The D + s → K + π + π − reaction and the scalar f 0 (500) , f 0 (980) and K ∗ 0 (700) resonances

We develop a model to reproduce the mass distributions of pairs of mesons in the Cabibbo-suppressed D + s → K + π + π − decay. The largest contributions to the process comes from the D + s → K + ρ 0 and D + s → K ∗ 0 π + decay modes, but the D + s → K ∗ 0 (1430) π + and D + s → K + f 0 (1370) modes also play a moderate role and all of them are introduced empirically. Instead, the contribution of the f 0 (500), f 0 (980) and K ∗ 0 (700) resonances is introduced dynamically by looking at the decay modes at the quark level, hadronizing q ¯ q pairs to give two mesons, and allowing these mesons to interact to ﬁnally produce the K + π + π − ﬁnal state. These last three modes are correlated by means of only one parameter. We obtain a fair reproduction of the experimental data for the three mass distributions as well as the relative weight of the three light scalar mesons, which we see as further support for the nature of these states as dynamically generated from the interaction of pseudoscalar mesons.


I. INTRODUCTION
The hadronic weak decays of D, D s mesons are an excellent source of information on the interaction of hadrons [1,2].In particular, decays of D, D s into three mesons allow one to study the interaction of pairs of particles at different invariant masses and observe hadronic resonances.One case which has attracted much attention is the decay with one kaon in the final state, D → Kππ(η) [3][4][5][6][7], and the simultaneous study of the D + → K − π + π + and D + → K 0 S π 0 π + reactions is done in [8].Related work on D → KKπ is also addressed in [9][10][11][12], D + s → π + K 0 S K 0 S in [13] and D → KKK is also addressed in [14,15].In the present work we study the singly Cabibbo-suppressed D s → K + π + π − decay.The reaction has been measured in [16] and more recently, with better statistics, in [17].In addition to the dominant mode D + s → K + ρ, ρ → π + π − and D + s → K * (892) 0 π + , K * (892) 0 → K + π − , the experiment finds traces of the f 0 (500), f 0 (980) and f 0 (1370) resonances.No theoretical work on this particular channel is available to the best of our knowledge, and we wish to address this problem here.The procedure followed follows the line of related work in which the dominant weak decay modes at the quark model are investigated and hadronization of quark pairs is considered to convert the first step weak decay into the production of three mesons.After this first step, the different meson pairs are allowed to interact to lead to the final observed channel [4,[9][10][11][12].In the process of interaction some resonances are generated, and in the light meson sector this task is undertaken using the chiral unitary approach [18].We shall see that in this process we generate the f 0 (500), f 0 (980) and K * 0 (700) scalar resonances from the ππ, K K and Kη, Kπ interaction respectively, providing support to the dynamical generation of these resonances.

II. FORMALISM
In the D + s → K + π + π − reaction one can guess that both the K * 0 decaying to K + π − and the ρ 0 decaying to π + π − are formed.We shall see that this is the case.To understand the process we look at the D + s decay at the quark level which shows that the process proceeds via a Cabibbo-suppressed mechanism.Instead of having the Cabibbo-allowed W + → cs, u d vertices, we have now W + → c d, us as one can see in Figs.1-6.many diagrams that we have obtained leading to this singly Cabibbo-suppressed decay mode.We shall give a weight to the different diagrams according to the following scheme: 1) weight α for K * 0 production 2) weight α h, the h factor accounting for the mechanism of hadronization 3) weight γ for ρ 0 production 4) weight γ h since it involves an extra hadronization as in the case of 2) 5) weight α h since it has the same topology as the case of 2) 6) weight γ h since it involves an extra hadronization with respect to case of 3) Next we proceed to look in detail into the different hadronization processes.In Figs. 2, 4 we have the hadronization of the ss component and we add a qq pair with the quantum numbers of the vaccum.By writing the q i qj matrix of SU(3) in terms of the pseudoscalar mesons we have where we have taken the standard η and η mixing of Ref. [19] and neglected the η which does not play a role in the generation of the resonances that we shall discuss.Then In Fig. 5 we have the hadronization of ds as In Fig. 6 we have the hadronization of d d as The 4) and 6) cases correspond to the same topology and have the same weight and can be summed into We can see that in Fig. 6 we already obtain K + π − π + at the tree level, but we also get other intermediate states that upon rescattering lead to the same state, as depicted in Fig. 7 Given Eqs. ( 2), ( 3), ( 4), ( 5), we can write the production matrix for each mechanism of Figs. 1 to 6.
with i = K + K − , K 0 K0 , ηη, where the weights W i are given by means of Eq. (2) as In the case of the two identical particles ηη we have considered the factor 2 for the two particles and 1 √ 2 because we work with the unitary normalization where the state is normalized as 1 √ 2 ηη to avoid double counting in the G loop function.In Eq. ( 6) G(M inv ) is the diagonal loop function of two intermediate pseudoscalar mesons, which we regularize with a cut off of q max = 600 MeV, and t i,j are the transition scattering matrices for the six pseudoscalar pairs, π + π − , π 0 π 0 , K + K − , K 0 K0 , ηη, π 0 η obtained in a coupled channel formalism as with the transition potentials V ij obtained from [20].For the π − K + and π 0 K 0 interaction we use Eq. ( 7) with the coupled channels π − K + , π 0 K 0 , ηK 0 with the transition potentials of Ref. [4,6].Similarly, we obtain with i = π + π − , π 0 π 0 , K + K − , K 0 K0 , ηη, π 0 η and with i = π − K + , π 0 K 0 and Note that in Eqs.(8), we have the term 1 in the amplitude, which correspond to the tree level K + π + π − production.This term is absent in Eq. ( 6) since the primary production does not contain K + π + π − .

A. vector production
We look now to the mechanisms of Figs. 1 and 3 for K * 0 and ρ 0 production respectively.We show these processes in Figs. 8, 9 respectively, including the K * 0 and ρ 0 decays.
In order to obtain the K * 0 → K + π − and ρ 0 → π + π − vertices we use the standard Lagrangian with indicating the SU(3) trace and V µ given by The vertex D + s → K + ρ 0 has the same structure and we take Following the lines detailed in Ref. [6] we can write the amplitude in terms of the invariant masses s 12 , 13 , s 23 for the particles in the order π − (1), π + (2), K + (3) as and similarly and we use the relationship

B. Higher mass scalar resonances
Following the analysis of the experimental work [17], we also allow the contribution of two scalar resonances, the f 0 (1370) and K * 0 (1430).These resonances do not come from pseudoscalar-pseudoscalar interaction but, instead, they are obtained from vector-vector interaction, together with many other states with J = 0, 1, 2 [22,23].Yet, these are the two resonances which are obtained with less precision in [22,23], with 150 − 200 MeV difference in the mass with respect to the experiment, hence here we do not try to obtain them in the way we have dealt with the light scalar resonances and introduce them empirically with weights as free parameters.
The mechanisms for the production of these resonances are depicted in Figs. 10, 11, and their amplitudes can be parameterized by means of for K * 0 (1430) production and for f 0 (1370) production, where the factor m 2 Ds is introduced to have β, δ dimensionless.We take the masses and widths from the PDG [21], M = 1425 MeV, Γ = 270 MeV for K * 0 (1430) meson, and M = 1370 MeV, Γ = 350 MeV for f 0 (1370) meson.

III. RESULTS
We conduct a best fit to the three invariant mass distributions of Ref. [17] and we get the values for the parameters α = 14.67 , γ = 10.75 , h = 6.86 , β = −33.23 , δ = −58.84 The results for the mass distributions are shown in Fig. 12.The agreement with the data is fair and the K * 0 , ρ 0 peaks are prominent in the reaction.The K * 0 (1430) contribution is observed as a soft peak in K + π − mass spectrum of Fig. 12 around 1400 MeV and the f 0 (1370), which has a very large width, shows up in the region around 1200 − 1400 MeV, where otherwise there would be strength missing.On the other hand, the f 0 (500), f 0 (980), K * 0 (700) have been introduced dynamically here, through the interaction of pseudoscalar pairs, and one can see their contribution in the low energy part of the M inv (π + π − ) spectrum of Fig. 12, the sharp peak around 980 MeV in the same spectrum and the low energy part of the K + π − mass spectrum in the same figure, respectively.

FIG. 12: Invariant mass distributions
Technically, from the amplitude t (2) , since the K K, ηη come from the ss hadronization that has I = 0, we can expect to obtain a contribution from the f 0 (980), which couples strongly to K K but weakly to ππ, and to a minor extend a contribution from the f 0 (500) which couples to ηη but not strongly.On the other hand, from t (4+6) we get contribution both from f 0 (980) and f 0 (500) since now we have ππ intermediate states which couple strongly to f 0 (500).Furthermore, from t (5) we get a contribution from the scalar K * 0 (700) resonance which couples to Kπ.We should note that all these three resonance contributions have been included by means of a unique parameter, h, and the fair reproduction of the spectra obtained supports that these contributions are indeed correlated and our mechanism for production of these resonances produces a fair reproduction of their relative weight in these mass distributions.
It is interesting to see what distributions we obtain if we keep only the K * 0 or the ρ 0 terms.This is shown in Figs. 13 and 14.We observe in Fig. 13 that much of the strength in the K + π − mass distribution outside the K * 0 peak is not accounted for.On the other hand it produces a two peak structure in the K + π + distribution and also in the π + π − one.These peaks are well known as reflections in some channels of resonances in another channel and should not be confused with signals of a new resonance.In Fig. 14 we repeat the exercise putting only the contribution of the ρ.Once again, we show that much strength outside the ρ region is not accounted for and, similarly to the case of the K * resonance alone, the ρ peak generate reflections with two peaks, both in the K + π + and K + π − mass distributions.

IV. CONCLUSIONS
We have performed a fit to the three mass distributions of the D + s → K + π + π − reaction in which we have introduced empirically the contributions of the main decay channels, D + s → K + ρ 0 and D + s → K * 0 π 0 .In addition, we also introduce empirically two other contributions from channels of smaller relevance, the D + s → π + K * 0 (1430) and D + s → K + f 0 (1370).The novelty of our approach is that we introduce the contribution from the f 0 (500), f 0 (980) and K * 0 (700) resonances from the perspective that they are dynamically generated resonances, stemming the interaction of pseudoscalar mesons.For this purpose we look at the decay channels at the quark level, perform a hadronization of q q pairs to produce three pseudoscalar mesons in the final state, and allow these mesons to interact by pairs to produce the desired final state.In this way the three light scalar mesons are introduced dynamically and their contributions are correlated by means of just one free parameter.We obtain a fair reproduction of the π + π − , K + π − and K + π + mass distributions and the relative weight of the contribution of the light scalar mesons also agrees with the measured spectra.We see these features as extra support for the dynamically generated origin of these resonances, stemming from the interaction of pseudoscalar mesons, which in the present case is considered using the chiral unitary approach.