Average kinetic energy of heavy quark and virial theorem

Hwang, Dae Sung; Kim, C. S.; Namgung, W.

doi:10.1016/s0370-2693(97)00661-8

Cited by 23 publications

(10 citation statements)

References 45 publications

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“…Another way to work out the virial theorem is to follow references [13,14,16]. They show that the following equation describes the relativistic virial theorem for a particle moving in potential U(r), when averages over time:…”

Section: Methods Bmentioning

confidence: 99%

A simple argument that small hydrogen may exist

Va’vra

2019

Physics Letters B

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Section: Methods Bmentioning

confidence: 99%

A simple argument that small hydrogen may exist

Va’vra

2019

Physics Letters B

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“…respectively with the Laguerre polynomial L and the variational parameter µ. We estimated µ for each state, for the preferred value of A, using [50],…”

Section: Cornell Potential With O 1 M Correctionsmentioning

confidence: 99%

“…We employ the Cornell potential for bottomonium, as it works well for charmonium, proven by our previous work [47]. Here, we employ the following Hamiltonian [49,50,51,52] and quark-antiquark potential [41,48] ,…”

Section: Cornell Potential With O 1 M Correctionsmentioning

confidence: 99%

Bottomonium spectroscopy using Coulomb plus linear (Cornell) potential

Kher¹,

Chaturvedi²,

Devlani³

et al. 2022

Preprint

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The coulomb plus linear (Cornell) potential is used to investigate the mass spectrum of bottomonium. Gaussian wave function is used in position and momentum space to estimate values of potential and kinetic energies, respectively. Based on our calculations, we study newly observed Υ (10860) as an admixture of 43 S 1 with 4 3 D 1 , and Υ (10753) as an admixture of 6 3 S 1 with 4 3 D 1 also, we try to assign Υ(11020) as a pure 43 D 1 bottomonium state. We also study the Regge trajectories in the (J, M 2 ) and (n r , M 2 ) planes to help prove our association. We estimate the pseudoscalar and vector decay constants, the radiative (Electric and Magnetic Dipole) transition rates, and the annihilation decay width for bottomonium states. PACS. XX.XX.XX No PACS code given 1 Introduction Bottomonium (bb meson) was discovered as Υ, Υ ′ and Υ ′′ by the E288 Collaboration at Fermilab in 1977, and were substantially studied at various e + e − storage rings[1,2]. In 1982, χ bJ (2P ) states and in 1983, χ bJ (2P ), (J = 1, 2, 3) states were discovered in E1 transition from Υ ′′ [3,4] and Υ ′ [5,6], respectively. In 1984, Υ(4S), Υ(5S) and Υ(6S) states were observed [7,8]. In 2008, after three decade of discovery of Υ(nS) resonance, the BaBar Collaboration found η b , the pseudoscalar partner of the triplet state Υ(1S)[9]. In 2012, the Belle collaboration reported the first evidence of the η b (2S) with mass 9999 ± 3.5 +2.8 −1.9 ,M eV /c 2 using h b (2P ) → γη b (2S) transition and first observation of h b (1P ) → γη b (1S) and h b (2P ) → γη b (1S) with mass 9402 ± 1.5 ± 1.8 M eV /c 2 [10]. In 2004, CLEO Collaboration presented the first evidence for the production of Υ(1S)(1D) states in the fourphoton cascade, such as in a two-photon cascade starting from the Υ(3S) → γχ b (2P J ), χ b (2P J ) → γΥ(1D) and then select events with two more subsequent photon transitions, Υ(1D) → γχ b (1P J ), χ b (1P J ) → γΥ(1S), followed by the Υ(1S) annihilation into either e + e − or µ + µ − [11]. In 2010, the BABAR Collaboration reported the observation of the J = 2 state of Υ(1 3 D J ) in the hadronic π + π − Υ(1S) decay channel, with Υ(1S) → e + e − or µ + µ − [12]. In 2011, BABAR Collaboration reported evidence for the h b (1P ) state in the decay Υ(3S) → π0h b (1P ), with data sample corresponds to 28 f b −1 of integrated luminosity at a center of mass energy of 10.355 GeV, the mass of the Υ(3S) resonance and in the same year, the Belle Collaboration reported the first observation of the spin-singlet bottomonium states h b (1P ) and h b (2P ) produced via e + e − → h b (nP )π + π − transition corresponds to 121.4 f b −1 of integrated luminosity near the peak of the Υ(5S) resonance at a center-of-mass energy √ s ∼ 10.865 GeV[13]. Again in 2011, ATLAS Collaboration observed the χ b (nP ) states, recorded by the ATLAS detector during the proton-proton collisions at the LHC, run at a center-of-mass energy √ s ∼ 7 T eV and these states were reconstructed through their radiative decays to Υ(1S, 2S) with Υ → µ + µ − . In addition to the mass pea...

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“…Regarding quantum mechanics, the relativistic virial theorem holds, by canonical quantization of Hamiltonian systems, as said before. Indeed, quantum relativistic versions of the virial theorem have appeared in the literature [17,18]. However, they are meant to be applied in nuclear physics and they only consider a simple two-quark problem with a phenomenological scalar potential U (notice that the fundamental theory of strong interactions, namely, QCD, includes a vector potential, like QED).…”

Section: Virial Theorem For the Electromagnetic Interactionmentioning

confidence: 99%

“…At any rate, the strong interaction Hamiltonian employed by Refs. [17,18] is not fully relativistic and only has relativistic kinematics (the potential U can be interpreted as the lowest order slow-motion approximation of a relativistic interaction). The formulation of a fully relativistic virial theorem for quark bound states requires a QFT framework and is presented in Sec.…”

Section: Virial Theorem For the Electromagnetic Interactionmentioning

confidence: 99%

The relativistic virial theorem and scale invariance

Gaite¹

2013

Usp. Fiz. Nauk

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The virial theorem is related to the dilatation properties of bound states. This is realized, in particular, by the Landau-Lifshitz formulation of the relativistic virial theorem, in terms of the trace of the energy-momentum tensor. We construct a Hamiltonian formulation of dilatations in which the relativistic virial theorem naturally arises as the condition of stability against dilatations. A bound state becomes scale invariant in the ultrarelativistic limit, in which its energy vanishes. However, for very relativistic bound states, scale invariance is broken by quantum effects and the virial theorem must include the energy-momentum tensor trace anomaly. This quantum field theory virial theorem is directly related to the Callan-Symanzik equations. The virial theorem is applied to QED and then to QCD, focusing on the bag model of hadrons. In massless QCD, according to the virial theorem, 3/4 of a hadron mass corresponds to quarks and gluons and 1/4 to the trace anomaly.

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Average kinetic energy of heavy quark and virial theorem

Cited by 23 publications

References 45 publications

A simple argument that small hydrogen may exist

A simple argument that small hydrogen may exist

Bottomonium spectroscopy using Coulomb plus linear (Cornell) potential

The relativistic virial theorem and scale invariance

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