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PHYSICAL REVIEW D 99, 094006 (2019)

Heavy pentaquark spectroscopy in the model

Floriana Giannuzzi* Universit`a degli Studi di Bari,via Orabona 4, I-70126 Bari, Italy

(Received 18 March 2019; published 7 May 2019)

QQ0qqq¯ pentaquarks are studied in a potential model, under the hypothesis that they are composite objects of two diquarks and one antiquark. The interaction between two colored objects includes two contributions, one based on the qq¯ potential in QCD, computed in the gauge/string duality approach, and another describing the spin-spin interaction. The model has been extended to investigate pentaquarks with different content, as Qqqqq¯ and QqqqQ¯ , the latter including the states observed by LHCb, þ þ þ þ Pcð4380Þ and Pcð4450Þ , later updated, with a new data sample, to Pcð4312Þ , Pcð4440Þ , and þ Pcð4457Þ .

DOI: 10.1103/PhysRevD.99.094006

I. INTRODUCTION [15], ITEP in Russia [16], Jefferson Lab in Virginia [17], and from the ELSA accelerator in Germany [18] announced (QCD) foresees the exist- the observation of the Θþ pentaquark, consisting of four ence of quark-antiquark states () and three-quark light and a strange antiquark, but such evidence has states (), as well as multiquark states [1,2] such as not been confirmed by later experiments [19]. , comprising two quarks and two antiquarks, and Considering that many experimental results will be pentaquarks, comprising four quarks and one antiquark. allowed in the next few years by the increasing luminosity The only requirement for these states is to be color singlets. at experiments like LHCb at CERN and Belle-II at Although most of the ground state mesons and baryons are SuperKEKB, in this paper we compute the of experimentally well known, many recently observed states heavy pentaquarks using a potential model. Our results can are under discussion since their quark content and/or spin/ be compared with outcomes of different studies that parity are uncertain [3,4]; for a review on possible exotic appeared in the past few years in this sector. For example, states see [5,6,7]. One of the most intriguing cases is the the masses of QQqqq¯ , Q being a heavy quark and q a light Xð3872Þ, first observed by the Belle Collaboration [8]. The quark, have been computed in [20,21], while [22,23,24,25] spin parity assignment 1þþ is compatible with a focus on the hidden-charm ccqqq¯ pentaquarks, having the state in the [9], however its decay channels same quark content as the states observed by LHCb. In [26] suggested a possible interpretation as a four-quark state a classification of all possible QQqqq¯ states and quantum [10,11]. In 2015, LHCb observed two resonances in the numbers has been presented. QQqqq¯ states have been J=ψp channel in Λ0 decay, labeled Pþ, with 4380 b c considered in, e.g., [27,28]. 8 29 MeV and 4449.8 1.7 2.5 MeV, opposite parity In the investigation of multiquark states, one of the most 3=2 5=2 and spin and , compatible with heavy pentaquark discussedissuesistheexistenceofpossibleinternalstructures. ccuud¯ Rencontres states [12]. Later on, in 2019, during the In this respect, the main hypotheses for pentaquarks are that de Moriond conference, the LHCb Collaboration they could be compact states, relyingonthe interaction among announced [13] the observation, in the same energy region, two diquarks and an antiquark [29], or molecular states, 4312 þ 4440 þ 4457 þ of the resonances Pcð Þ , Pcð Þ , and Pcð Þ relying on the interaction between a and a meson [14]. According to this new analysis, the previously [30,31]. Following the former approach, in Sec. III A we 4450 þ reported state Pcð Þ contains two narrow peaks, compute the masses of QQ0qqq¯ pentaquarks using the model 4440 þ 4457 þ corresponding to Pcð Þ and Pcð Þ . Previously, introduced in Sec. II. An attempt to study pentaquarks with a in 2003 researchers from the SPring-8 laboratory in Japan different quark content is put forward in Secs. III B–III C. Section IV contains discussions and conclusions. *[email protected]

Published by the American Physical Society under the terms of II. MODEL the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to We study pentaquarks in the potential model introduced the author(s) and the published article’s title, journal citation, in [32], in which meson masses are computed by solving and DOI. Funded by SCOAP3. the wave equation:

2470-0010=2019=99(9)=094006(7) 094006-1 Published by the American Physical Society FLORIANA GIANNUZZI PHYS. REV. D 99, 094006 (2019)

FIG. 1. Dendrogram and picture of the quark content of the pentaquark in the diquark-diquark-antiquark model (model A).

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi FIG. 2. Blue plain line: VQCDðrÞ from Eq. (3) for c ¼ 2 2 2 2 0 30 2 2 75 ð m1 − ∇ þ m2 − ∇ þ VðrÞÞψðrÞ¼MψðrÞ; ð1Þ . GeV and g ¼ . . Orange dashed line: Cornell potential − a 0 63 0 18 2 VðrÞ¼ r þ br þ C for a ¼ . , b ¼ . GeV , and C ¼ −0.22 GeV, fixed from the requirement that the Cornell where m1 and m2 are the masses of the constituent quark and antiquark, VðrÞ is the quark-antiquark potential, M and potential has the same asymptotic behavior as VQCDðrÞ. ψ are the mass and wave function of the meson. Equation (1) arises from the Bethe-Salpeter equation in diquarks can attract each other forming a four-quark state QCD by considering an instantaneous local potential of (in the 3 representation of the group), and finally a four- interaction. Differently from the Schrödinger equation, it quark state plus an antiquark form a color singlet (the has relativistic kinematics. pentaquark). We adopt the one--exchange approxima- Equation (1) can be also used to study pentaquarks if a tion, in which the potential of interaction between two pentaquark is considered as the of two diquarks quarks (3 ⊗ 3) is equal to half the qq¯ potential (3 ⊗ 3¯) [33]. and an antiquark. The strategy consists in computing at first The qq¯ potential used in (1) for each two-body inter- diquark masses and wave functions from interactions action has three terms: between single quarks, then the mass and wave function of the four-quark state formed by two diquarks, and finally V r V r V r V0; the mass and wave function of the pentaquark resulting from ð Þ¼ QCDð Þþ spinð Þþ ð2Þ the interaction between the four-quark state and one anti- quark, as shown in Fig. 1. We call this model A. In this where V0 is a constant term, VQCDðrÞ represents the color picture, each interaction is between two objects, as in Eq. (1). interaction and VspinðrÞ the spin-spin interaction. A Model is based on the diquark-diquark-antiquark descrip- For VQCDðrÞ we use the potential found in [34] in a tion of pentaquarks [29], and on SU(3) color group argu- phenomenological model inspired by the AdS/QCD cor- ments, according to which two quarks (in the 3 respondence by computing the expectation value of a representation of the group) can attract each other forming rectangular Wilson loop. The potential is given in para- a diquark (in the 3¯ representation), and similarly two metric form:

8 rffiffiffi Z  1 > g c 2 2 > λ −1 −2 λv =2 1 − 4 λð1−v Þ −1=2 − 1 <> VQCDð Þ¼ þ dv v ½e ð v e Þ π λ 0 rffiffiffi Z ð3Þ > 1 > λ 2 2 −1 2 : rðλÞ¼2 dv v2eλð1−v Þ=2ð1 − v4eλð1−v ÞÞ = ; c 0

where r is the distance between the quark and antiquark, c The term VspinðrÞ is given by [36]: and g are parameters. A comparison between V ðrÞ and QCD   the Cornell potential VðrÞ¼− a þ br þ C [35] is shown in ˜ 3 r δðrÞ σ 2 2 2 S S δ˜ ffiffiffi −σ r Fig. 2, for values of parameters c ¼ 0.30 GeV and g ¼ VspinðrÞ¼A 1 · 2 with ðrÞ¼ p e ; m1m2 π 2.75 in Eq. (3), and a ¼ 0.63, b ¼ 0.18 GeV2, C ¼ −0.22 GeV for the Cornell potential, these latter ensuring ð4Þ the two potentials have the same asymptotic behavior at large and small distances. One can notice that the Cornell where σ is a parameter defining the smeared delta function potential is slightly lower than VQCDðrÞ in the middle- and S is the spin of the interacting . As usual, we use distance region. the trick

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A 1 σ 3 TABLE I. Masses (GeV) and spin couplings (GeV) of diquarks. Qq diquark has spin 0, and Qq has spin 1. κ pffiffi S1 · S2, ½ f g ¼ 2 m1m2 ð πÞ while κ¯ is defined in Eq. (8).

½cqfcqg½bqfbqg½csfcsg½bsfbsg Mass 2.118 2.168 5.513 5.526 2.237 2.276 5.619 5.630 κ −1.799 0.600 −0.236 0.079 −1.197 0.399 −0.157 0.052 κ¯ −0.053 0.009 −0.012 0.003 −0.055 0.008 −0.009 0.002

1 However, diquarks are extended objects, so we take into S1 · S2 ¼ ðSðS þ 1Þ − S1ðS1 þ 1Þ − S2ðS2 þ 1ÞÞ; ð5Þ 2 account the structure of the diquarks by defining a smeared potential [32]: S being the total spin. The parameter A is proportional to Z Z α 1 the strong coupling constant s in the one-gluon-exchange ˜ 2 2 VðRÞ¼ dr1 dr2jψ ðr1Þj jψ ðr2Þj VðjR þ r1 − r2jÞ: approximation. N d d A cutoff at small distance is introduced to cure the ð6Þ singularity of the wave function, fixing the potential (2) at k the value Vðr Þ for r ≤ r , with r ¼ in case m1 ¼ m2, M M M M In this equation ψ is the diquark wave function, N is a k0 0 d and rM ¼ in case m1 ≠ m2 [37,38]. k and k are two 2 M normalization factor. Since jψ dðrÞj is strongly peaked at parameters and M is the mass of the final state. r ∼ 0, we cut the integral at the peak value of the function Notice that both the one-gluon-exchange approximation udðrÞ¼rψ dðrÞ [32]. In the last step, the potential produc- and the use of an instantanous potential can be properly ing a singlet state is obtained from a convolution of (2) with applied only to heavy states, in which at least one of the two the diquark-diquark ψ dd wave function [41]: interacting is heavy, i.e., contains a charm or Z . Therefore, we compute masses of penta- 1 Vˆ ðRÞ¼ dr jψ ðrÞj2VðjR þ rjÞ ð7Þ quarks containing at least one heavy quark. Moreover, at N0 dd each step we only consider states with orbital angular momentum l ¼ 0. with N0 a normalization factor. We solve the Salpeter equation (1) through the Multhopp method [38], which allows one to transform an integral III. RESULTS equation into a set of linear equations containing variables called Multhopp’s angles. We fix the parameters of the A. QQ0qqq¯ model as in [32], where the masses of heavy mesons have Using the model A introduced in the previous section, been fitted to their experimental values: we compute the masses of pentaquarks comprising two heavy quarks. Each heavy quark forms a diquark with one 2 c ¼ 0.300 GeV g ¼ 2.750 V0 ¼ −0.488 GeV light quark, therefore these states can be well described in κ A ¼ 7.920 A ¼ 3.087 σ ¼ 1.209 GeV this framework. In Table I the masses and spin couplings c b of heavy diquarks are shown, in which the spin coupling κ 0 302 0 454 1 733 2 2 mq ¼ . GeV ms ¼ . GeV mc ¼ . GeV is defined as the coefficient multiplying e−σ r in the spin- A 1 σ 3 5 139 spin interaction potential (4), i.e., κ pffiffi S1 · S2, mb ¼ . GeV: ¼ 2 m1m2 ð πÞ where the factor 1=2 is due to the one-gluon exchange Two values for the parameter A in (4) have been introduced, approximation for the quark-quark interaction; Table I also O in order to take into account the two scales, ðmcÞ and contains the values of κ¯ defined as O α ðmbÞ, at which s must be computed: Ab is used for states Z comprising at least a beauty quark and A otherwise. The 1 c κ¯ dr ψ r 2 V r ; 8 model, with this choice of parameters, has been able to ¼ dð Þ 2 spinð Þ ð Þ predict with very good accuracy the mass of ηb [39], ψ observed soon after by the BABAR Collaboration [40]. with VspinðrÞ from Eq. (4) and d the diquark wave As a first step, diquark masses are obtained by solving function. We adopt the following notation: ½Qq diquark Eq. (1) with potential (2) divided by a factor 2 and a cutoff has spin 0, and fQqg has spin 1. at r ¼ rM, as done in [32]. In the second step, we use the Pentaquark masses are shown in Table II, where Salpeter equation to study the interaction between two q ¼fu; dg, q0 ¼fu; d; sg, and Q; Q0 ¼fc; bg. Since we diquarks. The diquark-diquark potential is assumed to be set l ¼ 0 in all the cases, the states have negative parity. As the same as between two quarks in a diquark: this suggests for tetraquarks [32], a large number of states with different that we adopt again the potential (2) divided by a factor 2. spin is found when combining two diquarks and an

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TABLE II. Masses (GeV) of QQ0qqq¯ pentaquarks, where q ¼ u, d and Q, Q0 ¼fc; bg.

Mass (Q; Q0 ¼ c) Mass (Q; Q0 ¼ b) Mass (Q ¼ b; Q0 ¼ c) Content JP q0 ¼ u, dq0 ¼ sq0 ¼ u, dq0 ¼ sq0 ¼ u, dq0 ¼ s ¯ 0 0 1− q½Qq½Q q 2 4.54 4.66 11.15 11.25 7.85 7.96 ¯ 0 0 1− qfQqg½Q q 2 4.57 4.68 11.16 11.26 7.86 7.97 ¯ 0 0 1− q½QqfQ q g 2 4.57 4.66 11.16 11.25 7.92 8.01 ¯ 0 0 1− qðfQqgfQ q gÞs¼1 2 4.64 4.73 11.19 11.28 7.94 8.04 ¯ 0 0 1− qðfQqgfQ q gÞs¼0 2 4.69 4.78 11.20 11.29 7.96 8.05 ¯ 0 0 3− qðfQqgfQ q gÞs¼2 2 4.62 4.72 11.18 11.27 7.94 8.03 ¯ 0 0 3− qfQqg½Q q 2 4.65 4.77 11.18 11.28 7.89 8.00 ¯ 0 0 3− q½QqfQ q g 2 4.65 4.75 11.18 11.28 7.95 8.04 ¯ 0 0 3− qðfQqgfQ q gÞs¼1 2 4.72 4.82 11.21 11.30 7.97 8.06 ¯ 0 0 5− qfQqgfQ q g 2 4.75 4.85 11.22 11.31 7.98 8.07 antiquark. There are five spin-1=2, four spin-3=2, and one (see Sec. III C), the candidates for the peaks observed by spin-5=2 states, as expected when combining five spin-1=2 LHCb in the hidden charm sector. particles. For the sake of completeness we should mention The strategy of the computation is described in the that not all the states can be considered in the spectrum following scheme, in which each step consists in solving since one must take spin, flavor and color representations Eq. (1) for the indicated particles: such that the total wavefunction of identical (1) Q þ q → Qq () is antisymmetric (symmetric). More details about (2) Qq þ q¯ → Qqq¯ this topic can be found in [26–28]. (3) Qqq¯ þ q → Qqqq¯ (4) Qqqq¯ þ q → Qqqqq¯. B. Qqqqq¯ The first three interactions are between states in the same representation of the color group (3 or 3¯), while only the last Let us consider pentaquarks with one heavy quark and one produces a color singlet. We treat the last interaction in four light quarks. Model A introduced in Sec. II cannot be the same way as the one between a quark and an antiquark, used here, since only one heavy diquark can be constructed, i.e., by solving Eq. (1) with a qq¯ potential, m1 being the mass while the other diquark would contain two light quarks. of the quark and m2 the mass of the four-quark state. Nevertheless, we try to determine the masses of these states Equation (1) has been used to study the first three inter- by considering the singlet state resulting from subsequent actions as well, with a potential equal to half the qq¯ potential interaction of the heavy quark with a light quark, as a [33]. The correction (7) to the potential has been considered. sequence of the two-body interactions sketched in Fig. 3. The masses of pentaquarks with one charm or one beauty This model, labeled B, is described hereinafter. Although are shown in Table III. Since l ¼ 0, all the states have such a configuration of quarks inside the pentaquark is negative parity. Different states correspond to different spin not usually expected, it is interesting to investigate this combinations, in which ½ indicates the combination having ¯ possibility, which can also be applied to QqqqQ states the lowest spin, while fg is the one with the highest.

FIG. 3. Dendrogram and sketch of the quark content of the pentaquark in the model B described in Sec. III B.

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TABLE III. Masses of Qqqqq¯ pentaquarks, where q ¼ u, d, TABLE V. Masses (GeV) of QqqqQ¯ pentaquarks, with q ¼ u, computed in model B described in Fig. 3. d, computed in model B of Fig. 3.

Mass Mass Mass Mass Content JP (GeV) Content JP (GeV) Content JP (GeV) Content JP (GeV) ¯ 1− ¯ 1− ¯ 1− ¯ 1− ½f½½cqqqgq 2 3.36 ½f½½bqqqgq 2 6.69 ½½½½cqcqq 2 4.57 ½½½½bqbqq 2 11.19 ¯ 1− ¯ 1− ¯ 1− ¯ 1− ½½½½cqqqq 2 3.37 ½½½½bqqqq 2 6.70 ½f½½cqcqgq 2 4.57 ½f½½bqbqgq 2 11.19 ¯ 1− ¯ 1− ¯ 1− ¯ 1− ½f½fcqgqqgq 2 3.39 ½f½fbqgqqgq 2 6.71 ½½ffcqgcgqq 2 4.58 ½½ffbqgbgqq 2 11.21 ¯ 1− ¯ 1− ¯ 1− ¯ 1− ½½ffcqgqgqq 2 3.39 ½½ffbqgqgqq 2 6.71 ½f½fcqgcqgq 2 4.64 ½f½fbqgbqgq 2 11.22 ¯ 1− ¯ 1− ¯ 1− ¯ 1− ½½½fcqgqqq 2 3.40 ½½½fbqgqqq 2 6.71 ½½½fcqgcqq 2 4.65 ½½½fbqgbqq 2 11.22 ¯ 3− ¯ 3− ¯ 3− ¯ 3− ff½½cqqqgqg 2 3.44 ff½½bqqqgqg 2 6.71 ff½½cqcqgqg 2 4.64 ff½½bqbqgqg 2 11.20 ¯ 3− ¯ 3− ¯ 3− ¯ 3− ½fffcqgqgqgq 2 3.45 ½fffbqgqgqgq 2 6.72 ½fffcqgcgqgq 2 4.66 ½fffbqgbgqgq 2 11.22 ¯ 3− ¯ 3− ¯ 3− ¯ 3− ff½fcqgqqgqg 2 3.48 ff½fbqgqqgqg 2 6.73 f½ffcqgcgqqg 2 4.67 f½ffbqgbgqqg 2 11.22 ¯ 3− ¯ 3− ¯ 3− ¯ 3− f½ffcqgqgqqg 2 3.48 f½ffbqgqgqqg 2 6.73 ff½fcqgcqgqg 2 4.71 ff½fbqgbqgqg 2 11.23 ¯ 5− ¯ 5− ¯ 5− ¯ 5− ffffcqgqgqgqg 2 3.57 ffffbqgqgqgqg 2 6.75 ffffcqgcgqgqg 2 4.76 ffffbqgbgqgqg 2 11.24

B A In order to compare models and , we have computed in Table V. The states with hidden charm and spin 1=2 have ¯ again the spectra of QQqqq, now assuming that the masses in the range 4.57–4.65 GeV, spin-3=2 states have B interaction among quarks works as in model , with results masses in the range 4.64–4.71 GeV, the spin-5=2 one has shown in Table IV. By looking at Tables II and IV, we find mass 4.76 GeV. The LHCb Collaboration has argued [12] that the mass difference between spin-1=2 states is at most þ that the first observed pentaquarks Pcð4380Þ and 50 MeV in the charm sector and 30 MeV in the bottom P ð4450Þþ have masses 4380829MeV and 4449.8 3 2 c sector, while it is at most 10 MeV for states with spin = 1.7 2.5 MeV, respectively, and opposite parity, while 5 2 and = in the charm and bottom sectors. subsequent analyses [14] have shown that in this mass þ þ region there are three resonances, Pcð4312Þ , Pcð4440Þ , C. QqqqQ¯ 4457 þ 4311 9 0 7þ6.8 4440 3 and Pcð Þ , with mass . . −0.6 MeV, . B ¯ 1 3þ4.1 4457 3 0 6þ4.1 Model can be used to study QqqqQ states, in order to . −4.7 MeV, and . . −1.7 MeV, respectively. We compare the outcomes with the masses of the states can try to compare these data with our theoretical pre- observed at LHCb. The values of the masses are shown dictions. The mass differences between theoretical and experimental results are equal to 260–340 MeV for þ − Pcð4312Þ (assuming it has spin-parity 1=2 [13]), 130– þ − TABLE IV. Masses of QQqqq¯ pentaquarks, with q ¼ u, d, 210 MeV for Pcð4440Þ (assuming it has spin-parity 1=2 B þ computed in model (see Fig. 3). [13]), and 180–250 MeV for Pcð4457Þ (assuming it has spin-parity 3=2− [13]). Regarding the previously observed Mass Mass state P 4450 þ, if we assume it has spin-parity 5=2−, its Content JP (GeV) Content JP (GeV) cð Þ mass is different from the theoretical result by 310 MeV. ¯ 1− ¯ 1− ½½½½cqqqc 2 4.57 ½½½½bqqqb 2 11.18 Finally, the discrepancy between the previously observed þ 1 1 state P ð4380Þ and masses in Table V of spin 3=2 ½½½fcqgq¯qc − 4.60 ½½½fbqgq¯qb − 11.19 c 2 2 pentaquarks is in the range 260–330 MeV. Therefore, ¯ 1− ¯ 1− ½f½½cqqqgc 2 4.61 ½f½½bqqqgb 2 11.19 the present study suggests that the new experimental results ¯ 1− ¯ 1− for P ð4440Þþ and P ð4457Þþ are more compatible with a ½½ffcqgqgqc 2 4.63 ½½ffbqgqgqb 2 11.20 c c pentaquark spectrum with the predicted spin-parity assign- cq q¯ q c 1− 4.64 bq q¯ q b 1− 11.20 ½f½f g g 2 ½f½f g g 2 ment. If we estimate the theoretical error ≲80 MeV, as the ¯ 3− ¯ 3− ff½½cqqqgcg 2 4.63 ff½½bqqqgbg 2 11.19 one found when studying meson spectra with the same ff½fcqgq¯qgcg 3− 4.66 ff½fbqgq¯qgbg 3− 11.20 model and set of parameters [32], the masses predicted in 2 2 þ the first LHCb paper [12] and the mass of Pcð4312Þ [14] cq q¯ q c 3− bq q¯ q b 3− f½ff g g g 2 4.66 f½ff g g g 2 11.20 are significantly lower than the theoretical ones, while the ¯ 3− ¯ 3− 4440 þ 4457 þ ½fffcqgqgqgc 2 4.72 ½fffbqgqgqgb 2 11.22 newly observed states Pcð Þ and Pcð Þ [14] get a ffffcqgq¯gqgcg 5− 4.74 ffffbqgq¯gqgbg 5− 11.23 better comparison, even though their masses are system- 2 2 atically lower.

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FIG. 4. Dendrogram and sketch of the quark content of the pentaquark in the model described in Sec. III D.

D. More on QQqqq¯ picture. In particular, the scheme has been used for 0 ¯ Within this framework, i.e., finding pentaquark masses pentaquarks with two heavy quarks (QQ qqq). Then, we ¯ by a sequence of two-body interactions involving at least have studied Qqqqq pentaquarks introducing a different ¯ scheme of interaction, depicted in Fig. 3. The model has one heavy particle, another configuration for QQqqq states ¯ is allowed. Indeed, one can consider the interaction also been applied to QQqqq pentaquarks, with the same 4380 þ 4450 þ between a Qq diquark and a Qqq¯ state. A similar quark content as the states Pcð Þ and Pcð Þ configuration has been studied in [42–43] to compute observed by LHCb in the hidden-charm sector, recently updated to P 4312 þ, P 4440 þ, and P 4457 þ. pentaquark masses in the hidden charm sector, comprising cð Þ cð Þ cð Þ Comparing the predicted mass of cqqqc¯ states with the a cq diquark interacting with the triquark c¯ðqqÞ. The spirit experimental ones, we have found values higher than those of the computation is similar to what has been done in measured by LHCb, but with a better agreement for the previous sections, and is explained by Fig. 4. The results for þ þ newly observed Pcð4440Þ and Pcð4457Þ states. Further the pentaquark masses are in Table VI. A comparison investigations could help to shed light on this discrepancy, among Tables II, IV, VI shows that the masses found in this and clarify if it is due to the approximations involved in the configuration are larger than the ones found in models A B model. As a future study, it would be interesting to improve and . this potential model, making it more suitable for the description of exotic states, for instance by considering IV. CONCLUSIONS known masses of tetraquarks or pentaquarks as inputs when fixing the parameters, or by improving the choice of the We have computed pentaquark masses in a potential potential of interaction, going beyond the one-gluon- model. We have exploited a relativistic wave equation exchange approximation. A possible modification in this describing the interaction between two states, and tried to direction can consist in introducing, for pentaquark spec- accommodate pentaquarks in this framework by consider- troscopy, a new value for the constant term V0 of the quark- ing them as emerging from three subsequent interactions, antiquark potential, different from the one found when as shown in Fig. 1, in the diquark-diquark-antiquark studying meson spectra. Indeed, it has been stated that constituent quark masses in potential models can get different values in baryon and meson spectroscopy [10]. TABLE VI. Masses of QQqqq¯ pentaquarks in the diquark- This discrepancy can be taken into account in this model by triquark configuration, where q ¼ u, d. using a different offset for the potential, so a new value for V0. If we assume that the mass of the lightest spin-1=2 Mass Mass pentaquark with hidden charm is 4312 MeV, we find Content JP (GeV) Content JP (GeV) V0 ¼ −0.594 GeV. Using this value for studying the other l 0 ½cq½½cqq¯ 1− 4.59 ½bq½½bqq¯ 1− 11.20 states in the hidden-charm sector with ¼ and negative 2 2 parity, we find that the mass of the heaviest spin-1=2 ¯ 1− ¯ 1− ½cq½fcqgq 2 4.62 ½bq½fbqgq 2 11.21 pentaquark is 4.39 GeV and the mass of the heaviest ¯ 1− ¯ 1− 3 2 fcqg½½cqq 2 4.68 fbqg½½bqq 2 11.23 spin- = pentaquark is 4.45 GeV, getting a better agree- ment with experimental observations. Notice that spin fcqg½fcqgq¯ 1− 4.71 fbqg½fbqgq¯ 1− 11.25 2 2 splittings are not modified by a change in V0, while a ¯ 1− ¯ 1− fcqgffcqgqg 2 4.77 fbqgffbqgqg 2 11.26 different value of quark masses could, in principle, also fcqg½½cqq¯ 3− 4.69 fbqg½½bqq¯ 3− 11.23 affect the spin-spin interaction, so a proper investigation of 2 2 this aspect would deserve a dedicated study based on ¯ 3− ¯ 3− ½cqffcqgqg 2 4.70 ½bqffbqgqg 2 11.23 baryon spectroscopy. ¯ 3− ¯ 3− fcqg½fcqgq 2 4.72 fbqg½fbqgq 2 11.25 ACKNOWLEDGMENTS ¯ 3− ¯ 3− fcqgffcqgqg 2 4.78 fbqgffbqgqg 2 11.26 fcqgffcqgq¯g 5− 4.80 fbqgffbqgq¯g 5− 11.27 I would like to thank P. Colangelo and F. De Fazio for 2 2 valuable suggestions and discussions.

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