PHYSICAL REVIEW D 99, 094006 (2019) Heavy pentaquark spectroscopy in the diquark 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 quark 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 Quantum chromodynamics (QCD) foresees the exist- the observation of the Θþ pentaquark, consisting of four ence of quark-antiquark states (mesons) and three-quark light quarks and a strange antiquark, but such evidence has states (baryons), as well as multiquark states [1,2] such as not been confirmed by later experiments [19]. tetraquarks, 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 masses 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 meson focus on the hidden-charm ccqqq¯ pentaquarks, having the state in the quark model [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 mass 4380 Æ b c considered in, e.g., [27,28]. 8 29 MeV and 4449.8 1.7 2.5 MeV, opposite parity Æ Æ Æ 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 baryon 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-gluon-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 bound state 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 particle. As usual, we use distance region. the trick 094006-2 HEAVY PENTAQUARK SPECTROSCOPY IN THE DIQUARK … PHYS. REV. D 99, 094006 (2019) 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.