The positronium analogy

Suppose we want to treat quarkonium using QCD. How do we go about it Bound state problems cannot be treated by conventional perturbation theory. And so far, field theories can only be evaluated in perturbation theory. It seems as if there is a mismatch between field theory and the bound state problem. 

Let us first recall how bound state problems like positronium m-e treated in the simpler theory of QED. First, we use perturbation theory to calculate a scattering amplitude between electron and positron. Then we identify a potential which will give the amplitude in Born approximation. Last, we substitute the potential in the wave equation, for example, the 

Schrodinger equation and solve for the bound-state energy levels. To obtain decay rates, we again use perturbation theory to calculate transition matrix elements between the bound-state levels. 

In lowest-order perturbation theory, the QED potential arises from one-photon exchange. This potential contains a static part (i.e. a velocity independent part), which is the Coulomb potential, and non-static corrections. These non-static terms are most commonly treated in the Fermi-Breit approximation, which gives the corrections to order w /c. The Fermi-Breit terms include a spin-spin interaction, a spin-orbit interaction, and a tensor interaction. They also include a spin-independent part which depends on the particle momenta. 

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