Study of the $D^0\to K^-\mu^+\nu_\mu$ dynamics and test of lepton flavor universality with $D^0\to K^-\ell^+\nu_\ell$ decays

Using $e^+e^-$ annihilation data of $2.93~\mathrm{fb}^{-1}$ collected at center-of-mass energy $\sqrt{s}=3.773$ GeV with the BESIII detector, we measure the absolute branching fraction of $D^{0}\to K^{-}\mu^{+}\nu_{\mu}$ with significantly improved precision: ${\mathcal B}_{D^{0}\to K^{-}\mu^{+}\nu_{\mu}}=(3.413\pm0.019_{\rm stat.}\pm0.035_{\rm syst.})\%$. Combining with our previous measurement of ${\mathcal B}_{D^0\to K^-e^+\nu_e}$, the ratio of the two branching fractions is determined to be ${\mathcal B}_{D^0\to K^-\mu^+\nu_\mu}/{\mathcal B}_{D^0\to K^-e^+\nu_e}=0.974\pm0.007_{\rm stat.}\pm0.012_{\rm syst.}$, which agrees with the theoretical expectation of lepton flavor universality within the uncertainty. A study of the ratio of the two branching fractions in different four-momentum transfer regions is also performed, and no evidence for lepton flavor universality violation is found with current statistics. Taking inputs from global fit in the standard model and lattice quantum chromodynamics separately, we determine $f_{+}^{K}(0)=0.7327\pm0.0039_{\rm stat.}\pm0.0030_{\rm syst.}$ and $|V_{cs}| = 0.955\pm0.005_{\rm stat.}\pm0.004_{\rm syst.}\pm0.024_{\rm LQCD}$.

B. X. Yu 1,42,46 , C. X. Yu 33  Using e + e − annihilation data of 2.93 fb −1 collected at center-of-mass energy √ s = 3.773 GeV with the BESIII detector, we measure the absolute branching fraction of D 0 → K − µ + νµ with significantly improved precision: B D 0 →K − µ + νµ = (3.413 ± 0.019stat. ± 0.035syst.)%. Combining with our previous measurement of B D 0 →K − e + νe , the ratio of the two branching fractions is determined to be B D 0 →K − µ + νµ /B D 0 →K − e + νe = 0.974 ± 0.007stat. ± 0.012syst., which agrees with the theoretical expectation of lepton flavor universality within the uncertainty. A study of the ratio of the two branching fractions in different four-momentum transfer regions is also performed, and no evidence for lepton flavor universality violation is found with current statistics. Taking inputs from CKMFitter and LQCD separately, we determine f K + (0) = 0.7327 ± 0.0039stat. ± 0.0030syst. and |Vcs| = 0.955 ± 0.005stat. ± 0.004syst. ± 0.024LQCD. The semileptonic (SL) decays D 0 → K − ℓ + ν ℓ (ℓ = e or µ) provide an ideal test-bed to explore strong and weak effects in D decays. In the Standard Model (SM), their differential decay rates can be written as [1,2] dΓ where V cs is the Cabibbo-Kobayashi-Maskawa (CKM) matrix element parametrizing the mixing between the quark flavors in the weak interaction; f K ± (q 2 ) are the hadronic form factors (FFs) describing the strong interaction between the final state quarks; G F is the Fermi coupling constant; m D , m K and m ℓ are the masses of D, K and ℓ particles, respectively; q = p D − p K is the four-momentum transfer; | p K | and E K are the momentum and energy of kaon in the D rest frame; . Experimental studies of D 0 → K − ℓ + ν ℓ decays are important to precisely determine |V cs | and f K + (0), which are critical to test CKM matrix unitarity and validate the lattice quantum chromodynamics (LQCD) calculation.
In recent years, some abnormal branching fraction (BF) ratios of various SL B decays between different lepton flavors, which imply hints of lepton flavor universality (LFU) violation, were reported [3][4][5][6][7], and various new physics models, such as the two Higgs Doublet Model and the leptoquark model [8][9][10][11][12][13], were developed to explain these deviations. Ref. [14] applies these methods to c → s transitions and suggests that there may be a measurable deviation from the SM in D → Kℓν ℓ decays. Measurements of the ratios of the total or partial widths of D 0 → K − ℓ + ν ℓ decays, R µ/e = Γ D 0 →K − µ + νµ /Γ D 0 →K − e + νe , are important to test LFU. Any deviation from the SM prediction may indicate new physics contribution in these decays.
The D 0 → K − e + ν e dynamics was studied by CLEOc, Belle, BaBar, and BESIII [15][16][17][18]. However, D 0 → K − µ + ν µ dynamics was only investigated by Belle and FOCUS [15,19], with relatively poor precision. In this Letter, we report an improved measurement of the BF of D 0 → K − µ + ν µ decay [20]. From an analysis of its decay dynamics, we determine |V cs | and f K + (0) incorporating the inputs from CKMFitter [21] and LQCD [22]. We also search for LFU violation between D 0 → K − µ + ν µ and D 0 → K − e + ν e decays. This analysis is performed using 2.93 fb −1 of data taken at 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. [23]. The Monte Carlo (MC) events are generated with a geant4based [24] detector simulation software package, boost.
An inclusive MC sample, which includes the D 0D0 , D + D − and non-DD decays of ψ(3770), the initial state radiation (ISR) production of ψ(3686) and J/ψ, and the qq (q = u, d, s) continuum process, along with Bhabha scattering, µ + µ − and τ + τ − events, is produced at √ s = 3.773 GeV to determine the detection efficiencies and to estimate the potential backgrounds. The production of the charmonium states is simulated by the MC generator kkmc [25]. The measured decay modes of the charmonium states are generated using evtgen [26] with BFs from the Particle Data Group (PDG) [21], and the remaining unknown decay modes are generated by lundcharm [27]. The D 0 → K − µ + ν µ decay is simulated with the modified pole model [28].
At √ s = 3.773 GeV, the ψ(3770) resonance decays predominately into D 0D0 or D + D − meson pairs. If aD 0 meson is fully reconstructed byD 0 → K + π − , K + π − π 0 and K + π − π − π + , a D 0 meson must exist in the recoiling system of the reconstructedD 0 (called the single-tag (ST)D 0 ). In the presence of the STD 0 , we select and study D 0 → K − µ + ν µ decay (called the double-tag (DT) events). The BF of the SL decay is given by where N tot ST and N DT are the ST and DT yields, ε SL = ε DT /ε ST is the efficiency of reconstructing D 0 → K − µ + ν µ in the presence of the STD 0 , and ε ST and ε DT are the efficiencies of selecting ST and DT events.
All charged tracks must originate from the interaction point with a distance of closest approach less than 1 cm in the transverse plane and less than 10 cm along the z axis. Their polar angles (θ) should satisfy | cos θ| < 0.93. Charged particle identification (PID) is performed by combining the time-of-flight information and the specific ionization energy loss measured in the main drift chamber (MDC). The information of the electromagnetic calorimeter (EMC) is also included to identify muon candidates. Combined confidence levels for electron, muon, pion and kaon hypotheses (CL e , CL µ , CL π , CL K ) are calculated individually. Kaon (pion) and muon candidates should satisfy CL K(π) > CL π(K) and CL µ > 0.001, CL e and CL K , respectively. In addition, the deposited energy in the EMC of the muon is required to be within (0.02, 0.29) GeV. The π 0 meson is reconstructed via π 0 → γγ decay. The energy deposited in the EMC of each photon is required to be greater than 0.025 GeV in the barrel (| cos θ| < 0.80) region or 0.050 GeV in the end-cap (0.86 < | cos θ| < 0.92) region, and the shower time should be within 700 ns of the event start time. The π 0 candidates with both photons from the end-cap are rejected because of poor resolution. The γγ combination with an invariant mass (M γγ ) in the range (0.115, 0.150) GeV/c 2 is regarded as a π 0 candidate, and a kinematic fit by constraining the M γγ to the π 0 nominal mass [21] is performed to improve the mass resolu-tion. ForD 0 → K + π − , the backgrounds from cosmic ray events, radiative Bhabha scattering and dimuon events are suppressed with the same requirements as used in Ref. [29].
The STD 0 mesons are identified by the energy difference ∆E ≡ E beam − ED0 and the beam-constrained where E beam is the beam energy, and ED0 and pD0 are the total energy and momentum of the STD 0 in the e + e − rest frame. If there are multiple combinations in an event, the combination with the smallest |∆E| is chosen for each tag mode and for D 0 andD 0 . For one event, there may be up to six ST D candidates selected. To determine the ST yield, we fit the M BC distributions of the accepted candidates after imposing mode dependent ∆E requirements. The signal is described by the MC simulated shape convolved with a double-Gaussian function accounting for the resolution difference between data and MC simulation, and the background is modeled by an ARGUS function [30]. Fit results are shown in Figs  Candidates for D 0 → K − µ + ν µ must contain two oppositely charged tracks which are identified as a kaon and muon, respectively. The muon must have the same charge as the kaon on the ST side. To suppress the peaking backgrounds from D 0 → K − π + (π 0 ), the K − µ + invariant mass (M K − µ + ) is required to be less than 1.56 GeV/c 2 , and the maximum energy of any photon that is not used in the ST selection (E max extra γ ) should be less than 0.25 GeV.
The kinematic quantity U miss ≡ E miss − | p miss |c is cal-culated for each event, where E miss and p miss are the energy and momentum of the missing particle, which can be calculated by E miss ≡ E beam − E K − − E µ + and p miss ≡ p D 0 − p K − − p µ + in the e + e − center-of-mass frame, where E K − (µ + ) and p K − (µ + ) are the energy and momentum of the kaon (muon) candidates. To improve the U miss resolution, the D 0 energy is constrained to the beam energy and p D 0 ≡ −pD0 E 2 beam − m 2 D 0 , wherê pD0 is the unit vector in the momentum direction of the STD 0 and mD0 is theD 0 nominal mass [21].
The SL decay yield is obtained from an unbinned fit to the U miss distribution of the accepted events of data, as shown in Fig. 1 (d). In the fit, the signal, the peaking background of D 0 → K − π + π 0 decay and other backgrounds are described by the corresponding MCsimulated shapes. The former two are convolved with the same Gaussian function to account for the resolution difference between data and MC simulation. All parameters are left free. The fitted signal yield is N DT = 47100±259.
The efficiencies of finding D 0 → K − µ + ν µ for different ST modes are summarized in Table 1. They are weighted by the ST yields and give the average efficiency ε SL = (58.93 ± 0.07)%. To verify the reliability of the efficiency, typical distributions of the SL decay, e.g., momenta and cos θ of K − and µ + , are checked and good consistency between data and MC simulation has been found (See Fig. 1 of Ref. [31]).
The systematic uncertainties in the BF measurement are described as follows. The uncertainty in N tot ST is taken as 0.5% by examining the changes of the fitted yields by varying the fit range, signal shape and endpoint of the ARGUS function. The efficiencies of muon and kaon tracking (PID) are studied with e + e − → γµ + µ − events and DT hadronic events, respectively. The uncertainties of tracking and PID efficiencies each are assigned as 0.3% per kaon or muon. The differences of the momentum and cos θ distributions between D 0 → K − µ + ν µ and the control samples have been considered. The uncertainty of the E max extra γ requirement is estimated to be 0.1% by analyzing the DT hadronic events. The uncertainty in the M K − µ + requirement is estimated with the alternative M K − µ + requirements of 1.51 or 1.61 GeV/c 2 , and the larger change on the BF 0.4% is taken as the systematic uncertainty. The uncertainty of the U miss fit is estimated to be 0.5% by applying different fit ranges, and signal and background shapes. The uncertainty of the limited MC size is 0.1%. The uncertainty in the MC model is estimated to be 0.1%, which is the difference between our nominal DT efficiency and that determined by reweighting the q 2 distribution of the signal MC events to data with the obtained FF parameters (See below). The total uncertainty is 1.02% obtained by adding these The BFs of D 0 → K − µ + ν µ andD 0 → K + µ −ν µ are measured separately. The results are B D 0 →K − µ + νµ = (3.433 ± 0.026 stat. ± 0.039 syst. )% and BD0 →K + µ −ν µ = (3.392 ± 0.027 stat. ± 0.034 syst. )%. The BF asymmetry is determined to be A = )%, and no asymmetry in the BFs of D 0 → K − µ + ν µ andD 0 → K + µ −ν µ decays is found. All the systematic uncertainties except for those in the E max extra γ requirement and MC model are studied separately and are not canceled out in the BF asymmetry calculation.
The D 0 → K − µ + ν µ dynamics is studied by dividing the SL candidate events into various q 2 intervals. The measured partial decay rate (PDR) in the i-th q 2 interval, ∆Γ i msr , is determined by where N i pro is the SL decay signal yield produced in the i-th q 2 interval, τ D 0 is the D 0 lifetime and N tot ST is the ST yield. The signal yield produced in the i-th q 2 interval in data is calculated by where the observed DT yield in the j-th q 2 interval N j obs is obtained from the similar fit to the corresponding U miss distribution of data (See Fig. 2 of Ref. [31]). ε is the efficiency matrix (Table 1 of Ref. [31]), which is obtained by analyzing the signal MC events and is given by where N ij rec is the DT yield generated in the j-th q 2 interval and reconstructed in the i-th q 2 interval, N j gen is the total signal yield generated in the j-th q 2 interval, and the index k denotes the k-th ST mode. The measured PDRs are shown in Fig. 2 (a) and details can be found in Table 2 of Ref. [31].
The FF is parametrized as the series expansion parameterization [32] (SEP), which has been shown to be consistent with constraints from QCD [16,18,33]. The 2-parameter SEP is chosen and is given by (6) Here, P (t) = z(t, m 2 D * s ) and Φ is given by s is the mass of the lowest lying meson D * s [21] and χ V can be obtained from dispersion relations using perturbative QCD [34].
The PDRs are fitted by assuming the ratio f K + (q 2 )/f K − (q 2 ) to be independent of q 2 , and minimizing the χ 2 constructed as where ∆Γ i exp is the expected PDR in the i-th q 2 interval, and C ij = C stat ij + C syst ij is the covariance matrix of the measured PDRs among q 2 intervals. The statistical covariance matrix (Table 3 of Ref. [31]) is constructed as The systematic covariance matrix (Table 4 of Ref. [31]) is obtained by summing all the covariance matrices for each source of systematic uncertainty. In general, it has the form where δ(∆Γ i msr ) is the systematic uncertainty of the PDR in the i-th q 2 interval. The systematic uncertainties in N tot ST , τ D 0 and E max extra γ requirement are considered to be fully correlated across q 2 intervals while others are studied separately in each q 2 interval with the same method used in the BF measurement. Figures 2(a) and (b) show the fit to the PDRs of D 0 → K − µ + ν µ and the projection to f K + (q 2 ). The goodness-offit is χ 2 /NDOF = 15.0/15, where NDOF is the number of degrees of freedom. From the fit, we obtain the product of f K + (0)|V cs | = 0.7133 ± 0.0038 stat. ± 0.0030 syst. , the first order coefficient r 1 = −1.90 ± 0.21 stat. ± 0.07 syst. and the FF ratio f K − /f K + = −0.6 ± 0.8 stat. ± 0.2 syst. . The nominal fit parameters are taken from the results obtained by fitting with the combined statistical and systematic covariance matrix, and the statistical uncertainties of the fit parameters are taken from the fit with only the statistical covariance matrix. For each parameter, the systematic uncertainty is obtained by calculating the quadratic difference of uncertainties between these two fits.
Combining B D 0 →K − µ + νµ with our previous measurement B D 0 →K − e + νe = (3.505±0.014 stat. ±0.033 syst. )% [18] gives R µ/e = 0.974 ± 0.007 stat. ± 0.012 syst. , which agrees with the theoretical calculation in a SM quark model [35]. Additionally, we determine R µ/e in each q 2 interval, as shown in Fig. 2(c), where the error bars include both statistical and the uncanceled systematic uncertainties. In the R µ/e calculation, the uncertainties in N tot ST , τ D 0 as well as the tracking and PID efficiencies of the kaon cancel. In the q 2 region (0.2, 1.5) GeV 2 /c 4 , the R µ/e is close to 1, which is consistent with the SM prediction, and no deviation larger than 2σ is observed.
In summary, by analyzing 2.93 fb −1 of data collected at center-of-mass energy √ s = 3.773 GeV with the BE-SIII detector, we present an improved measurement of the absolute BF of the SL decay D 0 → K − µ + ν µ . Our result is consistent with the PDG value [21] and improves its precision by a factor of three. By fitting the PDRs of this decay, we obtain f K + (0)|V cs | = 0.7133 ± 0.0038 stat. ± 0.0029 syst. . Using |V cs | given by CKMFitter [21] yields f K + (0) = 0.7327 ± 0.0039 stat. ± 0.0030 syst. , while using the f K + (0) calculated in LQCD [22] results in |V cs | = 0.955 ± 0.005 stat. ± 0.004 syst. ± 0.024 LQCD . These results are consistent with our previous measurements using D 0 → K − e + ν e [18] within uncertainties and are important to test the LQCD calculation of f K + (0) [22,36,37] and CKM matrix unitarity with better accuracy. We also search for LFU violation in D 0 → K − ℓ + ν ℓ decays and for an asymmetry of the D 0 → K − µ + ν µ andD 0 → K + µ −ν µ BFs, but no evidence is found with current statistics.
The BESIII collaboration thanks the staff of BEPCII and the IHEP computing center for their strong sup- data, where the dots with error bars are data, the blue solid curve is the best fit, the red dotted curve is the D 0 → K − π + π 0 peaking background and the red dashed curve is the combinatorial background. Table 1: Weighted efficiency matrix for all three single tag modes, where εij represents the efficiency of events generated in the j-th q 2 interval and reconstructed in the i-th q 2 interval.