Flavor-singlet charm pentaquark

A new type of charm pentaquark $P_{cs}$ with quark content $c\bar{c}uds$ in light-flavor singlet state is studied in the quark model. This state is analogous to the $P_{c}$ with $c\bar{c}uud$ in light-flavor octet, which was observed in LHC in 2015. Considering various combinations of color, spin and light flavor as internal quantum numbers in $P_{cs}$, we investigate the mass ordering of the $P_{cs}$'s by adopting both the one-gluon exchange interaction and the instanton-induced interaction in the quark model. The most stable configuration of $P_{cs}$ is identified to be total spin $1/2$ in which the $c\bar{c}$ is combined to be color octet and spin $1$, while the $uds$ cluster is in a color octet state. The other color octet configurations, the total spin $1/2$ state with the $c\bar{c}$ spin 0 and the state with total spin $3/2$ and $c\bar{c}$ spin 1, are found as excited states. We also discuss possible decay modes of these charm pentaquarks.


I. INTRODUCTION
Studying exotic hadrons, so called X, Y , Z, is one of the most interesting topics in the present hadron physics [1][2][3][4][5][6][7][8]. In 2015, a new type of exotic hadron, a pentaquark with hidden charm P c , was observed in LHC experiment [9]. P c is considered to be ccuud as a minimal quark configuration, and hence this is the first discovery of pentaquark including charm quarks. P c was observed in J/ψp channel in the weak decay from Λ b baryon, and the two states were identified: Let us briefly summarize studies of charm pentaquarks. As an early work, existence of charm pentaquark was pointed out in the framework of the Skyrmion model, where η c meson is bound to the hedgehog configuration of pion [10]. Afterwards, hadron molecule model was analyzed in Refs. [11][12][13][14][15]. Coupled-channel calculation was considered in Refs. [11,12], but the obtained masses of charm pentaquark were much smaller (less than 4 GeV) than the values observed in LHCb. Other coupled-channel calculations gave the masses close to the observed ones [13][14][15]. Effect of the direct quark exchange in the hadronic molecule was considered in Ref. [16]. As a compact state, diquark model was analyzed in Ref. [17]. QCD sum rules were applied and the mass values close to the observed ones were reported [18].
As other possibities, the cusp effect by a triangle anomaly was discussed [19], and new experimental setup for pion beam was proposed [20]. More references will be found in Ref. [5].
Among many candidates of internal structure, we will consider the compact multiquark state. We focus on a new possible structure of charm pentaquark with quark configuration ccuds, which will be denoted by P cs , and investigate the mass ordering of P cs for different quantum numbers.
Let us consider the color structure and the light flavor structure in the pentaquark ccqqq with q = u, d or s. We assume that the pentaquark is a compact quark state, and consider quantum number of ccqqq clusters cc and qqq separately. We note that, due to the color singlet condition for hadrons, cc and qqq can be not only color singlet but also color octet.
Let us consider the decomposition of the flavor-spin multiplet for qqq in terms of SU (6) symmetry including SU(3) flavor symmetry and SU(2) spin symmetry: where the subscripts stand for totally asymmetric where the first term and second term in the parentheses represent the flavor multiplet and the multiplicity of spin, respectively.
Let us consider the simplest case. In the following, we consider all the particles are in S-wave, when cc is color singlet and qqq is also color singlet. Then, the light flavor of qqq is given by 56 S , because the color part of qqq is totally antisymmetric. Thus, we obtain the well-known multiplet, flavor octet with spin 1/2 and flavor decuplet with spin 3/2.
In contrast, the situation is different for the case that cc is color octet. In this case, the color of qqq should be color octet. According to the decomposition of the color multiplet for three particles,  (8,4), (10,2), (8,2) and (1,2). In the present study, we will focus on (1,2), because this multiplet becomes most stable in the color-spin interaction.
In the literature, there have been studies of internal configurations of charm pentaquark ccqqq as a compact state [21,22]. In this reference, the hyperfine splitting was provided by each of color-spin interaction, flavor-spin interaction and instanton-induced interaction, while the confinement potential for quarks was provided by the harmonic oscillator potential.
In a similar idea, we also will use the color-spin interaction and the instanton-induced interaction at short distance, but we adopt the linear potential as quark confinement potential.
In our case, we include the simultaneous combination of the color-spin interaction and the instanton-induced interaction, and consider the three-body force in the instanton-induced interaction which has not been considered so far. Furthermore, we investigate the details of the internal spatial structure in P cs .
The paper is organized as follows. In Section 2, we summarize the quark wave function of the charm pentaquark and introduce the setup of the quark model with color-spin interaction, the instanton-induced interaction and quark confinement potential. We prepare two models. One is given by the color-spin interaction, and another is given by the combination of the color-spin interaction and the instanton-induced interaction. In Section 3, we perform the variational calculation for mass of P cs , and investigate the internal color, spin and spatial structures of the obtained states. In Section 4, we discuss the possible decay modes of P cs .
The final section is devoted to our conclusion.

A. Wave function of charm pentaquark
For P cs (ccuds), we consider that the total wave function is given by a product of the spatial part (φ), the spin and color part of cc (ψ s,c cc ), and the spin, color and flavor part (ψ s,c,f uds ): The spatial part φ depends on the variables R and r i (i = 1, 2, 3) (Fig. 1). Here R is the position vector from the c quark to thec quark, and r i are the vectors for light quarks i = 1, 2, 3. We assume for simplicity that the internal angular momenta are S-wave because we focus on the ground states.
It is known that the Jacobi coordinates are very useful to solve many-body problems in general. In the present discussion, however, we simplify the situation in the following way.
We assume that the c andc quarks are sufficiently heavy, and that the midpoint of c and c, i.e. R/2, represents the center-of-mass of ccqqq system. In this limiting case, we can assign the original points of the vectors r i to be the center-of-mass of the system. We notice that in this treatment the motion of the cc (or uds) cluster to the total system is neglected.
Nevertheless, we expect that this would be a reasonable approximation as long as the mass of charm quark is much larger than those of light quarks.
As for the spatial wave function, we here consider only compact systems of five quarks and assume the Gaussian type with extension parameters a for R and b for r i . We use the common value b for r 1 , r 2 , r 3 , because the wave function of the light quarks will be distributed uniformly in space. In fact, as will be discussed later, the stability of the pentaquark considered here seems irrelevant to the diquark correlation between two light quarks, but rather sensitive to the cc correlations. In this sense, we may justify to treat the common variational parameter b.
With the simplifications stated above, we assume the spatial part of the wave function as φ(R, r 1 , r 2 , r 3 ) = 1 which is normalized by integrating over the space. The values of a and b will be determined by variational calculation. Note that all the orbital angular momenta are zero.
As for the spin-color part of cc (ψ s,c cc ) and spin-color-flavor part of uds (ψ s,c,f uds ), we consider several combinations of quantum numbers as summarized in Table I. As for spin, we consider the cases where the spin of cc is either 0 or 1, and the spin of uds is 1/2. Then, the total spin and parity of the the charm pentaquark is J P = 1/2 − with cc spin 0 or 1, and J P = 3/2 − with cc spin 1. We notice that cc with spin 0 and cc with spin 1 should be regarded as the independent states which are not mixed with each other. This observation is supported by the fact that the spin of charm quark is conserved in the heavy quark mass limit, as known in the heavy quark effective theory. In reality, however, there is a small correction TABLE I. Combinations of internal color states of P cs with isospin I = 0 and spin-parity J P .
(I, J P ) octet type (8) singlet type (1) component color spin flavor isospin component color spin flavor isospin term which breaks the heavy quark spin symmetry with an order of 1/m c , and it induces the mixing of cc spin 0 and cc spin 1.
First, we consider ψ s,c cscc . This is composed of the spin part (χ s cc ) and the color part (ψ c cc ): with c = 1 for color singlet and c = 8 for color octet, and s = 0 for spin singlet and s = 1 for spin triplet.
Second, as for ψ s,c,f uds , we consider the following combinations of color part (ψ c uds ), spin part (χ s uds ), and flavor part (ψ f uds ). In the case of three particles uds, we have to pay a special attention to the antisymmetriation of the wave functions. Because all the internal angular momenta are S-wave, the combination of color, spin and flavor of uds should be antisymmetric. We consider the color octet case and the color singlet case for uds. Let us first consider the case of flavor singlet f = 1. In this case, the combination of color and spin needs to be totally symmetric, because the flavor part is totally antisymmetric. Then the allowed combination of the color and spin is where the subscript λ (ρ) in 8 λ (8 ρ ) and 1/2 λ (1/2 ρ ) means that the first two light quarks are symmetric (antisymmetric) under exchange of the two light quarks. The product of ρ state and λ state makes the totally symmetric state under exchange of any two light quarks [? ].
Then, we have the uds wave function for light flavor singlet, f = 1: where uds is the flavor singlet wave function. We add an upper script I = 0, because we will consider isospin singlet I = 0 only. Second, we consider the light flavor octet, f = 8.
In this case, by combining the light flavor and the spin for light quarks, we may consider the totally symmetric state for flavor and spin, where λ (ρ) is the same notation as before. We add I = 0 for flavor wave function, because we will consider I = 0 only. The color part should be totally antisymmetric, ψ c=1 uds . Hence we obtain the uds wave function for flavor octet, f = 8: By combining P s,c cscc in Eq. (8) and P s,c,f cs in Eqs. (10) and (12), we will have four states in (I, J P ) = (0, 1/2 − ) and two states in (0, 3/2 − ). Their explicit forms are for cc spin 0 and (I, J P ) = (0, 1/2 − ), for cc spin 1 and (I, for cc spin 1 and (I, J P ) = (0, 3/2 − ), where the subscripts 8 and 1 indicate that the color representation of the components, cc and uds, and the square brackets indicate the composition of total spin. Notice that the spatial wave functions are different for each color and spin, as denoted by φ 8,1 , φ 8,1 and φ * 8,1 .

B. Model A: hamiltonian without instanton interaction
We consider the Hamiltonian for ccuds. It is given as sum of the kinetic term (K), the color-Coulomb term (V Coulomb ), the color-magnetic interaction (CMI) term (V CMI ) and the confinement term (V conf ): where each term is given by where we define ∇ R = ∂/∂R and ∇ k = ∂/∂r k (k = 1, 2, 3), λ i and σ i the Gell-Mann matrices for color and the Pauli matrices for spin for quarks i = c,c, q 1 , q 2 and q 3 , r ij = |r i − r j | the distance between the quark i and j. The hadron mass is given by the sum of the expectation value of H A and a constant term C: E = H A + C. As parameters we use α s for the coupling constant in the Coulomb potential and the CMI potential, µ cc = m c /2 with charm quark mass m c and m k the mass for light quark k = 1, 2, 3, and σ the string tension of the linear confinement potential. As for α s and σ, we use different values for light-light quark pairs and for light-heavy and heavy-heavy quark pairs. The parameters in the former are denoted by α s1 and σ 1 , and the ones for the latter are by α s2 and σ 2 . We use the onethird of the nucleon mass for m u = m d , and m s is from the mass ratio m u /m s = 0.6 so that they reproduce the masses of the light ground-state baryons, as summarized in Table III.
The constant term C Λ is adjusted to the Λ baryon. In the heavy sector, the values of m c for c quark mass, α s2 for the coupling constant between two heavy quarks (or a heavy quark and a light quark), σ 2 for the string tension between two heavy quarks (or a heavy quark and a light quark), and the constant C ηc for η c are taken from Ref. [23], which reproduce the masses of η c and J/ψ.
In the model A, we have considered the one-gluon exchange potential at short distance.
However, there can be additional interaction which originates from the instanton. The instanton is responsible for the U(1) A breaking in QCD vacuum, and can be seen in several mass spectrum of hadrons. One of the most prominent effects is seen in η mass, whose mass is much larger than the other Nambu-Goldstone bosons (π, η, K). Another example can be seen in H-dibaryons (uuddss) [25]. The instanton couples to massless quarks strongly through zero modes, and generates a six-quark vertex given by a three-body force in the flavor singlet channel. Indeed, the instanton has the property that there exists a zero-energy bound state of massless fermion around the instanton [26]. In our case, uds in the charm pentaquark ccuds can be flavor singlet (cf. Table I), and hence the instanton may play an for spin singlet (s = 0) and 1 for spin triplet (s = 1). color interesting role.
Let us summarize briefly the properties of the instanton. The instanton configuration is given by as the classical solution of QCD in four-dimensional Euclidean space. The parameter ρ is the instanton size. It is estimated as about 0.3 fm in the instanton liquid model [27]. This size is smaller than the typical hadron size, 1/Λ QCD ∼ 1 fm for Λ QCD 200 MeV. Therefore, it is possible to regard the instanton as a point-like object and the effective interaction between quarks via instanton can be represented by a point-like interaction.
The non-relativistic form of Hamiltonian of the instanton-induced interaction for quarks via instanton can be given by with ψ R (i) = 1 2 (1 + γ 5 )ψ(i) and ψ L (i) = 1 2 (1 − γ 5 )ψ(i) for light quark i = 1, 2, 3 [25,28]. This is a six-quark vertex, namely the three-body force (Fig. 2). The three flavors of quarks should be different, because the projection operator for ansisymmetrization of light flavor, A f 3 , is introduced to pickup the flavor singlet component. The parameter V 0 is the coupling constant, whose value can be determined phenomenologically. It is noted that the second term in the r.h.s., the hermitian conjugate to the first term, represents the contribution from the anti-instanton.
The three-body force of the instanton-induced interaction can be transformed to the twobody force. This is indeed accomplished by closing one pair of quarks (q 3 in Fig. 2) with a quark condensate ψ ψ , and the obtained interaction is given by as the effective interaction for q 1 and q 2 . The effective coupling constant V 0 (1, 2) is a product of V 0 in the three-body force and the loop of q 3 , namely the chiral condensate of q 3 , and the explicit form is given by It should be noted that the current mass m Then, the effective coupling constant can be eventually represented as The value of K should be in principle dependent on quark flavor. Nevertheless, we assume the SU (3) Finally, by defining the effective coupling by we obtain the effective two-body interaction which is much compactly represented in the form that the flavor dependence appears only in 1/(m i m j ).
From the above results for the three-body force and the two-body force, we derive the effective potentials [28], and where the spatial dependence between two quarks (three quarks) are represented by the delta-type potentials, δ (3) (r) with a distance between two quarks r. As we use the variational method with a single Gaussian extension parameter, we do not smear the delta function in this study. For complete solutions, we need to smear the delta according to the size of the Fermion zero modes around the instanton. A f 2 and A f 3 are the projection operators to pickup anti-symmetric representation for two-quark i, j and three-quark i, j, k, respectively.
It is interesting to notice that the two-body potential, V III2 , has the factor 1/m i m j and the spin dependence, and hence that V III2 resembles the spin-dependent part of the one-gluon exchange potential, Eq. (19). In this sense, it leaves some ambiguity in phenomenology about whether the spin-dependent interaction is supplied by the one-gluon exchange or by the instanton-induced interaction. We here introduce a new parameter p to control the contributions from the one-gluon exchange and the instanton-induced interaction in the Hamiltonian: where the subscripts LL, HL and HH indicate the operated pairs of two quarks, light-light quarks (LL), heavy-light quarks (HL) and heavy-heavy quarks (HH). The light baryon spectroscopy can not fix this value because the total strength of the spin dependent interaction is independent of p. On the other hand, we can determine p phenomenologically in the light meson sector so that the η mass is reproduced, giving p = 0.4. Note that p affects only the short range interaction among light quarks (LL). The confinement potential V conf is independent of p. The interactions between heavy-light quarks (HL) and heavy-heavy quarks (HH) are not affected by the instanton-induced interaction, because this interaction acts only on light quarks. We notice also that the three-body force V III3 is also weighted by p. It is clear that the one-gluon exchange (instanton-induced interaction) is recovered for p = 0 (p = 1).
The new parameters U 0 , V 0 and p in the instanton-induced interaction as well as the parameters in the one-gluon exchange are summarized in Table II. The parameters for heavy quarks, m c , α s2 , σ 2 and C ηc are the same as those in the model A. As for the light quark sector, m u , m s and σ 1 are the same also, because they should not depend on the details of the interaction at short distance. The parameters in the one-gluon exchange and the instanton-induced interaction, α s1 , U 0 , are determined by the mass splitting between N and ∆ baryons. It is useful to adopt the relation for a single Gaussian wave function with the size parameter b (cf. Eq. (7)). The value of V 0 is determined from U

A. Variational calculation
The masses of ccuds charm pentaquark are given by with P cs = P cs8,1 , P cs8,1 and P * cs8,1 and the constant term C = C Λ + C ηc . The values of a and b in φ 8,1 , φ 8,1 and φ * 8,1 are determined by the variational calculation for minimizing P cs |H|P cs .
To perform the variational calculation, we need to know several matrix elements of λ i · λ j and λ i · λ j σ i · σ j for a pair of quark i and j in the Hamiltonian (16). We will show the procedure of the calculations in the followings. The color octet channel is especially important because it gives the lower energy state than the color singlet one. In the following, therefore, we will show the matrix elements of the color octet channel, namely P cs8 (s cc = 0), P cs8 (s cc = 1) and P * cs8 (s cc = 1). Similar calculations can be performed for the color singlet channel.
As for P cs8 , we evaluate P cs8 |λ q · λ q |P cs8 = 1 2 The first equation is obtained by noting that uds color 8 state has the color3 and 6 with the same wieght. The second equation is given by the color octet representation of cc. , with subscripts "c :" the color representations.
The color-spin operators can be calculated by The first equation can be obtained by noting that the symmetric states and the antisymmetric states both in spin and in color exist with the same weight in uds. As for P cs8 and P * cs8 , we perform the similar calculations for the matrix elements of λ i · λ j and λ i · λ j σ i · σ j . The matrix elements of λ i · λ j in P cs8 and P * cs8 should be the same as those in P cs8 . We show the matrix elements of λ i · λ j σ i · σ j for heavy-light i, j pairs as for cq andcq pairs in P cs8 and for cq andcq pairs in P * cs8 . So far we have treated that the spin of charm quark pairs, s cc = 0 and s cc = 1, are conserved quantities, and regarded that P cs8 and P cs8 (or P cs1 and P cs1 ) are independent states with each other. However, this is not necessarily correct. It is important to comment that P cs8 and P cs8 (or P cs1 and P cs1 ) can be mixed by the color-spin mixing term λ i ·λ j σ i ·σ j for a heavy (anti)quark i and a light quark j, because both states have the common quantum number J P = 1/2 − irrespective to the difference of the spin of charm quark pairs, s cc = 0 and s cc = 1, respectively. The mixing effect is not so large because the spin-flip process should be suppressed by the factor 1/m Q with the heavy quark mass m Q , and it can be treated as the corrections. Therefore, we will ignore the mixing effect for simple presentation in most cases in the text, and we will treat P cs8 and P cs8 (or P cs1 and P cs1 ) as the independent states. will turn out that the octet gives the ground state of the charm pentaquark ccuds. For that purpose, we will use the matrix elements as . (50) For the model B, it is also necessary to calculate the matrix elements of λ i · λ j and λ i · λ j σ i · σ j . They are the same as those calculated for the model A. A special attention should be paid for the three-body force in the instanton-induced interaction: it vanishes for color singlet configuration (i.e. light flavor octet) and does not vanish for the color octet configuration (i.e. light flavor singlet).

B. Energy spectrum
The obtained numbers of the variational parameters (a and b) and the masses of charm pentaquarks are shown in Table V. The masses are shown also in Fig. 3. Notice that the mixing between P cs8 and P cs8 (P cs1 and P cs1 ) are not considered in those results.
First, let us compare the three states P cs1 (s cc = 0), P cs1 (s cc = 1) and P * cs1 (s cc = 1). We notice immediately that they are much above the threshold states η c Λ or J/ψΛ, and the splitting between P cs1 and P cs1 P * cs1 is almost identical to the η c -Jψ mass difference. This can be understood easily because in the present quark model there is no interaction between the color singlet cc and uds, and thus these P cs1 states are nothing but non-interacting η c Λ or J/ψΛ plus kinetic energy. However, this simple explanation cannot applied to P cs8 , P cs8 and P * cs8 due to the complicated color structure. Second, one of the most interesting observations is that, in color octet, the instantoninduced interaction reduces very much the mass of charm pentaquarks than the one-gluon exchange, while there is no large change in color singlet. Let us understand why the large reduction of mass in color octet arises. Based on the above observation, one may expect that the mass reduction in color octet is in fact supplied by the instanton-induced interaction.
However, the actual mechanism may not be so simple. We can check the attraction and c (M ) with several (I, J P ) and color combinations: P cs8 and P cs1 with cc spin 0 for (0, 1/2 − ), P cs8 and P cs1 with cc spin 1 for (0, 1/2 − ), and P * cs8 and P * cs1 with cc spin 1 for (0, 3/2 − ). The determined values of a and b are displayed also. The model A contains the one-gluon exchange only at short distance force, and the model B contains both the one-gluon exchange and the instanton-induced interaction.
color configuration P cs8 P cs1 P cs8 P cs1 P * Hamiltonian. Then, we find that the two-body interaction part (V III2 in Eq. (30)) gives an attraction, while the three-body part (V III3 in Eq. (31)) gives a repulsion. Because the uds flavor is singlet in color octet channel, the anti-symmetry of any two pairs of quarks gives a strong attraction in V III2 . In fact, the attraction in color octet (uds singlet) is stronger than the attraction in color singlet (uds octet). At the same time, however, it give also a strong repulsion in V III3 . As a result, the attraction in V III2 is almost canceled by the repulsion in V III3 , and hence the instanton-induced interaction does not provide much attraction. We should consider rather that the attraction is mainly provided by the one-gluon exchange rather than the instanton-induced interaction.
It is interesting to compare the size of inter-quark distance for color-octet configuration P cs8 , P cs8 , P * cs8 ) and color-singlet configuration (P cs1 , P cs1 , P * cs1 ). As a general tendency, in Table. V, we notice that the sizes between c andc (a) in color-octet configuration is larger than those in color-singlet configuration. This behavior can be understood in the following way. The important role is played by the λ i · λ j operators, which are included in the color Coulomb potential and the linear confinement potential. As for the cc potential, we find λ c · λc = 2/3 for color-octet configuration (P cs8 , P cs8 , P * cs8 ) and λ c · λc = −16/3 for color-singlet configuration (P cs1 , P cs1 , P * cs1 ). Due to the repulsion and attraction in each configuration, the cc sizes in color-octet are larger than those in color-singlet (see Fig. 4). On the other hand, the sizes of wave functions of light quarks (b) in color-octet configuration is smaller than those in color-singlet configuration. This is also understood from the values of λ i ·λ j , though the situation is a bit cumbersome. When we compare the value of λ q ·λ q for a pair of light quarks, we find from Table VI that both color-octet and -singlet configurations feel attraction provided that the former attraction is less attractive. Hence we may expect that the size of b in color-octet is larger than that in color-singlet. However, this is not the case. The trick is that the attraction by c (c) and q, λ c · λ q (λc · λ q ), exists only for color- and b (thin black arrow) in the model B for P cs8 , P cs8 and P * cs8 (cf. Table V).
octet configuration. This provides the shrinkage of the wave function of the light quarks in color-octet configuration (Fig. 4).
In Fig. 3, we notice that the masses of P cs8 , P cs8 and P * cs8 , M P cs8 , M P cs8 and M P * cs8 , are in order as given by both for the model A and the model B. This is naturally understood from the color-spin interaction part containing s i · s j part. We consider the color clusters cc with color octet and spin 0 or 1 and uds with color octet and spin 1/2. When the cc cluster has spin 0, there is no spin-spin interaction. When the cc cluster has spin 1, the compound states ccuds are split to the two states with total spin 3/2 and 1/2. The spin-spin operator gives the energy splitting for those two states, a repulsion for the former and an attraction for the latter (the strength fraction two-to-one), and hence the masses become different as shown in Fig. 5 C. Mixing between P cs8 and P cs8 Up to now, we have neglected the mixing of P cs8 and P cs8 . The mixing interaction is suppressed by the factor 1/m 2 c in the spin-spin interaction in Eq. (19) because the former contains the cc spin s cc = 0 and the latter does s cc = 1, and hence to ignore the mixing is a good approximation. We will investigate the accuracy of this approximation by considering the mixing of P cs8 and P cs8 . In this case, we consider the superposed state with coefficients c 1 and c 2 . The Schrödinger equation is schematically expressed as The energy E as an eigenvalue is given by E L for lower energy and by E H for higher energy, and the corresponding states will be denoted by P L cs8 and P H cs8 , respectively. In the variational calculation to obtain E L and E H , we use different size parameters in the spatial parts in the wave functions, (a 1 , b 1 ) for P cs8 and (a 2 , b 2 ) for P cs8 . However, we find that (a 1 , b 1 ) are only slightly different from (a 2 , b 2 ); a 1 = 0.  Table VII and VIII. Comparing the results of the masses of P cs8 and P cs8 summarized in Table V, we find that the mass of P L cs8 becomes smaller by about 20 MeV and P cs8 becomes larger by about 30 MeV. The mixing fractions are about 20 %. This value is consistent with the results in Ref. [21]. In this reference the state corresponding to ours is supplied by the combinations of |1 and |3 in [211] state, which contains a flavor singlet state. Notice that cc spin s cc = 0, 1 are mixed in each of |1 and |3 s.

IV. DISCUSSION
We investigate the possible decay modes of the charm pentaquark P cs8 , P cs8 and P * cs8 in the model B. The obtained masses are located above thresholds of several open channels, as shown in Fig. 3. The available decay channels are η c + Λ, J/ψ + Λ, D s + Λ c and D + Ξ c .
The most lowest threshold is given by η c + Λ, and the next lowest is by J/ψ + Λ. However, those two decay channels are suppressed by the effect of the light flavor SU(3) f breaking and the heavy quark HQS breaking. Because P cs8 , P cs8 and P * cs8 are flavor singlet, the decay to η c + Λ and/or J/ψ + Λ breals SU(3) f symmetry. Concerning P cs8 , the decay to J/ψ + Λ is further suppressed by the HQS breaking, because the spin of cc pair in P cs8 is predominantly singlet. Concerning P cs8 and P * cs8 , in contrast, the decay to η c + Λ is suppressed by the HQS breaking, because the spins of cc pair in P cs8 are approximately triplet. For P cs8 , P cs8 and P * cs8 , the decays to D s + Λ c and D + Ξ c are not suppressed both in the SU(3) f and in the HQS breaking. Though there may be some contributions which are not neglected for P cs8 , P cs8 because of S-wave decay, it may be possible that the emission energy is not so large, and hence the small phase space may make the decay widths small. The decay from P * cs8 (spin 3/2) is expected to be suppressed because it is D-wave decay.
We may consider the three-body state in the final state. The example is given by η c +π+Σ (threshold energy 4315 MeV). This decay process is not suppressed by the SU(3) f breaking.
However, the phase space of the three-body final states is smaller than that in two-body final sate, and hence the decay widths may not be so large. We may also consider that the decay widths could be suppressed because the color degrees of freedom should be recombined from the color octet in the initial state to the final state η c + Λ and J/ψ + Λ. To estimate the decay widths quantitatively is left as future works.

V. CONCLUSION
We investigate the internal structure of ccuds charm pentaquark, in which cc cluster is the color octet state. This is an exotic color configuration which cannot be realized in charmonia. The light flavor multiplet of this state is flavor-singlet. By adopting the colorspin interaction and the instanton-induced interaction, we have found that P cs8 with total spin 1/2 and cc spin 1 will be the most stable state, while the other states, P cs8 with total spin 1/2 and cc spin 0 and P * cs8 with total spin 3/2 and cc spin 1, are the excited states. The size of cc as well as the size of uds in those states are much less than one fm, and hence they are the compact multiquark states. We investigate also the mixing of the P cs8 and P cs8 due to the breaking of the heavy quark symmetry, but find that the mixing effect is not  Fig. 3). The decays to SU(3) f singlet final state is suppressed as indicated by "SU(3) f ", because P cs8 , P cs8 and P * cs8 are SU(3) f octet. The decay to the final state including η c (J/ψ) is suppressed for the initial state P cs8 and P * cs8 with s cc = 1 (P cs8 with s cc = 0), as denoted by "HQS". The decay channels in the last two rows are suppressed by the color recombination ("color recomb.").
Decay channels P cs8 (s cc = 0) P cs8 (s cc = 1) P * cs8 (s cc = 1) so large. We discuss several possible decay process of ccuds for the obtained masses, and find many channels should be suppressed by light flavor SU(3) symmetry or by the heavy quark symmetry or by both of them. Therefore, we conclude that the ccuds pentaquark is a candidate which should be searched in experimental studies. This is an interesting subject for experiments at high energy accelerator facilities.