Dirac equation in the confining SU (3)-Yang-Mills field and relativistic effects in the charmonium spectrum

Abstract

The recently obtained solutions of the Dirac equation in the confining SU(3)-Yang-Mills field in Minkowski spacetime are applied to describe the energy spectrum of charmonium. The nonrelativistic limit is considered for the relativistic effects to be estimated in a self-consistent way and it is shown that the given effects could be extremely important for both the energy spectrum and the confinement mechanism.

Estimates of the gluon concentrations in the confining SU(3)-Yang–Mills field for the first three states of charmonium

“…The gluon exchange between quarks is realized in such a way that at large distances it leads to the confining SU(3)-field which may be considered classically (the gluon concentration becomes huge and gluons form the boson condensate -a classical field) and is a nonperturbative solution of the SU(3)-Yang-Mills equations. Under the circumstances mesons are the relativistic bound states described by the corresponding wave functions -nonperturbative solutions of the Dirac equation in this confining SU(3)-field [1][2][3]. For each meson there exists its own set of real constants (for more details see below) a j , A j , b j , B j parametrizing the mentioned confining gluon field (the gluon condensate) and the corresponding wave functions while the latter ones also depend on µ 0 , the reduced mass of the current masses of quarks forming meson [1][2][3].…”

“…Under the circumstances mesons are the relativistic bound states described by the corresponding wave functions -nonperturbative solutions of the Dirac equation in this confining SU(3)-field [1][2][3]. For each meson there exists its own set of real constants (for more details see below) a j , A j , b j , B j parametrizing the mentioned confining gluon field (the gluon condensate) and the corresponding wave functions while the latter ones also depend on µ 0 , the reduced mass of the current masses of quarks forming meson [1][2][3]. It is clear that constants a j , A j , b j , B j , µ 0 should be extracted from experimental data and such a program has been just realized in Refs.…”

“…It is clear that constants a j , A j , b j , B j , µ 0 should be extracted from experimental data and such a program has been just realized in Refs. [2,3] for quarkonia.…”

Quark Confinement Mechanism and the Scale Λ QCD

“…As has been repeatedly explained in [11,12,13,14,15,16,17,18,19,20], parameters A 1,2 of solution (2) are inessential for physics in question and we can consider…”

“…of the euclidean Dirac operator D 0 on the unit sphere S 2 , while the coordinate r stands for the distance between quarks. The explicit form of Φ j is not needed here and can be found in [11,12,13,14,15,16,17,18,19,20]. One can only remark that spinors Φ j form an orthonormal basis in L 2 2 (S 2 ) (in what follows we denote L 2 (F ) the set of the modulo square integrable complex functions on any manifold F furnished with an integration measure, then L n 2 (F ) will be the n-fold direct product of L 2 (F ) endowed with the obvious scalar product).…”