Exploring the spectrum of the hidden charm strange pentaquark in an SU(4) flavor-spin model

We study the spectrum of the isoscalar pentaquark $udsc\overline{c}$, of either positive or negative parity, in a constituent quark model with linear confinement and a flavor-spin hyperfine interaction previously extended to SU(4) and used to describe the spectrum of the $uudc\overline{c}$ pentaquarks observed at LHCb in 2019. For positive parity we make a distinction between the case where one unit of angular momentum is located in the subsystem of four quarks and the case where the angular momentum is located in the relative motion between a ground state four-quark subsystem and the antiquark. The novelty is that we introduce the coupling between different flavor states, due to the breaking of exact SU(4)-flavor symmetry of the Hamiltonian model, both for positive and negative parity states. An important consequence is that the lowest state, located at 4404 MeV, has quantum numbers $J^P$ = $1/2^-$ while without coupling the lowest state has $J^P$ = $1/2^+$ or $3/2^+$.


I. INTRODUCTION
The 2019 LHCb observation of the narrow structures P + c (4312), P + c (4440) and P + c (4457) in the Λ 0 b → J/ψK − p decay [1] has given a new impetus to the study of hidden charm pentaquarks. The J/ψp component suggested that the pentaquark wave functions should have the flavor content uudcc.
A more general point of view has been adopted in Ref. [17] where the P c (4312) signal was analyzed by using some general principles of the S-matrix theory. In this way it was concluded that P c (4312) is more likely a virtual (unbound) molecular state.
The 2019 LHCb pentaquarks have also been analyzed in compact pentaquark models based on the chromomagnetic interaction of the one gluon exchange model, with quark/antiquark correlations [18] or without correlations [19,20]. In both cases the lowest state has negative parity.
Presently, the spin and parity of the narrow structures P + c (4312), P + c (4440) and P + c (4457) remains to be established experimentally.
Anticipating new experiments, the 2019 LHCb successful observation stimulated interest in the theoretical study of analogue pentaquarks in particular of the hidden charm pentaquarks with strangeness, the udscc system. For example, in Ref. [21] it has been analyzed in the framework of a molecular scenario with heavy quark symmetry constraints and in Ref. [22] within the chiral effective theory where the short range contact interaction, the long range one-pion-exchange and the intermediate range two-pion-exchange interaction were included. In Ref. [23] the hidden charm pentaquarks with strangeness have been considered in the hadrocharmonium model.
Predictions for the isoscalar udscc pentaquark have already been made previously. In Ref. [24] the spectrum of the udscc pentaquark was studied in the compact pentaquark picture in a quark model with either the chromomagnetic, the flavor-spin or the instanton induced interaction. In all cases it was found that the lowest state has the spin-parity J P = 1/2 − . In Ref. [25] the stability of several pentaquark systems has been analyzed in a constituent quark model with a simple chromomagnetic interaction, and the udscc pentaquark has been found among the most stable ones.
In an SU(4) classification of pentaquarks and its decomposition in SU (3)  The hidden charm pentaquarks having a strange quark are presently unknown. In principle they can be produced and observed, for example, in the study of the Ξ − b → J/ψΛK − reaction [21] or in the decay of Λ b into J/ψΛK 0 [28]. Their discovery would require much more data relative to the non-strange hidden charm pentaquarks observed at LHCb [29]. If discovered they may possibly distinguish between the various theoretical pictures.
Here we explore the spectrum of the pentaquark udscc within a quark model [30], which has a flavor dependent hyperfine interaction. The hyperfine splitting in hadrons is due to the short-range part of the Goldstone boson exchange interaction between quarks. The merit of the flavor-spin (FS) model is that it reproduces the correct ordering of positive and negative parity states of both nonstrange and strange baryons [30][31][32] in contrast to the one gluon exchange (OGE) model. However, it cannot explain the hyperfine splitting in mesons, because it does not explicitly contain a quark-antiquark interaction.
It is therefore useful to compare the spectrum of hidden charm nonstrange and hidden charm strange pentaquarks within the same model.
In a previous work [33] the model of Ref. [30] has been generalized from SU(3) to SU (4) in order to incorporate the charm quark. The extension has been made in the spirit of the phenomenological approach of Ref. [34] where, in addition to Goldstone bosons of the hidden approximate chiral symmetry of QCD, the flavor exchange interaction was augmented by an additional exchange of D mesons between u, d and c quarks and of D s mesons between s and c quarks. The model provided a satisfactory description of the heavy flavor baryons.
The extended SU(4) flavor-spin model has been applied to the study of uudcc pentaquarks. Presently we study the pentaquarks of structure udscc in the same framework considering both positive and negative parities.
The parity of the pentaquark is given by P = (−) ℓ + 1 , where ℓ is the orbital angular momentum. As shown in Ref. [33], there are two ways to introduce orbital excitations. For the lowest positive parity states one way is to introduce an angular momentum ℓ = 1 in the internal motion of the four-quark subsystem and the other is to introduce an unit of angular momentum in the relative motion between a ground state four-quark subsystem and the antiquark. According to the Pauli principle, in the first case the four-quark subsystem must be in a state of orbital symmetry [31] O . In the second case the four-quark subsystem is in the ground state [4] O .
In Ref. [33], in the context of a schematic flavor-spin interaction, i .e. exact SU(4) symmetry, it was shown that the lowest pentaquark state has a positive parity with the orbital excitation in the internal motion of the four-quark subsystem. Although the kinetic energy of such a state is higher than that of the totally symmetric [4] O state of negative parity, the flavor-spin interaction overcomes this excess and generates a lower eigenvalue for the [31] O state with an s 3 p configuration than for [4] O with an s 4 configuration.
In the exact SU(4) limit the strength of the interaction is the same for all pairs, independent of the quark masses, and it is a constant as a function of the relative distance between the interacting quarks. The model Hamiltonian introduced in the next section breaks the SU(4)-flavor symmetry through the quark masses and the radial dependence of the interaction potential. We calculate the masses of the lowest positive and negative parity states of the pentaquarks of structure udscc considering states with flavor symmetry [22] F , [31]

II. THE HAMILTONIAN
Here we closely follow the description of the model as presented in Ref. [33]. The parameters required by the incorporation of the strange quark were added.
The FS model Hamiltonian has the general form [30] with m i and p i denoting the quark masses and momenta respectively and r ij the distance between the interacting quarks i and j. The Hamiltonian contains the internal kinetic energy and the linear confining interaction The hyperfine part V χ (r ij ) has a flavor-spin structure extended to SU(4) in Ref. [33]. One with the SU(4) generators λ F i (F = 1,2,...,15)and λ 0 i = 2/3 1, where 1 is the 4 × 4 unit matrix.
In the SU(4) version the interaction (3) contains γ = π, K, η, D, D s , η c and η ′ mesonexchange terms. Every V γ (r ij ) is a sum of two distinct contributions: a Yukawa-type potential containing the mass of the exchanged meson and a short-range contribution of opposite sign, the role of which is crucial in baryon spectroscopy. For a given meson γ the meson exchange potential is In the present calculations we use the parameters of Ref. [31] to which we add the µ D and the µ Ds masses and the coupling constants and . These are The meson masses correspond to the experimental values from the Particle Data Group [35].
As discussed in the following, we ignore the η c -exchange.
The model of Ref. [31] has previously been used to study the stability of open flavor tetraquarks [36] and open flavor pentaquarks [37,38]. Accordingly, for the quark masses we take the values determined variationally in Refs. [36,37]  After integration in the flavor space, the two-body matrix elements containing contribu-tions due to light, strange and charm quarks are [33] In Eqs. (6) the pair of quarks ij is either in a symmetric [2] F or in an antisymmetric [11] Table I.
In the case of udscc pentaquarks there are also non-vanishing off-diagonal matrix elements. These are  indicates the flavor of the interacting quark pair.
Note that the integration in the orbital space is not yet performed in the diagonal and off-diagonal matrix elements presented above.
To reproduce the exact SU(4) limit one has to take Then, in the exact SU(4) limit, the flavor-spin interaction takes the following form [33] with C χ an equal strength constant for all pairs. Using Appendix A, one can check that the diagonal matrix elements of Table I are -27 C χ , -21 C χ , -15 C χ , -15 C χ and -7 C χ respectively. In the exact SU(4) limit the off-diagonal matrix elements of V χ vanish identically. Thus the lowest state of Table I is |1 because it acquires the largest attraction due to the FS interaction in the exact SU (4) limit. This implies that the lowest state has positive parity, conclusion which sometimes still hold at broken symmetry, as for example for the uuddc pentaquarks [38].

III. ORBITAL SPACE
The orbital wave functions are defined in terms of four internal Jacobi coordinates for pentaquarks chosen as where 1,2,3 and 4 are the quarks and 5 the antiquark so that t gives the distance between the antiquark and the center of mass coordinate of the four-quark subsystem.
For the lowest positive parity states having ℓ = 1, there are two ways to introduce orbital excitations [33]. One is to excite the four-quark subsystem, the other is to include the angular momentum in the relative motion between the four-quark subsystem and the antiquark. Both imply translational invariant states (no center of mass motion).

Liu:2019tjn
In this case one has to express the orbital wave functions of the four-quark subsystem of structure s 3 p in terms of the internal coordinates x, y, z for the specific permutation symmetry [31] O . The method of constructing translationally invariant states of definite permutation symmetry containing a unit of angular momentum was first given in Ref. [38] and recently revised in Ref. [33]. The three independent states denoted below by ψ i , which define the basis vectors of the irreducible representation [31] O in terms of shell model states r |nℓm where n = 0, ℓ = 1, are In this picture there is no excitation in the relative motion between the cluster of four quarks and the antiquark defined by the coordinate t. Then the pentaquark orbital wave functions ψ 5 i are obtained by multiplying each ψ i from above by the wave function t |000 which describes the relative motion between the four-quark subsystem and the antiquark c. Assuming an exponential behavior we introduce two variational parameters, a for the internal motion of the four-quark subsystem and b for the relative motion between the subsystem qqqc and c. We explicitly have where N = 2 3/2 a 11/4 b 3/4 3 1/2 π 5/2 (18) B. Excitation between the four-quark subsystem and the antiquark, P = +1 The authors of Ref. [24] have studied the qqqcc and the qqscc pentaquarks, in three different models, including the FS model. The orbital wave function of the four-quark subsystem has symmetry [4] O for both parities. Although the radial wave function was not specified, one can infer that the positive parity states of Ref. [24] were obtained by including a unit of orbital angular momentum in the relative motion between the four-quark subsystem and the antiquark. The states remains translationally invariant. In this case the orbital wave function takes the form where C. Negative parity states, P = -1 We also need the orbital wave function of the lowest negative parity state described by the s 4 configuration of symmetry [4] O which is with

IV. KINETIC ENERGY
The kinetic energy T of the Hamiltonian (1) can be calculated analytically. Below we present the expression of its expectation value for the three cases introduced above.
Case A. In this case the expectation value of the kinetic energy is defined by the average over the three wave functions defined by Eqs. (15)- (17). One obtains which is the generalization of Eq. (22) of Ref. [33] to include strange quarks and where µ 1 and µ 2 are the same as above. Again one can recover the identical particle limit when a = b but the contributions of the two terms are different because the coefficients 11/2 and 3/2 now become 9/2 and 5/2 respectively, which is natural because the unit of orbital excitation is no more located in the four quark subsystem but in the relative motion between the four-quark subsystem andc.
Case C. One deals with the symmetric state [4] O and no orbital excitation. The only orbital state has negative parity and Eq. (21) gives with µ 1 and µ 2 as above.

V. CONFINEMENT
By integrating in the color space, the expectation value of the confinement interaction (2) has the same form as that of the uudcc system [33] V conf = C 2 (6 r 12 + 4 r 45 ) where r ij is the interquark distance and the coefficients 6 and 4 account for the number of quark-quark and quark-antiquark pairs, respectively, for all cases A, B and C, but with different expressions for r ij in each case.
Case A. Here one has where i, j = 1,2,3,4,5 (i = j). An analytic evaluation gives and Case B. The expectation value of the confinement interaction is given by Eq. (28) with and Case C. In this case the four quarks are in the s 4 configuration described by the states |3 , |3 ′ or |4 and there is no orbital excitation at all. The expectation value of the confinement interaction is given by Eq. (28) as well, with and

VI. FLAVOR-SPIN INTERACTION
In order to integrate the expressions of Table I and Eqs. For states of type B or C there are two-body matrix elements between single particle s-states, ss . In every term q a q b is a pair of quarks from Eq. (6).

VII. RESULTS AND DISCUSSION
We have looked for variational solutions of the Hamiltonian of Sec. II using the orbital part of the wave functions as described in Sec. III, which contain the parameters a and b. The wave functions are the product of the four quarks subsystem states of flavor-spin structure defined in Table I  We have neglected the contribution of V uu ηc , V uc ηc and V sc ηc because little uū, dd and ss are expected in η c . We have also neglected V uc η ′ and V sc η ′ assuming a little cc component in η ′ . Thus, in the expressions of Table I we took For Case A the numerical results are presented in Table II. The eigenvalues of |1 |c and For Case B the masses and the mixing coefficients of the 1/2 + and 3/2 + states, obtained from the combination of the basis vectors |3 |c , |3 ′ |c and |4 |c are presented in Table III.
The optimal variational parameters are the same as in  (9) where the dominant π-and K-meson exchanges contribute with the same sign.
The Case C corresponding to negative parity 1/2 − state is shown in Table IV. The mixing coefficients are the same as those of Table III, because they result from the diagonalization of a hyperfine interaction identical to that of Case B. The difference between these cases appears only in the kinetic and the confinement matrices, which are diagonal. Hence, in Case C the masses can be obtained from those of Table III by  Looking at Tables II, III and IV one can see that the lowest mass is 4404 MeV. Thus the lowest pentaquark udscc has quantum numbers J P = 1/2 − , in contrast to the lowest pentaquark uudcc for which it was found J P = 1/2 + in Ref. [33].  Table I  The mixing of states |3 |c , |3 ′ |c and |4 |c has been first discussed in Ref. [24] with the corresponding notation |3 → |1 , |3 ′ → |1 ′ and |4 → |2 where the quark model of Ref.
[34] with a harmonic oscillator confinement and a simplified hyperfine interaction have been used. The mixing was introduced for J P = 1/2 − only, case C. There the J P = 1/2 − state appears at 4084 MeV and the J P = 1/2 + state at 4291 MeV, i. e. about 200 MeV above the lowest negative parity state. Thus the lowest J P = 1/2 − state of Ref. [24] is about 300 MeV lower than in the present case.
The J P = 1/2 − states found in this study are located within the energy range of the J P = 1/2 − resonances predicted in Ref. [21]. There only s-wave meson-baryon interactions were considered so that only negative parity states were discussed. Their coupling to the J/ψΛ channel was found to be small, but large enough to provide convenient production rates. The masses of hidden charm strange pentaquarks with J P = 1/2 − found in Ref. [22] within a chiral effective field theory are located as well in the energy range predicted in the present work. A similar mass range was found in Ref. [23] in a hadrocharmonium picture, with the difference that the lowest state has positive parity.

VIII. CONCLUSIONS
We have calculated a few of the lowest masses of the hidden charm strange pentaquarks udscc, in the SU(4) version of the flavor-spin model introduced in Ref. [33] where it was applied to uudcc pentaquarks. The model provides an isospin dependence and an internal structure of pentaquarks. For positive parity the angular momentum can be located in the internal motion of the four-quark subsystem, Case A, or in the relative motion between the four-quark subsystem and the antiquark, Case B.
According to the discussion presented in Ref. [33] at exact SU(4) symmetry the lowest positive pentaquark state has positive parity when the orbital excitation is located in the internal motion of the four-quark subsystem. For broken SU(4) such a result remained valid for the uudcc pentaquark. In the present analysis it was found that the lowest state of the udscc pentaquark has negative parity. This is due to the breaking of SU (4) In the uudcc pentaquark, there are only two flavor states which, in principle, can couple due to the breaking of SU(4). They are of type |3 and |4 with appropriate Weyl tableaux.
We found out that the coupling between the states of symmetry We recall that the parity sequence of the uudcc pentaquark studied in the hadrocharmonium model [40] was similar to ours [33], namely that the lowest pentaquark state has J P = 1/2 + quantum numbers. In the hadrocharmonium description of Ref. [23] the lowest state of the udscc pentaquark has positive parity, contrary to the present result.
Therefore, in the flavor-spin model the presence of the strange quark brings more richness to the flavor structure and changes the parity order of the lowest two state in the udscc pentaquark relative to the uudcc pentaquark.
The J P quantum numbers of the 2019 LHCb resonances are not yet known. Likewise, for possible future observations the spin and parity will be essential to discriminate between the existing interpretations of pentaquarks, or inspire new developments.
Appendix A: Exact SU(4) limit The exact SU(4) limit is useful in checking the integration in the flavor space, made in Table I. In this limit every expectation value of Table I reduces to the expectation value of Eq. (10) and one can use the following formula [27] i<j where n is the number flavors and k the number of quarks, here n = 4 and k = 4. C SU (n) 2 is the Casimir operator eigenvalues of SU(n) which can be derived from the expression [41] : where f ′ i = f i − f n , for an irreducible representation given by the partition [f 1 , f 2 , ..., f n ]. Eq. (A1) has been previously used for n = 3 and k = 6 in Ref. [41]. better. However, we prefer to use the same parametres as in Ref. [33] in order to make a comparison with the uudcc pentaquarks.  [24] for I = 0. We have checked them with the method of Ref. [42]. In Ref. [24] the flavor states were defined in the Young-Yamanouchi basis. The order of particles is always 1234 in every term.
In Table VI, except for [1111] F , not needed here, we give the correspondence between the Young-Yamanouchi basis and the notation of Ref. [24] for each Yamanouchi symbol which is a compact notation for a Young tableau. For a tableau with n particles it is defined by Y = (r n , r n−1 , ..., r 1 ) where r i represents the row of the particle i. The Weyl tableaux are indicated for each irreducible representation.
Here we write the flavor states in terms of products of symmetric φ [2] (q a q b ) = (q a q b + q b q a )/ √ 2 or antisymmetric φ [11] (q a q b ) = (q a q b − q b q a )/ √ 2 quark pair states for the pairs 12 and 34. This allows a straightforward calculation of the flavor integrated matrix elements (6) and in addition one can easily read off the isospin of the corresponding wave function.
For irrep [31] F the vectors [31] F 1 and [31] F 2 have to be combined in the so called Young-Yamanouchi-Rutherford basis first proposed in the context of nuclear physics [43,44]. It is defined such as the last two particles are either in a symmetric or an antisymmetric state The pair 12 is also in a symmetric or an antisymmetric state, which is very advantageous.