Amplitude analysis and branching fraction measurement of the decay Ds+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {D}_s^{+} $$\end{document}→ K+π+π−

Using 6.32 fb−1 of e+e− collision data collected at the center-of-mass energies between 4.178 and 4.226 GeV with the BESIII detector, we perform an amplitude analysis of the decay Ds+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {D}_s^{+} $$\end{document}→ K+π+π− and determine the amplitudes of the various intermediate states. The absolute branching fraction of Ds+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {D}_s^{+} $$\end{document}→ K+π+π− is measured to be (6.11 ± 0.18stat.± 0.11syst.) × 10−3. The branching fractions of the dominant intermediate processes Ds+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {D}_s^{+} $$\end{document}→ K+ρ0, ρ0→ π+π− and Ds+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {D}_s^{+} $$\end{document}→ K*(892)0π+, K*(892)0→ K+π− are determined to be (1.96 ± 0.19stat.± 0.23syst.) × 10−3 and (1.85 ± 0.12stat.± 0.13syst.) × 10−3, respectively. The intermediate resonances f0(500), f0(980), and f0(1370) are observed for the first time in this channel.


Introduction
One popular approach for studies of hadronic charm decays involves application of approximate flavor symmetries, such as flavor SU(3) F [1]. However, the SU(3) F flavor symmetry breaking effect has been observed in D 0 → K + K − and D 0 → π + π − for the first time, and later in the other singly Cabibbo-Suppressed (SCS) charm decays [2]. The SCS decay D + s → K + π + π − , with low contamination from other charm decays, is a promising channel to study the SU(3) F breaking effect. Furthermore, the measurements of the asymmetries of the branching fractions (BFs) of the charge conjugated decays of charmed mesons aid our understanding of charge-parity violation in the charm sector. To date, there have been a few measurements of charge-parity asymmetries, A CP , in the SCS D ± s decay modes [3][4][5]. Two-body charmed meson decays D ± s → V P , where V and P denote vector and pseudoscalar mesons, respectively, have been studied in various approaches. The theoretical predictions of the BFs of the D + s → K + ρ 0 (ρ 0 represents ρ(770) 0 throughout this paper) and D + s → K * (892) 0 π + processes are listed in table 1. References [6,7] studied these decay channels taking into account the SU(3) F flavor symmetry breaking effect, while ref. [8] uses a factorisation-assisted topological-amplitude approach with the ρ-ω mixing. Information about D + s → K 0 ρ + , D + s → K * (892) 0 π + and D + s → K * (892) + π 0 has been JHEP08(2022)196 Channel PDG [2] Cheng et. al [6] Wu et. al [7] Qin et. al [8] D extracted from the decay D + s → K 0 S π + π 0 [5], but is inconclusive regarding these models. More measurements are needed to confront the theoretical predictions.
The CLEO collaboration has reported the absolute BF of D + s → K + π + π − to be (0.654 ± 0.033 stat. ± 0.025 syst. )% [3], using 600 pb −1 of e + e − collisions recorded at a centerof-mass energy √ s = 4. 17 GeV. An amplitude analysis of this channel has been performed by the FOCUS collaboration with 567 ± 31 signal events [9]. Using 6.32 fb −1 of e + e − collision data collected with the BESIII detector at √ s = 4.178 − 4.226 GeV, we perform an amplitude analysis and BF measurement of the D + s → K + π + π − decay with the world's best precision. Charge conjugation is implied throughout this paper except when discussing CP violation.

Detector and data sets
The BESIII detector is a magnetic spectrometer [10,11] located at the Beijing Electron Positron Collider (BEPCII) [12]. A helium-based multilayer drift chamber (MDC), a plastic scintillator time-of-flight system (TOF), and a CsI(Tl) electromagnetic calorimeter (EMC) compose the cylindrical core of the BESIII detector, and they are all enclosed in a superconducting solenoidal magnet providing a 1.0 T magnetic field. The solenoid is supported by an octagonal flux-return yoke with resistive plate counter muon identifier modules interleaved with steel. The acceptance of charged particles and photons is 93% over a 4π solid angle. The charged-particle momenta resolution at 1.0 GeV/c is 0.5%, and the specific energy loss (dE/dx) resolution is 6% for the electrons from Bhabha scattering. The EMC measures photon energies with a resolution of 2.5%(5%) at 1 GeV in the barrel (end-cap) region. The time resolution of the TOF barrel part is 68 ps, while that of the end-cap part is 110 ps. The end-cap TOF was upgraded in 2015 with multi-gap resistive plate chamber technology, providing a time resolution of 60 ps [13][14][15]. About 83% of the data used in this paper benefits from this upgrade.
Data samples corresponding to a total integrated luminosity of 6.32 fb −1 are used in this analysis. The integrated luminosities of the data samples taken at different energy points are listed in table 2 [16][17][18]. These samples are classified into three sample groups, 4.178, 4.189-4.219, and 4.226 GeV according to the years of data taking and their running conditions. Since the D * ± s decays to γD ± s and π 0 D ± s with BFs of (93.5±0.7)% and (5.8±0.7)% [2], respectively, the signal events discussed in this paper are selected from the process geometric description of the BESIII detector and the detector response, and are used to determine detection efficiencies and to estimate backgrounds. The production of open charm processes, the initial-state radiation production of vector charmonium(-like) states and the continuum processes incorporated in kkmc [20,21] are included into the samples. The known decay modes are modeled with evtgen [22,23] using BFs taken from the Particle Data Group (PDG) [2], and the remaining unknown charmonium decays are modeled with lundcharm [24,25]. Final state radiation from charged final state particles is incorporated using photos [26].

Event selection
The tag method [27] is employed to select clean signal samples of e + e − → D * ± s D ∓ s → γD + s D − s in the following analyses. In this method, a single-tag (ST) candidate requires a reconstructed D − s decay to any of the ten hadronic final states listed in table 3. A double-tag (DT) candidate requires that the D + s is reconstructed in the signal mode D + s → K + π + π − in addition to the D − s decay to one of the tag modes. The selection criteria described here are the common requirements for both amplitude analysis and BF measurement. Further requirements for amplitude analysis and BF measurement are discussed in section 4.1 and section 5, respectively.
All charged tracks reconstructed in the MDC must satisfy |cosθ| < 0.93, where θ is the polar angle with respect to the direction of the positron beam. For charged tracks not originating from K 0 S decays, the closest distance to the interaction point is required to be less than ±10 cm along the beam direction and less than 1 cm in the plane perpendicular to the beam. Particle identification (PID) for charged tracks combines the measurements of the dE/dx in the MDC and the flight time in the TOF to form probabilities L(h) (h = K, π) for each hadron (h) hypothesis. The charged tracks are assigned as kaons or pions if their probabilities satisfy one of the two hypotheses, L(K) > L(π) or L(π) > L(K), respectively.
The K 0 S candidates are selected from all pairs of tracks with opposite charges whose distances to the interaction point along the beam direction are less than 20 cm. The selected tracks are assigned as π ± and no further PID requirements are applied. A primary vertex JHEP08(2022)196

Tag mode
Mass window (GeV/c 2 ) and a secondary vertex are reconstructed, and the decay length between the two vertexes is required to be greater than twice its uncertainty. The K 0 S candidates are required to have a π + π − invariant mass (M π + π − ) in the range [0.487, 0.511] GeV/c 2 .
The photon candidates are selected using the EMC showers. The minimum deposited energy of each shower in the barrel region (|cos θ| < 0.80) and in the end-cap region (0.86 < |cos θ| < 0.92) must be greater than 25 MeV and 50 MeV, respectively. The opening angle between the location of each shower in the EMC and the extrapolated position of the closest charged track must be greater than 10 degrees to reject showers originating from charged tracks. The shower is required to start within [0, 700] ns from the event time to suppress electronic noise and showers unrelated to the event.
The π 0 and η candidates are reconstructed from photon pairs with invariant masses in the ranges [0.115, 0.150] GeV/c 2 and [0.500, 0.570] GeV/c 2 , respectively, which correspond to about three standard deviations of the invariant mass resolutions. To improve their invariant mass resolutions, we require that at least one photon comes from the barrel region of the EMC. A kinematic fit constraining the γγ invariant mass to the π 0 or η known mass [2] is performed. The χ 2 of the kinematic fit is required to be less than 30. The η candidates are formed from π + π − η combinations with an invariant mass within the range of [0.946, 0.970] GeV/c 2 .
Ten tag modes are reconstructed and the corresponding mass windows on the tag D − s mass (M tag ) are listed in table 3 [5]. The quantity M rec is defined as where E cm is the energy of the initial state measured from the beam energy, p D − s is the three-momentum of the D

Event selection
Further selection criteria used to improve the signal purity for the amplitude analysis are described next. These criteria will not be used in the BF measurement. An extra photon candidate for the process D * ± s → γD ± s (satisfying the requirements in section 3) is selected in order to perform a six-constraint (6C) kinematic fit. This fit constrains the D − s decay to the utilized tag mode and the D + s decay to the signal mode with two hypotheses: the signal D + s comes from a D * + s or the tag D − s comes from a D * − s . The total four-momentum is constrained to the initial four-momentum of the e + e − system and the invariant masses of tag D − s and D * ± s candidates are constrained to the corresponding nominal masses. The combination with the minimum χ 2 is chosen. A χ 2 6C < 200 requirement is applied to suppress background from other processes. To ensure that all events fall into the phase-space (PHSP) boundary, a mass constraint on the signal D + s is added to the 6C kinematic fit. The four-momenta of the final state particles from this new 7C kinematic fit are used in the amplitude analysis.
The fits to the invariant-mass distributions of the selected signal D + s candidates (M sig ) for various data samples are shown in figure 1. In the fits, the signal is described by the MC-simulated shape convolved with a Gaussian function describing the data-MC resolution difference, and the background is described with the shape derived from the inclusive MC sample. Requiring M sig ∈ [1.950, 1.986] GeV/c 2 , we obtain 772, 444 and 140 signal candidates with purities of (95.7 ± 0.7)%, (95.2 ± 1.0)%, and (92.6 ± 2.2)% for the data samples at √ s = 4.178, 4.189-4.219, and 4.226 GeV, respectively.

Fit method
The amplitude analysis of D + s → K + π + π − is performed by an unbinned maximum likelihood fit. The likelihood function L is constructed with a signal-background combined probability density function (PDF). The log-likelihood is written as where i indicates the data sample groups. The p k denote the four-momenta of the final state particles K + , π + , and π − , where k denotes the k th event in data i. The The signal PDF f S (p) is given by where (p) is the detection efficiency and R 3 (p) is the three-body PHSP function. The R 3 (p) is defined as where j runs over the three daughter particles, E j is the energy of particle j and θ(E j ) is the step function. The total amplitude M is treated with the isobar model, which uses the coherent sum of the amplitudes of the intermediate processes, where c n = ρ n e iφn is the corresponding complex coefficient. The magnitude ρ n and phase φ n are the free parameters in the fit. The amplitude of the n th intermediate state

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The background PDF is given by where B (p) = B(p)/ (p) is the efficiency-corrected background shape. The shape of the background in data is modeled by the background events in the signal region derived from the inclusive MC samples. The comparisons of the M K + π + , M K + π − and M π + π − distributions of events outside the D + s mass signal region between data and MC simulation validate the description from the inclusive MC samples. We have also examined the distributions of the background events in the inclusive MC samples inside and outside the D + s mass signal region. Generally, they are compatible with each other within statistical uncertainties. The background shape B(p) is modeled using RooNDKeysPDF [29], which is a kernel estimation method [30] implemented in RooFit [29] to model the distribution of an input dataset as a superposition of Gaussian kernels.
In the numerator of eq. (4.2), the (p) and R 3 (p) terms are independent of the fitted variables, so they are regarded as constants in the fit. As a consequence, the log-likelihood becomes The normalization integrals of signal and background are evaluated by MC integration, where k is the index of the k th event of the MC sample and N MC is the number of the selected MC events. The M gen (p) is the signal PDF used to generate the MC samples in MC integration.
Tracking and PID differences between data and MC simulation are corrected for by multiplying the weight of the MC event by a factor γ , which is calculated as where n refers to tracking or PID, n,data (p) and n,MC (p) are the tracking or PID efficiency as a function of the momenta of the daughter particles for data and MC simulation, respectively. The tracking and PID efficiencies are studied using clean samples of e + e − → , e + e − → π + π − π + π − and e + e − → π + π − π + π − π 0 processes. By weighting each signal MC event with γ , the MC integration is given by JHEP08(2022)196

Propagator
The intermediate resonances f 0 (1370), K * (892) and K * (1410) 0 are parameterized with the relativistic Breit-Wigner (RBW) formulas, where s a , s b , and s c are the invariant-mass squared of particles a, b and c, respectively. The value of q 0 in eq. (4.10) is that of q when s a = m 2 a , where m a is the mass of particle a. The ρ 0 and ρ(1450) mesons are parameterized as the Gounaris-Sakurai (GS) line shape [32], which is given by where m π is the mass of π, and the normalization condition at P GS (0) fixes the parameter The f 0 (980) is parameterized with the Flatté formula [31]: , (4.17) where g ππ,KK are the constants coupling to individual final states. The parameters are fixed to be g ππ = (0.165±0.010 ± 0.015)GeV 2 /c 4 , g KK = (4.21±0.25 ± 0.21)g ππ and M =

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965 MeV/c 2 , as reported in ref. [31]. The Lorentz invariant PHSP factors ρ ππ (s) and ρ KK (s) are given by The resonance f 0 (500) is parameterized with the formula given in ref. [33]: where Γ tot (m) is decomposed into two parts: and where ρ ππ is the PHSP of the π + π − system and ρ 4π is the PHSP of the 4π system and is approximated by with the parameters fixed to the values given in ref. [34]. The K * 0 (1430) is parameterized with the Flatté formula: , (4.23) where ρ Kπ (m 2 ) and ρ η K (m 2 ) are the Lorentz invariant PHSP factors, and g K + π − ,η K are the constants coupling to individual final states. The parameters of the K * 0 (1430) are fixed to M = 1471.2 MeV/c 2 , g K + π − = 546.8 MeV/c 2 , and g η K = 230 MeV/c 2 , from CLEO [35].
The Kπ S-wave modeled by the LASS parameterization [36] is described by a K * 0 (1430) Breit-Wigner together with an effective range non-resonant component with a phase shift. It is given by with where the parameters F (φ F ) and R(φ R ) are the magnitudes (phases) for non-resonant state and resonance terms, respectively. The parameters a and r are the scattering length and effective interaction length, respectively. We fix these parameters (M, Γ, F, φ F , R, φ R , a, r) to the results obtained from the amplitude analysis to a sample of D 0 → K 0 S π + π − by the BABAR and Belle experiments [37]; these parameters are summarised in table 4.

Spin factors
The spin-projection operators are defined as [38] P 0 (a) = 1, (S wave) The p a , p b , and p c variables are the momenta of particles a, b and c, respectively, and r a = p b − p c . The covariant tensors are given bỹ The spin factors for S, P , and D wave decays are whereT has the same definition ast. The tensor describing the D + s decays is denoted byT and that of a decays is denoted byt.

Blatt-Weisskopf barriers
For a decay process a → b c, the Blatt-Weisskopf barriers factors [39] depend on the angular momenta L and the momentum q of the final-state particle b or c in the rest system of a. They are taken as X L=0 (q) = 1,

Figures 2(a) and 2(b)
show the Dalitz plots of M 2 K + π − versus M 2 π + π − of the selected DT candidates from the data samples and the signal MC samples generated based on the results of the amplitude analysis, respectively. In the fit, the magnitude and phase of the reference amplitude D + s → K + ρ 0 are fixed to 1.0 and 0.0, respectively, while those of the other amplitudes are left floating. The ω i values are fixed to the purities given in section 4.1.
The PHSP MC truth information without detector acceptance and resolution effects is used to calculate the fit fractions (FFs) for individual amplitudes. The FF for the n th JHEP08(2022)196 6.04 ± 0.14 ± 0.26 19.9 ± 2.9 ± 9.3 >10 D + s → K * (892) 0 π + 3.03 ± 0.08 ± 0.04 30.2 ± 1.8 ± 2.0 >10 D + s → K * (1410) 0 π + 5.60 ± 0.14 ± 0.09 4.5 ± 2.1 ± 2.5 5.2 D + s → K * 0 (1430) 0 π + 1.90 ± 0.19 ± 0.20 18.5 ± 2.5 ± 2.6 8.6 where N gen is the number of PHSP signal MC events at generator level. Interference IN between the n th and n th amplitudes is defined as The statistical uncertainties of FFs are obtained by randomly perturbing the fit parameters according to their uncertainties and covariance matrix and re-evaluating FFs. A Gaussian function is fit to the resulting distribution for each FF and the fitted width is taken as its statistical uncertainty. The phases, FFs and statistical significances for various amplitudes are listed in table 5. The interference between amplitudes is listed in table 6. The statistical significances for amplitudes tested but not included in the nominal fit are listed in table 7.
The mass projections of the nominal fit for the amplitude analysis are shown in figure 3. Their systematic uncertainties will be discussed in next section. The sum of the FFs is not unity due to interferences among amplitudes.

Systematic uncertainties for the amplitude analysis
The systematic uncertainties for the amplitude analysis are summarized in table 8, and are described below.
i Fixed parameters in the amplitudes. The masses and widths of K * (892) and K * (1410) 0 are shifted by their corresponding uncertainties [2]. The mass and width of f 0 (1370) are shifted according to the uncertainties from ref. [31]. The masses and coupling constants of the f 0 (980) and K * (1430) Flatté formulas are varied according to ref. [31] and ref. [35], respectively. The uncertainties of the lineshapes of ρ + and ρ(1450) 0 I  II  III  IV  V  VI  VII  VIII   I Table 6. Interference between amplitudes, in unit of % of total amplitude. I denotes D +

Amplitude
Statistical significance(σ) Table 7. Statistical significances for amplitudes tested, but not included in the nominal fit.
are estimated by replacing the GS with the RBW formula. The uncertainties of the lineshape of f 0 (500) are estimated by replacing the propagator with a RBW function with the mass and width fixed at 526 MeV and 534 MeV, respectively [34]. The changes of the phases φ and FFs are assigned as the associated systematic uncertainties.
ii R values. The estimation of the systematic uncertainty associated with the R parameters in the Blatt-Weisskopf factors is performed by repeating the fit procedure after varying the radii of the intermediate states and D + s mesons by ±1 GeV −1 .
iii Fit bias. An ensemble of 600 signal MC samples is generated according to the results of the amplitude analysis. The pull distribution, which is supposed to be a normal distribution, is used to validate the fit performance. The fitted pull values for FFs of D + s → K + ρ 0 , D + s → K + f 0 (980) and D + s → K * 0 (1430) 0 π + and the fitted pull values for phases of D + s → K + f 0 (500), D + s → K + f 0 (980) and D + s → K * (892) 0 π + deviate from zero by more than three, but less than five, standard deviations. Hence, the differences between input values and average fit results are taken as the systematic uncertainties.
iv Background estimation. The fractions of signal, i.e. ω i in eq. (4.1), are varied within their uncertainties and the largest difference from the fits is taken as the uncertainty from the background level. The uncertainty corresponding to the background shape is determined by replacing the input parameters (keeping M 2 K + π − but replacing M 2 π + π − with M 2 K + π + ) and changing the smoothing parameters in RooNDKeysPdf [29].

BF measurement
The BF measurements are based on the following equations: where N ST tag is the ST yield for the tag mode, N DT tag,sig is the DT yield, s is the total number of D * ± s D ∓ s pairs produced from the e + e − collisions, B tag and B sig are the BFs of the tag and signal modes, respectively. The ST tag is the efficiency to reconstruct the tag mode alone and DT tag,sig is the efficiency to reconstruct both the tag and signal modes. In the case of more than one tag mode and energy group, where α represents tag modes in the i th energy group. Solving for B sig ,

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Tag mode where N ST α,i and ST α,i are obtained from the data and inclusive MC samples, respectively. The DT α,sig,i is determined with signal MC samples in which D + s → K + π + π − events are generated according to the baseline model of the amplitude analysis.
In order to ensure that the DT sample is a subset of the ST sample in the BF measurement, the ST candidates are selected ahead of the selection of DT candidates. In addition to the selection criteria for final-state particles described in section 3, the requirement p(π) > 100 MeV/c is applied to all pions in order to exclude transition pions from D * decays. If there are multiple ST candidates, the combination with the M rec closest to the known mass of D * ± s [2] is kept. The yields for various tag modes are obtained by fitting the corresponding M tag distributions and listed in table 9. As an example, the fits to the M tag distributions of the selected ST candidates from the data sample at √ s = 4.178 GeV are shown in figure 4. In the fits, the signal is modeled by an MC-simulated shape convolved with a Gaussian function to take into account the data-MC resolution difference. The background is described by a second-order Chebyshev polynomial. For the tag modes D − s → K 0 S K − and D − s → π − η , there are peaking background contributions coming from D − → K 0 S π − and D − s → ηπ + π − π − decays, respectively. The D − → K 0 S π − and D − s → ηπ + π − π − background are estimated to be 1724 ± 34 and 89 ± 5 events according to the BFs given by PDG [2] and ref. [41], corresponding to about 0.3% and less than 0.1% of the total ST yields, respectively.
Once a tag mode is identified, we attempt to reconstruct the signal decay D + s → K + π + π − . If there are multiple candidates, the DT candidate with the average mass, (M sig + M tag )/2, closest to the D ± s known mass is retained. A 6C kinematic fit is also performed for the BF measurement, and the same K 0 S veto and χ 2 6C requirements as in section 4.1 are applied to suppress the background.
The DT yield is determined from the fit to the M sig distribution. The fit result is shown in figure 5, where the signal shape is modeled by an MC-simulated shape convolved JHEP08(2022)196 with Gaussian function, while the background shape is described with the shape derived from the inclusive MC sample. The DT yield obtained is 1415 ± 42. Based on this, we determine the BF to be B(D + s → K + π + π − ) = (6.11 ± 0.18 stat. ± 0.11 syst. ) × 10 −3 taking into account the differences in K + and π ± tracking and PID efficiencies between data and MC simulation.

The asymmetry of the two BFs is determined to be
The systematic uncertainties of tracking and PID have been canceled in the A CP calculation. The result is consistent with the hypothesis of CP symmetry [8].
The systematic uncertainties in the BF measurement are discussed below.
• ST yield. The uncertainty of the total yield of the ST D − s mesons is determined to be 0.5% by taking into account the background fluctuation in the fit, and examining the changes of the fit yields when varying the background shape.
• Background shape. To estimate the uncertainty due to the background shape of the signal D + s invariant mass distribution, a second-order Chebychev polynomial is used to replace the MC-simulated shape, and an uncertainty of 0.6% is obtained.

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• Amplitude model. The uncertainty from the amplitude model is determined by varying the amplitude model parameters based on their error matrix 600 times. A Gaussian function is used to fit the distribution of 600 DT efficiencies and the fitted width divided by the mean value is taken as an uncertainty. The related uncertainty is 0.5%.
• χ 2 6C requirement. The uncertainty of the χ 2 6C requirement is assigned to be the difference between the data and MC efficiencies of the D + s → K + K − π + candidates. The data and MC simulation control samples of D + s → K + K − π + including over 99% signal events are selected. Then, the efficiency corresponding to the χ 2 6C requirement is obtained, and the uncertainty is calculated by 1 − ε data ε MC , where ε data and ε MC are the selection efficiencies of data and MC simulation, respectively. The associated systematic uncertainty is assigned to be 1.0%.
• K 0 S rejection. The uncertainty of K 0 S rejection has been included in the uncertainty of the amplitude model, in which the inconsistency of the structure of π + π − spectrum between data and MC is estimated by varying the amplitude model parameters by 600 times. On the other hand, the remained D + s → K 0 S K + events are less than 0.1% after K 0 S rejection, thus the systematic uncertainty of K 0 S rejection can be negligible.
All of the systematic uncertainties are summarized in table 10. Adding them in quadrature results in a total systematic uncertainty of 1.8% in the BF measurement.

Summary
Using e + e − collision data equivalent to an integrated luminosity of 6.32 fb −1 recorded with the BESIII detector at the center-of-mass energies between 4.178 and 4.226 GeV, an amplitude analysis of the decay D + s → K + π + π − has been performed.  Table 11. The BFs for various intermediate processes decaying into the final state K + π + π − in this analysis and from the PDG [2], where the first and second uncertainties are statistical and systematic, respectively.
FFs and phases of the different intermediate processes are listed in table 5. The BF for the decay D + s → K + π + π − is measured to be (6.11 ± 0.18 stat. ± 0.11 syst. ) × 10 −3 , which is improved by about a factor of 2 compared to the world average value [2]. The BFs for the intermediate processes calculated with B i = FF i × B(D + s → K + π + π − ) in this analysis and from the PDG [2] are listed in table 11. The BFs of D + s → K + f 0 (500), D + s → K + f 0 (980), and D + s → K + f 0 (1370) are determined for the first time. The asymmetry of the BFs of D + s → K + π + π − and D − s → K − π − π + is determined to be (3.3 ± 3.0 stat. ± 1.3 syst. )%. No indication of CP violation is found.
The obtained BF of D + s → K + ρ 0 is in good agreement with the predictions in ref. [8], and the measured BF of D + s → K * (892) 0 π + is consistent with the prediction in ref. [6]. Meanwhile, our result deviates from the predictions of D + s → K + ρ 0 in refs. [6,7] and D + s → K * (892) 0 π + in refs. [7,8] over two standard deviations. Moreover, ref. [8] predicts the ratio of BF of D + s → K + ρ 0 to that of D + s → K + ω is far greater than one, while ref. [6] calculates that it should be close to one. The ratio is determined to be about two by taking the results in this analysis and in ref. [42]. More precise theoretical predictions are desirable to understand the D ± s → V P processes and SU(3) F flavor symmetry breaking effect.