Can discovery of hidden charm strange pentaquark states help determine the spins of $P_c(4440)$ and $P_c(4457)$?

The pentaquark states, $P_{c}(4312)$, $P_{c}(4440)$ and $P_{c}(4457)$, could be nicely arranged into a multiplet of seven molecules of $\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}$ dictated by heavy quark spin symmetry. However, the spins of $P_c(4440)$ and $P_c(4457)$ are not yet fully determined. In this work we employ the contact-range effective field theory to investigate the $SU(3)$-flavor counterparts of $\bar{D}^{(\ast)}\Sigma_{c}^{(\ast)}$, and study the possibility whether their discovery can help determine the spins of $P_c(4457)$ and $P_c(4440)$. We find the existence of a complete hidden charm strange multiplet of $\bar{D}^{(\ast)}\Xi_{c}^{(\prime\ast)}$ molecules irrespective of the spins of $P_c(4440)$ and $P_c(4457)$. On the other hand, we find that although molecules of $\bar{D}^{(\ast)}\Xi_{c}$ are also likely, depending on the realization of the underlying dynamics, their discovery can be more useful to determine the spins of $P_{c}(4440)$ and $P_{c}(4457)$ and to tell how the heavy quark and light quark interaction depends on the spin of the light quark pair.

Symmetry is a core concept in particle physics and plays an important role in studying heavy hadronic molecules. Two symmetries relevant to the present work are heavy quark spin symmetry (HQSS) and heavy antiquark diquark symmetry(HADS). The HQSS dictates that the strong interaction is independent of the spin of the heavy quark in the limit of heavy quark masses [34,35], which provides a natural explanation to the mass difference of (D, D * ) and (B, B * ), as well as those of their baryon counterparts. Applying HQSS to studying heavy hadronic molecules, D s0 (2317) and D s1 (2460) can be naturally interpreted as DK and D * K molecules of spin doublets [36,37]. The HQSS also plays an important role in describing the three pentaquark states, P c (4312), P c (4440), and P c (4457), asD * Σ c molecules. Particularly, we obtained a complete multiplet of hadronic molecules in theD ( * ) Σ ( * ) c system in both the EFT approach and OBE model [13,14], which has been later corroborated by many studies [15,[18][19][20][21]. In addition to HQSS, heavy anti-quark di-quark symmetry(HADS) has also been used to study heavy hadronic molecules [33,38] and to estimate the coupling of doubly charmed baryon to pion [39]. In Refs. [33,38], we extended theD ( * ) Σ ( * ) cc system via HADS, and predicted a complete multiplet of triply charmed hadronic molecules. In particular, we pointed out that the splitting of Ξ cc Σ c states are correlated with the spins of P c (4440) and P c (4457), which, given the fact that the former can be much easily simulated on the lattice [40], provides a possibility to determine the spins of the later in a model independent way.
Along this line, in the present work, we explore whether one can relate the pentaquark states P c (4440) and P c (4457) to other states via symmetries such that their discovery could shed light on the nature of the P c states, particularly their spin in the molecular picture. The symmetry of current interest is SU(3)-flavor symmetry, which relates the hidden charm states to hidden charm strange states. As a result, the discovery of the latter will shed light on the nature of the former.
Before P c (4380) and P c (4450) were discovered by the LHCb collaboration, Wu et al. had already predicted 4 hidden charm strange pentaquark states through local hidden gauge Lagrangian in combination with unitary techniques in couple channels [6]. After the discovery of three pentaquark states in 2019, the study was updated and 10 hidden charm strange pentquark states were predicted [41]. In Ref. [42] two partners of P c (4380) and P c (4450) were predicted in the OBE model. More recently, Wang et al. also predicted the existence of 10 hidden charm strange pentaquark states in the chiral effective field theory [43]. The discovery potential of hidden charm strange pentaquark states in the J/ψΛ invariant mass spectra of the Ξ b → J/ψΛK and Λ b → J/ψΛK decays have been explored [44,45], as well as their partial decay widths [46].
In this work we adopt an effective field theory (EFT) approach to study possible hidden charm strange molecules ofD ( * ) Ξ c andD ( * ) Ξ ′ * c , which can be regarded as the SU(3)-flavor counterparts ofD ( * ) Σ P c (4457) but also can help determine their spin ordering if the hidden charm strange pentaquark states are discovered by either experiments or lattice QCD calculations, analogous to the correlation dictated by HADS as shown in Refs. [33,38].
The manuscript is structured as follows. In Sec. II we present the details of the contact-range potential ofD ( * ) Ξ c andD ( * ) Ξ ′ * c according to heavy quark spin symmetry and SU(3) flavor symmetry. In Sec. III we give the full spectrum of hidden charm strange molecules. Finally we present the conclusions in Sec. IV

II. THEORETICAL FORMALISM
Here we explain how to determine theD ( * ) Ξ c andD ( * ) Ξ ′ * c interactions and study the likely existence of hidden charm strange pentaquark states. In the line of Refs. [13,33], their interactions can be determined in an EFT approach. One should note that we just consider the leading order contact-range potentials because our previous studies indicated that the pion exchange contributions are perturbative in the charm sector [47,48].
c are the same in the heavy quark mass and SU(3)-flavor symmetry limits. As a result, the same two low energy constants are needed to account for the contact-range There are seven combinations for theD ( * ) Ξ ′( * ) c system according to HQSS, whose potential can be written as The interaction of theD ( * ) Ξ c system is different from that ofD ( * ) Ξ ′ * c because the spin of the light quark pair in Ξ c is 0 and that in Ξ ′ * c is 1. In terms of HQSS, the contact-range potential betweenD ( * ) and Ξ ′ * c can be denoted as F 1/2 and F 3/2 via the coupling of the light quark spins, i.e., 1/2 ⊗ 1 = 1/2 ⊕ 3/2. Applying the same approach toD ( * ) Ξ c , the corresponding potential can be represented by one low energy constant F ′ 1/2 which is from the light quark spin coupling, 1/2 ⊗ 0 = 1/2.
In principle, F 1/2 and F ′ 1/2 can be different, and there is no reliable way to relate them. In this work, we will take two assumptions and rely on future experiments or lattice QCD simulations to verify which assumption is realized in nature.
First, we assume that F 1/2 in theD ( * ) Ξ ′ * c system and F ′ 1/2 in theD * Ξ c system are the same, which is denoted as Case I in the following. To find the relationship between theD ( * ) Ξ ′ * c system and theD ( * ) Ξ c system, the couplings of F 1/2 and F 3/2 can be rewritten as F 1/2 = C a − 2C b and Then the contact-range potentials ofD ( * ) Ξ c have the following form One should note that theD ( * ) Ξ ′( * ) c contact-range potential can also be denoted as F 1/2 and F 3/2 in terms of HQSS.
Second, we assume that F ′ 1/2 is not the same as F 1/2 , and turn to some phenomenological methods for help, which is denoted as Case II. One of such phenomenological methods is the local hidden gauge approach. According to Ref. [41], the contact-range potential of theD ( * ) Ξ c system is written as Clearly, the strength is the same for all the three channels, but it is different from that of Case I.
We hope that future experimental or lattice QCD data will tell which case is realized in nature.
To search for bound states, we solve the Lippmann-Schinwinger equation with contact range where φ(k) is the vertex function, B the binding energy, and µ the reduced mass. To solve the equation we have to regularize the contact potential with Λ the cutoff, f (x) a regular function, and C(Λ) the running coupling constant. A typical choice of the cutoff is Λ = 0.5 − 1 GeV, while for the regulator we choose a gaussian type f (x) = e −x 2 . In this work, we only consider S-wave contact contribution, thus the integral equation

III. NUMERICAL RESULTS AND DISCUSSIONS
From HQSS and SU(3)-flavor symmetry, we express the contact potentials ofD ( * ) and Ξ ′ * c by two coupling constants, C a and C b , which are the only two unknown inputs for us to obtain the binding energies of meson-baryon systems under consideration. In our previous works, we proposed that P c (4440) and P c (4457) are bound states ofD * Σ c with either spin 1/2 or 3/2 and negative party, and the two LECs C a and C b have been determined by fitting to the masses of P c (4440) and P c (4457). As a result, in the following, we study two scenarios, Scenario A where P c (4440) and P c (4457) have 1/2 − and 3/2 − , and Scenario B the other way around. The results are displayed in Table I. We find that all the seven states in theD ( * ) Ξ ′ * c system bind in both scenario A and B, consistent with Refs. [41] and [43]. It indicates that such kinds of hidden charm strange molecules must exist. In addition their results favor our scenario A, namely, P c (4440) and P c (4457) have spin 1/2 and 3/2, respectively.
From Table I  bound states [33], the cutoff dependence is much weaker, which implies that SU(3)-flavor symmetry is less broken than HADS. To estimate the breaking of SU(3)flavor symmetry, we supplement the potentials with a 15% uncertainty. The results taking into account the breaking show that all the states still bind and the spin orderings remain unchanged, which suggests that the hidden charm strange molecules must exist if P c (4312), P c (4440) and P c (4457) are (dominantly)D ( * ) Σ c molecules. Compared with their hidden charm partners, these states could be detected in the J/ψΛ invariant mass spectra of the Ξ b → J/ψΛK decay. If these states are discovered experimentally, it will not only further enrich the family of hadronic molecules but also helps to determine the spin ordering of P c (4440) and P c (4457).  For theD ( * ) Ξ c system, Case I assumes that the coupling F 1/2 is the same as the coupling F ′ 1/2 , and therefore the contact-range potential ofD ( * ) Ξ c can also be written as combinations of C a and C b . Thus we can calculate the binding energies of theD ( * ) Ξ c systems in the two scenarios A and B as well. The results are shown in Table II. Interestingly, we note that the binding energies in Scenario A are much larger than their counterparts in Scenario B. To estimate the uncertainty of SU(3)-flavor symmetry and the assumed equality of F 1/2 and F ′ 1/2 , we consider a larger uncertainty of 30% into the C a and C b values. We find that that theD ( * ) Ξ c systems in Scenario B can become unbound, while they still bind in Scenario A, which indicates that theD ( * ) Ξ c hidden charm strange molecules exist in Scenario A and may not necessarily exist in Scenario B. If such molecules are found experimentally, it implies that the spins of P c (4440) and P c (4457) are more likely to be 1/2 and 3/2, respectively.
In Case II we estimate the coupling F ′ 1/2 by turning to the local hidden-gauge approach. As shown above, the value is different from that of Case I. The corresponding results are tabulated in Table II. We find that the differences between the binding energies in scenario A and those in Scenario B become less extreme, which implies that we can not discriminate the spins of P c (4440) and P c (4457) in Case II. As a result, it may not help much to derive the spin ordering of P c (4457) and P c (4440) even these states are discovered experimentally. On the other hand, their discovery does help to reveal more the nature of the hidden charm pentaquark states.

IV. SUMMARY
In this work assuming that P c (4312), P c (4440) and P c (4457) areD ( * ) Σ ( * ) c molecules, we employed a contact-range effective field theory approach satisfying HQSS and SU(3)-flavor symmetry to study the likely existence of hidden charm strange molecules with the main purpose to determine the spin ordering of P c (4440) and P c (4457). To estimate the uncertainty caused by the breaking of these symmetries, we considered a 15% breaking forD ( * ) Ξ ′ * c and a 30% breaking forD ( * ) Ξ c in our study. Our results showed that there exists a complete multiplet of hadronic molecules in theD ( * ) Ξ ′( * ) c system irrespective of the spins of P c (4440) and P c (4457).
Assuming that the couplings F 1/2 and F ′ 1/2 are the same, the existence ofD ( * ) Ξ c molecules is only likely if P c (4457) has spin 3/2 and P c (4440) has spin 1/2. As a result, the discovery of these states can help to determine the spins of P c (4440) and P c (4457) in the molecular picture from an EFT perspecive. On the other hand, using the hidden gauge approach to infer the coupling of F ′ 1/2 , D ( * ) Ξ c molecules exist irrespective of the spins of P c (4440) and P c (4457), which thus offers little help in determining their spins.
Note added in proof: In a recent talk, the LHCb Collaboration has reported the likely existence of a hidden charm strange pentaquark P cs (4459) with a statistical significance of 3.1 sigma [49], which has inspired a number of theoretical studies [50][51][52].