal B}(D_{s}^{+}\to\phi e^{+}\nu_{e})=(2.26\pm0.45\pm0.09)$\%, ${\mathcal B}(D_{s}^{+}\to\phi \mu^{+}\nu_{\mu})=(1.94\pm0.53\pm0.09)$\%, ${\mathcal B}(D_{s}^{+}\to\eta \mu^{+}\nu_{\mu})=(2.42\pm0.46\pm0.11)$\% and ${\mathcal B}(D_{s}^{+}\to\eta'\mu^{+} \nu_{\mu}) = (1.06\pm0.54\pm0.07)$\%, where the first and second uncertainties are statistical and systematic, respectively. The branching fractions for the three semi-muonic decays $D_s^+\to\phi \mu^+\nu_\mu, \eta \mu^+\nu_\mu$ and $\eta' \mu^+\nu_\mu$ are determined for the first time and that of $D^+_s\to \phi e^+\nu_e$ is consistent with the world average value within uncertainties.


I. INTRODUCTION
The semi-leptonic (SL) decays of charmed mesons (D 0(+) and D + s ) provide an ideal window to explore heavy quark decays, as the strong and weak effects can be well separated in theory.The Operator Product Expansion (OPE) model predicts that the partial widths of the inclusive SL decays of D 0(+) and D + s mesons should be equal, up to non-factorizable components [1].However, the CLEO collaboration reported a deviation 18% for inclusive partial widths between D 0(+) and D + s SL decays, which is more than 3 times of the experimental uncertainties [2].Ref. [3] argues that this deviation may be due to that the spectator quark masses m u and m s differ on the scale of the daughter quark mass m s in the Cabibbo-favored SL transition.Therefore, comprehensive or improved measurements of the branching fractions (BFs) for the exclusive SL decays of D 0(+) and D + s will benefit the understanding of this difference.Also, these measurements can serve to verify the theoretical calculations on these decay rates.
In recent years, the D 0(+) SL decays have been well studied with good precision [4].However, the progress in the studies of the D + s SL decays is still relatively slow.Up to now, only D + s semi-electronic decays have been investigated by various experiments [5][6][7][8] and no measurements of D + s semi-muonic decays have been reported.We here report the first measurements of the BFs of the semi-muonic decays D + s → ηµ + ν µ , η ′ µ + ν µ and φµ + ν µ as well as a measurement of the BF of the semi-electronic decay D + s → φe + ν e .Charge-conjugate decays are implied throughout this paper, unless otherwise stated.Among them, the studies of D + s → η (′) µ + ν µ may also shed light on η − η ′ −glueball mixing [9], as their decay rates are expected to be sensitive to the η − η ′ mixing angle [10].
In this paper, all measurements are preformed by analyzing the same data set as used in our previous measurements of D + s → η (′) e + ν e [8].This data set, corresponding to an integrated luminosity of 482 pb −1 [11], was collected at the center-of-mass energy √ s = 4.009 GeV with the BESIII detector.

II. BESIII AND MONTE CARLO
BESIII is a cylindrical spectrometer that is composed of a Helium-gas based main drift chamber (MDC), a plastic scintillator time-of-flight (TOF) system, a CsI (Tl) electromagnetic calorimeter (EMC), a superconducting solenoid providing a 1.0 T magnetic field and a muon counter in the iron flux return yoke of the magnet.The momentum resolution of charged tracks in the MDC is 0.5% at a transverse momentum of 1 GeV/c, and the photon energy resolution is 2.5%(5.0%)at an energy of 1 GeV in the barrel (endcap) of the EMC.More details about BESIII detector are described in Ref. [12].
A GEANT4-based [13] Monte Carlo (MC) simulation software, which includes the geometric description of the BESIII detector and its response, is used to determine detection efficiencies and estimate background contributions.The simulation is implemented by the MC event generator KKMC [14] using EvtGen [15,16], including the beam energy spread and the effects of initial-state radiation (ISR) [17].Finalstate radiation (FSR) of the charged tracks is simulated with the PHOTOS package [18].An inclusive MC sample corresponding to an integrated luminosity of 11 fb −1 is generated at √ s = 4.009 GeV, which includes open charm production, ISR production of low-mass vector charmonium states, continuum light quark production, ψ(4040) decays and QED events.The open charm processes are simulated with cross sections taken from Ref. [19].The known decay modes of the charmonium states are produced by EvtGen with the BFs quoted from the Particle Data Group (PDG) [4], and the unknown decay modes are simulated by the LundCharm generator [20].The SL decays of interest are simulated incorporating with the ISGW2 form-factor model [3].s mesons, we can select the SL decays of interest (called double-tag (DT) events).For a specific ST mode i, the observed yields of ST (N i ST ) and DT (N i DT ) are given by and respectively.Where N D + s D − s is the total number of D + s D − s pairs produced in data, B i ST and B SL are the BFs for the ST mode i and the SL decay of interest, ǫ i ST is the efficiency of reconstructing the ST mode i (called the ST efficiency), and ǫ i DT is the efficiency of simultaneously finding the ST mode i and the SL decay (called the DT efficiency).The efficiency of ST and DT are determined by MC simulation.In this analysis, the ST D − s mesons are reconstructed in ten hadronic decay modes: and η ′ → γρ 0 decays, respectively.The ST modes are selected separately according to their charges.Based on Eq. ( 1) and Eq. ( 2), the BF of the SL decay can be determined according to by considering the multiple ST modes, where N tot DT and N tot ST are the total yields of ST and DT events for multiple ST modes, ǭSL = i (N i ST ǫ i DT /ǫ i ST )/N tot ST is the weighted efficiency of detecting the SL decay for the multi-ST mode according to the yields of different ST modes.
All charged tracks are required to be within a polar-angle (θ) range of | cos θ| < 0.93.The good charged tracks, except for those from K 0 S decays, are required to originate within an interaction region defined by V xy < 1.0 cm and |V z | < 10.0 cm, where V xy and |V z | are the distances of closest approach of the reconstructed track to the interaction point (IP) perpendicular to the beam direction and along the beam direction, respectively.Particle identification (PID) is implemented with the ionization energy loss (dE/dx) measured by the MDC and the time of flight recorded by the TOF.For each charged track, the combined confidence levels for pion and kaon hypotheses (CL π and CL K ) are calculated, respectively.A pion (kaon) is identified by requiring CL π > 0 and CL π > CL K (CL K > 0 and CL K > CL π ).The K 0 S candidates are reconstructed with two opposite charged tracks which satisfy |V z | < 20 cm and are assumed to be pions without PID.A vertex constrained fit is performed to the π + π − combinations, and the fitted track parameters are used in the further analysis.The distance L of the secondary vertex to the IP is also required to be positive with respect to the K 0 S flight direction.K 0 S candidates are required to have π + π − invariant mass within (0.485, 0.511) GeV/c 2 .Photon candidates are chosen from isolated clusters in the EMC.The deposited energy of a neutral cluster is required to be larger than 25 MeV in the barrel region (| cos θ| < 0.80) or 50 MeV in the end-cap region (0.86 < | cos θ| < 0.92).The angle between the photon candidate and the nearest charged track should be larger than 10 • .To suppress electronic noise and energy deposits unrelated to the events, the difference between the EMC time and the event start time is required to be within (0, 700) ns.The π 0 and η candidates are reconstructed with γγ pair with invariant mass within (0.115, 0.150) and (0.510, 0.570) GeV/c 2 .To improve momentum resolution, a kinematic fit is performed to constrain the γγ invariant mass to the nominal π 0 or η mass, and the fitted momenta of π 0 or η are used in the further analysis.To select candidates of φ, ρ − , η ′ π + π − η and η ′ γρ 0 mesons, the invariant masses of K + K − , π − π 0 , π + π − η and γρ 0 are required to be within (1.005, 1.040), (0.570, 0.970), (0.943, 0.973) and (0.932, 0.980) GeV/c 2 , respectively.For η ′ γρ 0 candidate, the π + π − invariant mass is additionally required to fall in (0.570, 0.970) GeV/c 2 to reduce combinatorial backgrounds.
The ST D − s meson is identified using the energy difference ∆E ≡ E D − s − E beam and the beam-constrained mass , where E beam is the beam energy, E D − s and | p D − s | are the total energy and momentum of the ST D − s candidate in the e + e − center-of-mass frame.For each ST mode, only the one with the minimum |∆E| is retained if there are multiple combinations in an event.To suppress combinatorial backgrounds, modes dependent ∆E requirements, which correspond to (−3.0, +3.0) times of the resolution around the fitted ∆E peak, are imposed on the ST D − s candidates.Figure 1 shows the M BC distributions of D − s candidates for individual ST mode.To obtain the ST yield (N i ST ), we perform a maximum likelihood fit on these M BC distributions.In the fits, we use the MC-simulated signal shape convoluted with a Gaussian function to model the D − s signals and an ARGUS function [21] to describe the combinatorial backgrounds.The events with M BC within a mass window of (−4.0, +5.0) times of the resolution around the D − s nominal mass [4] (called M BC signal region) are kept for further analysis.For each ST mode, the ST yield is obtained by integrating the D − s signal over the corresponding M BC signal region.The ST efficiency for the individual mode (ǫ i ST ) is determined by analyzing the inclusive MC sample.Table I summarizes the requirements on ∆E and M BC , the ST yields in data and the ST efficiencies.The total ST yield (N tot ST ) is 13092 ± 247.The SL decays D + s → φe + ν e , φµ + ν µ , ηµ + ν µ and η ′ µ + ν µ are selected recoiling against the ST D − s mesons.The charge of the electron (muon) candidate is required to be opposite to that of the ST D − s meson.For electron (muon) PID, the dE/dx, TOF and EMC information is used to form the combined confidence levels for electron, muon, pion and kaon hypotheses (CL e , CL µ , CL π and CL K ).The electron candidates should satisfy CL e /(CL e + CL π + CL K ) > 0.8 and CL e > 0.001, while the muon candidates are required CL µ > CL e , CL µ > CL K and CL µ > 0.001.It is required that there is no extra charged track except for those used in the DT event selection.For D + s → η (′) µ + ν µ decays, the energy deposited in the EMC by muon is required to be less than 300 MeV and the maximum energy (E max extraγ ) of the extra photons, which are not used in the DT event selection, is required to be less than 200 MeV.
The undetected neutrino in the SL decay is inferred by a kinematic variable U miss ≡ E miss − | p miss |, where E miss ≡ √ s − j E j is the missing energy and p miss ≡ − j p j is the missing momentum.Here, the index j runs over all the particles used in the DT event selection, E j and p j are the energy and momentum of the j-th particle in the e + e − rest frame.The U miss distribution of the SL decays candidates is expected to peak near zero.To further suppress backgrounds from the hadronic decays D + s → φ(η, η ′ )π + and φ(η, η ′ )π + π 0 for semi-muonic decays, we define a variable δE = E beam − (E φ(η,η ′ ) + E µ + as π + + E νµ as π 0 ), where E φ(η,η ′ ) is the energy of φ(η, η ′ ) candidate, E µ + as π + is the energy of µ + candidate by assuming it is pion, and E νµ as π 0 is the energy of missing particle by assuming to be π 0 (calculated with p miss ).The DT candidate events are required to have δE within (−0.080, −0.010), (−0.100, 0), (−0.070, −0.015) and (−0.060, −0.015) GeV for D + s → φµ + ν µ , ηµ + ν µ , η ′ ηπ + π − µ + ν µ and η ′ γρ 0 µ + ν µ , respectively.Figure 2 shows the U miss distributions of the accepted candidate events for the SL decays in data.The U miss signal  region is defined as (−0.10, 0.10) GeV, in which we observe 28.0 ± 5.3, 34.0 ± 5.8, 64.0 ± 8.0 and 28.0 ± 5.3 candidate events for D + s → φe + ν e , φµ + ν µ , ηµ + ν µ , and η ′ ηπ + π − andγρ 0 µ + ν µ , respectively.Some background events may also survive the selection criteria of the SL decays of interest.The backgrounds can be classed into two categories.Those background events, in which the ST D − s meson is reconstructed correctly but the SL decay is mis-identified, are defined as 'real-D − s ' background.The other background events, in which the ST D − s meson is reconstructed incorrectly, are called as 'non-D − s ' background.The number of 'real-D − s ' background events is estimated by analyzing the inclusive MC sample.While the 'non-D − s ' background yield is evaluated by using the events of data within the M BC sideband region, which is defined to be (1.920,1.950) and (1.990, 2.000) GeV/c 2 on the M BC distribution.The background yield in the M BC sideband region is then scaled by the ratio of the background integral areas between the M BC signal and sideband regions.) as well as the weighted efficiencies of detecting the SL decays according to the ST yields of data (ǭ SL ) are summarized in Table II, where the efficiencies ǭSL do not include the BFs of φ, η and η ′ in the SL decays.So, the BFs for  the SL decays are determined by where B sub denotes the BFs for the daughter particles φ, η and η ′ quoted from PDG [4].Inserting the numbers of ST , ǭSL and B sub in Eq. ( 4), we obtain the BFs for D + s → φe + ν e , φµ + ν µ , ηµ + ν µ and η ′ µ + ν µ , respectively.These results are summarized in Table II.

IV. SYSTEMATIC UNCERTAINTIES
In the BF measurements using DT method, the systematic uncertainties arising from the ST selection are almost canceled.Main systematic uncertainties in the measurements for BFs of SL decays are discussed below.
a. ST yield.The uncertainty of the total ST yield is estimated to be 1.8% by comparing the integrated and counted ST yields (calculated by subtracting the background yields from total events without performing a fit) in the M BC signal region.
b. Tracking and PID.The uncertainties in the tracking and PID for charged kaon and pion are investigated with the control sample of DT hadronic D D events and are assigned to be 1.0% and 1.0% per track individually.The efficiencies of the tracking and PID for electron and muon are studied by varying with the polar angle cos θ and momentum with the control samples e + e − → γe + e − and e + e − → γµ + µ − events, respectively.These efficiencies are weighted according to cos θ and momentum distributions of the electron and muon in the SL decays.The resultant differences of the two-dimensional weighted tracking and PID efficiencies for electron and muon between data and MC simulation are regarded as the relevant uncertainties.
c. E max extraγ requirement.The efficiency of E max extraγ requirement is investigated with fully reconstructed DT hadronic decays ψ(4040) → D * D + c.c..The difference of the efficiencies with the requirement of E max extraγ < 200 MeV between data and MC simulation is found to be (1.9 ± 0.6)%.To be conservative, we assign 2.5% to be the associated systematic uncertainty.d. φ (η, η ′ ) reconstruction.The reconstruction efficiencies for the φ, η and η ′ candidates, which include the mass window requirement and photon selection, are estimated with the control samples of D + → φπ + , D 0 → K 0 S η, D 0 → K 0 S η ′ π + π − η and K 0 S η ′ γρ 0 , respectively.The differences of efficiencies between data and MC simulation are estimated to be 0.4%, 2.3%, 2.5% and 2.8% for φ, η, η ′ π + π − η and η ′ γρ 0 , respectively, which are assigned as the associated uncertainties.
e. δE requirement.The uncertainties from δE requirements are estimated by varying the δE requirements by ±10%.
The individual systematic uncertainties discussed above are summarized in Table III and the total systematic uncertainties are the quadratic sum of the individual ones.The sources tagged with ' c ' are common systematic uncertainties between the two η ′ decay modes and the other sources are independent.Finally, we assign 7.1% as the total systematic uncertainty for D + s → η ′ µ + ν µ .

V. SUMMARY
In summary, by analyzing the 482 pb −1 data collected at √ s = 4.009 GeV with the BESIII detector, we determine the BFs for the SL decays D + s → φe + ν e , φµ + ν µ , ηµ + ν µ and η ′ µ + ν µ .Table IV presents the comparisons of the measured BFs with the world average values.The BFs of the semi-muonic decays D + s → φµ + ν µ , ηµ + ν µ and η ′ µ + ν µ are determined for the first time and are compatible with those of the corresponding semi-electronic decays [4].The BF of D + s → φe + ν e agrees with the world average value [4] within uncertainties.And, the results are consistent with previous experimental measurements and support that the ratio of D + s and D 0(+) differs from unity, there is an indication of difference between D 0(+) and D + s meson SL decay widths [2].Combining the previous BESIII measurements for semi-electronic decays [8] and this work, we calculate the ratios between the semi-electronic and semi-muonic decays, to be B(D + s → φµ + ν µ )/B(D + s → φe + ν e ) = 0.86 ± 0.29, B(D + s → ηµ + ν µ )/B(D + s → ηe + ν e ) = 1.05 ± 0.24 and B(D + s → ηµ + ν µ )/B(D + s → ηe + ν e ) = 1.14±0.68individually, where most of systematic uncertainties are canceled.The ratios are consistent with 1 within the uncertainties, and no obvious lepton universality violation is observed.Moreover, the ratio of B(D + s → ηµ + ν µ ) over B(D + s → η ′ µ + ν µ ) is calculated to be 0.44 ± 0.23, which is in agreement with those of previous measurements [5,7,8,23] within uncertainties and provides complementary data to probe the η − η ′ −glueball mixing.

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

j
Also at Key Laboratory for Particle Physics, Astrophysics and Cosmology, Ministry of Education; Shanghai Key Laboratory for Particle Physics and Cosmology; Institute of Nuclear and Particle Physics, Shanghai 200240, People's Republic of China k Government College Women University, Sialkot -51310.Punjab, Pakistan.
III. DATA ANALYSIS In e + e − collisions at √ s = 4.009 GeV, D + s and D − s mesons can only be produced jointly without additional hadrons.Thus in an event where a D − s meson (called singletag (ST) D − s meson) is fully reconstructed, the presence of a D + s meson is guaranteed.In the systems recoiling against the ST D −

FIG. 1 :
FIG.1:(Color online) Fits to the MBC distributions of the ST D − s decay modes.The dots with error bar are data, the red solid curves represent the total fits, and the blue dashed curves describe the fitted backgrounds.
The DT yields observed in data (N obs DT ), the expected number of 'real-D − s ' and 'non-D − s ' background (N bkg real−D − s and N bkg non−D − s

FIG. 2 :
FIG. 2: Distributions of Umiss of the candidate events for D + s → (a) φe + νe, (b) φµ + νµ, (c) ηµ + νµ and (d) η ′ µ + νµ where the pair of arrows represent the signal region.The dots with error bar are data, the red histograms are inclusive MC, and the yellow and oblique-line hatched histograms represent the scaled 'real-D − s ' and 'non-D − s ' backgrounds.

TABLE I :
Summary of the requirements on ∆E and MBC, the ST yields in data (NST) and the ST efficiencies (ǫST), which do not include the BFs for daughter particles π 0 , K 0 S , φ, η and η ′ for the individual ST mode.The uncertainties are statistical only.