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PHYSICAL REVIEW D 99, 094023 (2019)

Where is the stable ?

† ‡ Woosung Park,1,* Sungtae Cho,2, and Su Houng Lee1, 1Department of Physics and Institute of Physics and Applied Physics, Yonsei University, Seoul 03722, Korea 2Division of Science Education, Kangwon National University, Chuncheon 24341, Korea

(Received 5 December 2018; published 16 May 2019)

We systematically analyze the flavor color spin structure of the pentaquark q4Q¯ system in a constituent model based on the chromomagnetic interaction in both the SU(3) flavor symmetric and SU(3) flavor broken case with and without charm . We show that the originally proposed pentaquark state Qsqqq¯ by Gignoux et al. and by Lipkin indeed belongs to the most stable pentaquark configuration, but that when correction based on recent experiments are taken into account, a doubly charmed antistrange pentaquark configuration Pcc (udccs¯) could be the most attractive flavor exotic configuration þ − þ that could be stable and realistically searched for at present through the Pcc → ΛcK K π final states. The þ þ þþ − þ þ proposed final state is just reconstructing K instead of π in the measurement of Ξcc → ΛcK π π reported by the LHCb collaboration and hence measurable immediately.

DOI: 10.1103/PhysRevD.99.094023

I. INTRODUCTION In this work, we systematically analyze the color flavor The possible existence of mutiquark beyond the spin structure of the pentaquark configuration within a ordinary hadrons was first discussed for the states constituent based on chromomagnetic inter- in Refs. [1,2] and for the H-dibaryon in Ref. [3]. Later, action. We show that the originally proposed pentaquark Qsqqq¯ possible stable pentaquark configurations Qsqqq¯ were state indeed belongs to the most stable pentaquark proposed in Ref. [4] and in Ref. [5]. The long experimental configuration, but that when charm quark mass correction search for the H-dibaryon has not been successful so far but based on recent experiments is taken into account, a doubly P udccs¯ is still planned at JPARC [6]. The search by Fermilab E791 charmed antistrange pentaquark configuration cc ( ) [7] for the proposed pentaquark state also failed to find any could be the most attractive flavor exotic configuration that significant signal for the exotic configurations. could be stable and realistically searched for at present. ð4Þ On the other hand, starting from the Xð3872Þ [8], possible It is useful to view the classification in terms of SU F, which deserves a full analysis in our approach in the exotic configurations XYZ and the pentaquark Pc [9] were recently found. These states are not flavor exotic but future. It can be shown that the most stable configuration 140 are known to contain cc¯ quarks. Heavy quarks were for that we find belongs to a multiplet in such a scheme many years considered to be stable color sources that would [18]. Here, we are restricting our discussion to categoriz- allow for a stable multiquark configuration that does not ing the four quark structure in terms of the light flavors. fall into usual hadrons. In particular, with the recent This is so because the color spin interaction effects are experimental confirmation of the doubly charmed suppressed when a heavy quark is involved. Hence, for a [10–12], there is new excitement in the physics of exotics in first order estimate, our analysis should be a valid general and in hiterto unobserved flavor exotic states with starting point. more than one heavy quarks [13–17]. II. SYSTEMATIC ANALYSIS OF q4Q¯ We first discuss the classification of the flavor, color and spin wave function for the ground state of the pentaquark *[email protected] † q4 Q c¯ [email protected] composed of light quarks and one heavy antiquark ( or ‡ ¯ [email protected] b) assuming that the spatial parts of the wave function for all quarks are in the s-wave. We categorize them into the flavor Published by the American Physical Society under the terms of states in SUð3ÞF, and then examine the color ⊗ spin states. the Creative Commons Attribution 4.0 International license. q4 Further distribution of this work must maintain attribution to The flavor states for can be decomposed into the the author(s) and the published article’s title, journal citation, direct sum of the irreducible representation of SUð3ÞF as and DOI. Funded by SCOAP3. follows:

2470-0010=2019=99(9)=094023(9) 094023-1 Published by the American Physical Society WOOSUNG PARK, SUNGTAE CHO, and SU HOUNG LEE PHYS. REV. D 99, 094023 (2019)

TABLE I. The SUð6ÞCS representations containing the ½1C;S interaction relevant for the pentaquark configuration, which multiplet. is similar to that of a tetraquark in Ref. [2], by introducing the quadratic Casimir operator of SUð6Þ , which is denoted SUð3Þ ⊗ SUð2Þ state SUð6Þ representation CS C S CS by C6: 4 2 2 3 ½1C; 1=2 ½2 1, ½32 1 , [21], ½421 X5 3 2 2 2 3 3 c c 6 6 3 3 ½1C; 3=2 ½1 , ½32 1 , ½3 1 , ½421 − λi λj σ⃗i · σ⃗j ¼ 4C5 − 8C4 − 2C5 þ 4C4 2 2 i

094023-2 WHERE IS THE STABLE PENTAQUARK … PHYS. REV. D 99, 094023 (2019) eigenstates of the quadratic Casimir operator of SUð6ÞCS for The emphasis here is that jΨ1i and jΨ2i are both eigen- 6 6 four quarks, C4, with 26=3 being their eigenvalue. states of the quadratic Casimir operator C5 of the SUð6ÞCS In the other approach, the two states can be directly because jΨ1i and jΨ2i are themselves the eigenstates. The obtained from Eq. (2). As we can see in Eq. (2) and Table I, eigenvalue can be calculated using the following formula: 2 2 3 both the ½32 1 and the ½1 SUð6ÞCS representations have S ¼ 3=2 X4 the state of the color singlet and . Also, these states 6 1 c c 6 1 3 1 3 2 C ¼ − λ λ σ⃗i · σ⃗5 þ C þ C − C involve the ½21 multiplets in the SUð6ÞCS representation 5 4 i 5 4 2 5 2 4 among the four quarks, which are conjugate to the flavor i¼1 150 states, so that the two fully antisymmetric orthonormal 1 ⃗ ⃗ 1 ⃗ ⃗ þ ðS · SÞ5 − ðS · SÞ4 þ 2I: ð7Þ flavor ⊗ color ⊗ spin states can be constructed from the 3 3 SUð6Þ representation ½32212 and ½13, CS We can verify from Eqs. (6) and (7) that the following 6 eigenvalue equation holds for the Casimir operator C5 of the SUð6ÞCS:

ð8Þ

ð5Þ where the superscript of S indicates the total spin of 2 2 3 ½32 1 ð½1 Þ of SUð6ÞCS representation. Unlike Eq. (4), we did not add subscripts in the CS part of the Young- Yamanouchi bases in Eq. (5) as the total spins for the four quarks are not determined. It should be noted that jΨ1i (jΨ2i) in Eq. (5) not only shows the symmetry property with respect to the four quarks, but also the explicit dependence 2 2 3 on the multiplet ½32 1 ð½1 Þ of SUð6ÞCS representation for jΨ i 2 2 3 the pentaquark. This means that the color spin parts of 1 From Eq. (6), the multiplets of ½32 1 ð½1 Þ of SUð6ÞCS (jΨ2i) in Eq. (5) are the eigenstates of the quadratic Casimir representation in Eq. (5) become the eigenstates of the C6 49=4 21=4 6 operator, 5, with ( ) being its eigenvalue. quadratic Casimir operator of SUð6ÞCS for four quarks, C4, From the SUð6ÞCS representation point of view, we can having 26=3 as its eigenvalue, as follows: infer that the linear sum of two fully antisymmetric flavor ⊗ color ⊗ spin states coming from the coupling scheme must belong to either the ½32212 state or ½13 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 SUð6ÞCS, given by ð Þ pffiffiffi pffiffiffi 9 5 2 jΨ i¼pffiffiffi jψ iþpffiffiffi jψ i; 1 7 1 7 2 pffiffiffi pffiffiffi 2 5 jΨ2i¼− pffiffiffi jψ 1iþpffiffiffi jψ 2i: ð6Þ 7 7

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6 Also, this eigenvalue equation of C4 is true of the Young- Yamanouchi bases in ½13 multiplets. Therefore, we can calculate the matrix element of Eq. (3) in terms of jΨ1i and jΨ2i given in Eq. (5), by using Eqs. (9), (6), and (8). Following the same procedure, one can construct the flavor ⊗ color ⊗ spin states for the remaining flavor cases for S ¼ 3=2, which satisfy the antisymmetry property among four quarks. The number of independent color spin states can be obtained by noting that for the pentaquark, there are three independent color singlet states. Hence multiplying this to the spin degeneracy gives the total number of independent states for a given spin state. From ð Þ the result, it is found that there are all together 12 color ⊗ 11 spin states that are both color singlet and S ¼ 3=2. These are expressed by the Young-Yamanouchi bases of the SUð6ÞCS representation among the four quarks, together 4 ¯ with the SUð6ÞCS Young diagram for the full q Q penta- quark state,

Finally, in the S ¼ 5=2 case, there exists only one color ⊗ spin state coming from the ½32212 representation in Table I, ð10Þ

ð12Þ

It is straightforward to calculate the expectation value of c c c c λ1λ2σ⃗1 · σ⃗2 and λ4λ5σ⃗4 · σ⃗5 with respect to the 12 color ⊗ spin states for S ¼ 3=2 given in Eq. (10) in a 12 by 12 matrix from. Then, by applying the permutation operator c c c c on either λ1λ2σ⃗1 · σ⃗2 or λ4λ5σ⃗4 · σ⃗5, one can obtain all the expectation values in the 12 by 12 matrix from. Here, we use the formula given by B. Other spin states c c c c In analogy to the S ¼ 3=2 case, we can apply the same ðijÞλ1λi σ⃗1 · σ⃗iðijÞ¼λ1λj σ⃗1 · σ⃗j; ð13Þ procedure to the S ¼ 1=2 case. In this case, there are all together 15 color ⊗ spin states that are both color singlet where ðijÞ is a permutation operator of finite group, S4, and S ¼ 1=2, and that are expressed by the Young- which acts on the color ⊗ spin states in Eq. (10) as well as Yamanouchi bases of the SUð6ÞCS representation among on the Young-Yamanouchi states corresponding to the the four quarks, like Eq. (10), Young diagram of q4. The ðijÞ operator can be expressed

094023-4 WHERE IS THE STABLE PENTAQUARK … PHYS. REV. D 99, 094023 (2019) P 5 1 c c 4 ¯ TABLE II. The expectation value of − λ λ σ⃗i · σ⃗j of q Q in SUð3Þ limit, which means i

56 32 (15, 1=2) 2 þ 3m1 3m1m5 4 SUð6ÞCS ½2 1 88 Eigenvalue 3 56 16 (15, 3=2) 2 − 3m1 3m1m5 3 SUð6ÞCS ½1 40 Eigenvalue 3 0 4 20 4 4 4 4 4 28 (15 , 1=2) ð 2 − 2 þ 2 þ 2 þ Þ 3m1 3m1m5 3m1 3m1m5 3m1 3m1m5 3m1 3m1m5 2 2 SUð6ÞCS ½21, ½32 1 pffiffiffiffiffi pffiffiffiffiffi 8 8 Eigenvalue − 3 ð 10 − 1Þ, 3 ð 10 þ 1Þ pffiffiffiffi pffiffiffiffi pffiffiffiffi pffiffiffiffi 150 3=2 88 172 16 10 16 10 16 10 16 10 136 368 ( , ) ð 2 þ 2 þ 2 þ 2 − Þ 21m1 21m1m5 21m1 21m1m5 21m1 21m1m5 21m1 21m1m5 2 2 3 SUð6ÞCS ½32 1 , ½1 40 Eigenvalue −12, 3 0 8 16 (15 , 5=2) 2 þ m1 3m1m5 2 2 SUð6ÞCS ½32 1 40 Eigenvalue 3 3 1=2 − 14 − 22 − pffiffi2 − pffiffi 2 − pffiffi2 − pffiffi 2 − 46 þ 26 ( , ) ð 2 m m 2 2 2 3m m Þ m1 1 5 3m1 3m1m5 3m1 3m1m5 3m1 1 5 3 SUð6ÞCS ½21½421 pffiffiffiffiffi pffiffiffiffiffi 8 8 Eigenvalue − 3 ð 31 þ 8Þ, 3 ð 31 − 8Þ 40 20 (3, 3=2) − 2 þ 3m1 3m1m5 3 SUð6ÞCS ½421 20 Eigenvalue − 3 ¯ 16 40 (6, 1=2) − 2 − 3m1 3m1m5

SUð6ÞCS ½21 56 Eigenvalue − 3 ¯ 16 20 (6, 3=2) − 2 þ 3m1 3m1m5 2 3 SUð6ÞCS ½3 1 4 Eigenvalue 3

using the generator of SUð3ÞC and SUð2ÞS, given by each matrix element, we also show the relevant SUð6ÞCS c c ð1=3I þ 1=2λi λj Þ × ð1=2I þ 1=2σ⃗i · σ⃗jÞ [20] acting on representations for the pentaquark state as well as the the color and spin space. Furthermore, it is also block eigenvalue of Eq. (3). As can be seen in the table, the most diagonal in both spaces in a 12 by 12 matrix form. In the attractive channel is given by the 2 × 2 matrix valued c c same way, we can calculate the expectation value of λi λj σ⃗i · ðF; SÞ¼ð3; 1=2Þ state. Upon diagonalizing the matrix in σ⃗j with respect to the 15 color ⊗ spin states for S ¼ 1=2 the m5 → ∞ one finds the eigenvalues (−16, −40=3), given in Eq. (11) in a 15 by 15 matrix from. where the lowest one corresponds to the most attractive By using the flavor ⊗ color ⊗ spin states for S ¼ 1=2, pentaquark state discussed in Refs. [4,5]. It should be noted S ¼ 3=2, and S ¼ 5=2, the expectation values of Eq. (3) that the factor −16 in this case can also be naively obtained can be calculated, as given in Table II. In Table II, below by assuming two diquarks ðud; usÞ in the udusc¯

094023-5 WOOSUNG PARK, SUNGTAE CHO, and SU HOUNG LEE PHYS. REV. D 99, 094023 (2019)

C pentaquark. However, as noted from the case of H TABLE III. The value of mimj (unit MeV). dibaryon, SU(3) breaking effects together with the addi- C C C C C C tional attraction from the strong charm quarks are important uu us ss sc uc us¯ to the realistic estimate of the stability: The color spin 18.25 12.87 6.55 4.43 4.12 18.65 m − m Css¯ Csc¯ Cuc¯ Ccc¯ C ¯ C ¯ interaction from the J=ψ ηc is much stronger than from ub sb naively scaling the color spin splitting in the light quark 13.49 6.75 6.65 5.29 2.15 2.25 sector by the charm quark mass [14]. C C IV. PENTAQUARK BINDING IN THE We show the value of mimj in Table III, and for cc we 1=2C SUð3Þ BROKEN CASE take it to be cc¯ . It is important to extract the numbers F from experimental mass difference rather than assessing To analyze the stability of the pentaquarks against the the mass dependence through the naive relation 1 lowest threshold, we introduce a simplified form for the Cm m ∝ . For example, the mass difference between i j mimj matrix element of the hyperfine potential term, where we the J=ψ and ηc, which is around 113 MeV and comes approximate the spatial overlap factors by constants that dominantly from the color spin interaction, is much larger depend only on the constituent quark of the two than that between ρ and π, which is around 640 MeV, quarks involved, multiplied by the ratios in the constituent quark 5 2 2 X masses mu=mc ∼ 1=25, which gives a mass difference of H ¼ − C λcλcσ⃗ σ⃗: ð Þ hyp mimj i j i · j 14 26 MeV only. Therefore, although Table II is used to i

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TABLE IV. The quark configurations of I ¼ 1=2 and I ¼ 3=2 pentaquark states with their lowest threshold components and their binding energies, ΔE, in units of MeV. The numbers in the brackets show the multiplicities of the flavor ⊗ color ⊗ spin states. I ¼ 1=2 S ¼ 1=2 S ¼ 3=2 S ¼ 5=2 Quark Config. ΔE State ΔE State ΔE State ¯ uudsb −77 NBsð5Þ −45 NBs ð4Þ −4 Σ B ð1Þ uudsc¯ −99 NDsð5Þ −39 NDs ð4Þ −4 Σ D ð1Þ uudcs¯ 17 ΛcKð5Þ −88 ΣcKð4Þ −1 ΣcK ð1Þ uudcc¯ −34 Nηcð5Þ −15 NJ=ψð4Þ −9 ΣcD ð1Þ sssuc¯ 133 ΞDsð3Þ −17 ΞDs ð3Þ −34 Ξ Ds ð1Þ ¯ sssub 87 ΞBsð3Þ 73 ΞBs ð3Þ −34 Ξ Bs ð1Þ I ¼ 3=2 S ¼ 1=2 S ¼ 3=2 S ¼ 5=2 Quark Config. ΔE State ΔE State ΔE State uuusc¯ 214 ΣDð3Þ −42 ΔDsð3Þ 0 ΔDs ð1Þ ¯ uuusb 170 ΣBð3Þ 142 ΣB ð3Þ 0 ΔBs ð1Þ uuucs¯ 274 ΣcKð3Þ 186 ΣcKð3Þ 0 ΔDs ð1Þ uuucc¯ 191 ΣcDð3Þ −20 Δηcð3Þ 0 ΔJ=ψð1Þ

TABLE V. The quark configurations of I ¼ 0 and I ¼ 1 pentaquark states with the lowest threshold hadron components and their binding energies, ΔE, in units of MeV. The numbers in the brackets show the multiplicities of the flavor ⊗ color ⊗ spin states. I ¼ 0 I ¼ 1 S ¼ 1=2 S ¼ 3=2 S ¼ 5=2 S ¼ 1=2 S ¼ 3=2 S ¼ 5=2 Quark Config. ΔE State ΔE State ΔE State ΔE State ΔE State ΔE State udscc¯ −124 Ληcð7Þ −43 ΛJ=ψð5Þ −12 ΞcD ð1Þ −46 Σηcð8Þ −31 ΣJ=ψð7Þ −44 Σ J=ψð2Þ udssc¯ −117 ΛDsð4Þ −62 ΛDs ð3Þ −7 Ξ D ð1Þ 54 ΣDsð4Þ 1 ΣDs ð4Þ −17 Σ Ds ð1Þ udccs¯ −135 ΞccKð4Þ −94 ΞccKð3Þ −3 ΞccK ð1Þ 133 ΞccKð4Þ 85 ΞccKð4Þ 6 ΞccK ð1Þ udccc¯ −38 Λcηcð4Þ −43 ΛcJ=ψð3Þ −17 ΞccD ð1Þ 14 Σcηcð4Þ −31 Σcηcð4Þ 0 ΣcJ=ψð1Þ ¯ udssb −92 ΛBsð4Þ −67 ΛBs ð3Þ −7 Ξ B ð1Þ 24 ΣBsð4Þ 20 ΣBs ð4Þ −17 Σ Bs ð1Þ uudds¯ 98 NKð1Þ 74 NKð1Þ uuddc¯ 66 NDð1Þ 58 NDð1Þ uuddb¯ 54 NBð1Þ 52 NBð1Þ uuuds¯ 337 NKð2Þ −74 ΔKð2Þ 0 ΔKð1Þ uuudc¯ 223 NDð2Þ 79 NDð2Þ 0 ΔDð1Þ uuudb¯ 175 NBð2Þ 172 NBð2Þ 0 ΔBð1Þ

VI. RESULT AND SUMMARY [31] for the four quarks,1 for which the expectation value of Eq. (3) is −36, as can be seen in ðF ¼ 3;S¼ 1=2Þ sector of The result for the binding energy defined as the differ- Table II when m1 ¼ m5. In fact, the SUð6Þ representation ence between the hyperfine interaction of the pentaquark CS [21] state with S ¼ 1=2 gives the most attractive contri- against its lowest threshold values is given in Tables IV bution to the expectation value of Eq. (3) than any other and V. As can be seen in Tables IV and V, it is found state, and both the I ¼ 0 and I ¼ 1=2 comes from the that the most attractive pentaquark states are those with breaking of the flavor 3 state of this representation. ðI;SÞ¼ð0; 1=2Þ, apart from udccc¯, as well as the uudsc¯ In Table VI, we show the expectation value of Eq. (14) in with ðI;SÞ¼ð1=2; 1=2Þ. To understand the reason why terms of only a color ⊗ spin state coming from the SUð6Þ these could be bound states, we need to analyze CS representation [21] as well as the corresponding binding the expectation matrix value of Eq. (14) in terms of a dominant color ⊗ spin state among the possible states. For ⊗ 1 these states, the dominant color spin state comes from This state corresponds to the most stable Pcs¯ state discussed in the SUð6ÞCS representation [21] having the Young diagram Ref. [5].

094023-7 WOOSUNG PARK, SUNGTAE CHO, and SU HOUNG LEE PHYS. REV. D 99, 094023 (2019)

TABLE VI. The expectation value of Eq. (14) coming from the It should be noted that for the pentaquark to be stable dominant color ⊗ spin state for stable pentaquark candidate states against strong decay, the respective attraction obtained in (unit MeV). Table V should be large enough to overcome the additional kinetic energy needed to bring the pentaquark into a compact I ¼ 0, S ¼ 1=2 udccs¯ ðΔE ¼ −131Þ configuration. This additional repulsive energy can be Hhyp 11=4Ccc-11=2Ccs¯ -25=2Cuc estimated in a constituent quark model by comparing the -33=2Cus¯ -17=4Cuu total kinetic energy of a separated baryon meson configu-

ΞccK 8=3Ccc-32=3Cuc-16Cus¯ ration to that of the corresponding compact pentaquark configuration. The energy is related to p2 =2μ, where p is I ¼ 0, S ¼ 1=2 udscc¯ ðΔE ¼ −122Þ rel rel the relative momentum and thus the inverse size of the H 22=3C 44=3C 28=3C 14=3C hyp - sc¯ - uc¯ - us- uu compact configuration while μ is the reduced mass of Ληc -8Cuu-16Ccc¯ the lowest threshold baryon meson system [19]. For the P ðudccs¯Þ I ¼ 0, S ¼ 1=2 udssc¯ ðΔE ¼ −113Þ cc state, this is typically of order 100 MeV. Hence, the proposed pentaquark state could be a weakly stable H 11=4C 11=2C 25=2C hyp ss- sc¯ - us pentaquark or a resonance state slightly above the lowest -33=2Cuc¯ -17=4Cuu threshold, which is Ξcc þ K for this state. ΛD 8C 16C s - uu- sc¯ The Pccðudccs¯Þ could also be easily observed with the I ¼ 1=2, S ¼ 1=2 uudsc¯ ðΔE ¼ −92Þ present detection limits. Noting that Ξcc has been recently

Hhyp -33=4Csc¯ -55=4Cuc¯ -21=2Cus-7=2Cuu discovered, one can just add an additional to look for this possible resonance state. If the state is strongly bound, PDs -8Cuu-16Csc¯ þ − þ one could look at the Pccðudccs¯Þ → ΛcK K π decay or þ any hadronic decay mode similar to those of ΛcDs . The proposed final state is just reconstructing Kþ instead of πþ Ξþþ → Λ K−πþπþ energy against its threshold represented in the third row for in the measurement of cc c reported in each state. It should be noted that Hhyp for each state Ref. [10] and hence measurable immediately. Such a measurement would be the first confirmation of a flavor reduces to −36Cm m when the Cm m ’s are taken to be a i j i j exotic pentaquark state. quark mass independent constant. It should be noted that all these possible stable states are related to the attractive ACKNOWLEDGMENTS pentaquark states discussed in Ref. [4,5] in the flavor SU(3) symmetric limit. However, it should also be pointed out that This work was supported by the Korea National Research when the charm quark is also included, together with its Foundation under the Grants No. 2016R1D1A1B03930089 hyperfine contribution, it is the Pccðudccs¯Þ pentaquark and No. 2017R1D1A1B03028419(NRF), and by the configuration that is most attractive. This state has also National Research Foundation of Korea (NRF) grant been discussed recently in Ref. [24]. The next attractive funded by the Korea government (MSIP) (Grant state is udscc¯, which could also be stable. No. 2016R1C1B1016270).

[1] R. L. Jaffe, Phys. Rev. D 15, 267 (1977). [10] R. Aaij et al. (LHCb Collaboration), Phys. Rev. Lett. 119, [2] R. L. Jaffe, Phys. Rev. D 15, 281 (1977). 112001 (2017). [3] R. L. Jaffe, Phys. Rev. Lett. 38, 195 (1977); 38, 617(E) (1977). [11] R. Aaij et al. (LHCb Collaboration), Phys. Rev. Lett. 121, [4] C. Gignoux, B. Silvestre-Brac, and J. M. Richard, Phys. 052002 (2018). Lett. B 193, 323 (1987). [12] R. Aaij et al. (LHCb Collaboration), Phys. Rev. Lett. 121, [5] H. J. Lipkin, Phys. Lett. B 195, 484 (1987). 162002 (2018). [6] J. K. Ahn (J-PARC E42 Collaboration), J. Phys. Soc. Jpn. [13] H. X. Chen, Q. Mao, W. Chen, X. Liu, and S. L. Zhu, Phys. Conf. Proc. 17, 031004 (2017). Rev. D 96, 031501 (2017); 96, 119902(E) (2017). [7] E. M. Aitala et al. (E791 Collaboration), Phys. Lett. B 448, [14] M. Karliner and J. L. Rosner, Phys. Rev. Lett. 119, 202001 303 (1999). (2017). [8] S. K. Choi et al. (Belle Collaboration), Phys. Rev. Lett. 91, [15] E. J. Eichten and C. Quigg, Phys. Rev. Lett. 119, 202002 262001 (2003). (2017). [9] R. Aaij et al. (LHCb Collaboration), Phys. Rev. Lett. 115, [16] A. Francis, R. J. Hudspith, R. Lewis, and K. Maltman, Phys. 072001 (2015). Rev. Lett. 118, 142001 (2017).

094023-8 WHERE IS THE STABLE PENTAQUARK … PHYS. REV. D 99, 094023 (2019)

[17] J. Hong, S. Cho, T. Song, and S. H. Lee, Phys. Rev. C 98, [21] A. Park, W. Park, and S. H. Lee, Phys. Rev. D 98,034001 014913 (2018). (2018). [18] B. Wu and B. Q. Ma, Phys. Rev. D 70, 034025 (2004). [22] M. Oka and K. Yazaki, Prog. Theor. Phys. 66, 556 (1981). [19] W. Park, A. Park, S. Cho, and S. H. Lee, Phys. Rev. D 95, [23] W. Park, A. Park, and S. H. Lee, Phys. Rev. D 93, 074007 054027 (2017). (2016). [20] W. Park, A. Park, and S. H. Lee, Phys. Rev. D 92, 014037 [24] Q. S. Zhou, K. Chen, X. Liu, Y. R. Liu, and S. L. Zhu, Phys. (2015). Rev. C 98, 045204 (2018).

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