Improved measurements of $D^{0}\to K^{-}\ell^{+}\nu_{\ell}$ and $D^{+}\to\bar{K}^{0}\ell^{+}\nu_{\ell}$

Improved measurements of semileptonic decays of charmed mesons provide important inputs to further the understanding of weak and strong interactions in the charm sector. By analyzing their decay dynamics, one can determine the product of the modulus of the Cabibbo-Kobayashi-Maskawa (CKM) matrix element $|V_{cs(d)}|$ and the hadronic transition form factor. Taking $D\to\bar{K}e^{+}\nu_{e}$ as an example, the hadronic transition form factors at zero-momentum transfer $f^{K}_{+}(0)$Lubicz:2017syv ; Chakraborty:2021qav ; Parrott:2022rgu ; FermilabLattice:2022gku ; Wu:2006rd ; Verma:2011yw ; Ivanov:2019nqd ; Faustov:2019mqr ; Ke:2023qzc can be calculated via several theoretical approaches, e.g., lattice quantum chromodynamics (LQCD) Lubicz:2017syv ; Chakraborty:2021qav ; Parrott:2022rgu ; FermilabLattice:2022gku , QCD light-cone sum rules (LCSR) Wu:2006rd , covariant light-front quark model (LFQM)Verma:2011yw , the covariant confined quark model (CCQM) Ivanov:2019nqd , and the relativistic quark model (RQM) Faustov:2019mqr . Using the value of $|V_{cs}|$ provided by the CKMFitter group pdg2022 , the hadronic transition form factor $f_{+}^{K}(0)$ can be calculated, resulting in a stringent test of the theoretical predictions. Conversely, using the $f_{+}^{K}(0)$ value predicted by theory allows the determination of $|V_{cs}|$, which is important to test CKM matrix unitarity. Furthermore, measurements of the branching fractions of $D^{0}\to K^{-}\ell^{+}\nu_{\ell}$ and $D^{+}\to\bar{K}^{0}\ell^{+}\nu_{\ell}$ ($\ell=e$ or $\mu$) are important to test lepton flavor universality and isospin conservation in $D\to K\ell^{+}\nu_{\ell}$.

Previously, the branching fractions of $D^{0}\to K^{-}\ell^{+}\nu_{\ell}$ and $D^{+}\to\bar{K}^{0}\ell^{+}\nu_{\ell}$ were measured by BESII BES:2004rav ; BES:2004obp ; BES:2006kzp , BaBar BaBar:2007zgf , Belle Belle:2006idb , CLEO-c CLEO:2005rxg ; CLEO:2005cuk ; CLEO:2007ntr ; CLEO:2009svp , and BESIII BESIII:2021mfl ; BESIII:2015tql ; BESIII:2018ccy ; BESIII:2017ylw ; BESIII:2016hko ; BESIII:2015jmz ; BESIII:2016gbw . Studies of the decay dynamics of $D\to\bar{K}\ell^{+}\nu_{\ell}$ were reported by BaBar BaBar:2007zgf , CLEO-c CLEO:2009svp , and BESIII BESIII:2015tql ; BESIII:2018ccy ; BESIII:2017ylw ; BESIII:2015jmz . The previous BESIII analysis used 2.93 fb^-1 of $e^{+}e^{-}$ collision data taken at the center-of-mass energy $\sqrt{s}=3.773$ GeV. This paper reports improved measurements of the branching fractions and decay dynamics of $D^{0}\to K^{-}\ell^{+}\nu_{\ell}$ and $D^{+}\to\bar{K}^{0}\ell^{+}\nu_{\ell}$ by using 7.93 fb^-1 of $e^{+}e^{-}$ collision data collected by the BESIII detector at $\sqrt{s}=3.773$ GeV Luminosity . Throughout this paper, charge conjugate modes are implied.

The BESIII detector Ablikim:2009aa records symmetric $e^{+}e^{-}$ collisions provided by the BEPCII storage ring Yu:IPAC2016-TUYA01 in the center-of-mass energy range from 1.84 to 4.95 GeV, with a peak luminosity of $1.1\times 10^{33}\;\text{cm}^{-2}\text{s}^{-1}$ achieved at $\sqrt{s}=3.773~{}\text{GeV}$. BESIII has collected large data samples in this energy region Ablikim:2019hff ; Li:2021iwf . The cylindrical core of the BESIII detector covers 93% of the full solid angle and consists of a helium-based multilayer drift chamber (MDC), a time-of-flight system (TOF), and a CsI(Tl) electromagnetic calorimeter (EMC), which are all enclosed in a superconducting solenoidal magnet providing a 1.0 T magnetic field. The solenoid is supported by an octagonal flux-return yoke with resistive plate muon identification counters (MUC) interleaved with steel. The main function of the MUC is to separate muons from charged pions, other hadrons and backgrounds based on their hit patterns in the instrumented flux-return yoke. The charged-particle momentum resolution at $1~{}{\rm GeV}/c$ is $0.5\%$, and the ${\rm d}E/{\rm d}x$ resolution is $6\%$ for electrons from Bhabha scattering. The EMC measures photon energies with a resolution of $2.5\%$ ($5\%$) at $1$ GeV in the barrel (end-cap) region. The time resolution in the plastic scintillator TOF barrel region is 68 ps, while that in the end-cap region was 110 ps. The end-cap TOF system was upgraded in 2015 using multi-gap resistive plate chamber technology, providing a time resolution of 60 ps etof . Approximately 67% of the data used here was collected after this upgrade.

Simulated samples produced with geant4-based geant4 Monte Carlo (MC) software, which includes the geometric description Huang:2022wuo of the BESIII detector and the detector response, are used to determine detection efficiencies and to estimate backgrounds. The simulation models the beam energy spread and initial state radiation (ISR) in the $e^{+}e^{-}$ annihilations with the generator kkmc ref:kkmc . Signal MC samples of the decays $D\to\bar{K}\ell^{+}\nu_{\ell}$ are simulated with a specific two-parameter series expansion model Becher:2005bg . The background is studied using an inclusive MC sample that consists of the production of $D\bar{D}$ pairs from the $\psi(3770)$ (including quantum coherence for the neutral $D$ channels), the non-$D\bar{D}$ decays of the $\psi(3770)$, the ISR production of the charmonium states, and the continuum processes. These processes are also generated with kkmc. The known decay modes are modeled by evtgen ref:evtgen with branching fractions taken from the Particle Data Group (PDG) pdg2022 , while the remaining unknown charmonium decays are modeled with lundcharm ref:lundcharm . Final state radiation from charged final-state particles is incorporated using photos photos .

At $\sqrt{s}=3.773$ GeV, the $D$ and $\bar{D}$ mesons are produced in pairs from the $e^{+}e^{-}\to\psi(3770)\to D\bar{D}$ process, where $D$ stands for $D^{0}$ or $D^{+}$. This property allows us to do absolute branching fraction measurement with the well established double-tag (DT) method DTmethod . In this method, the single-tag (ST) candidate events are selected by reconstructing a $\bar{D}^{0}$ in the six hadronic final states $\bar{D}^{0}\to K^{+}\pi^{-}$, $K^{+}\pi^{-}\pi^{0}$, $K^{+}\pi^{-}\pi^{-}\pi^{+}$, $K^{+}\pi^{-}\pi^{0}\pi^{0}$, $K^{+}\pi^{-}\pi^{-}\pi^{+}\pi^{0}$, and $K^{0}_{S}\pi^{+}\pi^{-}$, or a $D^{-}$ in the six hadronic final states $D^{-}\to K^{+}\pi^{-}\pi^{-}$, $K^{0}_{S}\pi^{-}$, $K^{+}\pi^{-}\pi^{-}\pi^{0}$, $K^{0}_{S}\pi^{-}\pi^{0}$, $K^{0}_{S}\pi^{+}\pi^{-}\pi^{-}$, and $K^{+}K^{-}\pi^{-}$. These inclusively selected candidates are referred to as ST $\bar{D}$ mesons. In the presence of the ST $\bar{D}$ mesons, candidates for the signal decays are selected to form DT events. The branching fraction of the signal decay is determined by

$${\mathcal{B}}_{\rm sig}=N_{\mathrm{DT}}/(N_{\mathrm{ST}}^{\rm tot}\cdot\bar{\varepsilon}_{\rm sig}),$$

(1)

where $N_{\rm DT}$ is the total DT yield in data, $N_{\mathrm{ST}}^{\rm tot}$ is the total ST yield

$$N_{\mathrm{ST}}^{\rm tot}=\sum_{i=1}^{6}N_{\mathrm{ST}}^{i},$$

(2)

where $N_{\rm ST}^{i}$ is the ST yield of tag mode $i$, and $\bar{\varepsilon}_{\rm sig}$ is the weighted efficiency of detecting the semileptonic decay, calculated by

$$\bar{\varepsilon}_{\rm sig}=\sum_{i=1}^{6}\frac{N_{\mathrm{ST}}^{i}\varepsilon_{\rm sig}^{i}}{N_{\mathrm{ST}}^{\rm tot}}.$$

(3)

Here, $\varepsilon_{\rm sig}^{i}=\varepsilon_{\rm DT}^{i}/\varepsilon_{\rm ST}^{i}$, is the efficiency of detecting the semileptonic decay in the presence of the ST $\bar{D}$ meson of tag mode $i$, where $\varepsilon_{\rm DT}^{i}$ and $\varepsilon_{\rm ST}^{i}$ are the DT efficiency and ST efficiency, respectively.

For each charged track (except for those used for $K^{0}_{S}$ reconstruction), the polar angle ($\theta$) with respect to the MDC axis is required to satisfy $|\cos\theta|<0.93$, and the point of closest approach to the interaction point must be within 1 cm in the plane perpendicular to the MDC axis, $|V_{xy}|$, and within 10 cm along the MDC axis, $|V_{z}|$. Charged tracks are identified by using the $\mathrm{d}E/\mathrm{d}x$ and TOF information, with which the combined confidence levels under the pion and kaon hypotheses are computed separately. Charged tracks are assigned to the particle type that has the higher probability.

Candidate $K_{S}^{0}$ mesons are reconstructed from pairs of oppositely charged tracks. For these two tracks, their polar angles are required to satisfy $|\cos\theta|<0.93$ and the distance of closest approach to the interaction point is required to be less than 20 cm along the MDC axis. There is no requirement on the distance of closest approach in the transverse plane, and no particle identification (PID) criteria are required. The two charged tracks are constrained to originate from the same vertex, which is required to be away from the interaction point by a flight distance of at least twice the vertex resolution. The quality of the vertex fit is ensured by the requirement of $\chi^{2}<100$, and the invariant mass of the $\pi^{+}\pi^{-}$ pair is required to be within $(0.487,0.511)$ GeV/$c^{2}$.

Neutral pion candidates are reconstructed via the $\pi^{0}\to\gamma\gamma$ decay. Photon candidates are identified from EMC showers. The EMC time difference from the event start time is required to be within $(0,700)$ ns. The energy deposited in the EMC is required to be greater than 25 MeV in the barrel region ($|\cos\theta|<0.80$) and 50 MeV in the end-cap region ($0.86<|\cos\theta|<0.92$). The opening angle between the photon candidate and the nearest charged track in the EMC is required to be greater than $10^{\circ}$. For any $\pi^{0}$ candidate, the invariant mass of the photon pair is required to be within $(0.115,0.150)$ GeV$/c^{2}$. To improve the momentum resolution, a mass-constrained (1C) fit to the known $\pi^{0}$ mass pdg2022 is imposed on the photon pair, and the $\chi^{2}$ of the 1C kinematic fit is required to be less than 50. The four-momentum of the $\pi^{0}$ candidate from this kinematic fit is used for further analysis.

For the two-body tag mode of $\bar{D}^{0}\to K^{+}\pi^{-}$, the backgrounds originating from cosmic rays, Bhabha and dimuon events are vetoed with the following procedure defined in Ref. deltakpi . It is required that the TOF time difference between the two charged tracks is less than 5 ns, and at least one EMC shower with energy greater than 50 MeV or at least one additional charged track detected in the MDC survives in each event. Further, it is required that the two charged tracks are not consistent with being a muon pair or an electron-positron pair, identified using the TOF, $\mathrm{d}E/\mathrm{d}x$, EMC, and MUC measurement information with the combined confidence levels $\mathcal{L}_{e}$, $\mathcal{L}_{\mu}$, $\mathcal{L}_{K}$, and $\mathcal{L}_{\pi}$ for electron, muon, kaon, and pion hypotheses, respectively. To be identified as an electron, $\mathcal{L}_{e}$ is required to be greater than 0 and larger than $\mathcal{L}_{K}$, $\mathcal{L}_{\pi}$, as well as $0.8\cdot(\mathcal{L}_{e}+\mathcal{L}_{\pi}+\mathcal{L}_{K})$. To identify a track as a muon, $\mathcal{L}_{\mu}$ is required to be greater than 0, the deposited energy in the EMC should fall within the range of 0.15 to 0.30 GeV, and the hit depth in the MUC needs to be either greater than $(80\times|p_{\rm trk}|-60)$ cm or greater than 40 cm, where $p_{\rm trk}$ is the track momentum.

To identify the ST $\bar{D}$ mesons, we use two kinematic variables: the energy difference $\Delta E\equiv E_{\bar{D}}-E_{\mathrm{beam}}$ and the beam-constrained mass $M_{\rm BC}\equiv\sqrt{E_{\mathrm{beam}}^{2}/c^{4}-|\vec{p}_{\bar{D}}|^{2}/c^{2}}$, where $E_{\mathrm{beam}}$ is the beam energy, and $E_{\bar{D}}$ and $\vec{p}_{\bar{D}}$ are the total energy and momentum of the ST $\bar{D}$ meson in the $e^{+}e^{-}$ center-of-mass frame, respectively. If there are multiple $\bar{D}$ candidates for a specific tag mode, the one giving the least $|\Delta E|$ is chosen for further analysis. To suppress combinatorial backgrounds in the $M_{\rm BC}$ distribution, tag dependent $\Delta E$ requirements are imposed on the ST candidates. The detailed $\Delta E$ requirements and the ST efficiencies estimated by analyzing the inclusive MC sample are summarized in Table 1.

For each tag mode, the yield of ST $\bar{D}$ mesons is obtained by the maximum likelihood fit to the corresponding $M_{\rm BC}$ distribution. In the fit, the $\bar{D}$ signal shape is described by the sum of an MC-simulated signal shape, made by RooHistPdf in Rootclass RootClass , convolved with a double-Gaussian resolution function plus a single-Gaussian function, to account for resolution difference between data and MC simulation and initial state radiation (ISR) effects. The parameters of those functions are free. The background shape is described by an ARGUS function argus with the endpoint fixed at the $E_{\rm beam}$ value. Figure 1 shows the results of the fits to the $M_{\rm BC}$ distributions of the accepted ST candidates in data for different tag modes. The candidates with $M_{\rm BC}$ within $(1.859,1.873)$ GeV/$c^{2}$ for $\bar{D}^{0}$ tags and $(1.863,1.877)$ GeV/$c^{2}$ for $D^{-}$ tags are kept for further analysis. Summing over the tag modes gives the total yields of ST $\bar{D}^{0}$ and $D^{-}$ mesons ($N_{\rm ST}^{\rm tot}$) to be $(7922.7\pm 3.4_{\rm stat.})\times 10^{3}$ and $(4135.4\pm 2.4_{\rm stat.})\times 10^{3}$, respectively.

The candidates for $D^{0}\to K^{-}e^{+}\nu_{e}$, $D^{0}\to K^{-}\mu^{+}\nu_{\mu}$, $D^{+}\to\bar{K}^{0}e^{+}\nu_{e}$, and $D^{+}\to\bar{K}^{0}\mu^{+}\nu_{\mu}$ are selected from the remaining tracks in the presence of the ST $\bar{D}$ candidates. Candidates for $K^{-}$ and $K_{S}^{0}$ are selected with the same criteria as those used in the ST selection. The positron and muon are identified using the TOF, $\mathrm{d}E/\mathrm{d}x$, and EMC measurements with the combined confidence levels $\mathcal{L}_{e}$, $\mathcal{L}_{\mu}$, $\mathcal{L}_{K}$, and $\mathcal{L}_{\pi}$, which are calculated for electron, muon, kaon, and pion hypotheses, respectively. The positron candidate is required to satisfy $\mathcal{L}_{e}>0.8\cdot(\mathcal{L}_{e}+\mathcal{L}_{\pi}+\mathcal{L}_{K})$ and $\mathcal{L}_{e}>0.001$. The muon candidate is required to satisfy $\mathcal{L}_{\mu}>\mathcal{L}_{e}$ and $\mathcal{L}_{\mu}>0.001$, and the energy of the muon deposited in the EMC is required to be within $(0.1,0.3)$ GeV.

To suppress backgrounds associated with hadronic $D$ decays, it is required that there are no additional good charged tracks on the signal side ($N_{\rm extra}^{\rm trk}=0$). To reject the backgrounds from hadronic decays involving $\pi^{0}$, the maximum energy of extra photons ($E_{\text{extra~{}}\gamma}^{\rm max}$) not used in the event reconstruction is required to be less than 0.25 GeV. Requirements on the $\bar{K}$ and $\ell^{+}$ ($M_{\bar{K}\ell}$) invariant mass, which are $M_{K^{-}e^{+}}<1.83$ GeV/$c^{2}$ for $D^{0}\to K^{-}e^{+}\nu_{e}$, $M_{K^{-}\mu^{+}}<1.50$ GeV/$c^{2}$ for $D^{0}\to K^{-}\mu^{+}\nu_{\mu}$, $M_{K_{S}^{0}e^{+}}<1.84$ GeV/$c^{2}$ for $D^{+}\to\bar{K}^{0}e^{+}\nu_{e}$, and $M_{K_{S}^{0}\mu^{+}}<1.56$ GeV/$c^{2}$ for $D^{+}\to\bar{K}^{0}\mu^{+}\nu_{\mu}$, are used to suppress the backgrounds associated with the misidentification between $\pi^{+}$ and $\ell^{+}$.

The neutrino is not detectable by the BESIII detector. In order to determine the number of semileptonic $D$ candidates, we define $U_{\mathrm{miss}}\equiv E_{\mathrm{miss}}-|\vec{p}_{\mathrm{miss}}|c$, where $E_{\mathrm{miss}}$ and $\vec{p}_{\mathrm{miss}}$ are the missing energy and momentum of a DT event in the $e^{+}e^{-}$ center-of-mass frame, respectively. They are calculated by $E_{\mathrm{miss}}\equiv E_{\mathrm{beam}}-E_{\bar{K}}-E_{\ell^{+}}$ and $\vec{p}_{\mathrm{miss}}\equiv\vec{p}_{D}-\vec{p}_{\bar{K}}-\vec{p}_{\ell^{+}}$, where $E_{\bar{K}(\ell^{+})}$ and $\vec{p}_{\bar{K}(\ell^{+})}$ are the measured energy and momentum of the $\bar{K}(\ell^{+})$ candidate in an event. Here to improve the $U_{\mathrm{miss}}$ resolution, $\vec{p}_{D}$ is evaluated as $\vec{p}_{D}=-\hat{p}_{\bar{D}}\sqrt{E_{\mathrm{beam}}^{2}/c^{2}-m_{\bar{D}}^{2}c^{2}}$, where $\hat{p}_{\bar{D}}$ is the unit vector in the momentum direction of the ST $\bar{D}$ meson and $m_{\bar{D}}$ is the known $\bar{D}$ mass pdg2022 .