On the identification of asymptotically free in and out states for a certain interaction Lagrangian

Consider the Lagrangian $L= L_D + L_{KG} +L_{int}$ where the first term is the Dirac Lagrangian, the second is the Klein-Gordon and the interaction term is $L_{int} = g \bar \psi \tau \cdot \phi \psi$. The interaction term describes the interaction of Dirac spinors via a scalar; this model was used to describe the scattering of nucleons via a pion exchange.

The question can be put here by two different ways.

One way is to ask if we can construct out of this Lagrangian Feynman diagrams that will give the scattering of a nucleon with a pion or the scattering of two pions; Feynman diagrams for nπ and ππ interactions.

The other way, more general and interesting, is to ask if we can, from an interaction Lagrangian, derive what will be the possible in and out states for which we can have Feynman diagrams or in general scattering. That is, if we can understand which of the fields in the interaction Lagrangian can give us asymptotically free states or which fields cannot have in and out states, so that they cannot represent outer legs in a Feynman diagram; our Lagrangian cannot describe a scattering for these degrees of freedom; I am not sure if this can be simply shown by proving certain correlation functions as zero all the time, for example that all the correlation functions with odd number of $\phi$ fields are equal to zero.

I don' t know if the question is ill phrased; maybe I should consider the LSZ reduction formula somehow and show only specifically for this Lagrangian the outer and inner states. Any comments with suggestions for reading and any recommendations for studying are welcomed. If finally the post is ill- written please clarify the problem, since I am not sure how to search this specific subject. If it is only a matter of calculations, please inform me. Thanks in advance.

• I'm not sure what exactly you mean - can you give an example for a QFT where a field can only show up as internal lines of the Feynman diagrams? – ACuriousMind Feb 8 '17 at 12:40
• @ACuriousMind Hi. Consider the Lagrangian I give in the post. Regarding the first phrasing of the question, can I use this interaction Lagrangian for pion-pion scattering or nucleon-pion scattering? The examples I have seen in respect with the above Lagrangian contain the pion as an exchange between the two nucleons, that is the Feynman diagram has the nucleons as external lines. This is the one boson or pion exchange model. – Constantine Black Feb 8 '17 at 12:57
• @ACuriousMind To put it differently, can I calculate pion-nucleon and pion-pion scattering amplitudes with the posted Lagrangian, and is it just a coincidence the fact I have only seen it used for NN scattering or it's just a data fitting problem; the NN amplitudes come in agreement with the experiment but the πN and ππ don't. Thank you. – Constantine Black Feb 8 '17 at 13:00