$γ$ and $χ_{cJ}~(J=0,1)$ production in electron-positron annihilation at $\sqrt{s}=10.6$ GeV in the framework of Bethe-Salpeter equation

Abstract

In present work we study the production of ground and excited charmonium states in e − e + → γ + χ cJ (nP ) for J = 0, 1; n = 1, 2, through leading order (LO) (tree-level) diagrams, which proceed through exchange of a virtual photon that couples to γ and χ cJ through the triangular quark loop diagram, in the framework of 4 × 4 Bethe-Salpeter equation (BSE), at center of mass energy, √ s = 10.6GeV. With use of Covariant Instantaneous Ansatz, we have been able to express the 3D form of M f i as a linear superposition of terms involving combinations of ++, and −− components of Salpeter wave functions of final hadron, with each of the terms involving Ψ++ and Ψ−− being associated with a coefficient, I , 1,2 which are the results of pole integration in the complex σ-plane (σ = q.P P 2 being the component of internal momentum q of initial hadron that is longitudinal to external momentum, P ), which is again covariant. The cross sections for these processes with leading order (tree) level diagrams alone at √ s = 10.6GeV. are: e − e + → γχ c0 (1P ) = 0.6771f b, e − e + → γχ c0 (2P ) = 0.1943f b, e − e + → γχ c1 (1P ) = 11.057f b, and e − e + → γχ c1 (2P ) = 5.455f b, and provide a sizable contribution, which might be mainly due to the BSE being a fully relativistic approach that incorporates the relativistic effect of quark spins and can also describe internal motion of constituent quarks within the hadron in a relativistically covariant manner.