Where is the stable Pentaquark

We systematically analyze the flavor color spin structure of the pentaquark $q^4\bar{Q}$ system in a constituent quark model based on the chromomagnetic interaction in both the SU(3) flavor symmetric and SU(3) flavor broken case with and without charm quarks. We show that the originally proposed pentaquark state $\bar{Q}s qqq$ by Gignoux et al and by Lipkin indeed belongs to the most stable pentaquark configuration, but that when charm quark mass correction based on recent experiments are taken into account, a doubly charmed antistrange pentaquark configuration ($udc c \bar{s}$) is perhaps the only flavor exotic configuration that could be stable and realistically searched for at present through the $\Lambda_c K^+ K^- \pi^+$ final states. The proposed final state is just reconstructing $K^+$ instead of $\pi^+$ in the measurement of $\Xi^{++}_{cc} \rightarrow \Lambda_c K^- \pi^+ \pi^+$ reported by LHCb collaboration and hence measurable immediately.

We systematically analyze the flavor color spin structure of the pentaquark q 4Q system in a constituent quark model based on the chromomagnetic interaction in both the SU(3) flavor symmetric and SU(3) flavor broken case with and without charm quarks. We show that the originally proposed pentaquark stateQsqqq by Gignoux et al and by Lipkin indeed belongs to the most stable pentaquark configuration, but that when charm quark mass correction based on recent experiments are taken into account, a doubly charmed antistrange pentaquark configuration (udccs) is perhaps the only flavor exotic configuration that could be stable and realistically searched for at present through the ΛcK + K − π + final states. The proposed final state is just reconstructing K + instead of π + in the measurement of Ξ ++ cc → ΛcK − π + π + reported by LHCb collaboration and hence measurable immediately.
The possible existence of mutiquark hadrons beyond the ordinary hadrons were first discussed for the tetraquark states in Ref. [1,2] and for the H-dibaryon in Ref. [3]. Later, possible stable pentaquark configurations Qsqqq were proposed in Ref. [4] and in Ref. [5]. The long experimental search for the H-dibaryon was not successful so far but is still planned at JPARC [6]. The search by Fermilab E791 [7] for the proposed pentaquark state also failed to find any significant signal for the exotic configurations.
On the other hand, starting from the X(3872) [8], possible exotic meson configurations XY Z and the pentaquark P c [9] were recently found. These states are not flavor exotic but are known to containcc quarks. Heavy quarks were for many years considered to be stable color sources that would allow for a stable multiquark configuration that does not fall into usual hadrons. In particular, with the recent experimental confirmation of the doubly charmed baryon [10][11][12], there is a new excitement in the physics of exotics in general and in hiterto unobserved flavor exotic states with more than one heavy quarks [13][14][15][16][17].
In this work, we systematically analyze the color flavor spin structure of the pentaquark configuration within a constituent quark model based on chromomagntic interaction. We show that the originally proposed pentaquark stateQsqqq indeed belong to the most stable pentaquark configuration, but that when charm quark mass correction based on recent experiments are taken into account, a doubly charmed antistrange pentaquark configuration (sccud) is perhaps the only stable flavor exotic configuration that could be stable and realistically searched for at present.
Systematic analysis of q 4Q : We first discuss the classification of the flavor, color and spin wave function for the ground state of the pentaquark composed of q 4 light quarks and one heavy antiquark Q (c orb) assuming that the spatial parts of the wave function for all quarks are in the s-wave. We categorize them into the flavor states in SU(3) F , and then examine the color ⊗ spin states.
The flavor states for q 4 can be decomposed into the direct sum of the irreducible representation of SU(3) F as follows: Here, [4], [31], [21 2 ] and [2 2 ] indicate the Young tableau of the SU (3) F multiplet, with the subscripts representing the corresponding dimensions, while [15], [15 ′ ], [3] and [6] show the respective multiplicities. The 7776 dimensional color ⊗ spin states of q 4Q can be classified as the direct sum of the irreducible representations of SU(6) CS as follows: The Young Tableau and its subscript outside of the bracket respectively indicates the SU(6) CS representation and its dimension of the light quark sector q 4 while the Young Tableau inside the bracket is the SU(6) CS representation of the pentaquark consisting of q 4Q . By further decomposing the SU(6) CS into the sum of SU (3) C ⊗ SU (2) S multiplets, we can select out the physically allowed color singlet states. Table I shows the allowed color singlet states with the possible spin states, denoted by [1 C , S], allowed within each SU(6) CS representation.
Therefore, since the SU (6) CS representation of q 4Q as well as those of q 4 are given in Eq. (2), we can construct the flavor ⊗ color ⊗ spin states with color singlet, by using the fully antisymmetric property together with the conjugate relation between the flavor in Eq. (1) and the SU (6) CS representation in Eq. (2) among the four light quarks. Such combination will finally determine the allowed flavor and spin content of the pentaquarks in the flavor SU(3) symmetric limit. Color spin interaction for pentaquark system: In the constituent quark model based on the color spin interaction, the stability of a pentaquark depends critically on the expectation value of the interaction. Therefore, we derive the following elegant formula of the chromomagnetic interaction relevant for the pentaquark configuration, which is similar to that of a tetraquark in Ref. [2], by introducing the first kind of Casimir operator of SU(6) CS , which is denoted by C 6 : Here the lower index indicates the number of the participant quarks, C 3 the first kind of Casimir operator of SU (3) C , S the spin operator, and I the identity operator.
Spin=3/2: Let us discuss in detail the flavor [15 ′ ] case with S = 3/2. Here, there are two flavor ⊗ color ⊗ spin states which are orthonormal to each other. There are two methods to obtain these states.
In one approach based on the coupling scheme, the first (second) state comes from the coupling scheme of the color ⊗ spin state in which the spin among the four quarks is one (two), as given in Eq. (26) (Eq.(32)) in [18]. The fully antisymmetric flavor ⊗ color ⊗ spin states for S = 3/2 among the four quarks can be obtained by multiplying the color ⊗ spin state by their conjugate flavor [15 ′ ] state. In the other approach, the two states can be directly obtained from Eq. (2). As we can see in Eq. (2) and Table I From the SU (6) CS representation point of view, we can infer that the linear sum of two fully antisymmetric flavor ⊗ color ⊗ spin states coming from the coupling scheme must belong to either [32 2 1 2 ] state or [1 3 ] state. We find that the coefficients of the linear sum can be calculated from the condition that these are the eigenstates of the Casimir operator of SU(6) CS , given by, Following the same procedure, one can construct the flavor ⊗ color ⊗ spin states for the remaining flavor cases for S = 3/2, which satisfy the antisymmetry property among four quarks. From the result, it is found that there are all together 12 color ⊗ spin states that are both color singlet and S = 3/2. These are expressed by the Yamanouchi bases of the SU (6) CS representation among the four quarks, together with the SU (6) CS Young tableau for the full q 4Q pentaquark state: Other spin states: In analogy to the S = 3/2 case, we can apply the same procedure to the S = 1/2 case. In this case, there are all together 15 color ⊗ spin states that are both color singlet and S = 1/2, and that are expressed by the Yamanouchi bases of the SU (6) CS representation among the four quarks, like Eq. (5). Finally, in the S = 5/2 case, there exist only one color ⊗ spin state coming from the [32 2 1 2 ] representation in Table I .
By using the flavor ⊗ color ⊗ spin states for S = 1/2, S = 3/2, and S = 5/2, the expectation values of Eq. (3) can be calculated, as given in Table II. In Table II, below each matrix element, we also show the relevant SU (6) CS representations for the pentaquark state as well as the eigenvalue of Eq. (3). As can be seen in the table, the most attractive channel is given by the 2 × 2 matrix valued (F, S) = ([3], 1/2) state. Upon diagonalizing the matrix in the m 5 → ∞ one finds the eigenvalues (-16,-40/3), where the lowest one corresponds to the most attractive pentaquark state discussed in Ref. [4,5]. It should be noted that the factor -16 in this case can also be naively obtained by assuming two diquarks (ud, us) in the udusc pentaquark. However, as noted from the case of H dibaryon, SU(3) breaking effects together with the additional attraction from the strong charm quarks are important to the realistic estimate of the stability: The color spin interaction from the m J/ψ − m ηc are much stronger than from naively scaling the color spin splitting in the light quark sector by the charm quark mass [14].
Pentaquark binding in the SU (3) F broken case: To analyze the stability of the pentaquarks against the lowest    threshold, we introduce a simplified form for the matrix element of the hyperfine potential term, where we approximate the spatial overlap factors by constants that depend only on the constituent quark masses of the two quarks involved: We then assume that the difference between the pentaquark energy and the lowest threshold baryon meson states arise only from the hyperfine energy difference [19]. This is because other potential terms are linear in the number of quark involved so that assuming that all hadrons occupy the same size, the differences of their contribution to the pentaquark and baryon meson cancel.
To evaluate the binding energy of the pentaquark in terms of Eq. (6), we extract the C mimj values from the relevant mass differences between baryons and between mesons when involving one antiquark. The relations are given by, For C cc we take it to be 1/2C cc . Then, we calculate the binding energy, denoted by ∆E, by comparing the hyperfine potential energy between the pentaquark and its lowest decay channel. Isopin basis: We now investigate the stability of the pentaquark with respect to isospin (I) and spin (S), and allow the antiquark of the pentaquark to be eithers,c or b. The advantage of the Yamanuchi basis in the SU (6) CS representation to the pentaquark that characterizes the symmetry property among the four quarks makes it possible to find the flavor ⊗ color ⊗ spin states suitable for a certain symmetry, which is allowed by the Pauli principle. Since it is possible that there are several flavor ⊗ color ⊗ spin states, denoted by multiplicity, according to the symmetry property, the binding energy is obtained from diagonalizing the matrix element of the hyperfine potential energy.
We need to characterize isospin states of q 4 in order to classify the pentaquark with respect to I. As can be seen in [20], the iospin states to q 4 can be decomposed in the following way: I = 0 with Young tableau [2 2 ] consisting of uudd component, I = 1 with Young tableau [31] consisting of uuud component, and I = 2 with Young tableau [4] consisting of uuuu. The result for the binding energy defined as the difference between the hyperfine interaction of the pentaquark against its lowest threshold values are given in Table IV. As can be seen in Table IV, it is found that the most attractive pentaquark states are those with (I, S) = (0, 1/2), apart from udccc, and as well, the uudsc with (I, S) = (1/2, 1/2). To understand the reason why these particles could be bound states, we need to analyze the expectation matrix value of Eq. (6) in terms of a dominant color ⊗ spin state among the possible states. For these states, the dominant color ⊗ spin state comes from the SU (6) CS representation [21] having the Young tableau [31] for the q 41 , for which the expectation value of Eq. (3) is -36, as can be seen in (F = [3], S = 1/2) sector of Table II when m 1 = m 5 . In fact, the SU(6) CS representation [21] state with S = 1/2 gives the most attractive contribution to the expectation value of Eq. (3) than any other state, and both the I = 0 and I = 1/2 comes from the breaking of the flavor [3] state of this representation.
In Table III, we show the expectation value of Eq. (6) in terms of only a color ⊗ spin state coming from the SU (6) CS representation [21] as well as the corresponding binding energy against its threshold represented in the third row for each state. It should be noted that H hyp for each state reduces to -36C mimj when the C mimj 's are taken to be a quark mass independent constant. It should be noted that all these possible stable states are related to the attractive pentaquark states discussed in Ref. [4,5] in the flavor SU(3) symmetric limit. However, it should also be pointed out that when the charm quark is also included, together with its hyperfine contribution, it is the udccs pentaquark configuration that is most attractive. This state has also been discussed recently in Ref. [21]. The attraction obtained in Table IV should be large enough to overcome the additional kinetic energy, typically of order 100 MeV, to make the state compact of a hadron size.
Hence, the proposed pentaquark state is possibly the only stable pentaquark or a resonance state slightly above the lowest threshold, which is Ξ cc +K for this state. Noting that Ξ cc has been recently discovered, one can just add an additional Kaon to look for this possible resonance state. If the state is strongly bound, one could look at the udccs → Λ c K + K − π + decay or any hadronic decay mode similar to those of Λ c D + s . The proposed final state is just reconstructing K + instead of π + in the measurement of Ξ ++ cc → Λ c K − π + π + reported in Ref. [10] and hence measurable immediately. Such a measurement would be the first confirmation of a flavour exotic pentaquark state.