The Action
The action for a meson field φ interacting with a Dirac baryon field ψ is
where the integration is performed over d dimensions (typically 4 for four-dimensional spacetime). The meson Lagrangian is given by
Here, is a self-interaction term. For a free-field massive meson, one would have where is the mass for the meson. For a (renormalizable) self-interacting field, one will have where λ is a coupling constant. This potential is explored in detail in the article on the quartic interaction.
The free-field Dirac Lagrangian is given by
where m is the positive, real mass of the fermion.
The Yukawa interaction term is
where g is the (real) coupling constant for scalar mesons and
for pseudoscalar mesons. Putting it all together one can write the above more explicitly as
Read more about this topic: Yukawa Interaction