Editing Yukawa interaction (section)

==The action==
The Yukawa interaction is an interaction between a [[scalar field (quantum field theory)|scalar field]] (or [[pseudoscalar]] field) {{mvar|ϕ}} and a [[Dirac field]] {{mvar|ψ}} of the type
{{block indent | em = 1.5 | text = <math> V \approx g\,\bar\psi \,\phi \,\psi </math>{{pad|1em}} (scalar) {{pad|2em}} or {{pad|2em}}<math> g \,\bar\psi \,i\,\gamma^5 \,\phi \,\psi </math>{{pad|1em}} ([[pseudoscalar]]).}}

The [[action (physics)|action]] for a [[meson]] field <math>\phi</math> interacting with a [[Dirac field|Dirac]] [[baryon]] field <math>\psi</math> is

<math display="block">S[\phi,\psi]=\int \left[ \,
\mathcal{L}_\mathrm{meson}(\phi) +
\mathcal{L}_\mathrm{Dirac}(\psi) +
\mathcal{L}_\mathrm{Yukawa}(\phi,\psi)
\, \right] \mathrm{d}^{n}x
</math>

where the integration is performed over {{mvar|n}} dimensions; for typical four-dimensional spacetime {{math|1=''n'' = 4}}, and <math>\mathrm{d}^{4}x \equiv \mathrm{d}x_1 \, \mathrm{d}x_2 \, \mathrm{d}x_3 \, \mathrm{d}x_4 ~.</math>

The meson [[Lagrangian (field theory)|Lagrangian]] is given by
<math display="block">\mathcal{L}_\mathrm{meson}(\phi) = 
\frac{1}{2}\partial^\mu \phi \; \partial_\mu \phi - V(\phi)~.</math>

Here, <math>~V(\phi)~</math> is a self-interaction term. For a free-field massive meson, one would have <math display="inline">~V(\phi)=\frac{1}{2}\,\mu^2\,\phi^2~</math> where <math>\mu</math> is the mass for the meson. For a ([[renormalizable]], polynomial) self-interacting field, one will have <math display="inline">V(\phi) = \frac{1}{2}\,\mu^2\,\phi^2 + \lambda\,\phi^4</math> where {{mvar|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
<math display="block">\mathcal{L}_\mathrm{Dirac}(\psi) = 
\bar{\psi}\,\left( i\,\partial\!\!\!/ - m \right)\,\psi </math>

where {{mvar|m}} is the real-valued, positive mass of the fermion.

The Yukawa interaction term is 
<math display="block">\mathcal{L}_\mathrm{Yukawa}(\phi,\psi) = -g\,\bar\psi \,\phi \,\psi</math>

where {{mvar|g}} is the (real) [[coupling constant]] for scalar mesons and

<math display="block">\mathcal{L}_\mathrm{Yukawa}(\phi,\psi) = -g\,\bar\psi \,i \,\gamma^5 \,\phi \,\psi</math>

for pseudoscalar mesons. Putting it all together one can write the above more explicitly as

<math display="block">S[\phi,\psi] = \int \left[ \tfrac{1}{2} \, \partial^\mu \phi \; \partial_\mu \phi - V(\phi) + \bar{\psi} \, \left( i\, \partial\!\!\!/ - m \right) \, \psi - g \, \bar{\psi} \, \phi \,\psi \, \right] \mathrm{d}^{n}x ~.</math>