Physics – High Energy Physics – High Energy Physics - Phenomenology
Scientific paper
2004-06-04
Commun.Theor.Phys.43:113-118,2005
Physics
High Energy Physics
High Energy Physics - Phenomenology
4 pages, replace for shortening the manuscript
Scientific paper
We present an approach to solve a Bethe-Salpeter (BS) equation exactly without any approximation if the kernel of the BS equation exactly is instantaneous, and take positronium as an example to illustrate the general features of the solutions. As a middle stage, a set of coupled and self-consistent integration equations for a few scalar functions can be equivalently derived from the BS equation always, which are solvable accurately. For positronium, precise corrections to those of the Schr\"odinger equation in order $v$ (relative velocity) in eigenfunctions, in order $v^2$ in eigenvalues, and the possible mixing, such as that between $S$ ($P$) and $D$ ($F$) components in $J^{PC}=1^{--}$ ($J^{PC}=2^{++}$) states as well, are determined quantitatively. Moreover, we also point out that there is a problematic step in the classical derivation which was proposed first by E.E. Salpeter. Finally, we emphasize that for the effective theories (such as NRQED and NRQCD etc) we should pay great attention on the corrections indicated by the exact solutions.
Chang Chao-Hsi
Chen Jiao-Kai
Li Xue-Qian
Wang Guo-Li
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