Observation of $D^{+}\to K_{S}^{0}a_{0}(980)^{+}$ in the amplitude analysis of $D^{+} \to K_{S}^{0}\pi^+\eta$

We perform for the first time an amplitude analysis of the decay $D^{+}\to K_{S}^{0}\pi^+\eta$ and report the observation of the decay $D^{+}\to K_{S}^{0}a_{0}(980)^{+}$ using 2.93 fb$^{-1}$ of $e^+e^-$ collision data taken at a center-of-mass energy of 3.773 GeV with the BESIII detector. As the only W-annihilation free decay among $D$ to $a_{0}(980)$-pseudoscalar, $D^{+}\to K_{S}^{0}a_{0}(980)^{+}$ is the ideal decay to extract the contributions of the external and internal $W$-emission amplitudes involving $a_{0}(980)$ and study the final-state interactions. The absolute branching fraction of $D^{+}\to K_{S}^{0}\pi^+\eta$ is measured to be $(1.27\pm0.04_{\rm stat.}\pm0.03_{\rm syst.})\%$. The product branching fractions of $D^{+}\to K_{S}^{0}a_{0}(980)^{+}$ with $a_{0}(980)^{+}\to \pi^+\eta$ and $D^{+}\to \pi^+ K_0^*(1430)^0$ with $K_0^*(1430)^0\to K_{S}^{0}\eta$ are measured to be $(1.33\pm0.05_{\rm stat.}\pm0.04_{\rm syst.})\%$ and $(0.14\pm0.03_{\rm stat.}\pm0.01_{\rm syst.})\%$, respectively.

Perturbative quantum chromodynamics (QCD) approaches, such as QCD factorization and soft-collinear effective theory, have well explained physics of nonleptonic b-hadron decays [1][2][3][4].However, the charm quark mass is located between the perturbative and nonperturbative QCD regions, making neither of those approaches applicable.As a result, an accurate theoretical description of the underlying mechanism for exclusive hadronic decays of charmed mesons is still not available.A modelindependent analysis was then proposed in the so-called diagrammatic approach to phenomenologically describe charmed meson decays [5].The diagrammatic approach represents the W -emission and weak-annihilation (Wexchange or W -annihilation) amplitudes as topological quark-graph diagrams based on SU(3) flavor symmetry.With necessary experimental inputs, it enables us to extract the contribution and study the relative importance of each amplitude.
A large discrepancy between experimental results and theoretical predictions of branching fractions (BFs) for many D +,0 → a 0 (980)P decays have been found [19][20][21].The main reason could be ascribed to the contribution of the weak-annihilation amplitudes in D decays, which are hard to estimate accurately.Among D +,0 → a 0 (980)P , D + → K 0 S a 0 (980) + is the only decay free of weak-annihilation contributions, as depicted in Fig. 1, and mainly involves the internal W -emission in Fig. 2(a), while its BF is not theoretically predicted.Without the interference from weak annihilation, the study of D + → K 0 S a 0 (980) + will serve as a key experimental input and provide the most sensitive constraint to the contributions and phases of the internal W -emission amplitudes involving a 0 (980) in the diagrammatic approach method [19][20][21].It is worth noting that the external W -emission amplitude for this decay is naively expected to be rather suppressed compared to the internal one, due to G-parity violation [31,32].In addition, the light scalar particle a 0 (980) is commonly con-sidered as a candidate for exotic states, which are states other than typical quark-antiquark mesons, such as states for tetraquarks, K K bound states, and other possible states.The production of these exotic states essentially involves final-state interactions, such as quark exchange, resonance formation, etc. [31][32][33][34][35][36].For example, the production of the tetraquark a 0 (980) + state in D + decays can occur as a result of the fact that the seed tetraquark fluctuations u d → π + η, π + η ′ , K + K0 are dressed by strong resonance interactions in the final state [36].Figure 2(d) also illustrates an example of the production of the tetraquark a 0 (980) + state due to rescattering in the final state.Studying D + → K 0 S a 0 (980) + can experimentally constrain the contribution from these effects, which helps to pin down the nature of a 0 (980).The BESIII detector collected 2.93 fb −1 of e + e − collision data in 2010 and 2011 at √ s = 3.773 GeV [37], which corresponds to the mass of the ψ(3770) resonance.The ψ(3770) decays predominantly to D 0 D0 or D + D − without any additional hadrons.The excellent tracking, precision calorimetry, and the large D D threshold data sample provide an unprecedented opportunity to study the charmed meson decays.Based on this dataset, we present the first amplitude analysis of the decay D + → K 0 S π + η [38] and report the observation of the decay D + → K 0 S a 0 (980).Charge-conjugate states are implied throughout this Letter.
The BESIII detector [39] records the final-state particles of symmetric e + e − collisions provided by the BEPCII storage ring [40] in the center-of-mass energy range from 2.00 to 4.95 GeV, with a peak luminosity of 1 × 10 33 cm −2 s −1 achieved at √ s = 3.77 GeV.The cylindrical core of the BESIII detector covers 93% of the full solid angle and consists of a helium-based multilayer drift chamber, a plastic scintillator time-of-flight system, and a CsI(Tl) electromagnetic calorimeter, which are all enclosed in a superconducting solenoidal magnet providing a 1.0 T magnetic field.Simulated data samples produced with a geant4based [41] Monte Carlo (MC) toolkit, which includes the geometric description [42] 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 [43].The inclusive MC sample includes the production of D D pairs (including quantum coherence for the neutral D channels), the non-D D decays of the ψ(3770), the ISR production of the J/ψ and ψ(3686) states, and the continuum processes incorporated in kkmc.All particle decays are modelled with evtgen [44] using BFs either taken from the Particle Data Group (PDG) [6], when available, or otherwise estimated with lundcharm [45].Final-state radiation from charged final-state particles is incorporated using photos [46].By fully reconstructing the D + D − meson pairs, a double-tag (DT) method provides samples with high purity to perform amplitude analyses and measurements of absolute BFs of the hadronic D + meson decays.The DT candidates are required to be the D + meson decaying to the signal mode D + → K 0 S π + η and the D − meson decaying to six tag modes: The selection criteria for the final-state particles are the same as in Ref. [38].
The D ± mesons are selected using two variables, the energy difference ∆E = E D − E b and the beamconstrained mass , where E b is the beam energy and p D and E D are the momentum and the energy of the D ± candidate in the e + e − rest frame, respectively.The D − meson is reconstructed first through the six tag modes.In case of multiple candidates, the one with the minimum |∆E| is chosen.Once a tag is identified, the signal decay D + → K 0 S π + η is searched for at the recoiling side and the best signal candidate with the minimum |∆E| is selected.All D ± candidates are required to satisfy 1.865 < M BC < 1.875 GeV, and −0.055 < ∆E < 0.040 GeV for the tag modes containing π 0 in the final state, −0.025 < ∆E < 0.025 GeV for the others, and −0.020 < ∆E < 0.020 GeV for the signal side.In addition, the energy of the largest unused photons is required to be less than 0.23 GeV.There are 1113 DT events obtained for the amplitude analysis with a signal purity of (98.2 ± 0.4)%, which is determined from a two-dimensional unbinned maximum-likelihood fit to the distribution of M BC of the tag D − versus that of the signal D + .(Details of the fit are introduced in Ref. [38].) The amplitude analysis requires a sample with good resolution and all candidates falling within the phasespace boundary.Therefore, a four-constraint kinematic fit is performed, assuming D − candidates decaying to one of the tag modes and D + decaying to the signal mode.The invariant masses of (γγ) η , (π + π − ) K 0 S , and D ± candidates are constrained to their individual known masses [6].The Dalitz plot of the D + → K 0 S π + η candidates for the data sample is shown in Fig. 3(a).The intermediate-resonance composition is determined by an unbinned maximum-likelihood fit.The likelihood function L is constructed with a signal probability density function (PDF), which depends on the momenta p of the three final-state particles: , where j runs over the three final-state particles, k runs over each data event, N D is the number of candidate events in data, and f is the signal PDF.To take the background into account, its contribution is subtracted from the data by modifying the negative-likelihood function as follows [47]: where l runs over each background event obtained from the inclusive MC sample, N B is the number of these background events, and the weight w = (1 − purity)N D /N B is normalized to data based on the signal purity.
The signal PDF is written as where ǫ(p j ) is the detection efficiency and R 3 is the standard element of three-body phase space (see Sec. 49, "Kinematics," in Ref. [6]).The total amplitude, M(p j ), is the coherent sum of individual amplitudes of intermediate processes, ρ n e iφn A n , where ρ n and φ n are the magnitude and phase for the amplitude A n of the nth intermediate resonance, respectively.The amplitude A n is the product of the spin factor [48], the Blatt-Weisskopf barriers of the intermediate state and the D + meson [49], and the propagator for the intermediate resonance.The propagator of a 0 (980) + is parametrized with a Flatté formula and the parameters are fixed to the values given in Ref. [50].The propagator of K * 0 (1430) 0 is parametrized as a relativistic Breit-Wigner (RBW) function [51] using parameters obtained in Ref. [52].The normalization integral term in the denominator is handled by MC integration [12].
In the fit, the magnitude and phase of the amplitude D + → K 0 S a 0 (980) + are fixed to 1.0 and 0.0, respectively, while those of other amplitudes are left floating.Various combinations of amplitudes for intermediate resonances are tested.The statistical significance of each amplitude is calculated based on the change of the log-likelihood with and without this amplitude after taking the change of the degrees of freedom into account.Two dominant amplitudes with significance greater than 5σ, D + → K 0 S a 0 (980) + and D + → K * 0 (1430) 0 π + , remain as the nominal set.Other amplitudes for possible intermediate resonances, including and D + → (K 0 S π + ) S−wave η (using the LASS parametrization [53]), have no significant contribution.The contribution of the nth intermediate process relative to the total BF is quantified by a fit fraction (FF) defined as where N gen is the number of the generated phase-space MC events.The statistical uncertainties of FFs are determined by sampling randomly according to the Gaussian distributions of FF a0(980) + and FF K * 0 (1430) 0 and to their correlation through the covariance matrix.
The phases, FFs, and statistical significances for the amplitudes are listed in Table I.The Dalitz plot for the signal MC sample generated based on the amplitude analysis model is shown in Fig. 3(b).The Dalitz plot projections are shown in Fig. 4. The correlation matrix is provided in Table II.The systematic uncertainties of the amplitude analysis, as summarized in Table III, are estimated as described below.
The systematic uncertainty due to the a 0 (980) + line shape is estimated by varying the mass and coupling constant of the Flatté propagator by ±1σ and a threechannel-coupled Flatté formula which adds the πη ′ according to Ref. [50].The systematic uncertainty caused by the K * 0 (1430) 0 line shape is estimated by varying the mass and width of the RBW by ±1σ according to Ref. [52].The quadratic sum of these two uncertainties is taken to be the amplitude model systematic uncertainty.The systematic uncertainty due to the effective radius of the Blatt-Weisskopf barrier [49] is determined by varying the effective radius within the range [2.0, 4.0] GeV for intermediate resonances and [4.0, 6.0] GeV for D + mesons.The maximum variations are taken as the systematic uncertainties.The uncertainties caused by possible but insignificant intermediate resonances, such as K1 (1270) 0 π + , K * (1410) + η, K * 2 (1580) 0 π + , K 0 S a 0 (1450) + , and K 0 S a 2 (1320) + , are taken to be the differences of the phases and FFs with and without the intermediate resonances.
To determine the systematic uncertainty related to the background estimation, the sPlot technique [54] is instead employed on the M BC variable to eliminate the combinatorial background.An amplitude analysis, after applying sWeights, is performed, and deviations from the nominal results are assigned as the systematic uncertainties.The uncertainty associated with the detector acceptance difference between MC samples and data is determined by reweighting ǫ(p j ) in Eq. 2 with different reconstruction efficiencies of K 0 S , π + , and η according to their uncertainties.The changes of the fit results are taken as the systematic uncertainties.In addition, with the amplitude analysis results obtained in this work, 300 signal MC samples are generated with the same size as data.The pull value is calculated by V pull = (V fit − V input )/σ fit , where V input is the input value in the generator and V fit and σ fit are the output value and corresponding statistical uncertainty, respectively.The resulting pull distributions are consistent with the standardized normal distributions, showing no significant fit bias.
The BF of D + → K 0 S π + η was previously measured in Ref. [38].We update this BF to be (1.27 ± 0.04 stat.± 0.03 syst.)% with a signal MC sample generated based on the amplitude analysis model, which provides a more   precise estimation of the detection efficiency.The uncertainty associated with the amplitude analysis model, 0.7%, is estimated by varying the model parameters based on their error matrix.
In summary, we perform an amplitude analysis of D + → K 0 S π + η for the first time and report the observation of D + → K 0 S a 0 (980) + .The interferences between intermediate resonances are fully considered and a (15.83 ± 1.53 stat.± 1.65 syst.)% destructive interference is observed between the D + → K 0 S a 0 (980) + and D + → K * 0 (1430) 0 π + amplitudes.The amplitude analysis results are listed in Table I.With a detection efficiency obtained with MC samples according to the amplitude analysis model, we obtain B(D + → K 0 S π + η) = (1.27± 0.04 stat.± 0.03 syst.)%.This work uses the same datasets as the previous measurement of (1.31 ± 0.05)% did [6,38], whereas the small difference arises from the change of the signal model from a modified data-driven generator BODY3 [55] to the amplitude analysis model, which is used to estimate the detection efficiency.
The BESIII Collaboration thanks the staff of BEPCII and the IHEP computing center for their strong support.The authors thank Prof. Yu

WFIG. 1 .FIG. 2 .
FIG. 1. (a)The W -exchange and (b) the W -annihilation diagrams for D decays.For the D + meson, the W -exchange mechanism is simply absent.The W -annihilation mechanism cannot generate the hadronic mode with a K0 in the final state.

FIG. 3 . 2 K 0 S
FIG. 3. The Dalitz plots of M2 π + η versus M 2 K 0 S η of the D + → K 0S π + η candidates for (a) the data sample and (b) the signal MC sample generated according to the amplitude analysis results.

FIG. 4 .
FIG.4.Projections on the invariant masses of (a) K 0 S π + , (b) K 0 S η, and (c) π + η systems of the nominal fit.The data are represented by black dots with the error bars, the fit results by the solid red curves with the K 0 S a0(980) + yields by the dashed blue curves and the K * 0 (1430) 0 π + yields by the dashed magenta curves.The fit results take into account the relative phases and interferences between the two signal amplitudes, making a simple visual addition of the lines of the two signal amplitudes insufficient.The impact of the destructive interference is clearly visible in (b).Background is not shown as it is almost negligible graphically.

d
Also at Goethe University Frankfurt, 60323 Frankfurt am Main, Germany e Also at Key Laboratory for Particle Physics, Astrophysics and Cosmology, Ministry of Education; Shanghai Key Laboratory for Particle Physics and Cosmology; Institute of Nuclear and Particle Physics, Shanghai 200240, People's Republic of China f Also at Key Laboratory of Nuclear Physics and Ion-beam Application (MOE) and Institute of Modern Physics, Fudan University, Shanghai 200443, People's Republic of China g Also at State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, People's Republic of China h Also at School of Physics and Electronics, Hunan University, Changsha 410082, China i Also at Guangdong Provincial Key Laboratory of Nuclear Science, Institute of Quantum Matter, South China Normal University, Guangzhou 510006, China j Also at MOE Frontiers Science Center for Rare Isotopes, Lanzhou University, Lanzhou 730000, People's Republic of China k Also at Lanzhou Center for Theoretical Physics, Lanzhou University, Lanzhou 730000, People's Republic of China l Also at the Department of Mathematical Sciences, IBA, Karachi 75270, Pakistan

TABLE I .
Phases, FFs, and statistical significances for different amplitudes.The first uncertainty is statistical, while the second and third uncertainties are model and experimental systematic uncertainties, respectively.The total of the FFs is not necessarily equal to 100% due to interference effects.(1430) 0 π + 2.58 ± 0.06 ± 0.09 ± 0.01 10.83 ± 1.50 ± 1.27 ± 0.08

TABLE II .
Correlation matrix for the phases and FFs.
-Kuo Hsiao for helpful discussions.This work is supported in part by National Key R&D Program of China under Contracts Nos.2020YFA0406400, 2020YFA0406300; National Natural Science Foundation of China (NSFC) under Contracts Nos.11635010, 11735014, 11835012, 11875054, 11935015, 11935016, 11935018, 11961141012, 12022510, 12025502, 12035009, 12035013, 12061131003, 12192260, 12192261, 12192262, 12192263, 12192264, 12192265, 12221005, 12225509, 12235017, 12205384; 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 Nos.U2032104, U1832207; CAS Key Research Program of Frontier Sciences under Contracts Nos.QYZDJ-SSW-SLH003, QYZDJ-SSW-SLH040; 100 Talents Program of CAS; The Institute of Nuclear and Particle Physics (IN-PAC) and Shanghai Key Laboratory for Particle Physics and Cosmology; ERC under Contract No. 758462; European Union's Horizon 2020 research and innovation programme under Marie Sklodowska-Curie grant agreement under Contract No. 894790; German Research Foundation DFG under Contracts Nos.443159800, 455635585, Collaborative Research Center CRC 1044, 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