Study of the decay $D_s^+\to K_S^0K_S^0\pi^+$ and observation an isovector partner to $f_0(1710)$

Using $e^+e^-$ annihilation data corresponding to a total integrated luminosity of 6.32 $\rm fb^{-1}$ collected at center-of-mass energies between 4.178 and 4.226 GeV with the BESIII detector, we perform an amplitude analysis of the decay $D_{s}^{+} \to K_{S}^{0}K_{S}^{0}\pi^{+}$ for the first time. An enhancement is observed in the $K_{S}^{0}K_{S}^{0}$ mass spectrum near 1.7 GeV/$c^2$, which was not seen in $D_{s}^{+} \to K^+K^-\pi^{+}$ in an earlier work, implying the existence of an isospin one partner of the $f_0(1710)$. The branching fraction of the decay $D_{s}^{+}\to K_{S}^{0}K_{S}^{0}\pi^{+}$ is determined to be $\mathcal{B}(D_{s}^{+} \to K_{S}^{0}K_{S}^{0}\pi^{+})=(0.68\pm0.04_{\rm stat}\pm0.01_{\rm syst})\%$.

a Also at the Moscow Institute of Physics and Technology, Moscow 141700, Russia b Also at the Novosibirsk State University, Novosibirsk, 630090, Russia c Also at the NRC "Kurchatov Institute", PNPI, 188300, Gatchina, Russia Using e + e − annihilation data corresponding to a total integrated luminosity of 6.32 fb −1 collected at center-of-mass energies between 4.178 and 4.226 GeV with the BESIII detector, we perform an amplitude analysis of the decay D + s → K 0 S K 0 S π + for the first time.An enhancement is observed in the K 0 S K 0 S mass spectrum near 1.7 GeV/c 2 , which was not seen in D + s → K + K − π + in an earlier work, implying the existence of an isospin one partner of the f0(1710).The branching fraction of the decay D + s → K 0 S K 0 S π + is determined to be B(D + s → K 0 S K 0 S π + ) = (0.68 ± 0.04stat ± 0.01syst)%.
The constituent quark model has been successful in explaining the composition of hadrons in the past few decades.In this model, many of the observed light mesons can be described as q q states grouped into SU(3) flavor multiplets.In a recent work [1,2], the f 0 (500) and f 0 (980) mesons are considered to be the ground state SU(3) singlet and octet scalar isoscalar mesons, and the a 0 (980) meson is their isovector partner.The SU(3) singlet f 0 (1370) and octet f 0 (1500) are then considered to be the radial excited states of the f 0 (500) and f 0 (980) mesons, respectively, with an isovector partner in the a 0 (1450).However, in case of the next radial excitation in [1,2], for the singlet f 0 (1710) and the newly-identified octet state f 0 (1770), no corresponding isovector a 0 (1710) meson has been established yet.The BaBar collaboration recently claimed the observation of a new a 0 (1710) ± resonance in the decay to π ± η with a mass of approximately 1.7 GeV/c 2 in the η c → ηπ + π − decay [5].Constructing isospin eigenstates from kaon pairs, we obtain (|K + K − − K 0 K0 ) for isospin one, but (|K + K − + K 0 K0 ) for isospin zero.It follows that if the interference between an f 0 and an a 0 is constructive in decays to a K + K − pair, it is destructive in decays to a pair of neutral kaons and vice versa.A comparison between decays involving K + K − and K 0 S K 0 S pairs can thus give access to such interference terms and allows a search for the a 0 (1710) 0 in decays to two kaons.
The BESIII and the BaBar collaborations reported analyses of the D + s → K + K − π + decay [3,4] and observed contributions of the scalar mesons S(980) (where S(980) denotes an admixture of a 0 (980) 0 and f 0 (980)) and f 0 (1710).Both collaborations reported consistent results for the branching fractions (BF) B(D + s → S(980)π + , S(980) → K + K − ) = (1.05± 0.04 stat ± 0.06 syst )% and B(D Furthermore, analyses of D s decays are an important input for studies of the B 0 s meson, which predominantly decays to D s + X [6].In addition, hadronic D s decays probe the interplay of short-distance weak-decay matrix elements and long-distance QCD interactions.The measurement of BFs of hadronic D s decays provides valuable information to help understand strong force-induced amplitudes and phases [7][8][9][10]. The CLEO collaboration measured the absolute BF of the decay D + s → K 0 S K 0 S π + to be (0.77 ± 0.05 stat ± 0.03 syst )% [11], using a dataset corresponding to a luminosity of 586 pb −1 at a center-of-mass energy of 4.17 GeV.In this work, we present the first amplitude analysis and a more precise measurement of the BF of the D + s → K 0 S K 0 S π + decay using 6.32 fb −1 of data samples collected at center-of-mass energies of 4.178, 4.189, 4.199, 4.209, 4.219 and 4.226 GeV with the BESIII detector.We do not distinguish between the a 0 (1710) 0 and f 0 (1710) mesons, and denote the combined state as S(1710).Charge conjugation is implied throughout this paper.
The BESIII detector [12,13] records symmetric e + e − collisions provided by the BEPCII storage ring [14].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 [15].Simulated data samples produced with geant4based [16] Monte Carlo (MC) software, which includes the geometric description 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 [17].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 [17].The known decay modes are modeled with evtgen [18] using BFs taken from the Particle Data Group [6], and the remaining unknown charmonium decays are modeled with lundcharm [19].Final state radiation (FSR) from charged final state particles is incorporated using photos [20].
The process e s decays with a tag technique [21].There are two types of samples used in the tag technique: single tag (ST) and double tag (DT).In the ST sample, a D − s meson is reconstructed through a particular hadronic decay without any requirement on the remaining measured tracks and EMC showers.In the DT sample, a D + s , designated as the "signal", is reconstructed through D + s → K 0 S K 0 S π + , while a D − s , designated as "tag", is reconstructed through one of eight hadronic decay modes: A detailed description of selection conditions concerning charged and neutral particle candidates, the mass recoiling against D ± s candidates, and the mass of the tag candidates are provided in Refs.[22][23][24].
As in Refs.[23,24], an eight-constraint (8C) kinematic fit is performed to select signal events for the amplitude analysis.Besides the constraints arising from fourmomentum conservation, the invariant masses of the two K 0 S candidates, the tag D − s , and the D * +(−) s candidates are constrained to their known masses given in Ref. [6].If there are multiple signal combinations, the candidate with the minimum χ 2 of the 8C kinematic fit is chosen.Signal D + s candidates are selected if their invariant mass is in the interval [1.950, 1.990] GeV/c 2 .A further kinematic fit including a ninth constraint on the mass of the signal D + s is performed, and the updated four-momenta are used for the amplitude analysis.This ensures that all candidates fall within the phase space boundary.In total, 412 events are selected with a purity of f s = (97.3± 0.8)%.The purity is determined from a fit to the invariant mass distribution of the signal D + s candidates.
This analysis uses an isobar formulation in the covariant tensor formalism [25].The total amplitude M for the decay is described by a coherent sum of the amplitudes of all intermediate processes, M = n c n A n , where n indicates the n th intermediate state and c n = ρ n e iφn is the corresponding complex coefficient with magnitude ρ n and phase φ n .The model is symmetrized with respect to the two identical K 0 S mesons.The two-body decay amplitude A n is given by A n = P n S n F r n F D n , where S n and F r(D) n are the spin factor [25] and the Blatt-Weisskopf barrier factor of the intermediate state (the D ± s meson) [26], respectively, and P n is the relativistic Breit-Wigner amplitude [27] describing the propagator of the intermediate resonance.
Contributions of intermediate resonances are determined by an unbinned maximum-likelihood fit to data.A combined probability density function (PDF) for the signal and background hypotheses is constructed, depending on the momenta of the three final-state particles.See Refs.[23,24,28] for details.The signal PDF is constructed from the total amplitude M .The background PDF, B, is constructed from a background shape derived from the inclusive MC samples using the kernel estimation method RooNDKeysPdf [29,30].It models the distribution of an input dataset as a superposition of Gaussian kernels.This background PDF is then added to the signal PDF incoherently and the combined PDF is written as where B ǫ is defined as B/ǫ, ǫ is the acceptance in bins of the Dalitz plot determined with a MC sample of D + s → K 0 S K 0 S π + uniformly distributed over the Dalitz plot.The placeholder p j = {p 1 , p 2 , p 3 } represents the momenta of the final state particles, R 3 is the three-body phase-space element, and f s is the purity.The normalization integral in the denominator is determined by a MC technique as described in Ref. [23,24].1(a).The strong vertical and horizontal bands around 0.8 GeV 2 /c 4 are caused by the process D + s → K 0 S K * (892) + .We choose this process as a reference so that the magnitudes and phases of other amplitudes are to be understood as relative values with respect to this reference amplitude.The purity is a fixed quantity in the fit.Other possible contributions from resonances such as K 1 (1410) + , K * 0 (1430) + , a 0 (980), f 0 (980), f 2 (1270), a 2 (1320), f 0 (1370), a 0 (1450), f 0 (1500), f 2 (1525), a 2 (1700) and S(1710) are added to the fit one at a time.The masses and widths of all resonances are fixed to the known values [6] apart from those of the S(1710).The statistical significance of each new amplitude is calculated from the change of the loglikelihood taking the change in the number of degrees of freedom into account.Various combinations of these resonances are also tested.In addition to the reference amplitude D + s → K 0 S K * (892) + , the amplitude for the decay D + s → S(1710)π + is found to have a significance larger than 10σ.No other contribution has a significance of more than 3σ.The significance of a a 0 (980)/f 0 (980) contribution is less than 0.1σ.The Dalitz plot of the signal MC sample generated based on the result of the amplitude analysis is shown in Fig. 1(b).The mass projections of the fit are shown in Fig. 2. The goodness of fit is χ 2 /NDOF = 15.9/19= 0.8 for Fig. 2(a) and 28.8/32 = 0.9 for Fig. 2(b), where NDOF is the number of degrees of freedom.In the goodness of fit calculation, we merge neighboring bins until each bin has at least 10 entries.The contribution of the nth amplitude relative to the total BF is quantified by the fit fraction (FF) defined as The FFs for both amplitudes and the phase difference relative to the reference process are listed in Table I.The sum of the two FFs is 89.8%.The Breit-Wigner mass and width of the S(1710) are determined to be (1.723 ± 0.011 stat ± 0.002 syst ) GeV/c 2 and (0.140 ± 0.014 stat ± 0.004 syst ) GeV/c 2 , respectively.TABLE I. Fit Fractions (FF) for the two amplitudes, and phase difference to the reference process.The first and the second uncertainties are statistical and systematic, respectively.The sum of the two FFs is 89.8%.

Amplitude
Phase FF (%) D + s → K 0 S K * (892) + 0.0(fixed) 43.5 ± 3.9 ± 0.5 Systematic uncertainties for the results of the amplitude analysis, including the phase difference, FFs, and the mass and the width of the S(1710), are determined by differences between the results of the nominal fit and fits with the following variations.The mass and the width of the K * (892) + are shifted by their uncertainties [6].The radii of the Blatt-Weisskopf barrier factors are varied from their nominal values of 5 GeV −1 and 3 GeV −1 (for the D + s meson and the intermediate resonances, respectively) by ±1 GeV −1 .The uncertainties associated with the size of the background sample are studied by varying the purity within its statistical uncertainty.An alternative background sample is used to determine the background PDF, where the relative fractions of background processes from direct q q and non-D * ± s D ∓ s opencharm processes are varied by the statistical uncertainties of the known cross sections.To estimate the systematic uncertainty related to the reconstruction efficiency, the amplitude analysis is performed varying the particleidentification and tracking efficiencies according to their uncertainties.The total uncertainties are obtained by adding these contributions in quadrature.
The BF of D + s → K 0 S K 0 S π + is measured with the DT technique using the same tag modes and event selection criteria as in the amplitude analysis.However, the kinematic fit is not applied.We require the momentum of the isolated π + to be greater than > 0.1 GeV/c to remove soft pions from D * + decays.For each tag mode, the best ST candidate of the tag D − s is chosen as the combination with the recoiling mass closest to the known D * + s mass [6] and the best DT candidate is chosen as the combination with the average mass of the tag D − s (M tag ) and the signal D + s (M sig ) closest to the D * + s mass.The BF is given by [23,24] where α runs over the various tag modes, and i denotes the different center-of-mass energies.The ST yields in data N ST α,i and the DT yield N DT total,sig are determined by fits to the M tag and M sig distributions shown in Figs.3(af) and Fig. 3(i), respectively.The signal shape is modeled with the MC-simulated shape convolved with a Gaussian function.In the fits to the M tag distributions, the background is parameterized as a second-order Chebyshev polynomial.For the tag modes are added to the background to account for these peaking background contributions.In the fit to the M sig distribution, the background is described by the background MC.The corresponding efficiencies ǫ are obtained by analyzing the inclusive MC samples, with the signal events for D + s → K 0 S K 0 S π + generated based on the results of the amplitude analysis.The total ST yields of all tag modes and the DT yields are 531217 ± 2235 and 371 ± 21, respectively.The BF of D + s → K 0 S K 0 S π + is determined to be (0.68 ± 0.04 stat ± 0.01 syst )%.
We consider the following systematic uncertainties in the measurement of the BF.Varying the signal and background shapes and taking into account the background fluctuation, the uncertainty on the total number of ST D − s candidates is 0.4%.The uncertainty associated with the background shape in the fit to the DT M sig distribution is 0.3%, determined by replacing the nominal background shape with a second-order Chebyshev function and taking the difference between the two results.The uncertainty of the K 0 S reconstruction efficiency is examined using control samples of J/ψ → K 0 S K ± π ∓ and φK 0 S K ± π ∓ decays, and the data-MC efficiency ratio is (101.01 ± 0.53)% [31].We correct the signal efficiencies by this factor, and use the uncertainty of 0.53% as a systematic uncertainty.The π + tracking efficiencies are studied with e + e − → K + K − π + π − events.The data-MC efficiency differences of the π + particle-identification and tracking are both 1.0%.The uncertainty from the signal MC based on the results of the amplitude analysis is studied by varying the fit parameters according to the covariance matrix.The change of signal efficiency is estimated to be 0.5%.The uncertainty due to the limited MC sample size is obtained from α,i (f α,i δǫ α,i ǫα,i ) 2 , where f α,i is the tag yield fraction, and ǫ i and δ ǫi are the signal efficiency and the corresponding uncertainty of tag mode α and centerof-mass energy i, respectively.It is found to be 0.2%.In total, the systematic uncertainty on the branching fraction is 1.9%.
In summary, we present the first amplitude analysis of the decay D + s → K 0 S K 0 S π + using 6.32 fb −1 of e + e − annihilation data taken at center-of-mass energies between 4.178 and 4.226 GeV.The results are listed in Table I.The Breit-Wigner mass and width of the S(1710) are measured to be (1.723±0.011stat ±0.002 syst ) GeV/c 2 and (0.140±0.014 stat ±0.004 syst ) GeV/c 2 , respectively.These parameters are consistent with the PDG evaluation for the f 0 (1710) within 1.2σ and 0.7σ, respectively [6].
Because a significant D + s → f 0 (980)/a 0 (980) 0 π + contribution is observed in the amplitude analysis of D + s → K + K − π + [3], one would expect that about 10% of the signal comes from D + s → f 0 (980)/a 0 (980) 0 π + with f 0 (980)/a 0 (980) 0 → K 0 S K 0 S [6].However, almost no signal populates the region below 1.1 GeV/c 2 in the K 0 S K 0 S mass spectrum.This suppression can likely be attributed to destructive interference between a 0 (980) 0 and f 0 (980) in decays to two neutral kaons.The same interference term would then be constructive in decays to two charged kaons, explaining the large branching fraction observed there.On the other hand, an enhancement is seen in the K 0 S K 0 S mass spectrum around 1.7 GeV/c 2 .Ref. [3] reports B(D + s → f 0 (1710)π + , f 0 (1710) → K + K − ) = (0.10 ± 0.02 stat ± 0.03 syst )%.This corresponds to an expected BF of about 5 × 10 −4 for D + s → f 0 (1710)π + , f 0 (1710) → K 0 S K 0 S , based on isospin symmetry predicting the ratio B(f0(1710)→K + K − ) B(f0(1710)→K 0 S K 0 S ) to be two.In our amplitude analysis, we determine this BF to be (3.1±0.3 stat ±0.1 syst )×10 −3 , which is one order of magnitude larger than the expectation.Based on the same argument concerning the difference in interference between pairs of charged and neutral kaons in isospin one and isospin zero configurations, this observation implies the existence of an isospin one partner of the f 0 (1710) meson, the a 0 (1710) 0 , as proposed by Ref. [1] and as recently observed in Ref. [5].The f 0 (1710) and a 0 (1710) 0 amplitudes could then interfere constructively in decays to two neutral kaons and destructively in decays to two charged kaons, explaining the different observations made in this work and in Ref. [3].A simultaneous amplitude analysis of D + s → K + K − π + and D + s → K 0 S K 0 S π + can further clarify this situation.In addition, a charged partner of a 0 (1710) 0 is expected to be visible in the K 0 S K + mass spectrum in the related decay D + s → K 0 S K + π 0 [33].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.2020YFA0406400, 2020YFA0406300; National Natural Science Foundation of China (NSFC) under Contracts Nos. 11625523, 11635010, 11735014, 11822506, 11835012, 11875054, 11935015, 11935016, 11935018, 11961141012, 12022510, 12025502, 12035009, 12035013,

FIG. 1 . 2 K 0 S π + versus M 2 K 0 S
FIG. 1. Dalitz plot of M 2 K 0 S π + versus M 2 K 0 S π + for D + s → K 0 S K 0 S π + ,symmetrized for the indistinguishable K 0 S candidates (two entries per event), of (a) the sum of all data samples and (b) the signal MC samples generated based on the amplitude analysis result.The black curve indicates the kinematic boundary.

FIG. 2 .
FIG. 2. Distribution of (a) M K 0 S K 0 S and (b) M K 0 S π + from the nominal fit.The distribution of M K 0 S π + contains two entries per event, one for each K 0 S .The data samples are represented by points with uncertainties and the fit results by the blue lines.Colored dashed lines show the individual components of the fit model.Due to interference effects, the total PDF 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 and (i) the Msig distribution of the DT signal candidates.The data samples are represented by points with uncertainties, the total fit results by solid blue lines and the background contributions by dashed violet lines.The pairs of pink arrows indicate the signal regions.

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 Harvard University, Department of Physics, Cambridge, MA, 02138, USA h Also at State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, People's Republic of China i Also at School of Physics and Electronics, Hunan University, Changsha 410082, China j Also at Guangdong Provincial Key Laboratory of Nuclear Science, Institute of Quantum Matter, South China Normal University, Guangzhou 510006, China k Also at Frontiers Science Center for Rare Isotopes, Lanzhou University, Lanzhou 730000, People's Republic of China l Also at Lanzhou Center for Theoretical Physics, Lanzhou University, Lanzhou 730000, People's Republic of China m Currently at Istinye University, 34010 Istanbul, Turkey (Dated:January 24, 2022)

TABLE II .
BFs for amplitudes with the final state K 0 S K 0 S π + .The first and the second uncertainties are statistical and systematic, respectively.Amplitude BF (10−3 ) D + s → K 0 S K * (892) + → K 0 S K 0 S π + 3.0 ± 0.3 ± 0.1 D + s → S(1710)π + → K 0 S K 0 S π + 3.1 ± 0.3 ± 0.1 12061131003; the Chinese Academy of Sciences (CAS) Large-Scale Scientific Facility Program; Joint Large-Scale Scientific Facility Funds of the NSFC and CAS under Contracts Nos.U2032104, U1732263, U1832207; CAS Key Research Program of Frontier Sciences under Contract No. QYZDJ-SSW-SLH040; 100 Talents Program of CAS; INPAC and Shanghai Key Laboratory for Particle Physics and Cosmology; ERC under Contract No. 758462; European Union Horizon 2020 research and innovation programme under Contract No. Marie Sklodowska-Curie grant agreement No 894790; German Research Foundation DFG under Contracts Nos.443159800, Collaborative Research Center CRC 1044, FOR 2359, GRK 2149; Istituto Nazionale di Fisica Nucleare, Italy; Ministry of Development of Turkey under Contract No. DPT2006K-120470; National Science and Technology fund; Olle Engkvist Foundation under Contract No. 200-0605; STFC (United Kingdom); The Knut and Alice Wallenberg Foundation (Sweden) under Contract No. 2016.0157;The Royal Society, UK under Contracts Nos.DH140054, DH160214; The Swedish Research Council; U. S. Department of Energy under Contracts Nos.DE-FG02-05ER41374, DE-SC-0012069.