$\boldsymbol{\Omega_b^- \to (\Xi_c^+ \, K^-) \, \pi^-}$ and the $\boldsymbol{\Omega_c}$ states

We study the weak decay $\Omega_b^- \to (\Xi_c^+ \, K^-) \, \pi^-$, in view of the narrow $\Omega_c$ states recently measured by the LHCb collaboration and later confirmed by the Belle collaboration. The $\Omega_c(3050)$ and $\Omega_c(3090)$ are described as meson-baryon molecular states, using an extension of the local hidden gauge approach in coupled channels. We investigate the $\Xi D$, $\Xi_c \bar K$ and $\Xi_c^\prime \bar K$ invariant mass distributions making predictions that could be confronted with future experiments, providing useful information that could help determine the quantum numbers and nature of these states.


I. INTRODUCTION
The recent discovery of five narrow Ω c states by the LHCb collaboration [1] in pp collisions, also recently confirmed by the Belle collaboration [2] in e + e − collisions, motivated an increasing amount of theoretical work with different proposals for their structure. In particular, the correct assignment of quantum numbers J P remains an open question and it could be the key to understand the nature of these states.
Predictions using quark models for such states and related ones were done in Refs. [3][4][5][6][7][8][9][10][11][12][13][14][15]. Quark models mostly propose a diquark-quark structure (ss)c, where it is assumed that the diquark (ss) is in the ground state 1S bound by the attractive antitriplet color structure3; and in this case the identical flavor content of the diquark implies that it has spin S ss = 1, which then can combine with the spin of the c quark S c and with orbital momentum L of the diquark (ss) relative to the c quark. One of the typical assignments in the literature consists in the five 1P possibilities of combining angular momentum when L = 1, which yields two states with J P = 1/2 − , two with 3/2 − and one with 5/2 − , that split due to spin-dependent forces and could correspond to the five narrow Ω c states: Ω c (3000), Ω c (3050), Ω c (3066), Ω c (3090) and Ω c (3119), in the same order of increasing J. Alternative assignments also include the possibility of a radial excitation of the diquark (ss) relative to the c quark, where the last two Ω c states would be 2S excitations with quantum numbers J P = 1/2 + and 3/2 + , respectively, whereas the first three Ω c states would be the last 1P states with quantum numbers J P = 3/2 − and 5/2 − , in that order. Interesting discussions on this quark model picture can be found in Ref. [16] and similar approaches in Refs. [17][18][19][20].
On the other hand, some of these states could actually be pentaquark-like molecules, dynamically generated from meson-baryon interactions in coupled channels with charm C = 1, strangeness S = −2 and isospin I = 0. Predictions in the molecular picture using coupled channels of meson-baryon interactions were done in Refs. [39][40][41]. In this picture the interaction in S-wave of baryons with spin-parity J P = 1/2 + or J P = 3/2 + with pseudoscalar mesons leads to meson-baryon systems with J P = 1/2 − and J P = 3/2 − , respectively. Channels with vector mesons instead of pseudoscalars can also be included resulting in J P = 1/2 − , 3/2 − and 5/2 − . However, most of the recent works adopting this picture manage to relate two or three of the new Ω c states to meson-baryon systems with J P = 1/2 − and J P = 3/2 − , dominated by the pseudoscalar-baryon channels.
In Ref. [41] an SU(6) lsf × HQSS model (HQSS stands for heavy quark spin symmetry) extending the Weinberg-Tomozawa πN interaction was employed to make a systematic study of many possible meson-baryon systems. In Ref. [42] the renormalization scheme of Ref. [41] was reviewed, performing an update of the results of the C = 1, S = −2 and I = 0 sector in view of the new experimental data. The updated results indicate that one can relate the Ω c (3000) to a state with J P = 1/2 − and the Ω c (3050) to another state with J P = 3/2 − , with hints that the Ω c (3090) or Ω c (3119) could also have J P = 1/2 − . In Ref. [40] the molecular picture was developed using SU(4) symmetry to extend the interaction described by vector meson exchange in the local hidden gauge approach. This work was also reviewed under the light of the new experimental data and an updated study was made in Ref. [43], where it was shown that the Ω c (3050) and Ω c (3090) can be both related to meson-baryon resonances with J P = 1/2 − , stemming from pseudoscalar-baryon(1/2 + ) interaction.
A similar approach which also describes the meson-baryon interaction through vector meson exchange was recently developed in Ref. [44], using an extension of the local hidden gauge approach [45][46][47][48][49] and taking into account the spin-flavor wave functions of the baryons.
In the present work we will follow the description of the Ω c states of Ref. [44]; extensive discussions on the methods and results can be found there, as well as predictions of higher energy states of meson-baryon nature. In this framework, which will be discussed in more detail in the next section, the heavy quarks are treated as spectators, what implies that the dominant terms of the interaction come from light vector exchange, therefore the interaction is consistent with SU(3) chiral Lagrangians and heavy quark spin symmetry is preserved for the dominant terms in the (1/m Q ) counting [50][51][52]. A remarkable agreement of both masses and widths of the Ω c (3050) and Ω c (3090) was obtained from the pseudoscalarbaryon(1/2 + ) interaction, in accordance with results of Ref. [43]; also an extra sector of pseudoscalar-baryon(3/2 + ) could be related to the Ω c (3119), therefore assigning to it the quantum numbers J P = 3/2 − .
Other works on the molecular picture followed [53][54][55]. In Ref. [53] the authors propose that the broad structure found around 3188 MeV [1,2] could be related with a molecular ΞD state due to the proximity of its threshold around 3185 MeV. As we shall see in the next section, in our approach [44] the molecular state dominated by the ΞD channel corresponds to the Ω c (3090).
It is clear that any constraint on the quantum numbers of these new states is essential to discriminate between the different theoretical models. It is possible − or if we dare to say, likely − that some of the new Ω c states are 1P and/or 2S excitations as discussed in the quark model picture, and some are dynamically generated meson-baryon molecules.
However, answering these questions would automatically open new ones, since in both pictures, choosing one assignment over the other implies that there should be other partner states, which could be below or above the energy range searched experimentally; some of them might be detectable only in other decay channels. Understanding the properties of the observed states, like the narrowness, the couplings to one channel or another, the presence or absence of certain predicted states, and so on, require more information, not just on the quantum numbers, but also the measurement of such states in different reactions. Radiative transitions could play an important role on this search, such as the confirmation of the feeddown mechanism Ω c (3066) → Ξ ′ cK → Ξ c γK, which might be the cause of the enhancement near threshold on the LHCb data [1], even tough additional states have not been completely ruled out as an alternative explanation. The search for isospin-violating decays to Ω c π 0 or electric dipole transitions to Ω c γ could also bring valuable new information. In view of this exciting spectroscopy challenge, where any correct explanation brings with itself an equal or greater number of new questions, we will try to shed light on the discussion looking for different reactions where the new Ω c states could be spotted, and discuss peculiarities of each case that could help distinguish among the various different interpretations on the recent literature.
In the present work we propose the experimental study of these new states through the decay of Ω − b baryons, as suggested in Ref. [56]. The mass and lifetime of the Ω − b were recently measured by the LHCb collaboration [57], obtaining results compatible with the previous measurements of the same collaboration [58] and also with the ones of the CDF collaboration [59], but not with the results of the DØ collaboration [60]. We shall adopt the mass value listed by the Particle Data Group [61], which is quite close to the most recent measurement of LHCb.
We will discuss the decay Ω − b → (Ξ + c K − )π − , which could be performed by the LHCb collaboration [56] in the future and could be very useful to distinguish states with different structures and quantum numbers. First we present a brief summary of the main results of Ref. [44] and comment on the formalism employed there to obtain the Ω c (3050) and Ω c (3090) as meson-baryon molecules (further details can be found in the Appendix section). Next we will discuss the Ω − b → (Ξ + c K − )π − decay and how the coupled channels approach naturally accounts for the dynamical generation of the Ω c states from the hadronization that takes place after the conversion of the b quark into a c quark. Then we show the results of how these two states would be seen in the Ω − b decay if the molecular picture of Ref. [44] is correct, providing solid predictions that could easily be put to proof in the near future.

II. THE Ω c (3050) AND Ω c (3090) IN THE MOLECULAR PICTURE
In Ref. [44] a thorough discussion was made about the meson-baryon interaction due to the exchange of vector mesons, and an extension of the local hidden gauge approach [45][46][47][48][49] was used together with a method that takes into account the information of the spin-flavor wave functions of the baryons (see Appendix and Ref. [44] for details). It was shown that considering the heavy quarks as spectators, the interactions can be obtained using only SU(3) symmetry (without the need of SU(4) in the dominant terms), respecting heavy quark spin symmetry in the leading terms where only light vectors are exchanged [50][51][52].
The procedure begins with the choice of meson-baryon channels in the C = 1, S = −2 and I = 0 sector, interacting in S-wave, with a determined total spin J. In the case of J = 1/2, we can have the pseudoscalar-baryon interactions, where the baryons have J = 1/2, and also vector-baryon contributions that result in J = 1/2. However, in Ref. [44] it was shown that the coupling of vector-baryon channels with the pseudoscalar-baryon ones gives only a small contribution that can be safely neglected. Therefore, the states generated from the vector-baryon interaction decouple from the ones with pseudoscalars and can be treated separately. Since they do not play any role in the generation of the Ω c (3050) and Ω c (3090) in our approach, we will leave them aside in the present work. As presented in Ref. [44], a third state, the Ω c (3119), can also be related with another pseudoscalar-baryon system, where the baryons have J = 3/2, assigning the quantum numbers J P = 3/2 − to the Ω c (3119).
However, since in our approach this state does not mix with the channels of J = 1/2, its decay into Ξ cK calls for additional mechanisms with the exchange of pseudoscalars, which go beyond the scope of our approach here.
The channels chosen for the J P = 1/2 − pseudoscalar-baryon states and their respective threshold masses are shown in Table I. Using an extension of the local hidden gauge Lagrangians [45][46][47][48][49] and the baryon spinflavor wave functions with the heavy quark as spectator, the pseudoscalar-baryon interaction of each channel due to the exchange of vector mesons was calculated in Ref. [44], and then where V is the matrix describing the interaction between every channel (see Appendix for more details) and G is the meson-baryon loop function where k 0 + p 0 = √ s and ω l , E l , are the meson and baryon energies in the l-channel, respectively, and m l , M l the meson and baryon masses. As discussed in Ref. [44] a common sharp cutoff of 650 MeV was used to regularize the propagators. We will adopt the same cutoff value along the present work.
When solving Eq. (1) as a function of the center-of-mass energy √ s, one can go to the complex energy plane and look for poles in the second Riemann sheet, which correspond to the dynamically generated states such that the real part of the pole is the mass of the state and its width is given by twice the imaginary part of the pole. In Table II we show the poles found that correspond to the Ω c (3050) and Ω c (3090) states. In addition, we evaluate the couplings g i of the states obtained to the different channels and g i G i , which for S-wave gives the strength of the wave function at the origin [44,64,65].
As discussed in the previous sections and extensively in Ref. [44], the Ω c (3050) and Ω c (3090) are both strong candidates of meson-baryon molecules, dynamically generated from the coupled channels interaction of baryons with J P = 1/2 + and pseudoscalar mesons (J P = 0 − ) in S-wave, therefore in this picture their spin-parity assignment is J P = 1/2 − .
Since every quark has intrinsic J P = 1/2 + , in order to have Ξ + c K − in S-wave in the final state, the hadronization must occur between the c quark and one s quark, as shown in Fig. 1.
The reason for this approach is that the b quark must turn into a c quark, then both s quarks act as spectators in this decay, so their quantum numbers are fixed a priori as J P = 1/2 + , what implies that the c quark must be in L = 1 to account for the negative parity of the final state, and the hadronization must necessarily involve it so it can be deexcited (see Ref. [66] for details). That being said, we include theqq terms in such a way that the hadronization will occur between the c quark and an s quark.
css → c (ūu +dd +ss) ss ≡ H, where we have included the mixing between η and η ′ [67] for proper matching of Φ with the qq matrix.
Then replacing the cq meson terms we get where we have already neglected the heavy combination of D + s sss which could only contribute to states with J P = 3/2 − since sss corresponds to the Ω − . In terms of spinflavor wave functions, the ground state Ω − is flavor symmetric, hence it can only have symmetric spin ↑↑↑, i.e. J P = 3/2 + , since the color singlet provides the antisymmetric part of the wave function. Besides, D + s Ω − is far above (threshold at 3641 MeV) the energy range of the channel space considered to generate the meson-baryon molecules.
Next, we need to relate the uss and dss with the corresponding baryons, in this case, the Ξ baryons with J P = 1/2 + . For consistence with our approach in Ref. [44], we use the proper spin-flavor wave functions, inspired in Ref. [68], where with the Mixed-Symmetric and Mixed-Antisymmetric flavor and spin wave functions 1 : Thus, only the mixed symmetric will contribute and we get the following weight for Ξ 0 Equivalently, for the Ξ − wave function we only replace u → d and change the sign of φ M S and φ M A [44]. Again, only the mixed symmetric part will contribute and we get Note that we have dropped the common factor 1/ √ 2 from Eq. (8), which can be absorbed in the constant factor V P that we will introduce next.
Then after the hadronization we will have the combination where we have absorbed the global minus sign when we changed to the isospin 0 combination in the order baryon-meson 2 : 1 Note that the phase convention of Ξ 0 is changed in respect to Ref. [68]. This is necessary to be consistent with the chiral Lagrangians, as in Table III of Ref. [69]. See also the footnote of Ref. [70]. 2 Recall the isospin doublets: Now we can proceed to construct the amplitude of the process Ω − b → (Ξ cK ) π − . It is instructive to first look at the process Ω − b → (Ξ D) π − , as depicted in Fig. 2. From Eq. (11) we see that after the emission of a pion, the hadronization involving the css quarks generates a ΞD pair in isospin 0. Thus we can write this process as the sum of a tree-level contribution and the final state interaction of ΞD going through the molecular states of Table II. This information is contained in the diagonal t matrix element t I=0 ΞD→ΞD , with t the same as in Eq. (1), where all the coupled channels dynamics is included.

process. Tree-level (left) plus ΞD rescattering (right).
Then for the tree-level contribution (left diagram of Fig. 2) we simply write where the weight of ΞD production comes from Eq. (11) and V P contains all information related to the Ω − b weak decay, a common unknown factor in all process we will investigate. On the other hand, for the ΞD rescattering (right diagram of Fig. 2) we will have where G ΞD is the propagator of the baryon-meson loop, as in Eq. (2). Then the amplitude of the process Ω − b → π − ΞD is given by With this amplitude we can write the ΞD invariant mass distribution where p π − is the pion momentum in the Ω − b rest frame andp D is the D momentum in the ΞD rest framẽ For the Ω − b → (Ξ cK ) π − process there is no tree-level contribution, since the hadronization only produces a ΞD pair, as in Eq. (11), then the only contribution comes from the diagram in Fig. 3.
Due to our coupled channels approach, the transition ΞD → Ξ cK is already contained in the t matrix and the production of Ξ cK (also in isospin 0) appears naturally. The corresponding amplitude will be where t ΞD→ΞcK is the transition amplitude of ΞD → Ξ cK , from the same t matrix. Then the Ξ cK invariant mass distribution is also analogous, where Analogously, we can also calculate the invariant mass distribution for the final state Ξ ′ cK , replacing Ξ c by Ξ ′ c in the previous equations and taking the matrix element of the t matrix corresponding to the transition ΞD → Ξ ′ cK . It is also interesting to look at the case of coalescence, where the ΞD pair merge into the resonance regardless of the final decay channel, as depicted in Fig. 4.
The value of the amplitude in the process Ω − b → π − R i , where R i is one of the molecular states of Table II, is proportional to the coupling of that resonance to the ΞD channel, where the propagator is calculated at the resonance mass M R i . With this quantity we can calculate the equivalent of the integrated mass distribution around the R i resonance, which does not depend on its decay mode, where

IV. RESULTS
It is interesting to see how these processes show the importance of the coupled channels.
The decay of the Ω c states into ΞD is kinematically forbidden below the corresponding threshold at 3185 MeV, then we cannot see the corresponding peaks in the ΞD invariant  The Ξ cK threshold is at 2965 MeV, then we can see clearly, of course, the peaks of the Ω c states in the Ξ cK invariant mass distribution. According to Eq. (11), we expect only ΞD production from the hadronization that occurs right after the Ω − b decay, which means we have no tree-level contribution for Ξ cK production. However, the transition to Ξ cK through off-shell ΞD loops arises naturally from the coupled channels approach. In fact, both the Ω c (3050) and Ω c (3090) couple strongly to ΞD (see Table II), and their formation from the ΞD state formed in the first step of the Ω − b decay with subsequent transition to Ξ cK (going through the ΞD virtual state) is not only possible, but expected.
In Fig. 6 we show the Ξ cK invariant mass distribution. The only unknown quantity is the global factor V P , common to all amplitudes we investigate here. The ratio between the intensity of each peak does not depend on V P , so all ratios are predictions that could be confronted with future experiments. We can see that the intensity of the Ω c (3050) peak is about 65% higher than the Ω c (3090) peak. Note that we are using the same normalization for all the reactions, hence the ratio of the strength at the peaks in Fig. 6 to the strength of the ΞD mass distribution (solid line) in Fig. 5 is also a prediction. In arbitrary units, the ΞD distribution has a maximum of about 125 around 3240 MeV, whereas in the Ξ cK distribution the Ω c (3050) and Ω c (3090) peaks have an intensity of about 15.50 × 10 3 and 9.45 × 10 3 , respectively, therefore we predict that the Ξ cK distribution in the vicinity of the resonances peaks is roughly two orders of magnitude higher than the ΞD distribution. Cusp effects in the Ξ cK distribution also appear at the Ξ ′ cK and ΞD thresholds at 3074 MeV and 3185 MeV, respectively, but their intensity is very small compared to the peaks of the resonances and cannot be seen clearly in Fig. 6.
We also notice that apparently no significant interference pattern is seen between both states, even tough they have the same quantum numbers, a feature that also agrees with the fit performed by the LHCb collaboration [1,16].
As for the Ξ ′ cK invariant mass distribution, only the Ω c (3090) can be seen, as shown in Fig. 7, since this channel is also open for the decay of this state, whereas the Ω c (3050) is below the threshold of Ξ ′ cK . Again we can compare the intensity of the peaks. In the Ξ ′ cK distribution the Ω c (3090) peak has an intensity of 3.86 × 10 3 , which is about 40% of the intensity it has in the Ξ cK .
Finally, it is interesting to compare the results of Γ Ω − b →π − R i , given by Eq. (24), with the integrated invariant mass distribution of Eq. (20) around the peak of each resonance, In Table III we show the results of Eqs. (24) and (26) for the Ω c (3050) and Ω c (3090), and for the latter we also show the integrated Ξ ′ cK distribution. From these results we can draw some interesting conclusions. Let us look first at the Ω c (3050). In Table II we can see that this state is dominated by the Ξ ′ cK channel, and also has sizable contributions from the higher channels ΞD and Ω c η. However, the pole is at 3054 MeV, which is 20 MeV below the Ξ ′ cK threshold, so the only open channel is Ξ cK , to which the resonance couples very weakly and the phase space available is only about 90 MeV. This feature explains two points: 1) The narrowness of the state, whose upper limit of the width  On the other hand, the Ω c (3090) is dominated by the ΞD channel, with some contribution from the other channels. Even tough this state couples very strongly to ΞD, it is almost 100 MeV below the respective threshold. It can only decay to Ξ cK and Ξ ′ cK . In both cases the coupling is small, and in the latter channel the phase space available is less than 20 MeV (see Table II). This again, explains two main features: 1) The narrowness, with a width not so small as in the case of the Ω c (3050), but still very narrow, since the decay into Ξ cK is reasonable, with more than 120 MeV of phase space available, although the coupling to that channel is weak. The LHCb reports a width of 8.7±1.0±0.8 MeV [1], in fair agreement with our result of 10.24 MeV; 2) The fact that the integrated invariant mass distribution around the peak, 133482 in the Ξ cK distribution, is about 2/3 of the total given by the coalescence, 215237, which is also expected, since in Eq. (26) we are integrating only in the Ξ cK channel, whereas this state can also decay into Ξ ′ cK . The sum of both integrals, in Ξ cK and Ξ ′ cK , is 184556, close to the total given by the coalescence, but still below. This is also expected since Eq. (24) is actually an approximation that is valid in the limit of zero width, which works pretty well for the Ω c (3050) but is not so good for the Ω c (3090), with already 10 MeV of width. This can also be verified calculating the contribution of each channel to the total width, in terms of the coupling of the resonance to that channel. For Ξ cK where M R i is the mass of the resonance andp 2 is given bỹ Using this equation we get Γ ΞcK = 0.83 MeV for the Ω c (3050), in accordance with the value obtained from the pole position. For the Ω c (3090) we get Γ ΞcK = 7.24 MeV for the Ξ cK , and replacing Ξ c by Ξ ′ c and the corresponding coupling, we get Γ Ξ ′ cK = 2.56 MeV, which add up to 9.80 MeV, a bit below 10.24 MeV that we get from the pole position.
As a prediction we can also state, based on the coalescence results, that the ratio of the Ω c (3050) over the Ω c (3090) production is about 10% in the Ω − b decay:

V. CONCLUSIONS
We have studied the weak decay Ω − b → (Ξ + c K − ) π − , in view of the narrow Ω c states recently measured by the LHCb collaboration and later confirmed by the Belle collaboration.
Based on the previous work where the Ω c (3050) and Ω c (3090) are described as mesonbaryon molecular states, using an extension of the local hidden gauge approach in coupled channels, with results in remarkable agreement with experiment, we have investigated the ΞD, Ξ cK and Ξ ′ cK invariant mass distributions, and discussed the role of coupled channels in the process. Predictions that could be confronted with future experiments are presented, providing useful information that could help to determine the quantum numbers and nature of these states. Since Ω − b baryons have already been observed in several experiments, two of them performed by the LHCb collaboration, the present work should encourage such study in the near future, which would certainly bring novel key information for the understanding of these new states.

APPENDIX
The ingredients needed to obtain the molecular states presented here are the vector(V)pseudoscalar(P)-pseudoscalar(P) Lagrangian from the local hidden gauge approach, for the VPP vertex, where g = m V /2f π ; f π = 93 MeV is the pion decay constant, and m V is the mass of the light vector mesons. Φ is the pseudoscalar mesons matrix from Eq. (6) and V µ is the vector mesons matrix,  30)). As discussed in Ref. [44], this method yields the same results as the VBB Lagrangian of the local hidden gauge in SU (3), being consistent at the same time with this formalism and with the hypothesis of the heavy quark as spectator, ensuring heavy quark spin symmetry in the dominant terms where only light vectors are exchanged.
With these ingredients and the baryon spin-flavor wave functions, one can obtain the coefficients D ij of the V matrix that we plug in the Bethe-Salpeter equation in the form of Eq. (1), given by where M B i ,B j and E B i ,B j stand for the mass and the center-of-mass energy of the baryons, respectively, and the matrix D ij is given in Table IV.
The details of the calculation can be found in Ref. [44]. In Table IV we have the parameter λ in some non diagonal matrix elements, which involve transitions from one meson without charm to one with charm, likeK → D. In this case we have to exchange a heavy quark, and the propagator of the exchanged vector goes like and the ratio to the propagator of the light vectors is Then we take λ = 1/4 in all these matrix elements, as it was done in Ref. [71].
The diagonal matrix elements of Table IV coincide with those of Ref. [43], but not all the non diagonal. SU(4) symmetry is used in Ref. [43], but only SU(3) is effectively used in the diagonal terms. The same happens in our approach, with the difference that the heavy