Prospects of CKM elements $|V_{cs}|$ and decay constant $f_{D_{s}^+}$ in $D_s^+\to\mu^+\nu_\mu$ decay at STCF

We report a feasibility study of pure leptonic decay $D_s^+\to\mu^+\nu_\mu$ by using a fast simulation software package at STCF. With an expected luminosity of $1~\mathrm{ab}^{-1}$ collected at STCF at a center-of-mass energy of 4.009 GeV, the statistical sensitivity of the branching fraction is determined to be 0.3\%. Combining this result with the $c\rightarrow s$ quark mixing matrix element $|V_{cs}|$ determined from the current global Standard Model fit, the statistical sensitivity of $D_s^+$ decay constant, $f_{D_s^+}$, is estimated to be 0.2\%. Alternatively, combining the current results of $f_{D_s^+}$ calculated by lattice QCD, the statistical sensitivity of $|V_{cs}|$ is determined to be 0.2\%, which helps probe possible new physics beyond. The unprecedented precision to be achieved at STCF will provide a precise calibration of QCD and rigorous test of Standard Model.


I. INTRODUCTION
The proposed Super Tau-Charm Facility (STCF) [1] in China is a symmetric electron-positron collider designed to provide e + e − annihilation at center-of-mass (c.m.) energies √ s from 2.0 to 7.0 GeV. The peak luminosity is expected to be 0.5 × 10 35 cm −2 s −1 at √ s = 4.0 GeV and it will accumulate an integrated luminosity (L) of more than 1 ab −1 data each year. By operating at √ s = 4.009 GeV, the STCF will produce 2.0×10 8 D + s D − s with one year's data taking, which can be used to study the purely leptonic, semileptonic and hadronic decays of D + s with an unprecedent precision. Among these, the purely leptonic decay D + s → ℓ + ν ℓ (ℓ = e, µ or τ ) offers a unique window into both strong and weak effects in the charm sector. In the Standard Model (SM), the partial width of the decay D + s → ℓ + ν ℓ can be written as [2] Γ D + s →ℓ + ν ℓ = where G F is the Fermi coupling constant, |V cs | is the c → s Cabibbo-Kobayashi-Maskawa (CKM) matrix element, f D + s is the D + s decay constant that parameterizes the effect of the strong interaction, m ℓ and m D + s are the masses of lepton and D + s , respectively. The determination of Γ D + s →ℓ + ν ℓ can directly measure the product value of f D + s |V cs | since all other variables are known with high precision [3]. One can either extract |V cs | by using the predicted value of f D + s from lattice QCD (LQCD), or obtain f D + s by using the averaged experimental value of |V cs |.
Precise measurements of f D + s [4][5][6] and |V cs | are essential to probe new physics beyond the SM. Currently, the averaged f D + s from various experiments indicates a 1.5σ [7] difference from LQCD calculation [4], while the latter gives a negligible uncertainty comparing to the former. Besides, there are two standard deviations for the |V cs | extracted in D + s → l + ν l [7] and D → Klν l [7], which challenges the universality for the CKM elements. The up-to-date results of |V cs | and f D + s are still limited by statistics uncertainty in the measurment of D + s → ℓ + ν ℓ [8]. Future precise measurement of D + s → µ + ν µ is critical to calibrate various theoretical calculations of f D + s and test the unitarity of the CKM matrix. The SM predicts the ratio of decay widths for D + s → τ + ν τ and D + s → µ + ν µ to be 9.75, with negligible uncertainty. The lepton flavor universality (LFU) could be violated with some new physics mechanisms, such as a two-Higgs-doublet model with the mediation of charged Higgs bosons [9,10] or a Seesaw mechanism due to lepton mixing with Majorana neutrinos [11]. Using the most recent experimental results, the ratio Γ D + s →τ + ντ /Γ D + s →µ + νµ is obtained to be 9.98±0.52 [3], which is consistent with the SM prediction within uncertainty. However, high precise measurement of D + s → ℓ + ν ℓ decay is desirable to test LFU and other physics mechanisms beyond the SM.
In this paper, we present a feasibility study of D + s → µ + ν µ decay and estimate the sensitivity of various parameters at STCF [1], where D + s is from e + e − → D + s D − s at √ s = 4.01 GeV with a production cross section of σ 4.01 ≈ 0.2 nb. Though the production cross section of D + s D − s is higher, e.g. D + s D − s produced via e + e − → D + s D * − s + c.c with cross section σ 4.18 ≈ 0.9 nb at √ s = 4.18 GeV, the pair production of D + s D − s without additional particles at 4.009 GeV helps to reconstruct signal with better purity and free of additional systematic uncertainties coming from γ or π 0 reconstruction in D * − s decays. This paper is organized as follows. In Sec. II, the detector concept for STCF is introduced as well as the Monte Carlo (MC) samples used for this study. Section III is analysis of D s candidates. Section IV is optimization of detector response, and Sec. V is the results and discussion.

II. STCF DETECTOR AND MC SIMULATION
The STCF detector in design is a general purpose detector for e + e − collider. It includes a tracking system composed of inner and outer trackers, a particle identification (PID) system with 3σ charged K/π separation power up to 2 GeV/c, an electromagnetic calorimeter (EMC) with an excellent energy resolution and a good position resolution, a super conducting solenoid and a muon detector (MUD) that provides good π/µ separation. The detailed conceptual design for each subdetector can be found in Ref. [12].
Currently, the STCF detector and the corresponding offline software system are in the research and development [13]. A fast simulation software at STCF is therefore developed to access the physics reaches [12], which takes the most common event generator as input to perform a realistic simulation. It incorporates the effects from tracking efficiency of charged particles and their momentum resolution, the efficiency of PID, the detection efficiency of photon and its energy and position resolution, as well as kinematic fits. The fast simulation also provides flexibly interface for adjusting performance of each sub-system, which can be used to optimize the detector design according to physical requirements. The process D + s → µ + ν µ analysed here also serves as a benchmark process for the optimization of detector response, e.g. tracking efficiency, π/µ separation.
A pseudo-data sample, corresponding to an integrated luminosity of 0.1 ab −1 , is produced at √ s = 4.009 GeV, which includes all open charm processes, initial state radiation (ISR) production of the ψ(3770), ψ(3686) and J/ψ, and 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 [14]. The effects of ISR [15] and final state radiation (FSR) [16] are considered. The decay modes with known branching fraction (BF) are generated using evtgen [17] and the other modes are generated using lundcharm [18]. The passage of the particles through the detector is simulated by the fast simulation software [12].

III. ANALYSIS OF DS CANDIDATES
A double-tag technique is employed to measure the absolute BF of signal process D + s → µ + ν µ . In an event where a D − s meson (called the single-tag (ST) D − s meson) is fully reconstructed, the presence of a D + s meson is guaranteed. In the systems recoiling against the ST D − s mesons, we can select the leptonic decays of D + s → µ + ν µ (called the double-tag (DT) events).
In e + e − collision at √ s = 4.009 GeV, D + s mesons are produced from the process e + e − → D + s D − s . Using this threshold production characteristic, we can measure the absolute BF for D + s decays with a DT method. In this analysis, the ST D − s mesons are reconstructed from 14 hadronic decay modes, where the subscripts of η (′) represent the decay modes used to reconstruct η (′) . Throughout this paper, the charge conjugation is always implied.
Candidate charged tracks are selected when they pass the vertex and acceptance requirements in fast simulation. The K 0 S candidates are reconstructed from pairs of oppositely charged tracks, which satisfy a vertexconstrained fit to a common point. The two charged tracks with minimum χ 2 of vertex fit are assumed to be pions produced from K 0 S . The K 0 S is required to have an invariant mass in range 0.485 < M π + π − < 0.512 GeV/c 2 . Furthermore, the decay length of the reconstructed K 0 S is required to be larger than 2σ of the vertex resolution away from the interaction point. The π 0 and η mesons are reconstructed via γγ decays. For photon candidates, they are also required to pass the criteria for neutral showers in fast simulation. The γγ combinations with invariant masses M γγ ∈ (0.115, 0.150) and (0.500, 0.570) GeV/c 2 are regarded as π 0 and η mesons, respectively. A kinematic fit is performed to constrain M γγ to the π 0 or η nominal mass. The η candidates for the ηπ − ST channel are also reconstructed via π 0 π + π − candidates with its invariant mass within (0.530, 0.570) GeV/c 2 . The η ′ mesons are reconstructed via two decay modes, ηπ + π − and γρ 0 , whose invariant masses are required to be within (0.946, 0.970) and (0.940, 0.976) GeV/c 2 , respectively. In addition, the minimum energy of the γ from η ′ → γρ 0 decays must be greater than 0.1 GeV. The ρ 0 and ρ + mesons are reconstructed from π + π − and π + π 0 candidates, whose invariant masses are required to be larger than 0.5 GeV/c 2 and within (0.670, 0.870) GeV/c 2 , respectively. For π + π − π − and K − π + π − tags, the domi- are rejected by requiring the invariant mass of any π + π − combination to be more than 0.03 GeV/c 2 away from the nominal K 0 S mass [3]. Two kinematic variables (∆E, M BC ) reflecting energy and momentum conservation are used to identify the tagged D − s candidates. First, we calculate the energy difference where E D − s is the reconstructed energy of a tagged D − s meson and E beam is the beam energy. Correctly reconstructed signal events peak around zero in the ∆E distribution. To improve the signal purity, requirements of ∆E are applied, corresponding to ±3σ ∆E where σ ∆E is the resolution of ∆E for each tag mode. If there are multiple D − s candidates per tag mode, the one with minimum |∆E| is kept for further analysis. The second variable is the beam-energy constrained mass where − → p D − s is the three-momentum of the tagged D − s candidate. Figure 1 shows the M BC distributions for pseudo-data. The ST yields are obtained by fitting the M BC distributions where a MC-determined signal shape is used to model the signal and an ARGUS [19] function is for background. To select the signal process with a high purity, a mass window is required on M BC within ±3σ MBC , where σ MBC is resolution of M BC determined by fitting with a double-Gaussian function. The D + s → µ + ν µ candidate events are selected in the recoil side of the tagged D − s . We require that there is only one candidate charged track in the remaining particles whose charge is opposite to the tagged D − s . The charged track is identified as a muon candidate after passing the corresponding requirements in fast simulation [12]. To suppress the backgrounds with extra photon(s), the max-imum energy of the unused showers (E max extra γ ) is required to be less than 0.4 GeV.
The background events survived from above selection criteria can be categorized into two types. The first type, noted as BKGI, contains a correctly reconstructed D − s but the signal side is misreconstructed from D + s → τ + ν τ and other D + s decays. The contribution of BKGI is estimated from exclusive MC samples and the normalized number of events is fixed in the fit. The second type, noted as BKGII, contains the non-D + s background, which is expected to be a smooth distribution under the D − To characterize the signal of D + s → µ + ν µ , the missing mass squared (MM 2 ) is defined as where E µ + and − → p µ + are the energy and momentum of the muon candidate, respectively. The signal yield is extracted by fitting the combined MM 2 distribution from all 14 tag modes, where a shape extracted from signal MC sample is used to describe the signal and a firstorder Chebychev polynomial is used to describe the background, as shown in Fig. 2.
The BF of the D + s → µ + ν µ is calculated by where N sig is the number of the signal events determined by a fit to the MM 2 spectrum, and N tag is the number of events for all ST modes by fits to the M BC . The averaged detection efficiency for D + s → µ + ν µ can be expressed as where N i tag is the number of events for ST mode i, ǫ i tag,sig is the efficiency of detecting both the ST mode i and the pure leptonic decays, and ǫ i tag is the efficiency of detecting the ST mode i. The efficiencies of ST modes are determined with an independent generic MC sample, and the efficiencies of DT modes are determined with the signal MC sample of e + e − → D + s D − s , where D − s → 14 tag modes and D + s → µ + ν µ . With 0.1 ab −1 pesudo-data, the number of ST events for D − s to 14 decay modes is determined to be 3452605 ± 21268, and the number of DT events for D + s → µ + ν µ is 14687 ± 142. The averaged efficiencyǭ sig of signal process is calculated to be (75.78 ± 0.07)% by combining 14 tag modes. The corresponding BF of D + s → µ + ν µ is calculated to be (5.61 ± 0.05) × 10 −3 . The uncertainties are statistical only. The calculated BF agrees well with the input value. We can easily prospect the statistical sensitivity for the BF of D + s → µ + ν µ at STCF with 1 ab −1 data as it is proportional to 1/ √ L, to be B D + s →µ + νµ = (5.610 ± 0.016) × 10 −3 , where the statistical uncertainty is 0.3%.
Since a full systematic study requires both experimental data and MC, we are limited in our ability to estimate every possible source. A more precise estimation of systematic uncertainty will not be feasible until the design and construction of the detector is completed. Therefore, a rough systematic uncertainty is estimated by referring to measurement at BESIII [20], which includes reducible systematic uncertainties, σ red. , that can be scaled according to luminosity, and the irreducible systematic uncertainties, σ irred. , which mainly comes from theoretical input. It also includes predictably optimized systematic uncertainties, σ pre. , which mainly be optimized by control samples and corrections. The reducible systematic uncertainties include the tracking and PID of µ detection, other selection criteria such as extra energy requirement and D − s candidate ST by background function, as well as the uncertainty from the fitting procedure of MM 2 from background. The irreducible systematic uncertainties come from the effect of radiative correction.
The predictably optimized systematic uncertainties come from fit range, bin size, signal shape of fitting procedure of M BC and MM 2 , final state radiation (FSR) and tag bias.

IV. OPTIMIZATION OF DETECTOR RESPONSE
In the results presented above, a series of optimizations on detector responses have been performed, including efficiencies of charged particles and photons, momentum resolution of charged tracks, energy/position resolution of photons, and PID efficiencies. Following we will introduce the details of these optimizations one by one.
a. Tracking efficiency of charged particles: The response of tracking efficiency in fast simulation is characterized by its transverse momentum p T and polar angle cos θ. For high-momentum tracks, e.g. p T > 0.4 GeV/c of charged pions, the tracking efficiency within acceptance is over 99%. For low-momentum tracks, e.g. p T = 0.1 GeV/c of charged pions, the tracking efficiency is low due to various effects such as electromagnetic multiple scattering, electric field leakage, ionization energy loss etc.. Figure 3 shows the momentum distri- the tracker system. The spatial resolution of the charged track flight trajectory, determined by the accuracy of the track position, is studied for its influence on momentum resolution by applying a set of σ xy from 65 µm to 130 µm and σ z from 1240 µm to 2480 µm proportionally. Since the resolutions of kinematic variable ∆E and M BC are affected by the momentum resolutions of charged tracks, different mass windows are then applied to obtain the detection efficiencies of 14 tag modes. The results are shown in Fig. 4 where weak dependence from different spatial resolution is observed.
c. Detection efficiency of photons: The energy of photons from D − s decay ranges from less than 0.1 to 1.2 GeV as shown in Fig. 5. The detection efficiency of photons is studied with energy to be 200 MeV. Figure 4 shows the detection efficiency of 14 tag modes with the variation of detection efficiency, where the optimized point is found to be 97.32% for photons with energy of 200 MeV.
d. Energy/Position resolution of photon: The energy and position resolutions of photons are two key parameters for photon detection. Various resolutions of photons, i.e. energy resolution from 2.5% to 1.25% and position resolution from 6 mm to 3 mm at 1 GeV, have been tested for 14 tag modes of D − s decay. It is found that the detection efficiencies of decay modes containing photons are improved, especially for better energy resolution, as shown in Fig. 4.
The influence of energy and position resolutions of photons are also studied with the process e + e − → D + s D * − s + c.c, where the energy of photon in D * − s → γ/π 0 D − s locates within 200 MeV. Two key parameters of this process are discussed. One is the mass difference of reconstructed D * − s and D − s , ∆M , whose resolution is determined by resolution of photon energy. The other one is the π 0 /γ contamination in reconstruction of D * − s → γ/π 0 D − s . After a series of test with different energy/position resolutions, it is found that a better energy resolution will improve the resolution of ∆M as shown in Fig. 6. No obvious improvement is observed for the resolution of ∆M in the variation of position resolution, and the π 0 /γ contamination rate in different energy or position resolutions of photon keeps unchanged, to be about 28.1%. Although the volume of π 0 from D * − s is pretty small, it does make an impact.
e. π/K identification: The identifications of π and K are essential for the charm physics at STCF. Since the momenta of π/K are relatively low in this analysis, they can be mostly identified by the characteristic ionization energy loss (dE/dx) in the tracker system. Simulation indicates that with a dE/dx resolution of 6%, the π/K can be well separated when p < 0.8 GeV/c, which can meet the requirement for π/K separation in this analysis.
f. µ identification: The momentum of µ in this analysis is shown in Fig. 3, where the MUD is expected to provide a high identification efficiency for muon and low π/µ mis-identification rate. Reference [21] describes the details of the baseline design of MUD at STCF. With the performance of MUD provided in Ref. [21], three π/µ mis-identification rates are tested, to be 1%, 1.6% and 3%, corresponding to the identification efficiencies of muon to be 85%, 92% and 97% at p µ = 1 GeV/c. The optimized result is achieved at 3% of π/µ mis-identification rate.
From the discussion above, we use the following detector responses optimized from this analysis while others are kept the same in the fast simulation. The optimized responses include a tracking efficiency for low momentum charged particles, to be 72.19% at p T = 0.1 GeV/c, a detection efficiency of photons, to be 97.32% at 0.2 GeV, a π/µ mis-identification rate of 3% with the MUD performance provided from Ref. [21]. Comparing to the default response provided by fast simulation, the detection efficiency for ST is increased by a factor that in the range between 1.1 and 1.2, depending on the tag modes, and the efficiency for selecting D + s → µ + ν ν is increased by a factor of 1.3.

V. RESULTS AND DISCUSSION
With the expect sensitivity of B D + s →µ + νµ = (5.610 ± 0.016) × 10 −3 at STCF obtained in this analysis, and the world average values of G F , m µ , m D + s and the lifetime of D + s [3] as listed in Table 1 can be obtained according to Eq. (1) f D + s |V cs | = 248.9 ± 0.4 stat. MeV. Taking the CKM matrix element |V cs | = 0.97320 ± 0.00011 from the global fit in the SM [3] or the averaged decay constant f D + s = 249.9 ± 0.5 MeV of recent LQCD calculations [5,22] as input, f D + s and |V cs | can be determined separately  It is worth mentioning that the f D + s and |V cs | are from recent LQCD calculations, whose uncertainty will be reduced in the future, therefore the f D + s and |V cs | determined at STCF will also be reduced in the future. As shown in Figs. 7 and 8, the uncertainty of f D + s we determined is smaller than that from LQCD calculation, which calls for further improved LQCD calculation. The expected uncertainty of |V cs | is close to that from LQCD calculation. The accuracy of LFU test can be improved obviously experimentally, which makes it promising to search for the new physics beyond the SM.

VI. ACKNOWLEDGMENTS
The authors are grateful to the software group of STCF and the physics group of STCF for the profitable discussions. We express our gratitude to the supercomputing center of USTC and Hefei Comprehensive National Science Center for their strong support. This work is