The noncommutative parameter for $$ c{\bar{c}} $$ in nonrelativistic limit

The Effects of Noncommutativity on the Energy Spectra for Mass-Dependent Klein-Gordon and Shrödinger Equations with Vector Quark-Antiquark Interaction and Harmonic Oscillator Potential: Application to HLM Systems

“…In the physics literature, this is known as Bopp's shift, and this quantization procedure is called Bopp quantization. It is known to the specialists that Bopp's shift method [47][48][49][50] has been applied effectively and has succeeded in simplifying the three basic equations deformed Schrödinger equation DSE [51][52][53][54][55][56][57][58][59][60][61][62], the DKGE [63][64][65][66][67][68][69][70][71][72][73][74], the deformed Dirac equation DDE [75][76][77][78][79][80] and the deformed Duffin-Kemmer equation [81][82][83][84] with the notion of star product to the SE, the KGE, the DE and the Duffin-Kemmer equation with the notion of ordinary product, respectively. The results of the application of this method were very useful and yielded promising results in many physical and chemical fields.…”

On quarkonium masses in 3D non-commutative space

“…While it preserves the symmetries of the space, the Hamiltonian is constructed in a way that breaks the azimuthal symmetry of the problem, leading to results different from the ones presented here. In this framework, the hydrogen atom [34] has been considered, and recently also the spectrum of the b b [24] and cc [25] states. We will comment on these results where appropriate.…”

“…The setup of [24,25] allowed for a direct use of the standard perturbation theory in quantum mechanics, without any need of WKB or Pekeris-type approximations. Both of these works obtained leading correction to the energy spectrum at the second order in noncommutative parameter, as did we, but with a different n and l dependence.…”

“…So, we use its uncertainty to determine the upper value of λ. With the help of the correction to the mass spectrum σ 2 M Both [24] and [25] have considered the hyper-fine splitting of the quarkonium states to determine the upper bound on the characteristic distance of the non-commutativity, obtaining a value greater by two orders of magnitude than (4.4). Since the experimental error in the measurement of J/ψ(1S) is much smaller than the splitting, we obtain a stricter bound on λ.…”

“…Such states have been observed in particle accelerators, and masses of several of them are known [23]. Some attempts to treat quarkonium in NC space have been done in [24,25].…”