Interpretation of the $\Omega_c \to \pi^+ \Omega(2012) \to \pi^+(\bar{K} \Xi)$ relative to $\Omega_c \to \pi^+\bar{K} \Xi$ from the $\Omega(2012)$ molecular perspective

We present a mechanism for $\Omega_c \to \pi^+ \Omega(2012)$ production through an external emission Cabibbo favored weak decay mode, where the $\Omega(2012)$ is dynamically generated from the interaction of $\bar{K}\Xi^*(1530)$, $\eta\Omega$, with $\bar{K}\Xi$ as the main decay channel. The $\Omega(2012)$ decays later to $\bar{K}\Xi$ in this picture, with results compatible with Belle data. The picture has as a consequence that one can evaluate the direct decay $\Omega_c^0 \to \pi^+K^- \Xi^0$ and the decay $\Omega_c^0 \to \pi^+\bar{K} \Xi^*$, $\pi^+\eta\Omega$ with direct coupling of $\bar{K}\Xi^*$ and $\eta\Omega$ to $K^- \Xi^0$. We show that, within uncertainties and using data from a recent Belle measurement, all these three channels account for about (12-20)\% of the total $\Omega_c \to \pi^+K^- \Xi^0$ decay rate. The consistency of the molecular picture with all the data is established by showing that $\Omega_c \to \Xi^0 \bar{K}^{*0} \to \Xi^0K^- \pi^+$ together with $\Omega_c \to \pi^+ \Omega^* \to \pi^+K^- \Xi^0 $ account for about 85\% of the total $\Omega_c \to \pi^+K^- \Xi^0 $.


I. INTRODUCTION
The discovery of the Ω(2012) by the Belle collaboration using e + e − annihilation [1] prompted a fast and diverse reaction from the theoretical side, looking at it from the quark model perspective as the low-lying p-wave excited J P = 3/2 − state [2][3][4][5][6][7][8], or as a molecular state stemming from the KΞ * (1530) and ηΩ coupled channels interaction [9][10][11][12][13][14][15].There is much related work to Ω excited states in the literature which can be seen in the introduction of the references [7,8,14] and the recent review on the strange baryon spectrum in Ref. [16].Related work on new hadronic states can be seen in the recent review of Ref. [17].
We adopt the molecular perspective and in the present work we show the consistency of this picture with the latest result from Belle [18].The appealing feature of the molecular picture stems from the prediction of this state from the interaction of the KΞ * (1530) and ηΩ channels in Refs.[19,20].Even using quark models, in its version of the chiral quark model, a molecular structure of this type was claimed in the works of Refs.[21,22].By using the Weinberg compositeness condition and a coherent sum of KΞ * and ηΩ, the molecular picture was also advocated in Ref. [23].
The molecular picture was challenged by the Belle measurement of the Ω(2012) → π KΞ and Ω(2012) → KΞ, showing a rate for the first decay channel of the order or less than 12% of the second one [24].Since the π KΞ channel is associated to KΞ * (1530), such a small fraction did not speak in favor of the KΞ * component.Yet, the phase space for this decay is very small and in Refs.[13,14] the molecular picture was shown to be consistent with these data.A recent reanalysis of the Belle data, choosing also different cuts to determine the KΞ * content of the Ω(2012) [25] has established a value, rather than a boundary, for the π KΞ to KΞ ratio, which is R Ξπ K Ξ K = 0.97 ± 0.24 ± 0.07. ( The paper concludes that this ratio is "consistent with the molecular interpretation for the Ω(2012) − proposed in Refs.[9,11,12,23] (of Ref. [25]), which predicts similar branching fractions for Ω(2012) − decay to Ξ(1530) K and Ξ K".Indeed, in Table 5 of Ref. [11] the ratio of Eq. ( 1) ranges from 0.87 to 0.93 for different options, well within the range of the experiment.In Refs.[13,14], the molecular picture was pushed to the limit and showed that the ratio R Ξπ K Ξ K could be made as small as 10% but not smaller than that.We shall perform calculations with the parameters of Refs.[14] and [11] and compare the results with the experimental measurements of Ref. [18].
More recently the Ω(2012) has been observed in the Ω c decay [18].Anticipating the experiment on Ω c decay, the authors of Ref. [26] studied the decay Ω c → π + Ω(2012) → π + (π KΞ) and Ω c → π + Ω(2012) → π + KΞ, and concluded that the decay mode Ω c → π + Ω(2012) → π + KΞ was a good one to observe the Ω(2012) resonance, as was proved by its experimental measurement in Ref. [18].The new Belle measurements in Ref. [18] determine Ξ − .These ratios have been studied within a picture where the Ω(2012) is assumed to be the 1P , J P = 3/2 − excitation of the Ω in the quark model in Ref. [8] and consistency of the picture with data is found.The conclusion is based on the consistency with the Belle measurements of Ref. [18], Eqs. ( 2), ( 3), but the ratio of Eq. ( 1) on which the Belle collaboration establishes the support for the molecular picture in Ref. [25] is not discussed in Ref. [8].
In the present work we wish to look at these data from the molecular perspective.What we do is to make a picture for the Ω c → π + Ω(2012) → π + KΞ.Then we show that, associated to this signal within the molecular picture there is a background for Ω c → π + KΞ, without going through the resonance, which does not belong to Ω c → K * Ξ → π + KΞ reported in the PDG [27] nor to Ω c → π + Ω * → π + KΞ, which is evaluated in Ref. [8] within the constituent quark model.The data put indeed a strong constraint on the background from these extra sources, which could be very large depending on the picture for the Ω(2012).What we find is that the new contribution for the Ω c → π + KΞ background is compatible with present data considering the known Ω c → K * Ξ and Ω c → π + Ω * sources.

II. FORMALISM
In Ref. [18] the Belle collaboration reported on the ratios Let us see how this proceeds at the quark level.In Fig. 1 we show the external emission mechanism for Ω c → π + sss.
The picture of Fig. 1 offers a mechanism for π + Ω, which is one important decay channel.Our decay channels require three particles in the final state, hence some hadronization must occur to produce the extra particle.
In the molecular picture for Ω(2012) one must create the building blocks KΞ * and ηΩ.Both of them have negative parity in s-wave which we consider.The one body mechanism of the picture of Fig. 1 leaves the two lower s-quarks in the figure as spectators.We must hadronize the final state by introducing a qq pair with vacuum quantum numbers, and hence positive parity.Since parity is conserved after the weak vertex in Fig. 1, the upper s quark in the final state must be produced in L = 1 to have negative parity.Since finally we shall have the s quark in the K in the ground state, the hadronization must involve this quark.In the 3 P 0 picture of hadronization [28,29], the qq pair is created with L = 1 and S = 1.Then the L = 1 from the original s quark combines with the second L = 1 state to give zero orbital angular momentum, suited to the K and Ξ (or η, Ω) of the final state.The hadronization is depicted in Fig.  2: Hadronization of an ss pair.
After the hadronization the sss state becomes where P is the matrix of q q written in terms of pseudoscalar mesons where the η − η mixing of Ref. [30] is used.Thus, we obtain where we have neglected the η state which plays no role in the problem.Next, we must find the overlap of the three quark states with the physical states Ξ 0 , Ξ − , Ξ * 0 , Ξ * − , Ω.In terms of quarks these states are written as, where φ MS , φ MA are the mixed symmetric, mixed antisymmetric flavor wave functions and χ MS , χ MA the mixed symmetric, mixed antisymmetric spin wave functions [31].We have [32][33][34] 1 We shall only need the S z = 1/2 as we shall see below.We shall start from Ω c with S z = 3/2, ↑↑↑, and then we only need the Ξ * 0 , Ξ * − , Ω with S z = 3/2 and S z = 1/2, which we give below.For the Ω c state we single out the heavy quark and symmetrize the light quark pair [35,36].Thus, we have We take into account the weak interaction vertices.The W π + vertex is of the type [37,38] Together with the W q q coupling of the type [39,40] this leads to an interaction in the nonrelativistic approximation for the quark spins of the type [41] V in the Ω c rest frame [41] 2 , where q 0 is the energy of the π + and q its three-momentum, and C is a constant tied to the weak couplings and matrix elements of the radial wave functions of the baryon involved in terms of quarks.We shall find a way to eliminate this constant in the analysis of our results by constructing appropriate ratios.From Eq. ( 14) we can see that with the q 0 operator we can make transition from Ω c ↑↑↑ to Ξ * or Ω in S z = 3/2 and with σ • q to Ξ (S z = 1/2) or Ξ * , Ω in (S z = 3/2, 1/2).This assumes that the third component of the spin of the s quark to the right of the W cs vertex in Fig. 2 is the same as the one of the quark q from the hadronization, which together with the ss spectator quarks will make the Ξ, Ξ * or Ω final baryon.While this is intuitive since the K made from sq carries no spin nor angular momentum in the s-wave mode studied, we present a formal derivation in Appendix A.
This said, the derivation of the matrix element of the q 0 + σ • q operator between the Ω c original state and the final is straightforward.Defining with and taking into account the overlap of the states uss, dss, sss of Eq. ( 6) with the flavor wave functions of the baryon states of Eqs. ( 7) and (11) and the spin matrix operators of the q 0 + σ • q operator, we find the weight W for the matrix elements of Ω c ↑↑↑ going to π + and the different final states as with (q 0 , q ) the four momentum of the π + .Note that in our notation we have the isospin doublets ( K0 , −K − ), (Ξ 0 , −Ξ − ), and (Ξ * 0 , Ξ * − ), so the I = 0 states in the order of Ref. [20] are There is a global arbitrary normalization, which is the same for all channels and which disappears in the ratios that we evaluate.The last two equations of Eq. ( 18) allow the direct transition of Ω c → π + K − Ξ 0 , π + K0 Ξ − without going through the Ω(2012) resonance.They contribute to the denominators in Eqs. ( 2) and (3).On the other hand, the process Ω c → π + Ω(2012) → π + KΞ proceeds via the mechanism of Fig. 3, where one produces π + KΞ * , π + ηΩ, the KΞ * and ηΩ couple to the Ω(2012) resonance which later decay into KΞ.
The ratio of the width of the mechanism of Fig. 3 to the background of Ω c → π + KΞ coming from all different sources should be compared to the ratios of Eqs. ( 2), (3).We should note that in the case of Eq. ( 3) the final state is π + K0 Ξ − .We can also have K0 π + Ξ − , K0 π + Ξ * − production via internal emission as depicted in Fig. 4. We can have K0 π + Ξ − production for the background through this mechanism although suppressed by a color factor around 1 3 . The K0 π + Ξ * − production can also contribute to the Ω(2012) production, but its projection into I = 0 introduces an extra 1 √ 2 factor in the amplitude, but the mechanism has a different structure as to external emission, and in the incoherent sum of external and internal emission we could expect corrections of the order of 6%, We can accept larger uncertainties from this source, but in any case what is clear is that if we look at the π + K − Ξ 0 final state production we just have external emission and the comparison of Ω(2012) signal and background production is much cleaner.The former argument also hints at a smaller ratio of signal to background for π + K0 Ξ − production than π + K − Ξ 0 production, as seen in Eqs. ( 2), (3), although the ratios could be compatible within errors.For all these reasons we shall only look at the ratio of Eq. ( 2). A. The Ωc → π + Ω(2012) The process is depicted in Fig. 3 and the amplitude for the process is given considering Eqs. ( 19), ( 20) by where the weights W are given in Eq. ( 18), M inv stands for the invariant mass of the K − Ξ 0 final state, and R, M R , Γ R , g R, KΞ * , g R,ηΩ , g R, KΞ stand for the Ω(2012) resonance, its mass, width and couplings to KΞ * , ηΩ and KΞ.The functions G KΞ * and G ηΩ are the loop functions of KΞ * or ηΩ intermediate states, for which we use the cutoff regularized functions of Ref. [14].The couplings of the resonance to the different channels are also taken from the work of Ref. [14].
Next, we have to evaluate |t| 2 and sum over the polarizations of the Ω(2012).As we mentioned above, starting from Ω c ↑↑↑ we can reach Ω(2012) with S z = 3/2 and 1/2.Then, considering the weights of Eq. ( 18) we get We can take the z direction in the π + direction, and when integrating over the π + angles we shall get the angle averaged values of q 2 z , q 0 q z and |q Thus, The K − Ξ 0 mass distribution for the Ω c decay is then given by with B. Background for direct Ωc → π + K − Ξ 0 In Eq. ( 18) we also had the weights for Ω c → π + K − Ξ 0 direct transition.Following the same argumentation as before we obtain also dΓ for this direct transition, dΓ with p π , pK − as before and C. Background for Ωc → π + K − Ξ 0 through intermediate KΞ * and ηΩ states In Ref. [14] we had as coupled channels to obtain the Ω(2012), KΞ * , ηΩ and KΞ, the latter one in D-wave since Ω(2012) has J P = 3 2 − .A fit to the data of Ref. [24] in Ref. [14] rendered the coupling of KΞ * and ηΩ to KΞ, all of them in I = 0, in terms of the parameters α and β: with q K the K − momentum in the K − Ξ 0 rest frame (p K of Eq. ( 26)).
Then we have an additional mechanism to get background for Ω c → π + K − Ξ 0 as depicted in Fig. 5.The amplitude for Fig. 5 reads By using the same arguments as before and the values of W of Eq. ( 18), we obtain a new source of background for Ω c → π + K − Ξ 0 given by dΓ where, as in Eq. ( 22), we have summed over the S z = 3/2 and 1/2 components of KΞ * and ηΩ.

III. RESULTS
In order to estimate uncertainties we use three sets of parameters q max , that regularizes the G i loop functions, α, β, and the corresponding g R,i couplings and Γ R obtained in Ref. [14].The values of the parameters are tabulated in Table I 3 .We also use the set of parameters of Ref. [11] which provides a ratio of R Ξπ  Next, we define ratios in which the unknown constant C of the transition amplitude in Eq. ( 14) cancels where Γ bac , Γ bac are obtained integrating the differential distributions of Eqs. ( 27), (31) and Γ signal is the integral of the mass distribution of Eq. ( 24) for the Ω c → π + Ω(2012) → π + K − Ξ 0 process.Then we get the results for R 1 , R 2 shown in Tables II and III.
From Eqs. (33), (34) we obtain and the global π + K − Ξ 0 production branching fraction evaluated, equivalent to the magnitude of Eq. ( 2), but from the sources tied to the Ω(2012) production only, will be Note that the ratios R 1 , R 2 can be obtained in our approach as absolute numbers, since the unknown constant C cancelled in the ratios, but the magnitude of Eq. ( 36) implicitly includes the factor C and we must find some experimental magnitude to eliminate it.For this purpose we proceed as follows.If we divide Eq. ( 36) by the experimental Our aim is to show that there is no inconsistency of the measured ratio of Eq. ( 2) and our hypothesis for the Ω(2012) formed from the interaction of the KΞ * , ηΩ with the main decay channel as KΞ.Then we take for the ratio measured by Belle [24] (here is where the unknown constant C is implicitly evaluated) and find which ranges from (12.2 − 15.7)%, (39) with 38% uncertainty from Eq. (38) summing errors in quadrature.This means that we obtain with the three modes evaluated to produce π + K − Ξ 0 only about (12 − 16)% of the total production of π + K − Ξ 0 , with a maximum of about 20% counting uncertainties.This is a consequence of the molecular assumption made, with the properties derived from it.Note that both R 1 , R 2 depend on B signal according to Eqs. ( 33), (34) and we could have obtained in principle a fraction of π + K − Ξ 0 from these sources bigger than the experimental one, showing a clear contradiction.In what follows we show that, indeed, we should not get a bigger fraction of π + K − Ξ 0 production from our sources because there are two other sources not counted by us, which account for more than 80% of the total π + K − Ξ 0 production.One of the sources not included by us in [27]).We can see that where we have added relative errors in quadrature.This means that about 60% of the Ω c → Ξ 0 K − π + decay comes from the Ξ 0 K * 0 → Ξ 0 K − π + decay which is not a part of our calculation (we cannot relate it to the sources studied in our formalism).There is another source of Ξ 0 K − π + production.With the mechanism of Fig. 1 one can produce π + Ω * where Ω * are any kind of excited Ω(sss) states.The posterior decay of Ω * → Ξ 0 K − would be another source of Ξ 0 K − π + production not tied to the mechanism evaluated by us.There is no information on such decay in the scarce information on Ω * states in the PDG, but we can rely upon a theoretical quark model calculation for an approximate estimate of such contributions.The answer to this question is provided in Ref. [8].There we find Summing all these contributions we find a fraction of If we sum this fraction to the one of Eq. ( 40) we already find 100% of the Ω c → π + K − Ξ 0 decay.Counting errors, without considering the unknown ones from Eqs. ( 41), ( 42), ( 43), ( 44), we find a total fraction of 0.82 ± 0.16 66 − 98%.
The margin should be bigger considering uncertainties from the quark model, and this result matches well with the (12 − 16)% with 38% uncertainty fraction obtained in Eq. ( 39) from sources related to the Ω(2012) considered as a molecular state.We should stress that, should we have obtained a much smaller B signal than the one we obtained, the ratios obtained for R 1 and R 2 would have been much bigger, such as to make inconsistent the molecular picture of the Ω(2012).We certainly cannot see the results found as a proof of the molecular nature of the Ω(2012), but we see that the picture is consistent with this valuable experimental data through the nontrivial test done.

IV. CONCLUSIONS
We have carried out calculations of the Ω c → π + Ω(2012) → π + K − Ξ 0 decay from the perspective that the Ω(2012) is a molecular state build up from the KΞ * (1530), ηΩ channels which decay mostly in the KΞ channel.The process proceeds via external emission with a Cabibbo favoured mode, resulting in π + emission and the formation of a baryon system with sss quarks.We allow for the hadronization of an ss pair, leading to KΞ * , ηΩ, which interact giving rise to the Ω(2012) which later decays to K − Ξ 0 .However, the same operator from the weak transition and hadronization can lead to direct K − Ξ 0 production, hence contributing to a background process in Ω c → π + K − Ξ 0 decay.This means that we can relate these processes.At the same time, the analysis of the Ω(2012) in Ref. [14] with three channels KΞ * , ηΩ and KΞ leads to the coupling of KΞ * , ηΩ to KΞ such that when we produce the KΞ * , ηΩ in the first step, we can go from these states to KΞ without passing through the resonance, providing another source of background for Ω c → π + K − Ξ 0 .The three processes are tied to the properties of the Ω(2012) as a molecular state, consistent with the Belle data [24] and their ratios are very sensitive to these properties.We find that counting uncertainties, the three sources of Ω c → π + K − Ξ 0 decay evaluated account for about (12 − 20)% of the total Ω c → π + K − Ξ 0 branching ratio, but this quantity could be significantly larger should we have a different molecular picture with different couplings to the building channels.The consistency with experiment of this obtained fraction is established when we show that the Ω c → Ξ 0 K * 0 → Ξ 0 K − π + together with Ω c → π + Ω * → π + K − Ξ 0 decay channels account for about 85% of the total Ω c → π + K − Ξ 0 decay, as obtained in a quark model.We could show that the ratio of Eq. ( 2) from the Belle experiment [18] has been very useful to establish a new test for the nature of the Ω(2012) and look forward to other measurements that can put further constraints to the different pictures of the Ω(2012).

K
in very good agreement with the new Belle data [25].

TABLE II :
Results for R1, R2, R1 + R2 with different sets of parameters shown in TableI.