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

TL;DR

This study measures the decay rates and form factors of $D_s^+$ meson decays to $ a^{("}e^{+} u_e$ and determines the $ a- a'$ mixing angle, providing new insights into charm meson decay dynamics.

Contribution

First measurements of the decay dynamics of $D_s^+ o a^{("}e^{+} u_e$ decays and the $ a- a'$ mixing angle in the quark flavor basis.

Findings

Branching fractions: $ a e^{+} u_e$ = (2.323 ± 0.063 ± 0.063)% and $ a' e^{+} u_e$ = (0.824 ± 0.073 ± 0.027)%

Mixing angle $ a_{ m P} = (40.1 ± 2.1 ± 0.7)^ ext{o}$

Product of form factors and CKM matrix element: $f_+^{ a}(0)|V_{cs}| = 0.4455 ± 0.0053 ± 0.0044$ and $f_+^{ a'}(0)|V_{cs}| = 0.477 ± 0.049 ± 0.011$

Abstract

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 $f^{\eta^{\prime}}_+(0)|V_{cs}| = 0.477\pm0.049_{\rm stat}\pm0.011_{\rm syst}$.

Tables

Table 1. Table 1: Efficiencies ( ϵ γ ​ ( π 0 ) ​ SL subscript italic-ϵ 𝛾 superscript 𝜋 0 SL \epsilon_{\rm\gamma(\pi^{0}){\rm SL}} ), signal yields ( N DT tot superscript subscript 𝑁 DT tot N_{\rm DT}^{\rm tot} ), and the obtained BFs. Uncertainties on the least significant digits are shown in parentheses, where the first (second) uncertainties are statistical (systematic). The efficiencies do not include the BFs of η ( ′ ) superscript 𝜂 ′ \eta^{(\prime)} subdecays.

Decay	${\eta }^{(\prime )}$ decay	${ϵ}_{\gamma \left({\pi }^0\right)\mathrm{SL}}$ (%)	$N_{\mathrm{DT}}^{\mathrm{tot}}$	${ℬ}_{\mathrm{SL}}$ (%)
$\eta e^{+}{\nu }_e$	$\gamma \gamma$	41.11(27)	1834(47)	2.323(63)(63)
	${\pi }^0{\pi }^{+}{\pi }^{-}$	16.06(31)
${\eta }^{\prime }{e}^{+}{\nu }_e$	$\eta {\pi }^{+}{\pi }^{-}$	14.07(10)	261(22)	0.824(73)(27)
	$\gamma {\rho }^0$	18.98(10)

Table 2. Table 2: Results of the fits to Δ ​ Γ msr i Δ subscript superscript Γ 𝑖 msr \Delta\Gamma^{i}_{\rm msr} . Uncertainties on the least significant digits are shown in parentheses, where the first (second) uncertainties are statistical (systematic). N dof subscript 𝑁 dof N_{\rm dof} is the number of degrees of freedom.

Case	Simple pole			Modified pole			Series 2 Par.
	$f_{+}^{{\eta }^{(\prime )}}\left(0\right)\left|V_{cs}\right|$	$M_{\mathrm{pole}}$	${\chi }^2/N_{\mathrm{dof}}$	$f_{+}^{{\eta }^{(\prime )}}\left(0\right)\left|V_{cs}\right|$	$\alpha$	${\chi }^2/N_{d.o.f}$	$f_{+}^{{\eta }^{(\prime )}}\left(0\right)\left|V_{cs}\right|$	$r_1$	${\chi }^2/N_{d.o.f}$
$\eta e^{+}{\nu }_e$	$0.4505\left(45\right)\left(31\right)$	$3.759\left(84\right)\left(45\right)$	12.2/14	$0.4457\left(46\right)\left(34\right)$	$0.304\left(44\right)\left(22\right)$	11.4/14	$0.4465\left(51\right)\left(35\right)$	$-2.25\left(23\right)\left(11\right)$	11.5/14
${\eta }^{\prime }{e}^{+}{\nu }_e$	$0.483\left(42\right)\left(10\right)$	$1.88\left(60\right)\left(08\right)$	1.8/4	$0.481\left(44\right)\left(10\right)$	$1.62\left(91\right)\left(13\right)$	1.8/4	$0.477\left(49\right)\left(11\right)$	$-13.1\left(76\right)\left(10\right)$	1.9/4

Table 3. Table 3: Comparison of the measured f + η ( ′ ) ​ ( 0 ) superscript subscript 𝑓 superscript 𝜂 ′ 0 f_{+}^{\eta^{(\prime)}}(0) with the theoretical calculations. Errors on the least significant digits are shown in parentheses. For the LQCD model, the errors are statistical only, while A(B) assume M π = 470 ​ ( 370 ) subscript 𝑀 𝜋 470 370 M_{\pi}=470(370)  MeV.

	CLFQM Verma:2011yw	CQM Melikhov:2000yu	CCQM Soni:2018adu	3PSR Colangelo:2001cv	LCSR Azizi:2010zj	LCSR Offen:2013nma	LQCDA Bali:2014pva	LQCDB Bali:2014pva	LCSR Duplancic:2015zna	BESIII
$f_{+}^{\eta }\left(0\right)$	0.76	0.78	0.78(12)	0.50(4)	0.45(15)	0.432(33)	0.564(11)	0.542(13)	0.495(30)	0.4576(70)
$f_{+}^{{\eta }^{\prime }}\left(0\right)$	-	0.78	0.73(11)	-	0.55(18)	0.520(88)	0.437(18)	0.404(25)	0.558(47)	0.490(51)

Equations

$${\mathcal{B}}_{\mathrm{SL}}=N_{\mathrm{DT}}^{\mathrm{tot}}/(N_{\mathrm{ST}}^{\mathrm{tot}}\times {ϵ}_{\gamma ({\pi }^0)\mathrm{SL}}),$$

$$\frac{d\Gamma (D_s^{+}\to {\eta }^{(\prime )}{e}^{+}\nu )}{d{q}^2}=\frac{G_F^2|V_{cs}{|}^2}{24{\pi }^3}|f_{+}^{{\eta }^{(\prime )}}(q^2){|}^2|p_{{\eta }^{(\prime )}}{|}^3,$$

$$f_{+}(q^2)=\frac{f_{+}(0)}{(1-\frac{q^2}{M_{\mathrm{pole}}^2})(1-\alpha \frac{q^2}{M_{\mathrm{pole}}^2})},$$

$$f_{+}(q^2)=\frac{1}{A(q^2)}\frac{f_{+}(0)A(0)}{1+B(0)}(1+B(q^2)).$$

$${\chi }^2=\underset{ij=1}{\sum ^m}(\Delta {\Gamma }_{\mathrm{msr}}^i-\Delta {\Gamma }_{\exp }^i)C_{ij}^{-1}(\Delta {\Gamma }_{\mathrm{msr}}^j-\Delta {\Gamma }_{\exp }^j),$$

Full text

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 Riggio:2017zwh ; Zhang:2018jtm ; Fang:2014sqa . In the Standard Model, the rate of $D^{+}_{s}\to\eta e^{+}\nu_{e}$ and $D^{+}_{s}\to\eta^{\prime}e^{+}\nu_{e}$ 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) $f_{+}^{\eta^{(\prime)}}(q^{2})$, where $q$ is the momentum transfer to the $e^{+}\nu_{e}$ system. Unlike the final-state hadrons $K$ and $\pi$, the mesons $\eta^{(\prime)}$ 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 $\eta-\eta^{\prime}$-gluon Christ:2010dd ; Dudek:2011tt , whose mixing parameter can be determined from the SL decays, and, consequently, gives a deeper understanding of non-perturbative QCD confinement.

Recently, the FFs $f^{\eta^{(\prime)}}_{+}(0)$ were calculated using lattice quantum chromodynamics (LQCD) Bali:2014pva and QCD light-cone sum rules (LCSR) Offen:2013nma ; Duplancic:2015zna by assuming particular admixtures of quarks and gluons DiDonato:2011kr ; Ambrosino:2006gk ; Aaij:2014jna for $\eta$ and $\eta^{\prime}$ mesons. As information concerning the gluon content in the $\eta^{\prime}$ remains inconclusive, large uncertainties may be involved. Measurements of $f^{\eta^{(\prime)}}_{+}(0)$ are crucial to calibrate these theoretical calculations. Once the predicted $f^{\eta^{(\prime)}}_{+}(0)$ 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 $D_{s}^{+}\to\eta^{(\prime)}e^{+}\nu_{e}$ can shed light on $\eta-\eta^{\prime}$-gluon mixing. The $\eta-\eta^{\prime}$ mixing angle in the quark flavour basis, $\phi_{\rm P}$, can be related to the BFs of the $D$ and $D_{s}$ via $\cot^{4}\phi_{\rm P}=\frac{\Gamma_{D^{+}_{s}\to\eta^{\prime}e^{+}\nu_{e}}/\Gamma_{D^{+}_{s}\to\eta e^{+}\nu_{e}}}{\Gamma_{D^{+}\to\eta^{\prime}e^{+}\nu_{e}}/\Gamma_{D^{+}\to\eta e^{+}\nu_{e}}}$, in which a possible gluon component cancels DiDonato:2011kr . Determination of $\phi_{\rm P}$ gives a complementary constraint on the role of gluonium in the $\eta^{\prime}$, thus helping to improve our understanding of nonperturbative QCD dynamics and benefiting theoretical calculations of $D$ and $B$ decays involving the $\eta^{(\prime)}$.

Previous measurements of the BFs of $D^{+}_{s}\to\eta^{(\prime)}e^{+}\nu_{e}$ were made by CLEO Brandenburg:1995qq ; Yelton:2009aa ; Hietala:2015jqa and BESIII Ablikim:2016rqq , but these measurements include large uncertainties. This Letter reports improved measurements of the BFs and the first experimental studies of the dynamics of $D^{+}_{s}\to\eta^{(\prime)}e^{+}\nu_{e}$ ref:ChargeConjugation . Based on these, the first measurements of $f_{+}^{\eta^{(\prime)}}(0)$ are made, and measurements of $|V_{cs}|$ and $\phi_{\rm P}$ 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 $E_{\rm CM}=4.178$ GeV with the BESIII detector. A description of the design and performance of the BESIII detector can be found in Ref. Ablikim2010345 . 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 Lxin ; Gyingxiao . Monte Carlo (MC) simulated events are generated with a geant4-based Agostinelli:2002hh 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 $E_{\rm CM}=4.178$ GeV. It includes open charm processes, initial state radiation (ISR) production of $\psi(3770)$, $\psi(3686)$ and $J/\psi$, $q\bar{q}\,(q=u,\,d,\,s$) continuum processes, along with Bhabha scattering, $\mu^{+}\mu^{-}$, $\tau^{+}\tau^{-}$ and $\gamma\gamma$ events. The open charm processes are generated using conexc Ping:2013jka . 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) Tanabashi:2018oca by evtgen ref:evtgen , and the other modes are generated using lundcharm ref:lundcharm . The SL decays $D^{+}_{s}\to\eta^{(\prime)}e^{+}\nu_{e}$ are simulated with the modified pole model Chikilev:1999zn .

At $E_{\rm CM}=4.178$ GeV, $D_{s}^{+}$ mesons are produced mainly from the processes $e^{+}e^{-}\to D^{+}_{s}D_{s}^{*-}+c.c.\to D^{+}_{s}\gamma(\pi^{0})D_{s}^{-}$. 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 $\gamma(\pi^{0})$ from the $D_{s}^{*}$ [called double-tag (DT) $D_{s}^{+}$]. The BF of the SL decay is determined by

[TABLE]

where $N^{\rm tot}_{\rm ST}$ and $N^{\rm tot}_{\rm DT}$ are the ST and DT yields in data, $\epsilon_{\gamma(\pi^{0})\rm SL}$ is the efficiency of finding $\gamma(\pi^{0})\eta^{(\prime)}e^{+}\nu_{e}$ determined by $\sum_{k}\frac{N^{k}_{\rm ST}}{N^{\rm tot}_{\rm ST}}\frac{\epsilon^{k}_{\rm DT}}{\epsilon^{k}_{\rm ST}}$, where $\epsilon^{k}_{\rm ST}$ and $\epsilon^{k}_{\rm DT}$ 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 $\pi^{\pm}$ and $K^{\pm}$, are the same as those used in Ref. Ablikim:2018jun . 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 (${\mathcal{L}}_{\pi}$, ${\mathcal{L}}_{K}$ and ${\mathcal{L}}_{e}$) are formed. Positron candidates must satisfy ${\mathcal{L}}_{e}>0.001$ and ${\mathcal{L}}_{e}/({\mathcal{L}}_{e}+{\mathcal{L}}_{\pi}+{\mathcal{L}}_{K})>0.8$. The energy loss of the positron due to bremsstrahlung is partially recovered by adding the energies of the EMC showers that are within $10^{\circ}$ of the positron direction and not matched to other particles (FSR recovery).

Photon candidates are selected from the EMC showers that begin within 700 ns of the event start time and have an energy greater than 25 (50) MeV in the barrel (endcap) region of the EMC Ablikim2010345 . Candidates of $\pi^{0}$ or $\eta_{\gamma\gamma}$ are formed by photon pairs with an invariant mass in the range (0.115, 0.150) or (0.50, 0.57) GeV/$c^{2}$. To improve the momentum resolution, the $\gamma\gamma$ invariant mass is constrained to the $\pi^{0}$ or $\eta$ nominal mass Tanabashi:2018oca via a kinematic fit. Candidates of $\eta_{\pi^{0}\pi^{+}\pi^{-}}$, $\eta^{\prime}_{\eta_{\gamma\gamma}\pi^{+}\pi^{-}}$, $\eta^{\prime}_{\gamma\rho^{0}}$, $\rho^{0}$, and $\rho^{-}$ are formed from $\pi^{+}\pi^{-}\pi^{0}$, $\eta_{\gamma\gamma}\pi^{+}\pi^{-}$,$\gamma\rho^{0}_{\pi^{+}\pi^{-}}$, $\pi^{+}\pi^{-}$ and $\pi^{-}\pi^{0}$ combinations whose invariant masses fall in the ranges $(0.53,\,0.57)$, $(0.946,\,0.970)$, $(0.940,0.976)$, $(0.57,0.97)$, and $(0.57,\,0.97)$ GeV/$c^{2}$, respectively.

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 $D_{s}^{-}\to\pi^{+}\pi^{-}\pi^{-}$ and $K^{-}\pi^{+}\pi^{-}$, the contributions of $D^{-}_{s}\to K^{0}_{S}\pi^{-}$ and $K^{0}_{S}K^{-}$ are removed by requiring $M_{\pi^{+}\pi^{-}}$ outside $\pm 0.03$ GeV/$c^{2}$ around the $K^{0}_{S}$ nominal mass Tanabashi:2018oca .

The ST $D^{-}_{s}$ mesons are identified by the beam constrained mass $M_{\rm BC}\equiv\sqrt{(E_{\rm c.m.}/2)^{2}-|\vec{p}_{D_{s}^{-}}|^{2}}$ and the $D_{s}^{-}$ recoil mass $M_{\rm rec}\equiv\sqrt{(E_{\rm c.m.}-\sqrt{|\vec{p}_{D_{s}^{-}}|^{2}+M_{D_{s}^{-}}^{2}})^{2}-|\vec{p}_{D_{s}^{-}}|^{2}}$, where $\vec{p}_{D^{-}_{s}}$ is the 3-momentum of the ST candidate and $M_{D^{-}_{s}}$ is the nominal $D^{-}_{s}$ mass Tanabashi:2018oca . Non-$D^{+}_{s}D^{*-}_{s}$ events are suppressed by requiring $M_{\rm BC}\in(2.010,\,2.073)$ GeV/$c^{2}$. In each event, only the candidate with $M_{\rm rec}$ closest to the nominal $D_{s}^{*+}$ mass Tanabashi:2018oca is chosen. The ST yield is determined by fits to the $M_{\rm tag}$ spectra for each of the 14 tag modes shown in Fig. 1, where $M_{\rm tag}$ is the invariant mass of the ST candidate. Signals and the $D^{-}\to K^{0}_{S}\pi^{-}$ peaking background in the $D^{-}_{s}\to K^{0}_{S}K^{-}$ mode are described by MC-simulated shapes. 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 $N^{\rm tot}_{\rm ST}=395142\pm 1923$.

Once the $D_{s}^{-}$ tag has been found, the photon or $\pi^{0}$ from the $D_{s}^{*+}$ transition is selected. We define the energy difference $\Delta E\equiv E_{\rm CM}-E_{\rm tag}-E_{\gamma(\pi^{0})+D^{-}_{s}}^{\rm rec}-E_{\gamma(\pi^{0})}$, where $E_{\gamma(\pi^{0})+D^{-}_{s}}^{\rm rec}\equiv\sqrt{|-\vec{p}_{\gamma(\pi^{0})}-\vec{p}_{\rm tag}|^{2}+M^{2}_{D_{s}^{+}}}$, $E_{i}$ and $\vec{p}_{i}$ [$i=\gamma(\pi^{0})$ or tag] are the energy and momentum of $\gamma(\pi^{0})$ or $D_{s}^{-}$ tag, respectively. All unused $\gamma$ or $\pi^{0}$ candidates are looped over and that with the minimum $|\Delta E|$ is chosen. Candidates with $\Delta E\in(-0.04,\,0.04)$ GeV are accepted. The signal candidates are examined by the kinematic variable ${\rm MM}^{2}\equiv(E_{\rm CM}-E_{\rm tag}-E_{\gamma(\pi^{0})}-E_{\eta^{(\prime)}}-E_{e})^{2}-|-\vec{p}_{\rm tag}-\vec{p}_{\gamma(\pi^{0})}-\vec{p}_{\eta^{(\prime)}}-\vec{p}_{e}|^{2},$ where $E_{i}$ and $\vec{p}_{i}$ ($i=e$ or $\eta^{(\prime)}$) are the energy and momentum of $e^{+}$ or $\eta^{(\prime)}$. To suppress backgrounds from $D^{+}_{s}$ hadronic decays, the maximum energy of the unused showers ($E_{\rm\gamma~{}extra}^{\rm max}$) must be less than 0.3 GeV and events with additional charged tracks ($N^{\rm extra}_{\rm char}$) are removed. We require $M_{\eta^{\prime}e^{+}}<1.9$ GeV/$c^{2}$ for $D_{s}^{+}\to\eta^{\prime}e^{+}\nu_{e}$ and $\cos\theta_{\rm hel}\in(-0.85,\,0.85)$ for $D^{+}_{s}\to\eta^{\prime}_{\gamma\rho^{0}}e^{+}\nu_{e}$ to further suppress the $D_{s}^{+}\to\eta^{\prime}\pi^{+}$ and $D_{s}^{+}\to\phi e^{+}\nu_{e}$ backgrounds, where $\theta_{\rm hel}$ is the helicity angle between the momentum directions of the $\pi^{+}$ and the $\eta^{\prime}$ in the $\rho^{0}$ rest frame.

Figure 2 shows the MM2 distribution after all selection criteria have been applied. The signal yields are determined from a simultaneous unbinned maximum likelihood fit to these spectra, where $\mathcal{B}_{D_{s}^{+}\to\eta^{(\prime)}e^{+}\nu_{e}}$ measured using two different $\eta^{(\prime)}$ 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 $D_{s}^{+}\to\eta^{\prime}_{\gamma\rho^{0}}e^{+}\nu_{e}$, some peaking background from $D_{s}^{+}\to\phi e^{+}\nu_{e}$ still remains. This background is modeled by a separate component in the fit; its size and shape are fixed based on MC simulation.

Table 1 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 $M_{\rm tag}$ spectra with different signal shapes, background parameters, and fit ranges. The uncertainties in the tracking or PID efficiencies are assigned as 0.5% per $\pi^{\pm}$ by studying $e^{+}e^{-}\to K^{+}K^{-}\pi^{+}\pi^{-}$, and 0.5% per $e^{+}$ by radiative Bhabha process, respectively. The uncertainties of the $E_{\rm\gamma~{}extra}^{\rm max}$ and $N_{\rm char}^{\rm extra}$ requirements are estimated to be 0.5% and 0.9% by analyzing DT hadronic events. The uncertainties of the $\Delta E$ requirement, FSR recovery and $\theta_{\rm hel}$ 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 $J/\psi\to\pi^{+}\pi^{-}\pi^{0}$ Ablikim:2011kv and 1.0% per $\pi^{0}$ or $\eta$ by studying $e^{+}e^{-}\to K^{+}K^{-}\pi^{+}\pi^{-}\pi^{0}$. 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 ${\rm MM}^{2}$ fit is assigned as 0.9%, 1.3%, 1.2% and 1.2% for $D^{+}_{s}\to\eta_{\gamma\gamma}e^{+}\nu_{e}$, $\eta_{\pi^{0}\pi^{+}\pi^{-}}e^{+}\nu_{e}$, $\eta^{\prime}_{\eta\pi^{+}\pi^{-}}e^{+}\nu_{e}$ and $\eta^{\prime}_{\gamma\rho^{0}}e^{+}\nu_{e}$ (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 $K^{\pm}$ and $\pi^{\pm}$ as well as the selection of neutral particles between data and MC simulation in different environments. The uncertainty due to the $M_{\eta^{\prime}e^{+}}$ requirement is found to be negligible. The uncertainty due to peaking background is assigned to be 1.4% by varying its size by $\pm 1\sigma$ of the corresponding BF. The uncertainties due to the quoted BFs, 0.9%, 1.4%, 1.8% and 1.9% of $\eta^{(\prime)}$ decays Tanabashi:2018oca 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.

With the BFs measured in this work, we determine the BF ratio ${\mathcal{R}}^{D^{+}_{s}}_{\eta^{\prime}/\eta}={\mathcal{B}}_{D^{+}_{s}\to\eta e^{+}\nu_{e}}/{\mathcal{B}}_{D^{+}_{s}\to\eta^{\prime}e^{+}\nu_{e}}=0.355\pm 0.033_{\rm stat}\pm 0.015_{\rm syst}$, where the systematic uncertainties on the ST yield and due to the photon from $D_{s}^{*+}$, FSR recovery, tracking and PID of $e^{+}$ cancel. Using these BFs and $\mathcal{B}_{D^{+}\to\eta^{(\prime)}e^{+}\nu_{e}}$ reported in Ref. Zhangyu:Dptoetaenu , we determine the $\eta-\eta^{\prime}$ mixing angle to be $\phi_{P}=(40.1\pm 2.1_{\rm stat}\pm 0.7_{\rm syst})^{\circ}$. This result is consistent with previous measurements using $D\to\eta^{(\prime)}e^{+}\nu_{e}$ decays DiDonato:2011kr and $\psi\to\gamma\eta^{(\prime)}$ decays Ambrosino:2006gk within uncertainties.

To study the $D^{+}_{s}\to\eta^{(\prime)}e^{+}\nu_{e}$ dynamics, the candidate events are divided into various $q^{2}$ intervals. The measured partial decay width $\Delta\Gamma^{i}_{\rm msr}$ in the $i$th $q^{2}$ interval is determined by $\Delta\Gamma^{i}_{\rm msr}\equiv\int_{i}\frac{d\Gamma}{dq^{2}}dq^{2}=\frac{N_{\rm pro}^{i}}{\tau_{D_{s}^{+}}\times N^{\rm tot}_{\rm ST}}$, where $\tau_{D_{s}^{+}}$ is the lifetime of the $D_{s}^{+}$ meson Tanabashi:2018oca ; Aaij:2017vqj , and $N^{i}_{\rm pro}$ is the DT yield produced in the $i$th $q^{2}$ interval, calculated by $N_{\rm pro}^{i}=\sum^{m}_{j}(\epsilon^{-1})_{ij}N_{\rm obs}^{j}$. Here $m$ is the number of $q^{2}$ intervals, $N^{j}_{\rm obs}$ is the observed DT yield obtained from similar fits to the MM2 distribution as described previously, and $\epsilon_{ij}$ is the efficiency matrix determined from signal MC events and is given by $\epsilon_{ij}=\sum_{k}\left[(1/N_{\rm ST}^{\rm tot})\times(N^{ij}_{\rm rec}/N^{j}_{\rm gen})_{k}\times(N^{k}_{\rm ST}/\epsilon^{k}_{\rm ST})\right]$, where $N^{ij}_{\rm rec}$ is the DT yield generated in the $j$th $q^{2}$ interval and reconstructed in the $i$th $q^{2}$ interval, $N^{j}_{\rm gen}$ is the total signal yield generated in the $j$th $q^{2}$ interval, and $k$ sums over all tag modes. See Tables 1 and 2 of Ref. [30] for details about the range, $N^{i}_{\rm obs}$, $N^{i}_{\rm prd}$, and $\Delta\Gamma^{i}_{\rm msr}$ of each $q^{2}$ interval for $D^{+}_{s}\to\eta e^{+}\nu_{e}$ and $D^{+}_{s}\to\eta^{\prime}e^{+}\nu_{e}$, respectively.

In theory, the differential decay width can be expressed

[TABLE]

where $|p_{\eta^{(\prime)}}|$ 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 Becher:2005bg ,

[TABLE]

where $M_{\rm pole}$ is fixed to $M_{D^{*+}_{s}}$ and $\alpha$ is a free parameter. Setting $\alpha=0$ and leaving $M_{\rm pole}$ free, it is the simple pole model Becirevic:1999kt . In the two-parameter (2 Par.) series expansion Becher:2005bg

[TABLE]

Here, $A(q^{2})=P(q^{2})\Phi(q^{2},t_{0})$, $B(q^{2})=r_{1}(t_{0})z(q^{2},t_{0})$, $t_{0}=t_{+}(1-\sqrt{1-t_{-}/t_{+}})$, $t_{\pm}=(M_{D_{s}^{+}}\pm M_{\eta})$, and $r_{k}$ is a free parameter. The functions $P(q^{2})$, $\Phi(q^{2},\,t_{0})$, and $z(q^{2},\,t_{0})$ are defined following Ref. Becher:2005bg .

For each SL decay, the product $f_{+}(0)|V_{cs}|$ and one other parameter, $M_{\rm pole}$, $\alpha$, or $r_{1}$, are determined by constructing and minimizing

[TABLE]

with $\Delta\Gamma^{i}_{\rm msr}$ and the theoretically expected value $\Delta\Gamma^{i}_{\rm exp}$, where $C_{ij}=C_{ij}^{\rm stat}+C_{ij}^{\rm syst}$ is the covariance matrix of $\Delta\Gamma^{i}_{\rm msr}$ among $q^{2}$ intervals, as shown in Tables 3 and 4 in Ref. Supplement . For each $\eta^{(\prime)}$ subdecay, the statistical covariance matrix is constructed with the statistical uncertainty in each $q^{2}$ interval ($\sigma(N^{\alpha}_{\rm obs})$) as $C_{ij}^{\rm stat}=(\frac{1}{\tau_{D_{s}^{+}}N_{\rm ST}^{\rm tot}})^{2}\sum_{\alpha}\epsilon_{i\alpha}^{-1}\epsilon_{j\alpha}^{-1}[\sigma(N^{\alpha}_{\rm obs})]^{2}$. 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 ($\delta(\Delta\Gamma^{i}_{\rm msr})$) as $C_{ij}^{\rm syst}=\delta(\Delta\Gamma^{i}_{\rm msr})\delta(\Delta\Gamma^{j}_{\rm msr})$. Here, an additional systematic uncertainty in $\tau_{D^{+}_{s}}$ (0.8%) Tanabashi:2018oca ; Aaij:2017vqj is involved besides those in the BF measurements.

The $\Delta\Gamma^{i}_{\rm msr}$ measured by the two $\eta^{(\prime)}$ subdecays are fitted simultaneously, with results shown in Fig 3. In the fits, the $\Delta\Gamma^{i}_{\rm msr}$ becomes a vector of length $2m$. Uncorrelated systematic uncertainties are from tag bias, quoted BFs, $\eta$ (and $\pi^{0}$) reconstruction, and FF parametrization, while other systematic uncertainties are fully correlated. Table 2 summarizes the fit results, where the obtained $f_{+}^{\eta^{(\prime)}}(0)|V_{cs}|$ with different FF parameterizations are consistent with each other.

Combining $|V_{cs}|=0.97343\pm 0.00015$ from the global fit in the SM Tanabashi:2018oca with $f^{\eta^{(\prime)}}_{+}(0)|V_{cs}|$ extracted from the two-parameter series expansion, we determine $f^{\eta}_{+}(0)=0.4576\pm 0.0054_{\rm stat}\pm 0.0045_{\rm syst}$ and $f^{\eta^{\prime}}_{+}(0)=0.490\pm 0.050_{\rm stat}\pm 0.011_{\rm syst}$. Table 3 compares the measured FFs with various theoretical calculations within uncertainties. When combining $f^{\eta}_{+}(0)$ and $f^{\eta^{\prime}}_{+}(0)$ calculated from Ref. Offen:2013nma , we obtain $|V_{cs}|=1.031\pm 0.012_{\rm stat}\pm 0.009_{\rm syst}\pm 0.079_{\rm theo}$ and $0.917\pm 0.094_{\rm stat}\pm 0.021_{\rm syst}\pm 0.155_{\rm theo}$, respectively. These results agree with the measurements of $|V_{cs}|$ using $D\to\bar{K}\ell^{+}\nu_{\ell}$ Ablikim:2015ixa ; Ablikim:2015qgt ; Ablikim:2018evp ; Besson:2009uv ; Aubert:2007wg ; Widhalm:2006wz and $D_{s}^{+}\to\ell^{+}\nu_{\ell}$ decays Ablikim:2016duz ; Zupanc:2013byn ; delAmoSanchez:2010jg ; Onyisi:2009th ; Naik:2009tk within uncertainties.

In summary, by analyzing a data sample of 3.19 fb*-1* taken at $E_{\rm CM}=4.178$ GeV with the BESIII detector, we measure the absolute BFs of $D^{+}_{s}\to\eta^{(\prime)}e^{+}\nu_{e}$ with a DT method. The precision is improved by a factor of 2 compared to the world average values. Using these BFs and ${\mathcal{B}}(D^{+}\to\eta^{(\prime)}e^{+}\nu_{e})$ measured in our previous work Zhangyu:Dptoetaenu , we determine the $\eta-\eta^{\prime}$ mixing angle $\phi_{P}$, which provides complementary data to constrain the gluon component in the $\eta^{\prime}$ meson. From an analysis of the dynamics in $D^{+}_{s}\to\eta^{(\prime)}e^{+}\nu_{e}$, the products of $f_{+}^{\eta^{(\prime)}}(0)|V_{cs}|$ are determined for the first time. Furthermore, by taking $|V_{cs}|$ from a standard model fit (CKMfitter, Tanabashi:2018oca ) as input, we determine the form factor at zero momentum transfer $f^{\eta^{(\prime)}}_{+}(0)$ for the first time. The obtained FFs provide important data to distinguish various theoretical calculations Bali:2014pva ; Offen:2013nma ; Duplancic:2015zna ; Melikhov:2000yu ; Soni:2018adu ; Colangelo:2001cv ; Azizi:2010zj . Alternatively, we also determine $|V_{cs}|$ with $D_{s}^{+}\to\eta^{(\prime)}e^{+}\nu_{e}$ decays for the first time, by taking values for $f^{\eta^{(\prime)}}_{+}(0)$ calculated in theory. Our result on $|V_{cs}|$ together with those measured by $D\to\bar{K}\ell^{+}\nu_{\ell}$ and $D_{s}^{+}\to\ell^{+}\nu_{\ell}$ are important to test the unitarity of the CKM matrix.