Observation of the Semileptonic Decay $D^0 \to a_0(980)^- e^+ \nu_e$ and Evidence for $D^+ \to a_0(980)^0 e^+ \nu_e$

Using an $e^+e^-$ collision data sample of 2.93 fb$^{-1}$ collected at a center-of-mass energy of 3.773 GeV by the BESIII detector at BEPCII, we report the observation of $D^0 \to a_0(980)^- e^+ \nu_e$ and evidence for $D^+ \to a_0(980)^0 e^+ \nu_e~$ with significances of $6.4\sigma$ and $2.9\sigma$, respectively. The absolute branching fractions are determined to be $\mathcal{B}(D^0 \to a_0(980)^- e^+ \nu_e)\times\mathcal{B}(a_0(980)^- \to \eta \pi^-) = (1.33_{-0.29}^{+0.33}({\rm stat})\pm0.09({\rm syst}))\times10^{-4}~$ and $\mathcal{B}(D^+ \to a_0(980)^0 e^+ \nu_e)\times\mathcal{B}(a_0(980)^0 \to \eta \pi^0) = (1.66_{-0.66}^{+0.81}({\rm stat})\pm0.11({\rm syst}))\times10^{-4}~$. An upper limit of $\mathcal{B}(D^+ \to a_0(980)^0 e^+ \nu_e)\times\mathcal{B}(a_0(980)^0 \to \eta \pi^0)<3.0\times10^{-4}~$ is also determined at the $90\%~$ confidence level.

In this Letter, we present the first observation of the semileptonic decay D 0 → a 0 (980) − e + ν e and evidence for D + → a 0 (980) 0 e + ν e . The data sample used in this analysis was collected at center-of-mass energy √ s = 3.773 GeV (near the nominal mass of the ψ(3770)) by the BESIII detector at the BEPCII collider, and corresponds to an integrated luminosity of 2.93 fb −1 [8].
The BESIII detector is described in detail elsewhere [9]. The detector has a geometrical acceptance of 93% of 4π. It includes a multi-layer drift chamber (MDC) for measuring the momenta and specific ionization energy loss (dE/dx) of charged particles, a time-of-flight system (TOF) which contributes to charged particle identification (PID), a CsI(Tl) electromagnetic calorimeter (EMC) for detecting electromagnetic showers, and a muon chamber system designed for muon identification.
A detailed GEANT4-based [10] Monte Carlo (MC) simulation of the BESIII detector is used to determine the detection efficiencies and evaluate the possible background sources. Events are generated by the generator KKMC [11] using EVT-GEN [12], with the effects of beam energy spread and initialstate radiation (ISR) being taken into account. Final-state radiation is treated via the PHOTOS package [13].
A double-tag analysis technique [14] is employed; this takes advantage of D mesons produced via exclusive DD pair-production in the decay of the ψ(3770) resonance. We reconstructD mesons using specific hadronic decays, producing a sample of single tag (ST) events. We then search these ST events for the partner D meson undergoing the decay pro-cess of interest; successful searches result in our sample of double tag (DT) events. This strategy suppresses non-DD background effectively and provides a measurement of absolute branching fractions independent of the integrated luminosity and the DD production cross section. These absolute branching fractions are calculated as in which α denotes the different ST modes, N obs,α tag is the ST yield for tag mode α, N obs sig is the sum of the DT yields from all ST modes, and ǫ α tag and ǫ α tag,sig refer to the corresponding ST efficiency and the DT efficiency for the ST mode α determined by MC simulations. In this approach, most of the systematic uncertainties arising from the ST reconstruction are canceled.
The STD mesons are reconstructed with the following final states: The charged particles K ± and π ± , as well as the neutral particles π 0 and K 0 S , are selected with the same criteria as those in Ref. [15]. For theD 0 → K + π − final state, requirements on the opening angle and the difference of the time of flight of the two charged tracks are applied to reduce backgrounds from cosmic rays, Bhabha and di-muon events [16]. Throughout this Letter, charge-conjugate modes are implied, unless otherwise noted.
Two key kinematic variables, the energy difference ∆E ≡ E D − E beam , and beam-constrained mass M BC ≡ E 2 beam /c 4 − | p D | 2 /c 2 , are used to identify the STD candidates. Here, E beam is the beam energy, E D and p D are the reconstructed energy and momentum of theD candidate in the e + e − center-of-mass system. For trueD candidates, ∆E and M BC will peak at zero and the nominal mass of the D meson, respectively. We accept theD candidates with M BC greater than 1.83 GeV/c 2 and apply mode-dependent ∆E requirements of approximately three standard deviations. When multiple candidates exist, at most one candidate per tag mode per charm (i.e., D orD) is retained in each event by selecting the candidate with the smallest |∆E| [17]. The ST yields are determined by performing a maximum likelihood fit to the M BC distributions of the acceptedD candidates, as shown in Fig. 1. The signal shape is modeled by the MC simulated shape convolved with a Gaussian function with free parameters. The MC simulation includes the effects of beam energy spread, ISR, the ψ(3770) line shape, and experimental resolution, while the Gaussian allows for small imperfections in the Points with error bars represent data, the (red) solid lines are the total fits and the (blue) dashed lines represent the background contributions.
MC simulation. The combinatorial background is modeled by an ARGUS function [18]. The ST yield for each mode is calculated by subtracting the integrated ARGUS background yield from the total number of events contained in the signal regions defined as 1.858 < M BC < 1.874 GeV/c 2 for D 0 and 1.860 < M BC < 1.880 GeV/c 2 for D − . The ST yields in data and the corresponding ST efficiencies are listed in Table I.
We search in the selected ST events for the semi-leptonic decays D 0 → a 0 (980) − e + ν e and D + → a 0 (980) 0 e + ν e , using the remaining charged tracks and photon candidates not used for the ST candidate. Here, the a 0 (980) − and a 0 (980) 0 are reconstructed by their prominent decays to ηπ − and ηπ 0 , respectively. The PID of the charged hadrons (positrons) is accomplished by combining the dE/dx and TOF (dE/dx, TOF and EMC) information to construct a likelihood L i (L ′ i ) for each of the hypotheses i = e/π/K. The charged pion candidate is required to satisfy L π > L K and L π > 0.1%. The positron candidate is required to satisfy where E is the energy deposited in the EMC and p is the momentum measured by the MDC. A candidate signal event is required to have a single positron (electron) for signal D (D) decays. The π 0 and η candidates are formed from pairs of photon candidates with invariant two-photon masses within (0.115, 0.150) and (0.508, 0.572) GeV/c 2 , respectively. To improve the kinematic resolution, a oneconstraint (1-C) kinematic fit is performed by constraining the γγ invariant mass to the expected nominal mass [19]. Background consisting of a real photon paired with a fake one is suppressed by requiring the decay angle, defined as | cos θ decay,π 0 (η) | = |Eγ1−Eγ2| | p π 0 (η) c| , to be less than 0.80 and 0.95 for the π 0 and η candidates, respectively. Here, E γ1 and E γ2 are the energies of the two daughter photons of the π 0 (η), and p π 0 (η) is the reconstructed momentum of the π 0 (η). The a 0 (980) − candidate is formed with a charged pion and a selected η candidate. The a 0 (980) 0 candidate is formed from the combination of π 0 and η candidates with the least χ 2 1C,π 0 + χ 2 1C,η , where χ 2 1C,π 0 and χ 2 1C,η are the χ 2 values of the 1-C kinematic fits of the π 0 and η candidates, respectively. After the above selections, we veto any event with extra unused charged tracks. Events containing an additional unused π 0 candidate are also rejected. This π 0 veto suppresses the following backgrounds: for the D 0 → a 0 (980) − e + ν e mode, and D + → K 0 S e + ν e and D + →K * (892) 0 e + ν e (withK * (892) 0 → K 0 S π 0 ) for D + → a 0 (980) 0 e + ν e ; in all cases here, K 0 S → π 0 π 0 . Detailed MC studies show that D 0 → K * (892) − e + ν e and D + →K * (892) 0 e + ν e followed byK * → K 0 L π are prominent backgrounds, where the K 0 L signal in the EMC can mimic the higher-energy daughter of the η candidate. To suppress these background, the lateral moment [20] of EMC showers, which peaks around 0.15 for real photons but varies from 0 to 0.85 for K 0 L candidates, is required to be within (0, 0.35) for the higher-energy photon from the η decay. This requirement suppresses about 70% of the K 0 L backgrounds, while retaining 95% of the signal. Branching fractions of K 0 S → π + π − , π 0 → γγ and η → γγ are not included in the efficiencies. The first three rows are forD 0 candidates and the last six rows are for D − candidates.

Mode
N obs,α tag ǫ α tag (%) ǫ α tag,sig (%) K + π − 541 541 ± 753 65.92 ± 0.02 15.18 ± 0.20 K + π − π 0 1 040 340 ± 1209 34.66 ± 0.01 8.00 ± 0.08 K + π − π + π − 706 179 ± 982 38.96 ± 0.01 7.02 ± 0.09 K + π − π − 806 444 ± 953 51.08 ± 0.02 5.23 ± 0.07 K + π − π − π 0 252 088 ± 816 25.91 ± 0.02 2.40 ± 0.06 K 0 S π − 100 019 ± 337 54.33 ± 0.05 5.55 ± 0.21 K 0 S π − π 0 235 011 ± 759 29.63 ± 0.03 3.10 ± 0.08 K 0 S π + π − π − 131 815 ± 710 32.49 ± 0.05 2.66 ± 0.10 K + K − π − 69 642 ± 398 40.58 ± 0.06 4.09 ± 0.20 For the semileptonic signal candidate, the undetected neutrino is inferred by studying the variable U ≡ E miss −c| p miss |, where E miss and p miss are the missing energy and momentum carried by the neutrino from the semileptonic decay. These are calculated as E miss = E beam − E a0(980) − E e and p miss = −( p tag + p a0(980) + p e ), respectively, where E a0(980) (E e ) and p a0(980) ( p e ) are the energy and momentum of a 0 (980) (positron), p tag is the momentum of the STD in the center-of-mass frame. We calculate p tag = p tag E 2 beam /c 2 − M 2 D c 2 , wherep tag is the unit vector in the momentum direction of the STD and M D is the nominal D mass [19]. The signal candidates are expected to peak around To obtain the signal yields, we perform two-dimensional (2-D) unbinned maximum likelihood fits to the M ηπ versus U distributions, combining all tag modes. Projections of the 2-D fits are shown in Fig. 2. The signal shape in the U distribution is described by the MC simulation and that in the M ηπ distribution is modeled with a usual Flatté formula [21] for the a 0 (980) signal. The mass and two coupling constants, g 2 ηπ and g 2 KK are fixed to 0.990 GeV/c 2 , 0.341 (GeV/c 2 ) 2 and 0.304 (GeV/c 2 ) 2 [22], respectively. The backgrounds are divided into three classes: the residual background from certain specific D decay modes mentioned previously (Bkg I), the other D decay background (Bkg II), and the non-DD background (Bkg III). For each background source in Bkg I, the shape and yield are determined by the MC simulation incorporating the corresponding branching fraction [19]. The shape and yield for Bkg II are fixed based on the generic DD MC sample, in which all particles decay inclusively based on the branching fractions taken from the PDG [19], but with Bkg I modes removed. Bkg III, from the continuum processes e + e − → qq and τ + τ − , is modeled with a MC-determined shape generated with a modified LUND model [23], with the yield determined in the fit. The 2-D probability density functions (PDFs) of all these components are constructed by the product of the U and M ηπ distributions due to the negligible correlation between the two observables according to the exclusive background channel MC simulation.
The 2-D fits yield 25.7 +6.4 −5.7 signal events for D 0 → a 0 (980) − e + ν e and 10.2 +5.0 −4.1 signal events for D + → a 0 (980) 0 e + ν e . The statistical significance of signal, taken to be −2 ln(L 0 /L best ), where L best and L 0 are the maximum likelihood values with the signal yield left free and fixed at zero, respectively, is 6.5σ for D 0 → a 0 (980) − e + ν e and 3.0σ for D + → a 0 (980) 0 e + ν e . The corresponding DT efficiencies are presented in Table I.
The systematic uncertainties in the measurements are summarized in Table II and discussed below. The uncertainty due to the STD meson largely cancel in the DT analysis method. The uncertainties associated with the tracking and PID for the charged pion are estimated to be 1.0% and 0.5%, respectively, by investigating a control sample D + → K − π + π + based on a partial reconstruction technique. Similarly, the uncertainty related with the π 0 reconstruction, including the detection of two photons, is found to be 1.0% by studying the control sample D 0 → K − π + π 0 . Since η candidates are reconstructed similarly, the corresponding uncertainty is also assigned to be 1.0%. The uncertainties related to tracking and PID for the positron are investigated with a radiative Bhabha control sample in the different polar angle and momentum bins. The values for the tracking and PID are 1.0% and 0.6%, respectively, obtained after re-weighting according to the distributions of momentum and polar angle of the positron from the signal MC sample. The uncertainty arising from the choice of the best ηπ 0 combination in the D + decay is studied with a DT D hadronic decay sample, D 0 → K − π + π 0 versus D 0 → K + π − π 0 and is taken as 0.3% [24]. The systematic uncertainty from selecting the best ηπ 0 combination is assumed to be the same as the one from selecting the best π 0 π 0 combination in Ref. [24], considering the similar selection criteria of η and π 0 . The efficiency of the lateral moment requirement for photons is studied in different energy and polar angle bins using a control sample of radiative Bhabha events. The average data-MC efficiency difference, after re-weighting according to the energy and polar angle distributions of the signal MC sample, is taken as the systematic uncertainty. The form factor of the semileptonic decay for the nominal signal MC sample is parameterized with the model of Ref. [25]. An alternative MC sample based on the ISGW2 model [26] is produced to estimate the uncertainty associated with the signal model; the change in the detection efficiency is assigned as the corresponding systematic uncertainty. The uncertainties in the branching fractions of submodes are taken from the current world averages [19]. The effect of limited MC statistics is also included as a systematic effect. Uncertainties associated with the 2-D fits are estimated by varying the signal and background shapes and certain background contributions in Bkg I and Bkg II within their uncertainties. For the resolution of U , the distribution in U of the D 0 decay is convolved with a Gaussian function with free parameters and the fit is redone. Considering the limited statistics and large background contributions, the width of the Gaussian function for the D + decay is fixed to be FWHM+ FWHM0 · σ 0 , in which σ 0 is the output Gaussian width in the fit to the D 0 case, and FWHM + and FWHM 0 are the full width at half maximum of the nominal U shape for the D + and D 0 signal MC samples, respectively. Changes in the signal yields are assigned to be the corresponding uncertainties. For the a 0 (980) line shape, the mass and the two coupling constants in the Flatté formula are varied by one standard deviation, and the average change in the signal yield is taken to be the relevant uncertainty. The shapes of the DD and non-DD backgrounds are modeled using the Kernel PDF estimator [27] based on the MC samples with a smoothing parameter set to 1.5. The uncertainties of the shapes are determined by changing the smoothing parameter by ±0.5 and we take the relative changes on the signal yield as the associated uncertainties. We also shift the yields of Bkg I and Bkg II in the fits by 1σ, calculated from the corresponding branching fractions, luminosity measurements [8] and DD cross section [28]. The average changes on the signal yields are taken as the corresponding uncertainties.  Due to the limited statistical significance of the D + → a 0 (980) 0 e + ν e mode, an upper limit on the signal yield is also computed using a Bayesian method. The fit likelihood as a function of the number of signal events, denoted as f L (N ), is convolved with Gaussian functions that represent the systematic uncertainties. For all uncertainty sources except those from the 2-D fit, the effects are taken into account by Gaussian functions having widths equal to the corresponding uncertainties. Uncertainties due to the fit procedure are computed by varying choices of fit conditions to toy MC simulated events, sampled according to the shape of data. In each toy experiment, we perform a nominal fit and one alternative fit with the shape parameters varied in the fit procedure as described above. A Gaussian function is obtained with parameters taken from the mean and the root-mean-square of the resultant discrepancy between the two fitted yields. By integrating up to 90% of the physical region for the smeared f L (N ), we obtain an upper limit of N up < 18.5 at the 90% confidence level (C.L.) for the D + → a 0 (980) 0 e + ν e yield.
Since the branching fraction of a 0 (980) → ηπ has not been well-measured, we report the product branching fractions, obtaining where the first (second) uncertainties are statistical (systematic). The upper limit on the product branching fraction for D + decay is determined as B(D + → a 0 (980) 0 e + ν e ) × B(a 0 (980) 0 → ηπ 0 ) < 3.0 × 10 −4 at the 90% C.L. By convolving the likelihood value from the nominal fits with Gaussian functions whose widths represent the systematic uncertainties for the D 0 and D + decays, we calculate the signal significance including systematic uncertainties to be 6.4σ and 2.9σ for the D 0 and D + decays, respectively. To summarize, we present the observation of the semileptonic decay of D 0 → a 0 (980) − e + ν e and the evidence for D + → a 0 (980) 0 e + ν e . Taking the lifetimes of D 0 and D + [19] into consideration and assuming that B(a 0 (980) − → ηπ − ) = B(a 0 (980) 0 → ηπ 0 ), we find a ratio of partial widths of Γ(D 0 → a 0 (980) − e + ν e ) Γ(D + → a 0 (980) 0 e + ν e ) = 2.03 ± 0.95 ± 0.06, consistent with the prediction of isospin symmetry, where the shared systematic uncertainties have been canceled. The two branching fractions provide information about the dū and (uū − dd)/ √ 2 components in the a 0 (980) − and a 0 (980) 0 wave functions, respectively [4]. Along with the result of the branching fraction of D + → f 0 e + ν e , a result in preparation at BESIII, we will have a valuable input for understanding the nature of the light scalar mesons.
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