Research Article Hidden-Beauty Broad Resonanceπ‘Œπ‘(10890) in Thermal QCD J. Y. SΓΌngΓΌ ,1 A. TΓΌrkan,2 H. DaL,2,3 and E. Veli Veliev1,4 1Department of Physics, Kocaeli University, 41380 Izmit, Turkey 2ÖzyegΜ†in University, Department of Natural and Mathematical Sciences, CΜ§ekmeköy, Istanbul, Turkey 3Physik Department, Technische Universität München, D-85747 Garching, Germany 4Faculty of Education, Kocaeli University, 41380 Izmit, Turkey Correspondence should be addressed to J. Y. Süngü; jaleyil@gmail.com Received 2 December 2018; Revised 18 February 2019; Accepted 6 March 2019; Published 18 June 2019 Academic Editor: Alexey A. Petrov Copyright Β© 2019 J. Y. Süngü et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3. In this work, the mass and pole residue of resonanceπ‘ŒοΏ½ν‘ is studied by using QCD sum rules approach at finite temperature. Resonanceπ‘ŒοΏ½ν‘ is described by a diquark-antidiquark tetraquark current, and contributions to operator product expansion are calculated by including QCD condensates up to dimension six. Temperature dependencies of the mass π‘šοΏ½ν‘Œπ‘ and the pole residue πœ†οΏ½ν‘Œπ‘ are investigated. It is seen that near a critical temperature (𝑇�푐 ≃ 190 MeV), the values of π‘šοΏ½ν‘Œπ‘ and πœ†οΏ½ν‘Œπ‘ decrease to 87% and to 44% of their values at vacuum. 1. Introduction Heavy quarkonia systems provide a unique laboratory to search the interplay between perturbative and nonpertur- bative effects of QCD. They are nonrelativistic systems in which low energy QCD can be investigated via their energy levels, widths, and transition amplitudes [1]. Among these heavy quarkonia states, vector charmonium and bottomo- nium sectors are experimentally studied very well, since they can be detected directly in 𝑒+π‘’βˆ’ annihilations. In the past decade, observation of a large number of bottomonium-like states in several experiments increased the interest in these structures [2–6]. However, these observed states could not be conveniently explained by the simple π‘žπ‘ž picture of mesons. The presumption of hadrons containing quarks more than the standard quark content (π‘žπ‘ž or π‘žπ‘žπ‘ž) is introduced by a perceptible model for diquarks plus antidiquarks, which was developed by Jaffe in 1976 [7]. Later Maiani, Polosa, and their collaborators proposed that the X, Y, Z mesons are tetraquark systems, in which the diquark-antidiquark pairs are bound together by the QCD color forces [8]. In this color configuration, diquarks can play a fundamental role in hadron spectroscopy. Thus, probing the multiquark matter has been an intensely intriguing research topic in the past twenty years and it may provide significant clues to understand the nonperturbative behavior of QCD. In 2007, Belle reported the first evidence of 𝑒+π‘’βˆ’ 󳨀→Ξ₯(1𝑆)πœ‹+πœ‹βˆ’, Ξ₯(2𝑆)πœ‹+πœ‹βˆ’ and first observation for 𝑒+π‘’βˆ’ 󳨀→Ξ₯(3𝑆)πœ‹+πœ‹βˆ’, Ξ₯(1𝑆)𝐾+πΎβˆ’ decays near the peak of the Ξ₯(5𝑆) state at βˆšπ‘  = 10.87 GeV [2]. Assigning these signals toΞ₯(5𝑆), the partial widths of decays Ξ₯(5𝑆) 󳨀→ Ξ₯(1𝑆)πœ‹+πœ‹βˆ’ andΞ₯(5𝑆) 󳨀→ Ξ₯(2𝑆)πœ‹+πœ‹βˆ’ were measured unusually larger (more than two orders of magnitude) than formerly measured decay widths of Ξ₯(𝑛𝑆) states. Following these unusually large partial width measurement, Belle measured the cross sections of 𝑒+π‘’βˆ’ 󳨀→ Ξ₯(1𝑆)πœ‹+πœ‹βˆ’, Ξ₯(2𝑆)πœ‹+πœ‹βˆ’ and Ξ₯(1𝑆)πœ‹+πœ‹βˆ’ and reported that the resonance observed via these decays does not agree with conventional Ξ₯(5𝑆) line shape. These observations led to the proposal of existence of new exotic hidden-beauty state analogous to broad π‘Œ(4260) resonance in the charmonium sector, which is a Breit-Wigner shaped resonance with mass (10888.4+2.7βˆ’2.6 Β± 1.2) MeV/c2, and width (30.7+8.3βˆ’7.0 Β± 3.1) MeV/c2, and is called π‘ŒοΏ½ν‘(10890) [5]. In literature, there are several approaches to investigate the structure of exotic π‘ŒοΏ½ν‘ resonance. In [9], π‘ŒοΏ½ν‘ is considered as a Ξ› �푏Λ�푏 bound state with a highly large binding energy. In [10, 11], π‘ŒοΏ½ν‘ is interpreted as a tetraquark and its mass is estimated by using QCD sum rules at vacuum. Hindawi Advances in High Energy Physics Volume 2019, Article ID 8091865, 9 pages https://doi.org/10.1155/2019/8091865 https://orcid.org/0000-0002-1046-5755 https://creativecommons.org/licenses/by/4.0/ https://doi.org/10.1155/2019/8091865 2 Advances in High Energy Physics Moreover, it is likely that at very high temperatures within the first microseconds following the Big Bang, quarks and gluons existed freely in a homogenous medium called the quark-gluon plasma (QGP). One of the first quark-gluon plasma signals proposed in the literature is suppression of𝐽/πœ“ particles [12]. In 2011, CMS collaboration reported that charmonium states πœ™(2𝑆) and 𝐽/πœ“ melt or were suppressed due to interacting with the hot nuclear matter created in heavy-ion interactions [13, 14]. Following these observations in the charmonium sector, CMS collaboration also reported suppression of bottomonium states, Ξ₯(2𝑆) and Ξ₯(3𝑆) relative to the Ξ₯(1𝑆) ground state [15, 16]. The dissociation tempera- tures for theΞ₯states are expected to be related to their binding energies and are predicted to be 2𝑇�푐, 1.2𝑇�푐 and 𝑇�푐 for theΞ₯(1𝑆), Ξ₯(2𝑆), and Ξ₯(3𝑆) mesons, respectively, where 𝑇�푐 is the critical temperature for deconfinement [16–18]. In addition to these studies, one of the first investigations on thermal properties of exotic mesons was done in [19], in which the authors investigated in medium properties of 𝑋(3872) under the hypothesis that it is a 1++ state or2βˆ’+ state, but making no assumption on its structure. They estimated that the mass of the 1++ molecular state decreases with increasing temperature; the mass of charmonium or tetraquark state is almost stable. Inspired by these findings and motivated by the afore- mentioned discussions, we focus on the π‘ŒοΏ½ν‘ resonance and its thermal behavior. This paper is organized as follows. In Section 2, theoretical framework of Thermal QCD sum rules (TQCDSR) and its application to π‘ŒοΏ½ν‘ are presented, and obtained analytical expressions of the mass and pole residue of π‘ŒοΏ½ν‘ are given up to dimension six operators. Numerical analysis is performed and results are obtained in Section 3. Concluding remarks are discussed in Section 4. The explicit forms of the spectral densities are written in Appendix. 2. Finite Temperature Sum Rules for Tetraquark Assignment QCD sum rules (QCDSR) approach is based on Wilson’s operator product expansion (OPE) which was adapted by Shifman, Vainshtein, and Zakharov [24] and applied with remarkable success to estimate a large variety of properties of all low-lying hadronic states [25–27]. Later, this model is extended to its thermal version that is firstly proposed by Bochkarev and Shaposnikov and led to many successful applications in QCD at 𝑇 ΜΈ= 0 [28–35]. Very recently, in [36], the authors claimed that any QCDSR study on tetraquark states should contain 𝑂(𝛼2οΏ½ν‘  ) contributions to OPE, which are unknown. It is a very important and strong argument for studying tetraquark states within QCDSR; however, those terms and their thermal behaviors are not known, and their calculation is beyond the scope of this work. Thus we follow the traditional sum rules to investigate the thermal behavior of hadronic parameters of π‘ŒοΏ½ν‘, which were used very successfully in predicting properties of tetraquark states as well. In this work, we proceed with traditional sum rules analysis by using a tetraquark current, following several successful applications to exotic hadrons [37–42]. In this section, the mass and pole residue of the exotic π‘ŒοΏ½ν‘ resonance are studied by interpreting it as a bound [𝑏𝑠][𝑏𝑠] tetraquark via TQCDSR technique which starts with the two point correlation function Ξ οΏ½νœ‡] (π‘ž, 𝑇) = π‘–βˆ« 𝑑4π‘₯π‘’οΏ½ν‘–οΏ½ν‘žβ‹…οΏ½ν‘₯ ⟨Ψ 󡄨󡄨󡄨󡄨󡄨T {πœ‚οΏ½νœ‡ (π‘₯)πœ‚β€ ] (0)}󡄨󡄨󡄨󡄨󡄨 Ψ⟩ , (1) where Ξ¨ represents the hot medium state, πœ‚οΏ½νœ‡(π‘₯) is the interpolating current of the π‘ŒοΏ½ν‘ state, and T denotes the time ordered product. The thermal average of any operator 𝑂 in thermal equilibrium is given as βŸ¨π‘‚βŸ© = π‘‡π‘Ÿ (𝑒 βˆ’οΏ½ν›½H𝑂) π‘‡π‘Ÿ (π‘’βˆ’οΏ½ν›½H) , (2) where H is the QCD Hamiltonian, and 𝛽 = 1/𝑇 is inverse of the temperature, and 𝑇 is the temperature of the heat bath. Chosen current πœ‚οΏ½νœ‡(π‘₯) must contain all the information of the related meson, like quantum numbers, quark contents and so on. In the diquark-antidiquark picture, tetraquark current interpreting π‘ŒοΏ½ν‘ can be chosen as [10] πœ‚οΏ½νœ‡ (π‘₯) = π‘–πœ–πœ–βˆš2 {[𝑠 �푇 οΏ½ν‘Ž (π‘₯) 𝐢𝛾5𝑄�푏 (π‘₯)] [𝑠�푑 (π‘₯) π›ΎοΏ½νœ‡π›Ύ5𝐢𝑄�푇�푒 (π‘₯)] + [π‘ οΏ½ν‘‡οΏ½ν‘Ž (π‘₯)𝐢𝛾5π›ΎοΏ½νœ‡π‘„οΏ½ν‘ (π‘₯)] [𝑠�푑 (π‘₯)𝛾5𝐢𝑄�푇�푒 (π‘₯)]} , (3) where 𝑄 = 𝑏, 𝐢 is the charge conjugation matrix andπ‘Ž, 𝑏, 𝑐, 𝑑, 𝑒 are color indices. Shorthand notations πœ– = πœ–οΏ½ν‘ŽοΏ½ν‘οΏ½ν‘ andπœ– = πœ–οΏ½ν‘‘οΏ½ν‘’οΏ½ν‘ are also employed in (3). In TQCDSR, the correlation function given in (1) is cal- culated twice, as in two different regions corresponding two perspectives, namely the physical side (or phenomenological side) and the QCD side (or OPE side). By equating these two approaches, the sum rules for the hadronic properties of the exotic state under investigation are achieved. To derive mass and pole residue via TQCDSR, the correlation function is calculated in terms of hadronic degrees of freedoms in the physical side. A complete set of intermediate physical states possessing the same quantum number as the inter- polating current are inserted into (1), and integral over π‘₯ is handled. After these manipulations, the correlation function is obtained as Ξ PhysοΏ½νœ‡] (π‘ž, 𝑇) = ⟨Ψ σ΅„¨σ΅„¨σ΅„¨σ΅„¨σ΅„¨πœ‚οΏ½νœ‡σ΅„¨σ΅„¨σ΅„¨σ΅„¨σ΅„¨ π‘ŒοΏ½ν‘ (π‘ž)βŸ©οΏ½ν‘‡ βŸ¨π‘ŒοΏ½ν‘ (π‘ž) σ΅„¨σ΅„¨σ΅„¨σ΅„¨σ΅„¨πœ‚β€ ]󡄨󡄨󡄨󡄨󡄨 Ξ¨βŸ©οΏ½ν‘‡π‘š2 οΏ½ν‘Œπ‘ (𝑇) βˆ’ π‘ž2 + π‘ π‘’π‘π‘‘π‘Ÿπ‘Žπ‘π‘‘π‘’π‘‘ π‘‘π‘’π‘Ÿπ‘šπ‘ , (4) here π‘šοΏ½ν‘Œπ‘(𝑇) is the temperature-dependent mass of π‘ŒοΏ½ν‘ meson. Temperature-dependent pole residue πœ†οΏ½ν‘Œπ‘(𝑇) is defined in terms of matrix element as ⟨Ψ σ΅„¨σ΅„¨σ΅„¨σ΅„¨σ΅„¨πœ‚οΏ½νœ‡σ΅„¨σ΅„¨σ΅„¨σ΅„¨σ΅„¨ π‘ŒοΏ½ν‘ (π‘ž)βŸ©οΏ½ν‘‡ = πœ†οΏ½ν‘Œπ‘ (𝑇) π‘šοΏ½ν‘Œπ‘ (𝑇) πœ€οΏ½νœ‡, (5) where πœ€οΏ½νœ‡ is the polarization vector of the π‘ŒοΏ½ν‘ satisfying πœ€οΏ½νœ‡πœ€βˆ—] = βˆ’π‘”οΏ½νœ‡] + π‘žοΏ½νœ‡π‘ž]π‘š2 οΏ½ν‘Œπ‘ (𝑇). (6) Advances in High Energy Physics 3 After employing polarization relations, the correlation func- tion is written in terms of Lorentz structures in the form Ξ PhysοΏ½νœ‡] (π‘ž, 𝑇) = π‘š 2 οΏ½ν‘Œπ‘ (𝑇) πœ†2οΏ½ν‘Œπ‘ (𝑇)π‘š2 οΏ½ν‘Œπ‘ (𝑇) βˆ’ π‘ž2 (βˆ’π‘”οΏ½νœ‡] + π‘žοΏ½νœ‡π‘ž] π‘š2 οΏ½ν‘Œπ‘ (𝑇)) + . . . , (7) where dots denote the contributions coming from the contin- uum and higher states. To obtain the sum rules, coefficient of any Lorentz structure can be used. In this work, coefficients of π‘”οΏ½νœ‡] are chosen to construct the sum rules and the standard Borel transformation with respect to π‘ž2 is applied to suppress the unwanted contributions. The final form of the physical side is obtained as B (π‘ž2) Ξ Phys (π‘ž, 𝑇) = π‘š2οΏ½ν‘Œπ‘ (𝑇) πœ†2οΏ½ν‘Œπ‘ (𝑇) π‘’βˆ’οΏ½ν‘š2π‘Œπ‘(�푇)/οΏ½ν‘€2, (8) here 𝑀2 is the Borel mass parameter. In the QCD side,Ξ QCDοΏ½νœ‡] (π‘ž, 𝑇) is calculated in terms of quark-gluon degrees of freedom and can be separated into two parts over the Lorentz structures as Ξ QCDοΏ½νœ‡] (π‘ž, 𝑇) = Ξ QCD�푆 (π‘ž2, 𝑇) π‘žοΏ½νœ‡π‘ž]π‘ž2 + Ξ QCD�푉 (π‘ž2, 𝑇) (βˆ’π‘”οΏ½νœ‡] + π‘žοΏ½νœ‡π‘ž]π‘ž2 ) , (9) where Ξ QCD �푆 (π‘ž2, 𝑇) and Ξ QCD �푉 (π‘ž2, 𝑇) are invariant functions connected with the scalar and vector currents, respectively. In the rest framework of π‘ŒοΏ½ν‘ (q = 0), Ξ QCD�푉 (π‘ž20, 𝑇) can be expressed as a dispersion integral, Ξ QCD�푉 (π‘ž20, 𝑇) = ∫ οΏ½ν‘ 0(�푇) 4(οΏ½ν‘šπ‘+οΏ½ν‘šπ‘ ) 2 𝜌QCD (𝑠, 𝑇) 𝑠 βˆ’ π‘ž20 𝑑𝑠 + β‹… β‹… β‹… , (10) where corresponding spectral density is described as 𝜌QCD (𝑠, 𝑇) = 1πœ‹Im Ξ QCD�푉 (𝑠, 𝑇) . (11) The spectral density can be separated in terms of operator dimensions as 𝜌QCD (𝑠, 𝑇) = 𝜌pert. (𝑠) + πœŒβŸ¨οΏ½ν‘žοΏ½ν‘žβŸ© (𝑠, 𝑇) + 𝜌⟨�퐺2⟩+⟨Θ00⟩ (𝑠, 𝑇) + πœŒβŸ¨οΏ½ν‘žοΏ½νΊοΏ½ν‘žβŸ© (𝑠, 𝑇) + πœŒβŸ¨οΏ½ν‘žοΏ½ν‘žβŸ©2 (𝑠, 𝑇) . (12) In order to obtain the expressions of these spectral density terms, the current expression given in (3) is inserted into the correlation function given in (1) and then the heavy and light quark fields are contracted, and the correlation function is written in terms of quark propagators as Ξ QCDοΏ½νœ‡] (π‘ž, 𝑇) = βˆ’ 𝑖2 βˆ«π‘‘4π‘₯π‘’οΏ½ν‘–οΏ½ν‘žβ‹…οΏ½ν‘₯ β‹… πœ–πœ–πœ–οΏ½θ€ πœ–οΏ½θ€  ⟨{Tr [π›ΎοΏ½νœ‡π›Ύ5π‘†οΏ½ν‘ŽοΏ½ν‘ŽσΈ€ οΏ½ν‘ (βˆ’π‘₯) 𝛾5𝛾]𝑆�푏�푏󸀠�푠 (βˆ’π‘₯)] Γ— Tr [𝛾5𝑆�푑�푑󸀠�푠 (π‘₯) 𝛾5𝑆�푒�푒󸀠�푏 (π‘₯)] + Tr [𝛾5π‘†οΏ½ν‘ŽοΏ½ν‘ŽσΈ€ οΏ½ν‘ (βˆ’π‘₯) 𝛾5𝑆�푏�푏󸀠�푠 (βˆ’π‘₯)π›ΎοΏ½νœ‡] Γ— Tr [𝛾5𝑆�푑�푑󸀠�푠 (π‘₯) 𝛾5𝑆�푒�푒󸀠�푏 (π‘₯) 𝛾]𝛾5𝑆�푏�푏󸀠�푏 (π‘₯)] + Tr [𝛾5π‘†οΏ½ν‘ŽοΏ½ν‘ŽσΈ€ οΏ½ν‘ (βˆ’π‘₯) 𝛾5𝛾] Γ— 𝑆�푏�푏󸀠�푠 (βˆ’π‘₯)] β‹… Tr [𝛾5𝑆�푑�푑󸀠�푠 (π‘₯)𝛾5π›ΎοΏ½νœ‡π‘†οΏ½ν‘’οΏ½ν‘’σΈ€ οΏ½ν‘ (π‘₯)] + Tr [𝛾5π‘†οΏ½ν‘ŽοΏ½ν‘ŽσΈ€ οΏ½ν‘ (βˆ’π‘₯) 𝛾5 Γ— 𝑆�푏�푏󸀠�푠 (βˆ’π‘₯)] β‹… Tr [𝛾5𝑆�푑�푑󸀠�푠 (βˆ’π‘₯)𝛾5𝑆�푒�푒󸀠�푏 (π‘₯) 𝛾]]}⟩ �푇 , (13) where 𝑆�푖�푗 οΏ½ν‘ ,�푏 (π‘₯) are the full quark propagators and 𝑆�푖�푗 οΏ½ν‘ ,�푏 (π‘₯) = 𝐢𝑆�푖�푗�푇 οΏ½ν‘ ,�푏 (π‘₯)𝐢 is used. The quark propagators are given in terms of the quark and gluon condensates [25]. At finite temperatures, additional operators arise due to the breaking of Lorentz invariance by the choice of thermal rest frame. Thus, the residual𝑂(3) invariance brings additional operators to the quark propagator at finite temperature. The expected behavior of the thermal averages of these new operators is opposite of those of the Lorentz invariant old ones [43]. The heavy-quark propagator in coordinate space can be expressed as 𝑆�푖�푗 �푏 (π‘₯) = 𝑖 ∫ 𝑑4π‘˜(2πœ‹)4 𝑒 βˆ’οΏ½ν‘–οΏ½ν‘˜β‹…οΏ½ν‘₯ [ [ 𝛿�푖�푗 (οΏ½π‘˜ + π‘šοΏ½ν‘) π‘˜2 βˆ’ π‘š2 �푏 βˆ’ 𝑔𝐺 οΏ½ν›ΌοΏ½ν›½ οΏ½ν‘–οΏ½ν‘— 4 πœŽοΏ½ν›ΌοΏ½ν›½ (οΏ½π‘˜ + π‘šοΏ½ν‘) + (οΏ½π‘˜ + π‘šοΏ½ν‘) πœŽοΏ½ν›ΌοΏ½ν›½ (π‘˜2 βˆ’ π‘š2 �푏 )2 +𝑔212𝐺�퐴�훼�훽𝐺�훼�훽�퐴 π›ΏοΏ½ν‘–οΏ½ν‘—π‘šοΏ½ν‘ π‘˜2 + π‘šοΏ½ν‘οΏ½π‘˜ (π‘˜2 βˆ’ π‘š2 �푏 )4 + . . .] , (14) and the thermal light quark propagator is chosen as 𝑆�푖�푗�푠 (π‘₯) = 𝑖 οΏ½π‘₯2πœ‹2π‘₯4 𝛿�푖�푗 βˆ’ π‘šοΏ½ν‘ 4πœ‹2π‘₯2 𝛿�푖�푗 βˆ’ βŸ¨π‘ π‘ βŸ© 12 𝛿�푖�푗 βˆ’ π‘₯2 192 β‹… π‘š20 βŸ¨π‘ π‘ βŸ© [1 βˆ’ π‘–π‘šοΏ½ν‘ 6 οΏ½π‘₯] 𝛿�푖�푗 + 𝑖3 [οΏ½π‘₯ ( π‘šοΏ½ν‘ 16 βŸ¨π‘ π‘ βŸ© βˆ’ 1 12 βŸ¨π‘’οΏ½νœ‡Ξ˜οΏ½ν‘“οΏ½νœ‡]𝑒]⟩) + 13 (𝑒 β‹… π‘₯) �𝑒 βŸ¨π‘’οΏ½νœ‡Ξ˜οΏ½ν‘“οΏ½νœ‡]𝑒]⟩] 𝛿�푖�푗 βˆ’ 𝑖𝑔�푠𝐺�훼�훽�푖�푗 32πœ‹2π‘₯2 (οΏ½π‘₯πœŽοΏ½ν›ΌοΏ½ν›½ + πœŽοΏ½ν›ΌοΏ½ν›½οΏ½π‘₯) , (15) where 𝐺�훼�훽 οΏ½ν‘–οΏ½ν‘— ≑ 𝐺�훼�훽 �퐴 𝑑�퐴�푖�푗 is the external gluon field, 𝑑�퐴�푖�푗 = πœ†οΏ½ν΄οΏ½ν‘–οΏ½ν‘—/2 with πœ†οΏ½ν΄οΏ½ν‘–οΏ½ν‘— Gell-Mann matrices, 𝐴 = 1, 2, . . . , 8 symbolizes color indices, π‘šοΏ½ν‘  implies the strange quark mass, π‘’οΏ½νœ‡ is 4 Advances in High Energy Physics the four-velocity of the heat bath, βŸ¨π‘žπ‘žβŸ© is the temperature- dependent light quark condensate, and Ξ˜οΏ½ν‘“οΏ½νœ‡] is the fermionic part of the energy-momentum tensor. Furthermore, the gluon condensate related to the gluonic part of the energy- momentum tensor Ξ˜οΏ½ν‘” οΏ½ν›ΌοΏ½ν›½ is expressed via relation [43]: βŸ¨π‘‡π‘ŸοΏ½ν‘πΊοΏ½ν›ΌοΏ½ν›½πΊοΏ½νœ†οΏ½νœŽβŸ©οΏ½ν‘‡ = (π‘”οΏ½ν›ΌοΏ½νœ†π‘”οΏ½ν›½οΏ½νœŽ βˆ’ π‘”οΏ½ν›ΌοΏ½νœŽπ‘”οΏ½ν›½οΏ½νœ†) 𝐴 βˆ’ (π‘’οΏ½ν›Όπ‘’οΏ½νœ†π‘”οΏ½ν›½οΏ½νœŽ βˆ’ π‘’οΏ½ν›Όπ‘’οΏ½νœŽπ‘”οΏ½ν›½οΏ½νœ† βˆ’ π‘’οΏ½ν›½π‘’οΏ½νœ†π‘”οΏ½ν›ΌοΏ½νœŽ + π‘’οΏ½ν›½π‘’οΏ½νœŽπ‘”οΏ½ν›ΌοΏ½νœ†) 𝐡, (16) where 𝐴 and 𝐡 coefficients are 𝐴 = 124 βŸ¨πΊοΏ½ν‘ŽοΏ½ν›ΌοΏ½ν›½πΊοΏ½ν‘ŽοΏ½ν›ΌοΏ½ν›½βŸ©οΏ½ν‘‡ + 1 6 βŸ¨π‘’οΏ½ν›ΌΞ˜οΏ½ν‘”οΏ½ν›ΌοΏ½ν›½π‘’οΏ½ν›½βŸ©οΏ½ν‘‡ , 𝐡 = 13 βŸ¨π‘’οΏ½ν›ΌΞ˜οΏ½ν‘”οΏ½ν›ΌοΏ½ν›½π‘’οΏ½ν›½βŸ©οΏ½ν‘‡ . (17) In order to remove contributions originating from higher states, the standard Borel transformation with respect to π‘ž20 is applied in the QCD side as well. By equating the coefficients of the selected structure π‘”οΏ½νœ‡] in both physical and QCD sides, and by employing the quark hadron duality ansatz up to a temperature-dependent continuum threshold 𝑠0(𝑇), the final sum rules for π‘ŒοΏ½ν‘ are derived as π‘š2οΏ½ν‘Œπ‘ (𝑇) πœ†2οΏ½ν‘Œπ‘ (𝑇) π‘’βˆ’οΏ½ν‘š2π‘Œπ‘(�푇)/οΏ½ν‘€2 = βˆ«οΏ½ν‘ 0(�푇) 4(οΏ½ν‘šπ‘+οΏ½ν‘šπ‘ ) 2 𝑑𝑠 𝜌QCD (𝑠, 𝑇) π‘’βˆ’οΏ½ν‘ /οΏ½ν‘€2. (18) To find the mass via TQCDSR, one should expel the hadronic coupling constant from the sum rules. It is commonly done by dividing the derivative of the sum rule given in (18) with respect to (βˆ’π‘€βˆ’2) to itself. Following these steps, the temperature-dependent mass is obtained as π‘š2οΏ½ν‘Œπ‘ (𝑇) = βˆ«οΏ½ν‘ 0(�푇) 4(οΏ½ν‘šπ‘+οΏ½ν‘šπ‘ ) 2 𝑑𝑠 π‘ πœŒQCD (𝑠, 𝑇) π‘’βˆ’οΏ½ν‘ /οΏ½ν‘€2 βˆ«οΏ½ν‘ 0(�푇) 4(οΏ½ν‘šπ‘+οΏ½ν‘šπ‘ ) 2 𝑑𝑠 𝜌QCD (𝑠, 𝑇) π‘’βˆ’οΏ½ν‘ /οΏ½ν‘€2 , (19) where the thermal continuum threshold 𝑠0(𝑇) is related to continuum threshold 𝑠0 at vacuum via relation 𝑠0 (𝑇) = 𝑠0 [1 βˆ’ ( 𝑇𝑇�푐 ) 8] + 4 (π‘šοΏ½ν‘ + π‘šοΏ½ν‘ )2 ( 𝑇𝑇�푐 ) 8 (20) [44, 45]. For compactness, the explicit forms of spectral densities are presented in Appendix. 3. Phenomenological Analysis In this section, the phenomenological analysis of sum rules obtained in (18) and (19) is presented. First, the input parame- ters and temperature dependance of relevant condensates are given. Following, the working regions of the obtained sum rules at vacuum are analyzed. The behavior of QCD sum rules at 𝑇 = 0 is used to test the reliability of our analysis. 3.1. Input Parameters. During the calculations, input param- eters given in Table 1 are used. In addition to these input parameters, temperature-dependent quark and gluon con- densates, and the energy density expressions are necessary. The thermal quark condensate is chosen as βŸ¨π‘žπ‘žβŸ© = ⟨0 σ΅„¨σ΅„¨σ΅„¨σ΅„¨π‘žπ‘žσ΅„¨σ΅„¨σ΅„¨σ΅„¨0⟩1 + exp (18.10042 (1.84692 [1/GeV2] 𝑇2 + 4.99216 [1/GeV] 𝑇 βˆ’ 1)), (21) where ⟨0|π‘žπ‘ž|0⟩ is the light quark condensate at vacuum and which is credible up to a critical temperature 𝑇�푐 = 190 MeV. The expression given in (21) is obtained in [46, 47] from the Lattice QCD results given in [48, 49]. The temperature- dependent gluon condensate is parameterized via [46, 50] ⟨𝐺2⟩ = ⟨0󡄨󡄨󡄨󡄨󡄨𝐺2󡄨󡄨󡄨󡄨󡄨 0⟩ β‹… [1 βˆ’ 1.65 ( 𝑇𝑇�푐 ) 8.735 + 0.04967 ( 𝑇𝑇�푐 ) 0.7211] , (22) where ⟨0|𝐺2|0⟩ is the gluon condensate in vacuum state and 𝐺2 = 𝐺�퐴�훼�훽𝐺�훼�훽�퐴 . Additionally, for the gluonic and fermionic parts of the energy density, the following parametrization is used [46] βŸ¨Ξ˜οΏ½ν‘”00⟩ = βŸ¨Ξ˜οΏ½ν‘“00⟩ = 𝑇4 exp (113.867 [ 1 GeV2 ] 𝑇2 βˆ’ 12.190 [ 1 GeV ] 𝑇) βˆ’ 10.141 [ 1 GeV ] 𝑇5, (23) which is extracted from the Lattice QCD data in [51]. 3.2. Analysis of Sum Rules at 𝑇 = 0. In order to get reliable results, obtained sum rules should be tested at vacuum, and the working regions of the parameters 𝑠0 and 𝑀2 should be determined. Within the working regions of 𝑠0 and 𝑀2, convergence of OPE and dominance of pole contributions should be assured. In addition, the obtained physical results should be independent of small variations of these param- eters. Convergence of the OPE is tested by the following Advances in High Energy Physics 5 6 7 8 9 10 Pert M 2 (GeοΌ–2) 7 0.0 0.2 0.4 0.6 0.8 Po le x C on tin uu m 8 Pole Continuum 9 10 11 12 M 2 (GeοΌ–2) 0.0 0.5 1.0 1.5 2.0 O PE te rm s 2.5 + ⟨sＭ⟩ +⟨G2⟩ + ⟨ξ‹ͺ00⟩ + ⟨sοΌ‡οΌ­βŸ© + ⟨sＭ⟩2 Figure 1: The OPE convergence of the sum rules: the ratio of the sum of the contributions up to specified dimension to the total contribution is plotted with respect to 𝑀2 at 𝑠0 = 134 GeV2, 𝑇 = 0 (left). Pole dominance of the sum rules: relative contributions of the pole (blue) and continuum (red-dashed) versus to the Borel parameter 𝑀2 at 𝑠0 = 134 GeV2,𝑇 = 0 (right). Table 1: Input parameters [20–23]. π‘šοΏ½ν‘  = (0.13 Β± 0.03) MeVπ‘šοΏ½ν‘ = (4.24 Β± 0.05) GeVπ‘š2 0 = (0.8 Β± 0.2) GeV2 βŸ¨π‘ π‘ βŸ© = βˆ’0.8 Γ— (0.24 Β± 0.01)3 GeV3 ⟨0󡄨󡄨󡄨󡄨󡄨󡄨󡄨 1 πœ‹π›ΌοΏ½ν‘ πΊ2 󡄨󡄨󡄨󡄨󡄨󡄨󡄨 0⟩ = (0.022) GeV4 criterion. The contribution of the highest order operator in the OPE should be very small compared to the total contribution. In Figure 1, the ratio of the sum of the terms up to the specified dimension to the total contribution is plotted to test the OPE convergence. It is seen that all higher order terms contribute less than the perturbative part for 𝑀2 β‰₯ 6 GeV2. On the other hand, dominance of the pole contribution is tested as follows. The contribution coming from the pole of the ground state should be greater than the contribution of the continuum. In this work, the aforementioned ratio is PC = Ξ  (𝑀 2 max, 𝑠0) Ξ  (𝑀2max, ∞) > 0.50, (24) when 𝑀2 ≀ 10 GeV2 as can be seen in Figure 1. After checking these criteria, the working regions of the parameters𝑀2 and 𝑠0 are determined as 6 GeV2 ≀ 𝑀2 ≀ 10 GeV2; 132 GeV2 ≀ 𝑠0 ≀ 134 GeV2, (25) which is also consistent with 𝑠0 ≃ (π‘šοΏ½ν»+0.5 GeV)2 norm [10]. Within these working regions, the variations of the mass of π‘ŒοΏ½ν‘ with respect to 𝑀2 and 𝑠0 are plotted in Figure 2. It is seen that the mass is stable with respect to variations of 𝑀2 and 𝑠0. Table 2: Results obtained in this work for the mass of π‘ŒοΏ½ν‘ at 𝑇 = 0, in comparison with literature. π‘šοΏ½ν‘Œπ‘(MeV) Present Work 10735+122βˆ’107 Experiment 10889.9+3.2βˆ’2.6 [20] QCDSR 10880 Β± 130 [11] 10910 Β± 70 [10] 4. Results and Discussions Following the analysis presented in previous section, the mass of the ground state estimated by the tetraquark current given in (3) at 𝑇 = 0 is presented together with other results from literature in Table 2. It is seen that our results agree with other theoretical estimates and also with the experimental data onπ‘ŒοΏ½ν‘(10890) [10, 11, 20]. Thus, the broad resonance π‘ŒοΏ½ν‘ can be described by the tetraquark current given in (3), and our analysis can be extended to finite temperatures. To analyze the thermal properties of π‘ŒοΏ½ν‘(10890) reso- nance, the temperature dependencies of the mass and the pole residue of π‘ŒοΏ½ν‘ are plotted in Figure 3. It is seen that the mass and the pole residue of π‘ŒοΏ½ν‘ stay monotonous until 𝑇 β‰…0.12 GeV. However, after this point, they begin to decrease promptly with increasing temperature. At the vicinity of the critical (or so called deconfinement) temperature, the mass reaches nearly 87% of its vacuum value. On the other hand, the pole residue decreased to 44% of its value at vacuum as shown in Figure 3. Our predictions presented in Figure 3 are in good agreement with other QCD sum rules analysis on ther- mal behaviors of conventional or exotic hadrons [35–38]. However in [19], authors predicted a decrease of 5% in the mass of molecular 1++ state, and no change in the mass of charmonium 2βˆ’+ state, even beyond Hagedorn temperature𝑇�퐻 ∼ 177 MeV. Since the decay properties also depend on temperature, and while the mass and pole residue diminish, 6 Advances in High Energy Physics 6 9.0 9.5 10.0 10.5 11.0 11.5 7 8 9 10 132.0 9.0 9.5 10.0 10.5 11.0 11.5 132.5 133.0 133.5 134.0 M 2 (GeοΌ–2) s0=134 GeοΌ– 2 =132 GeοΌ–2s0 m Y  (G eV ) s0 (GeοΌ– 2) m Y  (G eV ) M 2=10 GeοΌ–2 M 2=8 GeοΌ–2 M 2=6 GeοΌ–2 Figure 2: Mass of π‘ŒοΏ½ν‘ as a function of 𝑀2 (left) and 𝑠0 (right). s0= 134 GeοΌ– 2 s0= 132 GeοΌ– 2 s0= 134 GeοΌ– 2 s0= 132 GeοΌ– 2 m Y  (T )/ m Y  (0 ) T/Tc 0.0 0.7 0.2 0.4 0.6 0.8 1.0 0.8 0.9 1.0 1.1 0.2 0.4 0.6 0.8 1.0 T/Tc 0.0 0.2 0.4 0.6 0.8 1.0 ξ‹­ Y  (T )/ ξ‹­ Y  (0 ) Figure 3: The mass (left) and pole residue (right) of π‘ŒοΏ½ν‘ as a function of temperature. the decay width might increase with increasing temperature [52] and decay widths at finite temperature should also be investigated. However the current status of π‘ŒοΏ½ν‘ resonance is very complicated, since it is very close to Ξ₯(5𝑆) state. Thus studying its decays requires establishment of a good model in the hidden-beauty sector. Finally, we would like to highlight the following remarks: (i) We observed that the mass (the pole residue) of exoticπ‘ŒοΏ½ν‘(10890) state starts to decrease near 𝑇/𝑇�푐 ≃ 0.7. (ii) Both quantities tend to diminish with increasing temperature up to critical temperature 𝑇�푐. (iii) Even though the sum rules at 𝑇 = 0 estimates the mass of π‘ŒοΏ½ν‘ consistent with experimental data, more theoretical efforts are required to discriminate π‘ŒοΏ½ν‘ andΞ₯(5𝑆). (iv) In order to get more reliable results on tetraquarks from QCD sum rules, O(𝛼2οΏ½ν‘  ) contributions suggested in [36] should be investigated. In summary, we revisited the hidden-beauty exotic stateπ‘ŒοΏ½ν‘ and studied its properties at vacuum and finite temper- atures. To describe the hot medium effects to the hadronic parameters of the resonance π‘ŒοΏ½ν‘, TQCDSR method is used considering contributions of condensates up to dimension six. Our results for 𝑇 = 0 are in reasonable agreement with the available experimental data and other QCD studies in the literature. Numerical findings show that π‘ŒοΏ½ν‘ can be well described by a scalar-vector tetraquark current. In the literature, remarkable drop in the values of the mass and the pole residue in hot medium was regarded as the signal of the QGP, which is called the fifth state of matter, phase transition. We hope that precise spectroscopic measurements in the exotic bottomonium sector can be done at Super-B factories, and this might provide conclusive answers on the nature and thermal behaviors of the exotic states. Appendix Thermal Spectral Density πœŒοΏ½ν‘„οΏ½νΆοΏ½ν·(𝑠,𝑇) for π‘ŒοΏ½ν‘ State In this appendix, the explicit forms of the spectral densities obtained in this work are presented. The expressions for𝜌pert.(𝑠) and 𝜌nonpert.(𝑠, 𝑇) areshown below as integrals Advances in High Energy Physics 7 over the Feynman parameters 𝑧 and 𝑀, where πœƒ is the step function, 𝜌pert. (𝑠) = 13072πœ‹6 ∫ 1 0 𝑑𝑧 ∫1βˆ’οΏ½ν‘§ 0 𝑑𝑀 1πœ…8πœ‰2 {[βˆ’πœ…π‘š2�푏 (𝑧 + 𝑀) + π‘ π‘§π‘€πœ‰] 2 [πœ…2π‘š4�푏𝑧𝑀 (𝑧 + 𝑀) [𝑀2 + (βˆ’1 + 𝑀) 𝑀 + 𝑧 (βˆ’1 + 4𝑀)] βˆ’2πœ…π‘š2�푏 [6π‘š2οΏ½ν‘ Ξ¦2 [7 (βˆ’1 + 𝑧) 𝑧 (βˆ’7 + 8𝑧) 𝑀 + 7𝑀2] + 𝑠𝑧2𝑀2 (12 (βˆ’1 + 𝑧) 𝑧 + (βˆ’12 + 25𝑧) 𝑀 + 12𝑀2)] πœ‰ +𝑠𝑧𝑀 (12π‘š2οΏ½ν‘ Ξ¦2 + 35𝑠𝑧2𝑀2) πœ‰3]} πœƒ [𝐿 (𝑠, 𝑀, 𝑧)] , (A.1) πœŒβŸ¨οΏ½ν‘ οΏ½ν‘ βŸ© (𝑠, 𝑇) = βŸ¨π‘ π‘ βŸ©128πœ‹4 ∫ 1 0 𝑑𝑧 ∫1βˆ’οΏ½ν‘§ 0 𝑑𝑀 1πœ…6 {[πœ…3π‘š5�푏𝑀 (𝑧 + 𝑀)2 + πœ…2π‘š4οΏ½ν‘π‘šοΏ½ν‘  (𝑧 + 𝑀) [19𝑧4 + 19 (βˆ’1 + 𝑀)2 𝑀2 + 2𝑧 (βˆ’1 + 𝑀) 𝑀 (βˆ’19 + 25𝑀) +𝑧3 (βˆ’38 + 50𝑀) + 𝑧2 [19 + 𝑀 (βˆ’88 + 81𝑀)]] βˆ’ πœ…2π‘š3�푏𝑧 (𝑧 + 𝑀) (π‘š2οΏ½ν‘ Ξ¦ + 2𝑠𝑀2) πœ‰ βˆ’πœ…π‘š2οΏ½ν‘π‘šοΏ½ν‘ π‘ π‘§π‘€ [22𝑧4 + 22 (βˆ’1 + 𝑀)2 𝑀2 + 𝑧3 (βˆ’44 + 111𝑀) + 𝑧 (βˆ’1 + 𝑀) 𝑀 (βˆ’44 + 111𝑀) +𝑧2 [22 + 𝑀 (βˆ’155 + 197𝑀)]] πœ‰ + πœ…π‘šοΏ½ν‘π‘ π‘§2𝑀 (π‘š2οΏ½ν‘ Ξ¦ + 𝑠𝑀2) πœ‰2 +3π‘šοΏ½ν‘ π‘ 2𝑧2𝑀2 (𝑧2 + (βˆ’1 + 𝑀) 𝑀 + 𝑧 (βˆ’1 + 21𝑀)) πœ‰3]}πœƒ [𝐿 (𝑠, 𝑀, 𝑧)] , (A.2) 𝜌⟨�퐺2⟩+⟨Θ00⟩ (𝑠, 𝑇) = 14608πœ‹2 ∫ 1 0 𝑑𝑧 ∫1βˆ’οΏ½ν‘§ 0 𝑑𝑀 1πœ…6πœ‰2 {192πœ‹2 βŸ¨Ξ˜οΏ½ν‘“00⟩ π‘§π‘€πœ‰2 [π‘š4�푏Φ2 (𝑧 + 𝑀) Γ— [3(βˆ’1 + 𝑧) 𝑧 + (βˆ’3 + 5𝑧) 𝑀 + 3𝑀2] + π‘š2�푏Φ𝑠𝑧𝑀 [βˆ’27 (βˆ’1 + 𝑧) 𝑧 + 27𝑀 βˆ’ 53𝑧𝑀 βˆ’ 27𝑀2] πœ‰ + 30𝑠2𝑧2𝑀2πœ‰3] βˆ’ 𝑔2οΏ½ν‘  βŸ¨Ξ˜οΏ½ν‘”00⟩ [3π‘š4�푏Φ2𝑧𝑀 (𝑧 + 𝑀) [2 (βˆ’1 + 𝑧)2 𝑧2 + (βˆ’1 + 𝑧) 𝑧 (βˆ’4 + 3𝑧) 𝑀 + (βˆ’2 + 𝑧) (βˆ’1 + 3𝑧) 𝑀2 + (βˆ’4 + 3𝑧) 𝑀3 + 2𝑀4] βˆ’ π‘š2�푏Φ𝑠𝑧2𝑀2 [24 (βˆ’1 + 𝑧)2 𝑧2 + (βˆ’1 + 𝑧) 𝑧 (βˆ’48 + 85𝑧) 𝑀 + [24 + 𝑧 (βˆ’133 + 121𝑧)]𝑀2 + (βˆ’48 + 85𝑧) 𝑀3 + 24𝑀4] πœ‰ βˆ’12π‘š3οΏ½ν‘π‘šοΏ½ν‘ Ξ¦3 (𝑧 + 𝑀)3 πœ‰2 + 12π‘šοΏ½ν‘π‘šοΏ½ν‘ Ξ¦2𝑠𝑧𝑧 (𝑧2 + 10𝑧𝑀 + 𝑀2) πœ‰3 + 30𝑠2𝑧3𝑀3 (𝑧 + 𝑀) πœ‰4] + βŸ¨π›ΌοΏ½ν‘ πΊ2πœ‹ ⟩ πœ‹2 [πœ…π‘š2�푏 {βˆ’6π‘š2οΏ½ν‘ Ξ¦2 [5 (βˆ’1 + 𝑧) 𝑧3 + 𝑧3𝑀 + (βˆ’5 + 𝑧) 𝑀3 + 5𝑀4] +𝑠𝑧2𝑀2 [2 (βˆ’1 + 𝑧) 𝑧2 (βˆ’18 + 11𝑧) + 𝑧 [72 + 𝑧 (βˆ’221 + 133𝑧)]𝑀 + [36 + 𝑧 (βˆ’221 + 231𝑧)]𝑀2 (βˆ’58 + 133𝑧) 𝑀3 + 22𝑀4]}πœ‰ βˆ’ 36πœ…3π‘š3οΏ½ν‘π‘šοΏ½ν‘  (𝑧 + 𝑀) (𝑧2 βˆ’ 6𝑧𝑀 + 𝑀2) πœ‰2 βˆ’ 60𝑠2𝑧3𝑀3 (𝑧 + 𝑀) πœ‰4 + πœ‰2π‘šοΏ½ν‘π‘§π‘€ [π‘š3�푏 (𝑧 + 𝑀) (4(βˆ’1 + 𝑧) 𝑧2 (βˆ’3 + 4𝑧) + 6𝑧 [4 + 𝑧 (βˆ’11 + 8𝑧)] 𝑀 + [12 + 11𝑧 (βˆ’6 + 5𝑧)]𝑀24 (βˆ’7 + 12𝑧)3 + 16𝑀4) +12π‘šοΏ½ν‘ π‘  (3𝑧2 βˆ’ 26𝑧𝑀 + 3𝑀2) πœ‰3]]}πœƒ [𝐿 (𝑠, 𝑀, 𝑧)] (A.3) πœŒβŸ¨οΏ½ν‘ οΏ½νΊοΏ½ν‘ βŸ© (𝑠, 𝑇) = π‘šοΏ½ν‘ π‘š20 βŸ¨π‘ π‘ βŸ©64πœ‹4 {∫ 1 0 𝑑𝑧 {3π‘š2�푏 + 𝑠 (𝑧 βˆ’ 1) π‘§πœƒ [𝐿�耠 (𝑠, 𝑧)]} + ∫ 1 0 𝑑𝑧 ∫1βˆ’οΏ½ν‘§ 0 𝑑𝑀 13πœ…5 Γ— {π‘§π‘€πœ‰ [πœ…π‘š2�푏 [5𝑧2 + 5 (𝑀 βˆ’ 1) 𝑀 + 𝑧 (11𝑀 βˆ’ 5)] βˆ’ 16π‘ π‘§π‘€πœ‰2]πœƒ [𝐿 (𝑠, 𝑀, 𝑧)]}} , (A.4) πœŒβŸ¨οΏ½ν‘ οΏ½ν‘ βŸ©2 (𝑠, 𝑇) = βŸ¨π‘ π‘ βŸ©25184πœ‹4 {∫ 1 0 𝑑𝑧 [648π‘š2οΏ½ν‘πœ‹2 + 𝑔2οΏ½ν‘  π‘šοΏ½ν‘π‘šοΏ½ν‘ π‘§ + 54πœ‹2 (5π‘š2οΏ½ν‘  + 4𝑠) (𝑧 βˆ’ 1) 𝑧] πœƒ [𝐿�耠 (𝑠, 𝑧)] + 𝑔2οΏ½ν‘  ∫ 1 0 𝑑𝑧 ∫1βˆ’οΏ½ν‘§ 0 π‘‘π‘€π‘§π‘€πœ‰ πœ…5 Γ— [3πœ…π‘š2�푏 [7𝑧2 + 7 (𝑀 βˆ’ 1) 𝑀 + 𝑧 (βˆ’7 + 15𝑀)] βˆ’ 64π‘ π‘§π‘€πœ‰2]πœƒ [𝐿 (𝑠, 𝑀, 𝑧)]} , (A.5) where explicit expressions of the functions 𝐿(𝑠, 𝑀, 𝑧) and𝐿�耠(𝑠, 𝑧) are 𝐿 [(𝑠, 𝑀, 𝑧) = πœ…βˆ’2 (βˆ’1 + 𝑀) [(βˆ’1 + 𝑀) 𝑀2 + 2 (βˆ’1 + 𝑀) 𝑀𝑧 + (βˆ’1 + 2𝑀) 𝑧2 βˆ’ π‘ π‘€π‘§πœ‰ + 𝑧3π‘š2�푏] , (A.6) 𝐿�耠 (𝑠, 𝑧) = 𝑠𝑧 (1 βˆ’ 𝑧) βˆ’ π‘š2�푏. 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Er, β€œB to 𝐷(𝐷*)𝑒]οΏ½ν‘’ transitions at finite tem- perature in QCD,” Physical Review D, vol. 81, no. 9, Article ID 096001, 2010. https://arxiv.org/abs/1901.03881 https://arxiv.org/abs/1901.03881 Hindawi www.hindawi.com Volume 2018 Active and Passive Electronic Components Hindawi www.hindawi.com Volume 2018 Shock and Vibration Hindawi www.hindawi.com Volume 2018 High Energy Physics Advances in Hindawi Publishing Corporation http://www.hindawi.com Volume 2013 Hindawi www.hindawi.com The Scientific World Journal Volume 2018 Acoustics and Vibration Advances in Hindawi www.hindawi.com Volume 2018 Hindawi www.hindawi.com Volume 2018 Advances in Condensed Matter Physics Optics International Journal of Hindawi www.hindawi.com Volume 2018 Hindawi www.hindawi.com Volume 2018 Astronomy Advances in Antennas and Propagation International Journal of Hindawi www.hindawi.com Volume 2018 Hindawi www.hindawi.com Volume 2018 International Journal of Geophysics Advances in Optical Technologies Hindawi www.hindawi.com Volume 2018 Applied Bionics and Biomechanics Hindawi www.hindawi.com Volume 2018 Advances in OptoElectronics Hindawi www.hindawi.com Volume 2018 Hindawi www.hindawi.com Volume 2018 Mathematical Physics Advances in Hindawi www.hindawi.com Volume 2018 Chemistry Advances in Hindawi www.hindawi.com Volume 2018 Journal of Chemistry Hindawi www.hindawi.com Volume 2018 Advances in Physical Chemistry International Journal of Rotating Machinery Hindawi www.hindawi.com Volume 2018 Hindawi www.hindawi.com Journal of Engineering Volume 2018 Submit your manuscripts at www.hindawi.com https://www.hindawi.com/journals/apec/ https://www.hindawi.com/journals/sv/ https://www.hindawi.com/journals/ahep/ https://www.hindawi.com/journals/tswj/ https://www.hindawi.com/journals/aav/ https://www.hindawi.com/journals/acmp/ https://www.hindawi.com/journals/ijo/ https://www.hindawi.com/journals/aa/ https://www.hindawi.com/journals/ijap/ https://www.hindawi.com/journals/ijge/ https://www.hindawi.com/journals/aot/ https://www.hindawi.com/journals/abb/ https://www.hindawi.com/journals/aoe/ https://www.hindawi.com/journals/amp/ https://www.hindawi.com/journals/ac/ https://www.hindawi.com/journals/jchem/ https://www.hindawi.com/journals/apc/ https://www.hindawi.com/journals/ijrm/ https://www.hindawi.com/journals/je/ https://www.hindawi.com/ https://www.hindawi.com/