Studies of the resonance structure in D 0 → K ∓ π ± π ± π ∓ decays.

Amplitude models are constructed to describe the resonance structure of D 0 → K - π + π + π - and D 0 → K + π - π - π + decays using pp collision data collected at centre-of-mass energies of 7 and 8 TeV with the LHCb experiment, corresponding to an integrated luminosity of 3.0  f b - 1 . The largest contributions to both decay amplitudes are found to come from axial resonances, with decay modes D 0 → a 1 ( 1260 ) + K - and D 0 → K 1 ( 1270 / 1400 ) + π - being prominent in D 0 → K - π + π + π - and D 0 → K + π - π - π + , respectively. Precise measurements of the lineshape parameters and couplings of the a 1 ( 1260 ) + , K 1 ( 1270 ) - and K ( 1460 ) - resonances are made, and a quasi model-independent study of the K ( 1460 ) - resonance is performed. The coherence factor of the decays is calculated from the amplitude models to be R K 3 π = 0.459 ± 0.010 ( stat ) ± 0.012 ( syst ) ± 0.020 ( model ) , which is consistent with direct measurements. These models will be useful in future measurements of the unitary-triangle angle γ and studies of charm mixing and C P violation.

Introduction

The decays1 and have an important role to play in improving knowledge of the Cabibbo–Kobayashi–Maskawa (CKM) unitarity-triangle angle . Sensitivity to this parameter can be obtained by measuring -violating and associated observables in the decay , where D indicates a neutral charm meson reconstructed in final states common to both and , of which are significant examples [1, 2]. A straightforward approach to such an analysis is to reconstruct the four-body D-meson decays inclusively, which was performed by the LHCb collaboration in a recent measurement [3]. Alternatively, additional sensitivity can be sought by studying the variation of the observables across the phase space of the D decays, a strategy that requires knowledge of the variation of the decay amplitudes of the charm mesons.

Studies of charm mixing and searches for violation in the – system, which for these final states have only been performed inclusively [4], will also benefit from an understanding of the variation of the decay amplitudes across their phase space. These decay modes are also a rich laboratory for examining the behaviour of the strong interaction at low energy, through studies of the intermediate resonances that contribute to the final states. All these considerations motivate an amplitude analysis of the two decays.

The decay has a branching ratio of  [5], which is the highest of all decay modes involving only charged particles, and is predominantly mediated by Cabibbo-favoured (CF) transitions. The decay is dominated by doubly Cabibbo-suppressed (DCS) amplitudes, with small contributions from mixing-related effects, and occurs at a rate that is suppressed by a factor of  [4] compared to that of the favoured mode. The favoured and suppressed modes are here termed the ‘right-sign’ (RS) and ‘wrong-sign’ (WS) decay, respectively, on account of the charge correlation between the kaon and the particle used to tag the flavour of the parent meson.

In this paper, time-integrated amplitude models of both decay modes are constructed using pp collision data collected at centre-of-mass energies of 7 and 8 TeV with the LHCb experiment, corresponding to an integrated luminosity of 3.0 fb. The RS sample size is around 700 times larger than the data set used by the Mark III collaboration to develop the first amplitude model of this decay [6]. An amplitude analysis has also been performed on the RS decay by the BES III collaboration [7] with around 1.6% of the sample size used in this analysis. This paper reports the first amplitude analysis of the WS decay.

The paper is organised as follows. In Sect. 2 the detector, data and simulation samples are described, and in Sect. 3 the signal selection is discussed. The amplitude-model formalism is presented in Sect. 4, and the fit method and model-building procedure in Sect. 5. Section 6 contains the fit results and conclusions are drawn in Sect. 7.

Detector and simulation

The LHCb detector [8] is a single-arm forward spectrometer covering the pseudorapidity range , designed for the study of particles containing or quarks. The detector includes a high-precision tracking system consisting of a silicon-strip vertex detector surrounding the pp interaction region, a large-area silicon-strip detector located upstream of a dipole magnet with a bending power of about , and three stations of silicon-strip detectors and straw drift tubes placed downstream of the magnet. The tracking system provides a measurement of momentum, , of charged particles with a relative uncertainty that varies from 0.5% at low momentum to 1.0% at 200. The minimum distance of a track to a primary vertex (PV), the impact parameter, is measured with a resolution of , where is the component of the momentum transverse to the beam, in . Different types of charged hadrons are distinguished using information from two ring-imaging Cherenkov (RICH) detectors. Photons, electrons and hadrons are identified by a calorimeter system consisting of scintillating-pad and preshower detectors, an electromagnetic calorimeter and a hadronic calorimeter. Muons are identified by a system composed of alternating layers of iron and multiwire proportional chambers.

The trigger [9] consists of a hardware stage, based on information from the calorimeter and muon systems, followed by a software stage, in which all charged particles with are reconstructed for 2011 (2012) data. At the hardware trigger stage, events are required to have a muon with high or a hadron, photon or electron with high transverse energy in the calorimeters. The software trigger requires a two-, three- or four-track secondary vertex with a significant displacement from the primary pp interaction vertices. At least one charged particle must have a transverse momentum and be inconsistent with originating from a PV. A multivariate algorithm [10] is used for the identification of secondary vertices consistent with the decay of a hadron.

In the simulation, pp collisions are generated using Pythia  [11] with a specific LHCb configuration [12]. Particle decays are described by EvtGen  [13]. The interaction of the generated particles with the detector, and its response, are implemented using the Geant4 toolkit [14, 15] as described in Ref. [16].

Signal selection and backgrounds

The decay chain with is reconstructed as a clean source of mesons for analysis. The mesons are reconstructed in the final states. The charged pion, , originating from the is referred to as ‘slow’ due to the small Q-value of the decay. The charge of the muon and slow pion are used to infer the flavour of the neutral D meson. Candidates are only accepted if these charges lead to a consistent hypothesis for the flavour of the neutral D meson. All other aspects of the reconstruction and selection criteria are identical between the RS and WS samples.

The two-dimensional plane vs. , where is the invariant mass of the meson candidate, and is mass difference between the and meson candidates, is used to define signal and sideband regions with which to perform the amplitude analysis and study sources of background contamination. The signal region is defined as of the signal peak in , which corresponds to about three times the width of the peak.

It is required that the hardware trigger decision is either due to the muon candidate or is independent of the particles constituting the reconstructed decay products of the candidate. For example, a high- particle from the other meson decay in the event firing the hadron trigger. The software trigger decision is required to either be due to the muon candidate or a two- three- or four-track secondary secondary vertex.

The WS sample is contaminated by a category of RS decays in which the kaon is mis-identified as a pion, and a pion as a kaon. To suppress this background, it is required that the kaon is well identified by the RICH detectors. The residual contamination from this background is removed by recalculating the mass of the candidate with the mass hypotheses of a kaon and each oppositely charged pion swapped, then vetoing candidates that fall within 12 of the nominal mass of the meson. As the majority of particles from the PV are pions, the particle identification requirements on the kaon also reduces the background from random combinations of particles.

Remaining background from random combinations of particles can be divided into two categories. Candidates where the is reconstructed from a random combination of tracks are referred to as combinatorial background. Candidates where the is correctly reconstructed but paired with an unrelated are referred to as mistag background. This latter source of background is dominated by RS decays. Both of these backgrounds are suppressed using a multivariate classifier based on a Boosted Decision Tree (BDT) [17-19] algorithm. The BDT is trained on RS data candidates from the signal region and the sidebands of the WS data, and uses 15 variables related to the quality of the reconstruction of the PV, and decay vertices, and the consistency of tracks in the signal candidate incoming from these vertices. Variables pertaining to the kinematics and its decay products are avoided to minimise any bias of the phase-space acceptance.

The signal and background yields in the signal region for each sample is determined by simultaneously fitting the two-dimensional vs. distribution for both samples. The , muon and slow pion candidates are constrained to originate from a common vertex in calculating the and masses. This requirement improves the resolution of the distribution by approximately a factor of two. The signal is modelled with a product of two Cruijff [20] functions. The Cruijff shape parameters are shared between both samples. The combinatorial background is modelled by a first-order polynomial in , and by a threshold function in ,where and the parameters p, a are determined by the fit. The background shape parameters, including those for the polynomial in , are allowed to differ between WS and RS samples. The mistag background component is a product of the signal shape in and the combinatorial background shape in . The optimal requirement on the output of the BDT classifier is selected by repeating the fit varying this requirement, and maximising the expected significance of the WS signal, which is defined aswhere is the background yield in the signal region. The expected number of WS candidates, , is estimated by scaling the number of RS signal candidates in the signal region by the ratio of branching fractions. The yields of the various contributions for both samples are listed in Table 1, and the and distributions, with the fit projections superimposed, are shown in Fig. 1. The purities of the RS and WS samples after selection are found to be 99.6 and , respectively, with of WS candidates arising from mistagged decays. Studies of simulated data indicate that the selected sample has a relatively uniform acceptance across the phase space, with approximately 30% reductions in acceptance near the edges of the kinematically allowed region. The samples also have a relatively uniform selection efficiency in decay time, being constant within for lifetimes greater than one average lifetime of the meson.

Table 1

Signal and background yields for both samples in the signal region, presented separately for each year of data taking

	Yield
	Signal	Combinatorial background	Mistag background
D^0→K^-π^+π^+π^-
2011	266368±490	977±10	–
2012	624332±765	2475±19	–
Total	890701±927	3452±24	–
D^0→K^+π^-π^-π^+
2011	875±32	151±3	47±6
2012	2154±51	340±5	108±9
Total	3028±61	491±7	155±11

Fig. 1

Invariant mass and mass difference distributions for RS (top) and WS (bottom) samples, shown with fit projections. The signal region is indicated by the filled grey area, and for each plot the mass window in the orthogonal projection is applied

For the amplitude analysis, a kinematic fit is performed constraining the mass to its known value [21], which improves the resolution in the phase space. This also forces all candidates to lie inside the kinematically allowed region. Candidates are only accepted if this kinematic fit converges.

Formalism of amplitude model

The amplitudes contributing to the decays are described in terms of a sequence of two-body states. It is assumed that once these two-body states are produced, rescattering against other particles can be neglected. Two-body processes are often referred to as isobars and this approximation as the isobar model. Isobars can be described in terms of resonances, typically using the relativistic Breit-Wigner amplitude for narrow vector and tensor states. For scalar states, there typically are multiple broad overlapping resonances, in addition to significant nonresonant scattering amplitudes between the constituent particles of the state. Such states cannot be described in terms of Breit-Wigner amplitudes and instead the K-matrix formalism [22, 23] is adopted, and will be denoted by and throughout for and S-waves, respectively.

The following decay chains are considered:Decay chains are described using a product of dynamical functions for each isobar and a spin factor. The amplitude for each decay chain is explicitly made to respect Bose symmetry by summing over both possible permutations of same-sign pions. The total amplitude is then modelled as a coherent sum of these processes. Spin factors are modelled using the Rarita–Schwinger formalism following the prescription in Ref. [24]; the details of this formulation are included in Appendix A.

Cascade decays have the topology – the meson decays into a stable pseudoscalar state and an unstable state X. The unstable state then decays to three pseudoscalars via another intermediate unstable state (Y). There are three distinct possibilities for cascade decays. The resonance X can either have isospin , and will therefore decay into the final state, or have isospin and therefore will decay into the final state. In the case, the next state in the cascade Y can either be in or , referred to as cases (1) and (2), respectively. In the case, there is only the state, referred to as case (3). Two complex parameters can be used to describe cascade decays: the coupling between the meson and the first isobar, and then the coupling between the first isobar and the second intermediate state. One of the couplings between isobars can be fixed by convention, typically the dominant channel. For example, for the resonance, the couplings for subdominant decay chains such as are defined with respect to the dominant decay.

Example: .

Example: .

Example: .

Quasi two-body decays have the topology – the meson decays into a pair of unstable states, which in turn each decay to a pair of stable pseudoscalar mesons. The only possibility where X, Y form resonances of conventional quark content is , with an example of a typical process being . The parameters to be determined describe the coupling between the initial state and the quasi two-body state. In the above example, there are three different possible orbital configurations of the vector–vector system, and hence this component has three complex parameters.

Resonances are modelled with the relativistic Breit-Wigner function unless otherwise stated, which as a function of the invariant-mass squared, s, takes the formwhere the mass of the resonance is and is the energy-dependent width. The form factor for a decay in which the two decay products have relative orbital angular momentum L is given by the normalised Blatt–Weisskopf function [25] , where q is the three-momentum of either decay product in the rest frame of the resonance, and is normalised to unity at zero momentum transfer. The factor k normalises the lineshape integrated over all values of s if the Blatt–Weisskopf form-factor and energy dependence of the width are neglected, and is included to reduce correlations between the coupling to the channel and the mass and width of the resonance.

For a resonance that decays via a single channel to two stable particles, such as , the width is given bywhere is the width at the resonance mass, and is the linear momentum of either decay product evaluated at the rest mass of the resonance. The energy-dependent width of a resonance that decays to a three-body final-state must account for the dynamics of the intermediate decay process, and follows that developed for the decay by the CLEO Collaboration in Ref. [26]. The width of a resonance R decaying into three bodies abc can be expressed in terms of the spin-averaged matrix element of the decay integrated over the phase space of the three-particle final state,where the matrix element consists of a coherent sum over the intermediate states in the three-body system, described using the isobar model and using the fitted couplings between the resonance and the intermediate isobars. In the example of the decay of the resonance, these are predominately the couplings to the and intermediate states. The width is normalised such that . In the three-body case, exponential form-factors are used rather than normalised Blatt–Weisskopf functions,where r characterises the radius of the decaying resonance.

The K-matrix formalism [22] provides a convenient description of a two-particle scattering amplitude, which is particularly useful in parameterising S-wave systems. This formulation can then be used in the description of multibody decays on the assumption that rescattering against the other particles in the decay can be neglected. The K-matrix formalism is used in this analysis to describe the and S-waves due to its relative success in parameterising the scalar contributions to three-body decays [27, 28] of the meson.

The S-wave (isoscalar) amplitude is modelled using the K matrix from Ref. [27, 29], which describes the amplitude in the mass range , considering the effects of five coupled channels, , , , , , and five poles with masses which generate the resonances. The K matrix also includes polynomial terms that describe nonresonant scattering between hadrons. The coupling to each of these poles and the direct coupling to each of the five channels depend on the production mode, which is modelled using the production vector or P-vector approach, in which the amplitude iswhere is the two-body phase-space matrix. The complex-valued vector function, , has one component for each of the coupled channels, and describes the coupling between the initial state and either one of the poles or a direct coupling to one of these channels. The generic P-vector for the isoscalar K-matrix therefore has 10 complex parameters. An additional complexity in the four-body case is that there are several initial states that couple to the S-wave, each of which has its own P vector. Several simplifying assumptions are therefore made to the P vector to avoid introducing an unreasonable number of degrees of freedom. The only direct production terms included in the P vector are to the and states, as the production of the final state via a direct coupling to another channel all have similar structure below their corresponding production thresholds. The couplings to poles 3, 4 and 5 (where the numbering of the poles is defined in Ref. [29]) are also fixed to zero, as production of these poles only has a small effect within the phase space. This choice reduces the number of free parameters per S-wave production mechanism to four complex numbers. The couplings to the poles are described by and , while the direct couplings to each channel by and . The production vectors used here should therefore be considered as a minimal simplified model. For production of S-wave states via resonances, such as the decay chain , improved sensitivity to the structure of the state can be achieved by studying a decay mode that produces the with a larger phase space. In several cases, one or more of these couplings are found to be negligible for a given production mode, and therefore are fixed to zero.

The S-wave is modelled using the K matrices from the analysis of by the FOCUS collaboration [28]. The K matrix considers two channels, and , and a single pole which is responsible for generating the resonance. Additionally, the K matrix includes polynomial terms that describe nonresonant scattering between the hadrons. The S-wave also contains a component. No poles or inelasticity are expected with this isospin, and therefore the associated amplitude can be modelled using a K matrix consisting of a single scalar term.

The amplitudes are constructed in the Q-vector [23] approximation. The P vector has the same pole structure as the K matrix, and therefore the approximationcan be made, where is a slowly varying complex vector. This is sometimes referred to as the Q-vector [23] approximation, and allows the insertion of into Eq. (7), and the rephrasing of the decay amplitude, , in terms of the T-matrix elements from scattering:wherewhich is the transition matrix associated with the scattering process. Given the relatively small energy range available to the system, it is reasonable to approximate as a constant. Inclusion of polynomial terms in is found not to improve the fit quality significantly. The coupling to the channel, , is defined with respect to the coupling to the channel, in all production modes. If the phase of is zero, the phase shift of the component matches that found in scattering experiments, which is the expected result if Watson’s theorem [30] holds for these decays. Similar to the S-wave, the components of and the coupling to the channel are allowed to differ between production modes.

Fit formalism and model construction

Independent fits are performed on the and data sets, using an unbinned maximum likelihood procedure to determine the amplitude parameters. The formalism of the fit is described in Sects. 5.1–5.3, and the method for systematically selecting plausible models is discussed in Sect. 5.4.

Likelihood

The probability density functions (PDFs) are functions of position in decay phase-space, , and are composed of the signal amplitude model and the two sources of background described in Sect. 3:The signal PDF is described by the function , where is the total matrix element for the process, weighted by the four-body phase-space density , and the phase-space acceptance, . The mistag component involving , is only present in the WS sample, and is modelled using the RS signal PDF. The combinatorial background is modelled by , and is present in both samples. The normalisation of each component is given by the integral of the PDF over the phase space, , where , weighted by the fractional yield, , determined in Sect. 3.

The PDF that describes the combinatorial background in the WS sample is fixed to the results of a fit to the two sidebands of the distribution, below and above . The components in this model are selected using the same algorithm to determine the resonant content of the signal modes, which is discussed in Sect. 5.4. In this case, the PDF incoherently sums the different contributions and assumes no angular correlations between tracks. The contamination from combinatorial background in the RS sample is very low, and hence this contribution can safely be assumed to be distributed according to phase space, that is .

The function to minimise isAs the efficiency variation across the phase space factorises in the PDF, these variations result in a constant shift in the likelihood everywhere except the normalisation integrals, and hence can be neglected in the minimisation procedure. Efficiency variations can then be included in the fit by performing all integrals using simulated events that have been propagated through the full LHCb detector simulation and selection. These events are referred to as the integration sample. The values of the normalisation integrals are independent of the generator distribution of the integration sample, however the uncertainties on the integrals are minimised when integration events approximate the function being integrated, which is known as importance sampling. Therefore, integration samples are generated using preliminary models that do not include efficiency effects.

Goodness of fit

The quality of fits is quantified by computing a metric. Candidates are binned using an adaptive binning scheme. Five coordinates are selected, and the phase space is repeatedly divided in these coordinates such that each bin contains the same number of candidates, following the procedure described in Ref. [4]. The division is halted when each bin contains between 10 and 20 entries. This procedure results in 32,768 approximately equally populated bins for the RS sample, and 256 for the WS sample. Five two- and three-body invariant mass-squared combinations are used as coordinates for the binning procedure, and . The is defined aswhere is the observed number of candidates in bin i and , the expected number of entries determined by reweighting the integration sample with the fitted PDF. The statistical uncertainty from the limited size of the integration sample, , is included in the definition of the , and is estimated aswhere is the weight of integration event j. The per degree of freedom is used as the metric to optimise the decay chains included in a model, using the model-building procedure described in Sect. 5.4.

Fit fractions

The values of coupling parameters depend strongly on various choices of convention in the formalism. Therefore, it is common to define the fractions in the data sample associated with each component of the amplitudes (fit fractions). In the limit of narrow resonances, the fit fractions are analogous to relative branching fractions. The fit fraction for component i isFor cascade processes, the different secondary isobars contribute coherently to the fit fractions. The partial fit fractions for each sub-process are then defined as the fit fraction with only the contributions from the parent isobar included in the denominator.

Model construction

The number of possible models that could be used to fit the amplitudes is extremely large due to the large number of possible decay chains (). A full list of the components considered is included in Appendix B.

A model of “reasonable” complexity typically contains different decay chains. Therefore, the number of possible models is extremely large, and only an infinitesimal fraction of these models can be tested. An algorithmic approach to model building is adopted, which begins with an initial model and attempts to iteratively improve the description by adding decay chains. For the initial model is that constructed by the Mark III collaboration [6], augmented by knowledge from other analyses, such as the additional decay channels of the found in the amplitude analysis of the decay performed by the FOCUS collaboration [31]. The two-body nonresonant terms in the Mark III model are replaced with the relevant K matrices, and the four-body nonresonant term replaced with a quasi two-body scalar–scalar term , modelled using a product of K matrix amplitudes.

For , where no previous study exists, the initial model is obtained by inspection of the invariant-mass distributions. There are clear contributions from the and resonances, and therefore combined with the expectation that the vector–vector contributions should be similar between WS and RS, the quasi two-body mode is included in all three allowed orbital states . The scalar–scalar contribution should also be comparable between WS and RS decay modes, and hence the quasi two-body term is also included.

The steps of the model-building procedure areThe model-building procedure therefore results in an ensemble of parametrisations of comparable fit quality.

Take a model and a set of possible additional decay chains, initially the complete set discussed in Appendix. B. Perform a fit to the data using this model adding one of these decay chains.

If adding this decay chain improves the per degree of freedom by at least 0.02, then retain the model for further consideration.

On the first iteration, restrict the pool of decay chains that are added to the model to those 40 contributions that give the largest improvements to the fit.

Reiterate the model-building procedure, using the 15 models with the best fit quality from step 2 as starting points. Finish the procedure if no model has improved significantly.

Fit results

This section presents fit results and systematic uncertainties, with the latter discussed first in Sect. 6.1. The model-building procedure outlined in Sect. 5.4 results in ensembles of parameterisations of comparable fit quality. The models discussed in this section, which are referred to as the baseline models, and are built to include all decay chains that are common to the majority of models that have a per degree of freedom differing from the best-fitting models by less than 0.1. The results for these baseline models are shown and their features discussed in Sects. 6.2 and 6.3 for the RS decay and the WS decay, respectively. The general features of models in the ensembles are discussed in Sect. 6.4. In Sect. 6.5 the models are used to calculate the coherence factor of the decays, and an assessment is made of the stability of the predicted coherence factors, strong-phase differences and amplitude ratios with respect to the choice of WS model in regions of phase space.

Systematic uncertainties

Several sources of systematic uncertainty are considered. Experimental issues are discussed first, followed by uncertainties related to the model and the formalism.

All parameters in the fit have a systematic uncertainty originating from the limited size of the integration sample used in the likelihood minimisation. This effect is reduced by importance sampling. The remaining uncertainty is estimated using a resampling technique. Half of the integration sample is randomly selected, and the fit performed using only this subsample. This procedure is repeated many times, and the systematic uncertainty from the finite integration statistics is taken to be of the spread in fit parameters.

There is an additional systematic uncertainty due to the imperfect simulation, which affects the efficiency corrections. The RS data are divided into bins in the transverse momentum, in which the efficiency corrections may be expected to vary, and the fit is performed independently in each bin. The results of these fits are combined in an uncertainty-weighted average, including the correlations between the different parameters, and the absolute difference between the parameters measured by this procedure and the usual fitting procedure is assigned as the systematic uncertainty. Additionally, the data is divided by data-taking year and software trigger category and independent fits performed using these subsamples. The fit results are found to be compatible within the assigned uncertainties between these samples, hence no additional systematic uncertainty is assigned.

The uncertainty associated with the determination of the signal fraction and mistag fraction in each sample is measured by varying these fractions within the uncertainties found in the fit to the vs. plane.

Parameters that are fixed in the fit, such as the mass and width, are randomly varied according to the uncertainties given in Ref. [21], and the corresponding spreads in fit results are assigned as the uncertainties. It is assumed that input correlations between these parameters are negligible. When performing fits to the WS sample, several parameters, such as the mass, width and couplings of the resonance, are fixed to the values found in the RS fit. The uncertainty on these parameters is propagated to the WS fit by randomly varying these parameters by their uncertainties. The radii of several particles used in the Blatt–Weisskopf form factor are varied using the same procedure. The radial parameter is varied by .

The uncertainty due to the background model in the WS fit is estimated using pseudo-experiments. A combination of simulated signal events generated with the final model and candidates from outside of the signal region is used to approximate the real data. The composite dataset is then fitted using the signal model, and differences between the true and fitted values are taken as the systematic uncertainties on the background parametrisation.

The choice of model is an additional source of systematic uncertainty. It is not meaningful to compare the coupling parameters between different parametrisations, as these are by definition the parameters of a given model. It is however useful to consider the impact the choice of parametrisation has on fit fractions and the fitted masses and widths. Therefore, the model choice is not included in the total systematic uncertainty, but considered separately in Sect. 6.4 and 6.5.

The total systematic uncertainty is obtained by summing the components in quadrature. The total systematic uncertainty is significantly larger than the statistical uncertainty on the RS fit, with the largest contributions coming from the form factors that account for the finite size of the decaying mesons. For the WS fit, the total systematic uncertainty is comparable to the statistical uncertainty, with the largest contribution coming from the parametrisation of the combinatorial background. A full breakdown of the different sources of systematic uncertainty for all parameters is given in Appendix C.

Results for the RS decay

Invariant mass-squared projections for are shown in Fig. 2 together with the expected distributions from the baseline model. The coupling parameters, fit fractions and other quantities for this model are shown in Table 2. The per degree of freedom for this model is calculated to be , which indicates that although this is formally a poor fit, the model is providing a reasonable description of the data given the very large sample size. Three cascade contributions, from , and resonances, are modelled using the three-body running-width treatment described in Sect. 4. The masses and widths of these states are allowed to vary in the fit. The mass, width and coupling parameters for these resonances are presented in Tables 3, 4 and 5. The values of these parameters are model dependent, in particular on the parametrisation of the running width described by Eq. (5) and of the form factors described by Eq. (6), and thus there is not a straightforward comparison with the values obtained by other experiments.

Fig. 2

Distributions for six invariant-mass observables in the RS decay . Bands indicate the expectation from the model, with the width of the band indicating the total systematic uncertainty. The total background contribution, which is very low, is shown as a filled area. In figures that involve a single positively-charged pion, one of the two identical pions is selected randomly

Table 2

Fit fractions and coupling parameters for the RS decay . For each parameter, the first uncertainty is statistical and the second systematic. Couplings g are defined with respect to the coupling to the channel . Also given are the and the number of degrees of freedom () from the fit and their ratio

	Fit fraction [%]	g	arg(g)[^∘]
K¯^∗(892)^0ρ(770)^0^L=0	7.34±0.08±0.47	0.196±0.001±0.015	-22.4±0.4±1.6
K¯^∗(892)^0ρ(770)^0^L=1	6.03±0.05±0.25	0.362±0.002±0.010	-102.9±0.4±1.7
K¯^∗(892)^0ρ(770)^0^L=2	8.47±0.09±0.67
ρ(1450)^0K¯^∗(892)^0^L=0	0.61±0.04±0.17	0.162±0.005±0.025	-86.1±1.9±4.3
ρ(1450)^0K¯^∗(892)^0^L=1	1.98±0.03±0.33	0.643±0.006±0.058	97.3±0.5±2.8
ρ(1450)^0K¯^∗(892)^0^L=2	0.46±0.03±0.15	0.649±0.021±0.105	-15.6±2.0±4.1
ρ(770)^0K^-π^+^L=0	0.93±0.03±0.05	0.338±0.006±0.011	73.0±0.8±4.0
α_3/2		1.073±0.008±0.021	-130.9±0.5±1.8
K¯^∗(892)^0π^+π^-^L=0	2.35±0.09±0.33
f_ππ		0.261±0.005±0.024	-149.0±0.9±2.7
β_1		0.305±0.011±0.046	65.6±1.5±4.0
a_1(1260)^+K^-	38.07±0.24±1.38	0.813±0.006±0.025	-149.2±0.5±3.1
K_1(1270)^-π^+	4.66±0.05±0.39	0.362±0.004±0.015	114.2±0.8±3.6
K_1(1400)^-K¯^∗(892)^0π^-π^+	1.15±0.04±0.20	0.127±0.002±0.011	-169.8±1.1±5.9
K2∗(1430)^-K¯^∗(892)^0π^-π^+	0.46±0.01±0.03	0.302±0.004±0.011	-77.7±0.7±2.1
K(1460)^-π^+	3.75±0.10±0.37	0.122±0.002±0.012	172.7±2.2±8.2
K^-π^+^L=0π^+π^-^L=0	22.04±0.28±2.09
α_3/2		0.870±0.010±0.030	-149.2±0.7±3.5
α_Kη^′		2.614±0.141±0.281	-19.1±2.4±12.0
β_1		0.554±0.009±0.053	35.3±0.7±1.6
f_ππ		0.082±0.001±0.008	-147.0±0.7±2.2
Sum of fit fractions	98.29±0.37±0.84
χ^2/ν	40483/32701=1.238

Table 3

Table of fit fractions and coupling parameters for the component involving the meson, from the fit performed on the RS decay . The coupling parameters are defined with respect to the coupling. For each parameter, the first uncertainty is statistical and the second systematic

a_1(1260)^+   m_0=1195.05±1.05±6.33MeV/c^2; Γ_0=422.01±2.10±12.72MeV/c^2
	Partial fractions [%]	g	arg(g)[^∘]
ρ(770)^0π^+	89.75±0.45±1.00
π^+π^-^L=0π^+	2.42±0.06±0.12
β_1		0.991±0.018±0.037	-22.2±1.0±1.2
β_0		0.291±0.007±0.017	165.8±1.3±3.1
f_ππ		0.117±0.002±0.007	170.5±1.2±2.2
ρ(770)^0π^+^L=2	0.85±0.03±0.06	0.582±0.011±0.027	-152.8±1.2±2.5

Table 4

Table of fit fractions and coupling parameters for the component involving the meson, from the fit performed on the RS decay . The coupling parameters are defined with respect to the coupling. For each parameter, the first uncertainty is statistical and the second systematic

K_1(1270)^-   m_0=1289.81±0.56±1.66MeV/c^2; Γ_0=116.11±1.65±2.96MeV/c^2
	Partial factions [%]	g	arg(g)[^∘]
ρ(770)^0K^-	96.30±1.64±6.61
ρ(1450)^0K^-	49.09±1.58±11.54	2.016±0.026±0.211	-119.5±0.9±2.3
K¯^∗(892)^0π^-	27.08±0.64±2.82	0.388±0.007±0.033	-172.6±1.1±6.0
K^-π^+^L=0π^-	22.90±0.72±1.89	0.554±0.010±0.037	53.2±1.1±1.9
K¯^∗(892)^0π^-^L=2	3.47±0.17±0.31	0.769±0.021±0.048	-19.3±1.6±6.7
ω(782)π^+π^-K^-	1.65±0.11±0.16	0.146±0.005±0.009	9.0±2.1±5.7

Table 5

Table of fit fractions and coupling parameters for the component involving the meson, from the fit performed on the RS decay . The coupling parameters are defined with respect to the coupling. For each parameter, the first uncertainty is statistical and the second systematic

K(1460)^-   m_0=1482.40±3.58±15.22MeV/c^2 ; Γ_0=335.60±6.20±8.65MeV/c^2
	Partial fractions [%]	g	arg(g)[^∘]
K¯^∗(892)^0π^-	51.39±1.00±1.71
π^+π^-^L=0K^-	31.23±0.83±1.78
f_KK		1.819±0.059±0.189	-80.8±2.2±6.6
β_1		0.813±0.032±0.136	112.9±2.6±9.5
β_0		0.315±0.010±0.022	46.7±1.9±3.0

The largest contribution is found to come from the axial vector , which is a result that was also obtained in the Mark III analysis [6]. This decay proceeds via the colour-favoured external W-emission diagram that is expected to dominate this final state.

There are also large contributions from the different orbital angular momentum configurations of the quasi two-body processes , with a total contribution of around . The polarisation structure of this component is not consistent with naive expectations, with the D wave being the dominant contribution and the overall hierarchy being . This result may be compared with that obtained for the study in Ref. [32], where the D-wave polarisation of the amplitude was also found to be dominant.

A significant contribution is found from the pseudoscalar state . This resonance is a excitation of the kaon [33]. Evidence for this state has been reported in partial-wave analyses of the process [34, 35], manifesting itself as a state with mass and width , coupling to the and channels. The intermediate decays of the meson are found to be roughly consistent with previous studies, with approximately equal partial widths to and . The resonant nature of this state is confirmed using a model-independent partial-wave analysis (MIPWA), following the method first used by the E791 collaboration [36, 37]. The relativistic Breit-Wigner lineshape is replaced by a parametrisation that treats the real and imaginary parts of the amplitude at 15 discrete positions in as independent pairs of free parameters to be determined by the fit. The amplitude is then modelled elsewhere by interpolating between these values using cubic splines [38]. The Argand diagram for this amplitude is shown in Fig. 3, with points indicating the values determined by the fit, and demonstrates the phase motion expected from a resonance.

Fig. 3

Argand diagram for the model-independent partial-wave analysis (MIPWA) for the resonance. Points show the values of the amplitude that are determined by the fit, with only statistical uncertainties shown

Four-body weak decays contain amplitudes that are both even, such as , where V and are vector resonances, and odd, such as , under parity transformations. Interference between these amplitudes can give rise to parity asymmetries which are different in and decays. These asymmetries are the result of strong-phase differences, but can be mistaken for asymmetries [39]. Both sources of asymmetry can be studied by examining the distribution of the angle between the decay planes of the two quasi two-body systems, , which can be constructed from the three-momenta of the decay products in the rest frame of the meson aswhere is the direction normal to the decay plane of a two-particle system ab,and is the direction of the combined momentum of the system.

The interference between P-even and P-odd amplitudes averages to zero when integrated over the entire phase space. Therefore, the angle is studied in regions of phase space. The region of the and resonances is studied as the largest P-odd amplitude is the decay . Selecting this region allows the identical pions to be distinguished, by one being part of the -like system and the other in the -like system. The data in this region are shown in Fig. 4, divided into quadrants of helicity angles, and , defined as the angle between the and the in the rest frame of the system. The distributions show clear asymmetries under reflection about , indicating parity nonconservation. However, equal and opposite asymmetries are observed in the -conjugate mode , indicating that these asymmetries originate from strong phases, rather than from -violating effects. Bands show the expected asymmetries based on the amplitude model, which has been constructed according to the -conserving hypothesis, and show reasonable agreement with the data.

Fig. 4

Parity violating distributions for the RS decay in the region defined by () mass windows about the nominal masses. Bands show the predictions of the fitted model including systematic uncertainties

Results for the WS decay

Invariant mass-squared distributions for are shown in Fig. 5. Large contributions are clearly seen in from the resonance. The fit fractions and amplitudes of the baseline model are given in Table 6. The per degree of freedom for the fit to the WS data is . If the true WS amplitude has a comparable structure to the RS amplitude, it contains several decay chains at the level that cannot be satisfactorily resolved given the small sample size, and hence the quality of the WS fit is degraded by the absence of these subdominant contributions.

Fig. 5

Distributions for six invariant-mass observables in the WS decay . Bands indicate the expectation from the model, with the width of the band indicating the total systematic uncertainty. The total background contribution is shown as a filled area, with the lower region indicating the expected contribution from mistagged decays. In figures that involve a single negatively-charged pion, one of the two identical pions is selected randomly

Table 6

Fit fractions and coupling parameters for the WS decay . For each parameter, the first uncertainty is statistical and the second systematic. Couplings g are defined with respect to the coupling to the decay . Also given are the and the number of degrees of freedom () from the fit and their ratio

	Fit fraction [%]	g	arg(g)[^∘]
K^∗(892)^0ρ(770)^0^L=0	9.62±1.58±1.03	0.205±0.019±0.010	-8.5±4.7±4.4
K^∗(892)^0ρ(770)^0^L=1	8.42±0.83±0.57	0.390±0.029±0.006	-91.4±4.7±4.1
K^∗(892)^0ρ(770)^0^L=2	10.19±1.03±0.79
ρ(1450)^0K^∗(892)^0^L=0	8.16±1.24±1.69	0.541±0.042±0.055	-21.8±6.5±5.5
K_1(1270)^+π^-	18.15±1.11±2.30	0.653±0.040±0.058	-110.7±5.1±4.9
K_1(1400)^+K^∗(892)^0π^+π^-	26.55±1.97±2.13	0.560±0.037±0.031	29.8±4.2±4.6
K^+π^-^L=0π^+π^-^L=0	20.90±1.30±1.50
α_3/2		0.686±0.043±0.022	-149.4±4.3±2.9
β_1		0.438±0.044±0.030	-132.4±6.5±3.0
f_ππ		0.050±0.006±0.005	74.8±7.5±5.3
Sum of fit fractions	101.99±2.90±2.85
χ^2/ν	350/239=1.463

Dominant contributions are found from the axial kaons, and , which are related to the same colour-favoured W-emission diagram that dominates the RS decay, where it manifests itself in the component. The contribution from the resonance is larger than that from the resonance. It is instructive to consider this behaviour in terms of the quark states, and , which mix almost equally to produce the mass eigenstates,where is a mixing angle. The mixing is somewhat less than maximal, with Ref. [40] reporting a preferred value of . In the WS decay, the axial kaons are produced via a weak current, which is decoupled from the state in the flavour-symmetry limit. If the mixing were maximal, the mass eigenstates would be produced equally, but a smaller mixing angle results in a preference for the , which is qualitatively consistent with the pattern seen in data. In the RS decay, the axial kaons are not produced by the external weak current, and hence there is no reason to expect either quark state to be preferred. The relatively small contribution from the is then understood as a consequence of approximately equal production of the quark states.

The coupling and shape parameters of the resonance are fixed to the values measured in the RS nominal fit. A fit is also performed with these coupling parameters free to vary, and the parameters are found to be consistent with those measured in the RS decay.

A large contribution is found from decays in all models that describe the data well. This contribution resembles a quasi nonresonant component due to the large width of the resonance, and is likely to be an effective representation of several smaller decay chains involving the resonance that cannot be resolved with the current sample size.

Alternative parametrisations

The model-finding procedure outlined in Sect. 5.4 results in ensembles of parametrisations of comparable quality and complexity. The decay chains included in the models discussed above are included in the majority of models of acceptable quality, with further variations made by addition of further small components. The fraction of models in this ensemble containing a given decay mode are shown in Table 7 for the RS decay mode with the average fit fraction associated with each decay chain also tabulated. The ensemble of RS models consists of about 200 models with per degree of freedom varying between 1.21 and 1.26. Many of the decay chains in the ensemble include resonances, such as the , decaying via radially excited vector mesons, such as the and mesons. In particular, the decay is included in the models discussed in Sects. 6.2 and 6.3 and is found in the majority of the models in the ensemble. This decay channel of the meson has a strong impact at low dipion masses due to the very large width of the resonance, of about 400. Models excluding this component are presented as alternative parametrisations in Appendix E as this decay mode has not been studied extensively in other production mechanisms of the resonance, and the ensemble contains models without this decay chain of similar fit quality to the baseline model. The situation can be clarified with independent measurements of the properties of these resonances. The resonance is also found in many models in the ensemble, and is likely to be present at some level despite only the low-mass tail of this resonance impacting the phase space. This resonance strongly interferes with the dominant component and, as the parameters of this resonance are poorly known, improved external inputs will be required to correctly constrain this component.

Table 7

Decay chains taken into account in alternative parametrisations of the RS decay mode . For each chain, the fraction of models in the ensemble that contain this decay, together with the associated average fit fraction, , are shown. Components are not tabulated if they contribute to all models in the ensemble, or if they contribute to less than 5% of the models

Decay chain	Fraction of models [%]	⟨F⟩[%]
K_1(1270)^-ρ(1450)^0K^-π^+	68.9	1.61
K_1(1400)^-ρ(1450)^0K^-π^+	33.4	0.34
a_1(1640)^+π^+π^-^L=0π^+K^-	23.1	2.47
K_1(1270)^-K¯^∗(1680)^0π^-π^+	18.4	0.38
K_1(1270)^-K¯^∗(1410)^0π^-π^+	12.0	0.29
K2∗(1430)^-K¯^∗(1410)^0π^-π^+	10.4	0.12
K^∗(1680)^-ρ(770)^0K^-π^+	10.4	0.07
K2∗(1430)^-ρ(1450)^0K^-π^+	10.4	0.10
K2∗(1430)^-K¯^∗(1680)^0π^-π^+	10.4	0.13
K_1(1400)^-ρ(770)^0K^-π^+	10.4	0.44
K_1(1400)^-K¯^∗(1410)^0π^-π^+	10.4	0.11
K(1460)^-K¯2∗(1430)^0π^-π^+	10.0	0.06

The coupling parameters cannot strictly be compared between different models, as in many cases these coupling parameters have a different interpretation depending on the choice of the model. However, it is instructive to consider how the fit fractions vary depending on the choice of model, which is shown in Table 8. It is also useful to consider how the choice of model impacts upon the fitted masses and widths, which is shown in Table 9. The values for the model described in Sect. 6.2 are also shown, which has compatible values with the ensemble. The variation with respect to the choice of model is characterised by the RMS of the parameters in the ensemble, and is of a comparable size to the combined systematic uncertainty from other sources on these parameters.

Table 8

Dependence of fit fractions (and partial fractions) on the choice of the RS model. This dependence is expressed as the mean value and the RMS of the values in the ensemble. Also shown is the fit fractions of the baseline model presented in Sect. 6.2

	(Partial) fraction [%]
	Baseline	Ensemble
		Mean±RMS
K¯^∗(892)^0ρ(770)^0^L=0	7.34±0.08±0.47	7.10±0.13
K¯^∗(892)^0ρ(770)^0^L=1	6.03±0.05±0.25	6.00±0.12
K¯^∗(892)^0ρ(770)^0^L=2	8.47±0.09±0.67	8.42±0.20
ρ(1450)^0K¯^∗(892)^0^L=0	0.61±0.04±0.17	0.65±0.13
ρ(1450)^0K¯^∗(892)^0^L=1	1.98±0.03±0.33	1.91±0.06
ρ(1450)^0K¯^∗(892)^0^L=2	0.46±0.03±0.15	0.46±0.05
ρ(770)^0K^-π^+^L=0	0.93±0.03±0.05	1.08±0.12
K¯^∗(892)^0π^+π^-^L=0	2.35±0.09±0.33	2.19±0.34
a_1(1260)^+K^-	38.07±0.24±1.38	38.06±2.08
ρ(770)^0π^+	89.75±0.45±1.00	86.66±4.52
π^+π^-^L=0π^+	2.42±0.06±0.12	3.01±1.02
ρ(770)^0π^+^L=2	0.85±0.03±0.06	0.80±0.10
K_1(1270)^-π^+	4.66±0.05±0.39	4.74±0.24
ρ(770)^0K^-	96.30±1.64±6.61	77.04±9.22
ρ(1450)^0K^-	49.09±1.58±11.54	34.13±8.19
ω(782)π^+π^-K^-	1.65±0.11±0.16	1.70±0.15
K¯^∗(892)^0π^-	27.08±0.64±2.82	26.95±2.52
K¯^∗(892)^0π^-^L=2	3.47±0.17±0.31	3.57±0.49
K^-π^+π^-	22.90±0.72±1.89	20.39±2.89
K_1(1400)^-K¯^∗(892)^0π^-π^+	1.15±0.04±0.20	1.23±0.10
K2∗(1430)^-K¯^∗(892)^0π^-π^+	0.46±0.01±0.03	0.44±0.04
K(1460)^-π^+	3.75±0.10±0.37	3.63±0.27
K¯^∗(892)^0π^-	51.39±1.00±1.71	53.18±1.52
π^+π^-^L=0K^-	31.23±0.83±1.78	30.46±1.19
K^-π^+^L=0π^+π^-^L=0	22.04±0.28±2.09	21.87±1.51

Table 9

Dependence of the fitted masses and widths on the final choice of the RS model. This dependence is expressed as the mean value and the RMS of the values in the ensemble. The values found for the baseline model presented in Sect. 6.2 are reported for comparison

	Baseline	Ensemble
m(a_1(1260)^+)(MeV/c^2)	1195.05±1.05±6.33	1196.85±6.21
Γ(a_1(1260)^+)(MeV/c^2)	422.01±2.10±12.72	420.92±8.70
m(K_1(1270)^-)(MeV/c^2)	1289.81±0.56±1.66	1287.77±3.97
Γ(K_1(1270)^-)(MeV/c^2)	116.11±1.65±2.96	114.27±7.57
m(K(1460)^-)(MeV/c^2)	1482.40±3.58±15.22	1474.60±12.28
Γ(K(1460)^-)(MeV/c^2)	335.60±6.20±8.65	333.89±12.88

The ensemble consists of 108 models, all of which have a per degree of freedom of less than 1.45, with the best models in the ensemble having a per degree of freedom of about 1.35. The fraction of models in this ensemble containing a given decay mode are shown in Table 10. In particular, there should be percent-level contributions from some of the decay chains present in the mode, such as and . In addition to the marginal decays of the present in the ensemble, the models suggest contributions from the , which resembles a nonresonant component due to its large width and position on the edge of the phase space. As is the case for the large component, this contribution is likely to be mimicking several smaller decay channels that cannot be resolved with the current sample size.

Table 10

Decay chains taken into account in alternative parametrisations of the WS decay mode . For each chain, the fraction of models in the ensemble that contain this decay, together with the associated average fit fraction, , are shown. Components are not tabulated if they contribute to all models in the ensemble, or if they contribute to less than 5% of the models

Decay chain	Fraction of models [%]	⟨F⟩ [%]
K_1(1270)^+ρ(770)^0K^+^L=2π^-	47.2	1.21
K^∗(1680)^+K^∗(1680)^0π^+π^-	38.0	2.89
K^∗(1680)^+ρ(770)^0K^+π^-	33.3	2.58
a_1(1640)^-π^+π^-^L=0π^-K^+	27.8	3.24
K^∗(1680)^+ρ(1450)^0K^+π^-	22.2	2.53
K_1(1270)^+K^∗(1410)^0π^+^L=2π^-	22.2	0.60
K_1(1270)^+π^+π^-^L=0K^+π^-	21.3	0.26
K^∗(1680)^+K^∗(1410)^0π^+π^-	17.6	1.98
ρ(770)^0K^+π^-^L=0	17.6	3.49
K^∗(1680)^+K2∗(1430)^0π^+π^-	16.7	0.82
K_1(1400)^+π^+π^-^L=0K^+π^-	13.0	0.29
K2∗(1430)^0K^+π^-ρ(770)^0	13.0	0.35
K^∗(1410)^0ρ(770)^0	10.2	3.50

Coherence factor

The coherence factor and average strong-phase difference are measures of the phase-space-averaged interference properties between suppressed and favoured amplitudes, and are defined as [41]where is the amplitude of the suppressed decay and is the favoured amplitude for decays. Additionally, it is useful to define the average ratio of amplitudes asKnowledge of these parameters is necessary when making use of this decay in transitions for measuring the -violating phase [41], and can also be exploited for charm mixing studies. Observables with direct sensitivity to the coherence factor and related parameters have been measured in collisions at the resonance with CLEO-c data [42], and through charm mixing at LHCb [4]. A global analysis of these results [42] yieldsThe baseline models presented in Sect. 6 can be used to calculate the model-derived coherence factorwhere the first uncertainty is statistical, the second systematic, and the third the uncertainty from the choice of WS model. This uncertainty is assigned by taking the spread in values from an ensemble of alternative models from the model-building algorithm, requiring that models have a per degree of freedom of less than 1.5, and that all unconstrained components in the fit have a significance of . This result is in good agreement with the direct measurement in Ref. [42]. This analysis has no sensitivity to and as each amplitude contains an arbitrary independent amplitude and phase.

The stability of the local phase description can also be verified by evaluating the model-derived coherence factor and associated parameters in different regions of phase space. This is equivalent to changing the definition of Eq. (19) such that integrals are performed over a limited region of phase space. In this case, it is also possible to determine the local values of and relative to the phase-space averaged values. Therefore, overall normalisation factors are fixed such that the central values of the direct measurement are correctly reproduced.

In order to define these regions, the phase space is divided into hypercubes using the algorithm described in Sect. 5.2. The division is done such that the hypercubes cannot be smaller in any dimension than . The hypercubes are grouped into bins of average phase difference between the two amplitudes in the bin, using the amplitude models described in Sect. 6. The range in phase difference between the two decay modes is split into eight bins. The division of this range is done such that each bin is expected to have an approximately equal population of WS events within the bin. The coherence factors, average strong phases and amplitude ratios and their RMS spread arising from the choice of WS model are summarised in Table 11. Good stability with respect to the choice of model is observed, which is a consequence of the dominant features of the amplitude being common for all models, and gives confidence to using the models presented in this paper to define regions of interest for future binned measurements of or studies of charm mixing. The relatively high coherence factor in some regions of phase-space demonstrates the potential improvements in sensitivity to measurements of -violating observables.

Table 11

Coherence factor and average strong-phase differences in regions of phase space. The spread of coherence factors, average strong-phase difference and ratio of amplitudes from choice of WS model characterised with the RMS of the distribution

Bin	R_K3π	δ_K3π[^∘]	r_K3π×10^-2
1	0.701±0.017	169±3	5.287±0.034
2	0.691±0.016	151±1	5.679±0.032
3	0.726±0.010	133±1	6.051±0.032
4	0.742±0.008	117±1	6.083±0.030
5	0.783±0.005	102±2	5.886±0.031
6	0.764±0.007	84±3	5.727±0.033
7	0.424±0.013	26±3	5.390±0.061
8	0.473±0.030	-149±7	4.467±0.065

Conclusions

The four-body decay modes have been studied using high-purity time-integrated samples obtained from doubly tagged decays. For the RS decay mode , the analysis is performed with a sample around sixty times larger than that exploited in any previous analysis of this decay. For the WS mode , the resonance substructure is studied for the first time with signal candidates.

Both amplitude models are found to have large contributions from axial resonances, the decays and for and , respectively. This is consistent with the general picture that W-emission topologies are crucial in describing these decays. Interference between the parity-even and parity-odd amplitudes causes large local parity violations, which are shown to be reasonably well modelled in the RS decay. A significant contribution from the pseudoscalar resonance is identified, which is validated using the model-independent partial waves method.

The coherence factor is calculated using the models, and is found to be consistent with direct measurements. It is found that the calculated value is relatively stable with respect to the parametrisation of subdominant amplitudes in the WS model. These models therefore provide a valuable input to future binned measurements of the -violating parameter and charm-mixing studies.

Table 12

Rules for calculating the current associated with a given decay chain in terms of the currents of the decay products. Where relevant, the spin projection operator and the orbital angular momentum operators L are those for the decaying particle

Topology	Current	Simplified current
S→[S_1S_2]	S_1S_2
S→[VS_1]^L=1	L^μV_μS_1
S→[V_1V_2]^L=0	g_μνV1μV2ν
S→[V_1V_2]^L=1	ε_μναβPSμL^νV1αV2β	ε_μναβPSμQSνV1αV2β
S→[V_1V_2]^L=2	L_μνV1μV2ν
S→[TS_1]^L=2	L_μνT^μνS_1
S→[TV]^L=1	L^μT_μνV^ν
S→[TV]^L=2	L^μνε_ναβγPSαTμβV^γ	ε_ναβγPSαQSνL_μT^βμV^γ
S→[T_1T_2]^L=0	T1μνT_2μν
V_μ→[S_1S_2]^L=1	S_μνL^νS_1S_2	L_μS_1S_2
V_μ→[V_1S]^L=0	S_μνV1νS
V_μ→[V_1S]^L=1	S_μνε^ναβγP_VαL_βV_1γS	-ε_μαβγPVαQVβV1γS
V_μ→[V_1S]^L=2	S_μνL^ναV_1αS	L_μαV1αS
V_μ→[TS]^L=1	S_μνL_αT^να
V_μ→[TS]^L=2	S_μνε^ναβγP_VαLβηT_γηS	-ε_μαβγPVαQVβT^γηL_η
V_μ→[TV_1]^L=0	S_μνT^ναV_1α
T_μν→[S_1S_2]^L=2	S_μναβL^αβS_1S_2	L_μνS_1S_2
T_μν→[VS]^L=1	S_μναβL1αV^βS	12L_μS_νβ+S_μβL_ν-13S_μνL_βV^β
T_μν→[VS]^L=2	S_μναβε^αγηλP_TγL2ηβV^λS	-12ε_μγηλL_ν+ε_νγηλL_μPTγQTηV^λ
T_μν→[T_1S]	S_μναβT1αβ

Table 13

Legend for systematic uncertainties, including whether this sources of uncertainty is considered on the RS/WS decay mode

	Description	RS	WS
I	Efficiency variations	✓
II	Simulation statistics	✓	✓
III	Masses and widths	✓	✓
IV	Form factor radii	✓	✓
V	Background fraction	✓	✓
VI	Background parameterisation		✓
VII	RS parameters		✓

Table 14

Systematic uncertainties on the RS decay coupling parameters and fit fractions for quasi two-body decay chains

			I	II	III	IV	V
K¯^∗(892)^0ρ(770)^0	F	7.340±0.084±0.637	0.426	0.050	0.063	0.466	0.025
	g	0.196±0.001±0.015	0.000	0.001	0.001	0.015	0.000
	arg(g)[^∘]	-22.363±0.361±1.644	1.309	0.239	0.119	0.955	0.075
K¯^∗(892)^0ρ(770)^0^L=1	F	6.031±0.049±0.436	0.358	0.029	0.061	0.239	0.006
	g	0.362±0.002±0.010	0.002	0.001	0.002	0.009	0.000
	arg(g)[^∘]	-102.907±0.380±1.667	1.431	0.224	0.321	0.760	0.025
K¯^∗(892)^0ρ(770)^0^L=2	F	8.475±0.086±0.826	0.492	0.051	0.059	0.659	0.023
ρ(1450)^0K¯^∗(892)^0	F	0.608±0.040±0.165	0.061	0.032	0.134	0.065	0.019
	g	0.162±0.005±0.025	0.007	0.004	0.018	0.015	0.003
	arg(g)[^∘]	-86.122±1.852±4.345	1.933	1.570	2.485	2.152	1.368
ρ(1450)^0K¯^∗(892)^0^L=1	F	1.975±0.029±0.351	0.115	0.017	0.315	0.103	0.003
	g	0.643±0.006±0.058	0.001	0.003	0.050	0.029	0.001
	arg(g)[^∘]	97.304±0.516±2.770	2.249	0.288	1.341	0.854	0.031
ρ(1450)^0K¯^∗(892)^0^L=2	F	0.455±0.028±0.163	0.078	0.016	0.090	0.110	0.004
	g	0.649±0.021±0.105	0.052	0.011	0.063	0.065	0.003
	arg(g)[^∘]	-15.564±1.960±4.109	1.208	1.323	2.631	2.484	0.762
ρ(770)^0K^-π^+^L=0	F	0.926±0.032±0.083	0.069	0.019	0.016	0.039	0.006
	g	0.338±0.006±0.011	0.000	0.004	0.002	0.010	0.002
	arg(g)[^∘]	73.048±0.795±3.951	3.567	0.469	0.481	1.549	0.185
α_3/2	g	1.073±0.008±0.021	0.018	0.005	0.005	0.009	0.003
	arg(g)[^∘]	-130.856±0.457±1.786	1.679	0.282	0.274	0.435	0.155
K¯^∗(892)^0π^+π^-^L=0	F	2.347±0.089±0.557	0.483	0.079	0.148	0.206	0.076
f_ππ	g	0.261±0.005±0.024	0.022	0.004	0.006	0.007	0.003
	arg(g)[^∘]	-149.023±0.943±2.696	2.275	0.540	1.176	0.617	0.196
β_1	g	0.305±0.011±0.046	0.040	0.010	0.013	0.013	0.007
	arg(g)[^∘]	65.554±1.534±4.004	3.017	0.857	2.322	0.771	0.455
K^-π^+^L=0π^+π^-^L=0	F	22.044±0.282±4.137	3.631	0.268	0.213	1.945	0.188
α_3/2	g	0.870±0.010±0.030	0.029	0.005	0.003	0.004	0.002
	arg(g)[^∘]	-149.187±0.712±3.503	3.467	0.350	0.250	0.194	0.157
α_Kη^′	g	2.614±0.141±0.281	0.263	0.063	0.041	0.062	0.018
	arg(g)[^∘]	-19.073±2.414±11.979	11.775	1.507	1.151	0.816	0.755
β_1	g	0.554±0.009±0.053	0.019	0.005	0.004	0.050	0.002
	arg(g)[^∘]	35.310±0.662±1.627	0.969	0.439	0.588	1.069	0.168
f_ππ	g	0.082±0.001±0.008	0.004	0.001	0.001	0.007	0.000
	arg(g)[^∘]	-146.991±0.718±2.248	1.849	0.463	0.593	1.003	0.252

Table 15

Systematic uncertainties on the RS decay coupling parameters, fit fractions and masses and widths of resonances for cascade topology decay chains

			I	II	III	IV	V
a_1(1260)^+K^-	F	38.073±0.245±2.594	2.198	0.155	0.171	1.356	0.053
	g	0.813±0.006±0.025	0.002	0.003	0.004	0.024	0.001
	arg(g)[^∘]	-149.155±0.453±3.132	2.628	0.321	0.531	1.579	0.162
ρ(770)^0π^+	F	89.745±0.452±1.498	1.116	0.298	0.596	0.720	0.192
π^+π^-^L=0π^+	F	2.420±0.060±0.202	0.165	0.043	0.037	0.102	0.010
β_1	g	0.991±0.018±0.037	0.005	0.015	0.012	0.031	0.006
	arg(g)[^∘]	-22.185±1.044±1.195	0.769	0.597	0.393	0.545	0.169
β_0	g	0.291±0.007±0.017	0.012	0.006	0.003	0.010	0.001
	arg(g)[^∘]	165.819±1.325±3.076	2.155	0.802	0.819	1.845	0.318
f_ππ	g	0.117±0.002±0.007	0.001	0.002	0.002	0.007	0.001
	arg(g)[^∘]	170.501±1.235±2.243	0.151	0.765	0.960	1.722	0.731
ρ(770)^0π^+^L=2	F	0.850±0.032±0.077	0.058	0.021	0.023	0.040	0.007
	g	0.582±0.011±0.027	0.020	0.007	0.008	0.015	0.002
	arg(g)[^∘]	-152.829±1.195±2.512	1.691	0.710	0.755	1.520	0.258
a_1(1260)^+	m_0MeV/c^2	1195.050±1.045±6.333	3.187	0.784	0.497	5.371	0.493
	Γ_0MeV/c^2	422.013±2.096±12.723	2.638	1.335	0.723	12.341	0.549
K_1(1270)^-π^+	F	4.664±0.053±0.624	0.485	0.037	0.285	0.268	0.012
	g	0.362±0.004±0.015	0.013	0.002	0.002	0.008	0.001
	arg(g)[^∘]	114.207±0.760±3.612	3.320	0.526	0.441	1.227	0.219
ρ(770)^0K^-	F	96.301±1.644±8.237	5.523	1.082	5.624	2.110	0.286
ρ(1450)^0K^-	F	49.089±1.580±13.727	7.467	1.062	11.159	2.611	0.452
	g	2.016±0.026±0.211	0.108	0.017	0.172	0.053	0.007
	arg(g)[^∘]	-119.504±0.856±2.333	1.597	0.489	1.102	1.190	0.146
K¯^∗(892)^0π^-	F	27.082±0.639±4.039	2.943	0.410	2.525	1.046	0.097
	g	0.388±0.007±0.033	0.025	0.004	0.017	0.011	0.001
	arg(g)[^∘]	-172.577±1.087±5.957	5.653	0.712	1.482	0.876	0.255
[K^-π^+]^L=0π^-	F	22.899±0.722±3.091	2.483	0.457	1.490	0.973	0.119
	g	0.554±0.010±0.037	0.033	0.007	0.005	0.015	0.001
	arg(g)[^∘]	53.170±1.068±1.920	1.564	0.659	0.401	0.735	0.323
K¯^∗(892)^0π^-^L=2	F	3.465±0.168±0.469	0.362	0.117	0.204	0.176	0.043
	g	0.769±0.021±0.048	0.035	0.014	0.011	0.027	0.004
	arg(g)[^∘]	-19.286±1.616±6.657	6.463	1.013	0.914	0.800	0.207
ω(782)π^+π^-K^-	F	1.649±0.109±0.228	0.161	0.083	0.120	0.069	0.007
	g	0.146±0.005±0.009	0.006	0.004	0.002	0.004	0.000
	arg(g)[^∘]	9.041±2.114±5.673	5.401	1.402	0.587	0.826	0.126
K_1(1270)^-	m_0MeV/c^2	1289.810±0.558±1.656	1.197	0.436	0.244	1.010	0.198
	Γ_0MeV/c^2	116.114±1.649±2.963	1.289	1.221	0.981	2.090	0.545
K_1(1400)^-K¯^∗(892)^0π^-π^+	F	1.147±0.038±0.205	0.079	0.022	0.181	0.049	0.003
	g	0.127±0.002±0.011	0.002	0.001	0.010	0.005	0.000
	arg(g)[^∘]	-169.822±1.102±5.879	2.052	0.687	5.343	1.124	0.270
K2∗(1430)^-K¯^∗(892)^0π^-π^+	F	0.458±0.011±0.041	0.031	0.007	0.010	0.024	0.001
	g	0.302±0.004±0.011	0.005	0.002	0.003	0.009	0.000
	arg(g)[^∘]	-77.690±0.732±2.051	0.898	0.409	1.174	1.360	0.051
K(1460)^-π^+	F	3.749±0.095±0.803	0.717	0.066	0.076	0.341	0.064
	g	0.122±0.002±0.012	0.002	0.001	0.002	0.012	0.001
	arg(g)[^∘]	172.675±2.227±8.208	6.826	2.235	2.413	2.619	1.761
K¯^∗(892)^0π^-	F	51.387±0.996±9.581	9.490	0.529	0.629	0.974	0.333
π^+π^-^L=0K^-	F	31.228±0.833±11.085	11.021	0.454	0.414	0.989	0.247
f_KK	g	1.819±0.059±0.189	0.180	0.027	0.030	0.036	0.025
	arg(g)[^∘]	-80.790±2.225±6.563	5.820	1.617	1.740	1.361	1.305
β_1	g	0.813±0.032±0.136	0.132	0.016	0.018	0.018	0.015
	arg(g)[^∘]	112.871±2.555±9.487	8.636	2.025	2.241	1.817	1.730
β_0	g	0.315±0.010±0.022	0.019	0.005	0.005	0.009	0.002
	arg(g)[^∘]	46.734±1.946±2.952	1.110	1.576	1.416	1.121	1.318
K(1460)^-	m_0MeV/c^2	1482.400±3.576±15.216	13.873	3.466	3.216	3.611	1.916
	Γ_0MeV/c^2	335.595±6.196±8.651	1.524	4.234	2.017	5.901	3.962

Table 16

Systematic uncertainties on the WS decay coupling parameters and fit fractions

			II	III	IV	V	VI	VII
K^∗(892)^0ρ(770)^0	g	0.205±0.019±0.010	0.002	0.006	0.003	0.001	0.005	0.006
	arg(g)[^∘]	-8.502±4.662±4.439	0.433	1.272	0.112	0.148	4.150	0.799
	F	9.617±1.584±1.028	0.134	0.436	0.344	0.069	0.567	0.637
K^∗(892)^0ρ(770)^0^L=1	g	0.390±0.029±0.006	0.002	0.003	0.000	0.001	0.004	0.003
	arg(g)[^∘]	-91.359±4.728±4.132	0.406	0.827	0.128	0.101	3.951	0.766
	F	8.424±0.827±0.573	0.069	0.091	0.210	0.020	0.458	0.249
K^∗(892)^0ρ(770)^0^L=2	F	10.191±1.028±0.789	0.089	0.130	0.255	0.018	0.658	0.314
ρ(1450)^0K^∗(892)^0	g	0.541±0.042±0.055	0.004	0.043	0.018	0.001	0.024	0.016
	arg(g)[^∘]	-21.798±6.536±5.483	0.573	4.532	0.547	0.254	0.254	2.960
	F	8.162±1.242±1.686	0.107	1.381	0.474	0.031	0.718	0.428
K_1(1270)^+π^-	g	0.653±0.040±0.058	0.004	0.017	0.009	0.001	0.049	0.024
	arg(g)[^∘]	-110.715±5.054±4.854	0.481	1.484	0.219	0.056	4.236	1.770
	F	18.147±1.114±2.301	0.104	0.800	0.423	0.021	1.788	1.125
K_1(1400)^+K^∗(892)^0π^+π^-	g	0.560±0.037±0.031	0.003	0.020	0.011	0.001	0.018	0.010
	arg(g)[^∘]	29.769±4.220±4.565	0.396	4.055	0.211	0.060	1.638	1.227
	F	26.549±1.973±2.128	0.190	1.715	0.469	0.046	0.940	0.667
K^+π^-^L=0π^+π^-^L=0	F	20.901±1.295±1.500	0.129	0.328	0.565	0.134	1.246	0.486
α_3/2	g	0.686±0.043±0.022	0.004	0.007	0.002	0.002	0.019	0.007
	arg(g)[^∘]	-149.399±4.260±2.946	0.502	0.277	0.181	0.082	2.809	0.651
β_1	g	0.438±0.044±0.030	0.004	0.006	0.010	0.001	0.026	0.010
	arg(g)[^∘]	-132.424±6.507±2.972	0.618	1.109	0.357	0.200	2.382	1.174
f_ππ	g	0.050±0.006±0.005	0.001	0.001	0.001	0.000	0.004	0.002
	arg(g)[^∘]	74.821±7.528±5.282	0.695	0.745	0.149	0.472	5.050	1.058

Table 17

Interference fractions for the RS mode , only shown for fractions . For each fraction, the first uncertainty is statistical and the second systematic

Decay chain a	Decay chain b	Interference fraction [%]
K¯^∗(892)^0ρ(770)^0	a_1(1260)^+K^-	5.74±0.03±0.1
K¯^∗(892)^0ρ(770)^0^L=2	K¯^∗(892)^0ρ(770)^0	-2.59±0.02±0.07
K^+π^-^L=0π^+π^-^L=0	a_1(1260)^+K^-	2.4±0.03±0.14
a_1(1260)^+K^-	ρ(770)^0K^-π^+^L=0	2.14±0.07±0.26
K¯^∗(892)^0ρ(770)^0^L=2	a_1(1260)^+K^-	1.76±0.01±0.08
K¯^∗(892)^0ρ(770)^0^L=1	ρ(1450)^0K¯^∗(892)^0^L=1	1.55±0.02±0.18
K_1(1270)^-π^+	K¯^∗(892)^0ρ(770)^0	-1.05±0.02±0.14
K_1(1400)^-K¯^∗(892)^0π^-π^+	K¯^∗(892)^0ρ(770)^0	0.96±0.02±0.1
K¯^∗(892)^0ρ(770)^0	ρ(1450)^0K¯^∗(892)^0	-0.83±0.05±0.11
K¯^∗(892)^0ρ(770)^0^L=2	ρ(1450)^0K¯^∗(892)^0^L=2	0.81±0.04±0.13
K(1460)^-π^+	K¯^∗(892)^0π^+π^-^L=0	0.78±0.03±0.1
K¯^∗(892)^0ρ(770)^0	K^-π^+^L=0π^+π^-^L=0	0.73±0.01±0.03
K¯^∗(892)^0π^+π^-^L=0	a_1(1260)^+K^-	-0.68±0.01±0.07
K_1(1270)^-π^+	K_1(1400)^-K¯^∗(892)^0π^-π^+	-0.67±0.02±0.12
K(1460)^-π^+	K¯^∗(892)^0ρ(770)^0	-0.66±0.02±0.05
a_1(1260)^+K^-	ρ(1450)^0K¯^∗(892)^0	-0.63±0.02±0.08
K¯^∗(892)^0ρ(770)^0^L=2	K(1460)^-π^+	-0.6±0.02±0.07
K(1460)^-π^+	ρ(1450)^0K¯^∗(892)^0	0.51±0.01±0.06

Table 18

Interference fractions for the WS mode , only shown for fractions . For each fraction, the first uncertainty is statistical and the second systematic

Decay chain a	Decay chain b	Interference fraction [%]
K_1(1400)^+K^∗(892)^0π^+π^-	K^∗(892)^0ρ(770)^0	5.09±0.49±0.56
K^∗(892)^0ρ(770)^0^L=2	K^∗(892)^0ρ(770)^0	-3.48±0.36±0.26
K_1(1270)^+π^-	ρ(1450)^0K^∗(892)^0	-2.17±0.24±0.37
K_1(1400)^+K^∗(892)^0π^+π^-	ρ(1450)^0K^∗(892)^0	-1.78±0.88±0.63
ρ(1450)^0K^∗(892)^0	K^∗(892)^0ρ(770)^0	1.59±0.69±0.77
K^∗(892)^0ρ(770)^0^L=2	ρ(1450)^0K^∗(892)^0	-1.49±0.29±0.30
K^∗(892)^0ρ(770)^0^L=2	K_1(1400)^+K^∗(892)^0π^+π^-	-1.36±0.13±0.12
K^∗(892)^0ρ(770)^0	K^+π^-^L=0π^+π^-^L=0	1.14±0.13±0.11
K_1(1400)^+K^∗(892)^0π^+π^-	K^+π^-^L=0π^+π^-^L=0	1.03±0.10±0.10
K_1(1270)^+π^-	K_1(1400)^+K^∗(892)^0π^+π^-	0.82±0.51±0.79
ρ(1450)^0K^∗(892)^0	K^+π^-^L=0π^+π^-^L=0	-0.65±0.11±0.09
K_1(1270)^+π^-	K^∗(892)^0ρ(770)^0	0.65±0.29±0.33

Table 19

Table of fit fractions and coupling parameters and other quantities for the RS decay , for a model excluding the decay chain . Also given is the per degree of freedom () for the fit. The first uncertainty is statistical and the second systematic. Couplings are defined with respect to the coupling to the channel

	Fit fraction [%]	g	arg(g)[^∘]
K¯^∗(892)^0ρ(770)^0	7.45±0.09±0.47	0.200±0.001±0.014	-26.4±0.4±1.4
K¯^∗(892)^0ρ(770)^0^L=1	6.03±0.05±0.25	0.366±0.002±0.020	-103.1±0.4±1.7
K¯^∗(892)^0ρ(770)^0^L=2	8.30±0.09±0.71
ρ(1450)^0K¯^∗(892)^0	0.11±0.02±0.06	0.068±0.006±0.013	-133.9±5.7±16.2
ρ(1450)^0K¯^∗(892)^0^L=1	1.99±0.03±0.33	0.652±0.006±0.064	97.3±0.5±3.8
ρ(1450)^0K¯^∗(892)^0^L=2	0.36±0.03±0.11	0.581±0.022±0.088	8.2±2.2±16.5
ρ(770)^0K^-π^+^L=0	1.29±0.04±0.09	0.318±0.007±0.012	69.8±0.9±6.0
α_3/2		1.227±0.011±0.015	-129.4±0.5±1.4
K¯^∗(892)^0π^+π^-^L=0	4.57±0.17±0.75
f_ππ		0.352±0.006±0.034	-148.5±0.8±1.7
β_1		0.507±0.012±0.045	69.7±1.1±3.0
a_1(1260)^+K^-	33.56±0.22±1.58	0.771±0.006±0.043	-151.6±0.5±3.6
K_1(1270)^-π^+	4.67±0.05±0.26	0.260±0.003±0.007	90.5±0.9±1.9
K_1(1400)^-K¯^∗(892)^0π^-π^+	0.87±0.03±0.14	0.112±0.002±0.011	-156.6±1.2±8.5
K2∗(1430)^-K¯^∗(892)^0π^-π^+	0.47±0.01±0.03	0.309±0.004±0.014	-79.0±0.7±2.6
K(1460)^-π^+	5.07±0.18±0.51	0.134±0.003±0.013	220.4±2.6±16.0
K^-π^+^L=0π^+π^-^L=0	30.20±0.45±3.20
α_3/2		0.897±0.009±0.020	-147.2±0.5±1.3
α_Kη^′		2.316±0.101±0.308	-2.6±2.1±6.1
β_1		0.656±0.008±0.067	33.0±0.6±2.3
f_ππ		0.093±0.001±0.009	-149.6±0.7±2.7
Sum of fit fractions	104.94±0.75±2.72
χ^2/ν	41896/32702=1.281

Table 20

Table of fit fractions and coupling parameters for the component involving the meson, from the fit performed on the RS decay . The coupling parameters are defined with respect to the coupling. For each parameter, the first uncertainty is statistical and the second systematic

a_1(1260)^+   m_0=1183.73±1.08±7.96MeV/c^2; Γ_0=423.36±2.20±12.89MeV/c^2
	Partial fractions [%]	g	arg(g)[^∘]
ρ(770)^0π^+	90.05±0.47±1.26
π^+π^-^L=0π^+	3.08±0.07±0.21
β_1		1.135±0.019±0.060	-17.7±1.0±1.0
β_0		0.312±0.007±0.016	157.3±1.4±2.9
f_ππ		0.159±0.003±0.011	176.8±1.0±2.3
ρ(770)^0π^+^L=2	0.84±0.04±0.07	0.584±0.012±0.024	-146.1±1.3±3.3

Table 21

Table of fit fractions and coupling parameters for the component involving the meson, from the fit performed on the RS decay . The coupling parameters are defined with respect to the coupling. For each parameter, the first uncertainty is statistical and the second systematic

K_1(1270)^-   m_0=1285.03±0.47±1.06MeV/c^2; Γ_0=90.79±1.12±2.54MeV/c^2
	Partial fractions [%]	g	arg(g)[^∘]
ρ(770)^0K^-	50.66±0.84±2.21
K¯^∗(892)^0π^-	25.25±0.56±1.80	0.520±0.008±0.024	-133.2±0.9±2.2
K^-π^+^L=0π^-	5.97±0.29±0.37	0.390±0.010±0.018	95.2±1.4±3.7
K¯^∗(892)^0π^-^L=2	2.73±0.14±0.24	0.946±0.028±0.147	8.8±1.7±2.4
ω(782)π^+π^-K^-	1.73±0.11±0.16	0.208±0.008±0.011	33.0±2.1±12.5

Table 22

Table of fit fractions and coupling parameters for the component involving the meson, from the fit performed on the RS decay . The coupling parameters are defined with respect to the coupling. For each parameter, the first uncertainty is statistical and the second systematic

K(1460)^-   m_0=1571.22±4.90±33.76MeV/c^2; Γ_0=376.64±7.43±25.30MeV/c^2
	Partial fractions [%]	g	arg(g)[^∘]
K¯^∗(892)^0π^-	45.73±1.09±1.95
π^+π^-^L=0K^-	37.71±1.06±1.98
f_KK		1.573±0.059±0.066	-102.6±2.7±10.0
β_1		0.875±0.032±0.042	85.3±2.6±8.3
β_0		0.323±0.010±0.023	25.6±2.1±8.4