Multi-Meson Model for the $D^+\to K^+K^-K^+$ decay amplitude

We propose a novel approach to describe the $D^+\to K^+K^-K^+$ decay amplitude, based on chiral effective Lagrangians, which can be used to extract information about $K\bar{K}$ scattering. Our trial function is an alternative to the widely used isobar model and includes both nonresonant three-body interactions and two-body rescattering amplitudes, based on coupled channels and resonances, for S- and P-waves with isospin $0$ and $1$. The latter are unitarized in the $K$-matrix approximation and represent the only source of complex phases in the problem. Free parameters are just resonance masses and coupling constants, with transparent physical meanings. The nonresonant component, given by chiral symmetry as a real polynomium, is an important prediction of the model, which goes beyond the (2+1) approximation. Our approach allows one to disentangle the two-body scalar contributions with different isospins, associated with the $f_0(980)$ and $a_0(980)$ channels. We show how the $K\bar{K}$ amplitude can be obtained from the decay $D^+\to K^+K^-K^+$ and discuss extensions to other three-body final states.


I. INTRODUCTION
Nonleptonic weak decays of heavy-flavoured mesons are extensively used in light meson spectroscopy. Owing to a rich resonant structure, these decays provide a natural place to study hadron-hadron interactions at low energies. In particular, almost 20 years ago, three-body decays of charmed mesons could confirm the existence of the controversial scalar states f 0 (600) (or sigma) [1] and kappa(800) [2]. More comprehensive investigations can be done nowadays, using the very large and pure samples provided by the LHC experiments, and still more data is expected in the near future, with Belle II experiments.
Three-body hadronic decays of heavy-flavoured mesons involve combinations of different classes of processes, namely heavy-quark weak transitions, hadron formation and final-state interactions (FSI), whereby the hadrons produced in the primary vertex are allowed to interact in many different ways before being detected. Final-state processes include both proper three-body interactions and a wide range elastic and inelastic coupled channels, involving resonances. In this framework, a question arises, concerning how to obtain information about two-body scattering amplitudes from the abundant data on three-body systems.
The key issue of this program is the modeling of the decay amplitudes. Most amplitude analyses have been performed using the so-called isobar model, in which the decay amplitude is represented by a coherent sum of both nonresonant and resonant contributions. This approach, albeit largely employed [3], has conceptual limitations. The outcome of isobar model analyses are resonance parameters such as fit fractions, masses and widths, which are neither directly related to any underlying dynamical theory nor provide clues to the identification of two-body substructures. Thus, the systematic interpretation of the isobar model results is rather difficult.
This situation motivated in the past decade efforts towards building models that are based on more solid theoretical grounds. Those models improve essentially the two-meson interaction description in the FSI, with the use of dispersion relations and chiral perturbation theory. Most of them work in the quasi-two-body (2+1) approximation, where interactions with the third particle are neglected. Recently, a collection of parametrizations based on analytic and unitary meson-meson form factors for D and B three-body hadronic decays within the (2+1) approximation was presented in Ref. [4]. Three-body FSIs were also considered and, in particular, shown to play a significant role in the D + → K − π + π + decay. In this process, three-body unitarity was implemented in different ways, by means of Faddeev-like decompositions [5][6][7], Khuri-Treiman equation [8] or triangle diagrams [9]. Whilst differing in methods and techniques, all these theoretical efforts have in common the attempt to include, in a systematic way, knowledge of two-body systems in the description of the decay amplitudes.
This work departs from the same broad perspective, but concentrates explicitly on the derivation of two-body scattering amplitudes from three-body decays. In order to do so, we suggest a new approach based on effective Lagrangians and apply it to the D + → K − K + K + decay. This process is interesting because there is very little information available on kaonkaon scattering, regarding both theory and experiment. Concerning the latter, one only has access to ππ elastic scattering data [10] and to the inelastic channel ππ → KK [10,11].
Information about KK interaction can be estimated by imposing unitarity constraints on the ππ data. On the theory side, KK amplitudes have been calculated in next-to-leading order chiral perturbation theory. Aiming at a full coupled-channel description, it was extended up to 1.2 GeV, using form factors [12] to describe the ηπ → KK contribution to η → πππ decay [13], or unitarized ressummation techniques [14], to include ππ → KK in the context of FSI of J/Ψ → φππ(KK) decays.
The main purpose of this work is disclose information about the dynamics of KK interactions by disentangling the two-body contributions contained in the D + → K − K + K + amplitude. In our model, the description of the KK interaction relies on a chiral Lagrangian with resonances, including all possible coupled channels for (J = 0, 1; I = 0, 1), extended to non-perturbative regimes by means of unitarization. A relevant feature of the model is that the relative contribution and phase of each component is fixed by theory, and this represents an important difference with the isobar model. Although the formalism is developed for a specific process it can be useful in other decays into three kaons. This paper extends and supersedes a previous version [15] and is organized as follows.
The motivation for building the amplitude is discussed in section II, whereas the model is presented in sections III and IV. The suggested amplitude for data fitting, together with a comparison between scattering and decay amplitudes is discussed in section V. Some simulations and general remarks are given in section VI. Details of the calculations are given in the appendices.

II. MOTIVATION FOR A NEW MODEL
The isobar model, widely used for describing heavy-meson decays into three pseudoscalars, relies on the assumption that these processes are dominated by intermediate states involving a spectator plus a resonance, and also includes non-resonant contributions.
In the decay H → P 1 P 2 P 3 , of a heavy meson H into three pseudoscalars P i , the isobar model emphasizes the sequence H → R P 3 , followed by R → P 1 P 2 .
The full H → P 1 P 2 P 3 decay amplitude is denoted by T and the isobar model employs a guess function to be fitted to data in the form of the coherent sum the subscript nr referring to the non-resonant term and the label k associated with resonances, as many of them as needed. The coefficients c k = e iθ k are complex parameters, to be determined by data. The choice τ nr = 1 is usual for the non-resonant term, whereas the sub amplitudes τ k depend on the invariant masses of the problem. For each resonance considered, the function τ k is given by F F stands for form factors, the angular factor is associated with angular momentum channels, and [line shape] k represents a resonance line shape, described by either a Breit-Wigner function such as (BW ) k = 1/[s − m 2 k + i m k Γ k ], m k and Γ k being the resonance mass and width, or by variations, such as the Flatté or Gounaris-Sakurai forms. The angular factor allows one to distinguish partial wave contributions and to employ the decomposition A good fit to decay data based on the structure given by eq.(1), would yield an empirical set of complex parameters c nr and c k . However, a question arises regarding the meaning of these parameters. Would they be useful to shed light into yet unknown two-body substructures of the problem? Can they provide reliable information about scattering amplitudes?
If we denote two-body scattering amplitudes by A this question may be restated as: can one extract A directly from T ? As we argue in the sequence, answers to these questions do not favour the isobar model.
On general grounds, there is no direct connection between a heavy-meson decay amplitude T and two-body scattering amplitudes A, involving the same particles. Their relationship involves several issues, which we discuss below.
a. dynamics -The dynamical contents of T and A are rather different, since the former must include weak vertices, which cannot be present in the latter. Specific features of Wmeson interactions are important to T and irrelevant to A. Therefore, although scattering amplitudes A may be substructures of T , there is no reason whatsoever for assuming that these A's are either identical or proportional to T . This is supported by case studies. For instance, some time ago, the FOCUS collaboration [16] produced a partial-wave analysis of the S-wave K − π + amplitude from the decay D + → K − π + π + . Several groups then compared [17] the phase of this empirical amplitude directly with that from the LASS K − π + scattering data [18] and the discrepancy found was seen as a puzzle. The fact that the FOCUS phase was negative at low energies was considered to be especially odd. In the language of this discussion, this kind of puzzle arose just because one was trying to compare T and A directly. The difference between observed S-wave decay and scattering phases was later explained by considering meson loops in the weak sector of the problem [5,6]. These loops account for the extra phases observed.
b. good quantum numbers: -Isospin is broken by weak interactions and is a good quantum number for A, but not for T . Scattering amplitudes A depend both on the angular momentum J and on the isospin I of the channel considered, whereas just a J dependence can be extracted from an empirical decay amplitude T . This point will be recast on more technical grounds while we discuss our model. For the time being, it suffices to stress that it is impossible to derive directly A (J,I) from T (J) simply because the former contains more structure than the latter. An extraction of A (J,I) from T (J) would amount to generating physical content about the isospin structure. in the former must also show up in the latter. In general, guess functions better suited for accommodating data should have structures similar to those used in meson-meson scattering Refs. [10,12,19]. In the case of the isobar model, the simple guess functions usually employed fail to incorporate these intermediate couplings.
d. unitarity -Good fits to Dalitz plots data may require several resonances with the same quantum numbers. At present, conceptual techniques are available which preserve unitarity while incorporating several resonances into amplitudes [20]. This allows one to go beyond the isobar model, where the amplitude is constructed as sums of individual line shapes (Breit-Wigner), as in eq.(1), a procedure known to violate unitarity, even in the case of scattering amplitudes [21].
e. non-resonant term -The non-resonant term may be important and involve less known interactions. In the case of heavy meson decays and some leptonic reactions, available energies can be large enough for allowing the simultaneous production of several pseudoscalars at a single vertex. Multi-meson dynamics then becomes relevant. For instance, the process e − e + → 4 π involves the matrix element ππππ|J µ γ |0 , J µ γ being the electromagnetic current [22]. A similar matrix element, with J µ γ replaced with the weak current (V − A) µ , describes the decay τ → ν 4π [22]. Interactions of this kind are also present in the model for f. lagrangians -Although the point of departure of the isobar model may be sound, the problems mentioned tend to corrode the physical meaning of parameters it yields from fits.
Thus, even if these fits are precise, the relevance of the parameters extracted remains restricted to specific processes. Moreover, in particular, one cannot rely on them for obtaining scattering information. The most conservative way of ensuring that the physical meaning of parameters is preserved from process to process is to employ lagrangians, which rely on just masses and coupling constants. Guess functions for heavy-meson decays constructed from lagrangians yield free parameters which allow the straightforward derivation of scattering amplitudes.

III. DYNAMICS
The fundamental QCD lagrangian for strong interactions is written in terms of gluons and quarks, the basic degrees of freedom. As the theory allows for gluon self-interactions, perturbative calculations hold at high energies only. At present, intermediate-energy reactions cannot be described in terms of quarks and gluons, and one is forced to rely on effective theories. At low energies, chiral perturbation theory (ChPT) [23][24][25] is highly successful.
It is ideally suited for describing interactions of pseudoscalar mesons in the SU (3) flavour sector, but can also encompass baryons. A prominent feature of ChPT is that it realizes the hidden symmetry of the QCD ground state, that manifests itself as a vacuum filled with uū, dd, and ss states. The lowest energy excitations of this vacuum are the pseudoscalar mesons, which are highly collective states. Another remarkable feature of the theory is that it yields multi-meson contact interactions. For instance, depending on the energy, reactions such as ππ → ππKK may involve a single interaction. On a more technical side, in ChPT, amplitudes are systematically expanded in terms of polynomials, involving both kinematical variables and quark masses. The orders of these polynomials, assessed at a scale Λ ∼ 1 GeV, determine a dynamical hierarchy and leading order (LO) contributions correspond to multimeson contact interactions, whereas resonance exchanges are next-to-leading order (NLO).
This understanding motivated an extension of the original chiral perturbation theory formalism, known as (ChPTR), in which resonances are explicitly included [26]. At present, ChPT yields the most reliable representation of the Standard Model at low energies.
Low-energy applications of ChPT are normally restricted to regions below the ρ mass whereas, in D decays, energies above 1.5 GeV are involved. Therefore, the description of hadronic interactions at those higher energies requires further extensions of the theory, which must include non-perturbative effects in a controlled way. A widely used and rather successful approach consists in ressumming a Dyson series based on chiral interactions, so as to obtain unitary scattering amplitudes [20]. In this work, we deal with the process D + → K − K + K + and, in principle, it should be described by a properly unitarized threebody amplitude. However, this is beyond present possibilities and, following the usual practice, we work in the so called (2 + 1) approximation, in which two-body unitarized amplitudes are coupled to spectator particles. Throughout the paper, we use the notation and conventions of Ref. [26]. If needed, another extension scheme for ChPT, based on the explicit inclusion of heavy mesons [27], is also available.
The theoretical description of a heavy meson decay into pseudoscalars involve two quite distinct sets of interactions. The first one concerns the primary weak vertex, in which a heavy quark, either c or b, emits a W and becomes a SU (3) quark. As this process occurs inside the heavy meson, it corresponds to the effective decay of a D or a B into a first set of SU (3) mesons. ChPT is fully suited for describing these effective processes. The primary weak decay is then followed by purely hadronic final state interactions (FSIs), in which the mesons produced initially rescatter in many different ways, before being detected. The decay D + → K − K + K + is doubly-Cabibbo-suppressed and any model describing it should involve a combination of these two parts, as suggested by Fig.1. 0000  0000  0000  0000  0000  0000  0000  0000   1111  1111  1111  1111  1111  1111  1111  1111   00000  00000  00000  00000  00000  00000  00000 00000  00000   11111  11111  11111  11111  11111  11111  11111 11111  11111   00000  00000  00000  00000  00000  00000  00000 00000  00000   11111  11111  11111  11111  11111  11111  11111  In this work we allow for the coupling of intermediate states and, within the (2 + 1) approximation, final state interactions are always associated with loops describing twomeson propagators. This provides a topological criterion for distinguishing the primary weak vertex from FSIs, namely that the former is represented by tree diagrams and the latter by a series with any number of loops. Each of these loops is multiplied by a tree-level scattering amplitude K and, schematically, this allows the decay amplitude T to be written The term within square brackets involves strong interactions only and represents a geometric series for the FSIs, which can be summed. Denoting this sum by S, one has S = 1/[1 − (loop × K)], which corresponds to the model prediction for the resonance line shape. The weak amplitude describes the process D → (P a P b )K + at tree level, where P i corresponds to a pseudoscalar with SU (3) label i. There are two competing topologies representing it, given by Fig.2. A peculiar feature of these vertices is that process (a) can yield P a P b = K − K + , whereas process (b) cannot. This can be seen by inspecting the quark structure of the latter, given in Fig.3, which shows that just a dd pair is available as a source of the two outgoing mesons at the strong vertex. Hence one could have P a P b = π 0 π 0 , π + π − , K 0K 0 , but not P a P b = K − K + . The production of a K − K + final state by mechanism (b) would thus require at least one FSI. In terms of the scheme depicted in eq.(2), this means that the first factor within the square bracket would be absent and the decay amplitude could be rewritten as Mechanism (b) is therefore suppressed when compared with mechanism (a). The Multi-Meson-Model (Triple-M) for the D + → K − K + K + amplitude proposed here assumes the dominance of process (a) of Fig.2, whereby the decay proceeds through the steps D + → Figure 3: Quark content of topology (b) of Fig.2.
Our model is based on the assumption that the weak sector of the doubly-Cabibbosuppressed decay D + → K − K + K + is dominated by the process shown in Fig.2 (a), in which quarks c andd in the D + annihilate into a W + , which subsequently hadronizes. The primary weak decay is followed by final state interactions, involving the scattering amplitude A. This yields the decay amplitude T given in Fig.4, which includes the weak vertex and indicates that the relationship with A is not straightforward.
This decay amplitude is given by where G F is the Fermi decay constant, θ C is the Cabibbo angle, the A µ are axial currents and P = p 1 + p 2 + p 3 . Throughout the paper, the label 1 refers to the K − , the label 3 the spectator K + and kinematical relations are given in appendix A.
Denoting the D + decay constant by F D , we write 0 |A µ | D + (P ) = −i √ 2 F D P µ and find a decay amplitude proportional to the divergence of the remaining axial current, given by were an exact symmetry, the axial current would be conserved and the amplitude T would vanish. As the symmetry is broken by the meson masses, one has the partial conservation of the axial current (PCAC) and T must be proportional to M 2 K . In the expressions below, this becomes a signature of the correct implementation of the symmetry.
The rich dynamics of the decay amplitude T is incorporated in the current A µ and displayed in Fig.5. Diagrams are evaluated using the techniques described in Refs. [25,26]. In chiral perturbation theory, the primary couplings of the W + to the K − K + K + system always where the functions M M (0,1) 11 with the K The imaginary propagatorsΩ of App.B are given bȳ θ being the Heaviside step function.
The dynamical meaning of the functionsΩ J ab and M (J,I) ab is indicated in Fig.6 The KK scattering amplitude, which is a prediction of the model, is derived in App.H and is written in terms of the denominators D (J,I) as C. decay amplitude The decay amplitude for the process D + → K − K + K + , given by eq.(5), has the general structure where T N R is the non-resonant contribution from diagrams (1A+1B) of Owing to chiral symmetry, all amplitudes are proportional to M 2 K , included in a common factor where F is the SU (3) pseudoscalar decay constant. Using the kinematical variables m 2 ij = (p i +p j ) 2 , the non-resonant contribution is the real polynomial corresponding to a proper three-body interaction. The amplitudes T (J,I) read the amplitude for the production of pseudoscalar mesons P a P b K + by a W + .
Comparing results (24)(25)(26)(27)(28)(29)(30)(31) and (17)(18)(19)(20), it is easy to see that the decay amplitudes T (J,I) and the scattering amplitudes A (J,I) are quite different objects, since the former include the weak interaction, which is encoded into the decay verticesΓ is our guess function, to be used in fits to data. As it is a blend of spin and isospin channels, attempts to compare it directly to the A (J,I) are meaningless.

D. free parameters
The free parameters of our function T derive from the basic lagrangian adopted [26] and consist basically of masses and coupling constants.
The former include m ρ , m φ , m a0 , m S1 , m So , whereas the latter involve F , the pseudoscalar decay constant, G Therefore, in decay analyses, the free parameters do not have the same meaning as their low-energy counterparts, since they are designed to be used into a mathematical structure which is different from ChPT. The former correspond to effective parameters describing the physics within the energy ranges defined by Dalitz plots and should not be expected to have the same values as the latter.

V. A TOY EXAMPLE: DECAY × SCATTERING AMPLITUDES
The Triple-M is aimed at predicting scattering amplitudes by using parameters obtained from fits to decay data. Even in the want of such fitted parameters at present, we explore the features of the lagrangian by using those suited to problems at low-energies,  [26], whereas the partial width Γ φ→KK ∼ 3.54 MeV [28] yields sin θ = 0.605. In the large N C limit, m S1 = m So [26] but, in order to perform the toy calculations, we choose m S1 = 1.370 GeV [28]. The discussion presented in the sequence makes it clear that there is no simple relation between the decay amplitude T and the scattering amplitudes A (J,I) .
The non-resonant contribution to the decay amplitude, eq.(23), corresponds to a genuine three-body interaction predicted by chiral symmetry. Nevertheless, in order to assess its relative importance, it is convenient to project it into the S-and P -waves suited to the other terms. Therefore, we rewrite it as so that the amplitude (21) can then be expressed as In the sequence, we discuss some aspects of this relationship, using the low-energy parameters of Ref. [26], as if they could explain decay data. In Figs In Fig.9 we present the phase shifts and inelasticity parameters associated with the scattering amplitudes A (J,I) . It important to stress that these figures correspond just to an exercise, since they are based on low-energy parameters. Nevertheless, they are instructive in showing the importance of the coupled channel structure, which is responsible for the inelasticities displayed. In the case of the I=1 P -wave, this related with the φ → πππ channel,

VI. SUMMARY
We propose a multi-meson-model (Triple-M) to describe the D + → K − K + K + decay, as a tool to extract information about KK scattering amplitudes. We depart from the The fitting parameters in the Triple-M are resonance masses and coupling constants, which have a rather transparent physical meaning. Although they entered the Triple-M through the ChPTR Lagrangian, their meanings change so as to accommodate nonperturbation effects of meson-meson interactions. To obtain realistic values for these parameters, they should be extracted from a Triple-M fit to data. As a lesser alternative, here we employ the low-energy parameters [26] values as if they resulted from data. In Fig.10  we show a toy Monte-Carlo Dalitz plot based on the Triple-M, where it is possible to see a destructive interference between the S-and P -waves on the low-energy sector of the φ(1020).
One of the φ(1020) lobes is depleted with respect to the other, resulting in a peak and a dip, a behaviour similar to that observed in LHCb preliminary data [29].
In our one-dimensional toy studies, Figs.7-8, we show that the Triple-M can track the hidden isospin signatures of two-body interactions in three-body data, allowing one to disentangle the relative contributions of resonances a 0 (980) and f 0 (980). By comparing results for the three-body amplitudes T J and the scattering amplitudes A (J,I) , it becomes clear that even though the later are present in the former, they cannot be extracted directly. However, with a model departing from a Lagrangian that includes a full two-body coupled channel dynamics, such as our Triple-M, fits to decay data can give rise to predictions for the KK scattering amplitudes A (J,I) .

ACKNOWLEDGMENTS
This work was supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq).

Appendix A: kinematics
Momenta are defined by D(P ) → K − (p 1 ) K + (p 2 ) K + (p 3 ), with P = p 1 + p 2 + p 3 . The invariant masses read and satisfy the constraint Their values are also limited by the boundaries of the Dalitz plot, by Appendix B: two-meson propagators and functions Ω Expressions presented here are conventional. They are displayed for the sake of completeness and rely on the the results of Ref. [25]. These integrals do not include symmetry factors, which are accounted for in the main text. One deals with both S and P waves and the corresponding two-meson propagators are associated with the integrals with p 2 = s. Both integrals I ab and I µν ab are evaluated using dimensional techniques [25]. For s ≥ (M a +M b ) 2 , the function I ab has the structure where Λ ab is a divergent function of the renormalization scale µ and of the number of dimensions n , which diverges in the limit n → 4 , whereas Π is regular component, given by The tensor integral is needed for a = b only, and one has where Λ pp aa and Λ g aa are divergent quantities. In the K-matrix approximation, one keeps only the imaginary parts of the loop integrals, which are contained in the function Π and has In the decay calculation, it is more covenient to use the functionsΩ, defined by These results are related with CM momenta bȳ where θ is the Heaviside step function.
Appendix C: partially dressed φ propagator The bare φ propagator, G αβγδ , is given by eq.(A.10) of Ref. [26]. It is dressed by both πρ andKK intermediate states and the corresponding self-energies are denoted respectively by Σ πρ and ΣK K . In this section we consider just contributions of the former kind. The full propagator is given by The φπρ interaction is extracted from the lagrangian where ω 1 = cos θ φ−sin θ ω is the singlet component. In the sequence, we write g = g 1 cos θ.
The self energy is given by with p = k/2 − , q = k/2 + and k 2 = s . Using the explicit form of G χηωζ and the definitions D π = p 2 − M 2 π , D ρ = q 2 − m 2 ρ , we find where we have used the fact that terms proportional to k µ and k λ do not contribute to eq.(C6). This integral is highly divergent, but the part regarding the Kρ cut is not. Terms containing factors D π and D ρ in the numerator do not contribute to the cut function and the relevant integral is Using the definition and the result the relevant component of I µλ becomes The on-shell contribution to eq.(C11) is given by where Q πρ is the CM threemomentum. We then have Numerically, Γ πρ φ = 0.1532 × Γ φ = 0.1532 × 0.004247 GeV [28]. Using this result into eq.(C1) and ressumming the series, we get the partially dressed propagator where the denominator D πρ φ (s) is given by In the evaluation of amplitudes involving aK(p 1 ) K(p 2 ) vertex, one encounters the product  (3) states, which are related to charged states by We need just two-meson intermediate states |ab , with the same quantum numbers as the K − K + system, which are given by |S KK = (1/2) |4 4 + 5 5 + 6 6 + 7 7 , (D11) The state |K − K + includes a conventional phase an reads and, therefore, Here, the function D πρ φ is a partially dressed φ propagator, discussed in App.C, eq.(C18), associated with the partial width of the decay φ → (ρπ + πππ). [J, The function D πρ φ is this expression represents a partially dressed φ propagator, discussed in App.C, eq.(C18), and accounts for the partial width of the decay φ → (ρπ + πππ). [J, and encompasses a coupled channel structure, which depends on the spin-isospin considered.
In order to display the meaning of the indices used in this structure, we label informally each (J, I) channel by its most prominent resonance and recall that ρ-channel: Γ The meanings of the indices used in the matrices M (J,I) , eq.(G4), are similar.
In this work, we need at most three coupled channels, which corresponds to The Multi-Meson-Model we consider in this work assembles a number of aspects that appear scattered in many calculations, but are normally absent in heavy meson decay analyses. The main unusual dynamical effects included into our model concern: i) the presence of a LO contact interaction in the two-body kernel, as indicated in Fig.6; ii) the introduction of two resonances in the (J = 0, I = 0) channel, preserving unitarity; iii) consideration of coupled channels. With the purpose of disclosing the role played by these features in the results, in this appendix we consider the scattering amplitude A (0,0) and show its behavior in a number of different scenarios. We begin by the simplest one, in which just the f 0 (980) is kept, and add the other contributions gradually, as described in  We begin by considering the artificial situation in which the kaon mass is lowered to M K = 0.4 GeV, so as to allow the f 0 (980) to be above threshold. The amplitude is shown in