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UAB–FT–534 hep-ph/0606314 November 2006 Scalar and vector meson exchange in 7 V P0P0γ decays 0 → 0 2 n R. Escribano1 a J 1Grup de F´ısica Teo`rica and IFAE, Universitat Auto`noma de Barcelona, 0 E-08193 Bellaterra (Barcelona), Spain 3 2 v Abstract 4 1 The scalar contributions to the radiative decays of light vector 3 6 mesons into a pair of neutral pseudoscalars, V P0P0γ, are studied → 0 within the framework of the Linear Sigma Model. This model has 6 the advantage of incorporating not only the scalar resonances in an 0 / explicitwaybutalsotheconstraints requiredbychiralsymmetry. The h experimental data on φ π0π0γ, φ π0ηγ, ρ π0π0γ and ω p → → → → - π0π0γ are satisfactorily accommodated in our framework. Theoretical p predictions for φ K0K¯0γ, ρ π0ηγ, ω π0ηγ and the ratio e → → → h φ f γ/a γ are also given. 0 0 : → v i X r a 1 Introduction The radiative decays of the light vector mesons (V = ρ,ω,φ) into a pair of neutral pseudoscalars (P = π0,K0,η), V P0P0γ, are an excellent labo- → ratory for investigating the nature and extracting the properties of the light scalar meson resonances (S = σ,a ,f ). The reason is the following: looking 0 0 at the quantum numbers of the initial vector and those of the final photon, both with JPC = 1 , the system made of the two neutral pseudoscalars are −− mainly in a 0++ state, i.e. a scalar state, or 2++. However, the lightest tensor resonances have masses of the order of 1.2 GeV and therefore their contributions to these processes are supposed to be negligible. In addition, there is also a vector meson contribution when one of the neutral pseudoscalars and the photon are produced by the exchange of an intermediate vector meson through the decay chain V VP0 P0P0γ. Fortunately, for most of the → → processes of interest the main contribution is by far the scalar one, thus mak- ing of the study of these radiative decays a very challenging subject in order to improve our knowledge on the lightest scalar mesons. This study also complements other analysis based on central production, D and J/ψ decays, etc. [1]. Particularly interesting are the so called golden processes, namely φ π0π0γ, φ π0ηγ and ρ π0π0γ, which, as we will see, can provide → → → us with valuable information on the properties of the f (980), a (980) and 0 0 σ(600) resonances, respectively. A further motivation for the present work is the renewed interest on these radiative decays from both the theoretical [2, 3, 4] and experimental [5, 6] sides. At present, there are two experimental facilities that provide measurements on the V P0P0γ decays. One is the VEPP-2M e+e collider in − → Novosibirsk with two experimental groups SND and CMD-2, and the other is the DAΦNE φ-factory in Frascati with the KLOE Collaboration. The Russian experiment operates at different center of mass energies having the advantage of not only measuring the processes φ π0π0γ and φ π0ηγ → → but also ρ π0π0γ and other similar decays. On the contrary, the Ital- → ian experiment operates at a fixed center of mass energy around the φ mass and is only able, at least in principle, to measure the φ-decay processes but not others. However, due to its higher luminosity the statistical accuracy of DAΦNE measurements is better than in VEPP-2M. Moreover, the good performance of the KLOE detector makes also the systematic error smaller and as a consequence the DAΦNE measurements on φ-decay processes are in general more precise. For φ π0π0γ, the first measurements of this decay → have been reported by the SND and CMD-2 Collaborations. For the branching ratio they obtain B(φ π0π0γ) = (1.221 0.098 0.061) 10 4 [7] − → ± ± × and (0.92 0.08 0.06) 10 4 [8], for m > 700 MeV in the latter case. − ππ ± ± × 1 More recently, the KLOE Collaboration has measured B(φ π0π0γ) = → (1.09 0.03 0.05) 10 4 [9] in agreement with VEPP-2M results but − ± ± × with a considerably smaller error. In all the cases, the spectrum is clearly peaked at m 970 MeV, as expected from an important f (980) con- ππ 0 ≃ tribution. For φ π0ηγ, the branching ratios measured by the SND and → CMD-2 Collaborations are B(φ π0ηγ) = (8.8 1.4 0.9) 10 5 [10] − → ± ± × and (9.0 2.4 1.0) 10 5 [8], and by the KLOE Collaboration are B(φ − ± ± × → π0ηγ) = (8.51 0.51 0.57) 10 5 fromη γγ and(7.96 0.60 0.40) 10 5 − − ± ± × → ± ± × from η π+π π0 [11]. The two values are in agreement and also agree with − → those of VEPP-2M. Again, in all the cases, the observed invariant mass distribution shows a significant enhancement at large π0η invariant mass that is interpreted as a manifestation of the dominant contribution of the a γ 0 intermediate state. For ρ π0π0γ, the only existing measurements in the → literature come from the study of the e+e π0π0γ process by the SND − → and CMD-2 experiments in the energy region 0.60–0.97 GeV. From the analysis of the energy dependence of the measured cross section they obtain for the branching ratio B(ρ π0π0γ) = (4.1+1.0 0.3) 10 5 [12] and B(ρ π0π0γ) = (5.2+1.5 0→.6) 10 5 [13], in−0a.9g±reement×wit−h the older → 1.3 ± × − result B(ρ π0π0γ) −= (4.8+3.4 0.2) 10 5 [14]. These values can be explained by→means of a signifi−c1a.8nt±contrib×utio−n of the σγ intermediate state together with the well-known ωπ contribution. For ω π0π0γ, the val- → ues of the branching ratio B(ω π0π0γ) = (6.6+1.4 0.6) 10 5 [12] and B(ω π0π0γ) = (6.4+2.4 0.8)→10 5 [13] are fo−u1n.3d±in agre×emen−t with the → 2.0± × − older GAMSresult B(−ω π0π0γ) = (7.2 2.6) 10 5 [15]. Finally, anupper − → ± × limit for ω π0ηγ has been obtained: B(ω π0ηγ) < 3.3 10 5 at 90% − CL [13]. Fo→r the rest of the processes, namely→φ K0K¯0γ a×nd ρ π0ηγ, → → the branching ratios are predicted to be very small and have not been yet measured. An early attempt to explain the V P0P0γ decays was done in Ref. [16] → using the vector meson dominance (VMD) model. In this framework, the V P0P0γ decays proceed through the decay chain V VP0 P0P0γ, → → → where the intermediate vector mesons exchanged are V = ρ for (ω,φ) π0π0γ, V = ω for ρ π0π0γ, V = K 0,K¯ 0 for φ K0K¯0γ, V = ρ,ω f→or ∗ ∗ → → (ρ,ω) π0ηγ, and V = ρ,ω for φ π0ηγ. The calculated branching ratios → → are BVMD = 1.2 10 5, BVMD = 5.4 10 6, BVMD = 1.1 10 5, φ π0π0γ × − φ π0ηγ × − ρ π0π0γ × − BVMD → = 2.8 10 5, BVMD → = 2.7 10 12, BVM→D = 4 10 10 and ω π0π0γ × − φ K0K¯0γ × − ρ π0ηγ × − BV→MD = 1.6 10 7 [16→]. The first four are foun→d to be substantially ω π0ηγ × − sma→ller than the experimental results quoted before. Later on, the same authors studied the V P0P0γ decays in a Chiral Perturbation Theory → (ChPT) context enlarged to include on-shell vector mesons [17]. In this for- 2 malism, Bχ = 5.1 10 5, Bχ = 3.0 10 5, Bχ = 9.5 10 6, φ π0π0γ × − φ π0ηγ × − ρ π0π0γ × − Bχ =→2.1 10 7, Bχ →= 7.6 10 9, Bχ →= 3.9 10 11 and ω π0π0γ × − φ K0K¯0γ × − ρ π0ηγ × − Bχ→ = 1.5 10 9. Takin→gintoaccountthesechira→lcontributionstogether ω π0ηγ × − wit→h the former VMD contributions, one finally obtains BVMD+χ = 6.1 φ π0π0γ × 10 5, BVMD+χ = 3.6 10 5, BVMD+χ = 2.6 10 5, BVMD+→χ BVMD , − φ π0ηγ × − ρ π0π0γ × − ω π0π0γ ≃ ω π0π0γ BVMD+χ→ Bχ , BVMD+χ→ BVMD andBVMD+χ→ BVMD ,→which, φ K0K¯0γ ≃ φ K0K¯0γ ρ π0ηγ ≃ ρ π0ηγ ω π0ηγ ≃ ω π0ηγ for→the first four→, are still b→elow the e→xperimental →results. Ad→ditional contributions are thus certainly required and the most natural candidates are the contributions coming from the exchange of scalar resonances (as stated in Sect. 1, the contributions from tensor and higher spin resonances are negligible). Needless to say that the previous two approaches do not contain the effect of scalar resonances in an explicit way. A first model including the scalar resonances explicitly is the no structure model, where the V P0P0γ → decays proceed through the decay chain V Sγ P0P0γ, with S a scalar → → state, and the coupling VSγ is considered as pointlike. This model seems to berule outby experimental dataonφ π0π0γ decays [7]. Asecond model is → the well-known kaon loop model (see Ref. [18] and references therein), where the initial vector decays into a pair of charged pseudoscalar mesons that after the emission of a photon rescatter into a pair of neutral pseudoscalars through the exchange of scalar resonances1. In Ref. [19], it was shown for the first time the convenience of studying the φ P0P0γ processes in order → to investigate the nature of the f and a scalar mesons. In this pioneer 0 0 work, B(φ f γ ππγ) = 2.5 10 4, B(φ a γ π0ηγ) = 2.0 10 4 0 − 0 − and B(φ →(f +→a )γ K0K¯0γ×) = 1.3 10→8, for →a four-quark str×ucture 0 0 − → → × of the scalar mesons involved, and 5.4 10 5, 2.4 10 5 and 2.0 10 9, − − − × × × respectively, for a two-quark structure. In addition, the contribution of the background processes is also calculated, B(φ ρπ ππγ) = 3.0 10 5 − → → × and B(φ ρ0π0 π0ηγ) = 0.8 10 5. Due to its success in predict- − → → × ing the branching ratios as well as the mass spectra, the kaon loop model is used at VEPP-2M and DAΦNE to extract information on the properties and couplings of the f and a from the analysis of experimental data 0 0 on φ π0π0γ and φ π0ηγ, respectively [7]–[11]. A third model that → → makes use of the characteristics of the two former formalisms is developed in Refs. [20]–[26]. The previous three models include the scalar resonances ad hoc, i.e. the pseudoscalar rescattering amplitudes, where the scalar resonances are exchanged, are not chiral invariant. This problem is solved in the next two models which are based not only onthe kaon loop model but also on 1It is named the kaon loop model because it was first applied to the φ π0π0γ and φ π0ηγ processes [19]. → → 3 chiral symmetry. The first one is the Unitarized Chiral Perturbation Theory (UChPT) where the scalar resonances do not appear explicitly but are gen- erated dynamically by unitarizing the one-loop pseudoscalar amplitudes. In this approach, taking into account only the contribution from unitarized chiral loops, BUχPT = 8 10 5, BUχPT = 8.7 10 5, BUχPT = 1.5 10 5, φ π0π0γ × − φ π0ηγ × − ρ π0π0γ × − BUχPT = →4.3 10 7, BUχPT →= 5 10 8, BUχPT →= 5.4 10 11 and ω π0π0γ × − φ K0K¯0γ × − ρ π0ηγ × − BU→χPT = 2.2 10 9 are o→btained [27, 28, 29]. A→later analysis including ω π0ηγ × − mo→re refined production mechanisms gives BUχPT = (1.2 0.3) 10 4 and φ π0π0γ ± × − BUχPT = (0.6 0.2) 10 4 [30]. The seco→nd model is the Linear Sigma φ π0ηγ ± × − Mo→del (LσM), a well-defined U(3) U(3) chiral model which incorporates ab × initio both the nonet of pseudoscalar mesons together with its chiral part- ner, the scalar mesons nonet. In this context, the scalar contributions to the V P0P0γ decays are conveniently parametrized in terms of LσM ampli- → tudes compatible with the corresponding ChPT rescattering amplitudes for low dimeson invariant masses. The advantage of the LσM is to incorporate explicitly the effect of scalar meson poles while keeping the correct behaviour at low invariant masses expected from ChPT. The purpose of the paper is threefold. Firstly, to compute the φ K0K¯0γ decay where the scalar effects are known to be dominant. This pr→o- cess is interesting to study, on one side, because it allows for a direct measurement of theKK¯ couplings to the f anda mesons thus avoiding a model 0 0 dependent extraction and, onthe other side, since it could posea background problem for testing CP-violation at DAΦNE. The direct measurement of the couplings seems to be feasible in the near future with the higher luminosity expected at DAFNE-2. Having 50 fb 1, the number of expected K0K¯0γ final − state is in the range 2 8 103 [5]. The analysis of CP-violating decays in φ K0K¯0 has been p÷ropo×sed as a way to measure the ratio ǫ/ǫ [31], but ′ be→cause this means looking for a very small effect, a B(φ K0K¯0γ) & 10 6 − → will limit the precision of such a measurement. Related to the φ K0K¯0γ → decay, there are also the processes φ (f ,a )γ which are the main contri- 0 0 butions to the former through the de→cay chain φ (f + a )γ K0K¯0γ. 0 0 → → An accurate measurement of the production branching ratio and of the mass spectra for φ f (980)/a (980)γ decays can clarify the controversial nature 0 0 → of these well established scalar mesons. Secondly, to update our previous works on the decays φ π0ηγ [32] and (ρ,ω) π0π0γ [33] in view of the → → new experimental data provided by the KLOE [11], SND [12] and CMD-2 [13] Collaborations. Concerning φ π0ηγ, the update consists in modifying → the needed scalar amplitude to fulfill the chiral constraints at low energy, to incorporate the complete one-loop a propagator, and to show the ex- 0 plicit dependence of that new amplitude on the pseudoscalar mixing angle. 4 The intermediate vector meson exchange contribution is also included for comparison with the more precise forthcoming data. The new analysis of (ρ,ω) π0π0γ includes the most recent values of the mass and width of the → σ meson together with a determination ofthe size of the f effects not consid- 0 ered previously. Finally, to calculate for the fist time the scalar contributions to the ρ π0ηγ and ω π0ηγ decays in the framework of the LσM where → → the scalar meson effects are included in an explicit way. Our calculation will allow to quantify such contributions with the aim of establishing their rel- ative impact as compared to the vector meson exchange contributions. As a matter of completeness, we have also included in this work an updated version of our former analysis on the φ π0π0γ decay [34]. → The paper is divided as follows. In Sect. 2, we discuss and update the contributions from chiral loops in which our work is inspired. Sect. 3 is de- voted to the analysis of the needed scalar amplitudes in the framework of the LσM, showing that these amplitudes can be considered as improved versions of their chiral counterparts. The VMD contributions that serve to complete our study are included in Sect. 4. The final results where our predictions for the different V P0P0γ decays are discussed in turn are given in Sect. 5. → Finally, in Sect. 6 we present our conclusions and comments. A detailed calculation of the scalar amplitudes as well as the complete expressions of the one-loopscalar propagatorsand theinvariant mass distributions are included in Appendix A. 2 Chiral-loop predictions The vector meson initiated V P0P0γ decays cannot be treated in strict → Chiral Perturbation Theory (ChPT). This theory has to be extended to incorporate on-shell vector meson fields. At lowest order, this may be easily achieved by means of the (p2) ChPT Lagrangian O f2 = D U DµU +M(U +U ) , (1) 2 µ † † L 4 h i where U = exp(i√2P/f), P is the usual pseudoscalar nonet matrix2 and, at this order, M = diag(m2,m2,2m2 m2) and f = f = 92.4 MeV. The π π K − π π covariant derivative, now enlarged to include vector mesons, is defined as D U = ∂ U ieA [Q,U] ig[V ,U] with Q = diag(2/3, 1/3, 1/3) being µ µ µ µ − − − − the quarkcharge matrixandV theadditionalmatrix containing thenonet of µ 2Inordertodescribetheprocessesinvolvingthephysicalηinthefinalstate,thesinglet term η0 has been added to the conventional octet part. 5 vector meson fields. We follow the conventional normalization for the vector nonetmatrixsuchthatthediagonalelements are(ρ0+ω )/√2,( ρ0+ω )/√2 0 0 and φ , where ω = (uu¯ +dd¯)/√2 and φ = ss¯ stand for the i−deally mixed 0 0 0 states. The physical ω and φ fields are approximately ω ǫφ and φ +ǫω 0 0 0 0 − respectively, where ǫ ϕ sinϕ tanϕ = +0.059 0.004 accounts for V V V ≡ ≃ ≃ ± the ω-φ mixing angle in the flavour basis [35]. One easily observes that there is no tree-level contribution from the La- grangian (1) to the V P0P0γ amplitudes and that, at the one-loop level, → one needs to compute the set of diagrams shown in Ref. [17]. A straightfor- ward calculation of V(q ,ǫ ) P0(p)P0(p)γ(q,ǫ) leads to finite amplitudes ∗ ∗ ′ → that are conveniently parametrized in the following way: 1 χ,K = χ = − χ = Aρ π0π0γ Aω π0π0γ √2Aφ π0π0γ → → → eg = − a L(m2 ) χ , (2) 2√2π2m2 { } π0π0 ×AK+K−→π0π0 K+ 1 χ = χ = − χ = Aρ π0ηγ Aω π0ηγ √2Aφ π0ηγ → → → eg = − a L(m2 ) χ , (3) 2√2π2m2 { } π0η ×AK+K−→π0η K+ eg χ,π = − a L(m2 ) χ , (4) Aρ→π0π0γ √2π2m2 { } π0π0 ×Aπ+π−→π0π0 π+ eg χ = a L(m2 ) χ , (5) Aφ K0K¯0γ 2π2m2 { } K0K¯0 ×AK+K− K0K¯0 → K+ → where a = (ǫ ǫ)(q q) (ǫ q)(ǫ q ) makes the amplitude Lorentz and ∗ ∗ ∗ ∗ { } · · − · · gauge invariant, m2 s (p+p)2 = (q q)2 is the invariant mass of P0P0 ≡ ≡ ′ ∗ − the final dimeson system, and L(m2 ) is the loop integral function defined P0P0 as [17, 19, 28, 36, 37] L(m2 ) = 1 2 f 1 f 1 + P0P0 2(a b) − (a b)2 b − a − − (6) + a g 1 g(cid:2) (cid:0)1 (cid:1) (cid:0) (cid:1)(cid:3) (a b)2 b − a − with (cid:2) (cid:0) (cid:1) (cid:0) (cid:1)(cid:3) 2 arcsin 1 z > 1 − 2√z 4 f(z) =  h (cid:16) (cid:17)2i  1 log η+ iπ z < 1  4 η− − 4 (7) (cid:16) (cid:17)  √4z 1arcsin 1 z > 1 − 2√z 4 g(z) =  1√1 4z log (cid:16)η+ (cid:17)iπ z < 1  2 − η− − 4 (cid:16) (cid:17)  6 and η = 1(1 √1 4z), a = m2 /m2 and b = m2 /m2 . The coupling ± 2 ± − V P+ P0P0 P+ constant g comes from the strong amplitude (ρ π+π ) = √2gǫ − ∗ A → − · (p p ) with g 4.2 to agree with Γ(ρ π+π ) = 150.3 MeV. + − exp − − | | ≃ → However, for the φ decays we replace g by g where g 4.5 to agree with s s | | ≃ Γ(φ K+K ) = 2.09 MeV [35] —in the good SU(3) limit one should − exp → have g = g . These couplings are the part beyond standard ChPT which s | | | | we have fixed phenomenologically. As seen from Eqs. (2) and (4), ρ π0π0γ proceeds through a loop of → charged kaons or pions, although the kaon contribution is suppressed by a factor of 103 [17]. (ω,φ) π0π0γ proceed only by kaon loops since pion loop → contributions are suppressed by G-parity for the ω case and, in addition, by the Zweig rule for the φ case —the same happens for the φ K0K¯0γ → case. (ρ,ω,φ) π0ηγ also proceed only by kaon loops because of isospin → conservation. The four-pseudoscalar amplitudes are found to depend linearly on the variable s = m2 only3: P0P0 s χ = , (8) AK+K− π0π0 4f f → π K 1 4 χ = s m2 (cφ +√2sφ )+ AK+K− π0η 4f f − 3 K P P → π K (cid:20)(cid:18) (cid:19) 4 sφ + (2m2 +m2) cφ P , (9) 9 K π P − √2 (cid:18) (cid:19)(cid:21) s m2 χ = − π , (10) Aπ+π− π0π0 f2 → π s χ = , (11) AK+K− K0K¯0 4f2 → K where4 f = 1.22f , φ is the pseudoscalar mixing angle in the quark- K π P flavour basis5 and (cφ ,sφ ) (cosφ ,sinφ ). In this analysis we use P P P P ≡ φ = 41.8 (θ = 12.9 ), a value obtained from the ratio φ η γ/ηγ [38] P ◦ P ◦ ′ − → and consistent with a fit to different decay channels [39]. It is worth noting that the amplitude in Eq. (9) includes not only the octet contribution (η ) to 8 the physical η, as done in Ref. [17], but also the contribution from the singlet (η ). In doing so, we have enlarged the initial SU(3)-flavour symmetry to 0 3 The χ have not to be understood as four-pseudoscalar amplitudes in standard CAhPP+TP−b→uPt0aPs0terms that factorize from χ when computed using the AV P0P0γ chiral-loop framework of Ref. [17]. → 4Strictly speaking, f =f at this order in the chiral expansion. K π 5 χ = 1 s 10m2 + 1m2 when the η-η mixing angle in the octet- AK+K−→π0η √6fπ2 − 9 K 9 π ′ singlet basis is fixed to θ =arcsin( 1/3) 19.5 as done in Ref. [32]. P (cid:0) (cid:1) ◦ − ≃− 7 B(V P0P0γ) chiral loops LσM → φ π0π0γ 4.2 10 5 1.00 10 4 − − → × × φ π0ηγ 2.9 10 5 7.8 10 5 − − → × × φ K0K¯0γ 4.1 10 9 7.5 10 8 − − → × × ρ π0π0γ 1.1 10 5 2.2 10 5 − − → × × ρ π0ηγ 6.3 10 11 1.3 10 10 − − → × × ω π0π0γ 1.5 10 7 1.7 10 7 − − → × × ω π0ηγ 1.6 10 9 3.4 10 9 − − → × × Table 1: Comparison between chiral-loop and LσM predictions of branching ratios for different V P0P0γ decays. → U(3) —nonet symmetry— in order to obtain the relevant couplings for the singlet, thus incorporating η-η mixing effects. A more general and rigorous ′ extension of ChPT accounting for the effects of the pseudoscalar singlet in a perturbative way [40] would coincide at this order with our treatment of the mixing effects. Integrating the invariant mass distribution for the different V P0P0γ → decays over the whole physical region one obtains the branching ratios shown in Table 1. These results improve the predictions for these processes given in Ref. [17] where SU(3)-breaking effects were ignored and the pseudoscalar singlet contributions not studied. 3 Scalar meson exchange We now turn to the contributions coming from scalar resonance exchange. From a ChPT perspective their effects are encoded in the low energy constants of the higher order pieces of the ChPT Lagrangian. However, the effects of the scalar states should manifest in the V P0P0γ decays not → as a constant term but rather through more complex resonant amplitudes. In this section we propose scalar amplitudes which not only obey the ChPT dictates in the lowest part of the P0P0 spectra, as they must, but also gen- erate the scalar meson effects for the higher part of the spectra where the resonant poles should dominate. The Linear Sigma Model (LσM) [41, 42, 43] will be shown to be particularly appropriate for our purposes. In this context, the V P0P0γ decays → proceed through a loop of charged pseudoscalar mesons emitted by the ini- 8 tial vector that, after the emission of a photon (in a gauge-invariant way), can rescatter into JPC = 0++ pairs of neutral pseudoscalars. The scalar contributions will be conveniently parametrized in terms of LσM amplitudes compatible with ChPT for low dimeson invariant masses. As an example of our procedure, we discuss in detail the K+K K0K¯0 rescattering ampli- − → tude needed for the φ K0K¯0γ process. The remaining P+P P0P0 − → → rescattering amplitudes are written afterwards. The K+K K0K¯0 amplitude in the LσM turns out to be − → LσM = g gσK+K−gσK0K¯0 gf0K+K−gf0K0K¯0 AK+K− K0K¯0 K+K−K0K¯0 − s m2 − s m2 → − σ − f0 g2 ga0K+K−ga0K0K¯0 a±0K∓K0(¯0) , (12) − s m2 − t m2 − a0 − a0 where the various coupling constants are fixed within the model and can be expressed in terms of f , f , the masses of the pseudoscalar and scalar π K mesons involved in the process, and the scalar meson mixing angle in the flavour basis φ [44, 45]. In particular, the g coupling accounting S K+K−K0K¯0 for the constant four-pseudoscalar amplitude can be expressed in a more convenient formbyimposingthatthe (K+K K0K¯0) vanishesinthe − LσM A → soft-pion(kaon) limit (either p 0 or p 0) —see Appendix A. Then, the ′ → → amplitude (12) can be rewritten as the sum of two terms each one depending only on s and t: s m2 LσM s (s)+ t (t) = − K AK+K− K0K¯0 ≡ ALσM ALσM 4f2 → K m2 m2 m2 m2 K − σ(cφ √2sφ )2 + K − f0(sφ +√2cφ )2 S S S S × D (s) − D (s) (cid:20) σ f0 m2 m2 t m2 m2 m2 K − a0 + − K K − a0 , (13) − D (s) 2f2 D (t) a0 (cid:21) K a0 where D (s) are the S = σ,f ,a propagators —similar expressions holds S 0 0 for D (t)— and (cφ ,sφ ) (cosφ ,sinφ ). A Breit-Wigner propagator is a0 S S ≡ S S used for the σ, while for the f and a complete one-loop propagators are 0 0 preferable (see App. A) [19, 46]. A few remarks on the four-pseudoscalar amplitude in Eq. (13) and its comparison with the chiral-loop amplitude in Eq. (11) are of interest: i) for m (S = σ,f ,a ), the LσM amplitude (13) reduces to S 0 0 → ∞ s+t 2m2 − K , 2f2 K 9