Yukawa Interaction - The Action

The Action

The action for a meson field φ interacting with a Dirac baryon field ψ is

S=\int d^dx \;\left[ \mathcal{L}_\mathrm{meson}(\phi) + \mathcal{L}_\mathrm{Dirac}(\psi) + \mathcal{L}_\mathrm{Yukawa}(\phi,\psi) \right]

where the integration is performed over d dimensions (typically 4 for four-dimensional spacetime). The meson Lagrangian is given by

\mathcal{L}_\mathrm{meson}(\phi) = \frac{1}{2}\partial^\mu \phi \partial_\mu \phi -V(\phi).

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

\mathcal{L}_\mathrm{Dirac}(\psi) = \bar{\psi}(i\partial\!\!\!/-m)\psi

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

S=\int d^dx \left[\frac{1}{2}\partial^\mu \phi \partial_\mu \phi -V(\phi) + \bar{\psi}(i\partial\!\!\!/-m)\psi -g \bar{\psi}\phi\psi \right].

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