First Observation of $D^+\to \eta\mu^+\nu_\mu$ and Measurement of its Decay Dynamics

By analyzing a data sample corresponding to an integrated luminosity of $2.93~\mathrm{fb}^{-1}$ collected at a center-of-mass energy of 3.773 GeV with the BESIII detector, we measure for the first time the absolute branching fraction of the $D^+\to \eta \mu^+\nu_\mu$ decay to be ${\mathcal B}_{D^+\to \eta \mu^+\nu_\mu}=(10.4\pm1.0_{\rm stat}\pm0.5_{\rm syst})\times 10^{-4}$. Using the world averaged value of ${\mathcal B}_{D^+\to \eta e^+\nu_e}$, the ratio of the two branching fractions is determined to be ${\mathcal B}_{D^+\to \eta \mu^+\nu_\mu}/{\mathcal B}_{D^+\to \eta e^+\nu_e}=0.91\pm0.13$, which agrees with the theoretical expectation of lepton flavor universality within uncertainty. Here, the uncertainty is the sum in quadrature of the statistical and systematic uncertainties. By studying the differential decay rates in five four-momentum transfer intervals, we obtain the product of the hadronic form factor $f^{\eta}_{+}(0)$ and the $c\to d$ Cabibbo-Kobayashi-Maskawa matrix element $|V_{cd}|$ to be $f_{+}^\eta (0)|V_{cd}|=0.087\pm0.008_{\rm stat}\pm0.002_{\rm syst}$. Taking the input of $|V_{cd}|$ from the global fit in the standard model, we determine $f_{+}^\eta (0)=0.39\pm0.04_{\rm stat}\pm0.01_{\rm syst}$. On the other hand, using the value of $f_+^{\eta}(0)$ calculated in theory, we find $|V_{cd}|=0.242\pm0.022_{\rm stat}\pm0.006_{\rm syst}\pm0.033_{\rm theory}$.

By analyzing a data sample corresponding to an integrated luminosity of 2.93 fb −1 collected at a center-of-mass energy of 3.773 GeV with the BESIII detector, we measure for the first time the absolute branching fraction of the D + → ηµ + νµ decay to be B D + →ηµ + νµ = (10.4 ± 1.0stat ± 0.5syst) × 10 −4 . Using the world averaged value of B D + →ηe + νe , the ratio of the two branching fractions is determined to be B D + →ηµ + νµ /B D + →ηe + νe = 0.91 ± 0.13, which agrees with the theoretical expectation of lepton flavor universality within uncertainty. Here, the uncertainty is the sum in quadrature of the statistical and systematic uncertainties. By studying the differential decay rates in five four-momentum transfer intervals, we obtain the product of the hadronic form factor f η + (0) and the c → d Cabibbo-Kobayashi-Maskawa matrix element |V cd | to be f η + (0)|V cd | = 0.087 ± 0.008stat ± 0.002syst. Taking the input of |V cd | from the global fit in the standard model, we determine f η + (0) = 0.39 ± 0.04stat ± 0.01syst. On the other hand, using the value of f η + (0) calculated in theory, we find |V cd | = 0.242 ± 0.022stat ± 0.006syst ± 0.033 theory .
PACS numbers: 13.20.Fc,12.15.Hh In the standard model (SM), the couplings between three families of leptons and gauge bosons are independent of lepton flavors. This property is known as lepton flavor universality (LFU) [1][2][3][4]. Semileptonic (SL) decays of pseudoscalar mesons, which are well understood in the SM, offer an ideal platform to test LFU and search for new physics effects. In the past decade, the BaBar, Belle, and LHCb collaborations reported anomalies in LFU tests with various SL B decays. The measured branching fraction (BF) ratios R τ /ℓ = B B→D ( * ) τ + ντ /B B→D ( * ) ℓ + ν ℓ (ℓ = µ, e) [5][6][7][8][9][10][11] deviate from the SM predictions by 3.1σ [12]. Various models [2,[13][14][15][16][17] were proposed to explain these differences. In view of this, scrutinizing the ratios of the semimuonic D decay BFs over their corresponding semielectronic counterparts offers important complementary tests of e-µ LFU. Recently, BESIII reported tests of LFU with the SL decays D →Kℓ + ν ℓ [18,19] and D → πℓ + ν ℓ [20], and no significant evidence of LFU violation is observed. Nevertheless, the knowledge of semimuonic D decays is still relatively limited. For example, although the D + → ηℓ + ν ℓ decay was predicted in the quark model 30 years ago [21], there is no experimental confirmation of D + → ηµ + ν µ yet. In this Letter, we report a complementary test of LFU with D + → ηℓ + ν ℓ decays based on the first measurement of the BF of D + → ηµ + ν µ . Throughout this Letter, charge conjugate channels are always implied. The BF obtained will also be important for the determination of the η-η ′ mixing angle, which will benefit the understanding of nonperturbative quantumchromodynamics (QCD) effects [22].
The investigation of D + → ηµ + ν µ decay dynamics allows the determination of the c → d Cabibbo-Kobayashi-Maskawa (CKM) matrix element |V cd | and the hadronic form factor (FF) f η + (0). The value of f η + (0) has been calculated with various approaches, e.g., QCD light-cone sum rules (LCSR) [23][24][25], lightfront quark model (LFQM) [26], covariant confined quark model (CCQM) [27,28], and relativistic quark model (RQM) [29]. The predicted values vary in a wide range from 0.36 to 0.71. According to Refs. [30,31], the predicted FFs of the SL D decays are expected to be insensitive to the spectator quark. Measurement of the hadronic FF in D + → ηµ + ν µ decay can be used to distinguish between these calculations. The FF prediction verified by experiment is useful to determine |V cd |. Measurements of SL D decay hadronic FFs help constrain lattice QCD calculations and lead to more reliable calculations of the hadronic FFs of SL D and B decays, which are crucial to accurately determine CKM parameters [30][31][32][33], and test the unitarity of the CKM matrix.
In this analysis, we use a data sample corresponding to an integrated luminosity of 2.93 fb −1 [34] taken at the center-of-mass energy √ s = 3.773 GeV with the BESIII detector. Details about the design and performance of the BESIII detector are given in Ref. [35]. Simulated samples produced with the geant4-based [36] Monte Carlo (MC) package which includes the geometric description of the BESIII detector and the detector response, are used to determine the detection efficiency and to estimate the backgrounds.
The simulation includes the beam energy spread and initial state radiation (ISR) in the e + e − annihilations modeled with the generator kkmc [37]. The inclusive MC samples consist of the production of D 0D0 , D + D − , and non-DD decays of the ψ(3770), the ISR production of the J/ψ and ψ(3686) states, and the continuum processes incorporated in kkmc [37]. The known decay modes are modeled with evtgen [38] using BFs taken from the Particle Data Group [39], and the remaining unknown decays from the charmonium states with lundcharm [40]. The final state radiation from charged particles is incorporated with the photos package [41]. The vector hadronic FF of the SL decay D + → ηµ + ν µ is simulated with the modified-pole model [42], where the parameter α of the vector hadronic FF is set by referring to that of D + → π 0 e + ν e measured by BESIII [43], and the pole mass is set at the nominal D * + mass [39].
The analysis is performed with the double-tag (DT) method, benefiting from the advantage of D + D − pair production at K 0 S π − π 0 , K 0 S π + π − π − , and K + K − π − , the presence of a D + meson is guaranteed. If the D + → ηµ + ν µ decay can be found in the system recoiling against an ST D − meson, the candidate event is called a DT event. The BF of the SL decay is determined by where N tot ST and N DT are the yields of the ST and DT candidates in data, respectively, and B η→γγ is the BF of the η → γγ decay. ε SL = ε DT /ε ST is the effective signal efficiency of finding D + → ηµ + ν µ decay in the presence of the ST D − meson, where ε ST and ε DT are the efficiencies of selecting the ST and DT candidates, respectively.
This analysis uses the same K ± , π ± , K 0 S , γ, and π 0 selection criteria as those employed in Refs. [18,20,[44][45][46][47]. The ST D − mesons are distinguished from combinatorial backgrounds by using the energy difference Here, E beam is the beam energy, and E D − and p D − are the total energy and momentum of the ST D − candidate in the e + e − centerof-mass frame. If multiple combinations for an ST mode are present in an event, the combination with the smallest |∆E| per tag mode per charge is retained for further analysis. The candidates are required to satisfy ∆E ∈ (−0.055, 0.045) GeV for the tags containing π 0 and ∆E ∈ (−0.025, 0.025) GeV for the other tags. For each tag mode, the yield of ST D − mesons is determined from the maximum likelihood fit of the M BC distribution of the accepted candidates. In the fit, the signal and background are described by an MC-simulated shape and an ARGUS function [48], respectively. To take into account the resolution difference between data and MC simulation, the MC-simulated signal shape is convolved with a double-Gaussian function. The widths and relative abundances of the Gaussian components are free parameters of the fit. The resulting fits of these M BC distributions are exhibited in Fig. 1. The candidates with M BC ∈ (1.863, 1.877) GeV/c 2 are kept for further analysis. The total yield of ST D − mesons is The D + → ηµ + ν µ candidates are selected in the sides recoiling against the ST D − mesons. It is required that there is only one charged track available for muon identification. The muon candidate is required to satisfy |V xy | < 1 cm and |V z | < 10 cm, where |V xy | and |V z | are the distances of closest approach to the interaction point of the reconstructed track in the transverse plane and along the axis of the drift chamber, respectively. Its polar angle (θ) with respect to the axis of the drift chamber must be within | cos θ| < 0.93.
Muon identification uses information from the timeof-flight and the electromagnetic calorimeter (EMC), as well as the specific ionization energy loss measured in the main drift chamber. The combined confidence levels for various particle hypotheses (CL i , i = e, µ, and K) are calculated. Muon candidates are required to satisfy CL µ > 0.001, CL µ > CL e , and CL µ > CL K . To reduce misidentification between hadrons and muons, the deposited energy in the EMC of the muon candidate is required to be within (0.105, 0.275) GeV.
The η candidates are reconstructed via the η → γγ decay. The invariant mass of the γγ candidate is required to be within (0.510, 0.570) GeV/c 2 . To improve momentum resolution, a one-constraint (1-C) kinematic fit is done on the selected photon pair, whose invariant mass is constrained to the η nominal mass (m η ) [39].
Due to the misidentification of pions as muons, some hadronic D + decays survive the above selection criteria. To suppress the peaking backgrounds from D + → ηπ + decays, we require the ηµ + invariant mass (M ηµ + ) to be less than 1.74 GeV/c 2 . To reject the backgrounds containing π 0 , e.g., D + → ηπ + π 0 , we require that the maximum energy of any extra photon (E max extra γ ) is less than 0.30 GeV and there is no extra π 0 (N extra π 0 ) in the candidate event. Here, the extra photon and π 0 denote the ones which have not been used in the DT selection.
The number of SL decays is determined using a kinematic quantity defined as U miss ≡ E miss − | p miss |, which is expected to peak around 0 for the correctly reconstructed signal events. Here, E miss ≡ E beam − E η − E µ + and p miss ≡ p D + − p η − p µ + are the missing energy and momentum of the DT event in the e + e − center-of-mass frame, in which E η (µ + ) and p η (µ + ) are the energy and momentum of the η (µ) candidates. The U miss resolution is improved by constraining the D + energy to the beam energy and p D + ≡ −p D − · E 2 beam − m 2 D + , wherep D − is the unit vector in the momentum direction of the ST D − and m D + is the D + nominal mass [39]. Figure 2 (a) shows the U miss distribution of the accepted DT events in data. The SL decay yield is obtained from an unbinned fit to the U miss distribution, where the SL signal, peaking backgrounds of D + → ηπ + π 0 , and non-peaking backgrounds (including a small contribution from wrongly reconstructed ST candidates) are described by the corresponding MC-simulated shapes. The yields of the signal and non-peaking backgrounds are free parameters of the fit, while the yield of the peaking background from D + → ηπ + π 0 decays is fixed based on MC simulation. The fit result is shown in Fig. 2(a). From the fit, we obtain the yield of DT events N DT = 234 ± 22 stat . The statistical significance, calculated by −2ln(L 0 /L max ), is found to be greater than 10σ. Here, L max and L 0 are the maximal likelihood of the nominal fit and that of the fit without signal component, respectively. The average efficiency of detecting D + → ηµ + ν µ decays, weighted by the yields of ST D − mesons in data, is ε SL = 0.3752 ± 0.0013. Here, the efficiency does not include the BF of the η → γγ decay. To verify the reliability of the efficiency determination, we have compared distributions of momenta and cos θ of the η and µ + of the selected D + → ηµ + ν µ candidate events between data and MC simulation, and they are in good agreement.
In the BF measurement, the systematic uncertainties arise from the following sources. The uncertainty in the total yield of ST D − mesons has been studied in Refs. [18,20,44], and is assigned as 0.5%. The muon tracking (PID) efficiencies are studied by analyzing e + e − → γµ + µ − events, and the muon tracking (PID) efficiency uncertainty is taken as 0.2% (0.2%) per muon, where the data/MC differences of the two-dimensional (momentum and cos θ) distributions of the control samples have been re-weighted by those of the D + → ηµ + ν µ signal decays. The uncertainty of η reconstruction is assumed to be 2.0%, the same as π 0 reconstruction, which was studied with DT DD hadronic decays of D 0 → K − π + , K − π + π + π − vs.D 0 → K + π − π 0 , K 0 S π 0 [18,44]. The uncertainties of the requirements of E max extra γ and N extra π 0 are estimated to be 2.3% by analyzing the DT candidate events of D + → ηπ + and π 0 e + ν e . The uncertainty due to the M ηµ + requirement is evaluated by replacing the nominal requirement with M ηµ + < 1.69 GeV/c 2 or M ηµ + < 1.79 GeV/c 2 , and the associated uncertainty is found to be negligible. The uncertainty in the U miss fit is assigned to be 3.7%, which is estimated with alternative signal and background shapes. The uncertainty due to the limited MC statistics is 0.5%. The uncertainty in the MC model, 0.3%, is assigned as the difference between our nominal DT efficiency and the DT efficiency determined by re-weighting the q 2 (q is the total four momentum of µ + ν µ ) distribution of the signal MC events using the FF parameters obtained from data. Adding these uncertainties quadratically yields the total systematic uncertainty to be 4.9%.
To study the dynamics in D + → ηµ + ν µ decay, the SL candidate events are divided into five q 2 intervals: (0.0, 0.25), (0.25, 0.5), (0.5, 0.75), (0.75, 1.0), and (1.0, (m D + − m η ) 2 ) GeV 2 /c 4 . Events with q 2 < 0, which may occur due to detector resolution, are not used in this analysis. The partial decay rate in the ith q 2 interval, ∆Γ i measured , is determined by where N i produced is the D + → ηµ + ν µ signal yield produced in the ith q 2 interval in data, τ D + is the lifetime of D + , N tot ST is the total yield of ST D − mesons, and where N j observed is the D + → ηµ + ν µ signal yield observed in the jth q 2 interval and ε is the efficiency matrix given by (4) Here, N ij reconstructed is the D + → ηµ + ν µ signal yield generated in the jth q 2 interval and reconstructed in the ith q 2 interval, N j generated is the total signal yield generated in the jth q 2 interval, and the index k sums over all ST modes. N i observed is obtained from the fit to the U miss distribution of the D + → ηµ + ν µ candidate events in the ith q 2 interval. The fit results of the U miss distributions in various intervals are shown in Figs. 2 (b)-(f), and the partial decay rates obtained are shown in Fig. 3.
With the ∆Γ i measured obtained above and the partial decay rate ∆Γ i expected predicted by theory, the χ 2 is constructed as where is the covariance matrix of the measured partial decay rates among q 2 intervals, and where G F is the Fermi coupling constant; m µ is the µ + mass; q = p D + − p η is the four-momentum transfer; | p η | and E η are the momentum and energy of η in the rest frame of D + , respectively; the vector hadronic FF f η + (q 2 ) is formulated following Ref. [49]. Here, the scalar hadronic FF f η 0 (q 2 ) has been ignored due to negligible sensitivity with limited data.
The statistical covariance matrix is constructed as where n sums from 1 to 5 intervals. The systematic covariance matrix is obtained by summing over that of each systematic uncertainty source, which is taken as where δ(∆Γ i measured ) is the systematic uncertainty of the partial decay rate in the ith q 2 interval. The systematic uncertainties arising from N tot ST , τ D + , muon tracking and PID, η reconstruction, as well as E max extra γ and N extra π 0 requirements are taken to be common across all the q 2 intervals; while the others are determined separately in each q 2 interval as above. Minimizing the χ 2 in Eq. (5) gives the product of f η + (0)|V cd | and the first order coefficient r 1 to be 0.087 ± 0.008 stat ± 0.002 syst and −0.9 ± 2.7 stat ± 0.2 syst , respectively. The nominal fit parameters are taken from the fit with the combined statistical and systematic covariance matrix, and their statistical uncertainties are taken from the fit only with the statistical covariance matrix. For each parameter, the systematic uncertainty is obtained by calculating the quadratic difference of uncertainties between these two fits. The fit result is shown in Fig. 3 and the goodness-of-fit is χ 2 /NDOF = 1.0/3, where NDOF is the number of degrees of freedom. The f η + (0)|V cd | measured in this work is consistent with the measurements using D + → ηe + ν e by CLEO [54] and BESIII [55].
In summary, the SL decay D + → ηµ + ν µ has been observed by analyzing 2.93 fb −1 of data collected at √ s = 3.773 GeV. The absolute BF of this decay is determined for the first time to be B D + →ηµ + νµ = (10.4 ± 1.0 stat ± 0.5 syst ) × 10 −4 . Using the world averaged value of B D + →ηe + νe = (11.4 ± 1.0) × 10 −4 gives the BF ratio R µ/e = B D + →ηµ + νe /B D + →ηe + νe = 0.91 ± 0.13, where the uncertainty is the sum in quadrature of the statistical and systematic errors, but dominated by the statistical error. This result agrees with the SM predictions (0.97-1.00) [23,28,56], thereby implying no LFU violation within current sensitivity. The obtained BF can be used to determine the η-η ′ mixing angle once B D + →η ′ µ + νe is measured with large data samples [57,58] in the near future.
The BESIII collaboration thanks the staff of BEPCII and the IHEP computing center for their strong support. This work is supported in part by National Key DPT2006K-120470; National Science and Technology fund; STFC (United Kingdom); Table 1. Comparison of our BF (in ×10 −4 ) and hadronic FF with various theoretical calculations for D + → ηℓ + ν ℓ . The first and second uncertainties are statistical and systematic, respectively. Theoretical calculations listed in the table assume no gluon component for η ′ . Numbers marked with * denote that the predicted B D + →ηe + νe is listed due to no predictions for D + → ηµ + νµ in Refs. [24,25].