Observation of the Semileptonic $D^+$ Transition into an Axial-Vector Meson $\bar K_1(1270)^0$

TL;DR

This paper reports the first observation of the semileptonic decay of the $D^+$ meson into an axial-vector meson ${ar{K}}_{1}(1270)^0$ with a measured branching fraction, using data from the BESIII detector.

Contribution

First observation of the $D^+ ightarrow ar{K}_1(1270)^0 e^+ u_e$ decay with a precise measurement of its branching fraction.

Findings

Decay observed with >10σ significance.

Branching fraction measured as (2.30 ± 0.26 ± 0.18 ± 0.25)×10^{-3}.

First experimental evidence of this semileptonic transition.

Abstract

By analyzing a 2.93~$\rm fb^{-1}$ data sample of $e^+e^-$ collisions, recorded at a center-of-mass energy of 3.773 $\rm \,GeV$ with the BESIII detector operated at the BEPCII collider, we have searched for the semileptonic $D^+$ transition into the axial-vector meson ${\bar{K}}_{1}(1270)^{0}$. The $D^{+} \rightarrow {\bar{K}}_{1}(1270)^{0}e^{+}\nu_{e}$ decay is observed for the first time with a statistical significance greater than $10\sigma$. Its decay branching fraction is determined to be ${\mathcal B}[D^+\to \bar K_1(1270)^0 e^+\nu_e]=(2.30\pm0.26\pm 0.18 \pm 0.25)\times10^{-3}$, where the first and second uncertainties are statistical and systematic, respectively, and the third originates from the input branching fraction of ${\bar{K}}_{1}(1270)^{0} \rightarrow K^{-} \pi^{+} \pi^{0}$.

Equations

$${\mathcal{B}}_{\mathrm{SL}}=N_{\mathrm{DT}}/(N_{\mathrm{ST}}^{\mathrm{tot}}\cdot {\epsilon }_{\mathrm{SL}}),$$

$${\mathcal{B}}_{\mathrm{SL}}\cdot {\mathcal{B}}_{\mathrm{sub}}=(1.06\pm 0.12_{-0.10}^{+0.08})\times 10^{-3},$$

$${\mathcal{B}}_{\mathrm{SL}}=(2.30\pm 0.26_{-0.21}^{+0.18}\pm 0.25)\times 10^{-3},$$

Full text

††footnotetext: Corresponding author: liuke@ihep.ac.cn

Studies of semileptonic (SL) $D$ transitions, mediated via $c\to s(d)\ell^{+}\nu_{\ell}$ at the quark level, are important for the understanding of nonperturbative strong-interaction dynamics in weak decays isgw ; isgw2 . Those transitions into S-wave states have been extensively studied in theory and experiment. However, there is still no experimental confirmation of the predicted transitions into P-wave states.

In the quark model, the physical mass eigenstates of the strange axial-vector mesons, $K_{1}(1270)$ and $K_{1}(1400)$, are mixtures of the ${}^{1}\rm P_{1}$ and ${}^{3}\rm P_{1}$ states with a mixing angle $\theta_{K_{1}}$. These mesons have been thoroughly studied via $\tau$, $B$, $D$, $\psi(3686)$ and $J/\psi$ decays, as well as via $Kp$ scattering 4 ; 5 ; 6 ; 7 ; 8 ; 14 ; jpsi ; chic0 ; 1 ; 2 . Nevertheless, the value of $\theta_{K_{1}}$ is still very controversial in various phenomenological analyses 18 ; 19 ; 20 ; 21 ; 22 ; 24 ; 26 ; 27 . Studies of the SL $D$ transitions into $\bar{K}_{1}(1270)$ provide important insight into the mixing angle $\theta_{K_{1}}$. The improved knowledge of $\theta_{K_{1}}$ is essential for theoretical calculations describing the decays of $\tau$ 18 , $B$ 20 ; B-K1g , and $D$ D-AP ; D-AP-1 particles into strange axial-vector mesons, and for investigations in the field of hadron spectroscopy Burakovsky .

Earlier quantitative predictions for the branching fractions (BFs) of $D^{0(+)}\to\bar{K}_{1}(1270)e^{+}\nu_{e}$ were derived from the Isgur-Scora-Grinstein-Wise (ISGW) quark model isgw and its update, ISGW2 isgw2 . ISGW2 implies that the BFs of $D^{0(+)}\to\bar{K}_{1}(1270)e^{+}\nu_{e}$ are about 0.1 (0.3)%. However, the model ignores the mixing between ${}^{1}\rm P_{1}$ and ${}^{3}\rm P_{1}$ states. Recently, the rates of these decays were calculated with three-point QCD sum rules (3PSR) Khosravi , covariant light-front quark model (CLFQM) theory , and light-cone QCD sum rules (LCSR) formfactor . In general, the predicted BFs range from $10^{-3}$ to $10^{-2}$ Khosravi ; theory ; formfactor , and are sensitive to $\theta_{K_{1}}$ and its sign. Measurements of $D^{0(+)}\to\bar{K}_{1}(1270)e^{+}\nu_{e}$ will be critical to distinguish between theoretical calculations, to explore the nature of strange axial-vector mesons, and to understand the weak-decay mechanisms of $D$ mesons.

Currently, there is very little experimental information available about semileptonic $D$ decays into axial-vector mesons, with the only result being the reported evidence for the process $D^{0}\to K_{1}(1270)^{-}e^{+}\nu_{e}$ from the CLEO Collaboration cleores . This Letter presents the first observation of $D^{+}\to\bar{K}_{1}(1270)^{0}e^{+}\nu_{e}$ charge by using an $e^{+}e^{-}$ data sample corresponding to an integrated luminosity of 2.93 fb*-1* lum recorded at a center-of-mass energy of $\sqrt{s}=3.773$ $\rm\,GeV$ with the BESIII detector Ablikim:2009aa .

Details about the design and performance of the BESIII detector are given in Ref. Ablikim:2009aa . Simulated samples produced with the geant4-based geant4 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 ref:kkmc . The inclusive MC samples consist of the production of the $D\bar{D}$ pairs, the non-$D\bar{D}$ decays of the $\psi(3770)$, the ISR production of the $J/\psi$ and $\psi(3686)$ states, and the continuum processes incorporated in kkmc ref:kkmc . The known decay modes are modeled with evtgen ref:evtgen using BFs taken from the Particle Data Group pdg2018 , and the remaining unknown decays from the charmonium states with lundcharm ref:lundcharm . The final-state radiation (FSR) from charged final-state particles are incorporated with the photos package photos . The $D^{+}\to\bar{K}_{1}(1270)^{0}e^{+}\nu_{e}$ decay is simulated with the ISGW2 model MPM , the $\bar{K}_{1}(1270)^{0}$ is set to decay into all possible processes containing the $K^{-}\pi^{+}\pi^{0}$ combination. The resonance shape of $\bar{K}_{1}(1270)^{0}$ is parameterized by a relativistic Breit-Wigner function, and the mass and width of $\bar{K}_{1}(1270)^{0}$ are fixed at the world-average values 1272$\pm$7 MeV and 90$\pm$20 MeV, respectively pdg2018 .

The measurement employs the $e^{+}e^{-}\to\psi(3770)\to D^{+}D^{-}$ decay chain. The $D^{-}$ mesons are reconstructed by their hadronic decays 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 events are referred to as single-tag (ST) $D^{-}$ mesons. In the presence of the ST $D^{-}$ mesons, candidate $D^{+}\to\bar{K}_{1}(1270)^{0}e^{+}\nu_{e}$ decays are selected to form double-tag (DT) events. The BF of $D^{+}\to\bar{K}_{1}(1270)^{0}e^{+}\nu_{e}$ is given by

[TABLE]

where $N_{\rm ST}^{\rm tot}$ and $N_{\rm DT}$ are the ST and DT yields in the data sample, $\varepsilon_{\rm SL}=\Sigma_{i}[(\varepsilon^{i}_{\rm DT}\cdot N^{i}_{\rm ST})/(\varepsilon^{i}_{\rm ST}\cdot N^{\rm tot}_{\rm ST})]$ is the efficiency of detecting the SL decay in the presence of the ST $D^{-}$ meson. Here $i$ denotes the tag mode, and $\varepsilon_{\rm ST}$ and $\varepsilon_{\rm DT}$ are the ST and DT efficiencies of selecting the ST and DT candidates, respectively.

We use the same selection criteria as discussed in Refs. epjc76_369 ; cpc40_113001 ; prl121_171803 . All charged tracks are required to be within a polar-angle ($\theta$) range of $|\rm{cos\theta}|<0.93$. All of them, except for those from $K^{0}_{S}$ decays, must originate from an interaction region defined by $V_{xy}<$ 1 cm and $|V_{z}|<$ 10 cm. Here, $V_{xy}$ and $|V_{z}|$ denote the distances of closest approach of the reconstructed track to the interaction point (IP) in the $xy$ plane and the $z$ direction (along the beam), respectively.

Particle identification (PID) of charged kaons and pions is performed using the specific ionization energy loss ($dE/dx$) measured by the main drift chamber (MDC) and the time-of-flight. Positron PID also uses the measured information from the electromagnetic calorimeter (EMC). The combined confidence levels under the positron, pion, and kaon hypotheses ($CL_{e}$, $CL_{\pi}$ and $CL_{K}$, respectively) are calculated. Kaon (pion) candidates are required to satisfy $CL_{K}>CL_{\pi}$ ($CL_{\pi}>CL_{K}$). Positron candidates are required to satisfy $CL_{e}>0.001$ and $CL_{e}/(CL_{e}+CL_{\pi}+CL_{K})>0.8$. To reduce the background from hadrons and muons, the positron candidate is further required to have a deposited energy in the EMC greater than 0.8 times its momentum in the MDC.

$K^{0}_{S}$ candidates are reconstructed from two oppositely charged tracks satisfying $|V_{z}|<$ 20 cm. The two charged tracks are assigned as $\pi^{+}\pi^{-}$ without imposing further PID criteria. They are constrained to originate from a common vertex and are required to have an invariant mass within $|M_{\pi^{+}\pi^{-}}-M_{K_{S}^{0}}|<$ 12 MeV$/c^{2}$, where $M_{K_{S}^{0}}$ is the $K^{0}_{S}$ nominal mass pdg2018 . The decay length of the $K^{0}_{S}$ candidate is required to be greater than twice the vertex resolution away from the IP.

Photon candidates are selected using the information from the EMC. It is required that the shower time is within 700 ns of the event start time, the shower energy be greater than 25 (50) MeV if the crystal with the maximum deposited energy in that cluster is in the barrel (end-cap) region Ablikim:2009aa , and the opening angle between the candidate shower and any charged tracks is greater than $10^{\circ}$. Neutral $\pi^{0}$ candidates are selected from the photon pairs with the invariant mass within $(0.115,0.150)$ GeV$/c^{2}$. The momentum resolution of the accepted photon pair is improved by a kinematic fit, which constrains the $\gamma\gamma$ invariant mass to the $\pi^{0}$ nominal mass pdg2018 .

The ST $D^{-}$ mesons are distinguished from the combinatorial backgrounds by two variables: the energy difference $\Delta E=E_{D^{-}}-E_{\rm beam}$ and the beam-energy constrained mass $M_{\rm BC}=\sqrt{E^{2}_{\rm beam}-|\vec{p}_{D^{-}}|^{2}}$, where $E_{\rm beam}$ is the beam energy, and $\vec{p}_{D^{-}}$ and $E_{D^{-}}$ are the measured momentum and energy of the ST candidate in the $e^{+}e^{-}$ center-of-mass frame, respectively. For each tag mode, only the one with the minimum $|\Delta E|$ is kept. The combinatorial backgrounds in the $M_{\rm BC}$ distributions are suppressed by requiring $\Delta E$ within $(-55,+40)$ MeV for the tag modes involving a $\pi^{0}$, and $(-25,+25)$ MeV for the other tag modes.

Figure 1 shows the $M_{\rm BC}$ distributions of the accepted ST candidates in the data sample for various tag modes. The ST yield for each tag mode is obtained by performing a maximum-likelihood fit to the corresponding $M_{\rm BC}$ distribution. In the fits, the $D^{-}$ signal is modeled by an MC-simulated $M_{\rm BC}$ shape convolved with a double-Gaussian function and the combinatorial-background shape is described by an ARGUS function ARGUS . The candidates in the $M_{\rm BC}$ signal region, $(1.863,1.877)$ GeV/$c^{2}$, are kept for further analysis. The total ST yield is $N^{\rm tot}_{\rm ST}=1522474\pm 2215$, where the uncertainty is statistical.

In the analysis of the particles recoiling against the ST $D^{-}$ mesons, candidate events for the $D^{+}\to\bar{K}_{1}(1270)^{0}e^{+}\nu_{e}$ channel are selected from the remaining tracks that have not been used for the ST reconstruction. The $\bar{K}_{1}(1270)^{0}$ meson is reconstructed using its dominant decay $\bar{K}_{1}(1270)^{0}\to K^{-}\pi^{+}\pi^{0}$. It is required that there are only three good charged tracks available for this selection. One of the tracks with charge opposite to that of the $D^{-}$ tag is identified as the positron. The other two oppositely charged tracks are identified as a kaon and a pion, according to their PID information. Moreover, the kaon candidate must have charge opposite to that of the positron. Other selection criteria, which have been optimized by analyzing the inclusive MC samples, are as follows. To effectively veto the backgrounds associated with wrongly paired photons, the $\pi^{0}$ candidates must have a momentum greater than 0.15 GeV/$c$ and a decay angle $|\cos\theta_{\rm decay,\pi^{0}}|=|E_{\gamma_{1}}-E_{\gamma_{2}}|/|\vec{p}_{\pi^{0}}|$ less than 0.8. Here, $E_{\gamma_{1}}$ and $E_{\gamma_{2}}$ are the energies of $\gamma_{1}$ and $\gamma_{2}$, and $\vec{p}_{\pi^{0}}$ is the momentum of the $\pi^{0}$ candidate. To suppress the potential backgrounds from the hadronic decays $D^{+}\to K^{-}\pi^{+}\pi^{+}\pi^{0}$, the invariant mass of the $K^{-}\pi^{+}\pi^{0}e^{+}$ combination, $M_{K^{-}\pi^{+}\pi^{0}e^{+}}$, is required to be smaller than 1.78 GeV/$c^{2}$.

Information concerning the undetectable neutrino is inferred by the kinematic quantity $U_{\mathrm{miss}}\equiv E_{\mathrm{miss}}-|\vec{p}_{\mathrm{miss}}|$, where $E_{\mathrm{miss}}$ and $\vec{p}_{\mathrm{miss}}$ are the missing energy and momentum of the SL candidate, respectively, calculated by $E_{\mathrm{miss}}\equiv E_{\mathrm{beam}}-\Sigma_{j}E_{j}$ and $\vec{p}_{\mathrm{miss}}\equiv\vec{p}_{D^{+}}-\Sigma_{j}\vec{p}_{j}$ in the $e^{+}e^{-}$ center-of-mass frame. The index $j$ sums over the $K^{-}$, $\pi^{+}$, $\pi^{0}$ and $e^{+}$ of the signal candidate, and $E_{j}$ and $\vec{p}_{j}$ are the energy and momentum of the $j$th particle, respectively. To improve the $U_{\mathrm{miss}}$ resolution, the $D^{+}$ energy is constrained to the beam energy and $\vec{p}_{D^{+}}\equiv-\hat{p}_{D^{-}}\sqrt{E_{\mathrm{beam}}^{2}-m_{D^{+}}^{2}}$, where $\hat{p}_{D^{-}}$ is the unit vector in the momentum direction of the ST $D^{-}$, and $m_{D^{+}}$ is the $D^{+}$ nominal mass pdg2018 . To partially recover the effects of FSR and bremsstrahlung (FSR recovery), the four-momenta of photon(s) within $5^{\circ}$ of the initial positron direction are added to the positron four-momentum measured by the MDC.

Events that originate from the process $D^{+}\to\bar{K}^{*}(892)^{0}[\to K^{-}\pi^{+}]e^{+}\nu_{e}$, in which a fake $\pi^{0}$ is wrongly associated to the signal decay, form a peaking background around $+0.02$ GeV in the $U_{\rm miss}$ distribution and around 1.15 GeV/$c^{2}$ in the $M_{K^{-}\pi^{+}\pi^{0}}$ distribution. To suppress these backgrounds, we define an alternative kinematic quantity $U_{\mathrm{miss}}^{\prime}\equiv E_{\mathrm{miss}}^{\prime}-|\vec{p}_{\mathrm{miss}}^{\prime}|$, where $E_{\mathrm{miss}}^{\prime}\equiv E_{\mathrm{beam}}-\Sigma_{j}E_{j}$ and $\vec{p}_{\mathrm{miss}}^{\prime}\equiv\vec{p}_{D^{+}}-\Sigma_{j}\vec{p}_{j}$, and $j$ only sums over the $K^{-}$, $\pi^{+}$ and $e^{+}$ candidates of the signal candidate. Since these backgrounds form an obvious peak around zero in the $U_{\mathrm{miss}}^{\prime}$ distribution, the $U_{\rm miss}^{\prime}$ values of the SL candidates are required to lie outside $(-0.09,0.03)$ GeV.

Figure 2 (a) shows the distribution of $M_{K^{-}\pi^{+}\pi^{0}}$ vs. $U_{\rm miss}$ of the accepted $D^{+}\to K^{-}\pi^{+}\pi^{0}e^{+}\nu_{e}$ candidate events in the data sample after combining all tag modes. A clear signal, which concentrates around 1.27 GeV/$c^{2}$ in the $M_{K^{-}\pi^{+}\pi^{0}}$ distribution and around zero in the $U_{\rm miss}$ distribution, can be seen. The DT yield is obtained from a two-dimensional (2-D) unbinned extended maximum-likelihood fit of the data presented by the distribution in Fig. 2(a). In the fit, the 2-D signal shape is described by the MC-simulated shape extracted from the signal MC events of $D^{+}\to\bar{K}_{1}(1270)^{0}e^{+}\nu_{e}$. The 2-D background shape is modeled by the MC-simulated shape obtained from the inclusive MC samples and the number of background events is a free parameter in the fit. The smooth 2-D probability density functions of signal and background are modeled by the corresponding MC-simulated shape roofit ; class . The projections of the 2-D fit on the $M_{K^{-}\pi^{+}\pi^{0}}$ and $U_{\rm miss}$ distributions are shown in Figs. 2 (b) and 2 (c). In the fit, we ignore the contributions from non-resonant decays $D^{+}\to K^{-}\pi^{+}\pi^{0}e^{+}\nu_{e}$, $\bar{K}^{*}(892)^{0}\pi^{0}e^{+}\nu_{e}$, $K^{*}(892)^{-}\pi^{+}e^{+}\nu_{e}$ and $K^{-}\rho(770)^{+}e^{+}\nu_{e}$, as well as the possible interference between them due to the low significance of these contributions with the limited size of the data set. The two decays $D^{+}\to\bar{K}_{1}(1400)^{0}e^{+}\nu_{e}$ and $D^{+}\to\bar{K}^{*}(1430)^{0}e^{+}\nu_{e}$ are indistinguishable, and as no significant contribution is found from either source, these components are not included in the fit. From the fit, we obtain the DT yield of $N_{\rm DT}=119.7\pm 13.3$, where the uncertainty is statistical. The statistical significance of the signal is estimated to be greater than 10$\sigma$, by comparing the likelihoods with and without the signal components included, and taking the change in the number of degrees of freedom into account.

For each tag mode, the DT efficiency is estimated with the corresponding signal MC events. The average signal efficiency is determined to be $\varepsilon_{\rm SL}=0.0742\pm 0.0007$. Compared to $\epsilon_{\rm SL}$, the signal efficiencies for individual tag modes vary within $\pm 10\%$. The reliability of the MC simulation is tested by examining typical distributions of the SL candidate events. The data distributions of momenta and $\cos\theta$ of $K^{-}$, $\pi^{+}$, $\pi^{0}$ and $e^{+}$ are consistent with those of MC simulations.

By inserting $N_{\rm DT}$, $\varepsilon_{\rm SL}$, and $N_{\rm ST}^{\rm tot}$ into Eq. (1), we determine the product of ${\mathcal{B}}_{\rm SL}$ and the BF of $\bar{K}_{1}(1270)^{0}\to K^{-}\pi^{+}\pi^{0}$ (${\mathcal{B}}_{\rm sub}$) to be

[TABLE]

where the first and second uncertainties are statistical and systematic, respectively.

The systematic uncertainties in the BF measurement, which are assigned relative to the measured BF, are discussed below. The DT method ensures that most uncertainties arising from the ST selection cancel. The uncertainty from the ST yield is assigned to be 0.5% epjc76_369 ; cpc40_113001 ; prl121_171803 , by examining the relative change in the yield between data and MC simulation after varying the $M_{\rm BC}$ fit range, the signal shape, and the endpoint of the ARGUS function.

The uncertainties associated with the efficiencies of $e^{+}$ tracking (PID), $K^{-}$ tracking (PID), $\pi^{+}$ tracking (PID) and $\pi^{0}$ reconstruction are investigated using data and MC samples of $e^{+}e^{-}\to\gamma e^{+}e^{-}$ events and DT $D\bar{D}$ hadronic events. Small differences between the data and MC efficiencies are found, which are $-(0.03\pm 0.15)\%$, $+(0.94\pm 0.27)\%$, $+(2.63\pm 0.32)\%$, $-(0.14\pm 0.18)\%$, $+(0.03\pm 0.13)\%$, $-(0.08\pm 0.18)\%$ for $e^{+}$ tracking, $e^{+}$ PID, $K^{-}$ tracking, $K^{-}$ PID, $\pi^{+}$ tracking and $\pi^{+}$ PID, respectively. The MC efficiency is then corrected by these differences and used to determine the central value of the BF. In the studies of $e^{+}$ tracking (PID) efficiencies, the 2-D (momentum and $\cos\theta$) tracking efficiencies of data and MC simulation of $e^{+}e^{-}\to\gamma e^{+}e^{-}$ events are re-weighted to match those of $D^{+}\to\bar{K}_{1}(1270)^{0}e^{+}\nu_{e}$ decays. After corrections, we assign the uncertainties associated with the $e^{+}$ tracking (PID), $K^{-}$ tracking (PID), $\pi^{+}$ tracking (PID) and $\pi^{0}$ reconstruction to be 1.0% (1.0%), 1.0% (0.5%), 0.5% (0.5%) and 2.0%, respectively.

The uncertainty associated with the $M_{K^{-}\pi^{+}\pi^{0}e^{+}}$ requirement is estimated by varying the requirement by $\pm 0.05$ GeV/$c^{2}$, and the largest change on the BF, 0.9%, is taken as the systematic uncertainty. Similarly, the systematic uncertainty in the $U_{\rm miss}^{\prime}$ requirement is estimated to be 1.7% by varying the corresponding selection window by $\pm 0.01$ GeV. The uncertainty of the input BFs of $\bar{K}_{1}(1270)^{0}$ is estimated by changing the BF of each subdecay by $\pm 1\sigma$. The largest variation in the detection efficiency, 0.5%, is assigned as the related systematic uncertainty. The uncertainty of the 2-D fit is estimated to be ${}^{+7.0\%}_{-8.2\%}$ by examining the BF changes with different fit ranges, signal shapes (dominated by varying the width of $\bar{K}_{1}(1270)^{0}$ by $\pm 1\sigma$) and background shapes. The uncertainty arising from background shapes is mainly due to unknown non-resonant decays, and is assigned as the change of the fitted DT yield when they are fixed by referring to the well known non-resonant fraction in $D^{+}\to\bar{K}^{*}(892)^{0}e^{+}\nu_{e}$ non-resonance . The uncertainty arising from the limited size of the MC samples is 1.0%.

The uncertainty due to FSR recovery is evaluated to be 1.3% which is the change of the BF when varying the FSR recovery angle to be $10^{\circ}$. The total systematic uncertainty is estimated to be ${}^{+8.0\%}_{-9.0\%}$ by adding all the individual contributions in quadrature.

When making use of the world average of ${\mathcal{B}}_{\rm sub}=0.467\pm 0.050$ pdg2018 ; K10 , we obtain

[TABLE]

where the third uncertainty, 10.7%, is from the external uncertainty of the input BF ${\mathcal{B}}_{\rm sub}$.

To summarize, by analyzing an $e^{+}e^{-}$ collision data sample of 2.93 $\rm fb^{-1}$ taken at $\sqrt{s}=3.773$ GeV, we report the observation of $D^{+}\to\bar{K}_{1}(1270)^{0}e^{+}\nu_{e}$ and determine its decay BF for the first time. The measured BF is 1.4% of the total semileptonic $D^{+}$ decay width, which lies between the ISGW prediction of 1% and the ISGW2 prediction of 2%. Our BF of $D^{+}\to\bar{K}_{1}(1270)^{0}e^{+}\nu_{e}$ agrees with the CLFQM and LCSR predictions when $\theta_{K_{1}}\approx 33^{\circ}$ or $57^{\circ}$ theory , and clearly rules out the predictions when setting $\theta_{K_{1}}$ negative formfactor . Making use of the measured value for the BF of $D^{0}\to K_{1}(1270)^{-}e^{+}\nu_{e}$ cleores and the world-average lifetimes of the $D^{0}$ and $D^{+}$ mesons pdg2018 , we determine the partial decay width ratio $\Gamma[D^{+}\to\bar{K}_{1}(1270)^{0}e^{+}\nu_{e}]/\Gamma[D^{0}\to K_{1}(1270)^{-}e^{+}\nu_{e}]=1.2^{+0.7}_{-0.5}$, which is consistent with unity as predicted by isospin conservation. This demonstration of the capability to observe $\bar{K}_{1}(1270)$ mesons in the very clean environment of SL $D^{0(+)}$ decays opens up the opportunity to conduct further studies of the nature of these axial-vector mesons. A near-future follow-up analysis of the dynamics of these SL decays with higher statistics will allow for deeper explorations of the inner structure, production, mass and width of $\bar{K}_{1}(1270)$ and $\bar{K}_{1}(1400)$, as well as providing access to hadronic-transition form factors.

The BESIII collaboration thanks the staff of BEPCII and the IHEP computing center for their strong support. Authors thank helpful discussions from Xianwei Kang and Haiyang Cheng. This work is supported in part by National Key Basic Research Program of China under Contract No. 2015CB856700; National Natural Science Foundation of China (NSFC) under Contract No. 11835012; National Natural Science Foundation of China (NSFC) under Contracts No. 11775230, No. 11625523, No. 11635010, No. 11735014; the Chinese Academy of Sciences (CAS) Large-Scale Scientific Facility Program; Joint Large-Scale Scientific Facility Funds of the NSFC and CAS under Contracts Nos. U1532257, U1532258, U1732263, U1832107, U1832207; CAS Key Research Program of Frontier Sciences under Contracts Nos. QYZDJ-SSW-SLH003, QYZDJ-SSW-SLH040; 100 Talents Program of CAS; INPAC and Shanghai Key Laboratory for Particle Physics and Cosmology; German Research Foundation DFG under Contract No. Collaborative Research Center CRC 1044, FOR 2359; Istituto Nazionale di Fisica Nucleare, Italy; Koninklijke Nederlandse Akademie van Wetenschappen (KNAW) under Contract No. 530-4CDP03; Ministry of Development of Turkey under Contract No. DPT2006K-120470; National Science and Technology fund; The Knut and Alice Wallenberg Foundation (Sweden) under Contract No. 2016.0157; The Royal Society, UK under Contract No. DH160214; The Swedish Research Council; U. S. Department of Energy under Contracts Nos. DE-FG02-05ER41374, DE-SC-0010118, DE-SC-0012069; University of Groningen (RuG) and the Helmholtzzentrum fuer Schwerionenforschung GmbH (GSI), Darmstadt.