The cross section of a scattering process in $QFT$ is computed in terms of the S-matrix elements. In perturbative $QFT$, the same is done by computing the S-matrix elements by using Feynman diagrammatic expansion. Is there a corresponding formalism for doing the same in the non-perturbative regime, like in the low energy limit of $QCD$?
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In general, for computing the scattering cross-section it is sufficient to use the pole approximation of the corresponding Green function in Heisenberg representation (details can be found in Weinberg's QFT Vol. 1, paragraph 10). This is nonperturbative approach, since poles are invisible in perturbation theory: they define bound states masses. To find them, schematically you need to solve equations of motion for field operator in Heisenberg representation, in which effects of other constituents of bound state are replaced to external field. The solution will give you energies of bound states.
In some cases, when there is symmetry breaking, you can extract the prime pole states by using Goldstome theorem, and replace quantities of underlying theory by these effective bound states using relations like PCAC. But in general this doesn't work, and we need to extract bound states explicitly.
Look also here.
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TY..in reference to weinberg's book, which chapter/page are you referring to? – Bruce Lee Feb 16 '16 at 10:51
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@BruceLee Chapter 10, nonperturbative methods, and chapter 14 (if I'm not wrong), bound stares in external fields. – Name YYY Feb 16 '16 at 12:01
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@BruceLee : sorry, chapter 10.2, polology. – Name YYY Feb 16 '16 at 14:42