Observation of $D^+ \to f_0(500) e^+\nu_e$ and Improved Measurements of $D \to\rho e^+\nu_e$

Using a data sample corresponding to an integrated luminosity of 2.93~fb$^{-1}$ recorded by the BESIII detector at a center-of-mass energy of $3.773$ GeV, we present an analysis of the decays $\bar{D}^0\to\pi^+\pi^0 e^-\bar{\nu}_e$ and $D^+\to\pi^-\pi^+ e^+\nu_e$. By performing a partial wave analysis, the $\pi^+\pi^-$ $S$-wave contribution to $D^+\to\pi^-\pi^+ e^+\nu_e$ is observed to be $(25.7\pm1.6\pm1.1)$% with a statistical significance greater than 10$\sigma$, besides the dominant $P$-wave contribution. This is the first observation of the $S$-wave contribution. We measure the branching fractions $\mathcal{B}(D^{0} \to \rho^- e^+ \nu_e) = (1.445\pm 0.058 \pm 0.039) \times10^{-3}$, $\mathcal{B}(D^{+} \to \rho^0 e^+ \nu_e) = (1.860\pm 0.070 \pm 0.061) \times10^{-3}$, and $\mathcal{B}(D^{+} \to f_0(500) e^+ \nu_e, f_0(500)\to\pi^+\pi^-) = (6.30\pm 0.43 \pm 0.32) \times10^{-4}$. An upper limit of $\mathcal{B}(D^{+} \to f_0(980) e^+ \nu_e, f_0(980)\to\pi^+\pi^-)<2.8 \times10^{-5}$ is set at the 90% confidence level. We also obtain the hadronic form factor ratios of $D\to \rho e^+\nu_e$ at $q^{2}=0$ assuming the single-pole dominance parameterization: $r_{V}=\frac{V(0)}{A_{1}(0)}=1.695\pm0.083\pm0.051$, $r_{2}=\frac{A_{2}(0)}{A_{1}(0)}=0.845\pm0.056\pm0.039$.

In the previous study at the CLEO-c experiment [11], no significant indication for the S-wave was seen. In this Letter, by performing a partial wave analysis (PWA) of D 0 → π − π 0 e + ν e and D + → π − π + e + ν e , we report the first observation of D + → f 0 (500)e + ν e , the measurements of the FF ratios for D → ρe + ν e and the related BFs. Charge conjugate states are implied throughout this Letter. The analysis is performed on a data sample corresponding to an integrated luminosity of 2.93 fb −1 [12,13] collected with the BESIII detector in e + e − annihilation at a center-of-mass energy ( √ s) of 3.773 GeV. The BESIII detector is described in detail elsewhere [14].
Monte Carlo (MC) simulation samples denoted as "generic MC" are described in Ref. [15]. The signal MC consists of exclusive decays ψ(3770) → DD, where the D decays to the SL signal modes, with the decay-product distribution determined by the results of our PWA, while theD decays inclusively, as in the generic MC.
For the BF measurement of SL decay, we use the double-tag technique that was first employed by the MARK-III collaboration [16]. The presence of a DD pair in an event allows a tag sample to be defined in which ā D is reconstructed in a hadronic decay mode. Then in the rest of the event that is recoiling against the tagged D meson, we require an positron and a set of hadrons as a signature of a SL decay.
A detailed description of the selection criteria for charged and neutral particle candidates is provided in Ref. [15]. The taggedD mesons are reconstructed by appropriate combinations of the charged tracks and π 0 candidates in the following hadronic final states: K + π − , K + π − π 0 , K + π − π 0 π 0 , K + π − π − π + , and K + π − π − π + π 0 for neutral tags, and K + π − π − , K + π − π − π 0 , K 0 S π − , K 0 S π − π 0 , K 0 S π − π + π − , and K + K − π − for charged tags. The tag samples are selected based on two variables calculated in the e + e − center-of-mass frame : ∆E ≡ ED − E beam and M BC ≡ E 2 beam − | pD| 2 , where ED and pD are the reconstructed energy and momentum of theD candidate, and E beam is the beam energy . If multiple candidates are present per taggedD mode, the one with the smallest |∆E| is chosen. The yield of each tag mode is obtained from a fit to the M BC distribution following Ref. [15]. We find (2759.6 ± 3.7)×10 3 and (1572.6 ± 1.5)×10 3 reconstructed neutral and charged tags, respectively.
After a tag is identified, we reconstruct the SL decay D 0(+) → π − π 0(+) e + ν e recoiling against the tag by requiring an e + candidate and a π − π 0(+) pair following Ref. [17]. The momentum reconstruction of the e + candidate is improved by recovering energy lost due to final-state radiation or bremsstrahlung in the inner detector region. If there are multiple π 0 candidates in an event, the γγ combination with its invariant mass closest to the nominal π 0 mass [18] is chosen. To suppress the background to the D + signal from the decay of D + → K 0 S e + ν e , K 0 S → π + π − , we veto events with a π + π − invariant mass within ±70 MeV/c 2 of the nominal K 0 S mass [18], which eliminates about 98.3% of such background. The reconstruction of the tag and SL decay candidates must include all charged tracks in the event and satisfy charge conservation. In addition, the maximum energy of extra photon candidates (E γ,max ), which are not used in the tag and SL decay reconstruction, is required to be less than 0.25 GeV to suppress the background events with extra π 0 .
Finally, we define the variable U miss ≡ E miss −| P miss | to identify the SL decay, which peaks at zero for the signal since the neutrino is undetected. Here E miss and P miss are the missing energy and momentum of the D meson; they are calculated in the e + e − center-of-mass frame by E miss = E beam − E ππ − E e and P miss = P SL − P ππ − P e , where E ππ and P ππ are the energy and momentum of ππ system, P SL is the momentum of SL candidate, which is estimated as P SL = −P tag E 2 beam − m 2 D to improve the U miss resolution. HereP tag denotes the unit momentum vector of theD tag and mD is the nominalD mass [18].
The main background contributions are from DD decays, the backgrounds from other processes are negligible. For the D 0 decay, the dominant background arises from D 0 → K * (892) − e + ν e , which results in U miss distribution that is predominantly greater than zero. The backgrounds that peak in U miss mostly arise from D 0 → K − e + ν e , K − → π − π 0 and D + → K 0 S e + ν e decays. For the D + decay, the background is dominated by D + →K * (892) 0 e + ν e , which peaks near zero and π 0 mass, depending on theK * (892) 0 decay mode. With all tag modes combined, we extract the signal yields by performing an unbinned-maximum-likelihood fit to the U miss distribution. The signal is described by the signal MC distribution convolved with a Gaussian function, and the background is modeled by the generic MC distribution convolved with the same Gaussian resolution function. The mean and standard deviation of the Gaussian function are left free to account for any difference between the U miss resolution in the MC simulation and the data. The fit results are shown in Fig. 1. We obtain signal yields of 1102 ± 45 and 1667 ± 50 for D 0 → π − π 0 e + ν e and D + → π − π + e + ν e , respectively, where the errors are statistical. Fits to the Umiss distributions for D 0 → π − π 0 e + νe (a) and D + → π − π + e + νe (b). The points with error bars are data, and the solid lines are the fits. The short-dashed lines are signals and the long-dashed lines are backgrounds.
To study the ππ system and measure the FF, we require |U miss | < 0.06 GeV to select samples for PWA; this leads to 498 ± 13 [480 ± 14] background events with a fraction of (33.28 ± 0.87)% [(23.82 ± 0.69)%] in the D 0 [D + ] mode. The differential decay rate for D 0(+) → π − π 0(+) e + ν e depends on five variables [19]: m, the invariant mass of the ππ system; q, the invariant mass of the e + ν e system; θ e (θ π ), the angle between the momentum of the e + (π − ) in the e + ν e (ππ) rest frame and the momentum of the e + ν e (ππ) system in the D rest frame; and χ, the angle between the normals of the decay planes defined in the D rest frame by the ππ pair and the e + ν e pair. The sign of χ should be changed when analyzing aD candidate in order to maintain CP conservation. In theory, the differential decay rate as a function of these variables is given in Ref. [20]. Neglecting the contributions from the positron mass, it depends on the hadronic FFs as defined in Ref. [21]. For the P -wave contribution, we use the Gounaris-Sakurai (GS) function [22] to describe ρ − and ρ 0 ; the ρ 0 − ω interference is taken into account by the form R ρ 0 −ω (m) = GS ρ 0 (m) × (1 + a ω e iφω RBW ω (m)), where RBW is a relativistic Breit-Wigner function with a constant width [23]. A Blatt-Weisskopf damping factor (r BW ) related to the meson radii is included in the decay amplitude. The q 2 dependence of the total FFs are parameterized in terms of one vector FF [V (q 2 )] and two axial vector FFs [A 1,2 (q 2 )] that are assumed to be dominated by a single pole: Here m V and m A are the pole masses and fixed to m D * (1 − ) ≃ 2.01 GeV/c 2 and m D * (1 + ) ≃ 2.42 GeV/c 2 [18] in the fit, respectively. At q 2 = 0, the FF ratios, r V = V (0) A1(0) and r 2 = A2(0) A1(0) , are determined from the fit to the differential decay rate. These ansätze are adequate according to the fit results shown in Fig. 2 (b, g). The S-wave contribution, characterized by the FF F 10 , is parameterized, assuming only f 0 (500) production, as where p ππ is the magnitude of the three-momentum of the ππ system in the D rest frame. Here the term A S (m) corresponds to the mass-dependent S-wave amplitude is modeled by the fixed resonant lineshape described in Ref. [24]; the parameters a S and φ S are the magnitude and phase of A S (m) relative to GS ρ 0 (m). We perform the PWA using an unbinned-maximumlikelihood fit. The negative log likelihood − ln L is defined as where ξ i denotes the five kinematic variables characterizing the i th event of N and η denotes the fit parameters; ω(ξ i , η) is the decay intensity, and B ǫ (ξ i ) is defined to be the background distribution corrected by the acceptance function ǫ(ξ i ) [25]. The background shape is parameterized using the generic MC and its fraction f b is fixed according to the result of the U miss fit. We model the background with a non-parametric function class RooND-KeysPdf [26] that uses an adaptive kernel-estimation algorithm [27]. The normalization integral in the denominator is determined using a MC technique [15].
A simultaneous PWA fit is performed on both isospinconjugate modes. The structure of the ππ system is only the ρ − in the D 0 mode and is dominated by the ρ 0 , with a small fraction of ω, in the D + mode. In the fit, the masses and widths of ρ and ω are fixed to those reported in Ref. [18]. We also consider other possible components in the D + mode, especially a π + π − S-wave contribution from the f 0 (500). We find that the cosθ π distribution of the fit can agree with data only after considering the S-wave contribution. The statistical significance of the f 0 (500) is determined to be more than 10σ from the change of −2 ln L in the PWA fits with and without this component, taking into account the change of the number of degrees of freedom. The projections of the five kinematic variables for the data are shown in Fig. 2. The difference of the cosθ π distribution between two modes is due to the π + π − S-wave interference contribution in D + decays. Based on this nominal solution, we obtain the fractions of the different components: f f0(500) = (25.7 ± 1.6 ± 1.1)%, f ρ 0 = (76.0 ± 1.7 ± 1.1)% and f ω = (1.28 ± 0.41 ± 0.15)%, as well as the FF ratios r V = 1.695±0.083±0.051 and r 2 = 0.845±0.056±0.039, with a correlation coefficient ρ rV ,r2 = −0.206, where the first and second uncertainties are statistical and systematic, respectively. To calculate the fractions and estimate the corresponding statistical uncertainties, we employ the same method described in Ref. [28]. As a cross check, we perform fits to the two modes separately, and the results are consistent with the simultaneous fit. Replacing the f 0 (500) component with a phase-space S-wave amplitude worsens the − ln L by 40.3. If the phase-space S-wave amplitude is added to the nominal solution on top of the f 0 (500) component, its statistical significance is only about 1σ, so this contribution is neglected. In addition, a possible f 0 (980) component contributing to the F 10 term is studied by adding it to the nominal solution, where f 0 (980) is parameterized by the Flatté formula with its parameters fixed to the BESII measurements [30]. The significance of this component is less than 2σ. By scanning the BF of the f 0 (980) component in the physical region, we obtain an upper limit at the 90% confidence level (CL), which is listed in Table I. To take the systematic uncertainty into account, the likelihood is convolved with a Gaussian function with a resolution equal to the systematic uncertainty.
We calculate the absolute BFs of both modes with the same method as described in Ref. [15]. For the D 0 mode, the only significant contribution observed is D 0 → ρ − e + ν e . For the D + mode, the absolute BFs of the different components are derived from B(D + → π − π + e + ν e ) × f i , where i denotes the different components of ππ system: f 0 (500), ρ 0 , and ω, and f i denotes the fraction obtained via the PWA. The BFs of π 0 → γγ and ω → π + π − [18] have been included in the calculation. All the results and comparisons with those reported by the Particle Data Group (PDG) [18] are summarized in Table I. In the measurements from PDG, ω decays to π + π − π 0 mode, and only ρ 0 is considered besides a small contribution from ω in the π + π − final state.
1.445 ± 0.058 ± 0.039 1.77 ± 0.16 D + → π − π + e + νe 2.449 ± 0.074 ± 0.073 -D + → ρ 0 e + νe 1.860 ± 0.070 ± 0.061 2.18 +0.17 −0.25 D + → ωe + νe 2.05 ± 0.66 ± 0.30 1.69 ± 0.11 D + → f0(500)e + νe, f0(500) → π + π − 0.630 ± 0.043 ± 0.032 -D + → f0(980)e + νe, f0(980) → π + π − < 0.028 -For the BF measurements, most systematic uncertain-ties related to the tag side are canceled when the doubletag technique is employed; therefore, systematic uncertainties arise mainly from the reconstruction of the SL decay. The systematic uncertainty associated with the tag yield for the D 0 (D + ) signal is estimated to be 0.2% (0.4%) by varying the M BC fit range. The uncertainties related to the π ± tracking efficiency, π ± particle identification (PID) efficiency and π 0 reconstruction efficiency are estimated to be 0.8% (1.2%), 0.2% (0.3%) and 0.6%, respectively, by studying the doubly-tagged DD hadronic decay samples. Using a sample of radiative Bhabha events, the uncertainty of the e ± PID efficiency is estimated to be 0.5% for both modes. The uncertainty from the e ± energy recovery is estimated to be 0.4% (0.7%) by comparing to the BFs obtained without recovery. The uncertainty from the K 0 S veto is estimated to be 1.8% by varying the size of the veto window. The fully reconstructed DD hadronic decays are used to show that the uncertainty due to the E γ,max requirement is negligible. We estimate the uncertainty in the signal yield of the U miss fit to be 1.5% (0.5%) by varying the fitting range. The uncertainty related to the modeling of the background shape is estimated to be 1.5% (1.4%) by changing the BFs of the dominant background channels by ±1σ, and σ is the uncertainty reported in Ref. [18]. We estimate the uncertainty due to the PWA model of the signal to be 0.3% (0.9%) by varying the parameters of the nominal solution by their statistical uncertainty. These estimates are added in quadrature to obtain the total systematic uncertainty of 2.5% (3.0%) for D 0 (D + ) mode.
The following sources of systematic uncertainties, as summarized in Table II, have been considered in the PWA procedure. The uncertainty related to variations to the fit are estimated by taking the difference between the alternative fit and the nominal fit. The uncertainty from the modeling of the background shape is assigned as for the BF measurement. The uncertainty due to the fixed background fraction f b is estimated by changing by ±1σ of its statistical error. The parameter of r BW is set to 3.0 GeV −1 in the nominal fit, the uncertainty related to this imperfect knowledge is estimated by varying the value within 2.0 ∼ 4.0 GeV −1 . We vary m V and m A by ±100 MeV/c 2 to estimate the uncertainties associated with the pole mass assumption. The uncertainty from the ρ or ω line shape is estimated by varying the mass and width of ρ or ω by ±1σ error [18]. The systematic uncertainty of the f 0 (500) modeling is considered by replacing with a conventional RBW function with the mass and width fixed to the BESII measurements [29]. The possible bias due to the fit procedure is studied with same method described in Ref. [28]. The mean bias is taken as a corresponding systematic uncertainty.
In summary, the SL decays D 0 → π − π 0 e + ν e and D + → π − π + e + ν e are studied using a data sample corresponding to an integrated luminosity of 2.93 fb −1 col- lected with the BESIII detector at √ s = 3.773 GeV. We measure the FF in D → ρe + ν e via a simultaneous PWA fit to both decay channels, and improve the absolute BFs for these decays. The FF measurements are consistent with the only measurement [11] but with improved precision. These measurements are compatible with the theoretical calculations [1,2] that have much larger uncertainty than experimental results. They also can aid the determination of V ub via a double-ratio technique [31]. The BFs results are consistent with isospin invariance: 2Γ(D + →ρ 0 e + νe) = 0.985±0.054±0.043. The BFs of different components contributing to the D + → π − π + e + ν e decay are also obtained. The hadronic system in this decay is dominated by the P -wave, which is mostly a ρ 0 contribution along with a much smaller one from the ω. Additionally, the S-wave process D + → f 0 (500)e + ν e is observed for the first time with a relative contribution of (25.7 ± 1.6 ± 1.1)%. This is compatible with the theoretical predictions reported in Refs. [5,6]. The process D + → f 0 (980)e + ν e is not significant and an upper limit on its BF is set at the 90% CL.
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 Basic Research Program of China under Contract No. 2015CB856700; National Natural Science Foundation of China (NSFC) under Contracts Nos. 11075174,