Observation of $a_0(1710)^+ \to K_S^0K^+$ in study of the $D_s^+\to K_S^0K^+\pi^0$ decay

Using $e^+e^-$ annihilation data corresponding to an integrated luminosity of 6.32 fb$^{-1}$ collected at center-of-mass energies between 4.178 GeV and 4.226 GeV with the BESIII detector, we perform the first amplitude analysis of the decay $D_s^+\to K_S^0K^+\pi^0$ and determine the relative branching fractions and phases for intermediate processes. We observe the $a_0(1710)^+$, the isovector partner of the $f_0(1710)$ and $f_0(1770)$ mesons, in its decay to $K_S^0K^+$ for the first time. In addition, we measure the ratio $\frac{\mathcal{B}(D_{s}^{+} \to \bar{K}^{*}(892)^{0}K^{+})}{\mathcal{B}(D_{s}^{+} \to \bar{K}^{0}K^{*}(892)^{+})}$ to be $2.35^{+0.42}_{-0.23\text{stat.}}\pm 0.10_{\rm syst.}$. Finally, we provide a precision measurement of the absolute branching fraction $\mathcal{B}(D_s^+\to K_S^0K^+\pi^0) = (1.46\pm 0.06_{\text{stat.}}\pm 0.05_{\text{syst.}})\%$.

Using e + e − annihilation data corresponding to an integrated luminosity of 6.32 fb −1 collected at center-of-mass energies between 4.178 GeV and 4.226 GeV with the BESIII detector, we perform the first amplitude analysis of the decay D + s → K 0 S K + π 0 and determine the relative branching fractions and phases for intermediate processes.We observe the a0(1710) + , the isovector partner of the f0(1710) and f0(1770) mesons, in its decay to K 0 S K + for the first time.In addition, we measure the ratio to be 2.35 +0.42 −0.23stat.± 0.10syst.. Finally, we provide a precision measurement of the absolute branching fraction B(D + s → K 0 S K + π 0 ) = (1.46 ± 0.06stat.± 0.05syst.)%.
The constituent quark model describes mesons as bound q q states grouped into SU(3) flavor multiplets.In this scenario, the f 0 (500) and f 0 (980) are often classified as the ground states of the isoscalar scalar mesons and the a 0 (980) meson is taken as their isovector partner.The f 0 (1370), f 0 (1500), and a 0 (1450) are then considered to be the corresponding radially excited states.Within the next higher set of excitations, however, which includes the f 0 (1710) and f 0 (1770), the isovector scalar meson (i.e. the a 0 (1710)) has been proposed but has not yet been well established [1][2][3][4].The f 0 (1710) is often considered to be a likely candidate for a glue-ball or K * K * molecule.Although the recent measurement of the branching fraction (BF) ratio B(f 0 (1710) → ηη ′ )/B(f 0 (1710) → ηη ′ ) [5,6] supports the hypothesis that the f 0 (1710) has a large glueball compenent, one decisive way to determine whether the f 0 (1710) is a glueball or a K * K * molecular is to search for an isovector partner, the a 0 (1710).The a 0 (1710) + meson was previously reported by the BaBar experiment in the process η c → a 0 (1710) + π − with a 0 (1710) + → π + η [7].In addition, the BESIII experiment reported interference between the f 0 (1710) and a 0 (1710) 0 in amplitude analyses of However, more studies of D + s meson decays are crucial to firmly establish the a 0 (1710) triplet.Ref. [2] predicts the product BF of D + s → a 0 (1710) + π 0 with a 0 (1710) + → K 0 S K + to be (2.0 ± 0.7) × 10 −3 .An amplitude analysis of D + s → K 0 S K + π 0 therefore provides an ideal opportunity to study the a 0 (1710) + → K 0 S K + decay.
The internal quark structure of the light scalar mesons, like the a 0 (980), have also been the source of much theoretical speculation.They have been considered to be q q, qq q q, K K, etc.The coupling constants, g a0πη and g a0KK , are predicted by various models [10][11][12] and therefore serve as important experimental constraints on theoretical models.Combining an analysis of we can determine the ratio B(a0(980)→K K) B(a0(980)→ηπ) .This is a key experimental input for the calculation of the coupling constants of the a 0 (980) and helps determine its quark composition [10][11][12][14][15][16].Furthermore, Ref. [17] predicts that B(D ), but the current experimental uncertainties are too large to verify this [18].In an analysis of D + s → K 0 S K + π 0 , we can measure the BFs of both modes simultaneously.Thus, the correlated systematic uncertainties arising from the masses and widths of the resonances, the model parameters, and the common backgrounds can be considered and reduced.
Because of its large BF, the Cabibbo-favored D + s → K 0 S K + π 0 decay is one of the golden decay channels of the D + s .This decay can be used as a reference channel for other decays of the D + s meson and it is important for our understanding of B 0 s decays to final states involving the D + s mesons [18].The CLEO experiment measured the absolute BF of the D + s → K 0 S K + π 0 decay to be (1.52 ± 0.09 stat.± 0.20 syst.)% [19], using 586 pb −1 of e + e − collisions recorded at a center-of-mass energy of 4.17 GeV.
In this Letter, we present the first amplitude analysis and a more precise measurement of the BF for the decay D + s → K 0 S K + π 0 using 6.32 fb −1 of data collected with the BESIII detector at center-of-mass energies between 4.178 and 4.226 GeV.Charge-conjugated modes are implied throughout this paper.
The BESIII detector [20,21] records symmetric e + e − collisions provided by the BEPCII storage ring [22].The cylindrical core of the BESIII detector covers 93% of the full solid angle and consists of a helium-based multilayer drift chamber (MDC), a plastic scintillator time-of-flight system (TOF), and a CsI(Tl) electromagnetic calorimeter (EMC), which are all enclosed in a superconducting solenoidal magnet providing a 1.0 T magnetic field.The end cap TOF system was upgraded in 2015 using multigap resistive plate chamber technology [23].
Simulated data samples produced with a geant4based [24] Monte Carlo (MC) package, which includes the geometric description of the BESIII detector and the detector response, are used to determine detection efficiencies and to estimate backgrounds.The beam energy spread and initial state radiation (ISR) in the e + e − annihilations are simulated with the generator kkmc [25].The inclusive MC sample includes the production of open charm processes, the ISR production of vector charmonium(-like) states, and the continuum processes incorporated in kkmc [25].The known decay modes are described with evtgen [26] using BFs taken from the Particle Data Group [18], and the remaining unknown charmonium decays are generated with lundcharm [27].Final state radiation (FSR) from charged final state particles is incorporated using photos [28].
We reconstruct the process e + e − → D * + s D − s → γD + s D − s using both single-tag (ST) and double-tag (DT) candidate events [29].An ST candidate is an event where only the D − s meson is reconstructed through particular hadronic decays (tag modes) without any requirement on the remaining measured tracks and EMC showers.A DT candidate is an event where the s being reconstructed through the tag modes.Eight tag modes are used: The selection criteria for the final state particles, transition photon, and the D ± s candidates are the same as Refs.[30][31][32].The K 0 S , π 0 , η and η ′ mesons are reconstructed through K 0 S → π + π − , π 0 → γγ, η → γγ and η ′ → π + π − η decays, respectively.
An eight-constraint kinematic fit is applied to the DT candidates to select signal events for the amplitude analysis.The total four-momentum is constrained to the four-momentum of the e + e − system, and the invariant masses of the K 0 S , π 0 , D − s , and D * +(−) s candidates are constrained to their corresponding known masses [18].Within each event, the candidate with the minimum χ 2 from the kinematic fit is chosen.The invariant mass of the signal D + s is then required to be within (1.930, 1.990) GeV/c 2 .A ninth constraint, on the mass of the signal D + s , is then added to the kinematic fit to guarantee all candidates lie inside the allowed phase-space.There are 1050 DT events obtained for the amplitude analysis with a signal purity of (94.7 ± 0.7)%, which is determined from a fit to the invariant mass distribution of the signal D + s candidates.
The intermediate resonance composition is determined using an unbinned maximum-likelihood fit.The likelihood function is described by a signal probability density function (PDF), |M(p j )| 2 , incoherently added to a background PDF, denoted as B [31][32][33].The signal amplitude M is constructed based on the isobar model formulation [34].The background PDF is constructed from in-clusive MC samples using RooNDKeysPdf [35].RooND-KeysPdf is a kernel estimation method [36] implemented in RooFit [35], which models the distribution of an input dataset as a superposition of Gaussian kernels.The likelihood function is then written as where ǫ is the acceptance function, the index k runs over selected events, p µ k represents the four-momenta of the final particles in the k th event, f s is the signal purity, and R 3 is an element of three-body phase space.The normalization integral in the denominator is calculated by MC integration [31].
The signal amplitude M is a coherent sum of the amplitudes for the intermediate processes, M = c n A n , where n indicates the n th intermediate state.The complex coefficient c n equals ρ n e iφn with magnitude ρ n and phase φ n .The amplitude A n is the product of the spin factor [34], the Blatt-Weisskopf barriers of the intermediate state and the D + s meson [37], and the relativistic Breit-Wigner function [38] to describe the propagator for the intermediate resonance.1, reveals there is a strong contribution from the process D + s → K * (892) 0 K + , which appears as the horizontal band around 0.8 GeV 2 /c 4 .Besides this dom- inant intermediate process, other possible intermediate resonances are considered, including the K * 0 (700), K * (892), K * (1410), K * 0 (1430), K * 2 (1430), K * (1680), a 0 (980), a 2 (1320), a 0 (1450), a 2 (1700), a 0 (1710), ρ(1700), and the (Kπ) S−wave (using the LASS parameterization [39] and the K-matrix [40]).Each possibility is added to the fit one at a time.Various combinations of these resonances are tested as well.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.If the significance of a newly added amplitude is larger than 5σ, this amplitude is kept, otherwise it is dropped.During the fit, f s is fixed and the magnitudes and phases of all intermediate processes are floating and are measured with respect to those of the D + s → K * (892) 0 K + .The mass and width of the a 0 (1710) + are free, those of the a 0 (980) + are fixed to the values given in Ref. [41], and those of all other resonances are fixed to their known values [18].and D + s → a 0 (1710) + π 0 , are eventually retained as the optimal set.The mass projections of the fit result are shown in Fig. 2. The contribution of the nth intermediate process relative to the total BF is quantified by a fit fraction (FF) defined as The ratio 42  −0.23stat.± 0.10 syst. is calculated as the quotient of their FFs, where correlations are accounted for in the systematic and statistical uncertainties.The phases and FFs for the intermediate processes are listed in Table I.The mass and width of the a 0 (1710) + are (1.817 ± 0.008 stat.± 0.020 syst. ) GeV/c 2 and (0.097 ± 0.022 stat.± 0.015 syst. ) GeV/c 2 , respectively.Some tests are made to further clarify the existence of the a 0 (1710) + .First, the recoil of the K * 0 (700) may cause an enhancement at the high end of the K 0 S K + spectrum, but the shape of the K * 0 (700) does not match the distribution of data and has a significance less than 3σ.Second, the log-likelihood value of a fit with the K * 0 (700) included and the a 0 (1710) + excluded decreases by 40 compared to the nominal fit.In addition, even though the ρ(1700) + and the a 2 (1700) + peak at the same position as the a 0 (1710) + in the K 0 S K + spectrum, the log-likelihood value is worse by 70 units when these resonances are included instead of the a 0 (1710) + .
The differences between the results of the nominal fit and the following alternative fits are assigned as the systematic uncertainties for the amplitude analysis.To estimate the systematic uncertainty related to resonances, the masses and widths of the K * (892) 0 , K * (892) + , a 0 (980) + , and K * (1410) 0 are varied by their uncertainties [18].The uncertainty associated with Blatt-Weisskopf barriers are studied by varying the radii by ±1 GeV −1 .The uncertainty caused by background is studied by increasing or decreasing f s within its statistical uncertainty, and by varying the proportion of MC background components according to the uncertainties of their cross section measurement.The intermediate resonances with statistical significances less than 5σ are included in the fit one by one and the largest difference with respect to the baseline fit is taken as the systematic uncertainty.The acceptance of the detector is exam-  ined by repeating the amplitude analysis fit with different particle-identification and tracking efficiencies according to their uncertainties.The total uncertainties are determined by adding all the contributions in quadrature.
To measure the absolute BF of the process D + s → K 0 S K + π 0 , we use the same event selection criteria as those for the amplitude analysis, except that the momentum of the final state π + originating from the signal D + s meson is required to be larger than 0.1 GeV/c to remove soft pions from D * + decays, and the best candidate strategy is changed.The best ST candidate from the tagged D − s is chosen using the recoiling mass closest to the known D * + s mass [18] per tag mode.The best DT candidate is chosen using the average mass of the tagged D − s (M tag ) and the signal D + s (M sig ) closest to the known D s mass per tag mode.The BF of the D + s → K 0 S K + π 0 decay is determined by where α represents various tag modes.The ST yield for tag mode α, N ST α , is obtained from fits to the M tag distributions of the ST candidates from the data sample, as shown in Figs.3(a-h).The MC-simulated shape convolved with a Gaussian function is used to model the signal shape while the background shape is parameterized by a second-order Chebyshev function.The MCsimulated shapes of decays are added to the Chebyshev functions in the fits to D − s → K 0 S K − and D − s → π − η ′ , respectively, to account for peaking background.The DT yield, N DT sig , is determined from the fit to the M sig distribution of the DT candidates from the data sample, as shown in Fig. 3(i), in which the signal shape is the MC-simulated shape convolved with a Gaussian function and the background shape is described by the MC-simulated background shape.The inclusive MC samples with D + s → K 0 S K + π 0 events generated based on the amplitude analysis are studied to determine the ST efficiencies ǫ ST α and DT efficiencies ǫ DT α,sig .The total ST yield of all tag modes and the DT yield are 531217 ± 2235 and 985 ± 40, respectively.The BF of the D + s → K 0 S K + π 0 decay is determined to be (1.46 ± 0.06 stat.± 0.05 syst.)%.The BFs for various intermediate processes are calculated with ) and the results are listed in Table I.
The systematic uncertainties on the BF measurement from the following sources are studied.The uncertainty in the total number of the ST D − s mesons is assigned to be 0.4%, including the changes of the fit yields when varying the signal shape, background shape, and taking into account the background fluctuation in the fit.The uncertainty associated with the background shape in the fit to the M sig distribution is estimated to be 1.9% by replacing the nominal background shape with a secondorder Chebyshev function.The uncertainty for the K 0  reconstruction efficiency is estimated to be 1.5% by using control samples of J/ψ → K 0 S K + π − and φK 0 S K + π − decays [42].The K + particle-identification and tracking efficiencies are studied with e + e − → K + K − π + π − events.The data-MC differences of the K + particleidentification and tracking efficiencies are assigned as systematic uncertainties, which are both 1.0%.The systematic uncertainty of the π 0 reconstruction efficiency is investigated by using a control sample of the process e + e − → K + K − π + π − π 0 and a 2.0% systematic uncertainty is assigned.The systematic uncertainty caused by the amplitude analysis model is studied by varying the parameters in the amplitude analysis fit according to the covariance matrix.The change of signal efficiency, 0.7%, is set as the corresponding systematic uncertainty.In summary, this Letter presents the first amplitude analysis of the decay D + s → K 0 S K + π 0 using 6.32 fb −1 of e + e − annihilation data taken at center-of-mass energies between 4.178 GeV and 4.226 GeV.The BF of D + s → K 0 S K + π 0 is determined to be (1.46 ± 0.06 stat.± 0.05 syst.)%, which is consistent with the CLEO result [19].The precision is improved by a factor of 2.8.

FIG. 1 . 2 K 0 S π 0 versus M 2 K
FIG. 1.The Dalitz plot of M 2 K 0 S π 0 versus M 2 K + π 0 for the decay D + s → K 0 S K + π 0 from (a)the data sample and (b) the inclusive MC sample generated based on the results of the amplitude analysis.The black curve indicates the kinematic boundary.

FIG. 2 .
FIG. 2. The projections of the Dalitz plot onto (a) M K 0 S K + , (b) M K 0 S π 0 , and (c) M K + π 0 .The data samples are represented by points with error bars, the fit results by blue lines, and backgrounds by black lines.Colored dashed lines show the components of the fit model.Due to interference effects, the total of the FFs is not necessarily equal to the sum of the components.

FIG. 3 .
FIG. 3. Fits to (a)-(h) the Mtag distributions of the ST candidates of various tag modes and (i) the Msig distribution of the DT candidates.The data samples are represented by points with error bars, the total fit results by blue solid lines, and backgrounds by violet dashed lines.The pairs of pink arrows indicate the signal regions.