Hidden charm pentaquarks with color-octet substructure in QCD Sum Rules

We study the hidden-charm pentaquark states udscc̄ with spins 1/2, 3/2, and 5/2 within the QCD sum rules approach. First, we construct the currents for the particular configuration of pentaquark states, that consist of the flavor singlet three-quark cluster uds of spins 1/2 and 3/2 and the twoquark cluster c̄c of spin 1, where both clusters are in color-octet state. To obtain QCD sum rules, the operator product expansion for the correlators is performed up to dimension-10 condensates. The extracted masses from QCD sum rules for the hidden-charm pentaquark states uds-c̄c are about 4.6 GeV (5.6 GeV) for spin 1/2±, about 5.1 GeV (6.0 GeV) for spin 3/2±, about 6.1 GeV (5.9 GeV) for spin 5/2±, where the masses of positive parity states given in parentheses. Additionally, based on the flavor singlet pentaquark states, it is also shown that other pentaquark states of the clusters as udc-c̄s and usc-c̄d lead to the similar masses to uds-c̄c case within error bars. Furthermore, in order to see whether any of the states, observed by LHCb Collaboration, could be understood as the pentaquark of the two clusters in the color-octet state, we study the pentaquark formed by the two clusters udc-c̄u, where the three-quark cluster is assumed to have the same flavor structure as the above uds cluster. We come to the conclusion that if the observed pentaquark will be found to have spin 1/2 and negative parity, then it could be described as a state of two color-octet clusters.


I. INTRODUCTION
Since the observation of the two exotic hidden-charm pentaquark P + c states of the quark content uudcc with the spins 3/2 and 5/2 through the decay Λ 0 b → J/ψK − p by LHCb collaboration [1], many studies on these states and other expected hidden-charm pentaquark states have been performed. Note that recently the LHCb collaboration has observed a three peak structure [2] using an updated analysis. The pentaquark states, including the above hidden-charm pentaquark states, were theoretically studied using quark models [3][4][5], diquark models [6][7][8][9][10][11][12][13][14][15], hadronic molecular states [16][17][18][19][20], the coupledchannel unitary approach [21][22][23], the contact-range effective field theory [24], and the hadroquarkonia model [25]. For a review on the hidden-charm multiquark states, see [26]. Among the expected hidden-charm pentaquark states, it is very intriguing to analyze the pentaquark state with the quark content udscc, which was considered in [3-5, 27, 28] within quarks models, since Λ 0 b could also decay into J/ψK − p via J/ψΛ * and could be observed through the decay Ξ − b → J/ψΛK − [4]. In this paper, we study first the flavor singlet hiddencharm pentaquark states of udscc with the spins 1/2, 3/2, and 5/2 using QCD sum-rules (SRs). We assume that these states consist of two colored clusters as discussed in [5,29,30] within quark models. So, we consider them as states consisting of the three-quark cluster uds and the two-quark clustercc. Additionally, we assume that all quarks are in an S-wave, the colors of both clusters are color octets, and the two-quark cluster has spin 1

II. INTERPOLATING CURRENTS
First, we consider the wave function of the flavor singlet uds cluster for constructing the three-quark interpolating current J m 3q . Then, we extend our current to the case of three arbitrary flavors. We will take the flavor structure of the interpolating current J m 3q from the flavor singlet wave function in the flavor SU (3) space as J flavor ∼ (ud − du)s + (su − us)d + (ds − sd)u . (1) We study both cases of an uds cluster: one with spin 1/2 and one with spin 3/2 for a total spin 1/2, 3/2, and 5/2 of the pentaquark states. To this end, we adopt the QCD SR method applied to the analysis of the baryon octet [31]. Therefore, we construct the current with the first two quarks contributing spin 0 to the total spin of the three-quark cluster with spin 1/2. On the other hand, in the current for the three-quark cluster with spin 3/2, the first two quarks give spin 1 to the total spin.
With these ingredients and Ioffe's current [31,32] with the definite chiralities which are well known to form a good basis, we consider the following structure of the spin part of the interpolating current of the three-quark cluster with spin 1/2 where the superscript A means that the current is antisymmetric under the exchange of the spinor indexes of the first two quarks. The first term in Eq. (1) is considered as an example, and then the rest is included in the final stage. From the above expression, Γ A must satisfy the following conditions in order to have no zero current where T means the transposition. These conditions limit the choices of Γ A to Γ A = 1, γ 5 . For a uds cluster of spin 3/2, we consider J S spin = 2(u T R CΓ S d L )Γ 2 γ 5 s + (R ↔ L) where the superscript S denotes that the current is symmetric under the exchange of the spinor indexes of the first two quarks. Similarly to the case of the above current for the spin 1/2 case, Γ S must satisfy the following conditions in order to have a nonzero current. Therefore, the only choice is Γ S = γ µ .
Before constructing the full current, we study the currents in color subspace. Using the adjoint representation of color SU (3), the color-octet structure of the current can be constructed as where m is a color index. Other choices for color tensors lead to zero currents or to the same full currents due to the symmetries in the spin and flavor subspaces.
To generalize the uds case, considered above, to other flavors of three-quark clusters, we distinguish the quarks in the three-quark cluster by (q 1 , q 2 , q 3 ). Combining the currents constructed in the flavor, spin and color subspaces, we get the interpolating currents for the considered structure of pentaquark states in the form of Here, the quark fields in the three-quark cluster carry flavor f i , color c i and spin l i indices as q i = q ficili to be contracted with the tensor which becomes These definitions reflect our choice for Γ A = γ 5 and Γ S = γ µ . We denote the flavor configuration by q 1 q 2 q 3 -q 5 q 4 , where q i is the flavor of the i-th quark q i . In this work, as mentioned in the introduction, we consider four cases for a given flavor configuration: uds-cc , udc-cs , usccd and udc-cu. Quarks fields are contracted with the antisymmetric tensor of flavor indices that corresponds to the flavor singlet configuration. Note that the free index l denotes the spinor component of the current and will be omitted in the following discussion. We mention here again that the spinor structure of the three-quark cluster in the full current Eq. (4) is chosen to have the particular structure of q R q R q L − q L q L q R for the antisymmetric case, Eq. (2), and q R q L q + q L q R q for the symmetric case Eq. (3).
The matrix Γ 2 , which can be considered as a factor of the current due to the following properties will be chosen according to the P-parity and the spin of the interpolating current under consideration. As for Γ 3 , following the analysis of [5], where the two-quark cluster with spin 1 in the uds-cc system yielded the most stable result, we will take Γ 3 = γ ν for most pentaquark states considered in this work. An alternative option for Γ 3 = 1 will also be considered.
To discuss the symmetry properties of the constructed three-quark currents in the color-spin subspace, we consider the six-dimensional fundamental representation of the SU (6) group [33,34] composed of the tensor product of the color SU (3) color and the spin SU (2) spin subgroups. Representing a quark by its dimension (3,2) where the first (second) corresponds to the dimension of SU (3) color (SU (2) spin ) subgroup, we have (3, 2)⊗(3, 2)⊗(3, 2) = (8, 2)⊕(8, 4)⊕ · · · , where only two irreducible representations are shown. The first term on the right-hand side has spin 1/2 and belongs to the fully symmetric 56-plet representation, while the second term has spin 3/2 and belongs to the mixed symmetric 76-plet of the full SU (6) group. The color-spin part of the constructed current J A , Eq. (4), represents (8,2) states studied in [5]. The current J S corresponds to (8,2) and (8,4), depending on the choice of Γ 2 .
In this section, we have constructed the general form for the pentaquark currents J 8 ∼ qqq-qq with two coloroctet compounds. The suggested currents are unique and can't be presented by the sum of any other currents considered previously. Nevertheless, omitting the flavor structure, we have related this type of current with the currents of other configurations: diquark-diquarkantiquark clustering J3 ∼ qq-qq-q with an anti-triplet color substructure suggested in [11][12][13], and a molecule form J 1 ∼ qqq-qq with color-singlet parts, see [20]. We conclude that the currents with color-octet parts J 8 , color-singlet parts J 1 , and color-anti-triplet parts J3 could be linearly dependent. For more details see Appendix C.
The correlator Π s (µ)(ν) (q 2 ) for the QCD sum-rule analysis of a pentaquark state is defined by with the interpolating current J (µ) for the considered pentaquark state of spin s. The subscript (µ) stands for the possible Lorentz indices of currents for the s = 3/2, 5/2 states. Since the current J (µ) can couple to the states with a spin lower than s, the phenomenological part of the SRs contains contributions from the lower spin states as well. Extracting the contribution from the state with spin s only, the correlator can be written as whereq = γ · q and · · · means the terms corresponding to the omitted contributions from states with spin s and also lower spins. Therefore, to construct SRs for the state of spin s, one needs to extract Π s 1,2 from the correlator. The ways of extracting Π s 1,2 for s = 3/2, 5/2 are summarized in Appendices A and B. Then, QCD SRs for the state of the spin s will be constructed by applying the dispersion relation [35] to the two scalar functions Π s 1,2 in Eq. (6) Here the spectral densities ρ s i (t) are defined in the physical t region by with i = 1, 2.
In the next subsections A, B, C, we present the relativistic interpolating currents for each state of spin 1/2, 3/2, and 5/2 states with proper choices of Γ 1 , Γ 2 , and Γ 3 . Then, in subsection D, we show how to calculate the spectral densities ρ s i (t) within the OPE for the QCD sum rules for each state.
We consider four types of the current for the spin 1/2 case: , where the upper index denotes the type of the current. The main results for the spin 1/2 case are obtained using the current J 1 , while currents J 2 , J 3 , J 4 are also studied as an alternative option. The interpolating current J 1 with the quantum numbers 1/2 + can be related to the spin-3/2 current as follows µ . The choice of Γ 2 = γ 5 γ µ insures that the spin-3/2 current J µ is projected by Γ 2 only on the 1/2-spin component so that 0|γ µ J µ |3/2 ± ∼ γ µ u µ = 0 thanks to the subsidiary condition for the 3/2 spinor u µ (see Eqs. (11) and (12)).
Since the relativistic interpolating current is considered, as discussed in [36,37], the current can couple to the state of negative parity as well. Denoting two such states by |1/2 + and |1/2 − , the current couples to the states through the following relations with the spinor u. The structure of the correlator becomes and then S 1/2 (µ)(ν) = 1 because there is no Lorentz index in the current. The two spectral densities can be obtained as Tr ImΠ 1/2 (s) .

B. 3/2 ± -states
For spin 3/2 states , we study two types of the current The main results will be obtained by using the current J 1 µ that has the quantum numbers 3/2 − . As in the spin-1/2 case, the interpolating current couples to the states of both parities through the relations with the corresponding spinors u µ [10,19] where the tensor T µν is Note that γ 5 in the first relation in Eq. (12) appears because the current has an intrinsic negative parity. The correlator has the structure Since it is known that the pure contributions from the S = 3/2 state to the correlator can be defined by the terms proportional to S 3/2 (µ)(ν) = −g µν [10,19,38], we show only the relevant terms here. The other terms that contribute to the correlator are given in Appendix A together with the derivation of the exact form for the projectors P 3/2,i µν . As in Appendix A, the two spectral densities can be obtained as for the construction of the SRs in one single form for all spin cases. This factor is related to the intrinsic negative parity of the current, see Eq. (12).
The only type of current studied here is with the choice Γ 2 = γ 5 corresponding to the quantum numbers 5/2 + . This current couples to the states of both parities through the relations [10,19]: T µν,αβ ≡g µαgνβ +g µβgνα 2 −g where symmetrization of the two indices in the curly brackets in the tensor t is imposed by t {µν} = t µν + t νµ .

D. OPE of correlators
In the previous subsections, we constructed various currents for spin 1/2, 3/2, and 5/2 pentaquark states. We specify the current by its three properties: (i) the spin of the pentaquark (1/2, 3/2, 5/2), (ii) the flavor clustering (uds-cc, udc-cs, usc-cd, udc-cu) of the current, (iii) the type of the current. For spin-1/2, we have introduced four options (type-1,2,3,4), for spin 3/2 -two (type-1,2), for spin 5/2 -only one current type-1. The following considerations of this subsection and the next section are based on the general definition of the correlator, Eq. (5), and are relevant to any current considered in the previous subsections.
In order to calculate the two functions Π s 1 and Π s 2 in Eq. (6) within the OPE for each current, we use the quark propagators for both the light quarks (u, d, s quarks) and the heavy quark (c quark) in the configuration space with dimension d = 4 − 2 to control ultraviolet divergences. The heavy quark propagator in the configuration space is given by the α-representation. Our technique for the OPE calculation is similar in some aspects to that discussed in [45]. We treat u, d quarks as massless quarks and include the linear effect of the strange quark mass m s in the OPE. With the hypothesis of the vacuum dominance (HVD) factorization, we perform the OPE up to the dimension-10 vacuum condensates so that where ρ s i,D is the contribution to the OPE from the dimension-D condensate for each case. The various vacuum condensates included in the OPE are listed in Tab. I with reference to the corresponding diagrams shown in Fig.1. It is found that the gluon-condensate contribution is tiny in comparison with the quark-condensate contribution. Therefore, we do not include the contributions The figures (b)-(h) are diagrams for the nonperturbative contributions. We use here nonlocal condensate notation [39][40][41][42][43][44] for the graphical representation of the various contributions originating from the standard (local) condensates. Some of the nonperturbative diagrams contribute to few terms of the operator OPE, as it is specified in Tab. I.
from the three-gluon condensate and the dimension-7 condensate GG qGq to the OPE. For the same reason, other contributions from the condensates to the OPE, which are given by the product of the gluon condensate and the quark condensate after the HVD factorization, are also not included. The calculated OPE contributions to the spectral density, Eq. (17), are given in the form of an integral with the integrand ρ s iD (t, α, β) where the integration boundaries are . A two-dimensional integration corresponds to a two heavy-quark propagator given in the form of the αrepresentation. Although we consider three cases of flavor configurations (uds-cc, udc-cs, usc-cd), the integrands ρ s i,k (s, α, β) are given in Appendix D, Eqs. (D1), (D2), (D3) only for the uds-cc configuration.

IV. SYSTEM OF QCD SRS AND NUMERICAL ANALYSIS
We construct the QCD SRs for the state with spin s using the scalar functions Π s 1 and Π s 2 in the correlators Eq. (6). As discussed in the previous section, since the relativistic interpolating current can couple to the two states with opposite parities, the physical parameters, masses and the decay constants for the two states are coupled together in the QCD SRs. First, we present the Term LO qq GG qGq qq 2 qq qGq qq 3 qGq 2  D  0  3  4  5  6  8  9  10  Diag. a  d b, c d, e  f  f, g  h  f   TABLE I: In the first row of the table, we list the vacuum condensates of the various operators that give a contribution to the OPE for the studied correlators. The second row provides the dimension of the operators. The dimension-7 condensate GG qGq is not included in our study due to the smallness of the gluon-condensate terms. The third row denotes the correspondence of the operators to the diagrammatic representations in Fig. 1. Note that here we denote contributions from both light and s quarks condensates by qq n .
system of the QCD SRs in the coupled forms and discuss how to decouple the system of the QCD SRs for each state of definite parity by using a proper combination of Π s 1 and Π s 2 . In this section, we omit for simplicity the index s in all formulas as far as the involved expressions are valid for any considered spin s.
In the framework of QCD SR [35], the Borel transfor-mationB is applied to both sides of Eq. (7). This transformation helps to reduce the SR uncertainties by suppressing the contributions from the excited resonances in the continuum and also higher-order OPE terms.
For the phenomenological part of the SR, we apply the phenomenological spectral densities, which are called by ρ ph i (t) and appear on the right-hand side in Eq. (7). For all considered states, we assume that these spectral densities can be decomposed into contributions from the resonances of the considered states and the contribution from the continuum starting from the threshold s 0 appealing to the quark-hadron duality hypothesis where the threshold s 0 is chosen to be the same for both parities and for both densities (ρ ph 1 and ρ ph 2 ). The OPE spectral densities ρ OPE i (t) = ρ s i (t) are defined by Eq. (17). The decay constants f ± and masses m ± are given in Eqs. (10), (12), (15). Then, the resonance contributions to the phenomenological part of the SR are defined as follows where we apply the Borel transformation to Eq. (7), as already discussed. Combining the full OPE results with the contribution from the continuum, we evaluate the theoretical part of the QCD SRs where the k-times derivatives with respect to −1/M 2 are taken after the Borel transformation. Finally, for each state of spin s=1/2, 3/2, 5/2, we obtain the following system of QCD SRs in the coupled form: where k ∈ Z + {0}.

A. Decoupled QCD SRs
This subsection is devoted to decoupling the SRs in Eqs. (19) into two QCD SR equations for each state of definite parity. It seems that there are four different ways to deal with this kind of coupled QCD SRs systems used in the pentaquark QCD SR studies. First, assuming that most of the contributions come from the lowest lying resonance of the considered parity, the contributions from the resonance of the opposite parity can be ignored and only the second equation in Eq. (19) has been considered. This approach has been applied to many studies on the states of S = 1/2 and to pentaquark states [13,26,46]. In a second way, used in [19], one resolves the systems (19) by taking into account the states of both parities without decoupling the system. The third way is to get the decoupled QCD SRs by using the old-fashioned correlator [37,47]. Here, we use a method that is similar to the fourth way [10], in which the system of SRs, see, Eq. (19), is decoupled into two QCD SRs for each state of definite parity.
To decouple the SRs given by Eqs. (19), we expand the region of validity for k to k ∈ {n/2|n ∈ Z}. This analytical continuation allows us to consider the following linear combination of Eqs.
As a result we can rewrite the SRs, given by Eq. (19), in decoupled form to read with R (res) where the reparameterized spectral densities ρ OPE ± are related to ρ OPE 1,2 (calculated by the OPE in Eq. (8)) as The decoupled QCD SRs, Eq. (20), can be written in explicit form

B. Numerical Analysis
In this subsection, we extract the masses and the decay constants from the constructed QCD SRs. The first step is to define the Borel window The low boundary M 2 − of the Borel window insures that the dimension-9 condensate qq 3 contributes less than 10% to the total value of the correlator. Here we use the following notation for the OPE contribution of dimension D The upper boundary M 2 + is determined by the above condition by setting ∆M 2 = 1 GeV 2 . We don't follow the common practice to define the upper boundary M 2 + by the condition that the resonance contribution gives at least 10% to the total value of the correlator, r i (s 0 ) > 1/10. for i = 1 , 2, where .
The values of this ratio are given in Tables II, III, (14). Central values of masses (2nd and 3rd columns) and decay constants (4th and 5th columns) given at the best-fit threshold (s0) (see column 8th). The first error bars from the third to the sixth column represent the variation with respect to the threshold value in the interval (s min 0 , s max 0 ) given in the 7th column. The second error bars in the columns from the third to the sixth represent the variation in the Borel window (M 2 − , M 2 + ) (see 9th column). The criteria values δ(s0) are given for each state in percentages in the 10th column. Additionally, the last column represents the criteria of the resonance contribution r1(s0).
We have also checked two extra choices: (0, 1/2) and (1/2, 1) for (k, ∆k) to confirm the small dependence of our results on k and ∆k. Similar decoupled QCD SRs have been considered in [10] with (k, ∆k) = (1/2, 1). Borel parameter dependencies of the masses m ± (s 0 , M 2 ) for the uds −cc case are shown in Fig. 2  ]. To find the best values of the five parameters f ± , m ± , s 0 , we demand the minimization of the Borel parameter dependence of the original coupled SRs, Eqs. (19) i.e., with masses and decay constants in R (res) fixed by Eqs. (22). The minimization of the Borel parameter dependence of the original coupled SRs instead of the decoupled SRs helps avoiding possible uncertainties related to the analytical continuation of the SRs. Finally, we  III: QCD SR results for the masses m± and the decay constants f± given for a pentaquark of both parities with spin 1/2, 3/2 (first column) and for a udc-cs flavor-clustering. The second column denotes the type of the current and the spin of thecs-part given in the parentheses. The types of the currents are defined in Eqs. (9) and (11). See the caption of Table II for more details.   combine the four criteria in one to get We use this combined criterion to define the best-fit value for the thresholds 0 and the threshold interval  Tables II, III, IV for all considered states. From these values we obtain the masses m ± (s 0 ) and the decay constant f ± (s 0 ) ats 0 given in Table II together with their variations in the threshold interval and the variations in the Borel window.
The central valuem of the mass and the uncertainty ∆ s m related to the threshold are defined bȳ Final results for the mass are given in Fig. 3 and Tab. V by the central value massm and the total uncertainty ∆m where the total uncertainty is the sum of the above uncertainties that includes only uncertainties stemming from the SR analysis and do not include the uncertainties of the condensates.
The lowest threshold value is taken to be s th = 6.5 GeV 2 , see Eq. (7). The QCD SR technique described above has been applied using various pentaquark currents. First, we studied the type-1 current for the three flavor configuration (uds-cc, udc-cs, usc-cd), see the results in Table II. Second, in Table III, we obtained results for some alternative currents to estimate their relevance. Finally, we used our method to study the udc-cu flavor configuration, in order to see whether P + c (4312), P + c (4440), and P + c (4457), observed by the LHCb Collaboration, can be understood as a pentaquark of two clusters in a color-octet state. The detailed results given in Table IV will be discussed in the next section.

V. DISCUSSION AND SUMMARY
In this section, we discuss the results obtained in the previous sections on the basis of the constructed QCD SRs for pentaquark states. We have constructed the currents for udscc pentaquarks of spin-1/2, 3/2, 5/2 that have two clusters of a color-octet. The first cluster consists of three quarks q 1 q 2 q 3 which has the same flavor structure as the flavor singlet state of uds, while the second cluster consists of quark-antiquarkcq 4 . There are four options for flavor clustering q 1 q 2 q 3 -cq 4 (uds-cc, udccs, usc-cd, dsc-cu). The results for dsc-cu and usc-cd are identical in our approach and, therefore, we present here only results for the dsc-cu configuration. The main predictions for pentaquarks are presented for the type-1 current, that has a spin-1cq 4 part. In section III, in addition to these main currents, we have also introduced the alternative currents for spin-1/2 states and spin-3/2 states, see Eq. (9) and Eq. (11). Particularly, we are interested in the alternative currents with a spin-0 quark-antiquark cluster: type-3 and type-4 for a spin-1/2 current and type-2 for a spin-3/2 current. In Table III, we presented the results for these alternative currents of a udc-cs configuration with a spin-0cs-cluster in comparison with the main currents that have a spin-1cs-cluster. One can see that these types of currents lead to larger masses compared to those for the spin-1 cases for both spin-1/2 and spin-3/2 pentaquarks. We have also checked that a similar conclusion is valid for other flavor configurations. This observation agrees with [5], where it has been shown that the two-quark cluster with spin 1 in uds-cc system yield the most stable result. In Table III, we have also considered the alternative current for a spin 1/2 state containing a spin-1cs-cluster (type-2 for spin-1/2 current) and found that this current gives the same result. Therefore, the main results in our paper are given for the hidden pentaquark states with a spin-1 quark-antiquark cluster.
Using type-1 currents, we have considered three types of flavor clustering (uds-cc, udc-cs, usc-cd) and found that they have similar masses and decay constants, see Table II. Therefore, we expect that these configurations have equal chances to be observed. The consideration of a possible mixing between these configurations is outside the scope of this work. Another observation is that the larger spin states give larger masses.
Our results are presented in comparison with other theoretical predictions [3,5,23] in Fig. 3. The masses from the effective Lagrangian framework [23] for the udscc flavor configuration with a color-singlet substructure, depicted by ♦, are lower for the spin 3/2 case and compa- and for the odd parity (blue color errorbars) are given for three types of flavor clustering: uds-cc (diamonds), udc-cs (squares), usc-cd (triangles). Central value and width of errorbars are given in Eq. (24) and Eq. (25). The result of our calculations are depicted by for uds-cc, by for udc-cs, by for usc-cd. The results of other theoretical predictions for udscc pentaquark are denoted by for the color-magnetic interaction based study [3], ♦ for the framework of the coupled channel unitary approach with the local hidden gauge formalism [21][22][23], for the quark model result [5].
rable consistently well with our predictions for the spin 1/2 case referring to a pentaquark state with a color-octet substructure. The quark model prediction for uds-cc [5], noted by in Fig. 3, is in very good agreement with our result for a spin-1/2 pentaquark, while the spin-3/2 case is different but still compatible. In order to see whether any of the pentaquarks observed by the LHCb Collaboration could be understood as a pentaquark composed of two clusters in the coloroctet state, we study the pentaquark formed by the two clusters udc-cu, where the three-quark cluster is assumed to have the same flavor structure as the flavor-singlet m, GeV The masses of a recently observed by LHCb [2] states are shown by the dashed lines in comparison to our QCD SR estimations (blue errorbars) for the lightest states with a color-octet substructure. For more details see Tab. IV. flavor 1/2 − 1/2 + 3/2 − 3/2 + 5/2 − 5/2 + uds-cc 4.6(5) 5.6(6) 5.1(4) 6.0(5) 6.1(3) 5.9(2) udc-cs 4.5(3) 5.4(4) 4.8(2) 5.9(4) 5.8(3) 5.8 (2) usc-cd 4.6(3) 5.4(4) 5.0(2) 5.9(4) 6.2(2) 5.8 (2) TABLE V: Final QCD SR results for udscc pentaquark masses for both parities with spin 1/2, 3/2, 5/2. Values are given according Eq. (24). For more details see Table II. structure of uds. QCD SR results for the masses m ± and decay constants f ± for such a pentaquark are presented in Table IV for spin 1/2, 3/2, 5/2 and both parities. To make a point, we present the lightest state masses from this table in Fig. 4 together with the states recently observed by the LHCb Collaboration. As shown in this figure, the obtained mass for a spin-1/2 udc-cu pentaquark, is in agreement with the experimental value. Therefore, we conclude that if the observed state has spin 1/2 and negative parity, then it could be described as a state with two color-octet clusters.
To summarize, we have estimated the masses of the various hidden-charm pentaquarks with color-octet substructure and with J P C =1/2 ± , 3/2 ± , 5/2 ± in the framework of QCD SRs. We have constructed the currents for a particular configuration of pentaquark states, which consists of a three-quark cluster with the same flavor structure as the flavor singlet combination uds, and, additionally, of a quark-antiquark cluster, where both clusters are in a color-octet state. In our work, three possible types of flavor-clustering of the currents has been considered.
To obtain QCD sum rules, the operator product expansion for the correlators with the constructed interpolating currents has been performed up to the level of dimension-10 condensates. From the constructed QCD SRs the masses and decay constants of the pentaquark states have been extracted. Numerical values are given in detail in Table II, and are briefly summarized in Table V. We follow the common practice to extract the g µνterm of the 3/2 correlator and consider only the largest spin contribution. Here, we formalize this extraction by introducing the appropriate projectors. The general form of the tensor can be written in the following way where we consider only P-even terms. The relation between two forms is given byt The linearly independent set t j µν of all possible structures is defined as follows A linear combination of tensorst µν can be used to construct the projectors as Then we can extract the coefficients C j from the expansion expressed by Eq. A1 The inverse of the matrix M is given by where s = q 2 . Using the projectors P Tr ImΠ 3/2 µν (s)q g µν + γ µ γ ν − 2q µ q ν q 2 + 1 24πs Tr Tr ImΠ 3/2 µν (s)q(q µ γ ν − q ν γ µ ) . (A2) To extract the terms of the largest spin state from the correlator, the projector method is applied. Similarly to Eq. (A1), the general form of the correlator can be written as follows where the relation between the two forms is given bỹ t 2i = t i µν,αβ ,t 2i−1 =qt i µν,αβ , C 2i = c 2i , C 2i−1 = c 1i with i = 1, · · · 14. We consider only P-even terms which are symmetric with respect to µν and αβ. The linearly independent set t i µν,αβ of all possible structures is defined as where the operatorŜ µν symmetrizes the tensor aŝ S µν t µν = t µν + t νµ . Linear combination of these tensors can be used as the projectors We provide only the first two rows of the inverse matrix, which define the projectors P 5/2,1 and P 5/2,2 applied to extract the spin-5/2 spectral densities, Eq. (16): Other rows of the inverse matrix are not used in our work but could be obtained from the above equations.

Appendix C: Currents in color subspace
Here, we consider the relation of the pentaquarks with different configurations: diquark-diquark-antiquark clustering J3 ∼ qq-qq-q with an anti-triplet color substructure suggested in [11][12][13], a molecule form J 1 ∼ qqq-qq with color-singlet parts, see [20], and the combination J 8 ∼ qqq-qq with color-octet compounds studied here. First, we consider only the color part of these currents Using a Fiertz identity, one can get the relation where the quark fields carry flavor f i , color c i and spin l i indices as q i = q ficili . Then, multiplying this relation with the same spinor tensor one can obtain a relation between the full currents where J t = J c t T l1l2l3l4l5l . The tensor has been introduced in such a way so that the definition for J 8 agrees with Sec. II: After performing a Fiertz transformation in the currents, we get: where the modified matricesΓ N i and Γ N i are defined in terms of the Fiertz identity where ∆ N = (1/2 , γ 5 /2 , γ ρ /2 , iγ 5 γ ρ /2 , iσ ρ,γ / √ 8) N . Then, the definition for the modified matricesΓ N i and Γ N i take the form Therefore, currents with color-octet parts J 8 , colorsinglet parts J 1 , and color-anti-triplet parts J3 are linearly dependent. Nevertheless, we would like to point out that the currents suggested in our work cannot be a linear combination of any other currents considered previously, for example in [11][12][13]20].