Electromagnetic multipole moments of the \(P_c^+(4380)\) pentaquark in light-cone QCD

Eur. Phys. J. C (2018) 78:379 https://doi.org/10.1140/epjc/s10052-018-5873-2 Regular Article - Theoretical Physics Electromagnetic multipole moments of the Pc+ (4380) pentaquark in light-cone QCD U. Özdem1,a , K. Azizi1,2,b 1 Department of Physics, Dogus University, Acibadem-Kadikoy, 34722 Istanbul, Turkey 2 School of Physics, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5531, Tehran, Iran Received: 19 March 2018 / Accepted: 4 May 2018 © The Author(s) 2018 Abstract We calculate the electromagnetic multipole moments of the Pc+ (4380) pentaquark by modeling it as the diquark–diquark–antiquark and D̄ ∗ c molecular state with − quantum numbers J P = 23 . In particular, the magnetic dipole, electric quadrupole and magnetic octupole moments of this particle are extracted in the framework of light-cone QCD sum rule. The values of the electromagnetic multipole moments obtained via two pictures differ substantially from each other, which can be used to pin down the underlying structure of Pc+ (4380). The comparison of any future experimental data on the electromagnetic multipole moments of the Pc+ (4380) pentaquark with the results of the present work can shed light on the nature and inner quark organization of this state. 1 Introduction Since the discovery of the X(3872), many charmonium/ bottomonium-like XYZ states have been reported in the experiment. Some of these hadrons were suggested to have internal structures more complex than the simple q̄q configuration for mesons or qqq/q̄ q̄ q̄ configuration for baryon/antibaryons in the conventional picture of the naive quark model, and they are good candidates of exotic hadrons. In the newly observed family of XYZ, there are some decay channels that break the isospin symmetry and affect the identification of the traditional charmonium/bottomonium states negatively. The investigation of the properties of these states is one of the most attractive and active branches of hadron physics. For some reviews on the theoretical and experimental progress on the properties of these new states see Refs. [1–12]. In 2015, the LHCb Collaboration discovered two candidates of the hidden-charm pentaquark states, a e-mail: b e-mail: Pc+ (4380) and Pc+ (4450), in the invariant mass spectrum of J/ψ p in the 0b → J/ψ K − p decay [13]. According to the LHCb measurements the Pc+ (4380) has a mass of 4380 ± 8 ± 29 MeV and a width of 205 ± 18 ± 86 MeV, while the Pc+ (4450) has a mass of 4449.8 ± 1.7 ± 2.5 MeV and a width of 39 ± 5 ± 19 MeV. The preferred spin-parity assignments of the Pc (4380) and Pc (4450) are J P = 3/2− and 5/2+ , respectively. The minimal quark content of the pentaquarks is cc̄uud because these states decay into J/ψ p, and hence they are good candidates of exotic hidden-charm pentaquarks. After the discovery of LHCb Collaboration there have been intensive theoretical studies to explain the properties of these states. The spectroscopic parameters and decays of the Pc+ (4380) and Pc+ (4450) pentaquarks have been studied with different models and approaches [14–50]. Different theoretical models give consistent mass results with the experimental observations. Hence, more spectroscopic and decay parameters are needed to be calculated and compared with the experimental data. In [46] it is shown that the molecular picture of D̄ ∗ c for Pc+ (4380) gives consistent results for both the mass and width with the experimental data. As we mentioned above, chasing the announcement of the observation of pentaquarks there have been extensive amount of studies on their features. However to acquire a deep understanding on their inner structure, which are still not precise yet, we are in need of more experimental and theoretical studies which may shed light on their features. In order to understand the internal structure of the hadrons in the nonperturbative regime of QCD, the essential challenges are the specification of the dynamical and statical properties of hadrons such as their electromagnetic multipole moments, coupling constants, masses and so on, both theoretically and experimentally. Many theoretical models precisely predict the mass and decay width of the multiquark states, but the internal structure of these states is still uncertain. In other words, the mass and decay width 123 379 Page 2 of 9 alone can not distinguish the internal structure of the multiquark states. Remember that the electromagnetic multipole moments are equally significant dynamical observables of the multiquark states. The electromagnetic multipole moments are directly related with the charge and current distributions in the hadrons and these parameters are directly connected to the spatial distributions of quarks and gluons inside the hadrons. Their magnitude and sign provide important information on structure, size and shape of hadrons. There are many studies in the literature committed to the study the electromagnetic multipole moments of the standard hadrons, but unfortunately relatively little are known about the electromagnetic multipole moments of the exotic hadrons. There are a few studies in the literature where the magnetic dipole moment of the pentaquarks are studied [17,51–57]. In this study, the magnetic dipole, electric quadrupole and magnetic octupole moments of the pentaquark state Pc+ (4380) (hereafter we will denote this state as Pc ) is extracted by using the diquark–diquark–antiquark and D̄ ∗ c molecular interpolating currents in the framework of the light cone QCD sum rule (LCSR). The LCSR has already been successfully applied to extract properties of hadrons for decades such as, form factors, coupling constants and the electromagnetic multipole moments. In this approach, the properties of the hadrons are expressed in terms of the light-cone distribution amplitudes (DAs) and the vacuum condensates [for details, see for instance [58–61]]. Since the electromagnetic multipole moments are expressed in terms of the features of the DAs and the QCD vacuum, any uncertainty in these parameters reflects the uncertainty of the estimations of the electromagnetic multipole moments. The rest of the paper is organized as follows: In section II, the calculation of the sum rules in LCSR will be presented. In the last section, we numerically analyze the sum rules obtained for the electromagnetic multipole moments and discuss the obtained results. The explicit expressions of the electromagnetic form factors defining the magnetic dipole, electric quadrupole and magnetic octupole moments are moved to the Appendix A. 2 The electromagnetic multipole moments of Pc pentaquark in LCSR In this section we derive the LCSR for the magnetic dipole, electric quadrupole and magnetic octupole moments of the Pc pentaquark. For this purpose, we consider a correlation function in the presence of the external electromagnetic field (γ ), 123 Eur. Phys. J. C (2018) 78:379  μν (q) = i d 4 xei p·x 0|T {Jμ (x) J¯ν (0)}|0γ , (1) where Jμ is (...truncated)