Ξc and Ξb excited states within a SU(6)lsf×HQSS model
Introduction
We study odd parity J = 1 / 2 and J = 3 / 2 Ξ c resonances using a unitarized coupled-channel framework based on a SU (6) lsf × HQSS-extended Weinberg–Tomozawa baryon–meson interaction, while paying a special attention to the renormalization procedure. We predict a large molecular Λ c K ¯ component for the Ξ c (2790) with a dominant 0 - light-degree-of-freedom spin configuration. We discuss the differences between the 3 / 2 - Λ c (2625) and Ξ c (2815) states, and conclude that they cannot be SU(3) siblings, whereas we predict the existence of other Ξ c -states, one of them related to the two-pole structure of the Λ c (2595) . It is of particular interest a pair of J = 1 / 2 and J = 3 / 2 poles, which form a HQSS doublet and that we tentatively assign to the Ξ c (2930) and Ξ c (2970) , respectively. Within this picture, the Ξ c (2930) would be part of a SU(3) sextet, containing either the Ω c (3090) or the Ω c (3119) , and that would be completed by the Σ c (2800) . Moreover, we identify a J = 1 / 2 sextet with the Ξ b (6227) state and the recently discovered Σ b (6097) . Assuming the equal spacing rule and to complete this multiplet, we predict the existence of a J = 1 / 2 Ω b odd parity state, with a mass of 6360 MeV and that should be seen in the Ξ b K ¯ channel.
The study of heavy baryons with charm or bottom content has been the subject of much interest over the past years in view of newly discovered states [[1]]. In particular, there has been a tremendous effort to understand the nature of the experimental states within conventional quarks models, QCD sum-rules frameworks, QCD lattice analysis or molecular baryon–meson models (see Refs. [[2]–[7]] for recent reviews).
The attention has been recently revived by the experimental observation of several excited states. Recent detections have been reported by the LHCb Collaboration regarding five excited states in the spectrum in pp collisions [[8]], and the excited state in and invariant mass spectra also in pp collisions [[9]]. Moreover, the Belle Collaboration has confirmed the observation of four of the excited states [[10]], and detected the state in its decay to in decays [[11]].
In view of these new observations, a large theoretical activity has been indeed triggered, in particular within dynamical approaches based on a molecular description of these states. Starting from the newly observed states, the molecular models of Refs. [[12]] have been reanalyzed in view of the new discoveries. While in Ref. [[14]] two resonant states at 3050 MeV and 3090 MeV with were obtained, being identified with two of the experimental states, in Ref. [[15]] two states and one were determined within an extended local hidden gauge approach, the first two in good agreement with [[14]]. Other theoretical works also examined the sector, trying to explain the extra broad structure observed by the LHCb around 3188 MeV [[8]]. In Ref. [[16]] it was shown that this bump could be interpreted as the superposition of two bound states, whereas in Ref. [[17]] a loosely bound molecule of mass 3140 MeV was determined.
Excited states below 3 GeV and the excited state found experimentally [[1]]. We show the assigned (when possible), the mass and width , as well as the decay channels
Baryon |
| M (MeV) | (MeV) | Decay channels |
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| , |
| ? |
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| , |
| ? |
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| , , , , |
| ? |
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| , |
With regards to , the theoretical analysis based on the local hidden gauge formalism has shown that not only the can have a molecular interpretation, but also other states around 3 GeV reported in the PDG [[1]]. In particular, the would be a molecular state, whereas , , and could be described as molecules with either or [[18]]. On the other hand, the same model has produced two states for with masses close to the experimental one with similar widths, being the spin–parity assignment either or [[18]]. The state has been also studied within a unitarized model that uses the leading-order chiral Lagrangian in Refs. [[19]], identifying the state as a S-wave molecule, with a preferred spin–parity assignment [[20]].
Over the past years, a unitarized coupled-channel scheme has been developed in Refs. [[21]–[27]] that implements heavy-quark spin symmetry (HQSS), which is a proper QCD symmetry that appears when the quark masses, such as that of the charm or bottom quark, become larger than the typical confinement scale. This approach is based on a consistent extension of the Weinberg–Tomozawa (WT) interaction, where "lsf" stands for light quark–spin–flavor symmetry, respectively. Within this framework, it has been identified a two-pole pattern for the resonance[1] [[21], [23]], similar to the [[29]–[31]]. The same scheme has also generated dynamically the and narrow resonances, discovered by LHCb [[32]], which turn out to be HQSS partners, naturally explaining their approximate mass degeneracy [[24]].
More recently, the work of Ref. [[23]] has been revisited in view of the newly discovered states, paying a special attention to the renormalization procedure used in the unitarized coupled-channel model and its impact on the sector. In Ref. [[33]] it was shown that some (probably at least three) of the states experimentally observed by LHCb would have or .
The discovery of the and has stimulated and motivated further research along this line. In the present work we follow a similar procedure as described in [[33]] and study the possible molecular interpretation of those states, revisiting the previous works on the [[23]] and [[24]] sectors. However, in the sector we do not restrict ourselves to the recent observation, but analyze all excited states found experimentally with masses up to 3 GeV [[1]]. The four excited states with masses below 3 GeV and the are collected in Table 1, showing the assigned spin–parity (when possible) as well as masses, widths, and decay channels. In this work we pay a special attention to the dependence on the renormalization scheme as well as to the flavor-symmetry content of the model, as we determine the possible HQSS partners and siblings among the experimental states while predicting new ones. Thus, we follow the discussion of Ref. [[23]] on its spin–flavor symmetry breaking pattern. Flavor SU(4) is not a good symmetry in the limit of a heavy charm quark, for this reason, instead of the breaking pattern SU(8) SU(4), we consider the pattern SU(8) SU(6), since the light spin–flavor group [SU(6)] is decoupled from heavy-quark spin transformations. This allows us to implement HQSS in the analysis and to unambiguously identify the corresponding multiplets among the resonances generated dynamically. At the same time, we are also able to assign approximate heavy [SU(8)] and light [SU(6)] spin–flavor multiplet labels to the states.
This work is organized as follows. In Sect. 2 we present the extension of the WT interaction, while in Sect. 3 we show our results for the and states,[2] respectively, and the possible experimental identification. Finally, in Sect. 4 we present our conclusions, emphasizing the possible classification of these experimental states according to the flavor-symmetry content of the scheme, while predicting new observations.
Formalism
We consider the sector with charm , strangeness and isospin quantum numbers, where the state has been observed by the Belle Collaboration [[11]]. Also, we examine the bottom , strangeness and isospin , where the has been found [[9]]. In order to do so, we revise the results in Ref. [[23]] for the states and in Ref. [[24]] for ones.
In the case of the , and sector, the building-blocks are the pseudoscalar ( ) and vector ( ) mesons, and the spin-1/2 ( , , , , , , , ), and spin-3/2 ( , , ) charmed baryons [[21], [23]]. For the bottom sector , and , one can substitute the c quark by a b quark, and we have the pseudoscalar ( ) and vector ( ) mesons, and the spin-1/2 ( , , , , , , , ), and spin-3/2 ( , , ) baryons [[24]]. All baryon–meson pairs with quantum numbers span the coupled-channel space for a given total angular momentum (J).
The S-wave tree level amplitudes between two baryon–meson channels are given by the WT kernel,
1
Graph
The and are the masses of the baryon and meson in the i channel, respectively, and is the center-of-mass energy of the baryon in the same channel,
2
Graph
The projection onto definite J is done as explained in Refs. [[23], [34]]. In [[34]], only baryons and pseudoscalar mesons are considered. The inclusion of spin baryons and vector mesons was discussed in Appendix A of Ref. [[23]], using results derived in [[35]]. The hadron masses, meson decay constants, , and matrices are taken from Refs. [[23]], where the underlying HQSS group structure of the interaction has been considered.
Starting from , we solve the Bethe–Salpeter equation (BSE) in coupled channels,
3
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where the is a diagonal matrix that contains the different baryon–meson loop functions ,
4
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with P the total momentum of the system such that . We omit the index J from here on for simplicity. The bare loop function is logarithmically ultraviolet (UV) divergent and needs to be renormalized. This can be done by separating the divergent and finite parts of the loop function,
5
Graph
with the finite part of the loop function, , given in Refs. [[33]]. The divergent contribution of the loop function, in Eq. (5) needs to be renormalized.
On the one hand, this can be done by one subtraction at certain scale ( )
6
Graph
where , with and the masses of the meson and the baryon, respectively, that belong to the channel with the smallest threshold for a given (C, S, I) or (B, S, I) sectors. This common scale is chosen to be independent of total angular momentum J [[13], [36]], and it is the approach used in the previous works of Refs. [[23]].
On the other hand, as discussed in our recent paper [[33]], we could also use a sharp-cutoff regulator in momentum space, so that
7
Graph
where is given in Refs. [[33], [37]].
Graph: Fig. 1 Evolution of the masses and widths of the dynamically generated Ξc states as we vary the renormalization scheme from using a subtraction constant to a common cutoff of Λ=1090 MeV. The cross symbolizes the position of the states in the subtraction constant scheme [or dimensional regularization (DR)] [[23]], while the triangle indicates the mass and width of the same states for the cutoff scheme
Note that if one uses channel-dependent cutoffs, the one-subtraction renormalization scheme is recovered by choosing in each channel in such a way that
8
Graph
However, we employ a common UV cutoff for all baryon–meson loops within reasonable limits. In this manner, we avoid a fictitious reduction or enhancement of any baryon–meson loop by using an unappropriated value of the cutoff [[28], [38]], as well as we prevent an arbitrary variation of the subtraction constants, as we correlate all of them with a UV cutoff [[33]].
The poles of the T matrix describe the odd-parity dynamically-generated and states, which appear in the first and second Riemann sheets (FRS and SRS). Poles of the scattering amplitude on the FRS below threshold are bound states, whereas poles on the SRS below the real axis and above threshold are resonances. The mass and the width of the bound state/resonance can be found from the position of the pole on the complex energy plane. At the complex pole, the T-matrix is given by
9
Graph
where , with the mass and the width of the state, and is the complex coupling of the state to the channel i, which is determined from Cauchy's theorem of residues. Thus, the dimensionless couplings are obtained by first assigning an arbitrary sign to one of them ( ), so
10
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The other couplings are calculated as,
11
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and can be used to analyze the contribution of each baryon–meson channel to the generation of the state.
Results
Ξc excited states
As mentioned in the Introduction, the first observation of the state was reported by the Belle Collaboration in Ref. [[11]]. This state was observed through its decay to with no assigned quantum numbers. Besides this recently discovered state, there are other three excited states with energies below 3 GeV [[1]]. As seen in Table 1, the state decays into , whereas the decays into and has also the decay chain , followed by [[40]]. Also, a with unknown quantum numbers has been observed decaying into , , , and .
Graph: Fig. 2 Evolution of the masses and widths of the dynamically generated Ξc states, as we vary the cutoff from Λ=1 GeV (triangles) to Λ=1.2 GeV (crosses). In a, b, the c1 (c8) state becomes virtual above (below) the ΛcK¯ (Ξc∗π) threshold. The squares and their associated errorbars show the masses and widths of the experimental Ξc(2790) and Ξc(2930) (a) and Ξc(2815) and Ξc(2970) (b) together with their experimental errors. The spin–parity of both Ξc(2930) and Ξc(2970) resonances are not experimentally determined [[1]], and we have displayed them here just for illustrative purposes
Masses and widths of the to states with or and odd parity in the , and sector, together with the couplings (in modulus) to the dominant baryon–meson channels ( ) and the couplings to the decay channels reported experimentally for the states. All results have been obtained for MeV. We also indicate the possible experimental identification as well as the HQSS, SU(6) and SU(3) irreducible representations of these states. We use the notation , where is the SU(6) irreducible representation (irrep) label (for which we use the dimension), is the spin carried by the quarks with charm (1/2 in all cases) and C the charm content (1 in all cases). In addition, we also use , where is the SU(3) irrep, with J the total angular momentum of the state (see Ref. [[23]] for details)
Irreps | State | M (MeV) | (MeV) |
| Couplings | Experiment |
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| 2773.59 | 10.52 | 1/2 | , , , , , , , , , |
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| 2627.5 | 38.84 | 1/2 | , , , , , , , | |
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| 2790.99 | 16.09 | 1/2 | , , , , , , , , , , | |
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| 2850.89 | 6.76 | 3/2 | , , , , , , |
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| 2715.23 | 12.28 | 1/2 | , , , , , , | |
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| 2807 | 1.82 | 1/2 | , , , , , , , , | |
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| 2922.5 | 2.48 | 1/2 | , , , , , , , , , , , , , |
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| 2792.06 | 22.79 | 3/2 | , , , , | |
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| 2942.05 | 1.46 | 3/2 | , , , , , , , , |
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We start by revising the results Ref. [[23]] in the sector in order to understand whether the experimental states can be accommodated in our model. The widths of our states as a function of their masses in the and sectors are shown in the upper and lower plots of Fig. 1, respectively, together with different baryon–meson thresholds, to which they can couple. The dynamically generated states of Ref. [[23]] are displayed with a cross and the "DR" legend, as those have been obtained using one subtraction at certain scale or dimensional regularization. In what follows, we label the states as , and they correspond to those given in Table V of Ref. [[23]]. They have either or and are ordered by their mass position. Hence, ( ) corresponds to the lightest (heaviest) state of mass 2699.4 MeV (2845.2 MeV), among those quoted in the mentioned table, where their SU(6) and SU(3) quantum numbers are also given. We observe that the masses of our states using one subtraction constant (DR) are below or close to the experimental or states, while being far below in mass with respect to or .
We then study the effect on masses and widths of the renormalization procedure so as to determine whether any our can be identified with a experimental state while assessing the dependence on the renormalization scheme, which might be significant (as shown in Ref. [[28]]). We proceed as described in Ref. [[33]] for the states, where we explore a different renormalization procedure, the cutoff scheme. In order to do so, we first need to determine how the masses and widths of our dynamically generated states change as we adiabatically move from the one subtraction renormalization scheme to the cutoff one. Thus, we change the loop functions by
12
Graph
where we slowly evolve x from 0 to 1 while following the evolution of the states, as seen in Fig. 1. The to states for a MeV are shown with a triangle. We find that most of these states move to higher energies, except for , and , whereas getting closer to the experimental values. Note that fot this cutoff, the state become virtual above the threshold.
Once we have identified our states in the cutoff scheme, we can assess the dependence of our results on this regulator, as well as their possible experimental identification. In Fig. 2 we show the evolution of the to states as we vary the cutoff from 1 GeV (triangles) to 1.2 GeV (crosses), and we also display different two-body thresholds. Moreover, in Table 2 we show masses and widths of the to states with or , together with the couplings to the dominant baryon–meson channels ( ) and the couplings to the decay channels reported experimentally for the states. All these results are obtained for MeV. In this table we also indicate the possible experimental identification as well as the HQSS, SU(6) and SU(3) irreducible representations (irreps), to which the to states belong (see Ref. [[23]] for group-structure details).
As we evolve the cutoff value from MeV to MeV, that is, from the right to left in Fig. 2, we observe that some of our dynamically generated resonances can be identified with the experimental states attending to the complex energy position.
In the sector, attending to the position in the complex plane, we observe that the could be one of the , , or even the states. The identification with the is possible because these states couple to , although this baryon–meson channel is not the dominant one for their dynamically generation, as seen in Table 2 for a MeV, except for . Indeed, this latter feature of disfavors its identification with the . This is because it would become too broad ( MeV) for UV cutoffs of around 1 GeV, that would lead the resonance to have masses closer to the experimental one, as seen in Fig. 2. In addition in the DR scheme, the mass of the state is close to 2790 MeV, but its width is approximately of 84 MeV [[23]] (see also Fig. 1), while experimentally MeV.
Looking at the behavior of the , , poles with the UV cutoff in Fig. 2, it seems natural to assign the to the pole. This state has a width of the order of 10 MeV for UV cutoffs in the region of 1.2 GeV, where it is located below the threshold. At the same time, the state has large and small couplings (see Table 2), respectively, which explains its small experimental width despite being placed well above the latter threshold, and it is natural to think that the channel should play an important role in the dynamics of the given its proximity to that threshold. Note that the light degrees of freedom (ldof) in the inner structure of the are predominantly coupled to spin–parity quantum numbers [[28]]. Thus with this identification, this first odd parity excited state would not have a dominant configuration consisting of a spinless light diquark and a unit of angular momentum between it and the heavy quark, as argued for instance in the Belle Collaboration paper [[40]]. This is to say, the will not be a constituent quark model -mode excited state [[28], [41]] with and hence it will not form part of any HQSS doublet, thus making the assignment to unlikely. Actually, if the spin–parity quantum numbers for the in the were predominantly , one would expect a larger width for this resonance, since its decay to the open channel is HQSS allowed. This is precisely the situation for the that is broader than the experimental state. In summary, we conclude a large molecular component for the that will have then a dominant configuration.
Our present identification with the pole differs from the previous assignments in Ref. [[23]], where the states were obtained using the one subtraction renormalization scheme. In this previous work, the state was assigned to due to its closeness in energy and the sizable coupling within the DR scheme.
With regards to the recently discovered , if we assume that this state has , we could identify it either with our or states, as they both couple to the channel, although not dominantly as seen in Table 2 for a MeV. The assignment to the pole is, however, disfavored because of the mass difference between this state and the experimental . As for , the small coupling of this state makes also somehow doubtful its identification with the . In the case of our and states, we should mention that we do not have any clear experimental candidate at this point for the dynamically generated state, whereas the state becomes broad and appears below 2650 MeV, thus not allowing for any reasonable experimental assignment.
For , the analysis of the evolution of the different states in Fig. 2 allows for the identification of the experimental with or . These states couple to in S-wave, although for , couplings to other baryon–meson states ( , or ) are larger as seen in Table 2. The experimental is quite narrow, MeV, despite the threshold being around 30 MeV below its mass. This hints to a subdominant molecular component in the inner structure of this resonance. Moreover, looking at the dependence of the pole masses and widths with the UV cutoff displayed in Fig. 2, it seems reasonable to assign the state to the resonance.
As for , assuming that it has , we could identify it with the state for values of the cutoff around GeV. In this case, we have to take into account that this state couples to and , and (though not dominantly), and those baryon–meson channels can decay into and , respectively. Nevertheless, the predicted width would be significantly smaller than the range of 20-30 MeV quoted in the PDG [[1]] and shown in Table 1. Compared to the results of Ref. [[23]], the was identified there with , assuming that and were the and HQSS partners.
In fact, we observe several HQSS partners among our states as well as possible siblings within the same SU(3) representation. The resonance belongs to an SU(3) antitriplet irrep, and it would be the HQSS (see Table 2) partner of a narrow state discussed in [[21], [23], [28]]. This latter state has large (small) ND and ( ) couplings, and depending on the renormalization scheme (one-subtraction or UV cutoff), it is part of a double pole pattern for the , similar to that found for the within unitarized chiral models [[29], [42]–[47]] (see related review in [[1]]), or it is located in the region of 2.8 GeV close to the ND threshold [[28]].
Masses and widths of the to states with or in the , and sector, together with the couplings to the dominant baryon–meson channels and the couplings to the experimental decay channels of the , using one-subtraction renormalization, as in Table IV of Ref. [[24]]. We also indicate the HQSS, SU(6) and SU(3) irreducible representations of these states, as in Table 2. States with are virtual states. Note that the lies in the real axis, but in a sheet that is not connected to the physical sheet, thus we are not showing the couplings indicating "R.S (real sheet) not connected"
Irreps | State | (MeV) | (MeV) | J | Couplings |
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| 5873.98 | 0 | 1/2 | , , , , , , |
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| 5940.85 | 35.59 | 1/2 | , , , , , |
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| 5880.76 | 0 | 1/2 | , , , , , , , |
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| 5880.27 | 0 | 3/2 | , , , , , , , |
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| 5949.93 | 0.7 | 1/2 | , , , , |
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| 6034.80 | 28.8 | 1/2 | , , , , , , |
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| 6035.39 | 0.02 | 1/2 | , , , , , |
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| 5958.20 | 0 | 3/2 | – R.S. not connected – |
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| 6043.28 | 0 | 3/2 | , , , , , |
On the other hand, the pole belonging to ( ) representation and the of the ( ) form a -HQSS doublet. As mentioned earlier, the can be identified with the , but we note that the is not the sibling of the because of the different coupling strengths to and , respectively. Whereas weakly couples to , the strongly does to . However, this latter state is narrow because the channel is closed (located around 30 MeV above the mass of the resonance). Indeed, recently it has been argued that the is probably a constituent three quark state [[28], [41]].
As for the and the states, those form part of a SU(6) -plet, belonging to the HQSS ( ) and ( ) irreps [[23]]. They form a HQSS doublet with and hence have large couplings to and , respectively. Indeed, as a good approximation, they are dynamically generated by the charmed baryon–Goldstone boson interactions. These moderately broad states are in the SU(3) and irreps, which should be completed by one and one resonances stemming from the and chiral interactions [[19], [28]], neglecting higher energy channels. The sibling is, however, not the . As mentioned before, the is probably a quark model ( -mode excitation) state [[28], [41]]. Another resonance with mass and width of around 2.7 GeV and 60 MeV [[19], [28]], that has not been discovered yet, would then be the SU(3) sibling of the state.
The features of the counterpart of in the sector are much more uncertain and depend on both the employed renormalization scheme and on the interplay between quark-model and baryon–meson degrees of freedom [[28]]. Thus, for instance neglecting the latter, it would appear around 2.6 GeV with a large width of 60-80 MeV because its sizable coupling to the pair. Within the UV cutoff RS, this state can be easily moved below the threshold and be identified with the narrow [[19]]. In the DR scheme advocated in Ref. [[23]], this broad state, together with the narrow state mentioned above in the discussion of the , gives rise to a double pole structure for the .
Within the UV cutoff renormalization scheme examined here, the HQSS-doublet might correspond to the experimental and states. The state, that we have tentatively assigned to the , exhibits (Table 2) moderate couplings to and , small ones to and , and finally large couplings to , and . It belongs to a SU(3) sextet, where there is also a state. The latter corresponds to the one labeled as d in our previous study of the odd-parity resonances [[33]], where it was tentatively assigned either to the or the observed by the LHCb Collaboration in the mode [[8]]. This is in fact consistent with what one might expect from its -sibling couplings. Assuming the equal spacing rule we could predict the possible existence of a state around 2800 MeV that will complete the sextet. The clearly fits into this picture since it is observed in the channel [[1]].
Graph: Fig. 3 Evolution of the masses and widths of the dynamically generated Ξb states, as we vary the cutoff from Λ=1000 MeV (triangles) to Λ=1400 MeV (crosses), with J=1/2 (upper panel) and J=3/2 (lower panel). The square and its bars represent the position of the Ξb(6227) resonance, and its errors in mass and width, respectively. We show the experimental result for both values of J due to its unknown quantum numbers. In b, the last five thresholds (not labelled in the figure because they are too close to each other) are: Ξb∗η(6492.45MeV), ΣB∗(6518.35MeV), ΩbK(6564.68MeV), Ξbρ(6565.04MeV) and Ξbω (6572.12MeV)
Recently there has been an analysis of the sector within a baryon–meson molecular model based on local hidden gauge that implements the interaction between the and ground-state baryons with and mesons [[18]]. The authors have found that five of their dynamically generated states can be identified with the experimental , , , and . Whereas the would be a state, the , , and could be either or ones. Compared to this approach, our model identifies the experimental and as states, and the and as . The different assignment is mainly due the distinct renormalization scheme used in the two approaches as well as the fact the interactions involving D and and light vector mesons with baryons are not completely fixed by HQSS or chiral symmetries, thus allowing for different assumptions.
Ξb excited states
With regards to the bottom sector, the resonance has been recently measured by the LHCb experiment [[9]], with MeV. Its quantum numbers, though, remain unknown, whereas the observed decay channels are and (see Table 1).
We start again by revising the previous results of Ref. [[24]] with , , ( sector). Masses and widths of the dynamically generated states within our model using the DR scheme, together with their irreps, spins and couplings to the dominant baryon–meson channels as well those for the experimental decay channels of are shown in Table 3. We obtain nine states, which are the bottom counterparts of the ones discussed in the previous subsection. Compared to Ref. [[24]], we report here five more poles, since in that reference only SU(3) flavor partners of states were searched (members of antitriplet irreps). Also, two of them, the state at 6035 MeV with and the one at 6043 MeV with were wrongly assigned in Ref. [[24]] to the SU(6) 15 representation. Instead, their should belong to the SU(6) 21 representation, as seen in Table 3. Moreover, there is a state at 6073 MeV in Table IV in Ref. [[24]] that does not appear in our present calculation. The differences between of them are due to the difficulty in determining the number of states and their representations as we break the HQSS symmetry to SU(3) in the bottom sector, as almost all states have zero width and states with widths closer to zero are more difficult to follow in the complex energy plane.
As Table 2, but for the sector ( MeV)
Irreps | State | (MeV) | (MeV) | J | Couplings | Experiment |
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| 6025.46 | 25.88 | 1/2 | , , , , , | |
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| 6152.61 | 15.29 | 1/2 | , , , , , , |
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| 6179.4 | 3.81 | 1/2 | , , , , , , , , , , | |
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| 6202.73 | 4.48 | 3/2 | , , , , , , | |
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| 6243.02 | 0.74 | 1/2 | , , , , | |
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| 6212.26 | 1.6 | 1/2 | , , , , , , | |
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| 6327.28 | 5.29 | 1/2 | , , , , , , , , | |
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| 6240.82 | 0.92 | 3/2 | , , , , | |
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| 6459.42 | 0.02 | 3/2 | , , , , , | |
As in the sector, our to states using one-subtraction renormalization are too low in energy so as to assign any of them to the experimental state. Thus, we proceed as in the previous subsection and vary the renormalization scheme from one-subtraction to cutoff. In this manner, we identify our to states using one-subtraction renormalization with the ones within the cutoff scheme, and we study their evolution as we change the value of the cutoff.
In Fig. 3 we display the evolution of the masses and widths of the dynamically generated states as we vary the cutoff from MeV (triangles) to MeV, for (upper plot) and (lower plot). The square and its bar represent the position of the resonance, and the error for its mass and width, respectively. We show the experimental result ( ) for both and because its quantum numbers have not been determined yet. Additionally, in Table 4, we collect the masses and the widths of the to states with or , together with the couplings to the dominant baryon–meson channels and the couplings to the decay channels of the , for MeV as in the charm sector. We also indicate the HQSS, SU(6) and SU(3) irreducible representations of these states.
We might try now to assign the experimental to any of our states, while determining the negative parity baryons with belonging to the same and SU(3) representations. The observed decay modes, , [[9]], of the resonance support that this state should have spin–parity, assuming S-wave. Moreover, the component should be also quite relevant, which according to the couplings collected in Table 4 makes plausible its identification either with the or states. The evolution displayed in the upper plot of Fig. 3 leads us to assign the to the state, as shown in Table 4. The pole would stem from a SU(6) 15-plet, composed of and SU(3) antitriplets and of a SU(3) sextet, where the would be accommodated. The and (LHCb [[32]]) would be part of the and multiplets forming a HQSS-doublet [[24]]. These antritriplets should be completed by another HQSS-doublet of and states, and , that according to Fig. 3 and Table 4 should have masses around 6250 MeV and could be seen in the and modes.
Coming back to the , it belongs to a sextet that should be completed by and states. The recent resonance seen by the LHCb Collaboration [[48]] in the channel nicely fits in this multiplet. Relying again in the equal spacing rule, we could foresee the existence of a odd parity state with a mass of around 6360 MeV that should be observed in the channel. Some molecular states were predicted previously in Ref. [[49]], but all of them above 6.4 GeV.
Graph: Fig. 4 Bottom baryon states classified within the J=1/2 (left diagram) and J=3/2 (right diagram) SU(3) 3∗ irreps. The question mark indicates states predicted in this work
Graph: Fig. 5 Charm and bottom resonances classified within SU(3) 6 irreps with J=1/2, which however stem from different SU(6)lsf×HQSS irreps: (120,21,62) and (168,15,62), respectively. The question mark indicates states predicted in this work
Previous works based on molecular approaches have also found the as a dynamically-generated state. In Refs. [[19]] a unitarized model using the leading-order chiral Lagrangian found the as a S-wave molecule, with a preferred spin–parity assignment [[20]]. In our present model the , and are the dominant channels in the generation of the , though it also couples (weakly) to . The main difference between models comes from the fact that our scheme has a more extensive number of channels, whereas the antitriplet and sextet multiplets of ground-state baryons mix when constructing the interaction matrices. Also, the work of Ref. [[18]] has also analyzed the sector. The authors have found two poles with masses close to the and widths MeV, close to the experimental one, with and spin–parity. In our model we identify the as a state and, again, the difference arises because of the renormalization scheme and the interaction matrices involving D, and light vector mesons.
Conclusions
In this work we have explored the possible molecular interpretation of several experimental excited and states. We have used a coupled-channel unitarized model, that is based on a HQSS-extended WT baryon–meson interaction, within the on-shell approximation. We have paid a special attention to the dependence of our predictions on the renormalization scheme, so as to assess the robustness of our results.
We have presented a molecular interpretation for the experimental , , , and states, and have predicted the spin–parity quantum numbers of the latter three resonances. We have found that the state has a large molecular component, with a dominant configuration, and discussed the differences between the and states, finding that they cannot be SU(3) siblings. We have also predicted the existence of other -states, not experimentally detected yet, being one of them related to the two-pole structure of the . Interestingly, the recently discovered and are found to be HQSS partners.
The flavor-symmetry content of the framework has also allowed us to understand the nature of the and states, for which we have determined their spin–parity. Moreover, we have predicted several states, some of them displayed in Figs. 4 and 5 (marked with a ? symbol). Among them, we stress the state, with a dominant contribution, in the sextet where the and are located, together with the and states, partners of the HQSS doublet and discussed in [[24]].
Acknowledgements
L.T. acknowledges support from Deutsche Forschungsgemeinschaft under Project Nr. 383452331 (Heisenberg Programme) and Project Nr. 411563442. R. P. Pavao wishes to thank the Generalitat Valenciana in the program GRISOLIAP/2016/071. This research is supported by the Spanish Ministerio de Economía y Competitividad and the European Regional Development Fund, under contracts FIS2017-84038-C2-1-P, FPA2013-43425-P, FPA2016-81114-P and SEV-2014-0398; the THOR COST Action CA15213; and by the EU STRONG-2020 project under the program H2020-INFRAIA-2018-1, Grant agreement no. 824093.
Data Availability Statement
This manuscript has no associated data or the data will not be deposited. [Authors' comment: The results of our work are already presented in the plots and tables of our paper, so there is no more data to be deposited elsewhere.]
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References
1
M. Tanabashi et al. (Particle Data Group). Phys. Rev. D 98, 030001 (2018)
2
Klempt E, Richard J-M. Rev. Mod. Phys.. 2010; 82: 10952010RvMP...82.1095K. 10.1103/RevModPhys.82.1095
3
Crede V, Roberts W. Rep. Prog. Phys.. 2013; 76: 0763012013RPPh...76g6301C. 10.1088/0034-4885/76/7/076301
4
Cheng H-Y. Front. Phys. (Beijing). 2015; 10: 101406. 10.1007/s11467-015-0483-z
5
Chen H-X, Chen W, Liu X, Zhu S-L. Phys. Rep.. 2016; 639: 12016PhR...639....1C3512406. 10.1016/j.physrep.2016.05.004
6
Chen H-X, Chen W, Liu X, Liu Y-R, Zhu S-L. Rep. Prog. Phys.. 2017; 80: 0762012017RPPh...80g6201C. 10.1088/1361-6633/aa6420
7
Guo F-K, Hanhart C, Meißner U-G, Wang Q, Zhao Q, Zou B-S. Rev. Mod. Phys.. 2018; 90: 0150042018RvMP...90a5004G. 10.1103/RevModPhys.90.015004
8
R. Aaij et al. (LHCb). Phys. Rev. Lett. 118, 182001 (2017). arXiv:1703.04639 [hep-ex]
9
R. Aaij et al. (LHCb). Phys. Rev. Lett. 121, 072002 (2018). arXiv:1805.09418 [hep-ex]
J. Yelton et al. (Belle). Phys. Rev. D 97, 051102 (2018). arXiv:1711.07927 [hep-ex]
Y. Li et al. (Belle). Eur. Phys. J. C 78, 252 (2018). arXiv:1712.03612 [hep-ex]
Jimenez-Tejero CE, Ramos A, Vidana I. Phys. Rev. C. 2009; 80: 0552062009PhRvC..80e5206J. 10.1103/PhysRevC.80.055206
Hofmann J, Lutz MFM. Nucl. Phys. A. 2005; 763: 902005NuPhA.763...90H. 10.1016/j.nuclphysa.2005.08.022
Montaña G, Feijoo A, Ramos A. Eur. Phys. J. A. 2018; 54: 642018EPJA...54...64M. 10.1140/epja/i2018-12498-1
Debastiani VR, Dias JM, Liang WH, Oset E. Phys. Rev. D. 2018; 97: 0940352018PhRvD..97i4035D. 10.1103/PhysRevD.97.094035
Wang C, Liu L-L, Kang X-W, Guo X-H, Wang R-W. Eur. Phys. J. C. 2018; 78: 4072018EPJC...78..407W. 10.1140/epjc/s10052-018-5874-1
Chen R, Hosaka A, Liu X. Phys. Rev. D. 2018; 97: 0360162018PhRvD..97c6016C. 10.1103/PhysRevD.97.036016
Yu QX, Pavao R, Debastiani VR, Oset E. Eur. Phys. J. C. 2019; 79: 1672019EPJC...79..167Y. 10.1140/epjc/s10052-019-6665-z
Lu J-X, Zhou Y, Chen H-X, Xie J-J, Geng L-S. Phys. Rev. D. 2015; 92: 0140362015PhRvD..92a4036L. 10.1103/PhysRevD.92.014036
Huang Y, Xiao C-J, Geng L-S, He J. Phys. Rev. D. 2019; 99: 0140082019PhRvD..99a4008H. 10.1103/PhysRevD.99.014008
Garcia-Recio C. Phys. Rev. D. 2009; 79: 0540042009PhRvD..79e4004G. 10.1103/PhysRevD.79.054004
Gamermann D, Garcia-Recio C, Nieves J, Salcedo LL, Tolos L. Phys. Rev. D. 2010; 81: 0940162010PhRvD..81i4016G. 10.1103/PhysRevD.81.094016
Romanets O, Tolos L, Garcia-Recio C, Nieves J, Salcedo L, Timmermans R. Phys. Rev. D. 2012; 85: 1140322012PhRvD..85k4032R. 10.1103/PhysRevD.85.114032
Garcia-Recio C, Nieves J, Romanets O, Salcedo L, Tolos L. Phys. Rev. D. 2013; 87: 0340322013PhRvD..87c4032G. 10.1103/PhysRevD.87.034032
Garcia-Recio C, Nieves J, Romanets O, Salcedo LL, Tolos L. Phys. Rev. D. 2013; 87: 0740342013PhRvD..87g4034G. 10.1103/PhysRevD.87.074034
Tolos L. Int. J. Mod. Phys. E. 2013; 22: 13300272013IJMPE..2230027T. 10.1142/S0218301313300270
Garcia-Recio C, Hidalgo-Duque C, Nieves J, Salcedo LL, Tolos L. Phys. Rev. D. 2015; 92: 0340112015PhRvD..92c4011G. 10.1103/PhysRevD.92.034011
J. Nieves, R. Pavao, (2019), arXiv:1907.05747 [hep-ph]
Oller J, Meissner UG. Phys. Lett. B. 2001; 500: 2632001PhLB..500..263O. 10.1016/S0370-2693(01)00078-8
Jido D, Oller J, Oset E, Ramos A, Meissner U. Nucl. Phys. A. 2003; 725: 1812003NuPhA.725..181J. 10.1016/S0375-9474(03)01598-7
Hyodo T, Weise W. Phys. Rev. C. 2008; 77: 0352042008PhRvC..77c5204H. 10.1103/PhysRevC.77.035204
R. Aaij et al. (LHCb). Phys. Rev. Lett. 109, 172003 (2012). arXiv:1205.3452 [hep-ex]
Nieves J, Pavao R, Tolos L. Eur. Phys. J. C. 2018; 78: 1142018EPJC...78..114N. 10.1140/epjc/s10052-018-5597-3
Nieves J, Ruiz Arriola E. Phys. Rev. D. 2001; 64: 1160082001PhRvD..64k6008N. 10.1103/PhysRevD.64.116008
Garcia-Recio C, Nieves J, Salcedo LL. Phys. Rev. D. 2006; 74: 0360042006PhRvD..74c6004G. 10.1103/PhysRevD.74.036004
Hofmann J, Lutz MFM. Nucl. Phys. A. 2006; 776: 172006NuPhA.776...17H. 10.1016/j.nuclphysa.2006.07.004
Garcia-Recio C, Geng L, Nieves J, Salcedo L. Phys. Rev. D. 2011; 83: 0160072011PhRvD..83a6007G. 10.1103/PhysRevD.83.016007
Guo F-K, Meißner U-G, Zou B-S. Commun. Theor. Phys.. 2016; 65: 5932016CoTPh..65..593G. 10.1088/0253-6102/65/5/593
Albaladejo M, Nieves J, Oset E, Sun Z-F, Liu X. Phys. Lett. B. 2016; 757: 5152016PhLB..757..515A. 10.1016/j.physletb.2016.04.033
J. Yelton et al. (Belle). Phys. Rev. D 94, 052011 (2016). arXiv:1607.07123 [hep-ex]
Yoshida T, Hiyama E, Hosaka A, Oka M, Sadato K. Phys. Rev. D. 2015; 92: 1140292015PhRvD..92k4029Y. 10.1103/PhysRevD.92.114029
Garcia-Recio C, Nieves J, Ruiz Arriola E, Vicente Vacas MJ. Phys. Rev. D. 2003; 67: 0760092003PhRvD..67g6009G. 10.1103/PhysRevD.67.076009
Hyodo T, Nam S, Jido D, Hosaka A. Phys. Rev. C. 2003; 68: 0182012003PhRvC..68a8201H. 10.1103/PhysRevC.68.018201
Garcia-Recio C, Lutz MFM, Nieves J. Phys. Lett. B. 2004; 582: 492004PhLB..582...49G. 10.1016/j.physletb.2003.11.073
Hyodo T, Jido D. Prog. Part. Nucl. Phys.. 2012; 67: 552012PrPNP..67...55H. 10.1016/j.ppnp.2011.07.002
Gamermann D, Garcia-Recio C, Nieves J, Salcedo L. Phys. Rev. D. 2011; 84: 0560172011PhRvD..84e6017G. 10.1103/PhysRevD.84.056017
Kamiya Y. Nucl. Phys. A. 2016; 954: 412016NuPhA.954...41K. 10.1016/j.nuclphysa.2016.04.013
R. Aaij et al. (LHCb). Phys. Rev. Lett. 122, 012001 (2019). arXiv:1809.07752 [hep-ex]
Liang W-H, Dias JM, Debastiani VR, Oset E. Nucl. Phys. B. 2018; 930: 5242018NuPhB.930..524L. 10.1016/j.nuclphysb.2018.03.008
]
[
Footnotes
The details of this double pole structure, generated by the , ND and coupled-channels dynamics, depend strongly on the adopted renormalization scheme, which could considerably enhance the role played by the two latter channels around the resonance energy. This is discuss in detail in Ref. [[28]].
From now on we refer to excited and independently of or spin–parity assignment.
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By J. Nieves; R. Pavao and L. Tolos
Reported by Author; Author; Author