Meaning of terms in Dirac Lagrangian

Question

If I have a look at the most generalized version of the Dirac Lagrangian I can not identify all terms with there contrubution in the Feynman diagrams.

$$\mathcal{L} = \sum_a (\overline{\psi}_a {\not}{\partial} \psi^a-{\not}{\partial} \psi^a \overline{\psi}_a+{\psi}_a {\not}{\partial} \overline{\psi}^a-{\not}{\partial} \overline{\psi}^a {\psi}_a -m \overline{\psi}^a\psi_a +m\psi^a \overline{\psi}_a)$$

The last two terms should represent the creation of a fermion and an antifermion but which two kinetic terms correspond to which propagators?

Answer 1

Integrating by parts, all the terms in $\mathcal L$ collapse into one, $$ \mathcal L=\bar\psi(i\not\partial+m)\psi+\text{boundary terms} $$

Therefore, in terms of Feynman diagrams, you have $$ ---\stackrel{p}{\blacktriangleright} --- \qquad=\qquad \frac{1}{\not p-m} $$

Your Lagrangian and the standard Dirac Lagrangian differ by a total derivative. Therefore, they give rise to the same action, same equations of motion, same dynamics, and same diagrammatics. Your "generalised" theory is just as general as the standard Dirac theory.

---

In principle, you could use the standard rules of perturbation theory to read off the Feynman rules of a total derivative in the Lagrangian; but the result is proportional to the sum of all momenta incoming to the vertex, and therefore such a term vanishes by conservation of momentum. The details are left to the reader.