Multi-Meson Model for the D + → K + K − K + decay amplitude
MMITP/18-041
Multi-Meson Model for the D + → K + K − K + decay amplitude R.T. Aoude,
1, 2
P.C. Magalh˜aes,
1, 3, ∗ A.C. dos Reis, and M.R. Robilotta Centro Brasileiro de Pesquisas F´ısicas, Rio de Janeiro, Brazil PRISMA Cluster of Excellence and Mainz Institute for Theoretical Physics,Johannes Gutenberg-Universit¨at Mainz, Germany Technical University of Munich, Germany Instituto de F´ısica, Universidade de S˜ao Paulo, S˜ao Paulo, Brazil (Dated: June 20, 2018)
Abstract
We propose a novel approach to describe the D + → K − K + K + decay amplitude, based on chiraleffective Lagrangians, which can be used to extract information about K ¯ K scattering. Our trialfunction is an alternative to the widely used isobar model and includes both nonresonant three-bodyinteractions and two-body rescattering amplitudes, based on coupled channels and resonances, forS- and P-waves with isospin 0 and 1. The latter are unitarized in the K -matrix approximation andrepresent the only source of complex phases in the problem. Free parameters are just resonancemasses 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, whichgoes beyond the (2+1) approximation. Our approach allows one to disentangle the two-body scalarcontributions with different isospins, associated with the f (980) and a (980) channels. We showhow the K ¯ K amplitude can be obtained from the decay D + → K − K + K + and discuss extensionsto other three-body final states. ∗ [email protected] a r X i v : . [ h e p - ph ] J un . INTRODUCTION Nonleptonic weak decays of heavy-flavoured mesons are extensively used in light mesonspectroscopy. Owing to a rich resonant structure, these decays provide a natural placeto 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 scalarstates f (600) (or sigma)[1] and kappa(800)[2]. More comprehensive investigations can bedone 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 differentclasses of processes, namely heavy-quark weak transitions, hadron formation and final-stateinteractions (FSI), whereby the hadrons produced in the primary vertex are allowed to inter-act in many different ways before being detected. Final-state processes include both properthree-body interactions and a wide range elastic and inelastic coupled channels, involvingresonances. In this framework, a question arises, concerning how to obtain informationabout 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 amplitudeanalyses have been performed using the so-called isobar model, in which the decay amplitudeis represented by a coherent sum of both nonresonant and resonant contributions. Thisapproach, albeit largely employed [3], has conceptual limitations. The outcome of isobarmodel analyses are resonance parameters such as fit fractions, masses and widths, whichare neither directly related to any underlying dynamical theory nor provide clues to theidentification of two-body substructures. Thus, the systematic interpretation of the isobarmodel results is rather difficult.This situation motivated in the past decade efforts towards building models that are basedon more solid theoretical grounds. Those models improve essentially the two-meson inter-action description in the FSI, with the use of dispersion relations and chiral perturbationtheory. Most of them work in the quasi-two-body (2+1) approximation, where interactionswith the third particle are neglected. Recently, a collection of parametrizations based on an-alytic and unitary meson-meson form factors for D and B three-body hadronic decays withinthe (2+1) approximation was presented in Ref.[4]. Three-body FSIs were also consideredand, in particular, shown to play a significant role in the D + → K − π + π + decay. In this2rocess, three-body unitarity was implemented in different ways, by means of Faddeev-likedecompositions[5–7], Khuri-Treiman equation[8] or triangle diagrams [9]. Whilst differingin methods and techniques, all these theoretical efforts have in common the attempt to in-clude, in a systematic way, knowledge of two-body systems in the description of the decayamplitudes.This work departs from the same broad perspective, but concentrates explicitly on thederivation of two-body scattering amplitudes from three-body decays. In order to do so, wesuggest 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 kaon-kaon scattering, regarding both theory and experiment. Concerning the latter, one onlyhas access to ππ elastic scattering data [10] and to the inelastic channel ππ → K ¯ K [10, 11].Information about K ¯ K interaction can be estimated by imposing unitarity constraints on the ππ data. On the theory side, K ¯ K amplitudes have been calculated in next-to-leading orderchiral perturbation theory. Aiming at a full coupled-channel description, it was extendedup to 1.2 GeV, using form factors [12] to describe the ηπ → K ¯ K contribution to η → πππ decay[13], or unitarized ressummation techniques[14], to include ππ → K ¯ K in the contextof FSI of J/ Ψ → φππ ( K ¯ K ) decays.The main purpose of this work is disclose information about the dynamics of K ¯ K in-teractions by disentangling the two-body contributions contained in the D + → K − K + K + amplitude. In our model, the description of the K ¯ K interaction relies on a chiral Lagrangianwith resonances, including all possible coupled channels for ( J = 0 , I = 0 , I. MOTIVATION FOR A NEW MODEL
The isobar model, widely used for describing heavy-meson decays into three pseu-doscalars, relies on the assumption that these processes are dominated by intermediatestates involving a spectator plus a resonance, and also includes non-resonant contributions.In the decay H → P P P , of a heavy meson H into three pseudoscalars P i , the isobar modelemphasizes the sequence H → R P , followed by R → P P .The full H → P P P decay amplitude is denoted by T and the isobar model employs aguess function to be fitted to data in the form of the coherent sum T = c nr τ nr + (cid:88) c k τ k , (1)the subscript nr referring to the non-resonant term and the label k associated with reso-nances, as many of them as needed. The coefficients c k = e iθ k are complex parameters, tobe determined by data. The choice τ nr = 1 is usual for the non-resonant term, whereasthe sub amplitudes τ k depend on the invariant masses of the problem. For each resonanceconsidered, the function τ k is given by τ k = [ F F ] × [ angular factor ] × [line shape] k , where F F stands for form factors, the angular factor is associated with angular momentum chan-nels, and [line shape] k represents a resonance line shape, described by either a Breit-Wignerfunction such as ( BW ) k = 1 / [ s − m k + i m k Γ k ], m k and Γ k being the resonance mass andwidth, or by variations, such as the Flatt´e or Gounaris-Sakurai forms. The angular fac-tor allows one to distinguish partial wave contributions and to employ the decomposition T = T S + T P + · · · .A good fit to decay data based on the structure given by eq.(1), would yield an empiricalset of complex parameters c nr and c k . However, a question arises regarding the meaning ofthese parameters. Would they be useful to shed light into yet unknown two-body substruc-tures 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 oneextract A directly from T ? As we argue in the sequence, answers to these questions do notfavour 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 relationshipinvolves several issues, which we discuss below.4 . dynamics - The dynamical contents of T and A are rather different, since the formermust include weak vertices, which cannot be present in the latter. Specific features of W -meson interactions are important to T and irrelevant to A . Therefore, although scatteringamplitudes A may be substructures of T , there is no reason whatsoever for assuming thatthese A ’s are either identical or proportional to T . This is supported by case studies. Forinstance, some time ago, the FOCUS collaboration[16] produced a partial-wave analysis ofthe 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 + scatter-ing data[18] and the discrepancy found was seen as a puzzle. The fact that the FOCUSphase was negative at low energies was considered to be especially odd. In the languageof 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 laterexplained by considering meson loops in the weak sector of the problem[5, 6]. These loopsaccount for the extra phases observed. b. good quantum numbers: - Isospin is broken by weak interactions and is a goodquantum number for A , but not for T . Scattering amplitudes A depend both on the angularmomentum J and on the isospin I of the channel considered, whereas just a J dependencecan be extracted from an empirical decay amplitude T . This point will be recast on moretechnical grounds while we discuss our model. For the time being, it suffices to stress thatit is impossible to derive directly A ( J,I ) from T ( J ) simply because the former contains morestructure than the latter. An extraction of A ( J,I ) from T ( J ) would amount to generatingphysical content about the isospin structure. c. coupled channels - It is well known that scattering amplitudes include importantinelasticities due to couplings of intermediate states. For instance, as Hyams et al.[10] pointout, K ¯ K intermediate states do influence elastic ππ scattering at some energies. Since scat-tering amplitudes A are substructures of the decay amplitude T , coupled channels presentin the former must also show up in the latter. In general, guess functions better suited foraccommodating data should have structures similar to those used in meson-meson scatter-ing Refs.[10, 12, 19]. In the case of the isobar model, the simple guess functions usuallyemployed fail to incorporate these intermediate couplings. d. unitarity - Good fits to Dalitz plots data may require several resonances with the samequantum numbers. At present, conceptual techniques are available which preserve unitarity5hile incorporating several resonances into amplitudes[20]. This allows one to go beyond theisobar 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 scatteringamplitudes[21]. e. non-resonant term -
The non-resonant term may be important and involve less knowninteractions. In the case of heavy meson decays and some leptonic reactions, available ener-gies can be large enough for allowing the simultaneous production of several pseudoscalarsat a single vertex. Multi-meson dynamics then becomes relevant. For instance, the pro-cess e − e + → π involves the matrix element (cid:104) ππππ | J µγ | (cid:105) , J µγ being the electromagneticcurrent[22]. A similar matrix element, with J µγ replaced with the weak current ( V − A ) µ ,describes the decay τ → ν π [22]. Interactions of this kind are also present in the model for D + → K − K + K + we discuss here. f. lagrangians - Although the point of departure of the isobar model may be sound, theproblems 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 re-stricted to specific processes. Moreover, in particular, one cannot rely on them for obtainingscattering information. The most conservative way of ensuring that the physical meaning ofparameters is preserved from process to process is to employ lagrangians, which rely on justmasses and coupling constants. Guess functions for heavy-meson decays constructed fromlagrangians yield free parameters which allow the straightforward derivation of scatteringamplitudes.
III. DYNAMICS
The fundamental QCD lagrangian for strong interactions is written in terms of gluonsand 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 reac-tions cannot be described in terms of quarks and gluons, and one is forced to rely on effectivetheories. At low energies, chiral perturbation theory (ChPT) [23–25] is highly successful.It is ideally suited for describing interactions of pseudoscalar mesons in the SU (3) flavoursector, but can also encompass baryons. A prominent feature of ChPT is that it realizesthe hidden symmetry of the QCD ground state, that manifests itself as a vacuum filled with6 ¯ u, d ¯ d , and s ¯ s states. The lowest energy excitations of this vacuum are the pseudoscalarmesons, which are highly collective states. Another remarkable feature of the theory is thatit yields multi-meson contact interactions. For instance, depending on the energy, reactionssuch as ππ → ππK ¯ K may involve a single interaction. On a more technical side, in ChPT,amplitudes are systematically expanded in terms of polynomials, involving both kinematicalvariables and quark masses. The orders of these polynomials, assessed at a scale Λ ∼ ρ masswhereas, in D decays, energies above 1 . D + → K − K + K + and, in principle, it should be described by a properly unitarized three-body amplitude. However, this is beyond present possibilities and, following the usualpractice, we work in the so called (2 + 1) approximation, in which two-body unitarizedamplitudes are coupled to spectator particles. Throughout the paper, we use the notationand conventions of Ref.[26]. If needed, another extension scheme for ChPT, based on theexplicit inclusion of heavy mesons[27], is also available.The theoretical description of a heavy meson decay into pseudoscalars involve two quitedistinct sets of interactions. The first one concerns the primary weak vertex, in which aheavy quark, either c or b , emits a W and becomes a SU (3) quark. As this process occursinside the heavy meson, it corresponds to the effective decay of a D or a B into a first setof SU (3) mesons. ChPT is fully suited for describing these effective processes. The primaryweak decay is then followed by purely hadronic final state interactions (FSIs), in which themesons 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 involvea combination of these two parts, as suggested by Fig.1.7 (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) KKK ++ − KKK ++ − KKK ++ − (a) = + ba (b) WW T Figure 1: Amplitude T for D + → K − K + K + : (a) primary weak vertex; (b) weak vertex dressedby final state interactions; the full line is the D , dashed lines are pseudoscalars. 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 two-meson propagators. This provides a topological criterion for distinguishing the primaryweak vertex from FSIs, namely that the former is represented by tree diagrams and thelatter by a series with any number of loops. Each of these loops is multiplied by a tree-levelscattering amplitude K and, schematically, this allows the decay amplitude T to be writtenas T = (weak tree) × (cid:2) × K ) + (loop × K ) + (loop × K ) + · · · (cid:3) . (2)The term within square brackets involves strong interactions only and represents a geometricseries 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. (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) = + K + ba K + ba K + ba(a) (b) W Figure 2: Competing topologies for the decay D + → K − K + K + ;as the pair P a P b is producedeither after (a) or before (b) the weak interaction. The weak amplitude describes the process D → ( P a P b ) K + at tree level, where P i cor-responds to a pseudoscalar with SU (3) label i . There are two competing topologies repre-senting 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 struc-ture of the latter, given in Fig.3, which shows that just a d ¯ d pair is available as a source of thetwo outgoing mesons at the strong vertex. Hence one could have P a P b = π π , π + π − , K ¯ K ,8ut not P a P b = K − K + . The production of a K − K + final state by mechanism (b) wouldthus require at least one FSI. In terms of the scheme depicted in eq.(2), this means thatthe first factor within the square bracket would be absent and the decay amplitude couldbe rewritten as T = (weak tree) × (loop × K ) × (cid:2) × K ) + (loop × K ) + (loop × K ) + · · · (cid:3) . (3)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 thedominance of process (a) of Fig.2, whereby the decay proceeds through the steps D + → W + → K + K − K + . −d−d−d K + c c q d−q D + Figure 3: Quark content of topology (b) of Fig.2.
IV. MULTI-MESON-MODEL FOR D + → K − K + K + Our model is based on the assumption that the weak sector of the doubly-Cabibbo-suppressed decay D + → K − K + K + is dominated by the process shown in Fig.2 (a), inwhich quarks c and ¯ d in the D + annihilate into a W + , which subsequently hadronizes. Theprimary 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 andindicates that the relationship with A is not straightforward.This decay amplitude is given by T = − (cid:20) G F √ θ C (cid:21) (cid:104) K − ( p ) K + ( p ) K + ( p ) | A µ | (cid:105) (cid:104) | A µ | D + ( P ) (cid:105) , (4)9 (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) = + KKK ++ − KKK ++ − KKK ++ − (a) (b) T Figure 4: Decay amplitude for D + → K − K + K + ; the weak vertex proceeds thought the interme-diate steps D + → W + and W + → K − K + K + and strong final state interactions are encompassedby the scattering amplitude A (full red blob). where G F is the Fermi decay constant, θ C is the Cabibbo angle, the A µ are axial currentsand P = p + p + p . Throughout the paper, the label 1 refers to the K − , the label 3 thespectator K + and kinematical relations are given in appendix A.Denoting the D + decay constant by F D , we write (cid:104) | A µ | D + ( P ) (cid:105) = − i √ F D P µ andfind a decay amplitude proportional to the divergence of the remaining axial current, givenby T = i (cid:20) G F √ θ C (cid:21) √ F D [ P µ (cid:104) A µ (cid:105) ] , (5)with (cid:104) A µ (cid:105) = (cid:104) K − ( p ) K + ( p ) K + ( p ) | A µ | (cid:105) . This result is important because, if SU (3)were an exact symmetry, the axial current would be conserved and the amplitude T wouldvanish. As the symmetry is broken by the meson masses, one has the partial conservationof the axial current (PCAC) and T must be proportional to M 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 (cid:104) A µ (cid:105) anddisplayed in Fig.5. Diagrams are evaluated using the techniques described in Refs.[25, 26]. Inchiral perturbation theory, the primary couplings of the W + to the K − K + K + system alwaysinvolve a direct interaction, accompanied by a kaon-pole term, denoted by (A) and (B) inthe figure. Only their joint contribution is compatible with PCAC. Diagrams (1A+1B) areLO and describe a non-resonant term, a proper three body interaction, which goes beyondthe (2 + 1) approximation, whereas Figs. (2A+2B) allow for the possibility that two of themesons rescatter, after being produced in the primary weak vertex. Diagrams (3A+3B) areNLO and describe the production of bare resonances at the weak vertex, whereas final staterescattering processes (4A+4B) endow them with widths.10
12 123 123321123 3 321 21312 (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) + (4A) + (3B) + (4B)(3A) + (2A) (2B) + KKK ++ − KKK ++ − += + (1B)(1A) + Figure 5: Dynamical structure of triangle vertices in Fig.4; the wavy line is the W + , dashed linesare mesons, continuous lines are resonances and the full red blob represent meson-meson scatteringamplitudes, described in Fig.6; all diagrams within square brackets should be symmetrized, bymaking 2 ↔ A. two-body unitarization and resonance line shapes
In the description of the two-body subsystem, we consider just S - and P - waves, corre-sponding to ( J = 1 , , I = 1 ,
0) spin-isospin channels. The associated resonances are ρ (770), φ (1020), a (980), and two SU (3) scalar-isoscalar states, S and S o , corresponding to a sin-glet and to a member of an octet, respectively. The physical f (980), together with a highermass f state, would be linear combinations of S and S o . Depending on the channel, theintermediate two-meson propagators may involve ππ , KK , ηη , and πη intermediate states,so there is a large number of coupled channels to be considered. = + + + ...= + (a)(b) Figure 6: (a) Tree-level two-body interaction kernel K ( J,I ) ab → cd - a NLO s -channel resonance, addedto a LO contact term. (b) Structure of the unitarized scattering amplitude. P a P b → P c P d are described by kernels K ( J,I ) ab | cd and their simple dynamical structure is shown in Fig.6, as LO four point terms, typicalof chiral symmetry, supplemented by NLO resonance exchanges in the s -channel. Just inthe ( J = 0 , I = 0) channel two resonances, S and S o , are needed. In these diagrams, all vertices represent interactions derived from chiral lagrangians[26]. Kernels are then functionsdepending on just masses and coupling constants. The mathematical structure of thesefunctions is displayed in App.F. In the case of the φ -meson, the kernel includes an effectivecoupling to the ( ρπ + πππ ) channel, which accounts for about 15% of its width. This effectiveinteraction is discussed in App.(C) and yields eq.(F6).All other resonance terms in the kernels contain bare poles. However, the evaluation ofamplitudes involves the iteration of the basic kernels by means of two-meson propagators, asin Fig.6(b). The propagators, denoted by ¯Ω, are discussed in App.B and, in principle, haveboth real and imaginary components. The former contain divergent contributions and theirregularization brings unknown parameters into the problem. This considerable nuisance isavoided by working in the K -matrix approximation, whereby just the imaginary parts ofthe two-meson propagators are kept. This gives rise to the structure sketched within thesquare bracket of eq.(2), where the terms (loop × K ) are realized by the functions M ( J,I ) ij given in eqs.(G10-G13). The ressummation of the geometric series, indicated in Fig.6(b),endows the s -channel resonances with widths. Thus among other structures, intermediatetwo-body amplitudes yield denominators D ( J,I ) , which are akin to those of the form D BW =[ s − m + i m Γ] employed in BW functions. These denominators, that correspond to thepredictions of the model for the resonance line shapes, are given in App.G and reproducedbelow. Explicit expressions read D ρ = (cid:104)(cid:16) − M (1 , (cid:17) (cid:16) − M (1 , (cid:17) − M (1 , M (1 , (cid:105) , (6) D φ = (cid:8) − M (1 , (cid:9) , (7) D a = (cid:104)(cid:16) − M (0 , (cid:17) (cid:16) − M (0 , (cid:17) − M (0 , M (0 , (cid:105) , (8) D S = [1 − M (0 , ][1 − M (0 , ][1 − M (0 , ] − [1 − M (0 , ] M (0 , M (0 , − [1 − M (0 , ] M (0 , M (0 , − [1 − M (0 , ] M (0 , M (0 , − M (0 , M (0 , M (0 , − M (0 , M (0 , M (0 , , (9)12here the functions M ( J,I ) ij read M (1 , = −K (1 , ππ | ππ [ ¯Ω Pππ / , M (1 , = −K (1 , ππ | KK [ ¯Ω PKK / ,M (1 , = −K (1 , ππ | KK [ ¯Ω Pππ / , M (1 , = −K (1 , KK | KK [ ¯Ω PKK / . (10) M (1 , = −K (1 , KK | KK [ ¯Ω PKK / . (11) M (0 , = −K (0 , π | π [ ¯Ω Sπ / , M (0 , = −K (0 , π | KK [ ¯Ω SKK / ,M (0 , = −K (0 , π | KK [ ¯Ω Sπ / , M (0 , = −K (0 , KK | KK [ ¯Ω SKK / . (12) M (0 , = −K (0 , ππ | ππ [ ¯Ω Sππ / , M (0 , = −K (0 , ππ | KK [ ¯Ω SKK / ,M (0 , = −K (0 , ππ | [ ¯Ω S / , M (0 , = −K (0 , ππ | KK [ ¯Ω Sππ / ,M (0 , = −K (0 , KK | KK [ ¯Ω SKK / , M (0 , = −K (0 , KK | [ ¯Ω S / ,M (0 , = −K (0 , ππ | [ ¯Ω Sππ / , M (0 , = −K (0 , KK | [ ¯Ω SKK / ,M (0 , = −K (0 , | [ ¯Ω S / , (13)with the K ( J,I ) ab | cd of App.F, whereas the subscripts 8 refer to the member of the SU (3) octetwith the quantum numbers of the η . The factor 1 / M (0 , and M (0 , because one is using the symmetrized π Sab = − i π Q ab √ s θ ( s − ( M a + M b ) ) , (14)¯Ω Paa = − i π Q aa √ s θ ( s − M a ) , (15) Q ab = 12 (cid:113) s − M a + M b ) + ( M a − M b ) /s , (16) θ being the Heaviside step function.The dynamical meaning of the functions ¯Ω Jab and M ( J,I ) ab is indicated in Fig.6(b). Theformer represents the two-body propagator for mesons a and b with angular momentum J , indicated by the dashed lines between two successive empty blobs, whereas the latterencompasses a blob and a two-body propagator. The functions M ( J,I ) ab correspond to thepaces of the the various geometric series entangled by the coupling of intermediate channels.13 . K ¯ K scattering amplitude The K ¯ K scattering amplitude, which is a prediction of the model, is derived in App.Hand is written in terms of the denominators D ( J,I ) as A (1 , KK | KK = 1 D ρ ( m ) (cid:104) M (1 , K (1 , ππ | KK + (cid:16) − M (1 , (cid:17) K (1 , KK | KK (cid:105) , (17) A (1 , KK | KK = 1 D φ ( m ) K (1 , KK | KK , (18) A (0 , KK | KK = 1 D a ( m ) (cid:104) M (0 , K (0 , π | KK + (cid:16) − M (0 , (cid:17) K (0 , KK | KK (cid:105) (19) A (0 , KK | KK = 1 D S ( m ) (cid:110)(cid:104) M (0 , (cid:16) − M (0 , (cid:17) + M (0 , M (0 , (cid:105) K (0 , ππ | KK + (cid:104)(cid:16) − M (0 , (cid:17) (cid:16) − M (0 , (cid:17) − M (0 , M (0 , (cid:105) K (0 , KK | KK + (cid:104) M (0 , (cid:16) − M (0 , (cid:17) + M (0 , M (0 , (cid:105) K (0 , | KK (cid:111) . (20) C. decay amplitude
The decay amplitude for the process D + → K − K + K + , given by eq.(5), has the generalstructure T = T NR + (cid:2) T (1 , + T (1 , + T (0 , + T (0 , + (2 ↔ (cid:3) , (21)where T NR is the non-resonant contribution from diagrams (1A+1B) of Fig.5 and the T ( J,I ) are the resonant contributions from diagrams (2A+2B+3A+3B+4A+4B), in the variousspin and isospin channels.Owing to chiral symmetry, all amplitudes are proportional to M K , included in a commonfactor C = (cid:26)(cid:20) G F √ θ C (cid:21) F D F M K ( M D − M K ) (cid:27) , (22)where F is the SU (3) pseudoscalar decay constant. Using the kinematical variables m ij =( p i + p j ) , the non-resonant contribution is the real polynomial T NR = C (cid:8)(cid:2) ( m − M K ) + ( m − M K ) (cid:3)(cid:9) , (23)14orresponding to a proper three-body interaction. The amplitudes T ( J,I ) read T (1 , = − (cid:104) ¯Γ (1 , KK − Γ (1 , c | KK (cid:105) ( m − m ) , (24)¯Γ (1 , KK = 1 D ρ ( m ) (cid:104) M (1 , Γ (1 , ππ + (cid:16) − M (1 , (cid:17) Γ (1 , KK (cid:105) , (25) T (1 , = − (cid:104) ¯Γ (1 , KK − Γ (1 , c | KK (cid:105) ( m − m ) , (26)¯Γ (1 , KK = 1 D φ ( m ) Γ (1 , KK , (27) T (0 , = − (cid:104) ¯Γ (0 , KK − Γ (0 , c | KK (cid:105) , (28)¯Γ (0 , KK = 1 D a ( m ) (cid:104) M (0 , Γ (0 , π + (cid:16) − M (0 , (cid:17) Γ (0 , KK (cid:105) , (29) T (0 , = − (cid:104) ¯Γ (0 , KK − Γ (0 , c | KK (cid:105) , (30)¯Γ (0 , KK = 1 D S ( m ) (cid:110)(cid:104) M (0 , (cid:16) − M (0 , (cid:17) + M (0 , M (0 , (cid:105) Γ (0 , ππ + (cid:104)(cid:16) − M (0 , (cid:17) (cid:16) − M (0 , (cid:17) − M (0 , M (0 , (cid:105) Γ (0 , KK + (cid:104) M (0 , (cid:16) − M (0 , (cid:17) + M (0 , M (0 , (cid:105) Γ (0 , (cid:111) , (31)where the various functions Γ ( J,I ) , given in App.E, are linear in the coefficient C . Thedynamical meaning of the functions Γ ( J,I )(0) ab can be inferred from Fig.5(b). They correspondto the tree diagrams (1A+1B) and (3A+3B) with the indices (1 , → ( a, b ) and representthe amplitude for the production of pseudoscalar mesons P a P b K + by a W + .Comparing results (24-31) and (17-20), 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 includethe weak interaction, which is encoded into the decay vertices ¯Γ ( J,I ) KK . Nevertheless, both A ( J,I ) KK | KK and ¯Γ J,I ) KK share the same denominators D ( J,I ) . The amplitude T , given by eq.(21)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.15 . 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 a , m S , m So , whereas the latter involve F , the pseudoscalar decay constant, G V ,the coupling constant of vector mesons to pseudoscalars, an angle θ , associated with ω − φ mixing, c d , c m , describing the couplings of both a and S o to pseudoscalars, and ˜ c d , ˜ c m , im-plementing the couplings of S to pseudoscalars. These lagrangian parameters first enterthe guess function through the functions Γ ( J,I )(0) ab and K ( J,I ) ab | cd in apps. E and F.In the strict framework of chiral perturbation theory, the values of the lagrangian pa-rameters are extracted by comparing results from field theoretical calculations performedto a given order to observables. As the former involve divergent loops, they are affected byrenormalization and values quoted in the literature depend on renormalization scales. Thiskind of procedure is theoretically consistent and yields a precise description of low-energyphenomena.In the case of heavy meson decays, this level of precision cannot be reached. The mainreason is that the problem involves necessarily a wide range of energies, both below andabove resonance poles, where perturbation does not apply and non-perturbative techniquesare needed. An instance is the resummation of the infinite series of diagrams indicated inFig.6, required by unitarization, which yields the denominators D ( J,I ) discussed in sect.IV A.Therefore, in decay analyses, the free parameters do not have the same meaning as theirlow-energy counterparts, since they are designed to be used into a mathematical structurewhich is different from ChPT. The former correspond to effective parameters describing thephysics within the energy ranges defined by Dalitz plots and should not be expected to havethe same values as the latter. V. A TOY EXAMPLE: DECAY × SCATTERING AMPLITUDES
The Triple-M is aimed at predicting scattering amplitudes by using parameters obtainedfrom fits to decay data. Even in the want of such fitted parameters at present, we ex-plore the features of the lagrangian by using those suited to problems at low-energies,which are: [ m ρ , m φ , m a , m So ] = [0 . , . , . , . F = 0 .
093 GeV,16 G V , c d , c m , ˜ c d , ˜ c m ] = [0 . , . , . , . , . φ → K ¯ K ∼ .
54 MeV[28] yields sin θ = 0 . N C limit, m S = m So [26] but,in order to perform the toy calculations, we choose m S = 1 .
370 GeV[28]. The discussionpresented in the sequence makes it clear that there is no simple relation between the decayamplitude T and the scattering amplitudes A ( J,I ) .The non-resonant contribution to the decay amplitude, eq.(23), corresponds to a genuinethree-body interaction predicted by chiral symmetry. Nevertheless, in order to assess itsrelative importance, it is convenient to project it into the S - and P -waves suited to theother terms. Therefore, we rewrite it as T NR = (cid:20) C M − M K + m ) + C m − m ) + (2 ↔ (cid:21) , (32)so that the amplitude (21) can then be expressed as T = (cid:2) T S + T P + (2 ↔ (cid:3) , (33) T S = (cid:20) C M D − M K + m ) + T (0 , + T (0 , (cid:21) , (34) T P = (cid:20) C m − m ) + T (1 , + T (1 , (cid:21) . (35)In the sequence, we discuss some aspects of this relationship, using the low-energy param-eters of Ref.[26], as if they could explain decay data. In Figs.7 and 8, we show the moduliand phases of the S - and P -wave decay amplitudes T S , eq.(34) and T P , eq.(35), togetherwith the moduli and phases of the corresponding K ¯ K scattering amplitudes A ( J,I ) . Thesefigures illustrate the usefulness of the lagrangian approach. Without it, one would be ableto determine just the full decay amplitudes T S and T P , represented by the continuous blackcurves in the figures, and would not have access to partial contributions in different isospinchannels. Moreover, it is also clear that one cannot guess the form of the K ¯ K scatteringamplitudes A ( J,I ) , represented by the red and blue dotted lines, from the decay components T S and T P .In Fig.9 we present the phase shifts and inelasticity parameters associated with the scat-tering amplitudes A ( J,I ) . It important to stress that these figures correspond just to anexercise, since they are based on low-energy parameters. Nevertheless, they are instructivein showing the importance of the coupled channel structure, which is responsible for the in-elasticities displayed. In the case of the I=1 P -wave, this related with the φ → πππ channel,17 s (GeV) ()*+ | T | SW(0,0)(0,1)NR2 s (GeV) ())*))+)),)) | A | (0,0)(0,1)2 s (GeV) ()(*+( p h a s e s ( d e g r ee s ) Figure 7: S -wave sector - top left: the continuous black curve (SW) is the modulus of the decayamplitude T S , eq.(34), in arbitrary units, whereas other curves are moduli of partial contributions;top right: moduli of the K ¯ K scatterig amplitudes A (0 , , red curve, and A (0 , , blue curve; bottom:the continuous black curve (SW) is the phase of the decay amplitude T S , eq.(34), and othercontinuous curves are phases of partial contributions; the dashed curves represent the phases ofthe K ¯ K scatterig amplitudes A (0 , (red) and A (0 , (blue). as discussed in App.C. In all cases, the bound η ≤ J = 0 , I = 0) channel, preserving unitarity; iii) inclusionof coupled channels. In App.J we discuss their piecemeal relevance, in the case of A (0 , .18 ,0 1,5 2,00246 | T | s (GeV) PW (1,0)(1,1) NR | A | s (GeV) (1,0)(1,1) pha s e ( deg r ee s ) s (GeV) T PW T(1,0)T(1,1) A(1,1) A(1,0)
Figure 8: P -wave sector - top left: the continuous black curve (SW) is the modulus of the decayamplitude T P , eq.(35), in arbitrary units, whereas other curves are moduli of partial contributions;top right: muduli of the K ¯ K scatterig amplitudes A (1 , , red curve, and A (1 , , blue curve; bottom:the continuous black curve (SW) is the phase of the decay amplitude T S , eq.(34), and othercontinuous curves are phases of partial contributions; the dashed curves represent the phases ofthe K ¯ K scatterig amplitudes A (1 , (red) and A (1 , (blue). 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 K ¯ K scattering amplitudes. We depart from the19 s (GeV) ()(*+(,-( δ S W s (GeV) (.*(.+(.,(.-..( η S W P W s (GeV) (1,1)(1,0) P W s (GeV) (1,1)(1,0) Figure 9: Phase shifts δ and inelasticity parameter η for K ¯ K scattering - top: S -waves; bottom: P -waves; blue and red curves correspond respectively to isospin I=0 and I=1. dominance of the annihilation weak topology, which allows one to describe the whole decayprocess within the SU (3) chiral symmetry framework. The non-resonant component is aproper three-body interaction that goes beyond the (2+1) approximation and is given bychiral symmetry as a real polynomium. Primary vertices describing the direct productionof mesons and of lowest SU(3) resonances, in S - and P -waves, with isospin 0 and 1, aredressed by FSIs involving coupled channels. The K ¯ K scattering amplitudes for each of the( J, I ) considered are derived from the ChPTR Lagrangian[26], unitarized by ressummationtechniques in the K -matrix approximation, in which particle propagators were kept on-shell,and include coupled-channels. They are the only source of imaginary terms in the decayamplitude and fix the relative phase between S - and P -waves in Triple-M. This representsan important improvement over the isobar model, where this phase is a fitting parameter.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-20 through the ChPTR Lagrangian, their meanings change so as to accommodate non-perturbation effects of meson-meson interactions. To obtain realistic values for these pa-rameters, they should be extracted from a Triple-M fit to data. As a lesser alternative, herewe employ the low-energy parameters[26] values as if they resulted from data. In Fig.10, c / GeV [ s0.811.21.41.61.82 ] c / G e V [ s Figure 10: Toy Dalitz plot for Triple-M in D + → K − K + K + decay with arbitrary normalization. we show a toy Monte-Carlo Dalitz plot based on the Triple-M, where it is possible to see adestructive 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 adip, 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 thehidden isospin signatures of two-body interactions in three-body data, allowing one to dis-entangle the relative contributions of resonances a (980) and f (980). By comparing resultsfor the three-body amplitudes T J and the scattering amplitudes A ( J,I ) , it becomes clear thateven 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 channeldynamics, such as our Triple-M, fits to decay data can give rise to predictions for the K ¯ K scattering amplitudes A ( J,I ) . 21 CKNOWLEDGMENTS
This work was supported by Conselho Nacional de Desenvolvimento Cient´ıfico e Tec-nol´ogico (CNPq).
Appendix A: kinematics
Momenta are defined by D ( P ) → K − ( p ) K + ( p ) K + ( p ), with P = p + p + p . Theinvariant masses read m = ( p + p ) = ( P − p ) , (A1) m = ( p + p ) = ( P − p ) , (A2) m = ( p + p ) = ( P − p ) , (A3)and satisfy the constraint M = m + m + m − m − m − m . (A4)Their values are also limited by the boundaries of the Dalitz plot, by( m + m ) ≤ m ≤ ( M − m ) , (A5)( m + m ) ≤ m ≤ ( M − m ) , (A6)( m + m ) ≤ m ≤ ( M − m ) . (A7) Appendix B: two-meson propagators and functions Ω Expressions presented here are conventional. They are displayed for the sake of com-pleteness and rely on the the results of Ref.[25]. These integrals do not include symmetryfactors, which are accounted for in the main text. One deals with both S and P waves andthe corresponding two-meson propagators are associated with the integrals { I ab ; I µνab } = (cid:90) d (cid:96) (2 π ) { (cid:96) µ (cid:96) ν } D a D b , (B1) D a = ( (cid:96) + p/ − M a , D b = ( (cid:96) − p/ − M b , (B2)22ith p = s . Both integrals I ab and I µνab are evaluated using dimensional techniques[25]. For s ≥ ( M a + M b ) , the function I ab has the structure I ab = i π [Λ ab + Π ab ] (B3)where Λ ab is a divergent function of the renormalization scale µ and of the number ofdimensions n , which diverges in the limit n → ab ( s ) = 1 + m a + m b m a − m b ln m a m b − m a − m b s ln m a m b − √ λs ln (cid:34) s − m a − m b + √ λ m a m b (cid:35) + i π √ λs , (B4) λ = s − s ( m a + m b ) + ( m a − m b ) . (B5)which, for a = b , reduces toΠ aa ( s ) = 2 − √ λs ln (cid:34) s − m a + √ λ m a (cid:35) + i π √ λs . (B6)The tensor integral is needed for a = b only, and one has I µνaa = i π (cid:26) p µ p ν s (cid:20) Λ ppaa + 112 (cid:2) s − m x (cid:3) Π aa (cid:21) − g µν (cid:20) Λ gaa + 112 (cid:2) s − m a (cid:3) Π aa (cid:21)(cid:27) , (B7)where Λ ppaa and Λ gaa 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 hasΠ ab → − π √ λs , (B8)Π µνaa → π (cid:20) g µν − p µ p ν s (cid:21) λ / s . (B9)In the decay calculation, it is more covenient to use the functions ¯Ω, defined byΠ ab → − i ¯Ω Sab , (B10)Π µνaa → i (cid:20) g µν − p µ p ν s (cid:21) ¯Ω Paa . (B11)23hese results are related with CM momenta by¯Ω Sab = − i π Q ab √ s θ ( s − ( M a + M b ) ) , (B12)¯Ω Paa = − i π Q aa √ s θ ( s − M a ) , (B13) Q ab = 12 (cid:113) s − M a + M b ) + ( M a − M b ) /s , (B14)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 and ¯ KK intermediate states and the corresponding self-energies are denoted respectively byΣ πρ and Σ ¯ KK . In this section we consider just contributions of the former kind. The fullpropagator is given by i ∆ αβγδ = i ∆ (0) αβγδ + i ∆ (1) αβγδ + i ∆ (2) αβγδ + i ∆ (3) αβγδ + · · · (C1) i ∆ (0) αβγδ = G αβγδ (C2) i ∆ (1) αβγδ = G αβab (cid:2) − i Σ abcd (cid:3) G cdγδ (C3) i ∆ (2) αβγδ = G αβab (cid:2) − i Σ abef (cid:3) G efgh (cid:2) − i Σ ghcd (cid:3) G cdγδ (C4)The φπρ interaction is extracted from the lagrangian L ω = i g (cid:15) µνρσ ∂ λ ω λµ (cid:2) ∂ ν π − ρ + ρσ + ∂ ν π + ρ − ρσ + ∂ ν π ρ ρσ (cid:3) (C5)where ω = cos θ φ − sin θ ω is the singlet component. In the sequence, we write g (cid:15) = g cos θ . φ φ ab cd Figure 11: Intermediate πρ contribution to the φ self-energy. The self energy is given by − i Σ abcdρπ = ( k a g bµ − g aµ k b )2 [ H µλ ] ( k c g dλ − g cλ k d )2 , (C6) H µλ = (cid:2) − g (cid:15) I µλ (cid:3) , (C7) I µλ = 1 i (cid:90) d (cid:96) (2 π ) p µ p λ p − M π (cid:15) µνχη G χηωζ ( q ) (cid:15) λξωζ , (C8)24ith p = k/ − (cid:96), q = k/ (cid:96) and k = s . Using the explicit form of G χηωζ and thedefinitions D π = p − M π , D ρ = q − m ρ , we find I µλ → m ρ (cid:90) d (cid:96) (2 π ) D π D ρ (C9) × (cid:26) g µλ (cid:20) − m ρ (cid:0) M π + D π (cid:1) + 14 (cid:0) s − M π − m ρ − D π − D ρ (cid:1) (cid:21) + (cid:96) µ (cid:96) λ (cid:2) k − D ρ (cid:3)(cid:27) , where we have used the fact that terms proportional to k µ and k λ do not contribute toeq.(C6). This integral is highly divergent, but the part regarding the Kρ cut is not. Termscontaining factors D π and D ρ in the numerator do not contribute to the cut function andthe relevant integral is I µλ → m ρ (cid:90) d (cid:96) (2 π ) D π D ρ (cid:110)(cid:104) s − s (cid:0) M π + m ρ (cid:1) + (cid:0) M π − m ρ (cid:1) (cid:105) g µλ + 4 s (cid:96) µ (cid:96) λ (cid:111) . (C10)Using the definition I πρ = (cid:90) d (cid:96) (2 π ) D π D ρ (C11)and the result (cid:90) d (cid:96) (2 π ) (cid:96) µ (cid:96) λ D π D ρ = − (cid:26) k (cid:2) s − s ( M π + m ρ ) + ( M π − m ρ ) (cid:3) I πρ (cid:27) g µλ + term proportional to k µ k λ , (C12)the relevant component of I µλ becomes I µλ → (cid:26) m ρ (cid:2) s − s ( M π + m ρ ) + ( M π − m ρ ) (cid:3) I πρ (cid:27) g µλ . (C13)The on-shell contribution to eq.(C11) is given by I πρ = − π (cid:112) λ πρ s , (C14)with λ πρ = (cid:2) s − s ( M π + m ρ ) + ( M π − m ρ ) (cid:3) = 4 s Q πρ , where Q πρ is the CM three-momentum. We then have H µλ = g µλ m φ s Γ πρφ ( s ) , (C15) m φ Γ πρφ ( s ) = g (cid:15) π m ρ s / Q πρ . (C16)25umerically, Γ πρφ = 0 . × Γ φ = 0 . × . i ∆ πραβγδ = G αβγδ (C17)+ (cid:34) i m φ Γ πρφ ( s ) /sD πρφ ( s ) (cid:35) (cid:2) g dα k β k c + g cβ k α k d − g cα k β k d − g dβ k α k c (cid:3) G cdγδ , where the denominator D πρφ ( s ) is given by D πρφ = s − m φ + i m φ Γ πρφ ( s ) . (C18)In the evaluation of amplitudes involving a ¯ K ( p ) K ( p ) vertex, one encounters the product i ∆ αβγδ (cid:0) p γ p δ − p γ p δ (cid:1) = − iD πρφ ( s ) ( p α p β − p α p β ) . (C19) Appendix D: SU(3) intermediate states
In the treatment of intermediate states, it is convenient to work with Cartesian SU (3)states, which are related to charged states by | π + (cid:105) = −| i (cid:105) / √ , | π − (cid:105) = | − i (cid:105) / √ , (D1) | π (cid:105) = | (cid:105) , | η (cid:105) = | (cid:105) , (D2) | K + (cid:105) = | i (cid:105) / √ , | K − (cid:105) = −| − i (cid:105) / √ , (D3) | K (cid:105) = | i (cid:105) / √ , | ¯ K (cid:105) = | − i (cid:105) / √ . (D4)We need just two-meson intermediate states | ab (cid:105) , with the same quantum numbers as the K − K + system, which are given by | V ππ (cid:105) = (1 / √ | − (cid:105) , (D5) | V KK (cid:105) = (1 / | − − (cid:105) , (D6) | V KK (cid:105) = (1 / | − − (cid:105) , (D7) | U π (cid:105) = (1 / √ | (cid:105) , (D8) | U KK (cid:105) = (1 / | − − (cid:105) , (D9)26 S ππ (cid:105) = (1 / √ | (cid:105) , (D10) | S KK (cid:105) = (1 / | (cid:105) , (D11) | S (cid:105) = | (cid:105) . (D12)The state | K − K + (cid:105) includes a conventional phase an reads | K − K + (cid:105) = − (1 / | (4 − i i (cid:105) = − (1 / | (cid:105) − i (1 / | − (cid:105) (D13)and, therefore, (cid:104) K − K + | = ( i/ (cid:104) V KK + V KK | − (1 / (cid:104) U KK + S KK | . (D14) Appendix E: tree decay sub-amplitudes
In the evaluation of intermediate state contributions shown in diagrams of Fig.5, we needtree level contribution for the process D → a b K + , denoted by T ( J,I )(0) , for spin J and isospin I . In the results displayed below, the first terms correspond to resonances in diagrams(3A+3B), whereas those within square brackets, labeled by c , represent contact interactionsin the top vertices of diagrams 2A and 2B. Using the constant C defined in eq.(22), we have[ J, I = 1 , → (cid:104) V ab K + | T (1 , | D (cid:105) = i m − m ) Γ (1 , a b , (E1)Γ (1 , ππ = C (cid:40)(cid:34) √ G V F (cid:35) m m − m ρ + (cid:20) − √ (cid:21) c (cid:41) , (E2)Γ (1 , KK = C (cid:26)(cid:20) G V F (cid:21) m m − m ρ + (cid:20) − (cid:21) c (cid:27) (E3)[ J, I = 1 , → (cid:104) V KK K + | T (1 , | D (cid:105) = i m − m ) Γ (1 , KK , (E4)Γ (1 , KK = C (cid:40)(cid:20) G V F sin θ (cid:21) m D πρφ ( m ) + (cid:20) − (cid:21) c (cid:41) , (E5)Here, the function D πρφ is a partially dressed φ propagator, discussed in App.C, eq.(C18),associated with the partial width of the decay φ → ( ρπ + πππ ).27 J, I = 0 , → (cid:104) U ab K + | T (0 , | D (cid:105) = Γ (0 , a b , (E6)Γ (0 , π = C (cid:40)(cid:34) √ √ F (cid:35) [ − c d P · p + c m M D ] m − m a (cid:2) c d (cid:0) m − M π − M (cid:1) + 2 c m M π (cid:3) + (cid:34) − √ √ (cid:2) M D / − P · p (cid:3)(cid:35) c (cid:41) , (E7)Γ (0 , KK = C (cid:26)(cid:20) F (cid:21) [ − c d P · p + c m M D ] m − m a (cid:2) c d (cid:0) m − M K (cid:1) + 2 c m M K (cid:3) + (cid:20) − (cid:2) M D − P · p (cid:3)(cid:21) c (cid:27) , (E8)[ J, I = 0 , → (cid:104) S ab K + | T (0 , | D (cid:105) = Γ (0 , a b , (E9)Γ (0 , ππ = C (cid:40)(cid:34) √ F (cid:35) [ − ˜ c d P · p + ˜ c m M D ] m − m S (cid:2) ˜ c d (cid:0) m − M π (cid:1) + 2 ˜ c m M π (cid:3) − (cid:20) √ F (cid:21) [ − c d P · p + c m M D ] m − m So (cid:2) c d (cid:0) m − M π (cid:1) + 2 c m M π (cid:3) + (cid:34) − √ (cid:2) M D − P · p (cid:3)(cid:35) c (cid:41) , (E10)Γ (0 , KK = C (cid:26)(cid:20) F (cid:21) [ − ˜ c d P · p + ˜ c m M D ] m − m S (cid:2) ˜ c d (cid:0) m − M K (cid:1) + 2 ˜ c m M K (cid:3) + (cid:20) F (cid:21) [ − c d P · p + c m M D ] m − m So (cid:2) c d (cid:0) m − M K (cid:1) + 2 c m M K (cid:3) + (cid:20) − (cid:2) M D − P · p (cid:3)(cid:21) c (cid:27) , (E11)Γ (0 , = C (cid:26)(cid:20) F (cid:21) [ − ˜ c d P · p + ˜ c m M D ] m − m S (cid:2) ˜ c d (cid:0) m − M (cid:1) + 2 ˜ c m M (cid:3) + (cid:20) F (cid:21) [ − c d P · p + c m M D ] m − m So (cid:2) c d (cid:0) m − M (cid:1) + c m (cid:0) − M π + 16 M K (cid:1) / (cid:3) + (cid:20) − (cid:2) M D / − P · p (cid:3)(cid:21) c (cid:27) . (E12)28ith P · p = 12 (cid:2) M D + M K − m (cid:3) . (E13) Appendix F: scattering kernels
The intermediate scattering amplitudes depend on interaction kernels in the four channelsconsidered, associated with
J, I = 1 ,
0. The kernel matrix elements for the reaction c d → a b are written as (cid:104) cd | K J,I | ab (cid:105) , in terms of the states defined in App.D, and displayed below.All kernels are written as sums of NLO resonance contributions and chiral polynomials,involving both LO and NLO terms. The NLO polynomials are derived by assuming thatthe LECs are saturared by intermedate vector and scalar resonances, with masses M V and M S , respectively. The kernel matrix elements read[ J, I = 1 , → (cid:104) V ab | K (1 , | V cd (cid:105) = ( t − u ) K (1 , ab | cd ) (F1) K (1 , ππ | ππ ) = − (cid:20) G V F (cid:21) ss − m ρ + (cid:20) F (cid:21) c (F2) K (1 , ππ | KK ) = −√ (cid:20) G V F (cid:21) ss − m ρ + (cid:34) √ F (cid:35) c (F3) K (1 , KK | KK ) = − (cid:20) G V F (cid:21) ss − m ρ + (cid:20) F (cid:21) c (F4)[ J, I = 1 , → (cid:104) V ab | K (1 , | V cd (cid:105) = ( t − u ) K (1 , ab | cd ) (F5) K (1 , KK | KK ) = − (cid:20) G V sin θF (cid:21) sD πρφ + (cid:20) F (cid:21) c (F6)The function D πρφ is this expression represents a partially dressed φ propagator, discussedin App.C, eq.(C18), and accounts for the partial width of the decay φ → ( ρπ + πππ ).[ J, I = 0 , → (cid:104) U ab | K (0 , | U cd (cid:105) = K (0 , ab | cd ) (F7) K (0 , π | π = − s − m a (cid:20) F (cid:21) (cid:2) c d ( s − M π − M ) + c m M π (cid:3) + (cid:20) M π F (cid:21) c (F8)29 (0 , π | KK ) = − s − m a (cid:34) √ √ F (cid:35) (cid:2) c d ( s − M π − M ) + c m M π (cid:3) (cid:2) c d s − ( c d − c m ) 2 M K (cid:3) + (cid:20) (3 s − M K ) √ F (cid:21) c (F9) K (0 , KK | KK ) = − s − m a (cid:20) F (cid:21) (cid:2) c d s − ( c d − c m ) 2 M K (cid:3) + (cid:104) s F (cid:105) c (F10)[ J, I = 0 , → (cid:104) S ab | K (0 , | S cd (cid:105) = K (0 , ab | cd ) (F11) K (0 , ππ | ππ ) = − s − m S (cid:20) F (cid:21) (cid:2) ˜ c d s − (˜ c d − ˜ c m ) 2 M π (cid:3) − s − m So (cid:20) F (cid:21) (cid:2) c d s − ( c d − c m ) 2 M π (cid:3) + (cid:20) s − M π F (cid:21) c (F12) K (0 , ππ | KK ) = − s − m S (cid:34) √ F (cid:35) (cid:2) ˜ c d s − (˜ c d − ˜ c m ) 2 M π (cid:3) (cid:2) ˜ c d s − (˜ c d − ˜ c m ) 2 M K (cid:3) + 1 s − m So (cid:20) √ F (cid:21) (cid:2) c d s − ( c d − c m ) 2 M π (cid:3) (cid:2) c d s − ( c d − c m ) 2 M K (cid:3) + (cid:34) √ s F (cid:35) c (F13) K (0 , ππ | = − s − m S (cid:34) √ F (cid:35) (cid:2) ˜ c d s − (˜ c d − ˜ c m ) 2 M π (cid:3) (cid:2) ˜ c d s − (˜ c d − ˜ c m ) 2 M (cid:3) + 1 s − m So (cid:20) √ F (cid:21) (cid:2) c d s − ( c d − c m ) 2 M π (cid:3) (cid:2) c d ( s − M ) + c m (16 M K − M π ) / (cid:3) + (cid:34) √ M π F (cid:35) c (F14) K (0 , KK | KK ) = − s − m S (cid:20) F (cid:21) (cid:2) ˜ c d s − (˜ c d − ˜ c m ) 2 M K (cid:3) − s − m So (cid:20) F (cid:21) (cid:2) c d s − ( c d − c m ) 2 M K (cid:3) + (cid:20) s F (cid:21) c (F15) K (0 , KK | = − s − m S (cid:20) F (cid:21) (cid:2) ˜ c d s − (˜ c d − ˜ c m ) 2 M K (cid:3) (cid:2) ˜ c d s − (˜ c d − ˜ c m ) 2 M (cid:3) − s − m So (cid:20) F (cid:21) (cid:2) c d s − ( c d − c m ) 2 M K (cid:3) (cid:2) c d ( s − M ) + c m (16 M K − M π ) / (cid:3) (cid:20) s − M K F (cid:21) c (F16) K (0 , | = − s − m S (cid:20) F (cid:21) (cid:2) ˜ c d s − (˜ c d − ˜ c m ) 2 M (cid:3) − s − m So (cid:20) F (cid:21) (cid:2) c d ( s − M ) + c m (16 M K − M π ) / (cid:3) + (cid:20) − M π + 16 M K F (cid:21) c (F17) Appendix G: channel dependent decay amplitudes - full results
The tree level decay amplitudes for channel with spin J and isospin I , given in App.E,are written as (cid:104) X ab K + | T ( J,I )(0) | D (cid:105) = i m − m ) Γ (1 ,I )(0) a b → ( X = V , V )= Γ (0 ,I )(0) a b → ( X = U , S ) (G1)The full amplitudes are obtained by including all possible final state interactions, as indicatedin Figs.5 and 6. The terms involving a single meson-meson interaction read (cid:104) X ab K + | T ( J,I )(1) | D (cid:105) = i m − m ) Γ (1 ,I )(1) a b → ( X = V , V )= Γ (0 ,I )(1) a b → ( X = U , S ) (G2)with Γ ( J,I )(1) ab = (cid:88) cd M ( J,I ) ab | cd Γ ( J,I )(0) cd , (G3) M ( J,I ) ab | cd = −K ( J,I ) ab | cd (cid:2) S F ¯Ω Jcd (cid:3) . (G4)where K ( J,I ) ab | cd are the scattering kernels displayed in App.F, ¯Ω Jcd are the two-meson propagatorsgiven in App.B, and the symmetry factor S F = 1 → c (cid:54) = d and S F = 1 / → c = d . Theterms Γ ( J,I )(2) a b , containing two meson-meson interactions are constructed in a similar way fromΓ ( J,I )(1) a b , and so on.The inclusion of all possible meson-meson interactions leads to the infinite geometricseries Γ ( J,I ) ab = σ ( J,I ) ab | cd Γ ( J,I )(0) cd , (G5)31 ( J,I ) ab | cd = (cid:8) M ( J,I ) + [ M ( J,I ) ] + · · · (cid:9) ab | cd , (G6)where σ ( J,I ) is its sum, given by σ ( J,I ) = (cid:2) − M ( J,I ) (cid:3) − . (G7)Thus, decay amplitude reads formallyΓ ( J,I ) = (cid:2) − M ( J,I ) (cid:3) − Γ ( J,I )(0) . (G8)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 informallyeach ( J, I ) channel by its most prominent resonance and recall that ρ -channel: Γ (1 , =Γ (1 , ππ , Γ (1 , =Γ (1 , KK ; φ -channel: Γ (1 , =Γ (1 , KK ; a -cannel: Γ (0 , =Γ (0 , π , Γ (0 , =Γ (0 , KK ; f -channel: Γ (0 , =Γ (0 , ππ , Γ (0 , =Γ (0 , KK , Γ (0 , =Γ (0 , .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 σ = 1det[1 − M ] × [1 − M ][1 − M ] − M M M [1 − M ]+ M M M [1 − M ]+ M M M [1 − M ]+ M M [1 − M ][1 − M ] − M M M [1 − M ]+ M M M [1 − M ]+ M M M [1 − M ]+ M M [1 − M ][1 − M ] − M M det(1 − M ) = [1 − M ][1 − M ][1 − M ] − [1 − M ] M M − [1 − M ] M M − [1 − M ] M M − M M M − M M M (G9)In the K -matrix approximation, the matrix elements M are purely imaginary, owingto the presence of the two-meson propagator. The explicit functions to be used in thecalculation are displayed below. M (1 , = −K (1 , ππ | ππ [ ¯Ω Pππ / , M (1 , = −K (1 , ππ | KK [ ¯Ω PKK / ,M (1 , = −K (1 , ππ | KK [ ¯Ω Pππ / , M (1 , = −K (1 , KK | KK [ ¯Ω PKK / . (G10) M (1 , = −K (1 , KK | KK [ ¯Ω PKK / . (G11)32 (0 , = −K (0 , π | π [ ¯Ω Sπ / , M (0 , = −K (0 , π | KK [ ¯Ω SKK / ,M (0 , = −K (0 , π | KK [ ¯Ω Sπ / , M (0 , = −K (0 , KK | KK [ ¯Ω SKK / . (G12) M (0 , = −K (0 , ππ | ππ [ ¯Ω Sππ / , M (0 , = −K (0 , ππ | KK [ ¯Ω SKK / ,M (0 , = −K (0 , ππ | [ ¯Ω S / , M (0 , = −K (0 , ππ | KK [ ¯Ω Sππ / ,M (0 , = −K (0 , KK | KK [ ¯Ω SKK / , M (0 , = −K (0 , KK | [ ¯Ω S / ,M (0 , = −K (0 , ππ | [ ¯Ω Sππ / , M (0 , = −K (0 , KK | [ ¯Ω SKK / ,M (0 , = −K (0 , | [ ¯Ω S / . (G13)The factor 1 / M (0 , and M (0 , because one is using the symmetrized π ( J,I ) c | KK and correspond to the contributions denoted by [ · · · ] c in App.E. Explicitexpressions for the vector channel read T (1 , = − (cid:104) ¯Γ (1 , KK − Γ (1 , c | KK (cid:105) ( m − m ) , (G14)¯Γ (1 , KK = 1 D ρ ( m ) (cid:104) M (1 , Γ (1 , ππ + (cid:16) − M (1 , (cid:17) Γ (1 , KK (cid:105) , (G15) D ρ = (cid:104)(cid:16) − M (1 , (cid:17) (cid:16) − M (1 , (cid:17) − M (1 , M (1 , (cid:105) . (G16) T (1 , = − (cid:104) ¯Γ (1 , KK − Γ (1 , c | KK (cid:105) ( m − m ) , (G17)¯Γ (1 , KK = 1 D φ ( m ) Γ (1 , KK , (G18) D φ = (cid:8) − M (1 , (cid:9) . (G19)The function D πρφ in these results is given by eq.(C18) and corresponds to the part of the φ propagator involving πρ intermediate states.33he scalar sector yields T (0 , = − (cid:104) ¯Γ (0 , KK − Γ (0 , c | KK (cid:105) , (G20)¯Γ (0 , KK = 1 D a ( m ) (cid:104) M (0 , Γ (0 , π + (cid:16) − M (0 , (cid:17) Γ (0 , KK (cid:105) , (G21) D a = (cid:104)(cid:16) − M (0 , (cid:17) (cid:16) − M (0 , (cid:17) − M (0 , M (0 , (cid:105) , (G22) T (0 , = − (cid:104) ¯Γ (0 , KK − Γ (0 , c | KK (cid:105) , (G23)¯Γ (0 , KK = 1 D S ( m ) (cid:110)(cid:104) M (0 , (cid:16) − M (0 , (cid:17) + M (0 , M (0 , (cid:105) Γ (0 , ππ + (cid:104)(cid:16) − M (0 , (cid:17) (cid:16) − M (0 , (cid:17) − M (0 , M (0 , (cid:105) Γ (0 , KK + (cid:104) M (0 , (cid:16) − M (0 , (cid:17) + M (0 , M (0 , (cid:105) Γ (0 , (cid:111) , (G24) D S = det (cid:2) − M (0 , (cid:3) . (G25) Appendix H: channel dependent scattering amplitudes - full results
The scattering amplitudes for channels with spin J and isospin I are given by (cid:104) X ab | A | X cd (cid:105) = ( t − u ) A (1 ,I ) ab | cd → ( X = V , V ) , (cid:104) X ab | A | X cd (cid:105) = A (0 ,I ) ab | cd → ( X = U , S ) , (H1)whereas the tree approximation reads (cid:104) X ab | A (0) | X cd (cid:105) = ( t − u ) K (1 ,I ) ab | cd → ( X = V , V ) , (cid:104) X ab | A (0) | X cd (cid:105) = K (0 ,I ) ab | cd → ( X = U , S ) , (H2)with the K given in App.F. The full amplitudes are obtained by including all loop contri-butions, as indicated in Fig.6. The terms involving a single loop read A ( J,I )(1) ab | cd = (cid:88) ef M ( J,I ) ab | ef A ( J,I )(0) ef | cd (H3) M ( J,I ) ab | ef = −K ( J,I ) ab | ef (cid:2) S F ¯Ω Jef (cid:3) . (H4)34here the ¯Ω Jef are the two-meson propagators given in App.B, with the symmetry factor S F = 1 → e (cid:54) = f and S F = 1 / → e = f . The inclusion of all possible intermediate loopsgives rise to the infinite geometric series A ( J,I ) ab | cd = σ ( J,I ) ab | ef A ( J,I )(0) ef | cd , (H5) σ ( J,I ) ab | ef = (cid:8) M ( J,I ) + [ M ( J,I ) ] + · · · (cid:9) ab | ef , (H6)which is very similar to that discussed in eq.(G5). In particular, the function σ ( J,I ) ab | ef is thesame as eq.(G6) and therefore we may rely on all the developments made in App.G. Explicitexpressions for the vector scattering amplitudes read A (1 , KK | KK = 1 D ρ ( m ) (cid:104) M (1 , K (1 , ππ | KK + (cid:16) − M (1 , (cid:17) K (1 , KK | KK (cid:105) , (H7) D ρ = (cid:104)(cid:16) − M (1 , (cid:17) (cid:16) − M (1 , (cid:17) − M (1 , M (1 , (cid:105) , (H8) A (1 , KK | KK = 1 D φ ( m ) K (1 , KK | KK , (H9) D φ = (cid:8) − M (1 , (cid:9) , (H10)where the function D πρφ is given by eq.(C18).The scalar sector yields A (0 , KK | KK = 1 D a ( m ) (cid:104) M (0 , K (0 , π | KK + (cid:16) − M (0 , (cid:17) K (0 , KK | KK (cid:105) (H11) D a = (cid:104)(cid:16) − M (0 , (cid:17) (cid:16) − M (0 , (cid:17) − M (0 , M (0 , (cid:105) , (H12) A (0 , KK | KK = 1 D S ( m ) (cid:110)(cid:104) M (0 , (cid:16) − M (0 , (cid:17) + M (0 , M (0 , (cid:105) K (0 , ππ | KK + (cid:104)(cid:16) − M (0 , (cid:17) (cid:16) − M (0 , (cid:17) − M (0 , M (0 , (cid:105) K (0 , KK | KK + (cid:104) M (0 , (cid:16) − M (0 , (cid:17) + M (0 , M (0 , (cid:105) K (0 , | KK (cid:111) , (H13) D S = det (cid:0) − M (0 , (cid:1) , (H14)with det (cid:0) − M (0 , (cid:1) given by eq.(G9). 35 ppendix I: phase shifts The partial wave expansion of the amplitude, for each isospin channel, reads A IKK | KK = 32 πρ ∞ (cid:88) J =0 (2 J + 1) P J (cos θ ) f ( J,I ) KK | KK ( s ) , (I1)where f ( J,I ) KK | KK is the non-relativistic scattering amplitude and ρ = (cid:112) − M K /s . Ouramplitudes are written as A IKK | KK = A (0 ,I ) KK | KK + ( t − u ) A (1 ,I ) KK | KK + · · · (I2)In the CM, one has ( t − u ) = ( s − M K ) cos θ and write A IKK | KK = A (0 ,I ) KK | KK + [( s − M K ) cos θ ] A (1 ,I ) KK | KK + · · · = 32 πρ (cid:104) f (0 ,I ) KK | KK ( s ) + 3 cos θ f (1 ,I ) KK | KK ( s ) + · · · (cid:105) (I3)with f (0 ,I ) KK | KK = ρ π A (0 ,I ) KK | KK , (I4) f (1 ,I ) KK | KK = ρ π s A (1 ,I ) KK | KK . (I5)In non-relativistic QM, the amplitude f is usually expressed [10] in terms of phase shifts δ and inelasticity parameters η as f ( J,I ) KK | KK = 12 i (cid:104) η ( J,I ) KK | KK e i δ ( J,I ) KK | KK − (cid:105) . (I6)In order to obtain (cid:104) δ ( J,I ) KK | KK , η ( J,I ) KK | KK (cid:105) from A ( J,I ) KK | KK , one drops all subscripts and super-scripts and write f = a + i b , with a = Re [ f ] , b = Im [ f ]. Using eq. (I6), one has1 + 2 i f = [1 − b ] + 2 i a = η [cos 2 δ + i sin 2 δ ] (I7)and thus η = (cid:112) [1 − b ] + 4 a (I8)tan δ = 2 a η − b (I9)As (1 + η − b ) ≥
0, the sign of δ is determined by the factor a .36 ppendix J: model structure The Multi-Meson-Model we consider in this work assembles a number of aspects thatappear scattered in many calculations, but are normally absent in heavy meson decay anal-yses. The main unusual dynamical effects included into our model concern: i) the presenceof a LO contact interaction in the two-body kernel, as indicated in Fig.6; ii) the introductionof two resonances in the ( J = 0 , I = 0) channel, preserving unitarity; iii) consideration ofcoupled channels. With the purpose of disclosing the role played by these features in theresults, in this appendix we consider the scattering amplitude A (0 , and show its behaviorin a number of different scenarios. We begin by the simplest one, in which just the f (980)is kept, and add the other contributions gradually, as described in table I. It indicates whena particular contribution, that was previously absent, has been turned ON. scenario A B C D MMMoctet resonance f (980)] ON ON ON ON ONcontact interaction x ON ON ON ONsinglet resonance f (1370) x x ON ON ON ππ coupled channel x x x ON ON ηη coupled channel x x x x ONTable I: Systematic investigation of the relative importance of A (0 , components. We begin by considering the artificial situation in which the kaon mass is lowered to M K = 0 . f (980) to be above threshold. The amplitude is shown inFig.12 and results are rather conventional. The vertical black line indicates the position ofthe empirical K ¯ K threshold and therefore, in actual scattering, one sees only the post-peakpart of the resonance, represented by the blue curves, for scenario A, in Fig. 13. Phasesin that figure follow general theorems in quantum scattering theory. In the absence ofinelasticities, the phase of a generic scattering amplitude A coincides with the usual phaseshift δ and, at low energies the latter → q (2 L +1) , where L is the angular momentum and q is the CM linear momentum.Inspecting these figures, one learns that the inclusion of the chiral contact term (A → B) and the second resonance (B → C) produces a strong impact on results. The influence37 ,0 1,5 2,0050100150 | A | s (GeV) pha s e ( deg r ee s ) s (GeV) Figure 12: Results for | A (0 , | - the kaon mass is artificially lowered to M K = 0 . f (980); the black vertical line indicates the actual K ¯ K threshold.left: modulus, right: phase. of the coupling to the ππ intermediate channel (C → D) is also rather large, especiallyat low energies, whereas ηη coupling (D → MMM) is much less important. In Fig.14 weshow the inelasticity parameter η . One must have η = 1 for elastic amplitudes, and wewould like to draw attention to the case of scenario C, that includes two resonances and nocoupled channels. In this case, the result for η stresses that our method for dealing withmultiple resonances is indeed consistent with unitarity. When the coupling to other channelsis allowed, η ≤ ππ intermediate states becomes clear. [1] E.M. Aitala et al. (E791), Phys. Rev. Lett.
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