Observation of $D^{0}\to b_{1}(1235)^{-}e^{+}\nu_{e}$ and evidence for $D^{+}\to b_{1}(1235)^{0}e^{+}\nu_{e}$

Experimental investigations of light hadron spectroscopy in semileptonic decays of the $D$ mesons ($D$ denotes $D^{0}$ or $D^{+}$) can be used to shed light on the role of non-perturbative strong interactions in weak decays. The first quantitative predictions for the partial widths of various semileptonic $D$ decays into $S$- or $P$-wave light meson states came from the quark model developed by Isgur, Scora, Grinstein, and Wise (namely ISGW) isgw . This model was later updated to include constraints from heavy quark symmetry, hyperfine distortions of wave functions, and form factors with more realistic behavior at high recoil masses (ISGW2) isgw2 . To date, studies of Cabibbo-suppressed semileptonic $D$ decays are not as advanced as their Cabibbo-favored counterparts, which have been extensively studied both theoretically and experimentally pdg2024 ; Ke:2023qzc ; Li:2021iwf . In general, these decays, which are mediated by the quark level process $c\to de^{+}\nu_{e}$, are expected to be dominated by the ground state pseudoscalar and vector mesons. Due to limited phase space, heavier mesons, such as $P$-wave states or the first radial excitations of the $d\bar{u}$ and $d\bar{d}$ mesons, are less likely to be produced. Among the heavier mesons, the most promising to be produced is the $P$-wave $b_{1}$ meson, which is accessible via $D\to b_{1}e^{+}\nu_{e}$. Throughout this Letter, $b_{1}$ denotes $b_{1}(1235)$. The ISGW2 model predicts the branching fractions of $D^{0}\to b_{1}^{-}e^{+}\nu_{e}$ and $D^{+}\to b_{1}^{0}e^{+}\nu_{e}$ to be $1.08\times 10^{-4}$ and $1.32\times 10^{-4}$ isgw2 , respectively. The covariant light-front quark model predicts the branching fraction of $D^{+}\to b_{1}^{0}e^{+}\nu_{e}$ to be $(7.4\pm 0.7)\times 10^{-5}$ cheng .

The axial-vector meson $b_{1}$, which is identified as a ${}^{1}P_{1}$ state in the quark model, was first observed by the HBC Collaboration in 1963 MA:1963 . Before 1994, it was studied in $p\bar{p}$ ppbar:1967 ; ppbar:1969 , $\gamma p$ gammap , and $p\pi$ ppi:1974_1 ; ppi:1974_2 ; ppi:1977 ; ppi:1981 ; ppi:1984 ; ppi:1984_1 ; ppi:1991 ; ppi:1992 ; ppi:1993 reactions, where the focus was on mass and width measurements. Later, it was widely observed in hadronic decays of $D$, $B$, and charmonium states pdg2024 . Historically, theorists and experimentalists usually considered the $\omega\pi$ decay mode to be dominant. However, this has never been validated experimentally. The $b_{1}$ mesons produced in semileptonic $D$ decays offer an ideal opportunity to gain insight into their nature bes3-white-paper . Furthermore, compared to the large backgrounds and complex interference patterns found in hadronic processes pdg2024 , the semileptonic $D$ meson decays are theoretically cleaner because leptons are not involved in strong interaction M.A . The verification of theoretical calculations of semileptonic $D$ decays into the $b_{1}$ help to constrain theoretical calculations in decays of the $\tau$ tau2:2018 ; tau1:2020 , $B$ B:2012 ; B:2012ew ; B:2017 , and charmonium states with $b_{1}$ involved in the final states jpsi:2001 ; jpsi:2004 ; jpsi:2019 .

To date, there is limited experimental information available on $D\to b_{1}e^{+}\nu_{e}$ decays. Only BESIII has previously searched for $D\to b_{1}\ell^{+}\nu_{\ell}$, using a portion of the same data used in this analysis, but no significant signal was obtained b1ev . This Letter reports an observation of $D^{0}\to b_{1}^{-}e^{+}\nu_{e}$ and the first evidence for $D^{+}\to b_{1}^{0}e^{+}\nu_{e}$, by analyzing 7.9 fb^-1 of $e^{+}e^{-}$ collision data taken at $\sqrt{s}=3.773$ GeV. Throughout this Letter, charge conjugate channels are always implied.

Details about the design and performance of the BESIII detector are given in Ref. BESIII . The end-cap time-of-flight system (TOF) system was upgraded in 2015 using multigap resistive plate chamber technology, providing a time resolution of 60 ps etof . Approximately 63% of the data used here was collected after this upgrade. 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 kkmc . The inclusive MC samples consist of the production of $D\bar{D}$ pairs with consideration of quantum coherence for all neutral $D$ modes, 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. The known decay modes are modeled with evtgen evtgen using branching fractions taken from the Particle Data Group  pdg2024 , and the remaining unknown charmonium decays are estimated with lundcharm lundcharm . Final state radiation from charged final state particles is incorporated using the photos package photos . The $D\to b_{1}e^{+}\nu_{e}$ signal MC events are simulated using the ISGW2 model isgw2 , where the $b_{1}$ is parameterized by a relativistic Breit-Wigner function with mass and width fixed to the world-average values of $1229.5\pm 3.2$ MeV/$c^{2}$ and $142\pm 9$ MeV, respectively pdg2024 .

At $\sqrt{s}=3.773$ GeV, the $D$ and $\bar{D}$ mesons are produced in pairs without accompanying particles in the final state. Because of this, one can study semileptonic $D$ decays with the double-tag (DT) method. Single-tag (ST) $\bar{D}^{0}$ mesons are first reconstructed using the hadronic decay modes $\bar{D}^{0}\to K^{+}\pi^{-}$, $K^{+}\pi^{-}\pi^{0}$, $K^{+}\pi^{-}\pi^{-}\pi^{+}$, $K_{S}^{0}\pi^{+}\pi^{-}$, $K^{+}\pi^{-}\pi^{0}\pi^{0}$, and $K^{+}\pi^{+}\pi^{-}\pi^{-}\pi^{0}$; while ST $D^{-}$ mesons are reconstructed via the decay modes $D^{-}\to K^{+}\pi^{-}\pi^{-}$, $K^{0}_{S}\pi^{-}$, $K^{+}\pi^{-}\pi^{-}\pi^{0}$, $K^{0}_{S}\pi^{-}\pi^{0}$, $K^{0}_{S}\pi^{+}\pi^{-}\pi^{-}$, and $K^{+}K^{-}\pi^{-}$. Semileptonic $D$ candidates are then reconstructed from the residual tracks and showers not used in the tag selection. Candidate events in which the $D$ decays into $b_{1}e^{+}\nu_{e}$ and the $\bar{D}$ decays into a tag mode are called DT events.

Because the branching fraction of the $b_{1}\to\omega\pi$ decay is unknown, the product branching fractions of the decay $D\to b_{1}e^{+}\nu_{e}$ (${\mathcal{B}}_{\rm SL}$) and its subsequent decay $b_{1}\to\omega\pi$ (${\mathcal{B}}_{b_{1}}$) are determined from

$${\mathcal{B}}_{\rm SL}\cdot{\mathcal{B}}_{b_{1}}=\frac{N_{\rm DT}}{N^{\rm tot}_{\rm ST}\cdot\bar{\varepsilon}_{\rm SL}\cdot{\mathcal{B}}_{\rm sub}},$$

(1)

where $N_{\rm ST}^{\rm tot}=\Sigma_{i}N^{i}_{\rm ST}$ and $N_{\rm DT}$ are the total ST and DT yields after summing over all tag modes $i$; ${\mathcal{B}}_{\rm sub}={\mathcal{B}}_{\omega}\cdot{\mathcal{B}}_{\pi^{0}}$ for $D^{0}\to b_{1}^{-}e^{+}\nu_{e}$ and ${\mathcal{B}}_{\rm sub}={\mathcal{B}}_{\omega}\cdot{\mathcal{B}}^{2}_{\pi^{0}}$ for $D^{+}\to b_{1}^{0}e^{+}\nu_{e}$, in which ${\mathcal{B}}_{\omega}$ and ${\mathcal{B}}_{\pi^{0}}$ are the branching fractions of $\omega\to\pi^{+}\pi^{-}\pi^{0}$ and $\pi^{0}\to\gamma\gamma$, respectively; and $\bar{\varepsilon}_{\rm SL}=\Sigma_{i}\frac{N^{i}_{\rm ST}}{N^{\rm tot}_{\rm ST}}\frac{\epsilon^{i}_{\rm DT}}{\epsilon^{i}_{\rm ST}}$ is the average signal efficiency of reconstructing $D\to b_{1}e^{+}\nu_{e}$ in the presence of an ST $D$, where $\epsilon^{i}_{\rm ST}$ and $\epsilon^{i}_{\rm DT}$ are the ST and DT efficiencies for the $i$-th tag mode, respectively.

All charged tracks detected in the multilayer drift chamber (MDC) must be within a polar angle ($\theta$) range of $|\rm{cos\theta}|<0.93$, where $\theta$ is defined with respect to the $z$-axis, which is the symmetry axis of the MDC. Except for charged tracks from $K^{0}_{S}$ decays, they 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) for charged tracks combines measurements of the specific ionization energy loss in the MDC (d$E$/d$x$) and the flight time in the TOF to form confidence levels for pion and kaon hypothesis ($CL_{\pi}$ and $CL_{K}$). Charged tracks with $CL_{K}>CL_{\pi}$ and $CL_{\pi}>CL_{K}$ are assigned as kaons and pions, respectively.

Each $K_{S}^{0}$ candidate is 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 required to have an invariant mass within $(0.487,0.511)$ GeV/$c^{2}$. The decay length of the $K^{0}_{S}$ candidate is required to be greater than twice the vertex resolution away from the IP. The quality of the vertex fits (primary-vertex fit and secondary-vertex fit) is ensured by a requirement of $\chi^{2}<100$.

Photon candidates are selected using showers in the electromagnetic calorimeter (EMC). The deposited energy of each shower must be more than 25 MeV in the barrel region ($|\!\cos\theta|<0.80$) and more than 50 MeV in the end-cap region ($0.86<|\!\cos\theta|<0.92$). To exclude showers associated with charged tracks, the angle subtended by the EMC shower and the position of the closest charged track at the EMC must be greater than $10^{\circ}$ as measured from the IP. To suppress electronic noise and showers unrelated to the event, the difference between the EMC time and the event start time is required to be within [0, 700] ns. For $\pi^{0}$ candidates, 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 one-constraint (1C) kinematic fit to the known $\pi^{0}$ mass pdg2024 is imposed on the photon pair. The four-momentum of the $\pi^{0}$ candidate updated by the 1C fit is kept for further analysis.

To separate the ST $\bar{D}$ mesons from combinatorial backgrounds, we define the energy difference $\Delta E\equiv E_{\bar{D}}-E_{\rm{beam}}$ and the beam-constrained mass $M_{\rm BC}\equiv\sqrt{E_{\rm{beam}}^{2}/c^{4}-|\vec{p}_{\bar{D}}|^{2}/c^{2}}$, where $E_{\rm{beam}}$ is the beam energy, and $E_{\bar{D}}$ and $\vec{p}_{\bar{D}}$ are the total energy and momentum of the $\bar{D}$ candidate in the $e^{+}e^{-}$ center-of-mass frame, respectively. If there is more than one $\bar{D}$ candidate in a given ST mode, the one with the minimal $|\Delta E|$ is kept for further analysis. The $\Delta E$ requirements and ST efficiencies for the different tag modes are listed in Table 1.

The ST yields are extracted by performing unbinned maximum likelihood fits to individual $M_{\rm BC}$ distributions. In the fit, the signal shape is derived from the MC-simulated signal shape convolved with a double-Gaussian function to consider the resolution difference between data and MC simulation. The background shape is described by the ARGUS function argus , with the endpoint parameter fixed at 1.8865 GeV/$c^{2}$ corresponding to $E_{\rm beam}$. Figure 1 shows 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 to be $(7896.0\pm 3.4_{\rm stat})\times 10^{3}$ and $(4149.9\pm 2.3_{\rm stat})\times 10^{3}$, respectively.

In the presence of the tagged $\bar{D}$ candidates, $D^{0}\to b_{1}^{-}e^{+}\nu_{e}$ and $D^{+}\to b_{1}^{0}e^{+}\nu_{e}$ candidates are selected from the residual tracks and photons not used in the tag reconstruction. The selection criteria of charged and neutral pions are the same as those used in the tag selection. For the selected candidates $D^{0}\to\pi^{+}\pi^{-}\pi^{-}\pi^{0}e^{+}\nu_{e}$ and $D^{+}\to\pi^{+}\pi^{-}\pi^{0}\pi^{0}e^{+}\nu_{e}$, two possible $\pi^{+}\pi^{-}\pi^{0}$ combinations can form the $\omega$ candidate. In order to veto backgrounds from $D\to a_{0}(980)e^{+}\nu_{e}$, the invariant masses of both $\pi^{+}\pi^{-}\pi^{0}$ combinations ($M_{\pi^{+}\pi^{-}\pi^{0}}$) are required to be greater than 0.6 GeV/$c^{2}$. This rejects almost all of the $D\to a_{0}(980)e^{+}\nu_{e}$ background events. One candidate is kept for further analysis if either of the $\pi^{+}\pi^{-}\pi^{0}$ combinations falls in the $\omega$ mass signal region $M_{\pi^{+}\pi^{-}\pi^{0}}\in$ (0.757,0.807) GeV/$c^{2}$. Events in the $\omega$ sideband region, defined as $M_{\pi^{+}\pi^{-}\pi^{0}}\in$ (0.697,0.742) and (0.822,0.867) GeV/$c^{2}$, are used to study potential combinatorial background in the $\omega$ signal region. To reject background events from $D^{0(+)}\to\bar{K}_{1}(1270)[\to K_{S}^{0}\pi^{+(0)}\pi^{-(0)}]e^{+}\nu_{e}$, we require the invariant masses of all $\pi^{+}\pi^{-}$ ($\pi^{0}\pi^{0}$) combinations satisfy $|M_{\pi^{+}\pi^{-}(\pi^{0}\pi^{0})}-M_{K_{S}^{0}}|>0.008$ GeV/$c^{2}$.

Identification of $e^{+}$ candidates is performed with combined d$E$/d$x$, TOF, and EMC information. Based on these, we calculate confidence levels for the positron, pion, and kaon hypotheses ($CL_{e}$, $CL_{\pi}$, and $CL_{K}$). Charged tracks satisfying $CL_{e}>0.001$ and $CL_{e}/(CL_{e}+CL_{\pi}+CL_{K})>0.8$ are assigned as $e^{+}$ candidates. To further reject background events from misidentified hadrons and muons, the deposited energy of any $e^{+}$ candidate in the EMC must be greater than 0.8 times its momentum reconstructed in the MDC.

The invariant mass of the $b_{1}e^{+}$ system ($M_{b_{1}e^{+}}$) is required to be less than 1.82 GeV/$c^{2}$ to suppress peaking background contributions from the decay $D\to b_{1}^{-(0)}\pi^{+}$. To suppress backgrounds with extra photons or $\pi^{0}$s, the maximum energy of any extra photon that has not been used in the event selection ($E^{\rm max}_{\rm extra~{}\gamma}$) must be less than 0.30 GeV, and there must be no additional $\pi^{0}$ candidates ($N^{\pi^{0}}_{\rm extra}$) in the candidate event. In addition, the opening angle between the missing momentum and the most energetic unused shower when found, $\theta_{\gamma,\rm miss}$, is required to satisfy $\cos\theta_{\gamma,\rm miss}<0.3$ and $\cos\theta_{\gamma,\rm miss}<0.4$ for $D^{0}\to b_{1}^{-}e^{+}\nu_{e}$ and $D^{+}\to b_{1}^{0}e^{+}\nu_{e}$, respectively. To suppress backgrounds due to final state radiation, the angle between the direction of the radiative photon and the $e^{+}$ momentum is required to be greater than 0.20 radians.

The presence of the missing neutrino is inferred using a kinematic variable defined as

$$U_{\rm miss}\equiv E_{\rm miss}-|\vec{p}_{\rm miss}|\cdot c,$$

(2)

with

$$E_{\rm{miss}}\equiv E_{\rm{beam}}-E_{b_{1}}-E_{e^{+}},$$

(3)

and

$$\vec{p}_{\rm{miss}}\equiv\vec{p}_{D}-\vec{p}_{b_{1}}-\vec{p}_{e^{+}},$$

(4)

where $E_{b_{1}\,(e^{+})}$ and $\vec{p}_{b_{1}\,(e^{+})}$ are the measured energy and momentum of the $b_{1}$ ($e^{+}$) candidates in the $e^{+}e^{-}$ center-of-mass frame, respectively. We calculate $\vec{p}_{D}\equiv-\hat{p}_{\bar{D}}\cdot\sqrt{E_{\rm{beam}}^{2}/c^{2}-m_{\bar{D}}^{2}\cdot c^{2}}$, where $\hat{p}_{\bar{D}}$ is a unit vector in the momentum direction of the ST $\bar{D}$ meson and $m_{\bar{D}}$ is the known $\bar{D}$ mass pdg2024 . The beam energy and the known $D$ mass are used to determine the magnitude of the momentum of the ST $D$ mesons in order to improve the $U_{\rm{miss}}$ resolution. The signal candidates are expected to peak around zero in the $U_{\rm miss}$ distribution and near the invariant mass of the $b_{1}$ in the $\omega\pi$ mass spectrum ($M_{\omega\pi}$).

To obtain the signal yields, we perform two-dimensional (2D) unbinned maximum likelihood fits to the $M_{\omega\pi^{-(0)}}$ vs. $U_{\rm miss}$ distributions. In the fits, the 2D signal and background shapes are modeled by the simulated shapes derived from the signal and inclusive MC samples, respectively, and the yields of the signal and background are left free. The projections of the 2D fit on the $M_{{}_{\omega\pi^{-(0)}}}$ and $U_{\rm miss}$ distributions are shown in Fig. 2. From the fits, the signal yields are $N_{\rm DT}$ = $35.6\pm 8.9$ and $N_{\rm DT}$ = $17.5\pm 6.7$ for $D^{0}\to b_{1}^{-}e^{+}\nu_{e}$ and $D^{+}\to b_{1}^{0}e^{+}\nu_{e}$, respectively, and the signal significances are estimated to be $5.2\sigma$ and $3.1\sigma$, by comparing the likelihoods with or without the signal involved and taking into account the change of degrees of freedom. Because the non-resonant component is highly suppressed in semileptonic $D$ decays and due to limited statistics, all $\omega\pi$ candidates are assumed to come from $b_{1}$ decay. The non-resonant component is considered as a source of systematic uncertainty. Based on MC simulations, the detection efficiencies $\bar{\varepsilon}_{\rm SL}$ are estimated to be $0.0715\pm 0.0005$ and $0.0417\pm 0.0004$ for $D^{0}\to b_{1}^{-}e^{+}\nu_{e}$ and $D^{+}\to b_{1}^{0}e^{+}\nu_{e}$, respectively.

The systematic uncertainties in the branching fraction measurements are discussed below. The uncertainty associated with the ST yield $N_{\rm ST}^{\rm tot}$ is estimated to be 0.1% Dtokkenu . The uncertainty from the quoted branching fraction of $\omega\to\pi^{+}\pi^{-}\pi^{0}$ is 0.8%. The efficiencies of the $e^{+}$ tracking and PID are investigated using a control sample of $e^{+}e^{-}\to\gamma e^{+}e^{-}$, and those of the $\pi^{\pm}$ tracking and PID as well as for $\pi^{0}$ reconstruction are estimated with control samples of $D^{0}\to K^{-}\pi^{+}$, $D^{0}\to K^{-}\pi^{+}\pi^{0}$, $D^{0}\to K^{-}\pi^{+}\pi^{+}\pi^{-}$, and $D^{+}\to K^{-}\pi^{+}\pi^{+}$ with the same tags. The systematic uncertainties in the tracking (PID) efficiencies are assigned as 1.0% (1.0%) per $e^{+}$ and 1.0% (1.0%) per $\pi^{\pm}$, respectively. The systematic uncertainty of the $\pi^{0}$ reconstruction efficiencies, including photon finding, the $\pi^{0}$ mass window, and the 1C kinematic fit, is assigned as 2.0% per $\pi^{0}$.

The systematic uncertainty associated with the $\omega$ mass window is assigned to be 1.9% using a control sample of $D^{0}\to K^{0}_{S}\omega$ reconstructed versus the same $\bar{D}^{0}$ tags as the nominal analysis. The systematic uncertainties arising from the $E_{\rm extra\,\gamma}^{\rm max}$ and $N_{\rm extra,\pi^{0}}$ requirements are estimated to be 1.8% and 2.0% for $D^{0}\to b_{1}^{-}e^{+}\nu_{e}$ and $D^{+}\to b_{1}^{0}e^{+}\nu_{e}$, respectively, which are estimated using the DT samples of $D^{0}\to K^{-}e^{+}\nu_{e}$ and $D^{+}\to K^{0}_{S}e^{+}\nu_{e}$ decays reconstructed versus the same tags as the nominal analysis. The systematic uncertainty related to the $b_{1}$ resonant parameters is estimated using alternative signal MC samples, which are produced by varying the mass and width of the $b_{1}$ by $\pm 1\sigma$. The maximum changes of the signal efficiencies, 1.4% and 4.8%, are assigned as the systematic uncertainties for $D^{0}\to b_{1}^{-}e^{+}\nu_{e}$ and $D^{+}\to b_{1}^{0}e^{+}\nu_{e}$, respectively. The uncertainty from limited MC statistics is estimated to be 0.3%. The systematic uncertainty due to the MC generator is assigned by examining the DT efficiency with the signal MC events produced with the phase space model. The maximum changes of the DT efficiencies, 1.4% and 0.5%, are assigned as the systematic uncertainties for $D^{0}\to b_{1}^{-}e^{+}\nu_{e}$ and $D^{+}\to b_{1}^{0}e^{+}\nu_{e}$, respectively. The systematic uncertainty associated with the cos$\theta_{\gamma,\rm miss}$ requirement is estimated to be 1.0% by analyzing the DT sample of $D^{0}\to K_{S}^{0}\pi^{-}e^{+}\nu_{e}$. The systematic uncertainty due to the sideband estimation of the $\omega$ background, which may be from $D^{0(+)}\to b_{1}^{-(0)}e^{+}\nu_{e}$ with $b_{1}^{-(0)}\to\pi^{+}\pi^{-}\pi^{0}\pi^{0(-)}$, is estimated by examining the measured branching fractions after considering this contribution. The changes in the branching fractions, 0.5% and 4.6%, are taken as the corresponding systematic uncertainties.

To estimate the systematic uncertainties of the signal and background shapes in the $U_{\rm miss}$ fit, a simulated shape convolved with a double Gaussian function with floating parameters is chosen as the alternative signal shape, and the simulated background shape derived from the inclusive MC sample is replaced with a 1st-order Chebyshev polynomial function. The difference in the fitted signal yields are taken as individual systematic uncertainties. 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 $D^{+}\to\bar{K}^{*}(892)^{0}e^{+}\nu_{e}$ kpi . The total systematic uncertainties are ${}^{+8.8\%}_{-10.7\%}$ and ${}^{+12.7\%}_{-13.9\%}$ by adding all the individual contributions in quadrature for $D^{0}\to b_{1}^{-}e^{+}\nu_{e}$ and $D^{+}\to b_{1}^{0}e^{+}\nu_{e}$, respectively.

The product branching fractions are determined to be ${\mathcal{B}}(D^{0}\to b_{1}^{-}e^{+}\nu_{e})\times{\mathcal{B}}(b^{-}\to\omega\pi^{-})=(0.72\pm 0.18^{+0.06}_{-0.08})\times 10^{-4}$ and ${\mathcal{B}}(D^{+}\to b_{1}^{0}e^{+}\nu_{e})\times{\mathcal{B}}(b_{1}^{0}\to\omega\pi^{0})=(1.16\pm 0.44\pm 0.16)\times 10^{-4}$ for $D^{0}\to b_{1}^{-}e^{+}\nu_{e}$ and $D^{+}\to b_{1}^{0}e^{+}\nu_{e}$, where the first and second uncertainties are statistical and systematic, respectively.

In summary, by analyzing $7.9~{}\rm{fb}^{-1}$ of $e^{+}e^{-}$ collision data taken at $\sqrt{s}=3.773$ GeV, we report the first observation of $D^{0}\to b_{1}^{-}e^{+}\nu_{e}$ with a significance of 5.2$\sigma$ and find a first evidence for $D^{+}\to b_{1}^{0}e^{+}\nu_{e}$ with a significance of 3.1$\sigma$ after considering systematic uncertainty. The product branching fractions are determined to be ${\mathcal{B}}(D^{0}\to b_{1}^{-}e^{+}\nu_{e})\times{\mathcal{B}}(b_{1}^{-}\to\omega\pi^{-})=(0.72\pm 0.18^{+0.06}_{-0.08})\times 10^{-4}$ and ${\mathcal{B}}(D^{+}\to b_{1}^{0}e^{+}\nu_{e})\times{\mathcal{B}}(b_{1}^{0}\to\omega\pi^{0})=(1.16\pm 0.44\pm 0.16)\times 10^{-4}$. Assuming ${\mathcal{B}}(b_{1}\to\omega\pi)=1$, these results are comparable with the theoretical predictions in Ref. cheng , thereby reinforcing the assumption that the $\omega\pi$ final state is the dominant decay mode of the axial-vector $b_{1}$ meson. Using the obtained branching fractions, the world-average lifetimes of the $D^{0}$ and $D^{+}$ mesons pdg2024 , and assuming that ${\mathcal{B}}(b_{1}^{-}\to\omega\pi^{-})={\mathcal{B}}(b_{1}^{0}\to\omega\pi^{0})$, we determine the partial decay width ratio $\frac{\Gamma(D^{0}\to b_{1}^{-}e^{+}\nu_{e})}{2\Gamma(D^{+}\to b_{1}^{0}e^{+}\nu_{e})}=0.78\pm 0.19^{+0.04}_{-0.05}$, which is consistent with unity within uncertainties after considering the shared systematic terms, and is thus consistent with isospin conservation. Observation of the $b_{1}$ in semileptonic $D$ decays opens a new avenue for experimental investigation. Future analysis of their decay dynamics, which will be made possible with the larger data samples currently being collected by BESIII bes3-white-paper and a potential future STCF stcf , will provide deep insights into the inner structure, production, mass, and width of the $b_{1}$, as well as provide access to hadronic-transition form factors.

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 R&D Program of China under Contracts Nos. 2023YFA1606000, 2020YFA0406300, 2020YFA0406400, 2023YFA1606704; National Natural Science Foundation of China (NSFC) under Contracts Nos. 12375092, 11635010, 11735014, 11935015, 11935016, 11935018, 11961141012, 12025502, 12035009, 12035013, 12061131003, 12192260, 12192261, 12192262, 12192263, 12192264, 12192265, 12221005, 12225509, 12235017; the Chinese Academy of Sciences (CAS) Large-Scale Scientific Facility Program; the CAS Center for Excellence in Particle Physics (CCEPP); Joint Large-Scale Scientific Facility Funds of the NSFC and CAS under Contract No. U1832207; 100 Talents Program of CAS; The Institute of Nuclear and Particle Physics (INPAC) and Shanghai Key Laboratory for Particle Physics and Cosmology; German Research Foundation DFG under Contracts Nos. 455635585, FOR5327, GRK 2149; Istituto Nazionale di Fisica Nucleare, Italy; Ministry of Development of Turkey under Contract No. DPT2006K-120470; National Research Foundation of Korea under Contract No. NRF-2022R1A2C1092335; National Science and Technology fund of Mongolia; National Science Research and Innovation Fund (NSRF) via the Program Management Unit for Human Resources & Institutional Development, Research and Innovation of Thailand under Contract No. B16F640076; Polish National Science Centre under Contract No. 2019/35/O/ST2/02907; The Swedish Research Council; U. S. Department of Energy under Contract No. DE-FG02-05ER41374.