Analysis of the \({\frac{1}{2}}^{\pm }\) pentaquark states in the diquark model with QCD sum rules

Eur. Phys. J. C (2016) 76:43 DOI 10.1140/epjc/s10052-016-3880-8 Regular Article - Theoretical Physics ± Analysis of the 21 pentaquark states in the diquark model with QCD sum rules Zhi-Gang Wang1,a , Tao Huang2,b 1 Department of Physics, North China Electric Power University, Baoding 071003, People’s Republic of China 2 Institute of High Energy Physics and Theoretical Physics Center for Science Facilities, Chinese Academy of Sciences, Beijing 100049, People’s Republic of China Received: 20 August 2015 / Accepted: 4 January 2016 / Published online: 25 January 2016 © The Author(s) 2016. This article is published with open access at Springerlink.com Abstract In this article, we present the scalar-diquark– scalar-diquark–antiquark type and scalar-diquark–axialvector-diquark–antiquark type pentaquark configurations in the diquark model, and study the masses and pole residues of ± the J P = 21 hidden-charm pentaquark states in detail with the QCD sum rules by extending our previous work on the + − J P = 23 and 25 hidden-charm pentaquark states. We calculate the contributions of the vacuum condensates up to dimension-10 in the operator product expansion by constructing both the scalar-diquark–scalar-diquark–antiquark type and the scalar-diquark–axialvector-diquark–antiquark type interpolating currents. The present predictions of the masses can be confronted to the LHCb experimental data in the future. 1 Introduction Recently, the LHCb collaboration observed two pentaquark candidates Pc (4380) and Pc (4450) in the J/ψ p mass spectrum in the 0b → J/ψ K − p decays with the significances of more than 9 σ [1]. The measured masses and widths are M Pc (4380) = 4380 ± 8 ± 29 MeV, M Pc (4450) = 4449.8 ± 1.7 ± 2.5 MeV,  Pc (4380) = 205 ± 18 ± 86 MeV and  Pc (4450) = 39 ± 5 ± 19 MeV, respectively. The Pc (4380) and Pc (4450) have the preferred spin–parity + − J P = 23 and 25 , respectively. The decays Pc (4380) → J/ψ p take place through relative S-wave while the decays Pc (4450) → J/ψ p take place through relative P-wave, the decays Pc (4450) → J/ψ p are suppressed in the phase space, so the Pc (4450) has smaller width. There have been several attempted assignments, such as the c D̄ ∗ , c∗ D̄ ∗ , χc1 p, J/ψ N (1440), J/ψ N (1520) molecule-like a e-mail: b e-mail: pentaquark states [2–8] (or not the molecular pentaquark states [9]), the diquark–diquark–antiquark type pentaquark states [10–14], the diquark-triquark type pentaquark states [15], re-scattering effects [16–18], etc. We can test their resonant nature by using photoproduction off a proton target [19–21]. In Ref. [14], we construct the scalar-diquark–axialvectordiquark–antiquark type interpolating currents, calculate the contributions of the vacuum condensates up to dimension-10 in the operator product expansion, and we extend the energyscale formula suggested in our previous works [22–25] to + − study the masses and pole residues of the J P = 23 and 25 hidden-charm pentaquark states with the QCD sum rules, − and assign the Pc (4380) and Pc (4450) to be the 23 and 5+ 2 pentaquark states, respectively. In this article, we extend ± our previous work to the study of the J P = 21 diquark– diquark–antiquark type hidden charm pentaquark state by calculating the contributions of the vacuum condensates up to dimension-10, and try to obtain the lowest masses based on the QCD sum rules. The article is arranged as follows: we choose the optimal pentaquark configurations in Sect. 2; in Sect. 3, we derive the ± QCD sum rules for the masses and pole residues of the 21 pentaquark states; in Sect. 4, we present the numerical results, and Sect. 5 is reserved for our summary and discussions. 2 Pentaquark configurations in the diquark model The diquarks q Tj Cqk have five structures in Dirac-spinor space, where C = Cγ5 , C, Cγμ γ5 , Cγμ and Cσμν for the scalar, pseudoscalar, vector, axialvector and tensor diquarks, respectively, and the j and k are color indices. The matrices Cγμ and Cσμν are symmetric, the matrices Cγ5 , C and Cγμ γ5 are antisymmetric. The attractive interactions 123 43 Page 2 of 14 Eur. Phys. J. C (2016) 76:43 of one-gluon exchange favor formation of the diquarks in color antitriplet 3c , flavor antitriplet 3 f and spin singlet 1s [26,27], while the favored configurations are the scalardiquark states (εi jk q Tj Cγ5 qk ) and axialvector-diquark states (εi jk q Tj Cγμ qk ) [28–30]. The calculations based on the QCD sum rules indicate that the heavy-light scalar and axialvectordiquark states have almost degenerate masses [28,29], while the masses of the light axialvector diquark states lie about (150–200) MeV above that of the light scalar-diquark states [30], if they have the same quark constituents. In this article, we take the diquark states as basic constituents, and we choose the scalar-diquark–scalar-diquark–antiquark type and scalar-diquark–axialvector-diquark–antiquark type pentaquark configurations. Now we illustrate how to construct the pentaquark states in the diquark model according to the spin–parity J P , − (1) 3− 1− 1− = ⊕ , (2) 2 c̄ 2 uudcc̄ 2 uudcc̄     + 1 1− 3+ + + + − + 0ud ⊗ 0uc ⊗ 1 ⊗ = 0ud ⊗ 0uc ⊗ ⊕ 2 c̄ 2 c̄ 2 c̄ + + 1 3 = ⊕ , (3) 2 uudcc̄ 2 uudcc̄     1+ 3+ 1− + + − + 0ud = 0 ⊗ 1+ ⊗ 1 ⊗ ⊗ 1 ⊗ ⊕ uc uc ud 2 c̄ 2 c̄ 2 c̄     + + + + 1 5+ 3 3 1 ⊕ ⊕ ⊕ ⊕ = , 2 uudcc̄ 2 uudcc̄ 2 uudcc̄ 2 uudcc̄ 2 uudcc̄ + 0ud ⊗ 1+ uc ⊗ (4) where the 1− denotes the contribution of the additional Pwave to the spin–parity, the subscripts ud, uc, c̄ and uudcc̄ denote the quark constituents. The quark and antiquark have opposite parity, we usually take it for granted that the quarks have positive parity while the antiquarks have negative parity, − so the c̄-quark has J P = 21 . − + The overlined states 23 uudcc̄ and 25 uudcc̄ are assigned to be the pentaquark states Pc (4380) and Pc (4450), respectively [14]. In previous work [14], we chose the scalardiquark–axialvector-diquark–antiquark type currents Jμ (x) and Jμν (x), T Jμ (x) = εila εi jk εlmn u Tj (x)Cγ5 dk (x) u m (x) ×Cγμ cn (x) C c̄aT (x), (5) 1 Jμν (x) = √ εila εi jk εlmn u Tj (x)Cγ5 dk (x) 2  T (x)Cγμ cn (x) γν C c̄aT (x) × um  T + um (x)Cγν cn (x) γμ C c̄aT (x) , (6) 123 + + pentaquark states, while their P-wave partners 21 uudcc̄ are supposed to be the lowest pentaquark states with the positive parity. In this article, we choose both the scalar-diquark–scalardiquark–antiquark type and the scalar-diquark–axialvectordiquark–antiquark type currents J jL j H (x), T (x) J00 (x) = εila εi jk εlmn u Tj (x)Cγ5 dk (x) u m ×Cγ5 cn (x) γ5 C c̄aT (x), (7) T (x) J01 (x) = εila εi jk εlmn u Tj (x)Cγ5 dk (x) u m μ T ×Cγμ cn (x) γ C c̄a (x), (8) ± 1− = , 2 c̄ 2 uudcc̄ 1 + + 0ud ⊗ 0uc ⊗ − to interpolate the 23 and 25 pentaquark states, respectively, where the i, j, k, . . . are color indices, the C is the charge conjugation matrix. − The underlined states 21 uudcc̄ are supposed to be the lowest to study the lowest pentaquark s (...truncated)